Scale factor (cosmology)

The scale factor, cosmic scale factor or sometimes the Robertson-Walker scale factor 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. 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: $13.799\pm 0.021\,\mathrm {Gyr}$ 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 ${\dot {d}}(t)=Hd(t)$ 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. 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.

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 $a(t)={\frac {1}{1+z}}$.

Chronology

Further information: Chronology of the universe

Radiation-dominated era

After Inflation), and until about 47,000 years after the Big Bang, the dynamics of the early universe were set by radiation (referring generally to the constituents of the universe which moved relativistically, principally photons and neutrinos).

For a radiation-dominated universe the evolution of the scale factor in the Friedmann–Lemaître–Robertson–Walker metric is obtained solving the Friedmann equations:

$a(t)\propto t^{1/2}.\,$

Matter-dominated era

Between about 47,000 years and 9.8 billion years after the Big Bang, the energy density of matter exceeded both the energy density of radiation and the vacuum energy density.

When the early universe was about 47,000 years old (redshift 3600), mass–energy density surpassed the radiation energy, although the universe remained optically thick to radiation until the universe was about 378,000 years old (redshift 1100). This second moment in time (close to the time of recombination) at which point the photons which compose the cosmic microwave background radiation were last scattered, is often mistaken as marking the end of the radiation era.

For a matter dominated universe the evolution of the scale factor in the Friedmann-Lemaitre-Robertson-Walker metric is easily obtained solving the Friedmann equations:

$a(t)\propto t^{2/3}$

Dark energy-dominated era

In physical cosmology, the dark-energy-dominated era refers to the last of the three phases of the known universe, the other two being the matter-dominated era and the radiation-dominated era. The dark-energy-dominated era began after the matter-dominated era, i.e. when the Universe was about 9.8 billion years old. As other forms of the matter – dust and radiation – dropped to very low concentrations, the cosmological constant term started to dominate the energy density of the Universe.

For a dark-energy-dominated universe, the evolution of the scale factor in the Friedmann–Lemaître–Robertson–Walker metric is easily obtained solving the Friedmann equations:

$a(t)\propto \exp(Ht)$

Here, the coefficient H in the exponential, the Hubble constant, is

$H={\sqrt {8\pi G\rho _{\mathrm {full} }/3}}={\sqrt {\Lambda /3}}.$

This exponential dependence on time makes the spacetime geometry identical to the de Sitter Universe, and only holds for a positive sign of the cosmological constant, the sign that was observed to be realized in Nature anyway. The current density of the observable universe is of the order of 9.44 x 10⁻²⁷kg m⁻³ and the age of the universe is of the order of 13.8 billion years, or 4.358 x 10¹⁷s. The Hubble parameter, H, is ~70.88 km s⁻¹Mpc⁻¹. (The Hubble time is 13.79 billion years.) The value of the cosmological constant, Λ, is ~2 x 10⁻³⁵s⁻².

See also

• Cosmological principle
• Lambda-CDM model
• Redshift

References

• Padmanabhan, Thanu (1993). Structure formation in the universe. Cambridge: Cambridge University Press. ISBN 0-521-42486-0.
• Spergel, D. N.; et al. (2003). "First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters". Astrophysical Journal Supplement. 148 (1): 175–194. arXiv:astro-ph/0302209. Bibcode:2003ApJS..148..175S. doi:10.1086/377226.

External links

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