Black hole formation from massive scalar field collapse in the Einstein–de Sitter universe

Abstract

We study the spherically symmetric collapse of a real, minimally coupled, massive scalar field in an asymptotically Einstein-de Sitter spacetime background. By means of an eikonal approximation for the field and metric functions, we obtain a simple analytical criterion-involving the physical size and mass scales ͑the field's inverse Compton wavelength and the spacetime gravitational mass͒ of the initial matter configurationfor generic ͑non-time-symmetric͒ initial data to collapse to a black hole. This analytical condition can then be used to place constraints on the initial primordial black hole spectrum, by considering spherical density perturbations that re-entered the horizon during an early matter-dominated phase that immediately followed inflation.

Primordial black holes from the preheating instability in single-field inflation

“…Inserting these expansions into the equations of motion leads, at leading order (namely order m 0 φ for the equations of motion), to the following expressions (restricting ourselves to a sum from j = 1 to j = 2 which, for the leading order, will be fully justified below): [31]. Notice that by time differentiating the last expression of m 0 , leading to zero since we have already shown thatṁ 0 = 0, one demonstrates that the expression obtained before, namely −R 0 3 (1 − k 0 )(2φ 0Ṙ0 + 2φ 0 R 0 + Φ 0 R 0Ṙ 0 /R 0 ) = 0, is identically satisfied and, therefore, does not lead to additional constraints.…”

“…In Ref. [31], it is claimed that one can go to next-to-leading order (namely order m −1 φ for the equations of motion), the solution at this order being given by the following…”

“…In this first appendix, we review (and correct a few typos in the work of) Ref. [31], that studies black hole formation from massive scalar field collapse. Let us consider an inhomogeneous massive scalar field φ(t, r) living in an inhomogeneous but isotropic (spherically symmetric) space time endowed with the metric ds 2 = −dt 2 + e −2Λ(t,r) dr 2 + R 2 (t, r) dθ 2 + sin 2 θdϕ 2 .…”

Primordial black holes from metric preheating: mass fraction in the excursion-set approach

“…As a consequence, an over-density δ R over the length scale R eventually collapses into a PBH, and in Ref. [22] (see also appendices A and B of Ref. [1]), it is shown that this occurs after a time…”

The wave equation in the Einstein and de Sitter spacetime

“…The EdeS model recently became a focus of interest for an increasing number of authors. (See, e.g., [2], [11], [12], [13], [14], [15], [22], [23], [29] and references therein.) We believe that the initial value problem and the explicit representation formulas obtained in the present paper fill the gap in the existing literature on the wave equation in the EdeS spacetime.…”