Generalised Dark Matter Parameters

• Generalised Dark Matter (GDM) parameters define an imperfect fluid model with an equation of state, rest-frame sound speed, and viscosity that extend the standard cold dark matter paradigm.
• They influence key cosmological observables by damping small-scale structure, modifying cosmic microwave background anisotropies, and potentially alleviating tensions in measurements like σ8.
• Future surveys and joint analyses are expected to constrain these parameters at the percent level, offering vital insights into alternative dark matter physics beyond the CDM limit.

Generalised Dark Matter (GDM) parameterizations extend the standard cold dark matter (CDM) paradigm by allowing nonzero internal pressure, sound speed, and viscosity for the cosmological dark matter component. These parameters encode the possibility that the dark matter fluid is imperfect—i.e., not completely cold and pressureless—and are designed to capture a wide array of fundamental models (thermal relics, scalar fields, self-interacting DM, effective field theory of large-scale structure, etc.) within a unified phenomenological framework. GDM models are now a standard testbed for planck-scale and next-generation cosmological data, as well as for novel astrophysical probes such as helioseismology and direct-detection limits.

1. Formal Definition of GDM Parameters

The GDM framework models dark matter as a general (possibly imperfect) fluid, with the energy-momentum tensor

$$T^{\mu}_{\;\nu} = (\rho + P) u^{\mu}u_{\nu} + P\delta^{\mu}_{\;\nu} + \Sigma^{\mu}_{\;\nu}$$

where $\rho$ is the dark matter energy density, $P$ is the pressure, $u^{\mu}$ the four-velocity with $u^{\mu}u_\mu = -1$, and $\Sigma^{\mu}_{\;\nu}$ the traceless anisotropic (shear) stress ($u^{\mu}\Sigma_{\mu\nu}=0$, $\Sigma^{\mu}_{\;\mu}=0$).

Three key phenomenological parameters are introduced:

• Equation of state (EoS): $w \equiv \bar P / \bar \rho$, defined for the homogeneous background.
• Rest-frame sound speed squared: $c_s^2 \equiv (\delta P / \delta\rho)_{\rm rest}$, entering the pressure perturbation closure.
• Viscosity parameter: $c_{\rm vis}^2$, regulating the anisotropic stress.

These parameters can be taken as constants for linear perturbation analyses, or allowed to vary with scale and time for more general treatments (Kopp et al., 2016).

The perturbative closure relations in synchronous gauge are: $$\Pi_g = c_a^2\,\delta_g + (c_s^2 - c_a^2)\,\hat\Delta_g \qquad \dot\Sigma_g + 3\mathcal{H}\Sigma_g = \frac{4}{1+w}c_{\rm vis}^2\,\hat\Delta_g$$ where $c_a^2 = \dot{\bar P}/\dot{\bar \rho}$ is the adiabatic sound speed, and $\hat\Delta_g$ is the comoving density perturbation (Kopp et al., 2016).

2. Physical and Theoretical Interpretation

Each parameter in the GDM framework maps onto specific physical properties of the dark matter fluid:

• $w$ (Equation of State): Controls the pressure-to-density ratio. For $w>0$, the energy density dilutes faster than $a^{-3}$; clustering is reduced at late times. In all viable models, $w$ must be $\ll 1$ to remain consistent with observations (Thomas et al., 2016, Pace et al., 2019).
• $c_s^2$ (Sound Speed): Encodes isotropic pressure support. Nonzero $c_s^2$ leads to a characteristic Jeans scale, $k_J^{-1} \sim c_s \tau$, below which density perturbations oscillate and growth is suppressed. This modifies the matter power spectrum on small scales and can alleviate the $\sigma_8$ tension (Thomas et al., 2016, Thomas et al., 2019).
• $c_{\rm vis}^2$ (Viscosity): Introduces scale-dependent shear damping, leading to suppression of structure growth below a viscous scale. In linear perturbation theory, CMB and large-scale structure are sensitive mainly to the degenerate combination $c_s^2 + (8/15)c_{\rm vis}^2$ (Thomas et al., 2016, Kopp et al., 2016).

