1 The Expanding Universe

In fact, it is straightforward to construct compact spaces for the $k=0$ and $k=-1$ cases. We simply need to impose periodicity conditions on the coordinates. For example, in the $k=0$ case we could identify the points $x^i=x^i+R^i$, $i=,1,2,3$ with some fixed $R^i$. This results in the torus ${𝐓}^3$.

Spaces constructed this way are homogenous, but no longer isotropic. For example, on the torus there are special directions that bring you back to where you started on the shortest path. This means that such spaces violate the cosmological principle. More importantly, there is no observational evidence that they do, in fact, describe our universe so we will not discuss them in what follows.

The Minkowski metric is appropriate for describing physics in some small region of space and time, like the experiments performed here on Earth. But, on cosmological scales, the Minkowski metric needs replacing so that it captures the fact that the universe is expanding. This is straightforward. We replace the flat spatial metric $d{𝐱}^2$ with one of the three homogeneous and isotropic metrics that we met in the previous section and write

$d{s}^2=-c^{2}d{t}^2+a^2(t)\left[{\displaystyle \frac{1}{1-k{r}^2/R^2}}d{r}^2+r^2(d{\theta }^2+{\sin }^2\theta d{\varphi }^2)\right]$

(1.12)

This is the Friedmann-Robertson-Walker, or FRW metric. The role of the dimensionless scale factor $a(t)$ is, as we shall see, to change distances over time.

There is a redundancy in the description of the metric. If we rescale coordinates as $a\to \lambda a$, $r\to r/\lambda$ and $R\to R/\lambda$ then the metric remains unchanged. We use this to set the scale factor evaluated at the present time $t_0$ to unity,

$a_0=a(t_0)=1$

where the subscript 0 will always denote the value of a quantity evaluated today.

Consider a galaxy sitting at some fixed point point $(r,\theta ,\varphi )$. We refer to the coordinates $(r,\theta ,\varphi )$ (or, equivalently, $(\chi ,\theta ,\varphi )$) on the 3d space as co-moving coordinates. They are analogous to the Lagrangian coordinates used in fluid mechanics. The physical (or proper) distance between the point $(r,\theta ,\varphi )$ and the origin is then

$d_{\mathrm{phys}}=a(t){\displaystyle {\int }_0^r}{\displaystyle \frac{1}{\sqrt{1-k{r}^{\prime \mathrm{\hspace{0.17em}2}}/R^2}}}𝑑r^{\prime }=a(t)R\chi$

(1.13)

However, there is nothing special about the origin, and the same scaling with $a(t)$ is seen for the distance between any two points. If we choose a function $a(t)$ with $\dot{a}>0$, then the distance between any two points is increasing. This is the statement that the universe is expanding: two galaxies, at fixed co-moving co-ordinates, will be swept apart as spacetime stretches.

Importantly, the universe isn’t expanding “into” anything. Instead, the geometry of spacetime, as described by the metric (1.12), is getting bigger, without reference to anything which sits outside. Similarly, a metric with describes a contracting universe. In Section 1.2, we will introduce the tools needed to calculate $a(t)$. But first, we look at some general features of expanding, or contracting universes.

The FRW metric is not invariant under Lorentz transformation. This means that the universe picks out a preferred rest frame, described by co-moving coordinates. We can still shift this rest frame by translations (in flat space) or rotations, but not by Lorentz boosts. Consider a galaxy which, in co-moving coordinates, traces a trajectory $𝐱(t)$. Then, in physical coordinates, the position is

${𝐱}_{\mathrm{phys}}(t)=a(t)𝐱(t)$

(1.14)

The physical velocity is then

${𝐯}_{\mathrm{phys}}(t)={\displaystyle \frac{d{𝐱}_{\mathrm{phys}}}{dt}}$

$=$

${\displaystyle \frac{da}{dt}}𝐱+a{\displaystyle \frac{d𝐱}{dt}}=H{𝐱}_{\mathrm{phys}}+{𝐯}_{\mathrm{pec}}$

(1.15)

Our own peculiar velocity is $v_{\mathrm{pec}}\approx 400$ $\mathrm{km}{\mathrm{s}}^{-1}$ which is pretty much typical for a galaxy. Meanwhile, the present day value of the Hubble parameter is

$H_0\approx 70\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$

This is, rather misleadingly, referred to as the Hubble constant. Clearly there is nothing constant about it. Although, in fairness, it is pretty much the same today as it was yesterday. It is also common to see the notation

$H_0=100h\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$

(1.16)

and then to describe the value of the Hubble constant in terms of the dimensionless number $h\approx 0.7.$ In this course, we’ll simply use the notation $H_0$.

The Hubble parameter has dimensions of ${\mathrm{time}}^{-1}$, but is written in the rather unusual units $\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$. This is telling us that a galaxy 1 Mpc away will be seen to be retreating at a speed of 70 $\mathrm{km}{\mathrm{s}}^{-1}$ due to the expansion of space. For nearby galaxies, this tends to be smaller than their peculiar velocity. However, as we look further away, the expansion term will dominate. The numbers above suggest that this will happen at distances around $400/70\approx 5$ Mpc.

