Fully viable DHOST bounce with extra scalar

Jin U Kang; Ok Song An; Ui Ri Mun; Un Gyong Ri; Yong Jin Kim

REVIEW 3 major objections 4 minor 1 cited by

Fully viable DHOST bounce with extra scalar

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper constructs the first two-field DHOST bouncing cosmology that claims to be free of ghost, gradient, and BKL instabilities and superluminality, while matching the observed nearly scale-invariant scalar spectrum and a negligible…

desk verdict A serious two-field DHOST bounce construction whose 'fully viable' label is one missing computation away from being earned. read the letter →

arxiv 2501.09985 v2 pith:P6KXVFQI submitted 2025-01-17 hep-th astro-ph.COgr-qchep-ph

Ok Song An , Jin U Kang , Yong Jin Kim , Ui Ri Mun , Un Gyong Ri This is my paper · ORCID

classification hep-thastro-ph.COgr-qchep-ph MSC 83F0583D05 PACS 98.80.-k98.80.Cq04.50.Kd

keywords DHOSTtheoriesbouncingcosmologyekpyroticcontractionentropicperturbationssuperluminalitynon-Gaussianitiesprimordialmodifiedgravity

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 aims to show that a non-singular bouncing universe can be a fully viable alternative to inflation, despite the no-go results that block healthy bounces in simpler scalar-tensor theories. It builds a two-field model whose adiabatic field is an exceptional subclass of Degenerate Higher-Order Scalar-Tensor (DHOST) theories and whose extra scalar supplies nearly scale-invariant perturbations through the entropic mechanism. The claim is that this model is free of BKL, ghost, and gradient instabilities, has no superluminal propagation, reduces to General Relativity plus a canonical scalar in the distant past and future, and produces a scalar spectrum with $n_s \simeq 0.965$, an amplitude matching Planck, and a negligible tensor-to-scalar ratio. If correct, it removes the last standard objections to bouncing cosmology as a competitor to inflation.

What carries the argument

The load-bearing object is the exceptional subclass of DHOST theories (Degenerate Higher-Order Scalar-Tensor gravity, a higher-derivative scalar-tensor class that avoids Ostrogradski ghosts) with a luminal extra scalar: the condition (2.17), $A_3 = \frac{2(XA_1 - 2F_2)(A_1 - 2F_{2X})}{X(3XA_1 - 4F_2)}$, makes the coupling function $f$ equal to $g$, which is necessary and, together with (2.20), sufficient for the sound speed of the coupled two-scalar system to stay at or below unity. The inverse-method reconstruction then turns the two-inequality set (2.22) and (2.23), namely $G_S \geq F_S > Y P_Y g^2$ together with $G_T \geq F_T > 0$, into a concrete Lagrangian by choosing the scale factor (3.5), the functions $g_1$ and $a_1$ in (3.19)-(3.20), $\Sigma$ in (3.27), $P_1$ in (4.14), $\chi$ in (4.19), and $P_{04}$ in (4.21). The conversion from entropic to curvature perturbations is driven by the coefficient $\Psi_1$ in (4.15), whose size is controlled by the acceleration of $\chi$ rather than its velocity, which is what keeps the trajectory inside the stable region.

What would settle it

Solve the coupled perturbation equations (4.17) past the conversion end for a range of wavenumbers and compute the final $\zeta$ power spectrum and bispectrum: if the resulting spectral index shifts from $n_s \simeq 0.965$ by more than the Planck error bars, or if $f_{\rm NL}$ exceeds the current observational bound, the 'fully viable' claim is refuted. A second, independent test: since the model gives a deep-blue tensor spectrum with negligible $r$ at the pivot scale, a primordial tensor signal near the inflation-scale upper bound would rule it out.

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Extended reading notes

Core claim

The central discovery is an explicit two-field construction, presented as the first of its kind, in the exceptional subclass of DHOST Ia theories with an extra scalar field $\chi$. In the DHOST sector the Lagrangian functions are fixed by the inverse method: a chosen scale factor (3.5) with ekpyrotic contraction ($\epsilon > 3$) and kinetic-dominated expansion, and the ansatz (3.7), with $g_0 = -g_1$ and $a_0 = -a_1$ so that the tensor sector has $G_T = F_T = 1$ and luminal gravitational-wave speed at all times. The extra scalar has luminal sound speed and $P(\chi,Y,\varphi) = P_1(\varphi)Y - P_{02}(\varphi)\chi^2 - P_{04}(\varphi)\chi^4$, with $P_1$ chosen as (4.14) so that $\chi$ develops a nearly scale-invariant spectrum during contraction, and a sech-shaped background trajectory (4.19) that bends sharply after the bounce and converts about 98% of $\delta\chi$ into curvature perturbations. The model functions (5.1)-(5.3) with parameters (4.26) satisfy the stability and non-superluminality conditions (2.22) and (2.23) throughout the whole background trajectory, and $\mu \sim O(10^{-2}) M_{\rm Pl}$ sets the observed scalar amplitude.

