4.6 A Taste of Cosmology

There are surprisingly few phenomena in Nature where we need to solve the Einstein equations sourced by matter,

$$G_{\mu \nu }+\Lambda g_{\mu \nu }=8\pi G{T}_{\mu \nu }$$

However there is one situation where the role of $T_{\mu \nu }$ on the right-hand side is crucial: this is cosmology, the study of the universe as a whole.

4.6.1 The FRW Metric

The key assumption of cosmology is that the universe is spatially homogeneous and isotropic. This restricts the our choices of spatial geometry to one of three: these are

• Euclidean Space ${\mathbf{R}}^3$ : This space has vanishing curvature and the familiar metric

$$d{s}^2=d{r}^2+r^2(d{\theta }^2+{\sin }^2\theta d{\varphi }^2)$$

• Sphere ${\mathbf{S}}^3$ : This space has uniform positive curvature and metric

$$d{s}^2=\frac{1}{1-r^2}d{r}^2+r^2(d{\theta }^2+{\sin }^2\theta d{\varphi }^2)$$

With this choice of coordinates, we have implicitly set the radius of the sphere to 1.

• Hyperboloid ${\mathbf{H}}^3$ : This space has uniform negative curvature and metric

$$d{s}^2=\frac{1}{1+r^2}d{r}^2+r^2(d{\theta }^2+{\sin }^2\theta d{\varphi }^2)$$

The existence of three symmetric spaces is entirely analogous to the the three different solutions we discussed in Section 4.2. de Sitter and anti-de Sitter both have constant spacetime curvature, supplied by the cosmological constant. The metrics above have constant spatial curvature. Note, however, that the metric on ${\mathbf{S}}^3$ coincides with the spatial part of the de Sitter metric in coordinates (4.12), while the metric on ${\mathbf{H}}^3$ coincides with the spatial part of the anti-de Sitter metric in coordinates (4.22)

We write these spatial metrics in unified form,

$$d{s}^2={\gamma }_{ij}d{x}^{i}d{x}^j=\frac{d{r}^2}{1-k{r}^2}+r^2(d{\theta }^2+{\sin }^2\theta d{\varphi }^2)\text{with}k=\{\begin{array}{ll}+1& {\mathbf{S}}^3\\ 0& {\mathbf{R}}^3\\ -1& {\mathbf{H}}^3\end{array}$$

In cosmology, we wish to describe a spacetime in which space expands as the universe evolves. We do this with metrics of the form

$$d{s}^2=-d{t}^2+a^2(t){\gamma }_{ij}d{x}^{i}d{x}^j\text{\hspace{1em}(4.63)}$$

This is the Friedmann-Robertson-Walker, or FRW metric. (It is also known as the FLRW metric, with Lemaˆıtre’s name a worthy addition to the list.) The dimensionless scale factor $a(t)$ should be viewed as the “size” of the spatial dimensions (a name which makes more sense for the compact ${\mathbf{S}}^3$ than the non-compact ${\mathbf{R}}^3$ , but is mathematically sensible for both.) Note that de Sitter space in global coordinates (4.21) is an example of an FRW metric with $k=+1$

Curvature Tensors

We wish to solve the Einstein equations for metrics that take the FRW form. Our first task is to compute the Ricci tensor. We start with the Christoffel symbols: it is straightforward to find ${\Gamma }_{00}^{\mu }={\Gamma }_{i0}^0=0$ and

$${\Gamma }_{ij}^0=a\dot{a}{\gamma }_{ij},{\Gamma }_{0j}^i=\frac{\dot{a}}{a}{\delta }_j^i,{\Gamma }_{jk}^i=\frac{1}{2}{\gamma }^{il}({\partial }_j{\gamma }_{kl}+{\partial }_k{\gamma }_{jl}-{\partial }_l{\gamma }_{jk})$$

To compute the Ricci tensor, we use the expression

$$R_{\mu \nu }={\partial }_{\rho }{\Gamma }_{\nu \mu }^{\rho }-{\partial }_{\nu }{\Gamma }_{\rho \mu }^{\rho }+{\Gamma }_{\nu \mu }^{\lambda }{\Gamma }_{\rho \lambda }^{\rho }-{\Gamma }_{\rho \mu }^{\lambda }{\Gamma }_{\nu \lambda }^{\rho }\text{\hspace{1em}(4.64)}$$

which we get from contracting indices on the similar expression (3.39) for the Riemann tensor

It’s not hard to see that $R_{0i}=0$ . The quick argument is that there’s no covariant 3-vector that could possibly sit on the right-hand side. The other components need a little more work.

