{"id":"1d0ff7f6-88d2-405e-8ddd-4334f11f1ce4","arxiv_id":"2507.09610","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"For power-law f(G) gravity, the matter power spectrum computed from 1+3 covariant perturbations is not scale invariant, with curves decaying and then rising above the GR flat line as k grows.","lead":"This paper computes the matter power spectrum for a power-law f(G) modified gravity model using covariant perturbation theory and dynamical-system background equations. It reports that the spectrum is not scale invariant unlike GR, but the key perturbation equations contain undefined symbols and the result depends on ad hoc initial conditions.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The solved dust system (48)–(49) is not a checkable consequence of the covariant equations: with w=0, Eq. (47) gives a source-free G equation, while Eq. (49) has an undefined parameter m; Eq. (48) also uses undefined m and q.","rationale":"The reader's weakest assumption is that Eqs. (48)–(49) may be an incorrect or unverifiable reduction of the covariant perturbation equations, with m and q undefined. My concern is exactly that, made concrete: the dust limit of Eq. (47) is inconsistent with Eq. (49) unless an unstated value of m is chosen, and Eq. (48) depends on both undefined m and q in its k-dependent terms. Therefore the numerical power spectra, and the abstract's not-scale-invariant claim, rest on an uncheckable or internally inconsistent system. This does not prove the physical claim false, but it means the paper does not provide a verifiable derivation. Since the reader already recommends REJECT, my finding does not change the verdict; it sharpens the reason for rejection.","tokens_in":9843,"tokens_out":11208,"duration_ms":138454,"concrete_test":"Perform the reduction from Eqs. (46)–(47) to redshift space by hand: set w=0 in Eq. (47), substitute \\dot G=-(1+z)HG' and \\ddot G=(1+z)^2H(HG''+H'G'), and use the dust Hubble ratio H'/H=3/[2(1+z)]. The result should be G''=-3/[2(1+z)]G'. Compare this term-by-term with Eq. (49), and then fix the values of m and q (e.g., from the G∝(1+z)^{-6/m} scaling) and re-derive Eq. (48). If Eq. (49) does not match, or if the derivation forces an m different from the one needed for the exponents in Eq. (48), then the solved system is not the stated f(G) perturbation equations, and the power-spectrum figures cannot support the central claim. A complementary numerical check is to solve Eqs. (48)–(49) with two different plausible m values and the stated initial conditions; if P(k) changes, the reported n-dependence is an artifact of an unspecified parameter.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is entirely carried by the numerical solution of the redshift-space pair (48)–(49), which is presented as the dust (w=0) limit of the covariant system (44)–(47). That reduction is not checkable from the text, and the check reveals a concrete problem. For dust, Eq. (47) has a vanishing right-hand side. Using the stated redshift rules \\dot G=-(1+z)HG' and \\ddot G=(1+z)^2H(HG''+H'G') with the dust background H'/H=3/[2(1+z)], Eq. (47) therefore yields G''=-3/[2(1+z)]G'. Eq. (49), by contrast, states G''=-[3/(2m)+1]/(1+z)G', with m never defined. These agree only if m=3 is silently assumed, but Eq. (48) also uses m in exponents (1+z)^{-6/m} and (1+z)^{-3/(2m)}, and q is never defined anywhere in Sections 2–4. Since m and q enter the k-dependent coefficients, the scale dependence of the computed P(k) is not fixed by the presented derivation. The five initial-condition sets are applied to a system whose coefficients are not a reproducible consequence of the model. Thus the claim that P(k) is not scale invariant cannot be verified from the paper.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"Using a power-law f(G)=βG^n model, the authors build a dynamical-system description of the background, write covariant first-order density perturbation equations in the 1+3 formalism, convert them to redshift space, solve them numerically for five sets of initial conditions at z_in=2000, and compute the matter power spectrum P(k)=|Δ(k)|^2 at z=0. The reported result is that P(k) is not scale invariant for n≠1, with the curves separating most clearly as n approaches 1, whereas n=1 is claimed to recover the scale-invariant GR case.","tokens_in":10141,"tokens_out":7677,"duration_ms":84730,"significance":"If correct, the claim that a power-law f(G) model breaks scale invariance of the matter power spectrum even from scale-invariant initial conditions would be an interesting, testable prediction. The paper includes useful checks, such as multiple initial-condition sets and the n=1 GR limit. However, the central result is carried entirely by Eqs. (48)–(49), whose reduction from the covariant equations is omitted and which contain undefined symbols m and q. As it stands, the numerical spectra in Figures 1–6 cannot be verified or reproduced, so the significance of the claimed result cannot be assessed.","major_comments":[{"comment":"The central numerical system is not reproducible: the symbols m and q appearing in Eq. (48) are never defined in Sections 2–4, and the algebraic reduction from Eqs. (44)–(47) to Eqs. (48)–(49) is not shown. Since