Based on a physical argument, we derive a new analytic formula for the amplitude of density perturbation at the threshold of primordial black hole formation in the Universe dominated by a perfect fluid with the equation of state $p=w\rho c^2$ for $w\ge 0$. The formula gives ${\delta }_{Hc}^{\mathrm{UH}}={sin}^2[\pi \sqrt{w}/(1+3w)]$ and ${\tilde{\delta }}_c=[3(1+w)/(5+3w)]{sin}^2[\pi \sqrt{w}/(1+3w)]$, where ${\delta }_{Hc}^{\mathrm{UH}}$ and ${\tilde{\delta }}_c$ are the amplitude of the density perturbation at the horizon crossing time in the uniform Hubble slice and the amplitude measure used in numerical simulations, respectively, while the conventional one gives ${\delta }_{Hc}^{\mathrm{UH}}=w$ and ${\tilde{\delta }}_c=3w(1+w)/(5+3w)$. Our formula shows a much better agreement with the result of recent numerical simulations both qualitatively and quantitatively than the conventional formula. For a radiation fluid, our formula gives ${\delta }_{Hc}^{\mathrm{UH}}={sin}^2(\sqrt{3}\pi /6)\simeq 0.6203$ and ${\tilde{\delta }}_c=(2/3){sin}^2(\sqrt{3}\pi /6)\simeq 0.4135$. We also discuss the maximum amplitude and the cosmological implications of the present result.

We construct cosmological long-wavelength solutions without symmetry in general gauge conditions which are compatible with the long-wavelength scheme. We then specify the relationship among the solutions in different time slicings. Nonspherical long-wavelength solutions are particularly important for primordial structure formation in the epoch of very soft equations of state. Applying this general framework to spherical symmetry, we show the equivalence between long-wavelength solutions in the constant mean curvature slicing with conformally flat spatial coordinates and asymptotic quasihomogeneous solutions in the comoving slicing with the comoving threading. We derive the correspondence relation between these two solutions and compare the results of numerical simulations of primordial black hole (PBH) formation in these two different approaches. To discuss the PBH formation, it is convenient and conventional to use ${\tilde{\delta }}_c$, the value which the averaged density perturbation at threshold in the comoving slicing would take at horizon entry in the lowest-order long-wavelength expansion. We numerically find that within (approximately) compensated models, the sharper the transition from the overdense region to the Friedmann-Robertson-Walker universe is, the larger the ${\tilde{\delta }}_c$ becomes. We suggest that, for the equation of state $p=(\mathrm{\Gamma }-1)\rho$, we can apply the analytic formulas for the minimum ${\tilde{\delta }}_{c,\min }\simeq [3\mathrm{\Gamma }/(3\mathrm{\Gamma }+2)]{\sin }^2[\pi \sqrt{\mathrm{\Gamma }-1}/(3\mathrm{\Gamma }-2)]$ and the maximum ${\tilde{\delta }}_{c,\max }\simeq 3\mathrm{\Gamma }/(3\mathrm{\Gamma }+2)$. As for the threshold peak value of the curvature variable ${\psi }_{0,c}$, we find that the sharper the transition is, the smaller the ${\psi }_{0,c}$ becomes. We analytically explain this intriguing feature qualitatively with a compensated top-hat density model. Using simplified models, we also analytically deduce an environmental effect that ${\psi }_{0,c}$ can be significantly larger (smaller) if the underlying perturbation of much longer wavelength is positive (negative).

For the past decade, gravitational lensing in the strong gravitational field has been studied eagerly. It is well known that, for the lensing by a black hole, an infinite number of Einstein rings are formed by the light rays which wind around the black hole nearly on the photon sphere, which are called relativistic Einstein rings. This is also the case for the lensing by a wormhole. In this paper, we study the Einstein ring and relativistic Einstein rings for the Schwarzschild black hole and the Ellis wormhole, the latter of which is an example of traversable wormholes of the Morris-Thorne class. Given the configuration of the gravitational lensing and the radii of the Einstein ring and relativistic Einstein rings, we can distinguish between a black hole and a wormhole in principle. We conclude that we can detect the relativistic Einstein rings by wormholes which have the radii of the throat a$\simeq${}0.5\,\,pc at a Galactic center with the distance 10~Mpc and which have a$\simeq${}10\,\,AU in our Galaxy using the most powerful modern instruments which have the resolution of 10-2\,\,arcsecond such as a 10-meter optical-infrared telescope. The black holes which make the Einstein rings of the same size as the ones by the wormholes are galactic supermassive black holes and the relativistic Einstein rings by the black holes are too small to measure with the current technology. We may test the hypotheses of astrophysical wormholes by using the Einstein ring and relativistic Einstein rings in the future.

Angular momentum plays very important roles in the formation of primordial black holes in the matter-dominated phase of the Universe if it lasts sufficiently long. In fact, most collapsing masses are bounced back due to centrifugal force, since angular momentum significantly grows before collapse. For masses with $q\le q_c\simeq 2.4{\mathcal{I}}^{1/3}{\sigma }_H^{1/3}$, where $q$ is a nondimensional initial quadrupole moment parameter, ${\sigma }_H$ is the density fluctuation at horizon entry $t=t_H$, and $\mathcal{I}$ is a parameter of the order of unity, angular momentum gives a suppression factor $\sim \exp (-0.15{\mathcal{I}}^{4/3}{\sigma }_H^{-2/3})$ to the production rate. As for masses with $q>q_c$, the suppression factor is even stronger as $\sim \exp (-0.0046q^4/{\sigma }_H^2)$. We derive the spin distribution of primordial black holes and find that most of the primordial black holes are rapidly rotating near the extreme value $a_{*}=1$, where $a_{*}$ is the nondimensional Kerr parameter at their formation. The smaller ${\sigma }_H$ is, the stronger the tendency towards the extreme rotation. Combining this result with the effect of anisotropy, we numerically and semianalytically estimate the production rate ${\beta }_0$ of primordial black holes. Then we find that ${\beta }_0\simeq 1.9\times {10}^{-6}{f}_q(q_c){\mathcal{I}}^6{\sigma }_H^2\exp (-0.15{\mathcal{I}}^{4/3}{\sigma }_H^{-2/3})$ for ${\sigma }_H\lesssim 0.005$, while ${\beta }_0\simeq 0.05556{\sigma }_H^5$ for $0.005\lesssim {\sigma }_H\lesssim 0.2$, where $f_q(q_c)$ is the fraction of masses whose $q$ is smaller than $q_c$ and we assume $f_q(q_c)$ is not too small. We argue that matter domination significantly enhances the production of primordial black holes despite the suppression factor. If the end time $t_{\mathrm{end}}$ of the matter-dominated phase satisfies $t_{\mathrm{end}}\lesssim (0.4\mathcal{I}{\sigma }_H{)}^{-1}{t}_H$, the effect of the finite duration significantly suppresses primordial black hole formation and weakens the tendency towards large spins.