Scale factor (cosmology)

The relative expansion of the universe is parametrized by a dimensionless scale factor a {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. The relative expansion of the universe is parametrized by a dimensionless scale factor a {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 matter and radiation changed and the universe entered a matter-dominated era. Recently results suggest that we have already entered an era dominated by dark energy, but examination of the roles of matter 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 matter 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 4 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 t {displaystyle t} to their distance at some reference time t 0 {displaystyle t_{0}} . The formula for this is: where d ( t ) {displaystyle d(t)} is the proper distance at epoch t {displaystyle t} , d 0 {displaystyle d_{0}} is the distance at the reference time t 0 {displaystyle t_{0}} and a ( t ) {displaystyle a(t)} is the scale factor. Thus, by definition, a ( t 0 ) = 1 {displaystyle a(t_{0})=1} . The scale factor is dimensionless, with t {displaystyle t} counted from the birth of the universe and t 0 {displaystyle t_{0}} set to the present age of the universe: 13.799 ± 0.021 G y r {displaystyle 13.799pm 0.021,mathrm {Gyr} } giving the current value of a {displaystyle a} as a ( t 0 ) {displaystyle a(t_{0})} or 1 {displaystyle 1} . The evolution of the scale factor is a dynamical question, determined by the equations of general relativity, which are presented in the case of a locally isotropic, locally homogeneous universe by the Friedmann equations. The Hubble parameter is defined: where the dot represents a time derivative. From the previous equation d ( t ) = d 0 a ( t ) {displaystyle d(t)=d_{0}a(t)} one can see that d ˙ ( t ) = d 0 a ˙ ( t ) {displaystyle {dot {d}}(t)=d_{0}{dot {a}}(t)} , and also that d 0 = d ( t ) a ( t ) {displaystyle d_{0}={frac {d(t)}{a(t)}}} , so combining these gives d ˙ ( t ) = d ( t ) a ˙ ( t ) a ( t ) {displaystyle {dot {d}}(t)={frac {d(t){dot {a}}(t)}{a(t)}}} , and substituting the above definition of the Hubble parameter gives d ˙ ( t ) = H d ( t ) {displaystyle {dot {d}}(t)=Hd(t)} which is just Hubble's law.