2  Expanding the Scale Factor

Let us build some intuition for the scale factor by expanding about the current time \(t_{0}\) : \[ \begin{aligned} a(t_{1}) = a(t_{0}) + (t_{1}-t_{0})\dot{a}(t_{0}) + \dots\\ \frac{a(t_{1})}{a(t_{0})} = 1 + (t_{1}-t_{0}) H(t_{0}) + \dots \end{aligned} \] The quantity \(t_{1}-t_{0}\) is known as the lookback time. We can also write this in terms of redshift \[ \frac{1}{1+z} = 1 + (t_{1}-t_{0}) H(t_{0}) + \dots \] or \[ z \approx H_{0} (t_{0}-t_{1}) \] which shows us how we can relate the lookback time to the redshift (at low redshift/small lookback time). We can also relate this back to Hubble’s law by multiplying both sides by \(c\) to get \[ v = c z = H_{0} [c (t_{0}-t_{1})] = H_{0} d \] We can go to higher orders as well \[ \frac{a(t)}{a({t_{0}})} = 1 + H_{0}(t-t_{0}) - \frac{1}{2} q_{0} H_{0}^{2} (t-t_{0})^{2} + \dots \] where \[ q_{0} \equiv -\frac{\ddot{a}}{a H^2}\Bigg|_{t=t_{0}} \] is the “deceleration parameter”. This particular choice reflects the assumption that the expansion rate of the Universe would be decelerating with time and pre-dates the discovery of the accelerated expansion of the Universe and dark energy. Our current best-guess \(q_{0} \approx -0.5\) (note the negative sign indicating acceleration).

One can invert this expression to get \[ z = H_{0}(t_{0} - t) + \frac{1}{2} (2 + q_{0})H_{0}^{2}(t_{0}-t)^{2} + \dots \] or \[ H_{0}(t_{0}-t) = z - \frac{1}{2}(2+q_{0})^{2}z^{2} + \dots \] We note that these expressions are really only useful at low redshift - we will develop the tools necessary to extend these to higher redshifts later. However, they are very useful to get some intuition for how to interpret the scale factor and redshift.