Primary gravitational waves at high frequencies II: Emergence of the exponential cut-off in the power spectrum

Author(s)

Hoory, Alipriyo, Martin, Jerome, Paul, Arnab, Sriramkumar, L.

Abstract

[Abridged] In slow roll inflation, the power spectrum (PS) of primary gravitational waves (PGWs) generated from the quantum vacuum rises as $k^2$ over wave numbers $k$ which never leave the Hubble radius. In fact, over such small scales, the PS exhibits a similar behavior at any time after inflation. In a recent work, we had argued that the PS of PGWs has to be regularized to truncate the unphysical quadratic rise at large wave numbers. Assuming instantaneous transitions from inflation to the epochs of radiation and matter domination, we had shown that the regularized PS oscillates with a constant amplitude about a vanishing mean over small scales during these epochs. We had also smoothed the transition (actually, the `effective potential' governing the equation of motion of GWs) from inflation to radiation domination using a linear function and evaluated the regularized PS of PGWs post inflation. In such a case, we had shown that, over small scales, while the regularized PS continues to oscillate about zero, its amplitude decreases as $k^{-1}$. In this work, using the Born approximation, we examine the behavior of the regularized PS of PGWs over small scales when they are evolved through smoother and smoother transitions from inflation to the epochs of radiation and matter domination. We illustrate that, at small scales or high frequencies, the suppression in the regularized PS of PGWs occurs more and more sharply as the transition is smoothed further and further. With the help of examples, we also show that, in the case of transitions described by infinitely differentiable `effective potentials', the regularized PS of PGWs exhibits an exponential suppression on small scales. We argue that the observation of the exponential drop in the PS of PGWs can help us determine the energy scale and the time of the end of inflation. We clarify related issues and discuss the wider implications.

Figures

We have plotted the `effective potential' $U(\eta)=a''/a$ in the cases of transition from de Sitter inflation [corresponding to $q=-1$ in Eq.~\eqref{eq:a-pli}] to the epoch of radiation domination when the quantity has been smoothed with the aid of linear, quadratic, cubic and quintic functions [cf. Eqs.~\eqref{eq:U-slt}, \eqref{eq:U-sqt1}, \eqref{eq:U-sqt2}, \eqref{eq:U-sct}, and~\eqref{eq:U-sqt}]. We have set $\Delta\ee=2 \vert\ee\vert$ in plotting this figure and we have indicated the beginning and the end of the smoother transitions (as dotted and dashed vertical lines). In addition, we have plotted the `effective potential' in the case of the infinitely smooth transition [in this regard, see Eq.~\eqref{eq:U-ict1}] for the value of $\Delta\ee$ mentioned above and $\gamma_{\e}=1$. In the case of the instantaneous transition, $U(\eta)$ has the maximum value of~$2/\ee^2$ at~$\ee$, i.e. at the end of inflation. When the transition is smoothed, we find that the maximum of $U(\eta)$ occurs {\it during}\/ the transition and the maximum value remains of the same order of magnitude. Therefore, the Born approximation can be employed for wave numbers $k \gtrsim \ke$ in all the cases of~$U(\eta)$ that we have considered.
Caption We have plotted the `effective potential' $U(\eta)=a''/a$ in the cases of transition from de Sitter inflation [corresponding to $q=-1$ in Eq.~\eqref{eq:a-pli}] to the epoch of radiation domination when the quantity has been smoothed with the aid of linear, quadratic, cubic and quintic functions [cf. Eqs.~\eqref{eq:U-slt}, \eqref{eq:U-sqt1}, \eqref{eq:U-sqt2}, \eqref{eq:U-sct}, and~\eqref{eq:U-sqt}]. We have set $\Delta\ee=2 \vert\ee\vert$ in plotting this figure and we have indicated the beginning and the end of the smoother transitions (as dotted and dashed vertical lines). In addition, we have plotted the `effective potential' in the case of the infinitely smooth transition [in this regard, see Eq.~\eqref{eq:U-ict1}] for the value of $\Delta\ee$ mentioned above and $\gamma_{\e}=1$. In the case of the instantaneous transition, $U(\eta)$ has the maximum value of~$2/\ee^2$ at~$\ee$, i.e. at the end of inflation. When the transition is smoothed, we find that the maximum of $U(\eta)$ occurs {\it during}\/ the transition and the maximum value remains of the same order of magnitude. Therefore, the Born approximation can be employed for wave numbers $k \gtrsim \ke$ in all the cases of~$U(\eta)$ that we have considered.
The behavior of the scale factor $a(\eta)$ (on top) and the Hubble parameter $H(\eta)$ (at the bottom) are plotted in the case of transition from de Sitter inflation to the epoch of radiation domination. We have plotted these quantities in all the cases (and in the same colors) for which we had plotted the `effective potential' $U(\eta)$ in the previous figure. We have also worked with the same values of the parameters we indicated before. During the transition, we have numerically solved for the scale factor in all the cases. Additionally, in the case of the smoother linear transition, we have plotted the scale factor we had obtained analytically in our previous work (in this regard, see Ref.~\cite{Hoory:2025qgm}).
