Cosmological perturbations in Energy-Momentum Squared Gravity

TL;DR

This work develops a fully covariant, gauge-invariant linear perturbation theory for Energy–Momentum Squared Gravity (EMSG) using the 1+3 formalism, deriving exact propagation equations for scalar, vector, and tensor modes on FLRW backgrounds in radiation and dust. By adopting an effective-fluid interpretation, the authors analyze two representative sub-models, Model A ($n=1$) and Model B ($n=1/2$), and reveal how non-linear matter terms modify the equation of state and sound speed, leading to distinctive features in growth, vorticity decay, and gravitational-wave propagation. The study identifies robust, testable signatures—such as early-time scalar tilts, altered Jeans scales, and modified tensor damping—that can be confronted with CMB and large-scale-structure data to constrain EMSG. The results recover GR smoothly in the limit $\eta\to0$, ensuring consistency with standard cosmology while highlighting potential deviations in high-density regimes relevant to the early universe and compact objects. The framework offers clear pathways to connect theoretical predictions with upcoming observational probes and motivates extensions to mixed epochs and non-flat geometries.

Abstract

We present a fully covariant and gauge-invariant analysis of linear cosmological perturbations in Energy-Momentum Squared Gravity. Working within the 1+3 formalism, we derive the exact propagation equations for scalar, vector, and tensor modes on FLRW backgrounds, in the case of radiation and dust. Two representative subclasses are examined in detail, in which non-linearity enters through $\mathcal{O}(ηρ^2)$ corrections or modifications in the equation-of-state parameter and the sound speed. For scalar perturbations, the density contrast can be enhanced or reduced relative to General Relativity, depending on the coupling parameter and the wavelength regime. A similar behaviour occurs for vector modes, allowing for a non-trivial vorticity at early-times. Tensor modes, described by the magnetic part of the Weyl tensor and the shear tensor propagate as damped waves with slowly varying effective masses. All sectors reduce continuously to their GR limits as $η\!\to\!0$. The framework isolates robust signatures - early-time scalar tilts, tensor damping shifts, and altered vorticity decay - that can be confronted with CMB and large-scale-structure observations to constrain these theories of gravity.

Figures (5)

• Figure 1: Effective equation-of-state parameter and sound speed for Models A and B, in the case of dust. For the sake of illustration, we set $\eta=1.0$ for both models.
• Figure 2: Model A (radiation). Physical density contrast $\delta^{(k)}(a)$ compared with GR for (left) $k_{\rm long}=0.01$ and (right) $k_{\rm short}=100$. Initial conditions at last scattering $a_\ast=1/1100$ use the same amplitude as GR, $\delta(a_\ast)=\delta_\ast=10^{-5}$, and the GR growing–mode slope $\delta'(a_\ast)=\delta_\ast/a_\ast$. Parameters: $\eta=0.01$, $\tilde{c}_0=1$. The final $a$ is chosen when $\delta\approx1$ (linear regime) in the long wavelength regime and also used for short wavelength, for the sake of comparison. In both panels, the Model A curve shows a early–time suppression relative to GR as encoded in Eq. (\ref{['eq:phys-vs-eff-rad']}).
• Figure 3: Model A (dust). Physical density contrast $\delta^{(k)}(a)$ compared with GR. Top–left:$k=0$, the dynamics in Model A drives super–Hubble growth that can exceed the GR growing mode when both are initiated with the same $(\delta,\delta')$ at $a_\ast$. Again, the final $a$ is chosen to keep the linear order of perturbations in the long wavelength regime. Top–right: subhorizon solutions ($k=100$) show oscillations with a $k$–dependent envelope; the mapping suppresses the amplitude relative to the effective mode and the curves approach GR as $a\to1$. Bottom: for $k=100$, the WKB solution closely tracks the numerical physical solution. Setup: $\eta=0.01$, $c_0=10^{-3}$, $a_\ast=1/1100$, $\delta(a_\ast)=10^{-5}$, and $\delta'(a_\ast)=\delta_\ast/a_\ast$.
• Figure 4: Model B (radiation). Physical contrast $\delta^{(k)}(a)$ compared with GR radiation. Left: long wavelength ($k=0.01$). Right: short wavelength ($k=100$). Initial conditions at last scattering $a_\ast=1/1100$ use the same amplitude as GR, $\delta(a_\ast)=\delta_\ast=10^{-5}$, and $\dot\delta(t_\ast)=0$. Parameters: $\eta=0.01$, $A=1$. For $\eta>0$ we have $\bar{w}_{\rm rad}\gtrsim 1/3$, hence a small Hubble friction and a short--wave envelope $\propto a^{(3\bar{w}_{\rm rad}-1)/2}$ that slowly grows, in agreement with the plotted evolution.
• Figure 5: Model B (dust). Physical contrast $\delta^{(k)}(a)$ compared with GR radiation. Left: long wavelength ($k=0$). Right: numerical subhorizon solutions for $k=100$ compared with the GR growing mode $\delta_{\rm GR}(a)=\delta_\ast(a/a_\ast)$. The envelope decay with $a$, while the increased Hubble friction further suppresses growth, explaining the rapid damping seen in the panel. Initial data at last scattering: $\delta(a_\ast)=10^{-5}$ and $\dot\delta(t_\ast)=p\,\delta_\ast/t_\ast$ with $p=2/[3(1+\bar{w}_{\rm dust})]$. Parameters: $\eta=0.1$, $A=1$; Scale factor range $0<a<1$.