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. Ω k = 0 and Ω Λ 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 assumes that the power spectrum of linear density perturbations P ( k ) becomes suppressed for comoving wavenumbers k > k fs , 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 k fs . 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” π / k fs in the range from 12 Mpc to 1 Mpc. Finally, we present four independent measurements of k fs , and of the mass m h 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].

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 a ≡ 1 . k = 2 π / λ is the comoving wavenumber. The normalization of P ( k ) is fixed, within uncertainties, by measurements of the Sachs-Wolfe effect in the range − 3.1 ≲ log 10 ( k / h   Mpc − 1 ) ≲ − 2.7 , and by measurements of σ 8 in the range − 1.3 ≲ log 10 ( k / h   Mpc − 1 ) ≲ − 0.6 [1]. The Planck satellite measurements of the Cosmic Microwave Background fluctuations accurately determine P ( k ) in the range − 3.5 < log 10 ( k / h   Mpc − 1 ) < − 0.6 corresponding to spherical harmonics with 3 < l < 2500 [1].

We consider warm dark matter (ΛWDM) with a power spectrum of linear density perturbations suppressed by a factor α for k > k fs . The suppression may be due to free-streaming, or to diffusion if elastic scattering between dark matter particles dominates over free-streaming, or to other causes. For warm dark matter, α ≈ 0 . k fs is the only new parameter for warm dark matter in addition to the parameters of the ΛCDM model. The transition mass corresponding to k fs is M fs ≡ 4 π R 3 Ω m ρ crit / 3 with R = 1.555 / k fs (the numerical factor 1.555 is obtained for a 3-dimensional Gaussian mass fluctuation, and its Fourier transform). For warm dark matter, the distribution of galaxy masses becomes suppressed by a factor β for M < M fs with respect to the cold dark matter case. β is a function of k.

We consider collisionless dark matter. The velocity of a dark matter particle at expansion parameter a has the form v h ( a ) = c / 1 + ( a / a ′ h NR ) 2 . We consider the case a ′ h NR ≲ a eq / 3 , i.e. warm dark matter. The comoving free-streaming distance of this particle until decoupling is

d fs = ∫ 0 dec v h ( a ) ⋅ d t a = ∫ 0 dec c 1 + ( a / a ′ h NR ) 2 d t a = η c a ′ h NR H 0 [ 1 Ω r ln ( 2 a eq a ′ h NR ) + 2 Ω m ( a eq − 1 / 2 − a dec − 1 / 2 ) ] , (1)

where η = 1 . The first term is the approximate integral from 0 to a eq , and the second term is the approximate integral from a eq to a dec . A numerical integration obtains η = 0.9118 .

If the free-streaming length d fs 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 d fs , for the net distribution of density fluctuations, that obtains the cut-off wavenumber k fs : 2 σ ( d fs ) = λ fs / 2 = π / k fs . We calculate σ ( d fs ) with Equation (1) with

a ′ h NR ≡ v h rms ( 1 ) c , (2)

and η = 0.2816 to obtain the root-mean-square σ ( d fs ) ≡ 〈 ( d fs − 〈 d fs 〉 ) 2 〉 over the distribution of the net density perturbations for an ideal gas (i.e. the distribution for temperature T h + ϵ minus the distribution for temperature T h [4] ), or η = 0.2628 for a gas of fermions with zero chemical potential, or η = 0.4115 for a gas of bosons with zero chemical potential. In the following we set η = 0.263 ± 0.053 for fermion warm dark matter with zero, or negative, chemical potential, or η = 0.412 ± 0.082 for bosons with zero chemical potential. v h rms ( 1 ) is the adiabatic invariant defined in References [4] or [5]. Comparisons with alternative calculations of the free-streaming cut-off wavenumber k fs are presented in Appendix A.

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 k I = 2 π I / L for I = 2 , 3 , ⋯ , I max , with I max = 69 .

At each “generation” I, starting at I = 2 , we calculate the relative density

δ ( x ) ≡ δ ρ ( x ) / 〈 ρ 〉 in the linear approximation by summing its Fourier components up to wavenumber k I . We then search maximums of δ ( x ) . If a maximum reaches (or exceeds) 1.69 in the linear approximation, which has already broken down, the exact solution (for spherically symmetric density perturbations) diverges, and we generate a galaxy if it “fits”. A galaxy i fits if its distance

to all generated galaxies j exceeds r j + 0.9 r i , where r i = π / k i and r j = π / k j are their radii [7] (the factor 0.9 was chosen to help “fill” space). After generating all galaxies of generation I, we step I → I + 1 , and generate the galaxies of generation I + 1 . Note that at generation I + 1 , corresponding to smaller galaxies, a “failed” galaxy that did not “fit” in a generation ≤ I , may fit at generation I + 1 , 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 M > M fs , and stripped down galaxies populate all masses, and are the only galaxies with M < M fs if α = 0 .

