Friedmann Equations - Density Parameter

Density Parameter

The density parameter, is defined as the ratio of the actual (or observed) density to the critical density of the Friedmann universe. The relation between the actual density and the critical density determines the overall geometry of the universe. In earlier models, which did not include a cosmological constant term, critical density was regarded also as the watershed between an expanding and a contracting Universe.

To date, the critical density is estimated to be approximately five atoms (of monatomic hydrogen) per cubic metre, whereas the average density of ordinary matter in the Universe is believed to be 0.2 atoms per cubic metre.

A much greater density comes from the unidentified dark matter; both ordinary and dark matter contribute in favor of contraction of the universe. However, the largest part comes from so-called dark energy, which accounts for the cosmological constant term. Although the total density is equal to the critical density (exactly, up to measurement error), the dark energy does not lead to contraction of the universe but rather accelerates its expansion. Therefore, the universe will expand forever.

An expression for the critical density is found by assuming Λ to be zero (as it is for all basic Friedmann universes) and setting the normalised spatial curvature, k, equal to zero. When the substitutions are applied to the first of the Friedmann equations we find:

The density parameter (useful for comparing different cosmological models) is then defined as:

This term originally was used as a means to determine the spatial geometry of the universe, where is the critical density for which the spatial geometry is flat (or Euclidean). Assuming a zero vacuum energy density, if is larger than unity, the space sections of the universe are closed; the universe will eventually stop expanding, then collapse. If is less than unity, they are open; and the universe expands forever. However, one can also subsume the spatial curvature and vacuum energy terms into a more general expression for in which case this density parameter equals exactly unity. Then it is a matter of measuring the different components, usually designated by subscripts. According to the ΛCDM model, there are important components of due to baryons, cold dark matter and dark energy. The spatial geometry of the universe has been measured by the WMAP spacecraft to be nearly flat. This means that the universe can be well approximated by a model where the spatial curvature parameter is zero; however, this does not necessarily imply that the universe is infinite: it might merely be that the universe is much larger than the part we see. (Similarly, the fact that Earth is approximately flat at the scale of the Netherlands does not imply that the Earth is flat: it only implies that it is much larger than the Netherlands.)

The first Friedmann equation is often seen in a form with density parameters.

Here is the radiation density today (i.e. when ), is the matter (dark plus baryonic) density today, is the "spatial curvature density" today, and is the cosmological constant or vacuum density today.