If we ignore the peculiar velocities, and further assume that we can approximate the Hubble parameter $H(t)$ as the constant $H_0$, then the velocity law (1.15) becomes a linear relation between velocity and distance

${𝐯}_{\mathrm{phys}}=H_0{𝐱}_{\mathrm{phys}}$

(1.17)

This linear relationship is referred to as Hubble’s law; some data is shown in the figures³³ 3 Both of these plots are taken from Ned Wright's cosmology tutorial.. At yet further distances, we would expect the time dependence of $H(t)$ to reveal itself. We will discuss this in Section 1.4.

There is no obstacle in (1.17) to velocities that exceed the speed of light, $|{𝐯}_{\mathrm{phys}}|>c$. This may make you nervous. However, there is no contradiction with relativity and, indeed, the entire framework that we have discussed above sits, without change, in the full theory of general relativity. The statement that “nothing can travel faster than the speed of light” is better thought of as “nothing beats light in a race”. Given two objects at the same point, their relative velocity is always less than $c$. However, the velocity ${𝐯}_{\mathrm{phys}}$ is measuring the relative velocity of two objects at very distant points and, in an expanding spacetime, there is no such restriction.

In a spacetime metric, light travels along null paths with $ds=0$. In the FRW metric (1.12), light travelling in the radial direction (i.e. with fixed $\theta$ and $\varphi$) will follow a path,

$cdt=\pm a(t){\displaystyle \frac{dr}{\sqrt{1-k{r}^2/R^2}}}$

(1.18)

If we place ourselves at the origin, the minus sign describes light moving towards us. Aliens on a distant planet, tuning in for the latest Buster Keaton movie, should use the plus sign.

Suppose that a distant galaxy sits stationary in co-moving coordinate $r_1$ and emits light at time $t_1$. We observe this signal at $r=0$, at time $t_0$, determined by solving the integral equation

$c{\displaystyle {\int }_{t_1}^{t_0}}{\displaystyle \frac{dt}{a(t)}}={\displaystyle {\int }_0^{r_1}}{\displaystyle \frac{dr}{\sqrt{1-k{r}^2/R^2}}}$

If the galaxy emits a second signal at time $t_1+\delta t_1$, this is observed at $t_0+\delta t_0$, with

$c{\displaystyle {\int }_{t_1+\delta t_1}^{t_0+\delta t_0}}{\displaystyle \frac{dt}{a(t)}}={\displaystyle {\int }_0^{r_1}}{\displaystyle \frac{dr}{\sqrt{1-k{r}^2/R^2}}}$

The right-hand side of both of these equations is the same because it is written in co-moving coordinates. We therefore have

${\displaystyle {\int }_{t_1+\delta t_1}^{t_0+\delta t_0}}{\displaystyle \frac{dt}{a(t)}}-{\displaystyle {\int }_{t_1}^{t_0}}{\displaystyle \frac{dt}{a(t)}}=0\mathit{\hspace{1em}\hspace{1em} }\Rightarrow \mathit{\hspace{1em}\hspace{1em} }{\displaystyle \frac{\delta t_1}{a(t_1)}}={\displaystyle \frac{\delta t_0}{a(t_0)}}=\delta t_0$

(1.19)

where, in the last equality, we’ve used the fact that we observe the signal today, where $a(t_0)=1$. We see that the expansion of the universe means that the time difference between the two emitted signals differs from the time difference between the two observed signals. This has an important implication when applied to the wave nature of light. Two successive wave crests are separated by a time

$\delta t_1={\displaystyle \frac{{\lambda }_1}{c}}$

with ${\lambda }_1$ the wavelength of the emitted light. Similarly, the time interval between two observed wave crests is

$\delta t_0={\displaystyle \frac{{\lambda }_0}{c}}$

The result (1.19) tells us that the wavelength of the observed light differs from that of the emitted light,

${\lambda }_0={\displaystyle \frac{a(t_0)}{a(t_1)}}{\lambda }_1={\displaystyle \frac{{\lambda }_1}{a(t_1)}}$

(1.20)

This is intuitive: the light is stretched by the expansion of space as it travels through it so that the observed wavelength is longer than the emitted wavelength. This effect is known as cosmological redshift. It shares some similarity with the Doppler effect, in which the wavelength of light or sound from moving sources is shifted. However, the analogy is not precise: the Doppler effect depends only on the relative velocity of the source and emitter, while the cosmological redshift is independent of $\dot{a}$, instead depending on the overall expansion of space over the light’s journey time.

The redshift parameter $z$ is defined as the fractional increase in the observed wavelength,

$z={\displaystyle \frac{{\lambda }_0-{\lambda }_1}{{\lambda }_1}}={\displaystyle \frac{1-a(t_1)}{a(t_1)}}\mathit{\hspace{1em}\hspace{1em} }\Rightarrow \mathrm{\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1}+z={\displaystyle \frac{1}{a(t_1)}}$

(1.21)

As this course progresses, we will often refer to times in the past in terms of the redshift $z$. Today we sit at $z=0$. When $z=1$, the universe was half the current size. When $z=2$, the universe was one third the current size.

