Primordial blackhole formation: Exploring chaotic potential with a sharp step via the GLMS perspective

Jobil Thomas; Minu Joy; Rinsy Thomas

REVIEW 5 major objections 6 minor 105 references

Primordial blackhole formation: Exploring chaotic potential with a sharp step via the GLMS perspective

T0 review · 5 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A sharp step on a chaotic inflationary potential can form primordial black holes that could be all dark matter in two mass windows.

desk verdict A competent parameter-scan demonstration of the known sharp-step chaotic inflation mechanism, whose headline f_PBH≈1 is exponentially sensitive to an untested threshold choice; deserves review but needs a sensitivity scan before the dark-matter claim can be used. read the letter →

arxiv 2411.10076 v1 pith:XMZEDNFO submitted 2024-11-15 astro-ph.CO

Rinsy Thomas , Jobil Thomas , Minu Joy This is my paper · ORCID

classification astro-ph.CO

keywords primordialblackholeschaoticinflationsharpsteppotentialultra-slowrollpeaktheoryGLMSapproximationPress-Schechterformalismdarkmatterabundance

topics Dark Matter

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

This paper argues that appending a small sharp step to the standard chaotic inflation potential can amplify primordial curvature fluctuations on small scales to about $10^{-2}$, enough to form primordial black holes while leaving the cosmic microwave background observables $n_s$ and $r$ unchanged. The step acts as a speed bump, briefly slowing the inflaton field and producing a spike in the curvature power spectrum at the scales that reenter during radiation domination. Using the GLMS approximation to peak theory, the authors compute that black holes of $10^{-13}M_\odot$ and $10^{-11}M_\odot$ would constitute nearly all of the dark matter, while $1M_\odot$ and $6M_\odot$ black holes would contribute fractions of 0.01 and 0.001. A sympathetic reader would care because this is a single-field inflation model in which the physics of the CMB scale is decoupled from the physics that produces PBHs, so the model makes concrete predictions that can be checked against gravitational-wave and microlensing constraints.

What carries the argument

The load-bearing object is the step potential $V(\phi)=\frac{1}{2}m^2\phi^2[1+c\tanh((\phi-\phi_{\rm step})/d)]$, where the dimensionless height $c$, width $d$, and position $\phi_{\rm step}$ are tuned to create a brief ultra-slow-roll phase that spikes the power spectrum at a chosen small scale. The abundance calculation then rests on the GLMS approximation to peak theory, specifically the analytic expression $\beta(M_{\rm PBH}) = \frac{1}{\sqrt{2\pi}}\left(\frac{R\sigma_1}{\sqrt{3}\sigma_0}\right)^3(\nu_{\rm th}^2-1)\exp(-\nu_{\rm th}^2/2)$, which replaces the full ten-dimensional peak statistics with two spectral moments $\sigma_0=\sigma_\delta$ and $\sigma_1$ of the smoothed density contrast. The smoothing is done with a Gaussian window function $W(k,R)=\exp(-k^2R^2/2)$ on the scale $R=1/k_{\rm PBH}$. Everything downstream depends exponentially on $\nu_{\rm th}^2=\delta_{\rm th}^2/\sigma_0^2$, which is why the choice $\delta_{\rm th}=0.414$ and the Gaussian assumption for the density field carry the argument.

What would settle it

Recompute $f_{\rm PBH}$ for the four parameter sets using $\delta_{\rm th}=0.33$ and $\delta_{\rm th}=0.66$ (or include the non-Gaussian corrections expected from the step) and check whether the near-unity abundances for the two lightest mass windows survive; because of the exponential dependence, a fractional change in $\nu_{\rm th}$ of order ten percent shifts $f_{\rm PBH}$ by orders of magnitude.

