RandallB said:
Can you help me understand a Cosmology (ie. GR?) based "red shift calculator" ?
The formulas to calculate the recession speed for a given redshift are not difficult to obtain, but you should be a little bit familiar with the Friedmann equation:
H^2 = \frac{8 \pi G} {3 c^2} (\rho_m + \rho_r) - \frac{k c^2} {a^2} + \frac{\Lambda c^2} {3}
with \inline{H = \frac{\dot{a}}{a}} the Hubble parameter, \inline{\rho} the energy density of matter and radiation (the energy density of radiation can be neglected), \inline{k} the curvature parameter and \inline{\Lambda} the cosmological constant.
The steps to obtain the expressions are, first, define the parameters which determine the cosmological model:
\Omega_m = \frac{8 \pi G} {3 c^2 H^2} \rho_m
\Omega_r = \frac{k c^2} {a^2 H^2}
\Omega_\Lambda = \frac{\Lambda c^2} {3 H^2}
Substitute in the formula and evaluate H for the current cosmological time as H
0 (which does also determine the cosmological model).
Further on, consider the evolution of matter as:
\frac{\rho_{m0}}{\rho_{m}} = (\frac{a}{a_0})^3
and consider the definition of redshift \inline{1+z = \frac{a}{a_0}}, to express da as a function of dz and, further on, dt as a function of dz considering \inline{dt = da/\dot{a}}...
One arrives to an equation:
dt = \frac{-dz}{(1+z) H_0 \sqrt{f(z)}}
You can integrate this between z = 0 and the given redshift of the galaxy to get the look-back time. The distance can be obtained measuring along the path of a radial photon considering that in the RW metric:
0 = - c^2 dt^2 + a^2dr^2
dr = \int \frac{c dt}{a}
substituting dt with the equation above for dt, and substituting a with the definition of redshift you should get:
D = \frac{c}{H_0} \int \frac{dz}{\sqrt{f(z)}}
For the general case, this must be integrated numerically. For special cases (such as a flat space) one can simplify things and calculate the integral.
If you have the distance, the recession speed can be calculated with the Hubble law v = H D, with H the current value of the Hubble parameter (H_0).
If you want to make a simple script to compute this an other values for the general case, another (similar) derivation can be used, which simplifies things. If you are interested, you may ask me. I have a javascript which is shorter, far more simple and more intuitive than Ned Wrights one – which is the one used by Siobahn Morgan (I also have no problem to post the code or the link here if others are also interested).
