Bruce Hoeneisen
Universidad San Francisco de Quito, Quito, Ecuador.
DOI: 10.4236/ijaa.2026.164020   PDF    HTML   XML   1 Downloads   19 Views   Citations

Abstract

Observed quasar light as a function of wavelength shows absorptions due to neutral hydrogen “clouds” along the line-of-sight. This “Lyman-$\alpha$ forest” has a power spectrum that has a cut-off. We study this cut-off. This cut-off may be due to warm dark matter free-streaming in the early universe, or to neutral hydrogen thermal velocity Doppler shift, or both. It is difficult to distinguish between these alternatives. If warm dark matter free-streaming dominates, we obtain the following density power spectrum cut-off wavevector: $k_{\text{fs}}=5.7\pm 0.3\pm 2.0$ Mpc⁻¹. This is a measure of the dark matter “warmness”. This result is in agreement with four independent published measurements, and in disagreement with published limits.

Keywords

Lyman-Alpha Forest, Warm Dark Matter, Free Streaming, Dark Matter

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1. Introduction

If dark matter is warm, then the comoving density power spectrum $P_{3\text{D}}\left(k\right)$ acquires a cut-off at comoving wavevector $k_{\text{fs}}$ due to warm dark matter free-streaming in the early universe. The challenge is to measure $k_{\text{fs}}$ to determine how warm or cold dark matter is. In the present article we highlight the difficulties of measuring $k_{\text{fs}}$ with quasar flux spectra, referred to as the “Lyman-$\alpha$ forest”.

Quasar light split into wavelength bins obtains the quasar flux spectra. This flux spectra shows absorption lines corresponding mainly to Lyman-$\alpha$ transitions of proper wavelength ${\lambda }_{\alpha }=1215.67$ Å (followed by isotropic emission) in neutral hydrogen “clouds” along the line-of-sight. These absorption lines are observed to be redshifted: $\lambda ={\lambda }_{\alpha }\left(1+z\right)$ , where $z$ is the redshift of the observed hydrogen “cloud”. Let $P_{1\text{D}}\left(k_z\right)$ be the observed power spectrum of the quasar flux spectra, with $k_z$ the wavevector along the line-of-sight (to be explained below). It turns out that this 1-dimesional Lyman-$\alpha$ power spectrum has a cut-off. The purpose of the present study is to try to understand the cause of this cut-off: is it due to i) warm dark matter free-streaming in the early universe, or ii) to baryon thermal effects?

The 3-dimensional density comoving power spectrum $P_{3\text{D}}\left(k\right)$ acquires a cut-off at comoving wavevector $k_{\text{fs}}$ if dark matter is warm. In Part I of this study [1] we assume that the observed Lyman-$\alpha$ power spectra cut-off is due to warm dark matter, and obtain $k_{\text{fs}}=5.0\pm 2.2$ Mpc⁻¹, in agreement with four independent measurements summarized in Table 3 of [2], but in disagreement with limits summarized in Figure 3 of [3]. The measurements of the dark matter “warmness” $k_{\text{fs}}$ are based on density runs of spheroidal dwarf galaxies, from the mass of first generation galaxies, from the stellar mass distributions of spiral galaxies, and from the number density of dwarf galaxies in the local field [2]. In the present article we continue the studies of Part I to try to understand these discrepancies.

Outline: In Section 2, we compare quasar flux power spectra with calculations in 16 bins of redshift $z$ . These power spectra have a cut-off. The calculations assume the observed Lyman-$\alpha$ forest cut-off is due to a density power spectrum cut-off (due to warm dark matter free-streaming in the early universe, Section 3), or to the thermal velocity Doppler shift of neutral hydrogen along the line-of-sight (Section 4). These two phenomena are degenerate: they can not be distinguished with Lyman-$\alpha$ forest observations alone, so independent data is needed. Comments on the observed quasar flux power spectra are presented in Sections 5 and 6. We try to break the degeneracy, i.e. find out which cut-off cause is dominant, in Sections 7 and 8. Section 9 presents a summary of data favoring warm or cold dark matter. Conclusions follow. For convenience of non-specialist readers (as myself), mathematical derivations are included in Appendices A-C. In Appendix D, we compare the present analysis with [4].

2. Data

We analyze Lyman-$\alpha$ power spectra obtained in [5], in 15 bins of redshift from $z=1.8$ to $z=4.6$ , from three data sources:

• Keck Observatory Database of Ionized Absorption toward Quasars (KODIAQ) Data Release 2 (DR2) ([6]-[8]) observed by HIRES ([9]) on the Keck I telescope.

• The Spectral Quasar Absorption Database (SQUAD) DR1 ([10]) observed by UVES ([11]) on the European Southern Observatory’s Very Large Telescope (VLT).

