Probability for Primordial Black Holes in Higher Derivative Theories

TL;DR

This paper evaluates the probability of primordial black hole formation in higher derivative gravity theories, revealing conditions under which such black holes can form during universe creation, with implications for inflationary cosmology.

Contribution

It introduces new solutions in $R^{3}$ theories allowing primordial black hole pairs without a cosmological constant, expanding understanding of universe creation in higher derivative gravity.

Findings

Probability depends on the sign and magnitude of $eta R^{2}$ and $eta R^{3}$ terms.

Existence of primordial black hole pairs without cosmological constant in specific $R^{3}$ solutions.

Constraints on parameters influence the likelihood of black hole pair creation during inflation.

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 ($\alpha R^{2}$) and/or cubic term ($\beta R^{3}$) in scalar curvature in addition to a cosmological constant ($\Lambda$) 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 $\alpha < - \frac{1}{8 \Lambda}$ in $R^{2}$-theory. In the $R^{3}$ theory, however, there exists a set of solutions without a cosmological constant when $\beta R^{2} = 1$ and $\alpha = - 3 \sqrt{\beta}$ which admit primordial black holes (PBH) pair in an inflationary universe scenario. We note further that when $\beta R^{2} \neq 1$, one gets PBH pairs in the two cases : (i) with $\alpha$ and $\Lambda$ both positive and (ii) with $\Lambda$ positive and $\alpha$ negative satisfying a constraint $6 | \alpha | \Lambda > 1$. However, the relative probability for creation of an inflationary universe with a pair of black holes in the $R^{3}$-theory suppresses when $\alpha > - 2 \sqrt{\beta} $ or $|\alpha| < 2 \sqrt{\beta} $. 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