Friedmann Robertson Walker Metric#

The Friedmann Robertson Walker Metric (FRW) describes the metric of the expanding flat universe. It is defined as,

\[\begin{split}\begin{gather} g_{\mu\nu} \equiv \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & a^2(t) & 0 & 0 \\ 0 & 0 & a^2(t) & 0 \\ 0 & 0 & 0 & a^2(t) \end{pmatrix}\\ \end{gather}\end{split}\]

Recall the Minkowski metric as,

\[\begin{split}\begin{gather} \eta_{\mu\nu} \equiv \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}\\ \end{gather}\end{split}\]

We can now see that the Minkowski metric takes \(a^2(t) = 1\) (constant scale factor). So as we expect, the Minkowski universe is a flat non-expanding universe where the only curvature exist in the 4-positional vector.

Applying this metric to the describe lengths \(ds\) is,

\[\begin{split} \begin{gather} ds^2 = g_{\mu\nu} dx^\mu dx^\nu\\ \boxed{ds^2 = -dt^2 + a(t)^2\left[dr^2 + S_k(r)^2d\Omega^2\right]}\\ S_k(t) = \begin{cases} R_0\sin\left(\dfrac{r}{R_0}\right) & k=1 \text{ (open)}\\ r & k=0 \text{ (flat)}\\ R_0\sinh\left(\dfrac{r}{R_0}\right) & k=-1 \text{ (closed)} \end{cases} \end{gather} \end{split}\]

  • \(k\) : Curvature constant

  • \(R_0\) : Radius of curvature

Proper Distance#

Considering a universe that follow the FRW metric of some scale factor \(a(t)\) that changes over time. The proper distance is the actual distance of the an observed object that emits light. Thus defined as,

\[ d_p(t) = \chi a(t) \]

  • \(\chi\): comoving distance

  • \(a(t)\): scale factor

This is given by a photon traveling in a null-space (it’s own straight path) where \(d\Omega = 0\) and \(ds = 0\). Therefore the metric becomes,

\[dt^2 = a(t)^2 dr^2\]

Consider a photon emitted at time \(t_e\) and observed at time \(t_o\). We can find the proper distance by:

\[\begin{split} \begin{gather} d_p(t) = \int{dr}\\ \boxed{d_p(t) = \int_{t_e}^{t_o}{\frac{dt}{a(t)}}} \end{gather} \end{split}\]

Additionally the comoving factor can be derived from the definition of proper distance

\[\begin{split} \begin{gather} \chi = \frac{1}{a(t_e)} \int_{t_e}^{t_o}{\frac{dt}{a(t)}}\\ \boxed{\chi = \left(1 + z\right)\int_{t_e}^{t_o}{\frac{dt}{a(t)}}} \end{gather} \end{split}\]