Quantum Cosmology in $f(R, T)$ Theory with Schutz's Perfect Fluid

TL;DR

This work develops a canonical quantum cosmology for FLRW universes in $f(R,T)$ gravity using Schutz's perfect fluid to obtain an internal time parameter $\tau$, thereby transforming the Wheeler–DeWitt equation into a Schrödinger–like SWDW equation. Focusing on the minimally coupled sector $f(R,T)=F^{0}(R)+G^{0}(T)$, the authors derive the full minisuperspace Hamiltonian, implement Hermitian factor ordering, and construct an explicit inner product, enabling probabilistic interpretation of the wave function. An explicit early-universe analysis shows that the asymptotic ratio $\varphi=B/\mathcal{A}$ becomes a constant, with $\varphi=0$ or $\varphi=2/7$ in representative models, and yields SWDW equations whose solutions produce nonsingular, bouncing quantum cosmologies in certain cases (e.g., $F^{0}(R)=b_{2}R^{2}$ or $G^{0}(T)=T$). Across three illustrative actions, the paper demonstrates how matter–geometry coupling shapes quantum dynamics, boundary conditions, and the emergence of a classical-like expansion at large times, while preserving a consistent quantum description at early times. The results offer a controlled setting to compare quantum cosmology across $f(R)$ and $f(R,T)$ theories and illuminate the role of internal time in modified gravity quantum cosmology.

Abstract

The $f(R, T)$ theory of gravity extends general relativity (GR) by allowing the gravitational Lagrangian to depend on both the Ricci scalar $R$ and the trace of the energy-momentum tensor $T$. The resulting matter-geometry coupling introduces additional dynamical effects that may account for the late-time acceleration of the universe without invoking dark energy. In the present work, we focus instead on the early-time regime and investigate the corresponding quantum cosmological dynamics. We analyze a Friedmann--Lemaitre--Robertson--Walker (FLRW) universe within the $f(R, T)$ framework, employing Schutz's perfect fluid formalism to extract a time parameter emerging from the matter sector itself. This approach is particularly well motivated in $f(R, T)$ gravity, where the coupling between geometry and the energy-momentum tensor's trace makes matter an active participant in the dynamics of spacetime and the evolution of cosmic time. The gravitational Hamiltonian, canonical momenta, and potential are derived, leading to the corresponding Schrödinger--Wheeler--DeWitt (SWDW) equation. The wave function of the universe is obtained for specific forms of $f(R, T)$, and the results are compared with previous studies in $f(R)$ and $f(R, T)$ models, highlighting the role of matter-geometry coupling in the emergence of quantum cosmological dynamics.