Most non-relativistic matter in the universe is in a “dark matter” form that has only been “observed” through its gravitational interaction [1] . Let us assume that this dark matter is a gas of a single particle species of mass $m_h$. The effect of this dark matter on cosmology, under quite general conditions, can be described with a single parameter $a_{h\text{NR}}$ [2] to be added to the six parameters of the standard cold dark matter ΛCDM cosmology [1] . $a_{h\text{NR}}$ is the characteristic universe expansion factor $a\left(t\right)$ at which the dark matter particles become non-relativistic. $a\left(t\right)$ is normalized so that $a\left(t_0\right)=1$ at the present time $t_0$. Several estimated limits on $a_{h\text{NR}}$ have been summarized in Table 3 of [2] . The limits are

$a_{h\text{NR}}<6\times {10}^{-5}$ (1)

from Big Bang Nucleosynthesis (BBN) observations,

$a_{h\text{NR}}\lesssim 7\times {10}^{-6}$ (2)

from observed Cosmic Microwave Background (CMB) radiation fluctuations,

$a_{h\text{NR}}\lesssim 7\times {10}^{-7}$ (3)

from observations of the Lyman-α forest of quasar light, and

$a_{h\text{NR}}\lesssim 6\times {10}^{-8}$ (4)

from the observed population of small galaxy halos. In contrast, $a_{h\text{NR}}$ has been measured with dwarf galaxy rotation curves [3] [4] :

$a_{h\text{NR}}=\left(1.39\pm 0.24\right)\times {10}^{-6}.$ (5)

There is a discrepancy between (4) and (5). Each of these measurements and limits has its own issues. The purpose of the present study is to try to understand the discrepancy between (4) and (5).

The general conditions mentioned above assume dark matter with or without interactions with the Standard Model sector or self-interactions, as long as the chemical potential is non-positive. In the present study, we further assume that dark matter was once in thermal equilibrium with the early Standard Model sector, i.e. we consider “thermal relic” dark matter.

The limit (4) is obtained from the minimum halo mass hosting Milky Way satellites [2] , [5] - [13] :

$M_h<5.4\times {10}^8{\text{M}}_{\odot },$ (6)

at $2\sigma$ confidence [2] . The corresponding observed stellar mass is of order $M_{\text{*}}\approx {10}^5{\text{M}}_{\odot }$ [2] (that agrees with the calculation in [14] ). The authors of [2] state that (4) is an estimate, and, due to non-linearities, “should be more rigorously studied by cosmological simulation”. Below we fill in this gap.

The present work is a continuation of [15] and [16] . These references may be consulted for details. To make the present article self-contained we list some definitions in Section 2. Warm dark matter introduces the “free-streaming cut-off” discussed in Section 3. Numerical integration of hydrodynamical equations is presented in Section 4. The non-linear regeneration of small scale structure is discussed in Section 5, and the bottom-up and top-down evolution of structure in the warm dark matter scenario is presented in Section 6. Conclusions follow.

We use the notation and parameter values of [1] . Our definitions follow. Let $v_{h\text{rms}}\left(a\right)$ be the 3-D root-mean-square thermal velocity of non-relativistic warm dark matter particles at expansion parameter $a\left(t\right)$ when the universe is nearly homogeneous. $v_{h\text{rms}}\left(a\right)$ scales as $a^{-1}$, so

$v_{h\text{rms}}\left(1\right)=a\cdot v_{h\text{rms}}\left(a\right)$ (7)

is an adiabatic invariant, and

$a_{h\text{NR}}\equiv 1.03\frac{v_{h\text{rms}}\left(1\right)}{c},$ (8)

where 1.03 is a threshold factor for bosons (0.98 for fermions) [4] . Warm dark matter free-streaming attenuates the linear power spectrum of density perturbations $P_{\text{CDM}}\left(k\right)$ of the ΛCDM cosmology, at large comoving wavevector $k$, by a factor [17]

${\tau }^2\left(k\right)\equiv \frac{P_{\text{WDM}}\left(k\right)}{P_{\text{CDM}}\left(k\right)}={\left[1+{\left(\alpha k\right)}^{2\nu }\right]}^{-10/\nu },$ (9)

where $\nu =1.12$, and

$\alpha =0.049{\left(\frac{m_h}{1\text{keV}}\right)}^{-1.11}{\left(\frac{{\Omega }_c}{0.25}\right)}^{0.11}{\left(\frac{h}{0.7}\right)}^{1.22}\frac{1}{h}\text{Mpc}.$ (10)

