Local effects of cosmological variations in physical “constants” and scalar fields. II. Quasispherical spacetimes

We investigate the conditions under which cosmological variations in physical “constants” and scalar fields are detectable on the surface of local gravitationally bound systems, such as planets, in nonspherically symmetric background spacetimes. The method of matched asymptotic expansions is used to deal with the large range of length scales that appear in the problem. We derive a sufficient condition for the local time variation of the scalar fields driving variations in constants to track their large-scale cosmological variation and show that this is consistent with our earlier conjecture derived from the spherically symmetric problem. We perform our analysis with spacetime backgrounds that are of Szekeres-Szafron type. They are approximately Schwarzschild in some locality and free of gravitational waves everywhere. At large distances, we assume that the spacetime matches smoothly onto a Friedmann background universe. We conclude that, independent of the details of the scalar-field theory describing the varying constant, the condition for its cosmological variations to be measured locally is almost always satisfied in physically realistic situations. The very small differences expected to be observed between different scales are quantified. This strengthens the proof given in our previous paper that local experiments see global variations by dropping the requirement of exact spherical symmetry. It 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 in the case where nonspherical inhomogeneities exist.

Figure 1

The surfaces $(t,r)$ are spheres, which are concentric to leading order in $\delta$ in both the exterior and interior limits.