_{1}

^{*}

We compare the observed galaxy stellar mass distributions in the redshift range
*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.

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 [_{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 < z ≲ 11 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 [

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 z ≳ 5 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 ≲ z ≲ 8.5 . Section 5 verifies the compatibility between predictions and the galaxy with largest observed spectroscopic redshift to date. We close with conclusions.

Let P ( k ) be the power spectrum of linear density perturbations in the cold dark matter ΛCDM model as defined in Reference [

In Reference [

M fs = 4 3 π r fs 3 Ω m ρ crit , (1)

where r fs = 1.555 / k fs . 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 M fs in the step function approximation [

In the present article we take

τ 2 ( k / k fs ) = e x p ( − k 2 / k fs 2 ) . (2)

This smooth cut-off is approximately the Born approximation of the calculation presented in Reference [

τ 2 ( k / k fs ) = { e x p ( − k 2 / k fs 2 ) if k ≤ k fs , e x p ( − k 2 / k fs 2 ) ⋅ k / k fs if k > k fs . (3)

All figures, except

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 z ≈ 5 , and thereafter seems to become dominated by the generation of stripped down galaxies. Irregular “clumpy galaxies”, that resemble beads on filaments or sheets [

The mean of the square of the fractional mass fluctuation in a sphere of comoving radius r 0 = 1.555 / k 0 (smoothed by a gaussian window function), and mass M ≡ 4 π r 0 3 Ω m ρ crit / 3 , at redshift z, is [

σ 2 ( M , z ) = f 2 ( 2 π ) 3 ( 1 + z ) 2 ∫ 0 ∞ 4 π k 2 d k P ( k ) e x p ( − k 2 k fs 2 ) e x p ( − k 2 k 0 2 ) , (4)

while density perturbations are still linear. For simplicity, we have assumed the cut-off factor (2). f is a correction due to the cosmological constant; f = 1,1.257,1.275 for z = 0,2,11 , respectively [

The Press-Schechter stellar mass function [

F ( M , z ) = 1 2 erfc ( ν 2 ) , (5)

where ν ≡ 1.686 / σ ( M , z ) . Then − ( ∂ F ( M , z ) / ∂ M ) d M is identified with the mass fraction in halos with masses between M and M + d M . 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 [

d n d l n M = ρ m M d l n ( σ − 1 ) d l n M f PS ( ν ) , (6)

where

f PS ( ν ) = 2 π ν e x p ( − ν 2 2 ) , (7)

and ρ m ≡ Ω m ρ crit . 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 [

f EC ( ν ) = 0.322 [ 1 + ν ˜ − 0.6 ] f PS ( ν ˜ ) , (8)

with ν ˜ = ν . Good fits to simulations are obtained with ν ˜ = 0.84 ν [_{f}_{s}.

Figures 1-3 present galaxy stellar mass function calculations for the ΛCDM model, and for ΛWDM with k fs = 1.6 Mpc − 1 and 0.8 Mpc − 1 , respectively . We have converted from the halo mass M to the stellar mass M s as follows: log 10 M s = log 10 M − 0.63 ± 0.19 [

We analyze Sloan Digital Sky Survey (SDSS) data release DR16 [^{3}). M s 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 z < 0.6 , and decreases at higher z [

In

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,

Let us compare the observed stellar mass function at z = 0 , e.g. ^{−}^{1} and 0.8 Mpc^{−}^{1}, respectively. At these z sat for M s = 10 10 M ⊙ the probability F ( M , z ) is of order 0.01, stripped down galaxies form, and the Press-Schechter formalism breaks down. Galaxy merging requires dissipation. The “saturation” observed at M s = 10 12 M ⊙ may be due to the long time required for “dry” mergers of galaxies with little gas content. In conclusion, to measure k fs , we need to compare observations with calculations at z ≳ 5 , before the saturation sets in.

Note that the predictions become insensitive to k fs for M > M fs . Therefore, to measure k fs , we verify that prediction and data are in agreement for M > M fs . For future convenience, l o g 10 ( M s fs / M ⊙ ) ≈ l o g 10 ( M fs / M ⊙ ) − 0.63 = 10.5,10.9,11.5 for k fs = 1.6,1.2,0.8 Mpc − 1 , respectively.

Reference [_{fs} summarized in

Taking the Ellipsoidal Collapse model with ν ˜ = 0.84 ν as the preferred prediction with an uncertainty Δ k fs = − 0.1 + 0.3 Mpc − 1 (see ^{−}^{1},

z | k fs [ Mpc − 1 ] Press-Schechter | k fs [ Mpc − 1 ] Ellipsoidal collapse, ν | k fs [ Mpc − 1 ] Ellipsoidal collapse, 0.84 ν |
---|---|---|---|

≈ 8 | 1.10 ± 0.30 | 1.10 ± 0.40 | 0.80 ± 0.30 |

≈ 7 | 1.10 ± 0.30 | 1.25 ± 0.35 | 0.85 ± 0.25 |

≈ 6 | 1.10 ± 0.30 | 1.25 ± 0.35 | 0.80 ± 0.30 |

an uncertainty due to P ( k ) , ±0.2, and statistical uncertainties, we obtain our final measurement: k fs = 0.90 − 0.34 + 0.44 Mpc − 1 . This result is insensitive to the “tail” in (3).

(Note: The present measurement of k_{fs} superceeds the estimate in Reference [

The galaxy with largest spectroscopically confirmed redshift to date is GN-z11 with z = 11.09 − 0.12 + 0.08 [

Comparing measurements of stellar mass distributions of galaxies in the redshift range 5.5 ≲ z ≲ 8.5 with expectations, we obtain the warm dark matter cut-off wavenumber k fs = 0.90 − 0.34 + 0.44 Mpc − 1 . 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 _{fs} [

Fermions Observable | v h rms ( 1 ) [ km / s ] | a ′ h NR × 10 6 | m h [ eV ] | k fs [ Mpc − 1 ] | l o g 10 ( M fs / M ⊙ ) |
---|---|---|---|---|---|

Spiral galaxies | 0.76 ± 0.29 | 2.54 ± 0.97 | 79 − 17 + 35 | 0.80 − 0.24 + 0.42 | 12.08 ± 0.50 |

No freeze-in/-out | 0.81 − 0.25 + 0.47 | 2.69 − 0.84 + 1.57 | 75 ± 23 | 0.76 ± 0.31 | 12.14 ± 0.52 |

M s distribution | 0.90 − 0.34 + 0.44 | 11.93 ± 0.56 | |||

Bosons Observable | v h rms ( 1 ) [ km / s ] | a ′ h NR × 10 6 | m h [ eV ] | k fs [ Mpc − 1 ] | l o g 10 ( M fs / M ⊙ ) |

Spiral galaxies | 0.76 ± 0.29 | 2.54 ± 0.97 | 51 − 11 + 22 | 0.51 − 0.15 + 0.28 | 12.66 ± 0.50 |

No freeze-in/-out | 0.26 − 0.08 + 0.16 | 0.88 − 0.28 + 0.52 | |||

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-IV acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS web site is http://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. (2020) Cold or Warm Dark Matter?: A Study of Galaxy Stellar Mass Distributions. International Journal of Astronomy and Astrophysics, 10, 57-70. https://doi.org/10.4236/ijaa.2020.102005