Scalar perturbation and density contrast evolution in $f(Q,C)$ gravity

TL;DR

The work develops a comprehensive framework for cosmological scalar perturbations in $f(Q,C)$ gravity, where $Q$ is the non-metricity scalar and $C$ the boundary term. By deriving the full linear perturbation equations and the connection-conservation structure, it identifies how modifications depart from General Relativity through an effective gravitational constant $G_{eff}$ and a modified density-growth dynamics. In the quasi-static and sub-horizon regimes, the density contrast obeys a generalized growth equation with a scale-dependent $G_{eff}$, and the growth rate $f_g$ and growth index $\\gamma$ are formulated, enabling confrontation with observations. The paper further demonstrates explicit models and shows how $f(Q,C)$ can reproduce or deviate from $\Lambda$CDM in both background and perturbation sectors, providing a practical route to constrain these theories with cosmological data.

Abstract

The symmetric teleparallel theory offers an alternative gravitational formulation which can elucidate events in the early and late universe without requiring the physical existence of dark matter or dark energy. In this formalism, $f(Q, C)$ gravity has been recently introduced by incorporating the boundary term $C$ with the non-metricity scalar $Q$. In this paper, we develop the theory of cosmological scalar perturbation for $f(Q, C)$ gravity, and retrieve that of $f(\mathring{R})$ and $f(Q)$ gravity from our result. The analysis assumes a model-independent approach within these theories that adheres to the conventional continuity equation at the background level. We derive the density contrast equation by employing some standard cosmological approximations, where the $f(Q,C)$ theory is encoded in the effective Newtonian constant $G_{eff}$. Finally, we derive the evolution equation of density growth $f_g$.

Figures (6)

• Figure 1: $G_{eff}/G$ vs redshift $z$ for $f(Q,C)=Q+\beta Q^2+\sigma C^2$
• Figure 2: $G_{eff}/G$ vs redshift $z$ for $f(Q,C)=Q+\lambda\sqrt{-Q}+\Xi C^2$
• Figure 3: Matter density contrast $\delta^m$ vs redshift $z$ for several $f(Q, C)$ models with $k=0.01 h/Mpc$.
• Figure 4: Dark energy equation of state parameter $\omega^{DE}$ vs redshift $z$
• Figure 5: Growth rate $f_g$ vs redshift $z$