Lecture 22

We examined features of FRW universes: their Hubble expansion, redshift, and causality structure.

First, we derived physical recession velocities for comoving objects, showing them to be receding with Hubble ``constant'' $H= \dot{R}/R$ , which is in fact not constant in time.

We then derived the relativistic Doppler shift, or redshift, by considering the propagation of light along a null geodesic from an emitter to an observer. We considered propagation of subsequent crests to a distant observer, and showed that the integral $\int\ cdt/a(t)$ for each, from emission to observation, must be the same. This led us to the conclusion that $\lambda/a$ must stay constant as the photon propagates; that is, that the photon's wavelength redshifts, or stretches, proportional to the expansion of the universe.

We concluded by studying the causality structure in a FRW universe. We defined two kinds of horizons, or boundaries of causal contact in the universe. The particle horizon measures the most distant point in the universe whose past behavior can have influenced me now; that is, the most distant point from which an emitted signal has had time to reach me since the universe began. The event horizon measures the most distant point in the universe whose behavior now can ever influence me in the future.

We solved for these two horizons by considering how light (the fastest signal) propagates between its time and position of emission and that of observation. Using the fact that light travels on null geodesics, we were able to solve for the comoving coordinate distance traveled by light in the specific time interval. Finally, by setting initial and final times to be 1) the beginning of the universe and now (for the particle horizon) or 2) now and the end of the universe (for the event horizon), we were able to solve for the comoving particle and event horizons $\sigma_p$ , $\sigma_e$ . We then converted these comoving horizon distances into physical proper distances now, using the metric and scale factor. We noted that the flat inflationary universe has infinite particle horizon, so that every point in the universe has been made visible to us; and that flat matter- and radiation-dominated universes have infinite event horizons, so that we will eventually see every point in the universe. We anticipated a few more interesting cases for the homework.

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2004-04-06