High-Frequency Thermal Graviton Remnant from the End of Inflation

Chen-Hao Wu; Xiao Liang; Ya-Peng Hu

REVIEW 3 major objections 5 minor 48 references

High-Frequency Thermal Graviton Remnant from the End of Inflation

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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

Pith's one-line read The paper argues that inflation ends with a thermal graviton bath that decouples into a Planckian gravitational-wave background peaking in the megahertz band.

desk verdict A novel idea with a credible qualitative prediction, but the headline numbers don't survive contact with the paper's own equations, and the release efficiency is a guess. read the letter →

arxiv 2504.17533 v3 pith:A7ZDTHDZ submitted 2025-04-24 gr-qc astro-ph.COhep-th

Chen-Hao Wu , Xiao Liang , Ya-Peng Hu This is my paper · ORCID

classification gr-qcastro-ph.COhep-th

keywords gravitationalwaveshigh-frequencydeSitterthermalstatehorizontemperatureinflationreheatingprimordialwavebackgroundPlanckianspectrum

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

The paper argues that at the end of inflation the universe is filled with a real thermal bath of sub-horizon gravitons at the de Sitter horizon temperature, and when the horizon disappears this bath is released as freely propagating radiation. Tracking that radiation through reheating and cosmological redshift, the authors predict a relic gravitational-wave background with a Planckian spectrum peaking in the megahertz band. For typical inflationary Hubble scales and reheating temperatures the peak abundance is around log10 Omega_G $h^{2}$ about -18 to -32, far below current sensitivity but within the reach of proposed high-frequency detectors. The signal would act as a thermometer for the reheating epoch, something the standard scale-invariant vacuum-fluctuation background cannot provide.

What carries the argument

The load-bearing object is the spectrum of the thermal graviton bath at temperature $T_H = H_\Lambda/(2\pi)$ (the horizon temperature of de Sitter space), with spectral energy density $d\rho_G/d\omega = (1/\pi^2) \omega^3/(e^{\omega/T_H} - 1)$ and total energy density $\rho_G = (\pi^2/15) T_H^4$. The mechanism is a quantum quench: the non-adiabatic end of inflation freezes the statistical distribution of sub-horizon modes before they can adjust to the new background. After that, radiation conservation ($\rho \propto a^{-4}$, $f \propto a^{-1}$) and entropy conservation across reheating fix today's spectrum. An efficiency factor $\gamma \le 1$ parameterizes any loss during conversion, with $\gamma \approx 1$ argued from the rapidity of the transition, the flatness of the local geometry, and the tiny graviton cross-section.

What would settle it

A full mode-by-mode quantum-field calculation of the transition from the de Sitter phase to a radiation-dominated universe, tracking the mode-mixing coefficients of sub-horizon graviton modes, would give the actual release spectrum and efficiency; if it yields a conversion efficiency much smaller than one, or a spectrum that is not Planckian, the predicted peak abundance and spectral shape fail. Observationally, a sufficiently sensitive broadband detector in the $10^6$ to $10^8$ Hz band that sees no Planckian peak at the predicted abundance would rule the mechanism out, assuming the inflationary Hubble scale is at the benchmark values.

Watch

Extended reading notes

Core claim

The central claim is that the thermal gravitons in equilibrium inside the de Sitter horizon during inflation do not adiabatically vanish when inflation stops; the rapid geometric transition acts as a quantum quench that releases them as free radiation on a timescale much shorter than a Hubble time. Because gravitons scatter only through Planck-suppressed interactions, the released spectrum keeps its Planckian shape, peaking at $f_{\rm peak} = (x/2\pi) T_H$ with $x \approx 2.8214$, and then redshifts as $a^{-1}$. The today spectral density parameter is $\Omega_G(f_0) \simeq (16\pi^2 \hbar / c^5 \rho_{\rm crit}) f_0^4 F(f_0)$, where $F$ is a redshifted Planck factor whose arguments involve the inflationary Hubble rate, the reheating temperature, and the effective number of entropy degrees of freedom. Numerically, the peak falls near $10^6$ to $10^8$ Hz with $\Omega_G h^2$ of order $10^{-18}$ for a high Hubble rate and efficient reheating, down to $10^{-32}$ for lower rates and inefficient reheating.

