Inferred Evidence for Dark Matter Kinematic Substructure with SDSS–Gaia

In this analysis, we use the SDSS DR9 (Ahn et al. 2012) data set, cross-matched to Gaia DR2 (Gaia Collaboration et al. 2018). The cross-match was performed inside the Whole Sky Database (WSDB), an archive providing SQL access to catalogs from all major wide-area surveys. In particular, we utilized Q3C, spatial indexing, and cross-matching plug-in (Koposov & Bartunov 2006) to select the nearest SDSS neighbor for each Gaia source within a 1'' aperture, while taking into account the proper motion of all objects and the time difference between the observations. The combination of the two data sets allows us to take advantage of the large number of halo stars in the SDSS spectroscopic data set, while simultaneously using the unprecedented accuracies of the proper motions provided by the Gaia survey. From the SDSS catalog, we select main-sequence (MS) stars, which are considered standard candles in this context, that satisfy $| b| \gt 10^\circ$, A_g < 0.5 mag, σ_RV < 50 km s⁻¹, ${\rm{S}}/{\rm{N}}\gt 10,3.5\,\lt \mathrm{log}(g)\lt 5$, 0.2 < g−r < 0.8, 0.2 < g − i < 4, $4500\,{\rm{K}}\,\lt {T}_{\mathrm{eff}}\lt 8000\,{\rm{K}}$, and 15 < r < 19.5. All stellar magnitudes are dereddened using the maps of Schlegel et al. (1998). Admittedly, the MS stars are not true standard candles, but accurate distances can be estimated using the so-called photometric parallax method, which was recently tested against Gaia parallaxes (see Deason et al. 2018). While other possible choices exist, e.g., (i) a mix of all stellar populations with Gaia parallaxes or (ii) red giants with photometric parallaxes and (iii) blue horizontal branch stars, the MS sample has a combination of advantageous properties making it a better choice for the analysis presented here. More precisely, the MS sample is not limited by Gaia's parallax error to a relatively short distance range and is not dominated by the disk stars. The photometric parallax uncertainties for the MS stars do not vary as strongly with metallicity as, e.g., those for red giants (whose distance error can easily reach 100% for the metal-poor giants compared to the typical 10%–20% for the MS stars). Finally, MS stars are much more numerous compared to the blue horizontal branch or red giant stars.

Distances to these predominantly MS stars are calculated using Equations (A2), (A3), and (A7) of Ivezic et al. (2008), with distance errors within the region of interest of ∼2%, or of the order of ∼100 pc. Recently, the validity of this photometric parallax calibration was verified by Deason et al. (2018) for a subset of stars with accurate Gaia DR2 parallaxes. We have also confirmed that the fractional errors on the distance as derived from the parallaxes are consistently larger than those derived using the photometric parallax. The celestial coordinates, heliocentric distances, proper motions, and radial velocities are then used to calculate stellar velocity components in spherical polar coordinates. We marginalize over the local standard of rest value assuming that it is described by a Gaussian with a center at 238 km s⁻¹ and a dispersion of 9 km s⁻¹ (Schönrich 2012). The components of the solar peculiar motion are those presented in Coşkunoǧlu et al. (2011). The measured parameters' uncertainties—including covariances in the proper motion components—are then propagated using Monte Carlo sampling to obtain estimates of the uncertainties in the spherical velocity components (see Belokurov et al. 2018b, for details). For the analysis described in the sections below, each star's Monte Carlo samples are modeled with a Gaussian distribution to obtain the full covariance matrix.

There are no strong reasons to believe that the SDSS spectroscopic sample of MS stars used in our analysis suffers from any appreciable kinematic bias (Belokurov et al. 2018b). However, as the sample uses a mixture of SDSS target categories, a moderate metallicity bias toward more metal-poor stars is expected (see Yanny et al. 2009, for details). We also repeated our benchmark analysis selecting only F/G stars, which are known to have no biases in either metallicity or kinematics. This reduces the sample size substantially, increasing the uncertainties on the recovered fit parameters. However, the overall results—most importantly the fractional contribution of the stellar components—remain essentially unchanged, within uncertainties.

Figure 1 shows the spatial distribution of stellar counts in SDSS–Gaia DR2, as a function of galactocentric radius, r, and vertical distance from the Galactic plane, z.