Caption The behavior of the scale factor $a(\eta)$ (on top) and the Hubble parameter $H(\eta)$ (at the bottom) are plotted in the case of transition from de Sitter inflation to the epoch of radiation domination. We have plotted these quantities in all the cases (and in the same colors) for which we had plotted the `effective potential' $U(\eta)$ in the previous figure. We have also worked with the same values of the parameters we indicated before. During the transition, we have numerically solved for the scale factor in all the cases. Additionally, in the case of the smoother linear transition, we have plotted the scale factor we had obtained analytically in our previous work (in this regard, see Ref.~\cite{Hoory:2025qgm}).
The regularized PS of PGWs $\ptr(\ye,\eta)$, evaluated soon after the transition to the epoch of radiation domination at the time $2\vert\ee\vert$, has been plotted for the cases wherein the `effective potential'~$U(\eta)$ during the transition from de Sitter inflation to radiation domination has been smoothed with the aid of linear, quadratic, cubic and quintic functions (in violet, aquamarine, brown and lime, respectively). We have also plotted (in red) the regularized PS for the case wherein~$U(\eta)$ is described by the infinitely differentiable function~\eqref{eq:U-ict1}. For comparison, we have included the actual and the regularized PS of PGWs, i.e.~$\pt(\ye,\eta)$ and $\ptr(\ye,\eta)$, that arise in the case of the instantaneous transition (in magenta and blue). Recall that, in the standard $\Lambda$CDM model of cosmology, the PS of PGWs depends {\it only}\/ on the reheating temperature~$\Tre$. We have plotted the PS assuming $g_{\ast,\mathrm{rh}}^{1/4} \Tre =5.7\times10^{15}\, \mathrm{GeV}$. This leads to a tensor-to-scalar ratio of $r\simeq 0.034$ over large scales, which is roughly the current upper bound from the CMB~\cite{Planck:2018jri,BICEP:2021xfz}. Moreover, as in the previous two figures, we have set $\Delta\ee =2\vert\ee\vert$ and $\gamma_{\e}=1$ in the cases wherein the transition has been smoothed. We should point out that it is only in the cases of the instantaneous and the smoother linear or quadratic transitions we can determine the PS over all the wave numbers~(in this regard, see Ref.~\cite{Hoory:2025qgm} and App.~\ref{app:qs}). In all the other cases, we have evaluated the PS in the Born approximation. Hence, we have plotted the PS in these cases only over $y_{\e}\gtrsim 1$ (i.e. around and beyond the vertical dotted line). Since, over $y_{\e} \gtrsim 1$, the regularized PS oscillates about zero in all the cases, we have plotted the absolute value of the quantity. Further, in the figure, we have delineated the envelope of the oscillations. It is clear that, as the `effective potential'~$U(\eta)$ is made smoother and smoother, the regularized PS of PGWs~$\ptr(\ye,\eta)$ is suppressed further and further over~$y_{\e} \gtrsim 1$. Importantly, in the case of the infinitely continuous transition, we find that the PS is, in fact, suppressed exponentially (in this regard, see Sec.~\ref{sec:ict}).
Caption The regularized PS of PGWs $\ptr(\ye,\eta)$, evaluated soon after the transition to the epoch of radiation domination at the time $2\vert\ee\vert$, has been plotted for the cases wherein the `effective potential'~$U(\eta)$ during the transition from de Sitter inflation to radiation domination has been smoothed with the aid of linear, quadratic, cubic and quintic functions (in violet, aquamarine, brown and lime, respectively). We have also plotted (in red) the regularized PS for the case wherein~$U(\eta)$ is described by the infinitely differentiable function~\eqref{eq:U-ict1}. For comparison, we have included the actual and the regularized PS of PGWs, i.e.~$\pt(\ye,\eta)$ and $\ptr(\ye,\eta)$, that arise in the case of the instantaneous transition (in magenta and blue). Recall that, in the standard $\Lambda$CDM model of cosmology, the PS of PGWs depends {\it only}\/ on the reheating temperature~$\Tre$. We have plotted the PS assuming $g_{\ast,\mathrm{rh}}^{1/4} \Tre =5.7\times10^{15}\, \mathrm{GeV}$. This leads to a tensor-to-scalar ratio of $r\simeq 0.034$ over large scales, which is roughly the current upper bound from the CMB~\cite{Planck:2018jri,BICEP:2021xfz}. Moreover, as in the previous two figures, we have set $\Delta\ee =2\vert\ee\vert$ and $\gamma_{\e}=1$ in the cases wherein the transition has been smoothed. We should point out that it is only in the cases of the instantaneous and the smoother linear or quadratic transitions we can determine the PS over all the wave numbers~(in this regard, see Ref.~\cite{Hoory:2025qgm} and App.~\ref{app:qs}). In all the other cases, we have evaluated the PS in the Born approximation. Hence, we have plotted the PS in these cases only over $y_{\e}\gtrsim 1$ (i.e. around and beyond the vertical dotted line). Since, over $y_{\e} \gtrsim 1$, the regularized PS oscillates about zero in all the cases, we have plotted the absolute value of the quantity. Further, in the figure, we have delineated the envelope of the oscillations. It is clear that, as the `effective potential'~$U(\eta)$ is made smoother and smoother, the regularized PS of PGWs~$\ptr(\ye,\eta)$ is suppressed further and further over~$y_{\e} \gtrsim 1$. Importantly, in the case of the infinitely continuous transition, we find that the PS is, in fact, suppressed exponentially (in this regard, see Sec.~\ref{sec:ict}).
