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{\bf ASTR 5720 Galaxies \& Cosmology Fall 1997.  Problem Set 2.\\
Due Thur 18 Sept}
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\noindent
{\bf 1. Cosmological Density of Radiation}
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\begin{itemize}
\item[(a)]
What is $\Omega_\gamma$ in the $2.73\,$K CMB radiation at the present time
(i.e.\ what is the mass-energy density in photons, divided by the critical
density)?
Express your answer in terms of the dimensionless Hubble constant
$h \equiv H_0/(100 \,{\rm km}\,{\rm s}^{-1}\,{\rm Mpc}^{-1}$.
\item[(b)]
At what redshift $z_{\rm eq}$ were the energy densities of matter and
radiation equal?
Express your answer in terms of $h$ and the present day $\Omega_M$ in matter.
Compare this redshift to the redshift $z_R \approx 1300$ of Recombination.
\end{itemize}

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\noindent
{\bf 2. The Age Problem}
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Derive formulae for the age $t$ of the Universe
as a function of the Hubble constant $H$ and the cosmological density
$\Omega_M$ of matter in:
\begin{itemize}
\item[(a)]
a matter-dominated Universe;
\item[(b)]
a flat ($\Omega_M + \Omega_\Lambda = 1$) Universe containing both matter
and vacuum energy.
\end{itemize}
Use your result to draw (with a computer, please) contours of constant age
on a contour diagram of $\Omega_M$ versus $H$,
with age in Gyr and the Hubble constant in units of
${\rm km}\,{\rm s}^{-1}\,{\rm Mpc}^{-1})$.
The Hipparcos-revised age of globular clusters estimated by
Chaboyer et al (1997) is $11.5 \pm 1.3$~Gyr.  Comment.

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\noindent
{\bf 3. Horizon Size at Recombination}
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How large was the horizon radius at Recombination?
Express your answer:
\begin{itemize}
\item[(a)]
as a comoving distance, expressed in $h^{-1} {\rm Mpc}$;
\item[(b)]
as an angle observed on the CMB today.
\end{itemize}
[Hint:
Assume a matter-dominated Universe.
Express your answers in terms of sensible things like
$\Omega_0$ and the redshift $z_R \approx 1300$ of Recombination.
To arrive at a simple answer
you will need to make some sensible approximations.
First, $\Omega = 1$ was an excellent approximation
at the time of Recombination, even though $\Omega_0$ is not necessarily 1 now.
Second, from our point of view,
the surface of last scattering is very close to our present horizon.]

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\noindent
{\bf 4. NOT FOR CREDIT: Charge Asymmetry of the Universe}
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From the constraint that the evolution of the Universe on large scales
is dominated by gravity rather than electricity,
estimate an upper limit to the global charge asymmetry
$(N_+ - N_-)/(N_+ + N_-)$ of electrically charged particles in the Universe.
What evidence would suggest that gravity rather than electricity dominates?

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