Simulations and Measurements of Warm Dark Matter Free-Streaming and Mass

We compare simulated galaxy distributions in the cold ΛCDM and warm ΛWDM dark matter models. The ΛWDM model adds one parameter to the ΛCDM model, namely the cut-off wavenumber kfs of linear density perturbations. The challenge is to measure kfs. This study focuses on “smoothing lengths” π/kfs in the range from 12 Mpc to 1 Mpc. The simulations reveal two distinct galaxy populations at any given redshift z: hierarchical galaxies that form bottom up starting at the transition mas Mfs, and stripped down galaxies that lose mass to neighboring galaxies during their formation, are near larger galaxies, often have filamentary distributions, and seldom fill voids. We compare simulations with observations, and present four independent measurements of kfs, and the mass mh of dark matter particles, based on the redshift of first galaxies, galaxy mass distributions, and rotation curves of spiral galaxies.


Introduction
The cold dark matter ΛCDM model is apparently in agreement with all current observations on scales greater than 10 Mpc, with only six parameters and a few ansatz, e.g. 0 k Ω = and Λ Ω constant [1]. However, at scales less than 10 Mpc, there appear to be discrepancies with observations, known as the "small scale crisis" (missing satellites, too big to fail, core vs. cusp, voids, dwarf galaxy distribution, dark matter adiabatic invariant, etc.) [2]. A modification of the ΛCDM model, proposed to address these issues, is warm dark matter (ΛWDM), which How to cite this paper: Hoeneisen, B. k k > , due to free-streaming of dark matter particles that become non-relativistic while the universe is still dominated by radiation. For the current status of warm dark matter research, and a list of references, see [1] and [2]. The challenge is to measure, or constrain, the cut-off wavenumber fs k . To this end, we present a comparison of galaxy distributions in the cold and warm dark matter models with simulations. This study focuses on "smoothing lengths" fs k π in the range from 12 Mpc to 1 Mpc. Finally, we present four independent measurements of fs k , and of the mass h m of dark matter particles. These measurements are based on: • The redshift of formation of first galaxies, • The distributions of masses of Sloan Digital Sky Survey SDSS DR15 galaxies, • Spiral galaxy rotation curves, and • The assumption that dark matter was once in thermal and diffusive equilibrium with the Standard Model sector, and decoupled while still ultra-relativistic.
Finally, the results of these measurements are presented in Table 2.
Our notation and the values of cosmological parameters are as in Reference [1].

Warm Dark Matter
Let ( ) P k be the power spectrum of linear density perturbations in the ΛCDM model as defined in Reference [3]. The normalization of ( ) P k refers to the present time with expansion parameter 1 a ≡ . corresponding to spherical harmonics with 3 2500 l < < [1].
We consider warm dark matter (ΛWDM) with a power spectrum of linear density perturbations suppressed by a factor α for

Free-Streaming
We consider collisionless dark matter. The velocity of a dark matter particle at where 1 η = . The first term is the approximate integral from 0 to eq a , and the second term is the approximate integral from eq a to dec a . A numerical inte- If the free-streaming length fs d were equal for all dark matter particles, P(k) would not have a cut-off: only the amplitudes in P(k) would change their phases without changing P(k). It is approximately the standard deviation of fs d , for the net distribution of density fluctuations, that obtains the cut-off wavenumber fs k : ( )  [4] or [5]. Comparisons with alternative calculations of the free-streaming cut-off wavenumber fs k are presented in Appendix A.

