The Big Picture
The biggest picture -- that large scale structure of the universe.
Once I was in an airport and I met someone I vaguely knew mainly as a friend of my brother in law -- Francesco Sylos Labini. He is an astrophysicist and was flying off to present a paper on the large scale structure of the universe. He kindly (end enthusiastically) presented his talk to me to pass the time.
Also, coincidentally, I met his father, the very famous Italian economists Paolo Sylos Labini the same day (and neither knew the other would be there).
So I once heard a lecture about the large scale structure of the Universe in an airport. The basic fact is that galaxies are not uniformly distributed in space. There is structure at the very humongogigantic scale. The talk was about one simple statistic which seems to capture this hint of a pattern. A very simple statistic.
Take ball of radius r around the galaxy i. How many other galaxies are in this ball as a function of r ? this gives G(i,r). Now average over i. This gives a function of r G(r). As r goes from unbelievably big to a gazillion times unbelievably big, does the function approach an asymptote ?
If Galaxies were really uniformly distributed and the patters were the sort of illusiory patterns that people see in structureless data, then the log of G(r) would approach the line of slope 3. In fact it goes to a slope of between 2 and 3 (I forget the exact number which the summary statistic which tells us about the position of the most atoms). So I will call the statistic X.
X = limit as r goes to the observable universe of log(G(r))/r. The estimate is for the currently observed universe.
There was something important about how this statistic seemed to be a consistent estimate of a population parameter based on how it was similar for different subsamples.
So my reaction was "hmm sounds vaguely like a Hausdorf dimension, like as a harmonic mean is to an arithmetic mean". Here the idea is how many galaxies in a sphere. For the Hausdorf dimension the question is how many spheres are needed to cover all of the galaxies. So Hausdorf has a function of r which is N(r) the number of spheres of radius r needed to cover a set. The Hausedorf dimention (if it exists) is the limit of log(N(r)) times r as r goes to zero. A line has Hausedorf dimension 1, a plane 2 so it's like dimension. However it can be a fraction for some sets thus called fractals.
Francesco told me that that was the original idea but it was too computationally burdensome. Now since there are a finite number of galaxies, if each is treated as a point the Hausedorf dimension is zero. As r gets to be small but much bigger than the width of a galaxy, the number of spheres will stick at one per galaxy (note I just read small but much bigger than a galaxy ... I'm thinking of taking the limit as r goes to zero but not really let's stop at a tiny number like a hillion jillion miles). So really the number like a Hausdorf dimension is the limit as r goes down not to zero but to a huge number such that each ball still contains a lot of galaxies.
The two calculations are different.
So I was thinking about how they are different. Imagine mixtures. Let's say there are two sets of galaxies half of galaxies are uniformly distributed and half are on a 2 dimensional manifold. The Hausdorf dimension is 3. As r gets small a tiny fraction of the spheres are needed to cover the manifold so they don't matter at all.
The galaxy statistic is not 3. Each galaxy on the manifold will be in many many spheres around other galaxies on the manifold. Almost all of the count of galaxies in a big sphere around another galaxy will be on the manifold. The statistic will coverge to 2.
So I have a question. Are galaxies distributed all on a set of one fractional dimension (say 2.7) or as a mixture some in one fractal and some in another ?
It seems to me that the calculations behind the statistic X can be used to give a meaningful answer to this question. If galaxies are a mixture of a bunch on say a 2 dimensional manifold and a bunch uniformly distributed, then G(i,r) will have a bimodal distribution for large r with G very big but growing like r squared if i is on the manifold and G very small but growth like r cubed if i is from the uniform distribution.
That is I think that the distibution of G(i,r) is interesting and not just the average.
I think that the claim "galaxies are distributed on a fractal of dimension 2.7 and within that fractal are uniformly distributed" has implications for the whole distribution of G(i,r) not just the mean. I wonder if it can be tested ?