Cosmological perturbations in Energy-Momentum Squared Gravity

This paper presents a fully covariant and gauge-invariant analysis of linear cosmological perturbations in Energy-Momentum Squared Gravity, deriving exact propagation equations for scalar, vector, and tensor modes that reveal distinct observational signatures—such as modified density contrasts, early-time vorticity, and altered tensor damping—relative to General Relativity.

The Gist

Imagine the universe as a giant, expanding balloon. For decades, scientists have used a standard set of rules (called General Relativity) to predict how this balloon behaves, how it stretches, and how the "dust" and "gas" floating inside it clump together to form stars and galaxies. This standard model works incredibly well, but it leaves some big questions unanswered, like what "dark energy" and "dark matter" actually are.

This paper explores a new set of rules called Energy-Momentum Squared Gravity (EMSG). Think of it as tweaking the recipe for how gravity works, specifically when things get very dense or energetic.

Here is a breakdown of what the authors did, using simple analogies:

1. The New Recipe: Adding a "Squared" Ingredient

In the standard recipe, gravity depends on how much energy and pressure matter has. In this new EMSG recipe, the authors add a "squared" ingredient.

• The Analogy: Imagine you are baking a cake. The standard recipe says the taste depends on the amount of sugar (energy). The new recipe says the taste also depends on the square of the sugar amount.
• The Effect: When there is very little sugar (low density, like today's empty space), the "squared" part is tiny, and the cake tastes just like the standard recipe. But when there is a massive pile of sugar (high density, like the early universe or inside a black hole), the "squared" part explodes in importance, changing how the cake rises and behaves.

2. The "Effective Fluid" Trick

To make the math manageable, the authors pretend that this new gravity rule doesn't change the laws of physics, but instead changes the properties of the stuff inside the universe.

• The Analogy: Imagine you are driving a car. Suddenly, the road gets sticky. Instead of saying "the road changed," you pretend the car's engine just got a little stronger and the tires got a little grippier. You call this a "new car" (an effective fluid) that behaves differently, even though the road is the same.
• The Result: They found that in this new theory, even "dust" (which usually has no pressure, like dry sand) starts acting like it has pressure and a "sound speed" (how fast a wave travels through it). This is a big deal because, in standard physics, dust doesn't push back against gravity.

3. Studying the Ripples (Perturbations)

The authors didn't just look at the smooth balloon; they looked at the ripples and waves on it. They studied three types of ripples:

• Scalar Modes (The Clumps): These are the ripples that turn into galaxies.
  • What they found: Depending on the specific version of the new theory, these clumps might grow faster or slower than in standard physics. In some cases, the new "pressure" from the dust stops small clumps from forming, acting like a safety net that prevents things from collapsing too easily.
• Vector Modes (The Swirls): These are like tiny whirlpools or vortices in the cosmic fluid.
  • What they found: In standard physics, these swirls usually die out very quickly as the universe expands. In this new theory, the "swirls" might last longer or die out at a different speed, depending on how "stiff" the new effective fluid is. This could leave a different fingerprint on the early universe.
• Tensor Modes (The Gravitational Waves): These are ripples in space-time itself, like waves on a pond.
  • What they found: These waves travel as "damped waves" (they get quieter as they travel). The new theory changes how fast they fade away. It's like changing the material of the pond; some materials absorb the wave faster than others.

4. Two Specific Versions (Model A and Model B)

The authors tested two specific ways to write this "squared" rule:

• Model A (The "Quadratic" Version): This is the direct "sugar squared" approach. Here, the behavior changes a lot depending on how dense the universe is. At high densities, the rules change dramatically, but as the universe expands and gets thin, it slowly returns to the standard rules we know.
• Model B (The "Square Root" Version): This is a slightly different mathematical twist. Interestingly, in this version, the "new car" (effective fluid) has constant properties. It behaves like a fluid with a fixed "stiffness" no matter how much it expands. This makes the math much cleaner and easier to predict.

5. The Bottom Line

The paper concludes that this new theory is a viable alternative to standard gravity.

