Viscous Cold Dark Matter Models

• Viscous Cold Dark Matter is a model that treats cold dark matter as a dissipative fluid, introducing an effective pressure via bulk viscosity.
• The approach employs both phenomenological formulations and causal thermodynamics frameworks, like Eckart and Müller–Israel–Stewart theories, to capture cosmic dynamics.
• vCDM models impact structure formation by damping perturbations and modifying halo growth, with constraints emerging from SNe, BAO, CMB, and galaxy-scale observations.

Viscous Cold Dark Matter (vCDM) denotes a class of cosmological descriptions in which cold dark matter is not treated as an exactly pressureless ideal dust, but as a dissipative medium whose effective pressure contains a viscous contribution, most commonly a bulk-viscous term of the form $\Pi=-3\xi H$. In this literature, the concept appears in several related but non-identical forms: phenomenological bulk-viscous dark matter in $\Lambda$-dominated FLRW cosmology, coarse-grained effective viscous descriptions of standard CDM in large-scale-structure theory, and, under a distinct nomenclature, “VCDM” models in minimally modified gravity that reproduce VCDM-like background dynamics without inserting an explicit viscous stress into the matter sector (Velten et al., 2012, Floerchinger et al., 2016, Arora et al., 5 Aug 2025).

1. Definition and scope

In the bulk-viscous fluid formulation, dark matter remains cold in the equilibrium sense, $p_k=0$, but acquires an effective pressure through dissipation. The basic replacement is

$p_{\rm eff}=p_k+\Pi=0-3\xi H,$

or, in equivalent notation, $P_{\rm eff}=p-3H\zeta$ (Velten et al., 2014, Villalobos et al., 15 Aug 2025). In homogeneous and isotropic cosmology, shear viscosity and heat conduction vanish at the background level, so bulk viscosity is the only dissipative channel compatible with the cosmological principle (Silva et al., 2018, Barbosa et al., 2015).

The same acronym is also used more broadly. In effective large-scale-structure theory, CDM is coarse-grained into a non-ideal fluid with effective viscosity and effective pressure generated by short-wavelength nonlinear modes rather than by microscopic collisions (Blas et al., 2015, Floerchinger et al., 2016). In a separate line of work, “VCDM” denotes a type-II minimally modified gravity theory with an auxiliary scalar potential $V(\phi)$; there the dynamics can mimic viscous-CDM phenomenology while matter itself remains minimally coupled and non-viscous (Arora et al., 5 Aug 2025, Mishra et al., 28 Mar 2026).

Usage	Core ingredient	Representative works
Bulk-viscous dark matter fluid	$\Pi=-3\xi H$ or $\Pi=-3\zeta H$ in FLRW	(Velten et al., 2012, Velten et al., 2014, Silva et al., 2018, Villalobos et al., 15 Aug 2025)
Effective viscous CDM in LSS	Coarse-grained stress tensor with effective pressure and viscosity	(Blas et al., 2015, Floerchinger et al., 2016, Natwariya et al., 2019)
“VCDM” in minimally modified gravity	Type-II MMG with variable $V(\phi)$	(Arora et al., 5 Aug 2025, Mishra et al., 28 Mar 2026)

This multiplicity of meanings is not merely terminological. It reflects three distinct physical interpretations: dissipation in the dark matter fluid itself, emergent transport in coarse-grained collisionless CDM, and modified-gravity realizations of the same late-time background behavior.

2. Relativistic fluid formulation and transport theory

The relativistic fluid description starts from an imperfect-fluid stress tensor. In one common notation,

$$T^{\mu\nu}_{\rm (dm)}= (\rho_{\rm dm}+p_{\rm dm}+\Pi)u^\mu u^\nu+(p_{\rm dm}+\Pi)g^{\mu\nu},$$

with $\Lambda$0 for cold dark matter, so that $\Lambda$1 (Avelino et al., 2013). In the nonextensive extension of the model, the stress tensor is written as

$\Lambda$2

with $\Lambda$3 and $\Lambda$4 (Silva et al., 2018).

