This article reviews current understanding of primordial black holes (PBHs), with particular focus on those massive examples ( ≳ 10 15 g ) which remain at the present epoch, not having evaporated through Hawking radiation. With the detection of gravitational waves by LIGO, we have gained a completely novel observational tool to search for PBHs, complementary to those using electromagnetic waves. Taking the perspective that gravitational-wave astronomy will make significant progress in the coming decades, the purpose of this article is to give a comprehensive review covering a wide range of topics on PBHs. After discussing PBH formation, as well as several inflation models leading to PBH production, we summarize various existing and future observational constraints. We then present topics on formation of PBH binaries, gravitational waves from PBH binaries, and various observational tests of PBHs using gravitational waves.

We carefully investigate the gravitational equations of the brane world, in which all the matter forces except gravity are confined on the 3-brane in a 5-dimensional spacetime with $Z_2$ symmetry. We derive the effective gravitational equations on the brane, which reduce to the conventional Einstein equations in the low energy limit. From our general argument we conclude that the first Randall-Sundrum-type theory predicts that the brane with a negative tension is an antigravity world and hence should be excluded from the physical point of view. Their second-type theory where the brane has a positive tension provides the correct signature of gravity. In this latter case, if the bulk spacetime is exactly anti–de Sitter spacetime, generically the matter on the brane is required to be spatially homogeneous because of the Bianchi identities. By allowing deviations from anti–de Sitter spacetime in the bulk, the situation will be relaxed and the Bianchi identities give just the relation between the Weyl tensor and the energy momentum tensor. In the present brane world scenario, the effective Einstein equations cease to be valid during an era when the cosmological constant on the brane is not well defined, such as in the case of the matter dominated by the potential energy of the scalar field.

The standard calculation of the spectrum of density perturbations produced during inflation assumes that there is only one real dynamical degree of freedom during inflation. However, there is no reason to believe that this is actually the case. In this paper we derive general analytic formulae for the spectrum and spectral index of the density perturbations produced during inflation.Comment: 10 pages, more explanation and references added, version to be published in Progress of Theoretical Physic

Without invoking a perturbative expansion, we define the cosmological curvature perturbation, and consider its behaviour assuming that the universe is smooth over a sufficiently large comoving scale. The equations are simple, resembling closely the first-order equations, and they lead to results which generalize those already proven in linear perturbation theory and (in part) in second-order perturbation theory. In particular, the curvature perturbation is conserved provided that the pressure is a unique function of the energy density.