Cold or Warm Dark Matter?: A Study of Galaxy Stellar Mass Distributions

We compare the observed galaxy stellar mass distributions in the redshift range 0 11 z <  with expectations of the cold ΛCDM and warm ΛWDM dark matter models, and obtain the warm dark matter cut-off wavenumber: 0.44 1 fs 0.34 0.90 Mpc k + − − = . This result is in agreement with the independent measurements with spiral galaxy rotation curves, confirms that k fs is due to warm dark matter free-streaming, and is consistent with the scenario of dark matter with no freeze-in and no freeze-out. Detailed properties of warm dark matter can be derived from k fs . The data disfavors the ΛCDM model.


Introduction
Most current cosmological observations are well described by the cold dark matter ΛCDM model with only six independent parameters, and a few assumptions that are consistent with present observations: flat space, a cosmological constant, and scale invariant adiabatic primordial density perturbations [1]. This economical description of the universe is apparently in agreement with all observations on large scales, but seems to have tensions with some small scale phenomena: the "cusp vs core" problem of spiral galaxies, i.e. simulations obtain a cusp while observations find a core, and the "missing satellite" problem [2]. The ΛCDM model assumes that dark matter has a negligible free-streaming length. However, fits to spiral galaxy rotation curves obtain a non-negligible dark matter free-streaming length [3]. This free-streaming cuts off the power spectrum of linear density perturbations at a comoving wavenumber k fs . Adding this parameter to the ΛCDM model obtains the warm dark matter model (ΛWDM).
We compare the observed galaxy stellar mass distributions in the redshift range 0 11 z <  with expectations of the cold and warm dark matter models, and obtain the cut-off wavenumber k fs . The notation and cosmological parameters are as in Reference [1]. The outline of this article is as follows. In Section 2 we obtain predictions, based on the Press-Schechter formalism, of the stellar mass distributions for the cold and warm dark matter models. This formalism is valid only at redshifts 5 z  as discussed in Section 3. In Section 4 we present measurements of k fs by comparing predictions with data in the redshift range 5. 5 8.5 z   . Section 5 verifies the compatibility between predictions and the galaxy with largest observed spectroscopic redshift to date. We close with conclusions. k k τ is a cut-off factor. The cut-off is due to freestreaming of the warm dark matter particles.

Predictions of the Stellar Mass Distributions
In Reference [3] we consider a step-function cut-off factor. In that approximation, the first galaxies to form have the transition mass where fs fs 1.555 r k = . Galaxies with larger masses form bottom up by hierarchical clustering. Once saturation is reached, galaxies that would have formed with mass M may "not fit", loose mass to neighboring galaxies, and collapse with mass less than M . These are stripped down galaxies, they populate all masses, and are the only galaxies that form with mass less than fs M in the step function approximation [3].
In the present article we take ( ) ( ) This smooth cut-off is approximately the Born approximation of the calculation presented in Reference [5]. The true cut-off factor has a longer tail at large k than the Born approximation [5]. To study the effect of the tail, we also consider the cut-off factor All figures, except Figure 13, include the tail: its effect is relatively small. As we shall see in the following, the smooth cut-off results in bottom up hierarchical clustering, as in the ΛCDM model, up to saturation at redshift 5 z ≈ , and thereafter seems to become dominated by the generation of stripped down galaxies. Irregular "clumpy galaxies", that resemble beads on filaments or sheets [6], that are dynamically unstable and break up, may also contribute to the galaxy stellar mass function [6] [7]. International Journal of Astronomy and Astrophysics , at redshift z, is [4] ( ) ( ) ( ) ( ) σ fixes the normalization of (4).
The Press-Schechter stellar mass function [8] is obtained from (4) as follows.
The mass fraction locked up in halos with mass greater than M at redshift z is identified with the probability that the relative fluctuation of mass M exceeds 1.686: . This identification is valid so long as the galaxies do not break up, or loose mass to neighboring galaxies, and have time to cluster. The Press-Schechter stellar mass function is then obtained after some algebra, and the inclusion of a "fudge factor" 2 [8], justified in [9]: and crit m m ρ ρ ≡ Ω . Equation (7) is valid in the spherical collapse approximation.
A calculation that takes into account the average ellipticity and prolateness of perturbations, is the ellipsoidal collapse approximation, pioneered by R.K. Sheth and G. Tormen [10] [11], that replaces . Good fits to simulations are obtained with 0.84 ν ν =  [11]. The factor 0.84 depends on the algorithm used to identify the collapsed halos, e.g. on the "link length" of the "friends-of-friends" algorithm, and also on the simulation volume. We note that Equations (6), (7) and (8), have no free parameters, except k fs .

