Pair creation of de Sitter black holes on a cosmic string background

We analyze the quantum process in which a cosmic string breaks in a de Sitter (dS) background, and a pair of neutral or charged black holes is produced at the ends of the string. The energy to materialize and accelerate the pair comes from the positive cosmological constant and, in addition, from the string tension. The compact saddle point solutions without conical singularities (instantons) or with conical singularities (submaximal instantons) that describe this process are constructed through the analytical continuation of the dS C metric. Then, we explicitly compute the pair creation rate of the process. In particular, we find the nucleation rate of a cosmic string in a dS background, and the probability that it breaks and a pair of black holes is produced. Finally we verify that, as occurs with pair production processes in other background fields, the pair creation rate of black holes is proportional to e(S), where the gravitational entropy of the black hole, S, is given by one-quarter of the area of the horizons present in the saddle point solution that mediates the process.