{"id":"20b3c45d-4c4b-43e4-b1a6-dd1da23995ef","arxiv_id":"2504.12656","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Spherical collapse in DHOST theories fails for the beyond-Horndeski parameter β1 above about 10^-7 because the scalar-field gradient becomes imaginary, and the halo mass function is suppressed relative to ΛCDM.","lead":"The paper applies the spherical collapse model to DHOST theories of modified gravity using the effective field theory of dark energy, and finds that tiny values of the beyond-Horndeski parameter β1 can prevent matter overdensities from collapsing when the scalar field becomes imaginary. It derives a new tight limit on β1 and shows that beyond-Horndeski effects suppress the predicted number of dark matter halos.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Loss of real x at D=0 is likely a quasi-static breakdown, not a physical halt: \\dot x diverges as D→0, so the β10<1e-7 constraint needs a full-dynamics check.","rationale":"The reader identified the quasi-static approximation and the physical interpretation of D<0 as the weakest assumption; my analysis agrees and sharpens it. The algebraic equation (33) is derived from the quasi-static action (2), so the absence of real roots is a statement about that truncated system. The two-root merger at D=0 generically makes \\dot x diverge as 1/√D, which is precisely the regime where quasi-staticity fails. Thus the claimed upper limit β10<1e-7 is not secured without a check of the full time-dependent dynamics or an explicit demonstration that the quasi-static approximation remains valid at the critical moment. The paper's own Sec. VI suggestion to confirm the no-collapse result with N-body simulations is an acknowledgment of this limitation. The generic-case mass-function suppression is less affected because it does not rely on the no-real-root interpretation, but it is also computed within the same quasi-static framework. The derivation is internally coherent and the appendices are explicit, so I do not see a reason to reject the paper; the correct status is a conditional acceptance pending the full-dynamics check.","tokens_in":17118,"tokens_out":7743,"duration_ms":90703,"concrete_test":"Evaluate the self-consistency of the quasi-static solution along the fiducial trajectory (β10=10^-4, αB0=-2×10^-3, Ai=2.30): compute \\dot x/(H x) and the ratio of the time-derivative terms that the degeneracy cancellation removes in deriving Eq. (33) to the leading quasi-static terms as D approaches zero. If either ratio exceeds O(1) before D=0, the loss of real roots is an artifact of the quasi-static truncation and the β10<1e-7 bound is not established. Alternatively, solve the full non-quasi-static scalar equation for the same parameters; if x remains real and R reaches zero, the concern lands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central constraint β10<1e-7 (Sec. IV.A) rests on interpreting the vanishing of the discriminant D of the algebraic equation (33) as the prevention of collapse. The self-consistency of that interpretation is the weak point. The quasi-static effective action (2) is used to derive the cubic/quadratic equation; but near the time when D→0, the two roots merge and the time derivative of the physical branch, \\dot x ≈ −\\dot D/(4C2√D), diverges as 1/√D. Hence the slow-evolution assumption underlying the quasi-static reduction is violated before the solution becomes imaginary. The same issue afflicts the generic cubic case when αH+2β1≠0: the discriminant can vanish, but the full time-dependent scalar dynamics (which the degeneracy cancellation removes only within the truncated action) may carry x smoothly through the would-be critical point. In that case no bound β10<1e-7 follows. The paper itself defers confirmation to N-body simulations in Sec. VI, effectively acknowledging that the spherical-collapse result is not decisive. The top-hat ansatz (Φ∝r², x(t) independent of r) is also least secure at the moment of merger. This is therefore a load-bearing, unresolved assumption.