College Cosmology
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EQUATIONS
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INFORMATION
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$$ \\
\large Z = \frac{v}{c} = \frac{\Delta \lambda}{\lambda_0} = \frac{\lambda - \lambda_0}{\lambda_0} $$
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\(
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\textbf{Z} = redshift \\
\textbf{v} = velocity \hspace{4 pt}(m \hspace{2 pt} s^{-1}) \\
\textbf{c} = speed \hspace{4 pt}of \hspace{4 pt} light \hspace{4 pt}(3.00 \hspace{2 pt} x \hspace{2 pt} 10^{8} \hspace{2 pt} m \hspace{2 pt} s^{-1}) \\
{\lambda} = observed \hspace{4 pt} wavelength \hspace{4 pt}(m \hspace{2 pt} s^{-1}) \\
{\lambda_0} = emitted \hspace{4 pt} wavelength \hspace{4 pt}(m \hspace{2 pt} s^{-1}) \\ \)
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$$ \large Z + 1 = \frac{\lambda}{\lambda_0} = \frac{D}D_0{} $$
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\( \\
\textbf{Z} = redshift \\
{\lambda} = wavelength \hspace{4 pt} now \hspace{4 pt} (m) \\
{\lambda_0} = wavelength \hspace{4 pt} when \hspace{4 pt} emitted \hspace{4 pt} (m) \\
\textbf{D} = distance \hspace{4 pt} to \hspace{4 pt} object \hspace{4 pt} now \hspace{4 pt} (ly) \\
\textbf{D_0} = distance \hspace{4 pt} to \hspace{4 pt} object \hspace{4 pt} back \hspace{4 pt} then \hspace{4 pt} (ly) \\ \)
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$$ \large v = H_0 d$$
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\( \\
\textbf{v} = velocity \hspace {4 pt} (km \hspace{2 pt} s^{-1}) \\
\textbf{H_0} = Hubble's \hspace {4 pt}constant \\ \hspace{12 pt}(\sim 68 \hspace{4 pt}km \hspace{2 pt} s^{-1} \hspace{2 pt} Mpc^{-1}) \\
\textbf{D} = proper \hspace{4 pt} distance \hspace{4 pt} (km) \)
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$$ \large t_p \equiv \sqrt{\frac{\hbar G}{c^5}} $$
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\( \textbf{t_p} = Planck \hspace{4 pt} time \hspace{4 pt} (5.391 \hspace{2 pt} x \hspace{2 pt} 10^{-44} \hspace{2 pt} s) \\ {\hbar} = reduced \hspace{4 pt} Planck \hspace{4 pt} constant \\ \hspace{5 pt} (1.055 \hspace{2 pt} x \hspace{2 pt} 10^{-34} \hspace{2 pt} J \hspace{2 pt} s) \\ \textbf{G} = gravitational \hspace{4 pt} constant \\ \hspace{8 pt} (6.674 \hspace{2 pt} x \hspace{2 pt} 10^{-11} \hspace{2 pt} m^3 \hspace{2 pt} kg^{-1} \hspace{2 pt} s^{-2}) \\ \textbf{c} = speed \hspace{4 pt} of \hspace{4 pt} light \hspace{4 pt} (3.00 \hspace{2 pt} x \hspace{2 pt} 10^8 \hspace{2 pt} m \hspace{2 pt} s^{-1}) \) |
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$$ \large \rho_{rad} = \frac{4 \sigma T^4}{c^3} $$
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\( {\rho_{rad}} = mass \hspace{4 pt} density \hspace{4 pt} of \hspace{4 pt} radiation \hspace{4 pt} (kg \hspace{2 pt} m^{-3}) \\ {\sigma} = Stefan-Boltzmann \hspace{4 pt} constant \hspace{4 pt} \\ \hspace{4 pt} (5.670 \hspace{2 pt} x \hspace{2 pt} 10^{-8} \hspace{2 pt} W \hspace{2 pt} m^{-2} \hspace{2 pt} K^{-4}) \\ \textbf{T} = temperature \hspace{4 pt} (K) \\ \textbf{c} = speed \hspace{4 pt} of \hspace{4 pt} light \hspace{4 pt} (3.00 \hspace{2 pt} x \hspace{2 pt} 10^8 \hspace{2 pt} m \hspace{2 pt} s^{-1}) \) |
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$$ \large \\
\rho_c = \frac{3 {H_0}^2}{8\pi G} $$
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\( {\rho_c} = critical \hspace{4 pt} density \hspace{4 pt} (kg \hspace{2 pt} km^{-3}) \\ \textbf{H_0} = Hubble's \hspace{4 pt} constant \hspace{8 pt} \\ hspace{4 pt} (\sim 68 \hspace{4 pt}km \hspace{2 pt} s^{-1} \hspace{2 pt} Mpc^{-1}) \\ \textbf {G} = gravitational \hspace{4 pt} constant \hspace{4 pt} \\ \hspace{8 pt} (6.67384 \hspace{2 pt} x \hspace{2 pt} 10^{-11} \hspace{2 pt} m^3 \hspace{2 pt} kg^{-1} \hspace{2 pt} s^{-2}) \) |
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$$ \large \Omega_0 = \frac{\rho_o}{\rho_c} $$
