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This study investigates the formation of primordial black holes (PBHs) resulting from the collapse of adiabatic fluctuations with large amplitudes and non-Gaussianity. Ref. \cite{Uehara:2024yyp} showed that fluctuations with large amplitudes lead to the formation of type B PBHs, characterized by the existence of the bifurcating trapping horizons, distinct from the more common type A PBHs without a bifurcating trapping horizon. We focus on the local type non-Gaussianity characterized by the curvature perturbation $\zeta$ given by a function of a Gaussian random variable $\zeta_{\rm G}$ as $\beta\zeta=-\ln(1-\beta \zeta_{\rm G})$ with a parameter $\beta$. Then we examine how the non-Gaussianity influences the dynamics and the type of PBH formed, particularly focusing on type II fluctuations, where the areal radius varies non-monotonically with the coordinate radius. Our findings indicate that, for $\beta>-2$, the threshold for distinguishing between type A and type B PBHs decreases with increasing $\beta$ similarly to the threshold for black hole formation. Additionally, for large positive values of $\beta$, the threshold for type B PBHs approaches that for type II fluctuations. We also find that, for a sufficiently large negative value of $\beta\lesssim-4.0$, the threshold value is in the type II region of $\mu$, i.e., there are fluctuations of type II that do not form black holes. Lastly, we calculate the PBH mass for several values of $\beta$. Then we observe that the final mass monotonically increases with the initial amplitude within the parameter region of type A PBHs, which differs from previous analytical expectations., Comment: 20pages, 14figures

This study investigates the formation of type II primordial black holes (PBHs) resulting from extremely large amplitudes of initial fluctuations in a radiation-dominated universe. We find that, for a sufficiently large initial amplitude, the configuration of trapping horizons shows characteristic structure due to the existence of bifurcating trapping horizons. We call this type of configuration of the trapping horizons type II-B PBH, while the structure without a bifurcating trapping horizon type II-A PBH. In Ref. [1], in the dust-dominated universe, the type B PBH can be realized by the type II initial fluctuation, which is characterized by a non-monotonic areal radius as a function of the radial coordinate (throat structure) in contrast with the standard case, type A PBH with a monotonic areal radius (type I fluctuation). Our research reveals that a type II fluctuation does not necessarily result in a type B PBH in the radiation-dominated case. We also find that for an initial amplitude well above the threshold value, the resulting PBH mass may either increase or decrease with the initial amplitude, depending on its specific profile rather than its fluctuation type.