Constraining the Halo Mass of Damped Lyα Absorption Systems (DLAs) at z = 2–3.5 Using the Quasar-CMB Lensing Cross-correlation

Damped Lyα systems (DLAs) are a class of absorbers along the QSO sightlines with high neutral hydrogen (H i) column densities of ${N}_{\mathrm{HI}}\geqslant 2\times {10}^{20}$ cm⁻² (Wolfe et al. 1986). They are featured by their damping wing absorption profile, a result of the quantum natural broadening of the H i Lyα transition. DLAs are thought to dominate the neutral-gas content of the universe in the redshift range of 0 ≤ z ≤ 5 (Wolfe et al. 2005), acting as important neutral gas reservoirs for star formation (e.g., Nagamine et al. 2004; Ota et al. 2014; Rudie et al. 2017). They can also be a powerful cosmological probe for related research, such as studying the hosts of quasars (e.g., Hennawi et al. 2009; Zafar et al. 2011; Cai et al. 2014).

Over the last three decades, a number of attempts have been made to discover the nature of DLAs by directly probing the emission of the DLA galaxies using the largest optical or infrared telescopes (e.g., Le Brun et al. 1997; Møller et al. 2002; Chen et al. 2005). These studies have been moderately successful at low redshift. Nevertheless, after the many deep observations with innovative techniques (e.g., Kulkarni et al. 2006; Fynbo et al. 2010; Fumagalli et al. 2015; Johnson-Groh et al. 2016), only about 20 DLA host galaxies at z ≳ 2 have been identified. These results suggest that DLA host galaxies have relatively small, sub-L^* galaxies with a star formation rate (SFR) of ∼0.1–10 ${M}_{\odot }$ yr⁻¹ (e.g., Krogager et al. 2017). With the Atacama Large Millimeter/submillimeter Array (ALMA), Neeleman et al. (2018, 2020) revealed a small sample (<10) of host galaxies of high-metallicity DLAs at z ≈ 4 using the submillimeter observations. They find that the SFR of DLA hosts is on the order of 10–100 M_⊙ yr⁻¹, indicating that metal-rich DLA hosts could be super-L^* galaxies, at odds with the previous findings.

Statistical studies of the DLA hosts also remain unclear and controversial. Several works measure the DLA bias by studying the cross-correlation between DLAs and other tracers, such as Lyman break galaxies (e.g., Cooke et al. 2006; Lee et al. 2011) and Lyα forests (e.g., Font-Ribera et al. 2012; Pérez-Ràfols et al. 2018). The latter imply that DLAs are hosted by massive halos with typical masses of ${10}^{11}\mbox{--}{10}^{12}{M}_{\odot }$. Mass–metallicity statistics reveal that the DLA hosts should have a stellar mass of 10^8.5 M_⊙ (e.g., Møller et al. 2013). This result is also supported by several hydrodynamical simulations (e.g., Pontzen et al. 2008; Berry et al. 2016), which suggest that DLA galaxies have stellar masses ranging from 10⁶M_⊙ to 10¹¹M_⊙, with the median value around 10⁸M_⊙. More statistical efforts are needed to provide more independent measurements on the DLA hosts.

Besides the detection techniques mentioned above, weak lensing of the cosmic microwave background (CMB) is a powerful tool to probe halo mass at high redshift. The cross-correlation between the CMB and other tracers of large-scale structure can be used to measure the tracer bias, with which one can infer the typical halo masses of the tracer populations. The cross-correlation measurement is not prone to systematic effects that are possibly present in the auto-correlation approach, such as incompleteness, random catalog generation, masking, etc. (Reid et al. 2016; Geach et al. 2019). This technique has been performed to analyze the clustering properties of quasars and galaxies and provide strong cosmological constraints (e.g., Sherwin et al. 2012; Bianchini et al. 2015; Krolewski et al. 2020). Likewise, CMB photons are deflected not only by the quasar hosts, but also by DLAs in the sightlines. The measured gravitational magnification of the background quasars would be strongly affected by the foreground DLAs if DLAs reside in massive halos comparable to those of the quasar hosts. The assumption has been confirmed by studies on the CMB lensing-quasar cross-correlations using quasars with and without DLAs in their sightlines (Alonso et al. 2018).

In this paper we carefully measure the cross-correlation between the Planck CMB lensing convergence map and two quasar overdensity maps. One of the two quasar maps ensures that each quasar sightline contains at least one DLA. Through comparing the cross-correlation results of the two maps, the properties of the DLA host are extracted. In Section 2 we introduce the data samples used in this work. Section 3 presents the estimator of the cross-correlation power spectrum and its error. In Section 4 we measure the biases of the quasar and DLA samples. The null test results are reported in Section 5. In Section 6 we discuss our measurements and their consistency with several previous works. We draw our conclusions in Section 7. Additionally, we apply simple tests by applying this approach to a quasar bias evolution model, a tSZ-free CMB lensing map and another DLA catalog in Appendices A, B, and C. Throughout the paper, we adopt a flat Λ cold dark matter (ΛCDM) cosmological model as described in Planck Collaboration et al. (2018a), with H₀ = 67.4, Ω_m = 0.315, Ω_bh² = 0.0224, and Ω_ch² = 0.120.

2.1. BOSS Quasars

2.2. Planck 2018 Data

2.3. DLA Catalog

2.4. Sample Selection

3.1. Map Construction and Mask Apodization

To match the resolution of the CMB lensing convergence map, we construct the quasar overdensity maps in the HEALPix format (Górski et al. 2005) with N_side = 2048, corresponding to 50,331,648 pixels for the whole sky:

$$\begin{eqnarray}&&{q}_{i}=\displaystyle \frac{{n}_{i}-\bar{n}}{\bar{n}},\end{eqnarray} \tag{ 1 }$$

where i is the pixel number, q_i is the quasar overdensity in the i-th pixel, and n_i is the quasar number counts in each pixel with $\bar{n}$ being its mean value.

