{"id":"c999542b-cdcf-47c3-ac93-edd2c7419654","arxiv_id":"2504.15005","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Quadratic Weyl gravity stays viable only when its coupling obeys 10^14 H0^2 less than about 1/alpha and 1/alpha far below the cutoff squared; smaller values suppress structure growth and negative values are classically unstable.","lead":"This paper calculates how a small extra term in Einstein's equations, quadratic in the Weyl curvature, would alter the growth of cosmic structure during matter domination, and finds the term must be neither too large nor negative. The result gives a concrete window for the coupling constant of these modified gravity models, bounded below by structure formation and above by the theory's cutoff scale.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Scalar-sector bound depends on unphysical initial conditions for the extra mode; varying them likely shifts the 10^14 H0^2 bound.","rationale":"The vector and tensor sectors robustly yield only alpha > 0, and the speed-of-light propagation result is consistent with constraints. The paper's distinctive contribution is the scalar-sector lower bound on alpha^-1, so that bound is the load-bearing element of the central claim. The reader correctly identifies that the bound is computed with a single, unvaried choice of initial conditions for the fourth-order scalar system. Because the scalar equation has four independent modes, and only the GR growing mode is physically tied to standard structure formation, the initialization of the two additional modes directly controls the present-day ratio used to set the 10% criterion. No physical principle selects those initial amplitudes and velocities; the paper's choice of 10^-5 for all derivatives is a gauge-invariant but arbitrary phase-space point. A concrete test that varies these initial conditions would settle whether the 10^14 H0^2 bound is robust or an artifact. Since the concern is real but addressable, the reader's CONDITIONAL verdict remains appropriate; no change is needed.","tokens_in":13418,"tokens_out":3394,"duration_ms":32093,"concrete_test":"Recompute Fig. 5 solving (A.1)-(A.2) or (A.3) with the same background and K0 grid, but vary the initial conditions for the two non-GR modes: set them to zero, and also to ±10^-7, ±10^-5, ±10^-3, while keeping the GR growing-mode initial amplitude at 10^-5. Record the maximum |delta_QG(0)/delta_GR(0)-1| over K0 in [10,100] for alpha_bar = 10^-13, 10^-14, 10^-15. If the alpha_bar=10^-14 curve crosses the 10% band under any of these choices, the paper's bound is an artifact of the chosen initialization.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The new lower bound 10^14 H0^2 ≲ 1/alpha (Sec. 7.2, Fig. 5) is obtained by integrating the fourth-order scalar equation (A.3) with all initial values of delta_m and its first three derivatives set to 10^-5 at N=-10. This excites the two extra homogeneous modes (the massive scalar and its partner) with the same amplitude as the GR growing mode. But the present-day ratio delta_QG/delta_GR that must lie within 10% of unity is a linear combination of four independent homogeneous solutions plus the particular solution. The amplitude and velocity of the extra modes are not fixed by the theory; they are the initial displacement and rate of the new scalar degree of freedom. The figure-5 constraint therefore measures a mixture of the physical alpha-dependent particular solution and an arbitrary initial-condition-dependent homogeneous component. For alpha_bar=10^-14, if the extra-mode amplitude were initialized to zero (with only the GR growing mode and the particular solution), the deviation could be smaller, weakening the bound; choosing a larger amplitude would strengthen it. The paper gives no physical or observational prescription for these four initial conditions. Thus the stated lower bound on alpha^-1 is conditional, not a property of the theory alone.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the effects of a term quadratic in the Weyl tensor, -α C^2, on linear cosmological perturbations during the matter-dominated era. The authors derive the equations of motion for vector, tensor, and scalar perturbations, show that α<0 leads to classical instabilities in the vector and tensor sectors, and require α>0. For the scalar sector, they reduce the two-field system to a fourth-order equation for the matter overdensity δm, use a perturbative expansion in α H_0^2 for small α, and impose a criterion that the matter power spectrum deviate from GR by at most 10% at present. Sampling wavenumbers K0 = k/(a0H0) in [10,100], they conclude that α H_0^2 ≲ 10^-14, i.e., 10^14 H_0^2 ≲ α^-1 ≪ M_cutoff^2. The central claim is that this window of α is required for classical stability and for recovering the growth of structure.","tokens_in":13658,"tokens_out":15777,"duration_ms":137273,"significance":"If the claimed bound holds, it provides a concrete constraint on the parameter space of quadratic Weyl gravity from structure formation, complementing the known stability requirement α>0. The paper's vector and tensor analyses are clear and the reduction of the scalar sector to a single fourth-order equation is a useful