Detection of Cosmological 21 cm Emission with the Canadian Hydrogen Intensity Mapping Experiment

Our analysis uses the CHIME stack data set acquired between 2019 January 1 and November 5. The stack data set is described in CHIME Collaboration et al. (2022b) and consists of the ${N}_{\mathrm{feed}}^{2}$ visibilities (with N_feed = 2048) after they have been integrated to Δt = 9.9405 s cadence, calibrated for complex gain variations, and compressed by averaging subsets of redundant baselines. We selected 102 nights from this period to include in the analysis, using criteria that will be described in Section 3.3.1. After masking intervals of poor data quality, these 102 nights contain 521 hr of total integration time on the relevant eBOSS field.

CHIME is sensitive to radio frequencies from 400 to 800 MHz, which corresponds to 21 cm emission from redshifts 2.55 to 0.78. However, frequencies from 400 to 500 MHz suffer from frequent narrowband, transient radio frequency interference (RFI). In addition, approximately 60% of frequencies between 488 and 584 MHz are corrupted by persistent RFI from locally broadcast TV channels. Hence, for this initial analysis we have restricted our attention to the CHIME data acquired in the relatively clean portion of the band between 587.5 and 800 MHz, corresponding to 21 cm emission from redshifts 1.42 to 0.78. The spectral resolution of the stack data set is Δν = 0.390625 MHz, resulting in 544 frequency channels within this range. We anticipate that the real-time, RFI excision algorithm that was deployed on the CHIME correlator in 2019 mid-October and recent improvements to the offline RFI excision algorithm will enable the inclusion of the lower half of the CHIME band in future analyses.

eBOSS (Dawson et al. 2016), the cosmological survey within the Sloan Digital Sky Survey IV (SDSS-IV; Blanton et al. 2017), was conducted over 4.5 yr using spectrographs previously used for BOSS (Smee et al. 2013), mounted on the Sloan Telescope (Gunn et al. 2006) at the Apache Point Observatory. eBOSS produced four distinct samples of objects, each of which has been used to measure large-scale clustering and place constraints on a variety of cosmological parameters (see Alam et al. 2021 for a summary of these results). In this work, we cross-correlate three of these samples, from SDSS Data Release 16 (Ahumada et al. 2020), with CHIME measurements.

The eBOSS ELG sample (Raichoor et al. 2021) selected targets using imaging from the Dark Energy Camera Legacy Survey (Dey et al. 2019), making special use of emission in the [O ii] double at (λ3727, λ3729) to obtain efficient and accurate redshift estimates. This resulted in a catalog of 173,736 unique objects over 0.6 < z < 1.1, spread across two fields: the 550 deg² North Galactic Cap (NGC) and the 620 deg² South Galactic Cap (SGC). These correspond to comoving volumes of 0.63 and 0.71 h⁻³ Gpc³, respectively. We show both fields, superimposed on a representative CHIME sky map, in Figure 1.

The LRG sample (Ross et al. 2020) is composed of objects from optical imaging taken during previous phases of SDSS (Albareti et al. 2017), along with infrared data from the Wide-field Infrared Survey Explorer satellite (Lang et al. 2016). Selection criteria were designed to target galaxies with z > 0.6, with the resulting final catalog containing 174,816 objects over 0.6 < z < 1.0, distributed between a 2566 deg² NGC field and a 1676 deg² SGC field (shown in Figure 1). The corresponding comoving volumes are 2.2 and 1.4 h⁻³ Gpc³, respectively.

The QSO sample (Lyke et al. 2020; Ross et al. 2020) is composed of objects observed during previous phases of SDSS and new objects selected from the same imaging data as the LRGs. The QSO catalog used for clustering (as opposed to the QSOs used for Lyα forest studies) contains 343,708 objects over 0.8 < z < 2.2, covering the same two fields as the LRGs (with comoving volume 12 h⁻³ Gpc³ for the NGC field and 8.0 h⁻³ Gpc³ for the SGC field).

Figure 2 shows the redshift distribution of the LRG, ELG, and QSO samples for the NGC field, along with vertical bands indicating redshift ranges that are outside of the CHIME band (dark gray) or excluded owing to persistent RFI (light gray). Persistent RFI obscures 14.2% of the 587.5–800 MHz band, but we mask a larger fraction of the band (not shown in Figure 2) in this analysis, for reasons that will be described in Section 4.2.

The stack of the SGC catalog on the CHIME data is a factor of 3–3.5 times noisier than the stack on the NGC catalog for the same tracer of LSS. There are several reasons for this. First, in the case of the LRG and QSO catalogs there are 50% fewer sources in the SGC field compared to the NGC field. Second, we have less integration time on the SGC field because the range of R.A. occupied by the SGC field transits at CHIME at night in the summer time, whereas the NGC field transits at night in the winter time, when the nights are longer. Finally, the SGC field is at a lower decl. where the CHIME primary beam response is reduced and where we are forced to use a more aggressive delay filter because of aliasing of foregrounds. For these reasons, we have only a modest detection (∼4σ) of 21 cm emission when stacking on the QSOs in the SGC field, and we do not have a detection for the ELGs and LRGs in the SGC field. In what follows, we present the results for the NGC field only. We note, however, that our measurements in the SGC field are consistent with the amplitude of the 21 cm signal inferred from the catalogs in the NGC field, given the increased noise.

Each object in the eBOSS clustering catalogs includes weight values that account for imaging systematics, close pairs (which can be affected by spectroscopic fiber collisions), and the probability of a catastrophic redshift failure. We found that incorporating these weights into our analysis had a negligible impact on our results. Therefore, we do not employ the eBOSS weights in what follows. The weight given to each object is determined entirely by the sensitivity of the CHIME data at that object's angular and spectral location.

The effective redshift, z_eff, of each catalog, when cross-correlated with CHIME data, is a combination of the redshift distribution of the sources in the catalog, the RFI mask used for the CHIME analysis, and the sensitivity of the CHIME data outside the masked regions. To determine z_eff, we first take each catalog, and for every source within it, we extract the inverse variance weights for that source in the processed CHIME data (these weights and how they are propagated through our pipeline will be described in Section 3). We then use these to construct a weighted median of the redshifts of the catalog. Similarly, to define an effective range of each catalog, we take the 16% and 84% weighted percentiles of the redshift distribution (i.e., the 68% equal-tailed interval), which gives a region within "1σ" of the effective redshift. This differs substantially from the minimum–maximum redshift range where the source number density drops at the edges of the redshift distribution, most notably for the low-redshift end of the QSO distribution and the high-redshift end of the LRG distribution. These are all summarized in Table 1.

Some sources have zero weight owing to RFI masking and outlier cuts. In Table 1 we give the total number of sources within the frequency band being analyzed and an effective source number, defined as the number of sources lying within a voxel with nonzero weight. Depending on the frequency range, this is typically ∼50% of the total source number.

In Table 1 we also list five additional catalogs that are subdivisions of the QSO catalog, the largest and broadest redshift sample. The three catalogs QSOb0, QSOb1, and QSOb2 divide the redshift span into three roughly equal parts from lowest to highest redshift; the two catalogs QSOb00 and QSOb01 further divide the lowest-redshift catalog into two more catalogs. These additional catalogs will be used in later analysis of the data.

CHIME is a transit instrument, and as such we are acutely sensitive to the precession of Earth's polar axis. Historically, the celestial coordinate system has been anchored to the vernal equinox, which makes the coordinate system sensitive to both an unavoidable precession of Earth's polar axis and an artificial shift in the zero-point of the R.A. coordinate (B1950 and J2000 coordinates are realizations of this anchored at their respective epochs).

The new system outlined in Petit & Luzum (2010) and Kaplan (2005) fixes some of these problems. The fundamental position of sources is given in International Celestial Reference System (ICRS) coordinates, which are fixed and unchanging coordinates that are essentially aligned with J2000 coordinates. Position as seen by an observer on Earth can be given in Celestial Intermediate Reference System (CIRS) coordinates, a frame in which the polar axis shifts with Earth's precession and the R.A. origin is minimally rotated. Unlike previous equinox-based coordinates, CIRS coordinates only contain the minimal shift required to keep the polar alignment. As such, they are much more suited to use in CHIME: over a 5 yr period a typical equinox R.A. position shifts by 43, or around a quarter of a CHIME pixel, whereas a CIRS position changes only by 15, about 1/10 of a pixel. This means that we are able to trivially align and average data products such as maps over much longer periods.

In this new system Greenwich Apparent Sidereal Time is replaced by Earth rotation angle. Instead of local sidereal time, we use local Earth rotation angle, which is equivalent to the current CIRS R.A. of the local meridian.

Throughout this paper the celestial coordinates we use will be CIRS coordinates at the average epoch of the data being analyzed, and any maps presented will be in those coordinates. In the absence of better terminology, we will use sidereal day to refer to the interval between transitions of the Earth rotation angle through zero.

We refer the reader to CHIME Collaboration et al. (2022b) for a description of the CHIME correlator, the real-time pipeline, and the archived data products. Below we highlight several aspects of the real-time processing that are relevant for interpreting what follows.

3.1.1. Real-time RFI Excision

3.1.2. Complex Gain Calibration

3.1.3. Compression

3.1.4. Weights

Here we describe the daily pipeline that applies additional processing to a copy of the archived visibility data. This includes correcting for clock drift, averaging over redundant baselines on different cylinder pairs, identifying and masking RFI, correcting common-mode thermal variations in the amplitude of the gain, interpolating the data onto a common grid in local Earth rotation angle, and finally masking ranges of time with poor data quality. We briefly describe each of these stages. The primary data product output by this processing is the visibility for all unique baselines on each local sidereal day as a function of local Earth rotation angle at Δν = 0.390625 MHz spectral resolution.

3.2.1. Timing Correction

The sampling rate of the analog-to-digital converters (ADCs) that digitize the signal measured by the CHIME feeds is derived from a 10 MHz clock that originates from a GPS-disciplined, oven-controlled crystal oscillator and is distributed to the circuit boards that house the ADCs through a hierarchical network consisting of coaxial cables, power splitters, and amplifiers. Thermal susceptibility of this distribution network results in copies of the clock drifting with respect to one another on timescales set by the different refrigeration cycles of the water chillers used to control the temperature of the electronics. The magnitude of this effect is particularly large between copies of the clock provided to ADCs in different receiver huts, which are temperature controlled by independent chillers.

The CHIME ADCs are housed in eight electronics crates. The thermal drift between the eight copies of the clock that are distributed to the eight crates is measured using a broadband noise source following the procedure described in CHIME Collaboration et al. (2022b). This yields a proxy for the drift, $\delta {\tau }_{{cd}}^{\mathrm{clock}}(t)$, between the copies of the clock provided to electronics crate c relative to electronics crate d. The visibility is then corrected as follows:

$$\begin{eqnarray}\begin{array}{rcl}{V}_{\mathrm{cal},1}({\boldsymbol{b}},\nu ,t) & = & \exp \left\{-2\pi j\nu \,\langle \delta {\tau }_{{cd}}^{\mathrm{clock}}(t){\rangle }_{{cd}\in {\boldsymbol{b}}}\right\}\\ & & \times \,{V}_{\mathrm{raw}}({\boldsymbol{b}},\nu ,t).\end{array}\end{eqnarray} \tag{ 1 }$$

Here $\langle \delta {\tau }_{{cd}}^{\mathrm{clock}}(t){\rangle }_{{cd}\in {\boldsymbol{b}}}$ is constructed by averaging the estimates of the relative clock drift between the pairs of crates that digitize the pairs of inputs that form every redundant baseline averaged by the real-time pipeline to obtain V_raw( b , ν, t). Applying this correction reduces the standard deviation of the delay noise on timescales less than 20 minutes from 4.25 to 1.5 ps on average, as inferred from the phase stability of the signal from bright point sources. Note that further improvements have been achieved by applying a more complicated, ADC-dependent correction in real time, but the analysis described in this work uses the simpler, offline correction described above.

3.2.2. Redundant Baseline Collation

3.2.3. RFI Excision

3.2.4. Thermal Calibration

Common-mode variations in the amplitude of the complex receiver gain are corrected using a linear regression model based on measurements of the outside temperature. Details of the model construction and an evaluation of its performance are provided in CHIME Collaboration et al. (2022b). To briefly summarize, fractional variations in the amplitude of the complex gain inferred from hundreds of bright point-source transits are regressed against the outside temperature as measured by the DRAO weather station at the time of transit. The resulting thermal susceptibility increases with frequency from 0.07% K⁻¹ at 400 MHz to 0.2% K⁻¹ at 800 MHz and varies across inputs at the 0.05% K⁻¹ level. The susceptibility is averaged over the 2048 inputs, and the frequency dependence is fit to a quadratic function. The visibility measured by baseline b at frequency ν and time t is then corrected as follows:

$$\begin{eqnarray}&&{V}_{\mathrm{cal},2}({\boldsymbol{b}},\nu ,t)={\left[1+\alpha (\nu )\left(T(t)-T({t}_{* })\right)\right]}^{2}\ {V}_{\mathrm{cal},1}({\boldsymbol{b}},\nu ,t),\end{eqnarray} \tag{ 2 }$$

where α(ν) is the quadratic model for the thermal susceptibility, T(t) is the outside temperature at time t, and T(t_*) is the outside temperature at time t_* at which the complex gain calibration was derived by the real-time pipeline. The quantity t_* is a step function that changes once per sidereal day to the most recent time of transit of the calibrator source. This procedure improves the stability from roughly 0.8% to 0.5% (standard deviation in fractional power units) by correcting the common-mode drift in the amplitude caused by changes in the outside temperature between daily point-source calibrations.

3.2.5. Weight Smoothing

3.2.6. Sidereal Regridding

The next stage of the daily processing pipeline interpolates the visibilities onto a fixed grid in local Earth rotation angle ϕ that ranges from 0° to 360° with 4096 samples, giving a spacing ${d}_{\phi }=5\buildrel{\,\prime}\over{.} 27$. The interpolation algorithm assumes that the processed visibilities, V_cal,2( b , ν, t), are sampled from some regularly gridded sky visibility, V_grid( b , ν, ϕ), and corrupted by both noise and RFI, denoted as n( b , ν, t). Since the sky visibility is band limited by its maximum fringe rate and the chosen sample rate is more than twice that, the following relation holds:

$$\begin{eqnarray}&&{V}_{\mathrm{cal},2}({\boldsymbol{b}},\nu ,{t}_{g})=\displaystyle \sum _{h}\ {K}_{{gh}}\ {V}_{\mathrm{grid}}({\boldsymbol{b}},\nu ,{\phi }_{h})+n({\boldsymbol{b}},\nu ,{t}_{g}),\end{eqnarray} \tag{ 3 }$$

where ${K}_{{gh}}=\mathrm{sinc}\left(\tfrac{{\rm{\Delta }}{\phi }_{{gh}}}{{d}_{\phi }}\right)$ is the interpolation kernel with Δϕ_gh = ϕ(t_g ) − ϕ_h , and the summation runs over the regular grid in local Earth rotation angle. The infinite support of the kernel is computationally problematic, so a common approximation involves truncating the kernel by multiplying it with a window function. We use the Lanczos kernel, which is given by

$$\begin{eqnarray}{K}_{{gh}}=\left\{\begin{array}{ll}\mathrm{sinc}\left(\displaystyle \frac{{\rm{\Delta }}{\phi }_{{gh}}}{{d}_{\phi }}\right)\mathrm{sinc}\left(\displaystyle \frac{{\rm{\Delta }}{\phi }_{{gh}}}{{{ad}}_{\phi }}\right) & | {\rm{\Delta }}{\phi }_{{gh}}| \leqslant {{ad}}_{\phi }\\ 0 & | {\rm{\Delta }}{\phi }_{{gh}}| \gt {{ad}}_{\phi }.\end{array}\right.\end{eqnarray} \tag{ 4 }$$

The parameter a controls the kernel width and couples at most 4a + 1 samples of V_grid.

We use a Wiener filter to invert Equation (3) and solve for the regularly gridded sky visibility, given the noisy, RFI contaminated data. Let v _cal,2 denote the vector containing the time-ordered visibility for a given frequency and baseline. The regularly gridded visibility v _grid for that frequency and baseline is estimated as

$$\begin{eqnarray}&&{\hat{{\boldsymbol{v}}}}_{\mathrm{grid}}={\boldsymbol{C}}{{\boldsymbol{K}}}^{T}{{\boldsymbol{N}}}^{-1}{{\boldsymbol{v}}}_{\mathrm{cal},2},\end{eqnarray} \tag{ 5 }$$

where

$$\begin{eqnarray}&&{\boldsymbol{C}}={\left({{\boldsymbol{S}}}^{-1}+{{\boldsymbol{K}}}^{T}{{\boldsymbol{N}}}^{-1}{\boldsymbol{K}}\right)}^{-1},\end{eqnarray} \tag{ 6 }$$

${{\boldsymbol{N}}}^{-1}={\left({{\boldsymbol{N}}}_{\mathrm{noise}}+{{\boldsymbol{N}}}_{\mathrm{RFI}}\right)}^{-1}$ is the inverse covariance of the noise and RFI, and S ⁻¹ is the inverse covariance of the sky visibility. The noise covariance is assumed to be diagonal and equal to the fast-cadence estimate of the variance described in Section 3.1.4. The RFI covariance is also assumed to be diagonal and equal to infinity for times and frequencies that are missing or have been masked by the procedure described in Section 3.2.3 and equal to zero otherwise. Finally, the sky covariance is assumed to be diagonal and constant as a function of baseline, frequency, and sidereal angle, such that ${{\boldsymbol{S}}}^{-1}={s}_{\max }^{-2}\,{\boldsymbol{I}}$, where I is the identity matrix and ${s}_{\max }=1\times {10}^{4}\,\mathrm{Jy}\,{\mathrm{beam}}^{-1}$ is chosen to be around the maximum flux observed on the sky. Each frequency and baseline is solved independently. This is made computationally tractable by utilizing the fact that C ⁻¹ is a band matrix, which is a consequence of the compact support of the Lanczos kernel. Choosing the kernel width, a, is a balance between the computational cost of the regridding (which is O(a²)) and the accuracy of the reconstruction. We use a = 5 in this work, which has deviations away from the ideal sinc transfer of ≲10⁻³ for the typical range of fringe rates in this analysis.

The covariance of the filtered signal is $\langle {\hat{{\boldsymbol{v}}}}_{\mathrm{grid}}{\hat{{\boldsymbol{v}}}}_{\mathrm{grid}}^{\dagger }\rangle ={\boldsymbol{C}}$ as given by Equation (6). The weight data set that tracks the inverse variance of the noise present in the visibilities is therefore updated to w( b , ν, ϕ_h ) = 1/C_hh ( b , ν). The interpolation scheme introduces ringing in the visibilities at the edge of any large gap of missing or masked data, with the post-interpolation weights at these edges gradually transitioning to ${s}_{\max }^{-2}$. In order to mask these artifacts, we apply a baseline-dependent threshold to the weight data set, setting it to zero if it is less than 50% of the average weight over all frequencies and sidereal angles. As these samples lie at the edge of large periods of missing data, the relative increase in the amount of data flagged is small.

3.2.7. Daytime, Moon, and Data Flags

After the individual days have been flagged and processed to a common grid in local Earth rotation angle, the days are averaged together to produce a high-sensitivity measurement of the sky.

This process is complicated by the presence of noise crosstalk, a bias in the zero-level of a non-autocorrelation visibility. Physically this requires both signal chains being correlated to share a common source of thermal noise. We expect there to be two major contributors to this: one is the leaking of thermal noise generated within the low-noise amplifier on one signal chain that is broadcast by the antenna and received (directly, or by an indirect path) by another antenna; another source is antennas seeing thermal emission from the ground by a direct path or by scattering or diffraction from nearby structures. The correlation of this common source of noise yields a bias in the visibility between the two antennas. A thorough investigation of the source of the crosstalk is beyond the scope of this paper; however, we will discuss the crosstalk briefly in Section 3.3.4.

We observe the crosstalk to be relatively stable in time, varying slowly over the course of 1 day (we quantify this in Section 3.3.4). In practice, this allows the crosstalk removal to be performed by estimating and removing a single time-independent signal from each day for each frequency and baseline. However, as the crosstalk signal is not known a priori and must be measured from the data, it is degenerate with any constant sky signal within the time period being used to estimate it.

As we use only nighttime data spread over a year, there is no single period in common between all days that we can choose as a reference. To account for this, we break the sidereal averaging into two stages: the first operates on data taken from each quarter of the year, and the second combines those into a full stacking of the data.

3.3.1. Sidereal Day Flags

3.3.2. Sidereal Averaging (Seasonal)

The first stage of sidereal averaging combines data from a single quarter of each calendar year and assigns each "good" day of data into alternating partitions of the data. By splitting into partitions per quarter, we are able to produce two jackknife splits of our data that have approximately the same sensitivity and sidereal coverage; these will be used for consistency tests in Section 7.2.

For each quarter, we pick a single hour-long range in local Earth rotation angle that is observed within the nighttime for the entire quarter and avoids the transits of bright point sources. This time range is used to reference the crosstalk signal for the entire quarter. We illustrate these ranges for each quarter, as well as how the quarters overlap with the eBOSS NGC field, in Figure 4.

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Every day we calculate the median over this time range for each visibility and subtract it from the data for that day. Assuming that the crosstalk signal is approximately constant across the day, this procedure will remove that day's crosstalk contamination and a small amount of the sky signal, which is the same across all days within the quarter. It is important to use consistent estimates of the crosstalk; therefore, if more than 70% of the data within this reference range are missing for a frequency on a given day, the entire frequency will be flagged out for the whole day. This differs from the initial selection discussed in Section 3.3.1, as it is determined from the full frequency-dependent missing data mask for that day, not just the frequency-independent data flags.

After the crosstalk has been removed consistently from all days within the quarter, the days within each partition are averaged together with an inverse variance weighting.

3.3.3. Sidereal Averaging (All)

The second stage of sidereal averaging is to combine the data for all quarters and partitions. As the crosstalk removal uses a different sky reference region for each quarter, a simple averaging would introduce discontinuities at the boundaries. To account for this, we exploit the overlap in local Earth rotation angle of the nighttime data for each quarter with its neighbors to solve for the differences and set a common reference.

To do this, we treat our estimate of the regularly gridded visibility v _grid,i (where we have dropped the $\,\hat{}\,$ symbol to simplify notation) for each frequency and baseline within a partition i (out of p total partitions) as being composed of a signal v that we are interested in that is constant for all partitions, a noise n _i , and a residual crosstalk contribution x _i that is different for each partition and also incorporates the bias from the per-partition crosstalk referencing. We write this as

$$\begin{eqnarray}&&{{\boldsymbol{v}}}_{\mathrm{grid},i}={\boldsymbol{v}}+{{\boldsymbol{n}}}_{i}+{{\boldsymbol{x}}}_{i}.\end{eqnarray} \tag{ 7 }$$

We model the statistics of each component as having zero mean with covariance matrices 〈 v v ^†〉 = S , $\langle {{\boldsymbol{n}}}_{i}{{\boldsymbol{n}}}_{i}^{\dagger }\rangle ={{\boldsymbol{N}}}_{i}$, and $\langle {{\boldsymbol{x}}}_{i}{{\boldsymbol{x}}}_{i}^{\dagger }\rangle ={\boldsymbol{X}}$. As the crosstalk has little time variation, we model the residuals as a low-rank contribution (with rank k), allowing us to factorize the covariance as X = U U ^†, where U is a rectangular matrix. Though the crosstalk referencing means that the modes x _i may be very different, we assume that the crosstalk statistics are the same across all partitions, so X does not depend on i. The noise matrix N _i is assumed to be diagonal and includes both the noise expected in the data (Section 3.1.4) and any masking that has been applied (Sections 3.2.3 and 3.2.7), encoded in the standard way of setting the inverse variance to zero for masked samples. Although the averaging over sidereal days has reduced the number of samples that are flagged entirely, ranges of R.A. observed during the daytime for the entire quarter and badly RFI-contaminated frequencies will still be masked.

To solve for the signal, we start by writing a Wiener estimator for s treating both n _i and x _i as a generalized noise

$$\begin{eqnarray}&&\hat{{\boldsymbol{v}}}={\boldsymbol{C}}\left[\displaystyle \sum _{i}{\left({{\boldsymbol{N}}}_{i}+{\boldsymbol{X}}\right)}^{-1}{{\boldsymbol{v}}}_{\mathrm{grid},i}\right],\end{eqnarray} \tag{ 8 }$$

where the covariance matrix C is defined by

$$\begin{eqnarray}&&{{\boldsymbol{C}}}^{-1}={{\boldsymbol{S}}}^{-1}+\displaystyle \sum _{i}{\left({{\boldsymbol{N}}}_{i}+{\boldsymbol{X}}\right)}^{-1}.\end{eqnarray} \tag{ 9 }$$

A naive application of this scheme would require tracking and inverting a large matrix for each frequency, but we can simplify it by repeated application of the Woodbury matrix identity. ²³ First, we expand the N _i + X term, allowing us to regroup Equation (9) as

$$\begin{eqnarray}&&{{\boldsymbol{C}}}^{-1}={{\boldsymbol{C}}}_{0}^{-1}+{\boldsymbol{W}}{{\boldsymbol{W}}}^{\dagger },\end{eqnarray} \tag{ 10 }$$

where

$$\begin{eqnarray}&&{{\boldsymbol{C}}}_{0}^{-1}={{\boldsymbol{S}}}^{-1}+\displaystyle \sum _{i}{{\boldsymbol{N}}}_{i}^{-1}\end{eqnarray} \tag{ 11 }$$

and W is a block matrix,

$$\begin{eqnarray}&&{\boldsymbol{W}}=\left({{\boldsymbol{W}}}_{0}\,| \,{{\boldsymbol{W}}}_{1}\,| \,\ldots \,| \,{{\boldsymbol{W}}}_{p-1}\right),\end{eqnarray} \tag{ 12 }$$

with one block for each partition, and within each block are k columns for each crosstalk mode and a row for each R.A. sample. The blocks are

$$\begin{eqnarray}&&{{\boldsymbol{W}}}_{i}={{\boldsymbol{N}}}_{i}^{-1}{\boldsymbol{U}}{\left({{\boldsymbol{I}}}_{k}+{{\boldsymbol{U}}}^{\dagger }{{\boldsymbol{N}}}_{i}^{-1}{\boldsymbol{U}}\right)}^{-1/2},\end{eqnarray} \tag{ 13 }$$

where I _k is the identity matrix of size k, and each W _i can be interpreted as a noise-weighted projection operator onto the crosstalk basis for each partition.

