pith. machine review for the scientific record. sign in

arxiv: 2604.27103 · v1 · submitted 2026-04-29 · 🌌 astro-ph.CO · gr-qc· hep-th

Recognition: unknown

Geodesically Complete Curvature-Bounce Inflation

Damien A. Easson
Authors on Pith no claims yet

Pith reviewed 2026-05-07 08:44 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qchep-th
keywords geodesically complete cosmologycurvature bounceinflationclosed FRWscalar fieldnonsingular universenull energy condition
0
0 comments X

The pith

Positive spatial curvature permits a nonsingular bounce followed by standard inflation using one scalar field in general relativity.

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

The paper constructs an early-universe model that begins with a curvature-supported bounce and transitions directly into inflation without encountering a singularity. It uses ordinary general relativity and a single canonical scalar field whose potential includes a positive vacuum offset. The full evolution stays below the Planck scale, obeys the null energy condition at all times, and produces inflationary predictions that align with current observations. Perturbations evolve regularly through the bounce and settle into the usual freezing behavior during inflation.

Core claim

A geodesically complete closed k=+1 bounce-plus-inflation cosmology is realized in ordinary general relativity, sourced by a single canonical scalar field with a positive vacuum offset. The bounce is supported by curvature rather than exotic stress energy, the matter content satisfies the NEC throughout and violates only the strong energy condition, and the solved branch remains sub-Planckian before evolving onto a curvature-diluted slow-roll phase whose observables match current constraints.

What carries the argument

The closed FRW metric with positive spatial curvature k=+1 combined with a scalar field potential that has a positive vacuum energy offset, allowing curvature to drive the bounce while the field later supports slow-roll inflation.

If this is right

  • The model yields n_s=0.9617 and r=0.0045 at N_*=55, and n_s=0.9650 and r=0.0037 at N_*=60, consistent with current data.
  • Both tensor and scalar perturbations propagate regularly through the bounce and inflationary phases, with the curvature perturbation freezing in the standard manner.
  • The entire evolution remains sub-Planckian and satisfies the null energy condition while violating only the strong energy condition.
  • Spatial curvature is rapidly diluted during inflation, producing an effectively flat universe at late times.

Where Pith is reading between the lines

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

  • Confirmation of positive curvature would make this a minimal way to obtain a complete early universe without introducing new fields or modified gravity.
  • The regular passage of perturbations through the bounce suggests similar constructions could address other cosmological singularities if positive curvature is present.
  • Direct numerical evolution of closed-universe modes offers a concrete testbed for studying how infrared perturbations behave in nonsingular cosmologies.

Load-bearing premise

Only positive spatial curvature permits a nonsingular, geodesically complete universe with ANEC-respecting matter in non-static FRW cosmology.

What would settle it

A measurement of negative spatial curvature or inflationary observables far outside n_s approximately 0.96 and r approximately 0.004 at 55-60 e-folds would rule out the model's viability.

Figures

Figures reproduced from arXiv: 2604.27103 by Damien A. Easson.

Figure 1
Figure 1. Figure 1: is therefore conventional: what matters physically is the existence of a single smooth global scalar history compatible with the solved background, not monotonicity through the bounce view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Background evolution of the curvature-bounce infla view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Inflation-era freeze of the physical curvature pertur view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Infrared tensor and scalar spectra for the exact closed view at source ↗
read the original abstract

The early universe need not be described by an incomplete inflationary phase connected to a separate, more exotic prehistory. Recent results show that, within non-static FRW cosmology, only positive spatial curvature permits a nonsingular, geodesically complete universe with ANEC-respecting matter. We construct a geodesically complete closed $k=+1$ bounce-plus-inflation cosmology in ordinary general relativity, sourced by a single canonical scalar field with a positive vacuum offset. The bounce is supported by curvature rather than exotic stress energy: the matter content satisfies the NEC throughout and violates only the strong energy condition, as in any accelerated expansion. The solved branch remains sub-Planckian and evolves onto a curvature-diluted slow-roll phase with inflationary observables consistent with current constraints. The pivot-scale predictions are $n_s=0.9617$, $r=0.0045$ at $N_*=55$ and $n_s=0.9650$, $r=0.0037$ at $N_*=60$. Direct evolution of closed-universe infrared perturbations shows regular tensor and scalar propagation through the bounce and inflationary era, with the physical curvature perturbation freezing in the standard way. This gives a minimal explicit realization of a complete early-universe cosmology in the closed FRW branch selected by completeness and ANEC compatibility.