Claim

$$R_{00}=-3\frac{\ddot{a}}{a}$$

Proof

Using the non-vanishing Christoffel symbols listed above, we have

$$R_{00}=-{\partial }_0{\Gamma }_{i0}^i-{\Gamma }_{i0}^j{\Gamma }_{j0}^i=-3\frac{d}{dt}\left(\frac{\dot{a}}{a}\right)-3{\left(\frac{\dot{a}}{a}\right)}^2$$

which gives the claimed result

Claim

$$R_{ij}=\left(\frac{\ddot{a}}{a}+2{\left(\frac{\dot{a}}{a}\right)}^2+2\frac{k}{a^2}\right)g_{ij}$$

Proof

This is straightforward to show for $k=0$ FRW metrics where the spatial metric is flat. It’s a little more annoying for the $k=\pm 1$ metrics. A trick that simplifies life is to compute the components of $R_{ij}$ at the spatial origin $\mathbf{x}=0$ where the spatial metric is ${\gamma }_{ij}={\delta }_{ij}$ , and then use covariance to argue that the right result must have $R_{ij}\sim {\gamma }_{ij}$ In doing this, we just have to remember not to set $\mathbf{x}=0$ too soon, since we will first need to differentiate the Christoffel symbols and then evaluate them at $\mathbf{x}=0$

We start by writing the spatial metric in Cartesian coordinates, on the grounds that it’s easier to differentiate in this form

$${\gamma }_{ij}={\delta }_{ij}+\frac{k{x}_i{x}_j}{1-k\mathbf{x}\cdot \mathbf{x}}$$

The Christoffel symbols depend on $\partial {\gamma }_{ij}$ and the Ricci tensor on ${\partial }^2{\gamma }_{ij}$ . This means that if we want to evaluate the Ricci tensor at the origin $\mathbf{x}=0$ , we only need to work with the metric to quadratic order in x. This simplifies things tremendously since

$${\gamma }_{ij}={\delta }_{ij}+k{x}_i{x}_j+\mathcal{O}(x^4)$$

Similarly, we have

$${\gamma }^{ij}={\delta }^{ij}-k{x}^i{x}^j+\mathcal{O}(x^4)$$

where $i,j$ indices are raised and lowered using ${\delta }^{ij}$ . Plugging these forms into the expression for the Christoffel symbols gives

$${\Gamma }_{jk}^i=k{x}^i{\delta }_{jk}+\mathcal{O}(x^3)$$

With this in hand, we can compute the Ricci tensor

$$\begin{array}{rl}& R_{ij}={\partial }_{\rho }{\Gamma }_{ij}^{\rho }-{\partial }_j{\Gamma }_{\rho i}^{\rho }+{\Gamma }_{ij}^{\lambda }{\Gamma }_{\rho \lambda }^{\rho }-{\Gamma }_{\rho i}^{\lambda }{\Gamma }_{j\lambda }^{\rho }\\ & =({\partial }_0{\Gamma }_{ij}^0+{\partial }_k{\Gamma }_{ij}^k)-{\partial }_j{\Gamma }_{ki}^k+({\Gamma }_{ij}^0{\Gamma }_{k0}^k+{\Gamma }_{ij}^k{\Gamma }_{lk}^l)-({\Gamma }_{ki}^0{\Gamma }_{j0}^k+{\Gamma }_{0i}^k{\Gamma }_{jk}^0+{\Gamma }_{li}^k{\Gamma }_{jk}^l)\end{array}$$

We can drop the ${\Gamma }_{ij}^k{\Gamma }_{lk}^l$ term since it vanishes at $\mathbf{x}=0$ . Furthermore, we can now safely replace any undifferentiated ${\gamma }_{ij}$ in the Christoffel symbols with ${\delta }_{ij}$ . What’s left gives

$$\begin{array}{rl}& R_{ij}=\left({\partial }_0(a\dot{a})+3k-k+3{\dot{a}}^2-{\dot{a}}^2-{\dot{a}}^2\right){\delta }_{ij}+\mathcal{O}(x^2)\\ & =\left(a\ddot{a}+2{\dot{a}}^2+2k\right){\delta }_{ij}+\mathcal{O}(x^2)\end{array}$$

We now invoke the covariance argument to write

$$R_{ij}=\left(a\ddot{a}+2{\dot{a}}^2+2k\right){\gamma }_{ij}=\frac{1}{a^2}\left(a\ddot{a}+2{\dot{a}}^2+2k\right)g_{ij}$$

as promised

With these results, we can now compute the Ricci scalar: it is

$$R=6\left(\frac{\ddot{a}}{a}+{\left(\frac{\dot{a}}{a}\right)}^2+\frac{k}{a^2}\right)$$

Finally, the Einstein tensor has components

$$G_{00}=3\left({\left(\frac{\dot{a}}{a}\right)}^2+\frac{k}{a^2}\right)\text{ and }{G}_{ij}=-\left(2\frac{\ddot{a}}{a}+{\left(\frac{\dot{a}}{a}\right)}^2+\frac{k}{a^2}\right)g_{ij}$$

Our next task is to understand the matter content in the universe.

