Fiducial wCDM: Dark Energy & ΛwDM Insights

• The fiducial wCDM model is defined by replacing the constant dark energy (w = -1) with a free parameter w to allow diverse cosmic expansion histories.
• It incorporates a modified dark matter sector (ΛwDM) with a tiny pressure (w_dm ~ 10⁻⁷) that helps reduce the S8 tension between early- and late-time observations.
• Advanced statistical techniques, including MCMC with modified Boltzmann solvers, constrain both w and w_dm using data from CMB, BAO, SNe Ia, RSD, and weak lensing.

The fiducial $w$CDM model generalizes the standard $\Lambda$CDM cosmological framework by replacing the cosmological constant (constant equation-of-state $w = -1$) with a dark energy component parametrized by a constant $w$, which is allowed to differ from $-1$. This extension enables exploration of a broader set of cosmic expansion histories and is instrumental in probing the nature of dark energy and potential deviations from cold dark matter. Recent analyses have further generalized the cold dark matter sector to allow for a dark matter equation of state $w_{\rm dm} \ne 0$, yielding the so-called $\Lambda w$DM model; key observational constraints on both $w$ and $w_{\rm dm}$ have been obtained using various cosmological probes including the CMB, BAO, SNe Ia, weak lensing, and redshift space distortions. The fiducial $w$CDM and $\Lambda w$DM models have become baselines for cosmological parameter inference in contemporary surveys and simulation suites.

1. Model Definition and Core Equations

In the $w$CDM model, the Universe’s expansion rate is governed by the Friedmann equation:

$$(H/H_0)^2 = \Omega_m a^{-3} + (1-\Omega_m)a^{-3(1+w)}$$

where $H$ is the Hubble parameter as a function of scale factor $a$; $\Omega_m$ is the present matter density parameter; and $w$ is the constant dark energy equation-of-state (pressure-to-density ratio).

The $\Lambda w$DM model further generalizes the matter sector. The dark matter density evolves as:

$\rho_{\rm dm}(a) \propto a^{-3(1 + w_{\rm dm})}$

with $w_{\rm dm}$ as the constant dark matter equation-of-state parameter. For $\Lambda$CDM, $w_{\rm dm}=0$; for $\Lambda w$DM, small $w_{\rm dm}>0$ is allowed.

The linear perturbation evolution for barotropic dark matter is described by (in the conformal Newtonian gauge): \begin{align*} \delta'{\rm dm} &= - (1+w{\rm dm})(\theta_{\rm dm} - 3\phi') \ \theta'{\rm dm} &= - \mathcal{H}(1-3w{\rm dm})\theta_{\rm dm} + k^2\psi + \frac{w_{\rm dm}}{1+w_{\rm dm}}k^2\delta_{\rm dm} \end{align*} where $\delta_{\rm dm}$ is the fractional density contrast, $\theta_{\rm dm}$ is the velocity divergence, and $\mathcal{H}$ is the conformal Hubble parameter.

A key observational parameter is $S_8 = \sigma_8 \sqrt{\Omega_m/0.3}$, quantifying both clustering amplitude ($\sigma_8$) and matter density. Tensions in $S_8$ between early- and late-time observations are sensitive to both $w$ and $w_{\rm dm}$.

2. Observational Constraints on $w$ and $w_{\rm dm}$

Recent cosmological analyses using CMB (Planck), BAO (SDSS, DESI), SNe Ia (Pantheon+), redshift-space distortion (RSD), and weak lensing (e.g., KiDS-1000) yield exceptionally tight constraints:

• For $\Lambda w$DM, Planck+BAO+SNe+RSD+WL (S8) give $w_{\rm dm} = 2.7^{+2.0}_{-1.9}\times 10^{-7}$ (95% CL); DESI BAO yields similar levels.
• Standard $w$CDM analyses typically find $w$ consistent with $-1$ to within $10\%$ (e.g., $w = -0.99^{+0.11}_{-0.13}$ from KiDS-1000 3$\times$2pt).

Likelihood-based model comparison methods (minimum $\chi^2$, AIC) show only marginal preference for nonzero $w_{\rm dm}$ with $\Delta \chi^2_{\rm min} < 1$ over $\Lambda$CDM; potential hints for $w_{\rm dm}>0$ are present but not highly significant.

