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arxiv: 2605.10656 · v1 · submitted 2026-05-11 · 🌌 astro-ph.CO · gr-qc· hep-ph· hep-th

Recognition: 2 theorem links

· Lean Theorem

Multifield stochastic inflation: Relevance of number of fields in statistical moments

Tomo Takahashi , Koki Tokeshi
Authors on Pith no claims yet

Pith reviewed 2026-05-12 05:09 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qchep-phhep-th
keywords multifield inflationstochastic inflationnumber of fieldse-folds momentsO(d) symmetryprimordial power spectruminflation termination
0
0 comments X

The pith

Stochastic effects in multifield inflation make the number of scalar fields affect the statistics of e-folds and primordial perturbations, unlike the classical case.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

In multifield inflation driven by d scalar fields with O(d) symmetry, the classical equations of motion and resulting predictions remain unchanged no matter how many fields are present. Once quantum fluctuations are treated as stochastic noise, this independence disappears and the moments of the duration of inflation become sensitive to d. The authors derive a general analytical expression for the nth statistical moment of the stochastic number of e-folds that holds at every perturbative order in the small-noise limit while keeping the d dependence explicit. Requiring that inflation ends successfully then imposes a finite upper limit on d. The same d dependence appears in the power spectrum and its scale dependence, as illustrated in several explicit O(d)-symmetric examples.

Core claim

While O(d) symmetry renders the number of fields irrelevant at the classical level, stochastic noise introduces an explicit dependence on d into all higher-order moments of the stochastic e-fold number N. A closed-form perturbative formula is obtained for arbitrary moments at all orders, preserving full analytical control over d. The requirement that the inflaton trajectory reaches the end of inflation then yields an upper bound on d.

What carries the argument

The general perturbative formula for arbitrary higher-order moments of the stochastic number of e-folds, which keeps the dependence on the number of fields d fully analytical at every order in the small-noise expansion.

If this is right

  • The mean, variance, and all higher moments of the stochastic e-fold number acquire explicit d dependence.
  • The primordial power spectrum and its spectral tilt receive stochastic corrections that scale with d.
  • Successful termination of inflation requires the number of fields to lie below an upper theoretical bound.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The closed-form d dependence allows any other O(d)-symmetric potential to be analyzed for its moment structure without repeating the full derivation.
  • The same machinery can be applied to compute stochastic corrections to other observables, such as the bispectrum, once the appropriate delta-N expansion is inserted.
  • The upper bound on d supplies a new consistency condition that can be checked against concrete string-theory constructions that realize large numbers of scalar fields.

Load-bearing premise

The small-noise perturbative expansion captures all relevant stochastic corrections without non-perturbative effects or higher-order noise terms becoming important.

What would settle it

Numerical solution of the stochastic differential equations for an O(d)-symmetric potential, extraction of the moments of N for several values of d, and direct comparison against the derived analytical formula.

Figures

Figures reproduced from arXiv: 2605.10656 by Koki Tokeshi, Tomo Takahashi.

Figure 1
Figure 1. Figure 1: shows the integrand in Eq. (2.21), the behaviour of which is determined by the ratio d/p. An interesting behaviour in ⟨N⟩ arises in the limit where the reflective boundary is sent to infinity. In this limit, Eq. (2.21) converges to give a finite number of the mean number of e-folds if and only if d/p < 1. When d/p = 1, on the other hand, it diverges logarithmically, and the divergence becomes worse for d/p… view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The summation over [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The effective force [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The bound on the number of fields as a function of [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. ( [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
read the original abstract

In multifield inflation driven by $d$ scalar fields, $O (d)$ symmetry renders the number of fields irrelevant at classical level. This ceases to be the case once stochastic effects are accommodated. The statistical quantities such as the mean number and the variance of $e$-folds as well as the primordial power spectrum and its scale dependence are perturbatively calculated in a small-noise regime. In particular, a general formula is derived for arbitrary higher-order statistical moments of the stochastic number of $e$-folds at all perturbative orders, keeping the dependence on the number of fields fully analytical. It is also discussed that the requirement for inflation to be successfully terminated puts a theoretical bound on the number of fields from above. Those general results are demonstrated for several $O (d)$-symmetric models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies multifield stochastic inflation with d scalar fields under O(d) symmetry. While the number of fields is irrelevant classically, stochastic noise makes d relevant. The authors perturbatively compute statistical quantities (mean and variance of e-folds, primordial power spectrum and tilt) in the small-noise regime, derive a general all-order formula for arbitrary higher moments of the stochastic number of e-folds that retains explicit analytic d dependence, and argue that successful termination of inflation imposes an upper theoretical bound on d. Results are illustrated on several O(d)-symmetric models.

