Collins in Wonderland

TL;DR

This paper explores the asymptotic behavior of scalar-field models in anisotropic, homogeneous cosmologies, demonstrating that the Wonderland solutions act as future attractors in certain Bianchi types within General Relativity.

Contribution

It extends the Wonderland exact solutions to include $p$-form fields in Bianchi types VI$_0$ and VI$_{ ilde{h}}$, showing they serve as future attractors.

Findings

Wonderland solutions are future attractors in Bianchi VI$_0$ and VI$_{ ilde{h}}$.

Explicit line-elements and gauge potentials for Wonderland are derived.

Simulation shows trajectories approach Wonderland in Bianchi type I.

Abstract

What is the asymptotic future of a scalar-field model if the assumption of isotropy is relaxed in generic, homogeneous space-times with general relativity? This paper is a continuation of our previous work on Bianchi cosmologies with a $p$-form field (where $p\,\in\,\{1,3\}$)---or equivalently: an inhomogeneous, mass-less scalar gauge field with a homogeneous gradient. In this work we investigate such matter sector in General Relativity, and restrict to space-times of the particular Bianchi types VI$_0$ and VI$_{\tilde{h}}$, where $\tilde{h}=h<0\,\cap\,\neq\,-1/9\,\cup\,-1$. We show that the previously found fabric of exact solutions named Wonderland are future attractors in $\mathcal{B}$(VI$_0$) and $\mathcal{B}$(VI$_{\tilde{h}}$), extending the Collins perfect-fluid equilibrium set to include a $p$-form (with $p\,\in\,\{1,3\}$). We also write down the line-element corresponding to Wonderland in VI$_{\tilde{h}}$ and give explicit expressions for the underling gauge-potential $\phi(t,\mathbf{x})$ corresponding to this solution. Simulation of a path approaching Wonderland in Bianchi type I is also given.