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 ) \propto k ^ { n } , with n \approx - 1.6 over the wavenumber range 0.04 { h } { Mpc } ^ { -1 } < k < 0.1 { h } { Mpc } ^ { -1 } .
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 { h } { Mpc } ^ { -1 } .
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 \mathrel { \hbox to 0.0 pt { \lower 3.0 pt \hbox { $ \mathchar 536 $ } \hss } \raise 2.0 pt% \hbox { $ \mathchar 318 $ } } 0.02 { h } { Mpc } ^ { -1 } .
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 .