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 longwavelength 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 useδc, the value which the averaged density perturbation at threshold in the comoving slicing would take at horizon entry in the lowestorder 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 theδc becomes. We suggest that, for the equation of state p = (Γ − 1)ρ, we can apply the analytic formulas for the minimumδc,min ≃ [3Γ/(3Γ+2)] sin 2 π √ Γ − 1/(3Γ − 2) and the maximumδc,max ≃ 3Γ/(3Γ + 2). As for the threshold peak value of the curvature variable ψ0,c, we find that the sharper the transition is, the smaller the ψ0,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 ψ0,c can be significantly larger (smaller) if the underlying density perturbation of much longer wavelength is positive (negative).

The standard deviation of the initial values of the nondimensional Kerr parameter a * of primordial black holes (PBHs) that formed in the radiation-dominated phase of the universe is estimated to the first order of perturbation for the narrow power spectrum. Evaluating the angular momentum at turnaround based on linearly extrapolated transfer functions and peak theory, we obtain the expression 〈 a * 2 〉 ≃ 4.0 × 10 − 3 M / M H − 1 / 3 1 − γ 2 1 − 0.072 log 10 ( β 0 ( M H ) / ( 1.3 × 10 − 15 ) ) − 1 , where M H , β 0(M H ), and γ are the mass within the Hubble horizon at the horizon entry of the overdense region, the fraction of the universe which collapsed to PBHs at the scale of M H , and a quantity that characterizes the width of the power spectrum, respectively. This implies that for M ≃ M H , the higher the probability of the PBH formation, the larger the standard deviation of the spins, while PBHs of M ≪ M H that formed through near-critical collapse may have larger spins than those of M ≃ M H . In comparison to the previous estimate, the new estimate has an explicit dependence on the ratio M/M H and no direct dependence on the current dark matter density. On the other hand, it suggests that the first-order effect can be numerically comparable to the second-order one.

Stability of a photon sphere, or stability of circular null geodesics on the sphere, plays a key role in its applications to astrophysics. For instance, an unstable photon sphere is responsible for determining the size of a black hole shadow, while a stable photon sphere is inferred to cause the instability of spacetime due to the trapping of gravitational waves on the radius. A photon surface is a geometrical structure first introduced by Claudel, Virbhadra and Ellis as the generalization of a photon sphere. The surface does not require any symmetry of spacetime and has its second fundamental form of pure trace. In this paper, we define the stability of null geodesics on a photon surface. It represents whether null geodesics perturbed from the photon surface are attracted to or repelled from the photon surface. Then, we define a strictly (un)stable photon surface as a photon surface on which all null geodesics are (un)stable. We find that the stability is determined by Riemann curvature. Furthermore, it is characterized by the normal derivative of the second fundamental form. As a consequence, for example, a strictly unstable photon surface requires nonvanishing Weyl curvature on it if the null energy condition is satisfied.

Sonic point/photon sphere (SP/PS) correspondence is a theoretical phenomenon which appears in fluid dynamics on curved spacetime and its existence has been recently proved in quite wide situations as theorems. The theorems state that a sonic point (SP) of radiation fluid flow must be on an unstable photon sphere (PS) when the fluid flows radially or rotationally on an equatorial plane in spherically symmetric spacetime of arbitrary dimensions. In this paper, we investigate SP/PS correspondence in spherically, planar and hyperbolically symmetric spacetime. As the corresponding objects of photon spheres in nonspherically symmetric spacetime, we consider photon surfaces introduced by Claudel et al. (2001) in the spacetime. After formulating the problem of radial fluid flows, we prove there always exists a correspondence between the sonic points and the photon surfaces, namely, SP/PS correspondence in nonspherically symmetric spacetime.

We give the formulation and the general analysis of the rotational accretion problem on Ddimensional spherical spacetime and investigate sonic points and critical points. First, we construct the simple two-dimensional rotating accretion flow model in general D-dimensional static spherically symmetric spacetime and formulate the problem. The flow forms a two-dimensional disk lying on the equatorial plane and the disk is assumed to be geometrically thin and has uniform distribution in the polar angle directions. Analyzing the critical point of the problem, we give the conditions for the critical point and its classification explicitly and show the coincidence with the sonic point for generic equation of state (EOS). Next, adopting the EOS of ideal photon gas to the analysis, we reveal that there always exists a correspondence between the sonic points and the photon spheres of the spacetime. Our main result is that the sonic point of the rotating accretion flow of ideal photon gas must be on (one of) the unstable photon sphere(s) of the spacetime in arbitrary spacetime dimensions. This paper extends this correspondence for spherical flows shown in the authors' previous work to rotating accretion disks.