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Mask apodization is commonly used in the cross-correlation calculation to avoid Gibbs-ringing induced by a sharp cutoff in a map during the harmonic transformation. We give the brief procedures on mask construction and apodization below, and visualize the apodized mask in Figure 2.

• 1.  
  We adopt the CMB lensing mask released by the Planck Collaboration as part of the Planck 2018 CMB lensing products. As for the BOSS quasar masks, we lower the resolution of the quasar overdensity maps to N_side = 32 to identify the region covered by the SDSS survey. N_side = 32 corresponds to 12,288 pixels or an angular resolution of 1099. This relatively low resolution is also chosen by Han et al. (2019) and Vielva & Sanz (2010), to represent the continuous sky coverage allowed by the survey. We label the empty pixels 0 and the remaining 1 to construct the quasar mask. A more accurate mask could be obtained from the random catalogs. We tried to use the CMASS random catalog provided by BOSS and get ${f}_{\mathrm{sky}}^{\kappa q}=0.184$ (the fraction of the sky shared by the quasar overdensity map and the CMB lensing convergence map. See Equation (2) for more details), very close to the original ${f}_{\mathrm{sky}}^{\kappa q}=0.179$ by downgrading the quasar map, making negligible difference to the final results.
• 2.  
  These masks are upgraded to N_side = 2048 again before performing the cross-correlation, to match the resolution of quasar maps and the CMB lensing map.
• 3.  
  We apodize the masks by smoothing both the CMB mask and the quasar mask using a Gaussian kernel. To decide a proper kernel size, we generate 200 mocks based on a theoretical template, and apply masks smoothed by Gaussian kernels with FWHM of 10', 30', and 1°, as well as a mask without smoothing. Then we calculate the mean squared error (MSE) of the residuals. A more detailed description for this procedure is given in Appendix D. We find that the mask smoothed by a Gaussian kernel with FWHM of 10' produces the minimal residual and thus should be the optimal choice.

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3.2. Angular Cross-power Spectrum Estimator

We compute the CMB lensing-quasar overdensity angular cross-power spectrum using a pseudo-C_ℓ estimator (Wandelt et al. 2001):

$$\begin{eqnarray}&&{\hat{C}}_{{\ell }}^{\kappa q}=\displaystyle \frac{1}{{f}_{\mathrm{sky}}^{\kappa q}}\displaystyle \sum _{m=-{\ell }}^{{\ell }}{\kappa }_{{\ell }m}^{* }{q}_{{\ell }m},\end{eqnarray} \tag{ 2 }$$

where ${f}_{\mathrm{sky}}^{\kappa q}$ is the fraction of the sky shared by the quasar overdensity map and the CMB lensing convergence map. ${\kappa }_{{\ell }m}$ and ${q}_{{\ell }m}$ are the spherical harmonic transforms of the CMB lensing convergence and the quasar overdensity maps, respectively. Note the 1/${f}_{\mathrm{sky}}^{\kappa q}$ estimator is known to be slightly biased (Wandelt et al. 2001). However, while approximate, this should be good enough for our setup due to the noise level of the spectra. The transform can be computed readily by the anafast function within healpy.⁸ We bin both the BOSS QSO and QSO with DLAs cross-power spectra into 10 bands over the range of $100\lt {\ell }\lt 1200$. We apply a low-ℓscale cut with ℓ_min = 100 because the cross-correlation on ℓ < 100 is deficient due to potential systematics shared by the quasar overdensity map and the CMB lensing map. This deficit of power at ℓ < 100 is also found in Han et al. (2019), Pullen et al. (2016), Ferraro et al. (2016), for which we have not found compelling explanations. Some large-scale systematics tracing the quasars may account for this (e.g., Geach et al. 2019).

Although Planck Collaboration et al. (2018b) has restricted the lensing auto-spectrum to the range of 8 ≤ ℓ ≤ 400 in the likelihood, ensuring robustness of the reconstruction only over this range, we perform a measurement on a higher multipole range with ℓ_max = 1200,⁹ expecting both noise and systematics to be subdominant over this range as discussed in Giannantonio et al. (2016).

3.3. Cross-power Spectrum Covariance Matrix

There are two common ways to estimate the statistical errors on the cross-power spectrum. One is through analytical calculation assuming both fields are Gaussian (Hivon et al. 2002; Efstathiou 2004; Cabré et al. 2007):

$$\begin{eqnarray}&&\displaystyle \frac{1}{{\sigma }_{i}^{2}(A)}=\displaystyle \sum _{{l}_{\min }(A)\lt l\lt {l}_{\max }(A)}\displaystyle \frac{{f}_{\mathrm{sky}}^{\kappa q}(2l+1)}{{\left({C}_{l}^{\kappa q}\right)}^{2}+{C}_{l}^{\kappa \kappa }{C}_{l}^{{qq}}},\end{eqnarray} \tag{ 3 }$$

where ${\sigma }_{i}^{2}$ is the error of the cross power in the ith bin. ${C}_{{\ell }}^{\kappa \kappa }$ and ${C}_{{\ell }}^{{qq}}$ are the expected CMB lensing and quasar auto-power spectra, including both signal and noise. The estimators of the auto-power spectra are

$$\begin{eqnarray}&&{\hat{C}}_{{\ell }}^{\kappa \kappa }=\displaystyle \frac{1}{{f}_{\mathrm{sky}}^{\kappa }(2{\ell }+1)}\displaystyle \sum _{m=-{\ell }}^{{\ell }}{\left|{\kappa }_{{\ell }m}\right|}^{2},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{\hat{C}}_{{\ell }}^{{qq}}=\displaystyle \frac{1}{{f}_{\mathrm{sky}}^{q}(2{\ell }+1)}\displaystyle \sum _{m=-{\ell }}^{l}{\left|{q}_{{\ell }m}\right|}^{2}.\end{eqnarray} \tag{ 5 }$$