The redshift is something that we can directly measure. Light from far galaxies come with a fingerprint, the spectral absorption lines that reveal the molecular and atomic makeup of the stars within. By comparing the frequencies of those lines to those on Earth, it is a simple matter to extract $z$. As an aside, by comparing the relative positions of spectral lines, one can also confirm that atomic physics in far flung places works the same as on Earth, with no detected changes in the laws of physics or the fundamental constants of nature.

We can get an estimate for the age of the Universe by Taylor expanding $a(t)$ about the present day, and truncating at linear order. Recalling that $a(t_0)=1$, we have

$a(t)\approx 1+H_0(t-t_0)$

(1.22)

This rather naive expansion suggests that the Big Bang occurs at

$t_0-t_{BB}=H_0^{-1}\approx 4.4\times {10}^{17}\mathrm{s}\approx 1.4\times {10}^{10}\mathrm{years}$

(1.23)

This result of 14 billion years is surprisingly close to the currently accepted value of around 13.8 billion years. However, there is a large dose of luck in this agreement, since the linear approximation (1.22) is not very good when extrapolated over the full age of the universe. We’ll revisit this in Section 1.4.

Strictly speaking, we should not trust our equations at the point $a(t_{BB})=0$. The metric (1.12) is singular here, and any matter in the universe will be squeezed to infinite density. In such a regime, our simple minded classical equations are not to be trusted, and should be replaced by a quantum theory of matter and gravity. Despite much work, it remains an open problem to understand the origin of the universe at $a(t_{BB})=0$. Did time begin here? Was there a previous phase of a contracting universe? Did the universe emerge from some earlier, non-geometric form? We simply don’t know.

Understanding the Big Bang is one of the ultimate goals of cosmology. In the meantime, the game is to push as far back in time as we can, using the classical (and semi-classical) theory of gravity that we trust. We will be able to reach scales $a\ll 1$, even if we can’t get all the way to $a=0$, and follow the subsequent evolution of the universe from the initial hot, dense state to the world we see today. This set of ideas, is often referred to as the Big Bang theory, even though it tells us nothing about the initial “Big Bang” itself.

The distance $d_H$ is sometimes referred to as the particle horizon. The name mimics the event horizon of black holes. Nothing inside the event horizon of a black hole can influence the world outside. Similarly, nothing outside the particle horizon can influence us today.

One option is that the universe begins collapsing in the future, and there is a second time $t_{BC}>t_0$ where $a(t_{BC})=0$. This is referred to as the Big Crunch. In this case, there is a limit on how far we can communicate before the universe comes to an end, given by

$c{\displaystyle {\int }_t^{t_{BC}}}{\displaystyle \frac{d{t}^{\prime }}{a(t^{\prime })}}={\displaystyle {\int }_0^{r_{\max }(t)}}{\displaystyle \frac{dr}{\sqrt{1-k{r}^2/R^2}}}$

Perhaps more surprisingly, even if the universe continues to expand and the FRW metric holds for $t\to \mathrm{\infty }$, then there could still be a maximum distance that we can influence. The relevant equation is now

$c{\displaystyle {\int }_t^{\mathrm{\infty }}}{\displaystyle \frac{d{t}^{\prime }}{a(t^{\prime })}}={\displaystyle {\int }_0^{r_{\max }(t)}}{\displaystyle \frac{dr}{\sqrt{1-k{r}^2/R^2}}}$

(1.25)

The maximum co-moving distance $r_{\max }$ is finite provided that the left-hand side converges. For example, this happens if we have $a(t)\sim e^{Ht}$ as $t\to \mathrm{\infty }$. As we will see later in the course, this seems to be the most likely fate of our universe. As Schrödinger described, it is quite possible that two friends who once played together as children could move apart from each other, only to find that they’ve travelled too far and can never return as they are inexorably swept further apart by the expansion of the universe. It’s not a bad metaphor for life.

In this context, the distance $r_{\max }(t)$ is called the (co-moving) cosmological event horizon. Once again, there is the analogy with the black hole. Regions beyond the cosmological horizon are beyond our reach; if we choose to sit still, we will never see them and never communicate with them. However, there are also important distinctions. In contrast to the event horizon of a black hole, the concept of cosmological event horizon depends on the choice of observer.

Suppose that we sit at some conformal time $\tau$. A signal can be emitted no earlier than ${\tau }_{BB}$ where the Big Bang singularity occurs. This then puts a restriction on how far we can see in space, defined to be the particle horizon

$R{\chi }_{\mathrm{ph}}=c(\tau -{\tau }_{BB})$

This is shown in Figure 8.

Looking forward, the issue comes because the end of the universe, at $t\to \mathrm{\infty }$, corresponds to a finite conformal time ${\tau }_{\mathrm{end}}$. This means that nothing we can do will be seen beyond a maximum distance which defines the cosmological event horizon,

$R{\chi }_{\mathrm{eh}}={\tau }_{\mathrm{end}}-\tau$

This is shown in Figure 9.

