Cosmological Averaging in Nonminimally Coupled Gravity

TL;DR

This paper investigates how inhomogeneities affect large-scale cosmology in gravity theories with nonminimal matter–geometry couplings, focusing on $f(R,T)=R+F(T)$. By employing global K-monopole particles as a toy model, it shows that spatial averaging of $F(T)$ generally differs from applying $F$ to the averaged trace, and that dust in these theories acquires a nonzero proper pressure, making proper coarse-graining essential. The authors derive that the theory is dynamically equivalent to GR with a modified matter Lagrangian, and they reveal both violations of the standard von Laue condition and a modified averaging prescription that can preserve standard dust evolution when treated consistently. The work cautions against naive averaging in nonminimally coupled gravity and underscores the importance of backreaction considerations for realistic cosmology in such theories, pointing to future research directions on backreaction and observational viability.

Abstract

We address the challenge, commonly referred to as the cosmological averaging problem, of relating the large-scale evolution of an inhomogeneous Universe to that predicted by a homogeneous matter distribution in theories of gravity with nonminimal matter-gravity couplings. To this end, we focus on the class of $f(R,T)$ models defined by $f(R,T)=R+F(T)$, which provide a simple yet theoretically consistent realization of nonminimal matter-gravity interactions and can be reformulated as general relativity minimally coupled to a modified matter Lagrangian. Using nonstandard global monopole solutions as a toy model for realistic particles, we show that the spatial average of $F$ typically differs significantly from $F$ evaluated at the spatially averaged trace of $T$, implying that homogeneous cosmological models generally fail to capture the correct large-scale dynamics of the Universe. We further show that dust in these theories generally exhibits a non-vanishing proper pressure. Our results underscore the necessity of properly accounting for spatial averaging when modeling cosmology in theories with nonminimal matter-gravity couplings.

Figures (5)

• Figure 1: The solid lines show the value of $\widebar{w}_ r\equiv \langle p \rangle_r / \langle \rho \rangle_r$ as a function of the distance to the monopole center $r$ for static global K-monopole solutions obtained numerically considering various values of $\alpha$ and $F(T) = -0.1T$. The colored horizontal dashed lines represent the corresponding asymptotic limits $\widebar{w}_\infty$, defined in Eq. \ref{['f(R,T):barw_infty']} for each value of $\alpha$.
• Figure 2: The solid lines display the behavior of $\widebar{w}_r$ as a function of the radial coordinate $r$, considering various values of $\alpha$ and $F(T) = -0.1T^2$. The dashed black line indicates the asymptotic value of $\widebar{w}$ implied by the standard von Laue condition ($\widebar{w}_\infty = 0$), which is not satisfied by any of models considered.
• Figure 3: The solid lines display the behavior of $\widebar {\mathcal{W}}_r \equiv \langle \mathscr{P} \rangle_r / \langle \varrho \rangle_r$ as a function of the radial coordinate $r$, considering various values of $\alpha$ and $F(T) = -0.1T^2$. indicates the asymptotic value of $\widebar{w}$ implied by the modified von Laue condition ($\widebar {\mathcal{W}}_\infty = 0$). Note that the value of all models considered (with $\alpha > 3/2$) $\widebar {\mathcal{W}}_r$ approaches zero at large distances from the global K-monopole center in agreement with the modified von Laue condition.
• Figure 4: The solid lines show $T/T(0)$ as a function of the distance from the monopole center $r$, for various values of $\alpha$ and $F(T) = -0.1\,T^2$ (the dashed line corresponds to $T/T(0) = 0$). The dotted lines, representing the ratio $T_{\rm TH}/T(0)$ obtained considering the simplified top-hat profile for the radial dependence of $T$, provide only a rough representation of the actual behavior of $T$.
• Figure 5: The solid lines display the behavior of $\langle F(T)\rangle_r / F(\langle T\rangle_r)$ as a function of the distance from the monopole center $r$, for several values of $\alpha$ and $F(T) = -0.1\,T^2$ (the dashed black line corresponds to $\langle F(T)\rangle_r / F(\langle T\rangle_r) = 1$). The dotted lines represent the behavior of $\langle F(T_{\rm TH})\rangle_r / F(\langle T_{\rm TH}\rangle_r)$ obtained using the analytical approximation, which shows excellent agreement with the numerical results at large $r$.