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A scaling invariance of the perturbations in $k$-inflation models

T0 review · 2 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read k-inflation models admit a parameter rescaling that fixes the CMB normalization without changing the spectrum shape.

desk verdict The scaling symmetry is real, but the paper applies it with c0 upside down, so the numerical demonstration doesn't follow from its own equations. read the letter →

arxiv 2502.06456 v2 pith:XVSS3MKE submitted 2025-02-10 gr-qc astro-ph.CO

Neven Bilić , Dragoljub D. Dimitrijević , Goran S. Djordjevic , Milan Milošević , Marko Stojanović This is my paper · ORCID
classification gr-qcastro-ph.CO MSC 83F05 PACS 98.80.Cq
keywords k-inflationk-essencescalinginvariancecurvaturepowerspectrumMukhanov-SasakiequationBunch-Daviesvacuumprimordialblackholestachyonmodel
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 shows that k-inflation models with noncanonical kinetic terms carry a scaling invariance: multiplying the Lagrangian, the field, and the conjugate momentum by powers of one constant $c_0$ leaves the background Hamilton equations exactly unchanged. For the two model classes studied --- the $(X/2)^\alpha - U(\lambda\varphi)$ class with the PLLS inflection-point potential and the $-U(\lambda\varphi)F(X)$ class with the tachyon Lagrangian --- the same rescaling redefines the input parameters $V_0$, $\lambda$, $\phi_0$, $\phi_{\rm in}$, and $\eta_{\rm in}$ without changing the shape of the curvature power spectrum. The authors use this freedom to absorb the normalization mismatch that the standard Bunch-Davies vacuum (the usual vacuum choice for the quantum fluctuations) produces, so that the spectrum at the CMB pivot scale can satisfy $P_S(q_{\rm CMB})=A_s=2.1\times10^{-9}$ while preserving the predicted shape and the primordial-black-hole peak. If the claim is correct, the observed amplitude fixes the single constant $c_0$, and no additional normalization constant is needed.

What carries the argument

The engine of the argument is the simultaneous rescaling symmetry of the background Hamilton equations: $L\to c_0^{-1}L$, $\lambda\to c_0^{-1/2}\lambda$, $\varphi\to c_0^{1/2}\varphi$, $\eta\to c_0^{-1}\eta$ for a general k-essence, with the model-specific exponents (44) and (56) for classes A and B. The second key object is the Mukhanov-Sasaki equation (91), the second-order equation for the curvature perturbation mode $v_q=z\zeta_q$; each term in it is invariant under multiplying the Hubble rate $H$ by a constant, because ratios $H/H_0$, the slow-roll parameters $\varepsilon_i$, and the sound speed $c_s$ are all unchanged. This 'background rescaling passes through to the perturbation equation' property is what allows the normalization constant to be moved into a redefinition of input parameters.

What would settle it

Recompute the curvature power spectrum for the redefined parameters in Table 1 by directly integrating Eq. (91) with the Bunch-Davies initial conditions (96)-(99) and check whether $P_S(q_{\rm CMB})$ equals $2.1\times10^{-9}$; if the ratio differs from unity, or if it varies with $q$, the claimed invariance fails. A cheaper numerical test is to apply the rescaling (44) or (56) with two different values of $c_0$ and compare the full spectra pointwise.

Watch

Extended reading notes

Core claim

The paper's central claim is that the Hamilton equations (31)-(32), together with the class-specific systems (41)-(42) and (54)-(55), are invariant under the simultaneous rescalings (33), (44), and (56). Concretely, for Model A with $\alpha=1.5$ (the PLLS potential) and for the tachyon/DBI model, redefining $V_0$, $\lambda$, $\phi_0$, $\phi_{\rm in}$, and $\eta_{\rm in}$ by powers of a single constant $c_0$ preserves the background dynamics in rescaled variables, and the Mukhanov-Sasaki equation (91) is unchanged when $H$ is multiplied by a constant. The normalization error introduced by the Bunch-Davies initial conditions, $c_0=9.4732\times10^{10}$ for the PLLS model and $c_0=10.972$ for the tachyon model, can therefore be absorbed exactly by a parameter redefinition. The result is a curvature power spectrum with the correct amplitude at the CMB pivot scale and an unchanged shape, including the enhancement peak that can seed primordial black holes.

