DM0 in Cosmology: Present-Day Dark-Matter EOS

• DM0 is the current dark-matter equation-of-state parameter in a CPL parameterization, defining deviations from pressureless cold dark matter.
• It modifies cosmic expansion and structure by altering redshifting, CMB acoustic peaks, and small-scale clustering via nonzero pressure and sound speed.
• Empirical constraints (e.g., w_dm0 < 1.43×10⁻³) link DM0 to dark matter warmness and effective thermal-relic mass, refining cosmological inferences.

Searching arXiv for the primary and related DM0 usages in cosmology to ground the article in the cited papers. In extended dark-matter fluid phenomenology, DM0 denotes the present-day dark-matter equation-of-state parameter, usually written as $w_{\rm dm0}$, in the Chevallier–Polarski–Linder form $w_{\rm dm}(a)=w_{\rm dm0}+w_{\rm dm1}(1-a)$. It quantifies the deviation of dark matter from the strictly pressureless cold-dark-matter limit at $a=1$, and is commonly analyzed together with a time-variation parameter $w_{\rm dm1}$ and a constant rest-frame sound speed $\hat{c}^2_{\rm s,dm}$. In the benchmark study “Testing the warmness of dark matter,” these quantities were constrained using Planck cosmic microwave background data, baryonic acoustic oscillations, and a local $H_0$ prior, yielding limits consistent with the standard CDM paradigm and no evidence for nonzero dark-matter pressure or sound speed (Kumar et al., 2019).

1. Definition and parameter space

In this usage, DM0 is the present value of the dark-matter equation-of-state parameter in the CPL-like parameterization

$w_{\rm dm}(a)=w_{\rm dm0}+w_{\rm dm1}(1-a).$

Here $w_{\rm dm0}$ is the present value, while $w_{\rm dm1}$ controls the linear-in-$(1-a)$ time variation. The same framework introduces a constant rest-frame sound speed,

$w_{\rm dm}(a)=w_{\rm dm0}+w_{\rm dm1}(1-a)$0

as a second extension beyond the pressureless CDM idealization (Kumar et al., 2019).

The physical interpretation is twofold. At the background level, $w_{\rm dm}(a)=w_{\rm dm0}+w_{\rm dm1}(1-a)$1 alters dark-matter redshifting, the timing of matter–radiation equality, and the Hubble rate $w_{\rm dm}(a)=w_{\rm dm0}+w_{\rm dm1}(1-a)$2. At the perturbative level, nonzero pressure or sound speed suppresses clustering below the dark-matter sound or free-streaming scale. In particular, $w_{\rm dm}(a)=w_{\rm dm0}+w_{\rm dm1}(1-a)$3 and $w_{\rm dm}(a)=w_{\rm dm0}+w_{\rm dm1}(1-a)$4 modify CMB acoustic peak heights and the large-scale integrated Sachs–Wolfe contribution, while suppressing the small-scale matter power spectrum and lowering $w_{\rm dm}(a)=w_{\rm dm0}+w_{\rm dm1}(1-a)$5 (Kumar et al., 2019).

A common misconception is that DM0 is, by itself, a direct particle-physics mass parameter. In this framework it is instead a fluid-level phenomenological parameter. The mass interpretation enters only indirectly through a warmness mapping discussed below.

2. Fluid description and perturbation theory

The underlying model treats dark matter as a perfect fluid with small but nonzero pressure and sound speed, with energy-momentum tensor

$w_{\rm dm}(a)=w_{\rm dm0}+w_{\rm dm1}(1-a)$6

and no anisotropic stress. In a spatially flat FLRW background, the evolution is governed by

$w_{\rm dm}(a)=w_{\rm dm0}+w_{\rm dm1}(1-a)$7

together with dark-matter conservation,

$w_{\rm dm}(a)=w_{\rm dm0}+w_{\rm dm1}(1-a)$8

(Kumar et al., 2019).

The adiabatic sound speed is

$w_{\rm dm}(a)=w_{\rm dm0}+w_{\rm dm1}(1-a)$9

More generally, in an arbitrary frame the pressure perturbation is

$a=1$0

In conformal Newtonian gauge, the linear perturbation equations are

$a=1$1

$a=1$2

These equations make explicit that $a=1$3 both damps velocity and generates pressure support, while the non-adiabatic term $a=1$4 couples density and velocity (Kumar et al., 2019).

