[Paper Review] Dynamical reconstruction of the $Λ$CDM model in scalar-tensor $f(R,T)$ gravity

Tiago B. Gonçalves, João Luís Rosa+1 more|arXiv (Cornell University)|May 9, 2023

Cosmology and Gravitation Theories113 references7 citations

TL;DR

This paper applies dynamical systems analysis to scalar-tensor $f(R,T)$ gravity, proposing two models with distinct geometry-matter couplings: additive ($R + T$) and multiplicative ($RT$). It finds that both models enable a natural transition from decelerated to accelerated expansion without exotic fluids, with the additive model successfully achieving asymptotic de-Sitter behavior, suggesting weaker curvature-matter coupling is more viable for late-time cosmic acceleration.

ABSTRACT

In this work, we use the dynamical system approach to explore the cosmological background evolution of the scalar-tensor representation of $f(R,T)$ gravity, where $R$ is the Ricci scalar and $T$ is the trace of the stress-energy tensor. The motivation for this work resides in finding dynamical cosmological behaviors comparable with the $Λ$CDM model without the necessity of recurring to a dark energy component. We introduce a set of dynamical variables that allow for a direct comparison with the cosmological standard model and the current experimental measurements, and develop a dynamical system framework to analyze the cosmological evolution of Friedmann-Lemaître-Robertson-Walker (FLRW) universes within this theory. In this framework, we obtain the critical points in the cosmological phase space and perform fully numerical integrations of the dynamical system to extract the cosmological behavior, subjected to initial conditions compatible with the measurements by the Planck satellite. The phase space of the theory is proven to feature fixed points associated with cosmological behaviors analogous to those of GR, whereas variations in the scalar field associated to the dependency in $T$ affect the phase space structure only quantitatively. Our results indicate that cosmological solutions featuring a radiation dominated epoch, followed by a transition into a matter dominated epoch, and finally a transition into an exponentially accelerated epoch, are allowed by the theory, while maintaining a present state compatible with the current measurements from the Planck satellite and solar system dynamics, and preserving the regularity of the scalar fields and their interaction potential.

Motivation & Objective

• To investigate whether $f(R,T)$ gravity with scalar-tensor representation can naturally produce a transition from decelerated to accelerated cosmic expansion without requiring dark energy or exotic fluids.
• To compare two classes of geometry-matter coupling—additive ($R + T$) and multiplicative ($RT$)—in terms of their cosmological phase space structure and stability.
• To assess the role of stress-energy tensor non-conservation ($\nabla^\mu T_{\mu\nu} \neq 0$) in enabling de-Sitter attractor solutions.
• To determine which model structure better reproduces the observed late-time de-Sitter-like behavior of the universe.
• To explore the implications of different on-shell matter Lagrangians ($\mathcal{L}_m = p, -\rho, T$) on cosmological dynamics, particularly in non-minimally coupled gravity.

Proposed method

• Formalism is based on the scalar-tensor representation of $f(R,T)$ gravity, introducing auxiliary scalar fields $\varphi$ and $\psi$ to recast the action into a more tractable form.
• The dynamical systems approach is employed by defining dimensionless variables from the Hubble parameter, scalar fields, and their derivatives to reduce the field equations to a system of autonomous ODEs.
• Critical points of the dynamical system are identified by solving the vanishing of the right-hand side of the evolution equations, corresponding to fixed points in phase space.
• Stability of critical points is analyzed via linearization and eigenvalue evaluation, determining whether trajectories converge to or diverge from them.
• Phase space trajectories are numerically integrated to trace cosmological evolution from initial decelerated states to accelerated regimes.
• The analysis is performed under two assumptions: $\nabla^\mu T_{\mu\nu} = 0$ (matter conservation) and $\nabla^\mu T_{\mu\nu} \neq 0$ (non-conservation), to assess the impact on attractor solutions.

Experimental results

Research questions

• RQ1 Can the additive $R + T$ coupling in scalar-tensor $f(R,T)$ gravity produce a natural transition from a decelerated to an accelerated expansion phase without exotic matter?
• RQ2 Does the multiplicative $RT$ coupling also support a transition to accelerated expansion, and how does its phase space structure compare to the additive case?
• RQ3 Is there a critical point in the phase space corresponding to a de-Sitter solution in either model, and is it a global attractor?
• RQ4 How does the non-conservation of the stress-energy tensor ($\nabla^\mu T_{\mu\nu} \neq 0$) affect the existence and stability of de-Sitter attractors?
• RQ5 Which matter Lagrangian choice ($\mathcal{L}_m = p, -\rho, T$) is most consistent with the dynamics in non-minimally coupled $f(R,T)$ gravity?

Key findings

• The additive model ($V = V_1\varphi^\alpha + V_2\psi^\beta$) features a critical point corresponding to a de-Sitter solution, and trajectories in its phase space evolve from decelerated to accelerated expansion, with some approaching asymptotic de-Sitter behavior.
• The multiplicative model ($V = V_0\varphi^\alpha\psi^\beta$) does not support a stable de-Sitter attractor; trajectories remain accelerated but deviate from de-Sitter even when close to it.
• Both models exhibit a wide range of cosmological solutions, including decelerated, accelerated, and coasting phases, with no global attractor in either case.
• The additive model allows for a natural transition from a dust-dominated decelerated phase to an accelerated phase, even when $w = 0$ and $\alpha = \beta = 2$, indicating robustness without exotic fluids.
• Relaxing the matter conservation condition ($\nabla^\mu T_{\mu\nu} \neq 0$) increases the number of critical points, including a de-Sitter attractor, suggesting non-conservation enhances the possibility of late-time de-Sitter evolution.
• The results imply that weaker curvature-matter coupling (additive form) is more favorable for modeling the observed asymptotic de-Sitter phase of the universe than stronger coupling (multiplicative form).