Pith. sign in

REVIEW 5 major objections 6 minor 2 cited by

Observable Primordial Gravitational Waves from Non-minimally Coupled $R^2$ Palatini Modified Gravity

T0 review · 5 major / 6 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A non-minimally coupled R^2 Palatini gravity model with a runaway inflaton potential predicts a primordial gravitational-wave spectrum with two flat plateaus—one from hyperkination and one from reheating—joined by a kination-era boost…

desk verdict Interesting extension of hyperkination GW predictions, but the spectra are built on a post-inflation background that is inconsistent with the paper's own energy bound and convergence times, undermining the central BBN-rescue claim. read the letter →

arxiv 2502.03573 v2 pith:J4MWUZKP submitted 2025-02-05 astro-ph.CO gr-qc

Hardik Jitendra Kuralkar , Sukanta Panda , Archit Vidyarthi This is my paper · ORCID
classification astro-ph.COgr-qc
keywords primordialgravitationalwavesPalatinigravityR^2hyperkinationkinationreheatingstochasticgravitational-wavebackgroundquintessentialinflation
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

This paper tries to establish that primordial gravitational waves can reveal the $R^{2}$ term in Palatini gravity. For a non-minimally coupled theory with a runaway inflaton potential, the gravitational-wave spectrum is controlled by a short hyperkination phase followed by kination and reheating, producing two plateaus joined by a boost in energy density that scales as a power of frequency. The hyperkination phase truncates the boost, preventing the gravitational-wave background from overproducing energy during Big Bang Nucleosynthesis; without the $R^{2}$ term there is no hyperkination and the theory fails. Lower coupling orders shift the spectra toward the sensitivity of planned detectors, so a future detection could distinguish the theory.

What carries the argument

The load-bearing object is the Palatini action L ∝ (1+ξχ/M_Pl)^t(R+$αR^{2}$) with a runaway potential. A Weyl transformation moves to the Einstein frame, where the $R^{2}$ term becomes a quartic kinetic term; that quartic term drives hyperkination (equation of state w=1/3), the canonical quadratic term then drives kination (w=1), and the decay mechanism brings radiation domination. The scale factor is solved piecewise and matched at the transitions, and tensor perturbations obey the Mukhanov-Sasaki equation whose Hankel-function solutions are matched at the same times. The final spectrum is Ω_GW with the two plateaus and the boost, with p=(Γ_τ+4−t√6)/2 controlling the slope.

What would settle it

Compute the inflationary spectral index and tensor-to-scalar ratio for t = -3, -2, -1, and 1 with the same runaway potential and compare with Planck; if any case cannot produce 50–60 e-folds or violates the Planck (ns,r) contours, the corresponding gravitational-wave spectra in Figures 10 and 11 lose their inflationary initial conditions.

Watch

Extended reading notes

Core claim

The central claim is that in Palatini gravity with L ∝ h(χ)(R+$αR^{2}$), h(χ)=(1+ξχ/M_Pl)^t, and a runaway potential, the post-inflationary gravitational-wave energy density has a distinctive three-epoch shape: a flat plateau during hyperkination, a rising segment Ω_GW ∝ $f^{{2/p}}$ during kination, and a second flat plateau during reheating. The exponent p depends on the coupling order t and the decay rate of the inflaton, so the slope of the boost encodes the theory. Because hyperkination dilutes the field energy as $a^{{-4}}$, it truncates the kination boost before it violates BBN bounds; the authors take this as evidence that $R^{2}$ terms are necessary in any Palatini model with a runaway potential. They also show that decreasing t makes the spectra more accessible to future experiments, and that for t>0 a supplementary inflaton-decay mechanism with T_reh ∼ $10^{8}$ GeV is needed to achieve reheating.

Load-bearing premise

The load-bearing premise is that slow-roll inflation works for every coupling order t with the runaway potential; only the V∝χ², h∝χ² case is checked against Planck, so if another t cannot deliver 50–60 e-folds and a viable (ns,r), that t's gravitational-wave spectrum lacks a consistent inflationary starting point.

Editorial extensions

If this is right

  • If the model is correct, a measurement of the reheating plateau determines the Hubble scale H and radiation fraction Ω_r^end at the end of inflation.
  • The slope of the kination boost, proportional to f^{2/p}, identifies the coupling order t (and inflaton decay rate), thereby distinguishing among the theories in this class.
  • The hyperkination plateau places the BBN-saving bound in the kHz–GHz band, motivating high-frequency gravitational-wave detectors such as resonant cavities.
  • Lower-order couplings (t<0) and the minimal Starobinsky t=0 case lie closer to future detector sensitivity curves, making them prime targets for LISA, DECIGO, BBO, ET, CE, and pulsar timing arrays.
  • For t<0, the flat reheating plateau can be matched against PTA and NANOGrav observations of the stochastic background to constrain the parameter space.

Reading between the lines

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

  • If the boost slope is ever measured, it could be inverted to extract the coupling order t and the inflaton decay parameter Γ_τ, turning the stochastic gravitational-wave background into a spectrometer for the theory's structure.
  • The necessity argument likely generalizes: any non-oscillatory (quintessential) inflation model in Palatini gravity that includes R^2 will require a hyperkination phase of at least the BBN-safe duration, so models without such a phase should be viewed with suspicion.
  • The reheating mechanism for t>0 introduces model-dependent couplings (b, n, m_χ), so low-frequency predictions may shift if those parameters differ; comparisons with PTA-band data should be treated as conditional on the decay model.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 6 minor

Summary. This paper considers a Palatini modified-gravity action L ∝ (1+ξχ/M_Pl)^t (R+αR^2) with a runaway inflaton potential, and computes the stochastic gravitational-wave background generated during the post-inflationary epochs of hyperkination, kination, and reheating. The authors derive the background evolution, solve the Mukhanov-Sasaki equation with mode matching, and obtain a spectrum with two plateaus connected by a kination boost Ω_GW ∝ f^{2/p}. They argue that the R^2-induced hyperkination epoch truncates the kination boost and prevents violation of the BBN bound, and they compare the resulting spectra with current and future detector sensitivities, identifying observable parameter regions and proposing bounds on H and Ω_r^end at the end of inflation. For t>0, a reheating mechanism with inflaton mass mχ around 10^4 GeV is added to achieve radiation domination.