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

0 major / 3 minor

Summary. The paper constructs an explicit geodesically complete closed (k=+1) bounce-plus-inflation cosmology in ordinary GR sourced by a single canonical scalar field with positive vacuum offset. The bounce is curvature-supported at finite a_min with rho=3/a_min^2, the matter satisfies the NEC (rho+p = dot{phi}^2 >=0) throughout while violating only the SEC, the field remains sub-Planckian, and the evolution transitions to a curvature-diluted slow-roll phase. Direct numerical integration yields ns=0.9617, r=0.0045 at N*=55 and ns=0.9650, r=0.0037 at N*=60, with regular scalar and tensor perturbation propagation across the bounce and standard freezing of the curvature perturbation.

Significance. If the explicit construction and numerics hold, the result supplies a minimal, observationally consistent realization of a nonsingular early-universe cosmology that avoids geodesic incompleteness by using positive spatial curvature and standard matter, without exotic stress-energy. Credit is due for providing the concrete scalar potential, the numerical background solution exhibiting the curvature bounce and NEC compliance, and the direct integration of perturbation equations through the bounce. This strengthens the case for closed FRW models selected by completeness and ANEC compatibility.

minor comments (3)
  1. [Introduction] The abstract and introduction cite the foundational result that only k=+1 permits ANEC-compatible geodesic completeness from prior work; a one-sentence recap of the key theorem statement would improve self-contained readability without altering the central construction.
  2. The specific potential parameters (including the vacuum offset value) that produce the quoted ns and r at N*=55 and 60 are not listed explicitly alongside the observables; adding them would aid reproducibility of the slow-roll phase.
  3. The description of the numerical integration of the perturbation equations across the bounce could clarify the choice of initial conditions and time-stepping method to confirm absence of singularities.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our explicit construction of a geodesically complete closed FRW cosmology with curvature-supported bounce and canonical scalar inflation. The recommendation for minor revision is noted; we will address any editorial or presentational improvements in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity; explicit construction stands independently

full rationale

The manuscript supplies an explicit scalar potential with positive vacuum offset, a numerical background solution exhibiting curvature-supported bounce (H=0 at finite a_min with rho=3/a_min^2), NEC compliance via rho+p=dot{phi}^2 >=0, sub-Planckian evolution, and direct integration of perturbation equations across the bounce yielding ns=0.9617, r=0.0045 at N*=55 and ns=0.9650, r=0.0037 at N*=60. These observables are computed from the solved dynamics rather than fitted by construction or renamed inputs. The statement that only k=+1 permits ANEC-compatible geodesic completeness is cited from prior work as motivation for the branch choice, but the paper's central derivation chain (potential choice, bounce solution, slow-roll phase, and perturbation freezing) is self-contained against the FRW equations and does not reduce to that citation or to any self-definitional loop.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on a domain assumption about curvature and ANEC compatibility drawn from prior work, plus an ad-hoc choice of scalar potential with positive vacuum offset whose parameters are selected to achieve the desired bounce and slow-roll. No new entities are postulated.

free parameters (1)
  • Scalar potential parameters including vacuum offset
    The potential shape and offset are chosen so that the bounce occurs, the evolution stays sub-Planckian, and slow-roll produces the quoted ns and r; these are free parameters tuned to the target cosmology.
axioms (1)
  • domain assumption Only positive spatial curvature permits a nonsingular, geodesically complete universe with ANEC-respecting matter in non-static FRW cosmology
    Invoked in the opening sentence as a recent result on which the construction is built.

pith-pipeline@v0.9.0 · 5529 in / 1499 out tokens · 65584 ms · 2026-05-07T08:44:19.573912+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

66 extracted references · 41 canonical work pages · 2 internal anchors

  1. [1]

    12 Here, V, V,ϕ, andV,ϕϕbelow denote the numerically nondimensionalized potential and its derivatives, not quantities with un-restored explicit mass scales

    Conventions and reconstructed background quantities We work in the closedk= +1FRW background ds2 =−dt2 +a(t) 2dΩ 2 3,(B1) with 8πG= 1, H≡ ˙a a,H≡ a′ a =aH,(B2) and scalar harmonics labeled by integers n≥3.(B3) The closed-universe wavenumber reported in the numerics is qn = √ n2−1 rc ,(B4) with curvature radiusrc = 1by default. 12 Here, V, V,ϕ, andV,ϕϕbelo...