Conventions differ by gauge, and the paper also gives a representative synchronous-gauge form used in Boltzmann codes. The dynamical role of DM0 remains the same across gauges: positive pressure and sound speed slow growth and alter metric driving.

3. Observable signatures

The principal observables are the CMB anisotropy spectra, BAO distances, and the inferred late-time structure-growth amplitude. If $a=1$5, dark matter redshifts differently from CDM, affecting horizon entry during radiation domination and shifting the angular diameter distance to last scattering. This slightly suppresses or reshapes the acoustic peaks and shifts features to smaller angular scales for larger $a=1$6 (Kumar et al., 2019).

A nonzero $a=1$7 further reduces acoustic peak amplitudes relative to large-scale anisotropy and enhances the late-time ISW contribution at $a=1$8 through additional potential decay after recombination. In the matter sector, both $a=1$9 and $w_{\rm dm1}$0 suppress small-scale clustering and reduce $w_{\rm dm1}$1.

The analysis also identified characteristic parameter degeneracies. The parameters $w_{\rm dm1}$2 and $w_{\rm dm1}$3 are mildly anti-correlated with $w_{\rm dm1}$4 and positively correlated with $w_{\rm dm1}$5, while $w_{\rm dm1}$6 is strongly anti-correlated with $w_{\rm dm1}$7. This parameter geometry explains why allowing DM0 to vary can shift the inferred Hubble constant upward and the clustering amplitude downward (Kumar et al., 2019).

This suggests that DM0 is most informative not as an isolated scalar but as part of a coupled deformation of both the background and perturbation sectors.

4. Data sets and inference methodology

The baseline analysis extended $w_{\rm dm1}$8CDM to a model denoted $w_{\rm dm1}$9WDM, implemented in CLASS, with parameter inference performed using MontePython and Metropolis–Hastings MCMC. Uniform priors were imposed on the extended parameters,

$\hat{c}^2_{\rm s,dm}$0

with all three restricted to be nonnegative. Chain convergence was checked with the Gelman–Rubin criterion $\hat{c}^2_{\rm s,dm}$1 (Kumar et al., 2019).

The CMB likelihood used Planck 2015 low-$\hat{c}^2_{\rm s,dm}$2 TT, low-$\hat{c}^2_{\rm s,dm}$3 polarization, high-$\hat{c}^2_{\rm s,dm}$4 TT, TE, EE power spectra, plus the CMB lensing likelihood. The BAO compilation consisted of 6dFGS at $\hat{c}^2_{\rm s,dm}$5, SDSS-MGS at $\hat{c}^2_{\rm s,dm}$6, and BOSS DR11 LOWZ and CMASS at $\hat{c}^2_{\rm s,dm}$7 and $\hat{c}^2_{\rm s,dm}$8. The local distance-ladder prior was the Hubble Space Telescope value

$\hat{c}^2_{\rm s,dm}$9

The tightest bounds came from the combined CMB+BAO likelihood, which constrained deviations in both the expansion history and the clustering sector. BAO is especially effective at limiting $H_0$0 and $H_0$1 because it sharply restricts allowed departures in $H_0$2 (Kumar et al., 2019).

5. Empirical constraints on DM0

The principal numerical results are 95% confidence upper limits on the three dark-matter extension parameters. All are consistent with zero, implying no detected deviation from CDM.

Data combination	$H_0$3	$H_0$4	$H_0$5
Planck	$H_0$6	$H_0$7	$H_0$8
Planck+BAO	$H_0$9	$w_{\rm dm}(a)=w_{\rm dm0}+w_{\rm dm1}(1-a).$0	$w_{\rm dm}(a)=w_{\rm dm0}+w_{\rm dm1}(1-a).$1
Planck+HST	$w_{\rm dm}(a)=w_{\rm dm0}+w_{\rm dm1}(1-a).$2	$w_{\rm dm}(a)=w_{\rm dm0}+w_{\rm dm1}(1-a).$3	$w_{\rm dm}(a)=w_{\rm dm0}+w_{\rm dm1}(1-a).$4
Planck+BAO+HST	$w_{\rm dm}(a)=w_{\rm dm0}+w_{\rm dm1}(1-a).$5	$w_{\rm dm}(a)=w_{\rm dm0}+w_{\rm dm1}(1-a).$6	$w_{\rm dm}(a)=w_{\rm dm0}+w_{\rm dm1}(1-a).$7

The tightest bound is therefore

$w_{\rm dm}(a)=w_{\rm dm0}+w_{\rm dm1}(1-a).$8

from Planck+BAO, accompanied by

$w_{\rm dm}(a)=w_{\rm dm0}+w_{\rm dm1}(1-a).$9

at 95% confidence (Kumar et al., 2019).