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Figure 2 shows the distribution of the spherical velocity components as a function of stellar metallicity⁵ for the SDSS–Gaia DR2 subsample within r ∈ [7.5, 8.5] kpc and $| z| \gt 2.5\,\mathrm{kpc}$. For Figure 2 (and subsequent figures), we sample from the posterior distributions of the mean and covariance matrices to construct the velocity components v_r, v_θ, v_ϕ for the best-fit model.

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Several features in Figure 2 are apparent by visual inspection. The first is the disk population, which is centered at [Fe/H] ∼ −0.8, v_r,θ ∼ 0 km s⁻¹, and v_ϕ ∼ 130 km s⁻¹. The second is a population with [Fe/H] ∼ −1.4 with large radial anisotropy. The third is a population that extends down to [Fe/H] ≲ −1.8 with nearly isotropic velocities.⁶

In this work, we will refer to the second population as "substructure" and the third population as the "halo." We are envisioning that both originate from the disruption of accreted satellites in the Milky Way. What we refer to as the "halo" is intended to encapsulate the tidal debris from the oldest mergers, which will typically be the most metal-poor and fully well mixed in phase space. What we call "substructure" constitutes tidal debris that is not fully phase mixed; such a component may exhibit interesting features in spatial and/or velocity coordinates, such as streams or debris flow. The prefix "sub-" suggests that this population is less dominant than the halo population; we adopt this terminology, as it is standard in the DM literature, but make no assumptions on its relative dominance in our study.

Evidence has been building for a multicomponent inner stellar halo that is dominated by the tidal debris of one massive merger (Deason et al. 2015; Fiorentino et al. 2015; Belokurov et al. 2018a; Helmi et al. 2018; Myeong et al. 2018a). The large radial anisotropy of the stars with [Fe/H] ≳ −1.7 in Figure 2 was first identified using the SDSS–Gaia DR1 sample (Belokurov et al. 2018b).⁷ This work noted that the "sausage"–like feature in the data appears to be non-Gaussian and estimated its contribution to be ∼66% of the nondisk population over the full SDSS footprint. They found that the radial anisotropy of the sample drops markedly at [Fe/H] ≲ −1.7, suggesting that a separate isotropic and metal-poor population is also present. It is unlikely that the radial and isotropic populations originated in the Milky Way, as their iron abundances are in line with those observed in Milky Way dwarf spheroidal galaxies (Venn et al. 2004) and their velocities are distinct from disk stars.

Recent work using local MS stars from SDSS–Gaia DR2, as well as a separate sample of more distant blue horizontal branch stars, demonstrated that the orbits of the most highly eccentric stars share a common apocenter at r ∼ 20 kpc (Deason et al. 2018). This radius is coincident with the observed break in the Milky Way's stellar density distribution (Deason et al. 2013), suggesting that the radial stellar population is the tidal debris of a recent and large merger that dominates the inner halo. Simulations have shown that the density of stars from a particular merger may exhibit a break at the average apocenter of the stars (Deason et al. 2013).

If the radial substructure is indeed associated with a large merger, one might expect that globular clusters were also stripped from the satellite progenitor as it was disrupted. Indeed, a number of globular clusters on highly radial orbits were recently identified that may be associated with the large merger(s) causing the sausage-like feature in the SDSS–Gaia data (Myeong et al. 2018b). The number of these clusters suggests a total progenitor mass of ∼10¹⁰ M_⊙; their tracks in age–metallicity space bound the maximum infall redshift to less than ∼3.

These new results have direct implications for the local DM distribution. Previous work demonstrated that the virialized DM component is traced by the most metal-poor stars in the Milky Way—this corresponds to the population that we refer to as the halo here (Herzog-Arbeitman et al. 2018a; Necib et al. 2018). The virialized DM is well tracked by stars with metallicities [Fe/H] ≲ −3 in the Eris simulation (Herzog-Arbeitman et al. 2018a) and for [Fe/H] ≲ −2 in the Fire simulation (Necib et al. 2018). The relevant metallicity range depends on the time when the earliest mergers occurred, which depends on a galaxy's detailed merger history. Additionally, the kinematic substructure observed in the data is highly reminiscent of debris flow (Kuhlen et al. 2012; Lisanti & Spergel 2012). Indeed, a study of the stellar halo in Via Lactea (where star particles were painted onto the most bound DM particles in subhalos) found precisely the same kind of radial substructure becoming apparent in the SDSS–Gaia data, and that the kinematics of the accreted stars correlate with that of the DM debris (Lisanti et al. 2015). The stellar–DM correlation for debris flow has since been verified using the Fire simulations (Necib et al. 2018). Therefore, if we want to infer the kinematic properties of the local DM, we will need to model the velocities of the halo and substructure populations.

