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Flat Energy Spectrum of Primordial Gravitational Waves vs Peaks and the NANOGrav 2023 Observation

V. K. Oikonomou

arXiv:2306.17351v3astro-ph.COgr-qc

TL;DR

The paper asks which gravity and particle-physics scenarios can produce observable primordial gravitational-wave spectra compatible with the 2023 NANOGrav signal. It analyzes flat and peak-like spectra from modified gravity, Einstein-Gauss-Bonnet models, and axion-Higgs reheating effects, finding that compatibility generally requires blue tensor tilts, low reheating temperatures, abnormal reheating, or combinations thereof.

  • Problem

    The paper addresses which primordial gravitational-wave spectra and reheating conditions can explain the 2023 NANOGrav stochastic signal.

    Source: §I · p. 1, §V · p. 12

  • Method

    The authors examine f(R) gravity, Einstein-Gauss-Bonnet gravity, and blue-spectrum inflation with axion-Higgs-induced abnormal reheating.

    Source: §II · p. 2, §III · p. 7, §V · p. 12

  • Results

    Compatibility requires either geometrically abnormal f(R) reheating, or blue tensor spectra with low reheating temperatures, including TR ∼400 GeV with nT ≥0.97 or TR ∼40 GeV with nT ≥1.1.

    Source: §V · p. 13

  • Takeaways & Limitations

    The viable explanations are constrained by the combined requirements of reheating history, reheating temperature, and tensor spectral tilt.

    Source: §V · pp. 13–14

Abstract

from arXiv · show

In this work we present several characteristic examples of theories of gravity and particle physics scenarios that may yield an observable energy spectrum of stochastic primordial gravitational waves, compatible with the 2023 NANOGrav observations. The resulting theories yield a flat or a peak-like energy spectrum, and we further seek the conditions which if hold true, the energy spectrum can be compatible with the recent NANOGrav stochastic gravitational wave detection. As we show, in most cases a blue tilted spectrum combined with a relatively low reheating temperature is needed, the scale of which is determined by whether the radiation domination era is ordinary or it is an abnormal radiation domination era. One intriguing Higgs-axion model, which predicts short slow-roll eras for the axion field at the post-electroweak breaking epoch, which eventually change the total equation of state parameter at the reheating era, can explain the NANOGrav signal, if a blue tilted tensor spectral index inflationary era precedes the reheating era, and a reheating temperature of the order $\mathcal{O}(400)\,$GeV. This specific model produces an energy spectrum of primordial gravitational waves with a characteristic peak that is detectable from both the NANOGrav and future LISA experiment, but not from the future Einstein telescope.

I. INTRODUCTION

The paper examines whether cosmological primordial gravitational-wave models can explain the 2023 NANOGrav stochastic signal and identifies conditions enabling compatibility. It focuses on modified-gravity and particle-physics scenarios producing flat or peak-like spectra.

  • The 2023 NANOGrav collaboration detected an isotropic stochastic gravitational-wave signal near f = 32 nHz using 15 years of data from 67 pulsars [16].Source: §I · p. 1
  • The signal’s origin remains uncertain because it may be astrophysical, cosmological, exotic, or a combination of sources.Source: §I · p. 1
  • The paper studies modified-gravity and axion-Higgs scenarios as possible cosmological explanations for the signal.Source: §I · pp. 1–2
  • The proposed models produce either flat spectra enhanced by abnormal reheating or peak-like spectra from blue tensor tilts and reheating effects.Source: §I · pp. 1–2
  • The authors emphasize that compatibility generally requires a blue spectrum, with the reheating-temperature requirement depending on whether reheating is ordinary or abnormal.Source: §I · p. 2

II. FLAT PRIMORDIAL WAVES ENERGY SPECTRUM: MODIFIED GRAVITY WITH AN ABNORMAL REHEATING ERA

This section develops an f(R)-gravity mechanism in which an abnormal reheating-era equation of state amplifies an otherwise nearly flat primordial gravitational-wave spectrum. For a representative model, intermediate and low reheating temperatures remain compatible with NANOGrav, while the highest tested temperature violates LIGO-Virgo constraints.