Load-bearing premise

The load-bearing premise is that a constant rescaling of $H$ leaves the perturbation equation (91) exactly unchanged and that the Bunch-Davies vacuum remains the correct initial condition for the rescaled parameters, so the normalization factor $c_0$ can be absorbed without altering the spectrum shape.

Editorial extensions

If this is right

  • For any parameter set in these two model classes, the predicted spectral shape, including the primordial-black-hole peak, is preserved when input parameters are redefined according to (44) or (56).
  • The Bunch-Davies vacuum remains the valid initial condition after the redefinition, so the rescaled model is a legitimate physical model rather than a purely formal relabeling.
  • The single constant $c_0$ is fixed by the observed CMB amplitude $A_s=2.1\times10^{-9}$, so no extra spectrum-normalization constant is needed.
  • The same rescaling applies to the slow-roll approximation (103), allowing fast approximate spectra that reproduce the shape of the exact numerical result, although with a factor 3-5 lower amplitude at the minimum.

Reading between the lines

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

  • Because the invariance is a property of the Hamilton equations and of Eq. (91) rather than of the specific potentials, the same normalization-absorbing redefinition should extend to other k-essence Lagrangians that admit multiplication by a constant; the paper only demonstrates classes A and B.
  • The rescaling changes the field values but not the horizon-crossing combination, so it offers a clean way to scan parameter space for primordial-black-hole abundances: normalize the CMB amplitude once, then vary only the shape parameters.
  • The paper's remark on one-loop backreaction implies that if small-scale loop corrections do alter the large-scale amplitude, the inferred $c_0$ would shift; the scaling symmetry would survive because the background is unaffected, but the physical prediction would need the loop correction included.
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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

2 major / 5 minor

Summary. The paper studies k-essence inflation models and claims a scaling invariance of the background Hamilton equations and of the associated curvature perturbation problem. Two classes are treated: Model A with Lagrangian L=(X/2)^α-U(λφ), exemplified by the PLLS inflection-point potential with α=1.5, and Model B with L=-U(λφ)F(X), exemplified by the tachyon/DBI Lagrangian. For each class the authors write a scaling of U, λ, φ, and η that leaves the background equations invariant, and they use this scaling to redefine input parameters so that the curvature power spectrum at the CMB pivot scale is forced to As=2.1×10^-9. Numerical spectra are plotted, approximate and exact spectra are compared, and nS versus r is compared with Planck 2018 data.

Significance. If the scaling invariance is correct, the paper offers a practical and inexpensive way to normalize k-inflation spectra to the CMB amplitude without altering the spectral shape, which is relevant for inflection-point models used in primordial black hole studies. The paper explicitly states that the pivot amplitude is fixed by construction rather than predicted, so the genuine external checks are the spectral shape, the spectral index and tensor-to-scalar ratio against Planck, and the comparison between the exact and approximate spectra. A concrete strength is the numerical work, including a documented comparison of integration methods. However, the current presentation contains a sign error in the stated scaling transformations that makes the numerical demonstration irreproducible as written; this must be corrected before the technical claim is assessable.

