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.
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?