To isolate the accreted stellar population, we can place a hard upper cutoff on the metallicity of the sample. The downside to this conservative approach is that it ignores the high-metallicity tail of the accreted stellar distribution that overlaps with disk stars. For this reason, we use a mixture model analysis to statistically identify the individual populations of accreted stars over the full metallicity range of the sample.

Each star, labeled by the index i, is associated with a set of observable quantities such as its velocity and metallicity, ${O}_{i}=\left({{\boldsymbol{v}}}_{i},{[\mathrm{Fe}/{\rm{H}}]}_{i}\right)$, as well as the variance for each. We assign each star a flag j = d, h, or s that designates whether it belongs to the disk, halo, or substructure population, respectively. The likelihood of observing O_i for a disk star is

$$\begin{eqnarray}&&\begin{array}{l}{p}_{d}({O}_{i}\,| \,\theta )\\ \quad =\ { \mathcal N }\left({{\boldsymbol{v}}}_{i}\,| \,{{\boldsymbol{\mu }}}^{d},{{\boldsymbol{\Sigma }}}_{i}^{d}\right){ \mathcal N }\left({[\mathrm{Fe}/{\rm{H}}]}_{i}\,| \,{\mu }_{[\mathrm{Fe}/{\rm{H}}]}^{d},{\sigma }_{[\mathrm{Fe}/{\rm{H}}],i}^{d}\right),\end{array}\end{eqnarray} \tag{ 1 }$$

where θ is the set of free parameters and ${ \mathcal N }$ denotes the normal distribution. The set θ includes the velocity distribution mean ${{\boldsymbol{\mu }}}^{d}$ and covariance matrix ${{\boldsymbol{\Sigma }}}_{i}^{d}$, as well as the metallicity distribution mean ${\mu }_{[\mathrm{Fe}/{\rm{H}}]}^{d}$ and dispersion ${\sigma }_{[\mathrm{Fe}/{\rm{H}}],i}^{d}$. The covariance matrix depends on the individual velocity dispersions σ_r,θ,ϕ and the correlation coefficients ρ_rθ, ρ_rϕ, and ρ_θϕ. Note that the velocity covariance matrix and the metallicity dispersion vary between stars because the observed covariance depends on the true value and the measurement error—specifically, ${{\boldsymbol{\Sigma }}}_{\mathrm{obs}}={{\boldsymbol{\Sigma }}}_{\mathrm{true}}+{{\boldsymbol{\Sigma }}}_{\mathrm{err}}$. There are 11 parameters associated with this model. The likelihood for a halo star is also given by Equation (1), except with $d\to h$, and thus comes with an additional 11 parameters.

Modeling the substructure population is more challenging, as initial evidence suggests that its radial velocities are non-Gaussian (Belokurov et al. 2018b). Therefore, we assume that the velocities are a sum of two multivariate normal distributions with equivalent parameters, except for equal and opposite mean in v_r:

$$\begin{eqnarray}\begin{array}{rcl}{p}_{s}\,({O}_{i}\,| \,\theta ) & = & \displaystyle \frac{1}{2}\,\left[{ \mathcal N }\left({{\boldsymbol{v}}}_{i}| {{\boldsymbol{\mu }}}^{\tilde{s}},{{\boldsymbol{\Sigma }}}_{i}^{s}\right)+{ \mathcal N }\left({{\boldsymbol{v}}}_{i}| {{\boldsymbol{\mu }}}^{s},{{\boldsymbol{\Sigma }}}_{i}^{s}\right)\right]\\ & & \times \ { \mathcal N }\left({[\mathrm{Fe}/{\rm{H}}]}_{i}\,| \,{\mu }_{[\mathrm{Fe}/{\rm{H}}]}^{s},{\sigma }_{[\mathrm{Fe}/{\rm{H}}],i}^{s}\right),\end{array}\end{eqnarray} \tag{ 2 }$$

