Almost Unified Dark Fluid Models

• Almost unified dark fluid models describe the entire dark sector as a single component using generalized equations of state and microphysical frameworks.
• These models employ methods such as bulk viscosity, generalized Chaplygin gas, and superfluid dynamics to replicate ΛCDM background behavior and introduce novel perturbative features.
• Despite closely mimicking ΛCDM at the background level, they face challenges with perturbative discrepancies and nonlinear structure formation that require precise parameter tuning.

An almost unified dark fluid model postulates that the entire dark sector—responsible for cosmic structure formation and late-time accelerated expansion—can be described by a single physical component, subject to either a generalized equation of state, dissipative processes (e.g., bulk viscosity), scalar field dynamics, or superfluid/composite behavior. This unified approach aims to replace or reduce the distinction between dark matter and dark energy by encoding their phenomenology in the evolution and microphysics of a single cosmic fluid, potentially also encompassing aspects such as dark radiation. The conceptual motivation is to address the “dark degeneracy” problem: multiple split-dark-sector models yield observationally indistinguishable cosmological signatures, so a unified framework offers a more parsimonious and, in some cases, less fine-tuned description. The almost unified qualifier reflects the presence of perturbative subtleties, small deviations, or additional parameters (e.g., entropy perturbations, nonzero sound speed) that prevent perfect unification at all perturbative orders or cosmic epochs.

1. Formulation of Unified Dark Fluid Models

Unified dark fluid models employ a variety of theoretical frameworks, with key distinctions in equation of state structure, underlying microphysics, and inclusion of dissipative or nonadiabatic effects.

Bulk Viscous Fluid Approaches

A core example is a cosmological model in which the dark sector is described as a single imperfect fluid with bulk viscosity. The pressure is parameterized as

$p_D = (γ - 1)\rho_D - 3α H \rho_D^m,$

where $\rho_D$ is the dark fluid density, $H$ the Hubble rate, $α$ and $m$ are model parameters, and $γ$ is set to unity for dust-like early-time behavior. Choices such as $m \approx -0.4$ and $β \approx 0.236$ (with $β \equiv α H_0 \rho_{D0}^{m-1}$) can produce background expansion indistinguishable from ΛCDM (0902.3163).

Generalized Chaplygin Gas and Extensions

Models inspired by the generalized Chaplygin gas posit an equation of state

$p = -A/\rho^\alpha,$

with $A, \alpha > 0$, yielding a cosmic evolution that interpolates from dust-like at early times to a negative-pressure regime mimicking dark energy at late times. Some extensions introduce an additional constant (cosmological constant-like) term to achieve an effective equation of state of the form

$P = -A_\mathrm{eff}/\rho^\alpha - V_0,$

combining the GCG behavior with a vacuum energy offset (Dunsby et al., 2023). Alternatively, logotropic or Murnaghan-inspired models introduce further flexibility in the pressure-density relation, particularly impactful in shaping late-time perturbation behavior.

Models with Constant or Parameterized Sound Speed

Another prominent formulation involves assuming a constant adiabatic sound speed $c_s^2 = \alpha$. The dark fluid's energy density then evolves as

$$\rho_d/\rho_{d0} = (1 - B_s) + B_s\, a^{-3(1+\alpha)},$$

where $B_s$ is an interpolation parameter and $\alpha$ tunes the deviation from pure matter-like evolution (Xu et al., 2011). While small $\alpha$ values ($\lesssim 10^{-5}$) are observationally favored, larger values are constrained by matter power spectrum data.

Superfluid, Scalar Field, and Composite Scenarios

Unified models also encompass frameworks wherein the dark fluid is a superfluid mixture of distinguishable quantum states (e.g., ground and excited states), coupled by Josephson/Rabi-type cosine potentials. These models naturally suppress the sound speed at the relevant cosmological epochs and can interpolate between dark matter and dark energy behavior dynamically (Ferreira et al., 2018). Scalar-tensor unifications (Aguilar et al., 2020) and string-inspired thin tube gas constructions (Lelyakov et al., 10 Jul 2025) further expand the microphysical landscape of unified dark sector models.

2. Background Evolution and Mimicking ΛCDM

A defining property of almost unified dark fluid models is their ability to emulate the background cosmic expansion history predicted by the ΛCDM paradigm, where separate cold dark matter and cosmological constant components govern early-time structure growth and late-time acceleration, respectively.

By tuning relevant model parameters—such as the viscous term exponent $m$ and normalization $β$ in viscous models or the coefficients $A$, $\alpha$ in GCG-type models—the total Hubble rate can be arranged to follow

$$3H^2 = 8\pi G (\rho_\mathrm{dark} + \rho_\mathrm{baryon} + \rho_\mathrm{rad}),$$

with the effective dark sector density mimicking a blend of matter- and cosmological-constant-like behavior at the background level (0902.3163, Hipólito-Ricaldi et al., 2010, Wang et al., 9 May 2024).

