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REVIEW 2 major objections 5 minor 71 references

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.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-01 23:36 UTC pith:WYQNB56N

load-bearing objection 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. the 2 major comments →

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

Tunnelling out of Starobinsky inflation: Raising the spectral tilt

Jose A. R. Cembranos , Jes\'us Luque , Javier Rubio This is my paper
classification astro-ph.CO gr-qchep-ph PACS 98.80.Cq
keywords Starobinsky inflationfirst-order phase transitionvacuum decayspectral tilttensor-to-scalar ratioCMB observablesstochastic gravitational wavesalpha-attractors
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

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.

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

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

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.

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 this falsifier — get emailed when new claim-graph text bears on it.

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.

Where Pith is reading between the lines

These are 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.

Editorial analysis

A structured set of objections, weighed in public.

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

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)
  1. [§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.
  2. [§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)
  1. [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.
  2. [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.
  3. [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).
  4. [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.
  5. [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

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.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 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.
axioms (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.

pith-pipeline@v1.3.0-alltime-deepseek · 21829 in / 16342 out tokens · 164881 ms · 2026-08-01T23:36:23.594052+00:00 · methodology

0 comments
read the original 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.

discussion (0)

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Reference graph

Works this paper leans on

71 extracted references · 56 linked inside Pith

  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]

    Pith/arXiv 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]

    Pith/arXiv 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 ...

    Pith/arXiv 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...

    Pith/arXiv arXiv 2022

  9. [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]

    arXiv 2025

  10. [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 –

    arXiv 2025

  11. [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]

    arXiv 2025

  12. [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]

    arXiv 2025

  13. [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]

    Pith/arXiv arXiv 2024

  14. [20]

    Del Grosso, A

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

    Pith/arXiv arXiv

  15. [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

  16. [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

  17. [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

  18. [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]

    Pith/arXiv arXiv 2022

  19. [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 physics interpretation,JCAP10(2024) 020 [2403.03723]

    Pith/arXiv arXiv 2020

  20. [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]

    Pith/arXiv arXiv 1994

  21. [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]

    Pith/arXiv arXiv 2009

  22. [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]

    Pith/arXiv arXiv 2015

  23. [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]

    Pith/arXiv arXiv 1995

  24. [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]

    Pith/arXiv arXiv 2011

  25. [32]

    Linde,Inflation with variable Omega,Phys

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

    Pith/arXiv arXiv 1995

  26. [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]

    Pith/arXiv arXiv 2017

  27. [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]

    Pith/arXiv arXiv 2019

  28. [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]

    Pith/arXiv arXiv 2022

  29. [36]

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

    Pith/arXiv arXiv 2026

  30. [37]

    Bettoni, G

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

    Pith/arXiv arXiv 2019

  31. [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 –

    Pith/arXiv arXiv 2018

  32. [39]

    Bettoni and J

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

    Pith/arXiv arXiv 2020

  33. [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]

    Pith/arXiv arXiv 2022

  34. [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]

    Pith/arXiv arXiv 2023

  35. [42]

    Laverda and J

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

    Pith/arXiv arXiv 2024

  36. [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]

    Pith/arXiv arXiv 2024

  37. [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]

    arXiv 2025

  38. [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]

    Pith/arXiv arXiv 2025

  39. [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]

    arXiv 2025

  40. [47]

    Rubio and G

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

    Pith/arXiv arXiv 2025

  41. [48]

    Laverda, T

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

    Pith/arXiv arXiv 2026

  42. [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]

    Pith/arXiv arXiv 2010

  43. [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]

    Pith/arXiv arXiv 2022

  44. [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]

    Pith/arXiv arXiv 2021

  45. [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]

    Pith/arXiv arXiv 2010

  46. [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]

    Pith/arXiv arXiv 2017

  47. [56]

    Martin, C

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

    Pith/arXiv arXiv 2014

  48. [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]

    Pith/arXiv arXiv 2025

  49. [58]

    Barman, N

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

    Pith/arXiv arXiv 2024

  50. [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

  51. [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 –

    Pith/arXiv arXiv 2013

  52. [61]

    Azatov and M

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

    Pith/arXiv arXiv 2021

  53. [62]

    Azatov, M

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

    Pith/arXiv arXiv 2021

  54. [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]

    Pith/arXiv arXiv 2025

  55. [64]

    Bernal, J

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

    Pith/arXiv arXiv 2020

  56. [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]

    Pith/arXiv arXiv 2024

  57. [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]

    Pith/arXiv arXiv 2017

  58. [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]

    Pith/arXiv arXiv 2019

  59. [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]

    Pith/arXiv arXiv 2016

  60. [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]

    Pith/arXiv arXiv 2018

  61. [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]

    Pith/arXiv arXiv 1993

  62. [71]

    Huber and T

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

    Pith/arXiv arXiv 2008

  63. [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]

    Pith/arXiv arXiv 2016

  64. [73]

    Jinno, T

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

    Pith/arXiv arXiv 2019

  65. [74]

    Repond and J

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

    Pith/arXiv arXiv 2016

  66. [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]

    Pith/arXiv arXiv 2015

  67. [76]

    Artymowski and J

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

    Pith/arXiv arXiv 2016

  68. [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]

    Pith/arXiv arXiv 2016

  69. [78]

    Kallosh, A

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

    Pith/arXiv arXiv 2013

  70. [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]

    Pith/arXiv arXiv 2011

  71. [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