Watch

Extended reading notes

Core claim

The central claim is that the potential $V(\phi)=\frac{1}{2}m^2\phi^2[1+c\tanh((\phi-\phi_{\rm step})/d)]$ with a sharp step can generate a peak in the primordial scalar power spectrum of order $10^{-2}$ at small scales, while the power spectrum on CMB scales remains nearly scale invariant with $n_s=0.96$ and $r=0.02$. Solving the Mukhanov-Sasaki equation numerically for four parameter sets, the paper finds peaks at wavenumbers corresponding to PBH masses $10^{-13}M_\odot$, $10^{-11}M_\odot$, $1M_\odot$, and $6M_\odot$. The fractional abundance is then computed with the GLMS formula $\beta(M_{\rm PBH}) = \frac{1}{\sqrt{2\pi}}\left(\frac{R\sigma_1}{\sqrt{3}\sigma_0}\right)^3(\nu_{\rm th}^2-1)\exp(-\nu_{\rm th}^2/2)$, with $\nu_{\rm th}=\delta_{\rm th}/\sigma_0$ and $\delta_{\rm th}=0.414$. The result is $f_{\rm PBH}\approx 1$ for the two lightest windows and $0.01$, $0.001$ for the heavier ones, while the Press-Schechter formalism gives values two to three orders of magnitude smaller. The paper presents the GLMS values as consistent with current observational constraints.

Load-bearing premise

The abundances assume that the smoothed density contrast is a Gaussian random field with a fixed collapse threshold $\delta_{\rm th}=0.414$; the result depends exponentially on this threshold and on the amplitude of the tail of the distribution.

Editorial extensions

If this is right

The same single potential produces PBHs in four distinct mass windows, from $10^{-13}M_\odot$ to $6M_\odot$, by moving the step position $\phi_{\rm step}$.
The step leaves the CMB observables $n_s$ and $r$ unchanged, so small-scale PBH physics can be tuned independently of large-scale cosmological parameters.
In the GLMS approximation, the two lightest mass windows have $f_{\rm PBH}\approx 1$, meaning PBHs could constitute the entirety of the dark matter in those windows.
The heavier windows, $1M_\odot$ and $6M_\odot$, yield $f_{\rm PBH}\approx 0.01$ and $0.001$, consistent with LIGO/Virgo merger-rate constraints.
The Press-Schechter formalism underestimates $f_{\rm PBH}$ by two to three orders of magnitude relative to the GLMS peak-theory calculation, so the choice of abundance estimator is decisive here.

Reading between the lines

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

If the step-induced non-Gaussianity is significant, the Gaussian GLMS estimate likely overstates the high-density tail; a calculation using the step-generated bispectrum or a numerical collapse simulation would bracket the true $f_{\rm PBH}$.
The same small-scale power enhancement that forms PBHs necessarily sources a stochastic gravitational-wave background at second order; the paper does not compute this, but it is a testable corollary for PTA and LISA observations.
Because $f_{\rm PBH}$ depends so steeply on $\delta_{\rm th}$, the 'PBH as all dark matter' conclusion should be read as contingent on the threshold and Gaussianity assumptions rather than as a robust prediction of the potential alone.
The decoupling of CMB-scale from small-scale parameters suggests the step parameters could be pinned down independently by future measurements of the scalar power spectrum at small scales, such as 21-cm observations.

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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,

Short version: this is a parameter-scan paper built on a known mechanism. The sharp-step chaotic potential, the ultra-slow-roll enhancement, and the GLMS/Press-Schechter abundance formulas are all established; the same authors' 2023 paper covers the same territory. What is actually new here is four tuned parameter sets that place the scalar power spectrum peak so that PBHs form near 1e-13, 1e-11, 1, and 6 solar masses, with f_PBH from GLMS near 1 in the two light windows and 0.01/0.001 in the heavy ones. That is a modest but legitimate demonstration, and the claim that CMB-scale ns and r are unaffected by the late step is plausible and supported by the plots.

The paper does several things well. It shows the fine-tuning sensitivity of the step parameters (c changed by 1e-7 shifts the peak amplitude by two orders of magnitude), it explicitly compares GLMS and PS results, and it is transparent about the formation-threshold literature, quoting the 0.33–0.66 range rather than hiding it. It also flags one-loop backreaction as a remaining issue. This is honest work.

