Friday, 16 April 2021

Luminosity distance

In section 8.5 we are looking at redshifts and distances. We started in an FLRW universe with metric
\begin{align}{ds}^2=-{dt}^2+a^2\left(t\right)\left[\frac{{\rm dr}^2}{1-\kappa r^2}+r^2{d\theta}^2+r^2\sin^2{\theta}{d\phi}^2\right]&\phantom {10000}(1)\nonumber\end{align}or
\begin{align}{ds}^2=-{dt}^2+a^2\left(t\right){R_0}^2\left[{d\chi}^2+{S_k}^2\left(\chi\right){d\Omega}^2\right]&\phantom {10000}(2)\nonumber\end{align}where $k\in\left\{-1,0,1\right\}$ and
\begin{align}S_k\left(\chi\right)\ \equiv\left\{\begin{matrix}\sin{\chi},&k=+1&\rm{closed}\\\chi,&k=0&\rm{flat}\\\sinh{\chi}&k=-1&\rm{open}\\\end{matrix}\right.&\phantom {10000}(3)\nonumber\end{align}(2) is Carroll's 8.106, a sort of hybrid FLRW metric.

This starts at Carroll's 8.110 where he defines distance luminosity $d_L$ which you can get if you know the absolute luminosity of a star (or galaxy) and can measure the amount of light that reaches you. We then correct that for the expansion of the universe and further correct it because the universe deviates from a perfect sphere according to $S_k\left(\chi\right)$. We arrive at the celebrated and complicated formula at (4) which uses (5) which uses (6) .
\begin{align}d_L=\left(1+z\right)\frac{{H_0}^{-1}}{\sqrt{\left|\Omega_{c0}\right|}}S_k\left[\sqrt{\left|\Omega_{c0}\right|}\int_{0}^{z}\frac{dz^\prime}{E\left(z^\prime\right)}\right]&\phantom {10000}(4)\nonumber\end{align}where
\begin{align}E\left(z\right)=\left[\sum_{i(c)}{\Omega_{i0}\left(1+z\right)^{n_i}}\right]^\frac{1}{2}&\phantom {10000}(5)\nonumber\end{align}and there are four elements in the summation, one each for matter, radiation, curvature and vacuum energy density. $\Omega_i$ is the density parameter which we met before and was defined as
\begin{align}\Omega_i=\frac{8\pi G}{3H^2}\rho_i&\phantom {10000}(6)\nonumber\end{align}where the $\rho_i$ are the energy densities for each of those four things. $\Omega_i$ and $H$, the Hubble factor acquire a 0 subscript for their values in the present epoch.

I then try to calculate (4) from known values. $\Omega_{c0}$ the pseudo-density parameter for curvature comes from Friedmann's equation and the other three density parameters. That radiation energy is negligible, matter energy density corresponds to what is usually called ordinary (baryonic) plus dark matter and vacuum energy density is dark energy. The integral in (4) must be calculated numerically and I plotted below (page 10 in the pdf). After that we have plots of luminosity distance vs redshift for the three kinds of universe determined by $k$ in (3). Luminosity distances greater 100 billion light years are quite easy to get. I thought everything was a failure until I found some distance calculators which gave more or less the same! According to Carroll, these distances are compared with those measured from absolute and apparent luminosity. Luminosity distance is mighty peculiar. Coming soon: Proper motion distance and Angular diameter distance.

I also found that there is a missing divide by sign in Carroll's equation 8.122.
Read all about it at Commentary 8.5.2 Redshifts and Distances.pdf (12 pages)