Kinematic Substructure of Dark Matter and Stars in the Eris Simulation and Implications for Direct Detection Experiments

Jonah Herzog-Arbeitman

Jonah Herzog-Arbeitman,1 Lina Necib,2 Mariangela Lisanti1
1Department of Physics, Princeton University, Princeton, NJ, USA
2Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA, USA


ABSTRACT

In this paper, we study the effectiveness of low-metallicity stars as a readily observable tracer population for Dark Matter (DM) in the \LambdaCDM model. Using the hydrodynamic N-Body simulation Eris, we show that the DM velocity distribution exhibits systematic departures from various theoretical models even in regions where virialization is expected. We demonstrate that the kinematics of stars passing low metallicity cuts converge to those of the local DM, which has promising ramifications for direct detection experiments. We also study the relationship between the features of stellar and DM substructure in velocity space, specifically debris flows and streams. We find unexpectedly that the correlation is not as strong as anticipated based on studies of DM-only simulations with algorithmically introduced stars. We conclude by summarizing our predictions for the heliocentric DM distribution at Earth-like radii and demonstrate explicitly the strong correlation between DM and appropriate tracer stars, noting the importance of this result to experimentalists in the search for a DM particle.


 I. INTRODUCTION

The \LambdaCDM Model is standard in modern Cosmology. It posits that General Relativity is the correct description of gravitational interactions and describes our universe using a composite energy-mass density that includes baryons, Dark Energy, and non-relativistic (cold) Dark Matter. Although this model is convincing theoretically and explains otherwise anomalous experimental results, the evidence for Dark Matter (DM) is almost entirely gravitational and therefore indirect.1–3 Detection of the elusive DM particle itself is essential for an unequivocal conformation of the theory.4 Numerous direct detection experiments are searching for individual DM particles at lower masses and cross sections than ever before.5–7 Locating a potential DM signal in their data is impossible without knowing a variety of astrophysical factors, e.g. the local DM density and velocity distribution, as well as model-dependent factors which are determined by the possible DM interactions and the target material.8–12 There is currently no way to experimentally determine the local DM phase space distribution, creating model-independent uncertainties associated with the astronomical contribution to the form factor. In this paper, we will study empirical models that estimate the DM phase space distribution. In particular, we are interested in local deviations (streams and debris) from the classical galactic models which do not address kinematic substructure and thus lack predictive power over smaller length scales.

We hypothesize that stellar kinematics provide a reasonable estimate of their DM counterparts. This is an actionable claim because there are many astronomical experiments gathering a wealth of stellar data, potentially providing an unprecedented amount of detail about local DM behavior. One motivation for this claim comes from our understanding of galactic structure formation.13 During the galaxy’s formation, DM forms bound states and begins to accrete baryons in its halos. These baryons, initially gas, develop into generations of stars which remain co-bound with DM and retain its kinematics. These early stars then provide a tracer population for the otherwise invisible DM. Because these stars were formed from almost entirely hydrogen and helium, lacking heavy metals such as iron, we can use cuts on iron fraction [Fe/H] to cull good tracers from the total population of stars. In this way, we can use the chemical composition of stars to estimate their age.

We test this claim using data from the hydrodynamic N-Body Eris Simulation,14,15 one of the first simulations to include gas, stars, and DM. Different possible structures are expected: a virialized (gravitationally equilibrated) halo where the DM and star particles orbit as predicted for particles in spherical (axisymmetric) potential, streams where DM and stars exhibit both a characteristic velocity and a spatially localized feature, and debris in which DM and stars exhibit a characteristic speed without any special spatial features.16–18 The latter two represent substructure which current models used in direct detect experiments lack, thereby limiting their predictive power.19–24

Previous works have studied these velocity structures in DM-only simulations,9,16 while Ref. [25] among others analyzed the velocity structures in a DM-only N-body simulation repopulated with the stars. They have shown that there is a sizable amount of debris in the inner galactic core. In this work, we investigate a fully hydrodynamic simulation with in situ stars and gas to understand the different velocity distributions and their effects on direct detection limits.