The actual and the regularized PS of PGWs, i.e. $\pt(\ye,\eta)$ and $\ptr(\ye,\eta)$, evaluated at the time of radiation-matter equality~$\eeq$, has been plotted for the different cases of interest and for the same set of values as in the previous figure. We have indicated the wave numbers $\keq$ and $\ke$~(as vertical dashed and dotted lines) and we have also included an inset highlighting the behavior of the PS around~$y_{\e} \simeq 1$. The suppression in the PS over $y_{\e} \gtrsim 1$ is evident.
Caption The actual and the regularized PS of PGWs, i.e. $\pt(\ye,\eta)$ and $\ptr(\ye,\eta)$, evaluated at the time of radiation-matter equality~$\eeq$, has been plotted for the different cases of interest and for the same set of values as in the previous figure. We have indicated the wave numbers $\keq$ and $\ke$~(as vertical dashed and dotted lines) and we have also included an inset highlighting the behavior of the PS around~$y_{\e} \simeq 1$. The suppression in the PS over $y_{\e} \gtrsim 1$ is evident.
The regularized PS of PGWs $\ptr(\ye,\eta)$, evaluated today at the conformal time~$\eta_0$, has been plotted for the cases wherein the `effective potential' $U(\eta)$ during the transitions from de Sitter inflation to the epoch of radiation domination and the transition from radiation to matter domination is described by linear functions and an infinitely differentiable function~(in aquamarine and red, respectively). We have plotted the PS for the same values of $g_{\ast,\mathrm{rh}}^{1/4} \Tre$, $\Delta\ee$ and $\gamma_{\e}$ that we had worked with in the previous two figures. Also, we have set $\Delta\eta_{\mathrm{eq}}=0.15\eta_{\mathrm{eq}}$ and $\gamma_{\mathrm{eq}}=1$. We should stress again that, since the PS in these cases have been evaluated in the Born approximation, they have been plotted only over the domain $y_{\e}\gtrsim 1$. Moreover, as in the previous figure, we have indicated the actual and regularized PS of PGWs, i.e. $\pt(\ye,\eta)$ and $\ptr(\ye,\eta)$, in the case wherein the two transitions are assumed to be instantaneous (in magenta and blue). These quantities have been plotted over a wide range of wavenumbers. Further, we have plotted the PS as a function of frequency~$f$ rather than the dimensionless variable~$y_{\e}$. For our choice of the reheating temperature, in the case of instantaneous transitions, the different wave numbers can be estimated to be~$(k_0/a_0,\keq/a_0,\ke/a_0)\simeq (1\times 10^{-4}, 0.007,1.02\times 10^{23})\,\mathrm{Mpc}^{-1}$, which correspond to the frequencies $(f_0,f_\mathrm{eq},f_{\e})=(1.94\times10^{-19},1.14\times10^{-17}, 1.92\times10^{8})\,\mathrm{Hz}$ (indicated by the vertical dotted-dashed, dashed and dotted lines; in this regard, see Ref.~\cite{Hoory:2025qgm}). In addition, we have included the sensitivity curves of a variety of ongoing and forthcoming GW observatories that are operating (or are expected to operate) over a wide range of frequencies~\cite{Moore:2014lga,Kanno:2023whr,Franciolini:2022htd}. It is evident that smoothing the transitions lead to a suppression in the regularized PS at high frequencies, as we have discussed.
Caption The regularized PS of PGWs $\ptr(\ye,\eta)$, evaluated today at the conformal time~$\eta_0$, has been plotted for the cases wherein the `effective potential' $U(\eta)$ during the transitions from de Sitter inflation to the epoch of radiation domination and the transition from radiation to matter domination is described by linear functions and an infinitely differentiable function~(in aquamarine and red, respectively). We have plotted the PS for the same values of $g_{\ast,\mathrm{rh}}^{1/4} \Tre$, $\Delta\ee$ and $\gamma_{\e}$ that we had worked with in the previous two figures. Also, we have set $\Delta\eta_{\mathrm{eq}}=0.15\eta_{\mathrm{eq}}$ and $\gamma_{\mathrm{eq}}=1$. We should stress again that, since the PS in these cases have been evaluated in the Born approximation, they have been plotted only over the domain $y_{\e}\gtrsim 1$. Moreover, as in the previous figure, we have indicated the actual and regularized PS of PGWs, i.e. $\pt(\ye,\eta)$ and $\ptr(\ye,\eta)$, in the case wherein the two transitions are assumed to be instantaneous (in magenta and blue). These quantities have been plotted over a wide range of wavenumbers. Further, we have plotted the PS as a function of frequency~$f$ rather than the dimensionless variable~$y_{\e}$. For our choice of the reheating temperature, in the case of instantaneous transitions, the different wave numbers can be estimated to be~$(k_0/a_0,\keq/a_0,\ke/a_0)\simeq (1\times 10^{-4}, 0.007,1.02\times 10^{23})\,\mathrm{Mpc}^{-1}$, which correspond to the frequencies $(f_0,f_\mathrm{eq},f_{\e})=(1.94\times10^{-19},1.14\times10^{-17}, 1.92\times10^{8})\,\mathrm{Hz}$ (indicated by the vertical dotted-dashed, dashed and dotted lines; in this regard, see Ref.~\cite{Hoory:2025qgm}). In addition, we have included the sensitivity curves of a variety of ongoing and forthcoming GW observatories that are operating (or are expected to operate) over a wide range of frequencies~\cite{Moore:2014lga,Kanno:2023whr,Franciolini:2022htd}. It is evident that smoothing the transitions lead to a suppression in the regularized PS at high frequencies, as we have discussed.