These equations, often used in the literature, define the “standard thermal relic mass” $m_h$. Limits in the literature often obtain a lower bound to this $m_h$. The actual dark matter particle mass is model-dependent, see Table 4 of [4] . Equation (9) can be approximated by

${\tau }^2\left(k\right)\approx \text{exp}\left(-\frac{k^2}{k_{\text{fs}}^2}\right),\text{where}{k}_{\text{fs}}=\frac{1}{2.59\cdot \alpha },$ (11)

except for a “tail” to be discussed below. $k_{\text{fs}}$ is the comoving free-streaming wavevector. At this wavevector, ${\tau }^2\left(k_{\text{fs}}\right)=1/e$ (other definitions in the literature are 1/2 or 1/4). An effective warm dark matter comoving free-streaming length can be defined as

$l_{\text{fs}}=0.26\frac{2\pi }{k_{\text{fs}}},$ (12)

where the factor 0.26 comes from an integration over angles. $l_{\text{fs}}$ is approximately proportional to $v_{h\text{rms}}\left(1\right)$:

$l_{\text{fs}}\approx 4.9\left[\frac{\text{Mpc}}{\text{km}/\text{s}}\right]\cdot v_{h\text{rms}}\left(1\right).$ (13)

An analytic expression that relates directly $v_{h\text{rms}}\left(1\right)$ and $k_{\text{fs}}$ (that does not include free-streaming during radiation domination while inside the horizon) is [18] [19]

$k_{\text{fs}}\left(t_{\text{eq}}\right)=\frac{1.455}{\sqrt{2}}\sqrt{\frac{4\pi G{\Omega }_m{\rho }_{\text{crit}}{a}_{\text{eq}}}{v_{h\text{rms}}^2\left(1\right)}}.$ (14)

This comoving wavevector approximately separates growing from decaying modes at the time $t_{\text{eq}}$ of equal radiation and matter densities. Thereafter, $k_{\text{fs}}$ grows as $\propto a^{1/2}$, allowing regeneration of small scale structure, giving $\tau \left(k\right)$ a “tail” that depends on $a\left(t\right)$. The 3-D Fourier transform of (11) defines a linear free-streaming mass scale

$M_{\text{fs}}\approx \frac{4}{3}\pi {\left(\frac{1.555}{k_{\text{fs}}}\right)}^3{\Omega }_m{\rho }_{\text{crit}},$ (15)

where ${\Omega }_m{\rho }_{\text{crit}}$ is the present mean matter density of the universe. In conclusion, each of the observables $m_h$, $k_{\text{fs}}$, $l_{\text{fs}}$, $v_{h\text{rms}}\left(1\right)$, $a_{h\text{NR}}$ and $M_{\text{fs}}$ determines all the others.

Astronomers obtain the galaxy “stellar mass” $M_{\text{*}}$ and “halo mass” $M_h$ (also called “virial mass”, which is a missnomer in view of [20] ). The “halo mass” $M_h$ is often defined as the galaxy mass contained inside a radius $r_{200}$ at which the dark matter density reaches 200 times the mean dark matter density of the universe at the observed redshift of the galaxy [21] . The galaxy stellar mass $M_{\text{*}}$ is obtained from measurements of the relative luminosities of the galaxy with several filters, the measurement of its redshift, and from stellar synthesis models. The galaxy halo mass $M_h$ is estimated from $M_{\text{*}}$, stellar images, (warm) dark matter simulations and abundance matching [21] , or exceptionally, with gravitational lensing measurements [20] .

The Press-Schechter prediction [22] , or its Sheth-Tormen extensions [23] [24] , depend on the variance of the relative density perturbation $\delta \left(x\right)\equiv \left(\rho \left(x\right)-\bar{\rho }\right)/\bar{\rho }$ on the linear total (dark matter plus baryon) mass scale $M_{\text{PS}}$, at redshift $z$ [14] [25] :

${\sigma }^2\left(M_{\text{PS}},z,k_{\text{fs}}\right)=\frac{f^2}{{\left(2\pi \right)}^3{\left(1+z\right)}^2}{\displaystyle {\int }_0^{\infty }4\pi k^2\text{d}k}{P}_{\text{CDM}}\left(k\right){\tau }^2\left(k\right)W^2\left(k\right),$ (16)

and so depends on the free-streaming cut-off factor ${\tau }^2\left(k\right)$, and on the window function $W\left(k\right)$ that defines the linear mass scale $M_{\text{PS}}$. For a Gaussian window function

