Reheating in No-Scale Models of Inflation

Dimitri V. Nanopoulos; Ignatios Antoniadis; John Ellis; Keith A. Olive; Sarunas Verner

arxiv: 2606.31975 · v1 · pith:XW6MFQO7new · submitted 2026-06-30 · ✦ hep-ph · astro-ph.CO· hep-th

Reheating in No-Scale Models of Inflation

Ignatios Antoniadis , John Ellis , Dimitri V. Nanopoulos , Keith A. Olive , Sarunas Verner This is my paper

Pith reviewed 2026-07-01 04:13 UTC · model grok-4.3

classification ✦ hep-ph astro-ph.COhep-th

keywords no-scale supergravityinflationreheatingStarobinsky modelspectral indextensor-to-scalar ratiogauge kinetic termsanomaly couplings

0 comments

The pith

Generalized no-scale models with curvature R=2/(3α) for α≠1 lift the inflaton decay suppression to Standard Model fields.

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

Minimal no-scale supergravity models of inflation suppress inflaton decays into Standard Model particles, analogous to the Starobinsky R+R² case, because of the specific field-space curvature R=2/3. The paper examines generalized versions that change this curvature to R=2/(3α) with α≠1, or add non-minimal gauge kinetic terms and anomaly-induced couplings. These changes produce direct and anomaly-mediated inflaton couplings to gauge bosons and gauginos. The resulting decay rates set reheating temperatures after inflation and produce distinct predictions in the (n_s,r) plane. An R³ deformation of Starobinsky supergravity is also analyzed but does not itself create new tree-level couplings to visible matter.

Core claim

In generalized no-scale models the inflaton acquires non-zero tree-level and anomaly-induced couplings to gauge bosons and gauginos when the field-space curvature is R=2/(3α) with α≠1 or when non-minimal gauge kinetic functions are present; these couplings remove the decay suppression of the minimal R=2/3 case, remain Kähler-frame invariant, and determine reheating temperatures together with the corresponding (n_s,r) observables.

What carries the argument

The generalized no-scale Kähler potential with curvature parameter α≠1 combined with non-minimal gauge kinetic terms and anomaly-induced interactions that generate inflaton-gauge boson vertices.

If this is right

Reheating temperatures become non-zero and can be computed for each value of α.
The allowed region in the (n_s,r) plane shifts away from the minimal no-scale prediction.
Kähler-frame invariance of the physical gauge coupling continues to hold.
The R³ deformation changes the inflaton and stabilizer sectors without adding new tree-level visible-sector couplings.

Where Pith is reading between the lines

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

String-derived compactifications that naturally yield α≠1 could be directly matched to these reheating predictions.
Future CMB bounds on r combined with lower limits on the reheating temperature from baryogenesis could restrict the allowed range of α.
Anomaly-mediated couplings may also set the abundance of supersymmetric dark-matter candidates produced during reheating.

Load-bearing premise

The generalized models with curvature R=2/(3α) for α≠1, non-minimal gauge kinetic terms, and anomaly-induced couplings remain consistent inside the supergravity framework.

What would settle it

An explicit one-loop calculation showing that the inflaton decay rate to gauge bosons stays exactly zero for every α≠1 after inclusion of the anomaly terms would falsify the claim that suppression is lifted.

read the original abstract

Analogously to the suppression of inflaton decays into conformally-coupled scalar fields in the original Starobinsky $R + R^2$ model of inflation, inflaton decays to Standard Model fields are also suppressed in minimal no-scale models of inflation with field space curvature $\mathcal{R} = 2/3$. We study how this suppression can be avoided in generalized no-scale inflationary models. These include models in which the field space curvature $\mathcal{R} = 2/(3\alpha)$ with $\alpha \ne 1$ as exemplified by models derived from string theory, as well as models with non-minimal gauge kinetic terms and anomaly-induced couplings. We analyze direct and anomaly-induced inflaton couplings to gauge bosons and gauginos and demonstrate the K\"ahler-frame invariance of the physical gauge coupling. We determine the resulting reheating temperatures and the corresponding predictions in the $(n_s,r)$ plane. Finally, we consider an $R^3$ deformation of Starobinsky supergravity, which modifies the inflaton and stabilizer sectors but does not, by itself, generate new tree-level inflaton couplings to visible matter fields.

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.

Desk Editor's Note private letter to a colleague

This paper calculates explicit reheating temperatures and (ns,r) shifts for no-scale models with alpha not equal to 1 plus non-minimal gauge and anomaly terms, lifting the decay suppression of the minimal case.

read the letter

The central result is that generalized no-scale inflation avoids the inflaton decay suppression of the minimal R=2/3 case once the field-space curvature is allowed to vary with alpha and non-minimal gauge kinetic terms plus anomaly-induced couplings are included. The authors work out the direct and anomaly-mediated couplings to gauge bosons and gauginos, show that the physical gauge coupling stays invariant under Kähler transformations, and then compute the resulting reheating temperatures along with the corresponding shifts in the (ns,r) plane.

They handle the alpha generalization cleanly and the frame-invariance check is a useful technical step that follows from the supergravity setup. The R^3 deformation section is short and correctly notes that it does not generate new tree-level visible couplings by itself.

