Density Fluctuations in Inflationary Models with Multiple Scalar Fields

Making use of the primordially isocurvature fluctuations, which are generated in inflationary models with multiple scalar fields, we make a phenomenological model that predicts formation of primordial black holes which can account for the massive compact halo objects recently observed. In inflationary universe models [1] with multiple scalar fields, not only adiabatic [2] but also primordially isocurvature fluctuations [3] are generated in general. For example, if we consider ordinary slow roll-over inflation models in Brans-Dicke gravity [4], the Brans-Dicke dilaton acquires isocurvature fluctuation out of quantum fluctuation in the inflationary regime. In this particular model, however, the isocurvature fluctuation does not play any important role in structure formation because the energy density of the Brans-Dicke field remains as small as ∼ ω times the total energy density, where ω > ∼ 500 is the Brans-Dicke parameter [5]. On the other hand, primordially isocurvature fluctuations can be cosmologically important if energy density of its carrier becomes significant in a later epoch [3]. Furthermore it is relatively easier to imprint a nontrivial feature on the spectral shape of the isocurvature fluctuations. Making use of this property here we construct a model which possesses a peak in the spectrum of total density fluctuation at the horizon crossing. Then we can produce a significant amount of primordial black holes (PBHs) around the horizon mass scale when the fluctuation at the horizon crossing becomes maximal. To be specific, we choose the model parameters so that one can account for the observed massive compact halo objects (MACHOs) [7] in terms of PBHs thus produced. That is, our goal here is to produce PBHs of mass ∼ 0.1M⊙ with the abundance of ∼ 20% of the halo mass, corresponding to about 10 times the critical density [8]. PBHs are formed if initial density fluctuations grow sufficiently and a high density region collapses within its gravitational radius. First let us review its formation process. The background spacetime of the early universe dominated by radiation is satisfactorily described by the spatially-flat Friedmann universe, ds = −dt + a(t) [