It has been suggested that the overdense region as a result of inhomogeneities in the early Universe would have undergone a collapse into the primordial black holes (PBH). In this work, we discuss a possible constraint on the PBH formation in the radiation dominated epoch by imposing the generalized second law of thermodynamics in the context of spherically collapsing scenario. It is found that both the critical temperature Tc over which the formation of PBH is not possible and the lower bound on the mass of PBH depend on the number of degrees of freedom at the time of PBH formation. In the standard model, one can show that the lower bound on the mass of PBH known in the literature, of order Planck mass, is consistent with the thermodynamic constraint constructed in this work. We also pointed out the possibility that the critical temperature(lower bound on PBH mass) can be lowered(increased) provided the number of relativistic degrees of freedom of the Universe is increasing substantially beyond the standard model.There are many observational indications , for example, the existence of the galaxies and the fluctuations observed in CMBR, which imply that the early Universe must have been inhomogeneous. It has been suggested that the overdense region as a result of inhomogeneities in the early Universe underwent into the primordial black holes(PBH). The existence of PBH has been considered to have interesting cosmological consequences, for example, on the cosmic microwave background radiation, primordial nucleosynthesis and dark matter candidate. The mass distribution of PBH can provide a useful information on the spectrum of density fluctuations in the early Universe. Hence the formation as well as the evolution of PBH have been interesting subjects of investigations [3] [4] [5]. Since PBH is an issue in the early Universe, the discussion depends on the theory of gravity, which might be different from that of Einstein. For example, the formation and evolution of PBH has been discussed also in scalar-tensor theories of gravity [15].One of the remarkable developments in black hole physics is the thermodynamic understanding of a black hole [6] [7]. The entropy of a black hole is proportional to the surface area of horizon and the temperature to the surface gravity. Then the second law of thermodynamics can be stated in a generalized way that the total entropy of black hole plus external Universe never decreases. Hence it is interesting to see whether the generalized second law of thermodynamics can impose any constraint on the physical processes in which black holes are involved. The purpose of this work is to discuss how the formation of PBH can be constrained by imposing the generalized second law of thermodynamics, which is believed to be valid during the spherically collapsing process.The evolution of the overdense gravity-dominated-regions is supposed to be described by the Friedman-RobertsonWalker(FRW) metric. Since overdense region is characterized with the spatial curvature, k > 0, in the background of expan...
