Inhomogeneous Initial Data and Small-Field Inflation
  We consider the robustness of small-field inflation in the presence of scalar
field inhomogeneities. Previous numerical work has shown that if the scalar
potential is flat only over a narrow interval, such as in commonly considered
inflection-point models, even small-amplitude inhomogeneities present at the
would-be onset of inflation at $\tau = \tau_i$ can disrupt the accelerated
expansion. In this paper, we parametrise and evolve the inhomogeneities from an
earlier time $\tau_{IC}$ at which the initial data were imprinted, and show
that for a broad range of inflationary and pre-inflationary models,
inflection-point inflation withstands initial inhomogeneities. We consider
three classes of perturbative pre-inflationary solutions (corresponding to
energetic domination by the scalar field kinetic term, a relativistic fluid,
and isotropic negative curvature), and two classes of exact solutions to
Einstein's equations with large inhomogeneities (corresponding to a stiff fluid
with cylindrical symmetry, and anisotropic negative curvature). We derive a
stability condition that depends on the Hubble scales $H(\tau_ i)$ and
$H(\tau_{IC})$, and a few properties of the pre-inflationary cosmology. For
initial data imprinted at the Planck scale, the absence of an inhomogeneous
initial data problem for inflection-point inflation leads to a novel, lower
limit on the tensor-to-scalar ratio.
