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REVIEW 3 major objections 3 minor 71 references

Primordial black holes through preheating instabilities in $\alpha$-attractor models

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper argues that preheating self-resonance in α-attractor models produces 10^2–10^7 g primordial black holes, that the Press–Schechter mass fraction overproduces them and is excluded by Hawking-evaporation constraints, while the…

desk verdict Useful alpha-attractor extension of the preheating PBH program, but the PS-vs-KP exclusion is undercut by an epoch inconsistency and an underspecified threshold. read the letter →

arxiv 2505.17790 v2 pith:QRVJAQHA submitted 2025-05-23 astro-ph.CO gr-qchep-th

Daniel del-Corral , Paolo Gondolo , K. Sravan Kumar , João Marto This is my paper
classification astro-ph.COgr-qchep-th
keywords primordialblackholespreheatingalpha-attractorinflationself-resonanceHawkingevaporationconstraintsPress-SchechterformalismKhlopov-Polnarevnonsphericalcollapse
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 tries to show that the matter-like preheating phase of α-attractor inflation, driven by self-resonant amplification of curvature perturbations, is a genuine source of primordial black holes with masses of order $10^{2}$ to $10^{7}$ grams. It argues that the standard Press–Schechter mass fraction overproduces these black holes and is excluded by Hawking-evaporation constraints, while the Khlopov–Polnarev mass fraction, which suppresses collapse anisotropy, stays below those constraints. The authors conclude that nonspherical effects cannot be neglected in preheating PBH calculations, and that small-α T- and E-models allowed by prior gravitational-wave bounds remain viable against evaporation limits. The reason to care is that this decides whether an early matter-dominated PBH channel is open at all, and which abundance estimator is reliable before horizon re-entry.

What carries the argument

The load-bearing objects are the instability band—the range of comoving wavenumbers whose Hill-equation Floquet exponents are positive during self-resonance—and the $k$-dependent collapse threshold $\delta_c(k)$, defined numerically as the minimum initial density perturbation that reaches $O(1)$ before the end of preheating. The instability band decides which modes grow; the Jeans/sound-speed condition decides which of those can collapse; the threshold decides which reach nonlinearity in time. The abundance formulas then convert the variance $\sigma_k^2$ into a mass fraction, with the Khlopov–Polnarev formula $\beta(k)\simeq 0.056\,\sigma_k^5$ replacing the Press–Schechter Gaussian-integral estimate and suppressing collapse for anisotropic perturbations.

What would settle it

Run a full numerical-relativity or high-resolution lattice simulation of self-resonant preheating in an E-model with $\alpha = 10^{-3}$, extract the true collapse threshold and mass fraction, and compare with the $\beta$ values this paper reports: if the simulated abundance matches Press–Schechter and violates the evaporation constraints, the KP viability conclusion fails; if it matches the KP curve, the PS exclusion is confirmed.

Watch

Extended reading notes

Core claim

Self-resonance during the preheating epoch in $\alpha$-attractor models amplifies scalar perturbations within a broad instability band, and three criteria—membership in that band, wavelength above the effective Jeans length and below the Hubble radius, and density contrast above a per-mode time-to-collapse threshold—select the modes that collapse into PBHs with masses $10^{2}$–$10^{7}$ g. Using the resulting power spectrum, the paper finds that the Press–Schechter formalism gives mass fractions that overshoot the Hawking-evaporation bounds, whereas the Khlopov–Polnarev formalism, based on the fraction of perturbations spherical enough to collapse, keeps the same models inside the bounds. This is the paper's central claim: the PS formalism is excluded by evaporation constraints and the KP formalism is viable, so the nonspherical suppression of collapse is decisive for PBH abundance during preheating.

Load-bearing premise

The calculation leans on the numerically computed collapse threshold $\delta_c(k)$, whose normalization and nonlinearity cutoff are not specified; a different choice could move the Press–Schechter exclusion and the Khlopov–Polnarev survival by orders of magnitude.