Load-bearing premise

The whole prediction stands on the premise that the horizon temperature of de Sitter space corresponds to a real, locally stored energy density of gravitons that is released as freely propagating radiation, with essentially unit efficiency, when inflation ends.

Editorial extensions

If this is right

The universe would contain a new high-frequency relic gravitational-wave background with a Planckian spectral shape, distinct from the nearly scale-invariant vacuum-fluctuation signal that dominates at lower frequencies.
A detection of the peak frequency would directly measure the reheating temperature, providing constraints complementary to CMB bounds on the spectral index.
The predicted abundance, with $\Omega_G h^2 \sim 10^{-18}$ at best, sits far below the dark-radiation bound from CMB and BBN, so the signal is not ruled out by existing cosmological constraints.
The peak frequency falls in the $10^6$ to $10^8$ Hz range, giving emerging high-frequency gravitational-wave detector concepts a concrete benchmark target.
The overall amplitude scales linearly with the release efficiency $\gamma$, while the spectral shape and peak frequency do not depend on $\gamma$.

Reading between the lines

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

If the thermal graviton bath is real, the same quantum-thermodynamics argument should apply to any effectively massless field present during inflation, so similar relic backgrounds could be predicted for axion-like particles or dark photons, possibly with different detection consequences.
A full mode-by-mode calculation of the de Sitter to radiation-dominated transition would supply an actual value for the release efficiency $\gamma$; if it comes out much smaller than one, the predicted peak abundance would drop proportionally, but the frequency and spectral shape would survive.
The mechanism is not limited to the end of inflation in principle: any cosmological epoch where a horizon-temperature bath is suddenly removed could leave a similar thermal snapshot, which is a testable extension connecting this work to late-time horizon dynamics.
Because the peak frequency is set by the reheating temperature, measuring the shape of the Planckian tail, not just the peak, could distinguish instantaneous reheating from reheating with a prolonged matter-dominated stage.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Colleague,

The paper's actual novelty is a concrete one: it asks what happens to the sub-horizon thermal gravitons in de Sitter when inflation ends, and treats their release as a snapshot of the Gibbons-Hawking bath. That's a different question from the usual frozen super-horizon spectrum, and I don't see a prior paper making this specific prediction of a present-day Planckian relic whose peak frequency tracks the reheating temperature. The authors build on Alicki et al.'s open-system derivation and produce a simple analytic formula for the redshifted spectrum, including the w≈0 reheating phase. That part is clear and well organized.

The soft spots are real, and they're in two places. First, the physical premise: the paper needs the dS detector response to be a real, uniform, drainable energy density ρ_G=(π²/15)T_H^4 that suddenly becomes free radiation. That's asserted, not derived. The Unruh analogy is a fair concern—detector thermalization doesn't guarantee an extractable reservoir. The paper itself concedes in Sec. V that a full Bogoliubov treatment is missing and that γ could be <1. So the amplitude is hostage to an efficiency factor that is currently a guess. The authors are honest about this, but it's still a load-bearing guess.

Second, and more immediate: the numbers in Sec. IV don't follow from their own equations. For the headline case H=10^14 GeV, T_reh=10^13 GeV, feeding Eq. (22) and the entropy ratio gives a peak frequency today around a few MHz, not 10^8 Hz. The energy density today comes out around log10(Ωh²) ~ -23, not -18. The scaling they quote (Ω∝H^4, f0∝H) is off; the correct scaling from their own formulas for fixed T_reh is Ω∝H^{4/3}, f0∝H^{1/3}. This looks like an algebraic slip in evaluating the redshift—possibly treating the reheating phase as radiation-like (ρ∝a^{-4}) rather than the stated w=0 (ρ∝a^{-3}). I won't flag the c^5 in Eq. (23) as dimensional trouble; that's actually consistent if ρ_crit is the mass density and the 1/c^2 is in the definition. The real issue is the arithmetic.

Bottom line: the qualitative idea—a HF Planckian relic from the dS thermal bath—survives a rederivation, but the headline values need to be recomputed and the physical premise needs either a supporting calculation or a clear 'speculative scenario' framing. I'd send it to a referee: it's worth one round of serious work, and the core question (can the dS vacuum thermal energy be released as on-shell gravitons?) is a good question for the community. I wouldn't cite it in its current form.