major comments (2)
  1. [Sec. 4.2, Eqs. (44), (56), and Table 1] The scaling direction in Eqs. (44) and (56) is inconsistent with the normalization procedure and with Table 1. For Model A, Eq. (44) gives U→c0^{-1}U and H→c0^{-1/2}H. Since the Bunch-Davies initial conditions (96)-(99) are independent of H and ε1, ε2, cs are invariant, the power spectrum amplitude scales as P∝H^2, so P→c0^{-1}P. Starting from PS=As/c0 in Eq. (100), this transformation yields As/c0^2, not As. The required transformation is the inverse, H→c0^{1/2}H, i.e. U→c0 U, λ→c0^{(α-1)/(2α)}λ, φ→c0^{(1-α)/(2α)}φ, η→c0^{(2α-1)/(2α)}η. Table 1 indeed uses the inverse: for α=1.5, V0 is multiplied by 9.4732×10^10, λ by 67.5≈c0^{1/6}, φ divided by 67.5, and η multiplied by 2.08×10^7≈c0^{2/3}. The same sign error appears in the sentence after Eq. (101) ('absorb c0 in H^2 and rescale the Hubble rate as H→c0^{-1/2}H'); the correct rescaling is H→c0^{1/2}H. This is a load-bearing error because the paper's numerical demonstration cannot be reproduced by following the stated equations.
  2. [Sec. 4.2, Eqs. (73), (76), (81), (91)] The claim that the rescaling 'does not affect Eq. (91)' is incomplete because the horizon-crossing and initial-condition relations involve the scale factor a0. Equation (76) fixes a0H0=ε cs0 q_CMB, Eq. (73) fixes ainHin=β cs q, and Eq. (81) is a0 e^N H=cs q. Under a constant rescaling of H and H0, the ratios H0/H in Eq. (91) remain invariant, but a0 in Eqs. (73), (76), and (81) does not transform. Unless a0 is rescaled inversely (or ε and the e-fold origin N0 are redefined consistently), the q-to-N map (92) and the initial subhorizon condition (73) change, so the shape-preservation claim is not guaranteed, in particular through the ultra-slow-roll phase. The paper should specify how a0, N0, or ε transform under (44)/(56) and verify that the resulting spectra are unchanged.
minor comments (5)
  1. [Sec. 4.2, Eq. (100)] The normalization to As is imposed by construction: c0 in Eq. (100) is chosen so that after redefinition the spectrum matches the observed pivot amplitude. This is a tuning consistency condition rather than a prediction, and the paper would benefit from stating this explicitly in Sec. 4.2 and in the conclusions to avoid implying the amplitude is derived.
  2. [Fig. 2 caption] The caption contains a typo: 'Tachyom model' should be 'Tachyon model'.
  3. [Sec. 4.4, paragraph after Eq. (107)] The statement that the approximate spectrum is 'consistently lower by a factor of 3-5' is not quantified with an error measure or a comparison at the pivot scale. Since the normalization is enforced in both cases, a quantitative measure (e.g., the ratio at the peak and at the pivot) would clarify the quality of the slow-roll approximation.
  4. [Sec. 5, one-loop backreaction paragraph] The paper correctly acknowledges that one-loop corrections could alter the curvature perturbation amplitude on CMB scales. This caveat is important and should also appear in the introduction or abstract, since the PBH-motivated peak amplitude is a key motivation.
  5. [General notation] The notation φ for a field of dimension length and ϕ for the physical field of dimension mass is introduced, but in several places (e.g., Sec. 3.1) the text uses φ′ and H without consistently recalling that derivatives are with respect to e-folds N; defining prime once after Eq. (45) and repeating it near Eq. (68) would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the scaling invariance is derived in-paper, and the CMB normalization is an openly fitted consistency condition, not a predicted amplitude.

full rationale

The derivation of the background scaling invariances is self-contained: Eqs. (33), (44), and (56) are verified by direct substitution into the Hamilton equations, and the perturbation-sector invariance of Eq. (91) under a constant rescaling of H is checked term-by-term in Sec. 4.2, with c_s, epsilon1, and epsilon2 invariant. The subsequent absorption of the normalization factor c0 is an openly acknowledged fitting step, not a hidden prediction: Eq. (100) defines c0 by the mismatch PS(q_CMB)=As/c0, and the redefined parameters in Table 1 are chosen so that the target amplitude is obtained by construction. Because the paper does not claim to derive As from first principles, but only that the scaling can be used to enforce the observed normalization while preserving the shape, this is not circular. The independent content (spectrum shape, peak position, and the nS/r comparison against Planck) is externally checked in Fig. 2. No load-bearing self-citation or imported uniqueness theorem was found. One technical caveat, not a circularity, is that the direction of the scaling as written in Eqs. (44)/(56) (H -> c0^{-1/2} H) is opposite to the direction needed for, and used in, Table 1 (H -> c0^{1/2} H); this appears to be a sign/direction error in the presentation of the scaling law rather than a circular reduction.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The central derivation relies on standard Friedmann cosmology, linear perturbation theory, and the Bunch-Davies vacuum. The free parameters are dominated by the normalization constants c0 chosen to impose the observed pivot amplitude, plus arbitrary numerical parameters and the chosen model inputs from Ref. [17]. No new particles, forces, or entities are introduced.

