On the initial value problem for second order scalar fluctuations in Einstein static

TL;DR

This paper demonstrates that homogeneous scalar perturbations in an Einstein static universe are unavoidable due to linearization stability, and these modes are exponentially unstable, challenging the idea of a long-lived Einstein universe.

Contribution

It shows that linear homogeneous scalar modes must be present in Einstein static universe perturbations, and their instability cannot be ignored for stability considerations.

Findings

Homogeneous scalar modes are required by linearization stability.

These modes are exponentially unstable.

Neglecting these modes does not yield a stable Einstein universe.

Abstract

We consider fluctuations in a perfect irrotational fluid coupled to gravity in an Einstein static universe background. We show that the homogeneous linear perturbations of the scalar and metric fluctuations in the Einstein static universe must be present if the second order constraint equations are to be integrable. I.e., the 'linearization stability' constraint forces the presence of these homogeneous modes. Since these linear homogeneous scalar modes are well known to be exponentially unstable, the tactic of neglecting these modes to create a long-lived, almost Einstein universe does not work, even if all higher order (L $>$ 1) modes are dynamically stable.