Spatial variation of fundamental couplings and Lunar Laser Ranging

If the fundamental constants of nature have a cosmic spatial variation, there will in general be extra forces with a preferred direction in space which violate the equivalence principle. We show that the millimeter-precision Apache Point Observatory Lunar Laser-ranging Operation provides a very sensitive probe of such variation that has the capability of detecting a cosmic gradient of the ratio between the quark masses and the strong interaction scale at the level with the gradient of ln (m_quark/Lambda_QCD) ~ 2.6 x 10^-6 Glyr^-1, which is comparable to the cosmic gradients suggested by the recently reported measurements of Webb et al. We also point out the capability of presently planned improved equivalence principle tests, at the Delta g/g < 10^-17 level, to probe similar cosmic gradients.