%Kiyoshi Shiraishi:Basic Cosmology %
03/03/1999
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Basic Cosmology


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\title{Basic Cosmology}
\author{Kiyoshi Shiraishi}
\date{\today}
\begin{document}
\maketitle
\begin{abstract}
This article is a summary of the standard cosmology.
\end{abstract}

\newpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Basic Equations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

{}~

{\bf Friedmann-Robertson-Walker metric:}
\be
ds^2=-dt^2+a^2(t)\left[\frac{dr^2}{1-kr^2}+r^2(d\theta^2+\sin^2 d\phi^2)\right]
\ee
\be
k=(+1,0,-1)=(closed,~flat,~open)
\ee

\smallskip

{\bf Einstein Eq.:}
\be
H^2\equiv\left(\frac{\dot{a}}{a}\right)^2=\frac{8\pi G\rho}{3}-\frac{k}{a^2}+
\frac{\Lambda}{3}
\ee
\be
\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}(\rho+3p)+\frac{\Lambda}{3}
\ee

\smallskip

{\bf Energy Conservation:}
\be
\dot{\rho}=-3\frac{\dot{a}}{a}(\rho+p)
\ee

\begin{quote} $\underline{\Lambda=0, k=0}$

Matter-dominated $p=0$: $\rho\propto a^{-3}$ and $a\propto t^{2/3}$

Radiation-dominated $p=\rho/3$: $\rho\propto a^{-4}$ and $a\propto 
t^{1/2}$

Equation-of-state $\gamma\equiv p/\rho$: $\rho\propto a^{-3(1+\gamma)}$ and 
$a\propto t^{2/[3(1+\gamma)]}$

Power-Law-Inflation: $a\propto t^n$ with $n>1$ or $\gamma<-1/3$
\end{quote}

For a while, we will assume $\Lambda=0$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Standard Model (matter-dominated, $\Lambda=0$)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

{}~

{\bf Hubble constant:}
\be
H_0=100 h_0 ~\km ~\s^{-1} ~\Mpc^{-1}=2.133 h_0\times 10^{-42} ~\GeV
\ee
\be
0.6T>m_e$, $N(T)=
g_{\gamma}+\frac{7}{8}(g_{e}+3g_{\nu})=43/4$

for $m_{\pi}>T>m_{\mu}$, $N(T)=57/4$
\end{quote}

\be
(T_{\nu}/T_{\gamma})^3=g_{\gamma}/(g_{\gamma}+\frac{7}{8}g_{e})=4/11
\ee

\smallskip

{\bf Cosmic Microwave Background:}
\be
T_0=2.728 \pm 0.002 ~\K~(95\% ~CL)
\ee
\be
n_{\gamma}\approx 412 ~\cm^{-3}
\ee
\be
\rho_{\gamma}=4.66 \times 10^{-34}~\g ~\cm^{-3}=0.262 ~\eV ~\cm^{-3}
\ee

{\bf Cosmic Neutrino Background:}
\be
T=1.947 \pm 0.003 ~\K~(95\% ~CL)
\ee

{\bf Present Entropy density:}
\be
s=7.0 ~n_{\gamma}
\ee

\be
\eta=n_B/n_{\gamma}
\ee
\be
2.8 \times 10^{-10}\leq\eta\leq 4.0 \times 10^{-10}~~~(from~BBN)
\ee
\be
\Omega_B=3.67 \times 10^7 \eta h_0^{-2} (T_0/2.728~\K)^3
\ee
\be
0.010\leq\Omega_B h_0^2\leq0.015
\ee

{\bf Matter-Radiation Equality:}
\be
a_{eq}=4.1707 \times 10^{-5} (\Omega_0 h_0^2)^{-1} ~a_0
\ee
\be
t_{eq}=3.2 \times 10^{10} (\Omega_0 h_0^2)^{-2} ~\s
\ee
\be
T_{eq}=5.6362 (\Omega_0 h_0^2) ~\eV
\ee

{\bf Decoupling (approx.):}
\be
a_{dec}=a_0/1100=21 (\Omega_0 h_0^2) ~a_{eq}
\ee
\be
t_{dec}=5.6384 \times 10^{12} (\Omega_0 h_0^2)^{-1/2} ~\s
\ee
\be
T_{dec}=0.26 ~\eV
\ee

\smallskip

{\bf Very Early Universe:}
\be
t=\frac{2.42}{\sqrt{N(T)}}\left(\frac{1~\MeV}{T}\right)^2 ~\s
\ee


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{$\Lambda\ne 0$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

{}~

\be
q_0\equiv -\frac{\ddot{a}a}{\dot{a}^2}=\frac{\Omega_0}{2}-\Omega_{\Lambda}
\ee

where
\be
\Omega_{\Lambda}=\frac{\Lambda}{3 H_0^2}
\ee

\be
-1<\Omega_{\Lambda}<2
\ee


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Basic Units}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

{}~

{\bf Natural Units:}
\be
c=\hbar=k_B=1
\ee

{\bf Planck Mass:}
\be
M_{P}\equiv (1/G)^{1/2}=1.221 \times 10^{19} ~\GeV
\ee

\be
1 ~\Mpc=3.085 \times 10^{24} ~\cm=1.5637 \times 10^{38} ~\GeV^{-1}=
3.2615 \times 10^6 ~\lighty
\ee

\bea
1 ~\GeV&=&1.1605 \times 10^{13} ~\K=1.6022 \times 10^{-3} ~\erg \nn
&=&
5.0676 \times 10^{13} ~\cm^{-1}=1.5192 \times 10^{24} ~\s^{-1}
\eea

\be
1 ~\GeV^{3}=1.3014 \times 10^{41} ~\cm^{-3}
\ee

\be
1 ~\GeV^{4}=2.3201 \times 10^{17} ~\g~\cm^{-3}
\ee

\end{document} 

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