Quantifying the uncertainty in the time-redshift relationship

The age of the Universe at a given redshift is a fundamental relationship in cosmology. For many years, the uncertainties in it were dauntingly large, close to a factor of 2. In this age of precision cosmology, they are now at the percent level and dominated by the uncertainty in the Hubble constant. The uncertainties due to the parameters that describe the current cosmological model, $\Lambda$CDM, are much less important. In decreasing order they are: uncertainty due to the matter density $\Omega_M$ around 0.9\% for $z> 3$; uncertainty due to the dark energy equation-of-state parameter $w$, less than 0.3\% for $z>3$; and uncertainty due to the curvature parameter $\Omega_k$, at most 0.07\%.