• XQ-100 observed using the X-Shooter spectrograph ([12]) under the European Southern Observatory Large Programme “Quasars and their absorption lines: a legacy survey of the high-redshift universe with VLT/XSHOOTER” ([13]).

In addition, we study Lyman-$\alpha$ power spectra obtained in [14] at redshift $z=4.2,4.6$ and 5.0.

The observed Lyman-$\alpha$ power spectra $P_{1\text{D}}\left(k_z\right)$ , obtained in [5] and [14], in 16 redshift bins are presented in Figures 1-3. Reference [5] notes that the measurements with $k_z<0.004$ s/km should be taken with caution because they are subject to large continuum errors, while measurements with $k_z>0.1$ s/km also need to be taken with caution because they are affected by noise, metal contamination and resolution effects. Several parameters obtained from Figures 1-3 are presented in Table 1.

Figure 1. Lyman-$\alpha$ power spectra from [5] in redshift bins $z=1.8$ to 2.8. $k_z$ is the line-of-sight wavevector in units s/km. $P_{1\text{D}}\left(k_z\right)$ is the 1-dimensional Lyman-$\alpha$ flux power spectrum. $k_z{P}_{1\text{D}}\left(k_z\right)/\pi$ is the dimensionless power spectrum. The Lyman-$\alpha$ flux is measured in bins of wavelength $\lambda$ . $k_z$ is defined as $k_z\equiv 2\pi /\Delta v$ with $\Delta v\equiv c\left(\lambda -{\lambda }_{\text{ref}}\right)/{\lambda }_{\text{ref}}$ . The reference ${\lambda }_{\text{ref}}$ is unspecified. The calculated curves are explained in Section 3.

Figure 2. Same as Figure 1 for redshift bins $z=3.0$ to 4.0.

To understand Figures 1-3, let us mention that Lyman-$\alpha$ flux measurements are made in wavelength bins. The Lyman-$\alpha$ power spectrum $P_{1\text{D}}\left(k_z\right)$ is expressed as a function of the wavevector $k_z\equiv 2\pi /\Delta v$ , defined in terms of the radial “velocity” $\Delta v\equiv c\left(\lambda -{\lambda }_{\text{ref}}\right)/{\lambda }_{\text{ref}}$ . ${\lambda }_{\text{ref}}$ is unspecified (it usually corresponds to ${\lambda }_{\text{ref}}={\lambda }_{\alpha }\left(1+z\right)$ with $z$ at the center of the redshift bin, or the quasar $z$ ). The comoving wavevector $k$ of the comoving density power spectrum $P_{\text{3D}}\left(k\right)$ is expressed in units of inverse comoving distance, e.g. Mpc⁻¹, while $k_z$ is expressed in inverse velocity units, e.g. s/km. The conversion factor for the radial wavevector $k_z$ in units s/km to radial Mpc⁻¹, for one bin of $a_i$ , is $H_i{a}_i$ , where

$a_i=1/\left(1+z_i\right)$ is the expansion parameter corresponding to the observed neutral hydrogen “cloud”, and $H_i\equiv \left(\text{d}{a}_i/\text{d}t\right)/a_i=H_0\sqrt{{\Omega }_m/a_i^3+{\Omega }_{\Lambda }}$ is the Hubble parameter at $a_i$ . This conversion factor can be understood as follows: $H_i^{-1}$ converts the radial velocity of an H_I atom at $a_i$ to an equivalent proper radial distance of an expanding “cloud” at $a_i$ , and the factor $1/a_i$ converts this proper radial distance to the corresponding comoving distance. For details, see derivations in Appendix A and Appendix B. Note that the conversion factor depends on the “cloud” expansion parameter $a_i$ .

Figure 3. Same as Figure 1 for redshift bins $z=4.2$ to 5.0. Power spectra from [5] (black squares) and [14] (blue triangles).

Table 1. At each redshift $z$ are presented $k_{\text{fs}}$ and $AB/{\left(1+z\right)}^4$ (defined in (1)-(3)), obtained from Figures 1-3. Their uncertainties are ±0.3 Mpc⁻¹ and ±0.002 km·s⁻¹·Mpc², respectively. These uncertainties include statistical uncertainties and uncertainties from the fit qualities. We note that $AB$ is approximately $\propto {\left(1+z\right)}^4$ so we present $AB/{\left(1+z\right)}^4$ .

