f(Q,T) Model in Symmetric Teleparallel Gravity

• f(Q,T) gravity is a modified framework that couples the nonmetricity scalar Q with the trace T to generate novel gravitational dynamics and energy–momentum exchange.
• It yields modified Friedmann equations that support accelerated cosmic expansion and accommodate exotic solutions like traversable wormholes under specific parameter regimes.
• Stability and perturbation analyses reveal challenges such as oscillatory growth and fine-tuned constraints, highlighting the need for comprehensive observational validation.

The $f(Q,T)$ model of symmetric teleparallel modified gravity generalizes the gravitational action by allowing non-minimal coupling between the nonmetricity scalar $Q$ and the trace $T$ of the energy–momentum tensor, thereby encoding gravity as a function of spacetime geometry (via $Q$) and explicit matter content (via $T$). This framework extends the purely geometric $f(Q)$ models and provides a broader, technically rich platform for addressing cosmological acceleration, energy conditions, dynamical evolution, perturbation theory, and astrophysical solutions such as wormholes. Below is an integrated review of the central features, formalism, empirical results, and open problems as developed in the literature.

1. Mathematical Foundations and Field Equations

The $f(Q,T)$ gravity theory is defined by the following action: $$S = \int \left[ \frac{1}{16\pi} f(Q, T) + \mathcal{L}_m \right] \sqrt{-g}\, d^4x$$ where $Q$ is the nonmetricity scalar, $T = g^{\mu\nu} T_{\mu\nu}$ is the trace of the energy-momentum tensor, and $\mathcal{L}_m$ denotes the matter Lagrangian. In symmetric teleparallel gravity, the connection is chosen so that both curvature and torsion vanish, leaving nonmetricity as the sole nontrivial geometric aspect.

The nonmetricity scalar $Q$ is given by: $$Q = -g^{\mu\nu}\left( L^{\alpha}_{\phantom{\alpha}\beta\mu} L^{\beta}_{\phantom{\beta}\nu\alpha} - L^{\alpha}_{\phantom{\alpha}\beta\alpha} L^{\beta}_{\phantom{\beta}\mu\nu} \right)$$ with the disformation tensor $L^{\alpha}_{\phantom{\alpha}\beta\gamma}$ constructed from covariant derivatives of the metric.

Variation of the action yields modified field equations: $$- \frac{2}{\sqrt{-g}} \nabla_\alpha \left( f_Q \sqrt{-g} P^\alpha_{\phantom{\alpha}\mu\nu} \right) - \frac{1}{2} f g_{\mu\nu} + f_T \left( T_{\mu\nu} + \Theta_{\mu\nu} \right) - f_Q\left[ P_{\mu\alpha\beta} Q_\nu^{\phantom{\nu}\alpha\beta} - 2 Q^{\alpha\beta}_{\phantom{\alpha\beta}\mu} P_{\alpha\beta\nu}\right] = 8\pi T_{\mu\nu}$$ where $f_Q = \partial f / \partial Q$, $f_T = \partial f / \partial T$, and $$\Theta_{\mu\nu} = g^{\alpha\beta} \delta T_{\alpha\beta} / \delta g^{\mu\nu}$$ encodes additional coupling between matter and geometry.

Crucially, the covariant divergence of the field equations implies $\nabla^\mu T_{\mu\nu} \neq 0$, leading to a dynamical exchange of energy and momentum between matter and geometry—manifested as extra forces on matter and the possibility of particle production.

2. Cosmological Dynamics and Phenomenology

2.1 Modified Friedmann Equations

For a flat FLRW background, $ds^2 = -dt^2 + a(t)^2 d\vec{x}^2$, the nonmetricity is $Q = 6H^2$ (with $H = \dot{a}/a$). The modified Friedmann equations generically acquire the form: $$3H^2 = 8\pi (\rho + \rho_{corr}), \qquad 2\dot{H} + 3H^2 = -8\pi (p + p_{corr})$$ where the correction terms $\rho_{corr}$ and $p_{corr}$ encapsulate contributions from $f(Q,T)$ derivatives and matter couplings. For example, in the model $f(Q,T) = \alpha Q^{n+1} + \beta T$, the Friedmann equation yields a power-law $a(t)$ and an accelerated cosmic expansion for suitable parameter ranges (Xu et al., 2019).

2.2 Exemplary $f(Q,T)$ Forms and Their Cosmological Implications

Model	Key Cosmological Features
$f(Q,T) = \alpha Q + \beta T$	Admits de Sitter solutions with exponential expansion; matter decays exponentially
$f(Q,T) = \alpha Q^{n+1} + \beta T$	Allows for accelerated power-law expansion; deceleration parameter can be negative
$f(Q,T) = -\alpha Q - \beta T^2$	Numerically produces late-time de Sitter behavior; theoretical models fit SN, Hubble data

For these choices, the universe transitions from decelerated to accelerated, with the effective equation of state $w_{\mathrm{eff}}$ crossing $-1$ (phantom divide) in some cases and yielding late-time de Sitter attractors. Many models can mimic $\Lambda$CDM at low redshift but yield differences at higher $z$ (Xu et al., 2019, Godani et al., 2021, Narawade et al., 2023).

