Two apparent problems with the cold dark matter ΛCDM cosmology are the “Missing Satellites Problem”, and the need of a rest-frame ultra-violet (UV) luminosity cut-off. The “Missing Satellites Problem” is the reduced number of observed Local Group satellites compared to the number obtained in ΛCDM simulations [1] . A UV luminosity cut-off is needed to not exceed the reionization optical depth τ = 0.054 ± 0.007 measured by the Planck collaboration [2] [3] [4] [5] . In the present study we consider warm dark matter as a possible solution to both problems.

The Press-Schechter formalism, when applied to warm dark matter, includes the free-streaming cut-off, but not the “velocity dispersion cut-off”, and is therefore only valid for total (dark matter plus baryon) linear perturbation masses M greater than the velocity dispersion cut-off mass M_vd (to be explained below). The purpose of the present study is to extend the Press-Schechter prediction to M < M vd , and compare this extension with the “Missing Satellites Problem”, and with the needed UV luminosity cut-off.

We continue the study of warm dark matter presented in [6] . Our point of departure is Figure 1 of [6] . Here we reproduce the panel corresponding to redshift z = 6 in Figure 1 (with one change: instead of the Gaussian window function in [6] , in the present article we use the sharp-k window function throughout, with mass parameter c = 1.555 as explained in [6] ). Figure 1 compares distributions, i.e. numbers of galaxies per decade (dex) and per Mpc³, of galaxy linear total (dark matter plus baryon) perturbation masses M, stellar masses M * , and rest-frame ultra-violet (UV) luminosities ν L UV , with the Press-Schechter prediction [7] , and its Sheth-Tormen ellipsoidal collapse extensions with parameter ν ≡ 1.686 / σ (not to be confused with the frequency above) and 0.84 ν [8] [9] . The data on M * is obtained from [10] [11] [12] [13] . The data on ν L UV , where ν is the frequency corresponding to wavelength 1550 Å, is obtained from [4] [14] [15] [16] . The UV luminosities have been corrected for dust extinction as described in [4] [17] . The predictions depend on the warm dark matter free-streaming comoving cut-off wavenumber k fs ( t eq ) , and the comparisons of predictions with

data provide a measurement of k fs ( t eq ) , see [6] for full details. In Figure 1 the predictions extend down to the velocity dispersion cut-offs indicated by red, blue and green dots [6] [18] . The purpose of the present study is to extend the predictions to smaller M * and ν L UV , and thereby address the “Missing Satellites Problem”, and the UV luminosity cut-off, respectively.

To obtain a self-contained article, we need to define the warm dark matter adiabatic invariant v h rms ( 1 ) , and the free-streaming cut-off factor τ 2 ( k ) . We consider non-relativistic warm dark matter to be a clasical (non-degenerate) gas of particles, as justified in [19] [20] . Let v h rms ( a ) be the root-mean-square velocity of non-relativistic warm dark matter particles in the early universe at expansion parameter a . As the universe expands it cools, so v h rms ( a ) decreases in proportion to a − 1 (if dark matter collisions, if any, do not excite particle internal degrees of freedom [21] ). Therefore,

v h rms ( 1 ) = v h rms ( a ) a = v h rms ( a ) [ Ω c ρ crit ρ h ( a ) ] 1 / 3 , (1)

is an adiabatic invariant. ρ h ( a ) = Ω c ρ crit / a 3 is the dark matter density. The warm dark matter velocity dispersion causes free-streaming of dark matter particles in and out of density minimums and maximums, and so attenuates the power spectrum of relative density perturbations ( ρ ( x ) − ρ ¯ ) / ρ ¯ of the cold dark matter ΛCDM cosmology by a factor τ 2 ( k ) . k is the comoving wavenumber of relative density perturbations. At the time t eq of equal radiation and matter densities, τ 2 ( k ) has the approximate form [22]

τ 2 ( k ) ≈ exp [ − k 2 / k fs 2 ( t eq ) ] , (2)

where the comoving cut-off wavenumber, due to free-streaming, is [22]

k fs ( t eq ) = 1.455 2 4 π G ρ ¯ h ( 1 ) a eq v h rms ( 1 ) 2 . (3)

