Generalized McVittie geometry in Horndeski gravity with matter
  We investigate McVittie and generalized McVittie solutions for Horndeski
gravity with a spatially homogeneous gravitational scalar field, which is
stealth at small scales near the central object but, at large scales, sources
the FLRW universe in which the central inhomogeneity is embedded. Unlike
previous studies, we include matter and obtain generalized McVittie solutions
in the extended cuscuton model. The possible configurations are classified
according to the time-dependence of the gravitational coupling, the radial
energy flow, the accretion rate onto the central object, and the Hubble rate.