where ${{\boldsymbol{\mu }}}^{\tilde{s}}={\left(-{\mu }_{r},{\mu }_{\theta },{\mu }_{\phi }\right)}^{s}$. This model can vary over unimodal and bimodal distributions in v_r. For example, when ${\mu }_{r}^{s}\to 0$, Equation (2) approaches a single Gaussian distribution peaked at zero. In the limit where ${\mu }_{r}^{s}\gg {\sigma }_{r}^{s}$ and ${\mu }_{r}^{s}\ne 0$, the radial lobes are very pronounced. If, in contrast, ${\mu }_{r}^{s}\ll {\sigma }_{r}^{s}$, then the overlap between the two lobes increases and the radial velocity distribution is more box-like. We assume that the radial lobes, if present, are symmetric about v_r = 0 km s⁻¹, as this would be expected if the tidal debris originates from a satellite as it moves toward (v_r < 0) and then away from (v_r > 0) the Galactic center.⁸

The likelihood for the complete set of N stars is

$$\begin{eqnarray}&&p\left(\{{O}_{i}\}\,| \,\theta \right)=\displaystyle \prod _{i=1}^{N}\,\displaystyle \sum _{j=d,h,s}{Q}_{j}\,{p}_{j}\left({O}_{i}\,| \,\theta \right),\end{eqnarray} \tag{ 3 }$$

where the braces around O_i indicate the full list of N values. Q_j is the probability that the star belongs to the jth population; these represent two additional parameters in the model, as Q_h = 1 − Q_d − Q_s.

We use emcee (Foreman-Mackey et al. 2013) to find the posterior distributions of all 35 free parameters. In particular, we use 250 walkers, with 5000 steps, and a burn-in period of 10,000 steps. The priors for the separate parameters are provided in Table 1. We perform the mixture analysis in separate regions within the dashed aqua box of Figure 1, which spans the range r ∈ [7.5, 10.0] kpc and $| z| \gt 2.5\,\mathrm{kpc}$. We find that the fit is well behaved in this radial span, as gauged primarily by its ability to reproduce the expected properties of the baryonic disk. Below $| z| \sim 2.5\,\mathrm{kpc}$, we find a persistent systematic bias in the fitting procedure that results from modeling the azimuthal disk velocities as a single Gaussian (Schönrich & Binney 2012), so we do not present those results here.

Table 1.  Parameters and Associated Prior Types/Ranges for the Disk, Halo, and Substructure Populations

Parameter	Type	Priors
		Disk	Halo	Substructure
μr	Linear	[−70, 70]	[−70, 70]	[0, 250]
μθ	Linear	[−70, 70]	[−70, 70]	[−70, 70]
μϕ	Linear	[0, 300]	[−70, 70]	[−70, 70]
σr,θ,ϕ	Linear	[0, 200]	[0, 200]	[0, 200]
ρrθ,rϕ,θϕ	Linear	[−1, 1]	[−1, 1]	[−1, 1]
${\mu }_{[\mathrm{Fe}/{\rm{H}}]}$	Linear	[−1.5, 0.5]	[−3, −1]	[−3, −1]
${\sigma }_{[\mathrm{Fe}/{\rm{H}}]}$	Linear	[0, 2]	[0, 2]	[0, 2]
Q	Linear	⋯	[0, 1]	[0, 1]

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Let us assume that the DM in the region of study can be divided into two populations. For the moment, we remain agnostic to the origin of these two populations, but we will return to this shortly. In this scenario, the total DM phase-space distribution satisfies

$$\begin{eqnarray}&&f(v)={\xi }_{1}\,{f}_{1}(v)+{\xi }_{2}\,{f}_{2}(v),\end{eqnarray} \tag{ 4 }$$

where f_i is the normalized velocity distribution and ξ_i is the relative fraction of the ith population. Note that ξ₁ + ξ₂ = 1. Starting from the distribution function in Equation (4), it is possible to derive the Jeans equations in spherical coordinates. To simplify the derivation and highlight the most important conceptual points, we make several assumptions about these two populations.

