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$\delta n$ formalism: A new formulation for the probability density of the curvature perturbation

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Counting e-folds forward from a fixed initial slice turns the curvature perturbation's probability density into a one-dimensional change of variables, $P_{\delta n}(\delta n)=P_{\delta\phi}(\delta\phi_0(\delta n))\,|d\delta\phi_0/d\delta…

desk verdict A genuinely simpler route to the curvature perturbation PDF, but the worked example has algebra errors and the error from the perfect-correlation assumption is unquantified. read the letter →

arxiv 2505.24590 v2 pith:YNI26QFJ submitted 2025-05-30 astro-ph.CO

Diego Cruces , Shi Pi , Misao Sasaki This is my paper
classification astro-ph.CO
keywords deltaNformalismcurvatureperturbationprobabilitydensityfunctionseparateuniverseapproachprimordialblackholesultra-slow-rollinflationconstant-rolle-foldingnumber
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 proposes a reformulation of the standard $\delta N$ formalism in which the e-folding number is counted forward in time rather than backward, calling the resulting object $\delta n$. Its central claim is that the probability density function of the curvature perturbation is then given by a one-dimensional change of variables, $P_{\delta n}(\delta n) = P_{\delta\phi}(\delta\phi_0(\delta n))\,|d\delta\phi_0/d\delta n|$, once the inflaton velocity fluctuation is treated as fully correlated with the field fluctuation after horizon exit. The practical gain is that computing the inverse function $\delta\phi_0(\delta n)$ is much easier than computing the forward function $\delta n(\delta\phi_0)$, which has limited the standard $\delta N$ formalism to simple potentials. A precise tail of the curvature PDF matters because it controls the predicted abundance of primordial black holes. The paper shows the formula reproduces known attractor and ultra-slow-roll PDFs and can accommodate trajectories in which inflation never ends.

What carries the argument

The central object is the $\delta n$ formalism: a version of the separate-universe ($\delta N$) calculation that counts e-folds $n$ forward from a fixed initial flat hypersurface to the comoving end-of-inflation hypersurface, so that $\delta n$ equals the curvature perturbation. The load-bearing identity is the full-correlation relation $\delta\pi_0 = \pm(\sigma_{\delta\pi,\delta\pi}/\sigma_{\delta\phi,\delta\phi})\,\delta\phi_0$, which replaces the two-dimensional initial-condition distribution with a single Gaussian variable. The key computational step is inverting the final-hypersurface condition $\phi(n_f, \phi_0+\delta\phi_0, \pi_0+\delta\pi_0) = \bar\phi_f$ to obtain $\delta\phi_0$ as a function of $\delta n$; this inverse feeds the change-of-variables formula for the PDF.

What would settle it

Take a single-field model in which the full function $\delta n(\delta\phi_0, \delta\pi_0)$ can be computed exactly, and compare the PDF from the exact two-dimensional integral using the full covariance matrix at finite $\sigma$ (for example $\sigma = 0.1$) with the PDF from the one-line formula. If the two disagree in the tail by more than a perturbative $\sigma^2$ correction, the perfect-correlation reduction is the culprit; repeating at several $\sigma$ values would map where the simplified formula breaks down.

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Extended reading notes

Core claim

The discovery is that the PDF of the curvature perturbation at the end of inflation can be computed from the field fluctuation alone, bypassing the velocity fluctuation. Because on superhorizon scales the quantum field and momentum fluctuations are fully correlated, $\delta\pi_0 = \pm(\sigma_{\delta\pi,\delta\pi}/\sigma_{\delta\phi,\delta\phi})\,\delta\phi_0$, the two-dimensional Gaussian integral defining $P_{\delta n}$ collapses to one dimension, giving $P_{\delta n}(\delta n) = P_{\delta\phi}(\delta\phi_0(\delta n))\,|d\delta\phi_0/d\delta n|$. The function $\delta\phi_0(\delta n)$ is obtained by solving the comoving-slice condition, which sets the value of the inflaton on the final hypersurface equal to its homogeneous background value; solving this condition for $\delta\phi_0$ in terms of $\delta n$ is the simplification that makes the method tractable. In the attractor and ultra-slow-roll limits the formula reduces to previously known PDFs, including the exponential-tail form, and it reproduces the exact equality $\mathcal{R} = \delta N = \delta n$.

Load-bearing premise

The shortcut rests on the premise that after a mode leaves the Hubble horizon the inflaton's velocity fluctuation is exactly proportional to its field fluctuation, so only one initial condition matters; in reality the proportionality is exact only in the strict superhorizon limit, and the error at finite scales is left unquantified.