Significance. If the central background calculation were consistent, the paper would be significant: it connects the coupling order t of non-minimal R^2 Palatini gravity to a specific spectral slope Ω_GW ∝ f^{2/p}, and it makes falsifiable predictions for future GW experiments in the kHz-GHz band. The analytic mode-matching derivations in Appendices C-E are detailed, and the use of external constraints such as the BBN bound and PLIC sensitivity curves is appropriate. The broad class of models (t = -3,...,2) and the parameter-space maps are useful. However, the main physical claim—that a post-inflationary hyperkination phase truncates the kination boost and rescues the spectrum from BBN overproduction—rests on a background that the paper's own equations appear to rule out, as detailed in the major comments.

major comments (5)
  1. [Section 2.4, Eqs. (2.45) and (2.47)] The bound in (2.45) gives \bar\chi_0 > 2 M_Pl at the end of inflation, while the quartic-dominated hyperkination regime requires \bar\chi < \sqrt{3} M_Pl according to (2.47). Since 2 M_Pl > \sqrt{3} M_Pl, the field at N=0 is already in the kination-dominated regime, not in a w=1/3 hyperkination phase. The scale factor (4.4), the mode functions (4.22), and the spectrum (4.41) are all built on the hyperkination branch of (2.42), so the central two-plateau structure and the claimed BBN rescue rest on a background that the paper's own equations rule out.
  2. [Section 4.1 and Table 1] The instantaneous-transition ansatz (2.42) is not the solution of the full equation (2.32): Table 1 lists convergence times of 0.76-6.18 e-folds, whereas (2.45) gives \Delta N_hyp \leq 0.20. Section 5.2 explicitly acknowledges this disagreement, yet the subsequent spectra are computed with the piecewise ansatz. The transition times and normalization of the mode functions therefore do not correspond to the dynamics of (2.32).
  3. [Section 2.3 and Appendix B] Successful slow-roll inflation is verified only for V = βχ^2 and h = γχ^2. The statement that inflation is "successfully achieved for all orders of couplings" is an assumption, and no (n_s, r) check is given for t = -3, -2, -1, 1, 2 with h=(1+ξχ/M_Pl)^t. Since the tensor normalization at η_end and the inflationary initial conditions enter the spectra in Figures 10 and 11, these spectra are not predictions of a consistent model unless the inflationary viability of each t is established.
  4. [Section 3 and Eq. (4.5)] The reheating parameter Γτ is introduced as a tuned power-law decay exponent, ρ ∝ a^{t√6-6-Γτ}, but Section 3's particle-decay mechanism yields a constant decay rate Γ, which gives exponential damping in cosmic time, not a power law in a. The mapping from mχ = 10^4 GeV to the tuned Γτ is not derived, and the spectral slope p = (Γτ + 4 - t√6)/2 depends on it. The predicted boost shape is therefore not tied to the microphysical reheating model.
  5. [Section 4.3 and Figures 10-11] The illustrative parameters α = 10^14 and H = 10^13 GeV used in the main observability plots, when inserted in (4.40), give \Delta N_hyp well above the allowed \Delta N_hyp \leq 0.20 derived from (2.45). The plotted spectra are thus displayed outside the model's valid parameter space, which undermines the visibility claims in Section 5.3.
minor comments (6)
  1. [Section 5] The phrase "sound-to-noise ratio" should read "signal-to-noise ratio".
  2. [Section 2.4 and Figure 2] The notation N^end_hyp = 0.2 and N^end_inf = 0 is confusing because N^end_hyp is a duration, not an e-fold coordinate; consider renaming to \Delta N_hyp and N_end = 0.
  3. [Eq. (4.13)] The general expression for ν in terms of p and w is introduced but never used, and the powers of w are not motivated; clarify or remove it.
  4. [Section 5.2] The text states that the convergence times "do not agree" with \Delta N_hyp \leq 0.20 and then asserts that a longer hyperkination "will not affect the spectrum much" without quantitative support; this needs either a calculation or removal.
  5. [Appendix B, Figure 12] The predictions are shown for N* = 170 and 200, but the text does not state whether these e-fold numbers are chosen to fit Planck or derived from the model; specify the reheating history used for N*.
  6. [Section 3] In the discussion of spontaneous symmetry breaking, the quadratic term in (3.3) is written with a "±" sign and the text appeals to "the minus sign"; this is ambiguous and should be stated explicitly.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the GW spectra are derived from an assumed expansion history and are checked against external BBN and detector bounds; the main caveats are internal-consistency and parameter-tuning issues, not circular reductions.

full rationale

The central spectrum (4.35)/(4.41) is not fitted to the target signal. It follows from the piecewise background (4.4), the Mukhanov-Sasaki solutions (4.22), and the matching coefficients (4.23)-(4.29); the two plateaus and the f^{2/p} kination boost are consequences of assuming w=1/3 hyperkination, w=1 kination, and radiation reheating, and the results are then compared with external BBN constraints and detector PLICs. The reheating parameter Gamma_tau (through p) and m_chi are chosen to make radiation domination possible, but they are not adjusted to reproduce any GW datum, so this is ordinary background input rather than a fitted prediction. The self-citations ([49], [97], [98]) support model viability, unitarity, and Planck compatibility; the GW derivation does not rest on them, and Appendix B provides an independent comparison, so they are not load-bearing. The paper's internal tension - Eq. (2.45) gives chi_bar_0 > 2 M_Pl while hyperkination in (2.47) requires chi_bar < sqrt(3) M_Pl, and Section 5.2 admits 'As Delta_Nhyp <= 0.2, it does not agree with the obtained convergence' - is a consistency and validity concern about whether the assumed hyperkination segment is the model's actual post-inflation background. If that background fails, the spectra are not valid predictions of the full dynamics, but that is not a circular equivalence: the spectrum is still computed from the stated assumptions rather than being defined as those assumptions. Accordingly, the circularity score is low.