$z$	1.8	2.0	2.2	2.4	2.6	2.8	3.0	3.2
$k_{\text{fs}}$ [Mpc−1]	4.5	4.3	4.3	4.3	4.5	4.5	4.8	5.1
$AB/{\left(1+z\right)}^4$ [km·s−1·Mpc2]	0.012	0.013	0.010	0.011	0.010	0.009	0.008	0.009
$z$	3.4	3.6	3.8	4.0	4.2	4.4	4.6	5.0
$k_{\text{fs}}$ [Mpc−1]	5.0	5.6	5.7	5.7	6.4	6.6	6.4	6.7
$AB/{\left(1+z\right)}^4$ [km·s−1·Mpc2]	0.009	0.009	0.008	0.008	0.009	0.011	0.012	0.010

Why does the observed Lyman-$\alpha$ power spectrum $P_{1\text{D}}\left(k_z\right)$ , presented in Figures 1-3, have a cut-off? We consider two alternatives: i) the density power spectrum cut-off due to warm dark matter free-streaming in the early universe, or ii) the neutral hydrogen H_I thermal velocity Doppler shift.

3. Dark Matter Warmness

Let us assume that the dark matter cut-off wavevector $k_{\text{fs}}$ (defined below) is the only source of the Lyman-$\alpha$ power spectrum $P_{\text{1D}}\left(k_z\right)$ cut-off. For comparison with the data, continuous lines in Figures 1-3 present Lyman-$\alpha$ forest power spectra $P_{\text{1D}}\left(k_z\right)$ calculated from the 3-dimensional comoving density power spectrum $P_{\text{3D}}\left(k\right)$ as follows:

$P_{1\text{D}}\left(k_z\right)=\frac{B}{2\pi }{\displaystyle {\int }_{k_z}^{\infty }{P}_{\text{3D}}\left(k\right)k\text{d}k},$ (1)

with $P_{\text{3D}}\left(k\right)$ approximated by (see Figure 22.5 of [15])

$P_{3\text{D}}\left(k\right)=A{\left(\frac{k}{6{\text{Mpc}}^{-1}}\right)}^b{\tau }^2\left(k\right),$ (2)

with $b=-2$ , and with the small-scale cut-off factor approximated to

${\tau }^2\left(k\right)=\text{exp}\left(-\frac{k^2}{k_{\text{fs}}^2}\right).$ (3)

These equations define $k_{\text{fs}}$ , $A$ , $B$ and $b$ . Note that this is our definition of the comoving cut-off wavevector $k_{\text{fs}}$ : ${\tau }^2\left(k_{\text{fs}}\right)=1/e$ . (Alternative definitions in the literature are ${\tau }^2\left(k_{\text{fs}}\right)=1/2$ or 1/4.) As shown in Figures 1-3, the adopted exponential ${\tau }^2\left(k\right)$ is in quite good agreement with the data for $z>2.2$ . We do not understand the disagreements for $z\le 2.0$ . For a dimensionless Lyman-$\alpha$ power spectrum $k_z{P}_{1D}\left(k_z\right)/\pi$ , $AB$ has units km·s⁻¹·Mpc². The factor $1/\left(2\pi \right)$ in (1) obtains the conventional normalization of $B$ .

In Equation (1), $P_{3\text{D}}\left(k\right)$ is the comoving 3-dimensional power spectrum of the quasar flux, which we approximate to the comoving 3-dimensional power spectrum of the matter density, an approximation valid in the linear regime [16].

Figures 1-3 and Table 1 present 16 measurements of the comoving wavevector $k_{\text{fs}}$ that range from 4.5 ± 0.3 Mpc⁻¹ at $z=1.8$ to 6.7 ± 0.3 Mpc⁻¹ at $z=5.0$ . It turns out that these measurements of $k_{\text{fs}}$ are in agreement, within uncertainties, with four independent measurements of $k_{\text{fs}}$ summarized in Table 3 of [2]. If this interpretation of the observed $P_{\text{1D}}\left(k_z\right)$ cut-off is correct, the measured $k_{\text{fs}}$ should be independent of $z$ . The observed dependence of $k_{\text{fs}}$ on $z$ may, or may not, be significant, see Figure 4, so we discuss this issue in Section 6 below. The dependence of the calculations on the slope $b$ is presented in Figure 5.