m appears in the exponents (1+z)^{-6/m} and (1+z)^{-3/(2m)} and in the coefficient of G' in Eq. (49), while q appears in the coefficients of G' and G in Eq. (48), the k-dependent coefficients used in the figures are not fixed by the derivation. Without definitions of these symbols, the reported power spectra cannot be checked.","section":"§3, Eqs. (48)–(49)"},{"comment":"For w=0, Eq. (47) has a vanishing right-hand side, so the Gauss-Bonnet perturbation satisfies \\ddot G=0. Using the stated redshift transformation and the dust background relation H'/H=3/[2(1+z)], this gives G''=-3/[2(1+z)]G'. Eq. (49) instead states G''=-[3/(2m)+1]/(1+z)G', which agrees only if m=3 is silently assumed. The text does not state this assumption, and Eq. (48) uses m in additional places. The system actually solved is therefore not established as the dust limit of the covariant equations.","section":"§3, Eq. (49)"},{"comment":"The initial conditions are imposed by setting the same amplitude for Δ_d, Δ'_d, G and G' for every k at z_in=2000, so the input spectrum is flat (scale invariant) by construction. In Eq. (48) the only explicit k dependence is the k^2/a^2 term; the claim that the output P(k) is not scale invariant is therefore substantially a consequence of that standard gradient term, rather than a demonstration that power-law f(G) itself generates scale dependence. To support the abstract's claim, the authors would need to compare with the GR limit under an identical numerical treatment.","section":"§4 and §5.1"},{"comment":"The numerical implementation is not described: the values of β and H0, the range and sampling of k, the integration scheme, and the handling of the (n-1) denominators in Eq. (48) are not given. In particular, n=1 is claimed to recover GR, but Eq. (48) contains 1/(n-1) terms; a limiting procedure is required and is not explained. Consequently Figures 1–6 cannot be reproduced from the text.","section":"§4, Figures 1–6"}],"minor_comments":[{"comment":"The matter continuity equation is written as \\dot ρ_m + 3H(1+3w_m)ρ_m = 0, which appears to be a typo; the standard form is \\dot ρ_m + 3H(1+w_m)ρ_m = 0.","section":"§2.1, Eq. (13)"},{"comment":"The text says 'fig. (5) for set IV', but the fifth set was labeled Set V in Section 4; the figure numbering/caption should be corrected.","section":"§5.1"},{"comment":"The caption states initial conditions at z0 rather than z_in=2000, inconsistent with the other figures and with the description in Section 5.1.","section":"Figure 6 caption"},{"comment":"There are several typographical errors, including 'decerelation', 'We derive' in the abstract, and inconsistent capitalization in the initial-condition set labels.","section":"§1 and Abstract"},{"comment":"The conclusion that the results 'support the ΛCDM and GR predictions' is not supported by any quantitative comparison with data or with a reference spectrum; this statement should be softened or substantiated.","section":"§5.2"},{"comment":"Reference [28] is incomplete: it gives only an arXiv identifier without an author list, title, or year.","section":"References"}],"recommendation":"reject","confidential_remarks":"To the editor: the main issue is verifiability of Eqs. (48)–(49). The undefined symbols m and q and the apparent inconsistency between Eq. (47) and Eq. (49) are load-bearing for the paper's central claim. A resubmission with the full reduction, definitions of m and q, and reproducible numerical details would be needed before the result could be evaluated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is the first paper I know that computes the matter power spectrum for a power-law f(G) model using the 1+3 covariant / dynamical-system framework, and the qualitative direction is plausible. But as it stands, the central numerical result is not checkable: equations (48) and (49) contain undefined parameters m and q, and the covariant-to-redshift reduction is mostly skipped. That is a load-bearing hole, not a cosmetic one.\n\nWhat the paper does well: it picks a well-defined model f(G)=β(G/H0^4)^n, sets up the background with dimensionless variables, follows the established pipeline from [23,24], runs five initial-condition sets, and plots P(k) for several n. The n=1 limit correctly reduces to a k-independent GR spectrum. The claim that f(G) introduces scale dependence is plausible on general grounds—a k^2/a^2 term acting on scale-invariant initial perturbations will generate k-dependence—and the paper does not fit any dataset, so there is no circularity at the level of parameter estimation.\n\nThe soft spots are serious. m and q appear in the solved system but are never defined anywhere in the text. Without them, the equations that determine the figures are not reproducible. The derivation from (44)–(47) to (48)–(49) is omitted; the stress-test note is right that for dust w=0, Eq. (47) has a vanishing right-hand side, and the paper's own redshift rules then give a specific second-order equation for G. Eq. (49) has a different coefficient with m in it. The two can only agree for a particular hidden value of m, and no such statement appears. Since m and q enter the k-dependent coefficients, the computed scale dependence is not fixed by the material you can check. The initial conditions are ad hoc, though the sensitivity study is at least transparent. The figures are unlabelled in the text, and the conclusion's last sentence says the results 'support the ΛCDM and GR predictions,' which sits oddly with the abstract's claim of non-scale-invariance.