How to calculate how much a galaxy moves from its coordinate at redshift 0 to redshift 1?

Question

I have two simulation snapshots in my hand at redshift 0 and 1. I know x, y, z coordinates of the galaxies in both redshifts 0 and 1, however there is no way for me to identify a single galaxy in both redshifts, i.e., there is no tracker/tracer for the galaxies that will trace the position of galaxies from redshift 1 to 0.

Now I have halo catalogue for both $z =0$ and $z=1$ and I am interested to trace a certain galaxy from z=0 to 1. I know that galaxy's exact x,y,z at $z =0$. My initial plan is to create a search with $\sqrt{x^2+y^2+z^2}\le r$ in kpc. So any galaxy at redshift 1 with coordinates less than r maybe a potential candidate.

My question is, is there any intelligible guess of what a potential value of r would be in kpc? In other words, I am interested to know with reduced Hubble parameter $h = 0.7$, how much far a galaxy might move from $z = 1$ to $z =0$?

Answer 1

The galaxies don't "move" (unless you have given them a peculiar velocity); space expands, such that the meaning of $x,y,z$ will change.

The Hubble parameter is simply defined as the rate of change of the scale parameter divided by the scale parameter $\dot{a}/a$. But $z$ is also related to the Hubble parameter, so if your epochs are defined by $z$, the value of the Hubble parameter at any particular epoch doesn't matter.

I am assuming that you have a coordinate system based on our Galaxy being at the "origin". In which case, the "initial" position of a galaxy (at $z=1$) is related to where it is "now" at $z=0$.

Redshift and scale factor are related by $ a = (1+z)^{-1}$. So taking your example of a galaxy at $x_0, y_0, z_0$ at $z=0$, then at $z=1$ all galaxies were closer together by a factor of two, and (ignoring any peculiar velocity) $x_1 = x_0/2$, $y_1 = y_0/2$ and $z_1 = z_0/2$.

I don't know what you mean by "halo catalogue". The simple relationship above completely breaks down on scales smaller than tens of Mpc, because galaxies then become influenced by their local gravitational potential. The relationship between galaxy separation and time then has little to do with cosmology and more to do with the dynamics of their local groups and clusters and no general answer can be given, other than to point out that $z=1$ corresponds to more than half the age of the universe - trying to predict the position of a galaxy within a cluster or group based on a snapshot 7 billion years ago is not possible!