Alternatively one can get the error as the diagonal of the covariance matrix of ${\hat{C}}_{{\ell }}^{\kappa q}$. To estimate the latter, we use 300 CMB lensing simulations and follow the steps described in Section 3.2 to calculate their cross-power spectra with our BOSS quasar sample and selected quasars with DLAs. The 300 simulations of CMB lensing convergence field we adopt are part of the Planck 2018 release, based on MV estimate from temperature and polarization of all 300 Planck Full Focal Plane (FFP10) simulations (Planck Collaboration et al. 2018b).¹⁰

The covariance matrix element between the ith and the jth bin, ${{ \mathcal C }}_{{ij}}$, can be estimated by

$$\begin{eqnarray}&&{{ \mathcal C }}_{{ij}}=\displaystyle \frac{1}{N-1}\displaystyle \sum _{n=1}^{N}({C}_{i,n}-\bar{{C}_{i}})({C}_{j,n}-\bar{{C}_{j}}),\end{eqnarray} \tag{ 6 }$$

where N = 300 is the total number of CMB lensing simulations, and ${C}_{i,n}$ denotes the ith bin of the cross-power spectrum between the nth mock CMB lensing simulations and our quasar sample. The error of each bin can be obtained from the square root of the variance, i.e., the diagonal of the covariance matrix:

$$\begin{eqnarray}&&{\sigma }_{i}=\sqrt{{{ \mathcal C }}_{{ii}}}.\end{eqnarray} \tag{ 7 }$$

With the covariance matrix, it is convenient to compute the correlation matrix, which describes the linear correlation between pairs of bins:

$$\begin{eqnarray}&&{r}_{{ij}}=\displaystyle \frac{{{ \mathcal C }}_{{ij}}}{\sqrt{{{ \mathcal C }}_{{ii}}{{ \mathcal C }}_{{jj}}}},\end{eqnarray} \tag{ 8 }$$

with $-1\leqslant {r}_{{ij}}\leqslant 1$. Two bins are tightly positively correlated when r is close to 1, negatively correlated when r is close to −1, and linearly uncorrelated when r = 0. We show the correlation matrices of lensing × quasar and lensing × DLA in Figure 3. As one can see, the off-diagonal blocks can be negligible in this case. First of all, the bins are fairly wide, and so the bin–bin covariance should be pretty small. Then, due to the big difference in the number count of the two catalogs, the overlapping part of the two catalogs is so small that their covariance can be ignored.

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We adopt Equation (7) in the bias measurement, and Equation (3) in the null test (Section 5).

3.4. Analytical Model

3.4.1. Measurement of Quasar Bias and DLA Bias

The CMB lensing convergence is defined as a weighted projection of the matter overdensity in the direction along the line of sight $\hat{n}$ (Lewis & Challinor 2006):

$$\begin{eqnarray}&&\kappa (\hat{n})={\int }_{0}^{{z}_{\mathrm{CMB}}}{dzW}(z){\delta }_{m}\left(\chi (z)\hat{n},z\right)\end{eqnarray} \tag{ 9 }$$

${\delta }_{m}(\chi (z)\hat{n},z)$ is the matter overdensity at redshift z in the direction $\hat{n}$, W(z) is the CMB lensing kernel:

$$\begin{eqnarray}&&W(z)=\displaystyle \frac{3{H}_{0}^{2}{{\rm{\Omega }}}_{m,0}}{2{cH}(z)}(1+z)\chi (z)\left(1-\displaystyle \frac{\chi (z)}{{\chi }_{\mathrm{CMB}}}\right),\end{eqnarray} \tag{ 10 }$$

where H₀ is the current Hubble parameter, H(z) is the Hubble parameter at redshift z, ${{\rm{\Omega }}}_{m,0}$ is the current matter density, c is the speed of light, χ(z) is the comoving distance to redshift z, and χ_CMB is the comoving distance to the last scattering surface at redshift z_CMB ≈ 1100.

To relate the quasar overdensity field to the matter overdensity field, the projected surface density of quasars is given by Peiris & Spergel (2000):

$$\begin{eqnarray}&&q(\hat{n})={\int }_{0}^{{z}_{\mathrm{CMB}}}{dzf}(z){\delta }_{m}(\chi (z)\hat{n},z),\end{eqnarray} \tag{ 11 }$$

where f(z) is its window function:

$$\begin{eqnarray}&&f(z)=\displaystyle \frac{b(z){dN}/{dz}}{\left(\int {{dz}}^{{\prime} }\tfrac{{dN}}{{{dz}}^{{\prime} }}\right)}+\displaystyle \frac{3}{2H(z)}{{\rm{\Omega }}}_{0}{H}_{0}^{2}(1+z)g(z)(5s-2)\end{eqnarray} \tag{ 12.1 }$$

$$\begin{eqnarray}&&g(z)=\displaystyle \frac{\chi (z)}{c}{\int }_{z}^{{z}_{{CMB}}}{{dz}}^{{\prime} }\left(1-\displaystyle \frac{\chi (z)}{\chi ({z}^{{\prime} })}\right)\displaystyle \frac{{dN}}{{{dz}}^{{\prime} }}.\end{eqnarray} \tag{ 12.2 }$$

The first term is the normalized, bias-weighted redshift distribution of the quasars; the second term is the magnification bias induced by the change in the density of sources due to lensing magnification, where $s\equiv d{\mathrm{log}}_{10}N/{dm}$ is the response of the number density to magnification bias. Due to the multiple selection criteria, we cannot simply histogram the sample to determine s. Instead, following Krolewski et al. (2020), we get s ≈ 0.078 for the BOSS subset and s ≈ 0.26 for the quasars with DLAs, and just neglect the uncertainty induced by the magnification bias.

