Blog #27: Size of the Universe

3) It is not strictly correct to associate this ubiquitous distance-dependent redshift we observe with teh 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 center 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 the 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 small amount of space in between grid points in the universe at time $t$. In the current epoch, $t = t_0$ and $a(t_0) \equiv 1$. $a(t)$ 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:

\begin{align*}
H(t) = \frac{1}{a(t)}\frac{da}{dt} \Big|_t
\end{align*}

and the Hubble Law is locally valid for any $t$: 

\begin{align*}
v = H(t)d
\end{align*}

where $v$ is the relative recessional velocity between two points and $d$ the distance that separates them. 

(a) Assume the rate of expansion, $\dot{a} \equiv \frac{da}{dt}$, 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.

In order to solve for the moment when the Big Bang occurred, we need to solve for $t_0$.

We can start with the equation given to us:


\begin{align*}
H(t) = \frac{1}{a(t)}\frac{da}{dt} \Big|_t
\end{align*}

We know that $\dot{a} \equiv \frac{da}{dt}$ and $a(t_0) = 1$, which we can substitute in this equation:

\begin{align*}
H(t) &= \frac{1}{a(t)}\frac{da}{dt} \Big|_t\\
H(t_0) &= \frac{1}{a(t_0)}\frac{da}{dt}\\
H_0 &=\frac{da}{dt}\\
H_0 dt &= da\\
\int_0^{t_0} H_0dt &= \int_0^{a(t_0) = 1} da\\
H_0 t_0 &= 1\\
t_0 = \frac{1}{H_0}
\end{align*}

A quick Google search shows that the value for $H_0$ is about $67.8 \frac{\frac{km}{s}}{Mpc}, which converted into seconds is: $H_0 = 2.3 \times 10^{-18} \frac{1}{s}. Using this information, we can solve for $t_0$ as follows:

\begin{align*}
t_0 &= \frac{1}{H_0}\\
t_0 &= \frac{1}{2.3 \times 10^{-18} \frac{1}{s}}\\
t_0 &\approx 4.4 \times 10^{17}\text{ seconds}\\
t_0 &\approx 14 \text{ Gyr}
\end{align*}

This shows that the Hubble Time, which is the time at the beginning of the Universe, is about 14 billion years, which is about 2 billion years earlier than the earliest globular clusters. 

(b) What is the size of the observable universe? What you will calculate is known as the Hubble Length.

Distance is measured by rearranging the equation for velocity as follows:

\begin{align*}
d = vt
\end{align*}

Since we know that the time in this equation is the Hubble Time, and $v$ is the speed of light, $c$, which gives us:

\begin{align*}
d &= vt\\
d &= H_0c\\
d &=   (4.4 \times 10^{17}) \times (3 \times 10^{10})\\
d &= 1.32 \times 10^{28} \text{ cm}\\
d &= 1.4 \times 10^{10} \text{ light years}
\end{align*}

This shows that the size of the observable universe is $1.4 \times 10^{10}$ light years, which is also known as the Hubble Length.