Abstract
We construct the general analytical solution for the mathcal {N}-field product-exponential potential in an expanding FLRW background. We demonstrate the relevance of this analytical solution in more general contexts for the derivation of estimates for the transitional time between an arbitrary initial state and the slow-roll solutions. In certain cases, these estimates can also be used to demonstrate the non-linear convergence towards the slow-roll solutions. In addition, we extend the solution to include stiff matter as well.
Highlights
Inflation is the leading paradigm for the origin of structure formation in an initially isotropic and homogeneous universe [1]
Scalar fields in cosmology have been extensively studied over the last decades mainly due to their applications in both the early and the late universe
Only a limited number of exact solutions exists in the literature
Summary
Inflation is the leading paradigm for the origin of structure formation in an initially isotropic and homogeneous universe [1] It was originally proposed as an attempt to evade the fine-tunings of the standard cosmological model by theorizing a period of accelerating expansion in the very early universe [2,3,4,5,6,7,8]. The aim of this work is the derivation of general solutions for an arbitrary number of fields. These solutions can be used to derive estimates for the transition time (or field displacement) from arbitrary initial states to inflation, or alternatively they can be used to demonstrate non-linear convergence towards the slow-roll solutions. Conventions: dots ( ̇) and primes ( ) indicate derivatives with respect to the cosmic time and number of e-folds respectively; Greek letters refer to spacetime indices while Latin ones to field-space indices; the spacetime metric has the mostly plus signature (−, +, +, +); we assume the Einstein’s convention for repeated indices; N is the e-folding number and N denotes the number of fields
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