  We present an analysis of the redshift-space power spectrum, $P(k)$, of rich
clusters of galaxies based on an automated cluster catalogue selected from the
APM Galaxy Survey. We find that $P(k)$ can be approximated by a power law,
$P(k)\proptok^{n}$, with $n\approx-1.6$ over the wavenumber range
$0.04\hr<k<0.1\hr$. Over this range of wavenumbers, the APM cluster power
spectrum has the same shape as the power spectra measured for optical and IRAS
galaxies. This is consistent with a simple linear bias model in which different
tracers have the same power spectrum as that of the mass distribution but
shifted in amplitude by a constant biasing factor. On larger scales, the power
spectrum of APM clusters flattens and appears to turn over on a scale $k \sim
0.03\hmpcrev$. We compare the power spectra estimated from simulated APM
cluster catalogues to those estimated directly from cubical N-body simulation
volumes and find that the APM cluster survey should give reliable estimates of
the true power spectrum at wavenumbers $k \simgt 0.02\hmpcrev$. These results
suggest that the observed turn-over in the power spectrum may be a real feature
of the cluster distribution and that we have detected the transition to a near
scale-invariant power spectrum implied by observations of anisotropies in the
cosmic microwave background radiation. The scale of the turn-over in the
cluster power spectrum is in good agreement with the scale of the turn-over
observed in the power spectrum of APM galaxies.
