Gravitational Waves Induced by Scalar Perturbations with a Lognormal Peak

Abstract

We study the stochastic gravitational wave (GW) background induced by the primordial scalar perturbation with the spectrum having a lognormal peak of width $\Delta$ at $k=k_{\ast }$. We derive an analytical formula for the GW spectrum ${\Omega }_{\text{GW}}$ for both narrow ($\Delta \ll 1$) and broad ($\Delta \gtrsim 1$) peaks. In the narrow-peak case, the spectrum has a double peak feature with the sharper peak at $k=2k_{\ast }/\sqrt{3}$. On the infrared (IR) side of the spectrum, we find power-law behavior with a break at $k=k_b$ in the power-law index where it chages from $k^3$ on the far IR side to $k^2$ on the near IR side. We find the ratio of the break frequency to the peak frequency is determined by $\Delta$ as $f_b/f_p\approx \sqrt{3}\Delta$, where $f_b$ and $f_p$ are the break and peak frequencies, respectively. In the broad-peak case, we find the GW spectrum also has a lognormal peak at $k=k_{\ast }$ but with a smaller width of $\Delta /\sqrt{2}$. Using these derived analytic formulae, we also present expressions for the maximum values of ${\Omega }_{\text{GW}}$ for both narrow and broad cases. Our results will provide a useful tool in searching for the induced GW signals in the coming decades.

15 pages, 3 figures. Some figures redrawn. References added and reorganized. Accepted for JCAP

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