\n\nWho is this for? Someone working in modified-gravity phenomenology might find the qualitative suggestion worth thinking about, but they would need to reconstruct the derivation themselves. As a referee, I would not accept the manuscript in this form; I would desk-reject with an invitation to resubmit after m/q are defined, the reduction is written out, and the code or parameter values are made available. The question is worth pursuing; this version does not let the reader verify it.","headline":"First f(G) matter-power-spectrum computation in this pipeline, but undefined parameters and an omitted derivation make the central numerical result uncheckable in the current form.","tokens_in":10690,"tokens_out":4020,"would_cite":false,"duration_ms":45947,"reading_group":"no","serious_thinker":"unclear","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"For a power-law $f(G)$ gravity model, this paper claims the matter power spectrum is not scale invariant, unlike in General Relativity: solving the coupled matter and Gauss-Bonnet perturbation equations in a dust-dominated universe…","keywords":["modified Gauss-Bonnet gravity","matter power spectrum","energy density perturbations","1+3 covariant formalism","dynamical system","scale invariance","dust-dominated universe","power-law f(G) model"],"falsifier":"Re-derive equations (48)-(49) from (44)-(45) with explicit definitions of $m$ and $q$ and repeat the stated integration: if the corrected equations produce a $k$-independent spectrum for $n\\neq1$, or if the $n=1$ run does not return a flat $P(k)$, the reported scale dependence is not established. A quick numerical check is to verify that the $n=1$ curve is exactly flat in the same solver that produces the $n\\neq1$ curves.","tokens_in":9603,"feed_emoji":"🌌","tokens_out":5997,"duration_ms":60209,"temperature":0.7,"pith_summary":"The paper tries to show that in a power-law $f(G)$ gravity model, the matter power spectrum at $z=0$ is not scale invariant, unlike in General Relativity. Starting from the $1+3$ covariant perturbation equations for a dust-dominated universe, the authors derive coupled redshift-space equations for the matter density contrast $\\Delta_m$ and the Gauss-Bonnet density perturbation $G$, solve them numerically for the model $f(G)=\\beta H_0^{4-4n}G^n$, and compute $P(k)=|\\Delta(k)|^2$. For $n=1$ the GR spectrum is recovered and is flat in $k$; for $n\\neq1$ the reported spectra decay and then rise above the GR line as $k$ increases, with the curves separating most clearly as $n$ approaches $1$. If correct, this means modified Gauss-Bonnet gravity imprints a scale-dependent clustering amplitude even when the input perturbations are scale invariant.","feed_headline":"Power-law f(G) gravity breaks the scale-invariant matter spectrum","feed_subtitle":"Dust-era perturbations in modified Gauss-Bonnet gravity yield spectra that dip and rise above the GR line as k grows.","key_machinery":"The machinery is the coupled linear perturbation system in redshift space, equations (48)-(49), for the matter density contrast $\\Delta_m$ and the Gauss-Bonnet density perturbation $G$, together with the dimensionless background variables $x$, $y$, and $\\Omega_d$ from the dynamical-system reduction of the $f(G)$ field equations. The paper solves the background equations (34), (36), and (37) first, then feeds those solutions into the perturbation system to obtain $\\Delta_m(z)$; the power spectrum is then formed as $P(k)=|\\Delta(k)|^2$ at $z=0$. The five initial-condition sets at $z_{in}=2000$ are the input that converts the differential system into concrete spectra, and they are what allow the paper to test how the shape of $P(k)$ responds to different starting amplitudes and velocities.","core_discovery":"For the power-law model $f(G)=\\beta H_0^{4-4n}G^n$ in a dust-dominated flat FRW universe, the paper claims that the linear matter density contrast $\\Delta_m$ is coupled to the Gauss-Bonnet perturbation $G$ through equations (48)-(49), and that the resulting power spectrum $P_{f(G)}(k)=|\\Delta(k)|^2$ evaluated today depends on wavenumber $k$. The central reported behavior is that the spectra decay below the scale-invariant GR line and then evolve above it as $k$ increases; the separation between curves for different $n$ grows as $n$ moves closer to $1$, while $n=1$ exactly reproduces the GR result. This non-scale-invariance is the paper's main discovery, and it is obtained without the quasi-static approximation, by integrating the perturbation system with five different sets of initial conditions at $z_{in}=2000$.","pith_inferences":["If the reported $k$-dependence survives a full treatment with radiation, baryons, and a transfer function, power-law $f(G)$ models with $n\\neq1$ would predict an effective tilt in the matter power spectrum that galaxy surveys could in principle measure.","Because the paper does not state the values of $m$ and $q$ used in the numerical integration, the concrete