Cosmologically active Brans-Dicke (or dilaton) scalar fields are generically
ruled out by solar system tests of gravity unless their couplings to ordinary
matter are much suppressed relative to gravitational strength, and this is a
major hindrance when building realistic models of light dilatons coupled to
matter. We propose a new mechanism for evading such bounds if matter also
couples to a light axion, that exploits nonlinear target-space curvature
interactions to qualitatively change how the fields respond to a gravitating
source. We find that dilaton-matter couplings that would be excluded in the
absence of an axion can become acceptable given an additional small
axion-matter coupling, and this is possible because the axion-dilaton
interactions end up converting the would-be dilaton profile into an axion
profile. The trajectories of matter test bodies are then controlled by the much
weaker axion-matter couplings and can easily be small enough to escape
detection. We call this mechanism Axion Homeopathy because the evasion of the
dilaton-coupling bounds persists for extremely small axion couplings provided
only that they are nonzero. We explore the mechanism using axio-dilaton
equations that are SL(2,R) invariant (as often appear in string
compactifications), since for these the general solutions exterior to a
spherically symmetric source can be found analytically. We use this solution to
compute the relevant PPN parameters, $\beta$ and $\gamma$, and verify that
their difference from unity can be much smaller than would have been true in
the absence of axion-matter couplings and so can therefore evade the
experimental bounds.