---
title: Generalised Dark Matter Parameters
url: https://www.emergentmind.com/topics/generalised-dark-matter-gdm-parameters
type: topic
---

# Generalised Dark Matter Parameters

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 [1605.00649].

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 [1605.00649].

## 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 [1601.05097, 1912.12250].
- **$c_s^2$ (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 [1601.05097, 1905.02739].
- **$c_{\rm vis}^2$ (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 $c_s^2 + (8/15)c_{\rm vis}^2$ [1601.05097, 1605.00649].

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 [1001.4634].
- Effective field theory of large-scale structure generically predicts $w, c_s^2, c_{\rm vis}^2\sim 10^{-6}$ [1605.00649].

## 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 [1601.05097, 1605.00649].
- In Planck data, the CMB is most sensitive to the degenerate combination $k_{\rm dec}^{-1} \sim \tau_* \sqrt{c_s^2+(8/15)c_{\rm vis}^2}$ [1605.00649].

**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 [1601.05097, 1905.02739].
- These parameters can reduce $\sigma_8$, possibly reconciling Planck CMB with weak-lensing measurements (addressing the $S_8$ tension) [1805.06202, 2601.16943].
- 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 [1912.12250].

**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 [1504.04378]. 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        | $< 2\times 10^{-3}$   | $< 3\times 10^{-6}$     | $< 6\times 10^{-6}$                | [1601.05097]     |
| Planck + BAO + LSS (MPS) | $< 1\times 10^{-3}$   | $< 1.2\times 10^{-6}$   | $< 1.9\times 10^{-6}$              | [1905.02739]     |
| Planck + WL              | $< 5.5\times 10^{-4}$ | $< 1.0\times 10^{-8}$   |  (fixed to 0 in study)             | [1805.06202]     |
| Euclid photometric proj. | $\sim 2\times 10^{-3}$| $< 1.8\times 10^{-9}$   |  (fixed to 0 in forecast)          | [1805.06202]     |
| Euclid all-probe proj.   | $1.9\%$ (rel. error)  | $2.0\%$ (rel. error)    |  (not forecasted)                  | [2601.16943]     |

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 [2601.16943, 2205.08070].

**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 [1905.02739].

## 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 [1601.05097].
- $c_s^2$ 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 [1905.02739, 2601.16943].
- 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 [1905.02739].

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 [1605.00649].

## 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 [1605.00649].
- **Effective field theory of structure formation:** Small-scale non-linearities induce effective $w, c_s^2, c_{\rm vis}^2 \sim 10^{-6}$ [1605.00649].
- **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 [1605.00649, 2205.08070].
- **DM–radiation or DM–baryon scattering:** Time-dependent $w(a)$ and $c_s^2(a)$ arise naturally in these scenarios [2205.08070].
- **Solar DM models:** Momentum- and velocity-dependent DM-nucleon cross-sections correspond to effective GDM-like effects in astrophysical environments [1504.04378].

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 [1605.00649].

## 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 $2\times10^{-9}$ [2601.16943, 1805.06202]. 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.

Source: https://www.emergentmind.com/topics/generalised-dark-matter-gdm-parameters