Referee Report

3 major / 5 minor

Summary. This paper proposes that thermal gravitons associated with the Gibbons-Hawking temperature T_H=H_Λ/(2π) during inflation, described as a local thermal bath in the quantum-thermodynamic framework of Alicki et al., are released at the end of inflation as free radiation with an assumed efficiency γ≈1. The authors derive the present-day spectral density Ω_G(f0) and peak frequency f_peak,0 by propagating the released radiation first through a matter-dominated reheating phase and then through the standard radiation- and matter-dominated eras. They present benchmarks for H_Λ=10^12,10^13,10^14 GeV and T_reh=10^9,10^13 GeV, concluding that the background peaks near MHz with log10(Ω_Gh^2)∼O(-18).

Significance. The proposal is conceptually interesting and, if correct, would provide a new observational window into the reheating temperature. The paper's framework is transparent, uses standard cosmological evolution, and makes falsifiable spectral-shape predictions without fitting to the putative signal. The main strength is the clean analytical propagation of a Planckian spectrum from the inflationary epoch to today. However, the quantitative benchmarks are inconsistent with the paper's own equations, and the assumed release efficiency γ≈1 is not derived; both issues must be addressed before the prediction can be taken at face value.

major comments (3)

[Sec. IV, Eqs. (16) and (22)] The numerical benchmarks do not follow from the stated formulas. From Eq. (16), ρ_reh_G=ρ*_G(a*/areh)^4 with a*/areh=(T_reh^4/(3H_Λ^2 m_p^2))^{1/3}, so for fixed T_reh the present-day abundance scales as Ω_G ∝ H_Λ^{4/3}; Eq. (22) gives f_peak,0 ∝ H_Λ^{1/3}. The table in Sec. IV instead lists Ω decreasing by 10^4 per decade in H_Λ (Ω ∝ H_Λ^4) and f_peak,0 ∝ H_Λ. A direct evaluation of the H_Λ=10^14 GeV, T_reh=10^13 GeV case with the paper's inputs gives f_peak,0 ~ 1 MHz and Ω_Gh^2 ~ 10^{-24}, rather than the quoted 10^8 Hz and 10^{-18}. The abstract's O(-18) claim and the Sec. IV benchmarks must be recomputed.
[Sec. II, Eqs. (11)-(12), and Sec. V] The central premise that the dS thermal state is a real, uniformly distributed, drainable graviton energy density that is released with γ≈1 is not established. Ref. [24] demonstrates KMS thermalization of a localized detector in the dS vacuum; it does not establish that the vacuum contains a reservoir of on-shell gravitons with ρ_G=(π²/15)T_H^4 that converts into propagating radiation when the horizon disappears. The Rindler/Unruh analogue shows that detector thermalization does not imply an extractable fluid. The three plausibility arguments given for γ≈1 (rapid end, flatness, small graviton cross-section) do not amount to a derivation of the conversion efficiency. Because the predicted amplitude is linear in γ and the authors themselves admit in Sec. V that a Bogoliubov calculation may yield γ<1, the paper should present γ as a free parameter and the spectrum as a template, or supply a microscopic estimate of γ.
[Eq. (21), middle expression] The intermediate formula Ω_G = f/(c²ρ_crit) dρ_G/dln f ... contains an extra factor f relative to the definition Ω_G = (1/(c²ρ_crit)) dρ_G,0/dln f0 in Eq. (19). Since dρ_G/dln f already includes the Jacobian factor f, this makes the intermediate expression dimensionally inconsistent; the final simplified expression is valid only if the extra f is removed. Please correct the derivation.

minor comments (5)

[Eq. (21)] The phrase 'using the relation 1/dlnf = f d/df' is incorrect notation; it should read 'd/dln f = f d/df'.
[Sec. III] The sentence 'ρ_crit = 3H0²/(8πG) represents the current critical energy density' is dimensionally a mass density in SI units; the c² factor in the definition of Ω resolves the units, but the wording should be clarified.
[Fig. 2] The figure caption does not state that all curves assume γ=1; this should be made explicit.
[Sec. IV] No detector sensitivity curves are shown despite the reference to Ref. [25]; adding a representative sensitivity curve or a quantitative statement of the required sensitivity would improve the discussion of observational prospects.
[Sec. V] The phrase 'captures the conservation of the horizon energy density at the moment of transition' is unclear; the released radiation is not conserved in density but redshifts, so the wording should be revised.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the relic spectrum follows from the assumed dS thermal state via standard redshift, with no fitted parameters or load-bearing self-citations.

