A Finite Measure for the Initial Conditions of Inflation

Abstract

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 $V=\lambda {\phi }^4+\Lambda$, 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 $10^{-29}$.

arXiv comment27+1 pages, 2 figures. V2: Corrected typos, clarified some wording, conclusions unchanged. V3: corrected typos, added clarifying footnotes. V4: corrected errors resulting from a bug in the code. Qualitative conclusions are unchanged, but significant changes to numerical results. This version incorporates the results of the published erratum

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