Calculating the mass spectrum of primordial black holes
  We reinspect the calculation for the mass fraction of primordial black holes
(PBHs) which are formed from primordial perturbations, finding that performing
the calculation using the comoving curvature perturbation $\mathcal{R}_{c}$ in
the standard way vastly overestimates the number of PBHs, by many orders of
magnitude. This is because PBHs form shortly after horizon entry, meaning modes
significantly larger than the PBH are unobservable and should not affect
whether a PBH forms or not - this important effect is not taken into account by
smoothing the distribution in the standard fashion. We discuss alternative
methods and argue that the density contrast, $\Delta$, should be used instead
as super-horizon modes are damped by a factor $k^{2}$. We make a comparison
between using a Press-Schechter approach and peaks theory, finding that the two
are in close agreement in the region of interest. We also investigate the
effect of varying the spectral index, and the running of the spectral index, on
the abundance of primordial black holes.