Two transport frameworks dominate the vCDM literature. The first is Eckart theory, in which the bulk viscous pressure responds instantaneously,

$\Lambda$5

or equivalently $\Lambda$6 (Velten et al., 2013, Velten et al., 2012). The second is Müller–Israel–Stewart causal thermodynamics, where $\Lambda$7 obeys a relaxation equation,

$\Lambda$8

with $\Lambda$9 the relaxation time and $p_k=0$0 (Cruz et al., 2018). The truncated MIS form,

$p_k=0$1

has also been used as a practical compromise between causality and tractability (Barbosa et al., 2015).

The thermodynamic viability condition is standard: entropy production requires $p_k=0$2 or $p_k=0$3 (Avelino et al., 2013, Villalobos et al., 15 Aug 2025). In the Eckart and Israel–Stewart approaches alike, acceleration driven entirely by viscous pressure tends to push the fluid away from near-equilibrium, since the relevant condition is $p_k=0$4 while acceleration without $p_k=0$5 typically needs $p_k=0$6 (Cruz et al., 2018, Villalobos et al., 15 Aug 2025). This is one reason many later studies keep a cosmological constant and treat viscosity as a correction rather than a full replacement for dark energy.

3. Background cosmology and model-building variants

At the background level, vCDM is usually embedded in flat or curved FLRW cosmology with baryons, radiation, a cosmological constant, and a viscous dark matter component. A standard form is

$p_k=0$7

with $p_k=0$8 determined by a modified continuity equation rather than the dust law $p_k=0$9 (Velten et al., 2014, Velten et al., 2012). In the recent $p_{\rm eff}=p_k+\Pi=0-3\xi H,$0-dominated formulation with curvature,

$p_{\rm eff}=p_k+\Pi=0-3\xi H,$1

and the viscous dark matter evolves through

$p_{\rm eff}=p_k+\Pi=0-3\xi H,$2

(Villalobos et al., 15 Aug 2025).

Several parameterizations recur. One family adopts

$p_{\rm eff}=p_k+\Pi=0-3\xi H,$3

with special cases $p_{\rm eff}=p_k+\Pi=0-3\xi H,$4 and $p_{\rm eff}=p_k+\Pi=0-3\xi H,$5 (Velten et al., 2012, Velten et al., 2014). Another uses

$p_{\rm eff}=p_k+\Pi=0-3\xi H,$6

studying both constant viscosity ($p_{\rm eff}=p_k+\Pi=0-3\xi H,$7) and variable viscosity ($p_{\rm eff}=p_k+\Pi=0-3\xi H,$8 free) in flat and curved universes (Villalobos et al., 15 Aug 2025). A further phenomenological form is

$p_{\rm eff}=p_k+\Pi=0-3\xi H,$9

which supplements the constant-viscosity case with an explicit redshift dependence (Herrera-Zamorano et al., 2020). In the nonextensive model, the generalized Friedmann equation becomes

$P_{\rm eff}=p-3H\zeta$0

so the same parameter $P_{\rm eff}=p-3H\zeta$1 controls both the entropic-gravity rescaling and the viscous sector through $P_{\rm eff}=p-3H\zeta$2 (Silva et al., 2018).

A central historical ambition was unification of the dark sector: a single viscous fluid behaving like matter at early times and dark energy at late times. Background fits to supernovae and $P_{\rm eff}=p-3H\zeta$3 can indeed be competitive, especially for $P_{\rm eff}=p-3H\zeta$4 with $P_{\rm eff}=p-3H\zeta$5, which reproduces the $P_{\rm eff}=p-3H\zeta$6CDM background expansion (Velten et al., 2011). However, linear perturbations, ISW constraints, and CMB large-angle behavior strongly challenge such unified viscous models, and the later consensus in this literature is that $P_{\rm eff}=p-3H\zeta$7CDM is substantially more viable than viscosity-only unification (Barbosa et al., 2015, Velten et al., 2011).