Significance. If the central derivations hold, the work supplies a useful closed-form perturbative framework for higher statistical moments in multifield stochastic inflation with explicit d dependence. The termination bound on d is a novel theoretical constraint that could inform model viability. The all-order moment formula is a clear technical strength, as is the retention of analytic d dependence rather than numerical fitting.

major comments (2)
  1. [§3] §3 (radial Langevin equation): the Itô correction produces an extra drift term ∼(d−1)(H/2π)^2/r. This term grows linearly with d even when the per-field noise amplitude is small, so the small-noise ordering assumed for the perturbative moment expansion can fail exactly in the regime where the termination bound on d is applied. The manuscript demonstrates results for several models but does not quantify the d range in which neglected higher-order noise corrections remain sub-dominant.
  2. [§5] §5 (termination bound): the upper bound on d is derived from the requirement that inflation ends successfully, yet this bound is applied without an accompanying check that the small-noise perturbative expansion remains valid at those d values. A concrete estimate of the d at which the Itô drift becomes O(1) relative to the classical drift would be needed to substantiate the bound.
minor comments (2)
  1. [Abstract] The abstract states that the primordial power spectrum and its scale dependence are calculated, but the main text emphasis is on moments of N; clarify whether these spectral quantities are derived from the same moment formula or treated separately.
  2. [Notation] Notation: the symbol for the number of fields (d) and the stochastic e-fold variable (N) should be introduced once with consistent font and usage throughout the equations and text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments on the validity of the small-noise perturbative expansion. We address the major comments point by point below. We agree that quantitative checks on the regime of validity are needed and will incorporate the requested estimates in the revised manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (radial Langevin equation): the Itô correction produces an extra drift term ∼(d−1)(H/2π)^2/r. This term grows linearly with d even when the per-field noise amplitude is small, so the small-noise ordering assumed for the perturbative moment expansion can fail exactly in the regime where the termination bound on d is applied. The manuscript demonstrates results for several models but does not quantify the d range in which neglected higher-order noise corrections remain sub-dominant.

    Authors: We agree that the Itô drift term scales linearly with (d−1) and that this could affect the small-noise ordering for sufficiently large d. Our perturbative expansion is performed in the noise amplitude with d kept explicit, but we acknowledge that an explicit quantification of the d range where the Itô term remains subdominant to the classical drift was not provided. In the revised manuscript we will add a dedicated paragraph (or subsection) deriving the condition under which the Itô correction is negligible compared with the classical radial drift for each of the O(d)-symmetric models considered. This will include an estimate of the maximum d for which higher-order noise corrections stay perturbatively small, thereby clarifying the domain of applicability of our results. revision: partial

  2. Referee: [§5] §5 (termination bound): the upper bound on d is derived from the requirement that inflation ends successfully, yet this bound is applied without an accompanying check that the small-noise perturbative expansion remains valid at those d values. A concrete estimate of the d at which the Itô drift becomes O(1) relative to the classical drift would be needed to substantiate the bound.

    Authors: We accept that the termination bound on d must be accompanied by a verification that the small-noise expansion remains valid at those values. In the revised version we will supply a concrete estimate by evaluating the ratio of the Itô drift term to the classical drift term at the d values obtained from the termination condition. For each model we will derive the inequality (d−1)(H/2π)^2/r ≪ |classical drift| and evaluate it numerically or analytically at the boundary d, thereby either confirming that the quoted bounds lie inside the perturbative regime or indicating the adjusted range where both the termination and small-noise conditions are simultaneously satisfied. revision: yes