For quasars with DLAs, DLAs contribute to the cross-correlation between CMB lensing map and the quasar overdensity. Simply assuming the weak lensing signal of CMB photons is the summation of the quasar and its foreground DLA contribution, the window function in Equation (12) should be written as:

$$\begin{eqnarray}\begin{array}{rcl}f(z) & = & \displaystyle \frac{{\bar{n}}_{\mathrm{DLA}}{b}_{\mathrm{DLA}}\cdot {{dN}}_{\mathrm{DLA}}/{dz}}{\left(\int {{dz}}^{{\prime} }\tfrac{{{dN}}_{\mathrm{DLA}}}{{{dz}}^{{\prime} }}\right)}+\displaystyle \frac{{b}_{\mathrm{QSO}}\cdot {{dN}}_{\mathrm{QSO}}/{dz}}{\left(\int {{dz}}^{{\prime} }\tfrac{{{dN}}_{\mathrm{QSO}}}{{{dz}}^{{\prime} }}\right)}\\ & & +\,\displaystyle \frac{3}{2H(z)}{{\rm{\Omega }}}_{0}{H}_{0}^{2}(1+z)g(z)(5s-2),\end{array}\end{eqnarray} \tag{ 13 }$$

${\bar{n}}_{\mathrm{DLA}}$ is the effective number count of DLAs for a single sightline. In this case it should be ${N}_{\mathrm{DLA}}/{N}_{\mathrm{QSO}}\approx 1.17$. Note that here N_QSO denotes the number of background quasars that have DLAs found in their spectra, different from N in Equation (12) referring to the number of the full quasar sample.

In a flat universe and under the Limber approximation (Limber 1953), the quasar-CMB lensing convergence angular cross-power spectrum is given by Lewis & Challinor (2006) and Sherwin et al. (2012):

$$\begin{eqnarray}&&{C}_{{\ell }}^{\kappa q}=\int \displaystyle \frac{{dz}}{c}\displaystyle \frac{H(z)}{{\chi }^{2}(z)}W(z)f(z){P}_{{mm}}\left(k=\displaystyle \frac{{\ell }}{\chi (z)},z\right),\end{eqnarray} \tag{ 14 }$$

where and ${P}_{{mm}}(k,z)$ is the 3D matter power spectrum.

3.4.2. The Fiducial Bias Model

We adopt the quasar bias measured in Eftekharzadeh et al. (2015) as our fiducial bias model. They report one of the most precise measurements of quasar bias for the BOSS sample using the auto-correlation approach. The quasar bias (see Table 1, repeating Table 5 in Eftekharzadeh et al. 2015) has no significant upward trend as z increases. We use $\bar{b}=3.60$, the average of their ${b}_{\mathrm{QSO}}$, as the fiducial value, and assume a scaling parameter a that depicts the level of deviation from the fiducial b_fid:

$$\begin{eqnarray}&&b=a\cdot {b}_{\mathrm{fid}}.\end{eqnarray} \tag{ 15 }$$

As a test, we also adopt the bias-redshift model derived in Laurent et al. (2017) to guarantee the robustness of our analysis. More details are shown in Appendix A.

Table 1.  Clustering Results for NGC-CORE Sample and Subsamples in Eftekharzadeh et al. (2015) (Table 5 in Their Paper)

Δz	ΔMi	$\bar{z}$	No. of Quasars	bQ	${\chi }_{\mathrm{red}}^{2}$
$2.20\leqslant z\leqslant 2.80$	$-28.74\leqslant {M}_{i}\leqslant -23.78$	2.434	55826	3.54 ± 0.10	1.06
2.20 ≤ z < 2.384	−28.70 ≤ Mi ≤ −23.95	2.297	24667	3.69 ± 0.11	1.55
2.384 ≤ z < 2.643	−28.74 ≤ Mi ≤ −24.11	2.497	24493	3.55 ± 0.15	0.61
2.643 ≤ z ≤ 3.40	−29.31 ≤ Mi ≤ −24.40	2.971	24724	3.57 ± 0.09	0.66
2.20 ≤ z ≤ 2.80	−28.74 ≤ Mi < −26.19	2.456	18477	3.69 ± 0.10	2.19
2.20 ≤ z ≤ 2.80	−26.19 ≤ Mi < −25.36	2.436	18790	3.56 ± 0.13	1.71
2.20 ≤ z ≤ 2.80	−25.36 ≤ Mi ≤ −23.78	2.411	18559	3.81 ± 0.19	0.4

Note. The first four columns are redshift and absolute magnitude range, the average redshift and the total number of NGC quasars. Columns 5 and 6 are the best-fitting bias values and the reduced χ² (over 7-DoF/9-DoF for the main sample/subsamples).

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3.4.3. From Bias to Halo Mass

We use the fitting formula in Tinker et al. (2010) to relate the bias to the peak height of the linear density field ν (Press & Schechter 1974; Sheth et al. 2001):

$$\begin{eqnarray}&&\begin{array}{l}\nu =\displaystyle \frac{{\delta }_{c}}{\sigma ({M}_{{\rm{\Delta }}})}\\ b(\nu )=1-A\displaystyle \frac{{\nu }^{\alpha }}{{\nu }^{\alpha }+{\delta }_{c}^{\alpha }}+B{\nu }^{\beta }+C{\nu }^{\gamma },\end{array}\end{eqnarray} \tag{ 16 }$$

where $A=1.0+0.24y\exp \left[-{\left(4/y\right)}^{4}\right]$, $\alpha =0.44y-0.88$, B = 0.183, $\beta =1.5$, $C=0.019+0.107y+0.19\exp \left[-{(4/y)}^{4}\right]$, and γ = 2.4 if we assume y = log₁₀Δ. δ_c = 1.686 is the critical overdensity for collapse, and σ(M_Δ) is the linear matter variance on the Lagrangian scale of the halo.