3. Energy Conditions and Stability

The modified field equations lead to changes in the effective energy density and pressure, impacting the classical energy conditions:

• Null, Weak, and Dominant Energy Conditions (NEC, WEC, DEC): Generally satisfied in most $f(Q,T)$ models for physically reasonable parameter choices, ensuring $\rho + p \geq 0$, $\rho \geq 0$, and $|\rho| \geq |p|$ (Arora et al., 2020, Godani et al., 2021).
• Strong Energy Condition (SEC): Typically violated in accelerating/cosmological constant–like regimes; this violation is closely tied to the presence of cosmic acceleration (Arora et al., 2020, Narawade et al., 2023).

Stability analysis via squared adiabatic sound speed $C_s^2 = dp/d\rho$ indicates that NEC and SEC violation near bounces or at late times can induce negative $C_s^2$, with potential for gradient instabilities that must be controlled by constraint on parameter space (Agrawal et al., 2021).

4. Cosmological Perturbations and Structure Growth

Linear perturbation theory in $f(Q,T)$ gravity introduces qualitatively new features:

• Tensor Modes (GWs): Propagation is unchanged compared to $f(Q)$ gravity when $f_T=0$; only an extra "friction" term occurs if $f_Q$ varies (Nájera et al., 2021).
• Scalar Sector: Coupling between $Q$ and $T$ leads to mixing of density and pressure perturbations; energy–momentum is not conserved ($$\delta \rho' + 3\mathcal{H} (\delta \rho + \delta p) \neq 0$$), introducing explicit source terms.
• Differential Density Equation: The sub-Hubble matter density contrast $\delta$ follows a scale-dependent second-order differential equation:

$$A(z, k)\, \delta'' + B(z, k)\,\delta' + C(z, k)\,\delta = 0$$

Modified by $f_T$ and including extra $k^2$-dependent terms absent in $\Lambda$CDM, leading to strong late-time oscillatory behavior and stringent phenomenological constraints (Filali et al., 5 Aug 2025, Nájera et al., 2021).

• Holographic Dark Energy (HDE): Integration of an HDE component partially damps oscillatory perturbations but does not fully reconcile the growth of structure with observations in standard parameter regimes (Filali et al., 5 Aug 2025).

5. Observational Constraints and Empirical Viability

$H_0$, $\Omega_m$, and EoS parameter values have been constrained using multi-probe datasets (OHD, SNeIa, BAO, growth data):

• Parametric Hubble ansatz (e.g., $H(z) = H_0 [\alpha + (1-\alpha)(1+z)^n]^{3/2n}$) recovers $\Lambda$CDM for $n=3$ and allows fitting to $H_0 \sim 65$–$70$ km/s/Mpc and $\Omega_m \sim 0.3$–$0.35$ (Narawade et al., 2023).
• Statefinder diagnostics and Om diagnostic reveal present-day quintessence–like behavior ($w_0 \gtrsim -1$).
• BBN constraints on $f(Q,T)=Q^{n+1}+mT$ require $-1.13 \lesssim n \lesssim -1.08$ and $-5.86 \lesssim m \lesssim 12.52$, with consistency for helium and deuterium but not lithium (Bhattacharjee, 2021).
• Dynamical system analyses yield late-time stable attractors corresponding to a dark energy dominated state and realistic density parameter evolution (e.g., $\Omega_m \approx 0.3$, $\Omega_{de} \approx 0.7$) (Narawade et al., 2023).

6. Special Solutions: Wormholes and Exotic Matter

$f(Q,T)$ gravity enables construction of static, spherically symmetric traversable wormholes with new properties relative to GR:

• Shape Functions: Power-law forms for $b(r)$ (throat geometry) derived analytically for linear models (e.g., $f(Q,T)=\alpha Q+\beta T$), with parameter domains where asymptotic flatness and flare-out conditions are satisfied (Tayde et al., 2022, Tayde, 8 May 2025).
• Energy Conditions: NEC is systematically violated near the throat (as required for traversable wormholes), with less severe violation than in GR depending on the magnitude and sign of matter–geometry coupling.
• Stability: TOV analysis demonstrates that hydrostatic and anisotropic force components balance, ensuring stability for constant redshift functions.
• Nonlinear Models: Full analytic solutions not tractable; numerical methods required. Noncommutative geometry extensions and inclusion of the MIT bag model for exotic matter sources have been analyzed (Tayde, 8 May 2025).

7. Open Challenges and Future Directions

Several technical and phenomenological facets remain under investigation:

• Fine-tuning and Model Selection: Determining whether the $f(Q,T)$ class can robustly resolve cosmological constant and coincidence problems without additional tuning.
• Perturbative Pathologies: Oscillatory growth and strong coupling in structure formation and the scalar sector can be problematic; full nonlinear analyses, or amended Lagrangians, may be necessary (Nájera et al., 2021, Filali et al., 5 Aug 2025).
• Broader Data Confrontation: Incorporating cosmic microwave background anisotropies, lensing, and the full growth factor dataset is needed to rule in/out viable $f(Q,T)$ scenarios.
• Astrophysical Predictions: Gravitational lensing by wormholes in $f(Q,T)$ can exhibit unique photon sphere behavior, suggesting observational discriminants between wormholes and black holes (Tayde, 8 May 2025).

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This synthesis shows that $f(Q,T)$ gravity represents a technically rich and observationally testable generalization of symmetric teleparallel gravity. It provides intrinsic geometric routes to cosmological acceleration, accommodates viable phenomenology at the background and perturbative levels, and yields novel astrophysical solutions—though stringent constraints on parameter space and stability must be respected for empirical viability.