A distinct complication appears once viscosity is combined with dark-sector interactions. In models with $P_{\rm eff}=p-3H\zeta$8 and $P_{\rm eff}=p-3H\zeta$9, complete cosmological dynamics requires either zero viscosity or negative viscosity, whereas the Local Second Law of Thermodynamics requires $V(\phi)$0 (Avelino et al., 2013). This creates a structural tension between early-time viability and fluid thermodynamics in that interacting class.

4. Perturbations, structure formation, and effective theory

The main phenomenological impact of vCDM is not on the homogeneous background but on the growth of inhomogeneities. Bulk viscosity introduces scale-dependent friction and pressure-like terms into the perturbation equations, suppressing structure growth most strongly at large $V(\phi)$1 (Velten et al., 2013, Velten et al., 2012). In Newtonian notation, the density contrast obeys a modified evolution equation of the schematic form

$V(\phi)$2

for cold viscous matter with $V(\phi)$3 and $V(\phi)$4 (Velten et al., 2013).

One precise result of this program is methodological: once dark matter carries even a small viscous pressure, the usual Newtonian approximation for sub-Hubble perturbations is no longer generically valid. For $V(\phi)$5, the neo-Newtonian formulation reproduces the relativistic dynamics much more accurately than the standard Newtonian treatment, and for $V(\phi)$6 the existence of nonlinear dwarf-scale structures requires $V(\phi)$7, corresponding to $V(\phi)$8 (Velten et al., 2013).

The small-scale constraints can be severe. Background data alone allow viscosities as large as $V(\phi)$9, but the existence of dwarf galaxies forces much smaller values. In the $\Pi=-3\xi H$0CDM analysis of halo formation, cluster-scale growth tracks standard CDM only if $\Pi=-3\xi H$1, while dwarf-scale collapse implies $\Pi=-3\xi H$2 in the conservative linear analysis; the paper’s final physical-unit summary states that dwarf galaxies require $\Pi=-3\xi H$3 (Velten et al., 2012). In the spherical-collapse and mass-function treatment, $\Pi=-3\xi H$4 suppresses the abundance of $\Pi=-3\xi H$5 halos by nearly an order of magnitude, whereas $\Pi=-3\xi H$6 remains close to $\Pi=-3\xi H$7CDM and still damps the smallest structures (Velten et al., 2014, Barbosa et al., 2015).

A separate line of work reformulates this suppression as an emergent effective fluid phenomenon. Coarse-graining CDM below a matching scale $\Pi=-3\xi H$8 generates effective viscosity and sound speed even if microscopic dark matter is collisionless. In this framework, $\Pi=-3\xi H$9–$\Pi=-3\zeta H$0 or, more broadly, $\Pi=-3\zeta H$1–$\Pi=-3\zeta H$2 defines the boundary between explicitly evolved mildly nonlinear modes and integrated-out short modes (Blas et al., 2015, Floerchinger et al., 2016). The effective parameters are matched to one-loop perturbation theory, with no free parameters besides $\Pi=-3\zeta H$3, and two-loop predictions for $\Pi=-3\zeta H$4, $\Pi=-3\zeta H$5, and $\Pi=-3\zeta H$6 agree with $\Pi=-3\zeta H$7-body simulations at the percent level up to $\Pi=-3\zeta H$8 at $\Pi=-3\zeta H$9 (Blas et al., 2015, Floerchinger et al., 2016).

Self-interacting dark matter adds yet another viscous channel. In kinetic theory, SIDM contributes microscopic shear and bulk viscosities, but these do not simply add to the effective large-scale-structure viscosities. Instead, the relaxation times combine in parallel,

$V(\phi)$0

which implies harmonic-mean addition for the viscosities,

$V(\phi)$1

At cluster scales and current SIDM bounds, the resulting effective viscosities are of order $V(\phi)$2–$V(\phi)$3, or dimensionless $V(\phi)$4–$V(\phi)$5 (Natwariya et al., 2019).

