Energy-Momentum Squared Gravity

• Energy-Momentum Squared Gravity is a modified gravity theory that incorporates quadratic matter terms in the action, altering gravitational dynamics in high-energy environments.
• The theory produces significant effects in cosmology and astrophysics, including singularity avoidance, modified FLRW evolution, and shifts in neutron star structure.
• EMSG implications are testable via gravitational wave observations and binary pulsar timing, which constrain the coupling parameters through precise phenomenological signatures.

Energy-Momentum Squared Gravity (EMSG) is a class of covariant metric theories extending General Relativity (GR) by introducing explicit dependence on quadratic scalars of the matter energy-momentum tensor, typically of the form $T_{\mu\nu}T^{\mu\nu}$. Unlike traditional modified gravities that generalize the curvature sector (e.g., $f(R)$ theories), EMSG posits that strong-field and early-universe deviations from GR originate from higher-order matter coupling terms. These terms significantly alter gravitational dynamics in regimes of high energy density, while they reduce to GR in vacuum. EMSG models have been systematically developed in several frameworks and have been shown to yield profound consequences for singularity avoidance, cosmological phase structure, neutron star physics, compact objects, gravitational collapse, and phenomenology accessible to both astrophysical and cosmological observations.

1. Formulation and Field Equations

The canonical EMSG action (minimal model) is constructed by augmenting the Einstein–Hilbert action with a function $f(T^2)$ of the matter-invariant $T^2 \equiv T_{\mu\nu}T^{\mu\nu}$: $$S = \int d^4x \sqrt{-g} \left[ \frac{1}{16\pi G} R + f(T^2) \right] + S_M$$ where $S_M$ represents the matter action, $g$ is the determinant of the metric, and $R$ is the Ricci scalar (Nazari et al., 2022, Roshan et al., 2016).

When specializing $f(T^2) = f'_{0}T^2$, the model introduces a single new coupling parameter $f'_{0}$ (SI: m s² kg⁻¹). Variation with respect to $g^{\mu\nu}$ yields the field equations: $$G_{\mu\nu} = 8\pi G (T_{\mu\nu} + T^{\mathrm{EMS}}_{\mu\nu})$$ with the EMSG correction,

$$T^{\mathrm{EMS}}_{\mu\nu} = f'_{0}\left( g_{\mu\nu} T^2 - 4T_\mu{}^\sigma T_{\nu\sigma} - 4\Psi_{\mu\nu} \right)$$

where $\Psi_{\mu\nu}$ depends on the matter Lagrangian (with $L_m = p$ for a perfect fluid, yielding explicit corrections in terms of $\rho, p, u^\mu$) (Nazari et al., 2022).

In more general models, the action may take the form $f(R, T_{\mu\nu}T^{\mu\nu})$ (Cipriano et al., 26 Aug 2024), $f(R, Q)$ with $Q \equiv T_{\mu\nu}T^{\mu\nu}$, or include nonminimal couplings to geometry such as $R_{\mu\nu}T^{\alpha \mu}T_\alpha^{\,\,\nu}$ (Shahidi, 2021). The field equations then possess additional structure and may include higher derivatives or non-trivial algebraic couplings between curvature and matter invariants.

A key property is that, for generic $f(T^2)$, the energy-momentum conservation law $\nabla_\mu T^{\mu\nu} = 0$ is generally violated, with only the effective combination $T_{\mu\nu} + T^{\mathrm{EMS}}_{\mu\nu}$ satisfying a generalized conservation law (Cipriano et al., 26 Aug 2024, Dunsby et al., 7 Nov 2025).

2. Cosmological Dynamics and Singularity Avoidance

EMSG yields a modified cosmological background evolution. For a spatially flat FLRW universe with barotropic fluid ($p = w\rho$), the modified Friedmann equation typically takes the schematic form

$$H^2 = \frac{\Lambda}{3} + \frac{\rho}{3} + \frac{f'_{0}}{6} (\rho^2 + 3p^2 + 8 \rho p)$$

or, for higher power models, more general functions of $\rho^{2n}$, leading to

$$H^2 = \frac{\Lambda}{3} + \frac{\kappa^2 \rho}{3} + \frac{\eta}{3} \rho^{2n} A(n, w)$$

with theory-specific coefficients $A(n, w)$ (Roshan et al., 2016, Board et al., 2017, Cipriano et al., 26 Aug 2024).