  2. [2]

    Evolved scalar variable The main evolved variable is the closed-universe gauge- invariant scalar mode Qn(t).(B9) This is the variable integrated directly in cosmic time. The solver evolves its real and imaginary parts as a first- order real system equivalent to the second-order complex equation ¨Qn +bn(t) ˙Qn +cn(t)Qn = 0.(B10) The friction coefficient is...

  3. [3]

    Canonicalized variable and effective frequency The canonicalized variable used only for adiabatic ini- tialization and frequency diagnostics is vn(η) =aQn√Un ≡snQn, s n≡a√Un .(B12) The canonicalization factor is Un = 1 + 3 n2−4 ( ˙ϕ H )2 reg .(B13) Here the subscript “reg” denotes the numerical regular- ization used in the implementation: the raw ratio is...

  4. [4]

    A simple analytic representative of the local bounce geometry It is useful to separate what is universal about a smooth closed-FRW bounce from what is model-dependent. Equa- tion (B17) shows that every smooth symmetric bounce of the type studied here has the same leading local geometric form, a(t) =a b + (t−tb)2 2ab +O ( (t−tb)4) , H(t) = t−tb a2 b +O ( (...

  5. [5]

    (B10) is imple- mented from the standard closed-universe gauge-invariant scalar formalism used in Refs

    Exact coefficientc n(t) The coefficientcn(t)appearing in Eq. (B10) is imple- mented from the standard closed-universe gauge-invariant scalar formalism used in Refs. [50–53], with the back- ground inputs ˙ϕ, V,ϕ, andV,ϕϕreconstructed from the solved branch as described above. It is written in the form cn = (8πG)Nn Dn ,(B35) with Dn =a 2 ˙a2[ 2(n2−4)˙a2 + 8...

  6. [6]

    (B17) with the finite-limit result Eq

    Local bounce expansion and analytic regularity of the scalar sector Combining the local expansion Eq. (B17) with the finite-limit result Eq. (B19), one finds Un =U n,b +O(x 2), ˙Un =O(x),(B44) and therefore bn(t) = 3H− ˙Un Un =O(x).(B45) Together with the removable finite bounce limit ofcn(t), this gives the local scalar mode equation ¨Qn +O(x) ˙Qn + ( cn...

  7. [7]

    Adiabatic initial data Adiabatic initial data are imposed only on the contract- ing branch. A sample is declared admissible only if all of the following hold: ω2 n >0, ⏐⏐⏐⏐ ω′ n ω2n ⏐⏐⏐⏐<ϵad,U n >0,U n finite, (B48) with default value ϵad = 5×10−2.(B49) The code requires at least three contiguous admissible grid points and chooses the earliest sample in t...

  8. [8]

    This definition is appropriate for the inflation-era freeze check, which is the physical scalar diagnostic used in the body of the paper

    Curvature perturbation and reported spectrum The curvature perturbation reported by the code is reconstructed from the evolved mode via ζn = (H ˙ϕ ) reg Qn,(B56) using the same threshold-and-interpolation regularization strategy described above. This definition is appropriate for the inflation-era freeze check, which is the physical scalar diagnostic used...

  9. [9]

    These checks were designed to test whether the scalar- sector conclusions depend sensitively on implementation choices rather than on the background itself

    Numerical robustness checks A focused robustness sweep was performed over repre- sentative modesn = 3, 5, 10, 30, 60by varying the interpo- lation threshold, the adiabatic-window criterion, and the initialization point within the first valid WKB window. These checks were designed to test whether the scalar- sector conclusions depend sensitively on impleme...

  10. [10]

    Normalizing this at the bounce by εσ≡ρσ,b/ρb, with ρb = 3/a2 b for the explicit background, one has ρσ(t) =εσρb ( ab a(t) )6

    Homogeneous shear estimate As a compact first-pass anisotropy check, one may model a small homogeneous shear contribution by an effective energy densityρσ∝a−6. Normalizing this at the bounce by εσ≡ρσ,b/ρb, with ρb = 3/a2 b for the explicit background, one has ρσ(t) =εσρb ( ab a(t) )6 . Since along the exact closed-FRW branchρtot(t) = 3(H2+ a−2)≥3a−2, it f...

