Comment on "Attractor solutions in scalar-field cosmology" and "How many e-folds should we expect from high-scale inflation?"

TL;DR

This paper reexamines the conserved measures in scalar-field cosmology, revealing infinitely many solutions and demonstrating that measure choice significantly impacts inflation predictions.

Contribution

It provides a general solution to the key differential equation, showing multiple measures exist and affect inflation estimates.

Findings

Multiple conserved measures exist for scalar-field cosmology.

Different measures lead to varying predictions of e-folds.

The previous unique measure assumption is challenged.

Abstract

In Ref. [1], it was claimed that in the spatially flat cosmological case there exists a unique conserved measure (up to normalization) on the $(\phi,\dot{\phi})$ phase space for scalar field with $m^2\phi^2$ potential by finding a unique solution to the differential equation (44) (in Ref. [1]) in the low-energy regime. In Ref. [2], it was also claimed that a unique solution to the same differential equation was found in the high-energy regime and using this solution the authors calculated the expected total number of e-folds of inflation. In this comment, we reanalyze the differential equation (44) and obtain general solutions both in the low-energy and high-energy regime, which can include the solution in Ref. [1] and the solution in Ref. [2] as a special case in the corresponding energy regime. In this way, we find that following the constructions in Ref. [1] there actually exist infinitely many nonequivalent conserved measures for the scalar-field cosmology with $m^2\phi^2$ potential on the $(\phi,\dot{\phi})$ phase space. Moreover, through specific calculations, we also show that different choices of measures can lead to quite different predictions of the expected total number of e-folds of inflation.