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 $H_0^2{c}^{-2}$ units, and a flat $\mathrm{\Lambda }$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 ${\mathrm{\Omega }}_{\mathrm{m},0}=\{0.206_{-0.061}^{+0.073},0.297_{-0.073}^{+0.085},0.412_{-0.086}^{+0.097}\}$} and ${\sigma }_8(z=0)=\{1.071_{-0.151}^{+0.213},0.867_{-0.097}^{+0.128},0.725_{-0.065}^{+0.080}\}$ for the coasting models, while for $\mathrm{\Lambda }$CDM ${\mathrm{\Omega }}_{\mathrm{m},0}=0.286_{-0.047}^{+0.053}$ and ${\sigma }_8(z=0)=0.764_{-0.035}^{+0.039}$. All models are consistent with the data, though the $\mathrm{\Lambda }$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 $\mathrm{\Lambda }$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 $\mathrm{\Delta }{S}_8^{\mathrm{C}\mathrm{o}\mathrm{a}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}}=\{2.12\sigma ,1.21\sigma ,0.62\sigma \}$ discrepancies with the standard $\mathit{\text{Planck}}$ 2018 value. A value of $S_8=0.746_{-0.039}^{+0.041}$ was obtained for the $\mathrm{\Lambda }$CDM model, indicating a tension level of $\mathrm{\Delta }{S}_8^{\mathrm{\Lambda }\mathrm{C}\mathrm{D}\mathrm{M}}=2.00\sigma$.