Editorial extensions

If this is right

  • If the formula is correct, the curvature PDF for any single-field model can be obtained by numerically evolving background trajectories with perturbed initial conditions until each hits the fixed end-of-inflation field value, then weighting each trajectory by the Gaussian PDF of $\delta\phi_0$.
  • The derived PDF reproduces the established attractor and ultra-slow-roll results, including the exponential tail, so the new method inherits those limits as consistency checks.
  • Because e-folds are counted forward, trajectories for which inflation never ends are naturally included in the integration, whereas the backward-time $\delta N$ formalism discards them.
  • The method turns the often difficult task of expressing $\delta n(\delta\phi_0, \delta\pi_0)$ into the simpler task of expressing $\delta\phi_0(\delta n)$, which is the main practical advantage claimed by the paper.
  • In the single-kick limit the forward-time computation is equivalent to the first-passage-time analysis of stochastic $\delta N$, suggesting a closer link between the classical and stochastic PDFs.

Reading between the lines

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

  • One testable extension the authors only gesture at is a quantitative error budget: for a mode at finite $\sigma$ the correlation coefficient $|\rho|$ is slightly below 1, and comparing the full two-dimensional integral with the one-line formula at fixed $\sigma$ would show how much of the tail is corrupted by the perfect-correlation approximation.
  • Because the new scheme reduces the PDF computation to a sequence of background integrations, it should make the non-Gaussian tail of $\mathcal{R}$ cheap to evaluate for potentials with nontrivial features, potentially enabling parameter scans aimed at primordial black hole abundances.
  • The forward-time logic aligns the classical computation with stochastic-$\delta N$ first-passage problems beyond the single-kick case, so one testable consequence is that multi-kick stochastic tails and the $\delta n$ tails agree when the stochastic kicks are taken one horizon crossing at a time.
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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 / 4 minor

Summary. The paper reformulates the δN formalism by counting e-folds forward in time. For single-field inflation, it argues that on superhorizon scales the inflaton field and momentum perturbations are exactly correlated, so the two-dimensional Gaussian initial PDF collapses to a one-dimensional Gaussian. This yields Eq. (22): the PDF of δn is the initial field PDF transformed by the inverse function δϕ0(δn). The method is illustrated on constant-roll and ultra-slow-roll models, where Eqs. (32) and (34) are claimed to reproduce known PDFs. The formalism is also connected to stochastic δN and first-passage-time problems.

Significance. The forward-in-time formulation is a conceptually useful reorganization of the δN formalism and, if correct, would simplify computations of curvature-perturbation PDFs, particularly for models where δn(δϕ0) is not analytic. The paper explicitly identifies the inverse-function simplification and the connection to stochastic inflation as selling points; these strengths are real. However, the validation is internally inconsistent and the key correlation reduction is uncontrolled, so the central claim is not yet established.

major comments (3)
  1. [The δn formalism, Eqs. (17)–(22)] The reduction of Eq. (12) to Eq. (22) rests on Eq. (20), which sets the conditional variance of δπ0 given δϕ0 to zero. For any finite σ ≡ k/(aH) at the initial time, det Σ/σϕϕ² is nonzero, and the residual δπ0 is dropped without an error estimate. This is not merely a formal point: in the flat-potential or near-critical regimes relevant for PBH formation, ∂δn/∂δπ0 can be exponentially large, so an O(σ) residual in δπ0 can shift δn by O(1) and change the tail probability by orders of magnitude. The conclusion acknowledges that higher-order gradient corrections break the complete correlation, but the paper does not bound the resulting error in Pδn. A quantitative estimate of the neglected term, or an explicit restriction of the claim to leading order in σ with a remainder estimate, is needed.
  2. [Constant-roll example, Eqs. (29)–(34)] The claimed recovery of known PDFs fails algebraically. Substituting the attractor assignments of Eq. (31) into Eq. (29) gives δϕ0/σδϕ = (ϕ̄f/|C_-^f|)(e^{λ_- δn} - 1). Inserting this into Eq. (30) yields a prefactor (λ_- ϕ̄f/|C_-^f|) e^{λ_- δn}/√(2π), whereas Eq. (32) has the reciprocal prefactor |C_-^f/(λ_- ϕ̄f)| e^{λ_- δn}/√(2π). The same reciprocal mismatch appears in the USR limit between Eq. (29) and Eq. (34). Since the constant-roll and USR limits are the only validation of the new formula, this inconsistency must be resolved before the central claim can be assessed.
  3. [A simplified method, Eq. (22) and Fig. 2] Equation (22) assumes that δϕ0 is a single-valued function of δn on the support of the initial PDF. In the attractor example this fails for δϕ0 below the threshold corresponding to δn → -∞: such trajectories never reach the comoving final slice, and the paper gives no prescription for their probability weight or for the branch structure of δϕ0(δn). The abstract claims these trajectories are naturally incorporated, but no formula or normalization condition for them is provided. This affects the normalization and the tail of Pδn and should be addressed explicitly.
minor comments (4)
  1. [Eq. (21)] The exponential factor in Eq. (21) is redundant once the delta function is imposed; if it is intended as a regularized representation of the delta, the limit σ → 0 should be stated explicitly.
  2. [Introduction and Eq. (9)] The sign convention for π is inconsistent: the Introduction defines π = −dϕ/dN, while Eq. (9) writes ∂ϕ/∂n = π. This should be reconciled because all subsequent formulas depend on it.
  3. [Text after Eq. (25)] The sentence 'which is generically very difficult δϕ0 [29]' contains a stray δϕ0; it should read 'which is generically very difficult to solve for δϕ0'.
  4. [Eq. (20) and Eq. (25)] Equation (20) leaves the sign as ±, but the text after Eq. (19) says it is fixed by Re[δϕδπ*]; the sign should be carried explicitly through Eq. (25) and Eq. (29), since it can change the resulting map.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the δn PDF formula is derived from the superhorizon correlation of δϕ and δπ and checked against external benchmark PDFs; self-citations are contextual, not load-bearing.

