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REVIEW 4 major objections 6 minor 261 references

Non-Standard Thermal History and Formation of Primordial Black Holes in Einstein-Gauss-Bonnet Gravity

T0 review · 4 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A single Einstein-Gauss-Bonnet inflation model can form primordial black holes from $10^{-14}$ to $10$ solar masses and source gravitational waves across current and future detector bands.

desk verdict A credible EGB-inflation PBH mechanism that overstates its predictive power and ignores a known loop-level threat; the numerics deserve scrutiny but the paper needs revision before its claims can be accepted. read the letter →

arxiv 2501.01867 v1 pith:UAZCA75C submitted 2025-01-03 gr-qc

Yogesh , Abolhassan Mohammadi This is my paper · ORCID
classification gr-qc MSC 83F0583C3583D05
keywords primordialblackholesEinstein-Gauss-Bonnetgravityultra-slow-rollinflationscalar-inducedgravitationalwavesnon-standardthermalhistoryMutatedHilltoppotentialcurvaturepowerspectrumenhancement
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 argues that in Einstein-Gauss-Bonnet gravity, the Mutated Hilltop inflation potential paired with a hyperbolic-tangent coupling between the scalar field and the Gauss-Bonnet term can generate an ultra-slow-roll phase without adding extra features to the potential. That phase amplifies the small-scale curvature power spectrum to about $10^{-2}$, enough to form primordial black holes with masses from roughly $10^{-14}$ to $10$ solar masses. The paper then shows how a non-standard post-inflationary epoch, with stiff equation of state $1/3<\omega\le1$, shifts both the masses and the abundances of those black holes, in some cases producing $f_{\rm PBH}>1$ and thereby excluding those parameter choices. It also derives the scalar-induced gravitational-wave spectrum, with peaks near $10^{-8}$ in $\Omega_{\rm GW}h^2$, and shows the signals fall in the reach of pulsar timing arrays and space-based interferometers. If correct, a single modified-gravity inflation model connects CMB observations, dark-matter candidates, stellar-mass black-hole merger events, and future gravitational-wave detectors.

What carries the argument

The load-bearing ingredient is the Gauss-Bonnet coupling function $\xi(\phi)=\xi_0\tanh[\xi_1(\phi-\phi_c)]$: near $\phi=\phi_c$ the combination $V_{,\phi}+4\xi_{,\phi}V^2/(3M_p^4)$ vanishes, creating a fixed point where $\dot\phi$, $\ddot\phi$, and $\dot H$ nearly vanish. The field stalls there for several e-folds, $\epsilon_1=-\dot H/H^2$ drops to about $10^{-7}$, and the slow-roll approximation breaks down, so the power spectrum must be obtained numerically from the Mukhanov-Sasaki equation $\nu_k''+(k^2-z_s''/z_s)\nu_k=0$ with Bunch-Davies initial conditions; Eq. (27) turns its solution into $P_\zeta(k)=k^3|\zeta_k|^2/(2\pi^2)$. This $P_\zeta(k)$ is the single output that feeds every primordial-black-hole and gravitational-wave prediction in the paper. For the non-standard thermal history the analysis is carried by the mass-scale relation of Eq. (33), the Gaussian density variance of Eq. (36), and the equation-of-state-dependent collapse threshold of Eq. (42); for the gravitational waves it is carried by the second-order source kernel of Eq. (48).

What would settle it

A decisive check is to compute the one-loop correction to the curvature power spectrum for the same parameter sets; if it suppresses the $\mathcal{O}(0.01)$ peak by a factor of ten or more, the black-hole masses, abundances, and gravitational-wave amplitudes built on that peak would no longer follow. The paper cites this concern without evaluating it, so the calculation is the clearest way to settle whether the predicted $\Omega_{\rm GW}h^2\sim10^{-8}$ signals are real. A null detection of that stochastic background by pulsar timing arrays or space-based interferometers would also put the benchmark peak in tension.

