Probing small-scale primordial power spectra with induced gravitational waves

TL;DR

This work studies small-scale primordial perturbations through second-order tensor modes induced by both scalar and tensor perturbations (TSIGW). It derives compact, analytic expressions for all three TSIGW energy-density components (scalar-scalar, tensor-scalar, tensor-tensor) and analyzes their observational consequences by combining PTA, LISA, and cosmological PBH/CMB/BAO constraints. The results show that TSIGW can dominate current PTA observations in monochromatic small-scale spectra while remaining consistent with cosmological bounds, offering a new pathway to constrain the amplitude and location of small-scale primordial perturbations and primordial gravitational waves. The study also assesses tensor-induced GWs in a Nieh-Yan modified Teleparallel Gravity framework, finding TIGW mainly affects high-frequency regions and has limited impact on PTA, with Bayes factors favoring TSIGW under realistic prior choices.

Abstract

Large-scale primordial perturbations have been well constrained by current cosmological observations, but the properties of small-scale perturbations remain elusive. This study focuses on second-order induced gravitational waves generated by large-amplitude primordial scalar and tensor perturbations on small scales. In this case, the induced gravitational waves include contributions from three types of source terms: scalar-scalar, tensor-scalar, and tensor-tensor. To distinguish them from second-order scalar induced gravitational waves (SIGWs), we refer to those generated by these three source terms as tensor-scalar induced gravitational waves (TSIGWs). We provide the analytical expressions for the kernel functions and the corresponding energy density spectra of second-order TSIGWs. By combining observations of stochastic gravitational wave background (SGWB) across different scales, TSIGWs can be used to constrain small-scale primordial curvature perturbations and primordial gravitational waves. Furthermore, we discuss the feasibility of TSIGWs dominating the current pulsar timing array (PTA) observations under various primordial power spectra scenarios. Our results indicate that TSIGWs generated by monochromatic primordial power spectra might be more likely to dominate the current PTA observations.

Figures (15)

• Figure 1: In the case of the monochromatic primordial power spectrum, the current energy density spectra of the three components of second-order TSIGW are shown as blue, green, and red curves, respectively. The corresponding energy density spectrum of PGW is represented by the purple curve. The parameters for the curves are $A_\zeta=1$ and $A_h=1$. To facilitate visualization, the LN power spectrum of PGW with $\sigma = 0.01$ is adopted in the figure as a substitute for the monochromatic primordial gravitational wave power spectrum.
• Figure 2: The current energy density spectra of SIGW and PGW$+$TSIGW for monochromatic spectra and LN spectra. The energy density spectra derived from the free spectrum of the NANOGrav 15-year are shown in blue. The blue and green curves represent the energy density spectra of GW with different line styles labeled in the figure. These parameters are selected based on the median values of the posterior distributions, with the median values shown as green numbers in Fig. \ref{['fig:corner_SIGW_mono']}, Fig. \ref{['fig:corner_TSIGW_mono']}, Fig. \ref{['fig:corner_SIGW_logn']} and Fig. \ref{['fig:corner_TSIGW_logn']}.
• Figure 3: The corner plot of the posterior distributions. The contours in the off-diagonal panels denote the $68\%$ and $95 \%$ credible intervals of the 2D posteriors. The numbers above the figures represent the median values and $1$-$\sigma$ ranges of the parameters. The blue and green solid curves correspond to the SIGW energy spectrum with or without SMBHB, assuming the monochromatic primordial power spectrum.
• Figure 4: The corner plot of the posterior distributions. The contours in the off-diagonal panels denote the $68\%$ and $95 \%$ credible intervals of the 2D posteriors. The numbers above the figures represent the median values and $1$-$\sigma$ ranges of the parameters. The blue and green solid curves correspond to the PGW$+$TSIGW energy spectrum with or without SMBHB, assuming the monochromatic primordial power spectrum.
• Figure 5: The SNR of LISA as a function of $f_*$, assuming fixed values $A_\zeta = 0.1$ and $A_h = 0.02$. The blue solid curve and green dashed curve represents the SNR for the SIGW and PGW$+$TSIGW, assuming the monochromatic primordial power spectrum.