What can the spatial distribution of galaxy clusters tell about their
  scaling relations?
  We aim to quantify the capability of the inhomogeneous distribution of galaxy
clusters, represented by the two-point statistics in Fourier space, to retrieve
information on the underlying scaling relations. We make a case study using the
mass X-ray luminosity scaling relation for galaxy clusters and study its impact
on the clustering pattern of these objects. We define the luminosity-weighted
power spectrum and introduce the luminosity power spectrum as direct assessment
of the clustering of the property of interest, in our case, the cluster X-ray
luminosity. Using a suite of halo catalogs extracted from N-body simulations
and realistic estimates of the mass X-ray luminosity relation, we measured
these statistics with their corresponding covariance matrices. By carrying out
a Fisher matrix analysis, we quantified the content of information (by means of
a figure-of merit) encoded in the amplitude, shape, and full shape of our
probes for two-point statistics. The full shape of the luminosity power
spectrum, when analyzed up to scales of k~0.2 h/Mpc, yields a figure of merit
which is two orders of magnitude above the figure obtained from the unweighted
power spectrum, and only one order of magnitude below the value encoded in
X-ray luminosity function estimated from the same sample. This is a significant
improvement over the analysis developed with the standard (i.e., unweighted)
clustering probes. The measurements of the clustering of galaxy clusters and
its explicit dependence on the cluster intrinsic properties can contribute to
improving the degree of knowledge regarding the underlying links between
cluster observables and the cluster masses (Abridged).
