Two-field Cosmological $\alpha$-attractors with Noether Symmetry

TL;DR

This paper explores Noether symmetries in two-field cosmological $$-attractors on hyperbolic surfaces, deriving potential forms, fixing the $$ parameter, and finding exact solutions that extend previous numerical studies.

Contribution

It identifies conditions for Noether symmetries in two-field $$-attractors on hyperbolic surfaces and derives exact solutions, generalizing previous numerical analyses.

Findings

Fixed $$-parameter for certain symmetries

Derived scalar potentials compatible with symmetries

Obtained exact solutions of equations of motion

Abstract

We study Noether symmetries in two-field cosmological $\alpha$-attractors, investigating the case when the scalar manifold is an elementary hyperbolic surface. This encompasses and generalizes the case of the Poincare disk. We solve the conditions for the existence of a `separated' Noether symmetry and find the form of the scalar potential compatible with such, for any elementary hyperbolic surface. For this class of symmetries, we find that the $\alpha$-parameter must have a fixed value. Using those Noether symmetries, we also obtain many exact solutions of the equations of motion of these models, which were studied previously with numerical methods.