A New Gravity Theory Could Rewrite the Early Universe’s Cosmic Clumpiness

A speculative modification of Einstein’s theory of gravity is offering cosmologists a new way to test what happened during the universe’s densest and most energetic moments. In a study of “energy-momentum squared gravity,” researchers have calculated how tiny ripples in matter, primordial rotation and gravitational waves would evolve when gravity responds not only to spacetime curvature but also to the square of the matter energy–momentum tensor. Their results suggest that the theory could leave distinctive fingerprints in the early universe: some density fluctuations may grow faster than expected, others may be suppressed, cosmic rotation may decay at an altered rate, and gravitational waves may experience modified damping. The predictions do not replace the standard cosmological model, which remains extraordinarily successful, but they identify specific observables that could be used to challenge it.

The work focuses on a class of theories in which the gravitational action depends on the scalar quantity (mathcal{T}=T_{munu}T^{munu}), formed by contracting the energy–momentum tensor with itself. In ordinary General Relativity, matter appears on the right-hand side of Einstein’s equations as the source of curvature, while the gravitational part of the action depends on the Ricci scalar (R). In energy-momentum squared gravity, or EMSG, the action is extended schematically to (F(R,mathcal{T})=R+etamathcal{T}^{n}). Here (eta) controls the strength of the new interaction and (n) determines how the correction scales with density. Because (mathcal{T}) grows rapidly in high-density environments, the modification is expected to be most important in the early universe, inside compact objects and during other extreme astrophysical events, while becoming negligible as the cosmos expands and matter thins out.

The researchers analyzed two representative versions of the theory. Model A uses (n=1), producing a correction proportional to the square of the matter density and pressure. Model B uses (n=1/2), a square-root dependence that produces a particularly simple effective description. Rather than treating the modified field equations as an entirely unfamiliar system, the study rewrites them in the language of an effective fluid. The extra gravitational terms are absorbed into an effective energy density (bar{rho}) and pressure (bar{p}), allowing the background expansion to be written in a form resembling the familiar Friedmann equations. This effective fluid has its own equation-of-state parameter (bar{w}=bar{p}/bar{rho}) and adiabatic sound speed (bar{c}_s^2=dbar{p}/dbar{rho}), both of which determine how perturbations propagate and whether gravity or pressure dominates on a given scale.

To follow the disturbances without introducing coordinate ambiguities, the study uses a fully covariant and gauge-invariant (1+3) formalism. In this approach, spacetime is split into the time direction defined by the four-velocity of the cosmic fluid and the three-dimensional spaces orthogonal to it. The central scalar variable is the comoving fractional density gradient, which measures how rapidly the effective density changes from place to place. It vanishes in an exactly homogeneous Friedmann–Lemaître–Robertson–Walker universe, making its first-order perturbation automatically gauge-invariant. The resulting evolution equation has the structure of a generalized Jeans equation: cosmic expansion damps perturbations, gravity encourages them to grow, and pressure generates a scale-dependent restoring force proportional to (bar{c}_s^2k^2/a^2), where (k) is the comoving wavenumber and (a) is the scale factor.

This framework reveals that EMSG can shift the boundary between growing and oscillating fluctuations. The instantaneous Jeans wavenumber is approximately (k_J^2=a^2mathcal{B}/bar{c}_s^2), where (mathcal{B}) contains the effective density, pressure, cosmological constant and spatial-curvature contributions. Modes with wavelengths smaller than the corresponding Jeans length are pressure-supported and tend to oscillate, while longer modes can become gravitationally unstable and grow. Since the effective density and sound speed depend on the matter density, the Jeans scale evolves as the universe expands. In Model A with dust, the study finds a particularly striking separation between scales: long-wavelength perturbations can grow faster than their General Relativity counterparts, while short-wavelength modes develop oscillations and strong damping. This scale dependence could alter the shape of the primordial matter spectrum and, after subsequent cosmic evolution, the distribution of galaxies and dark matter.

