An extensive investigation of the Generalised Dark Matter model

Abstract: The Cold Dark Matter (CDM) model, wherein the dark matter is treated as a pressureless perfect fluid, provides a good fit to galactic and cosmological data. With the advent of precision cosmology, it should be asked whether this simplest model needs to be extended, and whether doing so could improve our understanding of the properties of dark matter. One established parameterisation for generalising the CDM fluid is the Generalised Dark Matter (GDM) model, in which dark matter is an imperfect fluid with pressure and shear viscosity that fulfill certain closure equations. We investigate these closure equations and the three new parametric functions they contain: the background equation of state w, the speed of sound c_s^2 and the viscosity c_{vis}^2. Taking these functions to be constant parameters, we analyse an exact solution of the perturbed Einstein equations in a GDM-dominated universe and discuss the main effects of the three parameters on the Cosmic Microwave Background (CMB). Our analysis suggests that the CMB alone is not able to distinguish between the GDM sound speed and viscosity parameters, but that other observables, such as the matter power spectrum, are required to break this degeneracy. We also consider other descriptions of imperfect fluids that have a non-perturbative definition and relate these to the GDM model. In particular, we consider scalar fields, an effective field theory (EFT) of fluids, an EFT of Large Scale Structure, non-equilibrium thermodynamics and tightly-coupled fluids. These descriptions could be used to extend the GDM model into the nonlinear regime of structure formation, which is necessary if the wealth of data available on those scales is to be employed in constraining the model. We also derive the initial conditions for adiabatic and isocurvature perturbations and provide the result in a form ready for implementation in Einstein-Boltzmann solvers.

• The paper introduces a phenomenological GDM model that extends CDM by incorporating parameters for pressure, sound speed, and viscosity.
• It employs analytic and numerical techniques to quantify impacts on CMB anisotropies and matter power spectra, deriving constraints such as |w| < 10⁻³.
• The study maps diverse microphysical models to the GDM framework, offering a unified approach to understanding dark matter phenomenology in cosmology.

Detailed Summary of "An extensive investigation of the Generalised Dark Matter model" (1605.00649)

Introduction and Motivation

The paper systematically analyzes the Generalised Dark Matter (GDM) model as a phenomenological extension of the Cold Dark Matter (CDM) paradigm. Whereas CDM treats dark matter as a pressureless, perfect fluid—sufficient for current cosmological and galactic datasets—the GDM model parameterizes possible departures via three parameters: the background equation of state $w$, sound speed $c_s^2$, and viscosity $c_{\text{vis}}^2$. The investigation is motivated by the lack of non-gravitational detection of DM, hints from observed small-scale structure discrepancies, and the realization that a variety of particle and field models can be mapped to an imperfect fluid description.

GDM Model Formulation

The GDM model introduces three free functions: $w(a)$, $c_s^2(a,k)$, and $c_{\text{vis}}^2(a,k)$, unconstrained except for continuity and causality. These parameters control, respectively, the overall pressure, the sound speed of perturbations, and the anisotropic stress. The closure equations for GDM perturbations generalize those of perfect fluids, accommodating non-adiabatic pressure and shear:

• Pressure perturbation:

$\Pi_g = c_a^2 \delta_g + (c_s^2 - c_a^2) \Delta_g$

where $c_a^2$ is the adiabatic sound speed; $\Delta_g$ is the rest-frame density perturbation.

• Shear evolution:

$$\dot{\Sigma}_g = -3\mathcal{H}\Sigma_g + \frac{4}{1+w} c^2_{\text{vis}} \Delta_g$$

The phenomenological construction aims to systematically encompass both fundamental and effective models, including scalar fields, warm dark matter, and fluids with internal degrees of freedom.

Analytic and Numerical Phenomenology

Perturbation Regimes

The analysis categorizes the evolution of GDM perturbations across cosmic history, identifying key regimes:

• Large scales ($k\tau \ll 1$): The gravitational potential $\Phi$ remains constant unless GDM parameters deviate significantly from CDM.
• Sub-horizon scales: For $c_s^2$ or $c_{\text{vis}}^2$ nonzero, $\Phi$ decays below the scale $$k_{\text{dec}}^{-1} \sim \tau\sqrt{c_s^2 + (8/15)c_{\text{vis}}^2}$$. The matter power spectrum and secondary CMB anisotropies (ISW, lensing) are suppressed for modes inside the decay scale.

