  Shape Dynamics (SD) is a theory dynamically equivalent to vacuum General
Relativity (GR), which has a different set of symmetries. It trades refoliation
invariance, present in GR, for local 3-dimensional conformal invariance. This
contribution to the Loops 11 conference addresses one of the more urgent
questions regarding the equivalence: is it possible to incorporate normal
matter in the new framework? The answer is yes, in certain regimes. We present
general criteria for coupling and apply it to a few examples.The outcome
presents bounds and conditions on scalar densities (such as the Higgs potential
and the cosmological constant) not present in GR.
