Two-field cosmological α-attractors with Noether symmetry

Abstract

We study Noether symmetries in two-field cosmological α-attractors, investigating the case when the scalar manifold is an elementary hyperbolic surface.This encompasses and generalizes the case of the Poincaré 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 α-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.

Hidden symmetries of two-field cosmological models

“…This allows us to give a complete description of all time-independent Noether symmetries for models with rotationally-invariant scalar manifold metric and to classify Noether-symmetric models of this type. In particular, we find many new Noether symmetries that were never considered before, the vast majority of which do not satisfy the separation of variables Ansatz used in reference [20]. Most of these symmetries, which we describe explicitly, are not invariant under the U(1) group of rotations which preserves the scalar manifold metric.…”

“…Remark 4.1. In reference [20], we studied two-field rotationally invariant models with the separation of variables Ansatze:…”

“…The minisuperspace formulation allows one to study symmetries of such models using the Noether approach. We followed this method in reference [20] -where we restricted to scalar manifolds which are rotationally invariant and we considered only a special class of time-independent symmetries satisfying a certain separation of variables Ansatz. In the present paper, we still assume that the scalar manifold metric G is rotationally invariant 1 , but study the problem without making any other assumptions.…”

Noether symmetries of two-field cosmological models

“…The tangent space to the configuration space (1) and T (2) are the pullbacks of the vector bundles T R >0 and T N through the canonical projections. Accordingly, any vector field X ∈ X (N ) decomposes as:…”

“…The characteristic system for variational symmetries Theorem 5 [2] For the Lagrangian (5), the Noether symmetry condition amounts to the requirement that X (1) and X (2) have the following forms:…”