The estimator in Equation (8) can be rewritten as

$$\begin{eqnarray}&&\hat{{\boldsymbol{v}}}={\boldsymbol{C}}\left[\displaystyle \sum _{i}({{\boldsymbol{N}}}_{i}^{-1}-{{\boldsymbol{W}}}_{i}{{\boldsymbol{W}}}_{i}^{\dagger }){{\boldsymbol{v}}}_{\mathrm{grid},i}\right].\end{eqnarray} \tag{ 14 }$$

A second application of the Woodbury identity, this time to Equation (10), allows us to write C in a more easily applied form:

$$\begin{eqnarray}&&{\boldsymbol{C}}={{\boldsymbol{C}}}_{0}+{{\boldsymbol{C}}}_{0}{\boldsymbol{W}}{\left({{\boldsymbol{I}}}_{(k\times p)}-{{\boldsymbol{W}}}^{\dagger }{{\boldsymbol{C}}}_{0}{\boldsymbol{W}}\right)}^{-1}{{\boldsymbol{W}}}^{\dagger }{{\boldsymbol{C}}}_{0}.\end{eqnarray} \tag{ 15 }$$

To produce the final estimate for the stacked signal, we need to generate W _i for each day and retain it, accumulate $({{\boldsymbol{N}}}_{i}^{-1}-{{\boldsymbol{W}}}_{i}{{\boldsymbol{W}}}_{i}^{\dagger }){{\boldsymbol{v}}}_{\mathrm{grid},i}$ to generate ${{\boldsymbol{C}}}^{-1}\hat{{\boldsymbol{v}}}$, and then finally apply the deconvolving matrix C , which can be done efficiently by evaluating matrix-vector products from right to left in Equation (15) using the accrued W _i rather than explicit construction of C . Conceptually this final step uses the noise-weighted overlaps between the different partitions (in the I − W ^† C ₀ W term) to solve for a consistent bias and remove it.

In the implementation within our pipeline we model the crosstalk as a single time-independent constant mode per day (i.e., k = 1 and U ∝ 1, where 1 is a column vector filled with ones). We also assume that both X and S are much larger than the instrumental noise N for unmasked samples. This means that the estimator we use does not depend on S at all, nor on the scale of X (but it does depend on the form), and, importantly, means that C ₀ is a diagonal matrix. However, this does produce one singular mode, the sidereal average of each visibility, that must be regularized externally. Finally, rather than using the N for each baseline, we use an average over all baselines, which ensures that the same linear combinations of partitions are used for all baselines at a given frequency. We use these same linear combinations when updating the baseline-dependent weights in the final stack, although we drop the small correction to the weights that comes from removing the crosstalk, which primarily affects the off-diagonal elements of the noise covariance that we do not track in our analysis, for memory reasons.

As the sidereal-time-independent component of the sky is entirely degenerate with a constant noise bias, the mean of each sidereal stream is a singular mode. To regularize this degenerate mode, we add a constant offset to set the median in time of the full sidereal day to zero.

3.3.4. Crosstalk Properties

Throughout this analysis we have made the strong assumption that we can model the crosstalk as a constant mode per day. To test the validity of this, we use the estimate of the true sky $\hat{V}$ produced above and compare it with the data for each individual day V_grid,i . Assuming that the true sky estimate is faithful, this allows us to estimate the time-dependent crosstalk within that day. Practically, we do this by regressing these inputs against each other in short time intervals for each baseline and frequency, i.e., we fit a model

$$\begin{eqnarray}&&{V}_{\mathrm{grid},i}({\boldsymbol{b}},\nu ,\phi )=G({\boldsymbol{b}},\nu )\,\hat{V}({\boldsymbol{b}},\nu ,\phi )+X({\boldsymbol{b}},\nu ),\end{eqnarray} \tag{ 16 }$$

where G is an overall scaling factor and X is the crosstalk estimate for the interval. We use 64 intervals over the sidereal day, which gives a time resolution of ∼22 minutes. We exclude daytime intervals from our fit, as well as periods around bright source and lunar transits.

The accuracy of the inferred crosstalk is limited by the fact that we do not have an independent estimate of the true sky, and so we depend on the averaging over days that produced it to suppress time/R.A.-dependent crosstalk variation, and that assuming that the visibilities of the true sky have zero median does not significantly bias the crosstalk estimate.

Using the extracted crosstalk estimates across each day, we compute three summaries of the time dependence, which we show in Figure 5. The first is the mean crosstalk calculated over the full time range analyzed in this paper; the second is the interday variation, which is defined as the variance over days of the mean of each sidereal day; and the final summary is the intraday variation, which is the mean over days of the variance within each sidereal day. In all cases we have used a simple outlier cut on the intraday variance to remove anomalous time samples and frequencies, though some residual contributions remain that generate the baseline-independent horizontal banding visible in both the inter- and intraday variation. The following summary statistics we quote are all for instrumental Stokes I and at a frequency (≈613.7 MHz) observed to have low levels of interference.

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As we might expect, the crosstalk is significantly stronger for baselines within a cylinder, with the rms crosstalk taken over north–south (NS) baselines ∼30 times larger than between neighboring cylinders and ∼70 and ∼130 times larger than the two- and three-cylinder separations. For the intracylinder crosstalk, the signal is strongly concentrated in the shortest baselines, with 90% of the cumulative rms coming from baselines ≲5.5 m and 99% from ≲12.8 m. A delay-space analysis indicates that most of the crosstalk contributions correspond to specific path lengths. Within a single cylinder, the major contributions appear to be from the indirect paths between two feeds, with one reflection from the cylinder vertex (with smaller contribution from paths with multiple reflections); between cylinders, the major contribution is from the direct geometric path between the feeds, with smaller contributions from this path combined with a straight up-down reflection of the signal at either feed. These observations lead us to believe that the dominant crosstalk mechanism is from rebroadcast amplifier noise, though we cannot rule out a contribution from ground pickup. Regardless, the crosstalk removal and analysis that follows do not depend on the origin.

The importance of the time dependence of the crosstalk differs depending on the cylinder separation, with the fractional contributions being larger at longer cylinder separations, where the typical crosstalk is lower. Overall we find that the rms intraday crosstalk variations are ≈3%, 5%, 9%, and 15% of the mean level for the zero-to-three-cylinder separations, respectively, and the interday variations ≈5%, 8%, 10%, and 17%. We believe that the low intraday variation justifies our decision to use a single crosstalk reference region in each day, but the small separation between the inter- and intraday variation means that to make further improvements we will need to account for the time variation across each day.

The CHIME feeds have nonnegligible off-axis response. As a result, the signal from the four brightest point sources—Cygnus A, Cassiopeia A, Taurus A, and Virgo A—is significant whenever these sources are above the horizon. The first stage of the stacking pipeline performs a targeted removal of these four sources. All other foregrounds are removed from the data using a spectral filter that will be described in Section 4.5. Prior to filtering, the spectral structure introduced by the on-axis response of the instrument is calibrated (see Section 4.4) and deconvolved from the data (see Section 4.3). The effectiveness of this calibration will be quantified in Section 7.3 by comparing the recovered flux density of a set of standard calibrators to measurements made by other telescopes.

The following model is assumed for the contribution of the four brightest point sources to the visibility measured by baseline b at local Earth rotation angle ϕ and frequency ν:

$$\begin{eqnarray}&&{V}_{\mathrm{psrc}}({\boldsymbol{b}},\nu ,\phi )=\displaystyle \sum _{s=1}^{4}{a}_{s}(\nu ,\phi )\ {e}^{j2\pi \nu {\boldsymbol{b}}\cdot \hat{{\boldsymbol{n}}}({\theta }_{s},\phi -{\phi }_{s})/c},\end{eqnarray} \tag{ 17 }$$

where a_s (ν, ϕ), θ_s , and ϕ_s denote the primary-beam-modulated amplitude, decl., and R.A. of source s, respectively; $\hat{{\boldsymbol{n}}}$ is the unit vector pointing toward the source's location; and c is the speed of light. At every frequency and local Earth rotation angle we estimate the set of source amplitudes a using weighted linear regression:

$$\begin{eqnarray}&&\hat{{\boldsymbol{a}}}={\left({{\boldsymbol{Z}}}^{\dagger }{{\boldsymbol{N}}}^{-1}{\boldsymbol{Z}}\right)}^{-1}{{\boldsymbol{Z}}}^{\dagger }{{\boldsymbol{N}}}^{-1}{\boldsymbol{v}}.\end{eqnarray} \tag{ 18 }$$

Here v is a vector containing the visibilities for a selection of baselines,

$$\begin{eqnarray}&&{Z}_{{is}}={e}^{j2\pi \nu {{\boldsymbol{b}}}_{i}\cdot \hat{{\boldsymbol{n}}}({\theta }_{s},\phi -{\phi }_{s})/c}\end{eqnarray} \tag{ 19 }$$

is the geometric phase factor for baseline i and source s, and N is the noise covariance. As before, we assume that the noise covariance is diagonal and equal to the propagated fast-cadence estimate of the variance (see Section 3.1.4).

The amplitude a_s is equal to the spectral flux density of source s modulated by the power beam pattern of the instrument at the source's coordinates and is expected to vary slowly as a function of frequency and hour angle. To improve the signal-to-noise ratio, the best-fit amplitude for each source is smoothed in (ν, ϕ) by iteratively applying a 2D moving average window with size (1.2 MHz, 044) and number of iterations (12, 8). The model for the four brightest sources is then computed using Equation (17) and subtracted from the data. Note that only visibilities measured by baselines consisting of feeds on different cylinders are used to solve for the source amplitudes—because contamination from diffuse Galactic emission and noise crosstalk is significantly reduced for these intercylinder baselines—but the resulting model is subtracted from all baselines.

The inverse variance weights are multiplied by a global frequency mask that completely excludes certain frequency channels from the stacking analysis. The list below gives the conditions under which a frequency channel is masked and the fraction of the 587.5–800 MHz band that meets each condition.

• 1.  
  Mask any frequency channel that coincides with a known, persistent source of RFI. There were two sources of persistent RFI in the 587.5–800 MHz band: the mobile LTE bands, and the LO used by the Synthesis Telescope at DRAO (Landecker et al. 2000). These are shown in Figure 2 (14.2% masked).
• 2.  
  Mask any frequency channel where the sidereal stack is missing a subset (or all) of the full sidereal day, which prevents a straightforward application of the m-mode transform required for mapmaking. This could be due to a GPU node that was not operational for a significant portion of 2019, as one example (12.5% masked).
• 3.  
  Mask any frequency channel where the total integration time over the range of R.A. coinciding with the NGC field is less than 75% of the maximum over frequencies. Again, most often this is due to a temporarily nonoperational GPU node (5.9% masked).
• 4.  
  Mask any frequency channel where manual inspection of the foreground-filtered map in an initial iteration of the analysis revealed residuals that are large relative to the expected radiometric noise and corrupt a significant fraction of the NGC field. This procedure is described in greater detail in Section 4.6 (14.7% masked).

In total, these four conditions mask 47.2% of the 587.5–800 MHz band.

The next step in the data processing is to construct a map from the sidereal visibilities. We use a mapmaking technique that is tailored to CHIME, or effectively any transit radio interferometer consisting of cylindrical telescopes oriented in the NS direction with a close-packed array of antennas along the axis of each cylinder. The technique draws on the work presented in Shaw et al. (2014) and Masui et al. (2017) but is distinct and has not been described elsewhere, so we go into considerable detail in this section. Note that these types of maps are referred to as deconvolved ringmaps in CHIME Collaboration et al. (2022b).

4.3.1. Baseline Configuration

To good approximation, the CHIME baselines b _c are located on a 2D grid that lies in the plane tangent to Earth's surface at (latitude, longitude) ≡ (Λ, Φ) = (49320709, − 119623677). The 2D grid is given by

$$\begin{eqnarray}&&{{\boldsymbol{b}}}_{{\boldsymbol{c}}}={{xd}}_{x}\hat{{{\boldsymbol{x}}}_{c}}+{{yd}}_{y}{\hat{{\boldsymbol{y}}}}_{{\boldsymbol{c}}},\end{eqnarray} \tag{ 20 }$$

where $\hat{{{\boldsymbol{x}}}_{c}}$ is the unit vector that is orthogonal to the cylinder, ${\hat{{\boldsymbol{y}}}}_{{\boldsymbol{c}}}$ is the unit vector parallel to the cylinder, d_x = 22.0 m is the (center-to-center) cylinder spacing, d_y = 0.3048 m is the spacing of the feeds along the focal line, and the grid indices are denoted by x ∈ [−3, 3] and y ∈ [−255, 255].

The sidereal visibilities are arranged onto this 2D grid. Let ${V}_{{xy}}^{{pq}}(\nu ,\phi )$ denote the visibility measured at frequency ν and local Earth rotation angle ϕ by the baseline at the (x, y) grid position. The variables p, q ∈ {X, Y} refer to the polarizations of the two antennas that form the baseline, with the dipole of the X and Y polarizations oriented in the $\hat{{{\boldsymbol{x}}}_{c}}$- and ${\hat{{\boldsymbol{y}}}}_{{\boldsymbol{c}}}$-directions, respectively. The analysis presented in this work will only use the co-polar baselines, XX and YY, so that p = q, and we drop the redundant index in the notation going forward. Note that it is assumed that the visibilities have conjugate symmetry about the origin, specifically

$$\begin{eqnarray}&&{V}_{-x,-y}^{p}\equiv {\left({V}_{{xy}}^{p}\right)}^{* }.\end{eqnarray} \tag{ 21 }$$

The CHIME cylinders were aligned with the NS direction by design. However, we have empirically determined that the cylinders are rotated by ψ = − 0071 with respect to true astronomical north using observations of a large number of bright point sources (CHIME Collaboration et al. 2022b). Let b = R (ψ) b _c denote the baselines in a coordinate system where $\hat{{\boldsymbol{x}}}$ is aligned with the east–west (EW) direction, $\hat{{\boldsymbol{y}}}$ is aligned with the NS direction, and

$$\begin{eqnarray}{\boldsymbol{R}}(\psi )=\left[\begin{array}{cc}\cos \psi & -\sin \psi \\ \sin \psi & \cos \psi \end{array}\right]\end{eqnarray} \tag{ 22 }$$

is the rotation matrix that transforms between the cylinder-based coordinate system and the NS based coordinate system.

The measured visibility is the true sky visibility ${ \mathcal V }$ corrupted by noise n,

$$\begin{eqnarray}&&{V}_{{xy}}^{p}(\nu ,\phi )={{ \mathcal V }}_{{xy}}^{p}(\nu ,\phi )+{n}_{{xy}}^{p}(\nu ,\phi ).\end{eqnarray} \tag{ 23 }$$

The sky visibility ${ \mathcal V }$ is the integral of the spectral flux density, S, of the sky multiplied by the primary beam pattern, A, of the two feeds and a geometric phase factor set by the baseline between the feeds:

$$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal V }}_{{xy}}^{p}(\nu ,\phi ) & = & \displaystyle \int {\left|{A}^{p}(\nu ,\theta ^{\prime} ,\phi -\phi ^{\prime} )\right|}^{2}\ {e}^{j2\pi \nu {\boldsymbol{b}}\cdot \hat{{\boldsymbol{n}}}(\theta ^{\prime} ,\phi -\phi ^{\prime} )/c}\\ & & \times \,S(\nu ,\theta ^{\prime} ,\phi ^{\prime} )\cos \theta ^{\prime} d\theta ^{\prime} d\phi ^{\prime} .\end{array}\end{eqnarray} \tag{ 24 }$$

Here c is the speed of light and $\hat{{\boldsymbol{n}}}(\theta ^{\prime} ,\phi -\phi ^{\prime} )$ is the unit vector pointing toward decl. $\theta ^{\prime}$ and hour angle ${\rm\small{HA}}\equiv \phi -\phi ^{\prime}$ and is given by

$$\begin{eqnarray}&&\begin{array}{l}\hat{{\boldsymbol{n}}}(\theta ^{\prime} ,\phi -\phi ^{\prime} )=-\cos \theta ^{\prime} \sin (\phi -\phi ^{\prime} )\ \hat{{\boldsymbol{x}}}\\ \quad +\,(\cos {\rm{\Lambda }}\sin \theta ^{\prime} -\sin {\rm{\Lambda }}\cos \theta ^{\prime} \cos (\phi -\phi ^{\prime} ))\ \hat{{\boldsymbol{y}}}\\ \quad +\,(\sin {\rm{\Lambda }}\sin \theta ^{\prime} +\cos {\rm{\Lambda }}\cos \theta ^{\prime} \cos (\phi -\phi ^{\prime} ))\ \hat{{\boldsymbol{z}}},\end{array}\end{eqnarray} \tag{ 25 }$$

with Λ denoting the latitude of the telescope. Note that Equation (24) assumes that the primary beam pattern is the same for all feeds of a given polarization. It also assumes that there are no residual complex gain variations.

4.3.2. North–South Beamforming

The CHIME power beam, ∣A∣², is reasonably compact in the hour-angle direction. The FWHM of the main lobe is ≲21 (25) for the Y (X) polarization in the 587.5–800 MHz band, and the sidelobes are ≲1% (CHIME Collaboration et al. 2022b). If we restrict the integral in Equation (24) to the range of hour angles covering the main lobe of the primary beam, then we can expand the geometric phase to first order in the small angles HA and ψ to obtain

$$\begin{eqnarray}\begin{array}{rcl}{\boldsymbol{b}}\cdot \hat{{\boldsymbol{n}}}(\theta ^{\prime} ,\phi -\phi ^{\prime} ) & = & {{xd}}_{x}\left[\sin \left(\theta ^{\prime} -{\rm{\Lambda }}\right)\psi -\cos \theta ^{\prime} \ (\phi -\phi ^{\prime} )\right]\\ & & +\,{{yd}}_{y}\sin \left(\theta ^{\prime} -{\rm{\Lambda }}\right).\end{array}\end{eqnarray} \tag{ 26 }$$

The geometric phase due to the y component of the baseline is given by the second term in Equation (26). Since this term depends only on decl. and does not depend on hour angle, we can form a linear combination of all visibilities with the same x so that only signal from a specific decl., θ, adds coherently:

$$\begin{eqnarray}&&{V}_{x}^{p}(\nu ,\phi ,\theta )=\displaystyle \sum _{y}{W}_{{xy}}^{p}(\nu ,\phi )\ {V}_{{xy}}^{p}(\nu ,\phi )\ {e}^{-j2\pi \nu {{yd}}_{y}\sin (\theta -{\rm{\Lambda }})/c},\end{eqnarray} \tag{ 27 }$$

where

$$\begin{eqnarray}&&{W}_{{xy}}^{p}(\nu ,\phi )={w}_{{xy}}^{p}(\nu ,\phi )/\displaystyle \sum _{y^{\prime} }{w}_{{xy}^{\prime} }^{p}(\nu ,\phi )\end{eqnarray} \tag{ 28 }$$

denotes the relative weights, which are normalized to preserve point-source flux. This beamforming operation is repeated for a grid of pointings that span from horizon to horizon and are equally spaced in $\sin (\theta -{\rm{\Lambda }})$. This operation can be done efficiently with a fast Fourier transform (FFT), but, in practice, we evaluate the expression directly to ensure that the grid of pointings is the same for all frequencies.

We will refer to ${V}_{x}^{p}(\nu ,\phi ,\theta )$ as the hybrid beamformed visibility, since the NS component of the baseline has been beamformed to a specific decl., but there is still fringing associated with the EW component of the baseline. Combining Equations (24)–(27), we obtain the following theoretical expression for the hybrid beamformed visibilities:

$$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal V }}_{x}^{p}(\nu ,\phi ,\theta ) & = & \displaystyle \int {b}_{\mathrm{synth}}^{\hat{\theta },p}(\nu ,\theta ,\theta ^{\prime} ,\phi )\ {B}_{x}^{p}(\nu ,\theta ^{\prime} ,\phi -\phi ^{\prime} )\\ & & \times \,S(\nu ,\theta ^{\prime} ,\phi ^{\prime} )\ \cos \theta ^{\prime} d\theta ^{\prime} d\phi ^{\prime} ,\end{array}\end{eqnarray} \tag{ 29 }$$

where

$$\begin{eqnarray}&&\begin{array}{l}{B}_{x}^{p}(\nu ,\theta ^{\prime} ,\phi -\phi ^{\prime} )\\ \,\,=\,| {A}^{p}(\nu ,\theta ^{\prime} ,\phi -\phi ^{\prime} ){| }^{2}\ {e}^{j2\pi \nu {{xd}}_{x}\left[\sin (\theta ^{\prime} -{\rm{\Lambda }})\psi -\cos \theta ^{\prime} \ (\phi -\phi ^{\prime} )\right]/c}\end{array}\end{eqnarray} \tag{ 30 }$$

will be referred to as the beam transfer function and

$$\begin{eqnarray}&&{b}_{\mathrm{synth},x}^{\hat{\theta },p}(\nu ,\theta ,\theta ^{\prime} ,\phi )=\displaystyle \sum _{y}{W}_{{xy}}^{p}(\nu ,\phi )\ {e}^{j2\pi \nu {{yd}}_{y}\left[\sin (\theta ^{\prime} -{\rm{\Lambda }})-\sin (\theta -{\rm{\Lambda }})\right]/c}\end{eqnarray} \tag{ 31 }$$

is the synthesized beam in the $\hat{\theta }$-direction. The top panel of Figure 7 shows an example of ${b}_{\mathrm{synth}}^{\hat{\theta }}$ for the weighting scheme used in this analysis.

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The absolute weights in Equation (28) are set to the inverse variance of the corresponding visibility, i.e.,

$$\begin{eqnarray}&&{w}_{{xy}}^{p}(\nu ,\phi )={\left[\mathrm{Var}({V}_{{xy}}^{p}(\nu ,\phi ))\right]}^{-1},\end{eqnarray} \tag{ 32 }$$

which will maximize the point-source sensitivity since the amplitude of a true point source is the same for all baselines. We describe how the variance of the visibilities is estimated in Section 3.1.4. The inverse variance weights scale approximately as the number of redundant baselines that are averaged together by the real-time pipeline to produce ${V}_{{xy}}^{p}(\nu ,\phi )$, which scales with the NS baseline distance as (256 − ∣y∣). As a result, the inverse variance weights produce an approximately triangular window function in y. This yields a synthesized beam ${b}_{\mathrm{synth}}^{\hat{\theta }}$ that has an FWHM ranging from 035 at 585 MHz to 025 at 800 MHz and sidelobes that range from 0.05 to 10⁻⁴ of the peak. Note that, instead of inverse variance weights, we could set the weights to any window function that further suppresses the sidelobes at the expense of point-source sensitivity.

In principle, the synthesized beam ${b}_{\mathrm{synth}}^{\hat{\theta }}$ depends on both the EW baseline distance x and the local Earth rotation angle ϕ, because the inverse variance weights change with these parameters. However, the weights that are used in this analysis yield a synthesized beam that is quite stable with ϕ and similar across x. Indeed, the standard deviation of the synthesized beam over ϕ is at most 0.1% (relative to the peak) over all polarizations, frequencies, and decl., and the standard deviation over x is at most 2%. In order to simplify the derivation that follows, we will drop the dependence of the synthesized beam on both ϕ and x. This assumption can be enforced directly—while maintaining roughly the same sensitivity—by explicitly using the triangular window function, or, in other words, by setting w_xy (ν, ϕ) = 256 − ∣y∣. Doing so, we find no appreciable change in either the signal or noise in the stacks on the eBOSS catalogs.