Analytic solutions are presented for scalar perturbations in a GDM-dominated universe, highlighting that $c_s^2$ and $c_{\text{vis}}^2$ are highly degenerate in CMB polarization/temperature power spectra (for adiabatic initial conditions). The model's predictions are systematically compared to the $\Lambda$CDM baseline, elucidating parameter degeneracies.

Impact on Observables

• CMB: The GDM parameters primarily affect the ISW effect and CMB lensing, manifesting as reduced power at low and high $\ell$ due to decaying potentials. The equation of state $w$ alters the time of matter–radiation equality, producing shifts in the heights and locations of the acoustic peaks. Quantitative limits are derived for the Planck dataset: $|w| \lesssim 10^{-3}$, $c_s^2, c_{\text{vis}}^2 \lesssim 10^{-6}$.
• Matter Power Spectrum: GDM damps small-scale structure significantly below $k^{-1}_{\text{dec}}$, allowing for constraints complementary to the CMB, and providing a handle on degeneracies between $c_s^2$ and $c_{\text{vis}}^2$.
• Isocurvature Modes: The analysis derives adiabatic and isocurvature initial conditions in the presence of GDM, demonstrating non-degeneracy between GDM and baryon isocurvature for $w, c_s^2 \neq 0$.

Physical Interpretation and Model Connections

A significant portion of the work is devoted to relating the phenomenological GDM parameters to microphysical models:

• Thermodynamics: Non-equilibrium thermodynamics (Landau-Lifshitz framework) provides a natural interpretation of the non-adiabatic term and viscous contributions. Limits for vanishing or large heat conduction delineate when the GDM ansatz is a good approximation.
• EFTofLSS: Integrating out small-scale CDM fluctuations in the Effective Field Theory of Large Scale Structure generates a GDM-like imperfect fluid at large scales, with time-dependent but small $w$, $c_s^2$, and $c_{\text{vis}}^2$.
• Scalar Field Models: Both canonical and k-essence models (and oscillating axion-like fields) can be mapped onto the GDM template, with the sound speed inheriting a specific form depending on kinetic structure (e.g., $c_s^2 = 1$ for quintessence, $c_s^2 \approx k^2/(a^2 m^2)$ for ultralight axions).
• Two-Fluid Systems and Tight Coupling: The sum of two tightly coupled fluids (e.g., baryon-photon plasma, or DM–dark radiation interactions) exhibits non-adiabatic pressure in the GDM sense when the tight-coupling condition and adiabaticity are imposed.

Explicit mappings between GDM parameters and underlying model microphysics are derived in various limits for each scenario. The paper cautions against negative $c_s^2$ or $c_{\text{vis}}^2$ values, as these induce instabilities, and emphasizes that any mapping to GDM is only valid in regimes where the effective imperfect-fluid description holds.

Extensions and Theoretical Implications

• The GDM model provides a unifying language for testing a vast class of DM microphysical models with cosmological data. Its closure relations are shown to generally reproduce the linear evolution of perturbations in a host of physically-motivated contexts.
• Future extensions should focus on time- and scale-dependent parameterizations, non-linear structure formation, and explicit inclusion of more general entropy production (e.g., models with bulk viscosity).
• The observed near-degeneracy of $c_s^2$ and $c_{\text{vis}}^2$ in the CMB is expected to be broken by inclusion of small-scale LSS and Ly-$\alpha$ data, which probe the detailed evolution of matter fluctuations at late times.

Conclusion

The work provides an authoritative, systematic, and technical account of the GDM model as both a phenomenological tool and an effective field theory framework for DM phenomenology. The main results are:

• The GDM model robustly constrains departures from CDM on both background and perturbative levels using CMB and LSS observables.
• Strong, quantitative limits on GDM parameters have been obtained, enforcing the effective coldness, non-viscosity, and pressurelessness of DM within the sensitivity of current and near-future probes.
• Mapping between GDM parameters and microphysical (or effective) DM models is explicit in a wide class of scenarios, with the GDM formalism capturing the linear phenomenology of imperfect fluid models.
• Extensions to nonlinear regime, redshift- and scale-dependence, and inclusion of energy exchanges or more general non-adiabaticities are essential for leveraging future high-precision cosmological data to further constrain or reveal new DM microphysics.

The GDM paradigm thus provides a powerful bridge between data-driven and theoretical approaches to the nature of dark matter and its cosmological role.