The Age and Size of the Universe: Blog 27: Worksheet 8.1, Problem 3

It is not strictly correct to associate this ubiquitous distance-dependent redshift we observe with the velocity of the galaxies (at very large separations, Hubble’s Law gives ‘velocities’ that exceeds the speed of light and becomes poorly defined). What we have measured is the cosmological redshift, which is actually due to the overall expansion of the universe itself. This phenomenon is dubbed the Hubble Flow, and it is due to space itself being stretched in an expanding universe. Since everything seems to be getting away from us, you might be tempted to imagine we are located at the centre of this expansion. But, as you explored in the opening thought experiment, in actuality, everything is rushing away from everything else, everywhere in the universe, in the same way. So, an alien astronomer observing the motion of galaxies in its locality would arrive at the same conclusions we do. In cosmology, the scale factor, a(t), is a dimensionless parameter that characterizes the size of the universe and the amount of space in between grid points in the universe at time t. In the current epoch, t = \(t_0\) and \( a(t)_0 = 1\) a(t) is a function of time. It changes over time, and it was smaller in the past (since the universe is expanding). This means that two galaxies in the Hubble Flow separated by distance \(d_0 = d(t_0) \) in the present were \(d(t) = a(t)d_0 \)apart at time t. The Hubble Constant is also a function of time, and is defined so as to characterize the fractional rate of change of the scale factor: \[H(t) = \frac{1}{a(t)} \frac{ da}{ dt}|_t \] and the Hubble Law is locally valid for any t: \[v = H(t)d\] where v is the relative recessional velocity between two points and d the distance that separates them.

Part A: Assume the rate of expansion, a = da/dt, has been constant for all time. How long ago was the Big Bang (i.e. when a(t=0) = 0)? How does this compare with the age of the oldest globular clusters (= 12 Gyr)? What you will calculate is known as the Hubble Time.

Since we know \[v = H(t)d\] \[H(t) = \frac{v}{d} = \frac{1}{t} \] So \[ t_0 = \frac{1}{H(t)} = \frac{1}{68.816 \frac{km/s}{Mpc}} = 0.01453 \frac{Mpc \cdot s}{km} \] This is not a very useful number for us, so we can convert it to years: \[ 0.01453 \frac{Mpc \cdot s}{km} \cdot \frac{ 3.086 \times 10^{19} \: km}{Mpc} \cdot \frac{ 1 year }{3.15 \times 10^7}  = 1.385 \times 10^{10} \: years \] This is pretty close! The actual age of the universe is 13.82 billion years. This means the oldest globular clusters formed less than 2 billion years after the big bang.

Part B:  What is the size of the observable universe? What you will calculate is known as the Hubble Length.

Once again, we know \[ v = H(t)d\] So, if the universe is expanding at the speed of light and the speed of light in km/s is \( 3 \times 10^{5} \). \[ d = \frac{c}{H(t)} \] \[ d = \frac{3 \times 10^{5}}{68.816} = 4,356 Mpc \approx 4 \times 10^{3} Mpc \] The actual size of the universe is about \( 3 \times 10^{3} \) Mpc, so we aren't too far off.