The Local Effects of Cosmological Variations in Physical 'Constants' and   Scalar Fields I. Spherically Symmetric Spacetimes
  We apply the method of matched asymptotic expansions to analyse whether cosmological variations in physical `constants' and scalar fields are detectable, locally, on the surface of local gravitationally bound systems such as planets and stars, or inside virialised systems like galaxies and clusters. We assume spherical symmetry and derive a sufficient condition for the local time variation of the scalar fields that drive varying constants to track the cosmological one. We calculate a number of specific examples in detail by matching the Schwarzschild spacetime to spherically symmetric inhomogeneous Tolman-Bondi metrics in an intermediate region by rigorously construction matched asymptotic expansions on cosmological and local astronomical scales which overlap in an intermediate domain. We conclude that, independent of the details of the scalar-field theory describing the varying `constant', the condition for cosmological variations to be measured locally is almost always satisfied in physically realistic situations. The proof of this statement provides a rigorous justification for using terrestrial experiments and solar system observations to constrain or detect any cosmological time variations in the traditional `constants' of Nature. 
Douglas J. Shaw and John D. Barrow