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 ${ \Delta S_8^{ \Lambda\mathrm{ CDM}}=2.00\sigma}$.
Categories: astro-ph.CO