The soft spot is the one the stress-test note puts its finger on, and it is load-bearing. The f_PBH≈1 values are essentially determined by exp(−ν_th²/2) with ν_th = δ_th/σδ. The paper fixes δ_th = 0.414, reports no σδ values, and never scans δ_th over the 0.33–0.66 range it itself cites. Because the exponent is quadratic, that single choice changes the abundance by many orders of magnitude in either direction. The paper even notes that step-induced non-Gaussianity can alter the threshold, which makes the Gaussian integral in Eq. (25) an unvalidated assumption at exactly the point where the headline result sits. Also missing: numerical pipeline, code, convergence tests, so the P(k) curves cannot be independently reproduced. There are minor internal inconsistencies (Table 1 lists ns/r for one of four sets only; the conclusions quote 1e-10 and 10 M_sun instead of 1e-13 and 6 M_sun; Eq. (6) has a sqrt typo).

Overall: the central mechanism is not in doubt, but the specific abundance claims are not yet supported. This deserves a serious referee — a desk rejection would be wrong — and the referee should ask for a threshold and non-Gaussianity sensitivity scan, σδ for each window, and the code or a full numerical pipeline before the f_PBH≈1 claim is taken at face value. If the result survives a δ_th scan, it is a useful subfield contribution; if not, it is still a clean illustration of why threshold choice dominates step-model PBH abundances.

Referee Report

5 major / 6 minor

Summary. This paper studies single-field chaotic inflation with a hyperbolic-tangent sharp step in the potential. The authors solve the Mukhanov-Sasaki equation numerically, present four tuned parameter sets for which the scalar power spectrum is enhanced to O(10^-2) at small scales while the CMB-scale values of n_s and r are claimed to remain unchanged, and compute PBH abundances with the GLMS approximation of peak theory and with the Press-Schechter formalism. They report f_PBH ≈ 1 for the 10^-13 M_sun and 10^-11 M_sun mass windows and f_PBH ≈ 0.01 and 0.001 for the 1 M_sun and 6 M_sun windows, respectively. The central route—tuning a step to enhance P(k) and then applying the GLMS/PS abundance formulas—is standard, but the manuscript omits the numerical pipeline and the required sensitivity analysis, and it contains internal inconsistencies in the reported masses and CMB observables.

Significance. If the abundance results were robust, the model would be attractive because it decouples CMB-scale observables from small-scale PBH production and covers a wide mass range with a simple potential. The paper has several genuine strengths: it solves the full Mukhanov-Sasaki equation, explicitly studies the fine-tuning of c, d, and phi_step, and presents a side-by-side comparison of the GLMS and Press-Schechter formalisms. However, the headline f_PBH ≈ 1 claim is exponentially sensitive to the assumed density threshold and to the Gaussianity of the smoothed density field, neither of which is tested. The scientific value of the paper therefore depends on completing the robustness analysis rather than on the current numerical estimates alone.

major comments (5)

