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\begin{document}

\begin{center}
{\bf ASTR 5770 Cosmology Spring 2006.  Problem Set 4.  Due Mon 8 May}
\end{center}
\vskip4pt

\sect{Growth of baryon fluctuations after Recombination}

Before Recombination, baryons are tightly coupled to the oscillating
photon-baryon fluid, as a result of which their overdensity $\delta_b$
is small compared to the overdensity $\delta_d$ of non-baryonic CDM.
After Recombination, baryons are released from the photon-baryon
fluid, and are free to begin growing their fluctuations.
Both before and after Recombination,
the non-baryonic cold dark matter ($d$)
satisfies the continuity (mass)
and Euler (momentum) equations,
in conformal Newtonian gauge,
\begin{eqnarray}
  \dot \delta_i - k v_i
  &=&
  - 3 \dot \Phi
\nonumber
\\
  \dot v_i
  + {\dot a \over a} v_i
  &=&
  - k \Psi
\end{eqnarray}
with $i = d$.
After Recombination, the baryons satisfy the same equations with $i = b$.
Here overdots represent derivatives $d / d \eta$
with respect to conformal time $\eta$.
The relevant Einstein equations are the energy
and quadrupole pressure equations,
which in the matter-dominated regime are,
again in conformal Newtonian gauge,
\begin{eqnarray}
  k^2 \Phi + 3 {\dot a \over a} \left( \dot\Phi - {\dot a \over a} \Psi \right)
  &=&
  4 \pi G a^2 \rho_m \delta_m
\nonumber
\\
  k^2 ( \Psi + \Phi )
  &=&
  0
  \ .
\end{eqnarray}

\subsect{Differential equation for matter fluctuations}

In the Newtonian limit $k \eta \gg 1$,
the equations for matter fluctuations in the matter-dominated regime simplify to
(again, overdot means $d / d \eta$)
\begin{eqnarray}
  \dot \delta_m - k v_m
  &=&
  0
\nonumber
\\
  \dot v_m
  + {\dot a \over a} v_m
  &=&
  k \Phi
\nonumber
\\
  k^2 \Phi
  &=&
  4 \pi G a^2 \rho_m \delta_m
  \ .
\end{eqnarray}
First,
translate these equations into a single 2nd order differential equation
for the matter overdensity $\delta_m$.
Next,
change the time variable from conformal time $\eta$
to cosmic scale factor $a$,
to show that
\begin{equation}
  {d^2 \delta_m \over d a^2}
  + {3 \over 2 a} {d \delta_m \over d a}
  - {3 \over 2 a^2} \delta_m
  = 0
  \ .
\end{equation}
[Hint:
The Hubble parameter $H$ in the matter-dominated regime satisfies
\begin{equation}
  H^2 = {8 \pi G \over 3} \rho_m
  = \Omega_m H_0^2 a^{-3}
  \ .
\end{equation}
]

\pagebreak
\subsect{Growing and decaying solutions for matter fluctuations}

Find the two solutions for $\delta_m$.
[Hint:
Try a power law $\delta_m \propto a^n$.]

\subsect{Solution for $\delta_m$ after Recombination}

In terms of the CDM and baryon fluctuations,
the matter density fluctuation is
\begin{equation}
  \delta_m = f_d \delta_d + f_b \delta_b
\end{equation}
where $f_d$ and $f_b$ are constants, the CDM and baryon fractions
\begin{equation}
  f_d \equiv {\rho_d \over \rho_m}
  \ , \quad
  f_b \equiv {\rho_b \over \rho_m} = 1 - f_d
  \ .
\end{equation}
Assume that before Recombination at $a_\ast$,
the CDM fluctuations are growing according to the growing mode solution
\begin{equation}
  \delta_d = C_d a
  \quad (a \le a_\ast)
\end{equation}
where $C_d$ is some constant,
and that the baryonic fluctuations are negligibly small
\begin{equation}
  \delta_b = 0
  \quad (a \le a_\ast)
  \ .
\end{equation}
By imposing continuity of the value $\delta_m$
and its derivative $d \delta_m / d a$ at Recombination,
show that the matter fluctuation satisfies
\begin{equation}
\label{deltam}
  \delta_m = C a
\end{equation}
both before and after Recombination.
What is the constant $C$ in terms of the constant $C_d$?

\subsect{Green's function for CDM and baryons}

After Recombination,
the non-baryonic CDM and the baryons each individually grow as
\begin{equation}
\label{diffeqa}
  {d^2 \delta_i \over d a^2}
  + {3 \over 2 a} {d \delta_i \over d a}
  = {3 \over 2 a^2} \delta_m
\end{equation}
for $i = d, b$,
where $\delta_m$ on the RHS is given by the solution~(\ref{deltam}).
These equations can be solved using the Green's function $G(a;a^\prime)$,
which is the solution of
\begin{equation}
\label{Geq}
  \left( {d^2 \over d a^2}
  + {3 \over 2 a} {d \over d a} \right)
  G(a;a^\prime)
  = \delta_D ( a - a^\prime )
\end{equation}
subject to the constraint that
the Green's function vanishes before the impulse at $a = a^\prime$
\begin{equation}
  G(a;a^\prime)
  = 0
  \quad
  \mbox{for $a \le a^\prime$}
  \ .
\end{equation}
Find the Green's function $G(a;a^\prime)$.
[Hint:
What are the two homogeneous solutions,
for which the RHS of equation~(\ref{Geq}) vanishes?
After the impulse at $a = a^\prime$,
the Green's function must be some linear combination of these solutions.
Find the two constants in this solution by imposing
(a) continuity of $G(a;a^\prime)$ at $a = a^\prime$,
and (b) the change in the derivative $d G(a;a^\prime) / d a$ at $a = a^\prime$
that comes from integrating the Green's function equation~(\ref{Geq})
over a tiny interval about $a = a^\prime$.]

