The Robertson-Walker metrics

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

are applicable to universes that are both spatially homogenous and isotropic: That is, they have no preferred positions or directions. The spacetimes also feature a universal time \(t\) and a constant \(R\) with dimensions of length. There are three possible choices for the constant \(k: k=1,0\), or -1 , which correspond to three-dimensional spatial geometries that have constant positive curvature \((k=+1)\), constant negative curvature \((k=-1)\), or that are flat \((k=0\).) Here \(a(t)\) is called the "scale factor" of the universe; If \(a(t)\) grows with time distant galaxies become farther apart, or if \(a(t)\) shrinks distant galaxies come closer together. The function \(a(t)\) can be found using Einstein's field equations of general relativity, given the kind of matter, radiation, or other quantities that live in the universe. The result is the "Friedmann" equations, in which the universe is filled with a material having uniform mass density \(ho\) and uniform pressure \(p\); we can also add "dark energy" as represented by the constant \(\Lambda\). There are then two independent equations for \(a(t)\) :

\[\text { (1) } \frac{\dot{a}^{2}}{a^{2}}+\frac{k c^{2}}{a^{2}}=\frac{8}{3} \pi G ho+\frac{\Lambda}{3} c^{2}\]

and

\[\text { (2) } \frac{2 \ddot{a}}{a}+\frac{\dot{a}^{2}}{a^{2}}+\frac{k c^{2}}{a^{2}}=\frac{8 \pi G p}{c^{2}}+\Lambda c^{2}\]

where overdots represent time derivatives and \(G\) is Newton's gravitational constant.

(a) First suppose the universe is spatially flat, with \(k=0\), and that the cosmological constant \(\Lambda\) is also zero. Also suppose the energy density consists entirely of mass density \(ho\), which decreases as the universe expands so that \(ho a(t)^{3}=ho_{0} a_{0}^{3}\) where \(ho_{0}\) and \(a_{0}\) are the current mass density and scale factor. In that case solve the Friedmann equations to find \(a(t)\) in terms of \(t, G, ho_{0}\), and \(a_{0}\). It is thought that this is a good approximation to the situation for our universe in most of its history so far. It is called the "matter dominated" period.

(b) Repeat part (a) except suppose the energy density consists entirely of photons in thermal equilibrium, in which case the energy density obeys \(ho a(t)^{4}=ho_{0} a_{0}^{4}\). This situation is thought to be a good approximation for our universe for a hundred thousand years or so early on, and is called the "radiation dominated" period.

(c) Finally, repeat part (a) for the case \(k=0, ho=0, p=0\), but \(\Lambda=\) constant \(>0\). This may be a good approximation to our universe for a brief time after the big bang began; it is called the "inflationary" period for reasons that will be apparent from the solution. It may also be a good approximation for our universe in the distant future.

(d) At the current time the universe seems to be behaving as though it were driven by both dust-like matter and the cosmological constant \(\Lambda\). Sketch a graph of \(a(t)\) vs \(t\) extending from times long ago to times in the distant future, showing what happens as the universe gradually transitions from one form of dominance to the other.