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arxiv: 2606.28853 · v1 · pith:YPUNA6ENnew · submitted 2026-06-27 · 🌌 astro-ph.CO

Cosmological structure growth in energy-momentum squared gravity

Alvaro de la Cruz-Dombriz , Peter K. S. Dunsby , Payel Sarkar This is my paper

Pith reviewed 2026-06-30 08:37 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords f(R,T²) gravitycosmological perturbationsmatter density contrastgrowth indexfσ8modified gravitylarge-scale structurelate-time acceleration
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The pith

In f(R,T²) gravity the matter density contrast evolves with a growth index that decreases at late times but stays consistent with current fσ8 data for suitable coupling strengths.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates how cosmic structures grow in a modified gravity theory where the gravitational action depends on both the Ricci scalar and the square of the energy-momentum tensor. Using the gauge-invariant 1+3 covariant formalism, the authors derive the evolution equation for the matter density contrast and compute standard growth observables including the growth factor, growth index, and the product fσ8. For the representative cases n=1/2 and n=1/4, the growth index falls with increasing redshift yet recovers the expected matter-dominated value at early times, while late-time deviations from the standard model remain mild and scale-dependent. The model keeps predicted fσ8 values inside the current observational two-sigma envelope for appropriate choices of the coupling parameter α. A reader would care because the result tests whether a single matter-geometry coupling can simultaneously drive acceleration and preserve the observed pattern of galaxy clustering without invoking a separate dark-energy component.

Core claim

The central claim on the paper's own terms is that f(R,T²) gravity supplies a viable description of both late-time cosmic acceleration and large-scale structure formation: the growth index decreases with redshift and approaches standard matter-dominated behavior at early times, mild scale-dependent departures from ΛCDM appear at late times, and viable parameter choices for n=1/2 and n=1/4 keep the predicted fσ8 within the observational ±2σ bounds.

What carries the argument

The gauge-invariant 1+3 covariant formalism used to obtain the evolution equations for the matter density contrast in the presence of the T² term.

If this is right

  • The growth index decreases with increasing redshift and recovers the standard matter-dominated limit at early times.
  • Mild scale-dependent deviations from the ΛCDM growth rate appear only at late times.
  • For n=1/4 the departures from general relativity remain small; for n=1/2 and larger α they become stronger.
  • Viable parameter choices keep the model's fσ8 predictions inside the current observational ±2σ interval.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the coupling works as described, the same term that drives acceleration could also modify the growth rate enough to ease the σ8 tension without new fields.
  • Scale-dependent signatures at low redshift could be searched for with future surveys that bin galaxies by scale and redshift.
  • The same formalism might be applied to other quadratic scalars built from the energy-momentum tensor to generate different expansion histories.
  • Consistency with supernova distances or CMB lensing would provide an independent test not performed in this work.

Load-bearing premise

That the standard 1+3 covariant equations for density perturbations carry over to f(R,T²) without extra source terms generated by the direct matter-geometry coupling.

What would settle it

A future measurement of fσ8 at redshift z approximately 0.8 that lies outside the model's predicted band for the α values already compatible with present data.

Figures

Figures reproduced from arXiv: 2606.28853 by Alvaro de la Cruz-Dombriz, Payel Sarkar, Peter K. S. Dunsby.

Figure 1
Figure 1. Figure 1: FIG. 1. Evolution of density contrast for different wave numbers [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Evolution of density contrast for different wave numbers [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Growth factor rate for different values of [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Evolution of Growth index for different values of [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Growth index evolution for different [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Amplitude of matter density fluctuations for different values of [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The evolution of the weighted growth rate [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
read the original abstract

We investigate the cosmological evolution of matter perturbations in the modified gravity model $f(R,T^2)$, where $T^2=T_{\mu\nu}T^{\mu\nu}$ denotes the quadratic contraction of the energy--momentum tensor. Using the gauge-invariant 1+3 covariant formalism, we study the evolution of the matter density contrast and analyze several growth observables, including the growth factor, the growth index, and the weighted growth rate $f\sigma_8$. We consider representative values $n=1/2$ and $n=1/4$, which probe different regimes of the matter--geometry coupling. We show that the growth index decreases with increasing redshift and approaches the standard matter-dominated behavior at early times, while mild scale-dependent deviations from the $\Lambda$CDM model emerge at late times. The model predicts small departures from General Relativity for $n=1/4$, whereas stronger deviations appear for $n=1/2$ and larger values of the coupling parameter $\alpha$. We further compare the theoretical predictions for $f\sigma_8$ with current observational data and find that viable parameter choices remain within the observational $\pm2\sigma$ bounds. These results indicate that $f(R,T^2)$ gravity can provide a viable description of late-time cosmic acceleration and large-scale structure formation while remaining consistent with current growth observations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies the evolution of matter density perturbations in f(R,T²) gravity (with T² = T_μν T^μν) using the gauge-invariant 1+3 covariant formalism. For representative exponents n=1/2 and n=1/4 it computes the growth factor, growth index, and fσ8, reports mild scale-dependent deviations from ΛCDM at late times, and states that viable choices of the coupling α keep the predicted fσ8 within current observational ±2σ bounds, concluding that the model remains consistent with large-scale structure data while accommodating late-time acceleration.