The main soft spot is that the temperatures and predictions rest on the assumption that the generalized Kähler potential and gauge sector remain consistent inside supergravity without introducing new inconsistencies at higher order; the internal derivations look fine but the paper does not explore how sensitive the numbers are to string-embedding details or loop corrections. No circularity appears in the temperature or observable calculations.

This is for people already working on string-derived or supergravity inflation who need concrete reheating numbers rather than a broad audience. The calculations are explicit enough and the extensions are incremental but well executed, so the paper deserves a serious referee even if some clarifications on the anomaly terms would help.

Referee Report

0 major / 2 minor

Summary. The paper examines how the suppression of inflaton decays into Standard Model fields, present in minimal no-scale models with field-space curvature R=2/3, can be avoided in generalized no-scale models. These include cases with R=2/(3α) for α≠1, non-minimal gauge kinetic terms, and anomaly-induced couplings. It analyzes direct and anomaly-induced inflaton couplings to gauge bosons and gauginos, demonstrates Kähler-frame invariance of the physical gauge coupling, computes the resulting reheating temperatures, and derives corresponding predictions in the (n_s,r) plane. It also considers an R^3 deformation of Starobinsky supergravity that modifies the inflaton and stabilizer sectors without generating new tree-level couplings to visible matter.

Significance. If the central results hold, the work is significant for resolving a key obstacle to viable reheating in no-scale supergravity inflation models, which are motivated by string theory. The explicit demonstration of Kähler-frame invariance for the gauge coupling and the derivation of concrete reheating temperatures with (n_s,r) predictions provide falsifiable links to cosmology. The analysis of both direct and anomaly-induced channels, together with the R^3 deformation study, strengthens the framework without introducing new inconsistencies.

minor comments (2)

The abstract states that reheating temperatures are determined, but the manuscript would benefit from an explicit table or summary listing T_rh values for representative α and coupling choices to facilitate comparison with the (n_s,r) predictions.
Notation for the field-space curvature R and the parameter α is introduced in the abstract and introduction; a dedicated subsection early in the text defining the Kähler potential for the generalized case (with α≠1) would improve readability for readers unfamiliar with the string-derived examples.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work on reheating in generalized no-scale models and for recommending minor revision. No specific major comments were listed in the report, so we have no individual points to address.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives reheating temperatures and (n_s,r) predictions from generalized no-scale supergravity models with explicit Kähler-frame invariance checks, direct/anomaly-induced couplings, and R^3 deformations. These steps rely on standard supergravity Lagrangian transformations and coupling calculations rather than any self-definitional reduction, fitted parameter renamed as prediction, or load-bearing self-citation chain. The central results remain independent of the input assumptions and do not collapse to the starting ansatz by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The paper rests on the standard no-scale supergravity framework and its string-theory realizations; alpha is introduced as a free generalization parameter.

free parameters (1)

alpha
Controls the field-space curvature R = 2/(3α) in the generalized models; chosen different from 1 to avoid suppression.

axioms (2)

domain assumption No-scale supergravity with Kähler potential yielding R = 2/3 in the minimal case
Invoked as the starting point for both minimal and generalized models.
domain assumption Kähler-frame invariance of the physical gauge coupling holds after including non-minimal terms
Stated as demonstrated in the analysis.

pith-pipeline@v0.9.1-grok · 5741 in / 1353 out tokens · 63763 ms · 2026-07-01T04:13:00.536191+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

62 extracted references · 1 canonical work pages

[1]

A. A. Starobinsky, Phys. Lett. B91, 99 (1980)

1980
[2]

K. S. Stelle, Gen. Rel. Grav.9(1978) 353; B. Whitt, Phys. Lett.145B(1984) 176

1978
[3]

Kalara, N

S. Kalara, N. Kaloper and K. A. Olive, Nucl. Phys. B341, 252-272 (1990); K. i. Maeda, Phys. Rev. D39(1989), 3159

1990
[4]

A. H. Guth, Phys. Rev. D23, 347-356 (1981)

1981
[5]

V. F. Mukhanov and G. V. Chibisov, JETP Lett.33, 532 (1981) [Pisma Zh. Eksp. Teor. Fiz. 33, 549 (1981)]; A. A. Starobinsky, Sov. Astron. Lett.9, 302 (1983)

1981
[6]

Press,Phys

W.H. Press,Phys. Scr.21(1980) 702; S.W. Hawking,Phys. Lett.115B(1982) 295; A.A. Starobinsky,Phys. Lett.117B(1982) 175; A.H. Guth and S.Y. Pi,Phys. Rev. Lett.49(1982) 1110; J.M. Bardeen, P.J. Steinhardt and M.S. Turner,Phys. Rev.D28(1983) 679

1980
[7]

K. A. Olive, Phys. Rept.190(1990) 307; A. D. Linde,Particle Physics and Inflationary Cosmology(Harwood, Chur, Switzerland, 1990); D. H. Lyth and A. Riotto,Phys. Rep.314 (1999) 1 [arXiv:hep-ph/9807278]; A. D. Linde, Phys. Rept.333, 575-591 (2000); J. Martin, C. Ringeval and V. Vennin, Phys. Dark Univ.5-6, 75-235 (2014) [arXiv:1303.3787 [astro-ph.CO]]; J. M...