technical contribution. The authors are transparent about the heuristic 10% criterion and the perturbative nature of the small-α expansion, and they compare exact and approximate solutions. The central numerical claim, however, is conditional on choices that are not fully justified, so the headline inequality should be interpreted as a conditional result rather than a definitive property of the theory.","major_comments":[{"comment":"The fourth-order scalar equation (A.3) is integrated with all initial values of δ_m and its first three derivatives set to 10^-5 at N=-10, which fixes the amplitude and velocity of the additional scalar degree of freedom in an arbitrary way. Since the theory does not prescribe these initial conditions, the present-day ratio δ_m^QG/δ_m^GR (Fig. 5) is a mixture of the physical particular solution and an initial-condition-dependent homogeneous component, and the resulting bound ᾱ ≲ 10^-14 is conditional on this choice. The paper should either derive these initial conditions from a physical prescription (e.g., inflation or adiabatic vacuum) or explicitly test the sensitivity of the bound to varying the extra-mode amplitudes; without this, the abstract's inequality 10^14 H_0^2 ≲ α^-1 is not robust.","section":"Sec. 7.2, Figs. 4-5 captions"},{"comment":"The 10% power-spectrum deviation criterion is hand-set ('heuristic criterion') and no observational reference is given. Because the boundary value ᾱ ≲ 10^-14 is read off from where the ratio in Fig. 5 crosses the 10% band, the central numerical claim inherits the arbitrariness of this threshold. Please either connect the 10% level to a specific observational constraint on the matter power spectrum or show how the bound shifts for other thresholds (e.g., 5% and 20%) so that the reader can gauge the robustness of 10^14 H_0^2 ≲ α^-1.","section":"Sec. 7.2, paragraph before Fig. 5"}],"minor_comments":[{"comment":"The y-axis label reads δ_m^GR(0)/δ_m^QG(0) while the caption says the ratio is between δ_m^QG and δ_m^GR; the text's description of 'suppression' for ᾱ=10^-13 is inconsistent with the plotted ratio being less than unity at large K0. Please clarify the convention.","section":"Fig. 5"},{"comment":"The abstract counts 'two scalar degrees of freedom, δ_m and Φ'; in GR, Φ is not an independent propagating degree of freedom, and the theory adds one new scalar gravitational mode. The phrasing could be adjusted to avoid confusion about the number of new d.o.f.","section":"Abstract, Sec. 7"},{"comment":"The exact integrations of (A.3) for ᾱ=10^-13 and 10^-14 are performed in a stiff regime (coefficients diverge as ᾱ→0); a brief description of the numerical method and error control would increase confidence in the exact/approximate comparison in Fig. 4.","section":"Sec. 7.2, numerical integration"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the scope of JCAP, and the vector/tensor stability analysis is sound and well presented. The main reservation is the conditionality of the scalar-sector bound; I believe the authors can address this in revision by justifying or varying the initial conditions for the extra scalar mode and by grounding or testing the 10% threshold. No concerns about novelty or citation practices."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"If you only take one thing from this paper: it gives a new analytical lower bound on 1/alpha in quadratic-Weyl gravity, 10^14 H0^2 <~ 1/alpha << M_cutoff^2, from requiring that matter-era structure growth is recovered and that deviations from GR stay small. The vector and tensor stability parts are straightforward and convincing. The genuinely new piece is the scalar-sector bound, and the authors handle one obvious sensitivity well: they sample K0 over [10,100] instead of quoting a single wavenumber.\n\nThe soft spot is the initial conditions for the fourth-order scalar equation. In Sec. 7.2 and the captions to Figs. 4 and 5, they set delta_m and its first three derivatives all to 10^-5 at N=-10, without explaining why that is a representative choice. A fourth-order ODE has four homogeneous modes, and the present-day ratio delta_QG/delta_GR can in principle depend on the amplitude and velocity of the extra scalar degree of freedom. The stress-test note makes exactly this point. My own sense is that the worry is real but may not be load-bearing: the two extra modes are massive, with masses ~1/sqrt(alpha) ~ 10^7 H0 in the interesting window, and in an expanding universe their amplitudes should redshift away by z=0. But the paper never shows this, and the bound is advertised as a property of the theory. A referee should ask for a sensitivity scan or an analytic estimate showing that the present-day ratio is dominated by the particular solution, not by the initial data.\n\nThe 10% power-spectrum threshold is hand-set, but that is a lesser issue because it is an explicit criterion, not a fitted parameter, and the final bound is an order-of-magnitude window. The authors also flag where the perturbative expansion breaks down, and the exact solutions for alpha_bar=10^-13 and 10^-14 in Fig. 5 support the conclusion where it matters. Citation pattern is fine: the prior work by two of the same authors is used as context, not to manufacture novelty, and there is no circular reasoning.