4.6.2 The Friedmann Equations

We take the universe to be filled with perfect fluids of the kind that we introduced in Section 4.5.4. The energy momentum tensor is

$$T^{\mu \nu }=(\rho +P)u^{\mu }{u}^{\nu }+P{g}^{\mu \nu }$$

But we assume that the fluid is at rest in the preferred frame of the universe, meaning that $u^{\mu }=(1,0,0,0)$ in the FRW coordinates (4.63). As we saw in Section 4.5.4, the constraint ${\nabla }_{\mu }{T}^{\mu \nu }=0$ gives the condition (4.54)

$$u^{\mu }{\nabla }_{\mu }\rho +(\rho +P){\nabla }_{\mu }{u}^{\mu }=0\Rightarrow \dot{\rho }+\frac{3\dot{a}}{a}(\rho +P)=0$$

where we’ve used ${\nabla }_{\mu }{u}^{\mu }={\partial }_{\mu }{u}^{\mu }+{\Gamma }_{\mu \rho }^{\mu }{u}^{\rho }={\Gamma }_{i0}^i{u}^0$ , and the expression (4.64) for the Christoffel symbols. This is known as the continuity equation: it expresses the conservation of energy in an expanding universe. You can check that the second constraint (4.55) is trivial when applied to homogeneous and isotropic fluids. (It plays a role when we consider the propagation of sound waves in the universe.)

To make progress, we also need the equation of state. The fluids of interest have rather simple equations of state, taking the form

$$P=w\rho$$

with constant w. Of particular interest are the cases $w=0$ , corresponding to pressureless dust, and $w=1/3$ corresponding to radiation.

For a given equation of state, the continuity equation becomes

$$\frac{\dot{\rho }}{\rho }=-3(1+w)\frac{\dot{a}}{a}$$

So we learn that the energy density $\rho$ dilutes as the universe expands, with

$$\rho =\frac{{\rho }_0}{a^{3(1+w)}}\text{\hspace{1em}(4.65)}$$

with ${\rho }_0$ an integration constant. For pressureless dust, we have $\rho \sim 1/a^3$ which is the expected scaling of energy density with volume. For radiation we have $\rho \sim 1/a^4$ , which is due to the scaling with volume together with an extra factor from redshift.

Now we can look at the Einstein equations. The temporal component is

$$G_{00}+\Lambda g_{00}=8\pi G{T}_{00}\Rightarrow {\left(\frac{\dot{a}}{a}\right)}^2=\frac{8\pi G}{3}\rho +\frac{\Lambda }{3}-\frac{k}{a^2}\text{\hspace{1em}(4.66)}$$

This is the Friedmann equation. In conjunction with (4.65), it tells us how the universe expands.

We also have the spatial components of the Einstein equation,

$$\begin{array}{rlr}{G}_{ij}+\Lambda g_{ij}=8\pi G{T}_{ij}& \Rightarrow & 2\frac{\ddot{a}}{a}+{\left(\frac{\dot{a}}{a}\right)}^2+\frac{k}{a^2}-\Lambda =-8\pi GP\\ & \Rightarrow & \frac{\ddot{a}}{a}-\frac{\Lambda }{3}=-\frac{4\pi G}{3}(\rho +3P)\end{array}\text{\hspace{1em}(4.67)}$$

This is the acceleration equation, also known as the Raychaudhuri equation. It is not independent of the Friedmann equation; if you differentiate (4.66) with respect to time, you can derive Raychaudhuri.

There is plenty of physics hiding in these equations. Some particularly simple solutions can be found by setting $k=\Lambda =0$ and looking at a universe dominated by a single fluid with energy density scaling as (4.65). The Friedmann equation becomes

$${\left(\frac{\dot{a}}{a}\right)}^2\sim \frac{1}{a^{3(1+w)}}\Rightarrow a(t)={\left(\frac{t}{t_0}\right)}^{2/(3+3w)}$$

Picking $w=1/3$ we have $a(t)\sim t^{1/2}$ which describes the expansion of our universe when it was dominated by radiation (roughly the first 50,000 years). Picking $w=0$ we have $a(t)\sim t^{2/3}$ which describes the expansion of our universe when it was dominated by matter (roughly the following 10 billion years). You can find many more solutions of the Friedmann equations and a discussion of the relevant physics in the lectures on Cosmology.

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