After t eq , the Jeans mass decreases as a − 3 / 2 , so τ 2 ( k ) develops a non-linear regenerated “tail” when the relative density perturbations approach unity [23] . We will take τ 2 ( k ) , at the time of galaxy formation, to have the form

τ 2 ( k ) = { exp ( − k 2 k fs 2 ( t eq ) )     if   k < k fs ( t eq ) , exp ( − k n k fs n ( t eq ) )     if   k ≥ k fs ( t eq ) . (4)

The parameter n allows a study of the effect of the non-linear regenerated tail. If n = 2 , there is no regenerated tail. Agreement between the data and predictions, down to the velocity dispersion cut-off dots in Figure 1, is obtained with n in the approximate range 1.1 to 0.2 [6] .

A comment: In (4) we should have written k fs ( t gal ) instead of k fs ( t eq ) , where t gal is the time of galaxy formation. However, the measurement k fs ( t gal ) = 2.0 − 0.5 + 0.8   Mpc − 1 , with galaxy UV luminosity distributions and galaxy stellar mass distributions [6] , is in agreement with the measurement of k fs ( t eq ) = 1.90 ± 0.32   Mpc − 1 with dwarf galaxy rotation curves (from the measurement of the adiabatic invariant v h rms ( 1 ) = 0.406 ± 0.069   km / s in [21] , and Equation (3)). So we do not distinguish k fs ( t gal ) from k fs ( t eq ) (until observations require otherwise).

Let us now consider the velocity dispersion cut-off. In the ΛCDM scenario, when a spherically symmetric relative density perturbation ( ρ ( x ) − ρ ¯ ) / ρ ¯ reaches 1.686 in the linear approximation, the exact solution diverges, and a galaxy forms. This is the basis of the Press-Schechter formalism. The same is true in the warm dark matter scenario if the linear total (dark matter plus baryon) perturbation mass M exceeds the velocity dispersion cut-off M vd0 . For M < M vd0 , the galaxy formation redshift z is delayed by Δz due to the velocity dispersion. This delay Δz is not included in the Press-Schechter formalism. Δz is obtained by numerical integration of the galaxy formation hydro-dynamical equations, see [18] . The velocity dispersion cut-off mass M vd , indicated by the dots in Figure 1, corresponds, by definition, to Δ z = 1 . The values of M vd are presented in [6] . For M > M vd 0 = 10 0.67 M vd we take Δ z = 0 . For M < M vd 0 = 10 0.67 M vd we may approximate Δ z ≈ 1.5 [ log 10 ( M vd / M ⊙ ) + 0.67 − log 10 ( M / M ⊙ ) ] . The values of log 10 ( M vd / M ⊙ ) are summarized in Table 1.

The Press-Schechter prediction, and its extensions, are based on the variance σ 2 ( M , z , k fs ( t eq ) , n ) of the linear relative density perturbation ( ρ ( x ) − ρ ¯ ) / ρ ¯ at the total (dark matter plus baryon) mass scale M [6] [24] . This variance depends on the redshift z of galaxy formation, and on the parameters k fs ( t eq ) and n of the free-streaming cut-off factor τ 2 ( k ) of (4). Comparison of predictions and data for M > M vd 0 obtain a measurement of k fs ( t eq ) , see Figure 1, and [6] . The extension of the predictions to M < M vd 0 depends on two cut-offs: the free-streaming cut-off (through the parameters k fs ( t eq ) = 2.0 − 0.5 + 0.8   Mpc − 1 , that is already fixed by the measurements in [6] , and n), and the velocity dispersion cut-off. We illustrate the effect of n in Figure 2, without applying the velocity dispersion cut-off yet. The velocity dispersion cut-off is implemented by replacing σ 2 ( M , z , k fs ( t eq ) , n ) by σ 2 ( M , z + Δ z , k fs ( t eq ) , n ) , with Δz obtained from Table 1. We illustrate the effect of both n, and the velocity dispersion cut-off, in Figures 3-5, for galaxy formation at z = 8 , 6 , and 4, respectively.

The linear perturbation total (dark matter plus baryon) mass scale M, of the Press-Schechter formalism, cannot be measured directly. We find that the flat

portion of the rotation velocity of test particles in spiral galaxies, V flat , can be used as an approximate proxy for M.

