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. arXiv:1804.08577v2 [gr-qc] 29 Jun 20181 Apart from the induced second-order GWs, which are the topic of this paper, there are other mechanisms of GW production in the early Universe, including those associated with preheating [9-12], phase transitions [13][14][15], and topological defects such as cosmic strings [16,17]. In particular, it should be noted that the GWs are also produced from the primordial curvature perturbations in the subhorizon when shocks are formed [18]. It was reported in Ref.[18] that the resultant GW power spectrum is similar to that of the induced GWs we are considering, but the frequency is lowered by P 1/2 ζ , which would affect the constraints on PBH scenarios which aim to explain the merger rate of binary black holes of around 30 solar masses. For light PBH scenarios, GWs emitted by Hawking radiation are relevant [19][20][21].

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 matterdominated 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 TR is in the range TR 7 × 10 −2 GeV or 20 GeV TR 2 × 10 7 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.

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For a variety of on-going and planned gravitational-wave (GW) experiments, we study expected constraints on the fraction (fPBH) 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 the binary PBHs. The second one is induced through non-linear mode couplings of large primordial curvature perturbations inevitably associated with the generation of PBHs in the early Universe. In this paper, we focus on the PBHs with their masses of 10 −8 M ≤ MPBH < 1M , since they are not expected to be of a stellar origin. In almost all ranges of the masses, we show that the experiments are sensitive to constrain the fraction for 10 −5 fPBH 1 by considering the GWs from coalescing events and 10 −13 fPBH 1 by considering the GWs from curvature perturbations. Exceptionally, only in a narrow range of masses for MPBH 10 −7 M , the fraction cannot be constrained for fPBH 10 −13 by those two GW backgrounds.

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