full rationale

The central derivation is self-contained. Eq. (6) defines δn, Eq. (7) identifies it with R using standard slicing conditions, and Eq. (11) gives δn as a function of perturbed initial field data. Eq. (12) is an exact probability-conservation integral. The Gaussian initial PDF (14) comes from linear perturbation theory. The key reduction to a one-dimensional problem, Eq. (22), follows mathematically from the superhorizon full-correlation statement (17)-(20), which is attributed to established literature on squeezed inflationary perturbations, not to a self-citation. The constant-roll and ultra-slow-roll PDFs (32) and (34) are derived from the same EOMs and then compared with known literature results, which is an external benchmark check, not an input to the derivation. No fitted parameter is relabeled as a prediction, and no uniqueness theorem from the authors' prior work is invoked to force the formalism. The paper's explicit caveat that higher-order gradient corrections break the complete correlation and are left for future work is an acknowledged approximation, not a circular step. Self-citations such as [47], [61], and [66] appear in contextual statements or as comparisons and do not carry the derivation. Thus the paper is not circular; concerns about the uncorrelated residual at finite superhorizon scale are accuracy/validity issues, not circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are introduced. The method relies on standard single-field slow-roll machinery plus the separate universe approximation and the superhorizon correlation property. The assumption that the inverse function is single-valued is imposed without justification.

assumptions (5)
  • domain assumption Separate universe approximation (leading order in gradient expansion)
    The derivation in Eqs. (4)-(7) assumes SUA; the authors acknowledge in the Conclusion that the method fails if the scale is not far outside the Hubble horizon.
  • domain assumption Field and velocity fluctuations are fully correlated on superhorizon scales, ρ = ±1
    Used to reduce the 2D Gaussian to a 1D distribution in Eq. (20). It is derived from the perturbation equation in the σ→0 limit but is an approximation with unspecified sign.
  • domain assumption Initial field and velocity perturbations are Gaussian with covariance from linear perturbation theory
    The joint PDF in Eq. (14) assumes Gaussianity and no intrinsic non-Gaussianities, which are deferred to future work.
  • domain assumption Final comoving hypersurface is defined by a homogeneous field value ϕ = ϕ̄f
    Used to impose Eq. (24). Assumes inflation ends on a comoving slice where δϕ = 0, standard for single-field inflation but not always the end condition.
  • ad hoc to paper The map δϕ0(δn) is single-valued
    Eq. (22) requires a one-to-one relation to avoid summing over branches. The paper does not prove monotonicity or account for multiple roots.

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Cite this review

Pith. "Pith review of $\delta n$ formalism: A new formulation for the probability density of the curvature perturbation." pith.science (2026). https://pith.science/paper/YNI26QFJ

@misc{pith2026250524590,
  author       = {Pith},
  title        = {Pith review of: $\delta n$ formalism: A new formulation for the probability density of the curvature perturbation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YNI26QFJ}},
  note         = {Machine review of arXiv:2505.24590}
}
abstract

$\delta N$ formalism is a useful method to calculate the curvature perturbation. Contrary to what it is typically done in the literature, we re-formulate the $\delta N$ formalism by using the $e$-folding number $n$ counted forward in time. For a fixed initial time $\bar{n}_0$, the probability density function (PDF) of the field perturbation $\delta\phi_0$ and its velocity $\delta\pi_0$ are specified by the solutions of the perturbation equation on subhorizon scales. As $\delta\pi_0$ is fully correlated with $\delta\phi_0$ after horizon exit, we find a novel $\delta n$ formalism to calculate the curvature perturbation as well as its PDF. It can naturally incorporate those trajectories for which inflation never ends, which are lost when counting $N$ backward in time.

Figures

Figures reproduced from arXiv: 2505.24590 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic picture showing the equivalence of [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The procedure to obtain the PDF of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

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Forward citations

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