Isotropic and semi-isotropic observations in cosmology

We investigate the geometric consequences for dust universes of isotropies within the set of idealized galactic observations - -area distance, number counts, proper motions and image distortion. Anisotropy in the Hubble parameter is shown to indicate non-zero proper motions. If the proper motions and distortion are assumed to be zero (semi-isotropic observations), then the vorticity vanishes, the Hubble and deceleration parameters are isotropic, and Einstein's equations strongly constrain the area distance and the number counts. These constraints allow limited anisotropy, and in principle they provide indirect tests for the presence of proper motions and distortion. If, further, we assume isotropy of the area distance and number counts (isotropic observations), then we prove that Einstein's equations force the spacetime geometry to be isotropic. With the Copernican principle, this leads to an observational proof of the standard FLRW model.