Primordial black holes are considered to be pair created quantum-mechanically during inflation. In the context of General Relativity (GR), it has been shown that the pair creation rate is exponentially decreasing during inflation. Specifically, tiny black holes are favored in the early universe, but they can grow with the horizon scale, as inflation approaches its end. At the same time, cosmological, and not only, shortcomings of GR have triggered the pursuit for a new, alternative theory of gravity. In this paper, by using probability amplitudes from the No Boundary Proposal (NBP), we argue that any alternative gravity should have a black hole creation rate similar to that of GR; that is, in the early universe the creation of small black holes is in favor, while in the late universe larger black holes are being exponentially suppressed. As an example, we apply this argument in f (R)-theories of gravity and derive a general formula for the rate in any f (R)-theory with constant curvature. Finally, we consider well known f (R)-models and using this formula we put constraints on their free parameters.
I. INTRODUCTIONAccording to the concordance model in cosmology, ΛCDM, the universe is endowed with a positive cosmological constant, which has a current value of Λ ≃ 1.11 × 10 −52 m −2 ≃ 2.9 × 10 −122 in reduced Planck units [1]. However, it is expected that, in the beginning of the universe, and especially during inflation, it started out large enough and since then decreases, until its current value. In addition, the models of inflation predict cosmic density perturbations, as well as quantum fluctuations of the field, responsible for the inflation, which lead to topological changes of the spacetime and these respectively to the creation of primordial black hole pairs. Primordial black holes (PBHs) are theoretical objects with masses smaller than the solar mass down to the Planck mass. These objects are unlikely to form from the gravitational collapse of a star today, since low-mass black holes can form only if matter is compressed to enormously high densities by very large external pressures [2]. Such conditions of high temperatures and pressures can be found in the early stages of a violent universe and, thus, it is believed that PBHs may have been produced plentiful back then. Proposals for their formation have been addressed over the years, such as the collapse of primordial inhomogeneities [3], cosmological phase transitions [4,5] and close domain walls [6].Apart from the procedure of the spontaneous formation of black hole pairs, there should exist a force which would pull them apart, in order not to fall back together and annihilate. In our scenario, we assume that the cosmological constant plays the role of this force, and because of the rapid cosmological expansion in the early universe, the created black hole pair remains separate.During the nineties, the study of black hole pair creation in different backgrounds was of much interest [7][8][9][10][11][12][13][14][15]. Important results have been obtained c...
