Comments on Warm Dark Matter Measurements and Limits

Observed spiral galaxy rotation curves allow a measurement of the warm dark matter particle velocity dispersion and mass. The measured thermal relic mass 100 h m ≈ eV is in disagreement with limits, typically in the range 1 to 4 keV. We review the measurements, update the no freeze-in and no freeze-out scenario of warm dark matter, and try to identify the cause of the discrepancies between measurements and limits.


Introduction
The Λ cold dark matter (ΛCDM) cosmological model, based on just 6 parameters [1], is in spectacular agreement with precision measurements of the cosmic microwave background anysotropies, the power spectrum of large scale density fluctuations, and baryon acoustic oscillations. The ΛCDM model may have tensions with observations on scales smaller than the Galaxy, e.g. too few satellites of the Milky Way and Andromeda, and galaxies with cores instead of the cusps expected from simulations. An extension of the ΛCDM cosmology that addresses the small scale tensions is Λ warm dark matter (ΛWDM) that assumes that the dark matter has a non-negligible velocity dispersion. Since the non-relativistic velocity dispersion is the mean density of dark matter at the present time (we use the standard notation in cosmology, and astrophysical constants, as in [1]).
Let ( ) P k be the comoving power spectrum of linear relative density perturbations in the ΛCDM model after decoupling [2]. k is the comoving wavenumber. The corresponding power spectrum in the ΛWDM cosmology is is a cut-off factor due to warm dark matter freestreaming. The cut-off factor, at time eq t when matter begins to dominate, has the approximate form (see figure 5 of [3] k t is the comoving Jeans wavenumber at matter-radiation equality. This solution corresponds to adiabatic, i.e. thermal, initial fluctuations. After eq t , 3 2 J M a − ∝ decreases allowing non-linear regeneration of small scale structure.
An alternative fs k , obtained from simulations, is given in [4].
In summary, the ΛWDM extension of the ΛCDM model adds one parameter: the velocity dispersion  (6) and (7) of [4]). One purpose of the present study is to try to understand what may have gone wrong.

The No Freeze-In and No Freeze-Out Scenario
The measured adiabatic invariant is sufficient to acquire the NRTE in a very short time scale relative to the age of the universe, assuming quasi-degenerate dark matter [14]. For simplicity, we consider a single dark matter species.
Let h T T be the temperature ratio of dark matter and photons after decoupling of neutrinos, and after e e + − annihilation, and before dark matter becomes non-relativistic. This ratio is at decoupling of dark matter from the Standard Model sector [1].
for the dark matter [14]. Then, at the present time, determines the dark matter particle mass corresponding to no freeze-in and no freeze-out.
The expansion of dark matter, and the transition from the URTE momentum distribution to the NRTE momentum distribution, is assumed to occur with constant number of particles, i.e. the particle number density scales as 3 a − , and (arguably) with constant entropy, see [14] for full details. For example, the number density of dark matter particles is calculated as follows: where the particle energy is v T µ , and the dimensionless entropy per particle s k [14]. We note that these equations are valid for the entire range of h T , from non-relativistic . This scenario implies that dark matter has a dimensionless entropy per particle: 4.202 s k = for fermions, and 3.601 s k = for bosons, and that non-relativistic dark matter acquires a negative chemical potential µ [14]. For where h T T is the dark matter-to-photon temperature ratio after e e + − annihilation, and before dark matter becomes non-relativistic. Equation (12) is obtained from (11), (7) and (9). For bosons, from Equation (28) of [14], we obtain ( ) Note that the measurement of the adiabatic invariant  [15].
c Ω and 0 T . The (arguably) simplest extensions of the Standard Model for the present scenario, that include scalar, vector or spinor warm dark matter particles, are presented in [15].
Comments: Equations (11) to (14) update Equations (15) to (18) of [13] to the present scenario. Reference [13] assumes non-interacting dark matter (except for gravity), does not consider the URTE to NRTE transition, and assumes zero chemical potential for non-relativistic dark matter. Table 1 updates table 4 of [13]. Note that in Table 1 we no longer distinguish fermion from boson dark matter because they become indistinguishable (with the current level of precision) due to their negative non-relativistic chemical potential in the present scenario.  [14], in addition to the small scale power suppression factor ( ) 2 k τ due to free-streaming. Galaxies may form adiabatically without requiring relaxation or virialization [8]. The galaxy "virialized" mass (usually measured in simu-  [17], and smaller galaxies are "stripped-down" during their formation as they loose matter to neighboring galaxies [6]. This may be the main mechanism of the non-linear regeneration of the small scale structure.

