Age of The Universe - Cosmological Parameters

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 Ω_m, Ω_r, and Ω_Λ. 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 H₀, 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 a(t) 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 t₀ is then given by an expression of the form

where H₀ is the Hubble parameter and the function F 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, or .

The Hubble constant measures how fast the universe is expanding and is usually given in units of kilometers per second per megaparsec (km/s/mpc)—the further away an object is, the faster it recedes. One megaparsec = 3.086×1019 km, so if H₀ = 71 km/s/mpc, = 2.30×10−18 s−1, and the value of Hubble time is 4.35×1017 s or 13.8 billion years.

To get a more accurate number, the correction factor F 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 WMAP values (Ω_m, Ω_Λ) = (0.266, 0.732), shown by the box in the upper left corner of the figure, this correction factor is nearly one: F = 0.996. For a flat universe without any cosmological constant, shown by the star in the lower right corner, F = 2⁄₃ is much smaller and thus the universe is younger for a fixed value of the Hubble parameter. To make this figure, Ω_r is held constant (roughly equivalent to holding the CMB temperature constant) and the curvature density parameter is fixed by the value of the other three.

The Wilkinson Microwave Anisotropy Probe (WMAP) was instrumental in establishing an accurate age of the universe, though other measurements must be folded in to gain an accurate number. CMB measurements are very good at constraining the matter content Ω_m and curvature parameter Ω_k. It is not as sensitive to Ω_Λ directly, partly because the cosmological constant becomes important only at low redshift. The most accurate determinations of the Hubble parameter H₀ come from Type Ia supernovae. Combining these measurements leads to the generally accepted value for the age of the universe quoted above.

The cosmological constant makes the universe "older" for fixed values of the other parameters. This is significant, since before the cosmological constant became generally accepted, the Big Bang model had difficulty explaining why globular clusters in the Milky Way appeared to be far older than the age of the universe as calculated from the Hubble parameter and a matter-only universe. Introducing the cosmological constant allows the universe to be older than these clusters, as well as explaining other features that the matter-only cosmological model could not.