Scale factor (cosmology)

The scale factor or cosmic scale factor parameter of the Friedmann equations is a function of time which represents the relative expansion of the universe. It is sometimes called the Robertson-Walker scale factor.^[1] It is the (time-dependent) factor that relates the proper distance (which can change over time, unlike the comoving distance which is constant) for a pair of objects moving with the Hubble flow in an expanding or contracting FLRW universe—the distance between a pair of galaxies, for example—at any arbitrary time t to their distance at some reference time, generally taken to be the present, by the formula:

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

where d(t) is the proper distance at epoch t, d₀ is the distance at the reference time t₀ (the present epoch), and a(t) is the scale factor.^[2] Thus, by definition, a(t₀) = 1.

The scale factor could, in principle, have units of length^{[clarification needed]} or be dimensionless. Most commonly in modern usage, it is chosen to be dimensionless, with the current value equal to one: a(t₀) = 1, where t is counted from the birth of the universe and t₀ is the present age of the universe: $13.75\pm0.17\,\mathrm{Gyr}$.^[3]

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) = d₀a(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 given by the equation $a(t) = \frac{1}{1 + z}$.^[6][7]

See also

• Friedmann equations
• Friedmann-Lemaître-Robertson-Walker metric
• Redshift
• Cosmological principle
• Lambda-CDM model
• Hubble's law

References

1. ^ Steven Weinberg (2008). Cosmology. Oxford University Press. p. 3. ISBN 9780198526827. http://books.google.com/books?id=48C-ym2EmZkC&pg=PA3. 
2. ^ Schutz, Bernard (2003). Gravity from the Ground Up: An Introductory Guide to Gravity and General Relativity. Cambridge University Press. p. 363. ISBN 978-0521455060. 
3. ^ S. H. Suyu, P. J. Marshall, M. W. Auger, S. Hilbert, R. D. Blandford, L. V. E. Koopmans, C. D. Fassnacht and T. Treu. Dissecting the Gravitational Lens B1608+656. II. Precision Measurements of the Hubble Constant, Spatial Curvature, and the Dark Energy Equation of State. The Astrophysical Journal, 2010; 711 (1): 201 DOI: 10.1088/0004-637X/711/1/201
4. ^ Jones, Mark H.; Robert J. Lambourne (2004). An Introduction to Galaxies and Cosmology. Cambridge University Press. p. 244. ISBN 978-0521837385. 
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.

External links

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