The calculations also distinguish between an effective density contrast and the physical density contrast associated with ordinary matter. This distinction is essential because the effective fluid is a mathematical rearrangement of the modified gravitational equations, not a new material substance. The two perturbations are related by a density-dependent algebraic factor. For Model A, that factor approaches one as the density becomes small, recovering the General Relativity prediction at late times. At high density, however, it can suppress the physical contrast relative to the effective one by a substantial amount. During radiation domination, the researchers find that the physical density perturbation is slightly smaller than in General Relativity at early times and becomes nearly indistinguishable from it later. During a dust-dominated phase, the long-wavelength physical contrast can nevertheless grow faster than the standard growing mode once the modified effective dynamics is included. These results mean that the observable consequences cannot be inferred from the effective perturbation alone; the final mapping back to matter must be applied.

The theory changes vector and tensor disturbances as well. Vorticity, representing the local rotation of the cosmic fluid, ordinarily decays as the universe expands because of the stretching and dilution of the flow. In the covariant treatment, its evolution is governed by (omega_mupropto a^{3bar{c}_s^2-2}) for a barotropic, geodesic fluid without anisotropic stresses. General Relativity therefore predicts (a^{-2}) decay for dust and (a^{-1}) decay for radiation. EMSG preserves the basic dilution mechanism but changes the exponent through the effective sound speed. In Model A with dust, the sound speed is larger at high density and decreases with time, leading to slower early decay than in General Relativity before the standard behavior is recovered. In Model B, the effective sound speed is constant, so the vorticity follows a simple power law. Although some parameter ranges in intermediate expressions can suggest growing rotation, the study’s overall conclusion is that source-free vorticity does not remain growing on an expanding FLRW background in the physically relevant regimes. Even a modified decay rate, however, could affect the survival of primordial rotational signatures or the amplification of seed magnetic fields.

Primordial gravitational waves provide another potential test. The researchers track the shear of the cosmic flow and the magnetic part of the Weyl tensor, two covariant quantities that encode tensor perturbations. Both obey damped wave equations with effective mass terms that vary with the Hubble rate and the effective equation of state. In Model A during radiation domination, the leading behavior remains close to General Relativity, although high-density corrections can accelerate the decay of the shear and slightly shift the effective mass of the magnetic Weyl mode. In Model A with dust, the shear is again more strongly damped while the magnetic Weyl amplitude remains comparatively close to the standard prediction. Model B makes the effect more transparent: increasing the constant effective equation of state weakens the long-wavelength damping of the magnetic Weyl component but strengthens shear damping in a radiation-like background. For subhorizon waves, the modes oscillate with phases governed by the conformal-time integral, while their leading envelopes typically decay approximately as (a^{-3}) for the magnetic Weyl amplitude and (a^{-2}) for the shear, with model-dependent corrections.

The most important feature of the analysis is its continuous recovery of General Relativity. When the coupling (eta) tends to zero, the effective density, pressure, sound speed, perturbation equations and gravitational-wave behavior all return smoothly to their standard forms. That consistency makes the model testable rather than merely flexible: any deviation must be tied to the coupling and should become strongest where the density is high. The authors identify early-time scalar tilts, altered vorticity decay and shifted tensor damping as particularly robust signatures. Future comparisons could use the cosmic microwave background, its temperature and polarization transfer functions, the matter power spectrum, galaxy-growth measurements, baryon acoustic oscillations, primordial gravitational-wave backgrounds and B-mode polarization. The present study considers single radiation and dust fluids, so a realistic confrontation with data will require mixed radiation–matter evolution, baryons, cold dark matter, neutrino anisotropic stress, entropy perturbations and dark energy. Even so, the calculations turn an abstract modification of gravity into a set of concrete cosmic predictions—precisely the kind of fingerprints that could reveal whether Einstein’s theory remains complete under the extreme conditions of the young universe.