II. FLAT PRIMORDIAL WAVES ENERGY SPECTRUM: MODIFIED GRAVITY WITH AN ABNORMAL REHEATING ERA

  • The unmodified f(R) spectrum is practically flat and below the sensitivity curves of NANOGrav and future interferometers, but abnormal reheating can substantially enhance it.Source: §II · p. 2
  • The tensor-mode evolution is modified through the f(R) parameter aM, using a WKB treatment of the waveform deformation relative to general relativity.Source: §II · pp. 2–3
  • The model combines an inflationary R2 regime, an intermediate constant-EoS f(R) phase, and a viable late-time f(R) regime.Source: §II · pp. 3–4
  • The representative construction assumes w = −0.35 during reheating over z = [10^15, 4 × 10^16], corresponding to T = 10^9–10^11 GeV.Source: §II · p. 4
  • The second redshift integral dominates the amplification, yielding an overall factor of order e^−2D ∼ O(10^10).Source: §II · p. 4
  • The resulting spectrum is flat and compatible with NANOGrav for the tested model, while TR = 10^12 GeV is excluded by LIGO-Virgo constraints; the signal is also projected to be detectable by LISA and Einstein Telescope.Source: §II · pp. 4–5, Figure 1 · p. 4

III. PEAKS IN THE PRIMORDIAL WAVES ENERGY SPECTRUM I: EINSTEIN-GAUSS-BONNET THEORIES

Einstein-Gauss-Bonnet theories can produce peak-like primordial gravitational-wave spectra while maintaining luminal propagation through constrained scalar-potential and Gauss-Bonnet couplings. Compatibility with NANOGrav requires a strongly blue tensor spectrum and GeV-scale reheating.

Model construction

  • Luminal gravitational-wave propagation is imposed by constraining the scalar potential and Gauss-Bonnet coupling through a specific relation.The construction follows the Einstein-Gauss-Bonnet formalism developed in Refs. [62] [69] [77].Source: §III · p. 5
  • The Einstein-Gauss-Bonnet action uses a scalar field coupled to the Gauss-Bonnet invariant, whose curvature combination is explicitly defined from the Ricci and Riemann tensors.The Gauss-Bonnet invariant is G = R^2 − 4R_αβR^αβ + R_αβγδR^αβγδ.Source: §III · p. 5

Results

  • The model yields a characteristic peak-like gravitational-wave spectrum whose peak depends on reheating temperature and tensor spectral index.Spectra are shown for TR = 10^12, 10^7, and 10^4 GeV with nT ≈ 0.37, and for larger blue tilts in a second figure.Source: Figure 2 · p. 5, §III · p. 7, Figure 3 · p. 8

Model predictions

  • The analyzed Einstein-Gauss-Bonnet setup produces a blue tensor tilt, with nT = [0.378856, 0.379088] and r ≈ 0.003 for the stated parameter choices and 60 e-foldings.The parameters are μ = [22.09147657871, 22.09147657877], β = −1.5, and γ = 2.Source: §III · p. 7

Results

  • NANOGrav compatibility requires nT of order 1 and reheating temperatures of GeV order in Einstein-Gauss-Bonnet theories.For the displayed model, nT ≈ 1.1 and TR ≈ O(40) GeV are required; higher reheating temperatures violate the BBN bound for nT = 1.1.Source: §III · pp. 7–8

IV. PEAKS IN THE PRIMORDIAL WAVES ENERGY SPECTRUM II: AXION-HIGGS HIGHER ORDER INTERACTIONS WITH A BLUE TILTED TENSOR SPECTRAL INDEX

Higher-order axion-Higgs interactions modify the axion potential after electroweak symmetry breaking, producing repeated axion slow-roll and vacuum-decay episodes that alter the primordial gravitational-wave spectrum. The model can match NANOGrav with a blue tensor tilt and low reheating temperature, yielding a peak detectable by NANOGrav and LISA but not Einstein Telescope.

