We focus our attention on the number density of isolated dwarf galaxies in the Local Field because it might tell us something about the nature of dark matter. Most matter in the universe is in a “dark matter” form that has only been “observed” through its gravitational interaction [1]. In the current ΛCDM cosmology model, dark matter is a gas so cold that the dispersion velocity of the dark matter particles has a negligible effect on cosmological observations [1]. Nevertheless, we would like to know just how warm dark matter is to be able to make sensible extrapolations to the past, and also to perhaps understand several tensions of the ΛCDM cosmology on small scales [1], such as the large excess of predicted dwarf galaxies in the Local Field (see Section 5 below). We assume that dark matter is a gas of particles of mass m h . The “warmness” of dark matter can be specified by several related parameters, e.g., the comoving root-mean-square velocity v h rms ( 1 ) , the comoving linear density power spectrum cut-off wavevector k fs due to free-streaming, or the “standard thermal relic mass” m th . v h rms ( 1 ) is defined as

v h rms ( a ) is the root-mean-square non-relativistic thermal velocity of dark matter particles in the early, nearly homogeneous universe at expansion parameter a ( t ) . The corresponding dark matter density is ρ h ( a ) . The expansion parameter is normalized to a ( t 0 ) = 1 at the present time t 0 . At the present time, the dark matter density is ρ h ( 1 ) = Ω c ρ crit . Since v h rms ( a ) is proportional to a − 1 and ρ h ( a ) ∝ a − 3 , v h rms ( 1 ) is an adiabatic invariant.

Warm dark matter particles free-stream in and out of density minimums and maximums, thereby erasing small-scale density fluctuations. Let P Λ CDM ( k ) be the well-known comoving linear density power spectrum of ΛCDM cosmology, before the first galaxies, i.e., before non-linear re-generation of small-scale structure. k is the present time wavevector. Then, the comoving linear power spectrum in the warm dark matter model is P Λ WDM ( k ) = P Λ CDM ( k ) τ 2 ( k ) , where τ 2 ( k ) is a cut-off factor of the form τ 2 ( k ) = exp ( − k 2 / k fs 2 ) if dark matter has the Maxwell distribution of velocities [2]. This is our definition of k fs (alternative definitions in the literature are τ 2 ( k fs ) = 1 / 2 or 1/4). The relation between v h rms ( 1 ) and k fs (at a time t gal before the first galaxies and the corresponding non-linear re-generation of density fluctuations) is [2][3]

if dark matter particles have the Maxwell velocity distribution. The relation between the “standard thermal relic mass” m th and k fs is often defined by Equation (6) and Equation (7) of [4]. Note that the actual mass m h is model-dependent [5].

A summary of measurements of v h rms ( 1 ) can be found in [5]. A summary of limits on m th is presented in Figure 3 of [6], see references therein. These limits on m th place corresponding limits on k fs and v h rms ( 1 ) summarized in Table 2 of [7]. It turns out that there is a discrepancy between these measurements and limits that is not currently understood. The tightest limit on k fs is obtained from the number counts of satellites of the Milky Way. Here, we attempt to estimate k fs in the arguably simplest setting: we compare predictions with the observed number of isolated dwarf galaxies in the Local Field. Most of these dwarf galaxies are “red and dead”, i.e., old, and some are young and blue with stars in the main sequence, so their age has a wide distribution [8]. The isolated dwarf galaxies have generally not experienced mergers or accretions, and are not “stripped down” galaxies that have lost matter to neighboring hosts during their formation and hierarchical evolution. Counting isolated dwarf galaxies in the Local Field avoids the need to understand the formation and evolution of the Milky Way with its satellites in the warm dark matter scenario. Our main interest is to understand the discrepancy mentioned above. Therefore, we present this analysis in full detail, so the reader can check each step of the way (that is often not possible with the published limits). Isolated dwarf galaxies in the Local Field may be first-generation galaxies, a concept valid if dark matter is warm, and will surely teach us something.

