Energy-Momentum Squared Gravity (EMSG)

• EMSG is a modified gravity theory that extends General Relativity by incorporating a quadratic matter term, T₍μν₎T^(μν), to affect gravitational dynamics in high-density regimes.
• It introduces energy-momentum non-conservation in matter-rich environments, leading to interpretations of matter creation and altered thermodynamic evolution in cosmology.
• The theory’s corrections become significant at high densities, influencing early-universe bounces, compact star structures, and weak-field tests while reducing to GR in vacuum.

Energy-Momentum Squared Gravity (EMSG) is a class of modified gravity theories in which the gravitational action depends not only on the Ricci scalar $R$ but also on the scalar $T_{\mu\nu}T^{\mu\nu}$, or more generally on $f(R,T_{\mu\nu}T^{\mu\nu})$. In its standard quadratic realizations, the new term is proportional to $T_{\mu\nu}T^{\mu\nu}$, so deviations from General Relativity (GR) arise only in the presence of matter and become most relevant at high density and high curvature. This structure has made EMSG a framework for early-universe cosmology, compact-star interiors, strong-field astrophysics, and matter-coupled gravitational phenomenology, while preserving the GR limit in vacuum (Cipriano et al., 2024, Roshan et al., 2016).

1. Action principle and field equations

A general metric-formulation starting point is

$$S=\int d^4x\,\sqrt{-g}\,\Bigl[f\!\bigl(R,T_{\mu\nu}T^{\mu\nu}\bigr)+\mathcal L_m\Bigr],$$

with $R$ the Ricci scalar, $\mathcal L_m$ the matter Lagrangian, and $\mathcal T\equiv T_{\mu\nu}T^{\mu\nu}$. Varying with respect to $g^{\mu\nu}$ yields field equations of the form

$$f_R\,R_{\mu\nu}-\tfrac12\,g_{\mu\nu}\,f -\bigl(\nabla_\mu\nabla_\nu-g_{\mu\nu}\Box\bigr)f_R =2\kappa^2\,T_{\mu\nu}-f_{\mathcal T}\,\Theta_{\mu\nu},$$

where $T_{\mu\nu}T^{\mu\nu}$0, $T_{\mu\nu}T^{\mu\nu}$1, and $T_{\mu\nu}T^{\mu\nu}$2 is the metric variation of $T_{\mu\nu}T^{\mu\nu}$3 (Cipriano et al., 2024).

The original quadratic model introduced by Roshan and Shojai takes the Einstein-Hilbert action plus a term proportional to $T_{\mu\nu}T^{\mu\nu}$4,

$T_{\mu\nu}T^{\mu\nu}$5

or, in later notations,

$T_{\mu\nu}T^{\mu\nu}$6

The literature therefore uses $T_{\mu\nu}T^{\mu\nu}$7, $T_{\mu\nu}T^{\mu\nu}$8, and, in post-Minkowskian and lensing studies, $T_{\mu\nu}T^{\mu\nu}$9, for the new coupling (Roshan et al., 2016, Akarsu et al., 2018, Nazari et al., 2022).

For perfect fluids, many treatments recast the modified equations into an Einstein form with an effective stress-energy tensor. In compact-star applications this leads to effective densities and pressures such as

$f(R,T_{\mu\nu}T^{\mu\nu})$0

so that the matter sector is algebraically “dressed” by quadratic terms (Ghosh, 2024). A central structural property is that in vacuum the quadratic correction vanishes identically, and the theory reduces to the standard Einstein equations with cosmological constant; the modification is therefore intrinsically matter-supported (Nari et al., 2018).

2. Energy non-conservation, matter creation, and thermodynamic reading

Because $f(R,T_{\mu\nu}T^{\mu\nu})$1 depends explicitly on $f(R,T_{\mu\nu}T^{\mu\nu})$2, the ordinary matter tensor is generally not covariantly conserved: $f(R,T_{\mu\nu}T^{\mu\nu})$3 This non-conservation is one of the defining features of the metric $f(R,T_{\mu\nu}T^{\mu\nu})$4 framework and is the main reason EMSG differs conceptually from purely curvature-based modifications (Cipriano et al., 2024).