At high density, the quadratic and higher-order EMSG terms dominate. If the sign is suitable (e.g., $\eta < 0$ for Model I), these terms can enforce a maximum energy density $\rho_{\mathrm{max}}$ where $H=0$, realizing a cosmological bounce and resolving the initial big bang singularity. This mechanism has been confirmed in several minimal and extended models (Roshan et al., 2016, Cipriano et al., 26 Aug 2024, Khodadi et al., 2022). In anisotropic and Palatini formulations, viable bounces require positive matter coupling ($\eta > 0$) (Nazari et al., 2020).

The phase structure supports the full sequence of cosmic eras: nonsingular bounce $\rightarrow$ radiation-dominated era $\rightarrow$ matter era $\rightarrow$ de Sitter attractor, with transition between phases controlled by the strength of $f'_{0}$ or $\eta$ (Roshan et al., 2016, Khodadi et al., 2022).

3. Astrophysical and Compact Object Implications

EMSG introduces substantial modifications to the equilibrium structure and evolution of compact stars, especially neutron stars. The Tolman–Oppenheimer–Volkoff (TOV) equations are modified through $\rho^2$ and higher terms: $$\frac{dm}{dr} = 4\pi r^2 \rho \left[ 1 + \alpha \rho \left( 1 + 8\frac{P}{\rho} + 3\frac{P^2}{\rho^2} \right) \right]$$

$\frac{dP}{dr} = \ldots$

with additional nonlinearities in the effective density and pressure (Akarsu et al., 2018, Cipriano et al., 26 Aug 2024). Numerical studies utilizing modern nuclear equations of state have shown:

• The maximum neutron star mass and radius shift by several percent with $|\alpha| \sim 10^{-37}$ cm$^3$/erg.
• The allowed range from astrophysical observations is $-10^{-38}$ cm$^3$/erg $< \alpha < +10^{-37}$ cm$^3$/erg; outside this, either $M_{\mathrm{max}} < 2 M_\odot$ or the radius moves outside empirical bands.
• EMSG partially ameliorates the hyperon puzzle (hyperon-rich EoS can support $M_{\mathrm{max}} > 2 M_\odot$), but such configurations predict radii exceeding observational limits (Akarsu et al., 2018).
• EMSG generically yields negligible solar-system corrections, concentrating constraints to high-density regimes.

4. Relativistic Binaries, Gravitational Waves, and Observational Bounds

EMSG corrections to the dynamics and radiation from compact binaries, particularly neutron star binaries and binary pulsars, provide uniquely sensitive probes of $f'_0$. In the post-Minkowskian expansion, EMSG modifies the mass-quadrupole moment: $$\mathcal{I}^{jk}(\tau) = \int d^3x\, [\rho + c^2 f'_0 \rho^2] x^j x^k$$ which feeds into the radiated power and orbital period decay: $$\dot{P}_{\text{EMSG}} = \dot{P}_{\text{GR}} (1 + 2\alpha)$$ with $\alpha$ a small dimensionless EMSG parameter (Nazari et al., 2022). Direct and indirect GW events (e.g., GW170817, PSR J0737–3039A/B) constrain $|\alpha| \lesssim 10^{-5}$ for scale-independent models, and

$$-6\times 10^{-37}\,\text{m}\,\text{s}^2\,\text{kg}^{-1} < f'_0 < +10^{-36}\,\text{m}\,\text{s}^2\,\text{kg}^{-1}$$

when constrained by binary pulsar timing (Nazari et al., 2022, Akarsu et al., 2023). These constraints are compatible with those stemming from neutron star structure (Akarsu et al., 2018), and any EMSG deviation must remain subdominant at neutron star densities.

Future observational campaigns—continued double-pulsar timing, GW phase measurements with high-accuracy instruments—will further constrain $f'_0$ and probe strong-field deviations, potentially via measurements of additional post-Keplerian parameters such as Shapiro delay or periastron advance (Nazari et al., 2022).

