Expanding Universe#

Scale Factor and Comoving Distance#

To begin talking about the expanding universe, one must have the notion of the scale factor and comoving distance. Imagine a cartesian grid drawn on the universe. Each tick mark in the grid corresponds to the comoving distance and the distance of each tick marks corresponds to the scale factor.

We may interpret this with defining distance \(r\) in terms of the the scale factor \(a\) and co-moving distance \(\chi\). A stationary object \(\chi\) co-moving distance away is actually \(r\) away where \(r\) is,

\[ r(t) = a(t) \cdot \chi \]

Notice that \(a(t)\) is a function of time but not \(\chi\). If the object is not moving relative to the co-moving space \(\dot\chi = 0\) then the exansion of the universe keeps \(\chi\) constant.

Hubble’s Law#

Hubble’s law came from an empirical observation of the universe (like most cosmology), at which Hubble is credited with noticing (although poorly) the receding velocity (velocity going away from observer) of an object in space seems to be correlated with the distance away from the observer (in general, Earth; we will correct this misconception later). This relationship is written as the Hubble’s law:

\[ \begin{equation} \boxed{v_r(t) = H(t) \cdot r} \end{equation} \]

  • \(v_r\) : Receding velocity

  • \(r\) : Distance of the object

  • \(H(t)\) : Hubble’s constant, dependent on time

If time \(t\) is today then the Hubble constant becomes,

\[ H_0 = \frac{v_r(t_0)}{r} = 100 \frac{\text{km/s}}{\text{Mpc }\mathrm{h^{-1}}} \approx 72 \; \frac{\text{km/s}}{\text{Mpc}}\]

  • \(h\) : a empirical constant absorbed into the units, not to be confused with Planck’s constant. One may often use \(h = 0.72\)

This literally means “for every megaparsec, the object recedes away from the observer with a velocity of \(70\; \text{m/s}\) purely due to the expansion of space.”

A general intuition of the Hubble’s constant is the percentage of change in the universe’s space. We characterize the universe’s space or “length-like” by the scale factor such that,

\[ \begin{equation} \boxed{H(t) \equiv \frac{a'(t)}{a(t)}} \end{equation} \]