Density perturbations in nonminimally coupled gravity: symptoms of Lagrangian density ambiguity

TL;DR

The paper analyzes density perturbations in a nonminimally coupled $f(R)$ gravity and highlights how the choice of perfect-fluid Lagrangian, $\mathcal{L}_m=-\rho$ versus $\mathcal{L}_m=p$, crucially affects perturbation dynamics and the viability of late-time cosmologies. It first revisits the quasistatic treatment for $\mathcal{L}_m=-\rho$ and shows potential unphysical singularities, then extends to a full non-quasistatic perturbative framework where higher-order $k$-dependencies arise, constraining the magnitude of NMC terms. The analysis with $\mathcal{L}_m=p$ reveals that late-time dust domination mitigates many issues, restoring more GR-like background evolution while still admitting nontrivial perturbative modifications; inverse power-law $f_2(R)$ models illustrate how $G_{\text{eff}}$ and the weak-lensing parameter $\tilde{\Sigma}$ evolve, typically remaining within observational bounds. Collectively, the results place stringent perturbative constraints on NMC models and motivate further work on growth-rate observables, $\sigma_8$, and joint background-perturbation fits to test these theories against data. The work emphasizes that the broken degeneracy introduced by $\mathcal{L}_m$ is a critical facet of NMC gravity with tangible implications for structure formation and cosmological tests.

Abstract

The evolution of density perturbations is analysed in a modified theory of gravity with a nonminimal coupling between curvature and matter. We consider the broken degeneracy between the choices of matter Lagrangian for a perfect fluid, $\mathcal{L}_m=-ρ$ and $\mathcal{L}_m=p$, and determine the differences between their effects on the effective gravitational constant. We review the result for $\mathcal{L}_m=-ρ$ in the quasistatic approximation and show how it can lead to unphysical singular behaviour for late-time dominating models. This divergent regime can be avoided when considering the fully non-quasistatic perturbative equations, although the higher-order nature of the nonminimally coupled theory and the requirement of a physically viable effective gravitational constant strongly constrains the magnitude of these modifications to the action. We find that both of these issues can be removed when considering $\mathcal{L}_m=p$ at late times due to the pressureless nature of non-relativistic matter and provide predictions for inverse power-law models.

Figures (2)

• Figure 1: Weak lensing parameter $\tilde{\Sigma}$ (left) and effective gravitational constant $G_{\text{eff}}$ (right) in inverse power-law NMC models with $\mathcal{L}_m=p$. We have taken $R_n=3.3\times10^{-7} \ \text{Mpc}^{-2}$ for all models for comparative purposes.
• Figure 2: Normalised growth factor $D(z)$ (left) and growth rate $f_g(z)$ (right) in inverse power-law NMC models with $\mathcal{L}_m=p$. We have taken $R_n=3.3\times10^{-7} \ \text{Mpc}^{-2}$ for all models for comparative purposes.