In homogeneous cosmology, this departure from $f(R,T_{\mu\nu}T^{\mu\nu})$5 has been interpreted through the thermodynamics of open systems. For an adiabatic homogeneous fluid in FLRW spacetime,

$f(R,T_{\mu\nu}T^{\mu\nu})$6

and one defines the particle-production rate

$f(R,T_{\mu\nu}T^{\mu\nu})$7

The associated effective creation pressure is

$f(R,T_{\mu\nu}T^{\mu\nu})$8

the temperature evolution obeys

$f(R,T_{\mu\nu}T^{\mu\nu})$9

and the entropy production law is

$T_{\mu\nu}T^{\mu\nu}$0

Within this reading, the modified cosmological balance equations describe irreversible matter creation rather than a closed-fluid evolution (Cipriano et al., 2024).

This thermodynamic interpretation is not confined to the quadratic model. In scale-independent EMSG, the violation of local energy-momentum conservation and matter-current conservation is likewise interpreted as a process of matter creation or annihilation in an expanding universe, with the sign of the dimensionless parameter $T_{\mu\nu}T^{\mu\nu}$1 controlling the direction of the effect (Akarsu et al., 2023).

3. FLRW dynamics, bounce claims, and nucleosynthesis bounds

For the frequently studied special model

$T_{\mu\nu}T^{\mu\nu}$2

the flat-FLRW equations become

$T_{\mu\nu}T^{\mu\nu}$3

$T_{\mu\nu}T^{\mu\nu}$4

At high energy density, the quadratic term dominates. In the original EMSG cosmology this was presented as producing a maximum density $T_{\mu\nu}T^{\mu\nu}$5, a minimum scale factor $T_{\mu\nu}T^{\mu\nu}$6, and a nonsingular early-time bounce, together with a true sequence of cosmological eras leading from bounce to radiation, matter, and de Sitter expansion. In that treatment, the positive cosmological constant already plays a crucial role at the bounce through $T_{\mu\nu}T^{\mu\nu}$7 (Roshan et al., 2016).

A common misconception is that this conclusion is model-independently settled. A later analysis of radiation-dominated EMSG argued that the regular-bouncing branch is disconnected from the branch that reproduces the GR low-density limit. On that account, the branch relevant to our Universe is nonsingular in curvature but geodesically incomplete, and pure quadratic EMSG replaces the standard early-time singularity by a singular bounce rather than a regular one. Within the broader family with a $T_{\mu\nu}T^{\mu\nu}$8 correction, the case $T_{\mu\nu}T^{\mu\nu}$9 was identified as the unique power-law model that yields a viable regular radiation-dominated bounce on the physical branch (Barbar et al., 2019).

Independent early-universe constraints come from Big Bang Nucleosynthesis. In the $$S=\int d^4x\,\sqrt{-g}\,\Bigl[f\!\bigl(R,T_{\mu\nu}T^{\mu\nu}\bigr)+\mathcal L_m\Bigr],$$0 model,

$$S=\int d^4x\,\sqrt{-g}\,\Bigl[f\!\bigl(R,T_{\mu\nu}T^{\mu\nu}\bigr)+\mathcal L_m\Bigr],$$1

so the correction scales as $$S=\int d^4x\,\sqrt{-g}\,\Bigl[f\!\bigl(R,T_{\mu\nu}T^{\mu\nu}\bigr)+\mathcal L_m\Bigr],$$2 and decays faster than the GR term. Using updated nuclear abundances, the most stringent constraint comes from $$S=\int d^4x\,\sqrt{-g}\,\Bigl[f\!\bigl(R,T_{\mu\nu}T^{\mu\nu}\bigr)+\mathcal L_m\Bigr],$$3He,

$$S=\int d^4x\,\sqrt{-g}\,\Bigl[f\!\bigl(R,T_{\mu\nu}T^{\mu\nu}\bigr)+\mathcal L_m\Bigr],$$4

with D/H giving a slightly weaker range. In that specific $$S=\int d^4x\,\sqrt{-g}\,\Bigl[f\!\bigl(R,T_{\mu\nu}T^{\mu\nu}\bigr)+\mathcal L_m\Bigr],$$5 realization, a negative $$S=\int d^4x\,\sqrt{-g}\,\Bigl[f\!\bigl(R,T_{\mu\nu}T^{\mu\nu}\bigr)+\mathcal L_m\Bigr],$$6 is required both by stability and by BBN, and the allowed magnitude is so small that the correction is effectively negligible for most post-BBN cosmology (Jang et al., 2024).

