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A search for a problem free cosmology within the framework of an effectivenon - minimally coupled scalar tensor theory is suggested. With appropriatechoice of couplings in variants of a Lee - Wick model [as also in a modelsupporting Q - ball solutions], non topological solutions [NTS's], varying insize 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 thesedomains would be regions where effective gravitational effects would beindistinguishable from those expected in standard Einstein theory. For a largeclass of non - minimal coupling terms and the scalar effective potential, theeffective cosmological constant identically vanishes. The model passesclassical cosmological tests and we describe reasons to expect it to fare wellas regards nucleosynthesis and structure formation.Comment: 20 pages, Plain Tex, references added and expanded the previous  version of article, 2 figures available from dlohiya@ducos.ernet.i

A programme for a problem-free cosmology within the framework of a rich class of scalar-tensor theories