The Friedmann Equation

So that was the equation for Newtonian expansion. It surely can't look anything like the real thing, can it?

Actually, it can. Solving the Einstein field equations for an isotropic, homogeneous Universe yields a similar form for the dynamics equation:

which is what we had before, but with an additional term (Lambda) called the "cosmological constant" expressing the effects of "dark energy" (the vacuum energy density of space?).

Solving this we get the equation governing the expansion of space:

The Friedmann Equation

where k now corresponds to the curvature of space.

Given different values for rho, Lambda, and k, we can solve this differential equation to get the detailed form of R(t). But first, let's unpack this conceptually.

1. The Universe can accelerate

Look at the dynamics equation and note that if Lambda is positive, it counteracts the deceleration due to gravity and therefore can accelerate the expansion of the universe.

2. Space is curved

Think of our analogy of ants crawling around on an expanding balloon. Keep thinking along those lines.

We have three options for curvature:

    k=1: positive curvature (like a ball)
    k=0: zero curvature (flat like paper)
    k=-1: negative curvature (saddle-like)

(courtesy Syracuse University)

But remember, this is only an analogy to what is really going on with curved spacetime. Like all analogies, it can only be pushed so far. In non-Euclidian space, distances and sizes do non-intuitive things. This allows us to make observational tests of cosmology.

3. Density (rho) can take on different forms, and the universe behaves differently under each one.

We made the assumption of a matter dominated Universe (so that rho ~ R-3). This was not always true in the Universe. rho is actually the inertial mass density of the matter and radiation in the Universe. If the energy density in the universe is given by U, we can convert that into an equivalent matter density
and relate it to the photon density like this

So if we trace the history of the Universe back far enough, at some point the energy density of radiation exceeds that of matter, and we get to the radiation dominated era. At this point, the expansion dynamics are different, driven by radiation rather than gravity. This happens at a scale factor of R=10-3 - 10-4, so we can ignore it for the purposes of observational cosmology. But it does change the expansion rate in the early universe...

4. The future of expansion if Lambda is really a constant...
The energy density of matter and radiation drops as the universe expands. The density of dark energy does not. Eventually, Lambda wins. In this case, R ~ exp(t) -- exponential expansion.

5. The interplay between expansion history and curvature becomes very complex
Without lambda, spatial curvature and expansion history were locked together. With lambda, that connection is broken. Closed universes can expand forever, open universes can collapse, etc. Life becomes messy!