Assumptions & free parameters 8 free parameters · 7 assumptions · 0 invented entities

The central claim depends on the assumed runaway potential, the Palatini R^2 reformulation, the tuned kination decay exponent, and the unverified assumption that inflation works for all coupling orders. No new particle, force, or dimension is introduced beyond the generic boson and fermion fields used in the reheating mechanism.

free parameters (8)
  • t (coupling order) = varied from -3 to 2
    The order of the coupling is the key model parameter; the paper scans it and finds that lower t gives more observable spectra.
  • Gamma_tau (kination decay exponent) = not quoted; tuned to realize reheating for m_chi = 10^4 GeV
    Controls rho proportional to a^(t sqrt(6) - 6 - Gamma_tau) and the kination spectral slope 2/p; the paper tunes it rather than deriving it from the decay rate in Section 3.
  • H (Hubble parameter at end of inflation) = 10^13 GeV in most plots
    Scanned over the parameter space; sets the overall spectrum amplitude through H^2 / M_Pl^2.
  • Omega_end^r (radiation energy fraction at end of inflation) = 10^-10 in most plots
    Free input; sets the reheating plateau amplitude and frequency boundaries.
  • alpha (R^2 coefficient) = 10^14 in Figures 10 and 11
    Free parameter of the action, traded for Delta_N_hyp through Eq. (4.40).
  • m_chi (inflaton mass in reheating mechanism) = 10^4 GeV
    Fine-tuned so the inflaton decays before radiation domination; gives T_reh about 10^8 GeV.
  • xi (non-minimal coupling strength) = not specified in the text
    Appears in h(chi) = (1 + xi chi / M_Pl)^t; the paper never states the value used, although the attractor condition and spectra are written as if it drops out.
  • c (integration constant in kination attractor) = taken as 1
    Set to unity when solving for eta_reh; a free normalization of the attractor solution.
assumptions (7)
  • standard math The Palatini R^2 action can be recast as a scalar-tensor theory with a quartic kinetic term via the auxiliary-field and Weyl transformations.
    Standard f(R, phi) transformation; used in Eqs. (2.2) through (2.12).
  • domain assumption The inflaton potential is runaway and becomes negligible after inflation, with no oscillation minimum.
    Core setup of the paper, Section 2.1 and Figure 1; excludes oscillatory reheating models.
  • domain assumption The field derivative approaches a kinetic attractor requiring d < 0, which restricts t <= 2.
    Derived in Section 2.4, Eq. (2.37); used to truncate the coupling order at chi^2 R^2.
  • ad hoc to paper Successful slow-roll inflation is assumed for all coupling orders t with the runaway potential.
    Section 2.3 asserts this after verifying only one model in Appendix B; no slow-roll computation is given for t different from 2.
  • domain assumption Instantaneous transitions between inflation, hyperkination, kination, and reheating, with continuity of a and a'.
    Section 4.1 and Appendix C; standard matching approximation but not exact.
  • ad hoc to paper The kination energy density decays as rho proportional to a^(t sqrt(6) - 6 - Gamma_tau) with Gamma_tau tuned.
    Section 4.1, Eq. (C.8); the decay exponent is not derived from the microphysical decay rate in Section 3.
  • standard math Bunch-Davies vacuum for tensor modes at the start of inflation.
    Section 4.2 and Appendix D; standard choice for the initial tensor mode functions.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Observable Primordial Gravitational Waves from Non-minimally Coupled $R^2$ Palatini Modified Gravity." pith.science (2026). https://pith.science/paper/J4MWUZKP

@misc{pith2026250203573,
  author       = {Pith},
  title        = {Pith review of: Observable Primordial Gravitational Waves from Non-minimally Coupled $R^2$ Palatini Modified Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J4MWUZKP}},
  note         = {Machine review of arXiv:2502.03573}
}
abstract

We probe the spectrum of primordial gravitational waves (GWs) produced during the eras of hyperkination, kination, and reheating in a non-minimally coupled, $\mathcal{L} \propto (1+ \xi \chi /M_{\text{Pl}})^t (R+\alpha R^2)$, modified gravity using the Palatini formulation. We consider a runaway potential, which gives an era of kinetic domination after the end of inflation. The coupling order $t$ is varied to examine a large class of theories up to $\chi^2 R^2$. For models with $t>0$, reheating is not achieved naturally; hence, we supplement such theories with a reheating mechanism based on the interaction of inflaton and radiation produced at the end of inflation due to cosmological expansion. We demonstrate that the energy density of the GWs is enhanced as a function of the coupling during kination for all considered theories, and a short-lived phase of hyperkination truncates the boost and avoids the over-production of GWs. Hyperkination, and thus the $R^2$ term, should be deemed necessary in all theories with a runaway potential as it prevents the GW enhancement during kination from destabilizing the Big Bang Nucleosynthesis. The spectrum remains flat for the period of hyperkination and reheating. We examine the available parameter space for which the theories remain valid and place bounds on the Hubble parameter ($H$) and radiation energy density ($\Omega_r^{\text{end}}$) at the end of inflation. We find that as we decrease the order of the coupling, the spectra shift towards a more observable regime of future GW experiments. The observation of the plateau during reheating will constrain the $H$ and $\Omega_r^{\text{end}}$ values, while the spectral shape of the boost obtained during kination will confirm the nature of the theory. The bounds from hyperkination lie in the kHz-GHz frequency range.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Oscillon Formation in Palatini Modified Gravity Theories

    gr-qc 2026-05 unverdicted novelty 7.0 of 10

    Oscillons form in Palatini modified gravity with non-minimal coupling during preheating, yielding extended oscillon domination and ultra-high-frequency gravitational waves in the range of planned detectors.

