Tunnelling out of Starobinsky inflation: Raising the spectral tilt

Javier Rubio; Jes\'us Luque; Jose A. R. Cembranos

REVIEW 2 major objections 5 minor 71 references

Tunnelling out of Starobinsky inflation: Raising the spectral tilt

T0 review · 2 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Starobinsky inflation can end by tunnelling, and skipping the final stretch of its trajectory raises the spectral tilt to about 0.9725, the value recent CMB data prefer, with tensor-to-scalar ratio near 2e-3.

desk verdict Clever two-field mechanism that raises Starobinsky's n_s by ending inflation through a vacuum decay; the qualitative effect is solid, but the benchmark n_s depends on an unverified fixed-φ tunneling approximation. read the letter →

arxiv 2607.15354 v1 pith:WYQNB56N submitted 2026-07-16 astro-ph.CO gr-qchep-ph

Jose A. R. Cembranos , Jesús Luque , Javier Rubio This is my paper · ORCID

classification astro-ph.COgr-qchep-ph PACS 98.80.Cq

keywords Starobinskyinflationfirst-orderphasetransitionvacuumdecayspectraltilttensor-to-scalarratioCMBobservablesstochasticgravitationalwavesalpha-attractors

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

Starobinsky inflation predicts a spectral tilt of about 0.965, a value that sits at the edge of the higher tilt (about 0.974) preferred by recent cosmic-microwave-background measurements. This paper proposes a way to raise the tilt without altering the potential that generates the primordial fluctuations: end inflation by quantum tunnelling instead of slow roll. The field rolls along a metastable copy of the Starobinsky plateau whose minimum is shifted in field space, then tunnels to the true branch, landing beyond the slow-roll region and ending inflation abruptly. In the explicit benchmark the skipped interval is 12.15 e-folds, and the self-consistently computed observables become n_s = 0.9725 and r = 2.16e-3, inside the high-tilt region recent data prefer while the tensor signal stays as small as ever. The same vacuum transition leaves a stochastic gravitational-wave background peaking in the gigahertz range, and the construction extends to a wider family of plateau (alpha-attractor) models.

What carries the argument

Central is the branch-shift identity phi_TV = phi_FV - Delta-phi: an orthogonal tunnel leaves the global coordinate phi essentially unchanged while dropping the displacement from the branch minimum by Delta-phi, converting an inflating point on the false branch into a post-inflationary point on the true branch. From it follow the direct-exit condition Delta-phi >= phi_*^FV - phi_end and the skipped-e-folds parameter Delta-N_skip, which reparameterize the observables through N_eff = N_CMB + Delta-N_skip: n_s = 1 - 2/N_eff, r = 12/N_eff^2. The other workhorse is the O(4) Euclidean bounce action S4(phi), which locates the transition point phi_* as the first moment when percolation, a shrinking

What would settle it

Run a full two-field Euclidean bounce at the benchmark parameters (xi = -10^4, v_sigma = 9.695e-3 Planck masses, Lambda = 5.5e-3 Planck masses) without freezing phi, and read off the field displacement at the exit point. If the phi-component of the tunnelling path is even a few percent of the Planck mass, Delta-N_skip changes by several e-folds, shifting n_s by about 10^-3, the same scale as the gap between Starobinsky's standard prediction and recent data. A lattice simulation of percolation and reheating would independently confirm the assumed exit and thermalization timescales.

Watch

Extended reading notes

Core claim

The paper claims that Starobinsky inflation can be terminated by a first-order phase transition, and that this exit changes the model's CMB predictions. The potential holds two parallel Starobinsky branches: a metastable false branch whose minimum is displaced by Delta-phi from the true branch, with the two otherwise identical. Tunnelling in the orthogonal sigma direction leaves the global field phi nearly unchanged, so a point still inflating on the false branch lands on the true branch at a smaller displacement; if that landing point lies beyond the slow-roll endpoint (phi_* of about 0.94 Planck masses), inflation stops abruptly. The exit skips the final Delta-N_skip e-folds of the would-b

Load-bearing premise

The entire result rests on the tunnelling event moving the system almost purely in the extra field direction, leaving the main scalar field's value essentially unchanged; if the real two-field bounce shifts that field by more than a small fraction of the Planck mass, the landing point, the number of skipped e-folds, and the headline values of n_s and r all change.

Editorial extensions

If this is right

Starobinsky inflation is no longer tied to a single prediction: with a first-order exit the same plateau shape yields n_s = 0.9725 at r = 2.16e-3, inside the high-tilt region preferred by recent CMB combinations.
The old-inflation graceful-exit problem is avoided: the benchmark satisfies percolation, a decreasing physical false-vacuum volume, and a bubble abundance large enough for collisions while the background is still inflating.
A stochastic gravitational-wave background is produced by the end-of-inflation transition, peaking around 15 GHz in the benchmark (generically 10^7 to 10^10 Hz) with amplitude up to about 10^-9 to 10^-12, and it is not subsequently inflated away.
The construction is not specific to Starobinsky's potential: the shifted-branch mechanism embeds in E-model alpha-attractors, where the attractor parameter offers a further handle on the tensor-to-scalar ratio.
Observable dynamics remain effectively single-field (turning rate of order 10^-5, heavy transverse mode), so standard single-field CMB formulas apply and the only modification is the changed endpoint of inflation.

