Large Field Inflation and Gravitational Entropy
  Large field inflation can be sensitive to perturbative and nonperturbative
quantum corrections that spoil slow roll. A large number $N$ of light species
in the theory, which occur in many string constructions, can amplify these
problems. One might even worry that in a de Sitter background, light species
will lead to a violation of the covariant entropy bound at large $N$. If so,
requiring the validity of the covariant entropy bound could limit the number of
light species and their couplings, which in turn could severely constrain
axion-driven inflation. Here we show that there is no such problem when we
correctly renormalize models with many light species, taking the {\it physical}
Planck scale to be $M^2_{pl} \gtrsim N {\cal M}_{UV}^2$, where ${\cal M}_{UV}$
is the cutoff for the QFT coupled to semiclassical quantum gravity. The number
of light species then cancels out of the gravitational entropy of de Sitter or
near-de Sitter backgrounds at leading order. Working in detail with $N$ scalar
fields in de Sitter space, renormalized to one loop order, we show that the
gravitational entropy automatically obeys the covariant entropy bound.
Furthermore, while the axion decay constant is a strong coupling scale for the
axion dynamics, we show that it is {\it not} in general the cutoff of 4d
semiclassical gravity. After renormalizing the two point function of the
inflaton, we note that it is also controlled by scales much below the cutoff.
We revisit $N$-flation and KKLT-type compactifications in this light, and show
that they are perfectly consistent with the covariant entropy bound. Thus,
while quantum gravity might yet spoil large field inflation, holographic
considerations in the semiclassical theory do not obstruct it.
