Linear matter density perturbations in the Λ_sCDM model: Examining growth dynamics and addressing the S ₈ tension

We investigate linear matter density perturbations in the Λ_sCDM scenario, in which the Λ is replaced by one that undergoes a late-time (z ∼ 2) mirror AdS-dS transition, resulting in distinct growth dynamics that shape cosmic structure evolution. We begin our analysis by developing a systematic method to track perturbation growth using two complementary approaches: (i) determining the initial density contrast and its evolution rate for a given collapse scale factor, and (ii) computing the collapse scale factor for a specified initial density contrast and evolution rate. We derive analytical solutions for the growth rate f = Ω_m^γ and growth index γ in both models, reinforcing the theoretical foundation of our approach. Our analysis indicates that prior to the transition, during the AdS-like phase — the AdS-like Λ in Λ_sCDM reduces cosmic friction, causing linear matter density perturbations to grow more rapidly than in ΛCDM; this effect is most pronounced just before the transition, with a growth rate approximately 15% higher than that of ΛCDM around z ∼ 2. After the transition, Λ_sCDM behaves similarly to ΛCDM but features a larger cosmological constant, leading to higher H(z) and greater cosmic friction that more effectively suppresses growth. Before the transition, the growth index γ remains below both the ΛCDM and Einstein-de Sitter values (γ ≈ 6/11); during the transition, it increases rapidly and then grows gradually, paralleling ΛCDM while remaining slightly higher in the post-transition era-though overall, it stays near γ ∼ 0.55, as in the ΛCDM model. Using the Planck best-fit values, namely Ω_m0 = 0.28 for Λ_sCDM and Ω_m0 = 0.32 for ΛCDM, we find that the corresponding growth rates at z = 0 are f = 0.49 and f = 0.53, respectively. Notably, Λ_sCDM predicts a value closer to f = 0.48, recently obtained from LSS data when γ is treated as a free parameter in ΛCDM. This suggests that Λ_sCDM may naturally resolve the structure growth anomaly, without deviating from γ ∼ 0.55. The analysis of linear matter perturbations underscores Λ_sCDM's potential to resolve multiple cosmological tensions within a unified framework, motivating further exploration of its implications for nonlinear structure formation and observational tests.