The actual and the regularized PS of PGWs, viz. $\pt(k,\eta)$ and~$\ptr(\ye,\eta)$, evaluated soon after the transition to the epoch of radiation domination at the time $2\vert\ee\vert$, have been plotted for the case wherein the `effective potential'~$U(\eta)$ during the transition from de Sitter inflation to radiation domination has been smoothed with the aid of the quadratic function (in red and blue). For comparison, we have also plotted the corresponding PS in the case wherein $U(\eta)$ is smoothed with a linear function (in green and cyan). We should stress that, in these cases, we are able to evaluate the PS exactly over a wide range of wave numbers. It should be clear from the inset that, over $y_{\e} \gtrsim 1$, the regularized PS is suppressed as $k^{-1}$ in both these cases (as indicated in orange and brown). These exact results confirm the behavior of the PS in the domain $y_{\e}>1$ that we obtained earlier using the Born approximation.
Caption The actual and the regularized PS of PGWs, viz. $\pt(k,\eta)$ and~$\ptr(\ye,\eta)$, evaluated soon after the transition to the epoch of radiation domination at the time $2\vert\ee\vert$, have been plotted for the case wherein the `effective potential'~$U(\eta)$ during the transition from de Sitter inflation to radiation domination has been smoothed with the aid of the quadratic function (in red and blue). For comparison, we have also plotted the corresponding PS in the case wherein $U(\eta)$ is smoothed with a linear function (in green and cyan). We should stress that, in these cases, we are able to evaluate the PS exactly over a wide range of wave numbers. It should be clear from the inset that, over $y_{\e} \gtrsim 1$, the regularized PS is suppressed as $k^{-1}$ in both these cases (as indicated in orange and brown). These exact results confirm the behavior of the PS in the domain $y_{\e}>1$ that we obtained earlier using the Born approximation.
References
  • [1] V. F. Mukhanov, H. A. Feldman, and R. H. Brandenberger, Theory of cosmological perturbations. Part 1. Classical perturbations. Part 2. Quantum theory of perturbations. Part 3. Extensions, Phys. Rept. 215 (1992) 203–333.
  • [2] J. Martin, Inflation and precision cosmology, Braz. J. Phys. 34 (2004) 1307–1321, [astro-ph/0312492].
  • [3] J. Martin, Inflationary cosmological perturbations of quantum-mechanical origin, Lect. Notes Phys. 669 (2005) 199–244, [hep-th/0406011].
  • [4] B. A. Bassett, S. Tsujikawa, and D. Wands, Inflation dynamics and reheating, Rev. Mod. Phys. 78 (May, 2006) 537–589.
  • [5] L. Sriramkumar, An introduction to inflation and cosmological perturbation theory, Curr. Sci. 97 (2009) 868, [arXiv:0904.4584].
  • [6] D. Baumann and H. V. Peiris, Cosmological Inflation: Theory and Observations, Adv. Sci. Lett. 2 (2009) 105–120, [arXiv:0810.3022]. 10−2 10−1 100 ye 0.0 0.2 0.4 0.6 0.8 1.0 P T (k, 2|η e |), P reg T (k, 2|η e |) ×10−10
  • [7] D. Baumann, Inflation, in Theoretical Advanced Study Institute in Elementary Particle Physics: Physics of the Large and the Small, 7, 2009. arXiv:0907.5424.
  • [8] L. Sriramkumar, On the generation and evolution of perturbations during inflation and reheating, pp. 207–249. 2012.
  • [9] A. Linde, Inflationary Cosmology after Planck 2013, in 100e Ecole d’Ete de Physique: Post-Planck Cosmology, 2, 2014. arXiv:1402.0526.
  • [10] J. Martin, The Observational Status of Cosmic Inflation after Planck, Astrophys. Space Sci. Proc. 45 (2016) 41–134, [arXiv:1502.05733].
  • [11] Planck Collaboration, P. A. R. Ade et al., Planck 2015 results. XX. Constraints on inflation, Astron. Astrophys. 594 (2016) A20, [arXiv:1502.02114].
  • [12] Planck Collaboration, Y. Akrami et al., Planck 2018 results. X. Constraints on inflation, Astron. Astrophys. 641 (2020) A10, [arXiv:1807.06211].
  • [13] eBOSS Collaboration, S. Alam et al., Completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: Cosmological implications from two decades of spectroscopic surveys at the Apache Point Observatory, Phys. Rev. D 103 (2021), no. 8 083533, [arXiv:2007.08991].
  • [14] eBOSS Collaboration, C. Zhao et al., The completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: cosmological implications from multitracer BAO analysis with galaxies and voids, Mon. Not. Roy. Astron. Soc. 511 (2022), no. 4 5492–5524, [arXiv:2110.03824].
  • [15] DESI Collaboration, A. G. Adame et al., DESI 2024 VI: cosmological constraints from the measurements of baryon acoustic oscillations, JCAP 02 (2025) 021, [arXiv:2404.03002].