The Stellar Mass Distribution from SDSS Data
We analyze Sloan Digital Sky Survey (SDSS) data release DR16 [12]. We include all data in the right ascension range 145˚ to 230˚, and declination range 0˚ to 50˚. By eye inspection of each redshift bin of this sky patch, we see only mild extraneous features such as zones with different exposure. The galaxy properties, including stellar mass, stellar age, star formation rate (SFR), and metallicity, are obtained from the photon spectra in filters u, g, r, and i, by several stellar population synthesis (SPS) models. The results that we analyze are placed in the following SDSS DR16 classes: stellarMassFSPSGranWideDust [13], stellarMassStarformingPort [14], stellarMassPCAWiscBC03 [15], and stellarMassPCAWiscM11 [15]. The SPS of these classes are described in the cited references. The galaxy stellar mass distributions for these SPS are presented in Figures 4-7 s M is the galaxy stellar mass returned by the SPS. The reduction of the distributions at low mass are due to the relative luminosity threshold of the observations. To obtain the galaxy stellar mass functions it is still necessary to divide by the stellar mass completeness factor (which is over 80% at 0.6 z < , and decreases at higher z [16]). In Figure 4 we observe mass distributions that increase with redshift z at the high mass end. This top down evolution is also observed by the Dark Energy Survey (DES), see Figure 7 of Reference [17]. If we assume that the mass corresponding to a threshold factor 1/2 scales as the square of the luminosity distance, then the shift of the distributions to the right for 0.4 0.7 z < < should be even larger.    The top down evolution is observed even when the expected mass is replaced by the median mass minus one standard deviation, so the excess at high mass is not due to a statistical fluctuation. However, Figure 5 presents galaxy stellar mass distributions that do not change significantly with redshift. In Figure 6 and Figure 7 the evolution is slightly top down. In summary, at our current level of understanding, in the redshift range 0 0.7 z <  the galaxy stellar mass function either evolves top down, or is stationary within observational uncertainties.
Let us compare the observed stellar mass function at 0 z = , e.g. Figure 4, with the calculations in Figures 1-3. We find that at  may be due to the long time required for "dry" mergers of galaxies with little gas content. In conclusion, to measure fs k , we need to compare observations with calculations at 5 z  , before the saturation sets in. Note that the predictions become insensitive to fs k for fs M M > . Therefore, to measure fs k , we verify that prediction and data are in agreement for    Table 1), the contribution of correlated systematic uncertainties of the data obtained in Reference [18], ±0.15 Mpc −1 , Table 1. Measurements of the warm dark matter cut-off wavenumber k fs obtained from , approximations. The total uncertainties shown include statistical uncertainties, and systematic uncertainties estimated in [18].     [18] [19] [20] [21]. Note the onset of "saturation" at the high mass end (that is not understood).

Measurements of kfs from Stellar Mass
an uncertainty due to ( ) P k , ±0.2, and statistical uncertainties, we obtain our final measurement: . This result is insensitive to the "tail" in (3).
(Note: The present measurement of k fs superceeds the estimate in Reference [3] that was based on data in SDSS DR15, class stellarMassFSPSGranWideDust that shows strong top down galaxy evolution, see Figure 4.)

Estimate of kfs from Galaxy GN-z11
The galaxy with largest spectroscopically confirmed redshift to date is GN-z11 with 0.08 0.12 11.09 z + − = [22]. Its stellar mass is estimated to be 9 10 s M M ≈  . One such galaxy was found in a comoving search volume 6 3 1.2 10 Mpc V = × , for 1 z ∆ =. Figure 12 compares this single galaxy with expectations corresponding to the cut-off factor (3). To illustrate the effect of the cut-off factor tail, Figure 13 presents the expectations corresponding to the gaussian cut-off factor (2). From this single galaxy we obtain

Conclusion
Comparing measurements of stellar mass distributions of galaxies in the redshift range 5.5 8.5 z   with expectations, we obtain the warm dark matter cut-off wavenumber . This result is in agreement with the independent measurements obtained by fitting spiral galaxy rotation curves (demonstrating that the cut-off k fs is due to warm dark matter free-streaming), and is consistent with the scenario of dark matter with no freeze-in and no freeze-out, see Table 2 [3] [23] [24] [25]. Detailed properties of warm dark matter can be derived from k fs [3]. The observed stellar mass functions disfavor the ΛCDM model. , compared with one observed galaxy GN-z11 (assuming one similar galaxy per dex) [22]. The cut-off factor is given in Equation (3). This graph obtains k fs of the order of 1.1 Mpc −1 . Figure 13. Same as Figure 12, but with the gaussian cut-off factor (2).