Primary gravitational waves at high frequencies I: Origin of suppression in the power spectrum

TL;DR

This work addresses the high-frequency behavior of the primary gravitational-wave power spectrum (PGW PS) generated during inflation. It shows that the naive $k^2$ rise for modes inside the Hubble radius must be removed via adiabatic regularization, which yields a PS that oscillates about zero with fixed amplitude at large $k$. Furthermore, smoothing the transition from inflation to the radiation-dominated era introduces a power-law suppression of these oscillations, enhancing the physical behavior of real-space correlators. The combined use of regularization and smooth transitions provides well-behaved predictions across radiation and matter epochs and outlines extensions to more gradual transitions and reheating dynamics.

Abstract

[Abridged] The primary gravitational waves (PGWs) are generated in the early universe from the quantum vacuum during inflation. In slow roll inflation, the power spectrum (PS) of PGWs over large scales, which leave the Hubble radius during inflation, is nearly scale-invariant. However, over very small scales, which never leave the Hubble radius, the PS of PGWs behaves as k^2, where k denotes the wave number. We examine the PS of PGWs at such high wave numbers or frequencies when the PGWs are evolved post-inflation, through the epochs of radiation and matter domination. Firstly, we argue that the PS has to be regularized in order to truncate the unphysical k^2 rise at high frequencies. Assuming instantaneous transitions from inflation to the epochs of radiation and matter domination, we carry out the method of adiabatic regularization to arrive at the PS of PGWs over a wide range of frequencies. We show that the process of regularization truncates the k^2 rise and the PS of PGWs oscillates with a fixed amplitude about a vanishing mean value over small scales or, equivalently, at high frequencies. Secondly, we smooth the transition from inflation to radiation domination (to be precise, we smooth the 'effective potential' governing the equation of motion of PGWs) and examine the impact of the smoothing on the regularized PS of PGWs. With the help of a linear smoothing function, we explicitly show that the smoother transition leads to a power-law suppression in the amplitude of the oscillations (about the zero mean value) of the regularized PS of PGWs over small scales that never leave the Hubble radius during inflation. Our analysis indicates that, when transitions are involved, regularization as well as smooth transitions seem essential to ensure that the correlation functions of the PGWs in real space are well behaved. We discuss the directions in which our results need to be extended.

Figures (10)