$W\left(k\right)=\text{exp}\left(-\frac{k^2}{2k_0^2}\right)\text{and}{M}_{\text{PS}}=\frac{4}{3}\pi {\left(\frac{1.555}{k_0}\right)}^3{\Omega }_m{\rho }_{\text{crit}}.$ (17)

We calibrate the amplitude of $P_{\text{CDM}}\left(k\right)$ with ${\sigma }_8=0.811$ [1] with a “top-hat” window function of radius $8/h$ Mpc, and $f=1$. For our examples of Section 4, we take $f=1/0.79$ due to the recent accelerated expansion of the universe [14] . The resulting predictions of the galaxy stellar mass distributions with $M_{\text{PS}}\approx {10}^{1.5}{M}_{\text{*}}$, and the galaxy ultra-violet luminosity distributions, are excellent in a wide range of redshifts $z$ (provided $\tau \left(k\right)$ acquires a “tail” discussed in Section 6 below), see, for example [26] .

The formation of a galaxy with warm dark matter and baryons is described by hydrodynamical equations [15] . These equations are valid for collisionless, or collisional warm dark matter particles, so long as collisions are elastic. A numerical integration of these hydrodynamical equations is presented in Figure 1 , assuming spherical symmetry. We are interested in the limit of very small dark matter particle mass and large velocity dispersion, so, for all examples in this article, we choose

$m_h=0.15\text{keV}\text{and}{v}_{h\text{rms}}\left(1\right)=493\text{m}/\text{s}.$ (18)

This particular choice of parameters is justified in Table 4 of [4] . According to (8), (10) and (12) the comoving free-streaming cut-off wavevector is $k_{\text{fs}}\approx 0.67$ Mpc^-1, the comoving effective free-streaming length is $l_{\text{fs}}\approx 2.4$ Mpc, and the dark matter becomes non-relativistic at the expansion parameter $a_{h\text{NR}}\approx 1.7\times {10}^{-6}$. Note that this $a_{h\text{NR}}$ is in agreement with the measurement (5), and in disagreement with the limit (4). The linear total (dark matter plus baryon) mass corresponding to $k_{\text{fs}}$ is $M_{\text{fs}}\approx 2\times {10}^{12}{\text{M}}_{\odot }$.

The numerical integration in Figure 1 begins at redshift $z_i=65.9$, with initial dark matter and baryon densities

${\rho }_{hi}\left(r\right)=\frac{{\Omega }_h}{{\Omega }_b}{\rho }_{bi}\left(r\right)=\langle {\bar{\rho }}_{hi}\rangle \left[1+\delta \exp \left(-r^2/r_i^2\right)\right],$ (19)

with $\langle {\bar{\rho }}_{hi}\rangle =0.0119{\text{M}}_{\odot }/{\text{pc}}^3$ (corresponding to $z_i$), $r_i=2.72$ kpc and $\delta =0.0913$. The initial velocities are $v_{hi}\left(r\right)=v_{bi}\left(r\right)=H\left(t_i\right)r$ (and a tiny correction due to $\delta$). The linear “Press-Schechter” mass corresponding to $r_i$ is $M_{\text{PS}}={10}^9{\text{M}}_{\odot }$, and the relative mass fluctuation on this mass scale at $z_i$ is $\sigma =0.038$. The subscript $h$ stands for dark matter, and the subscript $b$ stands for “baryons”.

A moment past the last time presented in Figure 1 , the density ${\rho }_h\left(r\right)$ becomes a “cored isothermal sphere” [15] [16] , i.e. a solution of the static limit of the hydrodynamical equations, with a density run ${\rho }_h\left(r\right)\propto r^{-2}$ at $r\gg r_c$, and a core radius

$r_c=\sqrt{\frac{\langle v_{hr}^2\rangle }{2\pi G{\rho }_{hc}}},$ (20)

with the root-mean-square dark matter radial velocity

$\sqrt{\langle v_{hr}^2\rangle }\approx \frac{v_{h\text{rms}}\left(1\right)}{\sqrt{3}}{\left(\frac{{\rho }_{hc}}{{\Omega }_c{\rho }_{\text{crit}}}\right)}^{1/3},$ (21)

that is independent of $r$ and $t$ [16] . The radius of the isothermal sphere keeps growing due to the expansion of the universe [16] .

