Gravitational wave signal from massive gravity

Abstract

We discuss the detectability of gravitational waves with a time dependent mass contribution, by means of the stochastic gravitational wave observations. Such a mass term typically arises in the cosmological solutions of massive gravity theories. We conduct the analysis based on a general quadratic action, and thus the results apply universally to any massive gravity theories in which modification of general relativity appears primarily in the tensor modes. The primary manifestation of the modification in the gravitational wave spectrum is a sharp peak. The position and height of the peak carry information on the present value of the mass term, as well as the duration of the inflationary stage. We also discuss the detectability of such a gravitational wave signal using the future-planned gravitational wave observatories.

Figures (6)

• Figure 1: Schematic plot for the evolution of $a\,M_{GW}$ and $a\,H$. We show the critical momentum $k_c$, defined in (\ref{['kcdef']}), for which, the mass and momentum contributions are equal at the time of re-entry, while $k_0$, defined in (\ref{['k0def']}), is the momentum of the mode for which the mass term became important just today.
• Figure 2: Examples for each type of momenta based on the classification in the main text. For modes with short and intermediate wavelengths ($k_0<k$ and $k_c<k<k_0$), the horizon entry time is the same as in GR, with $a_k\sim a_k^{GR}$. However, for modes with long wavelength ($k<k_c$), the momentum term never becomes dominant and the horizon entry in the massive theory occurs earlier than in GR, i.e. $a_k < a_k^{GR}$.
• Figure 3: Schematic plot of the amplification factor ${\cal S}(\omega_0)$.
• Figure 4: Gravitational power spectrum in the massive gravity with $M_{GW,0}=10^{10} H_0\sim 2\times 10^{-7}\,\text{Hz}$ shown with respect to $\omega_0$. The solid (red) curve is the spectrum obtained by the numerical calculation, and the dashed (blue) curve is the semi-analytical spectrum given by Eq. (\ref{['eqn:def-A']}), where the GR spectrum is calculated numerically and the enhancement factor ${\cal S}(\omega_0)$ is given analytically by Eqs. (\ref{['S_short']}), (\ref{['S_int']}) and (\ref{['eqn:enhancement']}).
• Figure 5: The gravitational power spectra in the massive gravity theory with $M_{GW,0}=10^{10} H_0\sim 2\times 10^{-7}\,\text{Hz}$ shown with respect to $k/a_0H_0$. The spectra are obtained by the numerical calculation (solid red) and semi-analytic estimate by Eqs. (\ref{['eqn:def-A']}), (\ref{['S_short']}), (\ref{['S_int']}) and (\ref{['eqn:enhancement']}) (dotted blue), respectively. For the latter, the GR spectrum is calculated numerically and the enhancement factor ${\cal S}(\omega_0)$ is calculated analytically.