4. Perturbations and late-time cosmological phenomenology

Beyond homogeneous backgrounds, EMSG has been developed at the level of linear scalar and tensor perturbations. For the power-law class

$$S=\int d^4x\,\sqrt{-g}\,\Bigl[f\!\bigl(R,T_{\mu\nu}T^{\mu\nu}\bigr)+\mathcal L_m\Bigr],$$7

two cases have received particular attention: $$S=\int d^4x\,\sqrt{-g}\,\Bigl[f\!\bigl(R,T_{\mu\nu}T^{\mu\nu}\bigr)+\mathcal L_m\Bigr],$$8, the original quadratic model, and $$S=\int d^4x\,\sqrt{-g}\,\Bigl[f\!\bigl(R,T_{\mu\nu}T^{\mu\nu}\bigr)+\mathcal L_m\Bigr],$$9, the scale-independent model. In a dust background, the $R$0 case gives

$R$1

whereas the $R$2 case modifies the matter scaling to

$R$3

At first order, tensor perturbations propagate with

$R$4

exactly, but the friction term is modified, leading to a gravitational-wave luminosity distance

$R$5

Joint fits to simulated Einstein Telescope standard sirens and observed redshift-space-distortion data place the $R$6 and $R$7 couplings close to zero, with small positive central values and the $R$8CDM limit داخل the $R$9 region (Fu et al., 2024).

A different cosmological branch is the scale-independent model in which the action contains $\mathcal L_m$0. In one realization, photons and baryons couple as in GR, while cold dark matter and relativistic relics couple through EMSG. This induces “pseudo nonminimal” interactions, modifies the redshift scalings,

$\mathcal L_m$1

and shifts the effective number of relativistic species according to

$\mathcal L_m$2

Planck 2015 plus BAO data found $\mathcal L_m$3 consistent with zero, with no statistically significant deviation from $\mathcal L_m$4CDM (Akarsu et al., 2018).

The same scale-independent framework also admits nonstandard exact cosmologies. Static universes, de Sitter and steady-state solutions, power-law accelerated or decelerated dust and radiation eras, and Big-Rip behavior all appear depending on $\mathcal L_m$5 and the equation of state. The model further allows effectively ultra-stiff behavior near anisotropic singularities, so that isotropization need not require a physical stiff fluid (Akarsu et al., 2023).

5. Compact objects, black holes, and collapse

Because EMSG reduces to GR in vacuum and differs only where matter is present, compact stars are natural test systems. In the static spherically symmetric case, the hydrostatic balance equations become modified Tolman-Oppenheimer-Volkoff equations. One form, using the effective fluid variables, is

$\mathcal L_m$6

$\mathcal L_m$7

With realistic equations of state, comparison to neutron-star masses and radii gives

$\mathcal L_m$8

and within this range EMSG leaves standard cosmology safely unaltered back to $\mathcal L_m$9 s (Akarsu et al., 2018).

Using a polytropic equation of state, Nari and Roshan found that EMSG can yield larger or smaller neutron-star masses than GR depending on the central pressure and the magnitude of $\mathcal T\equiv T_{\mu\nu}T^{\mu\nu}$0, and that it can support $\mathcal T\equiv T_{\mu\nu}T^{\mu\nu}$1 stars with an ordinary polytrope. They also obtained analytic interior solutions, including a pressureless star supported by an effective pressure, although that solution is unstable to radial and local perturbations (Nari et al., 2018).

Asteroseismology extends this program. In EMSG neutron-star models based on soft, intermediate, and stiff hybrid-QCD equations of state, the $\mathcal T\equiv T_{\mu\nu}T^{\mu\nu}$2-mode frequency obeys a universal “$\mathcal T\equiv T_{\mu\nu}T^{\mu\nu}$3-Love” relation,

$\mathcal T\equiv T_{\mu\nu}T^{\mu\nu}$4

Using tidal deformabilities from GW170817 and GW190814, the canonical $\mathcal T\equiv T_{\mu\nu}T^{\mu\nu}$5 $\mathcal T\equiv T_{\mu\nu}T^{\mu\nu}$6-mode frequency for $\mathcal T\equiv T_{\mu\nu}T^{\mu\nu}$7 was inferred to be approximately $\mathcal T\equiv T_{\mu\nu}T^{\mu\nu}$8 and $\mathcal T\equiv T_{\mu\nu}T^{\mu\nu}$9, respectively, with similar bounds for $g^{\mu\nu}$0. The same study reported phase transitions in the stiff, intermediate, and soft equations of state and subluminal sound-speed profiles across the explored parameter range (Ghosh, 2024).

