Semianalytic Calculation of Gravitational Wave Spectrum Nonlinearly Induced from Primordial Curvature Perturbations

TL;DR

The paper addresses the problem of computing the second-order gravitational-wave spectrum sourced by primordial curvature perturbations across radiation- and matter-dominated eras. It introduces a semianalytic approach that isolates a universal time integral $I(v,u,x)$, allowing oscillation averaging to yield analytic or semi-analytic expressions for the induced GW power spectrum $\\mathcal{P}_h(\\eta,k)$ for simple spectra $\\mathcal{P}_\zeta(k)$. The authors provide explicit RD and MD results, including fully analytic formulas for monochromatic, scale-invariant, and power-law curvature spectra, and analyze transitions between RD and MD eras, deriving suppression and transfer effects at reheating and equality. These results reduce computational cost, clarify the physical origin of features in the GW spectrum, and aid in constraining inflation models and primordial black-hole scenarios via GW observations and associated BBN constraints.

Abstract

Whether or not the primordial gravitational wave (GW) produced during inflation is sufficiently strong to be observable, GWs are necessarily produced from the primordial curvature perturbations in the second order of perturbation. The induced GWs can be enhanced by curvature perturbations enhanced at small scales or by the presence of matter-dominated stages of the cosmological history. We analytically calculate the integral in the expression of the power spectrum of the induced GWs which is a universal part independent of the spectrum of the primordial curvature perturbations. This makes the subsequent numerical integrals significantly easy. In simple cases, we derive fully analytic formulas for the induced GW spectrum.

Figures (4)

• Figure 1: The energy density fraction $\Omega_{\text{GW}}$ of GWs produced in the RD era, eq. \ref{['Omega_GW_RD_delta']}, from the monochromatic source, eq. \ref{['P_zeta_delta']}.
• Figure 2: The energy density fraction $\Omega_{\text{GW}}$ of GWs produced in the MD era from the scale-invariant source. The brown lines (vanishing at $k=2 k_{\text{max}}$) show the case of an abrupt cutoff $k_{\text{max}}$\ref{['P_zeta_SI_cutoff']}, which may be interpreted as the scale corresponding to the beginning of inflaton oscillation or the scale where density perturbations become nonlinear. The dashed curve represents eq. \ref{['Omega_GW_MD_SI']} where reheating is not considered (the pure MD case). The effect of reheating (transition to the RD era) is included in the solid line by multiplying eq. \ref{['reheating_correction']} for the case $k_{\text{R}}= 10^{-3} k_{\text{max}}$ . These lines overlap for $k \gg k_{\text{R}}$. The above plot shows the spectra observed at late time ($\eta \gg \eta_{\text{R}}$) scaled back in time by the common redshift factor (independent of $k$) in such a way that they coincide with the spectra at $\mathcal{H}= k_{\text{R}}$ for modes $k \gg k_{\text{R}}$. In other words, we have taken into account the evolution of modes $k \ll k_{\text{R}}$ since reheating to their horizon entry. The green line shows the case of a MD era preceded by a RD era. $k_{\text{max}}$ is identified as $k_{\text{eq}}$. The standard radiation-matter transition corresponds to this case with $k_{\text{eq}}=1.0\times 10^{-2} \, \text{Mpc}^{-1}$Ade:2015xua. This line is obtained numerically using the interpolating transfer function \ref{['Phi_interpolation']}. Note that $\Omega_{\text{GW}}$ grows during the MD era. The above plot shows the spectra at $\mathcal{H}= 10^{-3} k_{\text{max}}$ (see footnote \ref{['fn:nonlinearity']}). The pink line (horizontal dotted dashed) is the standard in the RD era [see eq. \ref{['Omega_GW_RD_SI']}] shown for comparison.
• Figure 3: Simple examples of the energy density fraction $\Omega_{\text{GW}} h^2$ of the induced GWs. The brown lines show the case of the scale-invariant curvature perturbations with $A_\zeta = 2.2 \times 10^{-9}$. The horizontal part is the contribution from the RD era. The bottom left curves show the effect of the late-time MD era. The dashed line is in the nonlinear regime, and the solid line is a conservative one neglecting all of the contributions beyond the nonlinearity scale. The bottom right curve shows an example of an early MD era with the reheating temperature $T_{\text{R}}= 10^9 \, \text{GeV}$. The scale of the onset of the early MD era is assumed to be 200 times shorter than the reheating scale so that there is no nonlinearity issue. The green line shows the contribution from the RD era in the case of power-law curvature perturbations with $A_\zeta = 10^{-12}$, $n_{\text{s}}=2$, and $k_* = 0.05 \, \text{Mpc}^{-1}$. The blue lines denote existing pulsar timing array constraints from EPTA Lentati:2015qwp, NANOGrav Arzoumanian:2018saf, and PPTA Shannon:2015ect. The pink lines show sensitivity curves Sathyaprakash:2009xs of various future GW observations reproduced from Ref. Moore:2014lga. The observations are from SKA 5136190, eLISA Seoane:2013qna, LISA 2017arXiv170200786A, BBO Harry:2006fi, DECIGO Seto:2001qf, Einstein Telescope Punturo:2010zz, Cosmic Explorer Evans:2016mbw, and KAGRA Somiya:2011np. The gray line (dotted) shows the upper bound on the relativistic degrees of freedom from BBN, $\Omega_{\text{GW}}h^2 < 1.8 \times 10^{-6}$ (95% C.L.) derived in Appendix \ref{['sec:BBN']}.
• Figure 4: \ref{['sfig:chi2Nnu']}$\chi^2$ as a function of $N_{\nu, \rm eff}$ to fit abundances of D and $^4$He, respectively. \ref{['sfig:chi2OmegaGWv2']} Total $\chi^2$ as a function of $\Omega_{\rm GW}h_0^2$ to simultaneously fit both D and $^4$He.