Quantum Incompleteness of Inflation
  Inflation is most often described using quantum field theory (QFT) on a
fixed, curved spacetime background. Such a description is valid only if the
spatial volume of the region considered is so large that its size and shape
moduli behave classically. However, if we trace an inflating universe back to
early times, the volume of any comoving region of interest -- for example the
present Hubble volume -- becomes exponentially small. Hence, quantum
fluctuations in the trajectory of the background cannot be neglected at early
times. In this paper, we develop a path integral description of a flat,
inflating patch (approximated as de Sitter spacetime), treating both the
background scale factor and the gravitational wave perturbations quantum
mechanically. We find this description fails at small values of the initial
scale factor, because \emph{two} background saddle point solutions contribute
to the path integral. This leads to a breakdown of QFT in curved spacetime,
causing the fluctuations to be unstable and out of control. We show the problem
may be alleviated by a careful choice of quantum initial conditions, for the
background and the fluctuations, provided that the volume of the initial,
inflating patch is larger than $\gg H^{-1}$ in Planck units with $H$ the Hubble
constant at the start of inflation. The price of the remedy is high: not only
the inflating background, but also the stable, Bunch-Davies fluctuations must
be input by hand. Our discussion emphasizes that, even if the inflationary
scale is far below the Planck mass, new physics is required to explain the
initial quantum state of the universe.
