# Scale factor (cosmology)

### Scale factor (cosmology)

The scale factor, cosmic scale factor or sometimes the Robertson-Walker scale factor[1] parameter of the Friedmann equations is a function of time which represents the relative expansion of the universe. 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 to their distance at some reference time t_0. The formula for this is:

d(t) = a(t)d_0,\,

where d(t) is the proper distance at epoch t, d_0 is the distance at the reference time t_0 and a(t) is the scale factor.[2] Thus, by definition, a(t_0) = 1.

The scale factor is dimensionless, with t counted from the birth of the universe and t_0 set to the present age of the universe: 13.798\pm0.037\,\mathrm{Gyr}[3] giving the current value of a as a(t_0) or 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:

H \equiv {\dot{a}(t) \over a(t)}

where the dot represents a time derivative. From the previous equation d(t) = d_0 a(t) one can see that \dot{d}(t) = d_0 \dot{a}(t), and also that d_0 = \frac{d(t)}{a(t)}, so combining these gives \dot{d}(t) = \frac{d(t) \dot{a}(t)}{a(t)}, and substituting the above definition of the Hubble parameter gives \dot{d}(t) = H d(t) which is just Hubble's law.

Current evidence suggests that the expansion rate of the universe is accelerating, which means that the second derivative of the scale factor \ddot{a}(t) is positive, or equivalently that the first derivative \dot{a}(t) is increasing over time.[4] This also implies that any given galaxy recedes from us with increasing speed over time, i.e. for that galaxy \dot{d}(t) is increasing with time. In contrast, the Hubble parameter seems to be decreasing with time, meaning that if we were to look at some fixed distance d and watch a series of different galaxies pass that distance, later galaxies would pass that distance at a smaller velocity than earlier ones.[5]

According to the Friedmann–Lemaître–Robertson–Walker metric which is used to model the expanding universe, if at the present time we receive light from a distant object with a redshift of z, then the scale factor at the time the object originally emitted that light is a(t) = \frac{1}{1 + z}.[6][7]

## References

1. ^ Steven Weinberg (2008). Cosmology. Oxford University Press. p. 3.
2. ^ Schutz, Bernard (2003). Gravity from the Ground Up: An Introductory Guide to Gravity and General Relativity. Cambridge University Press. p. 363.
3. ^ Planck collaboration (2013). "Planck 2013 results. I. Overview of products and scientific results". Submitted to Astronomy & Astrophysics.
4. ^ Jones, Mark H.; Robert J. Lambourne (2004). An Introduction to Galaxies and Cosmology. Cambridge University Press. p. 244.
5. ^ Is the universe expanding faster than the speed of light? (see final paragraph)
6. ^ Davies, Paul (1992), The New Physics, p. 187.
7. ^ Mukhanov, V. F. (2005), Physical Foundations of Cosmology, p. 58.