  11. [11]

    A. H. Guth, The inflationary universe: A possible solution of the horizon and flatness problems, Phys. Rev. D23, 347 (1981)

    1981

  12. [12]

    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)

    1982

  13. [13]

    Albrecht and P

    A. Albrecht and P. J. Steinhardt, Cosmology for grand unified theories with radiatively induced symmetry break- ing, Phys. Rev. Lett.48, 1220 (1982)

    1982

  14. [14]

    Planck 2018 results. VI. Cosmological parameters

    N. Aghanimet al.(Planck), Planck 2018 results. vi. cosmological parameters, Astron. Astrophys.641, A6 (2020), [Erratum: Astron. Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO]

    work page internal anchor Pith review arXiv 2018

  15. [15]

    Borde, A

    A. Borde, A. H. Guth, and A. Vilenkin, Inflationary spacetimes are incomplete in past directions, Phys. Rev. Lett.90, 151301 (2003), arXiv:gr-qc/0110012

    work page arXiv 2003

  16. [16]

    The Ekpyrotic Universe: Colliding Branes and the Origin of the Hot Big Bang

    J. Khoury, B. A. Ovrut, P. J. Steinhardt, and N. Turok, The Ekpyrotic universe: Colliding branes and the origin of the hot big bang, Phys. Rev. D64, 123522 (2001), arXiv:hep-th/0103239

    work page Pith review arXiv 2001

  17. [17]

    Bouncing Cosmologies

    M. Novello and S. E. P. Bergliaffa, Bouncing cosmologies, Phys. Rept.463, 127 (2008), arXiv:0802.1634 [astro-ph]

    work page Pith review arXiv 2008

  18. [18]

    D. A. Easson, I. Sawicki, and A. Vikman, G-Bounce, JCAP11, 021, arXiv:1109.1047 [hep-th]

    work page Pith review arXiv

  19. [19]

    Y.-F. Cai, D. A. Easson, and R. Brandenberger, To- wards a Nonsingular Bouncing Cosmology, JCAP08, 020, arXiv:1206.2382 [hep-th]

    work page Pith review arXiv

  20. [20]

    A Critical Review of Classical Bouncing Cosmologies

    D. Battefeld and P. Peter, A critical review of classi- cal bouncing cosmologies, Phys. Rept.571, 1 (2015), arXiv:1406.2790 [astro-ph.CO]

    work page Pith review arXiv 2015

  21. [21]

    Bouncing Cosmologies: Progress and Problems

    R. Brandenberger and P. Peter, Bouncing cosmologies: Progress and problems, Found. Phys.47, 797 (2017), arXiv:1603.05834 [hep-th]

    work page Pith review arXiv 2017

  22. [22]

    Creminelli, D

    P. Creminelli, D. Pirtskhalava, L. Santoni, and E. Trincherini, Stability of geodesically complete cosmolo- gies, JCAP11, 047, arXiv:1610.04207 [hep-th]

    work page arXiv

  23. [23]

    Ijjas and P

    A. Ijjas and P. J. Steinhardt, Classically stable nonsingu- lar cosmological bounces, Phys. Rev. Lett.117, 121304 (2016), arXiv:1606.08880 [gr-qc]

    work page arXiv 2016

  24. [24]

    Ijjas and P

    A. Ijjas and P. J. Steinhardt, Fully stable cosmological solutions with a non-singular classical bounce, Phys. Lett. B764, 289 (2017), arXiv:1609.01253 [gr-qc]

    work page arXiv 2017

  25. [25]

    Y. Cai, Y. Wan, H.-G. Li, T. Qiu, and Y.-S. Piao, The effective field theory of nonsingular cosmology, JHEP01, 090, arXiv:1610.03400 [gr-qc]

    work page Pith review arXiv

  26. [26]

    Cai, H.-G

    Y. Cai, H.-G. Li, T. Qiu, and Y.-S. Piao, The effective field theory of nonsingular cosmology: Ii, Eur. Phys. J. C 77, 369 (2017), arXiv:1701.04330 [gr-qc]

    work page arXiv 2017

  27. [27]

    W.H.Kinney, S.Maity,andL.Sriramkumar,Borde-Guth- Vilenkin theorem in extended de Sitter spaces, Phys. Rev. D109, 043519 (2024), arXiv:2307.10958 [gr-qc]

    work page arXiv 2024

  28. [28]

    J. E. Lesnefsky, D. A. Easson, and P. C. W. Davies, Past- completeness of inflationary spacetimes, Phys. Rev. D 107, 044024 (2023), arXiv:2207.00955 [gr-qc]

    work page arXiv 2023

  29. [29]

