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arxiv: 2606.02660 · v1 · pith:XQECARV4new · submitted 2026-06-01 · ⚛️ physics.gen-ph

Late-Time Cosmology and Structure Formation in Quadratic f(Q) Gravity

G.G.L. Nashed , P.V. Tretyakov , A. Eid This is my paper

Pith reviewed 2026-06-28 12:01 UTC · model grok-4.3

classification ⚛️ physics.gen-ph
keywords f(Q) gravitysymmetric teleparallel gravityS8 tensionstructure formationgrowth ratelate-time cosmologymodified gravity
0
0 comments X

The pith

Quadratic f(Q) gravity weakens effective gravity at late times through a time-dependent G_eff, suppressing structure growth and lowering the predicted S8.

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

The paper studies the quadratic symmetric teleparallel model f(Q) = Q + α Q² + β, where Q scales with H² and adds an H⁴ term to the Friedmann equation. An exact algebraic solution for H(z) is derived and confronted with supernovae, BAO, and cosmic-chronometer data, showing only mild departures from ΛCDM at the background level. At linear order the quadratic term produces a reduced effective gravitational coupling G_eff(z) = G [1 + (2/3) A E²(z)]^{-1} for positive α, which damps the growth factor D(z), the growth rate f(z), and the observable fσ8(z). In the nonlinear regime the same weakening raises the spherical-collapse barrier and suppresses the halo mass function, yielding a lower S8 that can relieve the tension between Planck and low-redshift probes.

Core claim

The quadratic correction in f(Q) generates a time-dependent weakening of gravity that suppresses both linear growth and nonlinear halo abundance, thereby reducing the predicted value of S8 while the background expansion remains compatible with current distance data.

What carries the argument

The effective gravitational coupling G_eff(z) obtained by linearizing the quadratic f(Q) field equations around a Friedmann background and inserted into the Poisson equation.

If this is right

  • The background Hubble rate admits an exact algebraic expression that can be fitted directly to distance indicators.
  • Positive α produces a monotonic suppression of the linear growth factor and of fσ8(z) relative to ΛCDM.
  • The increased spherical-collapse threshold reduces the halo mass function at fixed mass, lowering the integrated S8.
  • The same parameter α that fits background data simultaneously drives the S8 reduction without additional dark-energy fields.

Where Pith is reading between the lines

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

  • If future weak-lensing surveys confirm a lower S8, the model supplies a concrete modified-gravity mechanism that does not require a separate dynamical dark-energy component.
  • The absence of an explicit screening mechanism suggests the predicted suppression may be testable in galaxy clusters or voids where the effective G_eff remains unscreened.
  • The algebraic H(z) solution could be used to forecast constraints from next-generation BAO or supernova surveys on the single parameter α.

Load-bearing premise

The linear modification to the Poisson equation and the spherical-collapse threshold remain valid descriptions of structure formation even when screening, higher-order perturbations, or baryonic effects are present.

What would settle it

A measurement of fσ8(z) at redshift 0.4–0.6 that lies above the ΛCDM prediction by more than the amount suppressed by the model for the α values allowed by background data would rule out the claimed alleviation of the S8 tension.

Figures

Figures reproduced from arXiv: 2606.02660 by A. Eid, G.G.L. Nashed, P.V. Tretyakov.