First, we assume that population 1 is isotropic and spherically symmetric and that its velocity components are uncorrelated. Most importantly, we take it to be in steady state, which allows us to ignore partial derivatives of f₁(v) with respect to time. When we state that a population is in equilibrium, we mean specifically that it is in steady state with ∂f_i/∂t = 0.

As a point of contrast, we will assume that population 2 has not reached steady state—or, at the very least, that we cannot confirm whether it has. Additionally, we will assume that its spatial density is spherically symmetric, that its velocity components are uncorrelated and have vanishing mean at present-day, and that the mean velocities and dispersions are spatially invariant in the region of interest.

The radial Jeans equation for this two-component model is

$$\begin{eqnarray}&&\begin{array}{l}{\xi }_{2}\,{\nu }_{2}\,\displaystyle \frac{\partial {\mu }_{r,2}}{\partial t}+\left[{\xi }_{1}\,{\sigma }_{r,1}^{2}\displaystyle \frac{\partial {\nu }_{1}}{\partial r}+{\xi }_{2}\,{\sigma }_{r,2}^{2}\,\displaystyle \frac{\partial {\nu }_{2}}{\partial r}\right]\\ \quad +\ \displaystyle \frac{{\xi }_{2}\,{\nu }_{2}}{r}\left[2{\sigma }_{r,2}^{2}-{\sigma }_{\phi ,2}^{2}-{\sigma }_{\theta ,2}^{2}\right]\\ \qquad =\ -\left({\xi }_{1}{\nu }_{1}+{\xi }_{2}{\nu }_{2}\right)\displaystyle \frac{\partial {\rm{\Phi }}}{\partial r},\end{array}\end{eqnarray} \tag{ 5 }$$

where ν_i is the number density, μ_r,i is the radial velocity mean, σ_r,i, σ_θ,i, σ_ϕ,i are the velocity dispersions for the ith population, and Φ is the gravitational potential.

Using Equation (5), one can recover the classical argument laid out in Drukier et al. (1986) that motivates a Maxwell–Boltzmann velocity distribution for DM. The observation of a flat rotation curve near the solar position suggests a logarithmic potential for the Milky Way halo of the form ${\rm{\Phi }}(r)\,={v}_{c}^{2}\mathrm{ln}(r)+\ \mathrm{constant}$, where v_c is the circular velocity. If we assume that all of the DM is in steady state (ξ₁ = 1, ξ₂ = 0) and that its number density is described as a falling power law, ν₁ ∝ r^−b, then Equation (5) predicts a tight link between the value of the power-law index b, the potential Φ, and the phase-space distribution. In this limit, the radial Jeans equation becomes

$$\begin{eqnarray}&&\displaystyle \frac{{\sigma }_{r,1}^{2}}{{\nu }_{1}}\displaystyle \frac{\partial {\nu }_{1}}{\partial r}=-\displaystyle \frac{{v}_{c}^{2}}{r}\,\longrightarrow \,b=\displaystyle \frac{{v}_{c}^{2}}{{\sigma }_{r,1}^{2}}.\end{eqnarray} \tag{ 6 }$$

Taking the local circular velocity to be v_c ∼ 235 km s⁻¹ and a dispersion of σ_r,1 ∼ 160 km s⁻¹ yields a power-law slope of b ∼ 2. A density distribution of the form ${\nu }_{1}\sim {r}^{-2}$ is isothermal, and one can use Poisson's equation to show that the associated velocity distribution is Maxwell–Boltzmann.

However, notice from Equation (5) that the simple derivation of the isothermal model breaks down in the presence of a second DM population. In particular, the tight predictive link between the potential Φ and the number density ν₁ no longer holds when ${\xi }_{2}\ne 0$. This is true regardless of whether the second DM population is in steady state or not. However, the challenge is exacerbated for the latter case, as we then need to quantify ∂μ_r,2/∂t.

This two-component model for the DM distribution¹⁰ is motivated in light of recent observations of the nearby accreted stellar distributions. As already discussed, the observation of a metal-poor and isotropic stellar halo is likely associated with tidal debris from the oldest luminous mergers that built up the Milky Way, while the anisotropic component at intermediate metallicities is due to tidal debris from a more recent merger. As DM would have also been stripped from these accreted satellites, it is reasonable to assume that it should also be separated into at least two populations as well. It is of course possible that the DM can be divided into more than two populations. This would be relevant if a significant fraction of the DM originated from nonluminous satellites or diffuse accretion, neither of which should be correlated with stellar distributions.