In this work we assume Δ = 200 and thus M refers to the total mass within the radius r₂₀₀ at which the enclosed mass density is 200 times the average matter density of the universe:

$$\begin{eqnarray}&&{M}_{200}=\displaystyle \frac{800\pi }{3}{\rho }_{{\rm{m}}}(z){r}_{200}^{3}.\end{eqnarray} \tag{ 17 }$$

COLOSSUS¹¹ (Diemer 2018) is a convenient tool to derive M from the peak height ν. Since the relation between bias and ν is slightly complicated, we invert the mass–bias relation and find the roots using scipy.

In this section we present our measurements of the quasar and DLA biases. We use the quasar bias measured from the BOSS sample as a prior to help us constrain the DLA bias.

4.1. Quasar Bias

First, we cross-correlate the BOSS quasar sample with the CMB lensing map to get the maximum-likelihood estimate of ${b}_{\mathrm{QSO}}$ using Equations (2) and (14). We reweight the redshift distribution of quasars in the BOSS subset following Alonso et al. 2018, so that it matches that of the QSOs with DLAs. The reweighting is to ensure the assumption that quasars in the two catalogs have similar bias. In fact, in this case reweighting the sample makes negligible difference to the final results. We run CAMB (Lewis et al. 2000) to calculate the nonlinear matter power spectrum using HALOFIT (Smith et al. 2003; Takahashi et al. 2012). In Figure 4, we present the cross-correlation (black point with error bar). We use the standard deviation of the fitting result as a proxy of its uncertainty, which can be derived from the resulted covariance from the curve fitting.

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With the parameterization in Equation (15), we estimate the parameter $a=0.71\,\pm \,0.19$ using Equation (12). This corresponds to ${b}_{\mathrm{QSO}}=a\cdot {b}_{\mathrm{fid}}=2.55\,\pm \,0.70$. Following Section 3.4.3, the bias value corresponds to a characteristic halo mass of $\mathrm{log}(M/{M}_{\odot }{h}^{-1})={11.79}_{-0.40}^{+0.63}$ at a median redshift of ${z}_{{\rm{m}}}\approx 2.51$.

4.2. DLA Bias

We cross-correlate the Planck 2018 CMB lensing map with the 17,774 quasars, which have DLAs in their sightlines. If we ignore the impact of DLAs and follow the steps introduced in Section 3.4.1, we obtain a = 1.00 ± 0.55, deviating from the value of ${b}_{\mathrm{QSO}}$ estimated in Section 4.1. Therefore, the contribution of DLAs in sightlines of quasars is not negligible.

We measure ${b}_{\mathrm{QSO}}$ and ${b}_{\mathrm{DLA}}$ with the Markov Chain Monte Carlo (MCMC) using emcee¹² (Foreman-Mackey et al. 2013). Since the luminosity distribution of the BOSS subset and the selected quasars with DLAs are not significantly different (see Figure 1), we assume that ${b}_{\mathrm{QSO}}$ of quasars with DLAs approximates that of the BOSS QSO subset. Then, we assume the DLA bias ${b}_{\mathrm{DLA}}$ is a constant over the selected redshift range at z = 2.2–3.4, with a prior uniform distribution over $0\lt {b}_{\mathrm{DLA}}\lt 15$ and further assume a Gaussian distribution of a as derived in Section 4.1, with the central value μ = 0.71 and the standard deviation σ = 0.19 as its prior distribution:

$$\begin{eqnarray}&&p(a)\sim N(\mu ,{\sigma }^{2}).\end{eqnarray} \tag{ 18 }$$

The likelihood function is:

$$\begin{eqnarray}&&p({\boldsymbol{C}}| a,{b}_{\mathrm{DLA}})=\displaystyle \frac{1}{| {\rm{\Sigma }}{| }^{\tfrac{1}{2}}{\left(2\pi \right)}^{\tfrac{n}{2}}}{e}^{-\tfrac{1}{2}{\left(\hat{{\boldsymbol{C}}}-{\boldsymbol{C}}\right)}^{T}{{\boldsymbol{\Sigma }}}^{-1}\left(\hat{{\boldsymbol{C}}}-{\boldsymbol{C}}\right)},\end{eqnarray} \tag{ 19 }$$

where $\hat{{\boldsymbol{C}}}$ is the data vector of the binned data points, ${\boldsymbol{C}}$ is the data vector of the calculated theoretical model for the bin, and ${\boldsymbol{\Sigma }}$ is covariance matrix as shown in Figure 3.

We have $a=0.68\,\pm \,0.18$, and ${b}_{\mathrm{DLA}}={1.37}_{-0.92}^{+1.30}$ corresponding to a characteristic DLA halo mass of $\mathrm{log}(M/{M}_{\odot }{h}^{-1})\,={10.60}_{-5.27}^{+1.44}$ at a median redshift of ${z}_{{\rm{m}}}\approx 2.30$. Despite the large uncertainty, the result provides a good constraint on the upper limit of DLA halo mass, yielding ${b}_{\mathrm{DLA}}\leqslant 3.1$ and log $(M/{M}_{\odot }{h}^{-1})\leqslant 12.3$ at a confidence level of 90%. The results are shown in Figures 4 and 5.

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We have also tried to simultaneously model the two catalogs, assuming that their ${b}_{\mathrm{QSO}}$ are exactly the same. The results show that ${a}_{\mathrm{QSO}}=0.67\pm 0.19$, ${b}_{\mathrm{DLA}}={1.57}_{-1.02}^{+1.33}$, and the 90% upper limit of ${b}_{\mathrm{DLA}}$ is 3.30, corresponding to a halo mass of log $(M/{M}_{\odot }{h}^{-1})\leqslant 12.38$. This is consistent with our analysis.