The regularly gridded baselines do not Nyquist sample the visibility of the sky in the $\hat{y}$-direction for frequencies $\nu \geqslant \tfrac{c}{2{d}_{y}}\approx 492\,\mathrm{MHz}$, which includes all frequencies considered in this analysis. As a result, the hybrid beamformed visibilities will suffer from aliasing. In this derivation, the effects of aliasing are encoded in the synthesized beam ${b}_{\mathrm{synth}}^{\hat{\theta }}$. Let $\beta (\nu )\equiv \tfrac{c}{\nu {d}_{y}}-1$. If $\sin \left(\theta -{\rm{\Lambda }}\right)\lt -\beta (\nu )$ or $\sin \left(\theta -{\rm{\Lambda }}\right)\gt \beta (\nu )$, then the synthesized beam will have two main lobes, one centered on the desired decl. ${\theta }^{{\prime} }=\theta$ and a duplicate centered on the frequency-dependent aliased decl. ${\theta }_{\mathrm{alias}}^{{\prime} }(\nu )$, given by the equation

$$\begin{eqnarray}&&\begin{array}{l}\sin \left({\theta }_{\mathrm{alias}}^{{\prime} }(\nu )-{\rm{\Lambda }}\right)\\ \,\,=\,\left\{\begin{array}{ll}\sin \left(\theta -{\rm{\Lambda }}\right)+\displaystyle \frac{c}{\nu {d}_{y}} & {\rm{if}}\,-1\leqslant \sin \left(\theta -{\rm{\Lambda }}\right)\leqslant -\beta (\nu )\\ \sin \left(\theta -{\rm{\Lambda }}\right)-\displaystyle \frac{c}{\nu {d}_{y}} & {\rm{if}}\,\beta (\nu )\leqslant \sin \left(\theta -{\rm{\Lambda }}\right)\leqslant 1.\end{array}\right.\end{array}\end{eqnarray} \tag{ 33 }$$

Hence, the hybrid beamformed visibility ${V}_{x}^{p}(\nu ,\phi ,\theta )$ will contain equal contributions from the sky (modulated by the beam transfer function) at ${\theta }^{{\prime} }$ and ${\theta }_{\mathrm{alias}}^{{\prime} }(\nu )$. This is illustrated in the top panel of Figure 7. At the upper edge of the band, β(800 MHz) = 0.23, which implies that there is a stripe of the sky centered on zenith (specifically θ ∈ [360, 626]) that is free from aliases at all CHIME frequencies. Outside of this stripe, the aliased sky is heavily attenuated in the intercylinder baselines by utilizing the fact that it will fringe at a different rate than the true sky. This is discussed further below.

The first sidelobe of the synthesized beam in the $\hat{\theta }$-direction has an amplitude that is 5% of the amplitude of the main lobe, the next sidelobe is 1%–2%, and beyond roughly 2° separation all sidelobes are below 0.5%. We therefore assume that a hybrid beamformed visibility is dominated by the sky at a narrow range of decl. centered on θ. This assumption will start to break down at right ascensions that coincide with bright foregrounds. This problem is mitigated to a certain extent by subtracting the four brightest point sources directly from the sidereal visibilities, as explained in Section 4.1, and using only intercylinder baselines that resolve out the bright, diffuse Galactic emission, which will be explained below.

4.3.3. Primary Beam Deconvolution

Since the beam transfer function does not change appreciably on scales less than the FWHM of the synthesized beam, it can be brought outside of the integral over $\theta ^{\prime}$ in Equation (29), resulting in the following equation:

$$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal V }}_{x}^{p}(\nu ,\phi ,\theta ) & = & \displaystyle \int {B}_{x}^{p}(\nu ,\theta ,\phi -\phi ^{\prime} )\ \cos \theta \ d\phi ^{\prime} \\ & & \times \,\displaystyle \int {b}_{\mathrm{synth}}^{\hat{\theta },p}(\nu ,\theta ,\theta ^{\prime} )\ S(\nu ,\theta ^{\prime} ,\phi ^{\prime} )\ d\theta ^{\prime} .\end{array}\end{eqnarray} \tag{ 34 }$$

Given a model for the primary beam A, the beam transfer function is computed using Equation (30) and then deconvolved from the hybrid beamformed visibilities at each decl. to recover the flux density of the sky S convolved with ${b}_{\mathrm{synth}}^{\hat{\theta }}$. The construction of the primary beam model will be described in Section 4.4. The FFT of the hybrid beamformed visibility is taken along the ϕ-axis,

$$\begin{eqnarray}&&{\tilde{V}}_{{xm}}^{p}(\nu ,\theta )=\displaystyle \frac{1}{{N}_{\phi }}\displaystyle \sum _{n}{V}_{x}^{p}(\nu ,{\phi }_{n},\theta ){e}^{-{jm}{\phi }_{n}}.\end{eqnarray} \tag{ 35 }$$

Here the sum runs over the N_ϕ = 4096 samples on the grid in local Earth rotation angle and the m-modes range over [−2047, 2048]. We will refer to this operation as the m-mode transform going forward. The same operation is performed on the beam transfer function. Figure 8 provides an example of the m-mode transform of both the hybrid beamformed visibility and the beam transfer function.

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The beam transfer function is then deconvolved from the data in m-space using a Tikhonov regularization scheme

$$\begin{eqnarray}&&{\tilde{M}}_{m}^{p}(\nu ,\theta )=\displaystyle \frac{{\sum }_{x}{W}_{x}^{p}(\nu )\ {\tilde{B}}_{{xm}}^{p\ * }(\nu ,\theta )\ {\tilde{V}}_{{xm}}^{p}(\nu ,\theta )}{\eta +{\sum }_{x}{W}_{x}^{p}(\nu )\ {\left|{\tilde{B}}_{{xm}}^{p}(\nu ,\theta )\right|}^{2}},\end{eqnarray} \tag{ 36 }$$

where

$$\begin{eqnarray}&&{W}_{x}^{p}(\nu )={w}_{x}^{p}(\nu )/\displaystyle \sum _{x}{w}_{x}^{p}(\nu )\end{eqnarray} \tag{ 37 }$$

denotes the relative weight given to each EW baseline and η is a regularization parameter. The different EW baselines measure a largely disjoint set of m-modes, with each baseline primarily sensitive to the range of m values centered on

$$\begin{eqnarray}&&{m}_{\mathrm{center},x}(\nu ,\theta )=-2\pi \nu \cos \left(\theta \right){{xd}}_{x}/c,\end{eqnarray} \tag{ 38 }$$

with width

$$\begin{eqnarray}&&{m}_{\mathrm{width}}(\nu ,\theta )=2\pi \nu \cos \left(\theta \right)w/c,\end{eqnarray} \tag{ 39 }$$

where w = 20 m is the width of the cylinder. However, there is some mild overlap that is dependent on the aperture illumination and accounted for by the m-mode transform of the primary beam pattern. Equation (36) first performs a weighted average of the measurements made by the different EW baselines and then deconvolves the primary beam by effectively dividing by the corresponding weighted average of the m-mode transform of the beam transfer function. The regularization parameter η is the assumed inverse signal-to-noise ratio. It defines which m-modes are signal dominated, and hence should be divided by the beam, and conversely which m-modes are noise dominated, and hence should not be amplified further by dividing by the beam.

We set

$$\begin{eqnarray}{w}_{x}^{p}(\nu )=\left\{\begin{array}{ll}0 & x=0\\ {\left[{\sigma }_{x}^{p}(\nu )\right]}^{-2} & | x| \gt 0,\end{array}\right.\end{eqnarray} \tag{ 40 }$$

where

$$\begin{eqnarray}&&{\left[{\sigma }_{x}^{p}(\nu )\right]}^{2}=\displaystyle \frac{1}{{N}_{\phi }^{2}}\displaystyle \sum _{n}{\left[{\sigma }_{x}^{p}(\nu ,{\phi }_{n})\right]}^{2}\end{eqnarray} \tag{ 41 }$$

is the variance of the noise in the m-mode transform of the hybrid beamformed visibility and

$$\begin{eqnarray}&&{\left[{\sigma }_{x}^{p}(\nu ,\phi )\right]}^{2}={\left(\displaystyle \sum _{y}{w}_{{xy}}^{p}(\nu ,\phi )\right)}^{-1}\end{eqnarray} \tag{ 42 }$$

is the variance of the noise in the hybrid beamformed visibility. This weighting scheme masks all intracylinder baselines and propagates the inverse variance weights through the beamforming and m-mode transform for intercylinder baselines. The redundancy of the array results in ${W}_{x}^{p}(\nu )\approx \left[0.0,0.5,0.333,0.166\right]$ for ∣x∣ = [0, 1, 2, 3], corresponding to the intracylinder autocorrelation that is removed, the three-fold redundancy in the one-cylinder separation, the twofold redundancy in the two-cylinder separation, and the single appearance of the three-cylinder separation.

The intracylinder baselines are masked for this analysis because they contain two sources of contamination that are significantly reduced in the intercylinder baselines: (1) large-scale diffuse Galactic emission, and (2) noise crosstalk (see Section 3.3). Since the noise crosstalk changes slowly with time, it contaminates only low m, which is where the signal from the sky resides for intracylinder baselines. Note that the signal from the sky at decl. near the north celestial pole (NCP) will also appear at low m, even for intercylinder baselines. However, the maximum decl. of sources in the eBOSS NGC field is 60°, which is far enough from the NCP that the crosstalk contamination in the intercylinder measurements is negligible.

The beam transfer function of the intercylinder baselines is largely insensitive to the range of m-modes occupied by the aliased sky for the decl. considered in this analysis. This can be shown in a rough way using Equations (33), (38), and (39). The NGC field spans decl. from 13° to 60°, and over this range there is zero overlap between ${m}_{\mathrm{center},x}(\nu ,\theta )\pm \tfrac{1}{2}{m}_{\mathrm{width}}(\nu ,\theta )$ and ${m}_{\mathrm{center},x}(\nu ,{\theta }_{\mathrm{alias}}(\nu ))\pm \tfrac{1}{2}{m}_{\mathrm{width}}(\nu ,{\theta }_{\mathrm{alias}}(\nu ))$ for all ν and for all x > 0. However, examining Figure 8, it is clear that for θ ≲ 20° our actual beam model does have some sensitivity to the aliased sky for the x = 1 baseline. Therefore, although the deconvolution procedure will heavily attenuate the aliased sky, it is still expected to introduce some nonnegligible contamination.

The regularization parameter is set to η = 10⁻⁴. This value was chosen by constructing a map for several different values of η between 10⁻⁶ and 10⁻³ and choosing the value that maximizes the point-source sensitivity. Note that smaller values of the regularization parameter result in better deconvolution of the primary beam in the $\hat{\phi }$-direction, but also higher noise, and were thus disfavored for the analysis presented in this work.

Finally, the deconvolved map is obtained by taking the inverse m-mode transform

$$\begin{eqnarray}&&{M}^{p}(\nu ,\theta ,\phi )=\displaystyle \sum _{m}{\tilde{M}}_{m}^{p}(\nu ,\theta ){e}^{{jm}\phi }.\end{eqnarray} \tag{ 43 }$$

4.3.4. Map Normalization

In order to determine the correct normalization for the map, we consider a radio sky that contains a single point source with unit flux density at decl. θ and local Earth rotation angle $\phi ^{\prime}$. The m-mode transform of the hybrid beamformed visibilities at that decl. is given by

$$\begin{eqnarray}&&{\tilde{{ \mathcal V }}}_{{xm}}^{p}(\nu ,\theta )={\tilde{B}}_{{xm}}^{p}(\nu ,\theta )\ {e}^{-{jm}\phi ^{\prime} }.\end{eqnarray} \tag{ 44 }$$

The source profile along the ϕ-axis of the resulting map is therefore

$$\begin{eqnarray}&&\begin{array}{l}{a}^{p}(\nu ,\theta ,\phi -\phi ^{\prime} )\\ \,\,=\,\displaystyle \sum _{m}\displaystyle \frac{{\sum }_{x}{W}_{x}^{p}(\nu )\ {\left|{\tilde{B}}_{{xm}}^{p}(\nu ,\theta )\right|}^{2}}{\eta +{\sum }_{x}{W}_{x}^{p}(\nu )\ {\left|{\tilde{B}}_{{xm}}^{p}(\nu ,\theta )\right|}^{2}}{e}^{{jm}(\phi -\phi ^{\prime} )},\end{array}\end{eqnarray} \tag{ 45 }$$

and the peak flux density of the source is a^p (ν, θ, 0), which in general is not equal to unity. Therefore, in order to preserve the point-source flux through the mapmaking process, the map is normalized as

$$\begin{eqnarray}&&{M}^{p}(\nu ,\theta ,\phi )\to \displaystyle \frac{{M}^{p}(\nu ,\theta ,\phi )}{{a}^{p}(\nu ,\theta ,0)}\end{eqnarray} \tag{ 46 }$$

and the synthesized beam in the $\hat{\phi }$-direction is given by

$$\begin{eqnarray}&&{b}_{\mathrm{synth}}^{\hat{\phi },p}(\nu ,\theta ,\phi -\phi ^{\prime} )=\displaystyle \frac{{a}^{p}(\nu ,\theta ,\phi -\phi ^{\prime} )}{{a}^{p}(\nu ,\theta ,0)}.\end{eqnarray} \tag{ 47 }$$

The bottom panel of Figure 7 shows an example of ${b}_{\mathrm{synth}}^{\hat{\phi }}$ for the weighting scheme, regularization parameter, and beam model employed in this analysis.

The resulting map is modeled as

$$\begin{eqnarray}&&{M}^{p}(\nu ,\theta ,\phi )={{ \mathcal M }}^{p}(\nu ,\theta ,\phi )+{n}^{p}(\nu ,\theta ,\phi ).\end{eqnarray} \tag{ 48 }$$

Here ${ \mathcal M }$ is related to the flux density of the sky through the relation

$$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal M }}^{p}(\nu ,\theta ,\phi ) & = & \displaystyle \int {b}_{\mathrm{synth}}^{\hat{\theta },p}(\nu ,\theta ,\theta ^{\prime} )\ {b}_{\mathrm{synth}}^{\hat{\phi },p}(\nu ,\theta ,\phi -\phi ^{\prime} )\\ & & \times \,S(\nu ,\theta ^{\prime} ,\phi ^{\prime} )\cos \theta d\theta ^{\prime} d\phi ^{\prime} ,\end{array}\end{eqnarray} \tag{ 49 }$$

where the synthesized beams in the $\hat{\theta }$- and $\hat{\phi }$-directions can be calculated directly from Equations (31) and (47), respectively. The quantity n^p (ν, θ, ϕ) represents the noise in the map.

4.3.5. Variance Estimation

The variance of the noise in the map is estimated as

$$\begin{eqnarray}&&{\left[{\sigma }_{\mathrm{map}}^{p}(\nu ,\theta ,\phi )\right]}^{2}={F}^{p}(\nu ,\phi )\ {\left[{\sigma }_{\mathrm{map}}^{p}(\nu ,\theta )\right]}^{2},\end{eqnarray} \tag{ 50 }$$

where ${\left[{\sigma }_{\mathrm{map}}^{p}(\nu ,\theta )\right]}^{2}$ is obtained by propagating the variance given by Equation (41) through the mapmaking (Equation (36)), inverse m-mode transform (Equation (43)), and normalization (Equation (46)) procedure. The integration time in the sidereal stack is nonuniform, primarily due to seasonal changes in the length of the day and the likelihood of rainfall. As a result, the variance of the noise depends on the local Earth rotation angle. This dependence is lost when propagating the variance through the forward and inverse m-mode transform. The factor F^p (ν, ϕ) approximately recovers this dependence and is given by the weighted average over EW baselines of the fractional change in the variance of the noise in the hybrid beamformed visibilities, i.e.,

$$\begin{eqnarray}&&{F}^{p}(\nu ,\phi )=\displaystyle \sum _{x}{W}_{x}^{p}(\nu )\displaystyle \frac{{\left[{\sigma }_{x}^{p}(\nu ,\phi )\right]}^{2}}{\tfrac{1}{{N}_{\phi }}{\sum }_{n}{\left[{\sigma }_{x}^{p}(\nu ,{\phi }_{n})\right]}^{2}},\end{eqnarray} \tag{ 51 }$$

where ${\sigma }_{x}^{p}(\nu ,\phi )$ is given by Equation (42).

This procedure for propagating the variance through the mapmaking has been validated as follows. We generate visibilities that have been randomly drawn from a circularly symmetric, complex Gaussian distribution with mean 0 and variance equal to the expected variance of the radiometric noise,

$$\begin{eqnarray}&&{V}_{{xy}}^{p}(\nu ,\phi )\sim { \mathcal N }\left(0,\displaystyle \frac{1}{2}{\sigma }_{{xy}}^{p}{\left(\nu ,\phi \right)}^{2}\right)+j{ \mathcal N }\left(0,\displaystyle \frac{1}{2}{\sigma }_{{xy}}^{p}{\left(\nu ,\phi \right)}^{2}\right),\end{eqnarray} \tag{ 52 }$$

where ${ \mathcal N }(\mu ,{\sigma }^{2})$ denotes a Gaussian distribution with mean μ and variance σ². Estimation of the expected radiometric variance ${\sigma }_{{xy}}^{p}{(\nu ,\phi )}^{2}={w}_{{xy}}^{p}{(\nu ,\phi )}^{-1}$ is described in Section 3.1.4. The mapmaking procedure is then applied to this Gaussian noise realization in an identical manner as to the data. The sample variance of the map pixels is calculated in a 2D rolling window in (θ, ϕ) and compared to the estimate given by Equation (50). In general, we find good agreement (≲5%) between the two. This technique of processing a Gaussian noise realization using the same pipeline that is applied to the data will be used in other comparisons below.

Our primary beam model is obtained by deconvolving a model for the radio sky that consists only of extragalactic point sources from the visibilities measured with baselines that have a large EW component. The long baselines resolve out the diffuse Galactic emission, making a point-source-only sky model a reasonable description of the data. There are several high-resolution, large-area sky surveys that can be interpolated to the 400–800 MHz CHIME band to construct this sky model. At lower frequencies we rely on the VLA Low-frequency Sky Survey (Cohen et al. 2007) at 74 MHz and the Westerbork Northern Sky Survey (Rengelink et al. 1997) at 326 MHz. At higher frequencies we rely on the NRAO VLA Sky Survey (Condon et al. 1998) at 1400 MHz and the Green Bank survey (Gregory et al. 1996) at 4850 MHz. The method used to deconvolve the sky model from the data is similar to the method used to construct a map, which was described in the previous section. Whereas the mapmaker deconvolves a model for the primary beam from the hybrid beamformed visibilities to estimate the intensity of the sky, the beam calibration deconvolves a model for the sky intensity from the same hybrid beamformed visibilities in order to estimate the primary beam. Appendix B describes this method in detail.

The resulting power beam, ∣A^Y(ν, θ, ϕ)∣², for the Y polarization array is shown in the top row of Figure 9. We briefly describe the main features of the CHIME primary beam pattern, referring the reader to CHIME Collaboration et al. (2022b) for a more in-depth discussion. The large (50%) ripples that are evident in the frequency and decl. direction are the result of multipath interference. Radiation from the sky can be absorbed and then reradiated by feeds or reflected off the ground plane. It then reflects off the cylinder and interferes with the primary path from the sky. The period of the ripple is ∼30 MHz and is set by the 5 m focal length of the CHIME cylinders. Harmonics at 60 and 90 MHz, which arise from multiple reflections off the focal line and cylinder, are also significant, although they may not be distinguishable by eye in Figure 9. The narrowing of the beam in the hour-angle direction as one moves toward higher frequencies is due to diffraction through the 20 m aperture. The apparent widening of the beam in the hour-angle direction as one approaches the NCP is simply due to the fact that a point at decl. θ travels $\cos \theta$ degrees on the sky for every degree in hour angle that elapses. The beam is normalized to 1.0 on meridian at the decl. of Cygnus A (4073392) at each frequency in order to match how the data are normalized during complex gain calibration. This imprints the interference pattern at the decl. of Cygnus A onto all other decl. The power beam for the X polarization array exhibits the same general features but is slightly wider in both the hour-angle and decl. direction and also has a lower response at zenith because the dipole illuminates the cylinder less efficiently.

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In order to characterize the effect that the ripples in the beam have on our final stacking result, we repeat our analysis with a "control" beam that has the same large-scale properties as the default beam model, but without the small-scale structure in the frequency and decl. direction. The control beam is a modified version of the analytical beam model proposed in Shaw et al. (2015, hereafter S15) for cylindrical telescopes. To briefly summarize the S15 model, the beam pattern of the antenna (henceforth "base" beam) is assumed to be that of a horizontal dipole mounted a distance λ/4 over a conducting ground plane. The response in the EW direction is the result of solving the Fraunhofer diffraction problem for a dipole illuminating an aperture with width equal to the 20 m cylinder width. The response in the NS direction is simply the reflected amplitude of the base beam. The primary beam of the telescope is then the outer product of these two 1D functions.

In this work, the S15 model for the base beam is modified to more accurately describe existing measurements of the CHIME primary beam. The assumption that the base beam for the X polarization is the base beam for Y polarization rotated by 90° is abandoned. The FWHM of the base beam in the EW direction is assumed to be polarization dependent but frequency independent and is obtained by performing a fit to holographic observations of several bright sources made in conjuction with the John A. Galt 26 m telescope (see CHIME Collaboration et al. 2022b for a description of these measurements). The FWHM of the base beam in the NS direction is assumed to be polarization and frequency dependent and is obtained by fitting a flattened Gaussian to the meridian profile of the default beam at each frequency and then fitting the resulting FWHM as a function of frequency to a third-order polynomial in order to smooth over the small-scale ripples while retaining large-scale variations observed in the width of the meridian beam with frequency. The rest of the procedure is unchanged: the beam model is given by the outer product of an EW response obtained by solving the Fraunhofer diffraction problem and a NS response obtained from the reflected base beam amplitude. The resulting beam model is shown in the bottom row of Figure 9.

The deconvolved map described in Section 4.3 is dominated by emission from extragalactic point sources, which is expected to be a factor of ∼10³–10⁵ brighter than the 21 cm signal of interest (Santos et al. 2005). This foreground contamination can be separated from the 21 cm signal on the basis of spectral scale; the foregrounds are expected to be spectrally smooth, whereas the 21 cm signal varies rapidly with frequency (Shaver et al. 1999; Oh & Mack 2003; Liu & Tegmark 2011). For each pixel in the map, we apply a high-pass filter along the frequency axis to supress the foregrounds while retaining some fraction of the 21 cm signal.

Designing an adequate filter is complicated by the fact that—as discussed in Section 4.2—47.2% of the band has been masked in order to remove RFI-like features and other narrowband, instrumental artifacts. The DAYENU technique (Ewall-Wice et al. 2021) is used to construct a linear filter for the irregularly sampled map spectra that achieves the required suppression at large spectral scales. In what follows, we briefly summarize this technique.

Let τ denote the delay, which is the Fourier transform dual to frequency ν. The following simple model is assumed for the covariance of the map as a function of τ:

$$\begin{eqnarray}\tilde{C}(\tau ,\tau ^{\prime} )=\left\{\begin{array}{ll}\frac{{\epsilon }^{-1}}{2{\tau }_{\mathrm{cut}}}\ {\delta }^{{\rm{D}}}(\tau ,\tau ^{\prime} ) & | \tau | \leqslant {\tau }_{\mathrm{cut}}\\ {\rm{\Delta }}\nu \ {\delta }^{{\rm{D}}}(\tau ,\tau ^{\prime} ) & | \tau | \gt {\tau }_{\mathrm{cut}},\end{array}\right.\end{eqnarray} \tag{ 53 }$$

where the region below τ_cut is the region of delay space contaminated by bright foregrounds, is a small number that corresponds to the ratio of the radiometric noise to foreground variance, Δν = 0.390625 MHz is the width of the frequency channel, and ${\delta }^{{\rm{D}}}(\tau ,\tau ^{\prime} )$ is the Dirac delta function. This model results in the following analytical formula for the covariance between frequency channel ν_m and ν_n :

$$\begin{eqnarray}&&{C}_{{mn}}={\epsilon }^{-1}\mathrm{sinc}\left[2\pi {\tau }_{\mathrm{cut}}({\nu }_{m}-{\nu }_{n})\right]+{\delta }_{{mn}}.\end{eqnarray} \tag{ 54 }$$

where $\mathrm{sinc}(x)\equiv \sin (x)/x$ and δ_mn is the Kronecker delta.

To construct the filter, the delay cut ${\tau }_{\mathrm{cut}}^{p}(\theta )$ and stop-band rejection are specified. Note that we allow the delay cut to vary as a function of polarization and decl. Equation (54) is then evaluated for each pair of frequency channels in the 587.5–800 MHz band. Rows and columns of the covariance matrix that correspond to masked frequencies are zeroed and the Moore–Penrose pseudo-inverse is calculated:

$$\begin{eqnarray}&&{{\boldsymbol{R}}}^{p}(\theta )={\left[{{\boldsymbol{m}}}^{T}{{\boldsymbol{C}}}^{p}(\theta ){\boldsymbol{m}}\right]}^{+},\end{eqnarray} \tag{ 55 }$$

where m is a vector that is 1 for valid frequencies and 0 for masked frequencies. The filter is then applied to each map pixel independently,

$$\begin{eqnarray}&&{M}_{\mathrm{hpf}}^{p}({\nu }_{m},\theta ,\phi )=\displaystyle \sum _{n}{R}_{{mn}}^{p}(\theta )\ {M}^{p}({\nu }_{n},\theta ,\phi ).\end{eqnarray} \tag{ 56 }$$

The weights are also propagated through the filtering operation according to

$$\begin{eqnarray}&&{w}_{\mathrm{hpf}}^{p}({\nu }_{m},\theta ,\phi )={\left[\displaystyle \sum _{n}{\left({R}_{{mn}}^{p}(\theta )\right)}^{2}\ {\left({\sigma }_{\mathrm{map}}^{p}({\nu }_{n},\theta ,\phi )\right)}^{2}\right]}^{-1},\end{eqnarray} \tag{ 57 }$$

where ${\sigma }_{\mathrm{map}}^{p}(\nu ,\theta ,\phi )$ is given by Equation (50).

In order to find an appropriate delay cut, the delay power spectrum of the map is estimated as

$$\begin{eqnarray}&&P(\tau ,\theta )={\mathrm{Var}}_{\phi }\left\{{\tilde{M}}^{p}(\tau ,\theta ,\phi )\right\},\end{eqnarray} \tag{ 58 }$$

where $\tilde{M}$ denotes the Fourier transform of the map along the frequency axis. Direct calculation of $\tilde{M}$ through the FFT will result in a point-spread function in delay space that has large sidelobes owing to the band-limited and irregularly spaced nature of our map spectra. This will leak power from the bright foreground to higher delays, thus biasing our determination of τ_cut. To address this, the delay power spectrum is estimated using a Gibbs sampling method, which is described in detail in Appendix A.