[Section 4.1, Eq. (25)] The headline abundance values are dominated by the factor exp(-nu_th^2/2) with nu_th = delta_th/sigma_delta, but the paper fixes delta_th = 0.414 after citing a permissible range of 0.33–0.66 and reports no values for sigma_delta or sigma_1 for the four parameter sets. Because f_PBH depends exponentially on nu_th^2, moving delta_th from 0.414 to 0.66 at the nu_th values implied by f_PBH ≈ 1 suppresses the abundance by roughly twenty orders of magnitude, while moving to 0.33 raises it. Without a reported sigma_delta and a delta_th sensitivity scan, the f_PBH ≈ 1 claim is not a robust prediction and the comparison with observational constraints in Figs. 10–12 is not meaningful.
[Section 4.1, Eqs. (22)–(25)] The GLMS abundance calculation assumes the smoothed density contrast is a Gaussian random field, and the paper itself states that delta_th depends on primordial non-Gaussianities. The step-induced ultra-slow-roll phase is precisely a regime known to generate non-Gaussianity, so the Gaussian integral in Eq. (25) is an unvalidated assumption at the point where the central result lives. The authors should at least quantify the expected non-Gaussian correction to the high-density tail or show that it is negligible; without that, the agreement with observational constraints may be an artifact of the Gaussian ansatz.
[Section 2, Table 1] The text states that all four parameter sets give the same n_s and r at the CMB pivot scale and that these are consistent with Planck, but Table 1 lists numerical entries for n_s and r only for set 2 (0.96 and 0.02); sets 1, 3, and 4 have empty entries. The authors should report n_s and r for every set, with uncertainties and the Planck reference values, or the claim of separate control of CMB observables is not supported by the presented data.
[Section 5 vs. Abstract and Table 2] The conclusions contain internal inconsistencies: the text refers to PBH masses of 10^-13 M_sun and 10^-10 M_sun and to windows of 1 M_sun and 10 M_sun, while the abstract and Table 2 give 10^-13 M_sun, 10^-11 M_sun, 1 M_sun, and 6 M_sun. The manuscript also refers to a previous mass range of 10^-17 M_sun, 10^-13 M_sun, and 30 M_sun in Section 3, which is not clearly distinguished from the current results. These discrepancies must be reconciled before publication.
[Sections 2–3, numerical method] The paper does not describe the numerical pipeline used to solve the Mukhanov-Sasaki equation: no initial conditions, number of e-folds, k-grid resolution, Runge-Kutta step size, or convergence tests are reported. Since the peak amplitude and position of P(k) are the fundamental inputs to f_PBH and since Fig. 6 shows that the peak amplitude changes by two orders of magnitude under c -> c + 10^-7, the absence of numerical details compromises reproducibility and prevents a reader from assessing whether the reported power spectra are converged.

minor comments (6)

[Section 1, Introduction] There is a typo: "Press-Scheter" should be "Press-Schechter."
[Global] Several stylistic errors appear, including "inturn," "extremely light weigh," and "e fold"; a careful proofreading pass is needed.
[Section 3, Eq. (10)] The notation g_s0* is confusing; the entropy degrees-of-freedom symbols should be defined consistently with the energy degrees-of-freedom g_*.
[Fig. 10–12] The figures for f_PBH versus mass are presented with shaded observational constraints, but the numerical values of f_PBH for the four points are not given; a table of f_PBH values, with and without the sensitivity test, would be more informative.
[Section 4.1, Eq. (25)] Please verify the prefactor in the GLMS expression; the standard GLMS result is often written with a factor 1/(2 pi) rather than 1/sqrt(2 pi), and the derivation leading to Eq. (25) should be shown or cited explicitly.
[Section 2, Table 1] The table header formatting for d/M_pl and phi_step/M_pl is not clear from the text; the units should be stated explicitly for every entry.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: PBH abundances are computed from the model power spectrum with a literature threshold; mass windows are tuned parameter choices but are presented as exploration, not independent predictions.

full rationale

The derivation chain is not circular. The authors solve the Mukhanov-Sasaki equation for the step potential, obtain P(k), smooth it with a Gaussian window (Eqs. 17-20), and insert sigma_delta and sigma_1 into the GLMS formula (Eq. 25) or the PS integral (Eq. 27), with delta_th = 0.414 stated as an adopted literature value. Nothing in the text indicates that c, d, or phi_step were adjusted to match f_PBH; the sensitivity plots (Figs. 6-8) vary parameters around a fixed set and recompute P(k). The four mass windows do follow from parameter choices (Section 3 states that increasing phi_step shifts the peak to larger scales and hence increases the PBH mass), so the masses are targets of the parameter scan rather than unique predictions; however, the paper describes this as a parameter-space exploration ("we have explored a broader parameter space, which shows how different sets of parameters leads to PBH formation across different mass ranges"), so this is model-building, not a hidden fit. The exponential sensitivity of f_PBH to delta_th and the absence of a threshold scan is a real robustness concern but not circularity: delta_th is an external input, and the computation is an honest application of the stated formula. Reference [57] is a self-citation but is not load-bearing, since the same statement is also supported by [49]. Thus the central abundance results retain independent content.