Editorial extensions

If this is right

  • The T- and E-models of $\alpha$-attractor inflation with small $\alpha$ can produce PBHs of $10^{2}$ to $10^{7}$ g during preheating, with masses growing as $\alpha$ decreases.
  • The Press–Schechter formalism overproduces these PBHs and violates the Hawking-evaporation constraints (Planck remnants, LSP, and dark-matter production), so it is excluded in this scenario.
  • The Khlopov–Polnarev formalism keeps the abundance below those constraints, leaving the small-$\alpha$ models viable.
  • Lengthening preheating to roughly 12–13 e-folds pushes the instability band into the nonlinear regime, causing overproduction in Press–Schechter but not necessarily in Khlopov–Polnarev.
  • The difference between the two formalisms means evaporation constraints cannot be used to rule out these models until nonspherical collapse is treated correctly.

Reading between the lines

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

  • Editorial extension: because $\beta_{\rm PS}$ depends on $\delta_c(k)$ through a complementary error function, the reported PS exclusion is sensitive to how "reaches $O(1)$" is implemented numerically; the authors do not quantify this sensitivity.
  • Editorial extension: the same instability band that forms PBHs should source a scalar-induced gravitational-wave background, so a future detection could be combined with the evaporation bounds to test whether the KP suppression is as strong as assumed.
  • Editorial extension: adding inhomogeneity and spin factors to the KP estimate, as the paper's product formula suggests, would lower abundances further and make PBH formation during preheating harder to observe.
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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 / 3 minor

Summary. This paper studies primordial black hole (PBH) formation during preheating in T- and E-model alpha-attractors. The authors use three criteria: a mode must lie in the self-resonance instability band, its wavelength must exceed the Jeans length, and its density contrast must exceed an alpha- and k-dependent threshold delta_c(k). With these criteria they compute the mass fraction beta(k) using both the Press-Schechter (PS) formalism, Eq. (5.2), and the Khlopov-Polnarev (KP) formalism, Eq. (5.3), and estimate PBH masses from the horizon mass at instability-band entry, Eq. (5.5). For alpha values near the authors' earlier gravitational-wave lower bounds, the resulting PBH masses lie in the roughly 10^2 to 10^7 g range. The authors find that the PS formalism overproduces PBHs relative to evaporating-PBH constraints while the KP formalism remains viable, and conclude that nonspherical effects are essential in preheating PBH calculations. The paper extends the authors' previous Starobinsky analysis to alpha-attractor models and imports the instability-band and Floquet results from their companion papers [26,27].

Significance. If the quantitative comparison is reliable, the paper makes a valuable point: PS-type estimates applied to matter-dominated preheating overproduce PBHs, while KP anisotropy suppression can bring predictions into agreement with evaporation constraints. The paper has real strengths: it uses numerical Mukhanov-Sasaki evolution cross-checked against Floquet theory (Fig. 5), validates the analytic sound speed against the full expression (Fig. 3), and compares explicitly with a wide set of observational constraints. However, the central quantitative claim is not currently reproducible. The numerical threshold delta_c(k) is defined only verbally, and the PS and KP formalisms are evaluated at different epochs without a stated convention for the relative normalization. These issues are load-bearing because they enter exponentially or through a fifth power in the abundances, so they must be resolved before the 'PS excluded, KP viable' conclusion can be accepted.