Reading between the lines

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

Editorial extension: the e-fold-shifting trick should work for any plateau model whose vacuum can be made metastable and shifted, Higgs-like plateau models included, turning each model's single (n_s, r) point into a one-parameter family rather than a fixed prediction.
Editorial extension: a genuine two-field bounce computation would turn the mechanism into a fully predictive model; the same numerical tools used for the sigma-direction action apply to the full field space, and the resulting phi-shift would set the error bar on the benchmark's n_s and r.
Editorial extension: the gravitational-wave band is the only place this mechanism is distinguishable from models that simply assume a larger e-fold number, since the CMB alone cannot fix the tunnelling duration; the reported peak frequency and amplitude define the sensitivity a future high-frequency gravitational-wave experiment would need.
Editorial extension: because the post-transition energy density first behaves as matter-like oscillations, the reheating history, and any relic produced during it, differs from ordinary Starobinsky reheating; the paper notes weakly coupled relics as a possibility but leaves their abundance unquantified.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This paper deserves a real look. It proposes a simple two-field modification of Starobinsky inflation where the observable stage runs on a shifted, metastable copy of the Starobinsky plateau, and a first-order phase transition in an orthogonal σ direction jumps the system to the true branch, ending inflation abruptly and skipping the last ~12 e-folds. That truncation raises n_s from the standard ~0.965 toward the ACT-preferred ~0.974 while keeping r tiny. The core idea is genuinely new as far as the cited literature goes—earlier papers modified Starobinsky via curvature corrections or reheating, not via a displaced false branch with a vacuum-decay exit.

What's good: the logic is transparent. The two branches have the same functional form in their own coordinates, so standard Starobinsky slow-roll expressions carry over; the effect is just a shift in the endpoint, parameterized by ΔN_skip. The benchmark is worked out carefully: they numerically solve the background, check the trajectory is effectively single-field (ω/H ~ 10^-5), compute the bounce with AnyBubble, and impose percolation, false-vacuum-volume, and bubble-abundance conditions simultaneously. They also verify the bubbles are subhorizon and Hawking–Moss is suppressed. That is more than most model papers do.

The soft spots are real but not fatal. The main one is the orthogonality assumption: tunneling is computed as a 1D σ-bounce at fixed φ, and Eq. (2.25) uses that to relate the landing point on the true branch. The potential couples σ and φ through A(σ)e^{-βφ/M_P}, and the two minima are separated by ~3.45 M_P in φ, so a genuine two-field bounce could shift φ by a non-negligible amount and move n_s and r by an amount comparable to the ACT preference. The paper states the assumption explicitly, but does not compute or bound the shift. That is a missing check, not necessarily an error—the mechanism is plausible because the σ barrier is steep and heavy. A full two-field bounce calculation would settle it.

The other assumptions are milder: the benchmark uses hand-set couplings (q=-0.94, Λ, v_σ) and instantaneous reheating. The reheating choice is clearly flagged, and the qualitative shift does not depend on it, only the numerical values. The slow-roll equations in the manuscript are standard Starobinsky expressions; the rendered form in the extract I saw is correct (ϵ_V decreases with φ), so I would not worry about that alleged mis-rendering.

Who this is for: anyone working on inflationary model-building, especially the ACT n_s tension, and people interested in vacuum-decay exits and gravitational waves from FOPTs. It deserves a serious referee. My recommendation: send it to peer review, and ask the authors to either provide a full two-field bounce computation or a bound on the φ-shift, plus a more detailed justification of the orthogonality approximation. Even if the exact numbers shift, the mechanism itself is worth publishing.

Referee Report

2 major / 5 minor

Summary. The paper proposes a two-field modification of Starobinsky inflation in which a symmetry-breaking field σ, with a double-well potential, shifts the Starobinsky minimum through A(σ)=1+ξσ²/M_P². At σ=v_σ the minimum is displaced by Δφ = −(M_P/β)ln(1+q), q=ξv_σ²/M_P²<0, giving a metastable 'false' branch; inflation proceeds there, and a first-order phase transition in σ transfers the system to the true branch (σ=0). Since the global field φ is assumed unchanged during tunnelling, the transition reduces the displacement from the branch minimum by Δφ. If this happens before the true-branch slow-roll endpoint, inflation ends abruptly, skipping ΔN_skip e-folds of the otherwise identical Starobinsky trajectory. This raises the effective number of e-folds at CMB horizon exit and hence n_s (and lowers r). The paper derives the direct-exit conditions, percolation and completion criteria, and performs self-consistent CMB matching (with instantaneous reheating in the benchmark). For the benchmark (ξ=−10^4, v_σ=9.695×10^-3 M_P, Λ=5.5×10^-3 M_P; q=−0.94) the transition occurs at φ_*=0.25 M_P, ΔN_skip=12.15, N_CMB=58.53, n_s=0.9725, r=2.16×10^-3; the associated gravitational-wave background peaks near 15 GHz. Appendices give a Jordan-frame and an α-attractor embedding.

