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REVIEW 3 major objections 4 minor 49 references

Spherical collapse in DHOST theories and EFT of dark energy

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2504.12656 v1 pith:6J7N7B4Q submitted 2025-04-17 astro-ph.CO gr-qc

Toshiki Takadera , Takashi Hiramatsu , Tsutomu Kobayashi This is my paper · ORCID
classification astro-ph.COgr-qc
keywords DHOSTtheoriessphericalcollapseEFTofdarkenergyhalomassfunctionbeyond-HorndeskigravityVainshteinmechanismlarge-scalestructuremodified
topics Dark Energy
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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 $\beta_{10}=2\times10^{-2}$.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [Sec. IV.A, Eqs. (33), (43), (44)] 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.
  2. [Sec. IV.A, Fig. 2 and Eq. (42)] 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.
  3. [Sec. IV.B and Sec. VI] 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.
minor comments (4)
  1. [Section IV heading] The heading reads 'EVOLUITON OF SPHERICAL OVERDENSITIES'; it should be 'EVOLUTION'.
  2. [Fig. 2 caption] The caption says 'real solutions for x cease to exit at some moment'; 'exit' should be 'exist'.
  3. [Sec. IV.A around Eq. (43)] 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.
  4. [Sec. V] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central beta10 bound follows from solving the model's own dynamical equations, not from fitted inputs or load-bearing self-citation.

full rationale

The paper's central results are derived rather than fitted: the spherical-collapse evolution is obtained by solving the coupled differential equations (28) and (35) together with the algebraic constraint (33), and the bound beta10 < 10^-7 follows from requiring the discriminant D of Eq. (43) to remain nonnegative, a condition derived from the model's coefficients and evolved variables. The critical density contrast and collapse time are computed from the model and then fed into the Press-Schechter formula, so no prediction reduces by construction to an input parameter. Self-citations appear in supporting roles only: the Boltzmann solver [11,12,43] is used for the illustrative linear power spectrum in the halo-mass-function section, the review [8] is background, and the screening discussion [18] motivates a case split that is in any case fixed algebraically by Eq. (34). The stability and Vainshtein restrictions are taken from external references [40,41,44]. The quasi-static approximation's possible breakdown near D=0 is a physical robustness concern, explicitly acknowledged in Sec. VI where the authors call for N-body confirmation; it is not a circularity because the paper does not assume the conclusion that collapse is prevented when deriving the condition D<0.

Assumptions & free parameters 5 free parameters · 9 assumptions · 0 invented entities

The central claim rests on the quasi-static approximation, the assumed ΛCDM background, the assumed time evolution of the EFT parameters (Eq. 16), the class-Ia degeneracy conditions, and the stability cuts of Appendix C. No genuinely new entities are introduced; the EFT parameters αB, αH, β1 are the model inputs whose effects are being studied. The observational inputs (Ωm0 = 0.3153, H0) are taken from prior measurements and are not fitted here.

