Quintom-like transit universe models in Metric-affine $f(R,T,Q,T_m)$ gravity

1. Problem Statement

The standard model of cosmology ($\Lambda$CDM) based on General Relativity (GR) faces challenges in explaining the rapid expansion of the universe during both early (inflation) and late (accelerated expansion) epochs without invoking a cosmological constant ($\Lambda$). While various modified gravity theories (e.g., $f(R)$, $f(T)$, $f(Q)$) have been proposed, they often rely on specific geometric assumptions (e.g., vanishing torsion or non-metricity).

This paper addresses the need for a more comprehensive geometric framework that unifies curvature, torsion, and non-metricity. Specifically, the authors aim to:

• Construct exact cosmological solutions within the Metric-affine $f(R, T, Q, T_m)$ gravity theory, which is the maximal extension of GR incorporating the Ricci scalar ($R$), torsion scalar ($T$), non-metricity scalar ($Q$), and the trace of the matter energy-momentum tensor ($T_m$).
• Investigate "transit" universe models (models that transition from deceleration to acceleration) without explicitly adding a cosmological constant to the Lagrangian.
• Constrain the model parameters using modern observational data to determine if the theory can reproduce a Quintom-like evolution (where the dark energy equation of state crosses the phantom divide, $\omega = -1$).

2. Methodology

Theoretical Framework

The authors utilize the Metric-affine gravity (MAG) formalism on a manifold $(M, g_{ij}, \Gamma^\rho_{ij})$. Unlike GR, where the connection is the Levi-Civita connection, MAG allows for independent torsion ($T^\alpha_{ij}$) and non-metricity ($Q_{\rho ij}$) tensors.

• Action: The action is defined as $S = \int \sqrt{-g} d^4x [f(R, T, Q, T_m) + \mathcal{L}_m]$.
• Field Equations: By varying the action with respect to the metric and the affine connection, the authors derive generalized field equations.
• Simplification: They assume a linear form for the function:
  $$f(R, T, Q, T_m) = \frac{1}{2}R + \alpha T + \beta Q + \gamma T_m$$
  where $\alpha, \beta, \gamma$ are coupling constants.
• Cosmological Setup: They apply the Friedmann-Lemaître-Robertson-Walker (FLRW) metric with a perfect fluid source ($p=0$, dust). They assume specific forms for the Weyl vector ($w_\mu$) and torsion vector ($A_\mu$) to simplify the field equations.

Analytical Solution

• Solving the connection equations yields two cases. The authors focus on the non-trivial case where torsion and non-metricity do not vanish, leading to a relationship between the coupling constants ($\alpha = \frac{\beta}{2\beta-1}$).
• Substituting the linear $f$-function into the Friedmann and Raychaudhuri equations results in a differential equation for the Hubble parameter $H(t)$.
• They derive a hyperbolic solution for the scale factor $a(t)$:
  $$a(t) \propto (\sinh[\sqrt{k_1 k_3}(t - c_0)])^{\frac{1}{k_3}}$$
  This solution naturally describes a universe transitioning from a decelerating phase to an accelerating phase.

Observational Constraints

To validate the model, the authors perform a Monte Carlo Markov Chain (MCMC) analysis using two datasets:

1. Cosmic Chronometers (CC): 31 data points of $H(z)$.
2. Pantheon Sample: 1048 Type Ia Supernovae (SNe Ia) apparent magnitude data.
   They minimize the $\chi^2$ statistic to constrain the Hubble constant ($H_0$) and the coupling parameter $\gamma$.

3. Key Contributions

1. Exact Hyperbolic Solution: The paper derives an exact analytical solution for the scale factor in Metric-affine $f(R, T, Q, T_m)$ gravity without introducing a cosmological constant. The acceleration is driven purely by the geometric modifications (torsion and non-metricity) and the matter coupling.
2. Quintom-A Behavior: The study demonstrates that the model naturally exhibits Quintom-A characteristics. In this scenario, the dark energy equation of state ($\omega_{de}$) starts in the quintessence regime ($\omega > -1$) in the early universe and crosses the phantom divide ($\omega = -1$) to enter the phantom regime ($\omega < -1$) in the late universe.
3. Transit Universe Model: The model successfully describes a "transit" phase where the universe evolves from a decelerating state ($q > 0$) to an accelerating state ($q < 0$), consistent with current cosmological observations.
4. Om Diagnostic: The authors apply the $Om(z)$ diagnostic test, confirming that the model behaves like a phantom dark energy scenario (positive slope of $Om(z)$) and converges toward the $\Lambda$CDM model in the far future.

4. Results

Parameter Constraints

Using MCMC analysis, the authors obtained the following constrained values:

• CC Dataset: $H_0 = 65.3^{+2.1}_{-2.5}$ km/s/Mpc, $\gamma = -0.6510^{+0.0065}_{-0.011}$.
• CC + Pantheon Dataset: $H_0 = 69.0 \pm 1.8$ km/s/Mpc, $\gamma = -0.6648^{+0.0022}_{-0.0027}$.
• The derived coupling constants $\alpha$ and $\beta$ are calculated based on $\gamma$, showing consistency with observational bounds.

Cosmological Parameters

• Equation of State (EoS):
  • Effective EoS ($\omega_{eff}$): Current value is $\approx -0.62$ (CC+Pantheon), consistent with $\Lambda$CDM.
  • Dark Energy EoS ($\omega_{de}$): The model shows a crossing of the phantom divide. Current values are $\omega_{de} \approx -1.26$ (CC) and $-1.63$ (CC+Pantheon).
• Deceleration Parameter ($q$):
  • Current value: $q_0 \approx -0.33$ (CC) and $-0.42$ (CC+Pantheon), confirming the current accelerating phase.
  • Transition Redshift ($z_t$): The redshift at which the universe switched from deceleration to acceleration is calculated as $z_t \approx 0.49$ (CC) and $z_t \approx 1.20$ (CC+Pantheon). These values are consistent with other literature estimates ($0.6 - 0.8$).
• Age of the Universe:
  • Calculated age: $t_0 = 13.92 \pm 1.3$ Gyr (CC) and $13.87 \pm 1.02$ Gyr (CC+Pantheon).
  • This aligns closely with the Planck 2018 estimate ($13.77 \pm 0.06$ Gyr).

5. Significance

• Geometric Unification: The paper provides a robust example of how a unified geometric framework (incorporating curvature, torsion, and non-metricity) can explain cosmic acceleration without the need for a cosmological constant or exotic scalar fields.
• Resolution of the Hubble Tension: The derived $H_0$ values ($65.3 - 69.0$ km/s/Mpc) fall within the range of recent measurements, bridging the gap between early-universe (CMB) and late-universe (SNe) measurements.
• Quintom Dynamics: The identification of a Quintom-A scenario is significant because it offers a theoretical mechanism for the dark energy equation of state to cross the "phantom divide" ($\omega = -1$), a feature difficult to achieve in standard scalar field theories without instabilities.
• Observational Viability: The model passes rigorous observational tests (CC, Pantheon, Om diagnostic, age of universe), suggesting that Metric-affine $f(R, T, Q, T_m)$ gravity is a viable candidate for describing the full evolution of the universe.

In conclusion, the authors successfully demonstrate that the maximal extension of GR via Metric-affine geometry can naturally generate a transit universe model with Quintom-like properties, offering a compelling alternative to the standard $\Lambda$CDM paradigm.