  • [16] DESI Collaboration, M. Abdul Karim et al., DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constraints, arXiv:2503.14738.
  • [17] L. P. Grishchuk, Amplification of gravitational waves in an istropic universe, Zh. Eksp. Teor. Fiz. 67 (1974) 825–838.
  • [18] A. A. Starobinsky, Spectrum of relict gravitational radiation and the early state of the universe, JETP Lett. 30 (1979) 682–685.
  • [19] M. C. Guzzetti, N. Bartolo, M. Liguori, and S. Matarrese, Gravitational waves from inflation, Riv. Nuovo Cim. 39 (2016), no. 9 399–495, [arXiv:1605.01615].
  • [20] C. Caprini and D. G. Figueroa, Cosmological Backgrounds of Gravitational Waves, Class. Quant. Grav. 35 (2018), no. 16 163001, [arXiv:1801.04268].
  • [21] G. Domènech, Scalar Induced Gravitational Waves Review, Universe 7 (2021), no. 11 398, [arXiv:2109.01398].
  • [22] R. Roshan and G. White, Using gravitational waves to see the first second of the Universe, Rev. Mod. Phys. 97 (2025), no. 1 015001, [arXiv:2401.04388].
  • [23] M. Giovannini, Gravitational wave astronomy and the expansion history of the Universe, arXiv:2412.13968.
  • [24] BICEP, Keck Collaboration, P. A. R. Ade et al., Improved Constraints on Primordial Gravitational Waves using Planck, WMAP, and BICEP/Keck Observations through the 2018 Observing Season, Phys. Rev. Lett. 127 (2021), no. 15 151301, [arXiv:2110.00483].
  • [25] N. Bernal and F. Hajkarim, Primordial Gravitational Waves in Nonstandard Cosmologies, Phys. Rev. D 100 (2019), no. 6 063502, [arXiv:1905.10410].
  • [26] N. Bernal, A. Ghoshal, F. Hajkarim, and G. Lambiase, Primordial Gravitational Wave Signals in Modified Cosmologies, JCAP 11 (2020) 051, [arXiv:2008.04959].
  • [27] M. R. Haque, D. Maity, T. Paul, and L. Sriramkumar, Decoding the phases of early and late time reheating through imprints on primordial gravitational waves, Phys. Rev. D 104 (2021), no. 6 063513, [arXiv:2105.09242].
  • [28] LIGO Scientific, Virgo Collaboration, B. P. Abbott et al., Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett. 116 (2016), no. 6 061102, [arXiv:1602.03837].
  • [29] LIGO Scientific, Virgo Collaboration, B. P. Abbott et al., Binary Black Hole Mergers in the first Advanced LIGO Observing Run, Phys. Rev. X 6 (2016), no. 4 041015, [arXiv:1606.04856]. [Erratum: Phys.Rev.X 8, 039903 (2018)].
  • [29] LIGO Scientific, Virgo Collaboration, B. P. Abbott et al., Binary Black Hole Mergers in the first Advanced LIGO Observing Run, Phys. Rev. X 6 (2016), no. 4 041015, [arXiv:1606.04856]. [Erratum: Phys.Rev.X 8, 039903 (2018)].
  • [30] LIGO Scientific, Virgo Collaboration, B. P. Abbott et al., The basic physics of the binary black hole merger GW150914, Annalen Phys. 529 (2017), no. 1-2 1600209, [arXiv:1608.01940].
  • [31] LIGO Scientific, VIRGO Collaboration, B. P. Abbott et al., GW170104: Observation of a 50-Solar-Mass Binary Black Hole Coalescence at Redshift 0.2, Phys. Rev. Lett. 118 (2017), no. 22 221101, [arXiv:1706.01812]. [Erratum: Phys.Rev.Lett. 121, 129901 (2018)].
  • [31] LIGO Scientific, VIRGO Collaboration, B. P. Abbott et al., GW170104: Observation of a 50-Solar-Mass Binary Black Hole Coalescence at Redshift 0.2, Phys. Rev. Lett. 118 (2017), no. 22 221101, [arXiv:1706.01812]. [Erratum: Phys.Rev.Lett. 121, 129901 (2018)].
  • [32] LIGO Scientific, Virgo Collaboration, B. P. Abbott et al., GW170814: A Three-Detector Observation of Gravitational Waves from a Binary Black Hole Coalescence, Phys. Rev. Lett. 119 (2017), no. 14 141101, [arXiv:1709.09660].
  • [33] LIGO Scientific, Virgo Collaboration, B. . P. . Abbott et al., GW170608: Observation of a 19-solar-mass Binary Black Hole Coalescence, Astrophys. J. Lett. 851 (2017) L35, [arXiv:1711.05578].
  • [34] LIGO Scientific, Virgo Collaboration, B. P. Abbott et al., Upper Limits on the Stochastic Gravitational-Wave Background from Advanced LIGO’s First Observing Run, Phys. Rev. Lett. 118 (2017), no. 12 121101, [arXiv:1612.02029]. [Erratum: Phys.Rev.Lett. 119, 029901 (2017)].
  • [34] LIGO Scientific, Virgo Collaboration, B. P. Abbott et al., Upper Limits on the Stochastic Gravitational-Wave Background from Advanced LIGO’s First Observing Run, Phys. Rev. Lett. 118 (2017), no. 12 121101, [arXiv:1612.02029]. [Erratum: Phys.Rev.Lett. 119, 029901 (2017)].