For the cold dark matter ΛCDM model the power spectrum of linear density perturbations is [3]

P ( k ) = 4 ( 2 π ) 3 c 4 N 2 C 2 k τ 2 ( 2 k / k eq ) 25 Ω m 2 H 0 4 ( k SW k ) 1 − n , (3)

with

k eq = 2 H 0 ( Ω m − Ω ν ) c Ω r . (4)

k is the comoving wavenumber. τ ( 2 k / k eq ) is a function given in Reference [3]. C is a function of Ω Λ / Ω m [3]. We take C = 0.767 [7]. The amplitude N 2 in (3) is related to A s ≡ Δ R 2 [1] by N 2 ≡ A s / ( 4 π ) ≡ Δ R 2 / ( 4 π ) . We take ln ( 10 10 Δ R 2 ) = 3.062 ± 0.029 [1], corresponding to N 2 = 1.70 × 10 − 10 . The pivot point is k SW = 0.001   Mpc − 1 . All other cosmological parameters are taken from Reference [1].

For warm dark matter we take the same P ( k ) for k < k fs . For k > k fs we replace P ( k ) by α P ( k ) , 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 k < k fs 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 α = ( Ω b / ( Ω b + Ω c ⋅ a dec / a eq ) ) 2 = 0.0033 . We set α = 0 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 1. The relative density δ ( x ) is the sum of Fourier components from I = 2 up to I fs in the case of warm dark matter (with α = 0 ). Each of these Fourier components grows in proportion to the expansion parameter a ∝ t 2 / 3 . The first galaxies have mass M fs , and form when, at certain locations, the sum of Fourier components of δ ( x ) up to k fs ≡ I fs 2 π / L reaches 1.69.

The simulations at redshift z = 0.5 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 M < M fs , but the reduction does not reach zero. M fs is not a cut-off mass. For warm dark matter, i.e. α = 0 , the reduction factor β is in the range 0.05 to 0.4, depending on the size L, and cut-off wavenumber k fs , 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].

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 m = 350   eV and 175 eV, respectively, that correspond to a characteristic mass, given by Equation (8) of Reference [9], M S = 6.5 × 10 11 M ⊙ and 7.1 × 10 12 M ⊙ , respectively. M S corresponds

to the total mass of the simulation particles in the collapsed object. The simulations are done in a cube of side 30 Mpc, with 256³ particles, they resolve objects down to mass, and the simulations are carried forward in time up to redshift 1 (limited by computing resources). The authors note that above (below) the characteristic mass scale, the galaxy formation is “bottom up” (“top down”). We quote from the conclusions of Reference [9] : “Below this mass scale (), objects are formed primarily by the fragmentation of pancakes and ribbons.

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

is reduced by a factor (0.06) for the (175 eV)

ΛWDM simulation, at redshift. increases by a factor 10 from to for, so the 's for are expected to be larger than for. With respect to the ΛCDM model, the ΛWDM simulations have respectively about 1/5 or 1/9 the number of satellites for a parent of Milky Way mass.

We note that these results are in agreement with our conclusions.

In the warm dark matter scenario, the first galaxies to form have mass. For a larger “smoothing length”, increases, and the first galaxies form at a later time, i.e. at smaller redshift. Therefore, the redshift of the first few galaxies, or the redshift of re-ionization, allows a measurement of. The galaxy with highest spectroscopically confirmed redshift, called GN-z11, has . The quasar with highest spectroscopically confirmed redshift has . The redshift of re-ionization is [1]. We will take the redshift of formation of the first (few) galaxies to be (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 is approximately at, corresponding to a comoving wavelength. Therefore, simulations with very large L are required to include the contributions to the relative density perturbation 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, and extrapolating from Figure 15 we estimate, corresponding to.

Figure 16 and Figure 17 present distributions of galaxy stellar masses. The data is from the Sloan Digital Sky Survey SDSS DR15 [10]. The simulations help us identify the peak marked “M_sfs” corresponding to the cut-off wavenumber k_fs. We obtain the transition stellar mass (the latter uncertainty is an upper bound to the SDSS measurement uncertainty). 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”.