For now, we solely focus on the subset of DM from luminous mergers, as that is where the results of this work have the most consequence. In this case, the recent observations from Gaia suggest that we should be moving to a two-component model for the DM, as per Equation (4). Population 1 would correspond to the DM from the oldest mergers (the "halo" component), and population 2 would correspond to the DM from the recent merger (the "substructure" component).

Numerical simulations have demonstrated that DM accreted from the oldest luminous mergers is well traced by metal-poor stars—this corresponds to the halo population (Herzog-Arbeitman et al. 2018a; Necib et al. 2018). Additionally, DM in debris flow is well traced by intermediate-metallicity stars (Lisanti et al. 2015; Necib et al. 2018). This suggests that one can use the stellar velocity distribution for the halo and substructure populations, recovered in Section 3, to model the individual DM populations.

As the halo population is the most metal-poor and is also isotropic, we expect that it consists of tidal debris from the earliest mergers that have reached stead state today. The metallicity range and velocity profiles that we recover share similar features to the distribution of stars accreted at redshifts z > 3 in a study of two separate Milky Way–like halos in the suite of Fire hydrodynamic simulations (Necib et al. 2018). We believe that this population of tidal material is accreted while the proto-galaxy is still forming, and is fully relaxed today.

The substructure population arises from a more recent merger, with estimated accretion redshift in the range of z ∼ 1–3 (Belokurov et al. 2018a; Myeong et al. 2018b). Necib et al. (2018) showed that the tidal debris from satellites accreted at z ≲ 3 leaves distinctive features in velocity space that stand out from the behavior of older tidal debris, which is fully relaxed. Further study is required to determine whether this new substructure has reached steady state.

In this section, we derive the heliocentric speed distribution for the halo and substructure populations, which we assume to trace the DM, and use it to calculate the scattering rate of a DM particle off a nuclear target. Ideally, we need the stellar velocity distribution at the solar position, but this is also the region where the disk contribution dominates. Any mismodeling of the disk may therefore strongly bias the fit results in this regime. For this reason, we restrict ourselves to $| z| \gt 2.5\,\mathrm{kpc}$. The results from Section 3 give us confidence that the halo and substructure distributions are likely invariant in z and that we can extrapolate them into the plane; however, this should be verified explicitly. Previous work (Belokurov et al. 2018b) also finds evidence for the radial substructure down to $| z| =1\,\mathrm{kpc}$.

The best-fit speed distribution in the heliocentric frame is shown in the left panel of Figure 8, for heliocentric distances of d_⊙ < 4 kpc and $| z| \gt 2.5\,\mathrm{kpc}$. To obtain this distribution, we draw values of v_r, v_θ, and v_ϕ from the full posterior distribution in the Galactic frame. Assuming that the stars are spatially uniform, we then transform to the heliocentric frame using the local rest-frame velocities v_⊙,pec = (U, V, W) = (8.50, 13.38, 6.49) km s⁻¹ (Coşkunoǧlu et al. 2011) and local circular velocity v_c = 235 km s⁻¹. The substructure and halo components are plotted separately (blue dotted and red dashed lines, respectively). An important subtlety arises when summing these two contributions, as we need to know the relative amount of DM that is contributed by the accreted satellites in each population—i.e., the fractions ξ_1,2 in Equation (4). For now, we make the simplifying assumption that the satellites have comparable mass-to-light ratios. In this case, the total contribution is shown as the solid black line. Necib et al. (2018) provide a more detailed prescription to estimate the relative DM fractions associated with each population.

