Black hole formation and slow-roll inflation

Black hole formation may occur if the spectrum of the curvature perturbation \zeta increases strongly as the scale decreases. As no such increase is observed on cosmological scales, black hole formation requires strongly positive running n' of the spectral index n, though the running might only kick in below the `cosmological scales' probed by the CMB anisotropy and galaxy surveys. The most obvious way of producing this running is through the running mass model of slow roll inflation. We obtain a new observational bound n'<0.026 on the running provided by this model, improving an earlier result by a factor two. We also discuss black hole production in more general scenarios. We show that the usual conditions \epsilon \ll 1 and |\eta| \ll 1 are enough to derive the spectrum P_{\zeta}(k), the introduction of higher order parameters \xi^{2} etc. being optional.