If dark matter is warm instead of cold, the power spectrum of density fluctuations becomes cut-off at short wavelengths due to dark matter particle free-streaming, first generation galaxies acquire a finite mass M h 1 , the number of galaxies per unit volume and mass range decreases at the low mass end, galaxy halos acquire a finite core density ρ c h (if no central black hole) with finite radius r c , the number density of isolated dwarf galaxies in the Local Field becomes reduced, reionization becomes delayed and the optical depth increases, the formation of low mass galaxies becomes delayed, and the number of satellite galaxies becomes reduced. Each of these effects offers an opportunity to measure the dark matter warmness.

Recently it has become possible to obtain the dark matter density ρ h ( r ) of dwarf spheroidal galaxies dSph by measuring the line-of-sight velocity of individual stars. This dark matter dominates the density ρ ( r ) of the dSph galaxies. Fitting ρ h ( r ) of 27 dwarf spheroidal galaxies [1] we arrive at the following tentative conclusions: i) dark matter is warm, ii) some dwarf spheroidal galaxies are first generation galaxies in the warm dark matter scenario, iii) rotating dwarf irregular galaxies dIrr and spiral galaxies have dark matter halos that are rotating strongly, and iv) the dark matter halo rotation of some dSph is negligible. With this new perspective, in the present article we update four published estimates of the dark matter warmness. Each of these estimates depends on assumptions and has its own delicate issues. It is the broad agreement of the four largely independent estimates that gives weight to the conclusion that dark matter is warm. The results of this study may also be applied to other extensions of the standard cold dark matter cosmology ΛCDM that have a power spectrum with a cut-off wavevector k fs [2].

The mass of a galaxy is often defined as the mass within a radius r 200 corresponding to a mean galaxy matter density 〈 ρ m ( r < r 200 ) 〉 = 200 Ω m ρ crit . Ω m ρ crit is the present day mean matter density of the universe. We assume that warm dark matter is a noble (i.e. not excitable), non-degenerate, non-relativistic, gas of particles of mass m h that are collisional or collisionless. It is in this scenario that we interpret the dSph data. The “warmness” of this gas may be specified by any of the equivalent parameters v h rms ( 1 ) , k fs , m th , m h or a NR . v h rms ( 1 ) is the comoving root-mean-square of the thermal velocity of the dark matter particles defined as

a is the expansion parameter normalized to a ( t 0 ) = 1 at the present time t 0 . v h rms ( a ) is the root-mean-square of the thermal velocity of the dark matter particles when the nearly homogeneous universe has expansion parameter a . v h rms ( a ) is proportional to a − 1 , so v h rms ( 1 ) is an adiabatic invariant. Due to the velocity dispersion v h rms ( a ) , dark matter particles move into, or out of, density minimums, or maximums, thereby attenuating short wavelength density perturbations. As a result of this dark matter free-streaming, the comoving density fluctuation power spectrum of the ΛCDM cosmology becomes multiplied by a cut-off factor τ 2 ( k ) , where k is the comoving wavevector [3]. We assume that the dark matter particles have a Maxwell distribution of velocities (for a justification of the negative chemical potential of the non-relativistic warm dark matter, see [4]). In this case τ 2 ( k ) has the form τ 2 ( k ) = exp ( − k 2 / k fs 2 ) [5]. This is our definition of k fs . (Other definitions in the literature are τ 2 ( k fs ) = 1 / 4 or 1/2.) The relation between v h rms ( 1 ) and k fs is given by Equation (38) of [5]. Instead of specifying k fs , it is customary in the literature to give the “standard thermal relic mass” m th (of spin 1/2) defined by Equations (6) and (7) of [6]. Note however, that the dark matter particle mass is model-dependent. For boson dark matter with N b = 1 (assumed to have zero chemical potential while ultra-relativistic) the mass m h is given by Equation (20) of [1] (see also Equation (26) of [7]). Finally, the expansion parameter a NR at which dark matter becomes non-relativistic, is defined as a NR ≡ v h rms ( 1 ) / c . For convenience, the relation between these parameters is presented in Table 1.

Table 1. The warmness of dark matter can be specified by any of the equivalent parameters v h rms ( 1 ) , k fs , m th , m h or a NR defined in Section 2. Their relations are presented in this Table.