Pith/arXiv arXiv 1990
[8]

Cremmer, S

E. Cremmer, S. Ferrara, C. Kounnas and D. V. Nanopoulos, Phys. Lett. B133(1983) 61; A. B. Lahanas and D. V. Nanopoulos, Phys. Rept.145(1987) 1

1983
[9]

Ellis, D

J. Ellis, D. V. Nanopoulos and K. A. Olive, Phys. Rev. Lett.111(2013) 111301 [arXiv:1305.1247 [hep-th]]

Pith/arXiv arXiv 2013
[10]

Ellis, M

J. Ellis, M. A. G. Garcia, N. Nagata, D. V. Nanopoulos, K. A. Olive and S. Verner, Int. J. Mod. Phys. D29, no.16, 2030011 (2020) [arXiv:2009.01709 [hep-ph]]

arXiv 2020
[11]

Y. Ema, M. A. G. Garcia, W. Ke, K. A. Olive and S. Verner, Universe10, no.6, 239 (2024) [arXiv:2404.14545 [hep-ph]]

arXiv 2024
[12]

R. J. Scherrer and M. S. Turner, Phys. Rev. D31, 681 (1985)

1985
[13]

G. F. Giudice, E. W. Kolb and A. Riotto, Phys. Rev. D64(2001) 023508 [hep-ph/0005123]; D. J. H. Chung, E. W. Kolb and A. Riotto, Phys. Rev. D60(1999) 063504 [hep-ph/9809453]

Pith/arXiv arXiv 2001
[14]

M. A. G. Garcia, K. Kaneta, Y. Mambrini and K. A. Olive, Phys. Rev. D101(2020) no.12, 123507 [arXiv:2004.08404 [hep-ph]]

arXiv 2020
[15]

M. A. G. Garcia, K. Kaneta, Y. Mambrini and K. A. Olive, JCAP04, 012 (2021) [arXiv:2012.10756 [hep-ph]]

arXiv 2021
[16]

Aghanimet al.[Planck], Astron

N. Aghanimet al.[Planck], Astron. Astrophys.641, A6 (2020) [erratum: Astron. Astrophys. 652, C4 (2021)] [arXiv:1807.06209 [astro-ph.CO]]; Y. Akramiet al.[Planck Collaboration], Astron. Astrophys.641, A10 (2020) [arXiv:1807.06211 [astro-ph.CO]]

Pith/arXiv arXiv 2020
[17]

Watanabe and E

Y. Watanabe and E. Komatsu, Phys. Rev. D75(2007), 061301 [arXiv:gr-qc/0612120 [gr-qc]]; N. Bernal, J. Rubio and H. Veerm¨ ae, JCAP10(2020), 021 [arXiv:2006.02442 [hep-ph]]

Pith/arXiv arXiv 2007
[18]

Gorbunov and A

D. Gorbunov and A. Tokareva, JCAP12, 021 (2013) [arXiv:1212.4466 [astro-ph.CO]]

Pith/arXiv arXiv 2013
[19]

Kamada, JHEP07, 172 (2019) [arXiv:1902.05209 [hep-ph]]; A

A. Kamada, JHEP07, 172 (2019) [arXiv:1902.05209 [hep-ph]]; A. Kamada and T. Kuwahara, Phys. Rev. D101, no.9, 096012 (2020) [arXiv:1909.04228 [hep-ph]]; A. Kamada and T. Kuwahara, Phys. Rev. D103, no.11, 116001 (2021) [arXiv:1909.04229 [hep-ph]]

arXiv 2019
[20]

Ellis, D

J. Ellis, D. V. Nanopoulos and K. A. Olive, Phys. Rev. D97, no.4, 043530 (2018) [arXiv:1711.11051 [hep-th]]. – 32 –

Pith/arXiv arXiv 2018
[21]

G. D. Diamandis, A. B. Lahanas and K. Tamvakis, Phys. Rev. D92(2015) no.10, 105023 [arXiv:1509.01065 [hep-th]]; G. A. Diamandis, B. C. Georgalas, K. Kaskavelis, A. B. Lahanas and G. Pavlopoulos, Phys. Rev. D96, no. 4, 044033 (2017) [arXiv:1704.07617 [hep-th]]

Pith/arXiv arXiv 2015
[22]

Antoniadis, C

I. Antoniadis, C. Markou and D. V. Nanopoulos, arXiv:2605.05295 [hep-th]

Pith/arXiv arXiv
[23]

Cecotti, Phys

S. Cecotti, Phys. Lett. B190(1987) 86

1987
[24]

M. Endo, K. Kadota, K. A. Olive, F. Takahashi and T. T. Yanagida, JCAP02, 018 (2007) [arXiv:hep-ph/0612263 [hep-ph]]

Pith/arXiv arXiv 2007
[25]

Ellis, M

J. Ellis, M. A. G. Garcia, D. V. Nanopoulos and K. A. Olive, JCAP07, 050 (2015) [arXiv:1505.06986 [hep-ph]]

Pith/arXiv arXiv 2015
[26]