A New Paradigm
In the ΛCDM scenario the first dark matter halos to collapse have arbitrarily

Comments on the Press-Schechter Galaxy Mass Distribution
The derivation of the Press-Schechter galaxy mass distribution [18], and its Sheth-Tormen ellipsoidal collapse extensions [19] [20], are valid for the hierarchical structure formation of the ΛCDM model, and for redshift 4.5 z  before saturation sets in. The derivation of the Press-Schechter relation is based on the variance of the linear relative density perturbation on the linear mass scale h M at redshift z [2]: f is a correction due to the accelerated expansion of the universe:  (17) ( ) 2 4 k W k π is ill-behaved: it is oscillatory and does not converge, and is not suited for warm dark matter with a cut-off factor For warm dark matter, the usual choice of window function, is a sharp cut-off of k at 0 k , i.e. [22]. The appropriate linear mass scale is where 2.7 c ≈ is calibrated with simulations at 0 z ≈ [21]. The Fourier transform of ( ) ( ) 2 4 r W r π is ill-behaved: it is oscillatory and does not converge, and has an ill defined volume in r-space.
Another window function that is considered [13] [23] is the Gaussian: with ( )  (2), is a good description of the data. Whether, or not, it is also a good description of warm dark matter is another question.
A comparison of the distributions with Gaussian and sharpk window functions is presented in Figure 1. With the Gaussian window function excellent agreement with the data is obtained, and fs k is measured:  (2) and (3) is valid at eq t . Free-streaming continues after eq t , but is complicated by gravity and by non-linear regeneration of small scale structure by the time of the formation of first galaxies. During their formation, proto-galaxies may loose matter to neighboring galaxies, or break up, and populate the low mass tail [6] [24]. By 3 z = there remains little memory of ( ) 2 k τ at eq t , see figure 2 of [5]. This regeneration has been described by Equations (13) and (14) of [25], and adds a long tail to   [23] for details), with Gaussian (top) and sharp-k (bottom) window functions. To match prediction to data at the high mass end we have set

