Multiple-scale analysis of modified gravitational-wave propagation

Abstract

We employ multiple-scale analysis to systematically derive analytical approximations describing the cosmological propagation of gravitational waves beyond general relativity, in a framework with two interacting spin-2 fields with time-dependent couplings. Such techniques allow us to accurately track the evolution of a system with slowly evolving time-dependent couplings over a large number of oscillation periods. We focus on tensor modes propagating on sub-horizon scales in a universe dominated by dark energy and explicitly derive solutions for a general class of models. To illustrate the possible applications of our general scheme and further corroborate our analytical results, we calculate the evolution of tensor perturbations in some phenomenological toy models and compare them with numerical simulations. We show that, generically, the interactions of independent spin-2 fields lead to non-trivial modifications to the amplitude and phase of the detected waveform, which are different from those obtained in other modified gravity theories with a single graviton. This provides an avenue to test and constrain gravitational models with new fundamental physical fields.

Figures (10)

• Figure 1: The numerical and approximate action, angle and $t$-derivative angle variables of Eq. \ref{['monomial time dependent model']}, evolving along the dimensionless time parameter $t$. We have chosen the parameters $k=10^3$, $\epsilon=10^{-3}$, $t_0=-10^{3}$ and $m=1$ . The thick, blue curves correspond to the approximate solutions, and the dashed, orange curves correspond to the numerical solutions.
• Figure 2: The numerical and approximate action, angle and $t$-derivative angle variables of Eq. \ref{['polynomial model']}, evolving along the dimensionless time parameter $t$. For these plots we have selected the values $k=10^3$, $\epsilon=10^{-3}$, $m=1$, $t_0=-10^3$ . The thick, blue curves correspond to the approximate solutions, and the dashed, orange curves correspond to the numerical solutions.
• Figure 3: The numerical and approximate action, angle and $t$-derivative angle variables for the model in Eq. \ref{['Chiral Model']}, evolving along the dimensionless time parameter $t$. We have chosen $k=10^3$, $\epsilon=10^{-3}$, $t_0=-10^3$. The thick, blue curves correspond to the approximate solutions, and the dashed, orange curves correspond to the numerical solutions.
• Figure 4: The numerical and approximate action angle and $t$-derivative angle variables of Eq. \ref{['velocity mixing model']}, evolving along the dimensionless time parameter $t$. For these plots we have chosen the parameters $k=10^3$, $\epsilon=10^{-3}$, $t_0=-10^3$ . The thick, blue curves correspond to the approximate solutions, and the dashed, orange curves correspond to the numerical solutions.
• Figure 5: The action, angle and $t$-derivative angle variables of Eq. \ref{['Friction mixing model']}, evolving along the dimensionless time parameter $t$. For these simulations we have chosen $k=10^3$, $\epsilon=10^{-3}$, $t_0=-10^3$, $\alpha=10^{-4}$ and $\Delta=10^{-2}$. The thick, blue curves correspond to the approximate solutions, and the dashed, orange curves correspond to the numerical solutions.