Figure 10 shows the delay power spectrum of the map as a function of decl. for each polarization. Note that the variance was calculated over ϕ ∈ [110°, 263°], which corresponds to the range of R.A. covered by the eBOSS NGC field. The delay power spectrum is normalized by the delay power spectrum of the expected radiometric noise. This is obtained by applying the mapmaking and delay power spectrum estimation to a Gaussian noise realization randomly drawn according to Equation (52).

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At high delays, the measured spectrum is in a good agreement with our expectation for the noise, and at low delays we are dominated by foreground emission. Ideally all foreground power would be contained within the bright peak centered on 0 ns. However, the ripples in the primary beam are imprinted on the foregrounds, leaking power to higher delays. The three additional peaks observed at integer multiples of ∼30 ns correspond to interference of the primary path through the telescope with secondary paths that have undergone one, two, and three additional reflections off the focal line and cylinder. The amplitude of these peaks has been reduced by deconvolving the model for the primary beam; however, they are still significant compared to our expectation for the noise. We are actively working on improving the accuracy of our beam model and implementing a deconvolution procedure that better addresses the off-axis response (see S15 for one example) to further reduce the amplitude of these peaks. The "U"-shaped tracks in the delay power spectrum correspond to the brightest point sources moving through the far sidelobes. In this case, there is a delay associated with the EW component of the baseline that is not corrected by the mapmaking procedure because it assumes that the instrument has no sensitivity outside the main lobe of the primary beam. For each bright source, there are three "U"-shaped tracks corresponding to the three intercylinder, EW baseline separations that extend out to progressively higher delays. Finally, at large zenith angle, or low and high decl., the foreground power extends out to higher delays owing to aliasing of the sky in the baselines with one-cylinder EW separation, as explained in Section 4.3.

The stop-band rejection is set to = 10⁻¹², which is much smaller than the inverse of the dynamic range of the delay power spectrum (∼5 × 10⁻⁹). This ensures that the brightest foreground features near 0 ns delay are attenuated to well below the radiometric noise level.

Our initial delay cut is defined as the minimum delay where the measured power spectrum is less than 3 times the power spectrum of the Gaussian noise realization. This is indicated by the dashed cyan line in Figure 10 and results in an aggressive filter that yields a map that is dominated by radiometric noise. However, the dominant contamination at delays just below our aggressive cut originates from a few bright sources in the far sidelobes, which are easily masked. In an attempt to maximize signal-to-noise ratio, we examine four different delay cuts that correspond to the aggressive cut minus [25 ns, 50 ns, 75 ns, 100 ns]. For each cut, a foreground filter is constructed and applied to both the data and a simulation of the 21 cm signal. Next, the regions around known bright point sources are masked. The foreground-filtered data and signal are then pushed through the rest of the analysis pipeline, which is described in the sections that follow. As the delay cut is reduced, the relative increase in the noise is compared to the relative reduction in the amplitude of the simulated 21 cm signal. The aggressive cut minus 75 ns results in the maximum signal-to-noise ratio of the four values tested. This is indicated by the solid blue line in Figure 10 and will be used as the delay cut ${\tau }_{\mathrm{cut}}^{p}(\theta )$ for the rest of the analysis.

The foreground filter heavily attenuates the signal at frequencies near the edges of the 587.5–800 MHz band and at frequencies neighboring large spans of masked frequencies. These heavily attenuated frequencies would be improperly upweighted when stacking on external catalogs because the pipeline accounts for the fact that the noise has been attenuated but does not account for the fact that the signal has also been attenuated. To address this, at each polarization and decl. the median value of the nonzero diagonal elements of the filter is calculated. Any frequency where the diagonal element of the filter is less than 20% of the median is masked. This removes approximately 4.2% of the 587.5–800 MHz band.

The simulations described in Section 5.3 predict that the rms of the radiometric noise is more than an order of magnitude larger than the rms of the 21 cm signal in the foreground-filtered, deconvolved map. The distribution of map pixel values is largely set by the radiometric noise, and the 21 cm signal is a small perturbation that is only evident after averaging over a large number of sources. The map does contain residual foregrounds, RFI, and instrumental artifacts that are large compared to the propagated fast-cadence estimate for the noise. However, this excess noise is for the most part restricted to specific frequency bins or localized to regions on the sky. The subset of the data that exceeds our expectation for the noise is masked using the following procedure.

The foreground-filtered, deconvolved map is standardized by dividing the value of each map pixel by the standard deviation from the fast-cadence estimate. These standardized maps are examined manually at each frequency channel. Any channel that contains residuals that both are large compared to the expected noise and corrupt a significant portion of the NGC field are masked. Note that the delay filter couples frequency channels, so a channel may show significant residuals due to the filter leaking some narrowband artifact from an adjacent channel. This can be disentangled for the most part by identifying artifacts with a common spatial profile across frequencies and then masking the channel where that artifact has the largest magnitude. It could also be automated through an iterative procedure of masking and foreground filtering. In the end, 14.7% of the 587.5–800 MHz band is discarded in this way. We note that newer versions of the pipeline with improved RFI excision have reduced this fraction to roughly 5%, and most of these frequency channels are believed to be recoverable in future analyses by making additional improvements to the RFI excision algorithm and by further vetting the time ranges that are included in the sidereal stack. The frequency mask generated through this procedure is applied to the unfiltered map, and the foreground filter is reapplied.

The total fraction of the 587.5–800 MHz band that remains after removing persistent RFI bands, frequencies that do not have complete sidereal coverage, frequencies that have low integration time for the NGC field, frequencies that show excess noise, and frequencies near the edges of the large gaps of missing data is 48.6%. Finally, any map pixel whose absolute value is greater than 6σ is masked, where σ is again obtained from the fast-cadence noise estimate. This removes 1.4% of the remaining map pixels within the NGC field. The choice of a 6σ threshold was informed by signal injection simulations that are described in Section 7.4. The threshold is large enough that the resulting bias in the amplitude of the 21 cm signal is small compared to the statistical uncertainty. Figure 11 shows the map at several stages of the pipeline processing for a typical frequency channel; the third and fourth panels depict the application of the 6σ mask.

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Figure 12 shows in black the standard deviation of the map pixels within the NGC field as a function of frequency after all masking has been applied. The noise for a single frequency channel is on average 0.6 mJy beam⁻¹, with an increase to 1.0 mJy beam⁻¹ in the upper 50 MHz of the band. The increased noise at high frequencies is driven by a reduction in the primary beam response on meridian when averaged over the decl. spanned by the NGC field, a mild increase in the system temperature, and frequent flagging of the highest frequencies (≳794 MHz) by the threshold applied in the sidereal regridding stage of the pipeline (see Section 3.2.6). This last item will be corrected in future revisions of the pipeline. The noise in the map is on average 50% greater than the expected radiometric noise, which is shown in red. To generate the expected radiometric noise, visibilities are drawn randomly according to Equation (52) and then propagated through the mapmaking and foreground filtering procedure. For comparison, we also show in blue the standard deviation of the map pixels in a jackknife of even and odd days. The procedure for constructing this jackknife will be described in Section 7.2. The noise in the jackknife is in better agreement with the expected radiometric noise, except in a few 6 MHz wide bands where there is still unmasked, transient RFI. This is due to the fact that the residual foregrounds are largely due to instrument chromaticity that is the same from day to day and thus cancels in the jackknife. Note that a slightly different frequency mask was used for the jackknife because the frequency channels at the upper edge of the band do not have full coverage of the sidereal day in the even or odd split. This results in the jackknife noise dropping below the expected radiometric noise because the large filter attenuation at the upper edge of the band is pushed to lower frequencies.

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Using a more aggressive mask (3σ) removes 6.6% of the map pixels within the NGC field and brings the measured noise to within 22% of the expected radiometric noise on average at the expense of introducing a significant nonlinearity into the analysis pipeline. Using a more aggressive mask (3σ) and more aggressive delay filter with a cutoff that is 75 ns larger (indicated by the dashed cyan line in Figure 10) brings the measured noise to within 12% of the expected radiometric noise on average. However, this results in a significant reduction in the amplitude of the stacked 21 cm signal in simulations, and a better signal-to-noise ratio is anticipated using the less aggressive delay cut. Note that with either mask or delay cut the excess noise from residual foreground, RFI, and instrumental artifacts is comparable to or less than the radiometric noise for the 102-day sidereal stack, assuming that they add in quadrature.

For each source in a given eBOSS catalog, a spectral cube centered on the source's location is extracted from the deconvolved, foreground-filtered map. First, the R.A. and decl. of the source are converted from ICRS to CIRS coordinates to account for the precession and nutation of Earth's polar axis. The redshift of the source is converted to the frequency of the redshifted 21 cm emission,

$$\begin{eqnarray}&&{\nu }_{21\mathrm{cm}}=\displaystyle \frac{1420.406\,\mathrm{MHz}}{1+z}.\end{eqnarray} \tag{ 59 }$$

The map pixel and frequency channel closest to these coordinates are found, and ±50 pixels (channels) are extracted in the angular (frequency) directions. This results in a spectral cube that spans ±3° in R.A./decl. and ±20 MHz in frequency.

The stacked signal is given by the weighted average of the spectral cubes over all sources in the catalog:

$$\begin{eqnarray}&&\begin{array}{l}{d}^{p}({\rm{\Delta }}\nu ,\ {\rm{\Delta }}\theta ,{\rm{\Delta }}\phi )\\ \,\,=\,\displaystyle \sum _{s}\ {W}_{\mathrm{hpf}}^{p}({\nu }_{s}+{\rm{\Delta }}\nu ,{\theta }_{s}+{\rm{\Delta }}\theta ,{\phi }_{s}+{\rm{\Delta }}\phi )\\ \,\,\,\,\times \,{M}_{\mathrm{hpf}}^{p}({\nu }_{s}+{\rm{\Delta }}\nu ,{\theta }_{s}+{\rm{\Delta }}\theta ,{\phi }_{s}+{\rm{\Delta }}\phi ),\end{array}\end{eqnarray} \tag{ 60 }$$

where (ν_s , θ_s , ϕ_s ) denote the frequency channel and map pixel closest to the coordinates of source s and

$$\begin{eqnarray}&&\begin{array}{l}{W}_{\mathrm{hpf}}^{p}({\nu }_{s}+{\rm{\Delta }}\nu ,{\theta }_{s}+{\rm{\Delta }}\theta ,{\phi }_{s}+{\rm{\Delta }}\phi )\\ \,\,=\,\displaystyle \frac{{w}_{\mathrm{hpf}}^{p}({\nu }_{s}+{\rm{\Delta }}\nu ,{\theta }_{s}+{\rm{\Delta }}\theta ,{\phi }_{s}+{\rm{\Delta }}\phi )}{\displaystyle \sum _{s}{w}_{\mathrm{hpf}}^{p}({\nu }_{s}+{\rm{\Delta }}\nu ,{\theta }_{s}+{\rm{\Delta }}\theta ,{\phi }_{s}+{\rm{\Delta }}\phi )}\end{array}\end{eqnarray} \tag{ 61 }$$

denotes the relative weight given to source s, with the absolute weight ${w}_{\mathrm{hpf}}^{p}$ given by Equation (57).

Note that we make no attempt to interpolate the spectral cubes onto a common grid relative to the coordinates of the source. Instead, we take a forward modeling approach where the stacking procedure is applied to simulations in order to characterize how the pixelization alters a stack of the 21 cm signal. This will result in a small degradation in signal-to-noise ratio because we are not stacking on the true peak, but given the pixelization used, we estimate this to be only ∼3% for the NGC field.

For simplicity, all model fitting and parameter estimation uses only the central pixel of the stack as a function of frequency offset. Going forward we will use d^p (Δν) ≡ d^p (Δν, 0, 0) to describe the stack of the pixels closest to the coordinates of the sources.

The probability distribution of the noise in the stack must be characterized in order to derive accurate uncertainties on the inferred model parameters. As discussed in Section 4.6, residual foregrounds and RFI are expected to be subdominant but significant contributors to the noise, and both are likely correlated between frequencies. More generally, the foreground filter couples all frequency channels, ensuring a nonzero correlation between frequency offsets in the stack. These factors are not accounted for in the propagated fast-cadence noise estimate, which only includes the radiometric contribution to the noise and does not account for the correlation between frequency channels. To develop an accurate noise model, we stack the data on a large number of random mock catalogs and examine the distribution of values.

Each eBOSS clustering catalog has a corresponding "random" catalog that approximates the 3D selection function of the clustering catalog and is more than 40 times as dense (Ross et al. 2020; Raichoor et al. 2021). We randomly sample the random catalog without replacement to generate a mock catalog that has the same number of sources as the true catalog. The deconvolved, foreground-filtered map is then stacked on the mock catalog following the same procedure described in Section 4.7. This process is repeated N_mock = 10,000 times.

The noise covariance of the stacked data is estimated using the sample covariance of the mocks

$$\begin{eqnarray}&&\hat{{\boldsymbol{\Sigma }}}=\displaystyle \frac{1}{{N}_{\mathrm{mock}}-1}\displaystyle \sum _{m=1}^{{N}_{\mathrm{mock}}}{\left({{\boldsymbol{d}}}_{m}-\hat{{\boldsymbol{\mu }}}\right)}^{T}\left({{\boldsymbol{d}}}_{m}-\hat{{\boldsymbol{\mu }}}\right),\end{eqnarray} \tag{ 62 }$$

where

$$\begin{eqnarray}&&{{\boldsymbol{d}}}_{m}=\left[{d}_{m}^{p}({\rm{\Delta }}{\nu }_{i},0,0):0\leqslant i\leqslant 100,p\in \{X,Y\}\right]\end{eqnarray} \tag{ 63 }$$

is a vector containing the stacked signal at the central pixel as a function of frequency offset for both polarizations for the mth mock catalog and

$$\begin{eqnarray}&&\hat{{\boldsymbol{\mu }}}=\displaystyle \frac{1}{{N}_{\mathrm{mock}}}\displaystyle \sum _{m=1}^{{N}_{\mathrm{mock}}}{{\boldsymbol{d}}}_{m}\end{eqnarray} \tag{ 64 }$$

is the sample mean of the mocks. An example of the sample covariance is shown in Figure 13.

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We find that the sample mean for a given frequency offset and polarization is nonzero at a level larger than expected given the standard error. The rms of the sample mean over all frequency channels and polarizations is ${\sigma }_{\hat{{\boldsymbol{\mu }}}}$ = 0.7−1.9 μJy beam⁻¹ depending on the tracer, which is roughly 20% of the sample standard deviation over mock catalogs and a factor of 20 times larger than the standard error. This sample mean over mocks is subtracted from the stack on the true catalog to ensure a consistent noise model.

We find that the distribution of values observed in the mocks is consistent with a multivariate Gaussian whose covariance and mean are set to the sample variance and mean as calculated above.

In this section we summarize improvements to the data processing that are expected in the short term. Real-time RFI excision was deployed on the CHIME correlator in 2019 October, and its performance is summarized in CHIME Collaboration et al. (2022b). In addition, we have developed more effective methods for offline RFI excision that can be applied retroactively to all past data. We have also improved the tests that are used to identify anomalous data in the individual sidereal days. As a result, we expect to recover most of the 14.7% of the 585–800 MHz band that was masked during manual inspection of the foreground-filtered maps. We also expect that these changes will mitigate artifacts responsible for the excess noise observed at high delays and enable an analysis of the 400–585 MHz band.

Our analysis must include intracylinder baselines in order to recover the large angular scales relevant for measuring the imprint of the baryonic acoustic oscillation (BAO) on the power spectrum of 21 cm emission. Preliminary work suggests that we can include the majority of intracylinder baselines in our analysis if we also apply a high-pass filter along the time axis to remove the additive noise crosstalk. This also removes the largest angular modes of the sky but retains those angular modes that probe the BAO.

Similarly, our analysis must use a less aggressive delay filter in order to recover large line-of-sight scales. Currently we are limited at low delays by sources in the far sidelobes and uncalibrated spectral structure in the primary beam. The development of a full-sky model for the primary beam and the deconvolution of the off-axis response has been a major focus of the CHIME collaboration over the past several years. Some of the progress on the full-sky beam model has been reported in CHIME Collaboration et al. (2022a, 2022b). An algorithm for performing the deconvolution is described in Shaw et al. (2014) and S15.

Although not currently limiting our results, the stability of the complex receiver gains is expected to improve in future analyses. In 2020 July an ADC-dependent correction for the clock drift was deployed on the CHIME correlator, which is expected to improve the delay noise on short timescales (≲20 minutes) by roughly a factor of 2. In addition, techniques are being developed to perform an offline correction for thermal expansion of the focal line, which is the dominant source of phase instability on longer timescales (CHIME Collaboration et al. 2022b).

Finally, the results presented in this work employed 521 hr of total integration time. The CHIME archive now contains almost 20 times this amount after accounting for the flags described in Sections 3.2.7 and 3.3.1. Inclusion of these data will significantly reduce the radiometric noise and certain types of systematic errors, for example, those due to RFI.

To set the stage for the modeling approach described later in this section, in Figure 14 we show the approximate range of physical scales probed by our stacking measurements, represented as comoving wavenumbers k_∥ (along the line of sight) and k_⊥ (transverse to the line of sight). This range depends on observing frequency due to the relationship between frequency and radial distance and also due to chromaticity of CHIME's beam response, so we show results at three frequencies within the portion of the band used in our analysis.

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The foreground filter described in Section 4.5 acts roughly as a high-pass filter in k_∥, with the minimum accessible k_∥ determined by the delay cut τ_cut; for Figure 14, we use τ_cut = 200 ns, reflective of the typical delay cut within the decl. covered by the eBOSS catalogs. The sensitivity at high k_∥ is attenuated by the finite width of CHIME's frequency channels, which we approximate as top hats with width 390.625 kHz.

Similarly, the sensitivity at high k_⊥ is determined by the profile of the synthesized beam associated with the maps described in Section 4.3. For Figure 14, we use the simplified 1D beamforming result from Masui et al. (2017) to obtain NS and EW synthesized beam profiles based on CHIME's feed layout and the analytical ("control") primary beam model discussed in Section 4.4, take the geometric mean of the FWHMs in each direction, and translate this into a comoving wavenumber at each plotted frequency. Finally, since intracylinder baselines are excluded from our analysis, we are not sensitive to any angular scales that are only probed by pure NS baselines; these scales are determined by the EW primary beam profile, and we translate the EW FWHM into a minimum accessible k_⊥.

Note that a more thorough treatment of the scales being probed is possible, in which the stacking measurements can be related to an integral of the galaxy–H i cross-power spectrum multiplied by a transfer function W(k_∥, k_⊥) that precisely encodes the sensitivity of our analysis to a given Fourier mode. Such a treatment is currently under development and will be presented in a forthcoming publication (CHIME Collaboration 2023, in preparation), but preliminary results are in good agreement with the estimates in Figure 14.

In this figure, we also show the maxima of the first three BAO wiggles in the matter power spectrum, located at multiples of k_BAO = 2π/r_drag ≈ 0.064 h⁻¹ Mpc. It is clear that our delay filter and exclusion of intracylinder baselines have effectively filtered out any sensitivity to BAO scales from our stacking measurements. The scales that remain are beyond the reach of analytical perturbative methods for LSS statistics in Fourier space (e.g., d'Amico et al. 2020; Ivanov et al. 2020; Chen et al. 2022); while these scales have some overlap with those accessible to hybrid simulation–perturbation theory methods (e.g., Kokron et al. 2021), the majority of our signal-to-noise ratio lies at even smaller scales, implying that we cannot immediately apply those methods in our present analysis.

Halo-based models for H i (e.g., Padmanabhan 2021) and galaxy clustering can in principle describe the full range of scales shown in Figure 14. However, we have found that a simpler model, which makes efficient use of our simulation framework described in Section 5.3, is fully capable of describing the observed signal while allowing for marginalization over hard-to-predict properties of nonlinear clustering. We describe this model and its application to our measurements in the following subsections.

Cosmological modeling of the distribution of galaxies ²⁴ and H i typically begins with the matter overdensity ${\delta }_{m}({\boldsymbol{x}},z)\equiv [{\rho }_{m}({\boldsymbol{x}},z)-{\bar{\rho }}_{m}(z)]/{\bar{\rho }}_{m}(z)$, where an overbar denotes a spatial average. In our modeling we assume that galaxies and H i are each linearly biased tracers of the total matter density. The overdensity corresponding to galaxy or H i number density, δ _g or δ_{H i} , can then be written in Fourier space as

$$\begin{eqnarray}&&{\delta }_{\alpha }({\boldsymbol{k}};z)=\left[{b}_{\alpha }(z)+f(z){\mu }^{2}\right]{\tilde{D}}_{\alpha }^{\mathrm{FoG}}(k\mu ,z){\delta }_{m}({\boldsymbol{k}},z)+{\epsilon }_{\alpha }(z),\end{eqnarray} \tag{ 65 }$$

with α ∈ [g, H i]. In Equation (65), b_α is the bias factor (assumed to be scale independent), and the f(z)μ² term encodes the effect of redshift-space distortions at linear order (Kaiser 1987), with f as the logarithmic growth rate and μ ≡ k_∥/k. We aim to capture the key nonlinear contributions to the two-point statistics of the fields: we include real-space nonlinear clustering in δ_m itself; the impact of small-scale velocities on redshift-space observations ("Finger of God"; Jackson 1972) is modeled with the damping function ${\tilde{D}}_{\alpha }^{\mathrm{FoG}};$ and finally, we include a term _α in Equation (65), which is uncorrelated with δ _m and represents the contribution of shot noise to δ_α .

In our analysis we will only require the two-point statistics of the correlated fields. These are captured entirely by the power spectrum of two fields:

$$\begin{eqnarray}&&\begin{array}{l}{P}_{\alpha \beta }({\boldsymbol{k}};{z}_{\alpha },{z}_{\beta })\\ \,\,=\,\left[{b}_{\alpha }({z}_{\alpha })+f({z}_{\alpha }){\mu }^{2}\right]\left[{b}_{\beta }({z}_{\beta })+f({z}_{\beta }){\mu }^{2}\right]\\ \,\,\,\,\times \,{\tilde{D}}_{\alpha }^{\mathrm{FoG}}(k\mu ,{z}_{\alpha }){\tilde{D}}_{\beta }^{\mathrm{FoG}}(k\mu ,{z}_{\beta })\,{P}_{m}({\boldsymbol{k}};{z}_{\alpha },{z}_{\beta })\\ \,\,\,\,+\,{P}_{\alpha \beta }^{\mathrm{shot}}({z}_{\alpha },{z}_{\beta }).\end{array}\end{eqnarray} \tag{ 66 }$$

The ingredients required to complete our model are functions for the nonlinear matter power spectrum P _m , the linear bias b_α , the Finger-of-God function ${\tilde{D}}_{\alpha }^{\mathrm{FoG}}$, and the shot noise ${P}_{\alpha \beta }^{\mathrm{shot}}$. We discuss our fiducial choices for these ingredients in the following sections.

5.2.1. Matter Power Spectrum

5.2.2. Linear Bias

We assume the following for the linear bias of each eBOSS sample:

$$\begin{eqnarray}&&{b}_{\mathrm{ELG}}(z)=1.5+0.7(z-0.85),\end{eqnarray} \tag{ 67 }$$

$$\begin{eqnarray}&&{b}_{\mathrm{LRG}}(z)=2.03+0.86(z-0.4)+0.13{\left(z-0.4\right)}^{2},\end{eqnarray} \tag{ 68 }$$

$$\begin{eqnarray}&&{b}_{\mathrm{QSO}}(z)=2.38+1.4(z-1.55)+0.28{\left(z-1.55\right)}^{2}.\end{eqnarray} \tag{ 69 }$$

The ELG bias uses the redshift evolution of the linear bias predicted by the simulations of Merson et al. (2019), normalized such that Equation (67) evaluates to the bias measurement from de Mattia et al. (2021) at the mean redshift of the eBOSS ELG sample. The LRG bias is based on Zhai et al. (2017), who fit a halo occupation distribution model to small-scale clustering of a combined BOSS+eBOSS LRG sample and computed the linear bias from this model. Specifically, Equation (68) is the result of a quadratic fit to the best-performing bias model from Figure 12 of Zhai et al. (2017). The QSO bias is taken from the fitting function in Laurent et al. (2017), based on measurements of the eBOSS QSO correlation function in four redshift bins.

For the linear bias of H i, we follow Cosmic Visions 21 cm Collaboration et al. (2018) in smoothly interpolating between measurements from the IllustrisTNG simulations (Villaescusa-Navarro et al. 2018) at z < 2 and the analytical model from Castorina & Villaescusa-Navarro (2017) at z > 2. ²⁵ We show our bias models for H i and each eBOSS sample in the left panel of Figure 15.