Although the extended parameters are statistically consistent with zero, allowing them to vary alters several baseline cosmological inferences. In all analyzed combinations, the mean $w_{\rm dm0}$0 increases relative to base $w_{\rm dm0}$1CDM, while the mean $w_{\rm dm0}$2 decreases. For example, for CMB+BAO, $w_{\rm dm0}$3WDM gives

$w_{\rm dm0}$4

versus

$w_{\rm dm0}$5

in $w_{\rm dm0}$6CDM, and

$w_{\rm dm0}$7

versus

$w_{\rm dm0}$8

The paper accordingly states that the well-known $w_{\rm dm0}$9 and $w_{\rm dm1}$0 tensions might be reconciled in the presence of extended dark-matter parameters within the $w_{\rm dm1}$1CDM framework (Kumar et al., 2019).

6. Warmness interpretation and particle-mass mapping

The fluid parameter DM0 can be related to a warmness measure through

$w_{\rm dm1}$2

In this sense, constraints on $w_{\rm dm1}$3 bound the present dark-matter temperature for a given mass (Kumar et al., 2019).

To translate this into an effective thermal-relic mass scale, the analysis used the standard Bode–Ostriker–Turok transfer function

$w_{\rm dm1}$4

with

$w_{\rm dm1}$5

Adopting $w_{\rm dm1}$6 and the best-fit cosmological parameters yields a lower bound

$w_{\rm dm1}$7

with reported values of $w_{\rm dm1}$8, $w_{\rm dm1}$9, $(1-a)$0, and $(1-a)$1 across the four data combinations. The resulting $(1-a)$2 bound is described as compatible with the Tremaine–Gunn bound and other warm-dark-matter constraints (Kumar et al., 2019).

A plausible implication is that DM0 functions as a phenomenological bridge between cosmological fluid inference and effective thermal-relic language, but the mapping remains model-dependent.

7. Later dynamical-dark-matter literature and interpretive context

Subsequent literature has used closely related notation for the present-day dark-matter equation-of-state parameter. In the 2025 “Evidence for Dynamical Dark Matter,” DM0 denotes $(1-a)$3 in the parameterization

$(1-a)$4

with a reported strong linear relation $(1-a)$5 and $(1-a)$6 evidence for evolution from combinations of CMB, DESI BAO, and Pantheon+ SN data (Wang, 30 Apr 2025). A later ACT DR6 analysis reported $(1-a)$7 and $(1-a)$8 for ACT+DESI+DESY5, described as a $(1-a)$9 evidence for dynamical dark matter and again consistent with $w_{\rm dm}(a)=w_{\rm dm0}+w_{\rm dm1}(1-a)$00 (Wang et al., 28 Jun 2025).

These later results should not be conflated with the 2019 warm-dark-matter bounds. The earlier study restricted $w_{\rm dm}(a)=w_{\rm dm0}+w_{\rm dm1}(1-a)$01, $w_{\rm dm}(a)=w_{\rm dm0}+w_{\rm dm1}(1-a)$02, and $w_{\rm dm}(a)=w_{\rm dm0}+w_{\rm dm1}(1-a)$03 to be nonnegative and found no evidence beyond CDM (Kumar et al., 2019), whereas the later DDM analyses allowed broad priors such as $w_{\rm dm}(a)=w_{\rm dm0}+w_{\rm dm1}(1-a)$04 and favored negative present-day values (Wang, 30 Apr 2025, Wang et al., 28 Jun 2025). This suggests that the phenomenological status of DM0 depends sensitively on parameterization, prior domain, and data combination.

Within the specific warm-dark-matter test that established the modern fluid-based use of the symbol, the encyclopedic conclusion is clear: DM0 is a tightly constrained present-day dark-matter equation-of-state parameter, observationally consistent with zero, but still useful because even very small allowed departures can shift inferred values of $w_{\rm dm}(a)=w_{\rm dm0}+w_{\rm dm1}(1-a)$05 and $w_{\rm dm}(a)=w_{\rm dm0}+w_{\rm dm1}(1-a)$06, quantify the warmness of dark matter, and motivate broader dynamical-dark-sector extensions (Kumar et al., 2019).