Can Primordial Black Holes Be a Significant Part of Dark Matter?

The computation of PBH (primordial black hole) production from primordial perturbations has recently been improved by considering a more accurate relation between the primordial power spectrum and the PBH mass variance. We present here exact expressions which are valid for primordial spectra of arbitrary shape and which allow accurate numerical calculations. We then consider the possibility to have a significant part of dark matter in the form of PBHs produced by a primordial spectrum of inflationary origin possessing a characteristic scale. We show that in this model the relevant PBH mass is constrained to lie in the range 5×1015 g . M . 10 g. This is much less than the mass range coming from the QCD phase transition, allowing the two mechanisms to be easily distinguished. PACS Numbers: 04.62.+v, 98.80.Cq Introduction: A consistent paradigm seems to emerge in cosmology, of which dark matter is an essential ingredient. However, the nature of dark matter remains to be one of the most important open problems. It is expected to account for about one quarter of the present critical energy density, most of the remaining three quarters being dark energy whose nature is also unknown. There exist many candidates which are classified into cold, hot, and warm dark matter. Some of these candidates would signal new physics beyond the standard model of particle physics, such as relic neutralinos, the supersymmetric cold dark matter (CDM) candidate. Another possible CDM candidate could be primordial black holes (PBHs). The advantage with them is that their existence rests on known physics, general relativity and the presence of primordial fluctuations, independent of the mechanism which generates them. Indeed, it is well known that PBHs can be produced in the early universe due to the collapse of overdense regions on the linear size of the Hubble radius at the time when this corresponds to their Schwarzschild radius [1]. These fluctuations in the energy density must exist one way or the other in order to explain the origin of all inhomogeneities we see in the universe. Their existence, and in particular their primordial origin, is reflected in the presence of acoustic peaks in the cosmological microwave background (CMB) anisotropy, and is now well established by observations. Therefore, the generation of PBHs is unavoidable. Inflationary fluctuations are produced on a huge range of scales and the observation of PBHs formed after inflation could probe the fluctuations on scales where the CMB gives no information. It would therefore be complementary to the results which are extracted from the CMB data. PBHs were produced in the very early universe. PBHs generated before 10 s, corresponding to masses M < M∗ ≈ 5× 10 g, have already evaporated by the present day due to Hawking radiation. PBHs with masses bigger than M∗ could, however, constitute a significant fraction, or even all, of the CDM. Unfortunately this possibility, though attractive, cannot be implemented with scale-free perturbations because they would lead to a negligible rate of PBH formation. However, in view of the naturalness of PBH generation, it is important to investigate whether other kinds of spectra could lead to a significant formation rate. General formalism: We will consider Gaussian primordial fluctuations, the kind of fluctuations expected in most inflationary models. The density contrast averaged over a sphere of radius R then reads p(δ) = 1 √ 2π σ(R) e − δ2 2σ2(R) , (1) where the dispersion σ(R) ≡ 〈(