The linear power spectrum of observed source number counts
  We relate the observable number of sources per solid angle and redshift to
the underlying proper source density and velocity, background evolution and
line-of-sight potentials. We give an exact result in the case of linearized
perturbations assuming general relativity. This consistently includes
contributions of the source density perturbations and redshift distortions,
magnification, radial displacement, and various additional linear terms that
are small on sub-horizon scales. In addition we calculate the effect on
observed luminosities, and hence the result for sources observed as a function
of flux, including magnification bias and radial-displacement effects. We give
the corresponding linear result for a magnitude-limited survey at low redshift,
and discuss the angular power spectrum of the total count distribution. We also
calculate the cross-correlation with the CMB polarization and temperature
including Doppler source terms, magnification, redshift distortions and other
velocity effects for the sources, and discuss why the contribution of redshift
distortions is generally small. Finally we relate the result for source number
counts to that for the brightness of line radiation, for example 21-cm
radiation, from the sources.
