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%TCIDATA{<META NAME="Title" CONTENT="Reference/Physics/Cosmology">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
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\section{Cosmology\label{Cosmology}}

If for the universe as a whole we assume that

\begin{enumerate}
\item  There exists a global time coordinate which acts as $x^{0}$ of a
Gaussian coordinate system

\item  The 3-dimensional spaces are isotropic for a certain value of $x^{0}$

\item  Each point is equivalent to every other point for a fixed $%
x^{0}\medskip $
\end{enumerate}

\noindent then the \textit{Robertson-Walker metric} can be derived for the
line element: 
\[
ds^{2}=-c^{2}dt^{2}+\frac{R^{2}(t)}{r_{0}^{2}\left( 1-\frac{kr^{2}}{%
4r_{0}^{2}}\right) }(dr^{2}+r^{2}d\Omega ^{2}) 
\]
For the \textit{scale factor} $R(t)$ the following equations can be derived 
\[
\frac{2\ddot{R}}{R}+\frac{\dot{R}^{2}+kc^{2}}{R^{2}}=-\frac{8\pi \kappa p}{%
c^{2}}+\Lambda ~~~\text{and}~~~\frac{\dot{R}^{2}+kc^{2}}{R^{2}}=\frac{8\pi
\kappa \varrho }{3}+\frac{\Lambda }{3} 
\]
where $p$ is the pressure and $\varrho $ the density of the universe. If $%
\Lambda =0$ can be derived for the \textit{deceleration parameter} $q$, 
\[
q=-\frac{\ddot{R}R}{\dot{R}^{2}}=\frac{4\pi \kappa \varrho }{3H^{2}} 
\]
where $H=\dot{R}/R$ is \label{Hubble's constant}\textit{Hubble's constant}.
This is a measure of the velocity with which galaxies far away are moving
away from each other, and has the value $\approx (75\pm 25)$ km$\cdot $s$%
^{-1}\cdot $Mpc$^{-1}$.

This gives three possible conditions for the universe (here, $W$ is the
total amount of energy in the universe):

\begin{enumerate}
\item  Parabolic universe: $k=0$, $W=0$, $q=\frac{1}{2}$. The expansion
velocity of the universe $\rightarrow 0$ if $t\rightarrow \infty $. The
density $\varrho _{\mathrm{c}}=3H^{2}/8\pi \kappa $ is the \textit{critical
density}.

\item  Hyperbolic universe: $k=-1$, $W<0$, $q<\frac{1}{2}$. The expansion
velocity of the universe remains positive forever.

\item  Elliptic universe: $k=1$, $W>0$, $q>\frac{1}{2}$. The expansion
velocity of the universe becomes negative after some time and the universe
starts collapsing.
\end{enumerate}

\subsection{\ Related Reference Topics}

{\small \hyperref{Calculus}{}{}{../Mathematics/cal_contents.tex}}

{\small \hyperref{Differential Equations}{}{}{../Mathematics/ODE_contents.tex%
}}

{\small \hyperref{Table of Integrals}{}{}{../Mathematics/TAB_integrals.tex}}

{\small \hyperref{Table of Derivatives}{}{}{%
../Mathematics/TAB_derivatives.tex}}

{\small \hyperref{Table of Physical Constants}{}{}{PhysicalConstants.tex}}

{\small \hyperref{Table of SI Units}{}{}{SIUnits.tex}}

\subsection{\ Related Scientific Notebook Topics}

{\small \hyperref{Basic Techniques for Doing Mathematics}{}{}{%
../../Help/DM1.tex}}

{\small \hyperref{Numbers, Functions, and Units}{}{}{../../Help/DM2.tex}}

{\small \hyperref{Algebra}{}{}{../../Help/DM3.tex}}

{\small \hyperref{Trigonometry}{}{}{../../Help/DM4.tex}}

{\small \hyperref{Function Definitions}{}{}{../../Help/DM5.tex}}

{\small \hyperref{Plotting Curves and Surfaces}{}{}{../../Help/DM6.tex}}

{\small \hyperref{Calculus}{}{}{../../Help/DM7.tex}}

{\small \hyperref{Matrix Algebra}{}{}{../../Help/DM8.tex}}

{\small \hyperref{Vector Calculus}{}{}{../../Help/DM9.tex}}

{\small \hyperref{Differential Equations}{}{}{../../Help/DM10.tex}}

{\small \hyperref{Vectors}{}{}{../../Help/DM9-1.tex#Vectors}}

{\small \hyperref{Gradient, Divergence, and Curl}{}{}{%
../../Help/DM9-2.tex#Gradient}}

{\small \hyperref{Plots of Vector Fields and Gradients}{}{}{%
../../Help/DM9-3.tex#Vector field}}

{\small \hyperref{Scalar and Vector Potentials}{}{}{%
../../Help/DM9-4.tex#Scalar potential}}

\rule{0.67in}{0.01in}

\quad {\small Index entries: \hyperref{Inner product}{}{}{%
../../Help/DMIndex.tex#Inner product}, \hyperref{Vector calculus}{}{}{%
../../Help/DMIndex.tex#Vector calculus}, \hyperref{Vector field}{}{}{%
../../Help/DMIndex.tex#Vector field}}\hfill

\noindent \hrulefill

\begin{center}
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\noindent \hrulefill

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