Primordial black holes and generalized constraints on chaotic inflation

It is argued that the main quantity of interest in chaotic inflation is the cosmological expansion rate $H$ expressed as a function of the inflaton field $\phi$. We derive a general prescription for realizing successful inflation in terms of a set of constraints on this function. The formalism is valid for all chaotic inflationary models based on a single scalar field which is minimally coupled to general relativity, so no restrictions on the dynamics of the field are necessary. This technique is used to investigate the possibility that primordial black holes (PBH's) may arise due to adiabatic quantum fluctuations in the inflaton. PBH formation can only be interesting if the amplitude of the fluctuations decreases with increasing mass scale and this is only possible if the field is accelerating or decelerating sufficiently fast. In this case, limits on the number of PBH's place very interesting constraints on the form of $H\left(\phi \right)$ since, together with the COBE measurement, they restrict the spectrum of fluctuations over 45 decades of mass. This corresponds to $35e$-foldings of inflationary expansion. If the amplitude of the fluctuations decreases as a power of mass, which is the most interesting situation, then $H\left(\phi \right)$ must have a trigonometric form and this allows the constraints to be expressed very simply.