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For comparison, we also plot the SHM as the gray dashed line. The SHM is the speed distribution associated with the Maxwell–Boltzmann distribution (after integrating over the angular coordinates) and is defined as

$$\begin{eqnarray}&&{f}_{\mathrm{SHM}}(v)=\displaystyle \frac{4{v}^{2}}{\sqrt{\pi }{v}_{c}^{3}}\exp \left[-\displaystyle \frac{{v}^{2}}{{v}_{c}^{2}}\right].\end{eqnarray} \tag{ 7 }$$

The dispersion of the SHM is closest to that of the halo posterior, albeit slightly higher; the SHM is isotropic with σ ∼ 156 km s⁻¹, while the best fits for the halo are $({\sigma }_{r},\,{\sigma }_{\theta },\,{\sigma }_{\phi })\,=\left({140.3}_{-4.9}^{+4.2},{114.2}_{-1.8}^{+3.3},{125.9}_{-3.4}^{+4.1}\right)$ km s⁻¹. The primary discrepancy with the SHM arises from the substructure population. When this component is included, the total speed distribution is discrepant with the SHM. In this case, the polar and azimuthal velocities of the substructure are Gaussian with means $({\mu }_{\theta },{\mu }_{\phi })=(-{3.1}_{-0.9}^{+0.9},{35.5}_{-1.8}^{+1.8})$ km s⁻¹ and dispersions $({\sigma }_{\theta },{\sigma }_{\phi })=({57.7}_{-0.8}^{+0.7},{61.2}_{-1.5}^{+1.5})$ km s⁻¹, but the radial distribution has peaks at $\pm {117.7}_{-2.1}^{+1.8}$ km s⁻¹, with a dispersion ${\sigma }_{r}={108.2}_{-1.3}^{+1.2}$ km s⁻¹ for each. The substructure component arises from a more recent merger, as underlined by the fact that it is more metal-rich and highly radial. Because it is not necessarily in steady state, there is no reason why the SHM should model it well.

The empirical distribution clearly underestimates the fraction of high-speed DM particles, as compared to the SHM, which affects DM models where the minimum scattering speed, v_min, needed to create a nuclear recoil of energy E_nr is high. For elastic scattering, the minimum speed depends on both the DM particle properties and the experiment as follows:

$$\begin{eqnarray}&&{v}_{\min }=\sqrt{\displaystyle \frac{{m}_{N}{E}_{\mathrm{nr}}}{2{\mu }^{2}}},\end{eqnarray} \tag{ 8 }$$

where m_N is the nuclear mass and μ is the DM–nucleus reduced mass. If the scattering is inelastic, then the minimum speed is even larger. The differential scattering rate, per unit detector mass, for the most common operators is then

$$\begin{eqnarray}&&\displaystyle \frac{{dR}}{{{dE}}_{\mathrm{nr}}}=\displaystyle \frac{{\rho }_{\chi }}{2\,{m}_{\chi }{\mu }^{2}}\,\sigma (q)\,g\left({v}_{\min }\right),\end{eqnarray} \tag{ 9 }$$

where m_χ is the DM particle mass and ρ_χ its local density, σ(q) is an effective scattering cross section that depends on the momentum transfer q, and the mean inverse speed is defined as

$$\begin{eqnarray}&&g({v}_{\min })={\int }_{{v}_{\min }}^{\infty }\displaystyle \frac{\tilde{f}(v)}{v}\,{dv},\end{eqnarray} \tag{ 10 }$$

where $\tilde{f}(v)$ is the heliocentric velocity distribution.

The right panel of Figure 8 shows the corresponding limits on the DM mass and DM–nucleon scattering cross section, σ_χ−n, assuming the simplest spin-independent operator. For this example, we assume a xenon target, energy threshold of 4.9 keVnr, and exposure of 1 kton × year. The 95% one-sided Poisson C.L. limit (three events) obtained using the velocity distribution inferred from SDSS–Gaia DR2 is shown in solid black and compared to the SHM in dashed gray. The substructure component drives the sensitivity at all masses, while the halo contribution is subdominant but becomes more important at lower masses. In both cases, the exclusion is significantly weakened for m_χ ≲ 30 GeV relative to that obtained using the SHM. For m_χ ≳ 100 GeV, the black and gray dashed lines approach each other because ${v}_{\min }\to 0$ in Equation (10).