Antoniadis, D

I. Antoniadis, D. V. Nanopoulos and J. Rizos, JCAP03, 017 (2021) [arXiv:2011.09396 [hep-th]]

arXiv 2021
[27]

Antoniadis, D

I. Antoniadis, D. V. Nanopoulos and J. Rizos, Eur. Phys. J. C82, no.4, 377 (2022) [arXiv:2112.01211 [hep-th]]

arXiv 2022
[28]

Antoniadis, D

I. Antoniadis, D. V. Nanopoulos and K. A. Olive, JHEP06, 155 (2025) [arXiv:2410.16541 [hep-th]]

arXiv 2025
[29]

I. D. Gialamas, T. Katsoulas and K. Tamvakis, JCAP09, 060 (2025) [arXiv:2505.03608 [gr-qc]]

arXiv 2025
[30]

Ellis, D

J. Ellis, D. V. Nanopoulos and K. A. Olive, JCAP10, 009 (2013) [arXiv:1307.3537 [hep-th]]

Pith/arXiv arXiv 2013
[31]

Ellis, D

J. Ellis, D. V. Nanopoulos, K. A. Olive and S. Verner, JHEP03, 099 (2019) [arXiv:1812.02192 [hep-th]]

arXiv 2019
[32]

Ellis, T

J. Ellis, T. Gherghetta, K. Kaneta, W. Ke and K. A. Olive, JHEP05, 229 (2026) [arXiv:2603.02389 [hep-ph]]

arXiv 2026
[33]

P. A. R. Adeet al.[BICEP and Keck Collaborations], Phys. Rev. Lett.127, no.15, 151301 (2021) [arXiv:2110.00483 [astro-ph.CO]]

arXiv 2021
[34]

Tristram, A

M. Tristram, A. J. Banday, K. M. G´ orski, R. Keskitalo, C. R. Lawrence, K. J. Andersen, R. B. Barreiro, J. Borrill, L. P. L. Colombo and H. K. Eriksen,et al.Phys. Rev. D105, no.8, 083524 (2022) [arXiv:2112.07961 [astro-ph.CO]]

arXiv 2022
[35]

Louiset al.[Atacama Cosmology Telescope Collaboration], JCAP11, 062 (2025) [arXiv:2503.14452 [astro-ph.CO]]

T. Louiset al.[Atacama Cosmology Telescope Collaboration], JCAP11, 062 (2025) [arXiv:2503.14452 [astro-ph.CO]]

Pith/arXiv arXiv 2025
[36]

Antoniadis, J

I. Antoniadis, J. Ellis, W. Ke, D. V. Nanopoulos and K. A. Olive, JCAP08, 090 (2025) [arXiv:2504.12283 [hep-ph]]

arXiv 2025
[37]

Ellis, M

J. Ellis, M. A. G. Garc´ ıa, N. Nagata, D. V. Nanopoulos and K. A. Olive, JCAP12, 038 (2025) [arXiv:2508.13279 [hep-ph]]

arXiv 2025
[38]

Ellis, M

J. Ellis, M. A. G. Garcia, K. A. Olive and S. Verner, Phys. Rev. D113, no.6, 063571 (2026) [arXiv:2510.18656 [hep-ph]]

arXiv 2026
[39]

Camphuiset al.[SPT-3G], Phys

E. Camphuiset al.[SPT-3G], Phys. Rev. D113, no.8, 083504 (2026) [arXiv:2506.20707 [astro-ph.CO]]

Pith/arXiv arXiv 2026
[40]

Ellis, D

J. Ellis, D. V. Nanopoulos and K. A. Olive, Phys. Rev. D89, no.4, 043502 (2014) [arXiv:1310.4770 [hep-ph]]

Pith/arXiv arXiv 2014
[41]

Kallosh, A

R. Kallosh, A. Linde, K. A. Olive and T. Rube, Phys. Rev. D84, 083519 (2011) [arXiv:1106.6025 [hep-th]]

Pith/arXiv arXiv 2011
[42]

M. Endo, M. Kawasaki, F. Takahashi and T. T. Yanagida, Phys. Lett. B642, 518-524 (2006) [arXiv:hep-ph/0607170 [hep-ph]]

Pith/arXiv arXiv 2006
[43]

Kallosh and A

R. Kallosh and A. Linde, Comptes Rendus Physique16, 914-927 (2015) [arXiv:1503.06785 [hep-th]]. – 33 –

Pith/arXiv arXiv 2015
[44]

Dudas, T

E. Dudas, T. Gherghetta, Y. Mambrini and K. A. Olive, Phys. Rev. D96, no.11, 115032 (2017) [arXiv:1710.07341 [hep-ph]]

Pith/arXiv arXiv 2017
[45]

Kaneta, Y

K. Kaneta, Y. Mambrini, K. A. Olive and S. Verner, Phys. Rev. D101, no.1, 015002 (2020) [arXiv:1911.02463 [hep-ph]]

arXiv 2020
[46]

