Initial data for rotating cosmologies

TL;DR

This paper investigates the existence, uniqueness, and stability of initial data solutions in rotating cosmologies with positive cosmological constant, focusing on bifurcation structures and conformal rescaling methods.

Contribution

It extends recent mathematical results to analyze bifurcation and stability of solutions in rotating cosmological models, including Kerr-deSitter and Bowen-York data.

Findings

Bifurcation structure of axially symmetric solutions clarified

Dynamical data constructed via conformal rescaling of Kerr-deSitter

Existence and stability conditions depend on coefficients of the Lichnerowicz equation

Abstract

We revisit the construction of maximal initial data on compact manifolds in vacuum with positive cosmological constant via the conformal method. We discuss, extend and apply recent results of Hebey et al. [19] and Premoselli [31] which yield existence, non-existence, (non-)uniqueness and (linearisation-) stability of solutions of the Lichnerowicz equation, depending on its coefficients. We then focus on so-called $(t,\varphi)$-symmetric data as "seed manifolds", and in particular on Bowen-York data on the round hypertorus $\mathbb{S}^2 \times \mathbb{S}$ (a slice of Nariai) and on Kerr-deSitter. In the former case, we clarify the bifurcation structure of the axially symmetric solutions of the Lichnerowicz equation in terms of the angular momentum as bifurcation parameter, using a combination of analytical and numerical techniques. As to the latter example, we show how dynamical data can be constructed in a natural way via conformal rescalings of Kerr-deSitter data.