$f(Q,T)$ gravity (1908.04760v2)

Abstract: We propose an extension of the symmetric teleparallel gravity, in which the gravitational action $L$ is given by an arbitrary function $f$ of the nonmetricity $Q$ and of the trace of the matter energy-momentum tensor $T$, so that $L=f(Q,T)$. The field equations of the theory are obtained by varying the gravitational action with respect to both metric and connection. The covariant divergence of the field equations is obtained, with the geometry-matter coupling leading to the nonconservation of the energy-momentum tensor. We investigate the cosmological implications of the theory, and we obtain the cosmological evolution equations for a flat, homogeneous and isotropic geometry, which generalize the Friedmann equations of general relativity. We consider several cosmological models by imposing some simple functional forms of the function $f(Q,T)$, corresponding to additive expressions of $f(Q,T)$ of the form $f(Q,T)=\alpha Q+\beta T$, $f(Q,T)=\alpha Q^{n+1}+\beta T$, and $f(Q,T)=-\alpha Q-\beta T^2$. The Hubble function, the deceleration parameter, and the matter energy density are obtained as a function of the redshift by using analytical and numerical techniques. For all considered cases the Universe experiences an accelerating expansion, ending with a de Sitter type evolution. The theoretical predictions are also compared with the results of the standard $\Lambda$CDM model.

• The paper introduces a novel coupling between nonmetricity Q and the trace T, resulting in the non-conservation of the energy-momentum tensor.
• The authors derive field equations using both metric and affine variations, extending Friedmann equations to yield de Sitter and power-law solutions.
• The results suggest alternative mechanisms for cosmic acceleration without dark energy, inviting further empirical validation against cosmological data.

Analysis of the Paper on $f(Q,T)$ Gravity

The paper "f(Q,T) gravity" by Yixin Xu et al. explores the extension of symmetric teleparallel gravity (STG) by proposing a novel coupling between nonmetricity $Q$ and the trace $T$ of the energy-momentum tensor of matter. Specifically, the authors introduce a gravitational action where the Lagrangian $L$ is given by an arbitrary function $f(Q, T)$.

Core Innovations and Theoretical Framework

The paper's core innovation lies in formulating the field equations by varying the gravitational action with respect to both the metric and the connection. This leads to the crucial implication that the covariant divergence of the field equations results in the non-conservation of the energy-momentum tensor due to the coupling between geometry and matter. The authors provide a detailed derivation of the field equations, considering both mathematical and physical aspects of the problem, including the metric and affine formalism, which is vital for addressing variational principles in modified gravity theories.

Cosmological Implications

A significant portion of the paper discusses cosmological applications of the proposed theory. The authors derive the cosmological evolution equations for a flat, homogeneous, and isotropic Universe, extending the Friedmann equations from general relativity (GR). The analysis is performed for multiple functional forms of $f(Q,T)$, including linear and quadratic dependencies on $Q$ and $T$.

Numerical and Analytical Results

The results are divided into several models with varying functional expressions of $f(Q,T)$:

1. Model 1: $f(Q,T) = \alpha Q + \beta T$: The field equations predict a de Sitter type universe with exponential expansion when a particular relationship between parameters holds.
2. Model 2: $f(Q,T) = \alpha Q^{n+1} + \beta T$: This model shows a power-law expansion of the universe and encapsulates the potential to depict both accelerating and decelerating phases depending on parameter values.
3. Model 3: $f(Q,T) = -\alpha Q - \beta T^2$: Presents a more complex behavior with redshift-dependent cosmological parameters. The authors provide a numerical analysis that shows deviations from the standard $\Lambda$CDM model, particularly at higher redshifts, indicating potential new physics.

Implications and Future Directions

The implications of this research are multifaceted:

• Theoretical: The $f(Q,T)$ gravity introduces another layer of complexity to possible geometric interpretations of gravitation. By demonstrating conventional energy-momentum tensor conservation laws may not hold, it challenges the standard usages of stress-energy in cosmological models.
• Cosmological: The models offer alternative explanations for the accelerated expansion of the Universe without invoking dark energy explicitly. The paper's frameworks suggest potential for explaining both late-time acceleration and potentially inflationary phases.
• Speculative Developments: Future investigations could focus on quantum gravity frameworks that might underpin these geometric structures of spacetime. Additionally, fitting the models presented with current cosmological data could help verify or constrain the theory's parameter space. The non-conservation implications might also warrant a reevaluation of matter-energy interactions in the high-energy regimes, such as the early Universe or near black hole environments.

Conclusion

The paper by Yixin Xu et al. provides a substantial theoretical extension to symmetric teleparallel gravity by coupling nonmetricity with the trace of the energy-momentum tensor. Although the current findings primarily address cosmological scenarios, the flexibility and generality of the $f(Q,T)$ framework have broader implications in both fundamental physics and potential astrophysical applications. This paper reinforces the importance of exploring geometrically distinct theories of gravity, which could bridge gaps between current observations and theoretical physics.