High Frequency Spectrum of Primordial Gravitational Waves

TL;DR

The paper addresses how primordial gravitational waves acquire a high-frequency tail from inflaton oscillations after inflation, extending the spectrum beyond the well-studied superhorizon modes. It develops a dual formalism—semi-classical evolution and Bogoliubov-coefficient methods—to compute the GW spectrum across all relevant scales, linking low-frequency and high-frequency behavior through the scales $k_1= a_e H_e$ and $k_2= a_e m_\phi$. Analytic estimates predict a $f^{-2}$ tail at low frequencies and a $f^{-1/2}$ tail at high frequencies for $w=0$, while the intermediate region requires numerical evaluation; the ratio $\Omega_{\rm GW}(f_2)/\Omega_{\rm GW}(f_1) \sim m_\phi/H_e$ can be large, revealing rich structure sensitive to the inflaton potential. By applying the framework to chaotic, Starobinsky, new inflation, and α-attractor T-models, the work demonstrates characteristic intermediate-region features that could distinguish inflationary scenarios, even though the overall amplitude remains challenging to observe.

Abstract

During inflation gravitational waves are produced in the superhorizon regime, which form stochastic background in the present universe with very wide range of frequencies. Higher frequency gravitational waves never experience superhorizon regime, but they are also amplified after inflation due to the inflaton oscillation. Taking account of the inflaton dynamics after inflation, the spectrum of primordial gravitational waves may extend to much higher frequencies than previously thought. In this paper we calculate the spectrum of high frequency gravitational waves produced during and after inflation in detail, in particular focusing on the connection between the low and high frequency regime, and show that the detailed spectrum can distinguish inflation models.

Figures (7)

• Figure 1: Schematic picture of the primordial GW spectrum for the case of $w=0$. Several characteristic comoving wavenumber are plotted: $k_{\rm eq}$ and $k_{\rm R}$ are the comoving Hubble scale at the matter-radiation equality and the end of reheating, respectively, and $k_{\rm cut}$ is the comoving inflaton mass scale at the end of reheating. For $k_1$ and $k_2$, see text. Our main focus is the behavior around the red shaded region.
• Figure 2: (Left) Time evolution of several length scales: Hubble scale $H^{-1}$ (black line), physical GW wavelength $a/k$ (red lines), the effective inflaton mass scale $m_\phi^{-1}$ (blue dashed line). The time at the end of inflation is represented by $t_{\rm e}$ and the star represents when the condition $k=am_\phi$ is satisfied for one choice of $k$ at $t=t_m$, at which particle production happens. (Right) Expected GW spectrum as a function of (present) GW frequency $f$: $f_1$ and $f_2$ correspond to the comoving wavenumber $k_1$ and $k_2$. The blue shaded region is hard to predict analytically and we need numerical simulation to derive the behavior in this region. We assumed $w<1/3$ in this figure, where $w$ is the equation of state parameter during the inflaton oscillation epoch.
• Figure 3: The same as Fig. \ref{['fig:smallw']}, but for $w>1/3$.
• Figure 4: (Left) GW spectrum for quadratic chaotic inflation. (Right) GW spectrum for Starobinsky inflation. We have taken $T_{\rm R}=10^{10}\,{\rm GeV}$. Solid line is the result of semi-classical method and the dashed line is the result of Bogoliubov method. Both give consistent results in the overlapping region where the both methods are applicable. It is clearly seen that the spectrum scales as $f^{-2}$ in the low frequency limit and $f^{-1/2}$ in the high frequency limit, as expected (see Fig. \ref{['fig:smallw']}).
• Figure 5: Time evolution of the homogeneous inflaton field $\phi/v$ (solid) and the Hubble parameter $H/H_{\rm inf}$ (dashed) in the new inflation model. We have taken $v/M_{\rm Pl} = 1$ in the left panel and $v/M_{\rm Pl} = 0.1$ in the right panel.