Energy-Momentum Squared Gravity Theory

• EMSG is a modified gravity theory that augments GR with quadratic energy-momentum tensor terms, introducing new dynamics in high-curvature regimes.
• The theory modifies field equations, impacting cosmological expansion, structure growth, and gravitational wave friction in matter-dense regions.
• EMSG predicts non-singular early universe models and alters compact object properties, offering alternative explanations for cosmic and astrophysical observations.

Energy-Momentum Squared Gravity (EMSG) is a covariant extension of General Relativity (GR) that augments the Einstein–Hilbert action with terms proportional to quadratic invariants of the energy-momentum tensor. Predominantly formulated as $f(R, T_{\mu\nu}T^{\mu\nu})$ gravity, where $R$ is the Ricci scalar and $T_{\mu\nu}$ the matter stress-energy tensor, EMSG generates deviations from GR only within matter distributions or high-curvature regimes. The theory is defined by new coupling constants and non-trivial field equations—yielding consequences across cosmology, astrophysics, and gravitational-wave physics.

1. Covariant Action and Field Equations

EMSG introduces an extra term in the Lagrangian proportional to $\mathcal{T}^n \equiv (T_{\mu\nu}T^{\mu\nu})^n$: $$S = \int d^4 x \sqrt{-g} \left[ \frac{1}{2\kappa} R + \eta \mathcal{T}^n + \mathcal{L}_m \right],$$ where $\kappa = 8\pi G$, $\eta$ is a new coupling constant, and $n$ the power index. The field equations derived from metric variation are

$$f_R R_{\mu\nu} - \frac{1}{2} f g_{\mu\nu} + (g_{\mu\nu}\Box - \nabla_\mu \nabla_\nu) f_R = \frac{1}{2} T_{\mu\nu} - f_\mathcal{T} \theta_{\mu\nu},$$

with $f_R = \partial f / \partial R$, $R$0, and

$R$1

Special choices include $R$2 (original EMSG), $R$3 (scale-independent EMSG), and arbitrary $R$4 for generalized power-law models (Fu et al., 2024).

2. Cosmological Evolution and Perturbations

In a spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW) background, the modified Friedmann equations for dust are:

• For $R$5 (scale-independent),

$R$6

with $R$7.

• For $R$8:

$R$9

where $T_{\mu\nu}$0.

Linear cosmological perturbation theory yields modified Poisson equations and matter growth equations. The density contrast evolution generalizes the GR form as: $T_{\mu\nu}$1 with $T_{\mu\nu}$2 dependent on $T_{\mu\nu}$3, and derivatives of $T_{\mu\nu}$4 (Fu et al., 2024).

3. Gravitational Wave Physics and GW Luminosity Distance

Tensor perturbations yield the EMSG gravitational wave equation: $T_{\mu\nu}$5 showing a modified friction term but $T_{\mu\nu}$6 (standard GW propagation speed). The GW luminosity distance differs from electromagnetic ($T_{\mu\nu}$7) by

$T_{\mu\nu}$8

supporting joint constraints using GW standard sirens and redshift space distortions (Fu et al., 2024).

4. Empirical Constraints and Growth of Structure

Best-fit cosmological parameters (with 1σ uncertainties, joint GW+RSD, $T_{\mu\nu}$9 and $\mathcal{T}^n \equiv (T_{\mu\nu}T^{\mu\nu})^n$0 cases; (Fu et al., 2024)): | Model | $\mathcal{T}^n \equiv (T_{\mu\nu}T^{\mu\nu})^n$1 [km/s/Mpc] | $\mathcal{T}^n \equiv (T_{\mu\nu}T^{\mu\nu})^n$2 | $\mathcal{T}^n \equiv (T_{\mu\nu}T^{\mu\nu})^n$3 | $\mathcal{T}^n \equiv (T_{\mu\nu}T^{\mu\nu})^n$4 or $\mathcal{T}^n \equiv (T_{\mu\nu}T^{\mu\nu})^n$5 | |--------------|------------------|-----------------|-----------------|-------------------------| | $\mathcal{T}^n \equiv (T_{\mu\nu}T^{\mu\nu})^n$6 | $\mathcal{T}^n \equiv (T_{\mu\nu}T^{\mu\nu})^n$7 | $\mathcal{T}^n \equiv (T_{\mu\nu}T^{\mu\nu})^n$8 | $\mathcal{T}^n \equiv (T_{\mu\nu}T^{\mu\nu})^n$9 | $$S = \int d^4 x \sqrt{-g} \left[ \frac{1}{2\kappa} R + \eta \mathcal{T}^n + \mathcal{L}_m \right],$$0 | | $$S = \int d^4 x \sqrt{-g} \left[ \frac{1}{2\kappa} R + \eta \mathcal{T}^n + \mathcal{L}_m \right],$$1 | $$S = \int d^4 x \sqrt{-g} \left[ \frac{1}{2\kappa} R + \eta \mathcal{T}^n + \mathcal{L}_m \right],$$2 | $$S = \int d^4 x \sqrt{-g} \left[ \frac{1}{2\kappa} R + \eta \mathcal{T}^n + \mathcal{L}_m \right],$$3 | $$S = \int d^4 x \sqrt{-g} \left[ \frac{1}{2\kappa} R + \eta \mathcal{T}^n + \mathcal{L}_m \right],$$4 | $$S = \int d^4 x \sqrt{-g} \left[ \frac{1}{2\kappa} R + \eta \mathcal{T}^n + \mathcal{L}_m \right],$$5 |