Watch

Extended reading notes

Core claim

On its own terms, the central discovery is that the fixed point of the Gauss-Bonnet coupling function $\xi(\phi)=\xi_0\tanh[\xi_1(\phi-\phi_c)]$ creates an ultra-slow-roll phase inside an otherwise standard Mutated Hilltop inflation, stalling the field and driving the first slow-roll parameter $\epsilon_1$ down to about $10^{-7}$. The curvature power spectrum computed from the Mukhanov-Sasaki equation then rises to $\mathcal{O}(0.01)$ at a scale set by $\phi_c$, while remaining about $2.1\times10^{-9}$ at the CMB pivot scale with $n_s=0.973$ and $r=0.004$. Feeding that spectrum into the $\omega$-dependent Gaussian collapse estimate gives primordial black hole masses spanning about $10^{-14}$ to $10$ solar masses; one benchmark set reaches $f_{\rm PBH}\simeq1$ for asteroid-mass black holes, so those black holes could be all of the dark matter. The same spectrum sources second-order gravitational waves whose peak sits near $\Omega_{\rm GW}h^2\sim10^{-8}$ and lands in different frequency bands depending on $\phi_c$. Including a stiff post-inflationary epoch $1/3<\omega\le1$ changes the black-hole mass-abundance relation and can push $f_{\rm PBH}$ above unity, which the paper reads as ruling out those combined parameter choices.

Load-bearing premise

The entire primordial-black-hole and gravitational-wave prediction rests on assuming that the usual linear perturbation calculation with Bunch-Davies initial conditions stays valid through the ultra-slow-roll phase, where the slow-roll approximation is known to break down.

Editorial extensions

If this is right

  • The same Einstein-Gauss-Bonnet setup that matches CMB pivot-scale observations can also produce primordial black holes from about $10^{-14}$ to $10$ solar masses, so a single inflation model can cover both stellar-mass merger events and lighter dark-matter candidates.
  • For the benchmark sets with the earliest peaks, asteroid-mass primordial black holes have $f_{\rm PBH}\simeq1$, meaning they could account for all of the dark matter without adding a bump or dip to the inflation potential.
  • A stiff post-inflationary epoch with $1/3<\omega\le1$ changes both the masses and the abundances of the black holes; several combinations give $f_{\rm PBH}>1$, which the model must count as excluded even though the same parameters are allowed in a radiation-dominated epoch.
  • The scalar-induced gravitational-wave spectrum peaks near $\Omega_{\rm GW}h^2\sim10^{-8}$, and different parameter sets place that peak in the reach of pulsar timing arrays or of space-based interferometers.
  • Increasing $\omega$ in the stiff epoch slightly enhances the gravitational-wave peak, so the non-standard thermal history is testable in the amplitude and shape of the stochastic background.

Reading between the lines

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

  • Because the quoted peak power spectrum is $\mathcal{O}(0.01)$, the perturbations are not weakly coupled, so the Gaussian collapse estimate used for the abundances is only a first approximation; the same peak would generate sizeable non-Gaussianity, which can change $f_{\rm PBH}$ by orders of magnitude and is not included in Eqs. (34)-(42).
  • The $f_{\rm PBH}>1$ results in stiff epochs can be inverted: instead of discarding those epochs, one could lower the power-spectrum peak until $f_{\rm PBH}=1$, turning the black-hole abundance into a constraint on the equation of state and the reheating temperature for each peak scale.
  • Because $\phi_c$ moves the peak scale while $\xi_0$ and $\xi_1$ set its height and width, future null detections by gravitational-wave observatories would directly constrain these three coupling parameters, independently of black-hole abundance bounds.
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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

4 major / 6 minor

Summary. The paper studies primordial black hole formation and secondary gravitational waves in the Mutated Hilltop inflation model coupled to an Einstein-Gauss-Bonnet term. A tanh coupling is chosen so that the field passes near a fixed point and enters an ultra-slow-roll phase, and the authors solve the background equations and the Mukhanov-Sasaki equation numerically. They report a curvature power spectrum peak of order O(0.01) on small scales while keeping the CMB pivot amplitude at the Planck value, then use the Press-Schechter formalism generalized to an equation of state 1/3 < ω ≤ 1 to compute PBH masses and abundances for two values of the potential parameter and four benchmark sets. The same power spectra are fed into the SIGWfast code to produce induced gravitational wave spectra. The headline claims are a PBH mass range from O(10^-14) to O(10) solar masses, compatibility with LIGO-Virgo and OGLE constraints, and induced GW signals within the reach of LISA, DECIGO, SKA, and pulsar timing arrays.

