N.b. This is a technical post, written to illustrate a question I believe to be interesting to some colleagues outside my particular discipline. I am accutely aware of its shortcomings as expository work, and pedagogical criticism is almost as welcome as an attempt to engage with the question at hand.
1. Preamble.
I have recently claimed that an important task for those working near and on the bridge between observational and theoretical cosmology is the classification of almost-Gaussian fields. The cosmological dark matter density field that we observe is thought to be a Gaussian random field that has been pushed away from this state by a number of distinct physical processes: i) the primordial density perturbation arising from cosmological inflation; ii) non-linear gravitational evolution; iii) the act of observing peaks (i.e. galaxies) of the field, rather than the field in its entirety. Here is what I imagine (as drawn for a different context by Roger Penrose):
This is a mathematical space where each point represents a configuration for the cosmological density field, modulo the operation by which all Gaussian random fields are identified with a single point. In the neighbourhood of this origin lie almost-Gaussian fields. I am interested in knowing whether regions of this space can be demarcated on the basis of the physical process causing the non-Gaussianity.
2. Classification of fields with the Euler characteristic.
I have also argued that the topological genus statistic (put forward by Gott and collaborators in 1986) provides a way to carry out this classification. The problem is to classify smooth functions on a topologically trivial three-dimensional manifold. To be explicit: the Euler characterstic is employed here not as an invariant of the underlying space, but of the level surfaces of a function defined on that space. Here’s what is meant by this:
Shown are 3 two-dimensional level surfaces (surfaces of constant value) cutting through the three-dimensional density field. In this example, the left-most surface shows the choice of a low value, the central surface a value close to the median and the right-most a high value. The surfaces separate regions of densities below their value from regions of densities above. Of course they needn’t look like this for a general function, but for the case of a Gaussian random field this will be how things appear. To quantify the morphology of each surface, one uses a slight generalisation of the topological genus. Informally,
This definition allows the genus to fall below zero for a surface with many disjoint segments (such as the voids and clusters above); I do not think this is what mathematicians would normally use, & I do not know where its use originates; perhaps it is an idea of Gott’s.
This is the genus curve for a Gaussian random field. Each point on the curve corresponds to a particular density value that defines a surface through the field, whose (generalised) genus is the value on the y-axis. The reflection symmetry of the curve is an important feature of Gaussianity; non-Gaussian fields display genus curves in which this symmetry does not appear.
It is an interesting, important and open question how distinct physical processes modify the genus curve. The determination of an analytic expression for the genus curve of a non-Gaussian field would be a significant milestone. However, that is not the question I wish to present today.
3. Abstraction of the genus as a tool for classifying functions.
As was belaboured previously, the genus calculated here is not for the underlying space, which is topologically trivial, but for the sequence of level surfaces through a function defined on that space. In particular, the function is scalar valued. The question I pose is whether an analogous quantity can be defined when the function is vector-valued, i.e. (in physicists’ terminology) when the function is a vector field, such as that corresponding to the flow of galaxies through the Universe; or to go further, for a tensor field, such as the cosmological gravitational wave background.
One idea is to convert the vector field back to a scalar one, by taking the divergence, and proceed as before. But I’m not really sure what this acheives. Alternatively, one could use the components of the vectors to define three fields (i.e. in 3 dimensions) and measure the connectedness along each component of that particular basis. But unless the result is invariant with respect to the choice of basis, this can’t be a useful quantity.
Question: What is the vector analogue to the notion of connectedness?
If this isn’t a coherent concept, why?
At least for the scalar case, the relevant mathematical framework is Morse theory, which aims at explaining the topology of the level sets of functions. You have to make some assumptions about the function for the theory to be applicable. The standard assumption is that the function has to be a Morse function, meaning that its critical points (points where the derivative vanishes) are nondegerate (the matrix of second derivatives is nonsingular there). You can relax this somewhat in what has become known as Morse-Bott theory.
Since the function you are interested in (the density) is empirically determined, I suppose that one can assume that it is Morse: since — informally speaking — any function can be perturbed infinitesimally to a Morse function.
I realise that this does not answer your question, which could now perhaps be rephrased as to whether there is a Morse theory for sections of bundles (or more generally maps) instead of functions. I don’t know the answer to this question, but I would be surprised if something along these lines did not exist. But at any rate, this is perhaps something that could be asked to mathematicians.
