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 = λϕ 4 + Λ, 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 −

Initial conditions of inflation in a Bianchi I Universe

“…We can therefore employ Laplace's principle of indifference [45] to argue that the initial conditions should be uniformally distributed over this manifold. This results in the well-defined, normalised measure (35).…”

“…We can use this result to describe any homogeneous scalar field theory in a geometric way, as was shown in [35,37]. Such a theory will have a Lagrangian of the form…”

“…Notice that, just as in F18, the normalisation constant N is finite provided the inflationary potential diverges quicker than ϕ 2 as ϕ → ∞ and there exists a non-zero cosmological constant. Therefore the measure (35) is well defined and requires no regularisation for this class of inflationary theories.…”

“…In [35] (hereafter F18) we constructed a measure of initial conditions that is finite for a large class of inflationary models and thus has no need for regularization. We achieved this by using the Eisenhart lift [36,37] to describe inflation as the geodesic motion on a field-space manifold.…”

Beyond the poles in attractor models of inflation

“…It has been argued that the presence of plateaus in attractor models (after canonicalisation) can resolve the problem of initial conditions, as it is more likely that the field will start on the infinite plateau as opposed to the finite smaller region [71]. Since it is not possible to imposing a uniform probability density on an infinite interval [72], it might make more sense to argue for a uniform distribution in the non-canonical field within the poles. However, in this case, we are tasked with explaining why the non-canonical field lives in that domain.…”