Significance. If the numerical power spectrum survives scrutiny, the paper demonstrates a useful mechanism in EGB gravity: a small-scale enhancement of Pζ without adding ad hoc features to the potential, obtained through a coupling-induced USR phase, with a full numerical treatment of the background and perturbations. Strengths include solving the Mukhanov-Sasaki equation rather than relying on slow-roll formulae, reporting explicit benchmark parameter sets, and using the public SIGWfast code for the GW spectra. The significance is currently limited by three issues: the unphysical f_PBH > 1 entries in the non-standard epoch tables, an unevaluated one-loop correction during USR, and the fact that the advertised mass range is largely set by the input choices of φc.

major comments (4)
  1. [§V, Tables III and IV] The abundance tables contain f_PBH values far above unity: for α = 1, set-1, ω = 0.9 the value is 7575.07, and for α = 2, set-1, ω = 0.9 it is 3402.79. Since f_PBH is defined in Eq. (40) as the fraction of dark matter in PBHs, values greater than one are internally inconsistent. The text in §V.B says only that this "seems unphysical," but the same numbers are then presented as predictions and plotted. The paper must either restrict the analysis to parameter and ω combinations with f_PBH ≤ 1, clearly label the remaining entries as excluded by self-consistency, or modify the abundance computation to account for the breakdown of the Press-Schechter/linear treatment in those regions. As written, the non-standard thermal history results are not quantitatively reliable.
  2. [§II, Eqs. (25)-(27), and ref. [44]] The central output is the tree-level linear spectrum through the USR phase. The paper cites Kristiano and Yokoyama (ref. [44]) but never computes or bounds the one-loop correction to Pζ. Because the same USR phase that produces the O(0.01) peak can also enhance interaction terms, an O(1) loop correction would change every PBH and induced-GW prediction built from Eqs. (33), (39), and (48). The authors should either evaluate the one-loop correction in this EGB model, derive a criterion showing it is subdominant, or explicitly state that all PBH and GW predictions are conditional on that correction being negligible. Simply citing the concern without engaging it is insufficient for the load-bearing claim.
  3. [Abstract and §IV, Tables I and II] The advertised wide mass range from O(10^-14) M⊙ to O(10) M⊙ is a direct consequence of choosing different φc values, which set k_peak in Tables I and II, followed by the mapping k → M in Eq. (33). The predicted mass is therefore a re-parameterization of the input rather than an independent model prediction. The authors should reframe the claim as a demonstration that the model can accommodate PBH masses across this range by tuning parameters, and they should discuss whether the required φc choices are natural or otherwise constrained by the CMB-scale fit and the fixed-point condition (24).
  4. [§IV.A and §IV.B] The text says the scalar power spectrum is obtained by "substituting the numerical solution of the Mukhanov-Sasaki equation in Eq. (12)" in both subsections. Eq. (12) is the slow-roll approximation and is explicitly invalid during USR, as the authors themselves note in §II.A. If Eq. (12) was actually used, the peak amplitudes in Figs. 2 and 4 are suspect; if Eq. (27) was used, the text must say so. This clarification is essential because every PBH and GW result in the paper is derived from those spectra.
minor comments (6)
  1. [§VI and §VII] The set-1 GW peak frequency is given as 10^-2 Hz in §VI but as 10^-1 Hz in the conclusion; please make the two statements consistent.
  2. [Throughout] "Mukhanov-Sasski" appears in §IV; the correct spelling is "Mukhanov-Sasaki."
  3. [Eq. (36)] The transfer function is written with the notation "Sin( l .√ω)" and the definition of l appears only in the same sentence; please define l = q/k explicitly and use standard mathematical notation.
  4. [Abstract] There is a typo "fromO(10^-14)" in the abstract; it should read "from O(10^-14)".
  5. [Fig. 9 caption] The constraint labeled "K" is referred to as "Kepler(k)" in the body text but is not expanded in the caption; please make the caption self-contained.
  6. [§V.A] The quantity M_PBH ψ(M_PBH) is used before the mass function ψ(M) is explicitly defined; please define the notation at first use near Eq. (37).

Circularity Check

2 steps flagged · score 6.0 of 10

The claimed wide PBH mass range and NANOGrav/GW frequency explanations reduce to the chosen φc values, which set k_peak and are then relabeled as masses and frequencies via Eq. (33) and the SIGW integral.

  1. fitted input called prediction [Section II A (after Eq. 23), with Eq. (33) and Tables I/II]
    "For the above-chosen coupling function, the parameter ϕc determines the position of the produced peak in the scalar power spectrum, while ξ0 and ξ1 specify the height and width of the peak."