The original EMSG paper also obtained exact charged black-hole solutions; in the small-$g^{\mu\nu}$1 limit the metric reduces to Reissner-Nordström plus order-$g^{\mu\nu}$2 corrections, including a term $g^{\mu\nu}$3 in $g^{\mu\nu}$4 (Roshan et al., 2016). By contrast, a Vaidya-collapse analysis in two EMSG models concluded that black-hole formation is not generic and that naked singularities can arise instead; in the additive $g^{\mu\nu}$5 model the resulting singularity is Tipler-weak, whereas in the product-coupling model it may be weak or strong depending on the initial data. This suggests that the end state of collapse is highly model-dependent within the broader EMSG family (Rudra, 2024).

6. Weak-field, post-Newtonian, and multimessenger tests

Weak-field analyses of EMSG are not uniform in emphasis. In the energy-momentum powered gravity formulation with $g^{\mu\nu}$6, the Newtonian and parametrized post-Newtonian limits were found to have the same potential form, the same PPN parameters $g^{\mu\nu}$7, and the same test-particle geodesics as GR for $g^{\mu\nu}$8. In that treatment, the main local effect is a renormalization of the inferred mass,

$g^{\mu\nu}$9

and the correct slow-motion criterion is $$f_R\,R_{\mu\nu}-\tfrac12\,g_{\mu\nu}\,f -\bigl(\nabla_\mu\nabla_\nu-g_{\mu\nu}\Box\bigr)f_R =2\kappa^2\,T_{\mu\nu}-f_{\mathcal T}\,\Theta_{\mu\nu},$$0, not merely $$f_R\,R_{\mu\nu}-\tfrac12\,g_{\mu\nu}\,f -\bigl(\nabla_\mu\nabla_\nu-g_{\mu\nu}\Box\bigr)f_R =2\kappa^2\,T_{\mu\nu}-f_{\mathcal T}\,\Theta_{\mu\nu},$$1 (Akarsu et al., 2022).

A dedicated lensing calculation, however, derived an additional density-squared potential

$$f_R\,R_{\mu\nu}-\tfrac12\,g_{\mu\nu}\,f -\bigl(\nabla_\mu\nabla_\nu-g_{\mu\nu}\Box\bigr)f_R =2\kappa^2\,T_{\mu\nu}-f_{\mathcal T}\,\Theta_{\mu\nu},$$2

which enters $$f_R\,R_{\mu\nu}-\tfrac12\,g_{\mu\nu}\,f -\bigl(\nabla_\mu\nabla_\nu-g_{\mu\nu}\Box\bigr)f_R =2\kappa^2\,T_{\mu\nu}-f_{\mathcal T}\,\Theta_{\mu\nu},$$3 at post-Newtonian order and modifies light bending and Shapiro delay. Solar-system data then imply

$$f_R\,R_{\mu\nu}-\tfrac12\,g_{\mu\nu}\,f -\bigl(\nabla_\mu\nabla_\nu-g_{\mu\nu}\Box\bigr)f_R =2\kappa^2\,T_{\mu\nu}-f_{\mathcal T}\,\Theta_{\mu\nu},$$4

and microlensing by neutron-star lenses can shift image positions by $$f_R\,R_{\mu\nu}-\tfrac12\,g_{\mu\nu}\,f -\bigl(\nabla_\mu\nabla_\nu-g_{\mu\nu}\Box\bigr)f_R =2\kappa^2\,T_{\mu\nu}-f_{\mathcal T}\,\Theta_{\mu\nu},$$5–$$f_R\,R_{\mu\nu}-\tfrac12\,g_{\mu\nu}\,f -\bigl(\nabla_\mu\nabla_\nu-g_{\mu\nu}\Box\bigr)f_R =2\kappa^2\,T_{\mu\nu}-f_{\mathcal T}\,\Theta_{\mu\nu},$$6 micro-arcseconds in the quoted estimates (Nazari, 2022). A plausible implication is that weak-field phenomenology depends sensitively on the specific EMSG subclass and on the post-Newtonian bookkeeping adopted.