We check our result using a simple null test (Sherwin et al. 2012), calculating the cross-power spectrum between the CMB lensing convergence map on one part of the sky and the quasar map on another part of the sky. The galactic longitude of the selected part of the CMB lensing map ranges from 0° to 100°, and that of the two quasar maps ranges from 120° to 220°. The error bar of the spectra is calculated by the analytical Gaussian error estimator described in Section 3.3. Almost all the bins in their cross-power spectra fall within 1σ of null, as shown in Figure 6, yielding a = 0.004 ± 0.007 for the BOSS subset and a = 0.001 ± 0.007 for the selected quasars with DLAs.

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We also correlate the 300 Planck FFP10 simulations with randomly positioned quasars and get the average of the 300 spectra as an additional null test, yielding a = −0.046 ± 0.131 for 300,000 randomly positioned quasars within 15 bins and a = −0.046 ± 0.296 for the other 100,000 quasars within 10 bins.

Our measurement of bias for the BOSS subset ($a=0.71\,\pm 0.19$, ${b}_{\mathrm{QSO}}=2.55\,\pm \,0.70$), albeit with large uncertainty, is lower than the result in Eftekharzadeh et al. (2015) measured by a two-point auto-correlation function. For the DLA-CMB lensing cross-correlation, Alonso et al. (2018) pioneered using the CMB lensing convergence map to constrain the DLA bias. They used the Planck 2015 CMB lensing map and two DLA catalogs labeled as N12 (Noterdaeme et al. 2012) and G16 (Garnett et al. 2017b), both created from SDSS-III DR12. They yield constraints on the QSO bias of 2.90 ± 0.69 (2.57 ± 0.52) and DLA bias of 2.66 ± 0.93 (1.92 ± 0.69) for their N12 (G16) sample by simultaneously modeling the DLA+QSO and QSO-only measurement. Despite large uncertainties, our DLA bias has a 1.5σ lower value than that of Alonso et al. (2018). Compared to Alonso et al. (2018), our study has several updates: (1) we adopt a Planck 2018 CMB lensing convergence map and an updated DLA catalog produced by a CNN model, currently the state-of-the-art finding algorithm for DLAs; (2) we apply similar sample selection criteria as described in Eftekharzadeh et al. (2015) so that the QSO sample we used for measuring ${b}_{\mathrm{QSO}}$ is more comparable to that of the auto-correlation results; (3) we apply the same sample selection criteria to both the BOSS quasars and the quasars with DLAs, including removing the radio-loud sample, so that the measured ${b}_{\mathrm{QSO}}$ of the BOSS subset is comparable to ${b}_{\mathrm{QSO}}$ of the selected quasars with DLAs; and (4) we have updated noise calculations, quantitatively considering the different apodization size and the tSZ effects (see Appendix B). Moreover, we provide a step-by-step procedure of the cross-correlation in detail for future reference.

Our results have also shown a smaller QSO bias comparing to the auto-correlation results. We have not yet found a convincing explanation for this discrepancy beyond statistical fluctuations. The systematics induced by the Planck and the lensing reconstruction may account for the fluctuation. Systematics related to incompleteness, random catalog generation, masking and so on may also contribute to this difference, which should be carefully considered when using the auto-correlation approach (e.g., Reid et al. 2016; Geach et al. 2019). However, the bias measurement of DLAs remains effectively immune to the unknown fluctuation of quasar bias, since ${b}_{\mathrm{DLA}}$ is subtracted from the combined lensing signal, i.e., a combination of weak lensing effect caused by both quasars and its foreground DLAs, and these quasars would share the same systematics as the selected BOSS quasars.

In this paper, we propose a meaningful upper limit for the DLA bias and our results favor a DLA halo mass of ${M}_{\mathrm{halo}}\lt {10}^{12.3}{M}_{\odot }{h}^{-1}$. Some observations indicate that metal-rich DLAs are hosted by massive halos with mass reaching ${10}^{12}\ {M}_{\odot }$ (e.g., Neeleman et al. 2018), while a number of hydrodynamical simulations and astrophysical models (e.g., Pontzen et al. 2008) prefer that DLAs generally reside in smaller halos. Resolving the discrepancy in the clustering strength of DLAs is important to determine the nature of DLAs. Font-Ribera et al. (2012) measured ${b}_{\mathrm{DLA}}$ by the cross-correlation of DLA and the Lyα forest. The value of the mean bias they measured for DLAs is ${b}_{\mathrm{DLA}}=2.17\pm 0.20$, indicating a halo mass of $\sim {10}^{12}{M}_{\odot }$. Using the same technique, Pérez-Ràfols et al. (2018) updated the result using the BOSS Data Release 12. Compared to Font-Ribera et al. (2012), Pérez-Ràfols et al. (2018) favor a lower bias, yielding ${b}_{\mathrm{DLA}}=2.00\pm 0.19$ and corresponding to a halo mass of $\sim 4\times {10}^{11}{h}^{-1}{M}_{\odot }$. Using the CMB lensing with a Planck 2018 convergence map, although the error bar of our measurement is much larger than the DLA-Lyα forest cross-correlation results, our measurement provides an independent and good constraint on the upper limit. The upper limit we pose is that the DLA halo mass is lower than ${10}^{12.3}\ {M}_{\odot }{h}^{-1}$ at a 90% confidence level, and this is consistent with the Lyα cross-correlation measurement. The relatively low resolution (5'–10'), relatively low signal-to-noise of Planck CMB lensing map, and the current limited sample sizes of selected quasars and DLAs may account for the large uncertainty. We expect new surveys such as DESI¹³ to provide a larger quasar and DLA database, and improvements in CMB lensing, such as ACT,¹⁴ Simons Observatory (SO),¹⁵ and CMB-S4,¹⁶ will also help a lot for a more precise bias measurement.