Let us relate $k_{\text{fs}}$ with the warm dark matter comoving root-mean-square free-streaming distance $l_{\text{fs}}$ . Consider the 3-dimensional power spectrum of equation (2). It is the square of the Fourier transform of the smoothed density field $\tilde{\rho }\left(x\right)$ . The square root of (2) is the product of two functions, i.e. $k^{-1}$ and $\text{exp}\left[-k^2/\left(2k_{\text{fs}}^2\right)\right]$ . The inverse Fourier transform of this product is the convolution of the two inverse Fourier transforms, i.e. $\propto \rho \left(x\right)$ and $\propto \text{exp}\left[-{\left|x\right|}^2{k}_{\text{fs}}^2/2\right]$ :

$\tilde{\rho }\left(x\right)=\frac{k_{\text{fs}}^3}{{\left(2\pi \right)}^{3/2}}{\displaystyle {\int }_{-\infty }^{+\infty }\rho \left(y\right)\text{exp}}\left[-\frac{{\left|x-y\right|}^2{k}_{\text{fs}}^2}{2}\right]{\text{d}}^{3}y.$(4)

(For convenience, Fourier transforms of gaussians are presented in Appendix C.) Note that this convolution obtains the weighted average of $\rho \left(x\right)$ over a comoving 3-dimensional root-mean-square distance that is proportianal to $l_{\text{fs}}$ : $l_{\text{fs}}\approx 1.31/k_{\text{fs}}$ , see (27) of [1]. (The relation is approximate: it depends on the distribution of velocities of the dark matter particles [1].)

Figure 4. Observed Lyman-$\alpha$ power spectra [5] [14] at $z=1.8$ to $z=5.0$ . Amplitudes have been adjusted so that all $k_z{P}_{1\text{D}}\left(k_z\right)/\pi$ coincide at ${\text{log}}_{10}\left(k_z/\text{s}/\text{km}\right)=-1.7$ .

Figure 5. The observed Lyman-$\alpha$ power spectrum [5] at $z=4.0$ is compared with calculation for $k_{\text{fs}}=5.7$ Mpc⁻¹, and $b=-1.7$ and $b=-2.3$ (red lines). For $b=-2.0$ , see Figure 2.

Consider a quasar flux as a function of $v\equiv c\Delta \lambda /\lambda$ . An absorption from an isolated “cloud” has the form $\text{exp}\left[-{\left(v-v_0\right)}^2/\left(2{\sigma }^2\right)\right]$ . If the cloud has temperature $T$ and is small, or non-expanding, the root-mean-square velocity of the neutral hydrogen atoms along the line-of-sight is $\sigma =\sqrt{k_{\text{B}}T/m_p}$ . If the cloud is cold and has a comoving root-mean-square length $l_{\text{fs}}/\sqrt{3}$ along the line-of-sight, and is expanding, then the proper root-mean-square thermal velocity along the line-of-sight is $\sigma =l_{\text{fs}}{a}_i{H}_i/\sqrt{3}$ . If the cloud is both hot and expanding, then

$\sigma =\sqrt{\left(k_{\text{B}}T/m_p\right)+{\left(l_{\text{fs}}{a}_i{H}_i\right)}^2/3}$ .

We may relate $l_{\text{fs}}$ to an equivalent temperature of a hydrogen “cloud” observed at expansion parameter $a_i$ :

$T=\frac{m_p{\left(l_{\text{fs}}{a}_i{H}_i\right)}^2}{3k_{\text{B}}}\approx 0.57\frac{m_p{\left(a_i{H}_i\right)}^2}{k_{\text{B}}{k}_{\text{fs}}^2}.$ (5)

Note that $T$ and $k_{\text{fs}}$ are degenerate, i.e. they can not be individually measured with Lyman-$\alpha$ forest data alone (assuming that both non-relativistic dark matter particles and neutral hydrogen atoms have the same thermal Maxwell velocity distribution). (Note that published limits on $k_{\text{fs}}$ , or the warm dark matter thermal relic mass, obtained solely with Lyman-$\alpha$ forest data, need to be taken with caution.) Our major challenge is to distinguish warm dark matter from hot neutral hydrogen, with data independent of the Lyman-$\alpha$ forest.

4. H_I Temperature

In Section 3, we assume that warm dark matter is the only cause of the observed Lyman-$\alpha$ forest power spectrum cut-off, implying that the intergalactic baryons are relatively cold. In this section we consider the opposite case: the baryon temperature is the only cause of the observed Lyman-$\alpha$ power spectrum cut-off, while dark matter is cold. There are several thermal effects. In this section we only consider the leading and simplest one, namely the neutral hydrogen thermal velocity Doppler shift. It is important to realize that, as long as the universe expands linearly, $\Delta v\equiv c\Delta \lambda /\lambda$ is independent of $a\left(t\right)$ .

From (5), for $z=5,4$ and 2, and $k_{\text{fs}}$ from Table 1, we obtain the needed temperature to cause the observed Lyman-$\alpha$ flux power spectrum cut-off assuming cold dark matter: $T\approx 13000$ K, 15,000 K and 16,000 K, respectively. In summary, if $T\approx 15000$ K, then the Lyman-$\alpha$ flux power spectrum cut-off is due to temperature alone, and the data is consistent with the cold dark matter ΛCDM scenario. On the other hand, if $T\lesssim 15000$ K, then the cut-off requires warm dark matter free-streaming.