Binary pulsars provide stronger strong-field radiative tests. In the post-Minkowskian treatment of gravitational-wave emission, the quadrupole moment is modified by $$f_R\,R_{\mu\nu}-\tfrac12\,g_{\mu\nu}\,f -\bigl(\nabla_\mu\nabla_\nu-g_{\mu\nu}\Box\bigr)f_R =2\kappa^2\,T_{\mu\nu}-f_{\mathcal T}\,\Theta_{\mu\nu},$$7, leading to a corrected orbital-decay rate $$f_R\,R_{\mu\nu}-\tfrac12\,g_{\mu\nu}\,f -\bigl(\nabla_\mu\nabla_\nu-g_{\mu\nu}\Box\bigr)f_R =2\kappa^2\,T_{\mu\nu}-f_{\mathcal T}\,\Theta_{\mu\nu},$$8. Fitting six binary pulsars yields

$$f_R\,R_{\mu\nu}-\tfrac12\,g_{\mu\nu}\,f -\bigl(\nabla_\mu\nabla_\nu-g_{\mu\nu}\Box\bigr)f_R =2\kappa^2\,T_{\mu\nu}-f_{\mathcal T}\,\Theta_{\mu\nu},$$9

in agreement with compact-star bounds (Nazari et al., 2022).

At first post-Newtonian order in quadratic EMSG, the $T_{\mu\nu}T^{\mu\nu}$00-body problem has also been analyzed for self-gravitating bodies. After introducing a suitable center-of-mass acceleration and using virial identities, including an EMSG-specific one, self-acceleration was shown to vanish, and a conserved total linear momentum was established. In the PPN language, $T_{\mu\nu}T^{\mu\nu}$01 and the conservation-law parameters $T_{\mu\nu}T^{\mu\nu}$02 vanish, so quadratic EMSG is semiconservative and compatible with binary-pulsar constraints on anomalous accelerations (Nazari, 21 Sep 2025).

7. Generalizations and unresolved problems

EMSG has been generalized in several directions. In the Palatini formalism one considers

$T_{\mu\nu}T^{\mu\nu}$03

with the simple specialization $T_{\mu\nu}T^{\mu\nu}$04. The independent connection becomes the Levi-Civita connection of the conformal metric $T_{\mu\nu}T^{\mu\nu}$05. Matter is non-conserved, test particles experience a fifth force, and the Newtonian potential satisfies a generalized Poisson equation containing both $T_{\mu\nu}T^{\mu\nu}$06 and $T_{\mu\nu}T^{\mu\nu}$07 terms. In this Palatini setting, viable cosmic bounces require $T_{\mu\nu}T^{\mu\nu}$08 (Nazari et al., 2020).

Another branch is energy-momentum log gravity, defined by

$T_{\mu\nu}T^{\mu\nu}$09

Dynamical-systems analysis shows that the exact cosmological solution previously found in this model is a future attractor. The EMLG contribution produces a constant effective inertial mass density, shifts the asymptotic Hubble rate to

$T_{\mu\nu}T^{\mu\nu}$10

and, for $T_{\mu\nu}T^{\mu\nu}$11, can screen $T_{\mu\nu}T^{\mu\nu}$12 in the past. The second law of thermodynamics requires $T_{\mu\nu}T^{\mu\nu}$13 in this model (Acquaviva et al., 2022).

Inflationary generalizations replace the quadratic term by a power law,

$T_{\mu\nu}T^{\mu\nu}$14

In the corresponding EMPG inflationary scenario, the presence of EMSG terms can bring monomial chaotic inflation into agreement with current observational constraints. For $T_{\mu\nu}T^{\mu\nu}$15, negative $T_{\mu\nu}T^{\mu\nu}$16 in the range

$T_{\mu\nu}T^{\mu\nu}$17

renders both $T_{\mu\nu}T^{\mu\nu}$18 and $T_{\mu\nu}T^{\mu\nu}$19 compatible with the BK18$T_{\mu\nu}T^{\mu\nu}$20Planck $T_{\mu\nu}T^{\mu\nu}$21 confidence region, while non-canonical matter Lagrangians can generate much larger equilateral non-Gaussianity than the canonical case (Mansoori et al., 2023). A more speculative extension combines EMSG with temporally varying $T_{\mu\nu}T^{\mu\nu}$22 and $T_{\mu\nu}T^{\mu\nu}$23, aiming to avoid both the initial singularity and inflation; an explicit example quoted in that setting is $T_{\mu\nu}T^{\mu\nu}$24 (Bhattacharjee et al., 2020).

Several problems remain open across the EMSG literature. The 2024 review isolates the microscopic origin of the quadratic matter coupling, stability and causality of perturbations, black-hole thermodynamics, gravitational-wave signatures in mergers, and the distinguishability of EMSG from other high-density frameworks such as brane-world models and loop quantum cosmology as central unresolved questions (Cipriano et al., 2024). Taken together, these lines of work place EMSG in a distinctive niche: it is a matter-sensitive modification of GR whose most consequential departures occur not in vacuum but in the densest known regimes of cosmology and astrophysics.