Recently it has become possible to derive the dark matter density ρ h ( r ) of dwarf spheroidal galaxies dSph from the observed line-of-sight velocities of individul stars. In [1] we fit the solution of hydrostatic equations to the density runs ρ h ( r ) of 27 dSph. The fits minimize a χ 2 by varying three parameters: the density ρ h ( r min ) of the first measured point (at r min ≈ 0.01 kpc), the variable v ′ h rms ( 1 ) / 1 − κ h , and the mass M BH of a possible central black hole. κ h is the fraction of gravity supported by dark matter halo rotation. We neglect κ h for some dSph (to be discussed below). The definition of v ′ h rms ( 1 ) is

〈 v r h 2 〉 is the root-mean-square of the radial component of the velocities of the dark matter particles, assumed to be independent of r (see, for example, [8]). The warm dark matter adiabatic invariant v h rms ( 1 ) is obtained from v ′ h rms ( 1 ) as follows:

f = 1 if dark matter is collisional. For collisionless dark matter, see discussion below. ϵ = 1 if the first generation galaxies formed without relaxation, or ϵ < 1 if there is relaxation. If the dSph has a black hole, then v ′ h rms ( 1 ) becomes dependent on the first measurement at r min so a correction η needs to be applied [1].

Equations (2) and (3) may be understood as follows. Consider collisional warm dark matter, and an observer in a density peak in the early universe. This observer feels no gravity, “sees” warm dark matter expand and then contract to form the core of a galaxy (as shown by hydrodynamical equations [9]). If the contraction is adiabatic, Equations (2) and (3) follow.

Figure 1. Measured dark matter density ρ h ( r ) of dwarf spheroidal galaxies (data from [13]-[17]). The continuous lines are solutions of hydrostatic equations [1] fitted with fixed M BH = 0 .

A comment on collisional vs collisionless dark matter. As and example take a core density ρ c = 10 8 M ⊙ / kpc 3 , core radius r c = 0.2 kpc (see Figure 1 and Figure 2), and, tentatively, a dark matter-dark matter collision cross-section per unit mass σ DM-DM / m h = 0.1 cm²/g (see Table 1 and Figure 13 of [10], and

[11]). For this example, 〈 v r h 2 〉 = 10 km/s, and the mean number of collisions of a particle with this velocity and permanently in the core (unrealistic) in the age of the Universe is of order 0.3. So dark matter may be significantly collisional if σ DM-DM / m ≈ 0.1 cm²/g, while σ DM-DM / m ≳ 1 cm²/g seems to be ruled out (see studies in [11]).

Figure 2. Measured dark matter density ρ h ( r ) of dwarf spheroidal galaxies (data from the link https://github.com/koreshk/Estimation-of-phase-space-density-in-dwarf-galaxies on 30 January 2026 given in [17]). The continuous lines are solutions of hydrostatic equations [1] fitted with fixed M BH = 0 .

Consider the case of collisionless warm dark matter. In this case f = x 1 / 6 / 2 ⋅ ln ( x ) , where x ≡ ρ c / ρ B , and the background density is ρ B = Ω c ρ crit / a 3 (see Appendix of [12]). For example, for galaxies observed at a = 1 , 0.3 , 0.1 , f = 2.2 , 1.4 , 0.9 , respectively. For our present estimate, we will take, as a bench-mark, f ≈ 3 .

To avoid the uncertainty of η due to the black hole, here we will study a subset of the 27 dwarf spheroidal galaxies that have a black hole mass that is not significantly different from zero, i.e. we require that the χ 2 of the fit increase by less than 3 units when M BH is fixed to zero. We also require that the galaxy have M h < 10 10 M ⊙ (to favor first generation galaxies). This selection leaves 11 dSph. Fitting these 11 dSph with M BH fixed to zero, obtains the results in Figure 1 and Figure 2, and summarized in Table 2.

Table 2. For dwarf spheroidal galaxies dSph with black hole mass M BH consistent with zero (see text), and M h ( < r 200 ) < 10 10 M ⊙ , we present the stellar mass M * and neutral hydrogen mass M HI (from the LVDB catalog [18]), and M h ( < r 200 ) and v ′ h rms ( 1 ) / ( 1 − κ h ) (from the fit with fixed M BH = 0 ). Some entries are not available.