We measure the bias of DLAs by studying the cross-correlation between the selected quasars with DLAs in their sightline and CMB lensing. We perform the measurement based on the bias measured for a BOSS subset. We find a quasar bias ${b}_{\mathrm{QSO}}=2.55\,\pm \,0.70$ for the BOSS subset, corresponding to a quasar halo mass of log $(M/{M}_{\odot }{h}^{-1})={11.79}_{-0.40}^{+0.63}$ at a median redshift of ${z}_{{\rm{m}}}\approx 2.51$. Our measurement of quasar bias is 2σ smaller than the fiducial bias model (Eftekharzadeh et al. 2015). Correlated systematics between the CMB lensing map and the quasar sample might account for such a low ${b}_{\mathrm{QSO}}$, but we have not yet found an explanation convincing enough. Despite its large uncertainty, our result provides a good constraint on the upper limit of DLA halo mass. We find ${b}_{\mathrm{DLA}}\leqslant 3.1$ and log $(M/{M}_{\odot }{h}^{-1})\,\leqslant 12.3$ at a confidence level of 90%, consistent with previous work (Font-Ribera et al. 2012; Pérez-Ràfols et al. 2018).

A simple null test is performed for the BOSS subset and the selected quasars with DLAs, by cross-correlating the CMB lensing convergence map on one part of the sky and the quasar map on another part of the sky. For the robustness of our measurement, we also repeat our analysis using a redshift-dependent bias model as the fiducial, on a CMB lensing map with thermal-SZ signal deprojected and using another DLA catalog (see Appendices A, B, and C). We expect future survey and CMB detection will improve the precision and accuracy of the DLA bias measurement.

We acknowledge the referee of this paper, David Alonso, for thoughtful and insightful comments that greatly improved the paper. Z.C. and X.L. are supported by the National Key R&D Program of China (grant No. 2018YFA0404503). We thank Jiashu Han, Dandan Xu, John Andrew Peacock, and Andreu Font-Ribera for helpful discussions. We also acknowledge the support of time and computing resources from DoA at Tsinghua University, and the kind reply of PLA Helpdesk.¹⁷ Our work is based on observations made by the Planck satellite and the Apache Point Observatory. The Planck project¹⁸ is funded by the member states of ESA and NASA. The SDSS¹⁹ project is funded by the participating institutions, the National Science Foundation, the United States Department of Energy and the Alfred P. Sloan Foundation.

Healpy (Górski et al. 2005), COLOSSUS (Diemer 2018), Scipy (Virtanen et al. 2020), CAMB (Lewis et al. 2000), emcee (Foreman-Mackey et al. 2013).

Adopting a model for the bias evolution with redshift is a good test for the robustness of our result. While Eftekharzadeh et al. (2015) found a roughly redshift-independent bias for the BOSS sample, other works (Shen et al. 2009; Laurent et al. 2017) found that the quasar bias sharply evolves with redshift. If the bias is redshift-dependent, the small mismatch in redshift distribution between the BOSS subset and the DLA quasars (Figure 1) could create a difference in their clustering, spuriously mimicking a change in the DLA bias. We test this possibility here by adopting an evolving bias model as an alternative to Equation (15). Laurent et al. (2017) measured the quasar correlation function of the first year of the eBOSS quasar sample, and provided a bias-redshift model by combining their results and the bias measured with the BOSS sample:

$$\begin{eqnarray}&&\left.{b}_{\mathrm{QSO}}(z)=\alpha \left[{\left(1+z\right)}^{2}-6.565\right)\right]+\beta ,\end{eqnarray} \tag{ A1 }$$

with $\alpha =0.278\pm 0.018,\beta =2.393\pm 0.042$. Despite the different sample selection, we adopt the central values of these parameters and use this bias model as the fiducial quasar bias in our analysis, and assume DLA bias remains a constant over the selected redshift range.

With this bias evolution model, we get a = 0.53 ± 0.16, corresponding to ${b}_{\mathrm{QSO}}=a\cdot {b}_{\mathrm{fid}}({z}_{{\rm{m}}})=2.13\pm 0.63$ and a characteristic halo mass of log $(M/{M}_{\odot }{h}^{-1})={11.45}_{-0.81}^{+0.48}$ at a median redshift of z_m ≈ 2.51. Furthermore, it gives ${b}_{\mathrm{DLA}}\,={1.76}_{-1.11}^{+1.40}$. The 90% upper limit of DLA bias is 3.59, corresponding to a halo of log $(M/{M}_{\odot }{h}^{-1})=12.50$.

This result is consistent with our previous analysis.

Thermal Sunyaev-Zel'dovich (tSZ) contamination is considered to be a potential systematic to CMB lensing cross-correlation analyses (van Engelen et al. 2014; Osborne et al. 2014). The Sunyaev-Zel'dovich (SZ) effect is caused by the inverse Compton scattering of CMB photons off hot electrons in the deep potential wells of galaxy clusters. Although the tSZ signal mainly comes from low-redshift structure, and is not expected to be a large source of contamination in our high-redshift analysis, it is worth testing the robustness of our result by repeating the bias measurement with the tSZ-free lensing map. The Planck Collaboration released a lensing reconstruction map on SMICA foreground-cleaned maps, where lensing signal is estimated from temperature (TT) only with tSZ signal deprojected.

In the absence of tSZ bias, we get a = 0.68 ± 0.19, ${b}_{\mathrm{QSO}}=a\cdot {b}_{\mathrm{fid}}=2.45\pm 0.70,{log}(M/{M}_{\odot }{h}^{-1})={11.72}_{-0.68}^{+0.42}$ for the BOSS quasars. The 90% upper limit of DLA bias is 3.78 and that of DLA halo mass is log $(M/{M}_{\odot }{h}^{-1})=12.58$. Figure 7 shows the data and best-fit spectra.