It turns out that conformal time is also a useful change of variable when solving the equations of cosmology. We’ll see an example in Section 1.3.2.

Furthermore, there is even an ambiguity in what we mean by “distance”. So far, we have defined the co-moving distance $R\chi$ and, in (1.13), the physical distance $d_{\mathrm{phys}}(t)=a(t)R\chi$. The latter is, as the name suggests, more physical, but it does not equate directly to something we can measure. Instead, $d_{\mathrm{phys}}(t)$ is the distance between two events which took place at some fixed time $t$, but to measure this distance, we would need to pause the expansion of the universe while we wheeled out a tape measure, typically one which stretches over several megaparsecs. This, it turns out, is impractical.

For these reasons, we need a more useful definition of distance and how to measure it. A useful measure of distance should involve what we actually see, and what we see is light that has travelled across the universe, sometimes for a long long time.

For objects that are reasonably close, we can use parallax, the slight wobble of a stars position caused by the Earth orbiting the Sun. The current state of the art is the Gaia satellite which can measure the parallax of sufficiently bright star sto an accuracy of $2\times {10}^{-5}$ arc seconds, corresponding to distances of $1/{10}^{\mathrm{th}}$ of a megaparsec. While impressive this is, to quote the classics, peanuts to space. We therefore need to turn to more indirect methods.

To resolve this degeneracy, cosmologists turn to standard candles. These are objects whose intrinsic brightness can be determined by other means. There are a number of candidates for standard candles, but some of the most important are:

• •

  Cepheids are bright stars which pulsate with a period ranging from a few days to a month. This periodicity is thought to vary linearly with the intrinsic brightness of the star. These were the standard candles originally used by Hubble.

• •

  A type Ia supernova arises when a white dwarf accretes too much matter from an orbiting companion star, pushing it over the Chandrasekhar limit (the point at which a star collapses). Such events are rare — typically a few a century in a galaxy the size of the Milky Way — but with a brightness that is comparable to all the stars in the host galaxy. The universal nature of the Chandrasekhar limit means that there is considerable uniformity in these supernovae. What little variation there is can be accounted for by studying the “light curve”, meaning how fast the supernova dims after the original burst. These supernovae were first developed as standard candles in the 1990s and resulted in the discovery of the acceleration of the universe.

• •

  The more recent discovery of gravitational waves opens up the possibility for a standard siren. The gravitational waveform can be used to accurately determine the distance. When these waves arise from the collision of a neutron star and black hole (sometimes called a kilanova), the event can also be seen in the electromagnetic spectrum, allowing identification of the host galaxy.

Given a standard candle, we can be fairly sure that we know the intrinsic luminosity $L$ of an object, defined as the energy emitted per unit time. We would like to determine the apparent luminosity $l$, defined by the energy per unit time per unit area, seen by a distant observer. In flat space, this is straightforward: at a distance $d$, the energy has spread out over a sphere ${𝐒}^2$ of area $4\pi d^2$, giving us

$l={\displaystyle \frac{L}{4\pi d^2}}\mathit{\hspace{1em}\hspace{1em} }\text{in flat space}$

(1.27)

The question we would like to ask is: how does this generalise in an FRW universe? To answer this, it’s best to work in the coordinates (1.11), so the FRW metric reads

$d{s}^2=-c^{2}d{t}^2+R^2\left[d{\chi }^2+S_k^2(\chi )(d{\theta }^2+{\sinh }^2\theta d{\varphi }^2)\right]$

with

There are now three things that we need to take into account. The first is that a sphere ${𝐒}^2$ with radius $\chi$ now has area $4\pi R^2{S}_k{(\chi )}^2$, which agrees with our previous result in flat space, but differs when $k\ne 0$. Secondly, the photons are redshifted after their long journey. If they are emitted with frequency ${\nu }_1$ then, from (1.20), they arrive with frequency

${\nu }_0={\displaystyle \frac{2\pi c}{{\lambda }_0}}={\displaystyle \frac{{\nu }_1}{1+z}}$

This lower arrival rate decreases the observed flux. Finally, the observed energy $E_0$ of each photon is reduced compared to the emitted energy $E_1$,

$E_0=\mathrm{\hslash }{\nu }_0={\displaystyle \frac{E_1}{1+z}}$

The upshot is that, in an expanding universe, the observed flux from a source with intrinsic luminosity $L$ sitting at co-moving distance $\chi$ is

$l={\displaystyle \frac{L}{4\pi R^2{S}_k{(\chi )}^2{(1+z)}^2}}$

Comparing to (1.27) motivates us to define the luminosity distance

$d_L(\chi )=R{S}_k(\chi )(1+z)$

(1.28)

For a standard candle, where $L$ is known, the luminosity distance $d_L$ is something that can be measured. From this, and the redshift, we can infer the co-moving distance $R{S}_k(\chi )$. In flat space, this is simply $R\chi =r$.

