The linear power spectrum of observed source number counts

This document is made available in accordance with publisher policies and may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the URL above for details on accessing the published version. Copyright and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners. To the extent reasonable and practicable, the material made available in SRO has been checked for eligibility before being made available. Copies of full text items generally can be reproduced, displayed or performed and given to third parties in any format or medium for personal research or study, educational, or not-for-profit purposes without prior permission or charge, provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way. 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 subhorizon 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.