Let us now discuss dwarf spheroidal galaxy rotation. For baryons, the fraction of gravity supported by stellar rotation with velocity V T b is κ b = V T b 2 / ( 2 〈 v r h 2 〉 ) . The measured value of κ b for Draco, Sextans, Umi, Fornax and Sculptor, assuming 〈 v r b 2 〉 ≈ 〈 v r h 2 〉 , is κ b ≈ 0.3 2 / 2 = 0.045 , see last column of Table 3 of [21]. See also [22], and Table 1 of [1]. For dark matter we expect κ h ≲ κ b , so neglecting κ h for our sample of 11 dSph may be justified. The upper panel of Figure 3 presents measured distributions of v ′ h rms ( 1 ) / 1 − κ h for the dSph (red), dwarf irregular dIrr (green) and spiral galaxies (blue). The three distributions in the top panel of Figure 3 become approximately overlaid if κ h ≈ 0 for dSph, κ h ≈ 0.90 for dIrr, and κ h ≈ 0.97 for spiral galaxies. So our interpretation is that the dark matter halos of dIrr and spiral galaxies are rotating strongly.

Figure 3. Top: Measurements of v ′ h rms ( 1 ) / 1 − κ h . v ′ h rms ( 1 ) is defined in (2). The distributions correspond to the 11 dwarf spheroidal galaxies dSph from Table 2 (red), rotating irregular dwarfs dIrr from Figure 2 and Table 2 of [19] (green), and spiral galaxies from Figure 4 and Table A3 of [20] (blue). Bottom: Distribution of the mass M h of the 11 dwarf spheroidal galaxies (from Table 2).

The distribution of the 11 measured v ′ h rms ( 1 ) / ( 1 − κ h ) in Table 2 with equal weights, has a mean 152 m/s and a standard deviation 64 m/s. This standard deviation is due to residual rotation, and experimental uncertainties. We seek the lower bound of the distribution of v ′ h rms ( 1 ) / ( 1 − κ h ) presented in Figure 3, to be able to set ϵ = 1 . We estimate

For comparison, the estimate in [1] is v h rms ( 1 ) = ( η ϵ / f ) ⋅ ( 88 ± 43 ) m / s . Note that we no longer have the correction η due to a central black hole, nor the correction ϵ due to relaxation.

From Figure 1 and Figure 2 we observe a tendency of ρ h ( r ) to increase at r ≲ 0.05 kpc. If this effect is significant, it may be due to a central black hole, and/or baryons, and/or non-isotropic velocities of collisionless dark matter. If dark matter velocities are not isotropic then there is a tendency of ρ h ( r ) to increase at r ≲ 0.05 kpc. To investigate this possibility we define

Now the spherically symmetric non-rotating hydrostatic equations become

Fits to the data for Leo I, for three values of β (assumed, for simplicity, to be independent of r ), are presented in Figure 4. This test shows that the dark matter particle velocities are approximately isotropic, so we should take f ≈ 1 . Note that dark matter may be collisional.

Figure 4. Three fits to the observed ρ h ( r ) of Leo I [13] are shown. The fits correspond, from top to bottom, to β = 0.11 , 0.14 , 0.17 with χ 2 = 14.04 , 2.58 , 11.79 , respectively, for 11 degrees of freedom. M BH = 0 is fixed. This figure tests the degree of anisotropy of dark matter particle velocities.

Distributions of galaxy mass exhibit a minimum mass M h 1 . For redshift z = 3 , 4 , 5 , 6 , 7 , 8 the minimum observed log 10 ( M h 1 / M ⊙ ) is ≈10.9, 8.7, 8.7, 8.7, 8.7, 8.7 respectively [23]. So first generation galaxies appear to have a mass M h 1 ≈ 5 × 10 8 M ⊙ or log 10 ( M h 1 / M ⊙ ) ≈ 8.7 . This minimum M h 1 is confirmed by Table 2 and Figure 3, so it appears that some dwarf spheroidal galaxies are first generation galaxes in the warm dark matter scenario.

We wish to obtain k fs from M h 1 . Simulations in [24] obtain M h 1 = 2 × 10 11 , 2 × 10 10 , 2 × 10 9 M ⊙ for m th = 0.25 , 0.5 , 1 keV, corresponding to k fs = 1.2 , 2.6 , 5.5 Mpc^-1, respectively. For M h 1 = 5 × 10 8 M ⊙ we estimate k fs ≈ 8.7 Mpc⁻¹, and m th ≈ 1.5 keV.