First, we integrate the path of a light-ray (1.18) to get an expression for the co-moving distance $\chi$ in terms of the “look-back time” $(t_0-t_1)$

	$R\chi =c{\displaystyle {\int }_{t_1}^{t_0}}{\displaystyle \frac{dt}{a(t)}}$	$=$	$c{\displaystyle {\int }_{t_1}^{t_0}}\left[1-H_0(t_0-t)+\mathrm{\dots }\right]𝑑t$		(1.30)
		$=$	$c(t-t_0)\left[1+{\displaystyle \frac{1}{2}}{H}_0(t-t_0)+\mathrm{\dots }\right]$

Next, we get an expression for the look-back time $t_0-t_1$ in terms of the redshift $z$. From (1.21), light emitted at some time $t_1$ suffers a redshift $1+z=1/a(t)$. Inverting the Taylor expansion (1.29), we have

$z={\displaystyle \frac{1}{a(t_1)}}-1\approx H_0(t_0-t_1)+{\displaystyle \frac{1}{2}}(2+q_0)H_0^2{(t_0-t_1)}^2+\mathrm{\dots }$

We now invert this to give the “look-back time” $t_0-t_1$ as a Taylor expansion in the redshift $z$. (As an aside: you could do the inversion by solving the quadratic formula, and subsequently Taylor expanding the square-root. But when inverting a power series, it’s more straightforward to write an ansatz $H_0(t_0-t_1)=A_{1}z+A_2{z}^2+\mathrm{\dots }$, which we substitute this into the right-hand side and match terms.) We find

$H_0(t_0-t_1)=z-{\displaystyle \frac{1}{2}}(2+q_0)z^2+\mathrm{\dots }$

(1.31)

Combining (1.30) and (1.31) gives

${\displaystyle \frac{H_{0}R\chi }{c}}=z-{\displaystyle \frac{1}{2}}(1+q_0)z^2+\mathrm{\dots }$

We can now substitute this into our expression for the luminosity distance (1.28). Life is easiest in flat space, where $R{S}_k(\chi )=R\chi$ and we find

$d_L={\displaystyle \frac{c}{H_0}}\left(z+{\displaystyle \frac{1}{2}}(1-q_0)z^2+\mathrm{\dots }\right)$

This expression is valid only for $z\ll 1$. By plotting the observed $d_L$ vs $z$, and fitting to this functional form, we can extract $H_0$ and $q_0$.

We treat all such sources as homogeneous and isotropic perfect fluids. This means that they are characterised by two quantities: the energy density $\rho (t)$ and the pressure $P(t)$. (If you’ve taken a course in fluid mechanics, you will be more used to thinking of $\rho (t)$ as the mass density. In the cosmological, or relativistic context, this becomes the total energy density.)

We will need the equation of state for two, different kinds of fluids. Both of these fluids contain constituent “atoms” of mass $m$ which obey the relativistic energy-momentum relation

$E^2=p^2{c}^2+m^2{c}^4$

(1.32)

The two fluids come from considering this equation in two different regimes:

• •

  Non-Relativistic Limit: $pc\ll m{c}^2$. Here the energy is dominated by the mass, $E\approx m{c}^2$, and the velocity of the atoms is $𝐯\approx 𝐩/m$.

• •

  Relativistic Limit: $pc\gg m{c}^2$. Now the energy is dominated by the momentum, $E\approx pc$, and the velocity of the atoms approaches the speed of light $|𝐯|\approx c$.

Suppose that there are $N$ such atoms in a volume $V$. In general, these atoms will not have a fixed momentum and energy, but instead the number density $n(p)$ will be some distribution. Because the fluid is isotropic, this distribution can depend only on the magnitude of momentum $p=|𝐩|$. It is normalised by

${\displaystyle \frac{N}{V}}={\displaystyle {\int }_0^{\mathrm{\infty }}}𝑑pn(p)$

The pressure of a gas is defined to be force per unit area. For our purposes, a better definition is the flux of momentum across a surface of unit area. This is equivalent to the earlier definition because, if the surface is a solid wall, the momentum must be reflected by the wall resulting in a force. However, the “flux” definition can be used anywhere in the fluid, not just at the boundary where there’s a wall. Because the fluid is isotropic, we are free to choose this area to be the $(x,y)$-plane. Then, we have

$P={\displaystyle {\int }_0^{\mathrm{\infty }}}𝑑p{v}_z{p}_{z}n(p)$

(If this is unfamiliar, an elementary derivation of this formula is given later in Section 2.1.2.) Because $𝐯$ and $𝐩$ are parallel, we can write

$𝐯\cdot 𝐩=vp=v_x{p}_x+v_y{p}_y+v_z{p}_z=3v_z{p}_z$

where the final equality is ensured by isotropy. This then gives us

$P={\displaystyle \frac{1}{3}}{\displaystyle {\int }_0^{\mathrm{\infty }}}𝑑pvpn(p)$

(1.33)

Now we can relate this to the energy density in the two cases. First, the non-relativistic gas. In this case, $p\approx mv$ so we have

$P_{\mathrm{non}-\mathrm{rel}}\approx {\displaystyle \frac{1}{3}}{\displaystyle {\int }_0^{\mathrm{\infty }}}𝑑pm{v}^{2}n(p)={\displaystyle \frac{1}{3}}{\displaystyle \frac{N}{V}}m⟨v^2⟩$

(1.34)

where $⟨v^2⟩$ is the average square-velocity in the gas.