5. Extensions: Dynamics, Phenomenology, and Theoretical Developments

EMSG encompasses a broader landscape of theories including nonminimal $R_{\mu\nu}T^{\alpha\mu}T_\alpha^{\,\,\nu}$ couplings (Shahidi, 2021), logarithmic forms (Acquaviva et al., 2022), power laws $f(R, T^2) = f_0 R^n (T^2)^m$ (Bahamonde et al., 2019), and even $f(R, T^2)$ models with Palatini variation (Nazari et al., 2020). Dynamical systems analysis yields the following:

• The phase space admits diverse critical points supporting matter, radiation, and de Sitter eras, late-time acceleration, phantom crossings, and bouncing regimes (Bahamonde et al., 2019, Board et al., 2017).
• EMSG models generate effective fluids with varying "dark-energy"–like behaviors, including screening mechanisms for the cosmological constant and shifting de Sitter attractor values, as in energy-momentum logarithmic gravity (EMLG) (Acquaviva et al., 2022).
• Nonminimal EMSG with conservation enforced by Lagrange multipliers remains observationally viable only for small effective couplings ($|\beta|\lesssim 1.1\times10^{-5}$), and closely mimics $\Lambda$CDM at $z\lesssim2$ (Shahidi, 2021).

Inflationary models sourced by EMSG (including “energy-momentum powered gravity” with $f(T^2)\propto (T^2)^\beta$) can bring otherwise excluded inflationary scenarios into accord with CMB data, reducing the tensor-to-scalar ratio $r$ and possibly enhancing non-Gaussianities depending on model parameters (Mansoori et al., 2023, Faraji et al., 2021).

Baryogenesis models exploit the non-conservation of $T_{\mu\nu}$ for successful gravitational baryogenesis in the radiation-dominated era, otherwise forbidden in GR (Pereira et al., 6 Sep 2024).

6. Cosmological Perturbations and Observable Signatures

EMSG modifies the dynamics of linear perturbations about FLRW backgrounds in all sectors—scalar, vector, and tensor—as shown in the manifestly covariant and gauge-invariant formalism of (Dunsby et al., 7 Nov 2025):

• Scalar Modes: The density contrast evolution equation acquires $\mathcal{O}(\eta \rho^2)$ coefficients, which can enhance or suppress growth depending on $\eta$, $\rho$, and $k$. For dust, even in the absence of microscopic pressure, EMSG induces an effective sound speed, yielding a finite Jeans length and suppressing small-scale structure.
• Vector Modes: The vorticity decay rate is altered, typically slowed at early times, enabling non-trivial vorticity to persist longer than in GR and affecting primordial magnetic field generation.
• Tensor Modes: Gravitational waves propagate as damped waves with effective time-varying masses. The decay rate of stochastic GW backgrounds and CMB $B$-modes are accordingly shifted.

All observable deviations reduce smoothly to their respective GR forms as the additional couplings vanish. Current, and especially next-generation, cosmological data (CMB, large-scale structure, GW backgrounds) provide robust constraints on $\eta$ of order $10^{-2}$–$10^{-3}$ (Dunsby et al., 7 Nov 2025).

7. Open Problems and Prospects

EMSG remains a highly constrained but fertile framework for exploring gravitational phenomena in the strong-matter regime. Its central distinguishing prediction—the breakdown of $\nabla_\mu T^{\mu\nu}=0$ and corresponding matter-curvature feedback—is amenable to falsification via neutron-star structure, GW phasing, cosmological perturbations, baryogenesis, and early-universe singularity avoidance (Cipriano et al., 26 Aug 2024, Dunsby et al., 7 Nov 2025, Akarsu et al., 2018, Nazari et al., 2022).

Key outstanding issues include:

• Microphysical origins—whether EMSG arises as the classical limit of quantum gravity, brane models, or is purely phenomenological.
• Non-perturbative dynamics in numerical relativity, including collapse, bounce, and possibly cosmic censorship violation (Rudra, 10 Feb 2024).
• The precise impact on late-time cosmological tensions (e.g., $H_0$ discrepancy) and integration with inflation, dark energy, and dark matter phenomenology.

Interdisciplinary efforts intersecting theory, numerical modeling, and observational campaigns are expected to further clarify the viability and implications of EMSG in the coming years.