Probability of primordial black hole pair creation in a modified gravitational theory

Abstract

We compute the probability for quantum creation of an inflationary universe with and without a pair of black holes in a modified gravity. The action of the modified theory of gravity contains $\ensuremath{\alpha}{R}^{2}$ and $\ensuremath{\delta}{R}^{\ensuremath{-}1}$ terms in addition to a cosmological constant ($\ensuremath{\Lambda}$) in the Einstein-Hilbert action. The probabilities for the creation of universe with a pair of black holes have been evaluated considering two different kinds of spatial sections, one which accommodates a pair of black holes and the other without black hole. We adopt a technique prescribed by Bousso and Hawking to calculate the above creation probability in a semiclassical approximation using the Hartle-Hawking boundary condition. We note a class of new and physically interesting instanton solutions characterized by the parameters in the action. These instantons may play an important role in the creation of the early universe. We also note that the probability of creation of a universe with a pair of black holes is strongly suppressed with a positive cosmological constant when $\ensuremath{\delta}=\frac{4{\ensuremath{\Lambda}}^{2}}{3}$ for $\ensuremath{\alpha}g0$ but it is more probable for $\ensuremath{\alpha}l\ensuremath{-}\frac{1}{6\ensuremath{\Lambda}}$. In the modified gravity considered here instanton solutions are permitted even without a cosmological constant when one begins with a negative $\ensuremath{\delta}$.