Circularity Check

0 steps flagged

Derivation of perturbative moments from stochastic equations is self-contained

full rationale

The paper starts from the standard multifield Langevin equations under O(d) symmetry, applies Ito calculus to obtain the radial drift, and performs a perturbative expansion in the small-noise parameter to obtain all-order moments of the e-fold number N_e while retaining explicit d dependence. No quoted step reduces a claimed prediction to a fitted input by construction, nor does any load-bearing premise rest on a self-citation chain whose validity is internal to the present work. The termination bound on d is presented as a separate consistency requirement applied to the derived moments rather than an input that forces the result. The derivation therefore remains independent of the target statistical quantities.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Limited to abstract; relies on O(d) symmetry as a domain assumption for the models and on the validity of perturbative expansion in small noise. No explicit free parameters or invented entities stated.

axioms (2)
  • domain assumption O(d) symmetry in the scalar potential for the multifield models
    Invoked to establish classical irrelevance of d and to keep calculations analytical.
  • domain assumption Validity of perturbative expansion in the small-noise regime
    Used to calculate mean, variance, power spectrum, and higher moments to all orders.

pith-pipeline@v0.9.0 · 5437 in / 1490 out tokens · 52083 ms · 2026-05-12T05:09:34.192359+00:00 · methodology

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Reference graph

Works this paper leans on

107 extracted references · 107 canonical work pages · 1 internal anchor

  1. [1]

    It then follows from Eq

    Zeroth order As was already confirmed near the end of Section III B, at zeroth order, only the combination of the dummy indices k1 = k2 = k3 = k4 = 0 is relevant, as can also be seen from N1(0) = 1. It then follows from Eq. (4.2) that d ⟨N⟩(0) (x)/dx≃ 1/f1(x). This appropriately reproduces the classical number ofe-folds, given by Eq. (3.9)

    work page

  2. [2]

    (4.2) must be −k3 + k4 = 1

    First order At first order in v, the exponent of v(x) in Eq. (4.2) must be −k3 + k4 = 1. There exist N1(1) = 3 terms in Eq. (4.2), and the combinations of integers are exhausted by (k1, k 2, k 3, k 4) =    (0,0,0,1), (0,1,1,2), (1,0,0,1). (4.4) The corresponding diagrammatic factors are D0,0 (x) = 1 andD 1,1 (x) = (−)2f2(x)/2! only. The result then...

    work page

  3. [3]

    Second order In deriving the second-order perturbative term in ⟨N⟩, there are N1(2) = 7 relevant terms that must be included in Eq. (4.2). The corresponding combinations of the integers are summarised in Appendix. Summing up all those contributions, the result is given by d⟨N⟩ (2) (x) dx = [v(x)]2 v(x) v′(x) 3 (d−1)(d−2) 4x2 + d−1 x v′(x) v(x) − 3 4 v(x) ...

    work page

  4. [4]

    (4.7) below

    Third order The third-order calculation includes N1(3) = 14 terms (again see Appendix), and a careful calculation leads to Eq. (4.7) below. As the pattern can be read off from Eqs. (4.5) and (4.6), the third-order result (4.7) involves the term cubic in the number of fields, which vanish at d= 1, 2, and also 3. d⟨N⟩ (3) (x) dx = [v(x)]3 v(x) v′(x) 4 (d−1)...

    work page

  5. [5]

    However, writing down the explicit forms of all the factors becomes more and more complicated when k gets larger

    General order In principle, the procedure employed until here applies to calculate O(vk) term in the mean number of e-folds for arbitrary k≥ 0, collecting the terms corresponding to (k1, k 2, k 3, k 4) that give rise to the desired order of v. However, writing down the explicit forms of all the factors becomes more and more complicated when k gets larger....

    work page

  6. [6]

    The combination of the indices in Eq

    Zeroth order At zeroth order in v, the evolution of the inflaton fields is deterministic, so that there is no fluctuations in the number of e-folds and its variance must vanish. The combination of the indices in Eq. (4.13) at this order is only k1 = · · · = k6 = 0, corresponding to N2(0) = 1. This gives rise to 1 2 d⟨N 2⟩(0) (x) dx ≃ ⟨N⟩(0) (x) f1(x) = 1 ...

    work page

  7. [7]

    First order Though Eq. (4.14) tells us that the number of the terms increases as k5, at first order, there are only 5 relevant combinations,i.e.,N 2(1) = 5, (k1, k 2, k 3, k 4, k 5, k 6) =    (0,0,0,0,0,1), (0,0,0,0,1,0), (0,0,0,1,0,1), (0,1,1,0,0,2), (1,0,0,0,0,1). (4.17) Summing up all the contributions, and then simplifying the expre...