Significance. Strengths: the mechanism is physically motivated and transparent; the paper uses explicit numerical bounce calculations (AnyBubble), states and checks three separate percolation conditions, and fixes M_φ from A_s rather than fitting n_s. If the fixed-φ tunneling assumption holds, the benchmark demonstrates a concrete way to move Starobinsky inflation toward the ACT-preferred high-n_s region while preserving small r. The GW prediction is falsifiable in principle. However, as the authors acknowledge in §2.2, the orthogonality assumption is an assumption; its validity is not demonstrated. A genuine two-field bounce would generically shift φ, and the quantitative benchmark—and the GW spectrum—are not robust. The paper is therefore a promising proof-of-principle whose headline numbers require further verification.

major comments (2)

[§2.2, Eq. (2.25); §4.1–4.2] The benchmark n_s=0.9725, r=2.16×10^-3 rests entirely on the relation φ_TV_* = φ_FV_* − Δφ, which assumes the bounce path is one-dimensional in σ with φ held fixed. The potential (2.2)–(2.5) couples σ and φ via A(σ)e^{-βφ/M_P}, and U_φ≠0 on the false branch; the Euclidean saddle generically moves in φ. The paper neither computes the escape point φ_out nor bounds |φ_out−φ_*|. Since ΔN_skip (Eq. 3.4) and the matching (Eq. 3.7) depend directly on φ_*, a shift of ~0.1 M_P changes ΔN_skip by ~1 e-fold and n_s by several×10^-4, comparable to the ACT preference motivating the model. The qualitative direction of the effect may survive, but the explicit numbers are not yet demonstrated.
[§5, Eqs. (5.3)–(5.13)] The gravitational-wave predictions inherit the same uncertainty. β_PT/H_* is computed as −dS_4/dN along the fixed-φ bounce (Eq. 5.4), and α uses the energy difference evaluated at φ_*. If the true two-field bounce moves in φ, both S_4(φ_*) and its derivative change, altering the quoted f_p≈15 GHz and Ω_p h²≈2.8×10^-12 (Eqs. 5.12–5.13). The GW signal is thus conditional on the same unverified orthogonality assumption; a computation or bound of the φ-shift is needed before the GW predictions can be taken at face value.

minor comments (5)

[Appendix C] The sufficient condition for a barrier, Λ^4 ≳ 12 M_φ² M_P², is not satisfied by the benchmark (Λ^4≈9.15×10^-10 vs 12M_φ²M_P²≈1.10×10^-9). The subsequent statement that this implies m_σ,FV²/H²≳96M_P²/v_σ² therefore does not apply to the benchmark. Since the numerically computed bounce confirms a barrier, this is a presentation issue, but it should be corrected or weakened.
[Section 3.2 / Fig. 2] The caption of Figure 2 should state explicitly that the gap between the standard Starobinsky point and the model curves is produced by the instantaneous-reheating assumption (N_RH=0), since this is not obvious from the figure alone.
[Eqs. (2.20)–(2.22)] The symbol φ_end is used for both branch endpoints; although the branch-adapted displacements are equal, the global coordinates differ. Please clarify to avoid confusion when reading Eq. (2.25).
[Sec. 4.1] The percolation integrals in Eq. (4.5) assume relativistic bubble walls (v_w=1) and quasi-de Sitter expansion. The paper should note that the benchmark does not compute the wall Lorentz factor or the effect of particle friction; this is a standard but non-trivial assumption.
[Sec. 4.2] For reproducibility, report S_4(φ) or Γ/H^4 as a function of N in addition to the derived percolation functions, since the benchmark relies on the numerical AnyBubble output.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: high-n_s result is derived from the model, not fitted to ACT; self-citations are contextual only.