Simulations
The Press-Schechter formalism with Gaussian window function, describes the observed stellar mass distributions at 8, 7, 6, 4.5 z = , and even 3 [13]. But does it describe warm dark matter? To investigate this question, we generate galaxies with a simple generator described in [16] and [17]. Briefly, we apply periodic International Journal of Astronomy and Astrophysics boundary conditions in a cube of comoving side 300 L = Mpc at a given ex- π (which is different from the often used "virialized" mass). A galaxy "fits" if it does not overlap previously generated galaxies. The integer I is then increased by 1 and galaxies of a smaller generation are formed. Note that a galaxy that did not fit at generation I may fit at a "generation" with larger I, and hence be created with a reduced mass. These are "stripped down" galaxies that have lost matter to neighboring galaxies in the course of their formation [6]. This is a simplified way to regenerate small scale structure in the warm dark matter scenario.
To compare the simulations with data it is necessary to make the transition from the "linear" halo mass h M to the stellar mass * M . Here we approximate the transformation as a fixed factor, which is adjusted so that data and simulation agree at the high stellar mass end (this factor is very sensitive to the power spectrum normalization 2 R ∆ ). The results for 4.5 z = , and 6 are presented in Figure 2, upper panels. The simulations shown are ΛCDM, ΛWDM with The bottom left panel of Figure 2 shows the cut-off factors ( ) 2 k τ of (2) and (22), as well as the "linear" cut-off factor (6) and (7) of [4], and the "non-linear" regenerated cut-off factor of (13) and (14) of [25].
An alternative way to estimate the stellar mass is to obtain the galaxy dark matter particle 1-dimensional dispersion velocity as where the factor ≈1.2 is calibrated from simulations described in [8], and ( ) J a λ is the proper Jeans length. The stellar luminosity in the R band is then obtained from the Tully-Fisher relation [26] Finally, to obtain the stellar mass The comparison is shown in Figure 2 bottom right panel.
We conclude that the data are in agreement with the Press-Schechter prediction with the Gaussian window function [13], and are also in agreement with B. Hoeneisen International Journal of Astronomy and Astrophysics , and the same plus a "tail", see equation (22). For references to the original data points see [13]. Bottom left: warm dark matter cut-off factor ( ) simulations of ΛWDM if the Gaussian ( ) 2 k τ develops a "tail". This tail need not be of primordial origin: it may be due to limitations of current galaxy generators. Let us mention that the simulations in [21] and [24] are warm dark matter only, they do not include baryons, and baryons act as a cold "tail". The needed extra tail is much smaller than the tail that current generators already obtain [5] [25]. The simulations in [24] include proto-galaxies, and go a long way between simulations in [21] (similar to the Press-Schechter prediction with a sharpk window function) and data. In any case, the "tail" needs to be in place by 12 z ≈ in order to obtain timely reionization even for

Comments on the Lyman-α Forest
Reionization is beautifully illustrated in [28]. During the dark ages the universe baryonic matter is mostly neutral hydrogen and neutral helium. At about 12 z ≈ , first quasars begin ionizing and heating the gas in bubbles expanding away from the quasar. As a result, peaks in the dark matter density correspond to minimums of the neutral hydrogen density. The bubbles overlap at about 6 z = .
Light from quasars is observed to have a "forest" of absorption lines due to The inter-galactic dark matter resembles a honey-comb of voids, surrounded by sheets, that meet at filaments, that meet at spheroidal nodes. Most of the Galaxies form in nodes and filaments.
"For a characteristic hydrogen number density of 55 m −3 , corresponding to 5 times the mean baryon density at 3 z = , and a characteristic temperature of 2 × 10 4 K, the pressure is 10 6 K·m −3 , and the baryon Jeans length is 320 kpc" [29]. In comparison, the warm dark matter Jeans length, at 3 z = with rms 0.67 The relation between the 1-dimensional and 3-dimensional flux power spectrum is [29] ( ) ( ) for z k in the range 0.0013 to 0.08 s/km, see Figure 3.
We consider 5.4 z = , so the range of z k is 0.12 to 7.  in [25] or [30] decrease with decreasing z , while the proper dark matter density power spectrum has the opposite behaviour?

Conclusions
A detailed no freeze-in and no freeze-out warm dark matter scenario has emerged from fits to approximately 60 relaxed spiral galaxy rotation curves, and from measurements of galaxy stellar mass distributions. The resulting thermal relic dark matter mass is of order 100 h m ≈ eV, depending on the spin and decoupling temperature of the dark matter particles, see Table 1 to be the warmest dark matter that may be consistent with reionization.
It is significant that the Particle Data Group quotes lower limits to the dark matter particle mass of 70 eV for fermions, and 10 −22 eV for bosons [1]. It is also interesting to note that the onset of degeneracy spoils the fits to spiral galaxy rotation curves, obtaining lower limits of 48 eV for fermions, see figure 5 of [13], and 45 eV for bosons [9]. We find that different treatments of the non-linear regeneration of small scale structure, within current uncertainties, may change a measurement of h m into a limit, and vice verse, see Figures 1-3. The core of galaxies is (arguably) evidence that dark matter is warm [8]. New studies, with data and simulations, are needed to better understand the warm dark matter tail.
These simulations need to have sufficient resolution to reliably generate "stripped down" galaxies, and need to include the baryon physics to obtain the observable stellar luminosity * L , as well as first stars to understand reionization.

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