full rationale

The derivation chain is explicit: (i) adopt the external result of Alicki et al. [24] that the dS vacuum has a KMS spectral density G~(ω)=(2π)^2 ω/(1−e^{−ω/T_dS}) (Eq. 7); (ii) convert this to a massless-boson density of states n_dS=ω²/(2π²) through the standard thermal-bath relation (Eqs. 8–9); (iii) integrate to ρ_G=(π²/15)T_H^4 (Eq. 12); (iv) assume a release efficiency γ≈1 at the end of inflation; and (v) redshift by standard a^{−4} evolution and entropy conservation (Eqs. 14–22). No parameter is fitted to the predicted Ω_G: H_Λ and T_reh are chosen benchmark inputs, and γ is declared rather than fit. The final Planckian shape is indeed the input thermal spectrum redshifted, but that is a model consequence, not a circular reduction: T_H is fixed independently by H_Λ, and the relic spectrum is not used to define T_H. The load-bearing physical premise, that the detector KMS response corresponds to a drainable homogeneous graviton energy density, is an assumption imported from external Ref. [24]; whether that premise is overstated is a physics/correctness question, not a derivation-level circularity. The only self-citation, Ref. [15] (Hu), appears in a non-load-bearing historical list and does not support any uniqueness or exclusion claim. The paper also explicitly flags its main limitation: 'a full Bogoliubov calculation would determine the exact efficiency of this conversion, likely introducing a factor <1' (Sec. V), so the amplitude is presented as conditional rather than forced. Therefore no circular step is exhibited. Separately, the Sec. IV benchmark scalings appear numerically inconsistent with Eqs. (21)–(22), but that is an internal consistency issue, not circularity.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

No new particles or fields are introduced. The paper's parameters are the release efficiency γ and benchmark choices for H_Λ and T_reh. The main load is carried by importing the Alicki et al. thermodynamic result and treating the dS thermal bath as a real fluid that can be released; this is an assumption not derived here.

free parameters (3)

γ (release efficiency) = 1 (assumed)
Set to 1 based on qualitative quench and decoupling arguments in Sec. II; the paper acknowledges the true value could be <1.
H_Λ (inflationary Hubble scale) = 10^12, 10^13, 10^14 GeV (benchmark choices)
Chosen as representative values within CMB constraints; sets the Gibbons-Hawking temperature.
T_reh (reheating temperature) = 10^9 or 10^13 GeV (benchmark choices)
Chosen to match R^2 and Higgs inflation; enters the redshift factors.

assumptions (4)

domain assumption The Alicki et al. quantum thermodynamics proof that a detector in de Sitter space thermalizes at T=H/2π, and that this thermal state is intrinsic to the vacuum and not a coordinate artifact.
This is the core premise; the paper does not re-derive it and treats it as established.
domain assumption The thermal bath energy density is given by the Stefan-Boltzmann law with two graviton polarizations, ρ_G = (π^2/15) T_H^4.
Derived in Eqs. (8)-(12) by equating the Alicki et al. spectral density to a thermal density of states; assumes the spectral density fully determines the energy density of the field itself.
ad hoc to paper At the end of inflation the horizon disappears quickly and the thermal gravitons decouple as free radiation with efficiency γ≈1, not undergoing adiabatic dilution or re-scattering.
Quantum quench argument in Sec. II; the paper explicitly says this is an assumption and future Bogoliubov calculation may reduce γ.
domain assumption The post-inflationary evolution is standard: reheating with w≈0, then radiation domination, entropy conservation, and free-streaming gravitons.
Sec. III uses ρ∝a^-3 during reheating and a^-4 after; standard cosmology.

how reviews work

0 comments

Cite this review

Pith. "Pith review of High-Frequency Thermal Graviton Remnant from the End of Inflation." pith.science (2026). https://pith.science/paper/A7ZDTHDZ

