A Finite Measure for the Initial Conditions of Inflation
  We investigate whether inflation requires finely tuned initial conditions in
order to explain the degree of flatness and homogeneity observed in the
Universe. We achieve this by using the Eisenhart lift, which can be used to
write any scalar field theory in a purely geometric manner. Using this
formalism, we construct a manifold whose points represent all possible initial
conditions for an inflationary theory. After equipping this manifold with a
natural metric, we show that the total volume of this manifold is finite for a
wide class of inflationary potentials. Hence, we identify a natural measure
that enables us to distinguish between generic and finely tuned sets of initial
conditions without the need for a regulator, in contrast to previous work in
the literature. Using this measure, we find that the initial conditions that
allow for sufficient inflation are indeed finely tuned. The degree of
fine-tuning also depends crucially on the value of the cosmological constant at
the time of inflation. Examining the example potential
$V=\lambda\varphi^4+\Lambda$, we find that we require percent-level fine tuning
if we allow the cosmological constant during inflation to be much larger than
it is today. However, if we fix the cosmological constant to its presently
observed value, the degree of fine tuning required is of order $10^{-29}$.