The overall effect of the empirical velocity distribution on the scattering limit depends on the details of the nuclear target, experimental threshold, and DM mass—all parameters that feed into the minimum scattering speed defined in Equation (8). A more model- and experiment-independent way of understanding these effects is to study the dependence of the time-averaged inverse speed, $\langle g({v}_{\min })\rangle$, as a function of the minimum speed, as this term captures the dependence of the scattering rate on the DM velocities. The left panel of Figure 9 plots this quantity for the empirical speed distribution obtained in this work (solid black) and the SHM (dashed gray). The scattering rate for the empirical distribution is reduced relative to that for the SHM at v_min ≳ 300 km s⁻¹; it is enhanced for lower minimum speeds. The scattering rate is completely suppressed for v_min ≳ 550 km s⁻¹, whereas the SHM continues to contribute events above this point.

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To better understand the implications of these results, let us consider the concrete example of a 10 GeV DM particle interacting in several detectors. Such a DM particle needs a minimum speed of ∼570 km s⁻¹ to scatter a xenon nucleus at an energy of ∼5 keV_nr in Xenon1T (Aprile et al. 2018) when requiring both S1 and S2 signals.¹¹ As seen from the left panel of Figure 9, this is highly suppressed relative to the SHM expectation.¹² In contrast, Xenon100 can detect recoils down to 0.7 keV_nr in the S2-only analysis (Aprile et al. 2016), similarly to the DarkSide-50 low-mass analysis (Agnes et al. 2018), which can detect argon recoils down to 0.6 keV_nr in energy. A 10 GeV DM particle only needs speeds of ∼130–200 km s⁻¹ to create such a recoil, and these speeds are well supported by the empirical distribution.

The empirical velocity distribution also impacts the time dependence of a signal. The DM scattering rate should modulate annually owing to Earth's motion around the Sun (Drukier et al. 1986). The right panel of Figure 9 compares the modulation amplitude assuming the newly derived velocity distribution, as compared to the SHM. To obtain the amplitude, we transform the velocities from the Galactic to the heliocentric frame, taking into account Earth's time-dependent velocity as defined in Lee et al. (2013). We do not include the effect of gravitational focusing, which may further affect the properties of the modulation signal (Lee et al. 2014).

The modulation amplitude for the SHM exhibits the expected features: a maximum when v_min ∼ 350 km s⁻¹ and a change in phase below ∼200 km s⁻¹ (see Freese et al. 2013, for a review). In comparison, the modulation amplitude obtained from the SDSS–Gaia DR2 distribution is maximal closer to v_min ∼ 250 km s⁻¹ and falls off faster toward higher v_min. This is due to the fact that the empirical velocity distribution f(v) is less broad than the SHM. Therefore, the differences in the heliocentric speed distribution over the year are typically more pronounced, but over a smaller range of v_min. The DAMA experiment, which claims an annually modulating signal, has an Na I target and a threshold energy of ∼3.3 keV_nr (Bernabei et al. 2018). An observable scatter of a 10 GeV DM particle off an Na nucleus requires a minimum speed of ∼270 km s⁻¹. We note that this falls in the region where the empirical distribution has a significant effect on the modulation amplitude and motivates a more careful study of the consistency with data.

The results shown here are specific to spin-independent interactions. However, the new velocity distribution will have an effect on other interaction operators—as the dependence of some of these operators on the DM momentum is nontrivial, the magnitude of the effects can vary from operator to operator (Fan et al. 2010; Fitzpatrick et al. 2012, 2013; Lisanti 2017). In general, any model that relies on the high-velocity tail of the velocity distribution to support the scattering rate will be affected by these results. One simple and illustrative case is that of inelastic (rather than elastic) spin-independent scattering off of nuclei. Let us return to the example of the 10 GeV DM particle scattering off a xenon target at a threshold of ∼0.7 keV_nr. The minimum scattering velocity for elastic scattering is ∼210 km s⁻¹, which is well supported by the velocity distribution as demonstrated in Figure 9. However, if one considers a case where there are nearly degenerate states separated by ∼15 keV in mass, then the inelastic scattering requires v_min ∼ 560 km s⁻¹, which is suppressed relative to the SHM expectation.

Similarly, the velocity distribution will also be relevant for the interpretation of DM–electron scattering interactions (see Battaglieri et al. 2017, for a review) and axion experiments (Ling et al. 2004; Hoskins et al. 2016; Sloan et al. 2016; Millar et al. 2017; Vergados & Semertzidis 2017; Foster et al. 2018).