Scale factor (cosmology)

The relative expansion of the universe is parametrized by a dimensionless scale factor ${\displaystyle a}$. Also known as the cosmic scale factor or sometimes the Robertson-Walker scale factor, this is a key parameter of the Friedmann equations.

In the early stages of the big bang, most of the energy was in the form of radiation, and that radiation was the dominant influence on the expansion of the universe. Later, with cooling from the expansion the roles of mass and radiation changed and the universe entered a mass-dominated era. Recently results suggest that we have already entered an era dominated by dark energy, but examination of the roles of mass and radiation are most important for understanding the early universe.

Using the dimensionless scale factor to characterize the expansion of the universe, the effective energy densities of radiation and mass scale differently. This leads to a radiation-dominated era in the very early universe but a transition to a matter-dominated era at a later time and, since about 5 billion years ago, a subsequent dark energy-dominated era.

Some insight into the expansion can be obtained from a Newtonian expansion model which leads to a simplified version of the Friedman equation. It relates the proper distance (which can change over time, unlike the comoving distance which is constant) between a pair of objects, e.g. two galaxy clusters, moving with the Hubble flow in an expanding or contracting FLRW universe at any arbitrary time ${\displaystyle t}$ to their distance at some reference time ${\displaystyle t_{0}}$. The formula for this is:

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