As another estimate we use Equation (8) of [25] to calculate the Jeans Mass M J given m th , and then estimate M h 1 ≈ M J / 2 (that is not critical, see Figure 2 of [25]). For M h 1 = 5 × 10 8 M ⊙ we obtain m th = 0.77 keV or k fs = 4.1 Mpc⁻¹. (For M h 1 = 2 × 10 8 M ⊙ we obtain m th = 0.98 keV or k fs = 5.4 Mpc⁻¹. For M h 1 = 1 × 10 9 M ⊙ we obtain m th = 0.65 keV, or k fs = 3.4 Mpc⁻¹).

So, from the mass M h 1 of first generation galaxies in the warm dark matter scenario, we estimate

The agreement with other determinations of k fs reinforce the conclusion that some dSph may be first generation galaxies in the warm dark matter scenario.

The observed stellar mass M * distributions as well as Press-Schechter predictions, and ellipsoidal collapse extensions, of the halo mass M h distributions are presented in [23]. The predictions are calculated with a Gaussian window function. The relation between M * and M h is assumed to be M h / M * = 10 1.5 , independently of M h . Here we wish to obtain a quantitative estimate of k fs from these distributions.

First we anchor the predictions to the Millenium simulations corresponding to the cold dark matter cosmology ΛCDM at M h = 10 10 M ⊙ . From Figure 2 of [26] we obtain, at redshift z = 5.72 and M h = 10 10 M ⊙ , [ M h 2 / ( Ω m ρ c r i t ) ] ⋅ d n / d M h = 0.022 , corresponding to d n / d log 10 ( M h ) = 0.20 dex⁻¹∙Mpc⁻³. Correcting this number from z = 5.72 to z = 6 obtains d n / d log 10 ( M h ) = 0.166 dex⁻¹∙Mpc⁻³. The Millenium simulation assumes σ 8 = 0.90 . We correct to σ 8 = 0.811 multiplying the power spectrum by (0.811/0.90)². The result is d n / d log 10 ( M h ) = 0.10 dex⁻¹∙Mpc⁻³, see small red star in Figure 5.

Secondly, we consider an improved relation between M * and M h which now becomes dependent on M h . For the star formation efficiency f * we use Equation (11) of [27] with f * 0 measured to be in the range 0.5 to 1.0, see Figure 4 of [27]. So at M h = 10 10 M ⊙ we obtain f * in the range 0.128 to 0.255. So M * is in the range M * ≡ M h f * Ω b / Ω m = 2 × 10 8 to 4 × 10 8 M ⊙ . Also log 10 ( M h / M * ) is in the range 1.7 to 1.4 (instead of 1.5 in Figure 1 of [23]). So we neglect this correction.

Finally, we switch from the Gaussian window function to the top-hat window function in r-space. We obtain Figure 5, calibrated at M h = 10 10 M ⊙ to the ΛCDM Millenium simulation, i.e. d n / d log 10 ( M h ) = 0.10 dex⁻¹∙Mpc⁻³, and estimate

Compare in detail with [23].

Figure 5. Comparison of predicted and observed distributions of M h / M ⊙ = 10 1.5 M * / M ⊙ at redshift z = 6 . Stellar mass data are from the Hubble Space Telescope [28] (black squares), from the continuity equation [29] (red triangles), and from the James Webb Space Telescope [30] (green triangles). The predictions are the ellipsoidal collapse extension of the Press-Schechter formalism (Equation (5) of [31] with the top-hat window function in r-space) normalized to ΛCDM at M h = 10 10 M ⊙ (small red star) as explained in the text.

In [32] we obtain k fs from the number density of isolated dwarf galaxies in the Local Field, i.e. within 3 Mpc of the Milky Way, excluding the Milky Way and Andromeda galaxies and satellites. We obtain k fs ≈ 5.6 before applying a large correction due to the non-linear regeneration of the density fluctuation power spectrum at large wave vector k . This regeneration occurs during the hierarchical formation of galaxies. I now realize that this non-linear regeneration correction should be applied to the universe at large, but not to first generation isolated dwarf galaxies in the Local Field that has a density approximately equal to the mean density of the universe. Consider and overdense region. This overdense region expands slower than a region with the mean density, and hence the comoving wavelength of density fluctuations in the overdense region decreases. This is a cause of the regeneration of the power spectrum at short wavelength or large wave vector. So, for first generation isolated dwarf galaxies in the Local Field, the non-linear regeneration correction should not be applied (only an uncertainty due to the uncertain density of the Local Field needs to be included). Omitting the non-linear regeneration correction in [32] obtains

