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.

About Phil Bull

I'm a theoretical cosmologist, currently working as a NASA NPP fellow at JPL/Caltech in Pasadena, CA. My research focuses on the effects of inhomogeneities on the evolution of the Universe and how we measure it. I'm also keen on stochastic processes, scientific computing, the philosophy of science, and open source stuff. View all posts by Phil Bull

2 responses to “Buchert deceleration parameter doesn’t care about the sign of the expansion rate

  • Roberto A Sussman

    On the application of Buchert’s formalism to LTB models, please look at my paper arXiv:1102.2663 [Back-reaction and effective acceleration in generic LTB dust models, CQG, vol 28, pp 235002 (2011) Roberto A Sussman]. The models by Bolejko and Andersson 2008 are cited and commented. At least I humbly request that you cite this paper, as it is the most comprehensive study of Buchert’s method in LTB models and corrects errors in previous papers (Paranjape and Singh, Rasanen, etc).

    Sincerely, Roberto A Sussman [please reply to sussman@nucleares.unam.mx]

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