To the stellar mass we need to add the mass of gas to obtain the mass of baryons, and then add the dark matter mass to obtain. To make these transitions using data, we select a large relaxed spiral galaxy in the SPARC sample [11], i.e. UGC11914. The observed circular velocity of rotation at the last observed radius () is km/s. The corresponding velocity contributions from gas, stars in the disk, and stars in the bulge, are 29.29 km/s, 121.61 km/s, and 82.73 km/s, respectively (taking the stellar mass-to-light ratios to be for the disk, and for the bulge [5] [11] ). These velocities contribute in quadrature. We obtain , and. (An alternative determination of, is to multiply by, that obtains.)

Note, in Figure 17, the distribution of stripped down galaxies for.

The root-mean-square (rms) velocity of non-relativistic dark matter particles in the early universe, when density perturbations are relatively small, can be written in the form

where a is the expansion parameter. Equation (5) assumes that dark matter decouples from the Standard Model sector, and from self-annihilation, while density perturbations are still relatively small. However, dark matter-dark matter elastic scattering is allowed. The “adiabatic invariant” has been measured [5] by fitting 40 spiral galaxy rotation curves in the SPARC sample [11]. We take

The factor is a correction for possible dark matter rotation. We take [4] [5] [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. This range is also consistent with 10 galaxies in the THINGS sample [5] [13]. (The different normalizations used for baryons in our analysis of the SPARC and THINGS samples is discussed in Reference [5], and accounts for the difference of between the measurement in Reference [4] and the present result.) The expansion parameter at which the dark matter becomes non-relativistic is defined as

(not to be confused with in Appendix A). Note that is less than the expansion parameter at matter-radiation equality, , i.e. we are dealing with warm dark matter. This corresponds to (the second uncertainty is due to the uncertainty of), , and for fermion dark matter; or, , and for boson dark matter.

The adiabatic invariant determines the ratio of dark matter temperature-to-mass, so one more constraint is needed to obtain and separately. It turns out that, if we assume that dark matter has zero chemical potential, and decouples (from the Standard Model sector, and from self-annihilation) while still ultra-relativistic, then dark matter is also in thermal equilibrium with the Standard Model sector in the early universe for the measured values of and [4] [12] !

Dark matter with chemical potential is special. It corresponds to dark matter that was once in diffusive equilibrium with the Standard Model sector and decoupled (from the Standard Model sector, and from self-annihilation) while still ultra-relativistic, and/or dark matter composed of equal number densities of fermions and anti-fermions, or may be the case in scenarios with sterile Majorana neutrinos, or bosons.

The mass of dark matter particles with, and with the observed mean dark matter density of the universe, is presented in Equation (18) of Appendix B. For fermions with chemical potential,

For bosons with chemical potential,

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 annihilation, while dark matter is still ultra-relativistic, is [4]

for fermions with, and

for bosons with. The temperature ratio (10) corresponds to dark matter with decoupling from the Standard Model sector, and from self-annihilation, while still ultra-relativistic, in the approximate temperature range from to [12]. Note that assuming obtains thermal equilibrium with the measured values of and! Is this a coincidence? This result is consistent with Big Bang Nucleosynthesis [12].

Let us now consider dark matter that was in both diffusive and thermal equilibrium with the Standard Model sector. Diffusive equilibrium implies chemical potential. The thermal relic mass for warm dark matter with, that decouples from the Standard Model sector, and from self-annihilation, while still ultra-relativistic, is [1]

where () is the number of fermion (boson) degrees of freedom, and is the ratio of dark matter-to-photon temperatures after annihilation while dark matter is still ultra-relativistic. if dark matter decouples from the Standard Model sector, and from self-annihilation, in the temperature range from to. ranges from 0.344 for decoupling in the temperature range to, to, say, 0.424 for decoupling in the temperature range to [12]. Setting, for the case, and allowing to be in the range 0.424 to 0.344, obtains a measurement of that is independent of spiral galaxy rotation curves.

A summary of the four independent measurements is presented in Table 2 for both fermions and bosons. Note that setting, and assuming that is in the range from 0.344 to 0.424, obtains in agreement with three independent measurements in Table 2. This agreement is either a coincidence,

or is evidence that indeed, and dark matter was in both diffusive 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) for fermions, or (9) for bosons.

