Probability for a primordial black hole pair in1/Rgravity

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 ($\ensuremath{\Lambda}$) in addition to $\ensuremath{\delta}{R}^{\ensuremath{-}1}$ term, the probability has been evaluated in a semiclassical approximation with Hartle-Hawking boundary condition. We obtain instanton solutions determined by the parameters $\ensuremath{\delta}$ and $\ensuremath{\Lambda}$ satisfying the constraint $\ensuremath{\delta}\ensuremath{\le}\frac{4{\ensuremath{\Lambda}}^{2}}{3}$. However, we note that two different classes of instanton solutions exists in the region $0<\ensuremath{\delta}<\frac{4{\ensuremath{\Lambda}}^{2}}{3}$. 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 $0<\ensuremath{\delta}<{\ensuremath{\Lambda}}^{2}$. It is also found that gravitational instanton solution is permitted even with $\ensuremath{\Lambda}=0$ but one has to consider $\ensuremath{\delta}<0$. However, in the later case a universe with a pair of black holes is less probable.