For cosmological purposes, our interest is in the total energy (1.32) and this is dominated by the contribution from the mass $E\approx m{c}^2+\mathrm{\dots }$. If we relate the pressure of a non-relativistic gas to this total energy $E$, we have

$P_{\mathrm{non}-\mathrm{rel}}={\displaystyle \frac{NE}{V}}{\displaystyle \frac{⟨v^2⟩}{c^2}}$

Since $⟨v^2⟩/c^2\ll 1$, we say that the pressure of a non-relativistic gas is simply

$P_{\mathrm{non}-\mathrm{rel}}\approx 0$

Note that this is the same pressure that keeps balloons afloat and your eardrums healthy: it’s not really vanishing. But it is negligible when it comes to its effect on the expansion of the universe. (We will, in fact, revisit this in Section 2 where we’ll see that the pressure does give rise to important phenomena in the early universe.)

Cosmologists refer to a non-relativistic gas as dust, a name designed to reflect the fact that it just hangs around and is boring. Examples of dust include galaxies, dark matter, and hydrogen atoms floating around and not doing much. We will also refer to dust simply as matter.

We can repeat this for a gas of relativistic particles with $v\approx c$ and $E\approx pc$. Now the formula for the pressure (1.33) becomes

$P_{\mathrm{rel}}\approx {\displaystyle \frac{1}{3}}{\displaystyle {\int }_0^{\mathrm{\infty }}}𝑑pvpn(p)\approx {\displaystyle \frac{1}{3}}{\displaystyle {\int }_0^{\mathrm{\infty }}}𝑑pEn(p)={\displaystyle \frac{N⟨E⟩}{3V}}$

with $⟨E⟩$ the average energy of a particle. The energy density is $\rho =N⟨E⟩/V$, so the relativistic gas obeys the equation of state

$P_{\mathrm{rel}}={\displaystyle \frac{1}{3}}\rho$

Cosmologists refer to such a relativistic gas as radiation. Examples of radiation include the gas of photons known as the cosmic microwave background, gravitational waves, and neutrinos.

Most of the equations of state we meet in cosmology have the simple form

$P=w\rho$

(1.35)

for some constant $w$. As we have seen, dust has $w=0$ and radiation has $w=1/3$. We will meet other, more exotic fluids as the course progresses.

There is an important restriction on the equation of state. The speed of sound $c_s$ in a fluid is given by

$c_s^2=c^2{\displaystyle \frac{dP}{d\rho }}$

We will derive this formula in Section 3.1.1, but for now we simply quote it. It’s important that the speed of sound is less than the speed of light. (Remember: nothing can beat light in a race.) This means that to be consistent with relativity, we must have $w\le 1$. In fact, the more exotic substances we will meet will have , suggesting an imaginary sound speed. What this is really telling us is that substances with do not support propagating sound waves, with perturbations decaying exponentially in time.

For example, starting from our expression, in (1.34) we derived an expression for the pressure of a non-relativistic gas: $P_{\mathrm{non}-\mathrm{rel}}\approx Nm⟨v^2⟩/3V$. This coincides with the ideal gas law if we relate the temperature to the average kinetic energy of an atom in the gas through

${\displaystyle \frac{1}{2}}m⟨v^2⟩={\displaystyle \frac{3}{2}}{k}_{B}T$

(1.37)

We will revisit this in Section 2.1 and gain a better understanding of this result and the role played by temperature.

A proper discussion of the continuity equation requires the machinery of general relativity. This is one of a number of places were we will revert to some simple Newtonian thinking to derive the correct equation. Such derivations are not entirely convincing, not least because it’s unclear why they would be valid when applied to the entire universe. Nonetheless, they will give the correct answer. A more rigorous approach can be found in the lectures on General Relativity.

Consider a gas trapped in a box of volume $V$. The gas exerts pressure on the sides of the box. If the box increases in size, as shown in the figure, then the change of volume is $dV=\mathrm{Area}\times dx$. The work done by the gas is $\mathrm{Force}\times dx=(PA)dx=pdV$, and this reduces the internal energy of the gas. We have

$dE=-PdV$

This is a simple form of the first law of thermodynamics, valid for reversible or adiabatic processes. It is far from obvious that we can view the universe as a box filled with gas and naively apply this formula. Nonetheless, it happily turns out that the final result agrees with the more rigorous GR approach so we will push ahead, and invoke the time dependent version of the first law,

${\displaystyle \frac{dE}{dt}}=-P{\displaystyle \frac{dV}{dt}}$

(1.38)

Now consider a small region of fluid, in co-moving volume $V_0$. The physical volume is

$V(t)=a^3(t)V_0\mathit{\hspace{1em}\hspace{1em} }\Rightarrow \mathit{\hspace{1em}\hspace{1em} }{\displaystyle \frac{dV}{dt}}=3a^2\dot{a}{V}_0$

Meanwhile, the energy in this volume is

$E=\rho a^3{V}_0\mathit{\hspace{1em}\hspace{1em} }\Rightarrow \mathit{\hspace{1em}\hspace{1em} }{\displaystyle \frac{dE}{dt}}=\dot{\rho }{a}^3{V}_0+3\rho a^2\dot{a}{V}_0$

The first law (1.38) then becomes

$\dot{\rho }+3H\left(\rho +P\right)=0$

(1.39)

This slightly unfamiliar equation is the expression of energy conservation in a cosmological setting.