In this paper, we will use the symbols r, \phi, \theta for radial, azimuthal and polar components in spherical coordinates, and R, T, z for radial, tangential and vertical components in cylindrical coordinates.

The remainder of this paper is organized as follows: First we introduce the Eris Simulation in Sec. II; we then analyze the different structures present in the nearest 35 kpc from the center in Sec. III; finally, we briefly study the implications in direct detection experiments in Sec. IV before concluding in Sec. V.


II. ERIS SIMULATION

Eris is a Milky Way-sized hydrodynamic zoom-in simulation, with N = 18.6 \times 10^{6} particles divided between gas, DM, and stars, a total virial mass M_{\text{vir}} = 7.9 \times 10^{11} M_{\astrosun}, and individual DM particle masses of 9.8 \times 10^4 M_{\astrosun} The force resolution of the simulation is 120 pc. Eris is a realization of the Milky Way, and therefore is a simulated spiral galaxy of radial scale length R_d = 2.5 kpc. Its mass budget in the different components (stars, gas, and DM) and its scaling relations between mass and luminosity are consistent with observations, allowing us to extrapolate results to our galaxy with cautious hope.26

In Eris, the center of the galaxy is defined as the point of maximum density in the gas distribution (though see Ref. [27] for a discussion of small misalignment in the different matter densities). We use the total angular momentum vector as calculated from the simulation to define the disk of the spiral galaxy and the sense of rotation. In these coordinates, the disk rotates with negative v_T and Earth is defined to lie in the plane at R = 8. Hereafter, the stellar disk is defined in the region |z| < 2 kpc where z is the vertical component, which corresponds to the distance perpendicular to the disk. The halo forms the complement of this region; stars in this region are referred to as halo stars. The two populations have very different kinematics as expected because the disk stars have a well-defined circular rotation. The roughly spherically symmetric DM halo extends well beyond on the disk and does not have a strongly differentiated disk of its own. Throughout the paper we will use radial bins [0,10], \, [10,20], \, [20,35] kpc; the kinematic behavior of each particle is roughly the same inside regions which allows us to convey the distributions using fewer plots, but of course none of our results depend on these partitions. The Earth, at 8 kpc from the center, lies in the first bin which is therefore of special importance.

In order to study the kinematics of DM and these two stellar populations, we will use the natural cylindrical coordinates v_R, v_T, v_z which measure radial and tangential velocity in the plane of the disk and velocity perpendicular to the disk. These components contain all of the information about the direction and magnitude of the velocity and thus are useful for differentiating kinematically distinct populations. However, for direction detection experiments, the speed |v| = \sqrt{ v_R ^2 + v_T ^2 + v_z ^2} is required.

In this analysis, we also look at different stellar populations as divided by metallicity as discussed in the introduction. Eris provides the iron (Z_\text{Fe}) and oxygen (Z_\text{O}) fractions in the stars which we use to compute the iron fraction [Fe/H] with respect to the Sun. Eris does not track higher elemental densities, so weighting coefficients are chosen to track the correlated metals groups.

(1)   \begin{equation*} Z = 2.09~ Z_\text{O} + 1.06 ~Z_{\text{Fe}}, \end{equation*}

from which we extract the fraction Y in helium and X in hydrogen as

(2)   \begin{eqnarray*} Y &=& 0.236 + 2.1~Z, \\ X &=& 1-Y - Z. \end{eqnarray*}

where .236 is the standard He mass fraction for primordial gas. Finally, the metallicity is computed as [\text{Fe}/\text{H}] with respect to the fraction of iron in the Sun which is 0.14\% and the fraction of hydrogen in the Sun which is 71.0\%.

(3)   \begin{equation*} [\text{Fe}/\text{X}] = \log_{10} \left( \frac{Z}{H}\right) - \log_{10} \left( \frac{0.14}{71.0}\right). \end{equation*}

It should be noted that Eris does not reproduce the experimental dependence of metallicity on distance to the disk quantitatively, but it consistently preserves the qualitative trend: lower metallicities skew towards the halo. Thus this calculation of metallicities is acceptable as a rough selection criterion but more careful multidimensional cuts are necessary to draw precise correspondences between the Eris simulation and physical data. We anticipate performing this analysis in a future paper. However, it is not too concerning here because applicable DM models would be generated from real stellar catalogues. We also anticipate this analysis in a future paper.