Deviation parameters are small and positive, consistent with $$S = \int d^4 x \sqrt{-g} \left[ \frac{1}{2\kappa} R + \eta \mathcal{T}^n + \mathcal{L}_m \right],$$6CDM at $$S = \int d^4 x \sqrt{-g} \left[ \frac{1}{2\kappa} R + \eta \mathcal{T}^n + \mathcal{L}_m \right],$$7 confidence. The growth rate $$S = \int d^4 x \sqrt{-g} \left[ \frac{1}{2\kappa} R + \eta \mathcal{T}^n + \mathcal{L}_m \right],$$8 is mildly suppressed for $$S = \int d^4 x \sqrt{-g} \left[ \frac{1}{2\kappa} R + \eta \mathcal{T}^n + \mathcal{L}_m \right],$$9 (deviation $\kappa = 8\pi G$0). Combination of standard-siren and large-scale structure data provides $\kappa = 8\pi G$1 tighter parameter constraints (Fu et al., 2024).

5. High-Energy and Early Universe Phenomena

Quadratic energy-momentum terms dominate at high densities. EMSG generically induces a cosmological bounce, replacing the big bang singularity with a finite minimum scale factor $\kappa = 8\pi G$2 and maximal density $\kappa = 8\pi G$3 (Roshan et al., 2016). In the radiation era $\kappa = 8\pi G$4 produces a non-singular, past-complete universe. Thermodynamic non-conservation of $\kappa = 8\pi G$5 allows for irreversible matter creation in an open-system framework, impacting entropy and temperature evolution (Cipriano et al., 2024). Baryogenesis is enabled by new CP-violating couplings based on $\kappa = 8\pi G$6, yielding successful baryon asymmetry in the early universe even with $\kappa = 8\pi G$7, where standard $\kappa = 8\pi G$8 gravity fails (Pereira et al., 2024).

Inflationary EMSG scenarios include power-law potentials for canonical and non-canonical scalar fields, reducing the tensor-to-scalar ratio $\kappa = 8\pi G$9 and accommodating non-Gaussianity (if the matter sector is non-canonical). Constraints from Planck/BICEP and joint Planck+BAO analyses require $\eta$0 for observationally viable inflation (Faraji et al., 2021, Mansoori et al., 2023).

6. Astrophysical Applications: Compact Objects and Stability

In vacuum, EMSG reduces identically to GR, but in the presence of matter, compact objects such as neutron stars exhibit mass-radius deviations. The generalized Tolman-Oppenheimer-Volkoff equations feature effective density and pressure with quadratic corrections. For positive $\eta$1, EMSG accommodates more massive neutron stars using standard polytropic equations of state, aiding the explanation of observed $\eta$2 pulsars without exotic matter (Nari et al., 2018). Observational stability bounds from pulsar timing enforce $\eta$3 m s$\eta$4 kg$\eta$5 (Nazari et al., 2022).

Jeans analysis in EMSG predicts modified instability thresholds for both non-rotating and disk systems, favoring $\eta$6 for stability—consistent with lack of observed fragmentation in hypermassive neutron stars (Kazemi et al., 2020). Light bending and gravitational lensing calculations deliver density-dependent corrections detectable at micro-arcsecond scales in dense lenses; solar-system tests restrict $\eta$7–$\eta$8 m s$\eta$9 kg$n$0 and show full agreement with current measurements (Nazari, 2022).

7. Dynamical, Thermodynamic, and Structural Properties

EMSG generically violates energy-momentum conservation, leading to irreversible particle creation and "thermodynamic" matter production in cosmological evolution (Cipriano et al., 2024). Scale-independent ($n$1) EMSG and power-law ($n$2) models exhibit modified scaling laws for matter and radiation densities, non-standard expansion histories, and the possibility of steady-state or big-rip type evolution depending on coupling parameters (Akarsu et al., 2023, Bahamonde et al., 2019). Dynamical-system studies reveal transitions through radiation-, matter-, and acceleration-dominated epochs, with de Sitter attractors in late times for suitable choices (Bahamonde et al., 2019, Marciu, 29 Jan 2026).

Thermodynamic consistency requires explicit accounting of second metric derivatives of the matter Lagrangian, and the scalar-tensor formulation allows a systematic analysis of stability and cosmic evolution. EMSG thus encompasses a broad spectrum of phenomenologically viable and theoretically rich scenarios—including emergent universe proposals (Einstein static states and graceful-exit mechanisms), violation or restoration of energy-momentum conservation (via Lagrange multipliers), and anisotropic cosmological models solvable by Noether symmetry techniques (Khodadi et al., 2022, Sharif et al., 2023, Sharif et al., 2021, Shahidi, 2021).

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EMSG remains a theoretically compelling extension of GR featuring non-minimal, quadratic matter-curvature couplings. It delivers robust early universe regularization, testable deviations in compact object structure, modified gravitational-wave propagation, and empirical compatibility with cosmological datasets (Fu et al., 2024, Roshan et al., 2016, Cipriano et al., 2024). The theory is under persistent scrutiny through dynamical analysis, astrophysical observation, GW detection, and large-scale structure, with future surveys expected to probe its mild but distinctive departures from standard cosmology.