Semianalytic calculation of the gravitational wave spectrum induced by curvature perturbations

TL;DR

This work derives a universal, analytic integration kernel I(u,v,x) for SGWs induced by primordial curvature perturbations, enabling efficient, model-agnostic computation of the SIGW spectrum across radiation- and matter-dominated eras. It provides fully analytic expressions for the RD and MD kernels and extends the formalism to transient RD/MD epochs with master formulas, including explicit results for delta, power-law, and top-hat curvature spectra. The top-hat curvature perturbation case is treated as a new minor update, yielding clear IR and UV limits and practical Padé-like fits. By clarifying the role of epoch transitions (MD↔RD) and their timescales, the work highlights when transitions enhance or suppress SIGWs (e.g., poltergeist mechanism for rapid MD→RD transitions) and supplies tools essential for rapid comparisons with PTA, PBH, and early-Universe scenarios. Collectively, these analytic/semi-analytic results accelerate parameter estimation and theory-to-data analyses in GW cosmology and early-Universe physics.

Abstract

The stochastic gravitational wave (GW) background is secondarily and inevitably induced by the primordial curvature perturbations beyond the first order of the cosmological perturbation theory. We analytically calculate the integration kernel of the power spectrum of the induced GWs, which is the universal part independent of the spectrum of the primordial curvature perturbations, in the radiation-dominated era and in the matter-dominated era. We derive fully analytic expressions of the GW spectrum when possible. As a minor update, we study the case of the top-hat function as the spectrum of the curvature perturbations. We also discuss generalization in the presence of multiple cosmological eras with different equations of state.

Figures (6)

• Figure 1: GW spectrum induced by the delta-function $\mathcal{P}_\zeta (k)$ during the RD era (solid black line). Also shown in dashed blue, orange, and bluish-green lines for comparison are those induced from the top-hat function with $\Delta = 10^{-1}$, $10^{-2}$, and $10^{-3}$, respectively. For the top-hat function case, $k_*$ should be regarded as $k_\text{med}\equiv \sqrt{k_\text{min}k_\text{max}}$.
• Figure 2: Dependence of the numerical coefficient $Q$ on the power $n_\text{s}$ of $\mathcal{P}_\zeta (k) = A (k/k_*)^{n_\text{s}-1}$. The solid black line is the numerical result, while the dashed green line is a fit by the Padé approximation \ref{['Q_fit']}.
• Figure 3: The normalized GW spectra induced by the top-hat $\mathcal{P}_\zeta(k)$. The left and right panels show the limits $k_\text{max} \to \infty$ and $k_\text{min}\to 0$. The solid black lines show the numerical results, while the dashed green lines show Padé-like fits [eqs. \ref{['Omega_GW_RD_top-hat_IR-fit']} and \ref{['Omega_GW_RD_top-hat_UV-fit']} in Appendix \ref{['sec:fits']}]. The thin horizontal gray lines show the scale-invariant case ($k_\text{min} \to 0$ and $k_\text{max} \to \infty$ with fixed $\widetilde{A}$). The dotted magenta lines show the approximations for the IR and UV limits [eqs. \ref{['Omega_GW_RD_top-hat_IR-limit']} and \ref{['Omega_GW_RD_top-hat_UV-limit']}].
• Figure 4: Comparison of the numerical results (solid darker lines) of the SIGW spectrum induced in an RD era by the top-hat spectrum of the curvature perturbations with finite widths and their approximations [eq. \ref{['Omega_GW_RD_top-hat_fit']}] (dashed lighter lines). From outside to inside, the width is $\Delta = 1$ (blue), $0.8$ (orange), $0.6$ (bluish green), $0.4$ (vermilion), $0.3$ (sky blue), $0.2$ (reddish purple), and $0.1$ (yellow). The fit is better for larger values of $\Delta$. The dashed lines are not plotted for $\Delta \leq 0.4$, where the fit is poor. The horizontal axis is normalized by $k_\text{med}\equiv \sqrt{k_\text{min}k_\text{max}}$.
• Figure 5: Dependence of the induced $\Omega_\text{GW}$ on the width of the top-hat $\mathcal{P}_\zeta (k)$ in an MD era. The normalizations are different between the left and right panels ($\widetilde{A} = A/(2\Delta)$). The blue, orange, bluish green, vermilion, and sky blue lines correspond to $\Delta = 10^{-4}$, $10^{-3}$, $10^{-2}$, $10^{-1}$, and $1$, respectively. The dashed reddish-purple line on the left panel is the limit $k_\text{min} \to 0$. The dashed black line on the right panel is the delta-function limit $k_\text{min}/k_\text{max} \to 1$.