Prospects for constraining cosmology with the extragalactic cosmic microwave background temperature

Observers have demonstrated that it is now feasible to measure the cosmic microwave background (CMB) temperature at high redshifts. We explore the possible constraints on cosmology which might ultimately be derived from such measurements. In addition to providing a consistency check on standard and alternative cosmologies, possibilities include constraints on the inhomogeneity and anisotropy of the universe at intermediate redshift $z\lesssim 10,$ an independent probe of peculiar motions with respect to the Hubble flow, constraining the epoch of reionization, and the time variability of the fine structure constant. We argue that the best possibility is as a probe of peculiar motions. We show, however, that the current measurement uncertainty $(\Delta T=\pm 0.002\mathrm{K})$ in the local present absolute CMB temperature imposes intrinsic limits on the use of such CMB temperature measurements as a cosmological probe. At best, anisotropies at intermediate redshift could only be constrained at a level of $\gtrsim 0.1\%$ and peculiar motions could only be determined to an uncertainty of $\gtrsim 311{\mathrm{km}\mathrm{}\mathrm{}\mathrm{s}}^{-1}.$ If the high z CMB temperature can only be measured with a precision comparable to the uncertainty of the local interstellar CMB temperature, then peculiar motions could be determined to an uncertainty of ${1101(1+z)\mathrm{}}^{-1}\left[\Delta T_{\mathrm{CMB}}\mathrm{}\right(z)/0.01\mathrm{}\mathrm{K}]{\mathrm{km}\mathrm{}\mathrm{s}}^{-1}.$