@misc{pith2026250417533,
  author       = {Pith},
  title        = {Pith review of: High-Frequency Thermal Graviton Remnant from the End of Inflation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A7ZDTHDZ}},
  note         = {Machine review of arXiv:2504.17533}
}

abstract

The standard inflationary theory focuses on the freezing of super-horizon fluctuations, which generate a scale-invariant spectrum, while the sub-horizon modes are expected to remain in thermal equilibrium. Building upon recent development of quantum thermodynamics of the de Sitter universe, we investigate the graviton remnant originating from this thermal horizon radiation released at the end of inflation. Unlike the stochastic background from super-horizon fluctuations, this signal represents a snapshot of the thermal dS state, which subsequently decouples and undergoes cosmological redshift. We present a semi-analytical approximation prediction for this relic background, typically peaking in near MHz band, with characteristic energy density of $\log_{10}(\Omega_{\rm G} h^2) \sim \mathcal{O}(-18)$. These signals occupy a High-Frequency band, offering a potential novel probe of the reheating temperature and the thermal history of the early universe.

Figures

Figures reproduced from arXiv: 2504.17533 by the authors.

**Figure 2.** FIG. 2. The dimensionless density parameter spectrum to [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

48 extracted references · 19 canonical work pages

[24]

Padmanabhan, Cosmological constant: The Weight of the vacuum, Phys

T. Padmanabhan, Cosmological constant: The Weight of the vacuum, Phys. Rept.380, 235 (2003), arXiv:hep- th/0212290

arXiv 2003
[1]

log 10(Ωpeak G,h h2)∼O(−18),H Λ∼10 14 GeV,

work page
[2]

log 10(Ωpeak G,h h2)∼O(−22),H Λ∼10 13 GeV,

work page
[3]

log 10(Ωpeak G,h h2)∼O(−26),H Λ∼10 12 GeV, at the corresponding peak frequency, written as

work page
[4]

A. D. Dolgov, P. D. Naselsky, and I. D. Novikov, Gravita- tional waves, baryogenesis, and dark matter from primor- dial black holes (2000), arXiv:astro-ph/0009407 [astro- ph]

arXiv 2000
[5]

log 10(fpeak,0,h)∼O(8),H Λ∼10 14 GeV,

work page
[6]

log 10(fpeak,0,h)∼O(7),H Λ∼10 13 GeV,

work page
[7]

Similarly, we can also get the prediction for the inefficient reheating case as

log 10(fpeak,0,h)∼O(6),H Λ∼10 12 GeV, respectively. Similarly, we can also get the prediction for the inefficient reheating case as

work page

Show all 48 references

[8]

log 10(Ωpeak G,l h2)∼O(−24),H Λ∼10 14 GeV, with log10(fpeak,0,l)∼O(6),
[9]

log 10(Ωpeak G,l h2)∼O(−28),H Λ∼10 13 GeV, with log10(fpeak,0,l)∼O(5),
[10]

log 10(Ωpeak G,l h2)∼O(−32),H Λ∼10 12 GeV, with log10(fpeak,0,l)∼O(4). HΛ=1012GeV,Treh=1013GeV HΛ=1012GeV,Treh=109GeV HΛ=1013GeV,Treh=1013GeV HΛ=1013GeV,Treh=109GeV HΛ=1014GeV,Treh=1013GeV HΛ=1014GeV,Treh=109GeV 10 1000 105 107 10910-40 10-35 10-30 10-25 10-20 10-15 f0(Hz) ΩGW...
[11]

S. W. Hawking, Black hole explosions, Nature248, 30 (1974)

1974
[12]

S. W. Hawking, Particle Creation by Black Holes, Com- mun. Math. Phys.43, 199 (1975), [Erratum: Com- mun.Math.Phys. 46, 206 (1976)]

1975
[13]

g⋆,s(T0) g⋆,s(Treh) 1/3 T0 Treh #4 ≃ 16π2ℏ c5ρcrit f4 e 2πℏf kBTH−1 a∗ areh 4

to the bouncing cosmologies of Barrau et al.[16]. But unlike earlier works that focus on the existence of a steady energy density associated with dS horizons, the signal studied here should be viewed as a snapshot of the thermal state at the moment inflation ends. One may noti...
[14]

G. W. Gibbons and S. W. Hawking, Cosmological Event Horizons, Thermodynamics, and Particle Creation, Phys. Rev. D15, 2738 (1977)

