Gravitational Waves Induced by Non-Gaussian Scalar Perturbations

We study gravitational waves (GWs) induced by non-Gaussian curvature perturbations. We calculate the density parameter per logarithmic frequency interval, ${\mathrm{\Omega }}_{\mathrm{GW}}(k)$, given that the power spectrum of the curvature perturbation ${\mathcal{P}}_{\mathcal{R}}(k)$ has a narrow peak at some small scale $k_{*}$, with a local-type non-Gaussianity, and constrain the nonlinear parameter $f_{\mathrm{NL}}$ with the future LISA sensitivity curve as well as with constraints from the abundance of the primordial black holes (PBHs). We find that the non-Gaussian contribution to ${\mathrm{\Omega }}_{\mathrm{GW}}$ increases as $k^3$, peaks at $k/k_{*}=4/\sqrt{3}$, and has a sharp cutoff at $k=4k_{*}$. The non-Gaussian part can exceed the Gaussian part if ${\mathcal{P}}_{\mathcal{R}}(k)f_{\mathrm{NL}}^2\gtrsim 1$. If both a slope ${\mathrm{\Omega }}_{\mathrm{GW}}(k)\propto k^{\beta }$ with $\beta \sim 3$ and the multiple-peak structure around a cutoff are observed, it can be recognized as a smoking gun of the primordial non-Gaussianity. We also find that if PBHs with masses of $10^{20}$ to $10^{22}\mathrm{g}$ are identified as cold dark matter of the Universe, the corresponding GWs must be detectable by LISA-like detectors, irrespective of the value of ${\mathcal{P}}_{\mathcal{R}}$ or $f_{\mathrm{NL}}$.

Figure 1

Typical gravitational wave density parameter induced by a non-Gaussian curvature perturbation at second order. The width of the peak is fixed at $\sigma ={10}^{-4}\mathrm{Hz}$. In the upper panel, $F_{\mathrm{NL}}$ is positive, where the abundance of the PBHs is fixed to be $f_{\mathrm{PBH}}=1$ for $M_{\mathrm{PBH}}={10}^{22}\mathrm{g}$. In the lower panel, $F_{\mathrm{NL}}$ is negative, where the amplitude of the peak is fixed to be ${\mathcal{A}}_{\mathcal{R}}={10}^{-3}$. In both cases, we have drawn $|F_{\mathrm{NL}}|=0$ (orange dashed), 10 (red), 20 (blue), and 50 (purple). The gray curve is the sensitivity bound of LISA from Ref. [132]. A reference line of the $k^3$ slope is also drawn for comparison.

Figure 2

The primordial black hole mass fraction at formation $\beta$ depicted as a function of $F_{\mathrm{NL}}$ and $F_{\mathrm{NL}}^2{\mathcal{A}}_{\mathcal{R}}$, for the positive $F_{\mathrm{NL}}$ (up) and the negative $F_{\mathrm{NL}}$ (down), respectively. The constant $\beta$ contours are drawn, where the upper bound given by $\beta <\{8.6\times {10}^{-16},8.6\times {10}^{-15}\}$ for the PBHs corresponding to PBH masses at $M_{\mathrm{PBH}}=\{{10}^{20},{10}^{22}\mathrm{g}\}$ can be seen as the border of the white and colored areas. The dashed lines are for ${\mathcal{A}}_{\mathcal{R}}={10}^{-2}$, ${10}^{-3}$, and ${10}^{-4}$ from left to right, while the shaded area is unphysical since ${\mathcal{A}}_{\mathcal{R}}>1$. The thick black curve is the absolute constraint that the GW energy density be smaller than the current density of radiation, while the red and blue curves are the sensitivity bound of LISA at $f_{\mathrm{GW}}=3\times {10}^{-2}$ and $3\times {10}^{-3}\mathrm{Hz}$, respectively; they correspond to PBH masses $M_{\mathrm{PBH}}={10}^{20}$ and $10^{22}\mathrm{g}$.