Bulk Viscous Unified Dark Matter

• The paper formulates bulk viscous unified dark matter as a single imperfect fluid that mimics ΛCDM background expansion with tuned parameters.
• It demonstrates that the added viscous pressure over-damps density perturbations, resulting in suppressed clustering and an enhanced ISW effect.
• Comparisons with barotropic UDM models underscore that relying solely on ρ and H dependence fails to concurrently satisfy background and structure formation observations.

Bulk viscous unified dark matter (BVUDM) models posit that a single imperfect fluid, characterized by a pressure contribution from bulk viscosity, acts as the source for both the gravitational effects attributed to dark matter and the accelerated expansion attributed to dark energy. In these frameworks, the negative effective pressure arising from bulk viscosity can, under specific parameterizations, closely mimic the expansion history of the standard ΛCDM model. However, such models also feature fundamentally distinct perturbation dynamics, leading to key observable consequences.

1. Theoretical Formulation and Parameterization

In BVUDM, the energy–momentum tensor incorporates bulk viscous effects via an effective pressure term,

$p_D = (γ - 1) \rho_D - 3 H \eta(\rho_D),$

where $γ$ parameterizes the intrinsic fluid EoS and $\eta(\rho_D)$ is the bulk viscosity coefficient. For dust-like behavior at early times, one sets $γ=1$. The effective pressure simplifies to

$p_D = -3\alpha H \rho_D^{m},$

with %%%%3%%%% and $\alpha, m$ as model parameters. The cosmic expansion, for a spatially flat universe, is governed by

$3H^2 = 8\pi G (\rho_D + \rho_B + \rho_R),$

with standard baryon ($\rho_B$) and radiation ($\rho_R$) components. The viscous fluid obeys the conservation equation,

$\dot{\rho}_D + 3H(\rho_D + p_D) = 0.$

For $m \simeq -1/2$ and suitably tuned $\alpha$, the background cosmology can virtually replicate ΛCDM.

2. Background Evolution and Structure Growth

Background expansion:

The effective pressure $p_D = -3\alpha H\rho_D^{m}$ enables the fluid to interpolate between cold dark matter–like behavior (at early, high-density epochs) and negative-pressure dark energy–like behavior at late times. The integrated energy density and pressure evolution can reproduce the observed history of decelerated and then accelerated expansion with high fidelity compared to ΛCDM.

Density perturbations:

Linear perturbation analysis substantially diverges from the cold dark matter case. For gauge-invariant density contrast $\Delta_D$,

$$\Delta_D'' + [C_1(a) + k^2 C_2(a)]\Delta_D' + [C_3(a) + k^2 C_4(a)]\Delta_D = 0,$$

where primes denote conformal time derivatives and coefficients $C_i(a)$ are functions of $H, \dot{H}$, and $w = p_D/\rho_D$. Crucially, terms proportional to $k^2$ (comoving wavenumber squared) appear in both the damping and restoring force coefficients. On subhorizon scales ($k \gg \mathcal{H}$), this scale dependence renders the equation over-damped, and $\Delta_D$ decays rapidly without oscillations, in contrast to the oscillatory or growing solutions characteristic of ΛCDM.

The rapid decay of density perturbations in the viscous fluid leads to a fast decay of the gravitational potential $\phi$, which has direct observational consequences (see below).

3. Observational Implications and Limitations

Observable	BVUDM Prediction	Tension With Data
Background Expansion	Mimics ΛCDM with tuned parameters	None (at background level)
Galaxy Power Spectrum	Weakened clustering; suppressed small-scale power	Structure formation can be insufficient in the viscous component
Integrated Sachs–Wolfe (ISW) Effect	Enhanced (φ decays quickly)	Prediction of excess low-$\ell$ CMB power
Weak Lensing	Altered growth and potentials	Lensing signal deviates from ΛCDM expectations

The primary challenge is that rapid, scale-dependent damping of density perturbations prevents the viscous fluid from supporting sufficient structure formation. While baryons (not affected by viscosity) may still cluster, the overall clustering power is reduced, as the viscous component dominates the total matter budget at late times. Simultaneously, the gravitational potential's accelerated decay increases the ISW effect, modifying the large-angle CMB power spectrum beyond observed levels.

4. Comparison to Barotropic Unified Dark Matter Models

The (generalized) Chaplygin gas model, a barotropic UDM with $p = -A/\rho^\alpha$, offers a contrasting approach. Here, the effective pressure lacks explicit $H$ dependence, and in the perturbation equation, the $k^2\Delta_D'$ term vanishes: $C_2 = 0.$ Consequently, small-scale perturbations often display under-damped, oscillatory, or even divergent solutions. This leads to a different set of problems, such as unobserved oscillations or amplification of small-scale power. In BVUDM, the explicit viscous damping prevents oscillations but over-damps structure formation, leading to rapid suppression instead.

5. Status of Viscous Models Without New Degrees of Freedom

Models that avoid introducing additional fields or dynamical degrees of freedom (restricting pressure to algebraic functions of $\rho$, $H$, or derivatives thereof) are highly constrained. For constant pressure, one regains the cosmological constant. For barotropic or $H$-dependent EoS, perturbation-level issues emerge—either through oscillatory/blow-up instabilities (barotropic) or excessive over-damping (viscous). The consistent inability to reconcile all cosmological observables suggests the need for alternative mechanisms, such as new fields or intrinsically non-trivial microphysics, if a viable unified dark sector is to be constructed within these frameworks.

6. Summary and Critical Assessment

Bulk viscous unified dark matter models, parameterized by $p_D = -3\alpha H\rho_D^m$, can reproduce the background expansion history to current observational precision for appropriately chosen parameters, closely paralleling ΛCDM. However, the introduction of scale-dependent and over-damped density perturbations fundamentally alters the growth of structure and the evolution of the gravitational potential. These modifications generate excessive suppression of clustering and an amplified ISW effect in the CMB, resulting in strong inconsistencies with observational data on cosmic structures, CMB angular power spectrum, and gravitational lensing.

Comparison with alternative UDM frameworks (e.g., Chaplygin gas) emphasizes that the problem is generic for models lacking new dynamical degrees of freedom: all such prescriptions produce dramatic departures at the perturbative level—either via oscillations, blow-up, or over-damping—none of which align with the full suite of cosmological observations.

The theoretical appeal of a unified description remains strong, but the combined evidence at the perturbation level underscores the severe challenge of constructing viable bulk viscous unified dark matter models without additional physics beyond an algebraic pressure prescription. Observational tests of the CMB and structure formation robustly constrain, and effectively exclude, the model class described here as a complete and accurate description of the dark sector.