Probability for Primordial Black Holes in Higher Derivative Theories

Abstract

The probability for quantum creation of an inflationary universe with a pair of black holes in higher derivative theories has been studied. Considering a gravitational action which includes quadratic ($αR^{2}$) and/or cubic term ($βR^{3}$) in scalar curvature in addition to a cosmological constant ($Λ$) in semiclassical approximation with Hartle-Hawking boundary condition, the probability has been evaluated. The action of the instanton responsible for creating such a universe, with spatial section with $S^{1}XS^{2}$ topology, is found to be less than that with a spatial $S^{3}$ topology, unless $α< - \frac{1}{8 Λ}$ in $R^{2}$-theory. In the $R^{3}$ theory, however, there exists a set of solutions without a cosmological constant when $βR^{2} = 1$ and $α= - 3 \sqrtβ$ which admit primordial black holes (PBH) pair in an inflationary universe scenario. We note further that when $βR^{2} \neq 1$, one gets PBH pairs in the two cases : (i) with $α$ and $Λ$ both positive and (ii) with $Λ$ positive and $α$ negative satisfying a constraint $6 | α| Λ> 1$. However, the relative probability for creation of an inflationary universe with a pair of black holes in the $R^{3}$-theory suppresses when $α> - 2 \sqrtβ $ or $|α| < 2 \sqrtβ $. However, if the above constraints are relaxed one derives interesting results leading to a universe with PBH in $R^{3}$-theory without cosmological constant. PACS No(s). : 04.20.Jb, 04.60.+n, 98.80.Hw