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: $13.799\pm0.021\,\mathrm{Gyr}$^[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]

See also

• Cosmological principle
• Lambda-CDM model
• Redshift

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 (2015). "Planck 2015 results. XIII. Cosmological parameters (See Table 4 on page 31 of pfd).". arXiv:1502.01589. Bibcode:2015arXiv150201589P.
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) Archived November 28, 2010, at the Wayback Machine.
6. Davies, Paul (1992), The New Physics, p. 187.
7. Mukhanov, V. F. (2005), Physical Foundations of Cosmology, p. 58.

External links

• Relation of the scale factor with the cosmological constant and the Hubble constant