Given M, the galaxy formation redshift z, and v h rms ( 1 ) , it is possible to obtain V flat by numerical integration of the galaxy formation hydro-dynamical equations [18] . Results for M = 2 × 10 10 M ⊙ are presented in Table 2. We note that for galaxy formation at redshift z between 6 and 10, and free-streaming cut-off wavenumber k fs ( t eq ) between 1 and 1000 Mpc⁻¹, we may approximate

V flat ≈ 45   km / s . Similarly, for several masses M, the corresponding rotation velocities V flat are summarized in Table 3. The data in Table 3 can be fit by the relation

M M ⊙ ≈ 2.1 × 10 8 ( V flat 10   km / s ) 3 , (5)

as shown in Figure 6. This becomes the Tully-Fisher relation, once M / M ⊙ is replaced by ≈ 10 1.5 M * / M ⊙ ≈ 10 1.5 L * / L ⊙ , see Figure 1.

L * L ⊙ ≈ 2.4 × 10 10 h − 2 ( V flat 200   km / s ) 3 , (6)

with h = 0.674 . L * is the stellar luminosity. The average bolometric luminosity of the sun is M ⊙ ≡ − 2.5 ⋅ log 10 ( L ⊙ / L ref ) = 4.74 , so (6) becomes

M bol ≈ − 3.95 + 5 ⋅ log 10 ( h ) − 7.5 ⋅ log 10 ( V flat km / s ) . (7)

This approximate relation, obtained from first principles, can be compared with the empirical Tully-Fisher relation (see Figure 1 of [25] with Δ V ≈ 3 V flat ).

The “Missing Satellites Problem” of the cold dark matter ΛCDM cosmology is described in [1] . The approximate number of observed satellites within 200 h − 1 kpc of the Local Group, per Mpc³, with V flat > V is [1]

n obs ( V flat > V ) ≈ 385 h 3 ( 10   km / s V ) 1.3 Mpc − 3 , (8)

while the corresponding number in the ΛCDM simulations is [1]

n sim ( V flat > V ) ≈ 5000 h 3 ( 10   km / s V ) 2.75 Mpc − 3 . (9)

The difference between (8) and (9) illustrates the “Missing Satellites Problem” of the cold dark matter ΛCDM cosmology. The ratio of simulation to observation at each V flat is

d n sim / d V flat d n obs / d V flat ≈ 5000 × 2.75 385 × 1.3 ( 10   km / s V flat ) 1.45 , (10)

for satellites within 200 h − 1 kpc of the Local Group. Similarly, for satellites within 400 h − 1 kpc of the Local Group, the ratio is [1]

d n sim / d V flat d n obs / d V flat ≈ 1200 × 2.75 55 × 1.4 ( 10   km / s V flat ) 1.35 . (11)

These ratios are approximately equal to 1 at V flat ≈ 70   km / s , corresponding to M ≈ 10 10.9 M ⊙ , see (5). These ratios are approximately equal to 14 at V flat ≈ 20   km / s , corresponding to M ≈ 10 9.2 M ⊙ , see (5).

Let us now consider the warm dark matter ΛWDM cosmology. We proceed as follows for each of the panels in Figures 3-5. We shift the ΛCDM prediction to the left until agreement with the data is obtained at log 10 ( x ) = 10.9 , where x is M / M ⊙ , or 10 1.5 M * / M ⊙ , or ν L UV / L ⊙ . We then follow the shifted ΛCDM prediction to log 10 ( x ) = 9.2 , and compare with the data. If the corresponding ratio is in the approximate range 14 to 7 (to account for satellites found since the publication of [1] ), and a good fit is obtained with k fs ( t eq ) = 2.0 − 0.5 + 0.8   Mpc − 1 [6] , we regard the parameter n of the prediction to be “good”. If there is some tension, we clasify n as “fair”. A summary is presented in Table 4. We conclude that for warm dark matter with 0.3 ≲ n ≲ 0.8 , the predicted and observed number of satellites are in agreement, for galaxies formed with redshift z ≳ 6 .