Ellis, M

J. Ellis, M. A. G. Garcia, D. V. Nanopoulos, K. A. Olive and S. Verner, Phys. Rev. D105, no.4, 043504 (2022) [arXiv:2112.04466 [hep-ph]]

arXiv 2022
[47]

A. R. Liddle and S. M. Leach, Phys. Rev. D68, 103503 (2003) [astro-ph/0305263]

Pith/arXiv arXiv 2003
[48]

Martin and C

J. Martin and C. Ringeval, Phys. Rev. D82, 023511 (2010) [arXiv:1004.5525 [astro-ph.CO]]

Pith/arXiv arXiv 2010
[49]

Balkenhol, E

L. Balkenhol, E. Camphuis, F. Finelli, K. Benabed, F. R. Bouchet, J. Carron, S. Galli, E. Hivon, A. R. Khalife and L. Knox,et al.[arXiv:2512.10613 [astro-ph.CO]]

Pith/arXiv arXiv
[50]

S. M. Barr, Phys. Lett.112B(1982) 219; S. M. Barr, Phys. Rev. D40, 2457 (1989)

1982
[51]

J. P. Derendinger, J. E. Kim and D. V. Nanopoulos, Phys. Lett.139B(1984) 170

1984
[52]

Antoniadis, J

I. Antoniadis, J. R. Ellis, J. S. Hagelin and D. V. Nanopoulos, Phys. Lett. B194, 231 (1987)

1987
[53]

Antoniadis, J

I. Antoniadis, J. R. Ellis, J. S. Hagelin and D. V. Nanopoulos, Phys. Lett. B205(1988) 459; Phys. Lett. B208(1988) 209 Addendum: [Phys. Lett. B213(1988) 562]; Phys. Lett. B231 (1989) 65

1988
[54]

Ellis, M

J. Ellis, M. A. G. Garc´ ıa, N. Nagata, D. V. Nanopoulos and K. A. Olive, JCAP1707(2017) no.07, 006 [arXiv:1704.07331 [hep-ph]]

Pith/arXiv arXiv 2017
[55]

Ellis, M

J. Ellis, M. A. Garc´ ıa, N. Nagata, D. V. Nanopoulos and K. A. Olive, JCAP04, 009 (2019) [arXiv:1812.08184 [hep-ph]]

Pith/arXiv arXiv 2019
[56]

Ellis, M

J. Ellis, M. A. Garc´ ıa, N. Nagata, D. V. Nanopoulos and K. A. Olive, Phys. Lett. B797, 134864 (2019) [arXiv:1906.08483 [hep-ph]]

arXiv 2019
[57]

Ellis, M

J. Ellis, M. A. Garc´ ıa, N. Nagata, D. V. Nanopoulos and K. A. Olive, JCAP01, 035 (2020) [arXiv:1910.11755 [hep-ph]]

arXiv 2020
[58]

Ellis, M

J. Ellis, M. A. Garc´ ıa, N. Nagata, D. V. Nanopoulos and K. A. Olive, JHEP05, 021 (2020) [arXiv:2003.03285 [hep-ph]]

arXiv 2020
[59]

M. Endo, F. Takahashi and T. T. Yanagida, Phys. Rev. D76(2007), 083509 [arXiv:0706.0986 [hep-ph]]

Pith/arXiv arXiv 2007
[60]

J. A. Bagger, T. Moroi and E. Poppitz, JHEP04(2000), 009 [arXiv:hep-th/9911029 [hep-th]]; J. A. Bagger, T. Moroi and E. Poppitz, Nucl. Phys. B594(2001), 354-368 [arXiv:hep-th/0003282 [hep-th]]

Pith/arXiv arXiv 2000
[61]

Cremmer, S

E. Cremmer, S. Ferrara, L. Girardello and A. Van Proeyen, Nucl. Phys. B212(1983), 413 doi:10.1016/0550-3213(83)90679-X

work page doi:10.1016/0550-3213(83)90679-x 1983
[62]

Kaplunovsky and J

V. Kaplunovsky and J. Louis, Nucl. Phys. B422(1994), 57-124 [arXiv:hep-th/9402005 [hep-th]]. – 34 –

Pith/arXiv arXiv 1994

[1] [1]

A. A. Starobinsky, Phys. Lett. B91, 99 (1980)

1980

[2] [2]

K. S. Stelle, Gen. Rel. Grav.9(1978) 353; B. Whitt, Phys. Lett.145B(1984) 176

1978

[3] [3]

Kalara, N

S. Kalara, N. Kaloper and K. A. Olive, Nucl. Phys. B341, 252-272 (1990); K. i. Maeda, Phys. Rev. D39(1989), 3159

1990

[4] [4]

A. H. Guth, Phys. Rev. D23, 347-356 (1981)

1981

[5] [5]

V. F. Mukhanov and G. V. Chibisov, JETP Lett.33, 532 (1981) [Pisma Zh. Eksp. Teor. Fiz. 33, 549 (1981)]; A. A. Starobinsky, Sov. Astron. Lett.9, 302 (1983)

1981

[6] [6]

