Gravitational Waves Induced by Scalar Perturbations with a Lognormal Peak

TL;DR

This paper derives analytical formulas for the gravitational wave spectrum induced by primordial scalar perturbations with a lognormal peak, covering narrow and broad cases, aiding future GW searches.

Contribution

It provides new analytical expressions for the induced GW spectrum with lognormal peaks, including features like double peaks and power-law behaviors, for both narrow and broad spectra.

Findings

Double peak feature at $k= 2k_*/\sqrt{3}$ in narrow peaks

Power-law behavior with a break at $k_b$ in the IR spectrum

Relation between break and peak frequencies $f_b/f_p\,\approx\sqrt{3}\Delta$

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 $k=k_*$. We derive an analytical formula for the GW spectrum $\Omega_\text{GW}$ for both narrow ($\Delta\ll1$) and broad ($\Delta\gtrsim1$) peaks. In the narrow-peak case, the spectrum has a double peak feature with the sharper peak at $k= 2k_*/\sqrt{3}$. 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 $k=k_*$ but with a smaller width of $\Delta/\sqrt2$. 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.