  A search for a problem free cosmology within the framework of an effective
non - minimally coupled scalar tensor theory is suggested. With appropriate
choice of couplings in variants of a Lee - Wick model [as also in a model
supporting Q - ball solutions], non topological solutions [NTS's], varying in
size upto 10 kpc to 1 Mpc can exist. We explore the properties of a ``toy''
Milne model containing a distribution of NTS domains. The interior of these
domains would be regions where effective gravitational effects would be
indistinguishable from those expected in standard Einstein theory. For a large
class of non - minimal coupling terms and the scalar effective potential, the
effective cosmological constant identically vanishes. The model passes
classical cosmological tests and we describe reasons to expect it to fare well
as regards nucleosynthesis and structure formation.
