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\begin{document}

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\begin{center}
{\bf ASTR 5720 Galaxies \& Cosmology Fall 1997.  Problem Set 5.\\
Due Th 20 Nov}
\end{center}
\vskip4truept

\sect{Inflation}

In this question,
suppose that the scalar inflaton field $\phi$ has a Coleman-Weinberg
type potential
\be
  V(\phi) = A \left\{ \phi^4
    \left [\ln \left( {\phi^2 \over \phi_0^2} \right) - {1 \over 2} \right]
    + {1 \over 2} \phi_0^4 \right\}
\ee
where $A$ is some unspecified dimensionless constant.
As usual, the inflaton field satisfies the equation of motion
\be
\label{eqmot}
  \ddot\phi + 3 H \dot\phi + {d V \over d \phi} = 0
\ee
and the energy density and pressure of the inflaton field are
(ignoring spatial derivatives)
\be
  \rho_\phi = \frac{1}{2} \dot\phi^2 + V(\phi)
  \ , \quad
  P_\phi = \frac{1}{2} \dot\phi^2 - V(\phi)
  \ .
\ee

\subsect{How big is $A$?}

During the slow-roll phase,
the inflaton field $\phi$ and the Hubble constant $H$
evolve according to
\be
\label{slowroll}
  3 H \dot\phi + {d V \over d \phi} = 0
  \ , \quad
  H^2 = {8\pi V(\phi) \over 3 m_p^2}
\ee
where the units are $\hbar = c = 1$,
and $m_p = (\hbar c/G)^{1/2}$ is the Planck mass.
If the inflaton field starts at the minimal value $\phi_i \approx H/(2\pi)$
from vacuum fluctuations,
determine an approximate constraint on $A$
required to accomplish $\ga 70$ e-foldings of inflation.
Compare your result to the value $A = (5625/64) \alpha_{\rm GUT}^2$
with $\alpha_{\rm GUT} \approx 1/42$ predicted by the spontaneous symmetry
breaking of minimal $SU(5)$,
where $\phi_0 \approx m_{\rm GUT} \approx 10^{-4} m_p$.
[Hint: we derived an equation in class for the number of e-foldings.
Do not attempt to solve the integral analytically
(the solution is an exponential integral),
but approximate it instead.
The number
$A = (5625/64) \alpha_{\rm GUT}^2$
is from Collins, Martin \& Squires (1989) p.\ 407,
which differs from the number
$A = (25/16) \alpha_{\rm GUT}^2$ given by
Kolb \& Turner (1990) p.\ 292.
Don't ask me who's right or why they differ.]

\subsect{Condition for inflation}

The condition for inflation to occur is that the Universe accelerate,
which requires $\rho + 3 P < 0$.
Translate this into a condition on the kinetic energy
$\frac{1}{2}\dot\phi^2$ versus potential energy $V$.
Show further that the slow roll condition
$|\ddot\phi| \ll |dV/d\phi|$
is equivalent to a condition of the form
$|\Delta\frac{1}{2}\dot\phi^2| \ll |\Delta V|$.
Assuming that equation (\ref{slowroll}) holds,
translate these conditions into conditions of the form
$|d V/d\phi| \la V/m_p$ and
$|d^2 V/d\phi^2| \la V/m_p^2$.
There are some constants in these inequalities,
which I have omitted here, but which you should keep.

\subsect{End of inflation}

Inflation ends when either of the conditions in (b) are violated.
Estimate the values of $\phi$ and $V$ at the end of inflation.
[Yes, you should find $V \approx V(\phi = 0)$.]

\subsect{Effective mass of inflaton oscillations}

After the end of inflation, the inflaton field $\phi$ undergoes
coherent simple harmonic oscillations about the minimum of the potential.
The kinetic and potential energies averaged over an oscillation are
$\langle V \rangle = \langle \frac{1}{2} \dot\phi^2 \rangle$,
so the effective pressure is zero, i.e.\ the oscillations behave like
massive particles.
The mass $m$ of these particles is equal to the frequency of oscillation,
given by $V = \frac{1}{2} m^2 \delta \phi^2$,
where $\delta\phi = \phi - \phi_0$ is the deviation of $\phi$ about the
minimum of the potential.
What is the mass $m$ of the inflaton particles?

\subsect{Inflaton-particle dominated regime}

By multiplying the equation of motion (\ref{eqmot}) by $\dot\phi$ and
averaging, argue that the oscillation-averaged energy density $\rho_\phi$
of the inflaton field satisfies the equation of motion
\be
\label{eqrho}
  {d \rho_\phi \over dt} + 3 H \rho_\phi = 0
  \ .
\ee
What familiar result does this imply for the behavior of $\rho_\phi$ with
cosmic scale factor $a$?

\subsect{Reheating}

Introduce a phenomenological constant decay rate $\Gamma$
into the equation of motion (\ref{eqrho}) of the inflaton density
\be
  {d \rho_\phi \over dt} + 3 H \rho_\phi + \Gamma \rho_\phi = 0
  \ .
\ee
Solve this equation for $\rho$ as a function of cosmic scale factor $a$
and time $t$.

\subsect{Reheat temperature}

Assume that the inflaton field decays into relativistic particles,
with energy density $\rho_r$.
The relativistic energy density satisfies
\be
  {d a^4 \rho_r \over dt} = \Gamma a^4 \rho_\phi
  \ .
\ee
Solve this equation assuming that the evolution of
$\rho_\phi$ and $H$ are given
by part (e) [not (f)!], which is valid for $\rho_r \ll \rho_\phi$,
and evaluate the result for $t = \Gamma^{-1}$,
which effectively defines the reheat epoch
(when most of the entropy has been created).
Hence estimate the reheat temperature $T_h$,
showing that $T_h \approx (m_p \Gamma)^{1/2}$.

\subsect{(Not for credit) Numerical solution}

Use Mathematica to solve the equation of motion (\ref{eqmot}) for $\phi$
numerically.
You will find that Mathematica cannot solve the equation (with NDSolve)
in the form given, so you will have to do some thinking.

\sect{Linear Growth of CDM and Baryon Fluctuations}

\subsect{CDM in a matter plus radiation universe}

Consider a flat Universe containing radiation
($\bar\rho_\gamma \propto a^{-4}$)
and Cold Dark Matter
($\bar\rho_M \propto a^{-3}$).
Find the growing mode solution for the linear evolution of Dark Matter
perturbations
$\delta \equiv ( \rho_M - \bar\rho_M ) / \bar\rho_M$.
You may assume the radiation remains completely smooth.
Interpret your result.

\subsect{Baryons in a CDM universe}

Consider a flat Universe containing Cold Dark Matter
and a sprinkling of baryons
(so $\bar\rho_b \propto \bar\rho_M \propto a^{-3}$
but $\bar\rho_b \ll \bar\rho_M$).
Just after Recombination, the CDM already contains perturbations,
but the baryons are practically smooth.
Find the growing mode solution for the linear evolution of baryonic
perturbations
$\delta_b \equiv ( \rho_b - \bar\rho_b ) / \bar\rho_b$
induced by growing mode CDM perturbations
$\delta \equiv ( \rho_M - \bar\rho_M ) / \bar\rho_M \propto a$.
Interpret your result.

\subsect{(Not for credit) Decaying solution}

Use the Wronskian to derive the decaying mode solution to (a).
Curiously, the decaying mode turns out to play the dominant role in
the growth of CDM fluctuations during the radiation-dominated regime.
The decaying mode has a logarithmic behavior $\sim \ln (a^{-1})$,
which contributes negatively, so ends up looking like a growing mode
$\sim \ln a$.

\end{document}