The mean temperature, at mean density, obtained from Lyman-$\alpha$ data and simulations that assume the cold dark matter ΛCDM scenario is approximately $T\approx 10000$ K [17]-[21]. The width of Lyman-$\alpha$ absorption features depend on the integalactic baryon temperature, and is compared with simulations to constrain the temperature [19]. However, this width is degenerate with the dark matter warmness.

At redshif $z\lesssim 6$ the intergalactic medium is highly ionized. The temperature of the intergalactic medium at redshift $z=4$ is highly variable, from ≈ 2000 K in underdense regions, to ≈ 10⁶ K in highly dense shock-heated gas. What is the temperature of the neutral hydrogen “clouds” (without using Lyman-$\alpha$ forest data assuming cold dark matter)? We do not know, see Section 7.

5. Amplitude of $P_{1D}\left(k_z\right)$

$AB$ , defined in (1)-(3), is expected to scale as $AB\propto {\left(1+z\right)}^{-2}\left(n_{\text{HI}}/n_{\text{H}}\right){\left(1+z\right)}^3$ . The first factor is due to the scaling of $P_{\text{3D}}\left(k\right)$ , and the last factor is due to density (to compensate the ionized plus neutral hydrogen density $n_{\text{H}}$ ). The fraction of neutral hydrogen $n_{\text{HI}}/n_{\text{H}}$ at low $n_{\text{H}}$ scales in proportion to density [22], i.e. $\propto {\left(1+z\right)}^3$ , because of the mean-free-path for the reaction ${\text{p}}^{+}+{\text{e}}^{-}\to {\text{H}}_{\text{I}}$ . The observed dependence of $AB$ on $z$ is $AB\propto {\left(1+z\right)}^4$ , see Table 1, so the dependence of $AB$ on $z$ is understood, at least in first approximation.

6. Comments on the Possible Dependence of the Observed $k_{fs}$ on $z$

If the cut-off of $P_{\text{1D}}\left(k_z\right)$ is due only to dark matter free-streaming, we expect $k_{\text{fs}}$ to be approximately independent of $z$ (because free-streaming is only effective in the early universe at expansion parameter of order $a_{\text{eq}}\approx 1/3402$ ). However, the observed $k_{\text{fs}}$ decreases from 6.4 ± 0.3 to 4.3 ± 0.3 from $z=4.6$ to $z=2.0$ . The significance of the dependence of $k_{\text{fs}}$ on $z$ may also be judged from Figure 4. What is the cause of this possible dependence?

• The large variation of the amplitude $AB$ from $z=5.0$ to 1.8 may result in an unknown experimental systematic dependence of the observed $k_{\text{fs}}$ on $z$ .

• If warm dark matter free-streaming is the only cause of the observed $P_{\text{1D}}\left(k_z\right)$ cut-off, we expect no dependence of $k_{\text{fs}}$ on $z$ so long as the density is linear and the universe is expanding homogeneously. However, a dependence of the observed $k_{\text{fs}}$ on $z$ will set in when density perturbations become non-linear. Consider an overdense region of the universe that expands slower than $a$ . The density peaks attract each other resulting in an increase of the observed comoving $k_{\text{fs}}$ . Some peaks coaless resulting in the bottom-up hierachical evolution of the density fluctuations, and a reduction of the observed $k_{\text{fs}}$ .

• Equations (1)-(3) are approximate: they assume that the neutral hydrogen density is proportional to the matter density, an approximation valid if density perturbations are linear. Non-linear density perturbations will introduce corrections to equations (1)-(3).

• The conversion factor $H_i{a}_i$ becomes approximate when density perturbations become non-linear.

• Perhaps intergalactic thermal velocity Doppler shift of neutral hydrogen, or Jeans smoothing, compete with warm dark matter free-streaming.

• Note that the zero point of $\text{Δ}v$ is not specified: we have assumed ${\lambda }_{\text{ref}}={\lambda }_{\alpha }\left(1+z\right)$ , with $z$ at the mid-point of the redshift bin.

7. Trying to Break the Degeneracy

The Lyman-$\alpha$ forest power spectrum has an observed cut-off that may be due to warm dark matter free-streaming or to the intergalactic neutral hydrogen temperature. The Lyman-$\alpha$ forest data apparently can not distinguish between these two alternatives. Which of these alternatives is dominant? One way to break this degeneracy is to estimate the intergalactic baryon temperature, as a function of mean hydrogen (ionized plus neutral) number density, based on the measured ultra-violet intensity. A summary of four calculations ([23]-[26]) can be found in [22]. From Figure 1 of [22] we obtain the temperatures listed in Table 2. From the large temperature uncertainties, we are unable to lift the degeneracy.