free parameters (5)
  • αB0 (fiducial kinetic braiding) = -2e-3
    The main constraint β10 < 10^-7 is derived for this fiducial present-day value of αB, not as a general bound. The paper fixes it in Secs. IV and V.
  • αK0 (kineticity) = 1
    Used in Appendix C to define the stability region (Fig. 7) and the sound-speed conditions (Eqs. C14-C16). Chosen by hand as a fiducial value.
  • Ai (initial amplitude) = 2.30
    Initial density amplitude used for the numerical examples in Figs. 1-4; the collapse time depends on it. The quoted β1 constraint is stated for 'any initial amplitudes' so Ai is not fitted.
  • γ0 (cosmological gravitational constant parameter) = 1 - αH0 - 3β10 or 1
    Two background models are defined by these choices for the relation between Gcos and GN, given in Eqs. (17)-(21). This is a modeling choice, not fitted to data.
  • (αH0, β10) for halo mass function = (2e-2, 0) and (0, 2e-2)
    Parameter pairs selected in Sec. V (Fig. 5 and 6) to demonstrate the halo mass function suppression; they lie in the stable region but are not compared with CMB constraints.
assumptions (9)
  • domain assumption Quasi-static (subhorizon) approximation is valid for the nonlinear spherical collapse problem
    The effective action (2) used throughout is derived under this approximation in Sec. II; the loss of real solutions for x is interpreted as a physical effect within this approximation.
  • domain assumption Background expansion is exactly ΛCDM after matter domination
    Eq. (11) sets H^2/H0^2 = Ωm0/a^3 + 1 - Ωm0 in Sec. III.A; the EFT coefficients are then specified on this background.
  • domain assumption Time dependence of the EFT functions αi(t) = αi0 (1 - Ωm(t))/(1 - Ωm0) = αi0 (H0/H)^2
    Eq. (16) in Sec. III.A; the entire analysis depends on this assumed time evolution, which is common in the literature but not derived.
  • standard math Class-Ia degeneracy conditions, especially β3 = -2β1[2(1+αH)+β1(1+αT)] (Eq. 7)
    These conditions remove the extra degrees of freedom in DHOST theory and are used to cancel time derivatives of x in deriving the algebraic equation (33). They are taken from the prior DHOST literature.
  • domain assumption GW170817 constraint: αT = αV + αH = 0 (Eq. 9)
    Imposed in Sec. II to enforce cGW = 1; this restricts the parameter space considered.
  • domain assumption Constant effective Planck mass, αM = 0 (Eq. 8)
    Assumed for simplicity in Sec. II, reducing the independent EFT parameters.
  • domain assumption The physical root of Eq. (33) is the one connected to the linear solution and obeying x^2 ∝ δ for δ >> 1 (Vainshtein regime); this requires β1 > 0 in the αH+2β1=0 case
    Used in Sec. IV and Appendix C to select the root and to exclude negative β1 (Eqs. C1-C8).
  • domain assumption Stability conditions (ghost-free, positive sound speed squared) hold throughout the evolution
    Appendix C imposes ĉs^2 > 0 and A_ζ > 0; the main text states these are checked at all times, but the analytic checks are at the matter-dominated epoch.
  • domain assumption Press-Schechter formalism remains valid in DHOST theories
    Used in Sec. V to compute the halo mass function; the authors note in Sec. VI that N-body simulations are needed to verify this assumption.

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Pith. "Pith review of Spherical collapse in DHOST theories and EFT of dark energy." pith.science (2026). https://pith.science/paper/6J7N7B4Q

@misc{pith2026250412656,
  author       = {Pith},
  title        = {Pith review of: Spherical collapse in DHOST theories and EFT of dark energy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6J7N7B4Q}},
  note         = {Machine review of arXiv:2504.12656}
}
abstract

We study the nonlinear evolution of matter overdensities using the spherical collapse model in degenerate higher-order scalar-tensor (DHOST) theories beyond Horndeski, employing the effective field theory (EFT) of dark energy approach. We investigate the impact of the EFT parameters characterising DHOST theories on the formation of large-scale structure. We identify the parameter space in which the collapse of the spherical overdensity is prevented by the scalar field turning imaginary at some moment, which allows us to place constraints on the model parameters. We show how the collapse time and the critical density contrast depend on the EFT parameters. To assess the observational implications, we compute the halo mass function using the Press-Schechter formalism. We find that the number density of halos is suppressed compared to the $\Lambda$CDM model due to ``beyond Horndeski'' effects, upon imposing the stability of linear perturbations.

Figures

Figures reproduced from arXiv: 2504.12656 by the authors.

Figure 1
Figure 1. FIG. 1. Evolution of [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Evolution of [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Evolution of [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Similar to the background model with γ0 ̸= 1 discussed above, it can be seen that the collapse time is delayed for larger values of αH0 and β10. However, the magnitude of the corrections is smaller as compared to the case with γ0 ̸= 1 because modification to GR manifes…
Figure 5
Figure 5. Figure 5: FIG. 5. Critical density contrast as a function of the redshift for the two background models. The parameters are given by [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Mass function and its deviation from ΛCDM model. The left panel shows the mass function Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Parameter region satisfying ˆc [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

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Reviewed August 16, 2026 · model on record in the stance chip above.