    The abstract's headline claim that the model 'is capable of predicting PBH formation in a wide range mass, from O(10^-14)M⊙ to O(10)M⊙' is obtained by scanning φc across four benchmark sets (Tables I and II), which moves k_peak from 2.084e13 to 8.044e5 Mpc^-1. Eq. (33) then converts each chosen k_peak into M_PBH. The paper itself states that 'by increasing the free parameter ϕc, the peak of the power spectrum is placed in the large scale k. This will result in a higher magnitude for the mass of PBH.' Thus the wide mass range is a relabeling of fitted input parameters, not an independent prediction.

  2. fitted input called prediction [Section VII (Conclusion), relying on Section II A and Eq. (48)]
    "For the third set of parameters, the peak of the induced GWs is at the frequency of about 10 −8 Hz, and its signal also crosses the SKA boundary range. The NANOGrav results can be explained by the model with the third set of parameters."

    The SIGW peak frequency is set by the scalar peak scale k_peak through the convolution in Eq. (48), and Section II A states that φc determines the position of the produced peak in the scalar power spectrum. Therefore, matching NANOGrav/SKA with 'set 3' is simply the consequence of choosing the φc (with ξ1 tuned to keep ~60 e-folds) that puts k_peak at 1.081×10^7 Mpc^-1. The explanatory match restates the parameter choice rather than testing the model.

full rationale

The core numerical computation is not itself circular: the paper solves the background equations (4)-(6) and the Mukhanov-Sasaki equation (25) through the USR phase, and the resulting Pζ(k) is a genuine dynamical output. The pivot-scale normalization of V0 to the Planck amplitude is a standard external fit, and the Press-Schechter mass function and induced-GW kernels are taken from established external literature. The circularity is confined to the presentation of scanned benchmark parameters as predictions: the 'wide mass range' of PBHs and the 'explanation' of NANOGrav are both controlled by the chosen φc values, which set k_peak; Eq. (33) and the SIGW frequency relation then relabel those choices as astrophysical predictions. Several non-circular correctness risks should be noted separately: the paper cites the Kristiano-Yokoyama one-loop concern (ref. [44]) but does not evaluate it, leaving the O(0.01) tree-level peak potentially vulnerable to O(1) loop corrections; several f_PBH entries in Tables III and IV exceed unity, which the paper itself calls 'unphysical'; and the USR regime violates slow-roll, so the linear, Bunch-Davies calculation has uncontrolled corrections. These are validity concerns, not additional circularity. No load-bearing self-citation or imported uniqueness theorem was found, and the central Pζ computation is independent enough that the overall score is moderate rather than maximal.

Assumptions & free parameters 9 free parameters · 6 assumptions · 1 invented entities

The model's flexibility comes from nine benchmark-dependent numbers and from assumptions about linear perturbations, Gaussian collapse, and a single-fluid post-inflationary era. The tanh coupling is an ad hoc input with no external evidence. No genuinely new particle or force is posited.

