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

We derive analytical expressions for the growth factor, $D(z)$, and density-weighted growth rate, $f\sigma_8(z)$, for cosmologies in which $a\propto t$ at late times. We fit $f\sigma_8(z)$ to data from redshift-space distortion measurements in the redshift range $z<2$ using the `dynesty` implementation of nested sampling. Three coasting models, with curvature parameters $k=\{-1, 0, +1\}$ in $H^2_0c^{-2}$ units, and a flat $\Lambda$CDM model are tested. We evaluate each model's consistency with the data by applying the Anderson--Darling test for normality on the normalized residuals. We obtained $\Omega_\mathrm{m,0}=\{ 0.206^{+0.073}_{-0.061},\, 0.297^{+0.085}_{-0.073},\, 0.412^{+0.097}_{-0.086}\}$} and $\sigma_{8}(z=0)=\{1.071^{+0.213}_{-0.151},\,0.867^{+0.128}_{-0.097},\,0.725^{+0.080}_{-0.065}\}$ for the coasting models, while for $\Lambda$CDM $\Omega_\mathrm{m,0}=0.286_{-0.047}^{+0.053}$ and $\sigma_{8}(z=0) = 0.764_{-0.035}^{+0.039}$. All models are consistent with the data, though the $\Lambda$CDM model is strongly favored over the coasting models, with $\log$ Bayes factors of $\log_{10}{\mathcal{B}} = \{1.79,\, 1.55,\,1.42\}$. A predictive performance metric and posterior predictive check confirmed that while $\Lambda$CDM achieves the highest predictive accuracy, it also shows the strongest indication of overfitting. We also examined whether the $S_8$ tension can be resolved by linear expansion for $z<2$. Curve fitting yielded $S_8 = \{0.890^{+0.024}_{-0.024},\,0.865^{+0.024}_{-0.024},\,0.850^{+0.026}_{-0.026}\}$ for the coasting models, resulting in $\Delta S^\mathrm{Coasting}_8=\{2.12\sigma,\,1.21\sigma,\,0.62\sigma\}$ discrepancies with the standard \textit{Planck} 2018 value. A value of $S_8=0.746^{+0.041}_{-0.039}$ was obtained for the $\Lambda$CDM model, indicating a tension level of ${\Delta S_8^{\Lambda\mathrm{CDM}}=2.00\sigma}$.