Gravitational waves induced by scalar perturbations with a lognormal peak

We study the stochastic gravitational wave (GW) background induced by the primordial scalar perturbation with the spectrum having a lognormal peak of width Δ at k=k_*. We derive an analytical formula for the GW spectrum Ω_GW for both narrow (Δ≪1) and broad (Δ≳ 1) peaks. In the narrow-peak case, the spectrum has a double peak feature with the sharper peak at k= 2k_*/√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 k³ on the far IR side to k² on the near IR side. We find the ratio of the break frequency to the peak frequency is determined by Δ as f_b/f_p≈√3Δ, 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 Δ/√2. Using these derived analytic formulae, we also present expressions for the maximum values of Ω_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.