free parameters (9)
  • α (Mutated Hilltop shape) = 1 and 2
    Appears in V(ϕ) = V0(1 - sech(αϕ)); two values are scanned by hand and change the background and spectra.
  • ϕc (coupling fixed-point position) = 3.095 to 4.050 for α=1; 2.250 to 2.690 for α=2
    Controls where the tanh coupling creates the USR phase and therefore where the Pζ peak and PBH mass sit.
  • ξ1 (coupling inverse width) = 28.1 to 73.0 for α=1; 61.3 to 140.5 for α=2
    Tuned so inflation lasts about 60 e-folds and the peak has the desired width.
  • ξ0 (coupling amplitude) = -2.986e6 to -2.272e7
    Derived from the fixed-point condition Eq. (24) once other parameters are chosen; still a benchmark-dependent quantity.
  • V0 (potential scale) = 1.29e-10 for α=1; 3.372e-11 for α=2
    Fixed by forcing Pζ(k*) to match the Planck value, i.e., a fit to the pivot amplitude.
  • ϕ⋆ (initial field value) = 4.482 for α=1; 2.91 for α=2
    Set by hand so that inflation lasts N=60 for each parameter set.
  • γ (collapse efficiency) = 0.33
    Assumed constant in M = γ M_H; directly scales PBH mass and abundance.
  • T_RD (reheating temperature) = 100 GeV
    Defines the transition into radiation domination in the non-standard history formulas; one representative value is used.
  • ω (post-inflation equation of state) = 1/3, 0.5, 0.7, 0.9
    Scanned input for the non-standard epoch; not derived from the model.
assumptions (6)
  • domain assumption The action (1) with a scalar field plus a Gauss-Bonnet coupling is a valid low-energy description of inflation.
    The entire computation starts from this action; no derivation from an underlying theory is provided.
  • domain assumption The spatially flat FLRW metric (3) and the slow-roll approximations (9)-(11) hold during the early part of inflation.
    Needed to set initial conditions and use the EGB slow-roll formulas.
  • ad hoc to paper The fixed-point condition (24) creates a stable ultra-slow-roll phase.
    Equation (24) is derived by setting derivatives to zero, but no stability analysis is given; the USR phase is a chosen feature of the model.
  • domain assumption Bunch-Davies vacuum and linear perturbation theory remain valid through the USR phase.
    Used for the Mukhanov-Sasaki initial conditions (26); the cited one-loop literature (ref. 44) questions this and is not addressed.
  • domain assumption PBH abundance follows Gaussian Press-Schechter theory with δc(ω) from Harada et al. (42) and γ=0.33.
    No non-Gaussianity or density-profile dependence is included; these choices enter directly into Tables III and IV.
  • domain assumption The universe is dominated by a single fluid with constant ω from PBH formation until T_RD, with entropy conservation and the matching (31)-(33).
    The non-standard thermal history calculation assumes the simplified abrupt-transition picture.
invented entities (1)
  • tanh coupling ξ(ϕ) = ξ0 tanh(ξ1(ϕ - ϕc))
    purpose: Creates a fixed point that induces the ultra-slow-roll phase and boosts Pζ to O(0.01).
    The functional form and its parameters are adopted from earlier EGB literature (refs. 88-92) and tuned to hit desired peak scales; there is no independent measurement that selects this coupling before the model is fitted.

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Pith. "Pith review of Non-Standard Thermal History and Formation of Primordial Black Holes in Einstein-Gauss-Bonnet Gravity." pith.science (2026). https://pith.science/paper/UAZCA75C

@misc{pith2026250101867,
  author       = {Pith},
  title        = {Pith review of: Non-Standard Thermal History and Formation of Primordial Black Holes in Einstein-Gauss-Bonnet Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UAZCA75C}},
  note         = {Machine review of arXiv:2501.01867}
}
abstract

Inflation provides a suitable environment for the formation of Primordial Black Holes(PBHs). In this article, we examine the formation of primordial black holes in the Mutated Hilltop inflation model coupled with the Einstein-Gauss-Bonnet term. A suitable choice of the coupling function with adjusted parameters can produce the USR regime during the inflationary phase, which lasts for some number of e-folds. The scalar field in this regime remains almost unchanged, and the first slow-roll parameter $\epsilon_1$ drops dramatically, leading to a significant enhancement to the curvature power spectrum for small scales so that it grows up to order of $\mathcal{O}(0.01)$; a crucial feature for producing PBH and secondary gravitational waves (GWs). We investigate the formation of PBHs for different sets of parameters. By considering the behavior of the scalar power spectrum, it is realized that the peak in the scalar power spectrum occurs in different scales. Then, the presented model is capable of predicting PBH formation in a wide range mass, from $\mathcal{O}(10^{-14})M_{\odot}$ to $\mathcal{O}(10)M_{\odot}$, which are compatible with the LIGO-Virgo data. PBHs with mass $\mathcal{O}(10^{-5})M_{\odot}$ can account for the micro-lensing event in OGLE as well as the asteroid masses $\mathcal{O}(10^{-15})M_{\odot}-\mathcal{O}(10^{-12})M_{\odot}$ PBH can be attributed to $100\% $ dark matter present in the universe. The generated perturbations during inflation re-enter the horizons after the end of inflation, where the universe may be in a non-standard epoch rather than the radiation-dominant phase, with an equation of state $1/3< \omega \leq 1$...

Figures

Figures reproduced from arXiv: 2501.01867 by the authors.

Figure 1
Figure 1. FIG. 1. The plot shows the scalar field, Hubble parameter, slow-roll parameter [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The power spectrum versus the mode [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The plot shows the scalar field, Hubble parameter, slow-roll parameter [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The power spectrum versus the mode [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The plot shows behavior of [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The plot shows the evolution of [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The plot shows behavior of [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The plot shows the evolution of [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. The plot shows the behavior of [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. The plot shows the produced induced GW for a) [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]

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