Readers: Jose has been kind enough to reformulate this question along Morse theoretic lines, and posted it to Math Overflow.
As Jose points out at MO, neither Gott nor the papers on this topic that have followed (incl. my own) have mentioned Morse theory by name. Yet it seems undeniable that this idea is a (rather straight-forward) application of it; getting to the bottom of this seems very likely to yield further insight.
In some off-blog discussion, Andrew Ranicki notes that Morse theory can be traced back to James Clark Maxwell, in an evocatively titled essay ‘On Hills and Dales.’ Further references include Milnor’s lecture notes on Morse theory & Bott’s papers.
A concern: how is the genus statistic computed? As you’ve described it, it appears to be quite scale-sensitive. If you use a scale on which the level-sets all appear linear, the statistic is zero everywhere. If you increase the scale, holes and isolated regions start to come into vision and the statistic appears interesting. If you back off further likely the entire surface will appear to be one large glob, and small imprecisions in your data could make it effectively impossible to compute. I’m curious what’s done in practice?
From your post it sounds like the purpose of this statistic is create some quantifiable data related to the surface and study how this data evolves in time under some standard physical processes. Are there results saying if process X is applied, the genus statistic varies by Y?
Thanks Ryan. I’d love to write a post on the algorithm for the computation of this statistic—it’s a really beautiful technique. It’s described in Gott’s original paper (link above), but maybe not as prettily as modern graphics could allow. [To-do list +1]
Regarding the scale sensitivity of the statistic, there are two parts to my answer. Firstly, the topology of the density field will definitely change as a function of the physical scale one is studying. Because gravity takes time to propagate, there is a largest physical scale above which the spatial distribution of dark matter is believed to be very close to something with the genus curve shown above; that is, above this scale the field is close to its primordial state. Below this scale, gravity has caused the formation of massive structures: galaxies, clusters, big filaments, voids, and the genus curve alters appreciably. Studying the genus curve as a function of physical scale is, broadly, the goal we are pursuing here.
Secondly, and this is more germane to your question, the numerical computation of the genus for a surface proceeds by embedding the surface within a data array, and the choice of voxel (cell) size for that array is indeed an important consideration. If one had a super-duper-computer, it would be possible to use as fine a mesh as one desired, interpolating the function smoothly between its known values—on the assumption that the dark matter fluid is smooth in just this way.
It’s certainly the case that the cell size should be below some value, or else one is washing out the interesting information. We set this depending on the physical scale of the structure that we are interested in studying.
I should also mention that the surfaces shown above are simulated data, just for illustration purposes. The data are a little messier; some recent examples of this analysis are here and here.
Your final paragraph gets to the heart of the matter succinctly. There are no results giving genus curve for a non-Gaussian field, & we do not know how process X changes the genus curve beyond some fairly vague statements. The modern approach is to generate realistic simulations of the density field under different physical assumptions and investigate the changes empirically; and to try to guess a form that way. Both approaches are important here.
Thanks. I’m going to need at least one more point of clarification. This surface that you’re studying, what exactly is it representing? It seems like you’re modelling the distribution of energy throughout space at some time slice. So I would be expecting a distribution or a stress-energy tensor, not a surface. Is the surface an artefact of a highly simplified model — you represent the energy as being constrained to an evolving surface? How do you allow for variable densities, like if the surface is stretched thin and the energy vacates a region, do you cut the surface and allow it to have boundary?
Sorry for the super naive questions. I’m trying to get a sense for what kind of invariants you might be able to use.
Thanks Ryan. The cube shown above is meant to represent a region of space in the Universe, at a particular moment in time. Inside that region is all kinds of stuff, but what is studied here is the density (i.e. the mass per very, very small volume element) of dark matter, which we believe to be the dominant type of matter shaping the formation of structures like galaxies and clusters.
One can of course do the same thing for something other than dark matter; for instance, there are vast regions of neutral hydrogen in the Universe that will be surveyed in the next few years. The same technique can be used to study the structure of these clouds.
Each surface is a surface of constant density, separating the regions of the Universe with higher-density dark matter from the lower-density regions.
I’m not quite sure I’ve followed the train of your later questions, but I’d be keen to answer them if you’d like to try asking them in a different way.
I’m very interested in considering other invariants that can be used; particularly from homology theory. This Morse theory application is, so far as I know, the most sophisticated application of topology to an observational science!