Table: Example Parameter Choices for Background Mimicry

Model Type	Key Parameters	Notes on ΛCDM Mimicry
Bulk viscous fluid	$m ≈ -0.4$, $β ≈ 0.236$	Closely reproduces ΛCDM curves
GCG	$\alpha \to 0$, $A$ set by Ω_DE	Reduces to dust + Λ in the limit
Constant $c_s^2$ model	$\alpha < 10^{-5}$, $B_s ≈ Ω_m$	Becomes indistinguishable at bg.

The PAge-like Unified Dark Fluid (PUDF) parameterization (Wang et al., 9 May 2024, su et al., 1 Apr 2025) uses cosmic age and deviation parameters to tune the background Hubble evolution, enforcing that at early times (high redshift) the effective $w \sim 0$ (matter-like), while at late times $w$ drops negative (accelerating phase).

3. Perturbation Dynamics and Observational Constraints

Although the background expansion can be accurately matched, unified dark fluid models typically yield distinctive signatures at the level of cosmological perturbations.

Viscous and Nonadiabatic Effects

In viscous models, equations governing the evolution of the density contrast $\Delta_D$ for the dark fluid feature additional scale-dependent terms: $$\Delta_D'' + [C_1(a) + k^2 C_2(a)] \Delta_D' + [C_3(a) + k^2 C_4(a)] \Delta_D = 0,$$ where $C_i(a)$ are model functions. The $k^2$-dependent damping terms—absent in ΛCDM—induce rapid, monotonic suppression of $\Delta_D$ on sub-horizon scales (0902.3163). This over-damped behavior markedly contrasts with the oscillatory or unstable modes encountered in some barotropic unified models (e.g., Chaplygin gas).

Chaplygin-type models with nonzero sound speed can exhibit oscillations or blow-up instabilities in the density contrast on small scales. In contrast, nonadiabaticity arising from bulk viscosity models suppresses such oscillations but at the cost of excessive decay of the gravitational potential $\phi$, leading to enhanced Integrated Sachs–Wolfe effect and altered weak lensing observables.

Effective and Adiabatic Sound Speed

Unified models with constant adiabatic sound speed $c_s^2 = \alpha$ (Xu et al., 2011, Xu, 2012) and, more generally, with an effective sound speed $c_{s,\mathrm{eff}}^2$ (distinct from the adiabatic limit), can retain near-perfect ΛCDM-like perturbative growth provided $\alpha$ (and/or $c_{s,\mathrm{eff}}^2$) is constrained to be very small ($< 10^{-5}$). When $c_{s,\mathrm{eff}}^2$ is allowed as a free parameter, cosmic data analyses favor nearly pressureless behavior, confirming the necessity for a nearly vanishing effective sound speed to comply with measured power spectra.

Superfluid models inherently suppress the sound speed through their non-relativistic phonon spectrum, ensuring that the effective inertia of the unified fluid matches that of cold dark matter throughout structure formation (Ferreira et al., 2018).

Relative Entropy and Baryon Power Spectrum

In two-component models with a viscous “unified” dark sector plus separately conserved baryons, nonadiabatic effects generate relative entropy perturbations (e.g., $$S_{MV} = \delta \rho_M/\rho_M - \delta \rho_V/(\rho_V + p_V)$$). This channel allows baryonic matter to cluster as in ΛCDM even as the unified dark component’s perturbations are damped. As a result, the predicted baryon power spectrum remains compatible with large-scale structure observations from 2dFGRS and SDSS, and joint statistical analyses yield best-fit baryon fractions close to the big-bang nucleosynthesis expectation ($\sim 4\%$) (Hipólito-Ricaldi et al., 2010, Zimdahl et al., 2010, Zimdahl et al., 2011).

4. Comparison to ΛCDM and Model Selection

Almost unified dark fluid models are built to match the precision successes of the ΛCDM model as closely as possible, particularly in background and linear observables. However, detailed statistical comparisons using comprehensive data sets (Planck 2018 CMB, BAO—including DESI, Pantheon+ SNe Ia, and Cosmic Chronometers) reveal a persistent, albeit sometimes slight, Bayesian preference for ΛCDM.