Model and potential

  • The axion-Higgs effective operators modify the axion potential after electroweak symmetry breaking, introducing a competing axion minimum alongside the Higgs vacuum.The model assumes ma ∼10^-10 eV, M = 20–100 TeV, λ ∼O(10^-20), g ∼O(10^-5), and electroweak symmetry breaking near T ∼100 GeV.Source: §IV · pp. 8–9

Axion dynamics

  • The axion’s disrupted oscillations can lead to slow-roll or rapid motion toward the new minimum, after which the energetically favored Higgs vacuum receives the decay energy.Rapid motion gives a stiff axion equation of state, whereas slow-roll lowers the total equation of state below radiation domination.Source: §IV · pp. 9–10, Figure 5 · p. 10
  • The analysis adopts the slow-roll scenario, changing the total equation of state from w = 1/3 to w = 0.28 because this case can explain the NANOGrav result.The axion slow-roll equation of state is wa ∼−1, while the rapid-roll alternative does not provide the required spectral behavior in this analysis.Source: §IV · pp. 9, 11

Spectrum modification

  • The model examines two or four axion slow-roll and decay episodes at distinct frequencies, with reheating temperatures TR = 10^12 GeV, 10^7 GeV, and 400 GeV.These episodes modify the h^2-scaled spectrum through a frequency-dependent multiplication factor.Source: §IV · p. 11

Gravitational-wave spectrum

  • For nT = 0.97 and r ∼0.003, the spectrum has characteristic peaks and matches NANOGrav only for TR ∼400 GeV with at least two axion decays.The four-decay case produces a more pronounced signal.Source: §IV · p. 11, Figure 6 · p. 12

Observational implications

  • The NANOGrav-compatible peak is also detectable by LISA but not by the Einstein Telescope, requiring nT ≥0.9 and TR ∼400 GeV.The peak frequency depends on when and how many times the axion slow-rolls and its vacuum decays to the Higgs vacuum; the required reheating temperature is four times the electroweak symmetry-breaking temperature.Source: §IV · pp. 11–12

V. CONCLUSIONS AND DISCUSSION

The paper finds that primordial gravitational-wave spectra compatible with NANOGrav can be flat or peak-like, but typically require blue tensor tilts, low reheating temperatures, or abnormal reheating dynamics. It also emphasizes that future multi-band observations are needed to distinguish cosmological models from astrophysical explanations.

  • Theories considered produce either flat or peak-like primordial gravitational-wave spectra compatible with the 2023 NANOGrav signal under different reheating conditions.The examples include f(R) gravity, Einstein-Gauss-Bonnet gravity, and Higgs-axion scenarios.Source: §V · p. 12
  • An abnormal reheating era geometrically realized by f(R) gravity can explain the NANOGrav signal with a flat spectrum detectable by future gravitational-wave experiments.Source: §V · pp. 12–13
  • The Higgs-axion scenario requires an abnormal reheating era, TR ∼400 GeV, and a blue tensor index nT ≥0.97 for NANOGrav compatibility.Detection also requires a blue tensor inflationary spectrum and two to four axion slow-roll eras.Source: §V · pp. 12–13
  • Einstein-Gauss-Bonnet models require TR ∼40 GeV and nT ≥1.1, although achieving such a blue tensor spectrum is highly non-trivial.Source: §V · pp. 12–13
  • Across most scenarios, low reheating temperature, a blue tensor spectrum, and abnormal reheating determine whether the model can reproduce the NANOGrav signal.Source: §V · p. 13
  • The source remains unresolved because cosmological models fit the NANOGrav background better in current Bayes-factor comparisons, while anisotropy measurements and future LISA, Einstein Telescope, and SKA data are still needed.The reported cosmological-model advantage ranges from Bayes factors of 10 to 100, but the paper states that it is too early for a definitive source attribution.Source: §V · pp. 13–14
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