full rationale

The derivation chain is self-contained for the central claim. The model potential (2.2)-(2.5) fixes two Starobinsky-shaped branches; the branch shift (2.10) and the explicitly stated assumption of a rapid, φ-orthogonal tunnel (Sec. 2.2: 'assume that tunnelling is both rapid compared to the Hubble timescale and approximately orthogonal to the ϕ direction') produce Eq. (2.25), φ_TV_* = φ_FV_* − ∆φ, and the direct-exit window (2.29). The observables n_s and r are then computed from the standard slow-roll formulas (3.1)-(3.4) after a self-consistent CMB matching (3.7), with the transition point φ* obtained from AnyBubble bounce actions and the percolation/completion conditions (4.2)-(4.5). Nothing in this chain is fitted to the ACT values; ACT is used only as motivation. The benchmark values n_s = 0.9725 and r = 2.16e-3 are consequences of the chosen (Λ, ξ, v_σ), not renamed fits. The free parameter q/φ_FV0 controls the outcome, which is model freedom/illustration, not circularity. Self-citations appear in contextual or side discussions (e.g., spectator-sector FOPTs, reheating, GW templates), but none is the load-bearing justification for the shift; the core ingredients are external (AnyBubble code, standard percolation criteria, standard slow-roll results). The most important caveat is the unquantified orthogonal-tunnelling approximation: the paper itself treats it as an assumption and does not bound the φ-shift of a genuine two-field bounce. That is a robustness/falsifiability gap, not a circular reduction—no equation defines the prediction in terms of its own output, and no fitted parameter is relabeled as a prediction. Hence score 0.

Assumptions & free parameters 4 free parameters · 5 assumptions · 1 invented entities

The central new physics is not derived from a unique UV theory; the two-field potential is an effective construction with three hand-set couplings plus an assumed reheating history. The benchmark is a proof-of-existence, not a parameter-free prediction.

free parameters (4)

ξ (via q) = ξ = -10^4, q = -0.94
Sets the branch separation Δφ = -(M_P/β)ln(1+q) ≈ 3.45 M_P. Chosen by hand; it directly controls ΔN_skip and therefore n_s.
vσ = 9.695 × 10^-3 M_P
Sets the σ vacuum position and transverse mass; chosen so that m_σ² ≫ H² and the transition completes at φ* ≈ 0.25 M_P.
Λ = 5.5 × 10^-3 M_P
Sets the double-well barrier height and the tunnelling action; chosen to satisfy percolation, false-volume decrease and bubble-abundance conditions.
N_RH = 0
Instantaneous reheating adopted for the benchmark. n_s and r depend on the reheating history through CMB matching, so this hand-set value is part of the result.

assumptions (5)

standard math Standard Euclidean O(4) vacuum-decay formalism with flat-space bounce and prefactor R_0^{-4}
Used in §4.1/§4.2; supported by the subhorizon condition R0H* = 5.9×10^-3, but gravitational corrections are not independently recomputed.
domain assumption The false-vacuum valley is well approximated by σ ≃ vσ; its small displacement is negligible for observables
Used throughout §§2–3; quantified in Appendix C as a few percent during the observable stage and O(10^-1) near the transition.
domain assumption The FOPT completes iff f > 0.34, df/dN > 3, and n_bub/H³ > 4
Adopted from Refs. [36,67]; these are phenomenological thresholds for percolation, false-vacuum volume decrease, and bubble collisions, not derived from first principles.
ad hoc to paper The two-field bounce is one-dimensional in σ with φ held fixed during tunnelling
Assumed in §2.2 and §4; no full two-field bounce is computed. This underpins the direct-exit window Eqs. (2.29)–(2.30) and the value of ΔN_skip.
domain assumption Post-transition energy is dominated by an oscillating σ field with w ≈ 0, then decays perturbatively; benchmark adopts instantaneous thermalization
Used for CMB matching (§3.2) and gravitational-wave dilution (§5); authors acknowledge that realistic dynamics may be richer.

invented entities (1)

Symmetry-breaking field σ with double-well potential independent evidence
purpose: Mediates the first-order transition between the displaced false Starobinsky branch and the standard true branch; sets the branch shift through A(σ)
If the model is correct, the transition produces a stochastic gravitational-wave background (benchmark: f_p ≈ 15 GHz, Ωh² ≈ 3×10^-12), a falsifiable but currently unobservable signature.

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Pith. "Pith review of Tunnelling out of Starobinsky inflation: Raising the spectral tilt." pith.science (2026). https://pith.science/paper/WYQNB56N

@misc{pith2026260715354,
  author       = {Pith},
  title        = {Pith review of: Tunnelling out of Starobinsky inflation: Raising the spectral tilt},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WYQNB56N}},
  note         = {Machine review of arXiv:2607.15354}
}

abstract

We propose a hybrid realization of Starobinsky inflation in which the inflationary epoch ends through vacuum decay. The model consists of an effective two-field system with a metastable Starobinsky branch shifted with respect to the true one. During the observable stage, the inflaton slow-rolls along the false branch, until a first-order phase transition in an orthogonal direction connects it to the true branch and ends inflation abruptly. This old-inflation-like exit skips the last part of the would-be Starobinsky trajectory. As a result, the Cosmic Microwave Background pivot scale exits the Hubble radius further from the minimum of the false branch than in ordinary Starobinsky inflation, raising the scalar spectral tilt $n_s$ while preserving the characteristic small tensor-to-scalar ratio. This provides a simple way of moving Starobinsky inflation towards the high-$n_s$ region favoured by recent ACT-related combinations. The same vacuum transition leaves a stochastic gravitational-wave relic whose peak frequency is controlled by the tunnelling timescale and the subsequent reheating history.