III. FEATURES OF THE VELOCITY DITRIBUTION 

A. Virialized Halo

1. Dark Matter Distribution

In this section, we provide simple standard models of the DM and stellar kinematic distribution. Because they incorporate no contributions from substructure, we will call them smooth models.

If we assume an isothermal sphere in which the mass density of the population of Dark Matter particles is \rho \propto r^{-\alpha}, we find that the solution for the velocity distribution of these particles is a Maxwell-Boltzmann (MB) distribution

(4)   \begin{equation*}  dn \propto \exp \left( - \frac{|v|^2}{2 \sigma^2}\right) d^3\vec{v}, \end{equation*}

where \sigma is a single parameter that defines the distribution and is related to the enclosed mass by

(5)   \begin{equation*}  M(r) = \frac{2 \sigma^2 r}{G}, \end{equation*}

with G the Newton gravitational constant.13 We computed the enclosed mass of the simulation and the corresponding velocity dispersion as a function of the radius and show the latter in Fig. 2. We find that the velocity dispersion is slowly falling outside the disk r > 10 kpc.

In the top row of Fig. 1, we compute the velocity dispersion in the middle of each bin to plot the theoretical MB distributions defined above. The DM distributions in the intervals [0, 10] kpc and [20, 35] kpc show a flat-peaked distribution, consistent with previous results from N-body simulations.29 Comparing the velocity distribution of DM to the Maxwell-Boltzmann distribution, we find that the MB distribution underpredicts the high tail of the distribution, but is close to matching the velocity distribution for low |v|. This is consistent with previous observations from other simulations.9 It is clear that MB models do not capture the DM distribution precisely, which provides further motivation for the search for tracer stars.

Figure 1: Comparison of theoretical and empirical |v| distributions of the Eris Simulation in the radial bins [0,10],[10,20],[20,35]. Row 1: The DM distribution is estimated with a Maxwellian according to (4). It peaks at lower velocities than the measured distribution and underestimates the high velocity tail. Row 2: The stellar distribution outside the disk is broken down into three metallicity bins as well to showcase that the low-metallicity stars have markedly different kinematics and match the DM much better at low radii. The smooth halo prediction given by (7) matches the higher metallicity stars well at low radii. The anomalies in [20,35] are discussed in the text.
Figure 2: Velocity Dispersion as a function of radius as defined by (5). It has a slowly falling tail characteristic of a DM halo.

It is also possible to compute the best-fit \sigma in each bin in order to fit the data to an MB model. The results for the three bins are \sigma = (215.437, 178.775,160.287) km/s, notably outside the range of velocity dispersions in Fig. 2 which makes the best-fit models physically questionable, underlining the weakness of MB models. The fit is of course better, but they are unrealizable as applicable models because to generate them, knowledge of the DM distribution itself is required!

2. Stellar Distribution

In order to compare the velocity distributions of the stellar component of the Eris simulation, we summarize the expected distribution of the stars in a smooth halo, i.e. a distribution with no disk. We choose not to include the stellar disk because we know that it has drastically different kinematics from the DM. We model the stellar distribution as a spherically symmetric density \rho \propto r^{-3.5}, where the velocity distribution is modeled as a multivariate normal distribution with mean and covariance matrix given by Refs. [25, 30–32].

(6)   \begin{equation*}  \mu_{r, \phi, \theta} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}, \qquad \Sigma_{r, \phi, \theta} = \begin{pmatrix} \sigma_r^2 & 0 & 0 \\ 0 & \sigma_\phi^2 & 0 \\ 0 & 0& \sigma_\theta^2 \end{pmatrix}. \end{equation*}

