Gurevich and Zybin
The whole paper is very nice to read, these people have some serious chops (although Andrei claims the cosmology part of it is a bunch of nonsense). The relevant sections I think are 4 and 5, and the scaling of the density they find is in eqn. 79. They have started with different initial conditions, but using an adiabatic invariant, they have derived the scaling of the density with x, which is not $\rho(x) \propto x^{-1/2}$ but rather $\rho(x) \propto x^{-4/7}$. I believe their analysis does not hold during the initial period of violent relaxation, but is valid thereafter.
I threw their scaling (which is slightly steeper) onto a plot, to compare to the numerical results:
By eye theirs looks a little steep, but it's not significant compared to our error bars. If we were going to try and say something about this quantitatively, all of a sudden this error bar discussion we have been having becomes really important. Jerry pointed out that in the scheme I have been using, the dispersion in adjacent bins is extremely correlated, because if a particle leaves a bin, it automatically shows up in the bin directly next to it. This means that treating the error bars as uncorrelated in a chi-square fit will result is an anomalously good fit. A possible error model (which I can check numerically): it is unlikely that a particle will ever "jump" a bin, that is, the dispersion always causes it to join the bin next to it. In the same way we measure the variance in each bin between all the simulations, we can measure the covariance between all adjacent bins. We can use a chi square estimator where the covariance matrix looks like the diagonal plus one off diagonal term either side of the diagonal. This will cause the chi square values to be about 3x what they are now (which is more consistent with the lousy K-S test values we were getting once upon a time). It may be that one of these two models will fit significantly better than the other.
Obviously finding this paper was exciting and also disappointing... I am still trying to figure out if what we have adds anything to this story, for example that the scaling is in place before the adiabatic approximation becomes valid (actually, I am not sure why they claim it becomes valid right after the first caustic, anybody understand this?)... The different behavior of very steep initial density profiles is also something new (but it would be nice to explain the origin of this). Anyhow, any thoughts about what I should do now would be greatly appreciated.
