Reading:

Remainder of Berry 6.1 -- 6.2 (attached).

Problems:

1. Consider the early time limit of FRW models with $w>-1/3$, where $R$ is small and the curvature term in the first Friedmann equation is negligible. We showed in class that at early times $R \propto t^{\frac{2}{3 (1+w)}}$, which grows more slowly than $t$ for $w>-1/3$. For such Friedmann-Robertson-Walker models, having $r \propto t^b$ with $b<1$,
   1. Assume the model describes a universe beginning at $t=0$. Calculate the maximum comoving coordinate value $\sigma_p (t)$ from which an observer at $\sigma = 0$ could have received signals at time $t$.

   2. Find the associated proper distance $d_p(t)$. Show that its early time limit is proportional to $t$, and find the coefficient.
      NOTE: $d_p(t)$, the limit on the total portion of the universe in principle visible to an observer at a given time, is called the ``particle horizon.''

   3. If $k=1$ (the model universe is closed), at what time does the full universe first become visible to the observer?

2. A model that does not meet the criteria of the above problem is the inflationary universe, where $p = -\rho$ ( or $w = -1$). We showed in class that the curvature term in the first Friedmann equation becomes negligible for late times, giving the solution $R \propto e^{Ht}$ where $H$ is constant.

   A comoving observer at $\sigma = 0$ will receive signals emitted from distant points now at some time in the future. Find the proper distance $d_e$ to the most distant signal emitted now which could ever reach the comoving observer in a flat inflationary universe.
   NOTE: $d_e$, the limit on the total portion of the universe which can ever be observed by a particular observer, is called the ``event horizon.''

3. Consider a flat matter-dominated universe beginning at $t=0$. It seems paradoxical that a photon emitted from $\sigma_p (t_o)$ at $t=0$ is just now (at $t_o$) becoming visible to us at $\sigma = 0$, since that photon had zero proper distance from us initially, at $t=0$.

   Solve for such a photon's proper distance from us as a function of time. Show that the photon initially recedes from us, then reaches a maximal proper distance and approaches us. At what value of $t/t_0$ does the photon reach this maximal proper distance?