1977
[15]

Anantua, R

R. Anantua, R. Easther, and J. T. Giblin, GUT-Scale Primordial Black Holes: Consequences and Constraints, Phys. Rev. Lett.103, 111303 (2009), arXiv:0812.0825 [astro-ph]

2009 arXiv
[16]

A. D. Dolgov and D. Ejlli, Relic gravitational waves from light primordial black holes, Phys. Rev. D84, 024028 (2011), arXiv:1105.2303 [astro-ph.CO]

2011 arXiv
[17]

R. Dong, W. H. Kinney, and D. Stojkovic, Gravitational wave production by Hawking radiation from rotating pri- mordial black holes, JCAP10, 034, arXiv:1511.05642 [astro-ph.CO]

arXiv
[18]

Hooper, G

D. Hooper, G. Krnjaic, J. March-Russell, S. D. McDer- mott, and R. Petrossian-Byrne, Hot Gravitons and Grav- itational Waves From Kerr Black Holes in the Early Uni- verse (2020), arXiv:2004.00618 [astro-ph.CO]

2020 arXiv
[19]

Arbey, J

A. Arbey, J. Auffinger, P. Sandick, B. Shams Es Haghi, and K. Sinha, Precision calculation of dark radiation from spinning primordial black holes and early matter- dominated eras, Phys. Rev. D103, 123549 (2021), arXiv:2104.04051 [astro-ph.CO]

2021 arXiv
[20]

Cheek, L

A. Cheek, L. Heurtier, Y. F. Perez-Gonzalez, and J. Turner, Redshift effects in particle production from Kerr primordial black holes, Phys. Rev. D106, 103012 (2022), arXiv:2207.09462 [astro-ph.CO]

2022 arXiv
[21]

Ireland, S

A. Ireland, S. Profumo, and J. Scharnhorst, Primordial gravitational waves from black hole evaporation in stan- dard and nonstandard cosmologies, Phys. Rev. D107, 104021 (2023), arXiv:2302.10188 [gr-qc]

2023 arXiv
[22]

Boluna, S

X. Boluna, S. Profumo, J. Bl´ e, and D. Hennings, Searching for Exploding black holes, JCAP04, 024, arXiv:2307.06467 [astro-ph.HE]

arXiv
[23]

J. R. Gott, Creation of Open Universes from de Sitter Space, Nature295, 304 (1982)

1982
[25]

Hu, Hawking radiation from the cosmological hori- zon in a FRW universe, Phys

Y.-P. Hu, Hawking radiation from the cosmological hori- zon in a FRW universe, Phys. Lett. B701, 269 (2011), arXiv:1007.4044 [gr-qc]

2011 arXiv
[26]

Barrau and L

A. Barrau and L. Linsefors, Our Universe from the cos- mological constant, JCAP12, 037, arXiv:1406.3706 [gr- qc]

arXiv
[27]

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

1981
[28]

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

1982
[29]

Albrecht and P

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

1982
[30]

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

1979
[31]

Caprini and D

C. Caprini and D. G. Figueroa, Cosmological Back- grounds of Gravitational Waves, Class. Quant. Grav.35, 163001 (2018), arXiv:1801.04268 [astro-ph.CO]

2018 arXiv
[32]

Christensen, Stochastic Gravitational Wave Back- grounds, Rept

N. Christensen, Stochastic Gravitational Wave Back- grounds, Rept. Prog. Phys.82, 016903 (2019), arXiv:1811.08797 [gr-qc]

2019 arXiv
[33]

E. W. Kolb and A. J. Long, Cosmological gravita- tional particle production and its implications for cos- mological relics, Rev. Mod. Phys.96, 045005 (2024), arXiv:2312.09042 [astro-ph.CO]

2024 arXiv
[34]

Alicki, G

R. Alicki, G. Barenboim, and A. Jenkins, Quantum ther- modynamics of de Sitter space, Phys. Rev. D108, 123530 (2023), arXiv:2307.04800 [gr-qc]

2023 arXiv
[35]

Aggarwalet al., Challenges and Opportunities of Gravitational Wave Searches above 10 kHz (2025), arXiv:2501.11723 [gr-qc]

N. Aggarwalet al., Challenges and Opportunities of Gravitational Wave Searches above 10 kHz (2025), arXiv:2501.11723 [gr-qc]

