Final considerations YouTube

Sphere of last scattering

Figure 9.4 - Sphere of last scattering

Figure 9.4 - Sphere of last scattering

From our perspective, we can chop up the universe into a sequence of spheres of larger and larger radii, from which light reaches us having started out at earlier and earlier times.

Nb. Remember, in homogeneous, isotropic space, we could have chosen anywhere to be the centre.

The absolute furthest out in space - or equivalently, back in time - we can see is called the sphere of last scattering.

We cannot see any photons from before this time, because everything in the universe would have been packed so close together that the temperature would have been similar to the surface of the sun, and hence opaque.

Photons leaving this sphere would have a similar wavelength as one leaving the sun, but since this sphere is moving away from us at a rate close to the speed of light, they will be hugely Doppler-shifted by the time they reach us. They are measured to have wavelengths $1000$ times larger than when they left the sphere, and are consequently $1000$ times less energetic.

Suppose we made the same measurements $5$ billion years ago, say. Then, the surface of last scattering would not have been so far away, and expanding at a slower rate. We would have detected less of a Doppler-shift and so the energy of a photon from this sphere would have been greater.

Conservation of energy

We can ask what happens to the energy from the photons coming from the sphere of last scattering that is lost en route to our telescopes.

The original equation that we used to generate the cosmologies we have considered so far is the energy equation, $$\frac{1}{2} v^2 - \frac{4 \pi G}{3} \rho R^3 = 0$$ which is just the usual difference between two kinds of energies, namely kinetic energy, $\frac{1}{2} v^2$ and potential energy, $\frac{4 \pi G}{3} \rho R^3$.

We then cast this expression in terms of the scale factor (rate of expansion), $$\left(\frac{\dot{a}}{a}\right)^2 - \frac{8 \pi G}{3} \rho= 0$$ where now $\left(\frac{\dot{a}}{a}\right)^2$ is the energy of expansion and $\frac{8 \pi G}{3} \rho$ is all other energy.

Neither of these latter two types of energy are conserved individually. Only the difference is conserved, and furthermore, under the assumption that space is flat, they cancel each other out, so that the total energy is zero. So as the energy associated with photons (from the surface of last scattering) decreases over time, the energy of expansion increases.

Nb. General relativity allows us to view total energy in two ways. We can include energy of expansion, in which case total energy is conserved. Alternatively, we could disregard energy of expansion and total energy isn't conserved.