Reionization begins in earnest at z ≈ 8 , and ends at z ≈ 6 . For each panel of Figure 3, corresponding to z = 8 , we integrate numerically the UV luminosity along the appropriate ellipsoidal collapse prediction (with parameter 0.84 ν ), that obtains excellent agreement with the data. The following procedure is followed in [4] : the observed UV luminosity distribution is extended (without the Δz velocity dispersion cut-off) to an assumed sharp UV magnitude cut-off M_UV, and the corresponding reionization optical depth τ is calculated. Here we obtain the equivalent sharp UV magnitude cut-off M_UV, and then the corresponding reionization optical depth τ from [4] . The results are summarized in Table 5. We note that, for the range 0.5 ≲ n ≲ 0.8 , we obtain agreement with the measured reionization optical depth τ = 0.053 ± 0.007 obtained by the Planck collaboration [2] [3] .

Comparisons of the rest frame galaxy UV luminosity distributions, and galaxy stellar mass distributions, with predictions for M > M vd , obtain the free-streaming cut-off wavenumber k fs ( t eq ) = 2.0 − 0.5 + 0.8   Mpc − 1 , with the non-linear regeneration of small scale structure parameter n in the wide approximate range 0.2 to 1.1 [6] . In the present work we have extended the predictions of the warm dark matter ΛWDM cosmology to M < M vd , including the free-streaming cut-off (4), and the velocity dispersion cut-off of Table 2. This extension is in agreement with the number of satellites of galaxies formed at z ≳ 6 , and with the needed UV cut-off (to not exceed the observed reionization optical depth), with n in the approximate range 0.5 to 0.8.

As a cross-check, we have obtained the adiabatic invariant in the core of dwarf galaxies dominated by dark matter, from their rotation curves. The result is v h rms ( 1 ) = 0.406 ± 0.069   km / s [21] , corresponding to a free-streaming cut-off wavenumber k fs ( t eq ) = 1.90 ± 0.32   Mpc − 1 , from Equation (1). This result confirms: 1) that the adiabatic invariant in the core of galaxies is of cosmological origin, as predicted for warm dark matter [18] , since several galaxies accurately share the same adiabatic invariant, and 2) confirms that k fs ( t eq ) is due to free-streaming. All of these results are data driven.

As a by-product of this study we obtain approximately the empirical Tully-Fisher relation from first principles, by integrating numerically the galaxy formation hydro-dynamical equations [18] . These hydro-dynamical equations predict that the core of first galaxies form adiabatically if dark matter is warm, i.e. conserves v h rms ( 1 ) .

Omitting the non-linear regeneration of small scale structure, i.e. setting n = 2 , or using the similar τ 2 ( k ) from the linear Equation (7) of [26] , and omitting the velocity dispersion cut-off, obtains strong disagreement with observations. These omissions have led several published studies to obtain lower warm dark matter particle “thermal relic mass” limits of several keV. Note that nature, and simulations [23] , re-generate non-linear small scale structure when relative density perturbations approach unity. May I suggest that these mass limits be revised, including the non-linear regeneration of small scale structure, and the velocity dispersion cut-off. We note that the Particle Data Group’s “Review of Particle Physics (2022)” quotes lower limits of 70 eV for fermion dark matter, or 10⁻²² eV for bosons [3] , not several keV.

To summarize, warm dark matter with an adiabatic invariant v h rms ( 1 ) = 0.406 ± 0.069   km / s [21] , a free-streaming comoving cut-off wavenumber k fs ( t eq ) = 2.0 − 0.5 + 0.8   Mpc − 1 [6] , and a non-linear small scale regenerated “tail” as in (4) with 0.5 ≲ n ≲ 0.8 , is in agreement with galaxy rotation curves [21] , galaxy stellar mass distributions, galaxy rest frame UV luminosity distributions [6] , the Missing Satellites Problem, and the UV luminosity cut-off needed to not exceed the measured reionization optical depth.

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

Hoeneisen, B. (2023) A Study of Warm Dark Matter, the Missing Satellites Problem, and the UV Luminosity Cut-Off. International Journal of Astronomy and Astrophysics, 13, 25-38. https://doi.org/10.4236/ijaa.2023.131002