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the spherical collapse of top-hat matter overdensities in DHOST theories using the EFT of dark energy action truncated to quasi-static, nonlinear derivative interactions. After imposing c_T=1, α_M=0, and a ΛCDM background, the authors derive an algebraic equation for the scalar gradient x (Eq. 33), which becomes quadratic when α_H+2β_1=0 and cubic otherwise. In the quadratic case they show that the discriminant D can become negative for β_10 above a threshold and interpret loss of real roots as a prevention of collapse, yielding the constraint β_10<10^{-7} for α_B0=-2×10^{-3}. In the generic cubic case they find that positive α_H0 and β_10 delay collapse and, using the Press–Schechter formalism, that the halo mass function is suppressed relative to ΛCDM for their selected parameters. The paper is careful about the degeneracy condition, the Vainshtein regime, and scalar stability conditions, and it explicitly notes that N-body simulations are needed to confirm the collapse behavior.","tokens_in":17394,"tokens_out":3314,"duration_ms":40712,"significance":"If the central interpretation is correct, the bound β_10<10^{-7} would be roughly five orders of magnitude stronger than the existing Hulse–Taylor constraint, and the predicted suppression of the halo mass function would be a sharp, falsifiable signature of beyond-Horndeski theories. The derivation is transparent and the stability and degeneracy conditions are applied carefully, including a useful separation of the two background models γ_0=1 and γ_0=1−α_H0−3β_10. The strength of the paper is its explicit algebraic control over the scalar-gradient equation and the clean identification of the discriminant as the quantity that controls the existence of real solutions. The weakness is that the physical interpretation of D=0 as a halt of collapse depends on the quasi-static and top-hat assumptions, which are precisely the assumptions that are least reliable near collapse; the authors themselves defer confirmation to N-body simulations in Sec. VI.","major_comments":[{"comment":"The central constraint β_10<10^{-7} rests on interpreting the vanishing of the discriminant D of the algebraic equation (33) as a physical prevention of collapse, but this interpretation is not established. Near a double root, the time derivative of the physical branch scales as \\dot x ≈ −\\dot D/(4 C_2 √D), which diverges as D→0. This violates the slow-evolution assumption that underlies the quasi-static reduction used to derive Eq. (33) from the action (2). The loss of real roots could therefore signal a breakdown of the quasi-static approximation rather than a fundamental obstruction to collapse. The paper itself acknowledges in Sec. VI that whether an overdense region 'indeed fails to collapse' must be confirmed with N-body simulations. Since the bound β_10<10^{-7} follows only under the disputed interpretation, this is a load-bearing unresolved assumption.","section":"Sec. IV.A, Eqs. (33), (43), (44)"},{"comment":"The claim that the bound β_10<10^{-7} is obtained 'with any initial amplitudes' is not supported by the presentation. The numerical demonstration in Figs. 1 and 2 uses the single initial amplitude A_i=2.30, and no scan over A_i is shown or described. Without such a scan, the statement that this is a universal requirement over all initial amplitudes is an assertion rather than a result. The authors should either provide the scan or soften the claim to 'for the representative amplitude A_i=2.30', which would change the strength of the constraint.","section":"Sec. IV.A, Fig. 2 and Eq. (42)"},{"comment":"The generic-case analysis (α_H+2β_1≠0) avoids the discriminant problem only by choosing parameters sufficiently far from the line α_H0+2β_10=0. The collapse-time delays reported in Figs. 3 and 4 are computed with the same quasi-static algebraic equation, so the same concern about the validity of the approximation near collapse applies there. The subsequent halo-mass-function suppression in Figs. 5 and 6 inherits this uncertainty. The paper would be strengthened by a quantitative statement of when the quasi-static approximation is expected to fail (for example, by comparing the magnitude of the neglected \\dot x terms with the retained terms along the numerical trajectories), rather than relying on the physical plausibility of the root-merging picture.","section":"Sec. IV.B and Sec. VI"}],"minor_comments":[{"comment":"The heading reads 'EVOLUITON OF SPHERICAL OVERDENSITIES'; it should be 'EVOLUTION'.","section":"Section IV heading"},{"comment":"The caption says 'real solutions for x cease to exit at some moment'; 'exit' should be 'exist'.","section":"Fig. 2 caption"},{"comment":"The sign discussion of D_4 would be easier to follow if the authors stated explicitly that D_4>0 is required for the negative