\subsect{Particular solution}

A particular solution to the differential equation~(\ref{diffeqa})
is an integral over the Green's function
$G(a;a^\prime)$
\begin{equation}
  \left. \delta_i \right|_{\rm particular}
  =
  \int^a
  G(a;a^\prime) f(a^\prime)
  \, d a^\prime
\end{equation}
of the forcing function $f(a^\prime)$
given by the RHS of equation~(\ref{diffeqa})
\begin{equation}
  f(a^\prime)
  = {3 \, C \over 2 \, a^\prime}
  \ .
\end{equation}
Find this particular solution.

\subsect{General solution}

The general solution to the differential equation~(\ref{diffeqa})
is a sum of the particular solution and a homogeneous solution.
Find the solutions for the baryon and CDM overdensities
$\delta_b$ and $\delta_d$ after Recombination.
[Hint:
Fix the two constants in the homogeneous solution for each species
by imposing continuity of the value and the derivative
of the overdensity at Recombination.]

Sketch the solutions for the matter, baryon, and CDM overdensities
through Recombination.

\subsect{Comment}

A common statement to be found in texts is
``Following Recombination, baryons fall into the dark matter potential wells''.
Comment, in the light of your solutions.

\sect{Isocurvature fluctuations (version of Dodelson Ex.~8.1)}

In addition to the ``natural'' adiabatic initial conditions set by inflation,
there is another set, called isocurvature initial conditions.
General initial conditions are a linear combination of
adiabatic and isocurvature conditions.
Isocurvature initial conditions are defined to have zero initial potential
\begin{equation}
  \Phi(0) = 0
  \ .
\end{equation}
Following the computation of the growth of fluctuations at super-horizon
scales in the notes pages 7.20-7.23,
one finds the super-horizon isocurvature solution,
as a function of the cosmic scale factor $a$
scaled to $1$ at matter radiation equality,
$a_{\rm eq} = 1$,
\begin{equation}
\label{isocurvaturePhi}
  \Phi
  =
%  \Phi(\mbox{late}) \left(
%  1 - {2 \over a} + {8 \over a^2} + {16 \over a^3} - {16 \sqrt{1 + a} \over a^3}
%  \right)
  \Phi(\mbox{late})
  {a ( 6 + a + 4 \sqrt{1 + a} ) \over ( 1 + \sqrt{1 + a} )^4}
\end{equation}
\begin{equation}
  \Theta_0 + \Phi = \mbox{constant} = 0
\end{equation}
\begin{equation}
  \delta + 3 \, \Phi = \mbox{constant} = \delta(0) = 5 \, \Phi(\mbox{late})
\end{equation}
which shows that the potential grows
even though it starts from zero.
Physically,
for isocurvature fluctuations
matter and radiation fluctuations are initially in opposite directions,
whereas for adiabatic fluctuations they are in the same direction.

%--------------------
% FIG
    \begin{figure}
    \begin{center}
    \leavevmode
    \includegraphics[scale=.7]{superhorizon.ps}
    \end{center}
\caption[1]{\small
    \label{bgrow}
Evolution of the potential $\Phi$ at super-horizon scales
as a function of the scale factor $a$
relative to the scale factor $a_{\rm eq}$ at matter-radiation equality,
for adiabatic and isocurvature initial conditions.
    }
    \end{figure}
%--------------------

\subsect{Temperature fluctuation}

Show that the monopole temperature fluctuation $\Theta_0$
evolves approximately as
\begin{equation}
  \left. \Theta_0 \right|_{\rm isocurvature}
  \approx
  \Phi(a > a_{\rm eq})
  \left[ 1 - 2 \cos(k \eta_s) \right]
  \ .
\end{equation}
Note that this is a different answer from the one given by Dodelson.
Comment on how this differs from the temperature fluctuation from
adiabatic initial conditions
\begin{equation}
  \left. \Theta_0 \right|_{\rm adiabatic}
  \approx
  \Phi(a > a_{\rm eq})
  \left[ 1 + \left( {5 \over 3 \, T(k)} - 2 \cos(k \eta_s) \right) \right]
  \ .
\end{equation}
[Hint:
Follow the simplified procedure given on pages 8.18 and 8.19 of the notes.
The equation to be solved is the low baryon limit, $R \rightarrow 0$,
of the equation on page 8.11, namely
\begin{equation}
  \left( {d^2 \over d \eta_s^2} + k^2 \right) \left( \Theta_0 + \Phi \right)
  =
  f
\end{equation}
with forcing function $f$
\begin{equation}
  f = k^2 ( \Phi - \Psi )
  \ .
\end{equation}
You may assume that the solution takes the form given on page 8.15,
a sum of a homogeneous solution and a particular solution
\begin{equation}
  \Theta_0 + \Phi
  =
  C_+ e^{i k \eta_s} + C_- e^{- i k \eta_s}
  +
  \int_0^{\eta_s}
  {\sin \left[ k ( \eta_s - \eta_s^\prime ) \right] \over k}
  f(\eta_s)
  \, d \eta_s
  \ .
\end{equation}
What are the constants $C_+$ and $C_-$ for isocurvature initial conditions?
Having figured out the constants,
for simplicity assume that the forcing function $f$ is constant,
given by its late-time value
\begin{equation}
  f \approx 2 k^2 \Phi ( a > a_{\rm eq} )
  \ .
\end{equation}

\end{document}