  2. ACT-DR6 consistent inflation in generalised entropic cosmology and $f(Q)$ gravity

    gr-qc 2026-07 conditional novelty 4.0 of 10

    Reconstruction produces explicit f(Q) and generalised-entropic inflation models (and scalar-coupled versions) whose slow-roll parameters match ACT-DR6 + Planck-BAO constraints on n_s and r.

Reference graph

Works this paper leans on

125 extracted references · 59 canonical work pages · cited by 2 Pith papers

  1. [95]

    Sánchez López, K

    S. Sánchez López, K. Dimopoulos, A. Karam and E. Tomberg,Observable gravitational waves from hyperkination in palatini gravity and beyond, The European Physical Journal C83 (2023)

    work page 2023

  2. [1]

    Albrecht and P.J

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

    1982

  3. [2]

    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

  4. [3]

    Kazanas,Dynamics of the Universe and Spontaneous Symmetry Breaking, Astrophys

    D. Kazanas,Dynamics of the Universe and Spontaneous Symmetry Breaking, Astrophys. J. Lett. 241 (1980) L59

    1980

  5. [4]

    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

  6. [5]

    Linde,Chaotic Inflation, Phys

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

    1983

  7. [6]

    Sato,First-order phase transition of a vacuum and the expansion of the Universe, Mon

    K. Sato,First-order phase transition of a vacuum and the expansion of the Universe, Mon. Not. Roy. Astron. Soc.195 (1981) 467

    1981

  8. [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. B 91 (1980) 99

    1980

Show all 125 references
  1. [8]

    Linde,Particle physics and inflationary cosmology, 2005

    A. Linde,Particle physics and inflationary cosmology, 2005

    2005

  2. [9]

    Dodelson,Modern cosmology, Academic Press, Amsterdam, nachdr

    S. Dodelson,Modern cosmology, Academic Press, Amsterdam, nachdr. ed. (20)

  3. [10]

    Rubakov and D.S

    V.A. Rubakov and D.S. Gorbunov,Introduction to the Theory of the Early Universe: Hot Big Bang Theory, WORLD SCIENTIFIC, 2 ed. (Aug., 2017), 10.1142/10447. – 35 –

    2017 doi

  4. [11]

    Kolb,The Early Universe, CRC Press, 1 ed

    E. Kolb,The Early Universe, CRC Press, 1 ed. (Mar., 2018), 10.1201/9780429492860

    2018 doi

  5. [12]

    Starobinsky,Spectrum of relict gravitational radiation and the early state of the universe, JETP Lett

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

    1979

  6. [13]

    Mukhanov and G.V

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

    1981

  7. [14]

    Hawking,The Development of Irregularities in a Single Bubble Inflationary Universe, Phys

    S.W. Hawking,The Development of Irregularities in a Single Bubble Inflationary Universe, Phys. Lett. B115 (1982) 295

    1982

  8. [15]

    Guth and S.Y

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

    1982

  9. [16]

    Collaboration, P.A.R

    B. Collaboration, P.A.R. Ade, Z. Ahmed, M. Amiri, D. Barkats, R.B. Thakur et al.,Bicep / keck xiii: Improved constraints on primordial gravitational waves using planck, wmap, and bicep/keck observations through the 2018 observing season,

    2018

  10. [17]

    Collaboration, Y

    P. Collaboration, Y. Akrami, F. Arroja, M. Ashdown, J. Aumont, C. Baccigalupi et al., Planck 2018 results. i. overview and the cosmological legacy of planck,

    2018

  11. [18]

    Collaboration, Y

    P. Collaboration, Y. Akrami, F. Arroja, M. Ashdown, J. Aumont, C. Baccigalupi et al., Planck 2018 results. x. constraints on inflation,

    2018

  12. [19]

    Mukhanov,Physical Foundations of Cosmology, Cambridge University Press, 1 ed

    V. Mukhanov,Physical Foundations of Cosmology, Cambridge University Press, 1 ed. (Nov., 2005), 10.1017/CBO9780511790553

    2005 doi

  13. [20]

    Baumann,Cosmology, Cambridge University Press, 1 ed

    D. Baumann,Cosmology, Cambridge University Press, 1 ed. (June, 2022), 10.1017/9781108937092

    2022 doi

  14. [21]

    Gorbunov and V.A

    D.S. Gorbunov and V.A. Rubakov,Introduction to the Theory of the Early Universe: Cosmological Perturbations and Inflationary Theory, World Scientific Publishing Company (Feb., 2011), 10.1142/7873

    2011 doi

  15. [22]

    Sahni,Energy density of relic gravity waves from inflation, Physical Review D42 (1990) 453–463

    V. Sahni,Energy density of relic gravity waves from inflation, Physical Review D42 (1990) 453–463

    1990

  16. [23]

    Allen,Stochastic gravity-wave background in inflationary-universe models, Physical Review D 37 (1988) 2078–2085

    B. Allen,Stochastic gravity-wave background in inflationary-universe models, Physical Review D 37 (1988) 2078–2085

    1988

  17. [24]

    Felder, L

    G. Felder, L. Kofman and A. Linde,Inflation and preheating in no models,

  18. [25]

    Gomez, S

    M.E. Gomez, S. Lola, C. Pallis and J. Rodriguez-Quintero,Quintessential kination and thermal production of gravitinos and axinos,

  19. [26]

    Joyce and T

    M. Joyce and T. Prokopec,Turning around the sphaleron bound: Electroweak baryogenesis in an alternative post-inflationary cosmology,