    work page

  8. [8]

    All those combinations of the indices can be found in Appendix, and subtraction of the total-derivative terms from ⟨N2⟩ (r) as was done in Eq

    Second order The second-order calculation requires collecting 16 terms,i.e., N2(2) = 16. All those combinations of the indices can be found in Appendix, and subtraction of the total-derivative terms from ⟨N2⟩ (r) as was done in Eq. (4.18) arrives at d dx δN2 (2) (x) = [v(x)]2 v(x) v′(x) 4 3(d−1) 2x + 3v′(x) v(x) − 5 2 v(x) v′(x) v′′(x) v(x) .(4.20) As in ...

    work page

  9. [9]

    Third order The third-order term includes the N2(3) = 41 combi- nations in Eq. (4.13). The computation of this term is cumbersome, but after some algebra it results in d dx δN2 (3) (x)≃[v(x)] 3 v(x) v′(x) 5 × (d−1)(6d−11) 4x2 + d−1 2x 17 v′(x) v(x) − 25 2 v(x) v′(x) v′′(x) v(x) + 35 4 v′(x) v(x) 2 − 29 2 v′′(x) v(x) + 8 v(x) v′(x) 2 v′′(x) v(x) 2 − 7 4 v(...

    work page

  10. [10]

    However, at third order, a large number of terms already appears

    General order The higher-order terms in the variance can also be enumerated order by order. However, at third order, a large number of terms already appears. Therefore, let us instead present the general-order formula for the variance, which comes from the total-derivative terms subtracted from ⟨N2⟩. Such total derivatives appear as the k4 = 0 term in Eq....

    work page

  11. [11]

    (4.23), it is understood that the field value x is implicitly evaluated at the horizon crossing of a relevant mode

    Power spectrum By virtue of the stochastic-δNformalism, it reads Pζ(x) = dδN 2(x) d⟨N⟩(x) = d⟨N⟩(x) dx −1 dδN 2(x) dx .(4.23) In Eq. (4.23), it is understood that the field value x is implicitly evaluated at the horizon crossing of a relevant mode. Provided that both factors ⟨N⟩ (x) and δN2(x) have been obtained in Sections IV A and IV B respectively, as ...

    work page

  12. [12]

    To calculaten S, let us introduce λi(x)≡P (i) ζ (x), σ i(x)≡ dP(i) ζ (x) dx .(4.30) After one writes Eq

    Tilt From the power spectrum, the spectral index nS is defined as nS −1≡ d lnPζ(k) d lnk k=aH ≃ − dx d⟨N⟩(x) · d lnPζ dx =− d lnPζ dx dN(x) dx =− 1 Pζ(x) dPζ(x) dx dN(x) dx .(4.29) Provided that the power spectrum has been obtained up to third order, the spectral index can be derived up to second order. To calculaten S, let us introduce λi(x)≡P (i) ζ (x),...

    work page

  13. [13]

    •N 1(0) = 1 term inO(v 0)

    Mean The relevant terms for the calculation of the mean number ofe-folds are listed below. •N 1(0) = 1 term inO(v 0). (k1, k 2, k 3, k 4) = (0,0,0,0).(A1) •N 1(1) = 3 terms inO(v 1). (k1, k 2, k 3, k 4) = (0,0,0,1),(0,1,1,2),(1,0,0,1).(A2) •N 1(2) = 7 terms inO(v 2). (k1, k 2, k 3, k 4) = (0,0,0,2),(0,1,1,3),(0,2,1,3),(0,2,2,4),(1,0,0,2),(1,1,1,3), (2,0,0...

    work page

  14. [14]

    Collecting those terms enables one to obtain the perturbative expansion of ⟨N2⟩, from which the variance δN2 can be derived

    Variance The relevant terms for the calculation of the second moment ofNare listed below. Collecting those terms enables one to obtain the perturbative expansion of ⟨N2⟩, from which the variance δN2 can be derived. Though not listed here, a similar counting can also be performed for the formula (4.22), which directly derives the perturbative expansion of ...

