### Scale factor (universe)

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 galaxies, 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:

- $d(t)\; =\; a(t)d\_0,\backslash ,$

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: $13.798\backslash pm0.037\backslash ,\backslash mathrm\{Gyr\}$^{[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:

- $H\; \backslash equiv\; \{\backslash dot\{a\}(t)\; \backslash over\; a(t)\}$

where the dot represents a time derivative. From the previous equation $d(t)\; =\; d\_0\; a(t)$ one can see that $\backslash dot\{d\}(t)\; =\; d\_0\; \backslash dot\{a\}(t)$, and also that $d\_0\; =\; \backslash frac\{d(t)\}\{a(t)\}$, so combining these gives $\backslash dot\{d\}(t)\; =\; \backslash frac\{d(t)\; \backslash dot\{a\}(t)\}\{a(t)\}$, and substituting the above definition of the Hubble parameter gives $\backslash dot\{d\}(t)\; =\; H\; d(t)$ 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 $\backslash ddot\{a\}(t)$ is positive, or equivalently that the first derivative $\backslash dot\{a\}(t)$ 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 $\backslash dot\{d\}(t)$ 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 $a(t)\; =\; \backslash frac\{1\}\{1\; +\; z\}$.^{[6]}^{[7]}

## See also

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

## References

## External links

- Relation of the scale factor with the cosmological constant and the Hubble constanteo:Universa krusta faktoro