Alternative ways to break the degeneracy are measurements of the dark matter $k_{\text{fs}}$ without the Lyman-$\alpha$ forest. Four independent measurements of $k_{\text{fs}}$ are presented in Table 3 of [2] (and references therein), that are in agreement with Table 1, but in disagreement, by a factor of order few, with limits summarized in Table 3 of [3]. Note that each of these measurements and limits have their own issues.

In summary, we are unable to clearly favor one or the other alternative. Perhaps both warm dark matter and baryon temperature contribute to the observed Lyman-$\alpha$ power spectrum cut-off?

Table 2. For redshifts $z=5$ and $z=2$ , are presented the mean hydrogen (neutral plus ionized) number density $\langle n_{\text{H}}\rangle ={\Omega }_b{\rho }_{\text{crit}}{\left(1+z\right)}^3/m_p$ , and the corresponding calculated temperatures from the measured UV intensity [22].

$z$	$\langle n_{\text{H}}\rangle$ [cm−3]	$T$ [K]
5	5.4 × 10−5	≈102 to105.5
2	6.8 × 10−6	≈101 to105.0

8. Both?

There are two total (dark matter plus baryon) masses of interest: the half-mode mass $M_{\text{hm}}$ and the Jeans mass $M_{\text{J}}$ [1]. $M_{\text{hm}}$ depends on the warm dark matter cut-off wavevector $k_{\text{fs}}$ , and the Jeans mass $M_{\text{J}}$ depends on the hydrogen temperature $T$ [1] [27]. The two masses are equal if

${\left(\frac{k_{\text{fs}}}{5.7{\text{Mpc}}^{-1}}\right)}^2=\frac{1+z}{1+4}\frac{12000\text{K}}{T}.$ (6)

If $M_{\text{hm}}\gg M_{\text{J}}$ , i.e. dark matter is sufficiently warm, then $M_{\text{hm}}$ is the smallest clump of total (dark matter plus baryon) mass that will collapse due to its own gravity, and this collapse includes the full complement of baryons. In this case there is no Jeans smoothing. If $M_{\text{hm}}\ll M_{\text{J}}$ , i.e. baryons are sufficiently hot, then clumps of total matter with mass greater than $M_{\text{hm}}$ , and less than $M_{\text{J}}$ , will have dark matter collapsing but without the full complement of baryons.

Before $z\approx 150$ , hydrogen is in thermal equilibrium with the cosmic microwave background radiation, and the total (dark matter plus baryon) $M_{\text{J}}\approx 6\times {10}^5{M}_{\odot }$ [27]. After reionization, the temperature jumps to ≈12,000 K and $M_{\text{J}}$ jumps to $\approx {10}^{10}{M}_{\odot }$ . Let us now consider the two scenarios.

If dark matter is warm: Assume $M_{\text{hm}}>M_{\text{J}}$ . To be specific consider $z\approx 4$ , $k_{\text{fs}}\approx 5.7$ Mpc⁻¹ and $T\lesssim 12000$ K after reionization. In this case $M_{\text{hm}}\approx {10}^{10}{M}_{\odot }$ [1]. First density perturbations to collapse have comoving wavevector $k$ of order $k_{\text{fs}}$ , and ride on top of larger and larger perturbations that have similar phases and so add up. The relative amplitudes of these perturbations grow in proportion to the expansion parameter $a\left(t\right)$ while linear. First clumps of mass to collapse have total (dark matter plus baryon) mass of order $\approx M_{\text{hm}}/10\approx {10}^9{M}_{\odot }$ (taking into account the tail of $P_{\text{3D}}\left(k\right)$ ). Galaxy formation is hierarchical, from bottom-up, starting at mass of order ${10}^9{M}_{\odot }$ , due to larger and larger perturbations becoming non-linear, resulting in the merging of galaxy halos; and also hierarchical formation from top-down, starting at mass of order ${10}^9{M}_{\odot }$ , due to halos loosing mass to neighboring halos. The corresponding dwarf galaxies do not have time to desintegrate due to the low dark matter particle velocities ($\approx \left(1+z\right)\times 120$ m/s). Is there any evidence for this scenario? Yes! I refer the reader to Section 4 of [2].

