Cosmic curvature on large-scale structures with homogeneous dark energy

This study explores the impact of cosmic curvature on structure formation through general relativistic first-order perturbation theory. We analyze continuity and Euler equations, incorporating cosmic curvature into Einstein equations. Emphasizing late-time dynamics, we investigate matter density contrast evolution in the presence of cosmic curvature, with a specific focus on sub-hubble scales. Solving the evolution equation, we conduct data analysis using cosmic chronometers, baryon acoustic oscillations from Dark Energy Spectroscopic Instrument Data Release 2 (DESI DR2) (including $r_d$ constraint from CMB), type Ia supernova observations from Pantheon+ sample, logarithmic growth rate $f$ data and $f\sigma_8$ data. While constraints on some parameters remain consistent, inclusion of cosmic curvature losens constraints on $\Omega_{\rm m0}$ and $\sigma_{\rm 8,0}$ in $\Lambda$CDM and $w_0$CDM models. The non-phantom behavior of dark energy proves more favorable in $w_0$CDM model, whereas in CPL model, there is certain evidence for non-phantom behaviour at lower redshift and phantom behaviour at higher redshift. Interestingly, we find certain evidence for non-zero cosmic curvature in $\Lambda$CDM model, but no evidence for it in both $w_0$CDM and CPL models, which indirectly proves the non-degeneracy between dynamical dark energy and cosmic curvature.