• It fits the past: As the universe gets less dense (which is what happened over billions of years), the new theory smoothly turns back into standard General Relativity. We wouldn't notice the difference in our local solar system.
• It changes the early universe: In the very beginning, when everything was packed tight, this new theory predicts different growth rates for galaxies, different fading rates for gravitational waves, and different behaviors for cosmic swirls.

Why does this matter?
The authors aren't saying this theory is definitely true. Instead, they have built a precise "map" of what the universe would look like if this theory were true. Now, astronomers can look at real data (like the Cosmic Microwave Background or the distribution of galaxies) and check: "Does the real universe match the Standard Model, or does it match this new EMSG map?" If the real data matches the EMSG map, it could solve some of the biggest mysteries in cosmology.

Technical Summary

Problem Statement

The standard $\Lambda$CDM model, based on General Relativity (GR), successfully explains a wide range of cosmological observations but leaves open conceptual and observational puzzles regarding dark energy, dark matter, and high-energy corrections to gravity. Modified theories of gravity that couple the gravitational Lagrangian explicitly to matter scalars constructed from the energy-momentum tensor ($T_{\mu\nu}$) offer a potential resolution. Specifically, Energy-Momentum Squared Gravity (EMSG), where the Lagrangian depends on the invariant $T \equiv T_{\mu\nu}T^{\mu\nu}$, introduces non-linear corrections that become significant in high-density regimes (early universe, compact objects).

While the background cosmology of EMSG has been studied, a fully covariant and gauge-invariant analysis of linear cosmological perturbations within this framework was lacking. The paper addresses the need to understand how the non-linear dependence on $T$ modifies the evolution of scalar (density), vector (vorticity), and tensor (gravitational wave) modes, and how these modifications compare to GR predictions.

Methodology

The authors employ the 1+3 covariant and gauge-invariant formalism (following Ellis et al.), which projects quantities orthogonal to the fluid four-velocity $u^\mu$. This approach avoids gauge ambiguities and uses physically transparent variables.

1. Model Definition: The study focuses on the class of models $F(R, T) = R + \eta (T)^n$, where $\eta$ is a coupling constant and $n \in \mathbb{R}$. Two representative subclasses are analyzed in detail:
   • Model A ($n=1$): Quadratic correction in $T$.
   • Model B ($n=1/2$): Square-root scaling in $T$.
2. Effective Fluid Interpretation: The modified field equations are recast into a form resembling GR but with an effective energy density ($\bar{\rho}$) and effective pressure ($\bar{p}$). This allows the use of standard cosmological tools while accounting for the non-minimal coupling. The effective equation-of-state parameter ($\bar{w}$) and adiabatic sound speed ($\bar{c}_s^2$) are derived as functions of the physical density $\rho$ and the coupling $\eta$.
3. Perturbation Analysis:
   • Scalar Modes: The evolution of the comoving fractional density gradient ($\bar{D}_\mu$) is derived, leading to a second-order propagation equation. The authors perform a harmonic decomposition to analyze the growth of density contrasts ($\delta$) for dust and radiation.
   • Vector Modes: The evolution of the vorticity vector ($\omega_\mu$) is integrated, examining how the effective sound speed modifies the decay of vorticity.
   • Tensor Modes: The propagation of gravitational waves is described via the magnetic part of the Weyl tensor ($H_{ab}$) and the shear tensor ($\sigma_{ab}$).
4. Physical vs. Effective Variables: A crucial step involves distinguishing between the "effective" density contrast ($\bar{\delta}$) used in the propagation equations and the "physical" matter density contrast ($\delta$) observable in the universe. An algebraic mapping is derived to relate these two quantities.

Key Contributions and Results

1. Background and Effective Fluid Properties

• The EMSG corrections introduce an effective pressure even for dust ($w=0$), leading to a non-zero effective sound speed $\bar{c}_s^2$.
• Model A: $\bar{w}$ and $\bar{c}_s^2$ are density-dependent. For dust, $\bar{c}_s^2 \approx \eta\rho$ at low densities, leading to a constant physical Jeans length at early times, potentially suppressing small-scale structure formation.
• Model B: For barotropic fluids, $\bar{w}$ and $\bar{c}_s^2$ are constant (shifted from their GR values by $\eta$). This simplifies the perturbation equations to power-law forms.