The Galaxy Generator in Fourier Space
We make use of the galaxy generator described in References [6] and [7]. This program generates galaxies, directly at any given redshift z, given the power spectrum of linear density perturbations ( ) P k . The hierarchical generation of galaxies is illustrated in Figure 1. We do not step particles forward in time, but rather work directly in Fourier space at a given redshift z, i.e. we generate galaxies in bins of the comoving wavenumber 2 I k I L = π for max 2,3, , At each "generation" I, starting at  When ( ) δ x reaches 1.69 in the linear approximation, the exact solution diverges and a galaxy forms. As time goes on, density perturbations grow, and groups of galaxies of one generation coalesce into larger galaxies of a new generation as shown on the right.
to all generated galaxies j exceeds and j j r k = π are their radii [7] (the factor 0.9 was chosen to help "fill" space). After generating all galaxies of generation I, we step 1 I I → + , and generate the galaxies of generation 1 I + . Note that at generation 1 I + , corresponding to smaller galaxies, a "failed" galaxy that did not "fit" in a generation I ≤ , may fit at generation 1 I + , and a galaxy is formed that has lost part of its mass to neighboring larger galaxies.
We find it convenient to distinguish two populations of galaxies at every redshift z: the hierarchical galaxies that fit, and the stripped down galaxies that did not fit, and were generated with a reduced radii. Note that stripped down galaxies have lost part of their mass to neighboring larger galaxies, they form near larger galaxies, seldomly in voids, and often are distributed in "filaments" and "sheets". Hierarchical galaxies have fs M M > , and stripped down galaxies populate all masses, and are the only galaxies with For the cold dark matter ΛCDM model the power spectrum of linear density perturbations is [3] ( ) . All other cosmological parameters are taken from Reference [1].
For warm dark matter we take the same ( ) , where α is a suppression factor. This is the only difference between the simulations for cold and warm dark matter. We use the same seed for the random number generator, so all generated galaxies for fs k k < are the same for the simulations with cold or warm dark matter. For warm dark matter, the expected value of α , for adiabatic initial conditions, is We set 0 α = for warm dark matter, but run simulations with several α to understand the onset of damping.
Let us describe the formation of galaxies in time, see Figure

Simulations with Redshift z = 0.5
The simulations at redshift 0.5 z = are presented in Table 1, and in Figures 2-13. We note that lowering α from 1 (for the ΛCDM model) to 0 (for the ΛWDM model), reduces the number of galaxies with fs M M < , but the reduction does not reach zero.
fs M is not a cut-off mass. For warm dark matter, i.e. 0 α = , the reduction factor β is in the range 0.05 to 0.4, depending on the size L, and cut-off wavenumber fs k , of the simulation, and decreases with increasing k. These are stripped down galaxies, and their quantitative simulation requires a more complete galaxy generation code, a large simulation size L, and a large dynamic range of galaxy masses. Note how stripped down galaxies cluster around the large hierarchical galaxies, often forming filament and sheet distributions, and seldom populating the voids. These characteristic features of warm dark matter were noted by P.J.E. Peebles [8].

Comparison with Simulations with Gravitating Particles Stepped Forward in Time
We briefly review a simulation carried out by P. Bode, J. P. Ostriker and N. Turok [9]. Three simulations are done: one ΛCDM simulation, and two ΛWDM simulations with thermal relic dark matter with 350 eV m = and 175 eV, respectively, that correspond to a characteristic mass, given by Equation (8)    The stripped down galaxies lose mass to neighboring galaxies during their formation, are near larger galaxies, often form filamentary distributions, and seldomly fill voids.    They are rarer and considerably less dense than halos of the same mass in ΛCDM. And their spatial distribution is very different-they are concentrated in sheets and ribbons running between the massive halos, an effect which has been noted for some time for dwarf galaxies in the local universe" [8]. "Likewise the apparent absence of dwarf systems in the voids noted by Peebles ..." [8]. With respect to the ΛCDM simulation, the number of halos with mass of order 10 10 M  is reduced by a factor 0.14 β ≈    We note that these results are in agreement with our conclusions.  The stripped down galaxies lose mass to neighboring galaxies during their formation, are near larger galaxies, often form filamentary distributions, and seldomly fill voids.

Estimate of kfs with the Redshift of First Galaxies
In the warm dark matter scenario, the first galaxies to form have mass fs M . For a larger "smoothing length" fs k π , fs M increases, and the first galaxies form at a later time, i.e. at smaller redshift first z . Therefore, the redshift of the first few galaxies, or the redshift of re-ionization, allows a measurement of fs k . The galaxy with highest spectroscopically confirmed redshift, called GN-z11, has 11.09 z = . The quasar with highest spectroscopically confirmed redshift has 6.6 z = . The redshift of re-ionization is 1.7 reion 1.4 8.8 z + − = [1]. We will take the redshift of formation of the first (few) galaxies to be first 9.5 2.0 z = ± (the redshift of the oldest galaxy is statistically uncertain, so in the simulations we extrapolate from galaxies with larger k down to zero counts).
The maximum of the power spectrum of linear density perturbations ( ) , corresponding to a comoving wavelength eq eq 2 609 Mpc k λ ≡ π = . Therefore, simulations with very large L are required to include the contributions to the relative density perturbation ( ) δ x of Fourier components of long wavelength. Such large simulations become prohibitive, so some extrapolation becomes necessary. Simulations corresponding to several redshifts are presented in Figure 14. Extrapolating to zero counts we obtain data points in Figure 15. From first z , and extrapolating from Figure 15 we estimate  Note that first galaxies, i.e. galaxies with old stellar populations, have a mass approximately equal to M sfs , which is an alternative way to identify "M sfs ".