As a cross-check with data, consider Figure 1, Figure 3 and Figure 7 of [33] for the pseudo-isothermal sphere ρ h ( r ) = ρ c h / ( 1 + r 2 / r c 2 ) . The dark matter temperature-to-mass ratio is T h ( r 200 ) / m h = ( 3.6 ± 0.5 ) × 10 − 6 K/eV at the “edge” of the halo, i.e. at r 200 defined such that ρ h ( r 200 ) = 200 ρ crit . This value of T h / m h is also valid in the center of halos with the least mass, i.e. M h ≈ 10 8.7 M ⊙ that are not rotating significantly (see Figure 1, Figure 3 and Figure 7 of [33]). Note that for the pseudo-isothermal sphere, 〈 v r h 2 〉 is approximately independent of r (see Figure 6 of [33]). The corresponding 〈 v r h 2 〉 is 5300 m/s for the lightest dwarfs with M h ≈ 10 8.7 M ⊙ , with non-rotating dark matter halo, and with core density ρ c ≈ 10 8 M ⊙ / kpc 3 (see Figure 1 and Figure 2 above). Then the dark matter comoving thermal velocity is v h rms ( 1 ) ≈ 64 / f m/s. This cross-check with data validates, within a factor ≈2, the measurement method of v h rms ( 1 ) presented in Section 3 for non-rotating first generation galaxies in the warm dark matter scenario.

Cross-check with simulations: The central dark matter densities of the dSph in Figure 1 and Figure 2 range from ρ c h ≈ 1 × 10 7 M ⊙ to ≈ 5 × 10 8 M ⊙ , or ρ c h / ρ crit = 8 × 10 4 to 4 × 10 6 . From Figure 2 of [34] we obtain m th ≳ 0.13 keV. From Figure 1 and Figure 2 the core radius is r c ≈ 0.1 (within a factor ≈2). From Figure 8 of [34] we obtain 0.4   keV ≲ m th ≲ 1.8   keV .

In [32] we obtain k fs from the number density of isolated dwarf galaxies in the Local Field. We repeat that analysis at redshift z = 7 that corresponds approximately to the redshift of half-reionization. We vary k fs to obtain the observed number density of galaxies at z = 7 . The result is k fs ≈ 7.0 ± 1.0 Mpc⁻¹. One simulation is presented in Figure 6. Corrections need to be studied. Never-the-less, k fs can not be very different from ≈7 Mpc⁻¹.

Figure 6. Warm dark matter density ρ h ( r ) , at redshift z = 7 with k fs = 7 Mpc⁻¹, and coordinate sg_zz = 3.3 Mpc, obtained from the comoving power spectrum of density fluctuations P Λ CDM ( k ) exp ( − k 2 / k fs 2 ) . The contours of relative overdensity are δ = 0 (red), 1.0 (green), and 1.686 (blue) corresponding to collapsed galaxy halos. For details, see [32].

We tentatively conclude that some dwarf spheroidal galaxies have negligible dark matter halo rotation, while rotating irregular dwarf and spiral galaxies have dark matter halos that rotate strongly (so rotation may be mainly due to the hierarchical formation of galaxies). We tentatively conclude that some dwarf spheroidal galaxies may indeed be first generation galaxies in the warm dark matter scenario. A summary of the four estimates discussed in this article is presented in Table 3. These estimates are in disagreement with published limits based on the number of Milky Way satellites, the Lyman-α flux power spectrum, and strong gravitational lensing, summarized in Figure 3 of [35]. Each of these estimates and limits has its own delicate issues, so more studies, measurements and cross-checks are needed to understand the disagreements. Comments on the number of Milky Way satellites, and on the Lyman-α flux power spectrum, are presented in Appendix A and Appendix B. Never-the-less, the broad agreement of these four largely independent estimates is evidence in favor of warm dark matter (or other extensions of ΛCDM with a power spectrum with a cut-off wavevector k fs ). These estimates allow an extrapolation of the dark matter temperature to the past [1] [7]. The suggestion is that the warm dark matter particles couple to a high-energy extension of the Standard Model of quarks and leptons [1].

Table 3. Summary of four estimates of k fs or v h rms ( 1 ) . f ≈ 1 (see Figure 4). These estimates imply 0.6   keV ≲ m th ≲ 1.6   keV , see Table 1.