Primordial density perturbation and primordial black holes

The black holes formed in the early universe are called primordial black holes. Primordial black holes provide a profound tool to detect the primordial cosmological structures and study nontrivial physics of the early Universe. By fitting the theoretical predictions about primordial black holes to the observational data, we can constrain the cosmological models. There exist several mechanisms about the formation of primordial black holes. In this review, we focus on the primordial black holes formed by the collapse of over-densed regions. This paper is organized as follows. The first section is the introduction. Then, in the second section, we introduce briefly several basic concepts on the primordial black holes formed by the collapse of the over-densed region, including the condition for an over-densed region to collapse into a primordial black hole, the mass, the formation time and the lifetime of a primordial black hole. About the primordial-black-hole mass, usually, in the over-densed-region-collapsing mechanism, the "Carr-Hawking" collapsing model is adopted to describe the formation. This collapsing model tells us that primordial black holes always form with the mass approximately equal to the mass contained by the Hubble horizon at the time when the over-densed regions enter the Hubble horizon. This is called the horizon-mass relation. But, recently, the numerical simulations show that the phenomenon of the critical collapse happens in the formation of primordial black holes. And primordial black holes with a range of masse are formed. The mass function of primordial black holes is given by a scaling relation. Here, both of the relations are introduced. Then, in the third section, based on the assumption that the primordial cosmological density perturbations obey the Gaussian distribution, we review the calculation of the mass spectrum of the primordial black holes. In this section, we, firstly, review the widely-used approximate expression of the initial mass spectrum of the primordial black holes. This approximate result is based on the horizon-mass relation that is implied by the "Carr-Hawking" collapsing model. Then by using the Press-Schechter method, we introduce the exact expression of the initial mass spectrum of the primordial black holes. This exact expression is based on the scaling relation that is implied by the critical collapse. Then by neglecting the effect of the Hawking radiation of primordial black holes, we express the present mass spectrum of the primordial black holes in terms of the initial mass spectrum of the primordial black holes. However, usually, the effect of the Hawking radiation of primordial black holes cannot be neglected. Then we also review how to calculate the mass spectrum of the primordial black holes by considering the effect of the Hawking radiation. Since the mainstream cosmologists believe that, up to now, inflation is the best mechanism to generate the primordial density perturbation, we assume that the initial over-densed regions also origin from inflation. Then in the subsection of the third section, we relate the primordial power spectrum generated by inflation to the initial mass spectrum of the primordial black holes. The fourth section is the summary and discussion. © 2018, Science Press. All right reserved.