    work page

  15. [15]

    A. A. Starobinsky, A New Type of Isotropic Cosmological Models Without Singularity, Phys. Lett. B91, 99 (1980)

    work page 1980

  16. [16]

    Sato, First Order Phase Transition of a Vacuum and Expansion of the Universe, Mon

    K. Sato, First Order Phase Transition of a Vacuum and Expansion of the Universe, Mon. Not. Roy. Astron. Soc. 195, 467 (1981)

    work page 1981

  17. [17]

    A. H. Guth, The Inflationary Universe: A Possible Solu- tion to the Horizon and Flatness Problems, Phys. Rev. D 23, 347 (1981)

    work page 1981

  18. [18]

    A. D. Linde, A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems, Phys. Lett. B108, 389 (1982)

    work page 1982

  19. [19]

    Albrecht and P

    A. Albrecht and P. J. Steinhardt, Cosmology for Grand Unified Theories with Radiatively Induced Symmetry Breaking, Phys. Rev. Lett.48, 1220 (1982)

    work page 1982

  20. [20]

    A. D. Linde, Chaotic Inflation, Phys. Lett. B129, 177 (1983)

    work page 1983

  21. [21]

    V. F. Mukhanov and G. V. Chibisov, Quantum Fluctu- ations and a Nonsingular Universe, JETP Lett.33, 532 (1981)

    work page 1981

  22. [22]

    A. A. Starobinsky, Dynamics of Phase Transition in the New Inflationary Universe Scenario and Generation of Perturbations, Phys. Lett. B117, 175 (1982)

    work page 1982

  23. [23]

    A. H. Guth and S. Y. Pi, Fluctuations in the New Infla- tionary Universe, Phys. Rev. Lett.49, 1110 (1982)

    work page 1982

  24. [24]

    J. M. Bardeen, P. J. Steinhardt, and M. S. Turner, Spon- taneous Creation of Almost Scale - Free Density Pertur- bations in an Inflationary Universe, Phys. Rev. D28, 679 (1983)

    work page 1983

  25. [25]

    A. A. Starobinsky, Spectrum of relict gravitational radia- tion and the early state of the universe, JETP Lett.30, 682 (1979)

    work page 1979

  26. [26]

    Polarski and A

    D. Polarski and A. A. Starobinsky, Semiclassicality and decoherence of cosmological perturbations, Class. Quant. Grav.13, 377 (1996), arXiv:gr-qc/9504030

    work page arXiv 1996

  27. [27]

    Lesgourgues, D

    J. Lesgourgues, D. Polarski, and A. A. Starobinsky, Quan- tum to classical transition of cosmological perturbations for nonvacuum initial states, Nucl. Phys. B497, 479 (1997), arXiv:gr-qc/9611019

    work page arXiv 1997

  28. [28]

    Planck 2018 results. X. Constraints on inflation

    Y. Akramiet al.(Planck), Planck 2018 results. X. Con- straints on inflation, Astron. Astrophys.641, A10 (2020), arXiv:1807.06211 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  29. [29]

    Planck 2018 results. IX. Constraints on primordial non-Gaussianity

    Y. Akramiet al.(Planck), Planck 2018 results. IX. Con- straints on primordial non-Gaussianity, Astron. Astro- 28 phys.641, A9 (2020), arXiv:1905.05697 [astro-ph.CO]

    work page Pith review arXiv 2018

  30. [30]

    Enqvist and M

    K. Enqvist and M. S. Sloth, Adiabatic CMB perturbations in pre - big bang string cosmology, Nucl. Phys. B626, 395 (2002), arXiv:hep-ph/0109214

    work page arXiv 2002

  31. [31]

    D. H. Lyth and D. Wands, Generating the curvature perturbation without an inflaton, Phys. Lett. B524, 5 (2002), arXiv:hep-ph/0110002

    work page Pith review arXiv 2002

  32. [32]

    Moroi and T

    T. Moroi and T. Takahashi, Effects of cosmological moduli fields on cosmic microwave background, Phys. Lett. B 522, 215 (2001), [Erratum: Phys.Lett.B 539, 303–303 (2002)], arXiv:hep-ph/0110096

    work page arXiv 2001

  33. [33]

