Cosmological structure growth in energy-momentum squared gravity

Technical Summary: Cosmological Structure Growth in Energy-Momentum Squared Gravity

Problem Statement
The standard $\Lambda$CDM cosmological model, while phenomenologically successful, faces theoretical challenges such as the cosmological constant and coincidence problems. This has motivated the exploration of modified gravity theories where the gravitational action depends on terms beyond the Ricci scalar $R$. Specifically, this work investigates $f(R, T^2)$ gravity, where the action includes a dependence on the quadratic contraction of the energy-momentum tensor, $T^2 = T_{\mu\nu}T^{\mu\nu}$. While the background cosmology of such theories has been studied, their implications for the growth of large-scale structure remain under-explored. The primary objective is to bridge this gap by analyzing the evolution of matter perturbations and comparing theoretical predictions for growth observables against current observational data.

Methodology
The authors employ the gauge-invariant $1+3$ covariant formalism to study linear cosmological perturbations. This approach utilizes geometrically defined quantities (such as the density gradient and expansion scalar) that vanish in the exact FLRW background, ensuring that first-order perturbations are automatically gauge-invariant.

1. Background Equations: The study focuses on the power-law model $f(R, T^2) = R + \alpha T^{2n}$, where $\alpha$ and $n$ are coupling parameters. The authors derive the background Friedmann equations using an effective fluid description, defining effective energy density ($\bar{\rho}$) and pressure ($\bar{p}$) that incorporate the matter-geometry coupling.
2. Perturbation Equations: Using the formalism developed in Ref. [39], the evolution equation for the effective comoving density gradient is derived. This equation is transformed into a differential equation for the density contrast $\delta(z)$ as a function of redshift.
3. Growth Observables: The study analyzes several key observables:
   • The growth factor $\delta(z)$.
   • The effective growth rate $\bar{f} = d \ln \bar{\delta} / d \ln a$.
   • The growth index $\gamma$, defined via $\bar{f} = \bar{\Omega}^\gamma$.
   • The weighted growth rate $f\sigma_8$.
4. Parameter Space: The analysis considers representative values $n = 1/2$ (linear dependence on matter density) and $n = 1/4$ (sub-linear coupling). The coupling parameter $\alpha$ is varied to probe different strengths of the matter-geometry interaction.
5. Observational Comparison: Theoretical predictions for $f(z)$ and $f\sigma_8(z)$ are compared against current observational data (compiled from Ref. [60]) using $\chi^2$ and reduced $\chi^2$ statistics.

Key Contributions and Results

• Evolution of Density Contrast: The numerical integration shows that matter perturbations grow monotonically with cosmic time. For $n=1/4$, deviations from General Relativity (GR) and the $\Lambda$CDM model are extremely small across the entire redshift range. For $n=1/2$, deviations are more pronounced, particularly for larger values of $\alpha$, where the growth of density fluctuations is slightly suppressed relative to GR.
• Scale Dependence: The model introduces explicit scale dependence via the effective sound speed term ($k^2/a^2$). However, the study finds that for viable parameter choices, this scale dependence is mild. Significant scale-dependent deviations from $\Lambda$CDM only emerge for relatively large values of $\alpha$, which are already disfavored by observational constraints.
• Growth Index Behavior: The growth index $\gamma$ decreases with increasing redshift, approaching the standard matter-dominated behavior ($\gamma \approx 0.55$) at early times. At late times, mild deviations emerge. Notably, for the $n=1/2$ case with large $\alpha$ (e.g., $\alpha = 0.5, 1$), the growth index becomes negative, a result inconsistent with current observational determinations.
• Statistical Viability:
  • The $n=1/4$ model provides an excellent fit to observational data, with a reduced $\chi^2$ nearly identical to that of the $\Lambda$CDM model. The difference in $\chi^2$ for $f(z)$ is negligible ($\Delta\chi^2 \approx 0.03$), and for $f\sigma_8(z)$ it is small ($\Delta\chi^2 \approx 0.08$). The authors conclude that current $f\sigma_8$ data cannot statistically distinguish the $n=1/4$ scenario from $\Lambda$CDM.
  • The $n=1/2$ model yields a less favorable fit (higher $\chi^2$) but remains within the observational $\pm 2\sigma$ bounds across the considered redshift range.

Significance and Claims
The paper claims that $f(R, T^2)$ gravity constitutes a viable extension of General Relativity capable of describing late-time cosmic acceleration and large-scale structure formation. The study demonstrates that the model's viability is highly sensitive to the exponent $n$ and the coupling strength $\alpha$.

• Viability: The $n=1/4$ model is shown to be observationally indistinguishable from $\Lambda$CDM given current data, suggesting it can reproduce the observed growth history while introducing only mild modifications to standard cosmology.
• Constraints: The analysis places meaningful restrictions on the parameter space; specifically, large values of $\alpha$ in the $n=1/2$ model are disfavored due to negative growth indices and poorer statistical fits.
• Future Prospects: The authors modestly note that while current observations constrain the model, future surveys (DESI, Euclid, Vera Rubin Observatory, Nancy Grace Roman Space Telescope) will provide the necessary sensitivity to potentially distinguish these models from $\Lambda$CDM and tighten constraints on the matter-geometry coupling. The paper does not claim to have definitively ruled out or confirmed the theory but establishes its consistency with existing growth data.