5. Observational constraints and statistical status

Observationally, vCDM has been confronted with supernovae, BAO, CMB angular scales, $V(\phi)$6 measurements, strong lensing, cosmic chronometers, and local $V(\phi)$7 priors. The constraints depend strongly on whether one probes only the background or also structure formation.

Early background analyses found that $V(\phi)$8CDM can fit SN, BAO, and CMB position data with a present-day bulk viscosity as large as $V(\phi)$9, and that the associated ISW amplification remained in the range $$T^{\mu\nu}_{\rm (dm)}= (\rho_{\rm dm}+p_{\rm dm}+\Pi)u^\mu u^\nu+(p_{\rm dm}+\Pi)g^{\mu\nu},$$0, much smaller than in unified viscous dark-fluid models (Velten et al., 2012). The same work concluded, however, that dwarf-galaxy formation forces the far tighter bound $$T^{\mu\nu}_{\rm (dm)}= (\rho_{\rm dm}+p_{\rm dm}+\Pi)u^\mu u^\nu+(p_{\rm dm}+\Pi)g^{\mu\nu},$$1 once halo growth is included (Velten et al., 2012).

In the nonextensive “extended $$T^{\mu\nu}_{\rm (dm)}= (\rho_{\rm dm}+p_{\rm dm}+\Pi)u^\mu u^\nu+(p_{\rm dm}+\Pi)g^{\mu\nu},$$2CDM” model, Bayesian analysis of Pantheon supernovae, BAO, and the CMB acoustic scale yielded, for the constant-viscosity case, $$T^{\mu\nu}_{\rm (dm)}= (\rho_{\rm dm}+p_{\rm dm}+\Pi)u^\mu u^\nu+(p_{\rm dm}+\Pi)g^{\mu\nu},$$3, $$T^{\mu\nu}_{\rm (dm)}= (\rho_{\rm dm}+p_{\rm dm}+\Pi)u^\mu u^\nu+(p_{\rm dm}+\Pi)g^{\mu\nu},$$4, $$T^{\mu\nu}_{\rm (dm)}= (\rho_{\rm dm}+p_{\rm dm}+\Pi)u^\mu u^\nu+(p_{\rm dm}+\Pi)g^{\mu\nu},$$5, and $$T^{\mu\nu}_{\rm (dm)}= (\rho_{\rm dm}+p_{\rm dm}+\Pi)u^\mu u^\nu+(p_{\rm dm}+\Pi)g^{\mu\nu},$$6, with $$T^{\mu\nu}_{\rm (dm)}= (\rho_{\rm dm}+p_{\rm dm}+\Pi)u^\mu u^\nu+(p_{\rm dm}+\Pi)g^{\mu\nu},$$7 relative to $$T^{\mu\nu}_{\rm (dm)}= (\rho_{\rm dm}+p_{\rm dm}+\Pi)u^\mu u^\nu+(p_{\rm dm}+\Pi)g^{\mu\nu},$$8CDM; for the $$T^{\mu\nu}_{\rm (dm)}= (\rho_{\rm dm}+p_{\rm dm}+\Pi)u^\mu u^\nu+(p_{\rm dm}+\Pi)g^{\mu\nu},$$9 case, $\Lambda$00 and $\Lambda$01, again statistically inconclusive (Silva et al., 2018). The same analysis allowed physical viscosities of order $\Lambda$02 and slightly alleviated the local-$\Lambda$03 tension, but did not require either nonextensivity or viscosity (Silva et al., 2018).

A later late-time analysis in flat and curved $\Lambda$04CDM with Pantheon+, CC, BAO-derived $\Lambda$05, and the 2022 SH0ES prior found that the constant-viscosity flat model gives $\Lambda$06 in the joint PPS+CC+R22 case, with present viscosity constrained to $\Lambda$07 in all scenarios (Villalobos et al., 15 Aug 2025). The paper reports partial alleviation of the Hubble tension to roughly $\Lambda$08 when local measurements are included, but BIC and Bayesian evidence still favor $\Lambda$09CDM even when AIC and DIC show mild support for some viscous variants (Villalobos et al., 15 Aug 2025).

