Extreme scenarios: the tightest possible constraints on the power
  spectrum due to primordial black holes
  Observational constraints on the abundance of primordial black holes (PBHs)
constrain the allowed amplitude of the primordial power spectrum on both the
smallest and the largest ranges of scales, covering over 20 decades from
$1-10^{20}/ \rm{Mpc}$. Despite tight constraints on the allowed fraction of
PBHs at their time of formation near horizon entry in the early universe, the
corresponding constraints on the primordial power spectrum are quite weak,
typically ${\cal P}_\mathcal{R}\lesssim 10^{-2}$ assuming Gaussian
perturbations. Motivated by recent claims that the evaporation of just one PBH
would destabilise the Higgs vacuum and collapse the universe, we calculate the
constraints which follow from assuming there are zero PBHs within the
observable universe. This extends the constraints right down to the horizon
scale at the end of inflation, but does not significantly tighten the existing
power spectrum constraints, even though the constraint on PBH abundance can
decrease by up to 46 orders of magnitude. This shows that no future improvement
in observational constraints can ever lead to a significant tightening in
constraints on inflation (via the power spectrum amplitude). The power spectrum
constraints are weak because an order unity perturbation is required in order
to overcome pressure forces. We therefore consider an early matter dominated
era, during which exponentially more PBHs form for the same initial conditions.
We show this leads to far tighter constraints, which approach ${\cal
P}_\mathcal{R}\lesssim10^{-9}$, albeit over a smaller range of scales and are
very sensitive to when the early matter dominated era ends. Finally, we show
that an extended early matter era is incompatible with the argument that an
evaporating PBH would destroy the universe, unless the power spectrum amplitude
decreases by up to ten orders of magnitude.