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Works this paper leans on

71 extracted references · 56 linked inside Pith

[1]

Guth,The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems,Phys

A.H. Guth,The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems,Phys. Rev. D23(1981) 347

1981
[2]

Guth and E.J

A.H. Guth and E.J. Weinberg,Could the Universe Have Recovered from a Slow First Order Phase Transition?,Nucl. Phys. B212(1983) 321

1983
[3]

Linde,A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems,Phys

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

1982
[4]

Linde,Chaotic Inflation,Phys

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

1983
[5]

Linde,Hybrid inflation,Phys

A.D. Linde,Hybrid inflation,Phys. Rev. D49(1994) 748 [astro-ph/9307002]

arXiv 1994
[6]

Baumann,Inflation, inTheoretical Advanced Study Institute in Elementary Particle Physics: Physics of the Large and the Small, pp

D. Baumann,Inflation, inTheoretical Advanced Study Institute in Elementary Particle Physics: Physics of the Large and the Small, pp. 523–686, 2011, DOI [0907.5424]

arXiv 2011
[7]

Starobinsky,A New Type of Isotropic Cosmological Models Without Singularity,Phys

A.A. Starobinsky,A New Type of Isotropic Cosmological Models Without Singularity,Phys. Lett. B91(1980) 99. [8]Planckcollaboration,Planck 2018 results. X. Constraints on inflation,Astron. Astrophys. 641(2020) A10 [1807.06211]. [9]BICEP, Keckcollaboration,Improved Constraints on Primordial Gravitational Waves using Planck, WMAP, and BICEP/Keck Observations ...

arXiv 1980
[10]

Tristram et al.,Improved limits on the tensor-to-scalar ratio using BICEP and Planck data, Phys

M. Tristram et al.,Improved limits on the tensor-to-scalar ratio using BICEP and Planck data, Phys. Rev. D105(2022) 083524 [2112.07961]. [11]Atacama Cosmology Telescopecollaboration,The Atacama Cosmology Telescope: DR6 constraints on extended cosmological models,JCAP11(2025) 063 [2503.14454]. [12]Atacama Cosmology Telescopecollaboration,The Atacama Cosmol...

arXiv 2022

Show all 71 references

[15]

Drees and Y

M. Drees and Y. Xu,Refined predictions for Starobinsky inflation and post-inflationary constraints in light of ACT,Phys. Lett. B867(2025) 139612 [2504.20757]

2025
[16]

Zharov, O.O

D.S. Zharov, O.O. Sobol and S.I. Vilchinskii,ACT observations, reheating, and Starobinsky and Higgs inflation,Phys. Rev. D112(2025) 023544 [2505.01129]. – 25 –

2025
[17]

Addazi, Y

A. Addazi, Y. Aldabergenov and S.V. Ketov,Curvature corrections to Starobinsky inflation can explain the ACT results,Phys. Lett. B869(2025) 139883 [2505.10305]

2025
[18]

Ketov, E.O

S.V. Ketov, E.O. Pozdeeva and S.Y. Vernov,Inflation in F(R) gravity models revisited after ACT,JCAP12(2025) 040 [2508.08927]

2025
[19]

Cecchini, M

C. Cecchini, M. De Angelis, W. Giarè, M. Rinaldi and S. Vagnozzi,Testing scale-invariant inflation against cosmological data,JCAP07(2024) 058 [2403.04316]

2024 arXiv
[20]

Del Grosso, A

L. Del Grosso, A. Urbano and M. Ziccolella,ACT stands for Awkward Cosmology Theories, 2607.09808

arXiv
[21]

Coleman,The Fate of the False Vacuum

S.R. Coleman,The Fate of the False Vacuum. 1. Semiclassical Theory,Phys. Rev. D15(1977) 2929

1977
[22]

Callan, Jr

C.G. Callan, Jr. and S.R. Coleman,The Fate of the False Vacuum. 2. First Quantum Corrections,Phys. Rev. D16(1977) 1762

1977
[23]

Coleman and F

S.R. Coleman and F. De Luccia,Gravitational Effects on and of Vacuum Decay,Phys. Rev. D 21(1980) 3305

1980
[24]

Devoto, S

F. Devoto, S. Devoto, L. Di Luzio and G. Ridolfi,False vacuum decay: an introductory review, J. Phys. G49(2022) 103001 [2205.03140]

2022 arXiv
[25]

Caprini et al.,Detecting gravitational waves from cosmological phase transitions with LISA: an update,JCAP03(2020) 024 [1910.13125]

C. Caprini et al.,Detecting gravitational waves from cosmological phase transitions with LISA: an update,JCAP03(2020) 024 [1910.13125]. [26]LISA Cosmology Working Groupcollaboration,Gravitational waves from first-order phase transitions in LISA: reconstruction pipeline and phy...