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5.2.3. Finger-of-God Models

We model the Finger-of-God damping in Fourier space as a Lorentzian:

$$\begin{eqnarray}&&{\tilde{D}}_{\alpha }^{\mathrm{FoG}}(k\mu ,z)=\frac{1}{1+{k}^{2}{\mu }^{2}{\sigma }_{P,\alpha }{\left(z\right)}^{2}/2},\end{eqnarray} \tag{ 70 }$$

where the damping scale σ_P,α can approximately be associated with the pairwise velocity dispersion of galaxies or H i emitters on nonlinear scales. The (constant-redshift) Fourier conjugate of this function is an exponential in comoving distance (e.g., Scoccimarro 2004),

$$\begin{eqnarray}&&{D}_{\alpha }^{\mathrm{FoG}}({x}_{\parallel },z)=\frac{{e}^{-\left|{x}_{\parallel }\right|\sqrt{2}/{\sigma }_{P,\alpha }(z)}}{\sqrt{2}{\sigma }_{P,\alpha }(z)},\end{eqnarray} \tag{ 71 }$$

and we implement the Finger-of-God effect by convolving our simulated maps with this kernel along the line-of-sight axis. This is equivalent to multiplying the 3D auto-power spectrum of tracer α by ${\tilde{D}}_{\alpha }^{\mathrm{FoG}}{\left(k\mu ,z\right)}^{2}$ and multiplying the cross-power spectrum of H i and tracer α by ${\tilde{D}}_{{\rm{H}}\,{\rm\small{I}}}^{\mathrm{FoG}}(k\mu ,z)\times {\tilde{D}}_{\alpha }^{\mathrm{FoG}}(k\mu ,z)$.

For each eBOSS sample, Fourier-space clustering measurements have been analyzed using Finger-of-God models similar to what we describe above. For ELGs and LRGs, de Mattia et al. (2021) and Gil-Marín et al. (2020) use a squared Lorentzian function multiplied into the 3D galaxy power spectrum, finding best-fit values of σ_P,ELG = 2.79 h⁻¹ Mpc at z_eff = 0.85 and σ_P,LRG = 3.64 h⁻¹ Mpc at z_eff = 0.7 (where we quote the average of separate fits to the NGC and SGC fields). For QSOs, Zarrouk et al. (2018) use a Gaussian Finger-of-God model and perform fits that isolate the contribution to this model from small-scale velocities (as opposed to QSO redshift errors, which have a similar effect on the observed clustering). Taking the average of their best-fit σ_P values for the "3-multipole" and "3-wedge" analyses yields σ_P,QSO = 1.3 h⁻¹ Mpc at z_eff = 1.48. We find that this is roughly equivalent to Lorentzian damping with σ_P,QSO = 1.12 h⁻¹ Mpc.

We use these values to fix the amplitude of our fiducial σ_P (z) models for each sample. We compute the redshift dependence from a simple model in which σ_P (z) scales like a weighted average of the velocity dispersion ${\sigma }_{v}^{2}(M,z)$ of a dark matter halo of mass M, weighted by the halo mass function dn/dM and the mean satellite occupation in a mass-M halo:

$$\begin{eqnarray}&&{\sigma }_{P,\alpha }(z)\propto \frac{1+z}{H(z)}{\left[\frac{\int {dM}\frac{{dn}}{{dM}}{N}_{\mathrm{sat},\alpha }(M){\sigma }_{v}^{2}(M,z)}{\int {dM}\frac{{dn}}{{dM}}{N}_{\mathrm{sat},\alpha }(M)}\right]}^{1/2}.\end{eqnarray} \tag{ 72 }$$

To evaluate Equation (72), we use the halo mass function from Tinker et al. (2008) and the eBOSS halo occupation distribution models from Alam et al. (2020). The final results for σ_P,α (z), incorporating the amplitude constraints described above, are well fit by quadratic functions of redshift, which we present below:

$$\begin{eqnarray}&&\frac{{\sigma }_{P,\mathrm{ELG}}(z)}{{h}^{-1}\,\mathrm{Mpc}}=2.79-0.77(z-0.85)+0.083{\left(z-0.85\right)}^{2},\end{eqnarray} \tag{ 73 }$$

$$\begin{eqnarray}&&\frac{{\sigma }_{P,\mathrm{LRG}}(z)}{{h}^{-1}\,\mathrm{Mpc}}=3.64+0.019(z-0.7)-0.19{\left(z-0.7\right)}^{2},\end{eqnarray} \tag{ 74 }$$

$$\begin{eqnarray}&&\frac{{\sigma }_{P,\mathrm{QSO}}(z)}{{h}^{-1}\,\mathrm{Mpc}}=1.12-0.14(z-1.48)-0.058{\left(z-1.48\right)}^{2}.\end{eqnarray} \tag{ 75 }$$

For H i, we choose the damping scale based on simulations from Sarkar & Bharadwaj (2019), who attempt to account for the motion of H i within galaxies in addition to the contribution from the velocity dispersion within dark matter halos. They assume that the Finger-of-God damping of the 3D H i power spectrum are given by a Lorentzian, and they fit a σ_P,Hi (z) relation to their simulations. We use these results, multiplied by a factor of 2^−1/2 to translate to the damping given by a squared Lorentzian (as implied by our Equation (70)). The adopted σ_{P, Hi} (z) model is well fit by a quadratic function of redshift, given by

$$\begin{eqnarray}&&\frac{{\sigma }_{P,{\rm{H}}\,{\rm{I}}}(z)}{{h}^{-1}\,\mathrm{Mpc}}=1.93-1.48(z-1)+0.81{\left(z-1\right)}^{2}.\end{eqnarray} \tag{ 76 }$$

Over the redshift range of interest, this σ_{P, Hi} (z) model is within 20% of the values obtained in Villaescusa-Navarro et al. (2018) from fits of a squared Lorentzian to measurements from the IllustrisTNG simulations.

We plot our models for eBOSS and H i damping scales in the middle panel of Figure 15.

5.2.4. 21 cm Brightness Temperature

We convert simulated maps of δ_{H i} into brightness temperature fluctuations by multiplying by the mean 21 cm brightness temperature ${\bar{T}}_{b}(z)$. Recall that, after the end of reionization, the spin temperature T_s is high compared to both the background CMB temperature and T_⋆ = h_Pl ν₂₁/k_B. In this limit, the 21 cm brightness temperature can be written as (e.g., Bull et al. 2015)

$$\begin{eqnarray}&&{T}_{b}({\boldsymbol{x}},z)=\frac{3{\rm{\hslash }}{c}^{3}{A}_{10}}{16{k}_{{\rm{B}}}{\nu }_{21}^{2}}\frac{{\left(1+z\right)}^{2}}{H(z)}{n}_{{\rm{H}}\,{\rm{I}}}({\boldsymbol{x}},z),\end{eqnarray} \tag{ 77 }$$

where n_{H i} is the comoving H i number density and A₁₀ is the Einstein coefficient for spontaneous emission in the 21 cm line. Using ${n}_{{\rm{H}}\,{\rm\small{I}}}({\boldsymbol{x}},z)={\bar{n}}_{{\rm{H}}\,{\rm\small{I}}}(z)[1+{\delta }_{{\rm{H}}\,{\rm\small{I}}}({\boldsymbol{x}},z)]$, we can write

$$\begin{eqnarray}\begin{array}{rcl}\delta {T}_{b}({\boldsymbol{x}},z) & \equiv & {T}_{b}({\boldsymbol{x}},z)-{\bar{T}}_{b}(z)\\ & = & {\bar{T}}_{b}(z){\delta }_{{\rm{H}}\,{\rm{I}}}({\boldsymbol{x}},z),\end{array}\end{eqnarray} \tag{ 78 }$$

which justifies our method of converting maps of δ_{H i} into T _b . Using ${\bar{n}}_{{\rm{H}}\,{\rm{I}}}(z)={{\rm{\Omega }}}_{{\rm{H}}\,{\rm{I}}}(z){\rho }_{c,0}/({m}_{\text{p}}+{m}_{\text{e}})$, where ${\rho }_{c,0}=3{H}_{0}^{2}/8\pi G$ is the critical density today, ²⁶ we can write ${\bar{T}}_{b}$ as

$$\begin{eqnarray}\begin{array}{rcl}{\bar{T}}_{{b}}(z) & = & \left[\displaystyle \frac{9{\rm{\hslash }}{c}^{3}{A}_{10}{H}_{100}}{128\pi {k}_{{B}}{\nu }_{21}^{2}G({m}_{{p}}+{m}_{{e}})}\right]\\ & & \times \,h\displaystyle \frac{{H}_{0}}{H(z)}\,{{\rm{\Omega }}}_{{H}\,{I}}(z)\,{\left(1+z\right)}^{2},\end{array}\end{eqnarray} \tag{ 79 }$$

where H₁₀₀ = 100 km s⁻¹ Mpc⁻¹ and h = H₀/100. The prefactor in square brackets is independent of cosmology, consisting only of fundamental constants and A₁₀. Using A₁₀ = 2.8843e − 15 s⁻¹ (Gould 1994), Equation (79) can be written more compactly as ²⁷

$$\begin{eqnarray}&&{\bar{T}}_{b}(z)\approx 191.06\left[h\frac{{H}_{0}}{H(z)}\,{{\rm{\Omega }}}_{{\rm{H}}\,{\rm{I}}}(z)\,{\left(1+z\right)}^{2}\right]\ \mathrm{mK}.\end{eqnarray} \tag{ 80 }$$

For Ω_{H i} (z), we use the fitting function from Crighton et al. (2015), which was determined from a compilation of Ω_{H i} estimates over 0 < z < 5:

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{{\rm{H}}\,{\rm\small{I}}}(z)=4\times {10}^{-4}{\left(1+z\right)}^{0.6}.\end{eqnarray} \tag{ 81 }$$

We plot Equations (80) and (81) in the right panels of Figure 15.

5.2.5. Shot Noise

The cross-correlation between maps of H i and the distribution of galaxies in a given sample will be sensitive to the H i content of the galaxies. Specifically, the 3D cross-power spectrum of δ T _b ( x , z) and δ _g ( x , z) contains a cross shot-noise contribution of the form (e.g., Wolz et al. 2017)

$$\begin{eqnarray}&&{P}_{Tg}^{\mathrm{shot}}({z}_{{\rm{H}}\,{\rm{I}}},{z}_{g})={C}_{{\rm{H}}\,{\rm{I}}}({z}_{{\rm{H}}\,{\rm{I}}}){\left\langle {M}_{{\rm{H}}\,{\rm{I}}}\right\rangle }_{g},\end{eqnarray} \tag{ 82 }$$

where ${\left\langle {M}_{{\rm{H}}\,{\rm{I}}}\right\rangle }_{g}$ is the mean H i mass per galaxy in the sample and

$$\begin{eqnarray}&&{C}_{{\rm{H}}\,{\rm\small{I}}}(z)=\displaystyle \frac{3{\rm{\hslash }}{c}^{3}{A}_{10}}{16{k}_{{\rm{B}}}{\nu }_{21}^{2}({m}_{{\rm{p}}}+{m}_{{\rm{e}}})}\displaystyle \frac{{\left(1+z\right)}^{2}}{H(z)}.\end{eqnarray} \tag{ 83 }$$

In principle, ${\left\langle {M}_{{\rm{H}}\,{\rm{I}}}\right\rangle }_{g}$ depends on redshift, but for simplicity we consider a single value that is averaged over the entire sample. We also write the shot-noise contribution as being constant for all z _g , but note that there is expected to be a gradual, scale-dependent decorrelation as $\left|{z}_{{\rm{H}}\,{\rm{I}}}-{z}_{g}\right|$ increases, due to relative displacements of sources between different time slices.

5.2.6. Model Parameters

To produce a parameterized model of the 21 cm signal, we use the ingredients presented in Sections 5.2.1–5.2.5 as a basis and introduce a finite number of parameters that will scale their magnitude but not their redshift dependence. In total, our model contains seven parameters that are used to model the contributions to the cross-power spectrum:

• Ω_{H i} : One of the key quantities controlling the stack signal is the total amount of neutral hydrogen in the universe. Although this quantity is expected to be redshift dependent, in this paper we use the model given in Equation (81) as a baseline and use a single redshift-independent parameter Ω_{H i} to scale the fiducial model about an effective redshift z_eff, which gives

  $$\begin{eqnarray}&&{{\rm{\Omega }}}_{{\rm{H}}\,{\rm\small{I}}}(z)={{\rm{\Omega }}}_{{\rm{H}}\,{\rm\small{I}}}\left[\displaystyle \frac{{{\rm{\Omega }}}_{{\rm{H}}\,{\rm\small{I}}}^{\mathrm{fid}}(z)}{{{\rm{\Omega }}}_{{\rm{H}}\,{\rm\small{I}}}^{\mathrm{fid}}({z}_{\mathrm{eff}})}\right].\end{eqnarray} \tag{ 84 }$$
• b_{H i} , b_g : To control the bias of the 21 cm field and galaxy density fields, which are again expected to be redshift dependent, we scale the models given in Section 5.2.2, giving

  $$\begin{eqnarray}&&{b}_{{\rm{H}}\,{\rm\small{I}}}(z)={b}_{{\rm{H}}\,{\rm\small{I}}}\left[\displaystyle \frac{{b}_{{\rm{H}}\,{\rm\small{I}}}^{\mathrm{fid}}(z)}{{b}_{{\rm{H}}\,{\rm\small{I}}}^{\mathrm{fid}}({z}_{\mathrm{eff}})}\right]\end{eqnarray} \tag{ 85 }$$

  for the 21 cm field and the equivalent definition for the galaxy density,

  $$\begin{eqnarray}&&{b}_{g}(z)={b}_{g}\left[\frac{{b}_{g}^{\mathrm{fid}}(z)}{{b}_{g}^{\mathrm{fid}}({z}_{\mathrm{eff}})}\right].\end{eqnarray} \tag{ 86 }$$
• M₁₀: The strength of the shot-noise contribution is governed by the mass of neutral hydrogen typically associated with a tracer galaxy ${\left\langle {M}_{{\rm{H}}\,{\rm{I}}}\right\rangle }_{g}$. We control this quantity with the parameter M₁₀ defined by

  $$\begin{eqnarray}&&{\left\langle {M}_{{\rm{H}}\,{\rm{I}}}\right\rangle }_{g}={M}_{10}\times {10}^{10}\ {M}_{\odot }.\end{eqnarray} \tag{ 87 }$$
• α_NL: The shape of the high-k real-space cross-power spectrum is uncertain because of nonlinear gravitational evolution and baryonic effects. We let this shape vary using a linear mode that interpolates from a linear to a nonlinear power spectrum

  $$\begin{eqnarray}&&P(k)={\alpha }_{\mathrm{NL}}\,{P}_{\mathrm{NL}}(k)+(1-{\alpha }_{\mathrm{NL}})\,{P}_{{\rm{L}}}(k).\end{eqnarray} \tag{ 88 }$$

  For P_NL(k) we use the model described in Section 5.2.1, and for P_L(k) we use a power spectrum with the same parameters but with the Halofit corrections turned off. This parameter is valid for α_NL > 0, where values above 1 correspond to increasing the power contributed by nonlinear evolution. Although this parameter is not physically motivated, we expect it to capture the effects of nonlinearities at the level that can be measured in this work.
• α_{FoG,H i} , α_FoG,g: To account for uncertainties in the Finger-of-God smoothing, we allow redshift-independent scaling of both the 21 cm and tracer velocity dispersion σ_P:

  $$\begin{eqnarray}&&{\sigma }_{P,{\rm{H}}\,{\rm{I}}}(z)={\alpha }_{\mathrm{FoG},{\rm{H}}\,{\rm{I}}}\ {\sigma }_{P,{\rm{H}}\,{\rm{I}}}^{\mathrm{fid}}(z),\end{eqnarray} \tag{ 89 }$$

  $$\begin{eqnarray}&&{\sigma }_{P,g}(z)={\alpha }_{\mathrm{FoG},g}\ {\sigma }_{P,g}^{\mathrm{fid}}(z).\end{eqnarray} \tag{ 90 }$$

Put together, these give a model for the cross-power spectrum of the 21 cm emission and the galactic tracer, controlled by the parameters given above. Written out fully, this gives

$$\begin{eqnarray}&&\begin{array}{c}{P}_{Tg}(k,\mu ;{z}_{1},{z}_{2})\\ \,\,=\,{{\rm{\Omega }}}_{{\rm{H}}\,{\rm{I}}}\left[\frac{{T}_{b}^{\mathrm{fid}}({z}_{1})}{{{\rm{\Omega }}}_{{\rm{H}}\,{\rm{I}}}^{\mathrm{fid}}({z}_{\mathrm{eff}})}\right]\,\frac{{D}^{+}({z}_{1})}{{D}^{+}({z}_{\mathrm{fid}})}\frac{{D}^{+}({z}_{2})}{{D}^{+}({z}_{\mathrm{fid}})}\\ \,\,\,\,\times \,\left({b}_{{\rm{H}}\,{\rm{I}}}\left[\frac{{b}_{{\rm{H}}\,{\rm{I}}}^{\mathrm{fid}}({z}_{1})}{{b}_{{\rm{H}}\,{\rm{I}}}^{\mathrm{fid}}({z}_{\mathrm{eff}})}\right]+f({z}_{1}){\mu }^{2}\right)\\ \,\,\,\,\times \,\left({b}_{g}\left[\frac{{b}_{g}^{\mathrm{fid}}({z}_{2})}{{b}_{g}^{\mathrm{fid}}({z}_{\mathrm{eff}})}\right]+f({z}_{2}){\mu }^{2}\right)\\ \,\,\,\,\times \,\left({P}_{{\rm{L}}}(k,{z}_{\mathrm{fid}})+{\alpha }_{\mathrm{NL}}\left[{P}_{\mathrm{NL}}(k,{z}_{\mathrm{fid}})-{P}_{{\rm{L}}}(k,{z}_{\mathrm{fid}})\right]\right)\\ \,\,\,\,\times {D}_{{\rm{H}}\,{\rm{I}}}^{\mathrm{FoG}}({\alpha }_{\mathrm{FoG},{\rm{H}}\,{\rm{I}}}\,k\mu ;{z}_{1})\ {D}_{{\rm{g}}}^{\mathrm{FoG}}({\alpha }_{\mathrm{FoG},g}\,k\mu ;{z}_{2})\\ \,\,\,\,+{M}_{10}\ \left[{C}_{{\rm{H}}\,{\rm{I}}}({z}_{1}){10}^{10}\ {M}_{\odot }\right],\end{array}\end{eqnarray} \tag{ 91 }$$

where we have highlighted the individual parameters in bold. Note that we evaluate the matter power spectrum at a fiducial redshift z_fid and apply the linear growth factor D⁺(z) to scale it to other redshifts.

We also require one more parameter to describe an apparent frequency or redshift offset between the H i and galaxies. This will be needed to account for systematic redshift errors in the eBOSS catalogs (see Section 8.1). As it is an observational effect, we do not include it in the cross-power spectrum description (where it would manifest itself as a phase rotation).

• Δν₀: This parameter shifts the stack signal away from being centered at zero frequency lag. Positive values of Δν₀ move the peak of the signal to higher frequencies and thus to lower redshifts.

We make extensive use of simulations in this work, for interpreting our stacking measurements in terms of physical models, determining the signal transfer function, and quantifying the linearity of our analysis pipeline via injection of simulated signals into the data. In this section, we describe our simulation methodology for generating sky maps of 21 cm emission and galaxy density (Section 5.3.1), propagating these through to mock galaxy catalogs (Section 5.3.2) and CHIME time streams (Section 5.3.3), and finally performing the stacking procedure (Section 5.3.4). The associated steps are schematically shown in Figure 16. Note that we do not attempt to simulate foregrounds, instead relying on several data-based tests to assess the contribution of residual foregrounds to our sky maps and cross-correlation measurements.

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5.3.1. Map Generation

5.3.2. Mock Catalogs

For each galaxy sample we consider, we create mock catalogs of N objects for each pair of simulated δ_{H i} and δ _g maps. To do so, we select the pixel indices and frequency channels from a probability density function given by

$$\begin{eqnarray}&&{ \mathcal P }({\boldsymbol{x}})\propto S({\boldsymbol{x}})\left[1+{\delta }_{g}({\boldsymbol{x}})\right],\end{eqnarray} \tag{ 92 }$$

where S( x ) is a sample-specific selection function. Once a voxel is selected, galaxies are assigned positions within it according to uniform random distributions and further displaced by simulated redshift errors as described below.

We obtain approximate galaxy selection functions from the public random catalogs associated with each eBOSS sample. In detail, for each sample, we build a histogram of object positions with 32 redshift bins from 0.8 < z < 2.5 and a HEALPix angular pixelization with N_side = 16 (roughly 37 resolution). We then form a rank-7 approximation to this distribution by performing a singular value decomposition of the histogram (represented as a N_z × N_pix matrix). Finally, we upsample this to the HEALPix resolution of the input maps and apply Gaussian smoothing in the angular direction (with width equal to the original pixel size) to apodize any sharp boundaries. Using this as the selection function for generating mocks ensures that we reproduce the large-scale footprint and modulations of each galaxy sample without introducing smaller-scale features of the catalogs into our simulations.

We generate random redshift errors using a separate scheme for each sample, based on estimates of redshift error distributions (represented as line-of-sight velocities) published by the eBOSS team. For LRGs, Ross et al. (2020) examined pairs of observations of the same target and found that the distribution of redshift differences was well fit by a Gaussian with σ = 91.8 km s⁻¹, corresponding to a redshift uncertainty of σ = 65.6 km s⁻¹ per object. For ELGs, Raichoor et al. (2021) quote three redshift error percentiles based on repeated observations; we find that these values are well fit by a Tukey lambda distribution with λ = − 0.4 and σ = 11.88 km s⁻¹.

For QSOs, Lyke et al. (2020) find that, over the entire QSO catalog, the distribution of redshift differences between repeated observations is well fit by a double Gaussian. This implies that the single-observation redshift errors are also described by a double Gaussian, with σ₁ = 150 km s⁻¹, σ₂ = 1000 km s⁻¹, and 18% of objects having errors drawn from the wider Gaussian. ²⁹ Though we use this model for our primary analysis, there is evidence that it does not completely capture the distribution of QSO redshift errors. We discuss the discrepancies and the effect on our analysis in Section 8.1.

We do not attempt to simulate catastrophic redshift errors, which the above references estimate to occur in less than 1% of the LRG and ELG samples and as much as 2% of the QSO sample. The effect of these errors on our stacking measurements is a simple suppression of the overall amplitude, by an amount equal to the catastrophic error fraction.

5.3.3. Time Streams

We make use of the m-mode formalism (S15) to translate simulated 21 cm maps into visibilities. In this formalism, the spherical harmonic coefficients ${a}_{{\ell }m}^{P}(\nu )$ of sky maps for Stokes parameter P ∈ {I, Q, U, V} are related to the sidereal-time Fourier transform of the visibility time stream, ${\tilde{V}}_{{xy},m}^{p}(\nu )$, via multiplication by a beam transfer matrix ${B}_{{xy};{\ell }m}^{p,P}(\nu )$:

$$\begin{eqnarray}&&{\tilde{V}}_{{xy},m}^{p}(\nu )=\displaystyle \sum _{P}\displaystyle \sum _{{\ell }}{B}_{{xy};{\ell }m}^{p,P}(\nu ){a}_{{\ell }m}^{P}(\nu ).\end{eqnarray} \tag{ 93 }$$

After performing this multiplication, we convert the result to a visibility time stream by inverse Fourier transforming in m, applying zero-padding to the Fourier transform such that the corresponding time resolution matches that of the observed sidereal stacks.

We carry out separate versions of this procedure with beam transfer matrices corresponding to the default or control beam models from Section 4.4. We compute these matrices using driftscan (Shaw et al. 2020a), with several performance optimizations: precision truncation using the bitshuffle library (Masui et al. 2015), omitting frequencies that fall outside of the mask described in Section 4.2, and only computing the P = I components (since the 21 cm signal is unpolarized).

Up to this point, the simulated data are in temperature units. To transform into spectral flux density units, we first compute the beam solid angle for the assumed beam model:

$$\begin{eqnarray}&&{{\rm{\Omega }}}^{p}(\nu )=\int d\theta \,\cos \theta \int d\phi \,| {A}^{p}(\nu ,\theta ,\phi ){| }^{2}.\end{eqnarray} \tag{ 94 }$$

We then multiply the visibilities by the standard Rayleigh–Jeans conversion factor and the beam solid angle, normalized by the power beam evaluated at ${\rm\small{HA}}=0^{\prime}$ and a reference decl. θ_ref:

$$\begin{eqnarray}&&\begin{array}{l}{\left.{V}_{{xy}}^{p}(\nu ,\phi )\right|}_{\mathrm{Jy}}\\ \,\,=\,\displaystyle \frac{2\times {10}^{26}{k}_{{\rm{B}}}{\nu }^{2}}{{c}^{2}}\displaystyle \frac{{{\rm{\Omega }}}^{p}(\nu )}{{\left|{A}^{p}(\nu ,{\theta }_{\mathrm{ref}},0)\right|}^{2}}{\left.{V}_{{xy}}^{p}(\nu ,\phi )\right|}_{{\rm{K}}}.\end{array}\end{eqnarray} \tag{ 95 }$$

With this normalization, a visibility corresponding to a point source that transits at θ = θ_ref has an amplitude equal to the flux of the source. For consistency with CHIME's beam and complex gain calibration, we set θ_ref to the decl. of Cygnus A.

From here, the simulated visibilities are processed in the same way as the real data: a global frequency mask and noise weights described in Sections 4.2 and 3.1.4 are applied, the contributions of the four brightest point sources are inferred and subtracted, beam-deconvolved maps are constructed as in Section 4.3, delay filtering is applied with the decl.-dependent delay cuts from Section 4.5, and the masking operations in Section 4.6 are applied. Just as we simulate visibilities for each of the default and control beam models, we also perform two versions of the mapmaking step, assuming either beam model; thus, we obtain four simulated data sets corresponding to each pair of assumed and deconvolved beam, and we compare the results in Section 7.3 in order to estimate the systematic uncertainty arising from our choice of beam model.