contribution to D, and if they clarified whether D_4 depends on time in the matter-dominated era where the early-time argument is made.","section":"Sec. IV.A around Eq. (43)"},{"comment":"The Press–Schechter formalism is applied to a theory with modified gravitational dynamics; the equality δ_c=δ_L(t_col) and the use of the linear power spectrum are standard, but the paper would benefit from a brief statement of the known limitations of this mapping in modified gravity and from an explicit caveat that the 10% suppression at 10^{14} M_⊙ is quoted at the present time for the chosen parameter set only.","section":"Sec. V"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is well within the scope of the journal and the derivation is generally careful. The single most important gate is the interpretation of D=0 as a physical halt; if the authors can supply a full-dynamics check or a quantitative quasi-static-validity assessment, the paper would be publishable. The 'any initial amplitudes' claim needs either a scan or a more modest formulation. I would not reject, because the issue is fixable by recasting the claims as conditional and by adding validation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The new thing here is real: this is the first spherical-collapse analysis covering the full class-Ia DHOST parameter space in the EFT basis, not just the shift-symmetric Galileon subset of Ref. [34]. The collapse-failure mechanism—the cubic/quadratic for the scalar gradient losing real roots when the discriminant D vanishes—is genuinely new, and the halo mass function suppression for positive αH0 and β10 is a concrete observational target. The paper is also careful about stability conditions, the Vainshtein regime, and the distinction between the two background choices for γ0. That is solid workmanship.\n\nThe soft spot is exactly where the reader and stress-test put it. The constraint β10 < 10^-7 in Sec. IV.A comes from interpreting D<0 as a physical halt to collapse. But the quasi-static action used to derive Eq. (33) is only valid while the field evolves slowly, and as D→0 the two roots merge and ẋ diverges like 1/√D. So the approximation is breaking down at the same moment the algebraic equation loses its real root. The full time-dependent dynamics might carry x smoothly through that point, in which case no bound follows. This is not a minor caveat; it is load-bearing for the headline claim. The paper itself effectively concedes the point by deferring to N-body simulations in Sec. VI. The top-hat ansatz is also least secure right at collapse, which adds to the uncertainty.\n\nThat said, the reader's circularity concern is off the mark. The results are computed from the model equations, not fit to an input, and the algebra in Appendices B and C looks internally consistent even if I did not verify every line. The failure to cross-check against the existing CMB constraints for the parameters used in the halo mass function is a minor omission, not a flaw: the mass-function plots are explicitly a demonstration.\n\nBottom line: this deserves a serious referee. The formalism and the qualitative effect are worth engaging with, but the central β1 bound should not be taken as a prediction until someone checks the quasi-static assumption against the full dynamics, analytically or in N-body. If that check goes through, this is a very strong new cosmological probe; if not, the paper still stands as a useful first step. I would send it out, with the quasi-static breakdown issue as the main referee request.","headline":"A careful spherical-collapse extension to DHOST with a striking β1 bound that rests on treating quasi-static loss of real roots as physical, and that needs a full-dynamics or N-body check.","tokens_in":17956,"tokens_out":1625,"would_cite":false,"duration_ms":21002,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"In DHOST theories, a beyond-Horndeski parameter $\\beta_{10}$ above $10^{-7}$ prevents spherical collapse because the scalar turns imaginary, while smaller positive values delay collapse and suppress massive halos.","keywords":["DHOST theories","spherical collapse","EFT of dark energy","halo mass function","beyond-Horndeski gravity","Vainshtein mechanism","large-scale structure","modified gravity"],"falsifier":"Run a fully time-dependent spherical top-hat collapse calculation in a class-Ia DHOST theory with $\\beta_{10}=10^{-4}$ and $\\alpha_{B0}=-2\\times10^{-3}$ from the same initial conditions; if the scalar-field gradient remains real and the physical radius reaches zero, the paper's identification of the vanishing discriminant with prevented collapse is refuted. An N-body simulation forming halos in a parameter region where the spherical model predicts collapse failure would be the same test at larger scales.","tokens_in":16916,"feed_emoji":"🌌","tokens_out":17256,"duration_ms":149360,"temperature":0.7,"pith_summary":"The paper asks whether matter overdensities can still collapse in degenerate higher-order scalar-tensor (DHOST) theories, the broadest scalar-tensor family that avoids extra degrees of freedom. It finds that once nonlinearity is included through a quasi-static spherical top-hat model, the modified-gravity scalar can lose its real nature before the overdense region reaches zero radius. In the special branch where graviton decay is forbidden, $\\alpha_H+2\\beta_1=0$, this failure is extremely sensitive: requiring real solutions for the scalar gradient up to the present time for any initial amplitude gives $\\beta_{10}<10^{-7}$ at the fiducial $\\alpha_{B0}=-2\\times10^{-3}$, a bound five orders of magnitude tighter than the earlier Hulse-Taylor limit. In the generic branch $\\alpha_H+2\\beta_1\\neq0$, positive $\\alpha_{H0}$ and $\\beta_{10}$ delay collapse and, through the critical density contrast, suppress the number of massive halos relative to $\\Lambda$CDM. The interest is that a purely nonlinear, time-dependent cosmological process becomes a sharp new test of beyond-Horndeski gravity.","feed_headline":"Collapse caps a modified-gravity parameter below one ten-millionth","feed_subtitle":"A tiny beyond-Horndeski coefficient can make the scalar go imaginary and halt structure formation.","key_machinery":"The load-bearing object is the cubic algebraic equation (33), $F(x,\\delta,\\dot{\\delta})=C_3x^3+C_2x^2+C_1x+C_0=0$, for the dimensionless scalar gradient $x=\\pi'/(H_0a^2r)$, obtained from the quasi-static effective action after eliminating the metric potentials $y=\\Phi'/(H_0^2a^2r)$ and $z=\\Psi'/(H_0^2a^2r)$. The coefficient $C_3=(\\alpha_H+2\\beta_1)(1-\\alpha_H-3\\beta_1)$ controls the structure: on the special line $\\alpha_H+2\\beta_1=0$ the equation degenerates to a quadratic, whose discriminant $D=D_1+D_2\\delta+D_3\\delta^2-D_4\\dot{\\delta}$ decides whether $x$ stays real. The negative term $-D_4\\dot{\\delta}$, proportional to $\\beta_1$, is the mechanism that can drive the discriminant negative during the evolution of an overdensity, and this loss of real roots is interpreted as the prevention of collapse.","core_discovery":"The paper claims that spherical collapse in DHOST theories fails when the algebraic equation governing the scalar-field gradient $x$ stops admitting real roots. In the case $\\alpha_H+2\\beta_1=0$ the equation is quadratic, and its discriminant $D$ contains a term $-D_4\\dot{\\delta}$ specific to time-dependent DHOST systems; for positive $\\beta_{10}$ with the fiducial negative $\\alpha_{B0}$, this term drives $D$ to zero, the two roots merge, and $x$ becomes imaginary, so the model no longer describes the collapsing region. Requiring that this never happens before the present day for any initial amplitude yields $\\beta_{10}<10^{-7}$, a limit five orders of magnitude stronger than the earlier Hulse-Taylor pulsar bound. When $\\alpha_H+2\\beta_1\\neq0$ the equation is cubic, the failure is avoided away from the special line, and the main effect is a delay of collapse for larger positive $\\alpha_{H0}$ and $\\beta_{10}$; in the background model with $\\gamma_0=1$, this delay raises the critical density contrast and suppresses the halo mass function relative to $\\Lambda$CDM, by roughly 10% at $10^{14}$ solar masses for $\\beta_{10}=2\\times10^{-2}$.","pith_inferences":["Beyond the paper's explicit conclusions, the same discriminant mechanism could be used to map exclusion regions for other EFT coefficient combinations, since the sign and size of $D_4$ depend on the time dependence of the $\\alpha$ functions and on $H(\\alpha_B+\\beta_1)-\\dot{\\beta}_1$.","A full time-dependent integration of the top-hat equations, without the quasi-static shortcut, would show whether the scalar gradient passes smoothly through the point where the quasi-static discriminant vanishes; if it does, the $\\beta_{10}$ bound would need to be relaxed.","Because the halo-abundance prediction uses the Press-Schechter