For instance, the PUDF and related “harmonic universe” models reproduce CMB TT and EE spectra to within $\lesssim 10^{-4}$ of ΛCDM over the full multipole range, and the growth of cosmic structure is matched to percent-level accuracy on linear scales (Wang et al., 9 May 2024, su et al., 1 Apr 2025). Despite this, Bayesian model comparison yields a negative evidence difference, e.g.,

$$\ln B_{\mathrm{PUDF},\,\Lambda\mathrm{CDM}} \approx -6.3,$$

strongly favoring standard ΛCDM over the unified alternative. These results underscore the importance of including perturbation and non-linear signatures, as well as careful numerical implementation (e.g., correct treatment of recombination physics in the Boltzmann code), in discriminating between cosmological models.

5. Microphysical Scenarios and Theoretical Generalizations

Unified dark fluid models have explored several microphysical realizations:

• Bulk viscosity: Single dissipative fluids with pressure $p = -\zeta \Theta$ (with $\Theta$ the expansion scalar), yielding background dynamics identical to a GCG with $\alpha = -1/2$ but distinct perturbative behavior. These models often require nonadiabatic pressure perturbations and encounter difficulties in reproducing the observed CMB power spectrum unless causal (e.g., Israel-Stewart) extensions are employed (Hipólito-Ricaldi et al., 2010, Zimdahl et al., 2010, Zimdahl et al., 2011).
• Superfluid dark sector: Systems with multiple superfluid states, coupled by time-dependent Josephson-like cosine potentials, can interpolate between dust-like clustering and late-time quasi-de Sitter acceleration, while maintaining low sound speed and providing a unified explanation for both cosmic acceleration and MOND phenomenology (Ferreira et al., 2018).
• Inhomogeneous scalar field configurations: The “thin tube of massless scalar field” (TToMSF) gas model extends the unified dark fluid framework via composite massless scalar configurations of variable thickness, leading to a locally varying equation of state $w \in (-1, 1)$ and providing a mechanism for accommodating both smooth acceleration and local inhomogeneities in matter clustering (Lelyakov et al., 10 Jul 2025).
• Scalar-tensor and Brans–Dicke models: Embedding the unified fluid in scalar-tensor gravity with dynamical Brans–Dicke parameters and self-interacting potentials further generalizes the unification scheme and allows for time variation of the gravitational “constant” and transitions across different cosmic epochs (Tripathy et al., 2020, Tripathy et al., 2014).
• Tachyon and logotropic fluids: Composite models combining a tachyonic sector with a symmetry-breaking scalar field yield equations of state resembling a logotropic fluid plus vacuum energy, bridging generalized Chaplygin gas and ΛCDM phenomenology (Dunsby et al., 2023).

6. Thermodynamic Properties and Theoretical Considerations

Unified dark fluid models with rapid transitions in the equation of state (e.g., those employing hyperbolic tangent parameterizations) can give rise to sharp oscillations in the first and second derivatives of the entropy for both the apparent horizon and the fluid components. While the total entropy always obeys the second law, the violent behavior of higher derivatives can raise doubts about the physical soundness of such models (Radicella et al., 2014). This scrutiny is especially relevant when examining the response to transitions from matter-like to dark-energy-like regimes, as well as the consistency with horizon thermodynamics.

Thermodynamic analogies—such as interpreting the unified EoS via the Murnaghan equation (solid-state physics) or adopting a logotropic description—provide further conceptual unification between the microphysics of the dark fluid and its macroscopic cosmological effects (Dunsby et al., 2023). These perspectives offer insight into the construction of fluid models that can simultaneously accommodate negative pressure for acceleration and sufficiently low sound speed to avoid structure suppression.

7. Outstanding Challenges and Future Prospects

While almost unified dark fluid models can successfully reproduce many background and large-scale linear observables, several challenges remain:

• Perturbation-level discrepancies: The distinguishing feature of scale-dependent damping or oscillatory behavior in the density contrast is often at odds with observed matter clustering and CMB signatures unless model parameters are fine-tuned.
• Nonlinear structure formation: Unified frameworks may predict different non-linear clustering and halo properties compared to ΛCDM due to small but finite residual pressure or viscosity in the dark fluid. Upcoming surveys and simulations capable of resolving these non-linear scales could probe such differences.
• Model selection and Bayesian evidence: Although background fits are nearly identical to ΛCDM and linear CMB spectra can be matched to $<10^{-4}$, statistical analyses consistently favor ΛCDM due to the additional phenomenological parameters or lack of clear improvement in fit quality.
• Microphysical foundations: Models based on symmetry principles (e.g., darkon fluids or TToMSF) offer deeper theoretical appeal, while others remain largely phenomenological or are constrained by the requirement to avoid unphysical features (e.g., negative entropy production or instability).

Continued progress in cosmological observations, improved perturbation calculations (including full causal dissipative treatments), and further exploration of microphysical scenarios compatible with unification will determine the viability and explanatory reach of almost unified dark fluid models.