Probability for Primordial Black Holes Pair in 1/R Gravity

Abstract

The probability for quantum creation of an inflationary universe with a pair of black holes in 1/R - gravitational theory has been studied. Considering a gravitational action which includes a cosmological constant ($\Lambda$) in addition to $ \delta R^{- 1} $ term, the probability has been evaluated in a semiclassical approximation with Hartle-Hawking boundary condition. We obtain instanton solutions determined by the parameters $\delta$ and $\Lambda$ satisfying the constraint $ \delta \leq \frac{4 \Lambda^{2}}{3}$. However, we note that two different classes of instanton solutions exists in the region $0 < \delta < \frac{4 \Lambda^{2}}{3}$. The probabilities of creation of such configurations are evaluated. It is found that the probability of creation of a universe with a pair of black holes is strongly suppressed with a positive cosmological constant except in one case when $0 < \delta < \Lambda^{2}$. It is also found that gravitational instanton solution is permitted even with $\Lambda = 0$ but one has to consider $\delta < 0$. However, in the later case a universe with a pair of black holes is less probable.