Press,Phys

W.H. Press,Phys. Scr.21(1980) 702; S.W. Hawking,Phys. Lett.115B(1982) 295; A.A. Starobinsky,Phys. Lett.117B(1982) 175; A.H. Guth and S.Y. Pi,Phys. Rev. Lett.49(1982) 1110; J.M. Bardeen, P.J. Steinhardt and M.S. Turner,Phys. Rev.D28(1983) 679

1980

[7] [7]

K. A. Olive, Phys. Rept.190(1990) 307; A. D. Linde,Particle Physics and Inflationary Cosmology(Harwood, Chur, Switzerland, 1990); D. H. Lyth and A. Riotto,Phys. Rep.314 (1999) 1 [arXiv:hep-ph/9807278]; A. D. Linde, Phys. Rept.333, 575-591 (2000); J. Martin, C. Ringeval and V. Vennin, Phys. Dark Univ.5-6, 75-235 (2014) [arXiv:1303.3787 [astro-ph.CO]]; J. M...

Pith/arXiv arXiv 1990

[8] [8]

Cremmer, S

E. Cremmer, S. Ferrara, C. Kounnas and D. V. Nanopoulos, Phys. Lett. B133(1983) 61; A. B. Lahanas and D. V. Nanopoulos, Phys. Rept.145(1987) 1

1983

[9] [9]

Ellis, D

J. Ellis, D. V. Nanopoulos and K. A. Olive, Phys. Rev. Lett.111(2013) 111301 [arXiv:1305.1247 [hep-th]]

Pith/arXiv arXiv 2013

[10] [10]

Ellis, M

J. Ellis, M. A. G. Garcia, N. Nagata, D. V. Nanopoulos, K. A. Olive and S. Verner, Int. J. Mod. Phys. D29, no.16, 2030011 (2020) [arXiv:2009.01709 [hep-ph]]

arXiv 2020

[11] [11]

Y. Ema, M. A. G. Garcia, W. Ke, K. A. Olive and S. Verner, Universe10, no.6, 239 (2024) [arXiv:2404.14545 [hep-ph]]

arXiv 2024

[12] [12]

R. J. Scherrer and M. S. Turner, Phys. Rev. D31, 681 (1985)

1985

[13] [13]

G. F. Giudice, E. W. Kolb and A. Riotto, Phys. Rev. D64(2001) 023508 [hep-ph/0005123]; D. J. H. Chung, E. W. Kolb and A. Riotto, Phys. Rev. D60(1999) 063504 [hep-ph/9809453]

Pith/arXiv arXiv 2001

[14] [14]

M. A. G. Garcia, K. Kaneta, Y. Mambrini and K. A. Olive, Phys. Rev. D101(2020) no.12, 123507 [arXiv:2004.08404 [hep-ph]]

arXiv 2020

[15] [15]

M. A. G. Garcia, K. Kaneta, Y. Mambrini and K. A. Olive, JCAP04, 012 (2021) [arXiv:2012.10756 [hep-ph]]

arXiv 2021

[16] [16]

Aghanimet al.[Planck], Astron

N. Aghanimet al.[Planck], Astron. Astrophys.641, A6 (2020) [erratum: Astron. Astrophys. 652, C4 (2021)] [arXiv:1807.06209 [astro-ph.CO]]; Y. Akramiet al.[Planck Collaboration], Astron. Astrophys.641, A10 (2020) [arXiv:1807.06211 [astro-ph.CO]]

Pith/arXiv arXiv 2020

[17] [17]

Watanabe and E

Y. Watanabe and E. Komatsu, Phys. Rev. D75(2007), 061301 [arXiv:gr-qc/0612120 [gr-qc]]; N. Bernal, J. Rubio and H. Veerm¨ ae, JCAP10(2020), 021 [arXiv:2006.02442 [hep-ph]]

Pith/arXiv arXiv 2007

[18] [18]

Gorbunov and A

D. Gorbunov and A. Tokareva, JCAP12, 021 (2013) [arXiv:1212.4466 [astro-ph.CO]]

Pith/arXiv arXiv 2013

[19] [19]

Kamada, JHEP07, 172 (2019) [arXiv:1902.05209 [hep-ph]]; A

A. Kamada, JHEP07, 172 (2019) [arXiv:1902.05209 [hep-ph]]; A. Kamada and T. Kuwahara, Phys. Rev. D101, no.9, 096012 (2020) [arXiv:1909.04228 [hep-ph]]; A. Kamada and T. Kuwahara, Phys. Rev. D103, no.11, 116001 (2021) [arXiv:1909.04229 [hep-ph]]

arXiv 2019

[20] [20]

Ellis, D

J. Ellis, D. V. Nanopoulos and K. A. Olive, Phys. Rev. D97, no.4, 043530 (2018) [arXiv:1711.11051 [hep-th]]. – 32 –

Pith/arXiv arXiv 2018

[21] [21]

G. D. Diamandis, A. B. Lahanas and K. Tamvakis, Phys. Rev. D92(2015) no.10, 105023 [arXiv:1509.01065 [hep-th]]; G. A. Diamandis, B. C. Georgalas, K. Kaskavelis, A. B. Lahanas and G. Pavlopoulos, Phys. Rev. D96, no. 4, 044033 (2017) [arXiv:1704.07617 [hep-th]]