The numerical integration of Figure 1 obtains the “pivot point” P with $r_{h\text{P}}=9$ kpc and ${\rho }_{h\text{P}}=2\times {10}^{-5}{\text{M}}_{\odot }/{\text{pc}}^3$. From these numbers, we obtain the radius at which the dark matter halo density is 200 times the present mean dark matter density of the universe (since satellites of the Milky Way are observed at $z\approx 0$): $r_{200}=16$ kpc, and the corresponding halo mass

$M_h\equiv M_{200}=\frac{2\langle v_{hr}^2\rangle }{G}{r}_{200}=3\times {10}^8{\text{M}}_{\odot }.$ (22)

This is the mass that needs to be compared to the minimum observed halo mass (6) [21] . Note that

${\rho }_h\left(r\right)r^2={\rho }_{hc}{r}_c^2={\rho }_{h\text{P}}{r}_{h\text{P}}^2=200{\Omega }_c{\rho }_{\text{crit}}{r}_{200}^2,$ (23)

and

$\sqrt{\langle v_{hr}^2\rangle }=\sqrt{2\pi G{\rho }_{h\text{P}}{r}_{h\text{P}}^2}.$ (24)

The origin of the core is explained in [16] , and is of cosmological origin, see (21). In the example of Figure 1 , the core radius and density are $r_c=2.0$ kpc and ${\rho }_{hc}=4\times {10}^{-4}{\text{M}}_{\odot }/{\text{pc}}^3$. $\sqrt{\langle v_{hr}^2\rangle }=6.6$ km/s, so the velocity of rotation of a test particle is $V_{\text{rot}}=\sqrt{2\langle v_{hr}^2\rangle }=9.4$ km/s. This is a “small” galaxy indeed!

The simulation for $M_{\text{PS}}={10}^8{\text{M}}_{\odot }$, with the listed parameters, does not form a halo, see Figure 2 . Simulations of more massive galaxies are presented in Figure 3 and Figure 4 . The halo masses $M_h$ are given in the figure captions. For each simulation, the value of $\sigma \left(M_{\text{PS}},z,k_{\text{fs}}\right)$ (with a top-hat window function) is also given in the figure caption.

Let us consider Figure 1 . How probable is the density fluctuation (19) at $z_i=65.9$? This density fluctuation has a Fourier transform (17) with a cut-off comoving wavevector $k_0=1.555/\left(2.72\text{kpc}\cdot 66.9\right)=8.6{\text{Mpc}}^{-\text{1}}$. For our example (18), $k_{\text{fs}}=0.67$ Mpc⁻¹. The linear $P\left(k\right)$ is exponentially attenuated for $k>k_{\text{fs}}$, so the density fluctuation in Figure 1 is improbable. This is the argument in [2] .

However, there are at least three phenomena not considered in [2] :

1. The linear power spectrum $P\left(k\right)$ of (11) develops a non-linear regenerated tail.

2. A majority of “small” galaxies have lost matter to their neighbors.

3. If dark matter particles are bosons that decouple while ultra-relativistic, i.e. are in thermal equilibrium with zero chemical potential while ultra-relativistic, then the excess of low momentum particles behaves as cold dark matter, see Figure 10 of [18] .

The non-linear regeneration of $P\left(k\right)$ for warm dark matter is studied in [27] - [31] . We note, in Figure 4 of [28] , or Figure 18 of [29] , or Figure 4 of [31] , that, if the ΛCDM power spectrum is cut-off by warm dark matter free-streaming, non-linear regeneration of $P\left(k\right)$ starts with first galaxies, and is a major first-order effect that should not be neglected! One of the origial reasons for considering warm dark matter is to reduce the counts of small galaxies with respect to the cold dark matter prediction that exceeds observations. This is the “missing satellites” problem. The suppression factor is obtained with warm dark matter simulations in [27] . Translating Equation (28) of [27] to our notation, we obtain

$\frac{n_{\text{ΛWDM}}\left(M_{\text{PS}}\right)}{n_{\text{ΛCDM}}\left(M_{\text{PS}}\right)}={\left(1+0.61\frac{M_{\text{fs}}}{M_{\text{PS}}}\right)}^{-1.16}.$ (25)

From the data in Figures 1, Figures 2 and Figures 4 of [26] (for $z=4,5$ or 6 that have sufficient data), we estimate the ratio of galaxy counts $n_{\text{data}}\left(M_{\text{PS}}\right)/n_{\text{ΛCDM}}\left(M_{\text{PS}}\right)\approx 0.01$ at $M_{\text{PS}}={10}^9{\text{M}}_{\odot }$ (corresponding to Figure 1 ). From (25) we obtain the estimate $M_{\text{fs}}\approx 9\times {10}^{10}{\text{M}}_{\odot }$, corresponding to $k_{\text{fs}}\approx 2$ Mpc⁻¹, $v_{h\text{rms}}\left(1\right)\approx 170$ m/s, $m_h\approx 0.4$ keV, and $a_{h\text{NR}}\approx 6\times {10}^{-7}$. This estimate of $a_{h\text{NR}}$ is a factor ≈10 above the limit (4) and a factor ≈2.3 below the measurement (5).