For specific microscopic models:

• Scalar field dark matter (e.g., axion or fuzzy DM) yields $w \sim 0$, $c_s^2 \sim (k/2am)^2$, $c_{\rm vis}^2 = 0$.
• Ghost condensation models yield $w(a) = c_s^2(a) = w_0 a^{-3}$, $c_{\rm vis}^2=0$, with $w_0$ set by the condensate scale (Furukawa et al., 2010).
• Effective field theory of large-scale structure generically predicts $w, c_s^2, c_{\rm vis}^2\sim 10^{-6}$ (Kopp et al., 2016).

3. Impact on Cosmological and Astrophysical Observables

GDM parameters influence an array of cosmological and astrophysical observables:

Cosmic Microwave Background (CMB):

• Nonzero $w$ shifts the time of matter-radiation equality, altering the relative heights and positions of acoustic peaks.
• $c_s^2$ and $c_{\rm vis}^2$ induce scale-dependent decay of gravitational potentials below a damping scale, enhancing the late-time ISW effect, suppressing CMB lensing power, and smoothing high-$\ell$ temperature anisotropies (Thomas et al., 2016, Kopp et al., 2016).
• In Planck data, the CMB is most sensitive to the degenerate combination $$k_{\rm dec}^{-1} \sim \tau_* \sqrt{c_s^2+(8/15)c_{\rm vis}^2}$$ (Kopp et al., 2016).

Large-Scale Structure (LSS):

• Positive $c_s^2$ and/or $c_{\rm vis}^2$ suppress the growth of density perturbations on scales $k \gtrsim k_{\rm dec}$, reducing the amplitude of the matter power spectrum $P(k)$ at small scales (Thomas et al., 2016, Thomas et al., 2019).
• These parameters can reduce $\sigma_8$, possibly reconciling Planck CMB with weak-lensing measurements (addressing the $S_8$ tension) (Tutusaus et al., 2018, Sakr et al., 23 Jan 2026).
• The halo mass function is suppressed for high-mass halos due to linear power suppression; nonlinear correction to the spherical collapse threshold is typically subdominant for allowed GDM parameter ranges (Pace et al., 2019).

Helioseismology and Solar Observables:

• In the solar environment, generalised momentum- and velocity-dependent cross sections $$\sigma(q,v_{\rm rel}) = \sigma_0 (q/q_0)^{2n_q}(v_{\rm rel}/v_0)^{2n_v}$$ yield modified DM capture and energy transport rates, affecting neutrino fluxes, sound speed profile, and the depth of the solar convective zone (Vincent et al., 2015). For spin-independent $q^2$ interactions at $m_\chi \sim 3$--$5$ GeV, a $>6\sigma$ improvement over the Standard Solar Model is obtained while respecting direct-detection and collider bounds.

4. Observational and Forecasted Constraints

A summary of current and projected bounds on GDM parameters from multiple probes:

Probe / Data	$|w|$ (95–99% CL)	$c_s^2$ (95–99% CL)	$c_{\rm vis}^2$ (95–99% CL)	Reference
Planck 2015 + BAO	$< 2\times 10^{-3}$	$< 3\times 10^{-6}$	$< 6\times 10^{-6}$	(Thomas et al., 2016)
Planck + BAO + LSS (MPS)	$< 1\times 10^{-3}$	$< 1.2\times 10^{-6}$	$< 1.9\times 10^{-6}$	(Thomas et al., 2019)
Planck + WL	$< 5.5\times 10^{-4}$	$< 1.0\times 10^{-8}$	(fixed to 0 in study)	(Tutusaus et al., 2018)
Euclid photometric proj.	$\sim 2\times 10^{-3}$	$< 1.8\times 10^{-9}$	(fixed to 0 in forecast)	(Tutusaus et al., 2018)
Euclid all-probe proj.	$1.9\%$ (rel. error)	$2.0\%$ (rel. error)	(not forecasted)	(Sakr et al., 23 Jan 2026)

These bounds indicate that the CDM limit ($w = c_s^2 = c_{\rm vis}^2 = 0$) remains consistent with all existing data. Excursions from the cold, pressureless limit are strongly limited by structure formation and CMB, with small positive $c_s^2$ and $c_{\rm vis}^2$ providing percent- to sub-percent-level suppression of $\sigma_8$ sufficient to ease moderate CMB–LSS tension (Sakr et al., 23 Jan 2026, Zhou et al., 2022).