2. Scalar Perturbations (Density Contrast)

• Mapping: The physical density contrast $\delta$ is related to the effective contrast $\bar{\delta}$ by a model-dependent factor. In Model A, this factor suppresses the physical amplitude at early times (approaching 1/2 for dust) and fades to unity at late times. In Model B, the mapping is trivial ($\delta = \bar{\delta}$).
• Growth Behavior:
  • Model A (Radiation): The physical density contrast grows slightly slower than in GR at early times but converges to GR behavior as the universe expands.
  • Model A (Dust): On super-Hubble scales, the physical mode can grow faster than the GR growing mode once the effective dynamics and the suppression mapping are combined. On sub-Hubble scales, the solutions exhibit $k$-dependent oscillations with a damping envelope distinct from GR.
  • Model B: The constant shift in $\bar{w}$ alters the Hubble friction and oscillation frequencies. For radiation ($\bar{w} > 1/3$), damping is weaker; for dust ($\bar{w} > 0$), damping is stronger, suppressing growth more effectively than in GR.

3. Vector Perturbations (Vorticity)

• The vorticity vector decays as $\omega_\mu \propto a^{-(2 - 3\bar{c}_s^2)}$.
• In GR, dust vorticity decays as $a^{-2}$ and radiation as $a^{-1}$.
• In EMSG, the decay rate is modified by the effective sound speed.
  • Model A (Dust): The decay is slower than $a^{-2}$ at early times (when $\eta\rho$ is large), potentially allowing vorticity to persist longer.
  • Model B: Depending on the sign and magnitude of $\eta$, vorticity can decay slower or faster than in GR. The paper notes that for certain parameters, vorticity could even grow, though this requires careful treatment of linear stability.

4. Tensor Perturbations (Gravitational Waves)

• Tensor modes propagate as damped waves with effective masses determined by $\bar{w}$ and the Hubble parameter.
• Model A: In the radiation era, tensor evolution matches GR exactly. In the dust era, the shear ($\sigma$) decays slightly faster than in GR due to the positive effective pressure, while the magnetic Weyl tensor ($H$) remains close to GR.
• Model B: The constant shift in $\bar{w}$ systematically modifies the damping. For radiation-like backgrounds ($\bar{w} > 1/3$), the magnetic Weyl amplitude decays more slowly, while the shear decays faster. For dust-like backgrounds, both channels are more strongly damped.
• In all cases, the limit $\eta \to 0$ continuously recovers the GR results.

Significance and Claims

The paper claims to provide the first fully covariant and gauge-invariant analysis of linear perturbations in EMSG, covering all three sectors (scalar, vector, tensor) simultaneously.

• Robust Signatures: The framework isolates specific, testable signatures that distinguish EMSG from GR:
  1. Early-time scalar tilts: Modifications to the growth rate of density perturbations, particularly the suppression of physical amplitudes in Model A radiation and the potential for enhanced super-Hubble growth in Model A dust.
  2. Tensor damping shifts: Altered decay rates for gravitational wave amplitudes (shear and magnetic Weyl tensor) driven by the effective equation of state.
  3. Altered vorticity decay: Modified power-law indices for the decay of primordial vorticity, which could impact the generation of primordial magnetic fields.
• Observational Relevance: The authors suggest these signatures can be confronted with CMB observations (temperature and polarization transfer functions, B-modes) and large-scale structure data (matter power spectra, growth indices) to constrain the coupling parameter $\eta$.
• Theoretical Utility: The work demonstrates that the 1+3 formalism is highly effective for analyzing modified gravity theories, cleanly separating kinematic effects from dynamic sources and providing a transparent path to the GR limit.

The paper concludes that while EMSG introduces significant phenomenological changes in high-density regimes, the theory remains well-behaved perturbatively (avoiding ghosts or tachyonic modes for $\eta > 0$) and offers a rich set of observational tests for future cosmological surveys.