Fitting for the total star velocity distribution in r \in [0,35] kpc, we find

(7)   \begin{equation*}  \mu_{r, \phi, \theta} = \begin{pmatrix} -18 \\ -25 \\ 4.0 \end{pmatrix}, \qquad \Sigma_{r, \phi, \theta} = \begin{pmatrix} 118^2 & 0 & 0 \\ 0 & 102^2 & 0 \\ 0 & 0& 125^2 \end{pmatrix}. \end{equation*}

where the units for \mu are km/sec and for \Sigma_{r, \phi, \theta} is in (km/sec)^2. These results are consistent with observational values cited in [25] where the velocity dispersions are \sigma_r = 120 km/sec, \sigma_\phi = 100 km/sec, and \sigma_\theta = 100 km/sec and the mean of the velocities are zero. Our means are slightly different from zero due to structures in the velocity distributions, especially in v_\theta and v_\phi

Figure 3: Metallicity distribution of stars in Eris. We have chosen our metallicity bins to separate the tail of older, low metallicity stars from the bulk of the distribution in [-2,.5].
In Fig. 1 we plot the expected smooth halo distribution in the same bins as determined by running a Monte Carlo simulation with the probability distribution defined by a multinormal distribution above with parameters given by (7). For comparison, we plot the stellar distributions as calculated from Eris broken down by metallicity and masking out the disk so the comparison is reasonable. We have chosen the metallicity bins to resolve the metal-poor tail of the distribution in Fig. 3. Immediately, the difference in kinematics is discernible. We find that the shape of the stellar distributions matches the smooth halo prediction well in [0,10] with the mean consistent with the stars of metallicities [Fe/H] \in [-3,-2]. This is the region where virialization is dominant, so there is little substructure. However, significant deviations from the overall shape occur in the radial bins with [10,20] and [20,35] kpc, the latter being especially dramatic. Unanticipated by the smooth halo, there are multiple peaks in the younger star population. These peaks are evidence of substructure, and are well explained in the following sections.

The correlation is strongest at low radii, suggesting that the stars of with low metallicity seem to have virialized with their host DM subhalos. The difference is more apparent here because most of the stars have formed a disk with coherent kinematics very different from that of the DM which has no corresponding disk. At large radii in and outside the disk, there is ostensibly less correspondence. However, this region is uniquely interesting because of the high incidence of kinematic substructure which makes such large scale distributions less telling than they might seem. We investigate the effect of debris and streams in the following two sections. After that discussion, we will analyze the convergence of low metallicity stars to the DM component by component and see the strong correlation explicitly.

Figure 4: Stellar mass distribution as a function of the radial distance from the center of the Eris galaxy simulation. The star distributions are broken by metallicities. In order to distinguish the stellar halo from the disk, we plot in dashed lines the stellar mass distribution with the cut |z| >2 kpc.

Comparing the distributions of DM and stars, we find that DM and the oldest stars are closest in shape as hoped. In Fig. 4, we show the stellar mass distribution (dM/dr) as a function of the radial distance r from the center. The stars are broken by metallicity [Fe/H]> -2, [Fe/H] \in [-3,-2] and [Fe/H] >-3. We find that the stellar mass is dominated by the youngest stars as expected, with the oldest stars composing \sim 1 \% of the total mass, indicating that tracer stars are fairly rare. Most of the Eris stars were born in the galaxy and thus inherit different kinematics; this suggests that studies of DM-only simulations, e.g. Ref. [33], with stars “painted on” post hoc to the DM, should be examined very carefully.16,34

B. STREAMS

Stellar streams are groups of particles which are localized both in velocity and space.19–24,35–38 We look in Eris to find examples of streams which we expect to be remainders other merging subhalos. In order to do so, we examine the velocity distributions of spheres with radius r = 2, 3, 5 \, \text{kpc} in the radial bins [0,10] kpc, [10,20] kpc, and [20,35] kpc respectively, distributed so that they randomly sample about 75\% of the volume. We proceed by fitting gaussians to each of the components of the velocity (v_R, v_T, v_z) in order to identify groups with specific velocity component in both the DM and stellar populations after making standard quality cuts to remove spheres with low statistics . This analysis succeeds in locating the stellar disk (with v_T strongly peaked at -200 km/s) and many local stellar streams but finds only two DM streams that match those in stars. We call these double streams. The results are visualized in Fig. 5. Despite being run many times on the data, this algorithm only finds these two streams (which are always added to Fig. 5). It also consistently reproduces the number of streams, so Fig. 5 is representative.