Load-bearing premise

The load-bearing premise is that the roughly 98% conversion of $\delta\chi$ into $\zeta$ preserves the nearly scale-invariant spectrum and produces only small non-Gaussianities, a step the paper does not compute directly: it derives $n_s$ for $\delta\chi$ before conversion and checks conversion efficiency, then says it expects small non-Gaussianities.

Editorial extensions

If this is right

Bounces no longer need to invoke strong gravity in the past: this model has GR-like asymptotics, so the strong-coupling worry and the associated large non-Gaussianities of earlier scenarios do not arise.
The predicted scalar spectral index $n_s \simeq 0.965$ and scalar amplitude $A_s = 2.105 \times 10^{-9}$ at the pivot scale match Planck data, while the tensor-to-scalar ratio is negligible.
Gravitational-wave speed equals the speed of light throughout the whole evolution, since $G_T = F_T = 1$.
Any future model with a luminal extra scalar can avoid superluminality by staying in the exceptional subclass and satisfying $F_S > Y P_Y g^2$; this is a checkable sufficient condition.
The nearly scale-invariant curvature perturbation is generated by the entropic field, not by the ekpyrotic adiabatic mode, which would otherwise be blue-tilted.

Reading between the lines

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

Editorial inference: the missing computation that would close the 'fully viable' claim is the final $\zeta$ power spectrum and bispectrum; the paper only checks the entropic spectrum and the conversion efficiency, and the size of $f_{\rm NL}$ from the nonlinear conversion remains open.
Editorial inference: because the model is fixed by parameterized template functions rather than derived from a symmetry, the same inequalities might be satisfied by other shapes of $\chi(t)$; a scan over $\lambda$, $q$, and $\chi_0$ with the same $\Psi_1$ mechanism could test the robustness of the roughly 98% efficiency.
Editorial inference: the deep-blue tensor spectrum (spectral index $4 - 2\nu_1$) is a generic ekpyrotic signature; a future detection of primordial B-modes at the level expected in slow-roll inflation would distinguish this class from inflation and falsify this particular model.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

I read the paper carefully. The central construction is real: they build an explicit two-field DHOST model with GR asymptotics, no ghost/gradient instability, no superluminality, and a roughly 98% efficient entropy-to-curvature conversion. The new sufficient condition (2.21) and the explicit Lagrangian functions (5.1)–(5.3) are not in the cited literature. The stability algebra in Secs. 2–3 is careful and internally consistent; the inverse method is standard for constructions like this, and choosing b and μ to match Planck does not make the model circular, it makes the perturbation amplitudes consistent by design.

That said, the headline claim “fully viable” outstrips the computed support. Three gaps, in decreasing order of severity. First, non-Gaussianity is explicitly deferred. Section 5 says only “we expect” small f_NL and promises future work. Planck bounds on f_NL are part of observational compatibility, and the cited references (56, 61, 62) show that even smooth, efficient conversion can generate large non-Gaussianities depending on the trajectory. The authors do not show their parameters lie in the safe region. Second, conversion is checked at the amplitude level for what looks like a single mode: Eq. (4.20) and Fig. 10 give |ζ/δχ| at one k, not the full ζ_k power spectrum over the observed range. A k-dependent transfer function during the conversion could alter n_s or add running, so the final curvature spectrum is not established. Third, there is a parameter mismatch: the stability plots in Sec. 3 (Figs. 4, 5) use ε = 5, while the final parameters in (4.26) use ε = 10 plus the new conversion parameters t_c, q, λ. Fig. 11 checks one key constraint for the full set, but the complete stability scan is not shown for the exact parameter set that produces the observables.

None of this destroys the construction. A stable, non-superluminal two-field DHOST bounce likely exists along these lines; what is missing is the demonstration of observational viability, which is exactly what the title promises. The fix is clear: compute the final curvature power spectrum and at least f_NL for the full parameter set, rerun the stability plots for those same parameters, and ideally provide code or data. This is refereeable, and a good referee would ask for exactly that before publication.