5.3.4. Mock Source Stacking

Finally, we stack the simulated observations on the associated mock catalogs, following the procedure in Section 4.7. Figure 17 shows stacking results corresponding to simulations of each eBOSS sample, for a single LSS realization but averaged over 100 mock catalogs of 400,000 objects each, in order to suppress shot noise associated with the catalog size. In the absence of delay filtering, the stacking amplitude inferred from these simulations for QSOs is greater than for ELGs and less than for LRGs; the former follows from ELGs having lower bias and higher Finger-of-God suppression than QSOs, while the latter is due to the higher bias of LRGs than QSOs, which wins over the more severe Finger-of-God effect for LRGs (see Figure 15).

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The delay filter significantly suppresses the signal level, reducing the zero-lag amplitude by around 80% for ELGs and LRGs and 63% for QSOs. We attribute the lower suppression for QSOs to their milder Finger-of-God suppression at small scales: the delay filter removes sensitivity to the largest scales (see Section 5.1), and the remaining smaller-scale contribution is larger for QSOs than for the other tracers owing to a smaller amount of suppression. Finally, redshift errors in the simulated catalogs reduce the zero-lag amplitude by no more than 10% for ELGs and LRGs but by 40% for QSOs, thanks to the much wider distribution of QSO redshift errors discussed above.

To interpret our results, we need to be able to calculate the expected signal from stacking on a given catalog for an underlying set of parameters θ . We call this quantity the template, denoted by s(Δν; θ ). Though the template is entirely determined by the cross-power spectrum in Equation (91), propagating this through the instrumental transfer function and our analysis procedure is challenging to do both efficiently and accurately, and so it will be left to a follow-up paper (CHIME Collaboration 2023, in preparation).

In this work, we instead use our simulation capability to calculate the templates. In brief, we generate LSS realizations corresponding to several modes, each of which is defined by a specific combination of model parameters; average over a sufficiently large number of random mock catalogs to estimate the stack signal for each mode; and calculate the full template for arbitrary parameter values by making linear combinations of the template modes and applying an effective treatment for the Finger of God. Overall the errors in this approach are ≲1%. We describe this approach in detail in Appendix D.

In Figure 18, we display the change in the H i-tracer cross-power spectrum (left panels) corresponding to variations of each of our eight model parameters, along with the corresponding change in the predicted stack signal (right panels). Access to the full k range shown in the left panels would allow nondegenerate constraints on several of these parameters, due to their different impacts on the cross-power spectrum. However, our filtering choices imply that the stack signal is only sensitive to nonlinear scales (k ≳ 0.3 h⁻¹ Mpc or so, as shown in Figure 14), and as a result, we are left with significant parameter degeneracies, which can be inferred from the similar variations in the right panels of Figure 18.

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The top left panel of Figure 19 shows the result of stacking the deconvolved, foreground-filtered maps on the 3D positions in the eBOSS NGC QSO catalog. It is shown as a function of R.A. offset and decl. offset at 0 MHz frequency offset, averaged over the two polarizations, in other words, d(0, Δθ, Δϕ) in the notation of Section 4.7. Also shown in the top row is our best-fit model for the 21 cm emission based on the simulations described in the preceding section and the residuals obtained by subtracting the best-fit model from the data. The residuals can be compared to the three panels in the second row, which correspond to three different techniques for estimating the noise present in the stack. The left panel is the result of applying the stacking procedure to a Gaussian noise realization generated according to Equation (52). The middle panel is the result of applying the stacking procedure to a jackknife of even and odd days (see Section 7.2). Finally, the right panel is the result of stacking the data on a random mock catalog.

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The noise in the residuals is consistent with that observed in the random mock catalog. Both are in excess of the noise in the even–odd jackknife, owing to the fact that residual foregrounds are highly correlated between even and odd days and therefore cancel in the jackknife. The noise observed in the even–odd jackknife is in excess of that observed in the Gaussian noise realization owing to unflagged RFI and variations in the foregrounds from day to day caused by instrument instability.

The third row of Figure 19 shows 1D slices of both data and the best-fit model. The negative shoulders in the R.A. direction that are observed in both the data and model are caused by the exclusion of intracylinder baselines from our analysis. The grating lobes in the R.A. direction, which are shown in Figure 7, have largely averaged away in the stack because their location varies with frequency and decl. It is important to note that the angular information displayed in Figure 19 was not used to constrain the model. For simplicity, the model is only fit to the central pixel of the stack as a function of frequency. A full 3D fit could further improve the signal-to-noise ratio and help break the degeneracy between the amplitude ${{ \mathcal A }}_{{\rm{H}}\,{\rm\small{I}}}$ and the Finger-of-God damping, but we leave that for a future analysis.

For all three tracers, the spatial extent of the signal is consistent with the synthesized beam computed directly from Equations (47) to (31) and averaged over sources, indicating that the 21 cm signal is unresolved. Figure 20 shows the central pixel of the stack as a function of frequency, i.e., d(Δν, 0, 0), for the three tracers in black. The-dark gray and light-gray contours indicate the central 68% and 95% of values, respectively, observed when stacking the maps on 10,000 random mock catalogs as outlined in Section 4.8. The red line indicates our best-fit model for the signal. Note that although the two polarizations are fit jointly, to simplify the figure we show only their weighted average, with the weights set to the inverse variance as measured by the random mock catalogs. Also note that the polarization- and frequency-dependent mean value of the noise has been characterized using the random mock catalogs and subtracted from both the stack on the true catalog and the stack on the mock catalogs that are shown in the figure.

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The best-fit model shown in both Figures 19 and 20 consists of fixing all nonlinear parameters at their fiducial values and allowing the parameters governing the large-scale clustering of H i to vary. This model has been described in Section 5.2.6. The bottom row shows the result of subtracting the best-fit model from the data and compares to the same gray mock catalog contours shown in the top row. For all tracers, the residuals are consistent with our noise model based on the random mock catalogs. This is also true for all QSO redshift bins, which are not shown.

We assume that the noise in the stacked source data is described by a Gaussian and that the signal is described by the model given in Section 5.4. This means that the likelihood function ${ \mathcal L }({\boldsymbol{d}}| {\boldsymbol{\theta }})$ of observing the stacked signal d given a template s ( θ ) with model parameters ${\boldsymbol{\theta }}=\left[{\rm{\Delta }}{\nu }_{0},{{\rm{\Omega }}}_{{\rm{H}}\,{\rm\small{I}}},{b}_{{\rm{H}}\,{\rm\small{I}}},{b}_{{\rm{g}}},{M}_{10},{\alpha }_{\mathrm{NL}},{\alpha }_{\mathrm{FoG},{\rm{H}}\,{\rm\small{I}}},{\alpha }_{\mathrm{FoG},{\rm{g}}}\right]$ is described by a multivariate Gaussian

$$\begin{eqnarray}&&{ \mathcal L }({\boldsymbol{\theta }})={ \mathcal P }({\boldsymbol{d}}| {\boldsymbol{\theta }})\end{eqnarray} \tag{ 96 }$$

$$\begin{eqnarray}&&=\displaystyle \frac{1}{{\left|2\pi {\boldsymbol{\Sigma }}\right|}^{1/2}}\exp \left(-\displaystyle \frac{1}{2}{\chi }^{2}\right),\end{eqnarray} \tag{ 97 }$$

with

$$\begin{eqnarray}&&{\chi }^{2}={\left({\boldsymbol{d}}-{\boldsymbol{s}}({\boldsymbol{\theta }})-{\boldsymbol{\mu }}\right)}^{T}{{\boldsymbol{\Sigma }}}^{-1}\left({\boldsymbol{d}}-{\boldsymbol{s}}({\boldsymbol{\theta }})-{\boldsymbol{\mu }}\right),\end{eqnarray} \tag{ 98 }$$

where s ( θ ) is the model for the 21 cm signal and μ and Σ⁻¹ are the mean and inverse covariance of the noise, respectively, which are estimated using the sample mean and covariance of the mock catalogs as outlined in Section 4.8.

We employ a Markov Chain Monte Carlo (MCMC) to sample from the joint posterior distribution,

$$\begin{eqnarray}&&{ \mathcal P }({\boldsymbol{\theta }}| {\boldsymbol{d}})=\displaystyle \frac{1}{{ \mathcal Z }}\,{ \mathcal L }({\boldsymbol{\theta }})\,\pi ({\boldsymbol{\theta }}),\end{eqnarray} \tag{ 99 }$$

where π( θ ) is the prior probability distribution over the model parameters and ${ \mathcal Z }$ is the normalization constant such that the posterior integrates to unity.

We use noninformative priors for most parameters, ascribing equal prior probability over large ranges. For the nonlinear parameters we choose to do this even where there is some external information either from simulations or more strongly from analysis of the eBOSS data themselves (e.g., on the Finger-of-God scale; see Section 5.2.3), as it is difficult to combine the different prescriptions for modeling the nonlinear scales. These analyses guide our choice of fiducial model, but we allow a wide range of variation around them when trying to fit the data.

The one exception to this is for the galactic bias b _g . As it is a large-scale parameter, it is less susceptible to systematic differences in the modeling, and we instead use a prior informed by modeling of the eBOSS tracers. For the QSOs, our fiducial model is that from Laurent et al. (2017), and to get an uncertainty on this, we fit a shift in the amplitude to the two lowest-redshift bins in their analysis (which overlap with that of this paper), which gives an uncertainty of 3% about the fiducial model. For the LRGs we translate the overall results of Zhai et al. (2017) of b = 2.30 ± 0.03 into a 1.3% uncertainty on the amplitude of the bias model used here. Finally, for the ELGs we symmetrize the measurements of b₁ from Tamone et al. (2020) to give an uncertainty of 10% for the ELG linear bias.

For Ω_{H i} , which gives an overall normalization to the signal, we use a prior symmetric about zero, despite the fact that physically Ω_{H i} ≥ 0. This is to ensure that our priors do not give an artificial bias toward positive signal and give a more robust estimation of the detection significance. However, we do enforce that b_{H i} ≥ 0 to exclude an unphysical mode of high probability with both Ω_{H i} < 0 and b_{H i} < 0.

We summarize our choice of priors in Table 2.

Table 2. The Prior Placed on Each Parameter during Our Analysis

Parameter	Type	Description
Standard parameters
ΩH i	Uniform	Range: −10−2 to 10−2
bH i	Uniform	Range: 0 to 10
bg	Gaussian	Mean: ${\bar{b}}_{{\rm{g}}}={b}_{{\rm{g}}}^{\mathrm{fid}}({z}_{\mathrm{eff}})$ standard deviation: QSOs 3%, LRGs 1.3%, ELGs 10%
Δν0	Uniform	Range: −0.8 MHz to 0.8 MHz
Nonlinear parameters
M10	Uniform	Range: 0 to 20; Fixed: 0
αNL	Uniform	Range: 0 to 5; Fixed: 1
αFoG,H i	Uniform	Range: 0 to 5; Fixed: 1
αFoG,g	Uniform	Range: 0 to 5; Fixed: 1

Note. There are two classes of parameters in our analysis, standard parameters that capture the large-scale quantities we hope to constrain, and nuisance parameters that model the signal on small, nonlinear scales. In our analysis the latter group of parameters will either be marginalized over or be fixed to their fiducial values in order to assess the contribution of modeling uncertainties to our constraints.

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The affine-invariant ensemble sampler from the emcee package (Foreman-Mackey et al. 2013) is used to sample from the joint posterior distribution. We run 32 samplers initialized from random locations within the region defined by Table 2. To analyze the chains, we need to reduce them down to an independent set of samples, but we must first determine how correlated the samples actually are. To do this, we first define the sample autocorrelation ${\hat{\rho }}_{i}^{s}(\tau )$ of a parameter θ_i within the sth chain as

$$\begin{eqnarray}&&{\hat{\rho }}_{i}^{s}[\tau ]=\displaystyle \frac{1}{N-\tau }\displaystyle \sum _{t=1}^{N-\tau }({\theta }_{i}^{s}[t]-{\bar{\theta }}_{i}^{s})({\theta }_{i}^{s}[t+\tau ]-{\bar{\theta }}_{i}^{s}),\end{eqnarray} \tag{ 100 }$$

where ${\bar{\theta }}_{i}^{s}$ is the sample mean for parameter θ_i , and then using this, we define the autocorrelation length for a single parameter and chain as

$$\begin{eqnarray}&&{\zeta }_{i}^{s}=\displaystyle \frac{1}{{\hat{\rho }}_{i}^{s}[0]}\displaystyle \sum _{\tau }{\hat{\rho }}_{i}^{s}[\tau ].\end{eqnarray} \tag{ 101 }$$

For a single aggregate autocorrelation length ζ summarizing the chain convergence we take the mean over the different samplers and then use the longest autocorrelation across the different model parameters. That is,

$$\begin{eqnarray}&&\zeta =\mathop{\max }\limits_{i}\left({\mathrm{mean}}_{s}{\zeta }_{i}^{s}\right).\end{eqnarray} \tag{ 102 }$$

This gives the number of MCMC steps over which the chains remain correlated. Only samples from each chain separated by ζ samples can be considered independent. The first 10 × ζ samples in each chain are discarded as burn-in. The chains are then thinned by ζ and concatenated. The parameter space is high dimensional and has complex degeneracies, which means that the correlation lengths are large, ζ ≳ 500 in the full parameter space. We also make extensive use of the GetDist package (Lewis 2019) for analyzing the MCMC chains.

In Figure 21 we show the constraint on the default model parameters for the QSO catalog. We show constraints for both a model where all parameters are allowed to vary and a model where the nonlinear parameters are fixed to their fiducial values (M₁₀ = 0, α_NL = 1, α_{FoG,H i} = 1, and α_FoG,g = 1). The constraints show that certain parameter combinations are highly degenerate, most notably Ω_{H i} –b_{H i} , but also correlations with the nonlinear parameters α_{FoG,H i} and α_FoG,g. As these degeneracies limit our ability to make a cosmological interpretation of our results, it is worth attempting to understand them.

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The most severe degeneracy in our model is between Ω_{H i} and b_{H i} , and it is clearly apparent in both the full and fixed models. The origin of this can be seen in Equation (91), which, simplified slightly down to the linear terms, has

$$\begin{eqnarray}&&{P}_{{\rm{H}}\,{\rm\small{I}},{\rm{g}}}(k,\mu )\propto {{\rm{\Omega }}}_{{\rm{H}}\,{\rm\small{I}}}({b}_{{\rm{H}}\,{\rm\small{I}}}+f{\mu }^{2})({b}_{{\rm{g}}}+f{\mu }^{2})P(k),\end{eqnarray} \tag{ 103 }$$

which contains a multiplicative Ω_{H i} b_{H i} term, responsible for the curved degeneracy seen in the Ω_{H i} –b_{H i} panel of Figure 21. Previous 21 cm cross-correlation analyses (Masui et al. 2013; Switzer et al. 2013; Wolz et al. 2022) gave constraints directly on the combination Ω_{H i} b_{H i} r, where r is a scale-independent cross-correlation parameter that absorbs modeling uncertainties on nonlinear scales; however, this is not sufficient for the analysis here. Although transforming our constraints to be in terms of Ω_{H i} b_{H i} removes the curved degeneracy, ³⁰ we find that a linear degeneracy against Ω_{H i} remains. This can be understood straightforwardly as the effect of the Kaiser redshift-space distortions. As CHIME has higher resolution in the frequency direction versus the angular direction, and we have removed low-k_∥ modes by foreground filtering, the sensitivity in this analysis is biased toward wavenumbers with higher μ (which is illustrated in Figure 14). As both b_{H i} and f are of order unity, the contribution of the Kaiser term is important and cannot be neglected.

To account for this, we transform to a plane of (Ω_{H i} b_{H i} )–Ω_{H i} and determine a linear combination of these parameters that minimizes their variance. For a single LRG, ELG, or QSO sample g, the solution for an exactly linear degeneracy can be found by using the MCMC samples to construct the covariance matrix between Ω_{H i} b_{H i} and Ω_{H i} , which we write as C _Ωb,g , and then finding the eigenvector with minimal eigenvalue, which gives the linear combination we are searching for. We will use this combination as our primary amplitude parameter

$$\begin{eqnarray}&&{{ \mathcal A }}_{{\rm{H}}\,{\rm\small{I}}}\equiv {10}^{3}\,{{\rm{\Omega }}}_{{\rm{H}}\,{\rm\small{I}}}\left({b}_{{\rm{H}}\,{\rm\small{I}}}+\langle f{\mu }^{2}\rangle \right),\end{eqnarray} \tag{ 104 }$$

where we make the interpretation that the coefficient 〈f μ²〉 is the sensitivity-weighted average f μ² that this CHIME analysis is probing. We perform this optimization on the chains with fixed nonlinear parameters, as this gives a cleaner separation from other degenerate parameters.

The 〈f μ²〉 coefficient preferred by each tracer differs slightly from ∼0.45 (QSOb00) to ∼0.62 (QSOb2), which we would expect, as both f and CHIME's sensitivity change with redshift. As we would like to be able to compare our measurements between tracers, we would instead like a single effective 〈f μ²〉. To do this, we minimize the covariance C _Ωb,all, defined by

$$\begin{eqnarray}&&{{\boldsymbol{C}}}_{{\rm{\Omega }}b,\mathrm{all}}^{-1}=\displaystyle \sum _{g}{{\boldsymbol{C}}}_{{\rm{\Omega }}b,g}^{-1},\end{eqnarray} \tag{ 105 }$$

where we sum over the tracers QSOb0, QSOb1, QSOb2, LRG, and ELG (we exclude the other QSO tracers to avoid double-counting the data). The form of C _Ωb,all is motivated by considering each tracer to be a different measurement in the (Ω_{H i} b_{H i} )–Ω_{H i} plane: if each distribution were Gaussian and all were consistent, the covariance on the combined distribution would be given by C _Ωb,all. After this procedure, we derive an effective 〈f μ²〉 ≈ 0.552, which we fix for the rest of this analysis. The overall loss of constraining power from fixing a single value is small, with a drop of ∼7% for the worst affected tracer (full QSO catalog).

The second degeneracy we focus on is between the Finger-of-God parameters. If we examine the cross-power spectrum given by Equation (91) and expand the Finger-of-God damping factors defined in Equation (70) assuming k_∥ σ_P ≫ 1, we find that

$$\begin{eqnarray}&&{P}_{{\rm{H}}\,{\rm\small{I}},{\rm{g}}}(k,\mu )\propto {D}_{{\rm{H}}\,{\rm\small{I}}}^{\mathrm{FoG}}({\alpha }_{\mathrm{FoG},{\rm{H}}\,{\rm\small{I}}}\mu k){D}_{{\rm{g}}}^{\mathrm{FoG}}({\alpha }_{\mathrm{FoG},{\rm{g}}}\mu k)\end{eqnarray} \tag{ 106 }$$

$$\begin{eqnarray}&&\mathop{\propto }\limits_{\sim }\displaystyle \frac{4}{{k}_{\parallel }^{4}{\sigma }_{P,{\rm{H}}\,{\rm\small{I}}}^{2}{\sigma }_{P,{\rm{g}}}^{2}}{\left({\alpha }_{\mathrm{FoG},{\rm{H}}\,{\rm\small{I}}}\,{\alpha }_{\mathrm{FoG},{\rm{g}}}\right)}^{-2}.\end{eqnarray} \tag{ 107 }$$

For most of the region of CHIME's k-space sensitivity (see Figure 14) we are close to this regime, and so we expect there to be an approximate degeneracy of the form α_{FoG,H i} α_FoG,g, which can be seen in the α_{FoG,H i} –α_FoG,g panel of Figure 21. This motivates us to transform to two new parameters

$$\begin{eqnarray}&&{\alpha }_{\mathrm{FoG},+}={\left({\alpha }_{\mathrm{FoG},{\rm{H}}\,{\rm\small{I}}}\,{\alpha }_{\mathrm{FoG},{\rm{g}}}\right)}^{1/2}\end{eqnarray} \tag{ 108 }$$

$$\begin{eqnarray}&&{\alpha }_{\mathrm{FoG},-}=\mathrm{log}({\alpha }_{\mathrm{FoG},{\rm{H}}\,{\rm\small{I}}}/{\alpha }_{\mathrm{FoG},{\rm{g}}}),\end{eqnarray} \tag{ 109 }$$

where in the large k_∥ limit α_FoG,+ controls the amount of Finger-of-God damping and α_FoG,− does not affect the cross-power spectrum. The logarithm in the definition of α_FoG,− is to limit the effect of small α_FoG,g values generating extremely large values for this parameter.

In Figure 22 we show these new parameters and how they are correlated with the parameters they are derived from. The new amplitude-like parameter ${{ \mathcal A }}_{{\rm{H}}\,{\rm\small{I}}}$ clearly flattens the degeneracy, capturing all the information in Ω_{H i} and b_{H i} . Similarly, the parameter α_FoG,+ correlates with the amplitude parameter, whereas the orthogonal combination α_FoG,− does not, although there is interesting behavior observed at low α_FoG,+ where we are even further from the regime where we can make the high-k_∥ expansion used in Equation (107).

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One of the key remaining degeneracies is that between the overall amplitude, ${{ \mathcal A }}_{{\rm{H}}\,{\rm\small{I}}}$, and the combined Finger-of-God strength, α_FoG,+. This can be understood physically: on the scales that CHIME observes, the Finger-of-God damping reduces the stacked signal amplitude, and so an increase in α_FoG,+ must be compensated by an increase in the underlying 21 cm signal amplitude to remain consistent with the measurements.

In Figures 23–25 we show the constraints for the QSO, ELG, and LRG tracers stacked over the full 585–800 MHz band for the amplitude parameter, ${{ \mathcal A }}_{{\rm{H}}\,{\rm\small{I}}}$, the frequency offset, Δν₀, the shot noise, M₁₀, and the two nonlinear nuisance parameters, α_FoG,+ and α_NL. In all cases we find an excellent goodness of fit with ${\chi }_{\min }^{2}$ being close to the 202 degrees of freedom we expect (this comes from the 101 frequency offsets in the stacked data for each of the X and Y polarizations).

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For all tracers, the amplitude constraints are significantly weakened by marginalizing over the nonlinear parameters compared to the case of fixed nonlinear parameters. However, the posteriors are non-Gaussian and highly skewed such that, despite the large credible interval, the probability that ${{ \mathcal A }}_{{\rm{H}}\,{\rm\small{I}}}\leqslant 0$ is negligible. Even though the nonlinear parameters are degenerate with the amplitude, the amplitude must be nonzero for a signal to be seen.

Interpretation of these constraints is complicated by a volume factor pushing the constraints toward larger Finger-of-God smoothing effects. The originally uniform prior on π(α_{FoG,H i} , α_FoG,g) transforms to a π(α_FoG,+, α_FoG,−) ∝ α_FoG,+. As the stack signal $\mathop{\propto }\limits_{\sim }{{ \mathcal A }}_{{\rm{H}}\,{\rm\small{I}}}/{\alpha }_{\mathrm{FoG},+}^{2}$, our broad noninformative priors give an unintentional upward pressure on ${{ \mathcal A }}_{{\rm{H}}\,{\rm\small{I}}}$, as there is more volume at higher FoG damping levels. This can be resolved by placing a flat prior on α_FoG,+, but as it is not a physical parameter, this is difficult to justify. Future analysis will need to have data that can break this degeneracy internally, or use better modeling that allows for the prior bounds on α_{FoG,H i} and α_FoG,g to be reduced.

Assessing the significance of the detection is difficult for two reasons. First, the posterior distributions of the full set of parameters are highly non-Gaussian, which means that a naive "mean over standard deviation" figure does not accurately represent the significance of a parameter being nonzero. Second, there is not a single amplitude-like parameter that we can use to assess significance. Although we are primarily interested in Ω_{H i} or ${{ \mathcal A }}_{{\rm{H}}\,{\rm\small{I}}}$, whose posterior distributions include the projected degeneracies with α_NL and α_FoG,x, they are also somewhat degenerate with M₁₀, and this should be captured, as any measurement of M₁₀ should also count toward a detection.

One way of describing the detection significance is by way of a Bayesian model comparison. In this case, we seek to compare two explanations of the data, one in which the signal is represented by the full signal model given above (${{ \mathcal M }}_{1}$), and a null model where the signal is exactly zero and the data are entirely noise (${{ \mathcal M }}_{0}$). To compare these, we need to calculate the marginal likelihood, or Bayesian evidence, ${ \mathcal Z }$, which is the normalization constant for the posterior distribution shown in Equation (99):

$$\begin{eqnarray}&&{ \mathcal Z }={ \mathcal P }({\boldsymbol{d}}| { \mathcal M })\end{eqnarray} \tag{ 110 }$$

$$\begin{eqnarray}&&=\int { \mathcal P }({\boldsymbol{d}}| {\boldsymbol{\theta }},{ \mathcal M })\,{ \mathcal P }({\boldsymbol{\theta }}| { \mathcal M })\,{d}^{n}\theta \end{eqnarray} \tag{ 111 }$$

$$\begin{eqnarray}&&=\int { \mathcal L }({\boldsymbol{\theta }})\,\pi ({\boldsymbol{\theta }})\,{d}^{n}\theta .\end{eqnarray} \tag{ 112 }$$

The evidence allows us to compare the relative probability of two models given the observed data

$$\begin{eqnarray}&&\displaystyle \frac{{ \mathcal P }({{ \mathcal M }}_{1}| {\boldsymbol{d}})}{{ \mathcal P }({{ \mathcal M }}_{0}| {\boldsymbol{d}})}=\displaystyle \frac{{ \mathcal P }({\boldsymbol{d}}| {{ \mathcal M }}_{1})}{{ \mathcal P }({\boldsymbol{d}}| {{ \mathcal M }}_{0})}\times \displaystyle \frac{{ \mathcal P }({{ \mathcal M }}_{1})}{{ \mathcal P }({{ \mathcal M }}_{0})}\end{eqnarray} \tag{ 113 }$$

$$\begin{eqnarray}&&=\displaystyle \frac{{{ \mathcal Z }}_{1}}{{{ \mathcal Z }}_{0}}\times \displaystyle \frac{{ \mathcal P }({{ \mathcal M }}_{1})}{{ \mathcal P }({{ \mathcal M }}_{0})},\end{eqnarray} \tag{ 114 }$$

where the ${ \mathcal P }({ \mathcal M })$ terms give the prior probabilities of the models. We assume that the model prior probabilities are equal from this point on, and we focus solely on the Bayesian evidence ratio ${{ \mathcal Z }}_{1}/{{ \mathcal Z }}_{0}$ (often termed the Bayes factor).