Assumptions & free parameters 5 free parameters · 5 assumptions · 1 invented entities

The central claim rests on one tuned potential, two standard abundance approximations, and a threshold choice. All numbers are computed rather than measured; no new particles or interactions are introduced. The main free parameters are the step geometry and the adopted threshold; the main axioms are the Gaussian field assumption, the single-wavenumber mass mapping, and the neglect of one-loop corrections.

free parameters (5)

c (step height, sets 1 to 4) = -7.91501e-3; -6.97333e-3; -2.70720e-3; -2.52303e-3
Tuned to control the amplitude of the scalar power spectrum peak; the sensitivity plot shows changes of 1e-7 altering the peak by a factor of 100.
d (step width, sets 1 to 4) = 0.029; 0.027; 0.003; 0.002
Tuned with c and phi_step to keep the peak at the target k range.
phi_step (step position, sets 1 to 4) = 9.6; 10.2; 12.2; 12.32
Sets the e-fold and scale where the ultra-slow-roll burst occurs, hence the PBH mass window.
delta_th (PBH formation threshold) = 0.414
Chosen from the cited range 0.33 to 0.66; the abundance is exponentially sensitive to this value.
gamma (collapse efficiency) = 0.2
Adopted from Carr 1975 and used in both the mass relation and the abundance prefactor; it is not fitted to this model, but it is a chosen numerical input.

assumptions (5)

domain assumption Gaussian random field for the density contrast
The peak-theory joint distribution in Eq (22) assumes Gaussian statistics, but ultra-slow-roll bursts and step features can generate non-Gaussianity; no non-Gaussian correction is applied.
domain assumption Fixed threshold delta_th=0.414 with radiation equation of state
Used in Eqs (25) and (27) for both formalisms; the cited literature range is 0.33 to 0.66 and f_PBH is exponentially sensitive to nu_th.
domain assumption Nearly monochromatic PBH mass function
The mass relation in Eq (11) maps one comoving wavenumber to one PBH mass; the actual spectra in Fig 9 are broad, so this is an approximation that is not quantified.
domain assumption Tree-level power spectrum is sufficient
The paper states that one-loop corrections and effective field theory backreaction are deferred to future work (Section 3, after Fig 9); if backreaction is significant, the peak height and f_PBH would change.
ad hoc to paper Sharp step ansatz for the potential
The tanh step in Eq (1) is inserted by hand into the chaotic potential to produce the desired ultra-slow-roll episode; physical motivations are cited but no concrete parent theory derivation is given.

invented entities (1)

None
purpose: No new particle, field, mediator, or dimension is introduced.
The tanh step is a modification of the inflaton potential, not a new physical entity, so no independent handle outside the paper exists.

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Cite this review

Pith. "Pith review of Primordial blackhole formation: Exploring chaotic potential with a sharp step via the GLMS perspective." pith.science (2026). https://pith.science/paper/XMZEDNFO

@misc{pith2026241110076,
  author       = {Pith},
  title        = {Pith review of: Primordial blackhole formation: Exploring chaotic potential with a sharp step via the GLMS perspective},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XMZEDNFO}},
  note         = {Machine review of arXiv:2411.10076}
}

abstract

A sharp step on a chaotic potential can enhance primordial curvature fluctuations on smaller scales to the $\mathcal{O}(10^{-2})$ to form primordial black holes (PBHs). The present study discusses an inflationary potential with a sharp step that results in the formation of PBHs in four distinct mass ranges. Also this inflationary model allows the separate consideration of observable parameters $n_s$ and $r$ on the CMB scale from the physics at small scales, where PBHs formation occur. In this work we computed the fractional abundance of PBHs ($f_{PBH}$) using the GLMS approximation of peak theory and also the Press-Schechter (PS) formalism. In the two typical mass windows, $10^{-13}M_\odot$ and $10^{-11}M_\odot$, $f_{PBH}$ calculated using the GLMS approximation is nearly equal to 1 and that calculated via PS is of $10^{-3}$. In the other two mass windows $1M_\odot$ and $6M_\odot$, $f_{PBH}$ obtained using GLMS approximation is 0.01 and 0.001 respectively, while $f_{PBH}$ calculated via PS formalism yields $10^{-5}$ and $10^{-6}$. The results obtained via GLMS approximation are found to be consistent with observational constraints. A comparative analysis of $f_{PBH}$ obtained using the GLMS perspective and the PS formalism is also included.