Significance. If the linearized perturbation equations are shown to be complete, the work would supply a concrete, observationally testable extension of energy-momentum-squared gravity to the growth sector, complementing existing background analyses and offering a falsifiable alternative to ΛCDM at the level of fσ8 and the growth index.

major comments (2)
  1. [§3 (perturbation equations) and §4 (growth observables)] The central fσ8 and growth-index results rest on the numerical integration of the density-contrast equation derived in the 1+3 formalism. The abstract and §3 (perturbation equations) state that the standard gauge-invariant formalism is applied directly, yet the action contains an explicit α f(T²) term whose variation produces additional contributions to the effective stress-energy tensor and to the linearized Raychaudhuri and momentum constraints. These extra source terms, proportional to f'(T²) and to first-order perturbations of T_μν T^μν, are not shown to have been re-derived and inserted; their omission would render the reported scale-dependent deviations and data comparison unreliable.
  2. [§4 and Table 1] Table 1 and the accompanying text select n=1/2 and n=1/4 as “representative” values after noting that they keep predictions inside observational bounds. This post-selection of parameters undermines the claim of a parameter-free or robust prediction; the manuscript should instead present the full α–n parameter space and the exclusion criteria used to define “viable” choices.
minor comments (2)
  1. [Abstract and §4] The abstract and §2 refer to “mild deviations” and “small departures” without quantifying the redshift or scale at which these become statistically distinguishable from ΛCDM; explicit Δfσ8(z) curves with error bands would clarify the claim.
  2. [§2] Notation for the coupling function f(T²) and the exponent n is introduced without an explicit functional form (e.g., whether f(T²) = (T²)^n or another power); this should be stated once in §2 for reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We respond to each major comment below, indicating the revisions that will be made.

read point-by-point responses

  1. Referee: [§3 (perturbation equations) and §4 (growth observables)] The central fσ8 and growth-index results rest on the numerical integration of the density-contrast equation derived in the 1+3 formalism. The abstract and §3 (perturbation equations) state that the standard gauge-invariant formalism is applied directly, yet the action contains an explicit α f(T²) term whose variation produces additional contributions to the effective stress-energy tensor and to the linearized Raychaudhuri and momentum constraints. These extra source terms, proportional to f'(T²) and to first-order perturbations of T_μν T^μν, are not shown to have been re-derived and inserted; their omission would render the reported scale-dependent deviations and data comparison unreliable.

    Authors: We agree that the additional source terms arising from the variation of the α f(T²) term must be explicitly derived and shown. Although the 1+3 covariant formalism was used to obtain the density-contrast equation, the manuscript did not display the modified linearized Raychaudhuri and momentum constraints with the extra contributions. In the revised version we will add the full derivation of these terms in §3 and confirm that they are included in the numerical integration of the growth observables. revision: yes

  2. Referee: [§4 and Table 1] Table 1 and the accompanying text select n=1/2 and n=1/4 as “representative” values after noting that they keep predictions inside observational bounds. This post-selection of parameters undermines the claim of a parameter-free or robust prediction; the manuscript should instead present the full α–n parameter space and the exclusion criteria used to define “viable” choices.

    Authors: The manuscript presents results for representative values of n that satisfy the observational bounds rather than claiming a parameter-free prediction. To strengthen the presentation we will revise §4 to explicitly state the exclusion criteria (consistency with background expansion history and fσ8 data) and add a figure or table showing the viable region in the α–n plane together with fσ8 predictions for a wider range of parameters. revision: partial

Circularity Check

0 steps flagged

No circularity; derivation uses standard formalism and external data benchmarks

full rationale

The provided abstract and context show the paper applies the gauge-invariant 1+3 covariant formalism to derive matter density contrast evolution equations, selects representative n values, and compares fσ8 to external observational data for viability within ±2σ bounds. No quoted equations or steps reduce by construction to fitted inputs, self-citations, or self-definitions. The consistency claim rests on external benchmarks rather than internal parameter forcing, satisfying the criteria for a self-contained result.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

Abstract-only review; free parameters and axioms are inferred at minimum from stated choices. No explicit ledger or independent evidence is provided in the text.

free parameters (2)
  • coupling parameter α
    Larger values produce stronger deviations; selected to remain within observational bounds.
  • exponent n
    Set to representative values 1/2 and 1/4 to probe different coupling regimes.
axioms (1)
  • domain assumption The gauge-invariant 1+3 covariant formalism applies without modification to the perturbation equations of f(R,T²)
    Invoked to study the evolution of the matter density contrast and growth observables.

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discussion (0)

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