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 Delta at k = k(*). We derive an analytical formula for the GW spectrum Omega(GW) for both narrow (Delta << 1) and broad (Delta greater than or similar to 1) peaks. In the narrow-peak case, the spectrum has a double peak feature with the sharper peak at k = 2k(*)/root 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) approximate to root 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/root 2 Using these derived analytic formulae, we also present expressions for the maximum values of Omega(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.