  In light of recent experimental proposals to measure the free fall
acceleration of antihydrogen in the earth's gravitational field, we investigate
the bounds that existing experiments place on any asymmetry between the free
fall of matter and antimatter. We conclude that existing experiments constrain
any such asymmetry to be less than about 10^-7. First we consider contributions
to the inertial masses of atoms that encode the presence of antimatter and use
precision Eotvos experiments to establish the level at which they satisfy the
equivalence principle. In particular we focus on vacuum polarization effects
and the antiquark content of nucleons. Second we consider a class of theories
that contain long range scalar and vector forces that cancel with one another
to some high precision. By construction such theories would be able to evade
detection in Eotvos experiments that utilize matter while still allowing for a
signal in antimatter experiments. Even taking such cancellation for granted,
however, we show that the radiative damping of binary pulsar systems constrains
these forces to be significantly weaker than gravity. Furthermore we show that
there are limits to the accuracy with which such cancellation can be arranged:
first by determining the precision to which scalar charges can track vector
charges in the best candidate theories; and, second, by showing that the
different velocity dependence of scalar and vector forces necessarily
introduces non-cancellation at a quantifiable level.
