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

Abstract

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 the resulting $f\sigma_8(z)$ predictions to redshift-space-distortion measurements in the 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 $\Lambda$CDM model are tested. We evaluate each model's consistency with the data using the Anderson--Darling test for normality applied to the uncertainty-normalised residuals, supplemented by posterior predictive checks. For the coasting models, we obtain ${\Omega_\mathrm{m,0}=\{0.206^{+0.073}_{-0.061},\,0.297^{+0.085}_{-0.073},\,0.412^{+0.097}_{-0.086}\}}$ and ${\sigma_{8}(z=0)=\{1.071^{+0.213}_{-0.151},\,0.867^{+0.128}_{-0.097},\,0.725^{+0.080}_{-0.065}\}}$, respectively. For $\Lambda$CDM, we obtain $\Omega_\mathrm{m,0}=0.286^{+0.053}_{-0.047}$ and $\sigma_{8}(z=0)=0.764^{+0.039}_{-0.035}$. All models are consistent with the data, although $\Lambda$CDM is favoured over the coasting models, with log Bayes factors ${\log_{10}{\mathcal{B}}=\{1.79,\,1.55,\,1.42\}}$. This preference is robust against alternative priors and prior parametrisations, but weakens when heavier-tailed likelihoods are adopted. Predictive performance is assessed using the expected log predictive density computed by leave-one-out cross-validation. The $\Lambda$CDM model has the largest predictive performance, but its advantage is statistically significant only relative to the ${k=\{-1,0\}}$ coasting models.