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{\large Astro 321: Problem Set 1}
Due April 7
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\section{\large Problem 1: Units}
Convert the following quantitites by inserting the appropriate factors
of $c$, $\hbar$, $k_B$ and unit conversions. You may find Peacock chapter 9
helpful.
\begin{itemize}
\item $H_0=100 h$km s$^{-1}$ Mpc$^{-1}$ into (a) eV, (b) Mpc$^{-1}$,
(c) Gyr$^{-1}$. [Corresponds to upper limit on the mass of
a dark energy particle, the inverse Hubble length, inverse
approximate age.]
\item $\rho_{\rm crit}= 3H_0^2/8\pi G$ into (a) g cm$^{-3}$, (b) GeV$^{4}$,
(c) eV cm$^{-3}$, (d) protons cm$^{-3}$, (e) $M_\odot$ Mpc$^{-3}$.
If the cosmological constant, has $\rho_\Lambda = 2\rho_{\rm crit}/3$,
what is its energy scale in $eV$ (i.e. $\rho_\Lambda^{1/4}$).
Compare that to the Planck mass.
\item $T_{\rm CMB} = 2.728$K to (a) eV. Assuming a black body distribution,
convert this to number density $n_\gamma$ in photons cm$^{-3}$
and energy density $\rho_\gamma$ in (a) eV cm$^{-3}$ (b)
g cm$^{-3}$, and $\Omega_\gamma = \rho_\gamma/\rho_{\rm crit}$.
\item $T_{\nu} = (4/11)^{1/3} T_{\rm CMB}$. Use this to express
$n_\nu$, $\rho_\nu$ and $\Omega_\nu$ in the above units assuming
that the neutrinos are relativistic (and fermions and have
three species!).
\item With the above relic number density, now consider the case where
one out of three neutrino species has a mass of 1 eV and the rest
are massless. What is the density of
relic neutrinos in units of the critical density
$\Omega_{\nu, \rm massive}$. For what mass is the density at the
critical value.
\end{itemize}
\section{\large Problem 2: Conformal Time}
\begin{itemize}
\item Assume the universe today is flat with both matter ($\Omega_m$)
and a cosmological constant ($\Omega_\Lambda$). (a) Compute the
conformal age or horizon of the universe
and plot your result for $H_0 \eta_0$
as a function of $\Omega_m$. (b) What is the current horizon size for
a universe with $\Omega_m=1/3$ and $h=1/\sqrt{2}$?
(c) What is the mass contained within the current horizon in solar
masses. If all objects were $10^{13} h^{-1}$ $M_{\rm \odot}$ in mass, how many
are in the observable universe.
\item Evaluate the conformal age as a function of the scale
factor in the above cosmology. What happens when $a \rightarrow
\infty$. Comment on the implications for establishing causal
contact between observers currently separated by much more than
a Hubble length.
\end{itemize}
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