---
title: Energy-Momentum Squared Gravity Theory
url: https://www.emergentmind.com/topics/energy-momentum-squared-gravity-theory
type: topic
---

# Energy-Momentum Squared Gravity Theory

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
$f_R = \partial f / \partial R$,
$f_\mathcal{T} = \partial f / \partial \mathcal{T}$, and
\[
\theta_{\mu\nu} = -2 \mathcal{L}_m [ T_{\mu\nu} - \frac{1}{2} T g_{\mu\nu} ] - T T_{\mu\nu} + 2 T_\mu^{\ \alpha} T_{\nu\alpha}.
\]
Special choices include $n=1$ (original EMSG), $n=1/2$ (scale-independent EMSG), and arbitrary $n$ for generalized power-law models [2408.01665].

## 2. Cosmological Evolution and Perturbations

In a spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW) background, the modified Friedmann equations for dust are:
- For $n=1/2$ (scale-independent),
  \[
  \rho(a) = \rho_0 a^{-3(1+\eta_t)},\qquad H^2 = H_0^2 \left[ \Omega_m (1+z)^{3(1+\eta_t)} + 1 - \Omega_m \right],
  \]
  with $\eta_t = \eta/(2\kappa)$.
- For $n=1$:
  \[
  \rho(a) = \rho_0 a^{-3},\qquad H^2 = H_0^2[ \Omega_m (1+z)^3 + \Omega_Q (1+z)^6 + \Omega_\Lambda ],
  \]
  where $\Omega_Q = \eta \rho_0^2 / (6 H_0^2)$.

Linear cosmological perturbation theory yields modified Poisson equations and matter growth equations. The density contrast evolution generalizes the GR form as:
\[
\ddot{\delta}_m + A(t) \dot{\delta}_m + B(t) \delta_m = 0,
\]
with $A(t), B(t)$ dependent on $a(t), H(t), \rho$, and derivatives of $f(R, \mathcal{T})$ [2408.01665].

## 3. Gravitational Wave Physics and GW Luminosity Distance

Tensor perturbations yield the EMSG gravitational wave equation:
\[
\ddot{h}_{ij} + 3H(1-\beta)\dot{h}_{ij} - \frac{1}{a^2} \nabla^2 h_{ij} = 0,\quad
\beta = -\frac{\dot{f}_R}{3H f_R},
\]
showing a modified friction term but $c_T=1$ (standard GW propagation speed). The GW luminosity distance differs from electromagnetic ($d_L^{EM}$) by
\[
d_L^{GW}(z) = d_L^{EM}(z) \exp \left[ -\frac{3}{2} \int_0^z \frac{\beta(z')}{1+z'} dz' \right],
\]
supporting joint constraints using GW standard sirens and redshift space distortions [2408.01665].

## 4. Empirical Constraints and Growth of Structure

Best-fit cosmological parameters (with 1σ uncertainties, joint GW+RSD, $n=1/2$ and $n=1$ cases; [2408.01665]):
| Model        | $H_0$ [km/s/Mpc] | $\Omega_m$      | $\sigma_8$      | $\eta_t$ or $\Omega_Q$  |
|--------------|------------------|-----------------|-----------------|-------------------------|
| $n=1/2$      | $67.80 \pm 0.81$ | $0.283^{+0.033}_{-0.038}$ | $0.730 \pm 0.022$ | $0.008 \pm 0.030$      |
| $n=1$        | $68.06 \pm 0.62$ | $0.271 \pm 0.019$ | $0.717^{+0.025}_{-0.022}$ | $0.00072 \pm 0.00060$  |

Deviation parameters are small and positive, consistent with $\Lambda$CDM at $68\%$ confidence. The growth rate $f\sigma_8(z)$ is mildly suppressed for $z \lesssim 1$ (deviation $<5\%$). Combination of standard-siren and large-scale structure data provides $\gtrsim 30\%$ tighter parameter constraints [2408.01665].

## 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 $a_{\min}$ and maximal density $\rho_{\max}$ [1607.06049]. In the radiation era $a(t) = a_{\min} \sqrt{\cosh(\alpha t)}$ produces a non-singular, past-complete universe. Thermodynamic non-conservation of $T_{\mu\nu}$ allows for irreversible matter creation in an open-system framework, impacting entropy and temperature evolution [2408.14106]. Baryogenesis is enabled by new CP-violating couplings based on $\partial_\mu(\mathcal{T}^2) J_B^\mu$, yielding successful baryon asymmetry in the early universe even with $w \simeq 1/3$, where standard $f(R)$ gravity fails [2409.04623].

Inflationary EMSG scenarios include power-law potentials for canonical and non-canonical scalar fields, reducing the tensor-to-scalar ratio $r$ and accommodating non-Gaussianity (if the matter sector is non-canonical). Constraints from Planck/BICEP and joint Planck+BAO analyses require $\zeta \lesssim 2 \times 10^{-5}$ for observationally viable inflation [2107.13547, 2306.09181].

## 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 $\eta$, EMSG accommodates more massive neutron stars using standard polytropic equations of state, aiding the explanation of observed $2M_\odot$ pulsars without exotic matter [1802.02399]. Observational stability bounds from pulsar timing enforce $-6 \times 10^{-37} < f_0' < 10^{-36}$ m s$^2$ kg$^{-1}$ [2201.08578]. 

Jeans analysis in EMSG predicts modified instability thresholds for both non-rotating and disk systems, favoring $\alpha > 0$ for stability—consistent with lack of observed fragmentation in hypermassive neutron stars [2001.04702]. Light bending and gravitational lensing calculations deliver density-dependent corrections detectable at micro-arcsecond scales in dense lenses; solar-system tests restrict $|f_0'| < 10^{-26}$–$10^{-27}$ m s$^2$ kg$^{-1}$ and show full agreement with current measurements [2204.11003].

## 7. Dynamical, Thermodynamic, and Structural Properties

EMSG generically violates energy-momentum conservation, leading to irreversible particle creation and "thermodynamic" matter production in cosmological evolution [2408.14106]. Scale-independent ($n=1/2$) EMSG and power-law ($n \neq 1$) 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 [2301.11204, 1906.00027]. Dynamical-system studies reveal transitions through radiation-, matter-, and acceleration-dominated epochs, with de Sitter attractors in late times for suitable choices [1906.00027, 2601.22333]. 

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 [2203.13496, 2305.09658, 2108.08682, 2104.07931].

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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 [2408.01665, 1607.06049, 2408.14106]. 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.

Source: https://www.emergentmind.com/topics/energy-momentum-squared-gravity-theory