free parameters (6)
  • c0 (PLLS model) = 9.4732e10
    Normalization constant chosen so that PS(q_CMB)=As=2.1e-9; absorbed by the parameter rescaling in Eq. (44).
  • c0 (Tachyon model) = 10.972
    Normalization constant chosen so that PS(q_CMB)=As; absorbed by the parameter rescaling in Eq. (56).
  • epsilon (initial horizon-crossing ratio) = 0.001
    Arbitrary small parameter setting a0H0 relative to cs0 qCMB; the paper notes the dependence can be removed by choosing a negative N0.
  • beta (subhorizon initial ratio) = 0.01
    Chosen from Ref. [28] to define the deep-subhorizon initial point; the final spectrum should be beta-independent, but numerics depend on it.
  • Original model inputs for PLLS (V0, lambda, phi0, phi_in, eta_in) = 1e-16, 1.961e-8, 0.835, 5.3, -1.53e-8
    Taken from the PLLS model [17]; chosen to produce an inflection-point ultra-slow-roll phase, not derived in this paper.
  • Original model inputs for Tachyon (V0, lambda, phi0, phi_in, eta_in) = 1e-16, 1.961e-8, 0.2586, 1.1, -3e-5
    Chosen by hand following the PLLS potential; required for the numerical demonstration but not part of the scaling theorem.
assumptions (6)
  • standard math Friedmann equations relate the Hamiltonian to H and ⁻H (Eqs. 19-20).
    Used throughout to derive background evolution, sound speed, and slow-roll parameters.
  • standard math Linear cosmological perturbation theory of Garriga and Mukhanov [26] governs the curvature perturbation, with the Mukhanov-Sasaki variable vq=zζq.
    Eqs. (65)-(68) and (91) rely on this framework.
  • domain assumption The Bunch-Davies vacuum state (94) provides the initial conditions in the deep subhorizon.
    Eqs. (96)-(99); if the physical vacuum were modified, the normalization transfer would change.
  • domain assumption The potential U(λφ)=V(λφ)-V(0) with the PLLS inflection-point form satisfies the same scaling as the full Lagrangian, including its constant subtraction.
    Required for Eq. (44) to apply to the PLLS potential; if V(0) breaks the scaling, the parameter redefinition changes the spectrum shape.
  • domain assumption Slow-roll formulas nS ≈ 1 - 2ε₁ - ε₂ and r ≈ 16cₛε₁ remain adequate for the comparison with Planck.
    Used in Sec. 4.4; during ultra-slow-roll near the inflection point, slow-roll may be violated, so these values are approximate.
  • domain assumption One-loop backreaction from small scales is assumed not to alter the background or the scaling property, though it may shift the amplitude.
    Explicitly flagged in Sec. 4.4 as beyond the scope; the authors state it could significantly alter curvature perturbations.

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Pith. "Pith review of A scaling invariance of the perturbations in $k$-inflation models." pith.science (2026). https://pith.science/paper/XVSS3MKE

@misc{pith2026250206456,
  author       = {Pith},
  title        = {Pith review of: A scaling invariance of the perturbations in $k$-inflation models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XVSS3MKE}},
  note         = {Machine review of arXiv:2502.06456}
}
abstract

We study the background and perturbations in $k$-essence inflation models and show that a general $k$-essence exhibits a simple scaling property. In particular, we study two classes of $k$-inflation models with the potential characterized by an inflection point. We demonstrate that these models enjoy scaling properties that could be used to redefine input parameters so that the perturbation spectra satisfy correct normalization at the CMB pivot scale. The background and perturbation equations are integrated numerically for two specific models.

Figures

Figures reproduced from arXiv: 2502.06456 by the authors.

Figure 1
Figure 1. The curvature power spectrum obtained by numerically solving Eq. (91) (full line) combined [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. r versus nS diagram with observational constraints from Ref. [31]. The dots represent the theoretical predictions of the PLLS model (blue) and Tachyom model (black). The dots are obtained by varying the parameters ϕ0 and λ of the potential and initial values ϕin and ηin as discussed in the text. r of each model with current observational constraints. To this end, we compute nS and r for each model for different sets… view at source ↗

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