Toroidal Perturbations of Friedmann-Robertson-Walker Universes

TL;DR

This paper derives explicit solutions for axisymmetric toroidal perturbations in all types of Friedmann-Robertson-Walker universes, using vector spherical harmonics and addressing special cases like closed universes.

Contribution

It provides a comprehensive set of explicit solutions for toroidal metric perturbations in FRW universes, including special treatment for closed models.

Findings

Explicit expressions for axisymmetric perturbations in all FRW models.

Solutions for odd-parity l-pole perturbations with prescribed angular velocities or momenta.

Special handling of Legendre equations in closed universes.

Abstract

Explicit expressions are found for the axisymmetric metric perturbations of the closed, flat and open FRW universes caused by toroidal motions of the cosmic fluid. The perturbations are decomposed in vector spherical harmonics on 2-spheres, but the radial dependence is left general. Solutions for general odd-parity $l$-pole perturbations are given for either angular velocities or angular momenta prescribed. In particular, in case of closed universes the solutions require a special treatment of the Legendre equation.