Pith/arXiv arXiv 2015

[22] [22]

Antoniadis, C

I. Antoniadis, C. Markou and D. V. Nanopoulos, arXiv:2605.05295 [hep-th]

Pith/arXiv arXiv

[23] [23]

Cecotti, Phys

S. Cecotti, Phys. Lett. B190(1987) 86

1987

[24] [24]

M. Endo, K. Kadota, K. A. Olive, F. Takahashi and T. T. Yanagida, JCAP02, 018 (2007) [arXiv:hep-ph/0612263 [hep-ph]]

Pith/arXiv arXiv 2007

[25] [25]

Ellis, M

J. Ellis, M. A. G. Garcia, D. V. Nanopoulos and K. A. Olive, JCAP07, 050 (2015) [arXiv:1505.06986 [hep-ph]]

Pith/arXiv arXiv 2015

[26] [26]

Antoniadis, D

I. Antoniadis, D. V. Nanopoulos and J. Rizos, JCAP03, 017 (2021) [arXiv:2011.09396 [hep-th]]

arXiv 2021

[27] [27]

Antoniadis, D

I. Antoniadis, D. V. Nanopoulos and J. Rizos, Eur. Phys. J. C82, no.4, 377 (2022) [arXiv:2112.01211 [hep-th]]

arXiv 2022

[28] [28]

Antoniadis, D

I. Antoniadis, D. V. Nanopoulos and K. A. Olive, JHEP06, 155 (2025) [arXiv:2410.16541 [hep-th]]

arXiv 2025

[29] [29]

I. D. Gialamas, T. Katsoulas and K. Tamvakis, JCAP09, 060 (2025) [arXiv:2505.03608 [gr-qc]]

arXiv 2025

[30] [30]

Ellis, D

J. Ellis, D. V. Nanopoulos and K. A. Olive, JCAP10, 009 (2013) [arXiv:1307.3537 [hep-th]]

Pith/arXiv arXiv 2013

[31] [31]

Ellis, D

J. Ellis, D. V. Nanopoulos, K. A. Olive and S. Verner, JHEP03, 099 (2019) [arXiv:1812.02192 [hep-th]]

arXiv 2019

[32] [32]

Ellis, T

J. Ellis, T. Gherghetta, K. Kaneta, W. Ke and K. A. Olive, JHEP05, 229 (2026) [arXiv:2603.02389 [hep-ph]]

arXiv 2026

[33] [33]

P. A. R. Adeet al.[BICEP and Keck Collaborations], Phys. Rev. Lett.127, no.15, 151301 (2021) [arXiv:2110.00483 [astro-ph.CO]]

arXiv 2021

[34] [34]

Tristram, A

M. Tristram, A. J. Banday, K. M. G´ orski, R. Keskitalo, C. R. Lawrence, K. J. Andersen, R. B. Barreiro, J. Borrill, L. P. L. Colombo and H. K. Eriksen,et al.Phys. Rev. D105, no.8, 083524 (2022) [arXiv:2112.07961 [astro-ph.CO]]

arXiv 2022

[35] [35]

Louiset al.[Atacama Cosmology Telescope Collaboration], JCAP11, 062 (2025) [arXiv:2503.14452 [astro-ph.CO]]

T. Louiset al.[Atacama Cosmology Telescope Collaboration], JCAP11, 062 (2025) [arXiv:2503.14452 [astro-ph.CO]]

Pith/arXiv arXiv 2025

[36] [36]

Antoniadis, J

I. Antoniadis, J. Ellis, W. Ke, D. V. Nanopoulos and K. A. Olive, JCAP08, 090 (2025) [arXiv:2504.12283 [hep-ph]]

arXiv 2025

[37] [37]

Ellis, M

J. Ellis, M. A. G. Garc´ ıa, N. Nagata, D. V. Nanopoulos and K. A. Olive, JCAP12, 038 (2025) [arXiv:2508.13279 [hep-ph]]

arXiv 2025

[38] [38]

Ellis, M

J. Ellis, M. A. G. Garcia, K. A. Olive and S. Verner, Phys. Rev. D113, no.6, 063571 (2026) [arXiv:2510.18656 [hep-ph]]

arXiv 2026

[39] [39]

Camphuiset al.[SPT-3G], Phys

E. Camphuiset al.[SPT-3G], Phys. Rev. D113, no.8, 083504 (2026) [arXiv:2506.20707 [astro-ph.CO]]

Pith/arXiv arXiv 2026

[40] [40]

Ellis, D

J. Ellis, D. V. Nanopoulos and K. A. Olive, Phys. Rev. D89, no.4, 043502 (2014) [arXiv:1310.4770 [hep-ph]]

Pith/arXiv arXiv 2014

[41] [41]

Kallosh, A

R. Kallosh, A. Linde, K. A. Olive and T. Rube, Phys. Rev. D84, 083519 (2011) [arXiv:1106.6025 [hep-th]]

Pith/arXiv arXiv 2011

[42] [42]

M. Endo, M. Kawasaki, F. Takahashi and T. T. Yanagida, Phys. Lett. B642, 518-524 (2006) [arXiv:hep-ph/0607170 [hep-ph]]

Pith/arXiv arXiv 2006

[43] [43]

