The Expanding Universe

We have known for almost 100 years that the universe is expanding. In the 1920s Edwin Hubble published his famous velocity–distance diagram showing that almost all galaxies are receding and that their velocity is proportional to their distance (Hubble 1929). This relationship is summarized in Hubble’s law,

$$v = H_0 D$$

where $H_0$ , the Hubble constant, is roughly $70\ {\rm km/s/Mpc}$ . This tells us that for every Megaparsec an object is away from us today, it tends to be receding at $70\ {\rm km/s}$ . So, for example, the Coma Cluster is $103\ {\rm Mpc}$ away and is receding at roughly $H_0 D = 70\ {\rm km/s/Mpc} \times 103\ {\rm Mpc} = 7210\ {\rm km/s}$ .

However, the fact that the universe is expanding now does not mean that it has always has been expanding or will continue to expand. How our universe will evolve in the future, and how it got to its current state, depends on its contents.

So what does the universe contain? An exhaustive list of stars, planets, people, etc would be hard to make, but fortunately at the scale of the universe we can gloss over many of those details. There are, we think, 3 fundamental things in the universe:

• Non-relativistic matter: Pretty much everything you can see and touch: stars, planets, books, air, etc are in this class. However, we don’t think that these everyday things make the majority of this type of matter. There is lots of evidence that roughly ~80% of matter acts slightly differently to “normal matter” and is very hard to detect directly. We don’t know exactly what this is yet but call it dark matter.
• Relativistic matter (or radiation): Anything that is traveling close to the speed of light. It is really hard to get massive objects to move that fast so most of the things traveling at the speed of light are photons (i.e., light).
• Dark energy: Something that is pushing everything apart. We don’t know what this is, though there are many propsed variants. For the examples here we will assume that dark energy is well modeled by a cosmological constant.

Working out how much of these three components there are (literally the number of kilograms of matter per cubic meter on average over the whole universe) has been one of the major goals of cosmology. We’re now fairly sure that, for matter, the density today ($\rho_{M, 0}$ ) is roughly $3 \times 10^{-27}\ {\rm kg / m^3}$ , or less than 2 hydrogen atoms per cubic meter. Note that the $0$ subscript in $\rho_{M, 0}$ indicates that this is the value now.

These units are not particularly useful though. Instead, the density is usually expressed in units of a critical density. This is the density at which, in a matter dominated universe, the universe’s expansion rate would asymptote to 0 at late times.

$$\rho_{crit, 0} = \frac{3 H_0^2}{8 \pi G} = 9 \times 10^{-27}\ {\rm kg / m^3}$$

We can define the amount of matter, radiation and dark energy relative to this critical density. For example, for our universe today, the densities are:

matter,

$$\Omega_{M, 0} = \frac{\rho_{M, 0}}{\rho_{crit, 0}} = 0.31$$

dark energy,

$$\Omega_{\Lambda, 0} = \frac{\rho_{\Lambda, 0}}{\rho_{crit, 0}} = 0.69$$

radiation, $$\Omega_{R, 0} = \frac{\rho_{R, 0}}{\rho_{crit, 0}} = 8 \times 10^{-5}$$

You’ll notice that the sum of these values is suspciously close to 1 and so the density is close to the critical density.

How different would our universe’s history be if these were changed? Below we show the size of the universe over time for a variety of cosmologies. The size is expressed relative to the size today – at the time of $H_0 = 70$ . This is called the scale factor ($a$ ) which is 1 today. Note that all cosmologies are consistent with our local observations – the expansion rate is the same at $t = 0$ .

Hover over each line to learn more about the cosmology of these universes.

Our universe (green) can be well approximated by a dark energy only universe (a de Sitter universe, grey) for the last few billion years, a radiation only universe for the first few thousand years (red), and a matter only universe (an Einstein-de Sitter universe, blue) in between. The reason for this is that the density of the components changes differently as the universe’s size changes.

The total amount of matter in the universe does not change (at least not significantly – a small amount of matter is converted to radiation and vice-versa). Thus, the density of matter decreases as the volume of the universe increases.

$$\rho_M(a) = \frac{\rho_{M, 0}}{a^3}$$

$$\Omega_M(a) = \frac{\rho_M(a)}{\rho_{crit}(a)} = \frac{\rho_{M, 0}}{a^3} \frac{8 \pi G}{3 H(a)^2} = \frac{\Omega_{M, 0}}{a^3} \frac{H_0^2}{H(a)^2}$$

Where $H(a)$ is the expansion rate at the time when the scale factor is $a$ .

The total amount of energy in radiation does change as the universe’s size changes. The energy of a photon is inversely proportional to its wavelength $E = hc / \lambda$ and as the universe expands the light is redshfited i.e., the wavelength ($\lambda$ ) increases. Thus, there is an additional factor of $a$ on top of the $a^3$ ratio. Other than that, the math is the same.

$$\rho_R(a) = \frac{\rho_{R, 0}}{a^4}$$

$$\Omega_R(a) = \frac{\Omega_{R, 0}}{a^4} \frac{H_0^2}{H(a)^2}$$

Dark energy on the other hand has (we think) a constant density. As the universe expands, more of it is created.

$$\rho_{\Lambda}(a) = \rho_{\Lambda, 0}$$

$$\Omega_{\Lambda}(a) = \Omega_{\Lambda, 0} \frac{H_0^2}{H(a)^2}$$

Let’s see how these densities, along with $H(a)$ , change over time for our universe.

