Scalar-Induced Gravitational Waves in Palatini $f(R)$ Gravity

TL;DR

This paper develops a general framework for Scalar-Induced Gravitational Waves in Palatini $f(R)$ gravity, focusing on a Starobinsky-like term and a radiation-dominated era with a subdominant dust component to enable perturbative corrections. By deriving the perturbation equations in the Palatini formalism and solving for the MG-induced transfer function, the authors obtain a kernel and a density spectrum for SIGWs, both analytically and numerically. They find a scale-invariant MG enhancement of the SIGW spectrum relative to GR, with a characteristic amplification factor $ riangle(eta,T_{ m reh})=2.76\alpha\left(T_{ m reh}/ ext{GeV}\right)^2\times 10^{-92}$ in the D-dominated regime, and potential suppression in the C-dominated regime; for plausible parameters this upgrade could be observable with upcoming GW experiments, enabling tests of Palatini gravity versus GR and metric $f(R)$ approaches. The results highlight the potential of SIGWs as probes of gravity's degrees of freedom at high frequencies and early times, and they outline future work on matter-dominated epochs, reheating histories, and broader $f(R)$ models.

Abstract

Primordial scalar perturbations that reenter the horizon after inflation may induce a second-order Gravitational Wave spectrum with information about the primordial Universe on scales inaccessible to Cosmic Microwave Background experiments. In this work, we develop a general framework for the study of Scalar-Induced Gravitational Waves in Palatini $f(R)$ gravity, a theory that was proven to successfully realise inflation and quintessence, and consider the case of the Starobinsky-like model as an example. A regime of radiation domination with a subdominant matter component is assumed, allowing for a well-motivated perturbative approach to the gravity modifications. We calculate the kernel function and the density spectrum numerically and find accurate analytical expressions. The spectral density, which may be tested across a wide range of frequencies by upcoming Gravitational Wave experiments, is shown to differ from the General Relativity and metric $f(R)$ gravity predictions under certain conditions. We comment on previous results in the literature regarding the metric formulation and make special emphasis on the potential of these distinctive features of the spectrum to probe the two formalisms of gravity.

Figures (10)

• Figure 1: Left: Source term (blue) with its subhorizon (dashed green) and superhorizon (dashed orange) limits. Right: Integrands of the first (blue) and second (orange) terms in the solution for the transfer function of $\tilde{\psi}^{\textrm{MG}}_k$.
• Figure 2: Left: Transfer function of $\tilde{\psi}_k^{\rm MG}$ for different values of $\mathcal{D}k^2$ in the parameter range close to the particular solution being negligible. The initial integration time is set to $x_{\rm reh}=10^{-3}$. We show the full numerical solution, without any approximations, in solid blue, orange, and green. For the green curve, the particular solution is negligible, meaning that the full solution is given by the homogeneous solution. In dashed black we show the analytical solutions corresponding to each numerical solution. Right: A zoom-in of the left panel after horizon crossing.
• Figure 3: Left: Transfer function of $\tilde{\psi}_k^{\rm MG}$ for different values of $\mathcal{D}k^2$ in the parameter range where the homogeneous solution is negligible. The initial integration time is set to $x_{\rm reh}=10^{-3}$. We show the full numerical solution, without any approximations, in solid blue, orange, and green. In dashed black we show the analytical solutions corresponding to each numerical solution. Right: A zoom-in of the left panel after horizon crossing.
• Figure 4: Left: Numerical (full blue) and analytical (dashed black) solutions for the modified gravity transfer function. Right: GR kernel (blue) and the modified gravity contribution (dashed orange), for $u=v=2$ and $\alpha = 10^{67}$. In both panels we use $k=10^{17}\text{Mpc}^{-1}$ and $\mathcal{C}=10^{30}$, so that the terms proportional to $\mathcal{C}$ dominate in the modified gravity transfer function.
• Figure 5: Left: Comparison between the exact solution for the transfer function of $\tilde{\psi}_k^{\rm MG}$ (solid blue) and its approximation (dashed orange), for $k=10^{15}\text{Mpc}^{-1}$. Right: Comparison between the exact solution for the modified gravity kernel $I_{\rm MG}$, evaluated at $u=v=2$ (solid blue) and its approximation (dashed orange), for $k=10^{15}\text{Mpc}^{-1}$.