  • [35] LIGO Scientific, Virgo Collaboration, B. P. Abbott et al., Search for the isotropic stochastic background using data from Advanced LIGO’s second observing run, Phys. Rev. D 100 (2019), no. 6 061101, [arXiv:1903.02886].
  • [36] KAGRA, Virgo, LIGO Scientific Collaboration, R. Abbott et al., Upper limits on the isotropic gravitational-wave background from Advanced LIGO and Advanced Virgo’s third observing run, Phys. Rev. D 104 (2021), no. 2 022004, [arXiv:2101.12130].
  • [37] NANOGrav Collaboration, G. Agazie et al., The NANOGrav 15 yr Data Set: Evidence for a Gravitational-wave Background, Astrophys. J. Lett. 951 (2023), no. 1 L8, [arXiv:2306.16213].
  • [38] NANOGrav Collaboration, G. Agazie et al., The NANOGrav 15 yr Data Set: Observations and Timing of 68 Millisecond Pulsars, Astrophys. J. Lett. 951 (2023), no. 1 L9, [arXiv:2306.16217].
  • [39] EPTA Collaboration, J. Antoniadis et al., The second data release from the European Pulsar Timing Array - I. The dataset and timing analysis, Astron. Astrophys. 678 (2023) A48, [arXiv:2306.16224].
  • [40] EPTA, InPTA: Collaboration, J. Antoniadis et al., The second data release from the European Pulsar Timing Array - III. Search for gravitational wave signals, Astron. Astrophys. 678 (2023) A50, [arXiv:2306.16214].
  • [41] A. Zic et al., The Parkes Pulsar Timing Array third data release, Publ. Astron. Soc. Austral. 40 (2023) e049, [arXiv:2306.16230].
  • [42] D. J. Reardon et al., Search for an Isotropic Gravitational-wave Background with the Parkes Pulsar Timing Array, Astrophys. J. Lett. 951 (2023), no. 1 L6, [arXiv:2306.16215].
  • [43] H. Xu et al., Searching for the Nano-Hertz Stochastic Gravitational Wave Background with the Chinese Pulsar Timing Array Data Release I, Res. Astron. Astrophys. 23 (2023), no. 7 075024, [arXiv:2306.16216].
  • [44] J. Yokoyama, Implication of pulsar timing array experiments on cosmological gravitational wave detection, AAPPS Bull. 31 (2021), no. 1 17, [arXiv:2105.07629].
  • [45] G. Janssen et al., Gravitational wave astronomy with the SKA, PoS AASKA14 (2015) 037, [arXiv:1501.00127].
  • [46] N. Bartolo et al., Science with the space-based interferometer LISA. IV: Probing inflation with gravitational waves, JCAP 12 (2016) 026, [arXiv:1610.06481].
  • [47] LISA Cosmology Working Group Collaboration, P. Auclair et al., Cosmology with the Laser Interferometer Space Antenna, Living Rev. Rel. 26 (2023), no. 1 5, [arXiv:2204.05434].
  • [48] W.-R. Hu and Y.-L. Wu, The Taiji Program in Space for gravitational wave physics and the nature of gravity, Natl. Sci. Rev. 4 (2017), no. 5 685–686.
  • [49] TianQin Collaboration, J. Mei et al., The TianQin project: current progress on science and technology, PTEP 2021 (2021), no. 5 05A107, [arXiv:2008.10332].
  • [50] Y. Gong, J. Luo, and B. Wang, Concepts and status of Chinese space gravitational wave detection projects, Nature Astron. 5 (2021), no. 9 881–889, [arXiv:2109.07442].
  • [51] J. Crowder and N. J. Cornish, Beyond LISA: Exploring future gravitational wave missions, Phys. Rev. D72 (2005) 083005, [gr-qc/0506015].
  • [52] V. Corbin and N. J. Cornish, Detecting the cosmic gravitational wave background with the big bang observer, Class. Quant. Grav. 23 (2006) 2435–2446, [gr-qc/0512039].
  • [53] G. M. Harry, P. Fritschel, D. A. Shaddock, W. Folkner, and E. S. Phinney, Laser interferometry for the big bang observer, Class. Quant. Grav. 23 (2006) 4887–4894. [Erratum: Class.Quant.Grav. 23, 7361 (2006)].
  • [53] G. M. Harry, P. Fritschel, D. A. Shaddock, W. Folkner, and E. S. Phinney, Laser interferometry for the big bang observer, Class. Quant. Grav. 23 (2006) 4887–4894. [Erratum: Class.Quant.Grav. 23, 7361 (2006)].
  • [54] J. Baker et al., Space Based Gravitational Wave Astronomy Beyond LISA, Bull. Am. Astron. Soc. 51 (2019), no. 7 243, [arXiv:1907.11305].
  • [55] DECIGO working group Collaboration, S. Kawamura, Primordial gravitational wave and DECIGO, PoS KMI2019 (2019) 019.
  • [56] S. Kawamura et al., Current status of space gravitational wave antenna DECIGO and B-DECIGO, PTEP 2021 (2021), no. 5 05A105, [arXiv:2006.13545].
  • [57] MAGIS-100 Collaboration, J. Coleman, Matter-wave Atomic Gradiometer InterferometricSensor (MAGIS-100) at Fermilab, PoS ICHEP2018 (2019) 021, [arXiv:1812.00482].