Calculating the evidence directly is challenging, as the region of high likelihood is typically much smaller than the prior volume, and so estimates tend to be dominated by sample noise. The standard techniques for evidence calculation are variants on nested sampling (Skilling 2006), but here we instead use the simpler process of thermodynamic integration (Gelman & Meng 1998), as we do not need the extra efficiency of nested-sampling-based techniques. To do this, we introduce the quantity

$$\begin{eqnarray}&&{ \mathcal Z }(\lambda )=\int { \mathcal L }{\left({\boldsymbol{\theta }}\right)}^{\lambda }\,\pi ({\boldsymbol{\theta }})\,{d}^{n}\theta .\end{eqnarray} \tag{ 115 }$$

Noting that ${ \mathcal Z }(0)=1$ and ${ \mathcal Z }(1)={ \mathcal Z }$, we can write the quantity we want to calculate as

$$\begin{eqnarray}&&\mathrm{ln}{ \mathcal Z }={\int }_{0}^{1}\displaystyle \frac{\partial \mathrm{ln}{ \mathcal Z }(\lambda )}{\partial \lambda }\,d\lambda .\end{eqnarray} \tag{ 116 }$$

This transformation is useful because we can write the integrand as

$$\begin{eqnarray}&&\displaystyle \frac{\partial \mathrm{ln}{ \mathcal Z }(\lambda )}{\partial \lambda }=\displaystyle \frac{1}{{ \mathcal Z }(\lambda )}\int \mathrm{ln}{ \mathcal L }({\boldsymbol{\theta }})\ { \mathcal L }{\left({\boldsymbol{\theta }}\right)}^{\lambda }\,\pi ({\boldsymbol{\theta }}){d}^{n}\theta \end{eqnarray} \tag{ 117 }$$

$$\begin{eqnarray}&&={\left\langle \mathrm{ln}{ \mathcal L }({\boldsymbol{\theta }})\right\rangle }_{\lambda },\end{eqnarray} \tag{ 118 }$$

where the ${\left\langle \ldots \right\rangle }_{\lambda }$ denotes an expectation evaluated against a posterior with the likelihood raised to the power λ. This gives us a straightforward way of calculating $\mathrm{ln}{ \mathcal Z }$: first, on a discrete grid in λ, we use a standard MCMC sampler to draw from the unnormalized distribution ${ \mathcal L }{({\boldsymbol{\theta }})}^{\lambda }\pi ({\boldsymbol{\theta }})$, and then we estimate ${\left\langle \mathrm{ln}{ \mathcal L }({\boldsymbol{\theta }})\right\rangle }_{\lambda }$ from these samples; second, we numerically integrate over these estimates to calculate $\mathrm{ln}{ \mathcal Z }$.

We calculate the evidence for the signal model, ${{ \mathcal M }}_{1}$, by multiple sampling runs (as described in Section 6.2) generated at different λ. As the bulk of the variation in the integrand is around λ ∼ 0, we use the common choice of a grid regularly spaced in λ^1/5 (Calderhead & Girolami 2009), and as the integrands are smooth and well behaved, we find that a Romberg integration over 33 samples achieves sufficient accuracy. For the evidence calculation, we use shorter chains per λ step than for the parameter estimation, with only 15,000 samples per chain. After removing the initial samples for burn-in and thinning to the independent samples, this leaves ∼700 samples for each λ step. To estimate the error on each evidence calculation, we bootstrap resample the set of points at each λ step, integrate over the resampled sets, and estimate the sample variance over bootstrap sets. This gives a typical error in $\mathrm{ln}{ \mathcal Z }$ of ∼0.1. In contrast, the null signal model, ${{ \mathcal M }}_{0}$, is a zero-parameter model, and so its evidence is simply the likelihood of the data evaluated at zero signal. That is,

$$\begin{eqnarray}&&\mathrm{ln}{{ \mathcal Z }}_{0}=-\displaystyle \frac{1}{2}\left[\mathrm{ln}\left|2\pi {\boldsymbol{\Sigma }}\right|+{\chi }_{0}^{2}\right],\end{eqnarray} \tag{ 119 }$$

with

$$\begin{eqnarray}&&{\chi }_{0}^{2}={\left({\boldsymbol{d}}-{\boldsymbol{\mu }}\right)}^{T}{{\boldsymbol{\Sigma }}}^{-1}({\boldsymbol{d}}-{\boldsymbol{\mu }}),\end{eqnarray} \tag{ 120 }$$

so that we do not need any MCMC scheme to calculate it.

With $\mathrm{ln}{{ \mathcal Z }}_{0}$ and $\mathrm{ln}{{ \mathcal Z }}_{1}$, we have both of the ingredients required to give the Bayes factor. To enable a comparison with other significance estimates, we can turn the evidence ratio into an effective "number of sigma." Assuming that the only two models that could explain the data are ${{ \mathcal M }}_{0}$ and ${{ \mathcal M }}_{1}$ and giving them equal prior probabilities, ${ \mathcal P }({{ \mathcal M }}_{0})={ \mathcal P }({{ \mathcal M }}_{1})=1/2$, we can write the probability of the null model as

$$\begin{eqnarray}&&{ \mathcal P }({{ \mathcal M }}_{0}| {\boldsymbol{d}})=\displaystyle \frac{1}{1+{{ \mathcal Z }}_{1}/{{ \mathcal Z }}_{0}}.\end{eqnarray} \tag{ 121 }$$

We turn this into an effective number of σ, ${N}_{{ \mathcal Z }}$, via

$$\begin{eqnarray}&&{N}_{{ \mathcal Z }}={{\rm{\Phi }}}^{-1}(1-{ \mathcal P }({{ \mathcal M }}_{0}| {\boldsymbol{d}})),\end{eqnarray} \tag{ 122 }$$

where Φ⁻¹(x) is the inverse cumulative distribution function of the standard normal distribution.

An alternative, frequentist method of estimating the detection significance is to use a likelihood ratio test. First, we compute the ratio of the maximum likelihood values between a model with no signal and one with the full signal model

$$\begin{eqnarray}&&\lambda =2\mathrm{ln}\left(\displaystyle \frac{{ \mathcal L }({\hat{{\boldsymbol{\theta }}}}_{\mathrm{ML}})}{{{ \mathcal L }}_{0}}\right)\end{eqnarray} \tag{ 123 }$$

$$\begin{eqnarray}&&={\rm{\Delta }}{\chi }^{2},\end{eqnarray} \tag{ 124 }$$

with ${\rm{\Delta }}{\chi }^{2}={\chi }_{0}^{2}-{\chi }_{\min }^{2}$. This quantity is asymptotically χ² distributed with degrees of freedom equal to the effective number of model parameters. As our model has several notable degeneracies, the effective number of model parameters will be less than the total number of parameters. We use the Bayesian model dimensionality (Handley & Lemos 2019)

$$\begin{eqnarray}&&{d}_{M}=2\left[\left\langle {\left(\mathrm{ln}{ \mathcal L }\right)}^{2}\right\rangle -{\left\langle \mathrm{ln}{ \mathcal L }\right\rangle }^{2}\right]\end{eqnarray} \tag{ 125 }$$

as an estimate of the number of parameters, where the expectation 〈...〉 is taken over the posterior. Taking an average of this over the set of tracers, we find d_M ∼ 4.4, and so we use 4 as the effective number of parameters. Using this, we can ascribe a detection significance via the probability for a ${\chi }_{\nu =4}^{2}$ distribution to exceed λ. We again turn this into an effective number of sigma using the inverse cumulative distribution function of a standard normal distribution:

$$\begin{eqnarray}&&{N}_{\mathrm{LR}}={{\rm{\Phi }}}^{-1}\left(1-{\int }_{\lambda }^{\infty }{\chi }_{4}^{2}(x){dx}\right).\end{eqnarray} \tag{ 126 }$$

As a final estimate of the detection significance, we take the best-fit (minimum χ²) template as a fixed single template and then fit that directly to the data with a varying amplitude, A. As the likelihood is Gaussian, the distribution of A can be computed exactly and is Gaussian with mean of 1 and variance ${({\rm{\Delta }}{\chi }^{2})}^{-1}$. This directly gives the number of sigma of detection, ${N}_{A}=\sqrt{{\rm{\Delta }}{\chi }^{2}}$. We expect this quantity to overestimate the detection significance, as the template has already been adjusted to fit the data.

Table 3 shows the detection significance for each tracer calculated by each method given above. The Bayes factors, $\mathrm{ln}({{ \mathcal Z }}_{1}/{{ \mathcal Z }}_{0})$, are ≳4.6 for all tracers, which corresponds to decisive evidence for a cross-correlation detection according to the interpretations of Jeffreys (1961) and Kass & Raftery (1995). The numbers of sigmas for each method are reasonably close, with the Bayesian-evidence-based number ${N}_{{ \mathcal Z }}$ the lowest of the three and the amplitude parameter the highest. The common criticism of evidence calculations is that they are dependent on the prior widths, and, as is the case here, a choice that is intended to be noninformative for the purpose of parameter estimation can significantly lower the evidence compared to a less conservative choice of prior. In our case, parameters like M₁₀ could be significantly narrower without influencing the parameter estimation, which would boost the Bayes factor. Although we do not attempt it here, one resolution to this for nested comparisons (of which this is one), advocated by Gordon & Trotta (2007), is to optimize the prior widths centered on the value implied by the nested model to maximize the Bayes factor.

Table 3. The Detection Significance for Each Tracer

Tracer	Bayesian			Likelihood Ratio		Amplitude
	$\mathrm{ln}({{ \mathcal Z }}_{1}/{{ \mathcal Z }}_{0})$	$\mathrm{ln}({{ \mathcal Z }}_{1}/{{ \mathcal Z }}_{2})$	${N}_{{ \mathcal Z }}$	Δχ2	NLR	NA
LRG	18.9	−1.5	5.7	60.3	7.1	7.8
ELG	10.8	−2.4	4.1	40.8	5.7	6.4
QSO	56.3	−2.2	10.3	133.5	11.1	11.6
QSOb0	23.9	−2.3	6.5	66.2	7.5	8.1
QSOb1	19.6	0.8	5.8	53.2	6.6	7.3
QSOb2	16.9	−0.9	5.3	50.0	6.4	7.1
QSOb00	7.6	1.5	3.3	27.8	4.5	5.3
QSOb01	14.6	−1.6	4.9	46.3	6.1	6.8

Note. We calculate the detection significance by three different methods, using the Bayesian evidence, a likelihood ratio test, and the amplitude constraints on the best-fit model. The log Bayes factors, $\mathrm{ln}({{ \mathcal Z }}_{1}/{{ \mathcal Z }}_{0})$, all exceed the highest threshold ≳4.6 for decisive evidence according to the scale of Jeffreys (1961), and the Δχ² values all have p-values ≲ 10⁻⁵. We also convert each to an effective number of sigmas for comparison, giving ${N}_{{ \mathcal Z }}$, N_LR, and N_A , respectively. The results are similar; however, ${N}_{{ \mathcal Z }}$ suffers from the choice of wide priors, and the amplitude ratio N_A is overly optimistic, as it does not account for the previous fitting of the template. We also give the evidence ratio $\mathrm{ln}({{ \mathcal Z }}_{1}/{{ \mathcal Z }}_{2})$ comparing the full signal compared to fixing the nonlinear parameters. For most catalogs there is no evidence in favor of the full model ($\mathrm{ln}{ \mathcal Z }\lt 0$), that is, the improved fit is not sufficient to support the additional free parameters. Only for the QSOb00 tracer is there moderate evidence to support the full model.

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We also calculate the evidence for the signal model where we fix the nonlinear parameters, which we call ${{ \mathcal Z }}_{2}$, and give the log Bayes factor relative to the full signal model, $\mathrm{ln}({{ \mathcal Z }}_{1}/{{ \mathcal Z }}_{2})$, in Table 3. In most cases $\mathrm{ln}({{ \mathcal Z }}_{1}/{{ \mathcal Z }}_{2})$ is negative, that is, there is not sufficient improvement in the fits to justify the expanded model from statistical arguments alone, and in the remaining cases the evidence is marginal.

Our rationale for varying the nonlinear parameters is to explore what our data tell us about the large-scale H i distribution while including the genuine uncertainties in the modeling. With that in mind, we do not take this as an indication that we should fix these nonlinear parameters, but as one that they are not meaningfully constrained, as they allow the model to overfit the data.

The following procedure is used to determine whether measurements made with the X and Y baselines are consistent given our model for the noise and 21 cm signal. The two polarizations are jointly fit to a restricted model and an unrestricted model. For the restricted model, both polarizations are described by the same set of parameters, θ , as outlined in Section 6.2. The version of the model that holds the nonlinear parameters fixed at their fiducial values is employed for this exercise, since the version that allowed them to vary did not yield a significantly better fit to the data for any tracer or QSO redshift bin. For the unrestricted model, the polarizations are described by a different set of parameters, θ _X and θ _Y . The maximum likelihood estimate of the parameters is obtained for each model using the L-BFGS-B optimization algorithm. The following test statistic is then calculated:

$$\begin{eqnarray}&&{\rm{\Delta }}{\chi }^{2}={\chi }^{2}({\hat{{\boldsymbol{\theta }}}}_{\mathrm{res}})-{\chi }^{2}({\hat{{\boldsymbol{\theta }}}}_{\mathrm{unres}}),\end{eqnarray} \tag{ 127 }$$

where χ² is given by Equation (98), ${\hat{{\boldsymbol{\theta }}}}_{\mathrm{res}}\equiv \hat{{\boldsymbol{\theta }}}$ denotes the maximum likelihood parameter estimates for the restricted model, and ${\hat{{\boldsymbol{\theta }}}}_{\mathrm{unres}}\equiv \left[{\hat{{\boldsymbol{\theta }}}}_{X},{\hat{{\boldsymbol{\theta }}}}_{Y}\right]$ denotes the maximum likelihood parameter estimates for the unrestricted model. The χ² values and the test statistic are quoted in Table 4 for all tracers and all QSO redshift bins.

The test statistic will follow a χ² distribution with ${\rm{\Delta }}\mathrm{DOF}={\mathrm{DOF}}_{\mathrm{res}}-{\mathrm{DOF}}_{\mathrm{unres}}$ degrees of freedom under the null hypothesis that the two polarizations are described by the same model. Naively we expect ΔDOF to be equal to the number of model parameters, since the unrestricted model has twice the number of parameters as the restricted model. However, the model parameters are highly degenerate, so that using the number of parameters would likely overestimate ΔDOF and bias the test toward accepting the null hypothesis.

To avoid this, the distribution of the test statistic under the null hypothesis is empirically measured using the random mock catalogs. We generate 10,000 realizations of our data by adding the best-fit, restricted model and a stack on a random mock catalog. We then fit each realization to the restricted and unrestricted models and calculate the test statistic. The probability to observe a value of the test statistic in excess of that observed in the data is then determined from the empirical cumulative distribution function. The results are presented in the last column of Table 4. For all tracers and QSO redshift bins, the null hypothesis that the two polarizations are described by the same set of model parameters is accepted with the probability to exceed (PTE) > 0.05. We also note that the empirical distributions are reasonably well described by a χ² distribution with ΔDOF ≈ 2.3 degrees of freedom.

The 102 sidereal days that were used to construct the sidereal stack are split into two subsets by chronologically ordering the days that went into each seasonal stack and then separating the even days into one set and the odd days into the other set (see Section 3.3.2). The two sets have size 50 and 52 sidereal days and a mean date that differs by 53 hr. Each set is averaged using the procedure outlined in Section 3.3. This yields two estimates of the visibilities that are then differenced according to

$$\begin{eqnarray}&&{\rm{\Delta }}V=\displaystyle \frac{1}{c}\left({V}_{\mathrm{even}}-{V}_{\mathrm{odd}}\right),\end{eqnarray} \tag{ 128 }$$

with

$$\begin{eqnarray}c=\left\{\begin{array}{ll}\displaystyle \frac{{w}_{\mathrm{even}}+{w}_{\mathrm{odd}}}{\sqrt{{w}_{\mathrm{even}}{w}_{\mathrm{odd}}}} & {\rm{if}}\,({w}_{\mathrm{even}}\gt 0)\wedge ({w}_{\mathrm{odd}}\gt 0){\rm{;}}\\ 0 & {\rm{otherwise.}}\end{array}\right.\end{eqnarray} \tag{ 129 }$$

Here $V\equiv {V}_{{xy}}^{p}(\nu ,\phi )$ and $w\equiv {w}_{{xy}}^{p}(\nu ,\phi )$ denote the visibilities and corresponding weights. The quantity c is a scale factor that will set the variance of the radiometric noise in the difference equal to that in the weighted average. In the limit that the even and odd splits have equal radiometric noise, and hence equal weight, then c = 2. In reality, the two splits have slightly different weights such that c = 2.0065 ± 0.018 over the baselines, frequency, and right ascensions examined. The processing described in Section 3.3 through Section 4.7 is applied to the differenced visibility, with the caveat that we use the global frequency mask, delay cut, and primary beam model that were previously derived from the weighted average of the full set of days.

The cosmological 21 cm signal is constant as a function of sidereal day and is expected to cancel in the difference. The radiometric noise, on the other hand, will be independent in the two subsets and therefore will remain in the difference. Transient RFI is also expected to be independent in the two subsets and remain in the difference. Residual foregrounds caused by spectral leakage due to a chromatic instrument transfer function will be the same from day to day and hence cancel in the difference. Residual foregrounds due to seasonal changes in the instrument transfer function will also cancel. On the other hand, residual foregrounds due to changes in the instrument transfer function from day to day will remain.

Since a significant portion of the noise in the stack is due to residual foregrounds that will be mitigated by the differencing procedure, the covariance matrix of the even–odd difference is expected to change relative to the covariance matrix of the full data set. We recalibrate the covariance matrix with mock catalogs as outlined in Section 4.8. We find better agreement between the even–odd difference covariance and the expected radiometric noise, suggesting that the majority of ∼50% excess noise in the full set is primarily due to foregrounds that are static from one day to the next.

Under the null hypothesis that the observed signal is the same on even and odd days, stacking the even–odd difference on the true catalog should be statistically equivalent to stacking on a random mock catalog. The distribution of the χ² test statistic for the random mock catalogs is well described by a theoretical χ² distribution with 202 degrees of freedom. Table 5 quotes the χ² value of the stack on each tracer and QSO redshift bin, as well as the fraction of the 10,000 random mock catalogs that have a χ² test statistic in excess of that observed for the true catalog. We find that all tracers and redshift bins have a PTE greater than 0.05, except for the LRG catalog, which has a PTE of 0.025, and the subset of the QSO catalog with a redshift between 0.91 and 1.03 (QSOb01), which has a PTE of 0.023. If the large χ² values are due to differences in the observed signal on even and odd days, then we would expect to see a copy of the signal in the jackknife. We recompute the test statistic using only frequencies ∣Δν∣ < 5 MHz, where the magnitude of the signal is largest. We find that the PTE for the LRG catalog increases to 0.12, the QSOb01 catalog decreases to 0.009, and all other tracers and QSO redshift bins have a value greater than 0.05. This leads us to conclude that the large χ² observed when stacking the jackknife on the LRG catalog originates from a rare noise fluctuation rather than differences in the signal on even and odd days. However, the large χ² for the QSOb01 catalog warrants additional investigation.

To explore this further, we perform a model-dependent analysis of the even and odd days that is similar to the analysis used to check for consistency between polarizations, which was described in Section 7.1. Each split is processed independently through the pipeline and then stacked on the true catalog and the random mock catalogs. We use the same random mock catalogs for both splits to ensure that the covariance matrix captures correlated noise between them. The two splits are jointly fit to both a restricted and unrestricted model. The restricted model describes both splits with the same set of parameters. The unrestricted model describes each split with a different set of parameters. We employ the version of our model where the nonlinear parameters are held fixed at their fiducial values (see Table 2). We compute Δχ² as given by Equation (127) and calibrate its distribution under the null hypothesis using the random mock catalogs. The results are presented in Table 6.

Table 6. Model-dependent Test for Consistency between Even and Odd Days

	χ2
Tracer	Restricted	Unrestricted	Δχ2	ΔDOF	PTE	$\tfrac{{\rm{\Delta }}{{ \mathcal A }}_{{\rm{H}}\,{\rm\small{I}}}}{2{{ \mathcal A }}_{{\rm{H}}\,{\rm\small{I}}}}$
LRG	484.0	483.0	1.0	2.4	0.71	$-{0.11}_{-0.14}^{+0.16}$
ELG	414.7	412.7	2.1	2.3	0.42	$-{0.15}_{-0.11}^{+0.22}$
QSO	425.2	422.4	2.8	2.6	0.34	$-{0.06}_{-0.10}^{+0.11}$
QSOb0	447.9	436.7	11.2	2.4	0.005	${0.26}_{-0.12}^{+0.12}$
QSOb1	370.6	368.3	2.4	2.5	0.41	$-{0.12}_{-0.12}^{+0.14}$
QSOb2	452.7	446.4	6.3	2.4	0.052	$-{0.16}_{-0.13}^{+0.16}$
QSOb00	378.7	378.1	0.6	2.3	0.81	${0.11}_{-0.17}^{+0.20}$
QSOb01	468.6	460.1	8.5	2.4	0.020	${0.25}_{-0.11}^{+0.13}$

Note. For each tracer and redshift bin, we report the minimum χ² obtained when fitting a model in which the even and odd splits are described by the same set of parameters (restricted) and a different set of parameters (unrestricted). The distribution of the difference, Δχ², under the null hypothesis that the splits are described by the same set of parameters is calibrated using random mock catalogs and approximately follows a theoretical χ² distribution with the quoted ΔDOF degrees of freedom. The PTE provides the fraction of random mock catalogs that exceed the value observed in the data. The last column provides the fractional error on the amplitude parameter, ${{ \mathcal A }}_{{\rm{H}}\,{\rm\small{I}}}$, inferred from this comparison.

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As anticipated, the LRG catalog passes the test (PTE = 0.71) and the QSOb01 catalog fails the test (PTE = 0.02). The QSOb0 catalog, which contains all QSOs with a redshift between 0.80 and 1.03 and is a superset of QSOb01, also fails the test (PTE = 0.005). The discrepancy appears primarily in the amplitude of the signal, with the even-day split exhibiting an approximately 50% larger amplitude than the odd-day split for these two redshift bins. We perform an MCMC fit of both the restricted and unrestricted models and use the posterior distributions of the amplitude parameter ${{ \mathcal A }}_{{\rm{H}}\,{\rm\small{I}}}$ to characterize the fractional error in ${{ \mathcal A }}_{{\rm{H}}\,{\rm\small{I}}}$ implied by this discrepancy. This is defined as half the difference in ${{ \mathcal A }}_{{\rm{H}}\,{\rm\small{I}}}$ between the even and odd splits as measured by the unrestricted model fit divided by the most likely value from the restricted model fit, and it is quoted in the last column of Table 6.

The discrepancy is suggestive of a 50% difference in our calibration between even and odd days at frequencies between 700 and 745 MHz. However, we have ruled out an error in the relative calibration of this magnitude by examining the spectra of 34 bright point sources in the NGC field extracted from the maps prior to foreground filtering. We find that the difference in spectra between the even and odd days is at most 1% over all sources and frequencies.

In order to account for the observed discrepancy, we will assume an additional 25% systematic error on the ${{ \mathcal A }}_{{\rm{H}}\,{\rm\small{I}}}$ constraint for the QSOb0 and QSOb01 catalogs.

In order to estimate the uncertainty on the default beam model described in Section 4.4, it is compared to independent measurements of the beam from observations of the Sun and holographic observations of bright point sources made in conjunction with the John A. Galt 26 m telescope (CHIME Collaboration et al. 2022b). Based on these comparisons, we estimate that within the main lobe the beam model is accurate to ≲5% relative to the beam on meridian at the decl. of Cygnus A. Currently our beam calibration technique is unable to constrain the sidelobes of the beam (for details see Appendix B). The solar and holographic data both suggest that the sidelobes are ≲1% at hour angles ≲30° and ≲0.1% at hour angles ≳30°. It is estimated that approximately 10% of the beam solid angle lies outside the region that we are able to measure with the default beam model.

The solar beam measurements are described in CHIME Collaboration et al. (2022a) and span −235 ≤ θ ≤ 235, which corresponds to the range of apparent decl. that the Sun travels between winter and summer solstice. The rms difference between the solar and default beam model is 4% (relative to the beam on meridian at the decl. of Cygnus A) in the region ∣HA∣ ≲ 3°, ∣θ∣ < 235, 587.5 MHz < ν < 800 MHz. However, the inferred amplitude of the 21 cm signal is primarily sensitive to the fractional error in the beam on meridian when averaged over the large range of decl. and frequencies covered by the eBOSS catalogs. In order to estimate the systematic error on the 21 cm amplitude due to beam uncertainties, the fractional difference between the default and solar beam model on meridian at the decl. and 21 cm frequency of each source in each catalog is extracted and then averaged using the same weights that are used in the stacking procedure described in Section 4.7. Only 3% of the QSOs and LRGs in the NGC field are at decl. that overlap with the solar data. However, 42% of the ELGs in the NGC field lie at decl. where there are two independent measurements of the beam, and the average fractional difference between these two measurements is 6%.