    Wands, Multiple field inflation, Lect

    D. Wands, Multiple field inflation, Lect. Notes Phys.738, 275 (2008), arXiv:astro-ph/0702187

    work page arXiv 2008

  34. [34]

    Baumann and L

    D. Baumann and L. McAllister,Inflation and String The- ory, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2015) arXiv:1404.2601 [hep- th]

    work page arXiv 2015

  35. [35]

    Adiabatic and entropy perturbations from inflation

    C. Gordon, D. Wands, B. A. Bassett, and R. Maartens, Adiabatic and entropy perturbations from inflation, Phys. Rev. D63, 023506 (2000), arXiv:astro-ph/0009131

    work page Pith review arXiv 2000

  36. [36]

    An Observational Test of Two-field Inflation

    D. Wands, N. Bartolo, S. Matarrese, and A. Riotto, An Observational test of two-field inflation, Phys. Rev. D66, 043520 (2002), arXiv:astro-ph/0205253

    work page Pith review arXiv 2002

  37. [37]

    B. A. Bassett, S. Tsujikawa, and D. Wands, Inflation dynamics and reheating, Rev. Mod. Phys.78, 537 (2006), arXiv:astro-ph/0507632

    work page Pith review arXiv 2006

  38. [38]

    C. M. Peterson and M. Tegmark, Testing Two-Field In- flation, Phys. Rev. D83, 023522 (2011), arXiv:1005.4056 [astro-ph.CO]

    work page arXiv 2011

  39. [39]

    C. M. Peterson and M. Tegmark, Testing multifield in- flation: A geometric approach, Phys. Rev. D87, 103507 (2013), arXiv:1111.0927 [astro-ph.CO]

    work page arXiv 2013

  40. [40]

    Pattison, V

    C. Pattison, V. Vennin, H. Assadullahi, and D. Wands, Quantum diffusion during inflation and primordial black holes, JCAP10(2017), 046, arXiv:1707.00537 [hep-th]

    work page arXiv 2017

  41. [41]

    Prokopec and G

    T. Prokopec and G. Rigopoulos, ∆ N and the stochastic conveyor belt of ultra slow-roll inflation, Phys. Rev. D 104, 083505 (2021), arXiv:1910.08487 [gr-qc]

    work page arXiv 2021

  42. [42]

    J. M. Ezquiaga, J. Garc´ ıa-Bellido, and V. Vennin, The exponential tail of inflationary fluctuations: con- sequences for primordial black holes, JCAP03(2020), 029, arXiv:1912.05399 [astro-ph.CO]

    work page arXiv 2020

  43. [43]

    Pattison, V

    C. Pattison, V. Vennin, D. Wands, and H. Assadullahi, Ultra-slow-roll inflation with quantum diffusion, JCAP 04(2021), 080, arXiv:2101.05741 [astro-ph.CO]

    work page arXiv 2021

  44. [44]

    Rigopoulos and A

    G. Rigopoulos and A. Wilkins, Inflation is always semi- classical: diffusion domination overproduces Primordial Black Holes, JCAP12(2021), 027, arXiv:2107.05317 [astro-ph.CO]

    work page arXiv 2021

  45. [45]

    Tada and V

    Y. Tada and V. Vennin, Statistics of coarse-grained cos- mological fields in stochastic inflation, JCAP02(2022), 021, arXiv:2111.15280 [astro-ph.CO]

    work page arXiv 2022

  46. [46]

    Hooshangi, M

    S. Hooshangi, M. H. Namjoo, and M. Noorbala, Rare events are nonperturbative: Primordial black holes from heavy-tailed distributions, Phys. Lett. B834, 137400 (2022), arXiv:2112.04520 [astro-ph.CO]

    work page arXiv 2022

  47. [47]

    Achucarro, S

    A. Achucarro, S. Cespedes, A.-C. Davis, and G. A. Palma, The hand-made tail: non-perturbative tails from multi- field inflation, JHEP05(2022), 052, arXiv:2112.14712 [hep-th]

    work page arXiv 2022

  48. [48]

    Animali and V

    C. Animali and V. Vennin, Primordial black holes from stochastic tunnelling, JCAP02(02), 043, arXiv:2210.03812 [astro-ph.CO]

    work page arXiv

  49. [49]

