Assignment 5

This assignment provides the background needed to study the simplest time-dependent solutions of Einstein's Field Equations. These are called the Friedman-Robertson-Walker (FRW) cosmological models. As in the previous problem set, you are expected to use mathematica or maple to obtain the relevant equations.

1. First, we will try to write out the most general three-dimensional (Euclidean) metric that is compatible with spatial isotropy and homogeneity. Convince yourself that this is satisfied by the following ansatz:

\begin{align} ds^2 = A(r) dr^2 + r^2 (d\theta^2 + \sin^2\theta d\varphi^2)\ . \end{align}
  • Compute the Christoffel connection, the Ricci tensor, Scalar curvature Einstein tensor for the above metric. Recall that in three dimensions, the Ricci tensor carries all the data contained in the Riemann tensor.
  • Impose the condition that the scalar curvature is constant $=6k$. Fix the integration constant by requiring $A(0)\neq0$.
  • Show that the above choice for $A(r)$ implies that the space is Einstein i.e., it solves Einstein's field equations $G_{ij}+\lambda g_{ij}=0$ with cosmological constant $\lambda$. Obtain the value of $\lambda$.

2. Next, consider the four-dimensional FRW metric

\begin{align} ds^2 = dt^2 - a(t)^2 \Big[ \frac{dr^2}{1-kr^2} + r^2 (d\theta^2 + \sin^2\theta d\varphi^2) \Big]\ , \end{align}

where $k=0,\pm1$. At a given time $t$, the spatial metric is the homogeneous isotropic metric that you obtained in the problem above.

  • Compute the Christoffel connection, the Riemann and Ricci tensors and hence the Einstein tensor for the above metric.
  • Take matter to be given by a perfect fluid with $T^{\mu\nu}=(p+\rho)\ U^\mu U^\nu - p g^{\mu\nu}$, where $g^{\mu\nu}$ to be given by the FRW metric. Argue (or simple assume) that $U^\mu=(1,0,0,0)$ in the frame where is space is isotropic and homogeneous. Show that the time component of the conservation of the energy-momentum tensor gives rise to the following equation:
\begin{align} \frac{d\rho}{dt}+ 3(p+\rho)\ \frac{\dot{a}}{a}=0\ . \end{align}
  • Show that the $tt$ and $rr$ component of Einstein's equations lead to the following two equations:
\begin{eqnarray} \Big[\frac{\dot{a}}{a}\Big]^2+ \frac{k}{a^2}&=&\frac{8\pi \rho}{3}\ .\\ 2\ \frac{\ddot{a}}{a}+\Big[\frac{\dot{a}}{a}\Big]^2 +\frac{k}{a^2}&=&- 8\pi p \ . \end{eqnarray}

The two equations may be simplified to obtain Friedman's equations:

\begin{eqnarray} \Big[\frac{\dot{a}}{a}\Big]^2+ \frac{k}{a^2}&=&\frac{8\pi \rho}{3}\ .\\ \frac{\ddot{a}}{a}&=&- \frac{8\pi}3 \Big[\rho+ 3p \Big]\ . \end{eqnarray}

These are the dynamical equations that determine the evolution of the scale factor $a(t)$. One defines two parameters from the scale factor. Let $H=\tfrac{\dot{a}}{a}$ and $q=-\tfrac{a\ddot{a}}{\dot{a}^2}$. The value of $H$ at any given time, say today, is called the Hubble constant and the value of $q$ is called the deceleration parameter.

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