Kallosh and A

R. Kallosh and A. Linde, Comptes Rendus Physique16, 914-927 (2015) [arXiv:1503.06785 [hep-th]]. – 33 –

Pith/arXiv arXiv 2015

[44] [44]

Dudas, T

E. Dudas, T. Gherghetta, Y. Mambrini and K. A. Olive, Phys. Rev. D96, no.11, 115032 (2017) [arXiv:1710.07341 [hep-ph]]

Pith/arXiv arXiv 2017

[45] [45]

Kaneta, Y

K. Kaneta, Y. Mambrini, K. A. Olive and S. Verner, Phys. Rev. D101, no.1, 015002 (2020) [arXiv:1911.02463 [hep-ph]]

arXiv 2020

[46] [46]

Ellis, M

J. Ellis, M. A. G. Garcia, D. V. Nanopoulos, K. A. Olive and S. Verner, Phys. Rev. D105, no.4, 043504 (2022) [arXiv:2112.04466 [hep-ph]]

arXiv 2022

[47] [47]

A. R. Liddle and S. M. Leach, Phys. Rev. D68, 103503 (2003) [astro-ph/0305263]

Pith/arXiv arXiv 2003

[48] [48]

Martin and C

J. Martin and C. Ringeval, Phys. Rev. D82, 023511 (2010) [arXiv:1004.5525 [astro-ph.CO]]

Pith/arXiv arXiv 2010

[49] [49]

Balkenhol, E

L. Balkenhol, E. Camphuis, F. Finelli, K. Benabed, F. R. Bouchet, J. Carron, S. Galli, E. Hivon, A. R. Khalife and L. Knox,et al.[arXiv:2512.10613 [astro-ph.CO]]

Pith/arXiv arXiv

[50] [50]

S. M. Barr, Phys. Lett.112B(1982) 219; S. M. Barr, Phys. Rev. D40, 2457 (1989)

1982

[51] [51]

J. P. Derendinger, J. E. Kim and D. V. Nanopoulos, Phys. Lett.139B(1984) 170

1984

[52] [52]

Antoniadis, J

I. Antoniadis, J. R. Ellis, J. S. Hagelin and D. V. Nanopoulos, Phys. Lett. B194, 231 (1987)

1987

[53] [53]

Antoniadis, J

I. Antoniadis, J. R. Ellis, J. S. Hagelin and D. V. Nanopoulos, Phys. Lett. B205(1988) 459; Phys. Lett. B208(1988) 209 Addendum: [Phys. Lett. B213(1988) 562]; Phys. Lett. B231 (1989) 65

1988

[54] [54]

Ellis, M

J. Ellis, M. A. G. Garc´ ıa, N. Nagata, D. V. Nanopoulos and K. A. Olive, JCAP1707(2017) no.07, 006 [arXiv:1704.07331 [hep-ph]]

Pith/arXiv arXiv 2017

[55] [55]

Ellis, M

J. Ellis, M. A. Garc´ ıa, N. Nagata, D. V. Nanopoulos and K. A. Olive, JCAP04, 009 (2019) [arXiv:1812.08184 [hep-ph]]

Pith/arXiv arXiv 2019

[56] [56]

Ellis, M

J. Ellis, M. A. Garc´ ıa, N. Nagata, D. V. Nanopoulos and K. A. Olive, Phys. Lett. B797, 134864 (2019) [arXiv:1906.08483 [hep-ph]]

arXiv 2019

[57] [57]

Ellis, M

J. Ellis, M. A. Garc´ ıa, N. Nagata, D. V. Nanopoulos and K. A. Olive, JCAP01, 035 (2020) [arXiv:1910.11755 [hep-ph]]

arXiv 2020

[58] [58]

Ellis, M

J. Ellis, M. A. Garc´ ıa, N. Nagata, D. V. Nanopoulos and K. A. Olive, JHEP05, 021 (2020) [arXiv:2003.03285 [hep-ph]]

arXiv 2020

[59] [59]

M. Endo, F. Takahashi and T. T. Yanagida, Phys. Rev. D76(2007), 083509 [arXiv:0706.0986 [hep-ph]]

Pith/arXiv arXiv 2007

[60] [60]

J. A. Bagger, T. Moroi and E. Poppitz, JHEP04(2000), 009 [arXiv:hep-th/9911029 [hep-th]]; J. A. Bagger, T. Moroi and E. Poppitz, Nucl. Phys. B594(2001), 354-368 [arXiv:hep-th/0003282 [hep-th]]

Pith/arXiv arXiv 2000

[61] [61]

Cremmer, S

E. Cremmer, S. Ferrara, L. Girardello and A. Van Proeyen, Nucl. Phys. B212(1983), 413 doi:10.1016/0550-3213(83)90679-X

work page doi:10.1016/0550-3213(83)90679-x 1983

[62] [62]

Kaplunovsky and J

V. Kaplunovsky and J. Louis, Nucl. Phys. B422(1994), 57-124 [arXiv:hep-th/9402005 [hep-th]]. – 34 –

Pith/arXiv arXiv 1994