    Tomberg, Stochastic constant-roll inflation and pri- mordial black holes, Phys

    E. Tomberg, Stochastic constant-roll inflation and pri- mordial black holes, Phys. Rev. D108, 043502 (2023), arXiv:2304.10903 [astro-ph.CO]

    work page arXiv 2023

  50. [50]

    Vennin and D

    V. Vennin and D. Wands, Quantum diffusion and large pri- mordial perturbations from inflation, inPrimordial Black Holes, edited by C. Byrnes, G. Franciolini, T. Harada, P. Pani, and M. Sasaki (Springer Nature Singapore, Sin- gapore, 2025) pp. 201–227

    work page 2025

  51. [51]

    Murata and Y

    T. Murata and Y. Tada, Stochastic tail of the curvature perturbation in hybrid inflation, Phys. Rev. D113, 023552 (2026), arXiv:2507.22439 [astro-ph.CO]

    work page arXiv 2026

  52. [52]

    Y. B. Zel’dovich and I. D. Novikov, The Hypothesis of Cores Retarded during Expansion and the Hot Cosmolog- ical Model, Soviet Astron. AJ (Engl. Transl. ),10, 602 (1967)

    work page 1967

  53. [53]

    Hawking, Gravitationally collapsed objects of very low mass, Mon

    S. Hawking, Gravitationally collapsed objects of very low mass, Mon. Not. Roy. Astron. Soc.152, 75 (1971)

    work page 1971

  54. [54]

    B. J. Carr and S. W. Hawking, Black holes in the early Universe, Mon. Not. Roy. Astron. Soc.168, 399 (1974)

    work page 1974

  55. [55]

    B. J. Carr, The Primordial black hole mass spectrum, Astrophys. J.201, 1 (1975)

    work page 1975

  56. [56]

    Vennin and A

    V. Vennin and A. A. Starobinsky, Correlation Functions in Stochastic Inflation, Eur. Phys. J. C75, 413 (2015), arXiv:1506.04732 [hep-th]

    work page arXiv 2015

  57. [57]

    A. D. Linde, Nonsingular Regenerating Inflationary Uni- verse (1982), Cambridge University preprint

    work page 1982

  58. [58]

    Vilenkin, The Birth of Inflationary Universes, Phys

    A. Vilenkin, The Birth of Inflationary Universes, Phys. Rev. D27, 2848 (1983)

    work page 1983

  59. [59]

    Winitzki,Eternal Inflation(World Scientific, 2009)

    S. Winitzki,Eternal Inflation(World Scientific, 2009)

    work page 2009

  60. [60]

    Takahashi and K

    T. Takahashi and K. Tokeshi, More fields are different: Stochastic view of multi-field inflationary scenario (2025), arXiv:2504.11158 [astro-ph.CO]

    work page arXiv 2025

  61. [61]

    D. S. Salopek and J. R. Bond, Nonlinear evolution of long wavelength metric fluctuations in inflationary models, Phys. Rev. D42, 3936 (1990)

    work page 1990

  62. [62]

    A General Analytic Formula for the Spectral Index of the Density Perturbations produced during Inflation

    M. Sasaki and E. D. Stewart, A General analytic formula for the spectral index of the density perturbations pro- duced during inflation, Prog. Theor. Phys.95, 71 (1996), arXiv:astro-ph/9507001

    work page Pith review arXiv 1996

  63. [63]

    Super-Horizon Scale Dynamics of Multi-Scalar Inflation

    M. Sasaki and T. Tanaka, Superhorizon scale dynamics of multiscalar inflation, Prog. Theor. Phys.99, 763 (1998), arXiv:gr-qc/9801017

    work page Pith review arXiv 1998

  64. [64]

    D. H. Lyth, K. A. Malik, and M. Sasaki, A General proof of the conservation of the curvature perturbation, JCAP 05(2005), 004, arXiv:astro-ph/0411220

    work page Pith review arXiv 2005

  65. [65]

    A. A. Abolhasani, H. Firouzjahi, A. Naruko, and M. Sasaki,Delta N Formalism in Cosmological Perturba- tion Theory(WSP, 2019)

    work page 2019

  66. [66]