Measurement of vhrms(1) with Spiral Galaxy Rotation Curves
The root-mean-square (rms) velocity of non-relativistic dark matter particles in the early universe, when density perturbations are relatively small, can be writ- where a is the expansion parameter. Equation  The factor 1 h κ − is a correction for possible dark matter rotation. We take [12]. Equation (6) is consistent with Figure 4 of Reference [5] for non-degenerate dark matter, and with Figure 7 for fermion dark matter with chemical potential 0 µ ≈ . This range is also consistent with 10 galaxies in the THINGS sample [5] [13].
If dark matter decouples from the Standard Model sector, and from self-annihilation, while ultra-relativistic, the ratio of dark matter-to-photon temperatures after e e + − annihilation, while dark matter is still ultra-relativistic, is [4] ( ) for fermions with 0 µ = , and ( ) for bosons with 0 µ = . The temperature ratio (10) corresponds to dark matter with 2 f N = decoupling from the Standard Model sector, and from self-annihilation, while still ultra-relativistic, in the approximate temperature range from c m to b m [12]. Note that assuming 0 µ = obtains thermal equilibrium with the measured values of ( )

Thermal Relic
Let us now consider dark matter that was in both diffusive and thermal equilibrium with the Standard Model sector. Diffusive equilibrium implies chemical potential 0 µ = . The thermal relic mass for warm dark matter with 0 µ = , that decouples from the Standard Model sector, and from self-annihilation, while still ultra-relativistic, is [1]  A summary of the four independent measurements is presented in Table 2   and thermal equilibrium with the Standard Model sector in the early universe (within uncertainties), and decoupled from the Standard Model sector, and from self-annihilation, while still ultra-relativistic; and, furthermore, that the mass of dark matter particles is indeed given by Equation (8)  h v and 0 T , implies that dark matter was never in thermal or diffusive equilibrium with the Standard Model sector, yet requires self-annihilation and freeze-out to obtain the observed dark matter density.

Conclusions
To understand warm dark matter, we find it convenient to classify galaxies, at any given redshift z , according to their origin: hierarchical galaxies, and stripped down galaxies. Hierarchical galaxies form from the bottom up: the first galaxies to form, in the warm dark matter scenario, have mass fs M , these galaxies cluster due to gravity, coalesce, and form galaxies of a new generation, in an ongoing hierarchical formation of galaxies, as illustrated in Figure 1. During their formation, stripped down galaxies lose part of their mass to neighboring galaxies. Hie- Hierarchical galaxies with fs M M < , but does not smooth the density perturbations of stripped down galaxies that are created highly non-linear.
In Table 2, we present a summary of four independent measurements of fs k and h m , separately for fermion and boson dark matter. These measurements are based on: 1) Spiral galaxy rotation curves (that obtain ( ) 2) The assumption that dark matter was once in thermal and diffusive equili- The conclusions of these studies are: • By construction, the generated galaxies are exactly the same for cold and warm dark matter for < cluster near neighboring larger galaxies, and often are distributed in filaments and sheets. These characteristic features of warm dark matter were noted by P. J. E. Peebles [8].
• In Table 2 we have presented four independent measurements. The first two determine NR = π is well satisfied, so the "smoothing length" fs k π is indeed due to free-streaming dispersion, and not to diffusion of self interacting dark matter, or to other causes.
• The mass of the Milky Way galaxy is approximately equal to the measured transition mass fs M . The warm dark matter scenario with this fs M solves, at least qualitatively, all problems of the "small scale crisis" mentioned in the Introduction.
• Several analysis of the Lyman-α forest, and of gravitational lensing, of light from distant quasars, have set lower limits on the thermal relic mass, typically in the range 2000 eV to 4000 eV. (Such thermal relics are assumed to self-annihilate and freeze out, to obtain the present mean dark matter density of the universe.) These limits are equivalent to setting lower limits on fs k in