major comments (3)
  1. [Sec. 3, after Eq. (3.10)] The numerical computation of the threshold delta_c(k) is not specified. The text says it is 'the minimum value of the density perturbation that reaches O(1)', but it does not state the target nonlinearity value (e.g., 1 versus 1.06 versus 1.69), the initial-condition normalization for delta_k, the time at which the amplitude is read, or the numerical method used. Since delta_c(k) enters exponentially in the PS mass fraction (Eq. 5.2) and also gates which modes are counted in the KP formalism, this omission makes the central PS-vs-KP comparison unreproducible and could change the abundances by orders of magnitude. Please provide the full algorithm, including the definition of O(1), the normalization of delta_k, and the numerical scheme, or give a convergence study against an explicit convention.
  2. [Sec. 5.1, Eq. (5.3) and Figs. 6-7] The evaluation epoch used for the two formalisms is inconsistent. The PS paragraph and the captions of Figs. 6 and 7 state that the mass fraction is evaluated 10 e-folds after the end of inflation (N=70), while the KP paragraph immediately after Eq. (5.3) states that the mass fraction is evaluated 'at the moment the perturbation crosses the IB and thus starts to amplify from a small value'. These epochs differ by roughly the duration of exponential self-resonant growth, during which sigma_k grows by many orders of magnitude. Because beta_KP is proportional to sigma_k^5 (Eq. 5.3) and beta_PS depends on erfc(delta_c/(sqrt(2) sigma_k)) (Eq. 5.2), the relative normalization of the two curves in Fig. 6 depends critically on which convention is used; moreover, if sigma_k at N=70 is used for the KP estimate, the condition sigma_k < 0.01 stated for the validity of Eq. (5.3) may be violated. Please specify one evaluation epoch for both formalisms, or quantify the sensitivity of the conclusion to this choice.
  3. [Sec. 6 and Abstract] The headline claim that 'the PS formalism is excluded by these constraints, which are based on Hawking radiation' is stronger than the analysis supports. The paper's own Sec. 6 states that for the mass range considered, the only active constraints are Planck remnants, LSP, and DM production, and it calls these 'highly theoretical' with the last two depending on the emitted particle mass. Since the remnant constraint rests on speculative end-of-evaporation physics, the conclusion should be reworded to reflect that the exclusion holds under specific assumptions about quantum gravity and particle physics, rather than presenting it as a robust Hawking-radiation constraint.
minor comments (3)
  1. [Captions of Figs. 6-7] The caption of Fig. 6 says the dashed and continuous curves correspond to 'KP (5.3) and PS (5.2) ... respectively', while the text of Sec. 5.1 says the PS formalism is shown by dashed curves and the KP formalism by continuous curves. These statements are mutually contradictory, making the central figure ambiguous. Please correct the caption and make it consistent with the text.
  2. [Sec. 5.2, caption of Fig. 7] The caption of Fig. 7 refers to Eq. (5.7) for the PBH mass, but the text explicitly does not use the critical-scaling formula (5.7) and instead computes masses from Eq. (5.5), arguing that perturbations are in the super-critical regime. The caption should cite Eq. (5.5).
  3. [Sec. 5.1 and Sec. 6] There are several small presentation errors: in Sec. 5.1 the reference appears as '[66?]'; in Sec. 6 'Einstein' is misspelled as 'Eintein'; and the statement that a preheating duration of 12-13 e-folds makes beta flat and equal to 1 is not supported by a figure or a quantitative derivation. These should be corrected or clarified.

Circularity Check

0 steps flagged · score 2.0 of 10

No definitional circularity: the PBH mass fractions are genuine outputs of the specified MS/Hill dynamics and the PS/KP formulas, and the external Hawking-evaporation constraints enter as benchmarks rather than as fitted inputs.

full rationale

Walking the derivation chain, I find no step in which a predicted quantity is identical by construction to an input, and no fitted parameter is renamed as a prediction. The instability band, Floquet exponents, and the allowed alpha range are imported from the authors' own prior work, e.g. Eq. (3.1) for l_inst is attributed to [27] and the alpha lower bounds are attributed to [26]; these are load-bearing self-citations, but they are independent prior physical results used as inputs, not fits to the evaporating-PBH constraints, and the paper also displays a full numerical Mukhanov-Sasaki solution in Fig. 5. The numerical threshold delta_c(k), defined as 'the minimum value of the density perturbation that reaches O(1)', is an operational collapse criterion calibrated from the same evolution that produces delta_k; this is a modeling assumption, not a parameter fitted to the PBH constraints, so the PS and KP abundances still emerge from the stated dynamics. The central PS-vs-KP comparison has a serious epoch ambiguity: the captions of Figs. 6 and 7 state that 'the evaluation is made 10 e-folds after the end of inflation', while the KP paragraph says 'the mass fraction is evaluated at the moment the perturbation crosses the IB and thus starts to amplify from a small value'. That inconsistency affects the reliability of the claimed exclusion of PS and viability of KP, but it is a correctness problem, not circularity: no equation in the paper reduces to its own input. The low score reflects the heavy self-reference in the input chain rather than any definitional or fitted-input circularity.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

