Slow-roll, acceleration, the Big Rip and WKB approximation in NLS-type formulation of scalar field cosmology

TL;DR

This paper explores the non-linear Schr"{o}dinger-type formulation of scalar field cosmology, analyzing slow-roll, acceleration, WKB approximation, and Big Rip singularity, providing new insights into the mathematical structure and behavior of such models.

Contribution

It introduces a novel NLS formulation for scalar (phantom) field cosmology, reexpresses key conditions, and analyzes the Big Rip singularity within this framework.

Findings

WKB approximation is valid for most slowly-varying Schr"{o}dinger potentials.

In the Big Rip scenario, two quantities diverge instead of three.

Approaching the Big Rip, the effective equation of state tends to -1 + 2/(3q).

Abstract

Aspects of non-linear Schr\"{o}dinger-type (NLS) formulation of scalar (phantom) field cosmology on slow-roll, acceleration, WKB approximation and Big Rip singularity are presented. Slow-roll parameters for the curvature and barotropic density terms are introduced. We reexpress all slow-roll parameters, slow-roll conditions and acceleration condition in NLS form. WKB approximation in the NLS formulation is also discussed when simplifying to linear case. Most of the Schr\"{o}dinger potentials in NLS formulation are very slowly-varying, hence WKB approximation is valid in the ranges. In the NLS form of Big Rip singularity, two quantities are infinity in stead of three. We also found that approaching the Big Rip, $w_{\rm eff}\to -1 + {2}/{3q}$, $(q<0)$ which is the same as effective phantom equation of state in the flat case.