  20. [27]

    Pallis,Quintessential kination and cold dark matter abundance,

    C. Pallis,Quintessential kination and cold dark matter abundance,

  21. [28]

    Pallis,Kination dominated reheating and cold dark matter abundance,

    C. Pallis,Kination dominated reheating and cold dark matter abundance,

  22. [29]

    Peebles and A

    P.J.E. Peebles and A. Vilenkin,Quintessential inflation,

  23. [30]

    Peloso and F

    M. Peloso and F. Rosati,On the construction of quintessential inflation models,

  24. [31]

    Huey and J.E

    G. Huey and J.E. Lidsey,Inflation, braneworlds and quintessence,

  25. [32]

    Dimopoulos,Towards a model of quintessential inflation,

    K. Dimopoulos,Towards a model of quintessential inflation,

  26. [33]

    Collaboration, P.A.R

    P. Collaboration, P.A.R. Ade, N. Aghanim, M. Arnaud, M. Ashdown, J. Aumont et al., Planck 2015 results. xiii. cosmological parameters,

    2015

  27. [34]

    Collaboration, N

    P. Collaboration, N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, M. Ballardini et al., Planck intermediate results. li. features in the cosmic microwave background temperature power spectrum and shifts in cosmological parameters, . – 36 –

  28. [35]

    Array, B

    K. Array, B. Collaborations, P.A.R. Ade, Z. Ahmed, R.W. Aikin, K.D. Alexander et al., Bicep2 / keck array vi: Improved constraints on cosmology and foregrounds when adding 95 ghz data from keck array,

  29. [36]

    Utiyama and B.S

    R. Utiyama and B.S. DeWitt,Renormalization of a classical gravitational field interacting with quantized matter fields, J. Math. Phys.3 (1962) 608

    1962

  30. [37]

    Stelle,Renormalization of Higher Derivative Quantum Gravity, Phys

    K.S. Stelle,Renormalization of Higher Derivative Quantum Gravity, Phys. Rev. D16 (1977) 953

    1977

  31. [38]

    Vilkovisky,Effective action in quantum gravity, Class

    G.A. Vilkovisky,Effective action in quantum gravity, Class. Quant. Grav.9 (1992) 895

    1992

  32. [39]

    Buchbinder, S.D

    I.L. Buchbinder, S.D. Odintsov and I.M. Lichtzier,The behaviour of effective coupling constants in ’finite’ grand unification theories in curved spacetime, Classical and Quantum Gravity 6 (1989) 605

    1989

  33. [40]

    Birrell and P.C.W

    N.D. Birrell and P.C.W. Davies,Quantum Fields in Curved Space, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, UK (1982), 10.1017/CBO9780511622632

    1982 doi

  34. [41]

    De Felice and S

    A. De Felice and S. Tsujikawa,f(r) theories, Living Reviews in Relativity13 (2010)

    2010

  35. [42]

    Sotiriou and V

    T.P. Sotiriou and V. Faraoni,f(r) theories of gravity, Reviews of Modern Physics82 (2010) 451–497

    2010

  36. [43]

    Nojiri and S.D

    S. Nojiri and S.D. Odintsov,Unified cosmic history in modified gravity: From f(r) theory to lorentz non-invariant models, Physics Reports 505 (2011) 59

    2011

  37. [44]

    Nojiri, S

    S. Nojiri, S. Odintsov and V. Oikonomou,Modified gravity theories on a nutshell: Inflation, bounce and late-time evolution, Physics Reports 692 (2017) 1–104

    2017

  38. [45]

    Woodard,Avoiding dark energy with 1/r modifications of gravity, inThe Invisible Universe: Dark Matter and Dark Energy, p

    R. Woodard,Avoiding dark energy with 1/r modifications of gravity, inThe Invisible Universe: Dark Matter and Dark Energy, p. 403–433, Springer Berlin Heidelberg (2007), DOI

    2007

  39. [46]

    Palatini,Deduzione invariantiva delle equazioni gravitazionali dal principio di Hamilton, Rend

    A. Palatini,Deduzione invariantiva delle equazioni gravitazionali dal principio di Hamilton, Rend. Circ. Mat. Palermo43 (1919) 203

    1919

  40. [47]

    Gialamas, A

    I.D. Gialamas, A. Karam, T.D. Pappas and E. Tomberg,Implications of palatini gravity for inflation and beyond, International Journal of Geometric Methods in Modern Physics20 (2023)

    2023

  41. [48]

    Shaposhnikov, A

    M. Shaposhnikov, A. Shkerin and S. Zell,Standard model meets gravity: Electroweak symmetry breaking and inflation,

  42. [49]

    Panda, A.A

    S. Panda, A.A. Tinwala and A. Vidyarthi,Ultraviolet unitarity violations in non-minimally coupled scalar-starobinsky inflation, Journal of Cosmology and Astroparticle Physics2023 (2023) 029

    2023

  43. [50]

    Mikura and Y

    Y. Mikura and Y. Tada,On uv-completion of palatini-higgs inflation,

  44. [51]

    McDonald,Does palatini higgs inflation conserve unitarity?,

    J. McDonald,Does palatini higgs inflation conserve unitarity?,

  45. [52]

    A. Ito, W. Khater and S. Rasanen,Tree-level unitarity in higgs inflation in the metric and the palatini formulation,

  46. [53]

    Bauer and D.A

    F. Bauer and D.A. Demir,Higgs-palatini inflation and unitarity,

  47. [54]

    Antoniadis, A

    I. Antoniadis, A. Guillen and K. Tamvakis,Ultraviolet behaviour of higgs inflation models,

  48. [55]

    Odintsov, V.K

    S.D. Odintsov, V.K. Oikonomou and F.P. Fronimos,Quantitative predictions for f(R) gravity primordial gravitational waves, Phys. Dark Univ.35 (2022) 100950 [2108.11231]