Figure 3 of [32] summarizes published limits on $m_h$ from several observables. The most stringent limit for each observable is presented in Table 1 . These limits are obtained at $z\lesssim 8$ and so are sensitive to the regenerated power spectrum.

Consider the compilation of measurements of $P\left(k\right)$ in Figure 2 of [28] . The primordial linear power spectrum $P\left(k\right)$ at $z_{\text{dec}}=1090$ is measured by the Planck mission for comoving wavevectors up to $k\lesssim 0.2$ Mpc⁻¹, i.e. does not reach warm dark matter free-streaming. All other measurements of $P\left(k\right)$ correspond to the regenerated power spectrum. A direct measurement of the linear $P\left(k\right)$ up to $k\approx 20$ Mpc⁻¹, before the first galaxies, will become possible with weak gravitational lensing of the cosmic microwave background (CMB) [28] .

The observed reionization of the universe requires a delayed galaxy formation compared to the ΛCDM scenario. This delay requires $m_h={0.51}_{-0.12}^{+0.22}$ keV [33] , or $k_{\text{fs}}\approx 2\pm 1$ Mpc⁻¹ [25] , or $m_h={1.3}_{-0.7}^{+0.3}$ keV [34] , or $m_h={0.66}_{-0.08}^{+0.07}$ keV [34] . These measurements are in tension with the limits in Table 1 and can be compared with the measurements summarized in [4] .

Small galaxies can develop due to the non-linear regeneration of small-scale structures, and also due to “stripping”, i.e. loss of matter to neighbors.

Linear relative density perturbations are both positive and negative. Relative to a homogeneous universe, regions with under-density grow faster, and regions with overdensity grow slower, and peaks turn around and collapse into halos. The result is underdense regions, surrounded by sheets, that meet at filaments, that meet nodes where large galaxy clusters form. Galaxies punctuate the nodes and filaments, and to a lesser extent, the sheets and voids.

Let us try to understand the formation of small galaxy satellites. In the warm dark matter scenario, the first galaxies have masses determined by the warm dark matter free-streaming length, i.e., a distribution of linear masses $M_{\text{PS}}$ of order $M_{\text{fs}}$. The dark matter density ${\rho }_h\left(r\right)$ of these halos becomes equal to the mean dark matter density of the universe at a radius $r_{\max }$ that grows in proportion to $a^{3/2}$, faster than the separation between galaxies $\propto a$ [16] . Therefore, as the universe expands, groups of galaxies overlap and coalesce, becoming larger galaxies. Smaller galaxies become caught up between larger galaxies, loosing matter to their expanding neighbors, and are either absorbed completely, or collapse as “stripped-down” satellites with large cores (and continue loosing matter to their neighbors, perhaps leaving behind a globular cluster). Thus, in the warm dark matter scenario, the first galaxies have masses of order $M_{\text{fs}}$, and, from there on, the structure forms bottom-up and top-down, as confirmed by simulations [30] .

Figure 1 of [35] can help us understand stripped-down galaxies. In the simulation of Figure 5 of [35] , $M_{\text{fs}}=3.6\times {10}^{11}{\text{M}}_{\odot }$ and $m_h=0.112$ keV, similar to our example (18). We find that the number of stripped-down satellites per unit volume and a decade of mass, for a linear mass $M_{\text{PS}}\ll M_{\text{fs}}$, is reduced to ≈15% of the corresponding density in the ΛCMD cosmology (some, however, do not collapse to a halo). Note that the Milky Way satellites are not formed as in Figure 1 , but are stripped-down galaxies. The distribution of mass of stripped-down galaxies extends all the way to zero, and does not exclude the measurement (5).