2020 arXiv
[27]

Copeland, A.R

E.J. Copeland, A.R. Liddle, D.H. Lyth, E.D. Stewart and D. Wands,False vacuum inflation with Einstein gravity,Phys. Rev. D49(1994) 6410 [astro-ph/9401011]

1994 arXiv
[28]

Cortes and A.R

M. Cortes and A.R. Liddle,Viable inflationary models ending with a first-order phase transition,Phys. Rev. D80(2009) 083524 [0905.0289]

2009 arXiv
[29]

Ashoorioon,Exit from Inflation with a First-Order Phase Transition and a Gravitational Wave Blast,Phys

A. Ashoorioon,Exit from Inflation with a First-Order Phase Transition and a Gravitational Wave Blast,Phys. Lett. B747(2015) 446 [1502.00556]

2015 arXiv
[30]

Bucher, A.S

M. Bucher, A.S. Goldhaber and N. Turok,An open universe from inflation,Phys. Rev. D52 (1995) 3314 [hep-ph/9411206]

1995 arXiv
[31]

Yamauchi, A

D. Yamauchi, A. Linde, A. Naruko, M. Sasaki and T. Tanaka,Open inflation in the landscape, Phys. Rev. D84(2011) 043513 [1105.2674]

2011 arXiv
[32]

Linde,Inflation with variable Omega,Phys

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

1995 arXiv
[33]

Jiang, T

H. Jiang, T. Liu, S. Sun and Y. Wang,Echoes of Inflationary First-Order Phase Transitions in the CMB,Phys. Lett. B765(2017) 339 [1512.07538]

2017 arXiv
[34]

Domènech, J

G. Domènech, J. Rubio and J. Wons,Mimicking features in alternatives to inflation with interacting spectator fields,Phys. Lett. B790(2019) 263 [1811.08224]

2019 arXiv
[35]

An, K.-F

H. An, K.-F. Lyu, L.-T. Wang and S. Zhou,Gravitational waves from an inflation triggered first-order phase transition,JHEP06(2022) 050 [2201.05171]

2022 arXiv
[36]

J. Zou, L. Bian and S.-J. Wang,Simulating first-order phase transition during inflation,Phys. Rev. D113(2026) 123514 [2602.05501]

2026 arXiv
[37]

Bettoni, G

D. Bettoni, G. Domènech and J. Rubio,Gravitational waves from global cosmic strings in quintessential inflation,JCAP02(2019) 034 [1810.11117]

2019 arXiv
[38]

Bettoni and J

D. Bettoni and J. Rubio,Quintessential Affleck-Dine baryogenesis with non-minimal couplings, Phys. Lett. B784(2018) 122 [1805.02669]. – 26 –

2018 arXiv
[39]

Bettoni and J

D. Bettoni and J. Rubio,Hubble-induced phase transitions: Walls are not forever,JCAP01 (2020) 002 [1911.03484]

2020 arXiv
[40]

Bettoni, A

D. Bettoni, A. Lopez-Eiguren and J. Rubio,Hubble-induced phase transitions on the lattice with applications to Ricci reheating,JCAP01(2022) 002 [2107.09671]

2022 arXiv
[41]

Kierkla, G

M. Kierkla, G. Laverda, M. Lewicki, A. Mantziris, M. Piani, J. Rubio et al.,From Hubble to Bubble,JHEP11(2023) 077 [2309.08530]

2023 arXiv
[42]

Laverda and J

G. Laverda and J. Rubio,Ricci reheating reloaded,JCAP03(2024) 033 [2307.03774]

2024 arXiv
[43]

Laverda and J

G. Laverda and J. Rubio,The rise and fall of the Standard-Model Higgs: electroweak vacuum stability during kination,JHEP05(2024) 339 [2402.06000]

2024 arXiv
[44]

Bettoni, G

D. Bettoni, G. Laverda, A. Lopez-Eiguren and J. Rubio,Hubble-induced phase transitions: gravitational-wave imprint of Ricci reheating from lattice simulations,JCAP03(2025) 027 [2409.15450]

2025
[45]

Goertz, M

F. Goertz, M. Hager, G. Laverda and J. Rubio,Phasing out of darkness: from sterile neutrino dark matter to neutrino masses via time-dependent mixing,JHEP02(2025) 213 [2407.04778]

2025 arXiv
[46]

Laverda and J

G. Laverda and J. Rubio,Higgs-induced gravitational waves: the interplay of non-minimal couplings, kination and top quark mass,JHEP08(2025) 203 [2502.04445]

2025
[47]

Rubio and G

J. Rubio and G. Laverda,Hubble-induced phase transitions in the Standard Model and beyond, PoSCORFU2024(2025) 073 [2505.00900]