Equation (6) is extraordinary: is it a coincidence, or is it telling us something? Equation (6) relates two minimum total (dark matter plus baryon) mass clumps that can collapse: $M_{\text{hm}}$ depending on the dark matter temperature-to-mass ratio, and $M_{\text{J}}$ depending on baryon temperature. The first generation of stars emit ultra-violet radiation that heat up and ionize the intergalactic hydrogen atoms. Let us speculate that, when the temperature reaches of order 12,000 K, during reionization, collapsing halos begin loosing their baryons, and the formation of stars is inhibeted, so the temperature approaches an equilibrium value. (Hot baryons also directly inhibit star formation, but such inhibition is not addressed by Equation (6).) Equation (6) is, perhaps, not a coincidence! It is perhaps telling us that the intergalactic temperature after reionization is self-regulating. Note that $k_{\text{fs}}$ in Equation (6) is the warm dark matter power spectrum cut-off wavevector, and has no relation to the neutral hydrogen thermal velocity Doppler shift. Note also that Equation (6) is in agreement with observations. If (6) is not a coincidence, it is telling us that dark matter is warm!

Another sealing to the baryon temperature is $T\approx 10000$ K to 4000 K due to hydrogen excitation followed by decay and radiation, see Figure 7 of [28].

If dark matter is cold: Assume dark matter is very cold, i.e. $k_{\text{fs}}\gtrsim 170$ Mpc⁻¹. Then first perturbations to collapse, before reionization, have dark matter mass smaller than $6\times {10}^5{M}_{\odot }$ and are nearly free of baryons. The collapse of matter is hierarchical, bottom-up. The first halos that collapse with the full complement of bayons, before reionization, have total (dark matter plus baryon) mass or order $6\times {10}^5{M}_{\odot }$ . At this point first stars begin forming, ionizing and heating up the baryons. Is an equilibrium reached?

9. Summary

Data that favor warm dark matter:

• Galaxies apparently form hierarchically, starting at $\approx {10}^9{M}_{\odot }$ , bottom-up and top-down, see Section 4 of [2] and references therein.

• Four independent measurements of $k_{\text{fs}}$ are in agreement with the warm dark matter Lyman-$\alpha$ power spectrum cut-off, see Table 3 of [2].

Data that favor cold dark matter:

• Limits on warm dark matter summarized in Figure 3 of [3]. (Limits that depend only on Lyman-$\alpha$ forest data need to be taken with caution, see Section 3.)

• The possible dependence of the measured $k_{\text{fs}}$ on $z$ , see Section 3.

10. Conclusions

The Lyman-$\alpha$ forest flux power spectrum $P_{\text{1D}}\left(k_z\right)$ has an observed cut-off. This cut-off may be due to warm dark matter free-streaming in the early universe, or to hot baryons, or both. These degenerate alternatives can not be distinguished with Lyman-$\alpha$ flux spectra alone. Limits on $k_{\text{fs}}$ obtained with Lyman-$\alpha$ data alone need to be taken with caution. Data favoring warm or cold dark matter are summarized in Section 9. If warm dark matter is the leading effect, i.e. if $T\lesssim 10000$ K, we obtain

$k_{\text{fs}}=5.7\pm 0.3\pm 2.0{\text{Mpc}}^{-1}.$ (7)

The first uncertainty is due to statistical uncertainties of the measured power spectra and the quality of the fits, and the second uncertainty is estimated from “other” phenomena listed in Section 6, the dispersion of measurements in Figure 4, and the apparent dependence of $k_{\text{fs}}$ on redshift $z$ in Table 1. This result is in agreement with [2] and [1], but in disagreement with measurements summarized in Figure 3 of [3]. A comparison of the present analysis and a limit obtained with the Lyman-$\alpha$ forest [4] is presented in Appendix D.

Appendix

A. Lyman-α Absorption

We consider pulses of light propagating radialy in a flat universe. The metric is $\text{d}{s}^2=c^2\text{d}{t}^2-a{\left(t\right)}^2\text{d}{r}^2$ . Note that the radial coordinate $r$ is the present, or comoving, distance to an event. For simplicity, we neglect the cosmological constant in these appendices. Then the evolution of the homogeneous universe is described by

$H\equiv \frac{\text{d}a/\text{d}t}{a}=H_0\sqrt{\frac{{\Omega }_m}{a^3}},$ (8)

$\frac{2}{3}{a}^{3/2}=H_0\sqrt{{\Omega }_m}t.$ (9)

Consider a neutral hydrogen “cloud” that absorbes Lyman-$\alpha$ photons at the proper wavelength ${\lambda }_i=1215.67$ Å, and at the expansion parameter $a_i$ . The universal time of the absorption is

$t_i=\frac{2}{3H_0\sqrt{{\Omega }_m}}{a}_i^{3/2},$ (10)

and the present day, or comoving, distance to the “cloud” is

$r_i=\frac{2c}{H_0\sqrt{{\Omega }_m}}\left(a_f^{1/2}-a_i^{1/2}\right).$ (11)