Figure 5: Characteristics of the velocity distributions of the sampled spheres in Sec. III B. For each sphere, the stellar and DM density is computed and the velocity characteristics are analyzed to determine whether the sphere contains a stream. The points are colored according to the radius of the sphere; there is a clear correlation with density as expected. All streams are found far from the center.

In Fig. 6, we show the local stellar and DM velocity distributions of the two streams located with this algorithm. The kinematic structure stays sharply and unimodally peaked up to a radius of 4 kpc around the local maximum of the stellar distribution although the density is peaked only out to approximately 1 kpc. We find distinctive peaks in the different components and a one-to-one correspondence between the location of these peaks for DM and the stars. These streams are of great interest because they represent significant deviations from the surrounding DM distribution. In fact, due to their significant number density which is orders of magnitude above the local average, these two streams dominate the distribution in the third panel of Fig. 1. The streams can even be identified in the total stellar mass, shown in Fig. 4 as the twin peaks in density at r = 30, 32 kpc. Due to the sharpness of their velocity structure, these two streams are likely subhalos in early stages of merging. We expect that these streams will be stripped as it orbits, eventually smoothing out its peak in the local phase space distribution.

 

Figure 6: The full velocity distributions of the 4 kpc spheres around each of the two streams, broken down into DM and stars. There is a clear correspondence between DM and stellar kinematics. This phase space localization is extremely rare and is indicative of a recent merger early in the process of being stripped.

As can be seen from Fig. 5, there are startlingly few double streams. Their absence is surprising because prior studies suggest that stellar streams are almost always accompanied by a corresponding DM stream.34 We attribute such a difference to the fact that our analysis is the first one to use a fully hydrodynamic simulation in the investigation of substructure. Due to the relative abundance of solely stellar streams, we find that DM virializes quite rapidly and its substructure is wiped out even where stellar substructure is largely preserved. This is due to the fact that stars are more tightly bound and, therefore, are harder to disrupt. Conversely, DM has wider orbits and is easier to strip.39,40 These findings mirror the conclusion of Ref. [41] about the relative time scales of stellar and DM halos.

C. DEBRIS

The kinematic substructure of Eris also differs from that of DM-only simulations like Via Lactea II in the location and prevalence of debris.16 DM debris is more common than DM streams but is still quite rare, again contrary to expectations derived from DM-only simulations.17 We employ the sphere-sampling technique introduced in Sec. IIIB to investigate the local kinematic characteristics of the DM halo not captured in broad distributions, being careful not to sample from the streams identified in the prior section \textit{or} from the disk. We filter the data to look for DM that is coherent in |v| but spatially diffuse, making cuts on the standard deviation of the Gaussian fits of each component of the local velocity distribution. In order to have reasonable sample sizes within spheres, we use a standard deviation cutoff of 90 \text{km/s}. About 33 \% of the total spheres exhibit the characteristics of concentration in the speed portion of phase space but dispersion in the spatial component. Notably, \textit{all} of these debris flows are located in the [20,45] kpc region, with 97 \% more than 30 kpc from the galactic center. This further supports the notion of virialization being driven in the core of the galaxy.

In Fig. 7, we plot the velocity distribution in the spheres analyzed above. In order to grasp the significant variation from the median, we show the inner 68\% of the distributions of the spheres in blue and the inner 99.7\% in grey. Note that neither the theoretical MB and smooth halo expectations capture the huge variance in the extremes of the distributions. Any sort of local prediction must take into account the possibility of debris; this is especially true at large radii but is also important in the tails of the distributions closer to the center of the galaxy. From the perspectives of Earth-bound experiments, it is very lucky that our planet falls in the first radial bin; otherwise, it would be very difficult to estimate the local distribution given its large variability.

Figure 7: The spheres in Sec. III C are used to generate the distributions of stars (\textit{Row 1}) and DM (\textit{Row 2}). The variance of these distributions is captured in the blue and grey regions which delineate the inner 68\% and 99.6\% respectively. The medians of these distributions match the smooth model predictions from Sec. III well at low radii, but of course do not capture local deviations which become very significant. Note that spatial cuts on the data avoid contamination from the disk and streams.