We study the “Local Field” dwarf galaxies in the Local Volume Database (LVDB) catalog [9] (in file read_comb_all.csv of October 11, 2025). A handy application (https://lvd-interactive.streamlit.app/) is available to visualize the distributions of the dwarf galaxies classified as Milky Way Dwarfs, M31 Dwarfs, Local Field Dwarfs, Distant Local Field Dwarfs, Globular Clusters, etc. The Local Field Dwarfs are defined to be at a distance less than 3 Mpc from the Sun, and are unassociated with the Milky Way or with M31. The catalog is complete for all known dwarf galaxies in the Local Field. Beyond the Local Field, the catalog is not complete. The multiple (and sometimes ambiguous) criteria to define a star cluster as a dwarf galaxy are listed in the catalog. The Local Field has 64 dwarf candidates, of which 55 are confirmed-real and confirmed-dwarf. We do not consider 11 of these dwarf galaxies because they are satellites of a host galaxy. Of the remaining 44 isolated dwarf galaxies, 42 are contained in the volume 6 Mpc × 6 Mpc × 2 Mpc defined in Figure 1, and within 3 Mpc of the Sun, and excluding the volumes assigned to the Milky Way and M31, leaving 56.5 Mpc³. The corresponding number density is 0.743 (1 ± 42^−1/2) isolated dwarf galaxies per Mpc³. The Milky Way lies at the center of Figure 1, but is not shown. The stellar mass distribution is presented in Figure 2. The stellar mass is obtained from the absolute magnitude M V in the V-band, assuming a mass-to-light ratio twice the solar ratio. We will assume that the distribution of stellar masses presented in Figure 2 is cut-off at the lower end, so the number density of dwarf galaxies in the Local Field is well defined (this assumption needs to be re-evaluated in the end). The volume of Figure 1 contains larger galaxies, notably the Milky Way and M31 with their satellites, as well as 11 smaller hosts, which are not shown nor counted.

A few of the dwarf galaxies in the Local Field have a measured line-of-sight velocity dispersion measurement. However, we do not use this information in the present analysis, we use only the number density in order to obtain an estimate of k fs that is independent of measurements based on rotation curves, line-of-sight velocity dispersions or stellar dynamics.

Figure 1. Supergalactic coordinates of 42 confirmed-real and confirmed-dwarf isolated galaxies in the Local Field, from the LVDB catalog [9]. The Milky Way (not shown) is at the center.

Figure 2. Stellar mass distribution of 42 isolated Local Field dwarf galaxies from the LVDB catalog [9] (that are confirmed-real and confirmed-dwarf).

What can we learn from Figure 1? The number density of isolated dwarf galaxies in the Local Field is 1/(1.1 Mpc)³. So a typical nearest neighbor distance is 1.1 Mpc. This means that the warm dark matter power spectrum P Λ CDM ( k ) exp ( − k 2 / k fs 2 ) is non-negligible at k = 2 π / ( 1.1   Mpc ) = 5.7 Mpc⁻¹, so k fs can not be very much smaller than 5.7 Mpc⁻¹.

Cold dark matter predicts a number density of dwarf galaxies much greater than observed (see Section 5). This is one of the reasons to consider warm dark matter. To reduce the number density of dwarf galaxies significantly, k fs can not be very much greater than 5.7 Mpc⁻¹.

Therefore, we make the following order-of-magnitude estimate:

Let us establish the notation [1]. The relative over-density at the present time is δ ( x ) ≡ ( ρ ( x ) − 〈 ρ 〉 ) / 〈 ρ 〉 . We apply periodic boundary conditions to a large cube of volume L 3 . Then, the wavevector has discrete values k = ( 2 π / L ) ( n e x + m e y + l e z ) , so k ≡ | k | = ( 2 π / L ) n 2 + m 2 + l 2 . The relative over-density may be written as

where the sums are over 8 octants of k . Taking δ k = δ − k * ≡ | δ k | e − i ϕ obtains

