_{1}

^{*}

In the warm dark matter scenario, the Press-Schechter formalism is valid only for galaxy masses greater than the “velocity dispersion cut-off”. In this work we extend the predictions to masses below the velocity dispersion cut-off, and thereby address the “Missing Satellites Problem” of the cold dark matter ΛCDM scenario, and the rest-frame ultra-violet luminosity cut-off required to not exceed the measured reionization optical depth. For warm dark matter we find agreement between predictions and observations of these two phenomena. As a by-product, we obtain the empirical Tully-Fisher relation from first principles.

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 [

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 [^{3}, 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 [

data provide a measurement of k fs ( t eq ) , see [

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 [

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 [

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

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

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 [

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

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 [

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 [

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 [

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

z | v h rms ( 1 ) | k fs ( t eq ) | log 10 ( M vd / M ⊙ ) |
---|---|---|---|

4 | 0.75 km/s | 1 Mpc^{−1} | 9.3 |

4 | 0.49 km/s | 1.53 Mpc^{−1} | 8.5 |

4 | 0.37 km/s | 2 Mpc^{−1} | 8.3 |

4 | 0.19 km/s | 4 Mpc^{−1} | 7.5 |

6 | 0.75 km/s | 1 Mpc^{−1} | 9.8 |

6 | 0.49 km/s | 1.53 Mpc^{−1} | 9.3 |

6 | 0.37 km/s | 2 Mpc^{−1} | 9.0 |

6 | 0.19 km/s | 4 Mpc^{−1} | 8.0 |

8 | 0.75 km/s | 1 Mpc^{−1} | 10.3 |

8 | 0.49 km/s | 1.53 Mpc^{−1} | 9.6 |

8 | 0.37 km/s | 2 Mpc^{−1} | 9.2 |

8 | 0.19 km/s | 4 Mpc^{−1} | 8.2 |

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 [^{−1}, we may approximate

V flat ≈ 45 km / s . Similarly, for several masses M, the corresponding rotation velocities V flat are summarized in

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

as shown in

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)

v h rms ( 1 ) [m/s] | 750 | 490 | 370 | 190 | 0.75 |
---|---|---|---|---|---|

k fs ( t eq ) [Mpc^{−1}] | 1 | 1.53 | 2 | 4 | 1000 |

z | |||||

4 | 38 | 33 | 36 | 37 | 34 |

5 | 41 | 35 | 37 | 37 | 37 |

6 | 47 | 44 | 40 | 49 | 42 |

8 | 45 | 42 | 45 | 46 | 49 |

10 | 51 | 49 | 47 | 53 | 51 |

M | V flat |
---|---|

3 × 10 12 M ⊙ | 255 km/s |

1 × 10 12 M ⊙ | 188 km/s |

1 × 10 11 M ⊙ | 82 km/s |

5 × 10 10 M ⊙ | 71 km/s |

2 × 10 10 M ⊙ | 45 km/s |

1 × 10 10 M ⊙ | 37 km/s |

5 × 10 9 M ⊙ | 28 km/s |

3 × 10 8 M ⊙ | 9 km/s |

This approximate relation, obtained from first principles, can be compared with the empirical Tully-Fisher relation (see

The “Missing Satellites Problem” of the cold dark matter ΛCDM cosmology is described in [^{3}, with V flat > V is [

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 [

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 [

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 [

Reionization begins in earnest at z ≈ 8 , and ends at z ≈ 6 . For each panel of _{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 [

Redshift z | Good | Fair | Poor |
---|---|---|---|

8 | 0.3, 0.5, 0.7 | 0.1, 1.1, 2.0 | |

6 | 0.3, 0.5, 0.7 | 0.1, 1.1, 2.0 | |

4 | 0.7 | 0.1, 0.3, 0.5, 1.1, 2.0 |

n | M_{UV} | τ | fit quality |
---|---|---|---|

0.9 | −18.6 | 0.050 ± 0.006 | fair |

0.8 | −18.3 | 0.050 ± 0.006 | good |

0.7 | −17.6 | 0.052 ± 0.006 | excellent |

0.6 | −16.9 | 0.053 ± 0.008 | excellent |

0.5 | −14.9 | 0.059 ± 0.008 | excellent |

0.4 | >−11.9 | >0.07 | good |

0.3 | >−11.9 | >0.07 | fair |

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 [

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 [

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 [

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 [^{−22} eV for bosons [

To summarize, warm dark matter with an adiabatic invariant v h rms ( 1 ) = 0.406 ± 0.069 km / s [

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