Cosmological Redshift
Cosmological Redshift#
Recall that redshift \(z\) is defined as,
\[\begin{split}
\begin{gather}
z \equiv \frac{\Delta \lambda}{\lambda}\\
z + 1 = \frac{\lambda_o}{\lambda}
\end{gather}
\end{split}\]
\(\lambda\): emitted wavelength
\(\lambda_o\): observed wavelength
\(\Delta \lambda\): naturally, \(\Delta \lambda = \lambda_o - \lambda\)
The expansion of the universe preserves the quantity,
\[
\begin{equation}
\frac{\lambda}{a} = \text{const}
\end{equation}
\]
One can imagine the wavelength of a photon is linearly proportional to the scale factor. This causes a cosmological redshift at the amount of,
\[\begin{split}
\begin{gather}
a(t) = \frac{1}{1+z}\\
\boxed{z(t) = \frac{1}{1 + a(t)}}
\end{gather}
\end{split}\]