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
$\Omega_\text{GW}$ 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.