  • [58] M. Evans et al., A Horizon Study for Cosmic Explorer: Science, Observatories, and Community, arXiv:2109.09882.
  • [59] M. Evans et al., Cosmic Explorer: A Submission to the NSF MPSAC ngGW Subcommittee, arXiv:2306.13745.
  • [60] B. Sathyaprakash et al., Scientific Objectives of Einstein Telescope, Class. Quant. Grav. 29 (2012) 124013, [arXiv:1206.0331]. [Erratum: Class.Quant.Grav. 30, 079501 (2013)].
  • [60] B. Sathyaprakash et al., Scientific Objectives of Einstein Telescope, Class. Quant. Grav. 29 (2012) 124013, [arXiv:1206.0331]. [Erratum: Class.Quant.Grav. 30, 079501 (2013)].
  • [61] M. Branchesi et al., Science with the Einstein Telescope: a comparison of different designs, JCAP 07 (2023) 068, [arXiv:2303.15923].
  • [62] A. Abac et al., The Science of the Einstein Telescope, arXiv:2503.12263.
  • [63] M. Goryachev, D. L. Creedon, S. Galliou, and M. E. Tobar, Observation of Rayleigh phonon scattering through excitation of extremely high overtones in low-loss cryogenic acoustic cavities for hybrid quantum systems, Phys. Rev. Lett. 111 (2013) 085502, [arXiv:1309.4830].
  • [64] S. Galliou, M. Goryachev, R. Bourquin, P. Abbé, J. P. Aubry, and M. E. Tobar, Extremely Low Loss Phonon-Trapping Cryogenic Acoustic Cavities for Future Physical Experiments, Sci. Rep. 3 (2013) 2132, [arXiv:1309.4832].
  • [65] D. J. Fixsen et al., ARCADE 2 Measurement of the Extra-Galactic Sky Temperature at 3-90 GHz, Astrophys. J. 734 (2011) 5, [arXiv:0901.0555].
  • [66] OSQAR Collaboration, R. Ballou et al., New exclusion limits on scalar and pseudoscalar axionlike particles from light shining through a wall, Phys. Rev. D 92 (2015), no. 9 092002, [arXiv:1506.08082].
  • [67] CAST Collaboration, K. Zioutas et al., First results from the CERN Axion Solar Telescope (CAST), Phys. Rev. Lett. 94 (2005) 121301, [hep-ex/0411033].
  • [68] M.-l. Tong, Y. Zhang, and F.-Y. Li, Using polarized maser to detect high-frequency relic gravitational waves, Phys. Rev. D 78 (2008) 024041, [arXiv:0807.0885].
  • [69] V. Domcke, C. Garcia-Cely, and N. L. Rodd, Novel Search for High-Frequency Gravitational Waves with Low-Mass Axion Haloscopes, Phys. Rev. Lett. 129 (2022), no. 4 041101, [arXiv:2202.00695].
  • [70] T. Bringmann, V. Domcke, E. Fuchs, and J. Kopp, High-frequency gravitational wave detection via optical frequency modulation, Phys. Rev. D 108 (2023), no. 6 L061303, [arXiv:2304.10579].
  • [71] Y. Kahn, J. Schütte-Engel, and T. Trickle, Searching for high-frequency gravitational waves with phonons, Phys. Rev. D 109 (2024), no. 9 096023, [arXiv:2311.17147].
  • [72] S. Kanno, J. Soda, and A. Taniguchi, Search for high-frequency gravitational waves with Rydberg atoms, Eur. Phys. J. C 85 (2025), no. 1 31, [arXiv:2311.03890].
  • [73] ADMX Collaboration, C. Bartram et al., Search for Invisible Axion Dark Matter in the 3.3–4.2 µeV Mass Range, Phys. Rev. Lett. 127 (2021), no. 26 261803, [arXiv:2110.06096].
  • [74] A. Berlin, D. Blas, R. Tito D’Agnolo, S. A. R. Ellis, R. Harnik, Y. Kahn, and J. Schütte-Engel, Detecting high-frequency gravitational waves with microwave cavities, Phys. Rev. D 105 (2022), no. 11 116011, [arXiv:2112.11465].
  • [75] A. Ringwald, J. Schütte-Engel, and C. Tamarit, Gravitational Waves as a Big Bang Thermometer, JCAP 03 (2021) 054, [arXiv:2011.04731].
  • [76] F.-Y. Li, M.-X. Tang, and D.-P. Shi, Electromagnetic response of a Gaussian beam to high frequency relic gravitational waves in quintessential inflationary models, Phys. Rev. D 67 (2003) 104008, [gr-qc/0306092].
  • [77] J. Beacham et al., Physics Beyond Colliders at CERN: Beyond the Standard Model Working Group Report, J. Phys. G 47 (2020), no. 1 010501, [arXiv:1901.09966].
  • [78] J. Ruz et al., Next Generation Search for Axion and ALP Dark Matter with the International Axion Observatory, in 2018 IEEE Nuclear Science Symposium and Medical Imaging Conference, p. 8824640, 2018.
  • [79] A. Hoory, J. Martin, A. Paul, and L. Sriramkumar, Primary gravitational waves at high frequencies I: Origin of suppression in the power spectrum, arXiv:2512.03959.