Figure 1
Figure 1. Figure 1: Background diagnostics of the quadratic f(Q) model. In this study we set H0 = 70, Ωm0 = 0.3, and Ωr0 = 9 × 10−5 . Panel (a) shows the expansion history E(z); panel (b) displays the effective dark-energy fraction Ωde(z); panel (c) shows the reconstructed effective dark-energy equation of state wde(z); and panel (d) presents the total effective equation of state wef f (z). The background expansion is a funda… view at source ↗
Figure 2
Figure 2. Figure 2: Growth diagnostics of the quadratic f(Q) model. The panels show the behaviour of key linear-perturbation quantities: (a) the linear growth factor D(z) normalized to D(0) = 1; (b) the logarithmic growth rate f(z); and (c) the redshift-space distortion observable fσ8(z). These quantities illustrate how the quadratic modification affects structure formation relative to ΛCDM. V. SLOW-ROLL DYNAMICS IN QUADRATIC… view at source ↗
Figure 3
Figure 3. Figure 3: Slow-roll diagnostics of the quadratic f(Q) model. The four panels show: (a) the first Hubble flow parameter ǫH(z); (b) the second Hubble flow parameter ηH; (c) the deceleration parameter q(z); and (d) the effective dark-energy equation of state wde(z). C. Effective Equation of State During Slow Roll Within the effective fluid description, the dark-energy equation-of-state parameter can be expressed as wde… view at source ↗
Figure 4
Figure 4. Figure 4: Redshift evolution of the effective gravitational [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
read the original abstract

We investigate the cosmological evolution associated with the quadratic symmetric teleparallel gravity framework, \( f(Q)=Q+\alpha Q^{2}+\beta \) where the relation \(Q\propto H^{2}\) generates an additional \(H^{4}\) contribution to the Friedmann equation. Using the exact algebraic solution for $H(z)$, we reconstruct the effective dark-energy sector and compare the background evolution with $\Lambda$CDM using Type Ia supernovae, BAO, and cosmic-chronometer data. At the perturbative level, the model modifies the Poisson equation through a time-dependent effective gravitational coupling $G_{\textrm eff}(z)=G\big[1+\tfrac{2}{3}A E^{2}(z)\big]^{-1}$, where $A=18\alpha H_{0}^{2}$. For $\alpha>0$ this produces a weakened gravitational interaction, suppressing the linear growth factor $D(z)$, the growth rate $f(z)$, and the RSD observable $f\sigma_{8}(z)$. In the nonlinear regime, the reduced gravitational strength increases the spherical-collapse threshold and suppresses the halo mass function, leading to a lower predicted value of $S_{8}=\sigma_{8}\sqrt{\Omega_{m}/0.3}$. Thus, the quadratic $f(Q)$ extension can reproduce mild deviations from $\Lambda$CDM at the background level while naturally alleviating the $S_{8}$ tension, offering a viable modified-gravity explanation for recent observational hints of dynamical dark energy.

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

3 major / 2 minor

Summary. The manuscript investigates quadratic f(Q) gravity with f(Q)=Q+αQ²+β, deriving an exact algebraic solution for H(z) that permits fitting background evolution to SNIa, BAO and cosmic-chronometer data with mild deviations from ΛCDM. It introduces a time-dependent effective gravitational coupling G_eff(z)=G[1+(2/3)A E²(z)]^{-1} (A=18α H0²) that weakens gravity for α>0, suppressing the linear growth factor D(z), fσ8(z) and, via an increased spherical-collapse threshold, the halo mass function, thereby lowering the predicted S8 and offering a modified-gravity resolution of the S8 tension.

Significance. If the G_eff derivation and the validity of the linear and spherical-collapse approximations hold without screening or higher-order effects, the exact H(z) solution and the linked background-plus-perturbation phenomenology would constitute a concrete, falsifiable modified-gravity explanation for both dynamical-dark-energy hints and the S8 discrepancy.

major comments (3)
  1. [Abstract / perturbative analysis] The G_eff(z) expression is stated in the abstract and perturbative discussion but no derivation from the quadratic term in the action or the resulting field equations is supplied; this is load-bearing for the claimed suppression of D(z), fσ8 and S8.
  2. [Nonlinear regime / structure formation] The assumption that the linear Poisson modification via G_eff remains accurate into the nonlinear regime, that the spherical-collapse threshold is simply rescaled, and that no screening, vector modes or extra degrees of freedom appear is not justified or tested; this directly underpins the predicted reduction in halo abundance and S8.
  3. [Background fit and parameter A] A is proportional to the same α that is adjusted to fit the background data, so the suppression of S8 is a direct consequence of that fit rather than an independent prediction; this weakens the claim that the model 'naturally alleviates' the tension.
minor comments (2)
  1. [Notation] The definition of the dimensionless Hubble parameter E(z) should be stated explicitly on first use.
  2. [References] Additional references to existing f(Q) literature and to other modified-gravity resolutions of the S8 tension would improve context.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript on quadratic f(Q) gravity. We address each major comment below, indicating where revisions will be made to improve clarity and rigor.