I would send it to peer review rather than desk-reject, with the explicit request for the missing non-Gaussianity computation. Once those pieces are filled, this will likely become a benchmark model in the bouncing-cosmology literature.

Referee Report

3 major / 4 minor

Summary. The paper constructs a two-field DHOST Ia bouncing cosmology by an inverse method: it first chooses a background with ekpyrotic contraction and kinetic-dominated expansion, then reconstructs the Lagrangian functions so that tensor modes are luminal and the scalar sector satisfies sufficient stability and non-superluminality conditions. An extra scalar field is added with luminal sound speed, and its entropic perturbations acquire a nearly scale-invariant spectrum during ekpyrotic contraction; the paper then numerically demonstrates an approximately 98% conversion of entropy perturbations into curvature perturbations and claims compatibility with Planck constraints on the scalar spectral index and amplitude, with a negligible tensor-to-scalar ratio. The abstract and conclusion assert that this is the first fully viable two-field DHOST bounce, free of BKL, ghost, gradient, and superluminality problems and compatible with observations.

Significance. If fully established, the model would be a significant explicit realization of a bouncing cosmology in the exceptional DHOST Ia subclass, circumventing known superluminality and no-go issues while providing a concrete Lagrangian and background. The paper's strengths are its detailed algebraic derivation of the sufficient stability conditions in Eq. (2.21), the explicit reconstruction in Eqs. (5.1)-(5.3), and the numerical verification of the local no-pathology conditions for the parameter set (4.26). However, the headline claim of observational viability currently rests on uncomputed quantities: the final curvature power spectrum after conversion and the non-Gaussianity parameter. The manuscript itself states in Section 5 that small non-Gaussianities are only 'expected,' not demonstrated. The significance is therefore conditional on completing or substantially revising that part of the claim.

major comments (3)

[§4.3–§4.4, §5] The claim that the model predicts a nearly scale-invariant curvature power spectrum compatible with Planck is not supported by the computations shown. Section 4.3 characterizes the conversion by the amplitude ratio |ζ(t_end)/δχ(t_begin)| in Eq. (4.20) and Figure 10, but Section 4.4 then assumes in Eqs. (4.27) and (4.32) that the δχ power spectrum is simply inherited by ζ. No computation of the final ζ_k power spectrum over the observed k-range, no k-dependent transfer function, and no bispectrum computation are given. Section 5 states only that non-Gaussianities are 'expected' to be small, citing Refs. [56,61,62], which in fact show that high efficiency and smoothness do not by themselves guarantee small f_NL. Since Planck bounds on non-Gaussianity are part of observational compatibility, the 'fully viable ... compatible with observations' claim outstrips the computed support. Please compute the converted curvature spectrum and f_NL (or a controlled estimate), or explicitly revise the claim to a conditional one.
[§4.3, Eq. (4.25)] The conversion demonstration is made for what appears to be a single Fourier mode with a single set of initial conditions: Eq. (4.20) uses a subscript k, but no k value is specified, and Eq. (4.25) fixes ζ and δχ amplitudes to 10^-5 without justifying that the amplitude ratio is independent of the mode and of the relative initial phase. A k-dependent conversion efficiency would change the final spectral index or introduce running/oscillatory features; the paper should either show the k-independence explicitly or at least state the representative mode and explain why the result is representative for the observable window.
[§3.2, §4.3] The no-ghost, no-gradient, and non-superluminality conditions (2.21) and (2.23) are verified numerically for the single parameter set (4.26), and the figures do not display the entire time range used for the checks. In particular, Figure 11 shows only 20 ≤ t ≤ 200, while the text claims the constraints hold 'throughout the whole time evolution.' The asymptotic behavior of the model functions is used to argue that the conditions hold in the far past and future, but the precise interval and resolution of the numerical verification should be stated. This is a curable gap, but it matters because the paper's central claim is that the model is 'completely free' of pathologies.

minor comments (4)

[§4.1, Eq. (4.21)] The text states that P04 > 0, but the function defined in Eq. (4.21) vanishes exactly at t = t_c because sech(0) = 1. Please clarify the behavior at that instant and explain why the scalar potential remains bounded from below there, referring to the numerical check in Figure 9.
[§4.4, Eq. (4.32)] The amplitude matching in Eq. (4.32) is only used to quote μ ~ O(10^-2) M_pl. The sensitivity of A_s to the parameters τ, ε, b, and μ should be stated, and the precise central value of μ used for the matching should be given rather than only an order of magnitude.
[§4.3, Fig. 10] Figure 10 should state the value of the wavenumber k (or k̄) used in the numerical integration, as well as the initial time t_i and the numerical method, so that the claim of 98% conversion is reproducible.
[Abstract, §5] The sentence 'We also expect our models to have sufficiently small non-Gaussianities' is an explicit admission that a central part of observational viability is not computed. This should be moved to a clearly qualified statement in the abstract and conclusion, or removed until the computation is performed.