Before we proceed, a warning: energy is a famously slippery concept in general relativity, and we will meet things later which, taken naively, would seem to violate energy conservation. For example, in Section 1.3.3, we will meet a fluid with equation of state $\rho =-P$. For such a fluid, $\dot{\rho }=0$ which means that the energy density remains constant even as the universe expands. Such is the way of the world and we need to get used to it. If this makes you nervous, recall that the usual derivation of energy conservation, via Noether’s theorem, holds only in time independent settings. So perhaps it’s not so surprising that energy conservation takes a somewhat different form in an expanding universe.

If we specify an equation of state $P=w\rho$, as in (1.35), then we can integrate the continuity equation (1.39) to determine how the energy density depends on the scale factor. We have

	${\displaystyle \frac{\dot{\rho }}{\rho }}=-3(1+w){\displaystyle \frac{\dot{a}}{a}}$	$\Rightarrow$	$\mathrm{ }\mathit{\hspace{1em}}\log (\rho /{\rho }_0)=-3(1+w)\log a$		(1.40)
		$\Rightarrow$	$\mathrm{ }\mathit{\hspace{1em}}\rho (t)={\rho }_0{a}^{-3(1+w)}$

with ${\rho }_0=\rho (t_0)$ and we’ve used the fact that $a(t_0)=1$.

We can look at how this behaves in simple examples. For dust (also known as matter), we have $w=0$ and so

${\rho }_m\sim {\displaystyle \frac{1}{a^3}}$

This makes sense. As the universe expands, the volume increases as $a^3$, and so the energy density decreases as $1/a^3$.

For radiation, we instead find

${\rho }_r\sim {\displaystyle \frac{1}{a^4}}$

(1.41)

This also makes sense. The energy density is diluted as $1/a^3$ but, on top of this, there is also a redshift effect which shifts the frequency, and hence the energy, by a further power of $1/a$.

The fact that the energy densities of dust and radiation scale differently plays a crucial role in our cosmological history. As we shall see in Section 1.4, our current universe has much greater energy density in dust than in radiation. However, this wasn’t always the case. There was a time in far past when the converse was true, with the radiation subsequently diluting away faster. We’ll see other contributions to the energy density of the universe that have yet different behaviour.

Finally we come to the main part of the story: we would like to describe how the perfect fluids which fill all of space affect the expansion of the universe. We start by giving the answer. The dynamics of the scale factor is dictated by the energy density $\rho (t)$ through the Friedmann equation

$H^2\equiv {\left({\displaystyle \frac{\dot{a}}{a}}\right)}^2={\displaystyle \frac{8\pi G}{3c^2}}\rho -{\displaystyle \frac{k{c}^2}{R^2{a}^2}}$

(1.42)

Here $R$ is some fixed scale, as in the FRW metric (1.12), $k=-1,0,+1$ determines the curvature of space, and $G$ is Newton’s gravitational constant

$G\approx 6.67\times {10}^{-11}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\mathrm{s}}^{-2}$

The Friedmann equation is arguably the most important equation in all of cosmology. Taken together with the continuity equation (1.39) and the equation of state (1.35), they provide a closed system which can be solved to determine the history and fate of the universe itself.

At this point, I have a confession to make. The only honest derivation of the Friedmann equation is in the framework of General Relativity Here we can only present a dishonest derivation, using Newtonian ideas. In an attempt to alleviate the shame, I will at least be open about where the arguments are at their weakest.

First, we work in flat space, with $k=0$. This, of course, is the natural habitat for Newtonian gravity. Nonetheless, we will see the possibility of a curvature term $-k/a^2$ in the Friedmann equation, re-emerging at the end of our derivation.

Our discussions so far prompt us to consider an infinite universe, filled with a constant matter density. That, it turns out, is rather subtle in a Newtonian setting. Instead, we consider a ball of uniform density of size $L$, expanding outwards away from the origin, and subsequently pretend that we can take $L\to \mathrm{\infty }$.

Consider a particle (or element of fluid) of mass $m$ at some position $𝐱$ with $r=|𝐱|\ll L$. It will experience the force of gravity in the form of Newton’s inverse-square law. But a rather special property of this law states that, for a spherically symmetric distribution of masses, the gravitational force at some point $𝐱$ depends only on the masses at distances smaller than $r$ and, moreover, acts as if all the mass is concentrated at the origin.