    2022 arXiv

  49. [56]

    Odintsov, V

    S. Odintsov, V. Oikonomou and R. Myrzakulov,Spectrum of primordial gravitational waves in modified gravities: A short overview, Symmetry 14 (2022) 729. – 37 –

    2022

  50. [57]

    Capozziello, M

    S. Capozziello, M. De Laurentis, S. Nojiri and S. Odintsov,Evolution of gravitons in accelerating cosmologies: The case of extended gravity, Physical Review D95 (2017)

    2017

  51. [58]

    Turner,Detectability of inflation-produced gravitational waves, Phys

    M.S. Turner,Detectability of inflation-produced gravitational waves, Phys. Rev. D55 (1997) R435

    1997

  52. [59]

    Harry and the LIGO Scientific Collaboration,Advanced LIGO: the next generation of gravitational wave detectors, Classical and Quantum Gravity27 (2010) 084006

    G.M. Harry and the LIGO Scientific Collaboration,Advanced LIGO: the next generation of gravitational wave detectors, Classical and Quantum Gravity27 (2010) 084006

    2010

  53. [60]

    Aasi, B.P

    The LIGO Scientific Collaboration, J. Aasi, B.P. Abbott, R. Abbott, T. Abbott, M.R. Abernathy et al.,Advanced LIGO, Classical and Quantum Gravity32 (2015) 074001

    2015

  54. [61]

    Acernese, M

    F. Acernese, M. Agathos, K. Agatsuma, D. Aisa, N. Allemandou, A. Allocca et al.,Advanced Virgo: a second-generation interferometric gravitational wave detector, Classical and Quantum Gravity 32 (2015) 024001

    2015

  55. [62]

    Akutsu, M

    T. Akutsu, M. Ando, K. Arai, Y. Arai, S. Araki, A. Araya et al.,Overview of KAGRA: Detector design and construction history, Progress of Theoretical and Experimental Physics 2021 (2021) 05A101

    2021

  56. [63]

    McLaughlin,The North American Nanohertz Observatory for Gravitational Waves, Classical and Quantum Gravity30 (2013) 224008

    M.A. McLaughlin,The North American Nanohertz Observatory for Gravitational Waves, Classical and Quantum Gravity30 (2013) 224008

    2013

  57. [64]

    Brazier, S

    A. Brazier, S. Chatterjee, T. Cohen, J.M. Cordes, M.E. DeCesar, P.B. Demorest et al.,The NANOGrav Program for Gravitational Waves and Fundamental Physics, Aug., 2019. 10.48550/arXiv.1908.05356

    2019 doi arXiv

  58. [65]

    Foster and D.C

    R.S. Foster and D.C. Backer,Constructing a pulsar timing array, The Astrophysical Journal 361 (1990) 300

    1990

  59. [66]

    Stappers, M

    B.W. Stappers, M. Kramer, A.G. Lyne, N. D’Amico and A. Jessner,The European Pulsar Timing Array, Chinese Journal of Astronomy and Astrophysics Supplement6 (2006) 298

    2006

  60. [67]

    Janssen, B.W

    G.H. Janssen, B.W. Stappers, M. Kramer, M. Purver, A. Jessner, I. Cognard et al.,European Pulsar Timing Array, inAIP Conference Proceedings, vol. 983, (Montreal (Canada)), pp. 633–635, AIP, 2008, DOI

    2008

  61. [68]

    Verbiest, L

    J.P.W. Verbiest, L. Lentati, G. Hobbs, R.v. Haasteren, P.B. Demorest, G.H. Janssen et al., The International Pulsar Timing Array: First Data Release, Feb., 2016. 10.48550/arXiv.1602.03640

    2016 doi arXiv

  62. [69]

    Perera, M.E

    B.B.P. Perera, M.E. DeCesar, P.B. Demorest, M. Kerr, L. Lentati, D.J. Nice et al.,The International Pulsar Timing Array: Second data release, Sept., 2019. 10.48550/arXiv.1909.04534

    2019 doi arXiv

  63. [70]

    Hobbs,The Parkes Pulsar Timing Array, Classical and Quantum Gravity30 (2013) 224007

    G. Hobbs,The Parkes Pulsar Timing Array, Classical and Quantum Gravity30 (2013) 224007

    2013

  64. [71]

    Bartolo, C

    N. Bartolo, C. Caprini, V. Domcke, D.G. Figueroa, J. Garcia-Bellido, M.C. Guzzetti et al., Science with the space-based interferometer LISA. IV: probing inflation with gravitational waves, Journal of Cosmology and Astroparticle Physics2016 (2016) 026

    2016

  65. [72]

    Amaro-Seoane, H

    P. Amaro-Seoane, H. Audley, S. Babak, J. Baker, E. Barausse, P. Bender et al.,Laser Interferometer Space Antenna, Feb., 2017. 10.48550/arXiv.1702.00786

    2017 doi arXiv

  66. [73]

    Caprini, D.G

    C. Caprini, D.G. Figueroa, R. Flauger, G. Nardini, M. Peloso, M. Pieroni et al., Reconstructing the spectral shape of a stochastic gravitational wave background with LISA, Journal of Cosmology and Astroparticle Physics2019 (2019) 017

    2019

  67. [74]

    Auclair, D

    P. Auclair, D. Bacon, T. Baker, T. Barreiro, N. Bartolo, E. Belgacem et al.,Cosmology with the Laser Interferometer Space Antenna, Living Reviews in Relativity26 (2023) 5. – 38 –

    2023

  68. [75]

    Punturo, M

    M. Punturo, M. Abernathy, F. Acernese, B. Allen, N. Andersson, K. Arun et al.,The Einstein Telescope: a third-generation gravitational wave observatory, Classical and Quantum Gravity 27 (2010) 194002