We have also compared the flux density of the brightest radio sources in the deconvolved map to their expected flux density in order to obtain an additional estimate of the systematic uncertainty on the 21 cm amplitude. The expected flux densities are obtained by interpolating recent measurements made by the VLA to frequencies in the CHIME band (P17). There are 14 sources in total used for this purpose, with an average decl. separation of 5°. These sources do not provide a dense sampling of the decl. axis like the solar data, but they do cover the full range of decl. spanned by the eBOSS catalogs. All 14 sources have interpolated flux densities that are accurate at the subpercent level and are greater than 10 Jy at 600 MHz. The rms of the fractional error in the flux density of these sources in the deconvolved map is 5.0%, 6.4%, and 7.4% for the range of frequencies and decl. spanned by QSO, LRG, and ELG catalogs in the NGC field, respectively. Taking instead the weighted average of the fractional error in the flux at the decl. and 21 cm frequencies nearest to the sources in each catalog yields 0.6%, 2%, and 0.5% for the QSO, LRG, and ELG catalogs in the NGC field. Note that this is an end-to-end test of our ability to recover the true flux of point sources and is sensitive to a variety of potential sources of systematic error, including beam calibration errors, but also complex gain errors and regridding artifacts.

As a final check, we simulate observations of the fiducial 21 cm signal using both the default beam and the control beam. For each of these simulations, we construct a map by deconvolving both the default beam and control beam and then stack the map on simulated catalogs. We then examine the fractional difference in the amplitude of the stacked signal for the four different pairs of (simulation beam model, deconvolution beam model) relative to the (default, default) pair that was used in the actual analysis. For all four pairs and all three tracers the observed difference is less than 6%. This provides an estimate of the systematic error due to uncertainty in the interference pattern that modulates the beam. Note that this is a very conservative estimate because the uncertainty on the interference pattern is roughly a factor of 10 less than than the amplitude of the interference pattern itself.

Based on the solar comparison, bright point-source comparisons, and simulations of different beam models, a conservative 8% systematic error on the amplitude of the 21 cm signal will be assumed for all fields.

Many of the elements in our analysis pipeline, such as the delay filtering, are explicitly linear, meaning that they operate independently on the 21 cm signal and foregrounds present in the data. To characterize the linearity of the entire analysis, we inject simulated 21 cm signal into the data, process the signal+data combination in the same way as the data, and stack the results on mock eBOSS catalogs that are correlated with the simulated signal. We also separately perform the stacking on mock catalogs using the data without injected signal and using mock observations containing only the injected signal. In a perfectly linear analysis, the difference of the signal+data and data stacks will be equal to the signal-only stacks, while nonlinearities will cause a violation of this equality. This method has previously been used to characterize signal loss in 21 cm analyses that rely on strongly nonlinear foreground filtering techniques (e.g., Masui et al. 2013; Paciga et al. 2013).

In detail, we generate correlated 21 cm and galaxy number density sky maps and propagate them through to simulated time streams and mock LRG, ELG, or QSO catalogs following the procedures in Section 5.3. The signal-only time stream is added to the sidereal stack derived from the data prior to subtraction of the brightest point sources (i.e., in the first box in Figure 6), and this combined time stream is passed through the same analysis pipeline as the data, culminating in the beam-deconvolved, filtered, masked map being stacked on the mock catalogs. Prior to delay filtering, the signal-only map has rms ∼ 0.3 mJy beam⁻¹, compared to ∼3 Jy beam⁻¹ for the data map; therefore, the addition of signal to the data map has negligible effect on the determination of the elevation-dependent delay cut (Section 4.5), or on which frequencies are identified as outliers (Section 4.6), so these aspects of the analysis are not regenerated for the signal+data combination.

However, the final masking step—which masks map pixels whose absolute value exceeds a chosen threshold, based on the estimated map noise level—is explicitly nonlinear, so we recompute this mask to determine the impact of this nonlinearity on the recovered signal. We find that this impact is significant, which can be explained as follows. The distribution of pixel values in the signal-only map is symmetric about zero, but this distribution is skewed positive if one only considers pixels containing an object in a given mock catalog, since these objects are more likely to occupy pixels corresponding to matter overdensities, which are also correlated with 21 cm emission. Thus, if a given pixel (containing a catalog object) in the data map is positive and just below the mask threshold, it is more likely to be perturbed above the threshold by the addition of the signal-only map; similarly, a given negative pixel that is just beyond the threshold is more likely to be perturbed within the threshold.

The net effect is that the signal injection increases the number of negative near-threshold unmasked pixels and decreases the number of such positive pixels, resulting in an artificial attenuation of the overall stacking amplitude. A lower threshold will result in a greater number of affected pixels and more severe attenuation, while a higher threshold will mitigate this, but at the expense of decreasing the signal-to-noise ratio owing to a greater number of anomalous pixels being included in the stack.

Figure 26 quantifies these two effects. For each threshold in the figure, we compute the difference of stacks on signal+data and data-only maps, form a "prediction" given by a stack on the corresponding signal-only map, and fit the overall amplitude of the prediction to the stack difference, using the data stack covariance matrix described in Section 4.8. This amplitude indicates the amount of attenuation (shown as solid lines in Figure 26) induced by the outlier mask, while the fractional uncertainty on this amplitude (shown as dashed lines) indicates the effect of the mask threshold on the statistical significance of the stacking measurement.

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Based on these results, we choose a mask threshold of 6σ, where σ is the estimated radiometric noise in the maps (see Section 3.1.4). For the fiducial 21 cm model assumed in our simulations, this results in signal attenuation of less than 4% for each eBOSS tracer, which is at least a factor of three smaller than the statistical uncertainty. Note that in our actual fits to data, presented in Section 6, the stacking amplitudes are factors of (1.9, 1.5, 1.4), for the (ELG, LRG, QSO) stacks, greater than in our simulations. We have rerun the test in Figure 26, modifying the amplitude of the injected signal accordingly, and have verified that the attenuation level is unchanged, while the fractional uncertainty decreases by the quoted factors. Even with this change, the attenuation is still less than half of the uncertainty for each tracer, which we deem to be acceptable for this analysis.

As illustrated in Figure 23, there is a statistically significant frequency offset in the QSO stacks of Δν₀ ≈ − 0.2 MHz, equal to roughly half the width of a CHIME frequency channel. As this is only seen in the QSO stacks and not within the overlapping LRG and ELG measurements, it is difficult to explain this as an instrumental issue within CHIME. Instead, we interpret this as being a systematic bias in the eBOSS QSO redshifts, stemming from the difficulty of determining a redshift from the complex processes producing a QSO spectrum (see Lyke et al. 2020). QSO emission lines such as C iv are frequently blueshifted from the host galaxy redshift by dynamical and radiative processes within the QSO's accretion disk and outflowing winds (Richards et al. 2011; Shen et al. 2016).

Similar to Lyke et al. (2020), we express the redshift error as a velocity that can be connected to our measured frequency offset

$$\begin{eqnarray}&&{\rm{\Delta }}v=c\displaystyle \frac{{\rm{\Delta }}z}{1+z}\end{eqnarray} \tag{ 130 }$$

$$\begin{eqnarray}&&=c(1+z)\displaystyle \frac{{\rm{\Delta }}{\nu }_{0}}{{\nu }_{21}}.\end{eqnarray} \tag{ 131 }$$

In Figure 27, we show the inferred velocity bias for the QSOb00, QSOb01, QSOb1, and QSOb2 stacks, which give nonoverlapping measurements in redshift. Overall, we measure Δv ∼ − 66 km s⁻¹ at ∼3.3σ, with individual bins ranging from 0.8σ (QSOb2) to 2.5σ (QSOb01). Our analysis does not account for the Doppler shift from Earth's motion around the solar system barycenter; however, while the shift on any individual source may be up to ∼30 km s⁻¹, on average this effect is small. Taking a weighted mean of the Doppler shift toward each source for each night of observation, we find an average Doppler correction of −3.1 km s⁻¹.

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Overall the results in Figure 27 are consistent with those of Lyke et al. (2020, Figure 3), who estimated the systematic bias in the z_PCA redshift estimates (which we used for stacking) as compared to redshifts of QSO host galaxies measured using stellar absorption lines. We anticipate that future QSO cross-correlation analyses with higher source numbers and improved processing of the CHIME data will be able to provide useful measurements of this bias across a broad range of redshifts.

The eBOSS QSO redshifts are significantly noisier than those of LRG and ELG samples owing to the broader emission lines, with significant long tails of poor redshift estimates (Lyke et al. 2020). This has a noticeable effect on the stack signal (recall Figure 17), as the convolutional effect of the redshift errors broadens and suppresses the peak of the stack signal. Uncertainties in the QSO redshift error model can therefore give sizable changes in the constraints on the signal amplitude.

In our primary analysis we use the "double-Gaussian" model of Lyke et al. (2020, Equation (A2)) to describe the QSO redshift uncertainties. However, the model as presented does not seem to match their measurements of the redshift errors in ways that are significant for our analysis. There are two clear differences: first, the fraction of observations that have errors drawn from the wider Gaussian component appears to be smaller in the data (as shown in Figure 4 of Lyke et al. 2020) than the ∼18% quoted in the model; second, there appears to be a significant reduction in the errors at low redshift compared to the rest of the sample (as shown in Figure 9 of Lyke et al. 2020), which is expected from the presence of O iii and Hβ in the wavelength range of the spectrograph at redshifts z ≲ 1 (Étienne Burtin, private communication).

To assess the importance of this, we modify the z_PCA redshift error model to capture these effects. This change is intended to give a plausible alternative consistent with the data presented in Lyke et al. (2020), though we do not claim that it is more realistic. Producing an improved model would require repeating the analysis of Lyke et al. (2020) and is beyond the scope of this paper. Our model is a straightforward modification of the published "double Gaussian" where we allow the coefficients to be redshift dependent. The redshift error on a single observation of a QSO, as given by a velocity error δ v, is drawn from a redshift probability distribution

$$\begin{eqnarray}\begin{array}{rcl}{ \mathcal P }(\delta v| z) & = & \displaystyle \frac{1}{1+{f}^{-1}(z)}\\ & & \times \,\left[{ \mathcal G }\left(\delta v,{\sigma }_{1}{\left(z\right)}^{2}\right)+{f}^{-1}(z){ \mathcal G }(\delta v,{\sigma }_{2}{\left(z\right)}^{2})\right],\end{array}\end{eqnarray} \tag{ 132 }$$

where ${ \mathcal G }$ is the standard normalized Gaussian with

$$\begin{eqnarray}&&{ \mathcal G }(x,{\sigma }^{2})=\displaystyle \frac{1}{\sqrt{2\pi {\sigma }^{2}}}{e}^{-\tfrac{{x}^{2}}{2{\sigma }^{2}}},\end{eqnarray} \tag{ 133 }$$

and σ₁(z) and f(z) are smooth step functions centered at z = 1.0 with σ₁ transitioning from 90 to 150 km s⁻¹,

$$\begin{eqnarray}&&{\sigma }_{1}(z)=\left[90+30\left(1+\tanh \left(\displaystyle \frac{z-1.0}{0.05}\right)\right)\right]\,\mathrm{km}\,{{\rm{s}}}^{-1},\end{eqnarray} \tag{ 134 }$$

and f⁻¹(z) changing from zero at low redshift (that is, no errors in the broad distribution) to ∼0.03 (a value that approximately matches the data of Lyke et al. 2020, Figure 4),

$$\begin{eqnarray}&&{f}^{-1}(z)=\displaystyle \frac{1}{35}\left(1+\tanh \left(\displaystyle \frac{z-1.0}{0.05}\right)\right).\end{eqnarray} \tag{ 135 }$$

It is necessary to change both σ₁ and f, as no single change is able to reproduce the observed low-redshift uncertainties. However, we do leave σ₂ unchanged with a redshift-independent σ₂(z) = 1000 km s⁻¹.

In Figure 28 we show the change in our constraints that occur if we switch to this modified model. The top panel compares the redshift dependence of our new model to the Lyke et al. (2020) model and the measured uncertainties in their Figure 9. The bottom panel shows the change in the inferred amplitude, ${{ \mathcal A }}_{{\rm{H}}\,{\rm\small{I}}}$, between the two models. We have fixed the nonlinear parameters in these constraints, which gives an indication of the statistical error on our constraints. At all redshifts the difference between the published model and our alternative is larger than the statistical uncertainty on ${{ \mathcal A }}_{{\rm{H}}\,{\rm\small{I}}}$ and suggests that redshift error distribution is a significant source of systematic uncertainty in our analysis. Future analyses will need to resolve the questions in this modeling to make precision constraints on ${{ \mathcal A }}_{{\rm{H}}\,{\rm\small{I}}}$.

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We use the differences observed in Figure 28 to estimate a systematic uncertainty from the redshift error modeling of $\left|{{ \mathcal A }}_{{\rm{H}}\,{\rm\small{I}}}^{\mathrm{alt}}-{{ \mathcal A }}_{{\rm{H}}\,{\rm\small{I}}}^{\mathrm{norm}}\right|/\sqrt{2}$, where the $\sqrt{2}$ comes from an argument that the models considered are samples from some distribution of plausible models.

We are interested in learning about the amplitude of fluctuations in the H i distribution, which is probed most effectively by the parameter ${{ \mathcal A }}_{{\rm{H}}\,{\rm\small{I}}}$ (see Equation (104)) in our analysis. In Section 6.3 we discuss the constraints on ${{ \mathcal A }}_{{\rm{H}}\,{\rm\small{I}}}$ in the case where we allow the full set of parameters to vary and where we pin the nonlinear parameters to their fiducial values.

The uncertainty in the case with fixed nonlinear parameters is dominated by the statistical uncertainty in the data and from the prior on the galactic bias. We assume that this statistical contribution is the same in the case where we allow the nonlinear parameters to vary, with the weaker constraints coming from modeling uncertainties. In this case, and assuming that the modeling errors are multiplicative within the degenerate regions of parameter space, we can roughly separate the uncertainty in the full parameter constraints into statistical and modeling contributions.

There are many potential additional sources of systematic errors in our analysis that have been discussed beyond the modeling uncertainty. These are listed, along with the statistical and modeling uncertainty breakdown, in Table 7. This error budget is dominated by the modeling uncertainties; however, both the systematic error added to cover unexpected validation failures (labeled "Consistency" in Table 7 and discussed in Section 7.2) and the error from uncertainties in the QSO redshift error model (Section 8.1) are larger than the statistical error and thus could be the limiting sources if the modeling of nonlinear scales can be improved.

In Table 8, we summarize the constraints on ${{ \mathcal A }}_{{\rm{H}}\,{\rm\small{I}}}$ for all the tracer catalogs. We show the case both with the nonlinear parameters fixed and when the full set is allowed to vary, illustrating again the substantial increase in the uncertainties from these parameters. We also give a final case including the total error budget from all the systematic contributions above (we have assumed that they are all multiplicative effects). As the modeling uncertainties are dominant, including these extra sources of error gives only marginal increases to the total uncertainty. The most severely affected catalogs are QSOb0 and QSOb01, due to the systematic error contributions from both issues in the QSO redshift error model (which is worse at low redshifts) and from the consistency test failures.

Table 8. Parameter Constraints for Each Tracer

Tracer	zeff	${{ \mathcal A }}_{{\rm{H}}\,{\rm\small{I}}}$
		Fiducial	Fixed NL	Full NL	Full + Systematics
LRG	0.84	1.13	${1.82}_{-0.25}^{+0.26}$	${1.51}_{-0.96}^{+3.60}$	${1.51}_{-0.97}^{+3.60}$
ELG	0.96	1.21	${2.35}_{-0.42}^{+0.43}$	${6.76}_{-3.74}^{+9.01}$	${6.76}_{-3.79}^{+9.04}$
QSO	1.20	1.37	${1.86}_{-0.17}^{+0.18}$	${1.68}_{-0.60}^{+1.06}$	${1.68}_{-0.67}^{+1.10}$
QSOb0	0.97	1.22	${2.27}_{-0.28}^{+0.31}$	${2.04}_{-0.94}^{+2.09}$	${2.04}_{-1.19}^{+2.21}$
QSOb1	1.12	1.31	${1.75}_{-0.25}^{+0.25}$	${2.89}_{-1.36}^{+3.13}$	${2.89}_{-1.44}^{+3.17}$
QSOb2	1.30	1.43	${1.81}_{-0.28}^{+0.27}$	${1.63}_{-0.86}^{+1.55}$	${1.63}_{-0.90}^{+1.57}$
QSOb00	0.84	1.14	${2.49}_{-0.54}^{+0.52}$	${1.49}_{-1.65}^{+4.06}$	${1.49}_{-1.69}^{+4.08}$
QSOb01	0.99	1.23	${2.23}_{-0.34}^{+0.35}$	${3.23}_{-1.56}^{+3.47}$	${3.23}_{-1.91}^{+3.64}$

Note. After reparameterization to avoid degeneracies, the physically interesting parameter is the 21 cm amplitude ${{ \mathcal A }}_{{\rm{H}}\,{\rm\small{I}}}$. We show the highest posterior density 68% credible intervals for these parameters for both a prior with the nonlinear parameters fixed and the full parameter space. Comparing the ${{ \mathcal A }}_{{\rm{H}}\,{\rm\small{I}}}$ constraints for the cases of fixed and varying nonlinear parameters, we can see that there is a substantial increase in the uncertainty from modeling the small-scale structure. We also show estimates for ${{ \mathcal A }}_{{\rm{H}}\,{\rm\small{I}}}$ including the effects of the systematic errors listed in Table 7. As the modeling errors are large, the additional uncertainty from this is small.

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To be able to compare our results directly to measurements of Ω_{H i} from other probes, we need to be able to break the degeneracy between Ω_{H i} and b_{H i} . Although our measurements are unable to do this internally and there are no external measurements of b_{H i} , we can use simulations as a guide.

As an indicator of the uncertainty on the bias, we use the bias measured at z = 1 from various simulations. Villaescusa-Navarro et al. (2018) use the IllustrisTNG hydrodynamic simulation and find that b_{H i} (z = 1) ≈ 1.49, and Ando et al. (2019) use another hydrodynamic simulation, the Osaka simulation, to find that b_{H i} (z = 1) ≈ 1.26 (from their b₀ measurements). Another approach uses semianalytic prescriptions on top of dark-matter-only simulations, such as Spinelli et al. (2020), who find b_{H i} (z = 1) ≈ 1.22 or 1.31 (depending on whether the Millennium I or II simulation is used), or Wang et al. (2021), who use an empirically calibrated star formation model to find b_{H i} (z = 1) ≈ 1.27. Collectively these prescriptions have a mean of ≈1.3 and a standard deviation of ≈0.1. With this in mind, we place a simulation-derived Gaussian prior on the bias with a conservative width of 20%, i.e., ${\sigma }_{{b}_{{\rm{H}}\,{\rm\small{I}}}}/{b}_{{\rm{H}}\,{\rm\small{I}}}^{\mathrm{fid}}=0.2$.

We reweight the MCMC chains from our analysis to apply the updated prior on b_{H i} and marginalize over all the other parameters to derive constraints on Ω_{H i} . We give our measurements as the highest posterior density credible interval about the mode of the distribution. In Figure 29 we show the measurements for the LRG and ELG samples, as well as the QSOs split across three redshift bins, compared to measurements from other experiments.

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There are four main methods for measuring Ω_{H i} that we include in Figure 29 for comparison:

• Direct H i surveys: At the lowest redshifts, blind surveys of the 21 cm line can measure the H i mass function directly, which can be integrated to obtain estimates of Ω_{H i} .
• H i stacking: At intermediate redshifts, it is difficult to detect individual galaxies in their 21 cm emission; to get around this, high-resolution radio data can be stacked on the positions of galaxies found in optical catalogs to get an estimate of the average amount of H i per galaxy in the sample. This can then be combined with an optical luminosity function for the sample and corrected for completeness to give an estimate of Ω_{H i} .
• H i intensity mapping: Another method is to cross-correlate H i intensity mapping data with optical catalogs. These are distinct from the H i stacking measurements described above in that they do not resolve the emission in the (average of) individual galaxies, but instead are sensitive to the correlated H i mass in the vicinity of the galaxy. Our results are an example of this technique.
• Damped Lyα: At the highest redshifts damped Lyα systems are detected in optical and UV QSO spectra, and the distribution of their observed column densities can be integrated to find Ω_{H i} .

For all measurements, we convert into the Planck 2018 cosmology used in this paper, using $H{(z)}^{2}\,={H}_{0}^{2}[{{\rm{\Omega }}}_{{\rm{m}}}{(1+z)}^{3}+1-{{\rm{\Omega }}}_{{\rm{m}}}]$, with Ω_m = 0.30964. In each case, the measurements are effectively a fluxlike quantity, which is multiplied by an area to give an H i mass, divided by a survey volume to give a density, and then divided by the critical density to give Ω_{H i} , though some of these steps are implicit (this is still true for the damped Lyα analysis, though the "area" in the mass is canceled with the one implicit in the volume). If we approximate the observations as coming from a narrow band in redshift, then the cosmology dependence is

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{{\rm{H}}\,{\rm\small{I}}}(z)\propto \displaystyle \frac{1}{{H}_{0}^{2}}\displaystyle \frac{{dz}}{{dr}},\end{eqnarray} \tag{ 136 }$$

where r is the comoving distance to redshift z. For the Wolz et al. (2022) intensity mapping points we convert their Ω_{H i} b_{H i} r measurements into Ω_{H i} constraints with the fiducial bias model we use in this paper to allow a consistent comparison.

In Figure 29 for the constraints both when varying the nonlinear parameters (top panel) and when fixing them (bottom panel) our results are in broad agreement with other Ω_{H i} constraints. As we would expect, the uncertainties are much larger when allowing the nonlinear parameters to vary, though the distributions are non-Gaussian and the probability of Ω_{H i} ≤ 0 is still negligible. We note that while we expect all points to be pushed toward higher values of Ω_{H i} by the prior volume effect in the FoG parameters discussed in Section 6.3, the CHIME+eBOSS ELG point is noticeably discrepant when the nonlinear parameters are varied. We believe that this is a chance fluctuation where the region further along the ${{ \mathcal A }}_{{\rm{H}}\,{\rm\small{I}}}$−α_FoG,+ degeneracy is preferred and leads to a ∼2σ shift from the Ω_{H i} values preferred by the other tracers. This can also be seen clearly in Figure 24, where the preferred range of α_FoG,+ values is higher than in the QSO and LRG cases (Figures 23 and 25, respectively). When fixing the nonlinear parameters, the ELG constraints are much more consistent with both the other CHIME tracers and the external data sets.

As the constraints with fixed nonlinear parameters do not include the full modeling uncertainties, they show the internal consistency and significance of our measurements but are not good indicators of the plausible range of Ω_{H i} determined from our data. In all cases we are showing constraints derived with the fiducial eBOSS QSO error model. As discussed in Section 8.1, we believe that this model may bias the QSO constraints (particularly the lowest-redshift bin) to higher values of Ω_{H i} .

As mentioned previously, our stacking analysis not only probes the correlated clustering of eBOSS catalog objects and H i but also is sensitive to the H i associated with the objects themselves, which sets the value of our M₁₀ parameter for each sample (recall that M₁₀ is defined as the mean H i mass per catalog object, in units of 10¹⁰ M_⊙). Figures 23–25 show that the posteriors for M₁₀ peak at nonzero values for the QSO and LRG stacks, while for the ELG stack the posterior peaks at M₁₀ = 0. In each case, however, the model where M₁₀ and the other nonlinear parameters (α_FoG,+ and α_NL) are allowed to vary is not strongly preferred over the case where these parameters are fixed to their fiducial values (see Section 6.4); thus, we cannot interpret the posteriors of M₁₀ as providing definitive information about the H i content of the objects in each catalog.

Nevertheless, the finite width of these posteriors indicates that future analyses may hold the promise of interesting constraints. In particular, for the ELG stack, the highest posterior density 68% credible interval is M₁₀ < 1.04. This is consistent with the simulations of Wolz et al. (2022), which were based on the DARK SAGE semianalytical galaxy evolution model (Stevens et al. 2016) and predicted a shot-noise contribution to the H i−ELG cross-power spectrum equivalent to M₁₀ ≈ 0.8 (as inferred from their Figure 12). It is also consistent with the analysis of Chowdhury et al. (2020), who stacked GMRT 21 cm observations on star-forming galaxies from the DEEP2 survey and found M₁₀ = 1.19 ± 0.26 at an effective redshift z_eff = 1.03. A cross-correlation analysis with greater power to break the parameter degeneracies in our model would likely improve the constraint on M₁₀ to a level where it could fruitfully be compared with these other values.

Empirical information on the H i content of LRGs at z ∼ 1 is scarce: direct stacking analogous to Chowdhury et al. (2020) has only been carried out at lower redshifts for such red galaxies (e.g., Rhee et al. 2018). Thus, constraints on M₁₀ for LRGs (and QSOs) would provide valuable information about the evolution and environments of these objects. On the other hand, inclusion of an external prior on M₁₀, obtainable from, for example, stacking GMRT observations on a subset of objects from each eBOSS catalog, would help to break the degeneracies in our model (or other, more detailed models of H i−galaxy cross-correlations), and we see this as a promising avenue for future investigation.