2025 arXiv
[48]

Laverda, T

G. Laverda, T. Mendes and J. Rubio,Tachyonic gravitational dark matter production after inflation,JCAP05(2026) 053 [2601.07670]

2026 arXiv
[49]

Espinosa, T

J.R. Espinosa, T. Konstandin, J.M. No and G. Servant,Energy Budget of Cosmological First-order Phase Transitions,JCAP06(2010) 028 [1004.4187]

2010 arXiv
[50]

Wang and Z.-Y

S.-J. Wang and Z.-Y. Yuwen,The energy budget of cosmological first-order phase transitions beyond the bag equation of state,JCAP10(2022) 047 [2206.01148]

2022 arXiv
[51]

Cutting, E.G

D. Cutting, E.G. Escartin, M. Hindmarsh and D.J. Weir,Gravitational waves from vacuum first order phase transitions II: from thin to thick walls,Phys. Rev. D103(2021) 023531 [2005.13537]

2021 arXiv
[52]

Punturo et al.,The Einstein Telescope: A third-generation gravitational wave observatory, Class

M. Punturo et al.,The Einstein Telescope: A third-generation gravitational wave observatory, Class. Quant. Grav.27(2010) 194002. [53]ETcollaboration,The Science of the Einstein Telescope,JCAP03(2026) 081 [2503.12263]

2010 arXiv
[54]

LISA Collaboration, Amaro-Seoane et al.,Laser Interferometer Space Antenna,.(2017) [1702.00786]

P. LISA Collaboration, Amaro-Seoane et al.,Laser Interferometer Space Antenna,.(2017) [1702.00786]. [55]AEDGEcollaboration,AEDGE: Atomic Experiment for Dark Matter and Gravity Exploration in Space,EPJ Quant. Technol.7(2020) 6 [1908.00802]

2017 arXiv
[56]

Martin, C

J. Martin, C. Ringeval and V. Vennin,Encyclopædia Inflationaris: Opiparous Edition,Phys. Dark Univ.5-6(2014) 75 [1303.3787]

2014 arXiv
[57]

Barman, N

B. Barman, N. Bernal and J. Rubio,Two or three things particle physicists (mis)understand about (pre)heating,Nucl. Phys. B1018(2025) 116996 [2503.19980]

2025 arXiv
[58]

Barman, N

B. Barman, N. Bernal and J. Rubio,Rescuing gravitational-reheating in chaotic inflation, JCAP05(2024) 072 [2310.06039]

2024 arXiv
[59]

Watkins and L.M

R. Watkins and L.M. Widrow,Aspects of Reheating in First Order Inflation,Nucl. Phys. B 374(1992) 446

1992
[60]

Falkowski and J.M

A. Falkowski and J.M. No,Non-thermal Dark Matter Production from the Electroweak Phase Transition: Multi-TeV WIMPs and ’Baby-Zillas’,JHEP02(2013) 034 [1211.5615]. – 27 –

2013 arXiv
[61]

Azatov and M

A. Azatov and M. Vanvlasselaer,Bubble wall velocity: heavy physics effects,JCAP01(2021) 058 [2010.02590]

2021 arXiv
[62]

Azatov, M

A. Azatov, M. Vanvlasselaer and W. Yin,Dark Matter production from relativistic bubble walls, JHEP03(2021) 288 [2101.05721]

2021 arXiv
[63]

Cembranos, J

J.A.R. Cembranos, J. Luque and J. Rubio,Scalar dark matter production through the bubble expansion mechanism: the role of the Lorentz factor and non-renormalizable interactions,Eur. Phys. J. C85(2025) 368 [2407.14592]

2025 arXiv
[64]

Bernal, J

N. Bernal, J. Rubio and H. Veermäe,UV Freeze-in in Starobinsky Inflation,JCAP10(2020) 021 [2006.02442]

2020 arXiv
[65]

Athron, C

P. Athron, C. Balázs, A. Fowlie, L. Morris and L. Wu,Cosmological phase transitions: From perturbative particle physics to gravitational waves,Prog. Part. Nucl. Phys.135(2024) 104094 [2305.02357]

2024 arXiv
[66]

Masoumi, K.D

A. Masoumi, K.D. Olum and B. Shlaer,Efficient numerical solution to vacuum decay with many fields,JCAP01(2017) 051 [1610.06594]

2017 arXiv
[67]

Ellis, M

J. Ellis, M. Lewicki and J.M. No,On the Maximal Strength of a First-Order Electroweak Phase Transition and its Gravitational Wave Signal,JCAP04(2019) 003 [1809.08242]

2019 arXiv
[68]

Kaiser,Nonminimal Couplings in the Early Universe: Multifield Models of Inflation and the Latest Observations,Fundam

D.I. Kaiser,Nonminimal Couplings in the Early Universe: Multifield Models of Inflation and the Latest Observations,Fundam. Theor. Phys.183(2016) 41 [1511.09148]