    2010

  69. [76]

    Reitze, R.X

    D. Reitze, R.X. Adhikari, S. Ballmer, B. Barish, L. Barsotti, G. Billingsley et al.,Cosmic Explorer: The U.S. Contribution to Gravitational-Wave Astronomy beyond LIGO, July, 2019. 10.48550/arXiv.1907.04833

    2019 doi arXiv

  70. [77]

    Phinney,The Big Bang Observer: Direct Detection of Gravitational Waves from the Birth of the Universe to the Present, NASA Mission Concept Study, 2004

    E.S. Phinney,The Big Bang Observer: Direct Detection of Gravitational Waves from the Birth of the Universe to the Present, NASA Mission Concept Study, 2004

    2004

  71. [78]

    Crowder and N.J

    J. Crowder and N.J. Cornish,Beyond LISA: Exploring future gravitational wave missions, Physical Review D72 (2005) 083005

    2005

  72. [79]

    Harry, P

    G.M. Harry, P. Fritschel, D.A. Shaddock, W. Folkner and E.S. Phinney,Laser interferometry for the Big Bang Observer, Classical and Quantum Gravity23 (2006) 4887

    2006

  73. [80]

    Dewdney, P

    P. Dewdney, P. Hall, R. Schilizzi and T. Lazio,The Square Kilometre Array, Proceedings of the IEEE 97 (2009) 1482

    2009

  74. [81]

    Kawamura, T

    S. Kawamura, T. Nakamura, M. Ando, N. Seto, K. Tsubono, K. Numata et al.,The Japanese space gravitational wave antenna—DECIGO, Classical and Quantum Gravity23 (2006) S125

    2006

  75. [82]

    Kawamura, M

    S. Kawamura, M. Ando, N. Seto, S. Sato, T. Nakamura, K. Tsubono et al.,The Japanese space gravitational wave antenna: DECIGO, Classical and Quantum Gravity28 (2011) 094011

    2011

  76. [83]

    Kawamura, M

    S. Kawamura, M. Ando, N. Seto, S. Sato, M. Musha, I. Kawano et al.,Current status of space gravitational wave antenna DECIGO and B-DECIGO, Progress of Theoretical and Experimental Physics 2021 (2021) 05A105

    2021

  77. [84]

    NI,Astrod-gw: Overview and progress, International Journal of Modern Physics D22 (2013) 1341004

    W.-T. NI,Astrod-gw: Overview and progress, International Journal of Modern Physics D22 (2013) 1341004

    2013

  78. [85]

    Sesana, N

    A. Sesana, N. Korsakova, M.A. Sedda, V. Baibhav, E. Barausse, S. Barke et al.,Unveiling the gravitational universe atµ-hz frequencies, Experimental Astronomy 51 (2021) 1333–1383

    2021

  79. [86]

    Badurina, E

    L. Badurina, E. Bentine, D. Blas, K. Bongs, D. Bortoletto, T. Bowcock et al.,Aion: an atom interferometer observatory and network, Journal of Cosmology and Astroparticle Physics 2020 (2020) 011–011

    2020

  80. [87]

    El-Neaj, C

    Y.A. El-Neaj, C. Alpigiani, S. Amairi-Pyka, H. Araújo, A. Balaž, A. Bassi et al.,Aedge: Atomic experiment for dark matter and gravity exploration in space, EPJ Quantum Technology 7 (2020)

    2020

  81. [88]

    Aggarwal, O.D

    N. Aggarwal, O.D. Aguiar, A. Bauswein, G. Cella, S. Clesse, A.M. Cruise et al.,Challenges and opportunities of gravitational-wave searches at mhz to ghz frequencies, Living Reviews in Relativity 24 (2021)

    2021

  82. [89]

    Gialamas and A.B

    I.D. Gialamas and A.B. Lahanas,Reheating in R2 Palatini inflationary models, Phys. Rev. D 101 (2020) 084007 [1911.11513]

    2020 arXiv

  83. [90]

    Gialamas, A

    I.D. Gialamas, A. Karam and A. Racioppi,Dynamically induced Planck scale and inflation in the Palatini formulation, JCAP 11 (2020) 014 [2006.09124]

    2020 arXiv

  84. [91]

    Gialamas, A

    I.D. Gialamas, A. Karam, T.D. Pappas and V.C. Spanos,Scale-invariant quadratic gravity and inflation in the Palatini formalism, Phys. Rev. D104 (2021) 023521 [2104.04550]

    2021 arXiv

  85. [92]

    Gialamas and K

    I.D. Gialamas and K. Tamvakis,Inflation in metric-affine quadratic gravity, JCAP 03 (2023) 042 [2212.09896]

    2023 arXiv

  86. [93]

    Gialamas, T

    I.D. Gialamas, T. Katsoulas and K. Tamvakis,Inflation and reheating in quadratic metric-affine gravity with derivative couplings, JCAP 06 (2024) 005 [2403.08530]. – 39 –

    2024 arXiv

  87. [94]

    Kaneda and S.V

    S. Kaneda and S.V. Ketov,Starobinsky-like two-field inflation, The European Physical Journal C 76 (2016)

    2016

  88. [96]

    Dimopoulos, A

    K. Dimopoulos, A. Karam, S. Sánchez López and E. Tomberg,Palatini r2 quintessential inflation, Journal of Cosmology and Astroparticle Physics2022 (2022) 076

    2022

  89. [97]

    Das and S

    N. Das and S. Panda,Inflation and reheating in f(r,h) theory formulated in the palatini formalism, Journal of Cosmology and Astroparticle Physics2021 (2021) 019

    2021

  90. [98]

    Panda, A

    S. Panda, A. Rana and R. Thakur,Constant-roll inflation in modifiedf (r, ϕ) gravity model using palatini formalism, The European Physical Journal C83 (2023)

    2023

  91. [99]

