# Buchert deceleration parameter doesn’t care about the sign of the expansion rate

This week’s task: debugging some code that calculates the Buchert spatial average in LTB models. It’s a Python code, using my homebrew LTB background solver (also in Python). I’m using the results reported in a few papers to help debug my code, but I’ve run into problems with reproducing one model in particular (model 8 from Bolejko and Andersson 2008, an overdensity surrounded by vacuum). Hmmm.

I’ll spare the gory details, but one potential problem was that I might have used the wrong sign for the transverse Hubble rate. The model, as specified in the paper, gives no clue as to the sign of the Hubble rates (i.e. whether the overdense region is in a collapsing or expanding phase), only specifying a density and spatial curvature profile. In the process of constructing the model, you need to take the square root of the LTB “Friedmann” equation, and of course there is a freedom in which sign of the root you take. Out of force of habit with LTB models, I was choosing the positive sign. So would choosing the negative sign solve the discrepancy I was seeing between my code and the Bolejko and Larsson paper?

As it turns out: No. I’ll have to keep trying. But it did lead to what I thought was an interesting little result: the Buchert averaged hypersurface deceleration parameter, usually written $q_\mathcal{D}$, is invariant under $\Theta \mapsto - \Theta$, where $\Theta$ is the expansion scalar for the dust congruence. This means that it doesn’t care whether your structures are collapsing or expanding, as long as the density profile and variance of the Hubble rates are the same. This is pretty trivial by inspection of the general form of the expression for $q_\mathcal{D}$, but it hadn’t crossed my mind before.