2016 arXiv
[69]

Cutting, M

D. Cutting, M. Hindmarsh and D.J. Weir,Gravitational waves from vacuum first-order phase transitions: from the envelope to the lattice,Phys. Rev. D97(2018) 123513 [1802.05712]

2018 arXiv
[70]

Kosowsky and M.S

A. Kosowsky and M.S. Turner,Gravitational radiation from colliding vacuum bubbles: envelope approximation to many bubble collisions,Phys. Rev. D47(1993) 4372 [astro-ph/9211004]

1993 arXiv
[71]

Huber and T

S.J. Huber and T. Konstandin,Gravitational Wave Production by Collisions: More Bubbles, JCAP09(2008) 022 [0806.1828]

2008 arXiv
[72]

Weir,Revisiting the envelope approximation: gravitational waves from bubble collisions, Phys

D.J. Weir,Revisiting the envelope approximation: gravitational waves from bubble collisions, Phys. Rev. D93(2016) 124037 [1604.08429]

2016 arXiv
[73]

Jinno, T

R. Jinno, T. Konstandin and M. Takimoto,Relativistic bubble collisions—a closer look,JCAP 09(2019) 035 [1906.02588]

2019 arXiv
[74]

Repond and J

J. Repond and J. Rubio,Combined Preheating on the lattice with applications to Higgs inflation,JCAP07(2016) 043 [1604.08238]

2016 arXiv
[75]

Galante, R

M. Galante, R. Kallosh, A. Linde and D. Roest,Unity of Cosmological Inflation Attractors, Phys. Rev. Lett.114(2015) 141302 [1412.3797]

2015 arXiv
[76]

Artymowski and J

M. Artymowski and J. Rubio,Endlessly flat scalar potentials andα-attractors,Phys. Lett. B 761(2016) 111 [1607.00398]

2016 arXiv
[77]

Karananas and J

G.K. Karananas and J. Rubio,On the geometrical interpretation of scale-invariant models of inflation,Phys. Lett. B761(2016) 223 [1606.08848]

2016 arXiv
[78]

Kallosh, A

R. Kallosh, A. Linde and D. Roest,Superconformal Inflationaryα-Attractors,JHEP11(2013) 198 [1311.0472]

2013 arXiv
[79]

Kallosh, A

R. Kallosh, A. Linde, K.A. Olive and T. Rube,Chaotic inflation and supersymmetry breaking, Phys. Rev. D84(2011) 083519 [1106.6025]

2011 arXiv
[80]

Hawking and I.G

S.W. Hawking and I.G. Moss,Supercooled Phase Transitions in the Very Early Universe,Phys. Lett. B110(1982) 35. – 28 –

1982

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

Guth,The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems,Phys

A.H. Guth,The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems,Phys. Rev. D23(1981) 347

1981

[2] [2]

Guth and E.J

A.H. Guth and E.J. Weinberg,Could the Universe Have Recovered from a Slow First Order Phase Transition?,Nucl. Phys. B212(1983) 321

1983

[3] [3]

Linde,A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems,Phys

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

1982

[4] [4]

Linde,Chaotic Inflation,Phys

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

1983

[5] [5]

Linde,Hybrid inflation,Phys

A.D. Linde,Hybrid inflation,Phys. Rev. D49(1994) 748 [astro-ph/9307002]

arXiv 1994

[6] [6]

Baumann,Inflation, inTheoretical Advanced Study Institute in Elementary Particle Physics: Physics of the Large and the Small, pp

D. Baumann,Inflation, inTheoretical Advanced Study Institute in Elementary Particle Physics: Physics of the Large and the Small, pp. 523–686, 2011, DOI [0907.5424]

arXiv 2011

[7] [7]

Starobinsky,A New Type of Isotropic Cosmological Models Without Singularity,Phys

A.A. Starobinsky,A New Type of Isotropic Cosmological Models Without Singularity,Phys. Lett. B91(1980) 99. [8]Planckcollaboration,Planck 2018 results. X. Constraints on inflation,Astron. Astrophys. 641(2020) A10 [1807.06211]. [9]BICEP, Keckcollaboration,Improved Constraints on Primordial Gravitational Waves using Planck, WMAP, and BICEP/Keck Observations ...

arXiv 1980

[8] [10]

Tristram et al.,Improved limits on the tensor-to-scalar ratio using BICEP and Planck data, Phys

M. Tristram et al.,Improved limits on the tensor-to-scalar ratio using BICEP and Planck data, Phys. Rev. D105(2022) 083524 [2112.07961]. [11]Atacama Cosmology Telescopecollaboration,The Atacama Cosmology Telescope: DR6 constraints on extended cosmological models,JCAP11(2025) 063 [2503.14454]. [12]Atacama Cosmology Telescopecollaboration,The Atacama Cosmol...

arXiv 2022