    Lloyd-Stubbs and J

    A. Lloyd-Stubbs and J. McDonald,Sub-Planckian ϕ2 inflation in the Palatini formulation of gravity with anR2 term, Phys. Rev. D101 (2020) 123515 [2002.08324]

    2020 arXiv

  92. [100]

    Kofman, A

    L. Kofman, A. Linde and A.A. Starobinsky,Reheating after inflation, Physical Review Letters 73 (1994) 3195–3198

    1994

  93. [101]

    Ford,Gravitational particle creation and inflation, Phys

    L.H. Ford,Gravitational particle creation and inflation, Phys. Rev. D35 (1987) 2955

    1987

  94. [102]

    Yahiro, G.J

    M. Yahiro, G.J. Mathews, K. Ichiki, T. Kajino and M. Orito,Constraints on cosmic quintessence and quintessential inflation,

  95. [103]

    Riazuelo and J.-P

    A. Riazuelo and J.-P. Uzan,Quintessence and gravitational waves, Physical Review D62 (2000) 083506

    2000

  96. [104]

    Giovannini,Spikes in the relic graviton background from quintessential inflation,

    M. Giovannini,Spikes in the relic graviton background from quintessential inflation,

  97. [105]

    Giovannini,Production and detection of relic gravitons in quintessential inflationary models,

    M. Giovannini,Production and detection of relic gravitons in quintessential inflationary models,

  98. [106]

    E.J. Chun, S. Scopel and I. Zaballa,Gravitational reheating in quintessential inflation,

  99. [107]

    Boyle and A

    L.A. Boyle and A. Buonanno,Relating gravitational wave constraints from primordial nucleosynthesis, pulsar timing, laser interferometers, and the cmb: implications for the early universe,

  100. [108]

    Abbott, E

    L.F. Abbott, E. Farhi and M.B. Wise,Particle production in the new inflationary cosmology, Physics Letters B117 (1982) 29

    1982

  101. [109]

    Dolgov and A.D

    A.D. Dolgov and A.D. Linde,Baryon asymmetry in the inflationary universe, Physics Letters B 116 (1982) 329

    1982

  102. [110]

    M. He, R. Jinno, K. Kamada, S.C. Park, A.A. Starobinsky and J. Yokoyama,On the violent preheating in the mixed higgs-r2 inflationary model, Physics Letters B791 (2019) 36–42

    2019

  103. [111]

    Tashiro, T

    H. Tashiro, T. Chiba and M. Sasaki,Reheating after quintessential inflation and gravitational waves, Classical and Quantum Gravity21 (2004) 1761–1771

    2004

  104. [112]

    Sahni, M

    V. Sahni, M. Sami and T. Souradeep,Relic gravity waves from braneworld inflation, Physical Review D 65 (2001) 023518

    2001

  105. [113]

    Giovannini,Production and detection of relic gravitons in quintessential inflationary models, Physical Review D60 (1999) 123511

    M. Giovannini,Production and detection of relic gravitons in quintessential inflationary models, Physical Review D60 (1999) 123511

    1999

  106. [114]

    Figueroa and E.H

    D.G. Figueroa and E.H. Tanin,Inconsistency of an inflationary sector coupled only to einstein gravity, Journal of Cosmology and Astroparticle Physics2019 (2019) 050–050

    2019

  107. [115]

    Artymowski, O

    M. Artymowski, O. Czerwińska, Z. Lalak and M. Lewicki,Gravitational wave signals and cosmological consequences of gravitational reheating, Journal of Cosmology and Astroparticle Physics 2018 (2018) 046–046. – 40 –

    2018

  108. [116]

    Thrane and J.D

    E. Thrane and J.D. Romano,Sensitivity curves for searches for gravitational-wave backgrounds, Physical Review D88 (2013)

    2013

  109. [117]

    Schmitz,New sensitivity curves for gravitational-wave signals from cosmological phase transitions, Journal of High Energy Physics2021 (2021)

    K. Schmitz,New sensitivity curves for gravitational-wave signals from cosmological phase transitions, Journal of High Energy Physics2021 (2021)

    2021

  110. [118]

    Herman, L

    N. Herman, L. Lehoucq and A. Fűzfa,Electromagnetic antennas for the resonant detection of the stochastic gravitational wave background, Phys. Rev. D108 (2023) 124009

    2023

  111. [119]

    Gertsenshtein,Wave resonance of light and gravitional waves, Sov Phys JETP 14 (1962) 84

    M. Gertsenshtein,Wave resonance of light and gravitional waves, Sov Phys JETP 14 (1962) 84

    1962

  112. [120]

    Harris, K.J

    C.R. Harris, K.J. Millman, S.J. van der Walt, R. Gommers, P. Virtanen, D. Cournapeau et al.,Array programming with NumPy, Nature 585 (2020) 357

    2020

  113. [121]

    Hunter,Matplotlib: A 2d graphics environment, Computing in Science & Engineering9 (2007) 90

    J.D. Hunter,Matplotlib: A 2d graphics environment, Computing in Science & Engineering9 (2007) 90

    2007

  114. [122]

    Team,pandas-dev/pandas: Pandas, Feb., 2020

    T.P.D. Team,pandas-dev/pandas: Pandas, Feb., 2020. 10.5281/zenodo.3509134

    2020 doi

  115. [123]

    56 – 61, 2010, DOI

    Wes McKinney,Data Structures for Statistical Computing in Python, inProceedings of the 9th Python in Science Conference, Stéfan van der Walt and Jarrod Millman, eds., pp. 56 – 61, 2010, DOI

    2010

  116. [124]

    Virtanen, R

    P. Virtanen, R. Gommers, T.E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau et al., SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python, Nature Methods 17 (2020) 261

    2020

  117. [125]

    Mathematica, Version 14.2

    W.R. Inc., “Mathematica, Version 14.2.” – 41 –

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.