Using OHD, Pantheon SNIa, and strong-lensing systems, the Eckart-based constant-viscosity model yielded the joint constraint $\Lambda$10, while the polynomial case gave $\Lambda$11 and $\Lambda$12 (Herrera-Zamorano et al., 2020). These models remained broadly consistent with $\Lambda$13CDM in $\Lambda$14 and $\Lambda$15 within $\Lambda$16, but reconstructed cosmographic parameters differed nontrivially: for the joint fit, the constant-viscosity case gave $\Lambda$17 and transition redshift $\Lambda$18, while the polynomial case gave $\Lambda$19 and $\Lambda$20 (Herrera-Zamorano et al., 2020). This suggests that even modest viscosity can shift the acceleration–deceleration transition and mimic some signatures of dynamical dark energy.

Not all fitted viscous models are thermodynamically acceptable. In interacting viscous dark-fluid cosmologies constrained by Union2.1 SNe, the CMB shift parameter, BAO, and $\Lambda$21, the preferred bulk-viscosity coefficient is consistently negative, with representative fits such as $\Lambda$22 or $\Lambda$23, in direct conflict with the Local Second Law of Thermodynamics (Avelino et al., 2013). This remains one of the clearest examples in which the statistical fit and fluid-level thermodynamics point in opposite directions.

6. Alternative “VCDM” nomenclature and present assessment

A distinct contemporary usage identifies VCDM not with a viscous matter fluid but with a type-II minimally modified gravity theory in which the cosmological constant is replaced by a function $\Lambda$24 of an auxiliary, non-propagating scalar. The background equations take the form

$\Lambda$25

with $\Lambda$26 reconstructed from $\Lambda$27, while ordinary matter remains minimally coupled and obeys the standard continuity equation (Arora et al., 5 Aug 2025, Mishra et al., 28 Mar 2026). In this framework, essentially arbitrary FLRW backgrounds, including phantom-like effective equations of state, can be realized without introducing extra propagating scalar degrees of freedom or modifying the subhorizon effective Newton constant (Arora et al., 5 Aug 2025).

Late-time data analyses in this MMG sense of VCDM can produce results rather different from fluid-vCDM studies. For the VCDM-inspired Hubble function

$\Lambda$28

the combined CC+RSD+DESI DR2+Union3 fit gave $\Lambda$29, $\Lambda$30, $\Lambda$31, $\Lambda$32, and $\Lambda$33 (Mishra et al., 28 Mar 2026). For the same data, the paper reports $\Lambda$34 against $\Lambda$35, with $\Lambda$36 and $\Lambda$37, indicating better late-time fit quality within that effective modified-gravity realization (Mishra et al., 28 Mar 2026). These results are not directly comparable to bulk-viscous fluid constraints, because the “viscosity-like” behavior there resides in the gravitational sector rather than in a dark matter pressure term.

A plausible synthesis of the broader literature is therefore threefold. First, bulk-viscous dark matter is a viable extension of $\Lambda$38CDM at the background level and can modestly shift late-time kinematics, but current data generally do not require nonzero viscosity and model-selection criteria usually continue to favor standard $\Lambda$39CDM (Silva et al., 2018, Villalobos et al., 15 Aug 2025). Second, attempts to make viscosity the primary driver of cosmic acceleration or a full dark-sector unifier are strongly constrained by ISW, CMB, and structure-growth considerations, and in some interacting models by the sign of entropy production itself (Velten et al., 2011, Avelino et al., 2013). Third, effective viscous descriptions remain technically important in large-scale-structure theory even when microscopic dark matter is collisionless, because coarse-graining unavoidably generates non-ideal transport at mildly nonlinear scales (Blas et al., 2015, Floerchinger et al., 2016).

The open problems identified across these lines of work are consistent: a microphysical origin for dark matter viscosity, a causal treatment of perturbations beyond Eckart theory, nonlinear structure formation in genuinely viscous or SIDM-based models, and the disentangling of viscosity from dynamical dark-energy or modified-gravity effects in high-precision cosmological data.