    Geshnizjani, E

    G. Geshnizjani, E. Ling, and J. Quintin, On the initial singularity and extendibility of flat quasi-de Sitter space- times, JHEP10, 182, arXiv:2305.01676 [gr-qc]. 17

    work page arXiv

  30. [30]

    D. A. Easson and J. E. Lesnefsky, Eternal universes, Phys. Rev. D112, 063545 (2025), arXiv:2404.03016 [hep-th]

    work page arXiv 2025

  31. [31]

    D. A. Easson and J. E. Lesnefsky, Inflationary resolution of the initial singularity, Phys. Lett. B875, 140370 (2026), arXiv:2402.13031 [hep-th]

    work page arXiv 2026

  32. [32]

    Garcia-Saenz, J

    S. Garcia-Saenz, J. Hua, and Y. Zhao, Geodesic complete- ness, cosmological bounces, and inflation, Phys. Rev. D 110, L061304 (2024), arXiv:2405.04062 [gr-qc]

    work page arXiv 2024

  33. [33]

    N. L. Burwig and D. A. Easson, Open case for a closed uni- verse, Phys. Rev. D113, 083530 (2026), arXiv:2510.13971 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  34. [34]

    R. M. Wald and U. Yurtsever, General proof of the aver- aged null energy condition for a massless scalar field in two-dimensional curved space-time, Phys. Rev. D44, 403 (1991)

    1991

  35. [35]

    L. H. Ford and T. A. Roman, Averaged energy conditions and quantum inequalities, Phys. Rev. D51, 4277 (1995)

    1995

  36. [36]

    Graham and K

    N. Graham and K. D. Olum, Achronal averaged null energy condition, Phys. Rev. D76, 064001 (2007), arXiv:0705.3193 [gr-qc]

    work page arXiv 2007

  37. [37]

    Hartman, S

    T. Hartman, S. Kundu, and A. Tajdini, Averaged Null Energy Condition from Causality, JHEP07, 066, arXiv:1610.05308 [hep-th]

    work page arXiv

  38. [38]

    A. C. Wall, Proving the Achronal Averaged Null Energy Condition from the Generalized Second Law, Phys. Rev. D81, 024038 (2010), arXiv:0910.5751 [gr-qc]

    work page arXiv 2010

  39. [39]

    A. D. Linde, Inflation with variable Omega, Phys. Lett. B351, 99 (1995), arXiv:hep-th/9503097

    work page arXiv 1995

  40. [40]

    A. D. Linde, Can we have inflation with Omega> 1?, JCAP05, 002, arXiv:astro-ph/0303245

    work page arXiv

  41. [41]

    G. F. R. Ellis and R. Maartens, The emergent universe: Inflationary cosmology with no singularity, Class. Quant. Grav.21, 223 (2004), arXiv:gr-qc/0211082

    work page arXiv 2004

  42. [42]

    G. F. R. Ellis, J. Murugan, and C. G. Tsagas, The emer- gent universe: An explicit construction, Class. Quant. Grav.21, 233 (2004), arXiv:gr-qc/0307112

    work page arXiv 2004

  43. [43]

    P. W. Graham, B. Horn, S. Kachru, S. Rajendran, and G. Torroba, A simple harmonic universe, JHEP02, 029, arXiv:1109.0282 [hep-th]

    work page arXiv

  44. [44]

    Matsui, F

    H. Matsui, F. Takahashi, and T. Terada, Non-singular bouncing cosmology with positive spatial curvature and flat scalar potential, Phys. Lett. B795, 152 (2019), arXiv:1905.01253 [hep-ph]

    work page arXiv 2019

  45. [45]

    S. F. Bramberger and J.-L. Lehners, Nonsingular bounces catalyzed by dark energy, Phys. Rev. D99, 123523 (2019), arXiv:1901.10198 [hep-th]

    work page arXiv 2019

  46. [46]

    Hwang and H

    J. Hwang and H. Noh, Non-singular big-bounces and evolution of linear fluctuations, Phys. Rev. D65, 124010 (2002), arXiv:astro-ph/0112079 [astro-ph]

    work page arXiv 2002

  47. [47]

    Gordon and N

    C. Gordon and N. Turok, Cosmological perturbations through a general relativistic bounce, Phys. Rev. D67, 123508 (2003), arXiv:hep-th/0206138 [hep-th]

    work page arXiv 2003

  48. [48]