The variance of the density defines the power spectrum P ( k ) :

so

with P ( k ) in units Mpc³. Brackets denote volume averages. In detail, at the present time, and in the linear approximation (that has already broken down), we take

where

For our simulations, we take L = 10 Mpc. δ B is the relative over-density of the Local Volume, and fills in for wavevectors less than 3 ⋅ 4 π / L . We take δ B = 0 , and in the end, add a systematic uncertainty on k fs to include the uncertainty of δ B . The ϕ i ( n , m , l ) with i = 1 , 2 , 3 , 4 are random phases between 0 and 2 π . These phases are generated with a common seed, so that runs with different coordinate sg_zz can be compared. The linear power spectrum P Λ CDM ( k ) extrapolated to the present time is taken from the analytical expression (Equation (8.1.42) of [10]) with a spectral index slope n s = 0.965 and an amplitude normalized to σ 8 = 0.811 [1]. As a cross-check, we present this power spectrum in Figure 3.

Figure 3. Present-day linear power spectrum of ΛCDM cosmology (from Equation (8.1.42) of [10] with n s = 0.965 and normalized to σ 8 = 0.811 ) [1].

Figure 4. Simulations at redshift z= 0 with k fs = 2 , 3 , 4 , 5 , 7 and 20 Mpc⁻¹. Shown are contours of constant relative overdensity: δ ( x ) = 0 red, 1.69 blue, 3 redish, 4 black, and 5 green, in the linear approximation (that has already broken down). Some of these contours may be omitted for clarity. The coordinates are supergalactic. sg_zz = 2.5 Mpc.

Here, we neglect non-linear re-generation of small-scale structure after the formation of the first galaxies. Therefore, the measurement in this section obtains an upper bound to k fs . Here, we also neglect a possible Local Volume over-density or under-density, i.e., we set δ B = 0 . To illustrate the sensitivity of the measurement of k fs , we present simulations with several k fs at redshift z = 0 in Figure 4, and at redshift z = 4 in Figure 5. Let us recall that a spherically symmetric perturbation that reaches a relative over-density δ = 1.686 in the linear approximation (that has already broken down) has an exact solution that diverges. Comparing the simulations in Figure 4 with Figure 1, we conclude that 4 ≲ k fs ≲ 7 Mpc⁻¹. Since more than half of the dwarf galaxies are “red and dead” [8], i.e., old, we estimate 4 ≲ k fs ≲ 10 Mpc⁻¹ from Figure 5 corresponding to z = 4. From Figure 4and Figure 5, we also confirm that the ΛCDM scenario, with k fs > 20 Mpc⁻¹, obtains too many dwarf galaxies in the Local Field, and therefore consider warm dark matter as a possible solution to this problem.

Figure 5. Simulations at redshift z = 4 with k fs = 4 , 6 , 8 , 10 , 15 and 20 Mpc⁻¹. Shown are contours of constant relative over-density: δ ( x ) = 0 red, 1.69 blue, and 3 redish, in the linear approximation (that has already broken down). The coordinates are supergalactic. sg_zz = 2.5 Mpc.

To be able to count the simulated isolated dwarf galaxies, we present in Figure 6 simulations with k fs = 5.5 Mpc⁻¹, at several supergalactic coordinates sg_zz. To count galaxies, we focus our attention on the blue contours with δ = 1.686 , and do not count very large perturbations with an irregular blue contour that would correspond to a large galaxy with satellites. Similar simulations are carried out for k fs = 4.0 , 5.0, 5.5, 6.0, 6.5 and 20 Mpc⁻¹. The corresponding number densities of simulated isolated dwarf galaxies are presented in Table 1. The measured k fs scales with the number of counted Local Field or simulated galaxies to the 1/3 power, and we add corresponding systematic uncertainties in the end. From Table 1, we obtain an upper bound to k fs :

Figure 6. Simulations with k fs = 5.5 Mpc⁻¹ and sg_zz = 2.0, 2.3, 2.6, 2.9, 3.1, and 3.4 Mpc. Shown are contours of constant relative over-density: δ ( x ) = 0 red, 1.69 blue, and 5 green, at redshift z = 0, in the linear approximation (that has already broken down).

Table 1. Comoving number densities of isolated dwarf galaxies obtained from linear simulations with several k fs , to be compared with 0.743 ± 0.115 Mpc⁻³ isolated dwarf galaxies in the Local Field. This is before non-linear re-generation of the power spectrum at high wavevector k .