3. Impact on Structure Formation and $S_8$ Tension

A small $w_{\rm dm}>0$ introduces nonzero sound speed in dark matter ($c_s^2=w_{\rm dm}$), increasing the Jeans length and producing scale-dependent suppression in the matter power spectrum. This is crucial for reconciling the observed $S_8$ discrepancy:

• In $\Lambda$CDM, $S_8$ tension between CMB and weak lensing is $>3\sigma$.
• Allowing $w_{\rm dm}\sim\mathcal{O}(10^{-7})$ in $\Lambda w$DM reduces this tension to $<1\sigma$.

This plausible mechanism emerges because a small pressure-like dark matter component dampens the growth of structure at small scales, lowering both $\sigma_8$ and $S_8$ as measured by late-time surveys.

4. Parameter Inference Methodologies

Leading analyses employ Markov Chain Monte Carlo (MCMC) frameworks, using Boltzmann solvers such as CLASS or CAMB modified to incorporate nonzero $w_{\rm dm}$ in both background and linear perturbation equations. The parameter set $$\mathcal{P} = \{\omega_b, \omega_{\rm dm}, \theta_s, A_s, n_s, \tau_{\rm reio}, w_{\rm dm}\}$$ is typically explored with flat priors, $w_{\rm dm}\ge 0$.

Observational likelihoods combine data from:

• Planck CMB power spectra and lensing reconstruction.
• BAO measurements from SDSS and DESI.
• SNe Ia (e.g., Pantheon+ luminosity distances).
• RSD (measuring $f\sigma_8$ growth rate).
• Weak lensing constraints via $S_8$ prior.

Due to the lack of nonlinear weak lensing predictions in $\Lambda w$DM, current analyses use an $S_8$ prior rather than full power spectrum modeling.

5. Comparison with $\Lambda$CDM and Statistical Significance

The difference between $\Lambda CDM$ and $\Lambda w$DM models is encapsulated by the inclusion of $w_{\rm dm}$ as a free parameter:

• $\Lambda$CDM: $w_{\rm dm}=0$, pressureless dark matter.
• $\Lambda w$DM: $w_{\rm dm}\sim 10^{-7}$ (most likely positive), barotropic dark matter.

While fits to current data marginally favor a tiny $w_{\rm dm}>0$, the improvement in fit (AIC, $\Delta \chi^2_{\rm min}$) is statistically weak. Model selection criteria do not decisively favor $\Lambda w$DM, but its ability to reduce $S_8$ tension is notable.

6. Datasets and Systematics

The full suite of relevant datasets includes: | Probe | Observable | Role in $\Lambda w$DM Constraints | |------------------|-----------------------------------|-------------------------------------------| | Planck CMB | $C_\ell$, lensing | Background, structure growth, $\sigma_8$ | | BAO (SDSS/DESI) | Distance measurements | Expansion history, indirect constraints | | SNe Ia (PP) | Luminosity distance | Background expansion | | RSD | $f\sigma_8$ growth rate | Linear perturbation evolution | | WL (KiDS-1000) | $S_8$ prior | Small-scale matter fluctuations |

Systematics such as nonlinear matter power spectrum modeling, scale dependence of $w_{\rm dm}$, and baryonic feedback remain challenging, particularly for interpreting weak lensing data.

7. Implications and Future Directions

A central implication of the fiducial $w$CDM and $\Lambda w$DM models is that even minute deviations from $w_{\rm dm}=0$ can have cosmologically observable effects across multiple probes. Marginal preference for $w_{\rm dm}\sim 10^{-7}$, together with reduced $S_8$ tension, suggests a possible role for noncold dark matter physics.

Future advances will require:

• Full nonlinear weak lensing likelihoods for $\Lambda w$DM.
• Improved simulations and emulators (e.g., CosmoGridV1).
• Higher precision joint analyses including Stage-IV CMB, DESI, Euclid, LSST datasets.

Resolution of the $S_8$ tension, robust inference of dark matter microphysics, and discrimination between dynamical dark energy and cosmological constant models remain open challenges that the fiducial $w$CDM framework is well-positioned to address (Yao et al., 1 Jul 2025, Xu et al., 2013, Yang et al., 2013).