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\( {\Omega_0} = density \hspace{4 pt} parameter \\ {\rho_0} = actual \hspace{4 pt} mass \hspace{4 pt} density \hspace{4 pt} of \hspace{4 pt} the \\ \hspace{9 pt} universe \hspace{4 pt} (kg \hspace{2 pt} m^{-2}) \\ {\rho_c} = critical \hspace{4 pt} density \hspace{4 pt} (kg \hspace{2 pt} m^{-2}) \) |
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$$ \large \\
\sigma = \left( \frac{I_s}{I_i} \right) \frac{1}{N} = \frac{P}{N} $$
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\( \\
{\sigma} = cross \hspace{4 pt} section \hspace{4 pt} (m^2) \\
\textbf{I_s} = intensity \hspace{4 pt} of \hspace{4 pt} scattered \hspace{4 pt} particles \\ \hspace{6 pt} (W \hspace{2 pt} m^{-2} \hspace{2 pt} s^{-1}) \\
\textbf{I_i} = intensity \hspace{4 pt} of \hspace{4 pt} incident \hspace{4 pt} particles \\
\hspace{6 pt} (W \hspace{2 pt} m^{-2} \hspace{2 pt} s^{-1}) \\
\textbf{N} = number \hspace{4 pt} density \hspace{4 pt} of \hspace{4 pt} target \hspace{4 pt} particles \\
\hspace{7 pt} (m^{-2}) \\
\textbf{P} = probability \hspace{4 pt} of \hspace{4 pt} interaction \)
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$$ \large \beta = \frac{v}{c} $$
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\( {\beta} = velocity \hspace{4 pt} coefficient \\ \textbf{v} = velocity \hspace{4 pt} (m \hspace{2 pt} s^{-1}) \\ \textbf{c} = speed \hspace{4 pt} of \hspace{4 pt} light \hspace{4 pt} (3.00 \hspace{2 pt} x \hspace{2 pt} 10^8 \hspace{2 pt} m \hspace{2 pt} s^{-1}) \) |
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$$ \large \\
t' = \gamma \left(t - \frac{vx}{c^2} \right ) \\
x' = \gamma \hspace{2 pt} (x - vt) \\
y' = y \\
z' = z $$
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\( \\
\textbf{t | t'} = time \hspace{4 pt} measured \hspace{4 pt} by \\
\hspace{14 pt} observer | observer' \hspace{4 pt} (s) \\
{\gamma} = Lorentz \hspace{4 pt} factor \\
\textbf{v} = measured \hspace{4 pt} scalar \hspace{4 pt} velocity \hspace{4 pt} (m \hspace{2 pt} s^{-1}) \\
\textbf{x|x'} = distance \hspace{4 pt} in \hspace{4 pt} x-direction \hspace{4 pt} as \\ \hspace{18 pt} measured \hspace{4 pt} by \hspace{4 pt} observer \hspace{4 pt} / \\
\hspace{17 pt} observer' \hspace{4 pt} (m) \\
\textbf{c} = speed \hspace{4 pt} of \hspace{4 pt} light \hspace{4 pt} (3.00 \hspace{2 pt} x \hspace{2 pt} 10^8 \hspace{2 pt} m \hspace{2 pt} s^{-1}) \\
\textbf{y|y'} = distance \hspace{4 pt} in \hspace{4 pt} y-direction \hspace{4 pt} as \\ \hspace{18 pt} measured \hspace{4 pt} by \hspace{4 pt} observer \hspace{4 pt} / \\ \hspace{16 pt} observer' \hspace{4 pt} (m) \\
\textbf{z|z'} = distance \hspace{4 pt} in \hspace{4 pt} z-direction \hspace{4 pt} as \\ \hspace{18 pt} measured \hspace{4 pt} by \hspace{4 pt} observer \hspace{4 pt} / \\ \hspace{16 pt} observer' \hspace{4 pt} (m) \\
\)
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$$ \large \Delta t' = \gamma \Delta t $$
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\( \\
{\Delta t} = time \hspace{4 pt} between \hspace{4 pt} two \hspace{4 pt} co-local \hspace{4 pt} events \\ \hspace{14 pt} for \hspace{4 pt} an \hspace{4 pt} observer \hspace{4 pt} in \hspace{4 pt} some \hspace{4 pt} intertial \\
\hspace{14 pt} frame \hspace{4 pt}(s)\\
{\Delta t'} = time \hspace{4 pt} between \hspace{4 pt} two \hspace{4 pt} co-local \\ \hspace{14 pt} events \hspace{4 pt} for \hspace{4 pt} another, \hspace{4 pt} inertially \\ \hspace{14 pt} moving \hspace{4 pt} with velocity \hspace{4 pt} \textbf{v}, w.r.t \hspace{4 pt} the \\
\hspace{11 pt} \hspace{4 pt} former \hspace{4 pt} observer \hspace{4 pt} (s)\\
{\gamma} = Lorentz \hspace{4 pt} factor\\ \)
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$$ \large L = \frac{L_0}{\gamma} $$
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\( \\ \\
\textbf{L} = length \hspace{4 pt} measured \hspace{4 pt} by \hspace{4 pt} observer \\
\hspace{6 pt} travelling \hspace{4 pt} at \hspace{4 pt} velocity \hspace{4 pt}\textbf{v} \hspace{4 pt} (m) \\
\textbf{L_0} = proper \hspace{4 pt} length \hspace{4 pt} (m) \\
{\gamma} = Lorentz \hspace{4 pt} factor \)
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