Figures

Figures reproduced from arXiv: 2411.10076 by the authors.

**Figure 1.** Plot of the chaotic inflationary potential with sharp step and enlarged inset showcasing the feature. For a spatially flat universe, the equation of motion of the scalar field and the scale factor is governed by, H 2 = 1 3M2 pl V (φ) + 1 2 φ˙2 (3) where Hubble parameter, H = a˙ a . The extend of inflation is given by the amount of e folds during inflation, Ne = ln a(tend) a(t0) (4) where t0 and tend corresponds … view at source ↗

**Figure 2.** Slow roll parameter ǫ1 plotted as a function of number of e-folds for the chaotic inflationary potential with sharp step The step feature modifies the potential locally and then the inflaton field dynamics quickly settle back into a stable slow-roll trajectory. This behavior confirms the robustness of the attractor solution in our model. Figures 2 and 3 shows the local violation of slowroll conditions due to the pre… view at source ↗

**Figure 3.** Slow roll parameter ǫ2 plotted as a function of number of e-folds for the chaotic inflationary potential with sharp step P(k) and the tensor power spectrum PT (k) on the CMB scale are shown in Figures 4 and 5 respectively. It is obvious that the scalar and tensor power spectra are nearly scale invariant on CMB scales for all the four sets. 1 1 1 11 1 1 1 1 1 1  1 [PITH_FULL_IMAGE:figures/… view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: Scalar power spectrum P(k) on large scale for the chaotic inflationary model with step, for four distinct parameter sets [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: Tensor power spectrum PT (k) for the chaotic inflationary model with step, for four distinct parameter sets In this chaotic inflationary model with sharp step, fine tuning of the potential parameters is required, for the observables ns and r to be consistent with CMB d…

**Figure 6.** Figure 6: Sensitivity of scalar power spectrum to the potential parameter c with φstep = 12.32 and d = 0.002 1 1 1 1 11 11 1 k 1 1 1 1 1 1 [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 7.** Figure 7: Sensitivity of scalar power spectrum to the potential parameter φstep with c = −2.52302 × 10−3 and d = 0.002 [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

**Figure 8.** Figure 8: Sensitivity of scalar power spectrum to the potential parameter d with c = −2.52302 × 10−3 and φstep = 12.32. inflationary period results in ǫ decreasing appreciably from its pivot scale. This leads to a substantial enhancement in the amplitude of the scalar power spec…

**Figure 9.** Figure 9: Scalar power spectrum P(k) on broader scales for four distinct parameter sets of PBHs during the radiative epoch [79, 80, 81]. In the radiative epoch [44], H 2 = Ω0rH 2 0 (1 + z) 4 g∗ g0∗ − 1 3 g s 0∗ g0∗ 4 3 . (10) Here g0∗ and g s 0∗ represent the effective deg…

**Figure 10.** Figure 10: Fractional abundance of PBHs in the framework of GLMS approximation in peak theory formalism is plotted as a function of PBH mass. This graphical representation pertains to the four distinctive cases discussed in [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]

**Figure 11.** Figure 11: Fractional abundance of PBHs in the framework of Press-Schechter formalism is plotted as a function of PBH mass. This graphical representation pertains to the four distinctive cases discussed in [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]

**Figure 12.** Figure 12: Comparative analysis of the fractional abundance of PBHs across four different PBH mass scales as in [PITH_FULL_IMAGE:figures/full_fig_p019_12.png]

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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