Look Back Time, the Age of the Universe, and the Case for a Positive Cosmological Constant

We present explicit expressions for the calculation of cosmological look back time, for zero cosmological constant and arbitrary density parameter $\Omega$, which, in the limit as redshift becomes infinite, give the age of the universe. The case for non-zero cosmological constant is most easily solved via numerical integration. The most distant objects presently known (approaching redshift z = 5) have implied ages of $\approx$ 1-2 Gyr after the the Big Bang. The range of such age is narrow, in spite of a variety of cosmological models one might choose. We give a graphical representation of a variety of cosmological models and show that a wide range of Hubble constants and values of the age and density of the universe compatible with modern studies are consistent with adoption of a positive cosmological constant.