DM shows less in the way of debris structure than the stellar population in Eris. We attribute such a discrepancy to the fact that DM orbits are hotter (see Ref. [39]) and therefore are easier to strip in recent mergers. As stated before, this makes the preservation of structure in DM much harder, and again points to the increased virialization DM undergoes which smoothes out substructure. Nevertheless, there are still significant peaks in [10, 35] and heavy high-velocity tails in [0, 10].

D. Full Component Comparison of DM and Stars

Having identified the characteristics of substructure, we now study the velocity components of each distribution. In Fig. 11, we show the velocity distributions for the stars and DM with the usual radial bins and metallicity cuts. By plotting the tangential, radial and vertical velocity components for stars and DM, we can test our claim: that the low metallicity stars (in red) converge in distribution to the DM.

The first striking feature of every component and radial bin is the smoothness of the DM line. It is consistently symmetric, peaked at zero, and smoothly and slowly decaying at the tails. There is some structure in tangential velocity for r \in [20,35], due to the aforementioned streams. (Fig. 12 demonstrates that the deviations might be contamination from a dark disk; see Ref. [42]). The smoothness of the distributions agrees with the notion of rapid virialization in the DM component. The flatness of the peaks and heaviness of the tails that characterizes DM kinematics lead to deviations from Maxwell-Boltzmann model described in Fig. 1.

Different velocity structures appear in the stellar component. Looking first at the radial velocities, we see agreement between the DM and stars that weakens at higher distances and metallicities. Below r = 20 kpc, the distributions are smooth and symmetric; for r \in [20,35], the distribution picks up strong peaks from the streams in this region. Their effect is reduced for stars with [Fe/H] < -3. In the tangential velocities, we notice that the stellar distributions have a double-peak structure composed of the peak at zero and another one at \sim -200 km/s which is due to contamination from the disk. Again, the stars passing the lowest metallicity cut have diminished deviations from the symmetric structure. In the vertical velocity, the agreement is again strong and the trend of low metallicity stars converging to DM continues. In the largest radial bin, the effects of the stream dominate.

Fig. 12 illustrates how the DM is nearly the same in the complement region, |z| < 2 kpc. A dark disk begins to emerge in v_T at larger radii, but otherwise the distributions are symmetric and heavy in the tails. The stars, however, are much different. The rotation of the disk is obvious from the sharply peaked distribution of the tangential velocities. The low metallicity stars have significantly diminished peaks, but are still far from resembling the DM. Thus, we argue that low metallicity stars in the halo are in fact better tracers of DM in disk than disk stars are. Although at first this may sound odd, it reflects the spherical symmetry of the DM distribution. We will show more explicit evidence for this in the following section.


IV. IMPLICATIONS FOR DIRECT DETECTION EXPERIMENTS 

DM direct detection constraints assume that DM in the Earth’s frame follows a Maxwell-Boltzmann distribution.43 It is important to emphasize that deviations from this assumption can lead to non-trivial changes of the DM constraints.5,6 Thus, refinements of the MB distribution in the Standard Halo Model can sharpen experimental predictions.

The relevant quantity in direct detection experiments is the differential detection rate of DM particles with mass m_\chi:

(8)   \begin{equation*} \frac{dR}{dQ} =N_T \Delta t \frac{\sigma_0 \rho_0}{\sqrt{\pi} v_0 m_\chi m_r^2} F(Q^2) T(Q), \end{equation*}

where N_T is the number of target nuclei, \Delta t is the exposure time, Q is the momentum transfer, \sigma_0 is the scattering cross section at zero momentum transfer, m_r is the reduced mass of the nucleon and DM particle, v_0 = 220 km/sec is circular velocity of the Sun around the Galactic Center, \rho_0 = 0.3 \text{GeV}/\text{cm}^3 is the local DM density, F(Q^2) is the atomic form factor, and T(Q) is the integral of the velocity distribution as a function of the energy transfer.43

The astrophysical contribution is isolated in the integral of the velocity distribution denoted T(Q):

(9)   \begin{equation*} T(Q) = \frac{\sqrt{\pi}}{2} v_0 \int_{v_\text{min}}^\infty \frac{f(v)}{v} dv. \end{equation*}

where v_{\text{min}}(Q) is the minimum velocity required for scattering which is determined in part by the mass of the DM particle.44 As experiments push possible masses lower, T(Q) depends increasingly upon the tail of f(v), making the contributions of debris flows all the more important.