Circularity Check

2 steps flagged · score 6.0 of 10

The 'observationally compatible' part of the claim is calibrated: b and mu are chosen so that Eqs. (4.11) and (4.32) reproduce Planck's n_s and A_s, while the final zeta power spectrum and f_NL are not computed.

fitted input called prediction [Section 4.2, Eq. (4.11); parameter choice in (4.26b)]
"ns = 1 − 2bǫ/(ǫ − 1). Since ǫ ≫ 1, for the value of b around 0.0158 the spectral tilt becomes ns ≃ 0.965, which is in good agreement with the Planck data [50]."

The parameter b is a free input in the constructed Lagrangian function P1(φ) (Eq. 4.14). Equation (4.11) then determines n_s in terms of b, and Eq. (4.26b) sets b = 0.0158 precisely so that n_s ≈ 0.965. The abstract and Section 5 nevertheless present 'nearly scale-invariant curvature perturbations ... compatible with observations' as an output or prediction of the model. The agreement with Planck is therefore imposed by the choice of b, not derived from the model; the spectral tilt of the entropic perturbations is an input target, not an independent consequence.
fitted input called prediction [Section 4.4, Eq. (4.32); characteristic scale mu fixed in (3.1)-(3.2)]
"It follows from (4.32) that for µ ∼ O(10−2)Mpl the amplitude of the scalar power spectrum is in good agreement with the Planck data (4.28)."

The scale mu is a free parameter introduced through the clock choice phi = mu^2 t and the rescaling (3.2). Equation (4.32) shows that the scalar amplitude scales as mu^4 (up to the pivot factor), and the text chooses mu ~ 10^-2 M_pl so that A_s matches the measured value (2.105 ± 0.030) × 10^-9. The amplitude is thus calibrated to the observation rather than predicted. When Section 5 lists 'nearly scale-invariant curvature perturbations ... compatible with observations' as a prediction of the model, that compatibility is a restatement of the parameter choice, especially since the final zeta power spectrum after entropy-to-curvature conversion is not computed.

full rationale

The paper is an explicit inverse-method construction: it selects the background H(t), adopts ansatze for the Lagrangian functions, and then verifies the stability and non-superluminality inequalities (2.21) and (2.23) numerically. Those checks are genuine constraints and give the construction substantial independent content: ghost/gradient stability, subluminality, GR asymptotics, and a blue tensor spectrum are not put in by hand but are checked against the model's own equations. The circularity is confined to the observational-compatibility part of the headline. Section 4.2 chooses b = 0.0158 so that Eq. (4.11) returns n_s ≈ 0.965, and Section 4.4 chooses mu ~ 10^-2 M_pl so that Eq. (4.32) returns the Planck amplitude; Section 5 then calls the result a prediction of nearly scale-invariant curvature perturbations compatible with observations. The tilt and amplitude are imposed inputs, so that part of the 'fully viable' claim reduces to construction targets rather than derived outputs. In addition, the remaining observational pillar, small non-Gaussianities, is explicitly not computed: Section 5 says only 'we expect' and defers confirmation to future work. That is a missing calculation rather than a circular step, but it reinforces that the headline claim outstrips the demonstrated support. No load-bearing self-citations occur; the cited entropic-mechanism and DHOST results are external to the authors.

Assumptions & free parameters 11 free parameters · 8 assumptions · 0 invented entities

The construction introduces a large number of hand-chosen functions and parameters (tau, epsilon, w, u, b, mu, tc, q, lambda, chi0, and the P04 prefactor). It relies on external results for DHOST degeneracy, the exceptional-subclass non-superluminality condition, ekpyrotic smoothing, and the entropic conversion mechanism. The ansatz forms (3.7) and (4.19) are ad hoc. No new particles, forces, or dimensions are introduced.

