It is known that the cosmological constant can be dynamically tuned to an arbitrary small value in classes of scalar tensor theories. The trouble with such schemes is that effective gravity itself vanishes. We explore the possibility of avoiding this “no-go” with a spatially varying effective gravity. We demonstrate this in principle with the non-minimally coupled scalar field having an additional coupling to a fermionic field. The expectation value of the scalar field gets anchored to a non-trivial value inside compact domains. But for the non-minimal coupling to the scalar curvature, these configurations are analogous to the non-topological solutions suggested by Lee and Wick. With non-minimal coupling, this leads to a peculiar spatial variation of effective gravity. As before, one can dynamically have the long distance (global) gravitational constant
and Λ, the cosmological constant, tending to zero. However, inside compact domains,
can be held to a universal (non-vanishing) value. Long distance gravitational effects turn out to be indistinguishable from those expected of general theory of relativity (GTR). There are two ways in which the ensuing theory may lead to a viable effective gravity theory: a) the compact domains could be of microscopic (sub-nuclear) size, or b) the domains could be large enough to accommodate structures as large as a typical galaxy. Aspects of effective gravity and cosmology that follow are described. A toy Freidman-Robertson-Walker (FRW) model free from several standard model pathologies and characteristic features emerges. Although this proposed model remained less studied but became an important aspect in initiating a viable approach in addressing the cosmological constant problem. Comparison with some additional observational constraints like quantitative constraints on gravitational constant. G, from some high precision tests will be included our forthcoming paper which will address the limitations of the present paper.
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