For earthbound experiments, velocities must be measured in the heliocentric frame.45 Thus we are interested in the local disk DM at Earth-like radii. For now, we neglect annual modulation effects that would exist in the geocentric frame (see Refs. [44, 46] for the scale of these deviations). This shift is performed using the transformation

(10)   \begin{equation*}  \vec{v} = \vec{v}_{\text{gal}} - \vec{v}_{\astrosun} \end{equation*}

where \vec{v}_{\astrosun} in Cartesian form is determined for a given earth location at (R,\phi, z) = (8 \text{kpc},\phi,0) by

(11)   \begin{equation*} \vec{v}_{\astrosun} = \begin{pmatrix} U \cos \phi - V \sin \phi \\ V \cos \phi + U \sin \phi \\ W \end{pmatrix}. \end{equation*}

Here, (U,V,W) = (-10,-5.23 - 205, 7.17) \, \text{km/s} are the standard cylindrical velocity coordinates of the sun (with the rotation of the disk explicitly separated out) as given by [13] for instance.

Figure 8: Comparison of DM with |z| > 5 and |z|<2 (the latter is shown in orange). The measures of spread are the inner 68 \% and 99.7 \% of the distributions, which were generated in the manner described in Sec. IV.We see that the distributions are nearly identical, justifying our choice to model disk DM with halo stars, which we have shown lack the strong peak in v_T at the disk rotation speed.

As motivated in the previous section, the DM kinematics are essentially the same in and out of the disk at small radii. We demonstrate this with a comparison of the DM distribution for |z| > 5 and |z| \leq 2 (in orange) in Fig. 8. This distributions were generated by drawing 15 random cylinders with centers at R = 8,z=0 kpc, radii of 3.5 kpc, and cut spatially so that all particles satisfy R \in [7,9] in addition to the z-cut. We then perform the boosts in (10) individually within each cylinder. The scatter for halo DM is shown in the same manner as
Fig. 7 and the median for disk DM is shown in orange. We have omitted the disk scatter for visual clarity, but it is similar in magnitude to that of the halo. The distributions are nearly identical with the disk DM having a slightly flatter peak. The most important feature of this distribution is the small scatter around the DM line. Because Earth is fairly close to the center of the galaxy, its local DM is well virialized so most debris has been wiped out. This means that there is minimal deviation from the median, allowing us adopt its best fit curve as an accurate measure of the distribution.

Figure 9: Heliocentric Velocity Distributions for the stars passing the cuts described in Sec. IV shown for (in order from left to right) stars with no metallicity cuts, stars with [Fe/H] < -3, and [Fe/H] < -3.5. The low metallicity stars converge to the disk DM distribution (in orange) well around the peak but negligibly in the high velocity tail. The large fluctuations in the scatter is due to the low number of stars passing these cuts.
Table 1: Parameters from the best fit generalized Maxwellian calculated from the distributions in Fig. 9.

Fig. 9 shows the heliocentric distributions computed similarly for low metallicity stars, using 15 cylinders of radius 3.5 kpc and cutting so that |z| > 5 kpc and R \in [7,9] kpc. This strong cuts on z lower the contamination from disk stars at the cost of small sample sizes. This inflates the scatter around the median, especially when we look at the very tail of the metallicity distribution. Thus, these results are convincing only regarding the convergence of low metallicity stars to DM; the cuts themselves require a more careful, multidimensional description. We have also fit the median line of the distributions with a best-fit generalized MB model defined by

(12)   \begin{equation*} f\big(|v|\big) = \frac{|v|^2}{\frac{1}{3} \sigma ^3 \Gamma \left(1+\frac{3}{2 \alpha }\right)} \exp \left\{ -\left(\frac{|v|^2}{\sigma^2}\right)^\alpha \right\} \end{equation*}