This statement is simplest to prove if we formulate the gravitational force law as a kind of Gauss’ law,

${𝐅}_{\mathrm{grav}}=-m\nabla \mathrm{\Phi }\mathit{\hspace{1em}\hspace{1em} }\mathrm{where}\mathit{\hspace{1em}\hspace{1em} }{\nabla }^2\mathrm{\Phi }={\displaystyle \frac{4\pi G}{c^2}}\rho$

with $\mathrm{\Phi }$ the gravitational potential. The (perhaps) unfamiliar factor of $c^2$ in the final equation arises because, for us, $\rho$ is the energy density, rather than mass density. We then integrate both sides over a ball $V$ of radius $x$, centred at the origin. Using the kind of symmetry arguments that we used extensively in the lectures on Electromagnetism, we have

${\displaystyle {\int }_S}\nabla \mathrm{\Phi }\cdot d𝐒={\displaystyle {\int }_V}{\displaystyle \frac{4\pi G}{c^2}}\rho 𝑑V\mathit{\hspace{1em}\hspace{1em} }\Rightarrow \mathit{\hspace{1em}\hspace{1em} }\nabla \mathrm{\Phi }(r)={\displaystyle \frac{GM(r)}{r^2}}$

where $M(r)=4\pi \rho r^3/3c^2$ is the mass contained inside the ball of radius $r$. This means that the acceleration of the particle at $𝐱$ is given by

$m\ddot{r}=-{\displaystyle \frac{GmM(r)}{r^2}}$

We multiply by $\dot{r}$ and integrate. As the ball expands with $\dot{r}\ne 0$, the total mass contained with a ball of radius $r(t)$ does not change, so $\dot{M}=0$. We then get

${\displaystyle \frac{1}{2}}{\dot{r}}^2-{\displaystyle \frac{GM(r)}{r}}=E$

(1.43)

where we recognise $E$ as the energy (per unit mass) of the particle. Finally, we describe the position $𝐱$ of the particle in a way that chimes with our previous cosmological discussion, introducing a scale factor $a(t)$

$𝐱(t)=a(t){𝐱}_0$

Substituting this into (1.43) and rearranging gives

${\left({\displaystyle \frac{\dot{a}}{a}}\right)}^2={\displaystyle \frac{8\pi G}{3c^2}}\rho -{\displaystyle \frac{C}{a^2}}$

(1.44)

where $C=-2E/{|{𝐱}_0|}^2$ is a constant. This is remarkably close to the Friedmann equation (1.42). The only remaining issue is why we should identify the constant $C$ with the curvature $k{c}^2/R^2$. There is no good argument here and, indeed, we shouldn’t expect one given that the whole Newtonian derivation took place in a flat space. It is, unfortunately, simply something that you have to suck up.

There is, however, an analogy which makes the identification $C\sim k$ marginally more palatable. Recall that a particle has reached escape velocity if its total energy $E>0$. Conversely, if , the particle comes crashing back down. For us, the case of means $C>0$ which, in turn, corresponds to positive curvature. We will see in Section 1.3.2 that a universe with positive curvature will, under many circumstances, ultimately suffer a big crunch. In contrast, a negatively curved space will keep expanding forever.

Clearly the derivation above is far from rigorous. There are at least two aspects that should give us pause. First, when we assumed $\dot{M}=0$, we were implicitly restricting ourselves to non-relativistic matter with $\rho \sim 1/a^3$. It turns out that in general relativity, the Friedmann equation also holds for any other scaling (1.40) of $\rho$.

However, the part of the above story that should make you feel most queasy is replacing an infinitely expanding universe, with an expanding ball of finite size $L$. This introduces an origin into the story, and gives a very misleading impression of what the expansion of the universe means. In particular, if we dial the clock back to $a(t)=0$ in this scenario, then all matter sits at the origin. This is one of the most popular misconceptions about the Big Bang and it is deeply unfortunate that it is reinforced by the derivation above. Nonetheless, the arguments that lead to (1.44) do provide some physical insight into the meaning of the various terms that can be hard to extract from the more formal derivation using general relativity. So let us wash the distaste from our mouths, and proceed with understanding the universe.

Life is considerably simpler if we restrict attention to a flat $k=0$ universe with just a single fluid component. In this case, using (1.46), we have

${\left({\displaystyle \frac{\dot{a}}{a}}\right)}^2={\displaystyle \frac{D^2}{a^{3(1+w)}}}$

(1.47)

where $D^2=8\pi G{\rho }_{w,0}/3c^2$ is a constant. The solution is

$a(t)={\left({\displaystyle \frac{t}{t_0}}\right)}^{2/(3+3w)}$

(1.48)

The various constants have been massaged into $t_0={(\frac{3}{2}(1+w)D)}^{-1}$ so that we recover our convention $a_0=a(t_0)=1$. There is also an integration constant which we have set to zero. This corresponds to picking the time of the Big Bang, defined by $a(t_{BB})=0$ to be $t_{BB}=0$. With this choice, $t_0$ is identified with the age of the universe.