No new particles, fields, or forces are introduced; PBHs and alpha-attractor potentials are pre-existing. The main inputs pulled from outside the paper are the self-resonance machinery from the authors' own [27], the R-delta relation from [9], the KP formula from [60], and the constraint curves from [6]. The papers own free choices are the O(1) threshold definition, the 10 e-fold preheating duration, gamma=1, and delta_max=1.

free parameters (4)
  • Nonlinearity threshold 'O(1)' = Unspecified (order unity)
    The k-dependent collapse threshold delta_c(k) is defined as the minimum initial density contrast that reaches O(1) during preheating; the exact value of the nonlinearity cutoff is not given and enters exponentially in Eq. (5.2).
  • Preheating duration = 10 e-folds after the end of inflation
    All mass fractions and constraint comparisons are evaluated at 10 e-folds after inflation; at 12-13 e-folds the PS beta saturates to 1, so the viability conclusion is sensitive to this choice (Sec. 5.1).
  • Collapse efficiency gamma = 1
    PBH mass is set to the horizon mass at IB entry, Eq. (5.5), with gamma=1; critical-scaling corrections are neglected because perturbations are claimed to be super-critical, but this is still a simplifying model choice.
  • Maximum density contrast delta_max = 1
    Upper cutoff in the Press-Schechter integral Eq. (5.2), chosen to avoid entering the nonlinear regime.
assumptions (7)
  • domain assumption The instability band and Floquet exponents from [27] apply to the T/E alpha-attractor potentials studied here.
    The IB scale l_inst (Eq. 3.1) and the Hill-equation/Floquet evolution (Eq. 4.5) are imported from the authors' prior paper; Fig. 5 is reproduced from [27] rather than re-derived in this work.
  • domain assumption Density contrast is related to curvature perturbation by Eq. (4.7), as established in [9] for Starobinsky inflation.
    The paper states this relation was shown numerically and analytically in [9] and uses it for all alpha values without a dedicated derivation here.
  • domain assumption The average sound speed approximation Eq. (3.7) is valid for oscillating scalar fields during preheating.
    Used for the Jeans criterion; validated only for two k values in Fig. 3.
  • domain assumption The collapse time for an overdensity is given by massive scalar field spherical collapse in Einstein-de Sitter, Eq. (3.9), and the threshold is the amplitude that reaches O(1).
    This converts the growth dynamics into a collapse threshold; the O(1) cutoff is not precisely defined.
  • domain assumption Density perturbations follow Gaussian statistics in the Press-Schechter integral, Eq. (5.2).
    Assumed without testing against the non-Gaussianities expected from parametric resonance.
  • domain assumption The KP abundance formula beta = 0.056 sigma^5 (Eq. 5.3) is valid for all relevant modes (sigma < 0.01).
    Taken from Harada et al. [60]; the paper asserts the condition is satisfied.
  • domain assumption Standard big bang cosmology applies to the evaporating-PBH constraints; early matter domination modifies them negligibly.
    Stated in Sec. 1 with a plausibility argument, but no quantitative calculation is given.

