Signatures of the Very Early Universe: Inflation, Spatial Curvature and
  Large Scale Anomalies
  A short inflationary phase may not erase all traces of the primordial
universe. Associated observables include both spatial curvature and "anomalies"
in the microwave background or large scale structure. The present curvature
$\Omega_{K,0}$ reflects the initial curvature, $\Omega_{K,\mathrm{start}}$, and
the angular size of anomalies depends on $k_\mathrm{start}$, the comoving
horizon size at the onset of inflation. We estimate posteriors for
$\Omega_{K,\mathrm{start}}$ and $k_\mathrm{start}$ using current data and
simulations, and show that if either quantity is measured to have a non-zero
value, both are likely to be observable. Mappings from
$\Omega_{K,\mathrm{start}}$ and $k_\mathrm{start}$ to present-day observables
depend strongly on the primordial equation of state; $\Omega_{K,0}$ spans ten
orders of magnitude for a given $\Omega_{K,\mathrm{start}}$ while a simple and
general relationship connects $\Omega_{K,0}$ and $k_\mathrm{start}$. We show
that current bounds on $\Omega_{K,0}$ imply that if $k_\mathrm{start}$ is
measurable, the curvature was already small when inflation began. Finally,
since the energy density changes slowly during inflation, primordial
gravitational wave constraints require that a short inflationary phase is
preceded by a nontrivial pre-inflationary phase with critical implications for
the expected value of $\Omega_{K,\mathrm{start}}$.
