Cosmological long-wavelength solutions and primordial black hole
  formation
  We construct cosmological long-wavelength solutions without symmetry in
general gauge conditions compatible with the long-wavelength scheme. We then
specify the relationship among the solutions in different time slicings.
Applying this general framework to spherical symmetry, we derive the
correspondence relation between long-wavelength solutions in the constant mean
curvature slicing with conformally flat spatial coordinates and asymptotic
quasihomogeneous solutions in the comoving gauge and compare the numerical
results of 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 compensated models, the sharper the
transition from the overdense region to the FRW universe is, the larger the
$\tilde{\delta}_{c}$ becomes. We suggest that, for the equation of state
$p=(\Gamma-1)\rho$, we can apply the analytic formulas for the minimum
$\tilde{\delta}_{c, {\rm min}}\simeq
[3\Gamma/(3\Gamma+2)]\sin^{2}\left[\pi\sqrt{\Gamma-1}/(3\Gamma-2)\right]$ and
the maximum $\tilde{\delta}_{c, {\rm max}}\simeq 3\Gamma/(3\Gamma+2)$. 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 feature. Using simplified models, we also
analytically deduce an environmental effect that $\psi_{0,c}$ can be
significantly larger (smaller) if the underlying density perturbation of much
longer wavelength is positive (negative).