2025
[36]

de Sitter, On Einstein’s Theory of Gravitation and its Astronomical Consequences

W. de Sitter, On Einstein’s Theory of Gravitation and its Astronomical Consequences. Third Paper., Mon. Not. Roy. Astron. Soc.78, 3 (1917)

1917
[37]

Akramiet al.(Planck), Planck 2018 results

Y. Akramiet al.(Planck), Planck 2018 results. X. Con- straints on inflation, Astron. Astrophys.641, A10 (2020), arXiv:1807.06211 [astro-ph.CO]

2020 arXiv
[38]

R. H. Brandenberger and R. Kahn, Hawking Radiation in an Inflationary Universe, Phys. Lett. B119, 75 (1982)

1982
[39]

F. L. Bezrukov and D. S. Gorbunov, Distinguishing be- tween R 2-inflation and Higgs-inflation, Phys. Lett. B 713, 365 (2012), arXiv:1111.4397 [hep-ph]

2012 arXiv
[40]

Bezrukov, D

F. Bezrukov, D. Gorbunov, and M. Shaposhnikov, On initial conditions for the Hot Big Bang, JCAP06, 029, arXiv:0812.3622 [hep-ph]

arXiv
[41]

Rubio, Higgs inflation, Front

J. Rubio, Higgs inflation, Front. Astron. Space Sci.5, 50 (2019), arXiv:1807.02376 [hep-ph]

2019 arXiv
[42]

Mukhanov,Physical Foundations of Cosmology(Cam- bridge University Press, Oxford, 2005)

V. Mukhanov,Physical Foundations of Cosmology(Cam- bridge University Press, Oxford, 2005)

2005
[43]

Husdal, On Effective Degrees of Freedom in the Early Universe, Galaxies4, 78 (2016), arXiv:1609.04979 [astro- ph.CO]

L. Husdal, On Effective Degrees of Freedom in the Early Universe, Galaxies4, 78 (2016), arXiv:1609.04979 [astro- ph.CO]

2016 arXiv
[44]

Aghanimet al.(Planck), Planck 2018 results

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

2020 arXiv
[45]

A. G. Riesset al., A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km s1 Mpc1 Uncertainty from the Hubble Space Telescope and the SH0ES Team, Astrophys. J. Lett.934, L7 (2022), arXiv:2112.04510 [astro-ph.CO]

2022 arXiv
[46]

E. W. Kolb,The Early Universe, Vol. 69 (Taylor and Francis, 2019)

2019
[47]

C. W. Misner, K. S. Thorne, and J. A. Wheeler,Gravi- tation(W. H. Freeman, San Francisco, 1973)

1973
[48]

B. P. Abbottet al.(LIGO Scientific, Virgo), Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett.116, 061102 (2016), arXiv:1602.03837 [gr-qc]

2016 arXiv

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [24]

Padmanabhan, Cosmological constant: The Weight of the vacuum, Phys

T. Padmanabhan, Cosmological constant: The Weight of the vacuum, Phys. Rept.380, 235 (2003), arXiv:hep- th/0212290

arXiv 2003

[2] [1]

log 10(Ωpeak G,h h2)∼O(−18),H Λ∼10 14 GeV,

work page

[3] [2]

log 10(Ωpeak G,h h2)∼O(−22),H Λ∼10 13 GeV,

work page

[4] [3]

log 10(Ωpeak G,h h2)∼O(−26),H Λ∼10 12 GeV, at the corresponding peak frequency, written as

work page

[5] [4]

A. D. Dolgov, P. D. Naselsky, and I. D. Novikov, Gravita- tional waves, baryogenesis, and dark matter from primor- dial black holes (2000), arXiv:astro-ph/0009407 [astro- ph]

arXiv 2000

[6] [5]

log 10(fpeak,0,h)∼O(8),H Λ∼10 14 GeV,

work page

[7] [6]

log 10(fpeak,0,h)∼O(7),H Λ∼10 13 GeV,

work page

[8] [7]

Similarly, we can also get the prediction for the inefficient reheating case as

log 10(fpeak,0,h)∼O(6),H Λ∼10 12 GeV, respectively. Similarly, we can also get the prediction for the inefficient reheating case as

work page