Density contrast in the scalar-tensor extension of non-metricity gravity

TL;DR

This work develops a full scalar cosmological perturbation theory for a scalar-tensor extension of non-metricity gravity with a non-minimally coupled scalar. By deriving the complete perturbed field equations and applying a quasi-static limit, it yields a scale-dependent Poisson equation with an effective gravitational constant $G_{ m eff}$ and a density-contrast evolution equation, enabling analysis of matter growth through the growth rate $f_g$ and growth index $ abla ext{gamma}$. Numerical exploration shows that the background evolution can resemble ΛCDM, while perturbations exhibit measurable deviations depending on scale, offering a pathway to constrain non-minimally coupled non-metricity cosmologies with large-scale structure data. The results lay a foundation for testing these theories against observations and motivate extending the perturbative treatment beyond the quasi-static regime and across broader model spaces and observables.

Abstract

We present a novel derivation of scalar cosmological perturbations in the scalar-tensor extension of non-metricity gravity, where the non-metricity scalar $Q$ is non-minimally coupled to a dynamical scalar field. While previous investigations of symmetric teleparallel gravity focused primarily on background evolution or specialised gauge choices, a complete treatment of scalar perturbations in this non-minimally coupled framework has remained unexplored. In this work, we derive the full set of perturbed field equations, impose the quasi-static approximation, and obtain the effective Poisson equation together with the corresponding modified gravitational constant $G_{\rm eff}$. These ingredients allow us to construct the density contrast evolution equation and analyse the matter growth rate and growth index. Through numerical analysis, we showed that the scalar non-metricity theory is comparable to the well-known $ΛCDM$ model to some extent. The results provide a foundation for testing scalar non-metricity theories against large-scale structure observations and open new avenues for constraining non-minimally coupled non-metricity cosmologies.

Figures (6)

• Figure 1: $H/H_0$ vs $z$, $n=0.21$, $f_0=10^{-8}$ and $U_0=0.7223$.
• Figure 2: Evolution of Dark Energy equation of state parameter $\omega^{DE}$.
• Figure 3: The ratio of effective gravitational constant $G_{ff}$ to that of Newtonian constant $G$, $G_{eff}/G$ vs redshift $z$.
• Figure 4: Matter density constrast $\tilde{\delta}^m{}_k$ vs redshift $z$ for several wavenumbers $k$. We took the initial condition for $\delta(z)=\frac{1}{1+z}$ and $\delta'(z)=-\frac{1}{(1+z)^2}$ at $z=1.5$.
• Figure 5: Evolution of matter growth rate $f_g$ vs redshift $z$ for various wavenumbers $k$.