Linear matter density perturbations in the $\Lambda_{\rm s}$CDM model: Examining growth dynamics and addressing the $S_8$ tension

Understand

We investigate linear matter density perturbations in the $\Lambda_{\rm s}$CDM, in which the $\Lambda$ is replaced by late-time ($z\sim2$) mirror AdS-dS transition, resulting in distinct growth dynamics.

• We use two complementary approaches: (i) determining the initial density contrast and its evolution rate for a given collapse scale factor, (ii) computing the collapse scale factor for a specified initial density contrast and evolution rate.
• We derive analytical solutions for the growth rate $f=\Omega_{\rm m}^\gamma$ and growth index $\gamma$ in both models.
• Prior to the transition, the AdS-like $\Lambda$ reduces cosmic friction, causing linear matter density perturbations to grow more rapidly than in $\Lambda$CDM; this effect is most pronounced just before the transition, with a growth rate around $15\%$ higher than $\Lambda$CDM around $z\sim2$.