    Martin and P

    J. Martin and P. Peter, Parametric amplification of metric fluctuations through a bouncing phase, Phys. Rev. D68, 103517 (2003), arXiv:hep-th/0307077

    work page arXiv 2003

  49. [49]

    Peter and J

    P. Peter and J. Martin, Propagating cosmological pertur- bations in a bouncing universe, Phys. Rev. D70, 063515 (2004), arXiv:hep-th/0402081

    work page arXiv 2004

  50. [50]

    Penrose, Gravitational collapse and space-time singu- larities, Phys

    R. Penrose, Gravitational collapse and space-time singu- larities, Phys. Rev. Lett.14, 57 (1965)

    1965

  51. [51]

    Hawking, The Occurrence of singularities in cosmology, Proc

    S. Hawking, The Occurrence of singularities in cosmology, Proc. Roy. Soc. Lond. A294, 511 (1966)

    1966

  52. [52]

    Hawking, The Occurrence of singularities in cosmology

    S. Hawking, The Occurrence of singularities in cosmology. II, Proc. Roy. Soc. Lond. A295, 490 (1966)

    1966

  53. [53]

    Hawking, The occurrence of singularities in cosmology

    S. Hawking, The occurrence of singularities in cosmology. III. Causality and singularities, Proc. Roy. Soc. Lond. A 300, 187 (1967)

    1967

  54. [54]

    S. W. Hawking and R. Penrose, The singularities of grav- itational collapse and cosmology, Proc. Roy. Soc. Lond. A314, 529 (1970)

    1970

  55. [55]

    S. W. Hawking and G. F. R. Ellis,The Large Scale Struc- ture of Space-Time, Cambridge Monographs on Mathe- matical Physics (Cambridge University Press, 1973)

    1973

  56. [56]

    A. A. Starobinsky, A new type of isotropic cosmological models without singularity, Phys. Lett. B91, 99 (1980)

    1980

  57. [57]

    Kofman, A

    L. Kofman, A. D. Linde, and A. A. Starobinsky, Re- heating after inflation, Phys. Rev. Lett.73, 3195 (1994), arXiv:hep-th/9405187

    work page arXiv 1994

  58. [58]

    Kofman, A

    L. Kofman, A. D. Linde, and A. A. Starobinsky, Towards the theory of reheating after inflation, Phys. Rev. D56, 3258 (1997), arXiv:hep-ph/9704452

    work page arXiv 1997

  59. [59]

    Shtanov, J

    Y. Shtanov, J. H. Traschen, and R. H. Brandenberger, Universe reheating after inflation, Phys. Rev. D51, 5438 (1995), arXiv:hep-ph/9407247

    work page arXiv 1995

  60. [60]

    Langlois, Hamiltonian formalism and gauge invariance for linear perturbations in inflation, Class

    D. Langlois, Hamiltonian formalism and gauge invariance for linear perturbations in inflation, Class. Quant. Grav. 11, 389 (1994)

    1994

  61. [61]

    Bonga, B

    B. Bonga, B. Gupt, and N. Yokomizo, Inflation in the closed flrw model and the cmb, JCAP10, 031, arXiv:1605.07556 [astro-ph.CO]

    work page arXiv

  62. [62]

    Bonga, B

    B. Bonga, B. Gupt, and N. Yokomizo, Tensor perturba- tions during inflation in a spatially closed universe, JCAP 05, 021, arXiv:1612.07281 [astro-ph.CO]

    work page arXiv

  63. [63]

    Kiefer and T

    C. Kiefer and T. Vardanyan, Power spectrum for pertur- bations in an inflationary model for a closed universe, Gen. Rel. Grav.54, 30 (2022)

    2022

  64. [64]

    Handley, Primordial power spectra for curved in- flating universes, Phys

    W. Handley, Primordial power spectra for curved in- flating universes, Phys. Rev. D100, 123517 (2019), arXiv:1907.08524 [astro-ph.CO]

    work page arXiv 2019

  65. [65]

    M. I. Letey, Z. Shumaylov, F. J. Agocs, W. J. Handley, M. P. Hobson, and A. N. Lasenby, Quantum initial con- ditions for curved inflating universes, Phys. Rev. D109, 123502 (2024), arXiv:2211.17248 [gr-qc]

    work page arXiv 2024

  66. [66]

    R. C. Tolman,Relativity, Thermodynamics, and Cosmol- ogy(Oxford University Press, Oxford, 1934)

    1934