  A cosmic axion, via the electromagnetic anomaly, induces an oscillating
electric dipole for the electron of frequency $m_a$ and strength
$\sim$(few)$\times 10^{-32}$ e-cm, two orders of magnitude above the nucleon,
and within a few orders of magnitude of the present standard model constant
limit. We give a detailed study of this phenomenon via the interaction of the
cosmic axion, through the electromagnetic anomaly, with particular emphasis on
the decoupling limit of the axion, $\partial_t a(t)\propto m_a \rightarrow 0$.
The analysis is subtle, and we find the general form of the action involves a
local contact interaction and a nonlocal contribution, analogous to the
"transverse current" in QED, that enforces the decoupling limit. We carefully
derive the effective action in the Pauli-Schroedinger non-relativistic
formalism, and in Georgi's heavy quark formalism adapted to the "heavy
electron" ($m_e>>m_a$). We compute the electric dipole radiation emitted by
free electrons, magnets and currents, immersed in the cosmic axion field, and
discuss experimental configurations that may yield a detectable signal.
