Gravitational waves induced by matter isocurvature in general cosmologies

TL;DR

This work addresses how matter isocurvature perturbations induce gravitational waves in a general two-fluid early-Universe background characterized by a constant equation of state $w$. By deriving an analytic GW kernel $I(x,k,u,v)$ from a Green’s-function solution and splitting the time integrals into superhorizon and subhorizon parts, the authors obtain a closed-form treatment of isocurvature-induced GWs, including a Dirac-delta isocurvature spectrum that yields an analytic $\Omega_{\rm GW}(k)$. The key findings are that the induced GW spectrum depends crucially on $b=(1-3w)/(1+3w)$, with a peak or dip at $k/k_p=2c_s$ ($c_s=\sqrt{w}$) and enhanced amplitudes for soft EoS ($b>0$); the low-frequency tail matches the universal adiabatic scaling. These results provide a benchmark for probing the pre-BBN expansion history, including early matter domination, and have potential applications to scenarios with PBH or soliton domination that generate isocurvature fluctuations. Overall, the work offers a theoretical framework and analytic tools to connect small-scale isocurvature dynamics to observable stochastic gravitational waves, enabling tests of reheating physics and exotic early-Universe phases.

Abstract

The expansion history and content of the Universe between the end of inflation and the onset of Big Bang Nucleosynthesis is mostly unknown. In this paper, we study gravitational waves (GWs) induced by matter isocurvature fluctuations in a generic perfect fluid background as a novel probe of the physics of the very early Universe. We analytically compute the induced GW kernel and analyze the spectral GW energy density for a sharply peaked isocurvature power spectrum. We show that the spectral shape of the GW signal is sensitive to the equation of state parameter $w$ of the perfect fluid dominating the early Universe after inflation. We find that the GW amplitude is enhanced for a soft equation of state. Our framework can be applied to dark matter isocurvature and models leading to early matter-dominated eras, such as primordial black holes and cosmological solitons.

Figures (6)

• Figure 1: The evolution of the gravitational potential $\Phi_{\rm iso}(k\tau)$ with isocurvature initial conditions, normalized by $\kappa^{b-1}$. We show some examples of soft and stiff equations of state ($b=-\frac{1}{3}, \, \frac{1}{3}, \, \frac{3}{5}$, corresponding to $w=\frac{2}{3}, \, \frac{1}{6}, \, \frac{1}{12}$), with fixed $\kappa = 10^5$. Solid lines represent the approximate solution \ref{['eq:Phi_Iso_GD']}, whereas the dotted line, which lies almost perfectly on top of the solid one, shows the exact analytical solution of \ref{['eq:Phi_iso_Green_sol']}. In \ref{['fig:Phi_iso_2']} in \ref{['app:einsteinequations']}, we also include the result of numerical integration.
• Figure 2: Here we show the effect of varying the equation of state $w$, encoded by the parameter $b$ defined in \ref{['eq:bDefinition']}, on the isocurvature-induced GW spectrum. Note how the peak amplitude is enhanced for soft EoS ($b>0$), and how the resonant scale located at $k/k_p = 2 c_s$ shifts with the varying speed of sound, $c_s=\sqrt{w}$. We show the normalized spectrum $\Omega_{\rm GW}^*$, cf. \ref{['eq:Omega_GW_norm']}.
• Figure 3: Comparison of the subhorizon and superhorizon contributions to the GW spectrum. We show the normalised spectrum $\Omega_{\rm GW}^*$. For numerical convenience, we evaluated the spectrum for $b\approx 0$ at $b=5\times 10^{-3}$.
• Figure 4: Comparison of the contributions of the $\Phi\Phi$-term stemming from $G_{ij}$ and the $(\Phi+\Phi'/\mathcal{H})^2$-term stemming from $T_{ij}$. The vertical gray dashed line marks $k/k_p=c_s=\sqrt{w}$ for the respective values $w=1/6$ and $w=2/3$, and the dashdotted line in \ref{['subfig:OmegaGW_G_vs_T_1by3']} marks $k/k_p=2 c_s$.
• Figure 5: \ref{['subfig:Phi_1']}: The evolution of the gravitational potential $\Phi_{\rm iso}(k\tau)$ for soft and stiff equation of state with $\kappa = 10^5$. Solid lines represent the approximate analytical solution \ref{['eq:Phi_Iso_GD']}, dotted lines show the exact analytical solution \ref{['eq:Phi_iso_Green_sol']}, and the dashed colored lines show a numerical solution of the full coupled system \ref{['eq:PhiEq', 'eq:SEq']}. The figure illustrates how the analytical approximation captures the exact numerical solution remarkably well for $\tau \ll \tau_{w \rm eq}$, with $k \tau_{w \rm eq}$ marked by the gray dotted, dash-dotted and dashed vertical lines for $b=-\frac{1}{3}, \, \frac{1}{3}$ and $\frac{3}{5}$, respectively. See how for soft EoS the approximation breaks down earlier, whereas for stiff EoS it is valid almost up to $\tau_{w \rm eq}$. \ref{['subfig:Phi_2']}: Here we show $| \Phi_{\rm iso}(k\tau)|$ for a very soft EoS with $b=0.8$ and $\kappa = 10^8$. We compare the analytical approximation \ref{['eq:Phi_Iso_GD']} with numerical solutions for isocurvature and adiabatic initial conditions. See how the approximate solution tracks the isocurvature solution before horizon entry, but starts behaving similarly to the adiabatic one with zero-crossing oscillations for $x\gtrsim 1$ (note that we are plotting the absolute value).