numbers behind the figures are not reproducible from the text; pinning those definitions down is the natural first step before comparing with data.","A testable extension is to run the same system with a logarithmic grid in $k$ and check whether the dip-and-rise feature converges; if it is an artifact of the initial-condition amplitudes, it should weaken as the initial velocity perturbations are varied systematically.","The paper only considers dust, so an immediate extension is to include radiation and see whether the scale dependence persists through matter-radiation equality, where the initial conditions are set."],"forward_implications":["For $n=1$ the model reduces to General Relativity and the power spectrum is scale invariant, so the GR limit is built into the calculation.","For $n\\neq1$, a scale-invariant input perturbation at $z_{in}=2000$ does not stay scale invariant by $z=0$; the clustering amplitude acquires a $k$-dependence.","The matter density contrast is coupled to the Gauss-Bonnet perturbation, so the modified-gravity sector directly shapes the matter power spectrum rather than only changing the background expansion.","Across the five initial-condition sets the reported spectra do not oscillate, which the paper takes as agreement with earlier $f(R)$ and modified-gravity power-spectrum studies.","The spread of the curves near $n=1$ gives a target for future observational constraints on the power-law index $n$."],"supporting_citations":[{"why":"Supplies the power-spectrum formula $P(k)=|\\Delta(k)|^2$ and the initial-condition strategy for modified-gravity perturbation equations.","marker":"[24]"},{"why":"Provides the scalar-tensor gravity power-spectrum calculation using dynamical-system analysis that this paper extends to $f(G)$ gravity.","marker":"[23]"},{"why":"Gives the $f(G)$ field equations and the curvature-fluid energy density and pressure used for the background.","marker":"[13]"},{"why":"Introduces the power-law model $f(G)=\\beta H_0^{4-4n}G^n$ that the paper adopts.","marker":"[26]"},{"why":"Establishes the covariant perturbation setup for modified Gauss-Bonnet gravity that the paper's equations (44)-(45) build on.","marker":"[17]"},{"why":"Constructs the dynamical system for modified Gauss-Bonnet gravity that supplies the background variables $x$, $y$, $\\Omega_d$.","marker":"[22]"}],"fun_headline_variants":["f(G) gravity breaks scale-invariance in matter spectrum","Modified Gauss-Bonnet gravity yields non-scale-invariant matter power","Power-law f(G) gravity: matter spectrum deviates from GR","Matter power spectrum in f(G) gravity: not scale-invariant","f(G) model produces k-dependent matter power spectrum"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The calculation stands on the assumption that equations (48) and (49) are the correct redshift-space reduction of the covariant perturbation equations (44) and (45), even though the symbols $m$ and $q$ appearing in them are never defined and the reduction is not shown.","fun_headline_variants_meta":{"raw":{"variants":["f(G) gravity breaks scale-invariance in matter spectrum","Modified Gauss-Bonnet gravity yields non-scale-invariant matter power","Power-law f(G) gravity: matter spectrum deviates from GR","Matter power spectrum in f(G) gravity: not scale-invariant","f(G) model produces k-dependent matter power spectrum"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000597,"raw_usage":{"total_tokens":2770,"prompt_tokens":901,"completion_tokens":1869,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":517,"completion_tokens_details":{"reasoning_tokens":1791}},"tokens_in":517,"tokens_out":1869,"duration_ms":11754,"temperature":1.0,"reasoning_tokens":1791,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:51:14.001037+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Re-derive equations (48)-(49) from (44)-(45) with explicit definitions of $m$ and $q$ and repeat the stated integration: if the corrected equations produce a $k$-independent spectrum for $n\\neq1$, or if the $n=1$ run does not return a flat $P(k)$, the reported scale dependence is not established. A quick numerical check is to verify that the $n=1$ curve is exactly flat in the same solver that produces the $n\\neq1$ curves.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the power-spectrum formula $P(k)=|\\Delta(k)|^2$ and the initial-condition strategy for modified-gravity perturbation equations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the scalar-tensor gravity power-spectrum calculation using dynamical-system analysis that this paper extends to $f(G)$ gravity."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the $f(G)$ field equations and the curvature-fluid energy density and pressure used for the background."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the power-law model $f(G)=\\beta H_0^{4-4n}G^n$ that the paper adopts."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the covariant perturbation setup for modified Gauss-Bonnet gravity that the paper's equations (44)-(45) build on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Constructs the dynamical system for modified Gauss-Bonnet gravity that supplies the background variables $x$, $y$, $\\Omega_d$."}],"review_version":1}