Testing Quasi-Linear Coasting Cosmologies with Late-Time Large-Scale Structure Growth

Technical Summary: Testing Quasi-Linear Coasting Cosmologies with Late-Time Large-Scale Structure Growth

Problem Statement
The standard $\Lambda$CDM cosmological model faces empirical challenges, most notably the "growth tension," where the weighted amplitude of matter fluctuations ($S_8$) inferred from Cosmic Microwave Background (CMB) data differs from values derived from late-time probes such as weak lensing and redshift-space distortions (RSD). Additionally, the model relies on poorly understood components like dark energy and dark matter. This study investigates whether "quasi-linear coasting cosmologies"—models where the scale factor evolves linearly with cosmic time ($a \propto t$) at late times, following a standard $\Lambda$CDM-like phase prior to recombination—can provide a consistent alternative explanation for the observed growth of large-scale structure (LSS). Specifically, the authors test whether these models can resolve the growth tension and how they compare to $\Lambda$CDM when fitted to RSD data.

Methodology
The authors developed a theoretical framework to test coasting cosmologies against observational data without relying on specific realizations of the linear expansion mechanism (e.g., K-matter or $R_h = ct$ specific conditions), relying solely on the functional form $a(t) \propto t$.

1. Analytical Derivation: Starting from the fluid treatment of matter perturbations and assuming isentropic initial conditions and negligible sound speed at late times, the authors derived a closed-form expression for the growth factor $D(z)$ and the density-weighted growth rate $f\sigma_8(z)$. For a coasting universe, the growth factor is expressed using modified Bessel functions of the second kind ($K_1$), dependent on the curvature parameter $k$ and the present-day matter density parameter $\Omega_{m,0}$.
2. Data Processing: The study utilized a dataset of 35 largely uncorrelated $f\sigma_8(z)$ measurements in the redshift range $z \in (0, 2)$, compiled by Skara and Perivolaropoulos. To account for the model-dependence of RSD measurements (which require a fiducial cosmology to convert redshifts to distances), the authors applied an Alcock-Paczyński correction factor, $q(z)$, to homogenize the data for both the coasting models and a baseline flat $\Lambda$CDM model.
3. Statistical Analysis: The models were fitted to the data using nested sampling (via the dynesty implementation) to obtain posterior distributions for $\Omega_{m,0}$ and $\sigma_8(z=0)$.
   • Consistency: Model consistency was evaluated using the Anderson-Darling (AD) test on uncertainty-normalized residuals and posterior predictive checks.
   • Model Preference: Bayesian evidence ($Z$) was used to calculate log Bayes factors ($\log_{10} B$) comparing coasting models to $\Lambda$CDM. Predictive performance was assessed via Leave-One-Out Cross-Validation (LOO-CV) to compute the expected log predictive density ($elpd_{loo}$).
   • Robustness: The analysis tested sensitivity to prior volumes, parameterizations (using $S_8$ vs. $\sigma_8$), and likelihood shapes (Gaussian vs. Student's $t$-distributions).

Key Contributions

• Generalized Growth Expressions: The paper provides a general analytical derivation for $D(z)$ and $f\sigma_8(z)$ in quasi-linear coasting cosmologies with arbitrary spatial curvature ($k = -1, 0, +1$), independent of specific dark energy mechanisms.
• Systematic Comparison: It offers a rigorous statistical comparison between these coasting models and the standard $\Lambda$CDM model using a large, updated dataset of RSD measurements, moving beyond previous analyses that used smaller datasets or different statistical metrics.
• Growth Tension Context: The study explicitly calculates the $S_8$ value implied by RSD data under the coasting assumption to determine if the "growth tension" persists or is alleviated relative to the Planck 2018 CMB baseline.

Results

• Consistency: All tested models (three coasting variants with $k \in \{-1, 0, +1\}$ and flat $\Lambda$CDM) are statistically consistent with the data. The Anderson-Darling tests failed to reject the null hypothesis that residuals follow a standard normal distribution for any model.
• Parameter Constraints:
  • $\Lambda$CDM: $\Omega_{m,0} = 0.286^{+0.053}_{-0.047}$, $\sigma_8 = 0.764^{+0.039}_{-0.035}$.
  • Coasting Models: $\Omega_{m,0}$ and $\sigma_8$ values varied by curvature:
    • $k=-1$: $\Omega_{m,0} = 0.206^{+0.073}_{-0.061}$, $\sigma_8 = 1.071^{+0.213}_{-0.151}$.
    • $k=0$: $\Omega_{m,0} = 0.297^{+0.085}_{-0.073}$, $\sigma_8 = 0.867^{+0.128}_{-0.097}$.
    • $k=+1$: $\Omega_{m,0} = 0.412^{+0.097}_{-0.086}$, $\sigma_8 = 0.725^{+0.080}_{-0.065}$.
• Model Preference: The $\Lambda$CDM model is strongly favored over all coasting models based on Bayesian evidence, with log Bayes factors of $\log_{10} B = \{1.79, 1.55, 1.42\}$ for $k=\{-1, 0, +1\}$, respectively. This preference remains robust against changes in priors and likelihood shapes, though it weakens with heavier-tailed likelihoods.
• Predictive Performance: $\Lambda$CDM exhibits the highest predictive performance ($elpd_{loo}$). The preference is statistically significant ($p_k > 0.975$) for $k=-1$ and $k=0$, but not for $k=+1$. Posterior predictive checks indicate that $\Lambda$CDM has the most conservative predictive uncertainty calibration (residual scatter is smaller than predicted).
• $S_8$ Discrepancy: When interpreted within the coasting framework, the $S_8$ discrepancy relative to the Planck 2018 $\Lambda$CDM baseline varies:
  • $\Lambda$CDM fit: $2.00\sigma$ discrepancy.
  • $k=-1$ coasting: $2.12\sigma$ discrepancy.
  • $k=0$ coasting: $1.21\sigma$ discrepancy.
  • $k=+1$ coasting: $0.62\sigma$ discrepancy.
    The flat and positively curved coasting models reduce the apparent $S_8$ tension relative to the CMB baseline, whereas the negatively curved model does not.

Significance and Claims
The paper concludes that while quasi-linear coasting cosmologies are compatible with current late-time growth data (they are not ruled out), they are not generally preferred over the standard $\Lambda$CDM model by model-comparison diagnostics. The authors emphasize that their results do not constitute a model-independent resolution of the growth tension, as the reference $S_8$ value is itself derived within the $\Lambda$CDM framework. A definitive assessment would require a self-consistent calculation of CMB anisotropies and matter power spectra within the specific quasi-linear models. The study serves as a motivation for future work that includes full covariance treatments of overlapping data, joint fitting of expansion and growth probes, and the modeling of the transition from the early $\Lambda$CDM phase to the late-time coasting regime.