Non-linear re-generation of small-scale structure is a major effect that needs to be included in the analysis [11]-[14].

According to Table 1 for k fs = 20 Mpc⁻¹, the number density of isolated dwarf galaxies in the ΛCDM cosmology is approximately 25/0.743 = 34 times the observed number density in the Local Field! To reduce the cold dark matter predicted number density, we assume dark matter is warm. The reduction in number counts, obtained from the halo model and simulations, taking account of non-linear evolution, has the form [14].

The “half-mode” mass is M hm ≡ 4 3 π ρ ¯ ( λ hm / 2 ) 3 with the half-mode length scale λ hm defined such that τ ( 2 π / λ hm ) ≡ 1 / 2 . From τ 2 ( k ) = exp ( − k 2 / k fs 2 ) , we obtain k fs = 2 π / ( ln ( 4 ) λ hm ) . M is the “halo mass”, often defined as the mass enclosed in a region where 〈 ρ 〉 = 200 ρ crit . Table 2 presents the relation between m th , M hm , λ hm and k fs .

Table 2. The standard thermal relic warm dark matter mass m th and the corresponding half-mode mass M hm , from Table 1 of [14]. Also presented are the corresponding half-mode length scale λ hm and the cut-off wavevector k fs before the formation of the first galaxies.

Let us consider a dwarf galaxy with stellar mass M * ≈ 2 × 10 6   M ⊙ , see Figure 2. The corresponding halo mass, according to an extrapolation of Figure 11 of [15], is M ≈ 10 10   M ⊙ . Then, from (11), the needed warm dark matter suppression factor 0.743/25 is obtained with M hm = 2 × 10 11   M ⊙ , λ hm = 2.1 Mpc, and

before the formation of the first galaxies. This is our final estimate. Half of the non-linear re-generation correction from (10) to (12) will be assigned to one of the systematic uncertainties, see Section 9. This estimated uncertainty includes the uncertainty of Equation (11) when applied to the Local Field, and the uncertainty on M (up to a factor 6). This non-linear re-generation correction is the dominating uncertainty of the present analysis.

We need to cross-check that the dark matter is sufficiently cold so that a halo of mass M ≈ 10 10   M ⊙ collapses. From Table 1 of [16], the needed v h rms ( 1 ) is ≲830 m/s, corresponding to the lower bound

We note that k fs ≈ 2.5 Mpc⁻¹ satisfies this criterion.

Figure 7 presents the superposition of two graphs. One graph shows stellar mass distributions in units 10 1.5 M * , and the other graph presents predicted distributions, for several values of k fs , as a function of the linear Press-Schechter mass M PS . The predictions are the Press-Schechter prediction and two ellipsoidal collapse extensions [17]-[20]. The superposition is useful if 10 1.5 M * ≈ M PS as we have assumed in the past. With this approximation, predictions are in surprisingly good agreement with observations of stellar mass distributions and of ultra-violet luminosity distributions in a wide range of redshifts [20]. However, the approximation 10 1.5 M * ≈ M PS breaks down at the high and low mass ends as shown in [15]. Dwarf galaxies do not follow the general distributions of larger galaxies. (This is also confirmed in [21]). In particular, to compare the predictions in Figure 7 with dwarf galaxies, the red X’s would have to be shifted to the right by about 2 orders of magnitude, as discussed in Section 6, see [15]. We might then arrive at the conclusion that dark matter is cold in agreement with the published limits on k fs summarized in Figure 3 of [6]. However, we also need to understand the stellar mass distributions of large galaxies, and furthermore, cold dark matter obtains a great excess of dwarf galaxies as shown in Table 1.

The excess of dwarf galaxies relative to large galaxies in Figure 7 may perhaps be understood by counting galaxies in Figure 6 (a jump between first-generation galaxies and galaxies formed by bottom-up hierarchical evolution?), or may be due to an over-density of the Local Volume that reduces its expansion relative to the rest of the universe. We consider this possibility next.

