ratio where the subscript is one of for baryons, for cold dark matter, for radiation (photons plus relativistic neutrinos), and for dark energy. Since the densities of various species scale as different powers of , e.g. for matter etc., the Friedmann equation can be conveniently rewritten in terms of the various density parameters as where is the equation of state parameter of dark energy, and assuming negligible neutrino mass (significant neutrino mass requires a more complex equation). The various parameters add up to by construction. In the general case this is integrated by computer to give the expansion history and also observable distance–redshift relations for any chosen values of the cosmological parameters, which can then be compared with observations such as supernovae and baryon acoustic oscillations. In the minimal 6-parameter Lambda-CDM model, it is assumed that curvature is zero and , so this simplifies to Observations show that the radiation density is very small today, ; if this term is neglected the above has an analytic solution where this is fairly accurate for or million years. Solving for gives the present age of the universe in terms of the other parameters. It follows that the transition from decelerating to accelerating expansion (the second derivative crossing zero) occurred when which evaluates to or for the best-fit parameters estimated from the Planck spacecraft. Historical development The discovery of the cosmic microwave background (CMB) in 1964 confirmed a key prediction of the Big Bang cosmology. From that point on, it was generally accepted that the universe started in a hot, dense state and has been expanding over time. The rate of expansion depends on the types of matter and energy present in the universe, and in particular, whether the total density is above or below the so-called critical density. During the 1970s, most attention focused on pure-baryonic models, but there were serious challenges explaining the formation of galaxies, given the small anisotropies in the CMB (upper limits at that time). In the early 1980s, it was realized that this could be resolved if cold dark matter dominated over the baryons, and the theory of cosmic inflation motivated models with critical density. During the 1980s, most research focused on cold dark matter with critical density in matter, around 95% CDM and 5% baryons: these showed success at forming galaxies and clusters of galaxies, but problems remained; notably, the model required a Hubble constant lower