Isocurvature Induced Gravitational Waves at Pulsar Timing Arrays

TL;DR

This work analyzes primordial isocurvature perturbations as sources of scalar-induced gravitational waves detectable by Pulsar Timing Arrays. By deriving initial conditions with coupled neutrinos and studying four isocurvature types (CDI, BDI, NDI, DRDI), the authors show DRDI produces qualitatively distinct SIGW spectra due to anisotropic stress, while NDI effectively maps onto CDI/BDI up to a rescaling. They employ delta-function, broken power-law, and log-normal isocurvature spectra and constrain them using NANOGrav 15-year data, finding strongest limits around $k\sim10^6\ \mathrm{Mpc}^{-1}$ and highlighting the complementary nature of PTA constraints to large-scale probes. The results demonstrate PTA as a powerful, independent window into beyond-ΛCDM physics and the role of early-universe perturbations in shaping the stochastic gravitational-wave background.

Abstract

Gravitational waves (GWs) are powerful probes of new physics in the early Universe. In particular, GWs induced by primordial isocurvature perturbations encode information of novel dynamics beyond the standard $Λ$CDM model. Existing studies of isocurvature induced GWs focus on a particular type: cold dark matter (CDM) isocurvature. In this work, we present a more comprehensive study of four kinds of isocurvature involving CDM, baryons, neutrinos and free-streaming dark radiation (DR). We first reformulate initial conditions of isocurvature with coupled neutrinos because modes relevant for observations at Pulsar Timing Arrays enter the horizon before neutrino decoupling. With these new initial conditions, neutrino isocurvature is phenomenologically similar to CDM isocurvature up to an overall coefficient, which leads to an interesting conversion of isocurvature between radiation and matter sectors. We then find that the spectrum of isocurvature induced GWs from free-streaming DR is qualitatively different than that from CDM due to the presence of anisotropic stress. Unlike GWs induced by CDM isocurvature that are suppressed at high frequencies due to matter density being suppressed at early times, DR isocurvature induced GWs is proportional to the constant ratio between DR density and total radiation. Finally, we utilize two general parametrizations of the isocurvature power spectrum: a delta function and a broken power law, and derive novel constraints with recent NANOGrav data. Our results set stringent constraints on isocurvature around $10^{6}\,\textrm{Mpc}^{-1}$, which are complementary to cosmological observations at large scales.

Figures (7)

• Figure 1: The absolute value of transfer functions of metric perturbations $|\Phi|$ and $|\Psi|$ in the Newtonian gauge (left) and curvature perturbation $|\zeta|$ (right) as a function of $k\tau$ for AD, CDI and DRDI. $\Phi=\Psi$ for AD and CDI due to no anisotropic stress. $k_{\rm eq}$ is the wavenumber of the mode that enters the horizon at the matter radiation equality, and $R_{\rm dr}\equiv \bar{\rho}_{\rm dr}/\bar{\rho}_{r}$ is the fraction of the DR density to the total radiation density. The scaling of the CDI (DRDI) case is valid for $k\gg k_{\rm eq}$ ($R_{\rm dr}\ll1$).
• Figure 2: The $f(x,u,v)$ function in the source term for induced GWs for AD (blue), CDI (orange) and DRDI (green) with $x = k\tau$. For simplicity we chose $u = v = 1$ but the main behavior is independent of such particular values.
• Figure 3: The spectrum of GWs ($\Omega_{\rm GW}$) induced by AD (blue), CDI (orange) and DRDI (green) perturbations. We consider three choices of power spectrum: a delta function (Eq. \ref{['eq:P_delta']}), a log-normal form with $\Delta=0.1$ (Eq. \ref{['eq:P_lognormal']}) and a broken power law (Eq. \ref{['eq:P_Broken_Powerlaw']}). For each power spectrum, $k_0$ denotes the characteristic scale and we have set the amplitude $A_{\rm ad/iso}=1$. Here we choose $k_0\gg k_{\rm eq}$ for CDI and $R_{\rm dr}\ll1$ for DRDI.
• Figure 4: The best fit and constraint of AD power spectrum from NANOGrav. We utilize two general parametrizations of the isocurvature power spectrum: a delta function (left, Eq. \ref{['eq:P_delta']}) and a broken power law (right, Eq. \ref{['eq:P_Broken_Powerlaw']}). Each spectrum has a characteristic scale $k_0$ and amplitude $A_{\rm ad}$. The constraint is shown in the gray shaded region, while $1\sigma,\ 2\sigma$ CL regions are depicted in progressively lighter shades of blue.
• Figure 5: The best fit and constraint of DRDI and CDI power spectrum from NANOGrav. We utilize two general parametrizations of the isocurvature power spectrum: a delta function (left, Eq. \ref{['eq:P_delta']}) and a broken power law (right, Eq. \ref{['eq:P_Broken_Powerlaw']}). Each spectrum has a characteristic scale $k_0$ and amplitude $A_{\rm iso}$. The constraint is shown in the gray shaded region, while $1\sigma,\ 2\sigma$ CL regions are depicted in progressively lighter shades of blue.