Figure 7. Superposition of two graphs at redshift z= 4. One graph presents stellar mass distributions in units 10 1.5 M * of isolated dwarf galaxies in the Local Field from the LSDB catalog [9] (red X’s) (assuming the Local Field dwarfs are already in place at z= 4), and distributions obtained with the Hubble Space Telescope [22] (black squares), the continuity equation [23] (red triangles), and the James Webb Space Telescope (green triangles) [24]. The second graph presents the Press-Schechter [17] prediction and two ellipsoidal collapse extensions [18][19] assuming a critical universe, for several values of k fs , as a function of the linear Press-Schechter mass M PS . See [20] for full details.

The mean total (dark matter plus baryon) density in the Local Volume 〈 ρ m , LV 〉 within 11 Mpc of the Milky Way is dominated by 21 large galaxies [25]. This mean is 〈 ρ m , LV 〉 ≈ 0.2 ρ crit , compared to the universe average 〈 ρ m 〉 ≈ 0.315 ρ crit . So, these measurements suggest the Local Field is underdense. The measured k fs assuming 〈 ρ m , LV 〉 = 〈 ρ m 〉 becomes multiplied by ( 〈 ρ m 〉 / 〈 ρ m , LV 〉 ) 1 / 3 due to the different expansion of the Local Volume with respect to the average universe.

However, an alternative estimate of 〈 ρ m , LV 〉 can be obtained by comparing the stellar mass distribution of the isolated dwarf galaxies in the Local Field with that of galaxies at large, see Figure 7. This comparison suggests that the Local Field may have a matter over-density of a factor ≈ 5.

To be conservative, we will take 〈 ρ m , LV 〉 to be in the range 0.15 ρ crit to 1.6 ρ crit , corresponding to a k fs correction factor between 1.3 and 0.58. This is the second dominating uncertainty of the present estimate of k fs , see Table 3.

We have estimated the cut-off wavevector k fs due to warm dark matter free-streaming. It is defined as follows: P Λ CDM ( k ) exp ( − k 2 / k fs 2 ) is the linear density power spectrum, before first galaxies, i.e., before re-generation of small-scale perturbations, referred to the present time. We have chosen the arguably simplest setting, namely the number density of isolated dwarf galaxies in the Local Field, that avoids a detailed, quantitative understanding of galaxy formation and evolution, including satellites, in the warm dark matter scenario. The simple order-of-magnitude estimate (3) obtains k fs ≈ 5.7 Mpc⁻¹. From the upper bound (10) and lower bound (13), we obtain

with an estimated confidence of 90%. Including the non-linear re-generation of small-scale structure obtains (12):

with 68% confidence. The breakdown of uncertainties is presented in Table 3. Combining these two results, we finally obtain

with 68% confidence. The corresponding comoving warm dark matter root-mean-square thermal velocity from (2) is

with 68% confidence. The measurement of v h rms ( 1 ) allows the extrapolation of dark matter cosmology to the past [5][26]. The corresponding “standard thermal relic mass” is m th = 0.36 − 0.26 + 0.23 keV. This result is in agreement with independent measurements summarized in [5], and in disagreement with published limits summarized in Table 3 of [6], and in Table 2 of [7], for example, m th ≳ 10 keV. Therefore, all limits and measurements, including the present one, need to be revised.

The bottom line is this: The ΛCDM prediction, e.g., the last panel in Figure 4, looks very different from the data in Figure 1.

The tightest limit on k fs comes from the number of observed satellites of the Milky Way [6]. Such limits are obtained with simulations. These same simulations should also obtain the number density of dwarf galaxies in the Local Field. This cross-check would settle the discrepancy.

Table 3. Uncertainties of k fs = 2.5 − 2.0 + 1.8 Mpc⁻¹ at 68% confidence.

All data in this article was obtained from the Local Volume Database (LVDB) catalog presented by Andrew B. Pace [9]. This catalog contains citations to each property of each dwarf galaxy, so a large community of astronomers have made the present investigation possible. Katya Gozman developed the handy application (https://lvd-interactive.streamlit.app/) to visualize the galaxy distributions. I thank Karsten Müller for his early interest in this work and for many useful discussions.