• Figure 1: The PS of PGWs $\mathcal{P}_{_{\mathrm{T}}}(k,\eta)$ evaluated at the time of radiation-matter equality $\eta_{\mathrm{eq}}$ has been plotted for the case of instantaneous transition from de Sitter inflation to the epoch of radiation domination [cf. Eq. \ref{['eq:ps-ds-rd-it']}]. If we assume the standard $\Lambda$CDM model of cosmology, the PS depends only on the reheating temperature $T_{\mathrm{rh}}$. For $g_{\ast,\mathrm{rh}}^{1/4} T_{\mathrm{rh}} =1.8\times10^{16}\,\mathrm{GeV}$, we find that the tensor-to-scalar ratio proves to be $r\simeq 0.034$ over large scales, which is roughly the current upper bound from the CMB Planck:2018jriBICEP:2021xfz. Keeping this constraint in mind, we have plotted the PS for $g_{\ast,\mathrm{rh}}^{1/4} T_{\mathrm{rh}}=1.8\times(10^{16}, 10^{14},10^{12})\, \mathrm{GeV}$ (in red, blue and green). Evidently, the PS is scale invariant over large scales such that $y_{\mathrm{e}}\ll 1$ and $x\ll 1$ [cf. Eq. \ref{['eq:ps-si']}]. These limits correspond to wave numbers $k\lesssim k_{\mathrm{eq}}$, which are outside the Hubble radius at the time of radiation-matter equality. Also, the fact that $\mathcal{P}_{_{\mathrm{T}}}(k,\eta_{\mathrm{eq}})\propto H_{\mathrm e}^2 \propto T_{\mathrm{rh}}^4$ on large scales should be clear from the comparison of the scale-invariant amplitudes for the different $T_{\mathrm{rh}}$. The PS behaves as $k^{-2}$ with superimposed oscillations over the intermediate scales such that $y_{\mathrm{e}}\ll 1$ and $x\gg 1$ [cf. Eq. \ref{['eq:ptlargescale']}]. These limits correspond to wave numbers $k$ such that $k_{\mathrm{eq}} \lesssim k \lesssim k_{\mathrm{e}}$, which renter the Hubble radius during the radiation-dominated era. The sharp dips in the PS over this domain correspond to wave numbers where the PS vanishes. Importantly, note that, when $y_{\mathrm{e}} > 1$ (with $y_{\mathrm{e}}=1$ indicated by the vertical dotted line), $\mathcal{P}_{_{\mathrm{T}}}(k,\eta_{\mathrm{eq}}) \propto k^2$ [cf. Eq. \ref{['eq:ptsmallscale']}]. These correspond to wave numbers $k > k_{\mathrm{e}}$ which never leave the Hubble radius. We should add that PS is scale invariant until $k_{\mathrm{eq}}/a_0 \simeq 0.007\,\mathrm{Mpc}^{-1}$. The change from the scale-invariant behavior to the $k^{-2}$ behavior occurs at different $y_{\mathrm{e}}$ because $k_{\mathrm{e}}$ is different for different $T_{\mathrm{rh}}$.
• Figure 2: The PS of PGWs $\mathcal{P}_{_{\mathrm{T}}}(k,\eta)$ evaluated today, i.e. at $\eta_0$, has been plotted for the case of instantaneous transition from de Sitter inflation to the epoch of radiation domination and subsequently to the epoch of matter domination [cf. Eq. \ref{['eq:generalptmatunregulated']}]. We have plotted the PS for the three values of $g_{\ast,\mathrm{rh}}^{1/4} T_{\mathrm{rh}}$ we had plotted in the previous figure. Note that, in addition to the different behavior of the PS in the three regimes mentioned in the previous figure, there arises a fourth regime when $x_{\mathrm{eq}} \ll 1$ and $z\gg 1$, where the PS behaves as $k^{-4}$ with superimposed oscillations [cf. Eq. \ref{['eq:pteqtodaycale']}]. These wave numbers correspond to $k_0 \lesssim k \lesssim k_{\mathrm{eq}}$ which re-enter the Hubble radius during the epoch of matter domination.
• Figure 3: The PS of PGWs $\mathcal{P}_{_{\mathrm{T}}}(k,\eta)$ evaluated today, i.e. at the conformal time $\eta_0$, has been plotted (in red) assuming $g_{\ast,\mathrm{rh}}^{1/4} T_{\mathrm{rh}}=1.8 \times 10^{16}\,\mathrm{GeV}$ (corresponding to a tensor-to-scalar ratio of $r\simeq 0.034$, which is consistent with the current constraints) for the case of instantaneous transitions from de Sitter inflation to the epochs of radiation and matter domination. In contrast to the previous two figures, we have plotted the PS as a function of frequency $f$ rather than the dimensionless ratio $y_{\mathrm{e}}$. For the above-mentioned scenario and reheating temperature, the different wave numbers can be estimated to be $(k_0/a_0,k_{\mathrm{eq}}/a_0,k_{\mathrm{e}}/a_0)\simeq (1.2 \times 10^{-4}, 0.007, 1.24\times 10^{23})\,\mathrm{Mpc}^{-1}$, which correspond to the frequencies $(f_0,f_\mathrm{eq},f_{\mathrm{e}})=(1.94\times10^{-19},1.14\times10^{-17}, 1.92\times10^{8})\,\mathrm{Hz}$ (indicated by the vertical dotted-dashed, dashed and dotted lines). In the figure, we have also included the sensitivity curves of various present and future GW observatories operating over a wide range of frequencies (in this context, see the discussion in the introductory section as well as Refs. Moore:2014lgaKanno:2023whrFranciolini:2022htd). While the curves for ADMX, SQMS, GB, JURA and IAXO represent projected future sensitivities, it is important to point out that the sensitivity curves of some detectors at high frequencies, such as BAW, ARCADE, OSQAR and CAST, are actually current upper bounds, not mere projections. The fact that CAST has not observed the signal implies that the PS of PGWs that we have plotted is ruled out. It should also be clear from the figure that, unless the $k^2$ rise in the PS of PGWs is truncated, the signal can, in principle, be observed by one or more of the detectors operating at high frequencies.
• Figure 4: The actual and the regularized PS of PGWs, viz. $\mathcal{P}_{_{\mathrm{T}}}(k,\eta)$ and $\mathcal{P}_{_{\mathrm{T}}}^\mathrm{reg}(k,\eta)$, evaluated during the early stages of the radiation-dominated epoch at $\eta=-2\eta_{\mathrm{e}}=2\vert\eta_{\mathrm{e}}\vert$ have been plotted (in red and blue) for the case of instantaneous transition from de Sitter inflation [cf. Eq. \ref{['eq:ps-ds-rd-it-r']}]. We have plotted the PS for the same value of $g_{\ast,\mathrm{rh}}^{1/4} T_{\mathrm{rh}}$ as in the previous figure. Note that, over $y_{\mathrm{e}}>1$ (indicated by the vertical dotted line), while the original PS rises as $k^2$ (with oscillations superimposed upon it), the regularized PS oscillates about zero with a constant amplitude. In other words, over $k\gtrsim k_{\mathrm{e}}$, $\mathcal{P}_{_{\mathrm{T}}}^\mathrm{reg}(k,\eta)$ can turn negative.
• Figure 5: The actual and the regularized PS of PGWs, viz. $\mathcal{P}_{_{\mathrm{T}}}(k,\eta)$ and $\mathcal{P}_{_{\mathrm{T}}}^\mathrm{reg}(k,\eta)$, evaluated today at the conformal time $\eta_0$, have been plotted (in red and blue) for the case of instantaneous transition from de Sitter inflation to the epochs of radiation and matter domination [cf. Eqs. \ref{['eq:generalptmatunregulated']} and \ref{['eq:generalptmatregulated']}]. We have chosen to work with $g_{\ast,\mathrm{rh}}^{1/4} T_{\mathrm{rh}}=1.8\times 10^{16}\, \mathrm{GeV}$ as in the previous two figures. Also, as in Fig. \ref{['fig:PT-hf']}, the vertical lines indicate the wave numbers $k_0/a_0$, $k_{\mathrm{eq}}/a_0$ and $k_{\mathrm{e}}/a_0$. Moreover, as in Fig. \ref{['fig:PT-hf']}, we have included the sensitivity curves of the various GW observatories. It is clear that the process of regularization truncates the $k^2$ rise for $k \gtrsim k_{\mathrm{e}}$. Also, note that the regularization procedure does not alter the PS over scales $k \lesssim k_{\mathrm{e}}$. We find that, over $k \gtrsim k_{\mathrm{e}}$, the regularized PS $\mathcal{P}_{_{\mathrm{T}}}^\mathrm{reg}(k,\eta)$ oscillates about zero as illustrated in the previous figure. However, the amplitude of the oscillations is significantly suppressed during the late stages of matter domination when compared to the values during the early stages of radiation domination.