The case of negative, with large, with the same measured and, can not be ruled out, but is not compelling: it requires a coincidence of with the measured and, 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.

To understand warm dark matter, we find it convenient to classify galaxies, at any given redshift, 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, 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. Hierarchical galaxies have masses, and dominate galaxies with. Stripped down galaxies populate all masses, and, and are the only galaxies with. The “smoothing length” suppresses Hierarchical galaxies with, 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 and, 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 equilibrium with the Standard Model sector, and decoupled while still ultra-relativistic: measures, and, (this analysis has no input from spiral galaxy rotation curves),

3) The redshift of formation of first galaxies (measures), and

4) The distributions of masses of SDSS galaxies (measures).

The conclusions of these studies are:

• By construction, the generated galaxies are exactly the same for cold and warm dark matter for (if the same random number seed is used), so all successes of the ΛCDM model are preserved.

• For warm dark matter, the distribution of galaxy masses becomes suppressed for by a factor. From the simulations, is in the approximate range 0.05 to 0.4. The value of is sensitive to the size L of the simulation and to, and requires a more detailed implementation of the galaxy generator, and hence is uncertain. From data, can reach approximately 0.68 at, see Figure 16.

• The distributions of galaxies with are very different for the cold and warm dark matter scenarios. For warm dark matter there are huge empty voids, the stripped down galaxies with 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 and hence, while the last two measurements obtain. The relation is well satisfied, so the “smoothing length” 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. The warm dark matter scenario with this 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 in the range 10 Mpc⁻¹ to 21 Mpc⁻¹, in disagreement with the measurements presented in Table 2. Note that these limits depend on accurate simulations of the hydrogen or dark matter densities of non-linear stripped down galaxies at redshift. These discrepancies between the limits and our measurements of need to be resolved.

• Let us consider fermion dark matter with. From the excellent agreement of the four independent measurements presented in Table 2, we conclude that dark matter was once in thermal and diffusive equilibrium with the Standard Model sector, and decoupled (from the Standard Model sector, and from self-annihilation) while still ultra-relativistic, and in the approximate temperature range from to. These dark matter particles have a mass. The case of Majorana sterile neutrino warm dark matter was illustrated in Figure 11 of Reference [12].

The simulations in this article were made possible by the High Power Computing Center of Universidad San Francisco de Quito.

Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS web site is www.sdss.org.

SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard-Smithsonian Center for Astrophysics, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU)/University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatário Nacional/MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.

The author declares no conflicts of interest regarding the publication of this paper.

Hoeneisen, B. (2019) Simulations and Measurements of Warm Dark Matter Free-Streaming and Mass. International Journal of Astronomy and Astrophysics, 9, 368-392. https://doi.org/10.4236/ijaa.2019.94026

In this article we obtain form with

with, and 0.4115 for negative chemical potential, fermions with, and bosons with, respectively. For an early thermal relic the ratio of dark matter-to-photon temperatures after annihilation, while dark matter is still ultra-relativistic, is. Then, for early thermal relics, for fermions, and for bosons, see Appendix B. Note that we extended the integral in (1) to, when baryon acoustic oscillations freeze, and gravity from baryons dominate for. In TableB we compare this reference calculation of with several alternatives.

As an alternative to the reference calculation, we could have integrated up to, and obtained

with, and 0.4031 for, fermions with, and bosons with, respectively.

Reference [14] obtains a lower limit of 2500 eV (at) for the mass of thermal relic dark matter from a study of the Lyman-α forest of distant quasar light. The free-streaming cut-off wavenumber in Equation (2) of [14] is obtained from Equation (6) and Equation (7) of [15], which were obtained with a code that solves the Boltzmann equations.

Reference [16] obtains by solving the linearized collisionless Boltzmann-Vlasov equation exactly!

An interesting alternative is based on the Jeans length for a collisionless fluid, taken from Lecture Notes by Frank van den Bosch, Theory of galaxy formation, Yale University, fall 2018:

See TableB for a comparison of these alternative calculations of.

In Reference [4] we use the approximation valid for. In this Appendix we present the exact equations, valid for all. We obtain

where, and

(upper sign for fermions, lower sign for bosons). Then Equation (8) of Reference [4] becomes

and Equation (16) of Reference [4] becomes

For, we obtain for fermions, , , , , and ; and for bosons, , , , , and . Einstein condensation sets in at.

For we obtain for fermions, , , , , and ; and for bosons, , , , , and .

For we obtain for both fermions and bosons , , , , , and.