# Scale factor (cosmology)

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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. ISBN 978-0-19-852682-7.
2. ^ Schutz, Bernard (2003). Gravity from the Ground Up: An Introductory Guide to Gravity and General Relativity. Cambridge University Press. p. 363. ISBN 978-0-521-45506-0.
3. ^ Planck collaboration (2013). "Planck 2013 results. I. Overview of products and scientific results". Submitted to Astronomy & Astrophysics. arXiv:1303.5062. Bibcode:2014A&A...571A...1P. doi:10.1051/0004-6361/201321529.
4. ^ Jones, Mark H.; Robert J. Lambourne (2004). An Introduction to Galaxies and Cosmology. Cambridge University Press. p. 244. ISBN 978-0-521-83738-5.
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.