deletions | additions       diff --git a/cosmology.tex b/cosmology.tex  index 32d28f9..88c192f 100644  --- a/cosmology.tex  +++ b/cosmology.tex 
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\begin{split}  \frac{d x}{d t} &= \frac{1}{a} \implies x(t) = \int \frac{\mathrm{d} t}{a(t)}\\  1 + z &= \frac{\lambda_0}{\lambda} = \frac{\omega_0}{\omega} = \frac{a(t_0 )}{a(t_e )} \\  v = &=  H_0 d \\ H_0 &= H(t_0) = \frac{\dot{a}(t_0)}{a(t_0)}  \end{split}  \end{equation}