  It is widely believed that the perovskite Sr$_2$RuO$_4$ is an unconventional
superconductor with broken time reversal symmetry. It has been predicted that
superconductors with broken time reversal symmetry should have spontaneously
generated supercurrents at edges and domain walls. We have done careful imaging
of the magnetic fields above Sr$_2$RuO$_4$ single crystals using scanning Hall
bar and SQUID microscopies, and see no evidence for such spontaneously
generated supercurrents. We use the results from our magnetic imaging to place
upper limits on the spontaneously generated supercurrents at edges and domain
walls as a function of domain size. For a single domain, this upper limit is
below the predicted signal by two orders of magnitude. We speculate on the
causes and implications of the lack of large spontaneous supercurrents in this
very interesting superconducting system.
