The expansion of the universe is parameterized by a dimensionless scale factor a = a ( t ) (with time t counted from the birth of the universe), defined relative to the present time, so a 0 = a ( t 0 ) = 1 ; the usual convention in cosmology is that subscript 0 denotes present-day values, so t 0 denotes the age of the universe. The scale factor is related to the observed redshift z of the light emitted at time t e m by