formula and a linear power spectrum, an N-body simulation resolving the Vainshtein mechanism is the natural independent check of whether the suppression of massive halos is real."],"forward_implications":["On the branch $\\alpha_H+2\\beta_1=0$, a positive $\\beta_{10}$ above $10^{-7}$ at $\\alpha_{B0}=-2\\times10^{-3}$ prevents the quasi-static spherical collapse from running to completion, so those parameters are excluded if collapse is to remain possible.","The resulting bound on $\\beta_{10}$ is five orders of magnitude stronger than the earlier Hulse-Taylor pulsar bound.","In the generic branch, larger positive $\\alpha_{H0}$ and $\\beta_{10}$ delay the collapse time for fixed initial amplitude, with most of the delay in the $\\gamma_0\\neq1$ background coming from weaker early-time linear growth.","The critical density contrast $\\delta_c$ is raised relative to $\\Lambda$CDM for positive $\\alpha_{H0}$ and $\\beta_{10}$, and the halo mass function is consequently suppressed, with a reduction of about 10% near $10^{14}$ solar masses for $\\beta_{10}=2\\times10^{-2}$.","These statements are made after imposing the stability of linear perturbations, so the constraints apply within the parameter region that is already stable."],"supporting_citations":[{"why":"supplies the quasi-static effective action and the coefficient map from which the field equations and the cubic for x are derived.","marker":"[17]"},{"why":"provides the EFT-based spherical collapse and halo abundance setup that this paper extends to DHOST theories.","marker":"[34]"},{"why":"motivates the special branch alpha_H + 2 beta_1 = 0 through the condition that gravitons do not decay into dark energy.","marker":"[40]"},{"why":"underpins the Vainshtein-regime behaviour and the choice of the physical root, including the sign restriction on beta_1.","marker":"[41]"},{"why":"gives the earlier Hulse-Taylor bound against which the new beta_10 constraint is compared.","marker":"[18]"},{"why":"supplies the sound-speed stability conditions used to delimit the acceptable parameter region.","marker":"[44]"},{"why":"provides the halo mass function formula that converts the critical density contrast into halo abundances.","marker":"[42]"},{"why":"furnishes the linear power spectrum through the DHOST-capable Boltzmann solver used in the mass-function calculation.","marker":"[43]"}],"fun_headline_variants":["DHOST collapse fails when scalar goes imaginary","Imaginary scalar halts structure formation in DHOST models","DHOST collapse bound five orders tighter than pulsars","Imaginary scalar field caps DHOST parameter at 10^-7","Beyond-Horndeski collapse suppressed by imaginary scalar"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that the quasi-static shortcut, which removes explicit time derivatives of the scalar gradient, stays valid all the way up to the moment of collapse, so the disappearance of real solutions is a genuine obstruction and not a symptom that the shortcut has broken.","fun_headline_variants_meta":{"raw":{"variants":["DHOST collapse fails when scalar goes imaginary","Imaginary scalar halts structure formation in DHOST models","DHOST collapse bound five orders tighter than pulsars","Imaginary scalar field caps DHOST parameter at 10^-7","Beyond-Horndeski collapse suppressed by imaginary scalar"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00049,"raw_usage":{"total_tokens":2422,"prompt_tokens":969,"completion_tokens":1453,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":585,"completion_tokens_details":{"reasoning_tokens":1372}},"tokens_in":585,"tokens_out":1453,"duration_ms":11327,"temperature":1.0,"reasoning_tokens":1372,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T12:26:20.269416+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a fully time-dependent spherical top-hat collapse calculation in a class-Ia DHOST theory with $\\beta_{10}=10^{-4}$ and $\\alpha_{B0}=-2\\times10^{-3}$ from the same initial conditions; if the scalar-field gradient remains real and the physical radius reaches zero, the paper's identification of the vanishing discriminant with prevented collapse is refuted. An N-body simulation forming halos in a parameter region where the spherical model predicts collapse failure would be the same test at larger scales.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the sound-speed stability conditions used to delimit the acceptable parameter region."}],"review_version":1}