  • [80] D.-G. Wang, Y. Zhang, and J.-W. Chen, Vacuum and gravitons of relic gravitational waves and the regularization of the spectrum and energy-momentum tensor, Phys. Rev. D 94 (2016), no. 4 044033, [arXiv:1512.03134].
  • [81] L. Parker, Amplitude of Perturbations from Inflation, hep-th/0702216.
  • [82] I. Agullo, J. Navarro-Salas, G. J. Olmo, and L. Parker, Reexamination of the Power Spectrum in De Sitter Inflation, Phys. Rev. Lett. 101 (2008) 171301, [arXiv:0806.0034].
  • [83] I. Agullo, J. Navarro-Salas, G. J. Olmo, and L. Parker, Revising the predictions of inflation for the cosmic microwave background anisotropies, Phys. Rev. Lett. 103 (2009) 061301, [arXiv:0901.0439].
  • [84] I. Agullo, J. Navarro-Salas, G. J. Olmo, and L. Parker, Revising the observable consequences of slow-roll inflation, Phys. Rev. D 81 (2010) 043514, [arXiv:0911.0961].
  • [85] Y. Urakawa and A. A. Starobinsky, Adiabatic regularization of primordial perturbations generated during inflation, in 19th Workshop on General Relativity and Gravitation in Japan, 2009.
  • [86] A. del Rio and J. Navarro-Salas, Spacetime correlators of perturbations in slow-roll de Sitter inflation, Phys. Rev. D 89 (2014), no. 8 084037, [arXiv:1401.6912].
  • [87] S. Pla and B. A. Stefanek, Renormalization of the primordial inflationary power spectra, Phys. Lett. B 856 (2024) 138926, [arXiv:2402.14910].
  • [88] M. Maggiore, Gravitational wave experiments and early universe cosmology, Phys. Rept. 331 (2000) 283–367, [gr-qc/9909001].
  • [89] S. Pi, M. Sasaki, A. Wang, and J. Wang, Revisiting the ultraviolet tail of the primordial gravitational wave, Phys. Rev. D 110 (2024), no. 10 103529, [arXiv:2407.06066].
  • [90] T. S. Bunch and P. C. W. Davies, Quantum Field Theory in de Sitter Space: Renormalization by Point Splitting, Proc. Roy. Soc. Lond. A 360 (1978) 117–134.
  • [91] F. Finelli, G. Marozzi, G. P. Vacca, and G. Venturi, The Impact of ultraviolet regularization on the spectrum of curvature perturbations during inflation, Phys. Rev. D 76 (2007) 103528, [arXiv:0707.1416].
  • [92] M. Maggiore, Gravitational Waves. Vol. 1: Theory and Experiments. Oxford University Press, 2007.
  • [93] S. A. Fulling and L. Parker, Renormalization in the theory of a quantized scalar field interacting with a robertson-walker spacetime, Annals Phys. 87 (1974) 176–204.
  • [94] L. Parker and S. A. Fulling, Adiabatic regularization of the energy momentum tensor of a quantized field in homogeneous spaces, Phys. Rev. D 9 (1974) 341–354.
  • [95] S. A. Fulling, L. Parker, and B. L. Hu, Conformal energy-momentum tensor in curved spacetime: Adiabatic regularization and renormalization, Phys. Rev. D 10 (1974) 3905–3924.
  • [96] T. S. Bunch, Calculation of the Renormalized Quantum Stress Tensor by Adiabatic Regularization in Two-Dimensional and Four-Dimensional Robertson-Walker Space-Times, J. Phys. A 11 (1978) 603–607.
  • [97] T. S. Bunch, ADIABATIC REGULARIZATION FOR SCALAR FIELDS WITH ARBITRARY COUPLING TO THE SCALAR CURVATURE, J. Phys. A 13 (1980) 1297–1310.
  • [98] P. R. Anderson and L. Parker, Adiabatic Regularization in Closed Robertson-walker Universes, Phys. Rev. D 36 (1987) 2963.
  • [99] N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, UK, 1982.
  • [100] L. E. Parker and D. Toms, Quantum Field Theory in Curved Spacetime: Quantized Field and Gravity. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 8, 2009.
  • [101] M. Zhu and Y.-F. Cai, Smoking-gun signatures of bounce cosmology from echoes of relic gravitational waves, arXiv:2603.13924.
  • [102] C. Wang, Y. Xu, and W. Zhao, Graviton Production from Inflaton Condensate: Boltzmann vs Bogoliubov, arXiv:2604.12687.
  • [103] Y. Wang, Q.-f. Wu, and X.-J. Xu, A Unified Bogoliubov Approach to Primordial Gravitational Waves: From Inflation to Reheating, arXiv:2604.17478.
  • [104] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Books on Advanced Mathematics. Dover Publications, New York, 1965. NBS Applied Mathematics Series 55.
  • [105] C. J. Moore, R. H. Cole, and C. P. L. Berry, Gravitational-wave sensitivity curves, Class. Quant. Grav. 32 (2015), no. 1 015014, [arXiv:1408.0740].
  • [106] G. Franciolini, A. Maharana, and F. Muia, Hunt for light primordial black hole dark matter with ultrahigh-frequency gravitational waves, Phys. Rev. D 106 (2022), no. 10 103520, [arXiv:2205.02153].