read point-by-point responses

  1. Referee: [Abstract / perturbative analysis] The G_eff(z) expression is stated in the abstract and perturbative discussion but no derivation from the quadratic term in the action or the resulting field equations is supplied; this is load-bearing for the claimed suppression of D(z), fσ8 and S8.

    Authors: We agree that an explicit step-by-step derivation of G_eff from the quadratic term in the action and the perturbed field equations is necessary for transparency. In the revised manuscript we will insert a dedicated subsection in the perturbative analysis that starts from the modified Einstein equations for f(Q) = Q + αQ² + β, linearizes the metric perturbations, and arrives at the time-dependent effective coupling G_eff(z) = G[1 + (2/3)A E²(z)]^{-1}. This addition will directly support the subsequent claims about suppressed growth. revision: yes

  2. Referee: [Nonlinear regime / structure formation] The assumption that the linear Poisson modification via G_eff remains accurate into the nonlinear regime, that the spherical-collapse threshold is simply rescaled, and that no screening, vector modes or extra degrees of freedom appear is not justified or tested; this directly underpins the predicted reduction in halo abundance and S8.

    Authors: We recognize that extending the linear G_eff modification into the nonlinear regime without explicit checks for screening, vector modes, or extra degrees of freedom constitutes an approximation whose validity range is not fully quantified in the present work. We will add a paragraph in the structure-formation section that states these assumptions explicitly, references analogous treatments in other modified-gravity models, and notes that N-body simulations would be required for definitive validation. The claims on halo abundance will be qualified accordingly. revision: partial

  3. Referee: [Background fit and parameter A] A is proportional to the same α that is adjusted to fit the background data, so the suppression of S8 is a direct consequence of that fit rather than an independent prediction; this weakens the claim that the model 'naturally alleviates' the tension.

    Authors: We disagree that the S8 suppression is merely a tautological consequence of the background fit. The single parameter α simultaneously sources the H⁴ term in the Friedmann equation (constrained by SNIa+BAO+CC data) and determines the functional form of G_eff(z) through the perturbed field equations; once α is fixed by background observations, the growth suppression follows deterministically from the model dynamics without additional free parameters. We will expand the discussion section to emphasize this direct linkage and thereby clarify why the alleviation can still be regarded as natural within the model. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper derives an algebraic H(z) from the quadratic f(Q) action, fits α (via A=18α H0²) to background data (SN+BAO+CC), then uses the identical α to compute G_eff(z) and the resulting suppression of D(z), fσ8(z) and halo abundance. This is a standard single-parameter model prediction linking background and perturbations; the S8 outcome is not forced by construction or by renaming a fit. No self-citations, no self-definitional steps, and no fitted-input-called-prediction pattern appear in the provided derivation chain. The analysis remains independent of the target S8 observable.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on two free parameters (α, β) that are adjusted to data and on the domain assumption that Q ∝ H² holds throughout the late-time FLRW evolution; no new entities are postulated.

free parameters (2)
  • α
    Coefficient of the Q² term; sets the strength of the H^4 correction and the deviation of G_eff from G.
  • β
    Constant term in f(Q); contributes to the effective dark-energy density.
axioms (1)
  • domain assumption Q is proportional to H² in a flat FLRW background
    Standard relation used to obtain the modified Friedmann equation from the chosen f(Q).

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discussion (0)

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Reference graph

Works this paper leans on

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