Whether or not the primordial gravitational wave (GW) produced during inflation is sufficiently strong to be observable, GWs are necessarily produced from the primordial curvature perturbations in the second order of perturbation. The induced GWs can be enhanced by curvature perturbations enhanced at small scales or by the presence of matter-dominated stages of the cosmological history. We analytically calculate the integral in the expression of the power spectrum of the induced GWs, which is a universal part independent of the spectrum of the primordial curvature perturbations. This makes the subsequent numerical integrals significantly easy. In simple cases, we derive fully analytic formulas for the induced GW spectrum.

We discuss the footprint of evaporation of primordial black holes (PBHs) on stochastic gravitational waves (GWs) induced by scalar perturbations. We consider the case where PBHs once dominated the Universe but eventually evaporated before the big bang nucleosynthesis. The reheating through the PBH evaporation could end with a sudden change in the equation of state of the Universe compared to the conventional reheating caused by particle decay. We show that this “sudden reheating” by the PBH evaporation enhances the induced GWs, whose amount depends on the length of the PBH-dominated era and the width of the PBH mass function. We explore the possibility to constrain the primordial abundance of the evaporating PBHs by observing the induced GWs. We find that the abundance parameter $\beta \gtrsim {10}^{-5}-{10}^{-8}$ for $\mathrm{????}({10}^3-10^5)\mathrm{g}$ PBHs can be constrained by future GW observations if the width of the mass function is smaller than about a hundredth of the mass.

For a variety of ongoing and planned gravitational-wave (GW) experiments, we study expected constraints on the fraction ($f_{\mathrm{PBH}}$) of primordial black holes (PBHs) in dark matter by evaluating the energy-density spectra of two kinds of stochastic GW backgrounds. The first one is produced from an incoherent superposition of GWs emitted from coalescences of all of the binary PBHs. The second one is induced through nonlinear mode couplings of large primordial curvature perturbations inevitably associated with the generation of PBHs in the early Universe. In this paper, we focus on PBHs with masses ${10}^{-8}{M}_{\odot }\le M_{\mathrm{PBH}}<1M_{\odot }$, since they are not expected to be of stellar origin. In almost all mass ranges, we show that the experiments are sensitive enough to constrain the fraction for ${10}^{-5}\lesssim f_{\mathrm{PBH}}\lesssim 1$ by considering the GWs from coalescing events and ${10}^{-13}\lesssim f_{\mathrm{PBH}}\lesssim 1$ by considering the GWs from curvature perturbations. Exceptionally, the fraction cannot be constrained for $f_{\mathrm{PBH}}\lesssim {10}^{-3}$ by these two GW backgrounds only in the narrow mass range around $M_{\mathrm{PBH}}\simeq {10}^{-7}{M}_{\odot }$.

We study gravitational waves induced from the primordial scalar perturbations at second order around the reheating of the Universe. We consider reheating scenarios in which a transition from an early matter-dominated era to the radiation-dominated era completes within a timescale much shorter than the Hubble time at that time. We find that an enhanced production of induced gravitational waves occurs just after the reheating transition because of fast oscillations of scalar modes well inside the Hubble horizon. This enhancement mechanism just after an early matter-dominated era is much more efficient than a previously known enhancement mechanism during an early matter era, and we show that the induced gravitational waves could be detectable by future observations if the reheating temperature $T_{\mathrm{R}}$ is in the range $T_{\mathrm{R}}\lesssim 7\times {10}^{-2}\mathrm{GeV}$ or $20\mathrm{GeV}\lesssim T_{\mathrm{R}}\lesssim 2\times {10}^7\mathrm{GeV}$. This is the case even if the scalar perturbations on small scales are not enhanced relative to those on large scales, probed by the observations of the cosmic microwave background.