free parameters (11)

b = 0.0158
Sets the ekpyrotic phase shape of P1 in Eq (4.14); chosen so that ns=1-2*b*epsilon/(epsilon-1) equals 0.965, matching Planck.
mu = ~10^-2 Mpl
Characteristic energy scale; set in Sec 4.4 so that the scalar amplitude in Eq (4.32) matches As=(2.105 +/- 0.030)*10^-9.
tau = 10
Controls the duration of the bounce in Eq (3.5); chosen by hand, following the value used in reference [27].
epsilon = 10 final (5 in earlier stability plots)
Ekpyrotic equation-of-state parameter in the scale factor (3.5); must exceed 3 for BKL suppression. The value changes between Sec 3.2 and the final parameter set (4.26).
w = 2
Coefficient in g1 in Eq (3.19), introduced to avoid simultaneous zero of H, Theta, and Xi; chosen to avoid fine-tuning.
u = 0.1
Shift parameter in g1 and a1; chosen to avoid fine-tuning of the zero crossings.
tc = 100 = 10*tau
Center of the conversion process in chi(t), Eq (4.19); chosen so conversion occurs well after the bounce.
q = 1/12
Shape parameter in chi(t), Eq (4.19), that helps make P02 positive in the asymptotic limit; chosen by educated guess.
lambda = 0.35
Amplitude of the chi(t) bump; controls conversion efficiency; chosen by tentative numerical analysis.
chi0 = 0.1
Bias in chi(t) to prevent zero crossing and singular P02; chosen by hand.
Prefactor 10 in P04 = 10
Arbitrary overall coefficient of P04 in Eq (4.21), chosen for potential boundedness and small dynamical role.

assumptions (8)

domain assumption DHOST Ia degeneracy conditions (2.3) eliminate the Ostrogradski ghost in the two-field action (2.1).
Invoked in Sec 2.1 after Eq (2.2); a result from cited literature [15,17,37,66], not proven here.
domain assumption The exceptional subclass condition (2.17) is necessary for a luminal extra scalar to avoid superluminality.
Adopted from [36,37] in Sec 2.3, Eq (2.17).
domain assumption Linearized perturbations on a spatially flat FLRW background capture ghost, gradient, and superluminality pathologies.
Used throughout Sec 2.2; standard in the literature [8].
domain assumption Ekpyrotic contraction with epsilon>3 robustly suppresses BKL instability and anisotropy growth.
Invoked in Sec 3.1 with references [46-49]; not derived in this paper.
domain assumption Efficient smooth conversion of entropic perturbations into curvature perturbations preserves a nearly scale-invariant spectrum and small non-Gaussianity.
Relied on in Sec 4.3 and Sec 5; supported only by references [56,61,62], not by a calculation in this paper.
ad hoc to paper The ansatz (3.7) for the Lagrangian functions is sufficiently general to accommodate a healthy model.
The quadratic-in-X ansatz for F and A1, linear-in-X F2, and K=0 is chosen for tractability in Sec 3.1; no completeness argument is given.
standard math Bunch-Davies vacuum initial conditions apply in the asymptotic past.
Imposed in Sec 4.2, Eq (4.8), for the mode functions of delta-chi; standard in inflationary and ekpyrotic perturbation theory.
domain assumption A single zero crossing of Theta is harmless for the solutions of the perturbation equations.
Assumed in Sec 3.2 after Eq (3.17); cites [27,69] for regularity of the solutions.

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Cite this review

Pith. "Pith review of Fully viable DHOST bounce with extra scalar." pith.science (2026). https://pith.science/paper/P6KXVFQI

@misc{pith2026250109985,
  author       = {Pith},
  title        = {Pith review of: Fully viable DHOST bounce with extra scalar},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P6KXVFQI}},
  note         = {Machine review of arXiv:2501.09985}
}

read the original abstract

In this paper we construct a class of Degenerate Higher-Order Scalar-Tensor (DHOST) theories with an extra scalar field, which admits viable solutions of bouncing universe satisfying the following requirements: (i) absence of Belinski-Khalatnikov-Lifshitz (BKL) instability, ghost and gradient instability, (ii) absence of superluminality, (iii) generation of nearly scale-invariant curvature perturbations and very small tensor-to-scalar ratio, and (iv) conventional asymptotics in the distant past and future, where gravity sector is described by General Relativity and the DHOST scalar has a canonical form of Lagrangian. We also expect our models to have sufficiently small non-Gaussianities of primordial curvature perturbations to be compatible with observations. As such, this work exemplifies for the first time the fully viable two-field DHOST bouncing cosmology, which is free of instability and superluminality problems as well as compatible with observations.

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Reference graph

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Pith tools

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