Interestingly, this shows that even though expansion is accelerating ($a(t)$ curves upward), $H(a)$ decreases over time. In the case of our cosmology, it asymptotes to a value of roughly $58\ {\rm km/s/Mpc}$ .

$$\frac{H(a)^2}{H_0^2} = \frac{\Omega_{R, 0}}{a^4} + \frac{\Omega_{M, 0}}{a^3} + \Omega_{\Lambda, 0} + \frac{1 - \Omega_{0}}{a^2}$$ $$\lim_{a \to \infty} H(a) = H_0 \sqrt{\Omega_{\Lambda, 0}}$$

This may seem counterintuitive, but comes from the definition of $H$ .

$$H = \frac{\dot{a}}{a}$$

If $H$ is a constant (as we can see that it is at late times in our universe, either from the plot or equation above), the solution to that equation is $a(t) = e^{H t}$ and so the scale factor increases exponentially.

Another way to think about this is to consider two objects separated by some distance $d_0$ . The velocity between them today will be $v_0 = Hd_0$ . At a later time, the distance between them will have increased and $v_1 = Hd_1$ . Even if $H$ is constant $v_1 > v_0$ as $d_1 > d_0$ . In fact, as $v \propto d \propto a = e^{H t}$ , the velocity will increase exponentially with time, even though $H$ is constant.

To really understand how changing the contents of the universe affects its expansion, play with the cosmology and see how it changes $a(t)$ , $H(a)$ , and $\Omega(a)$ .

Some questions you might have.

• Radiation collapses the universe more quickly than matter. Why?
  • It has an extra factor of $a$ in the denominator and therefore $H(a)$ decreases more quickly as $a$ grows. Why does it have this extra factor of $a$ ?
  • The intuitive answer is that as the wavelength is stretched, the energy of radiation decreases. However, it can also be explained from the equations of state (pressure – energy density relation). $P_{M} / \epsilon_{M} \approx 0$ while $P_{R}/\epsilon_{R} = 1/3$ . Why does this matter?
  • The acceleration equation (the combination of fluid and Friedmann equations) says that the second derivative (acceleration) of the scale factor $\ddot a \propto -(\epsilon + 3P)$ . The positive pressure of radiation results in a larger negative acceleration resulting in a faster collapse. But why does energy density have the same effect as 3 times the pressure?
  • Their units are the same ($\epsilon \rightarrow$ energy per volume, $P \rightarrow$ force per area, but force is energy per distance) which suggests they might be similar. The stress-energy tensor treats them similarly – the trace (sum of the elements on the diagonal) contains an energy density term and 3 pressure terms. I would need to understand GR better (or at all) to really explain this, or reach to deeper whys.
• What are the limiting behaviors of $\Omega$ ?
  • In a flat universe $\Omega(t) = 1$ . Flat at any time $\rightarrow$ flat at all times.
  • In a $\Lambda$ free, negatively curved universe, $\Omega_{R, M} \rightarrow 0$ . The amount of matter is insufficient to prevent the universe growing without bound and so as $t \rightarrow \infty$ it looks more like an empty universe.
  • Similarly, in a $\Lambda$ free, positively curved universe, $\Omega_{R, M} \rightarrow \infty$ at the stationary point. The matter halts the expansion and as $H(t) \rightarrow 0$ , $\rho_{crit} \rightarrow 0$ . During the contraction phase $\Omega_{R,M}$ switches sign as $H(t)$ and $\rho_{crit}$ go negative.
  • In an expanding universe with $\Lambda$ , $\Omega_{\Lambda} \rightarrow 1$ . This is because (as we have seen), $\lim_{a \to \infty} H(a) = H_0 \sqrt{\Omega_{\Lambda, 0}}$ and $\Omega_{\Lambda}(a) = \Omega_{\Lambda, 0} \frac{H_0^2}{H(a)^2}$ . I don’t have a great intuitive explanation for why though.
• Why are all universes with a stationary point ($H(t) = 0$ ) symmetrical?
  • This is most naturally explained with the acceleration equation, $\frac{\ddot{a}}{a} = - \frac{4\pi G}{2c^2} (\epsilon + 3P)$ . At a given $a$ , regardless of whether the universe is expanding or contracting, the contents ($\epsilon$ and $P$ ) are the same and therefore the acceleration must be the same. Thus the process of decelerating to the stationary point is mirrored by the acceleration after, thus the velocity ($\frac{\dot{a}}{a} = H(t)$ ) and dimensions ($a(t)$ ) must also be mirrored.

If you want to know more about this, I highly recommend Barbara Ryden’s “Introduction to Cosmology” (Ryden 2016). It might be the best textbook I’ve used for any subject, striking a really nice balance between readable, detailed, and even funny. Almost everything I’ve done here is in chapter 5.

References