Galileon Hairs of Dyson Spheres, Vainshtein's Coiffure and Hirsute
  Bubbles
  We study the fields of spherically symmetric thin shell sources, a.k.a. Dyson
spheres, in a {\it fully nonlinear covariant} theory of gravity with the
simplest galileon field. We integrate exactly all the field equations once,
reducing them to first order nonlinear equations. For the simplest galileon,
static solutions come on {\it six} distinct branches. On one, a Dyson sphere
surrounds itself with a galileon hair, which far away looks like a hair of any
Brans-Dicke field. The hair changes below the Vainshtein scale, where the extra
galileon terms dominate the minimal gradients of the field. Their hair looks
more like a fuzz, because the galileon terms are suppressed by the derivative
of the volume determinant. It shuts off the `hair bunching' over the `angular'
2-sphere. Hence the fuzz remains dilute even close to the source. This is
really why the Vainshtein's suppression of the modifications of gravity works
close to the source. On the other five branches, the static solutions are all
{\it singular} far from the source, and shuttered off from asymptotic infinity.
One of them, however, is really the self-accelerating branch, and the
singularity is removed by turning on time dependence. We give examples of
regulated solutions, where the Dyson sphere explodes outward, and its
self-accelerating side is nonsingular. These constructions may open channels
for nonperturbative transitions between branches, which need to be addressed
further to determine phenomenological viability of multi-branch gravities.