\n\nBottom line: this is a modest but genuine step. It will be useful to people building or constraining higher-curvature gravity, and it deserves a proper referee, not a desk reject. I would send it out, with the request to address the initial-condition sensitivity and to state the provenance of the 10% criterion. If that comes back clean, I would cite it.","headline":"A clean new lower bound on the Weyl-squared coupling from matter-era scalar growth, with a real but addressable sensitivity to initial conditions in the scalar sector.","tokens_in":14155,"tokens_out":8406,"would_cite":true,"duration_ms":77927,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["04.50.Kd","98.80.-k"],"model":"deepseek-v4-flash","headline":"The paper constrains the free coupling of a quadratic Weyl term, showing that viable matter-era cosmology requires $10^{14}H_0^2 \\lesssim 1/\\alpha \\ll M_{\\text{cutoff}}^2$.","keywords":["quadratic Weyl gravity","modified gravity","cosmological perturbations","Jeans instability","matter power spectrum","classical stability","extra scalar degree of freedom","effective field theory cutoff"],"falsifier":"Compute the present-day ratio $\\delta_m^{\\mathrm{QG}}/\\delta_m^{\\mathrm{GR}}$ by numerically integrating the full fourth-order scalar equation with initial amplitudes and velocities for $\\delta_m$ spanning, say, $10^{-7}$ to $10^{-3}$ at the initial epoch used in the paper ($N=-10$), while keeping the background fixed. If the ten-percent crossing moves outside the range $1/\\alpha \\sim 10^{14}H_0^2$, the paper's bound is not robust; conversely, if the crossing stays put for all these initial conditions, the bound is confirmed as a genuine property of the model.","tokens_in":13203,"feed_emoji":"🌌","tokens_out":13398,"duration_ms":115464,"temperature":0.7,"pith_summary":"Adding a term quadratic in the Weyl tensor, $-\\alpha C^2$, to Einstein gravity introduces new vector, tensor, and scalar degrees of freedom. This paper asks whether those extra modes can coexist with the standard matter-dominated expansion history and with the growth of large-scale structure. It finds that $\\alpha$ must be positive to keep the vector and tensor modes classically stable, and that the inverse coupling $1/\\alpha$ must be at least about $10^{14}H_0^2$ so that the extra scalar mode does not suppress the Jeans instability and distort the matter power spectrum by more than ten percent. Together with the effective-field-theory requirement $1/\\alpha \\ll M_{\\text{cutoff}}^2$, this leaves a single window for the coupling. If the paper is right, quadratic Weyl gravity can be a viable late-time extension of general relativity only inside that window, with the new modes sufficiently heavy to hide at low energies.","feed_headline":"Quadratic Weyl gravity survives only in one coupling window","feed_subtitle":"The extra scalar mode must be heavy enough to preserve structure growth, light enough to stay under the cutoff.","key_machinery":"The central object is the quadratic Weyl term $-\\alpha C^2$, whose coupling $\\alpha$ sets the mass scale of the new degrees of freedom: in the vector and tensor sectors the modes acquire an effective squared mass $1/(2\\alpha)$, which converts stability into the condition $\\alpha>0$. The scalar sector is reduced to a single fourth-order ordinary differential equation for the gauge-invariant matter overdensity $\\delta_m$ (equation A.3), obtained by algebraically eliminating the Bardeen potential $\\Phi$ from the coupled second-order system. Expanding $\\delta_m = \\delta_m^{\\mathrm{GR}} + \\alpha H_0^2\\, \\delta_m^{(1)} + \\dots$ makes the deviation from general relativity explicit as a source term proportional to $K_0^2(1+z)^2\\delta_m^{\\mathrm{GR}}$, and the demand that this correction stay small is what yields the lower bound on $1/\\alpha$. The same coefficient appears in the vector and tensor stability conditions, so the paper's whole argument hangs on tracking the single dimensionless quantity $\\alpha H_0^2$.","core_discovery":"On the authors' own terms, the discovery is a pair of inequalities on the coupling $\\alpha$ in the action $S = (M_{\\mathrm{Pl}}^2/2)\\int d^4x\\sqrt{-g}(R-2\\Lambda-\\alpha C^2)+S_{\\mathrm{mat}}$. In an FLRW background during matter domination, the vector and tensor perturbations carry a mass term proportional to $1/(2\\alpha)$, so $\\alpha>0$ is required to avoid classical instabilities; for $\\alpha<0$ the modes blow up near the present time. In the scalar sector, the gauge-invariant density contrast $\\delta_m$ obeys a fourth-order equation, and expanding around the GR growing mode shows that the correction is controlled by $\\alpha H_0^2$ times a redshift-weighted source. Requiring that the Jeans instability be recovered and that the present-day matter power spectrum differ from GR by no more than $10\\%$ gives $\\bar\\alpha = \\alpha H_0^2 \\lesssim 10^{-14}$, i.e. $10^{14}H_0^2 \\lesssim 1/\\alpha \\ll M_{\\text{cutoff}}^2$. The paper verifies this by exact numerical integration of the fourth-order equation and by a first-order perturbative solution.","pith_inferences":["An extension the authors do not pursue: the same fourth-order scalar equation could be integrated from different initial amplitudes and velocities for the new scalar mode; if the ten-percent power-spectrum criterion is sensitive to those choices, the precise value of the lower bound would shift.","One testable consequence of the window is that the matter power spectrum in this theory should show a characteristic scale-dependent suppression relative to $\\Lambda$CDM controlled by $\\alpha H_0^2$; future large-scale-structure surveys that measure $P(k)$ at the percent level could therefore directly measure or bound $\\alpha$.","The matter-era constraint does not by itself settle the behavior of the Weyl-squared term in other epochs; applying the same expansion around the appropriate background solutions during radiation domination or inflation could give independent, possibly tighter windows on $\\alpha$."],"forward_implications":["If the constraint holds, quadratic Weyl gravity with $0<1/\\alpha < 10^{14}H_0^2$ is observationally problematic during matter domination: the extra scalar suppresses the growing density mode and changes the matter power spectrum by more than ten percent.","Within the allowed window, the vector and tensor sectors are classically stable even though the new modes are ghosts; quantum consistency then requires the fakeon prescription, keeping the theory unitary.","The two-sided inequality is meaningful only when the effective-field-theory cutoff obeys $M_{\\text{cutoff}}^2 \\gg 10^{14}H_0^2$, so the result forces the cutoff of the low-energy theory to be far above the Hubble scale.","Gravitational waves in the subhorizon regime propagate at the speed of light in this model, consistent with current constraints, while scalar perturbations deviate from $\\Lambda$CDM by at most ten percent at present."],"supporting_citations":[{"why":"introduces the action with the quadratic Weyl term $-\\alpha C^2$ that the paper constrains.","marker":"[2]"},{"why":"established that the theory is renormalizable and asymptotically free, the motivation for studying viable parameter space.","marker":"[3]"},{"why":"found that the Weyl-squared term destabilizes an inflationary background, prompting the matter-era analysis here.","marker":"[8]"},{"why":"supplies the fakeon prescription used to keep the classically stable ghost modes unitary.","marker":"[9]"},{"why":"provides the standard GR growth of density perturbations used as the baseline for the scalar-sector comparison.","marker":"[11]"},{"why":"shows the Weyl-squared term vanishes on FLRW backgrounds, so the background dynamics remain identical to LambdaCDM.","marker":"[13]"},{"why":"gives the Schutz-Sorkin fluid action used to model pressureless matter and derive the perturbation equations.","marker":"[15]"}],"fun_headline_variants":["Quadratic Weyl gravity's coupling squeezed into one window","Matter growth and stability confine Weyl-squared coupling","Weyl-squared gravity passes only for alpha in narrow range","Stability and power spectrum fix alpha for Weyl-squared models"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that setting the density perturbation and all its derivatives to $10^{-5}$ at the initial time is a representative choice for the new scalar mode; if the extra mode starts with a different amplitude or velocity, the value of $1/\\alpha$ at which deviations from general relativity reach ten percent could change.","fun_headline_variants_meta":{"raw":{"variants":["Quadratic Weyl gravity's coupling squeezed into one window","Matter growth and stability confine Weyl-squared coupling","Weyl-squared gravity passes only for alpha in narrow range","Stability and power spectrum fix alpha for Weyl-squared models"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000242,"raw_usage":{"total_tokens":1543,"prompt_tokens":982,"completion_tokens":561,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":598,"completion_tokens_details":{"reasoning_tokens":493}},"tokens_in":598,"tokens_out":561,"duration_ms":5858,"temperature":1.0,"reasoning_tokens":493,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:35:33.760690+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the present-day ratio $\\delta_m^{\\mathrm{QG}}/\\delta_m^{\\mathrm{GR}}$ by numerically integrating the full fourth-order scalar equation with initial amplitudes and velocities for $\\delta_m$ spanning, say, $10^{-7}$ to $10^{-3}$ at the initial epoch used in the paper ($N=-10$), while keeping the background fixed. If the ten-percent crossing moves outside the range $1/\\alpha \\sim 10^{14}H_0^2$, the paper's bound is not robust; conversely, if the crossing stays put for all these initial conditions, the bound is confirmed as a genuine property of the model.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"introduces the action with the quadratic Weyl term $-\\alpha C^2$ that the paper constrains."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"established that the theory is renormalizable and asymptotically free, the motivation for studying viable parameter space."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the Schutz-Sorkin fluid action used to model pressureless matter and derive the perturbation equations."}],"review_version":1}