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The consistency shows that foreground contamination is subdominant. Since the tSZ-free analysis is based on temperature maps only and lacks polarization information, we take the measurement on the MV estimated lensing signal as our primary conclusion.

Ho et al. (2020) provided a revised DLA catalog using Gaussian processes. Their pipeline improved the ability to detect multiple DLAs along a single sightline. They analyzed 158,825 Lyα spectra selected from SDSS DR12 and present updated estimates for the statistical properties of DLAs, including the column density distribution function, line density, and neutral hydrogen density. We apply our sample selection criteria to their DLA catalog as described in Section 2.4 except the DLA confidence (they do not have such a parameter in their catalog), and get 12,368 quasars with 17,836 foreground DLAs. Figure 8 shows the properties of the selected sample. Also we repeat the steps in Section 3.3 to calculate the corresponding covariance matrix.

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With this new DLA catalog we get a = 0.78 ± 0.20, b_QSO = 2.79 ± 0.72 for the BOSS quasars. As for DLAs, we obtain ${b}_{\mathrm{DLA}}={2.71}_{-1.25}^{+1.28}$. This result is perfectly consistent with that in Alonso et al. (2018). The 90% upper limit of b_DLA is 4.34, corresponding to a halo of log(M/M_⊙ h⁻¹) = 12.69. Figure 9 shows the data and best-fit spectra.

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We use mocks to decide a proper kernel size. For each mock, we start with theoretical curves ${C}_{{\ell }}^{\kappa \kappa }$, ${C}_{{\ell }}^{{qq}}$, and ${C}_{{\ell }}^{\kappa q}$ given by:

$$\begin{eqnarray}&&\begin{array}{l}{C}_{{\ell }}^{\kappa \kappa }=\displaystyle \int \displaystyle \frac{{dz}}{c}\displaystyle \frac{H(z)}{{\chi }^{2}(z)}{W}^{2}(z){P}_{{mm}}\left(k=\displaystyle \frac{{\ell }}{\chi (z)},z\right),\\ {C}_{{\ell }}^{{qq}}=\displaystyle \int \displaystyle \frac{{dz}}{c}\displaystyle \frac{H(z)}{{\chi }^{2}(z)}{f}^{2}(z){P}_{{mm}}\left(k=\displaystyle \frac{{\ell }}{\chi (z)},z\right),\\ {C}_{{\ell }}^{\kappa q}=\displaystyle \int \displaystyle \frac{{dz}}{c}\displaystyle \frac{H(z)}{{\chi }^{2}(z)}W(z)f(z){P}_{{mm}}\left(k=\displaystyle \frac{{\ell }}{\chi (z)},z\right).\end{array}\end{eqnarray} \tag{ D1 }$$

See more details of these equations in Section 3.4. We draw a sample of 2 × 10⁸ quasars with the bias ${b}_{\mathrm{QSO}}=2.5$ uniformly distributed over the redshift range of 2.2 ≤ z ≤ 3.4. We then generate Gaussian-distributed correlated CMB lensing and quasar fields through healpy.synfast following Equations (13) and (14) in Serra et al. (2014):

$$\begin{eqnarray}&&\begin{array}{l}{a}_{{\ell }m}^{\kappa \kappa }={\xi }_{a}{\left({C}_{{\ell }}^{\kappa \kappa }\right)}^{1/2}\\ {a}_{{\ell }m}^{{qq}}={\xi }_{a}{C}_{{\ell }}^{\kappa {\rm{q}}}/{\left({C}_{{\ell }}^{\kappa \kappa }\right)}^{1/2}+{\xi }_{b}{\left({C}_{{\ell }}^{{qq}}-{\left({C}_{{\ell }}^{\kappa {\rm{q}}}\right)}^{2}/{C}_{{\ell }}^{\kappa \kappa }\right)}^{1/2},\end{array}\end{eqnarray} \tag{ D2 }$$

where ξ denotes a random amplitude so that $\left\langle \xi {\xi }^{* }\right\rangle =1$ and $\left\langle \xi \right\rangle =0$, yielding

$$\begin{eqnarray}&&\begin{array}{l}\left\langle {a}_{{\ell }m}^{\kappa \kappa }{a}_{{\ell }m}^{\kappa \kappa * }\right\rangle ={C}_{{\ell }}^{\kappa \kappa },\\ \left\langle {a}_{{\ell }m}^{\kappa \kappa }{a}_{{\ell }m}^{{qq}* }\right\rangle ={C}_{{\ell }}^{\kappa {\rm{q}}},\\ \left\langle {a}_{{\ell }m}^{{qq}}{a}_{{\ell }m}^{{qq}* }\right\rangle ={C}_{{\ell }}^{{qq}}.\end{array}\end{eqnarray} \tag{ D3 }$$

We multiply the synthetic Gaussian maps by the original binary masks, to ensure that pixels without data remain empty, such as regions around the Galactic plane. We then apply the apodized masks with different smoothing kernels before measurement of cross-power spectra. We repeat these steps on 200 Gaussian mocks, average them together and bin them into 10 bands. By comparing the binned average reconstructed spectra ${\hat{C}}_{i}^{\kappa q}$ and the binned theoretical template ${C}_{i}^{\kappa q}$ and calculating the mean squared error (MSE) of the residuals:

$$\begin{eqnarray}&&\mathrm{MSE}=\displaystyle \sum _{i=1}^{N}\displaystyle \frac{{\left({\hat{C}}_{i}^{\kappa q}-{C}_{i}^{\kappa q}\right)}^{2}}{N},\end{eqnarray} \tag{ D4 }$$

where N = 10. Table 2 shows the comparison of different smoothing kernels.