Observations in Cosmology

Observations of cosmological effects in anisotropic, inhomogeneous cosmological models are discussed in detail, with numerical estimates. The first and last sections of the paper form a self-contained unit for readers who are unfamiliar with Riemannian geometry. The other sections contain mathematical derivations. Three assumptions are made: (i) that the universe is described by a Riemannian space time with slowly varying metric tensor; (ii) that light travels along null geodesics and obeys the usual area4ntensity law; and (ili) that the gravitational field is related to the matter by Einstein's field equations for dust. The third assumption is not needed for many of the results; the second assumption is proved in the geometric optics limit, assuming a general relativistic model. The importance of trying to observe angular variations in the various cosmological effects is emphasized. It is shown that otherwise unobservable anisotropies or inhomogeneities can easily give the observed order of magnitude and either sign for the acceleration parameter measured in any one direction via redshifts. A model-independent law for the apparent area of a distant object is given. Detailed equations for number counts and for apparent proper motions are given. It is pointed out that observations to date do not exclude the possible presence of anisotropies and inhomogeneities whose dynamical effects are comparable to the dynamical effects of the expansion. A new effect, the "distortion effect," is discussed. In any anisotropic model, all distant objects in a particular direction on the celestial sphere may appear distorted, with a definite preferential direction for their longest dimension. The effect gives a direct measurement of space-time curvature and may be observable. But a positive result of the measurement would not favor general relativity over other theories in which assumptions (i) and (ii) above hold.