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Pith. "Pith review of Primordial black holes through preheating instabilities in $\alpha$-attractor models." pith.science (2026). https://pith.science/paper/QRVJAQHA

@misc{pith2026250517790,
  author       = {Pith},
  title        = {Pith review of: Primordial black holes through preheating instabilities in $\alpha$-attractor models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QRVJAQHA}},
  note         = {Machine review of arXiv:2505.17790}
}
abstract

In this work, we explore the production of primordial black holes (PBHs) within the context of $\alpha$-attractor inflationary models, focusing on the preheating phase following inflation. During this phase, self-resonance instabilities arise due to deviations of the inflationary potential from a quadratic form. PBH formation is analyzed using three criteria: (1) the perturbation must lie within the instability band, (2) its characteristic length must exceed the Jeans length, and (3) it must have sufficient time to collapse based on the estimations of massive scalar field spherical collapse in Einstein-de Sitter universe. Based on these criteria, we calculate the PBH mass fraction using the Press-Schechter (PS) and Khlopov-Polnarev (KP) formalisms. Our results show that the PS formalism tends to overestimate PBH abundance during preheating, as it neglects nonspherical effects. In contrast, the KP formalism yields more realistic predictions by incorporating such effects. We provide a detailed comparison with observational constraints from evaporating PBHs. Notably, the PS formalism is excluded by these constraints, which are based on Hawking radiation, while the KP formalism remains viable. These findings underscore the importance of accounting for nonspherical effects and accurate collapse dynamics in studies of PBH formation during preheating.

Figures

Figures reproduced from arXiv: 2505.17790 by the authors.

Figure 1
Figure 1. Normalized α-attractor potentials for (a) T-model and (b) E-model plotted for several values of α: 10−1 in blue, 10−2 in orange, 10−3 in green, and 10−4 in red. Press–Schechter formalism [28, 29], commonly used in radiation-dominated scenarios, and the Khlopov–Polnarev formalism [30–32], which is more appropriate for matter-dominated epochs. For estimating PBH masses, we adopt the standard approach of considering th… view at source ↗
Figure 2
Figure 2. Constraints on the mass fraction of evaporating PBH from Planck remnants (brown), LSP (magenta) CMB effects (entropy, distortions, and anisotropies, in blue), BBN (4He and D/H in orange), and γ-ray backgrounds (EGB in red and GGB in purple). The gray dashed vertical line marks the critical mass Mcrit. The figure is produced from the data presented in [6, 42] and references therein. We remark that these are just the … view at source ↗
Figure 3
Figure 3. Numerically averaged (red) and analytical (dashed black) sound speeds, eqns. (3.5) and (3.7), respectively, for two different modes: a) k = 50 kend and b) k = 350 kend. The underlying model is an E-model with α = 10−2.5 and the horizontal dotted lines mark the limiting values of the sound speed, i.e., ⟨c 2 s(k)⟩ = 1 during inflation and ⟨c 2 s(k)⟩ = 0 during preheating. In the high-k limit, both density and pressure… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Schematic representation of the collapse of perturbations during self-resonant preheating, fol￾Bo modes k1 and k2. See text for details. where gµν is the background space-time metric. To study the scalar perturbations, we introduce a fluctuation in the scalar field as …
Figure 5
Figure 5. Figure 5: Normalized curvature perturbations, k 3/2Rk, for a) T-model and b) E-model. Numerical computation is shown in continuous black and Floquet theory as described in [27] in dashed green. The vertical grey line marks the scale that exits the horizon at the end of inflation…
Figure 6
Figure 6. Figure 6: Mass fraction for (a) T-model and (b) E-model using both KP (5.3) and PS (5.2) formalisms in dashed and continuous, respectively. The evaluation is made 10 e-folds after the end of inflation and values of α go from 10−2 to 10−3.5 for the T-model and from 10−1.5 to 10−3…
Figure 7
Figure 7. Figure 7: Mass fraction for (a) T-model and (b) E-model using PS (dashed) and KP (continuous) formalisms as a function of the mass of the PBH formed, eqn. (5.7). Evaluations are made at 10 e-folds after the end of inflation. The red-shaded region represents the values of β(k) co…

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