    Fujita, M

    T. Fujita, M. Kawasaki, Y. Tada, and T. Takesako, A new algorithm for calculating the curvature pertur- bations in stochastic inflation, JCAP12(2013), 036, arXiv:1308.4754 [astro-ph.CO]

    work page arXiv 2013

  67. [67]

    Fujita, M

    T. Fujita, M. Kawasaki, and Y. Tada, Non-perturbative approach for curvature perturbations in stochastic δN formalism, JCAP10(10), 030, arXiv:1405.2187 [astro- ph.CO]

    work page arXiv

  68. [68]

    Ando and V

    K. Ando and V. Vennin, Power spectrum in stochastic inflation, JCAP04(04), 057, arXiv:2012.02031 [astro- ph.CO]

    work page arXiv 2012

  69. [69]

    A. A. Starobinsky, Stochastic de Siter (inflationary) stage in the early Universe, Lect. Notes Phys.246, 107 (1986)

    work page 1986

  70. [70]

    A. A. Starobinsky and J. Yokoyama, Equilibrium state of a selfinteracting scalar field in the De Sitter background, Phys. Rev. D50, 6357 (1994), arXiv:astro-ph/9407016

    work page Pith review arXiv 1994

  71. [71]

    Cruces, Review on Stochastic Approach to Inflation, 29 Universe8, 334 (2022), arXiv:2203.13852 [gr-qc]

    D. Cruces, Review on Stochastic Approach to Inflation, 29 Universe8, 334 (2022), arXiv:2203.13852 [gr-qc]

    work page arXiv 2022

  72. [72]

    Mollerach, S

    S. Mollerach, S. Matarrese, A. Ortolan, and F. Lucchin, Stochastic inflation in a simple two field model, Phys. Rev. D44, 1670 (1991)

    work page 1991

  73. [73]

    Assadullahi, H

    H. Assadullahi, H. Firouzjahi, M. Noorbala, V. Vennin, and D. Wands, Multiple Fields in Stochastic Inflation, JCAP06(2016), 043, arXiv:1604.04502 [hep-th]

    work page arXiv 2016

  74. [74]

    Vennin, H

    V. Vennin, H. Assadullahi, H. Firouzjahi, M. Noor- bala, and D. Wands, Critical Number of Fields in Stochastic Inflation, Phys. Rev. Lett.118, 031301 (2017), arXiv:1604.06017 [astro-ph.CO]

    work page arXiv 2017

  75. [75]

    Generation of fluctuations during inflation: comparison of stochastic and field-theoretic approaches

    F. Finelli, G. Marozzi, A. A. Starobinsky, G. P. Vacca, and G. Venturi, Generation of fluctuations during inflation: Comparison of stochastic and field-theoretic approaches, Phys. Rev. D79, 044007 (2009), arXiv:0808.1786 [hep-th]

    work page Pith review arXiv 2009

  76. [76]

    Stochastic growth of quantum fluctuations during slow-roll inflation

    F. Finelli, G. Marozzi, A. A. Starobinsky, G. P. Vacca, and G. Venturi, Stochastic growth of quantum fluctuations during slow-roll inflation, Phys. Rev. D82, 064020 (2010), arXiv:1003.1327 [hep-th]

    work page Pith review arXiv 2010

  77. [77]

    Pattison, V

    C. Pattison, V. Vennin, H. Assadullahi, and D. Wands, Stochastic inflation beyond slow roll, JCAP07(2019), 031, arXiv:1905.06300 [astro-ph.CO]

    work page arXiv 2019

  78. [78]

    Casini, R

    H. Casini, R. Montemayor, and P. Sisterna, Stochas- tic approach to inflation. 2. Classicality, coarse graining and noises, Phys. Rev. D59, 063512 (1999), arXiv:gr- qc/9811083

    work page arXiv 1999

  79. [79]

    Winitzki and A

    S. Winitzki and A. Vilenkin, Effective noise in stochastic description of inflation, Phys. Rev. D61, 084008 (2000), arXiv:gr-qc/9911029

    work page arXiv 2000

  80. [80]

    Matarrese, M

    S. Matarrese, M. A. Musso, and A. Riotto, Influence of superhorizon scales on cosmological observables gener- ated during inflation, JCAP05(2004), 008, arXiv:hep- th/0311059

    work page arXiv 2004

Showing first 80 references.