The formation of a stripped-down galaxy is shown in Figure 5 . This galaxy starts out as in Figure 4 with $M_{\text{PS}}={10}^{11}{\text{M}}_{\odot }$, but looses matter to neighbors (with spherical symmetry for convenience). The pivot point P has $r_{h\text{P}}=32$ kpc, and ${\rho }_{h\text{P}}=1.3\times {10}^{-5}{\text{M}}_{\odot }/{\text{pc}}^3$. So ${\rho }_{h\text{P}}{r}_{h\text{P}}^2=1.33\times {10}^4{\text{M}}_{\odot }/\text{pc}$, compared to $3.06\times {10}^4{\text{M}}_{\odot }/\text{pc}$ for the galaxy with no stripping in Figure 4 . The parameters scale as

${\rho }_{h\text{P}}{r}_{h\text{P}}^2\propto r_c^{-4},{\rho }_{hc}^{2/3},M_{hc}^{4/3},M_{200}^{2/3},r_{200}^2,\langle v_{hr}^2\rangle ,V_{\text{circ}}^2.$ (26)

So, with respect to the galaxy with no stripping in Figure 4 , the core mass $M_{hc}$ is reduced by a factor 0.54, $M_{200}\equiv M_h$ by a factor 0.29, and the circular velocity of a test particle is reduced by a factor 0.66.

A simple empirical way to account for both non-linear regeneration and stripped-down galaxies is to take

${\tau }^2\left(k\right)=\{\begin{array}{ll}\text{exp}\left(-\frac{k^2}{k_{\text{fs}}^2}\right)\hfill & \text{if}k<k_{\text{fs}},\hfill \\ \text{exp}\left(-\frac{k^n}{k_{\text{fs}}^n}\right)\hfill & \text{if}k\ge k_{\text{fs}},\hfill \end{array}$ (27)

where $n$ is measured to be in the range 0.5 to 1.1 [25] . This “tail” is sufficient to bring predictions in line with observations, see [26] . The effect of $n$ on the predicted stellar mass distributions and on the ultra-violet luminosity distributions is presented in Figure 3 of [25] .

Even the linear power spectrum $P\left(k\right)$ has a tail calculated in [18] . For our example in Section 5, with $k=8.6{\text{Mpc}}^{-1}$ and $k_{\text{fs}}=0.67{\text{Mpc}}^{-1}$, we estimate $n\approx 1.08$ from Figure 10 of [18] for boson dark matter that decouples while ultra-relativistic (this is because the enhancement of low momentum bosons behaves like cold dark matter). In comparison, ${\tau }^2\left(k\right)$ of (9) has a tail corresponding to $n\approx 1.36$.

Warm dark matter free-streaming attenuates small-scale linear density perturbations. However, non-linear regeneration of small-scale structures is a major effect. Furthermore, a majority of galaxies with linear mass $M_{\text{PS}}\ll M_{\text{fs}}$ are stripped-down galaxies that have lost matter to neighbors. The limit (4), based on the observation of small Milky Way satellite galaxies, does not take regeneration, or stripping, into account. With the warm dark matter simulations in [27] , an estimate of this limit becomes $a_{h\text{NR}}\lesssim 6\times {10}^{-7}$, which is a factor ≈10 above the limit (4), and a factor ≈2.3 below the measurement (5). An empirical way to deal with non-linear regeneration and stripping is to add to ${\tau }^2\left(k\right)$ a “tail” as in (27). For the measured $n$ [25] , we obtain agreement with observed galaxy stellar mass distributions, and observed galaxy ultra-violet luminosity distributions, over a wide redshift range, down to $M_{\text{PS}}=5\times {10}^8{\text{M}}_{\odot }$, or $M_h=2\times {10}^8{\text{M}}_{\odot }$, see [26] . So the minimum halo mass (6) is not excluded. Note that the mass distribution of stripped-down galaxies extends all the way to zero.

We are still far from a detailed and quantitative understanding of galaxy formation with warm dark matter. Each limit and measurement has its own issues, so caution and an open mind are called for. In any case, we are unable to rule out the measurement (5) beyond a reasonable doubt, in spite of all the published limits to the contrary.

All limits on $k_{\text{fs}}$ that depend on galaxies need to include in the analysis the regeneration of small-scale structure, and the formation of stripped-down galaxies. Limits on $k_{\text{fs}}$ from the Lyman-α forest need, in addition, to understand clouds of “left over” neutral hydrogen traces in the re-ionized universe [36] . On the other hand, measurements of $v_{h\text{rms}}\left(1\right)$ with galaxy rotation curves are direct, but have the issue of relaxation that is constrained by observations, see [4] and references therein.

I thank Karsten Müller for his early interest in this work and for many useful discussions.