Dinitz conjecture

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: $13.798\pm 0.037\,\mathrm {Gyr}$ 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 $a(t)={\frac {1}{1+z}}$.

See also

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

External links

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