the main sequence, so the lowest-mass stars that have evolved away from the main sequence set a minimum age). Cosmological parameters The problem of determining the age of the universe is closely tied to the problem of determining the values of the cosmological parameters. Today this is largely carried out in the context of the ΛCDM model, where the universe is assumed to contain normal (baryonic) matter, cold dark matter, radiation (including both photons and neutrinos), and a cosmological constant. The fractional contribution of each to the current energy density of the universe is given by the density parameters and The full ΛCDM model is described by a number of other parameters, but for the purpose of computing its age these three, along with the Hubble parameter , are the most important. If one has accurate measurements of these parameters, then the age of the universe can be determined by using the Friedmann equation. This equation relates the rate of change in the scale factor to the matter content of the universe. Turning this relation around, we can calculate the change in time per change in scale factor and thus calculate the total age of the universe by integrating this formula. The age is then given by an expression of the form where is the Hubble parameter and the function depends only on the fractional contribution to the universe's energy content that comes from various components. The first observation that one can make from this formula is that it is the Hubble parameter that controls that age of the universe, with a correction arising from the matter and energy content. So a rough estimate of the age of the universe comes from the Hubble time, the inverse of the Hubble parameter. With a value for around , the Hubble time evaluates to billion years. To get a more accurate number, the correction function must be computed. In general this must be done numerically, and the results for a range of cosmological parameter values are shown in the figure. For the Planck values (0.3086, 0.6914), shown by the box in the upper left corner of the figure, this correction factor is about For a flat universe without any cosmological constant, shown by the star in the lower right corner, is much smaller and thus the universe is younger for a fixed value of the Hubble parameter. To make this figure, is