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).

Time evolution of a black hole lattice toy model universe is simulated. The vacuum Einstein equations in a cubic box with a black hole at the origin are numerically solved with periodic boundary conditions on all pairs of faces opposite to each other. Defining effective scale factors by using the area of a surface and the length of an edge of the cubic box, we compare them with that in the Einstein\char21{}de Sitter universe. It is found that the behavior of the effective scale factors is well approximated by that in the Einstein\char21{}de Sitter universe. In our model, if the box size is sufficiently larger than the horizon radius, local inhomogeneities do not significantly affect the global expansion law of the Universe even though the inhomogeneity is extremely nonlinear.

Time evolution of a black hole lattice universe with a positive cosmological constant $\mathrm{$\Lambda${}}$ is simulated. The vacuum Einstein equations are numerically solved in a cubic box with a black hole in the center. Periodic boundary conditions on all pairs of opposite faces are imposed. Configurations of marginally trapped surfaces are analyzed. We describe the time evolution of not only black hole horizons, but also cosmological horizons. Defining the effective scale factor by using the area of a surface of the cubic box, we compare it with that in the spatially flat dust dominated Friedmann-Lema\^\itre-Robertson-Walker (FLRW) universe with the same value of $\mathrm{$\Lambda${}}$. It is found that the behavior of the effective scale factor is well approximated by that in the FLRW universe. Our result suggests that local inhomogeneities do not significantly affect the global expansion law of the Universe irrespective of the value of $\mathrm{$\Lambda${}}$.

This paper is a biased review of the primordial black hole (PBH) formation and abundance estimation. We first review the three-zone model for PBH formation to help an intuitive understanding of the PBH formation process. Then, for more accurate analyses, we introduce necessary tools such as cosmological long-wavelength solutions, the definition of the mass and compaction function in a spherically symmetric spacetime and peak theory. Combining all these tools, we calculate the PBH mass spectrum for the case of the monochromatic curvature power spectrum as a demonstration.