Constraints on the dark matter particle mass from the number of Milky Way satellites

We have conducted $N$-body simulations of the growth of Milky Way-sized halos in cold and warm dark matter cosmologies. The number of dark matter satellites in our simulated Milky Ways decreases with decreasing mass of the dark matter particle. Assuming that the number of dark matter satellites exceeds or equals the number of observed satellites of the Milky Way, we derive lower limits on the dark matter particle mass. We find with 95% confidence $m_s>13.3\mathrm{keV}$ for a sterile neutrino produced by the Dodelson and Widrow mechanism, $m_s>8.9\mathrm{keV}$ for the Shi and Fuller mechanism, $m_s>3.0\mathrm{keV}$ for the Higgs decay mechanism, and $m_{\mathrm{WDM}}>2.3\mathrm{keV}$ for a thermal dark matter particle. The recent discovery of many new dark matter dominated satellites of the Milky Way in the Sloan Digital Sky Survey allows us to set lower limits comparable to constraints from the complementary methods of Lyman-$\alpha$ forest modeling and x-ray observations of the unresolved cosmic x-ray background and of dark matter halos from dwarf galaxy to cluster scales. Future surveys like LSST, DES, PanSTARRS, and SkyMapper have the potential to discover many more satellites and further improve constraints on the dark matter particle mass.

Figure 1

Nonphysical halos formed along a filament and accreting onto a larger halo at $z=1$ in a WDM simulation ($m_{\mathrm{WDM}}=1\mathrm{keV}$). These halos are numerical artifacts.

Figure 2

Power spectra for our simulations. The dotted line is the power spectrum for an 11 keV standard sterile neutrino from Abazajian [51]. The neutrino spectrum is approximately the same as a 2 keV thermal particle, validating the scaling relation of Viel et al. [52]. The vertical dashed lines indicate the lattice cell size in our high and low resolution refinement levels.

Figure 3

Density profile of Milky Way halos in the set A and set B CDM and WDM simulations. Thick lines are the high resolution simulations. Set B simulations are at top, the set A and the WDM cosmologies in each set have been offset downward for clarity. The profiles are plotted starting from the convergence radius of Power et al. [128] for both resolutions (vertical lines).

Figure 4

Portraits of the simulated Milky Way halo at $z=0$ in the set B high resolution simulations. Structure within 500 kpc of the MW centers is shown.

Figure 5

Cumulative mass functions for subhalos within $R_{50}$ for the CDM set A (left) and set B (right) simulations. Subhalo masses have been normalized by the $M_{50}$ mass of the host. Poisson error bars have been added to the high resolution simulations and fit with a straight line. Both high and low resolution (dotted lines) turn away from the straight fit below about 200 times the mass resolutions of the simulations (short vertical lines). Mass functions for the set C halos have also been added (thin lines) and show the A and B abundances are within the halo to halo scatter.

Figure 6

Cumulative velocity functions for subhalos within $R_{100}$ in our low resolution set C (thin lines) and high resolution set A and set B (thick lines) MW halos. Subhalo circular velocities have been normalized to the maximum circular velocity of the host halo. The dashed line is the subhalo velocity function of Via Lactea II, the straight solid line is the fitting formula from the Bolshoi simulation applied to our A and B halos. The shaded regions show the minimum and maximum and $\pm 1\sigma$ from the mean of the 68 MW-sized halos of Ishiyama et al. [77].

Figure 7

Cumulative velocity functions for subhalos within $R_{50}$ in our low resolution set C (thin lines) and high resolution set A and set B (thick lines) simulations. Subhalo circular velocities have been normalized to the circular velocity of the host halo at a radius enclosing an overdensity of $\Delta =50$. The dashed line is the velocity function for Via Lactea II and the straight solid line is the average abundance from the Aquarius simulations [76]. The shaded region is the minimum/maximum range and $\pm 1\sigma$ about the mean for halos from the simulations of Ishiyama et al. [77].

Figure 8

Cumulative velocity functions for satellites in our high resolution set A and set B CDM and WDM simulations. The set A abundances (thin lines) have been increased by 7% to normalize the CDM abundances to those of the set B and show that the relative suppression of halos in WDM cosmologies compared to CDM is similar across both simulations. The straight gray line is the Bolshoi fitting formula applied to the set B halo.

Figure 9

(left) Velocity functions for set B high (top) and low (bottom) resolution simulations normalized with Eq. (12). Solid lines are $R=400$, 300, 250, and 200 kpc (thick), dotted line is $R=150\mathrm{kpc}$, dashed line is $R=100\mathrm{kpc}$, dot-dashed line is $R=75\mathrm{kpc}$. The WDM cosmologies have been shifted down vertically for clarity. The value $\alpha =0.55$ was set by the high resolution simulation and provides good normalization for the low resolution as well but the effects of incompleteness become apparent at much larger radii (150–200 kpc compared to 75–100 kpc for high resolution). (right) Same as the left panel, but for the set A high (top) and low (bottom) resolution simulations. The value $\alpha =0.55$ also provides good normalization for this set of halos.

Figure 10

(left) Number of satellites with distance from the Milky Way grouped into 50 kpc bins. The simulation satellites have been cut by circular velocity $>6\mathrm{km}/\mathrm{s}$. Solid lines show the data from the high resolution and dashed lines show the low resolution simulations, thick lines are the set B simulations. The bars with arrows are the observed satellites after correcting for the sky coverage of the SDSS but not including Willman 1. Observations are incomplete at distances greater than about 50–100 kpc (depending on the luminosity and surface brightness of the dwarf), while simulations have not converged for less than about 100 kpc for high resolution and 150 kpc for low resolution. (right) Number of satellites with distance from the Milky Way like the plot at left but calculated from the convergence equation [Eq. (12)]. The convergence correction affects mainly the 4 keV 0–50 kpc bin.

Figure 11

(top) Number of satellites from 0–50 kpc calculated from the convergence equation in the set B high resolution simulation for $v_{\max }>6\mathrm{km}/\mathrm{s}$ (sloped solid line) with $1\sigma$ and $2\sigma$ limits (shaded regions) compared to the observed number of Milky Way satellites, excluding Willman 1, with correction for sky coverage (solid horizontal line) and at 68% and 95% confidence (dashed and dotted horizontal lines). (bottom) The observed number of satellites minus the number of satellites in simulation with $1\sigma$ and $2\sigma$ limits (shaded regions). The number of satellites in simulation must be greater than or equal to the number of observed satellites therefore where the difference equals zero sets a lower limit to the dark matter particle mass (arrows).

Figure 12

Comparison of CDM power spectra calculated from the fitting formula of Eisenstein and Hu (EH97) and from the LINGER software normalized by BBKS. On scales $k>0.1h/\mathrm{Mpc}$ ($<14\mathrm{Mpc}$) the power spectra are nearly identical. The “MW” vertical line is the diameter of a spherical region with density ${\Omega }_m{\rho }_c$ enclosing a Milky Way-sized mass $2\times {10}^{12}{M}_{\odot }$ ($\sim 5\mathrm{Mpc}$). This scale is well within the range where the power spectra are nearly equal.

Figure 13

Subhalo velocity function comparison for CDM high resolution set B simulations using fitting formula from BBKS and Eisenstein and Hu (thick lines) and the LINGER using set C simulations (thin lines). (left) Subhalos within $R_{100}$ and velocities normalized by $v_{\max }$ of their host. The straight sloped line is the fitting formula from the Bolshoi simulation. (right) Subhalos within $R_{50}$, normalized by $v_{50}$ of their host. The subhalo abundances between the BBKS and EH97 simulations are in good agreement and within the scatter of the set C halos.