This absorption is observed at $r_f=0$ , and the present expansion parameter $a_f=1$ , at the universal time of observation

$t_f={\left\{t_i^{1/3}+\frac{r_i}{3c}{\left(\frac{3H_0\sqrt{{\Omega }_m}}{2}\right)}^{2/3}\right\}}^3.$ (12)

Note that the redshifted wavelength ${\lambda }_f$ is

$\frac{\partial t_f}{\partial t_i}=\frac{{\lambda }_f}{{\lambda }_i}=\frac{a_f}{a_i}.$ (13)

B. Conversion from Comoving Radial Distance $\Delta r$ to Radial “Velocity” $\Delta v\equiv c\Delta \lambda /\lambda$

(For simplicity we consider a homogeneous flat universe and neglect the cosmological constant.) Consider two Lyman-$\alpha$ lines from two “clouds” at $r_i$ and $r_i+\text{Δ}r$ . The received wavelength at the origin at the universal time $t$ is ${\lambda }_{\alpha }/a\left(t\right)$ for both clouds. We would like to compare the received wavelengths at the same arrival time

$t_f={\left\{{\left(t_i+\Delta t\right)}^{1/3}+\frac{r_i+\Delta r}{3c}{\left(\frac{3H_0\sqrt{{\Omega }_m}}{2}\right)}^{2/3}\right\}}^3,$ (14)

so $\Delta t=-\Delta r{a}_i/c$ .

$a\left(t+\Delta t\right)=a\left(t\right)+\frac{\text{d}a}{\text{d}t}\Delta t=a_i\left(1+\frac{2}{3t_i}\Delta t\right),$ (15)

${\lambda }_i=\frac{{\lambda }_{\alpha }}{a_i};{{\lambda }^{\prime }}_i=\frac{{\lambda }_{\alpha }}{{a^{\prime }}_i}=\frac{{\lambda }_{\alpha }}{a_i}\left(1-\frac{2}{3t_i}\Delta t\right),$ (16)

$\Delta v\equiv c\frac{{{\lambda }^{\prime }}_i-{\lambda }_i}{{\lambda }_i}=-c\frac{2}{3t_i}\Delta t=-\Delta r\frac{2a_i}{3t_i}.$ (17)

Note that $\text{Δ}v$ is a radial “velocity”. The last fraction is the conversion factor of interest:

$\frac{2a_i}{3t_i}=a_i{H}_0\sqrt{\frac{{\Omega }_m}{a_i^3}}\equiv a_i{H}_i.$ (18)

Note that the conversion depends on $a_i$ .

An alternative derivation is as follows: calculate $\text{d}{r}_i$ from (11) and compare with $\text{d}v$ .

C. 1D and 3D Gaussians

The following equations at hand are useful to understand the main text. Consider this normalized 1D Gaussian with standard deviation (or root-mean-square) $\sigma$ :

$f\left(x\right)=\frac{1}{\sqrt{2\pi }\sigma }\text{exp}\left(-\frac{x^2}{2{\sigma }^2}\right).$ (19)

Its Fourier transform is

$F\left(k\right)\propto \text{exp}\left(-\frac{k^2{\sigma }^2}{2}\right).$ (20)

Consider this normalized 3D Gaussian:

$f\left(x\right)=f\left(\left|x\right|\right)=\frac{1}{{\left(2\pi \right)}^{3/2}{\sigma }^3}\text{exp}\left(-\frac{{\left|x\right|}^2}{2{\sigma }^2}\right).$ (21)

Its fourier transform is

$F\left(k\right)=F\left(\left|k\right|\right)\propto \text{exp}\left(-\frac{k^2{\sigma }^2}{2}\right).$ (22)

D. Comparison of the Present Analysis with [4]

The present article, Part II, is based on Lyman-$\alpha$ data obtained in [5] and [14]. The article [4] is based on the same data obtained in [14]. Both articles try to understand the observed Lyman-$\alpha$ flux power spectrum cut-off. Both articles should obtain consistent results.

The two articles, Part II and [4], are in approximate agreement. If the baryonic temperature $T\approx 10000$ K [4] or $T\approx 15000$ K (Part II), then the observed cut-off is due to temperature alone, and dark matter is cold. If the temperature is lower, then warm dark matter free-streaming is needed to obtain the observed cut-off. If this is the case, i.e. $T\lesssim 10000$ K, the obtained warmness of dark matter, $k_{\text{fs}}$ or equivalently $m_{\text{th}}$ , turns out to be in agreement with the measurements summarized in [2] and in [1].

The main difference between Part II and [4] is the external data used to constrain the baryon temperature and break the $T\leftrightarrow k_{\text{fs}}$ degeneracy.

Conflicts of Interest

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

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