Figure 10: Comparing for the best-fit Maxwellians for a range of metallicity cuts in the heliocentric frame. The convergence to the DM line is clear and monotonic.

with parameters for DM and stars over varying chemical cuts displayed in Table 1. Although the harsh cuts introduce noise to the data, there is a clear trend of convergence at low metallicities. We see that the topmost distribution of stars passing all spacial cuts bears little resemblance to the DM, whereas the following low metallicity cuts push median much closer to the disk DM line. Furthermore, the low-statistics problem we encounter here is an artifact of the simulation; in real sky surveys there is no shortage of stars. The heavy tail of the DM is the hardest to capture, a fact that is reflected in Fig. 10. This graph demonstrates the behavior of the form factor T(Q) computed for median stellar distributions across a range of metallicities and the disk DM itself. The convergence here is strong support for our initial claim: that low metallicity stars (passing other spatial cuts) retain the kinematics of their DM host halos. Refining these cuts for real metallicity distributions makes these more accurate form factors experimentally realizable.


V. CONCLUSION 

In this paper, we studied the velocity distributions of stars and DM in the hydrodynamic Eris simulation. We have shown that there exist non-trivial features like streams and debris which have a strong impact on the local phase space distribution. We characterized these features and qualitatively explained their origins. We found that DM is more virialized than the stars and, therefore, stellar streams and debris should not always be interpreted as a location for DM streams and debris, a finding which is particularly unexpected. We suggest that this depends on the formation time of these streams and conjecture that only the most recent mergers like those in Fig. 5 retain a strong correspondence. We analyzed the correlation between the velocity distributions of stars with differing metal content and DM in order to identify a tracer population capable of extrapolating the DM velocity distributions from that of the measurable stellar component.

We found that there exists a convergence in correlation between the velocity distribution of low metallicity stars of the inner halo (r \in [0,10]) and DM that persists but weakens at larger distances due to the dark disk. Thus, old stars can be used as tracers for the local DM distribution; while our analysis was too rough for exact cuts to be proposed, the trend of convergence is persuasive. As we showed in Fig. 10, low metallicity stars mimic the surrounding DM very well, in support of our initial hypothesis. Our study of substructure makes us increasingly confident that scatter around the median DM distribution is minimal, but warns us that this property does not hold at larger distances. However, we have demonstrated that at earth-like radii there is a dearth of substructure in DM and stars, so the uncertainty of these unpredictable features is minimized. The overall result is a hopeful one with tangible experimental benefits. We hope to pursue this line of reasoning in a future paper to analyze real stellar distributions resulting from sky surveys like RAVE and GAIA to provide more accurate parameters for the local DM distribution in hopes of further aiding direct detection experiments in confirming the existence of a DM particle.48–50

Figure 11: Full velocity distribution in the Eris halo (|z| > 2) with distributions from regions divided spatially and by metallicity. Corresponding DM distributions are shown for comparison. Visually, the lowest metallicity stars have distributions that match the DM much better than the higher metallicity stars. A discussion of the various peaks in [20, 35] ard the double peaks in the tangential velocity is given in Sec. III C. All distributions were smoothed using a cubic Savitzky-Golay filter with a window size of seven points.[47]
Figure 12: Full velocity distribution in the Eris halo (|z| > 2) with distributions from regions divided spatially and by metallicity. Corresponding DM distributions are shown for comparison. The lowest metallicity stars are still better tracers of DM than high metallicity stars, but they retain more of the velocity features of the disk: sharper peaks overall and a clear sense of rotation. A more complete discussion is given in Sec. III C. All distributions were smoothed using a cubic Savitzky-Golay filter with a window size of seven points.[47]


 VI. ACKNOWLEDGMENTS 

We thank Piero Madau for the Eris simulation. We also thank Siddharth Swarathma for suggesting the yt toolkit and Michael Lin for numerous sanity checks.

 

Received: January 8, 2017
Revised: March 7, 2017
Accepted: March 24, 2017
Published: April 10, 2017


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