Nonlinear modeling caveats: Constraints from nonlinear scales (e.g., k > 0.1 h/Mpc) depend sensitively on the employed GDM halo model prescription; in the conservative (quasi-linear) regime, the bounds are robust (Thomas et al., 2019).

5. Methodological Implementation and Degeneracies

GDM phenomenology is implemented at the Boltzmann solvers level by modifying the fluid equations to include the closures given above. The parameter degeneracies in CMB and LSS data require careful handling:

• $w$ is anti-correlated with the DM density $\omega_g$ and positively correlated with $H_0$. Addition of BAO data strongly constrains $w$, restoring parameter precision to near-CDM levels (Thomas et al., 2016).
• $c_s^2$ and $c_{\rm vis}^2$ are essentially degenerate in CMB owing to their appearance in the decay scale $k_{\rm dec}^{-1}$.
• Adding large-scale structure probes (especially lensing and galaxy clustering) can break these degeneracies and yield improved joint constraints (Thomas et al., 2019, Sakr et al., 23 Jan 2026).
• Varying the neutrino mass introduces a three-way degeneracy with $c_s^2$ and $c_{\rm vis}^2$ due to their similar suppression of small-scale power (Thomas et al., 2019).

Most analyses to date assume constant-in-time and scale GDM parameters, but extensions to scale- or time-dependent forms are natural within the formalism (Kopp et al., 2016).

6. Relation to Fundamental Theories and Extensions

The GDM parametrization serves as an effective description for a variety of physical models:

• Scalar fields: Quintessence, k-essence, and "fuzzy DM" map onto GDM with specific $(w, c_s^2, c_{\rm vis}^2)$ prescriptions (Kopp et al., 2016).
• Effective field theory of structure formation: Small-scale non-linearities induce effective $w, c_s^2, c_{\rm vis}^2 \sim 10^{-6}$ (Kopp et al., 2016).
• Self-interacting and collisional DM: Can be encoded in nonzero $c_{\rm vis}^2$ and $c_s^2$, mimicking the phenomenology of pressure and viscosity (Kopp et al., 2016, Zhou et al., 2022).
• DM–radiation or DM–baryon scattering: Time-dependent $w(a)$ and $c_s^2(a)$ arise naturally in these scenarios (Zhou et al., 2022).
• Solar DM models: Momentum- and velocity-dependent DM-nucleon cross-sections correspond to effective GDM-like effects in astrophysical environments (Vincent et al., 2015).

Initial conditions for adiabatic and isocurvature modes in Einstein-Boltzmann codes are provided in systematic form, paralleling the standard CDM prescription but incorporating the three GDM parameters (Kopp et al., 2016).

7. Future Prospects and Open Questions

Forecasts for Stage IV cosmological surveys such as Euclid predict percent-level constraints on both $w$ and $c_s^2$ using joint photometric, spectroscopic, and weak-lensing data—with $c_s^2$ errors potentially reaching $2\times10^{-9}$ (Sakr et al., 23 Jan 2026, Tutusaus et al., 2018). Nonlinear prescription for GDM in the strongly-coupled regime remains an open challenge, as does mapping the degeneracies between GDM and massive neutrinos. Full exploitation of next-generation data will depend on robust modeling of GDM's impact on the nonlinear matter power spectrum and halo abundance.

A crucial implication is that GDM, while a powerful phenomenological framework for testing dark matter physics, is currently best constrained in the linear regime. Any future detection of nonzero $w, c_s^2$, or $c_{\rm vis}^2$ would point to specific microphysical properties of the dark matter sector, distinguishing it decisively from standard cold dark matter.