Nous n'irons pas sans vous, Mesdames, à travers les espaces non commutatifs
To construct a quantum theory of gravity coupled to matter, one can either neglect matter altogether (as for instance in Loop Quantum Gravity), or consider instead that the interaction between gravity and matter is the most important aspect of the dynamics. The latter is indeed the philosophy followed in the context of Noncommutative Geometry, aiming at obtaining matter and gravity from (noncommutative) geometry . The geometry is considered to be the tensor product of a continuous 4-dimensional geometry for space-time and an internal 0-dimensional geometry for the gauge content of the theory, namely the Standard Model of particle physics. The fruitful outcome of this approach is that considering gravity alone on the product space one obtains gravity and matter on the ordinary 4-dimensional space-time.
The conventional way to identify the geometry of a given object is by measuring distances between its points. There is however a different approach proposed more than a century ago by Weyl, who suggested to identify a membrane’s shape 4 through the way it vibrates as one bangs it. Studying the frequencies of the resulting fundamental tone and overtones, one may get information about the membrane’s geometry. Following Weyl’s law, the largest frequencies of the sound of a membrane are basically determined by the area of the membrane and not by its shape, thus verifying Lorentz’s conjecture.
In a paper under the intriguing title “Can one hear the shape of a drum?”, Kac has formulated the puzzle of whether one can reconstruct the geometry of a n-dimensional manifold (possibly with boundary) from the eigenvalues of the Laplacian on that manifold . In other words, the relevant question is whether one (assuming he/she had perfect pitch) can deduce the precise shape of a drum just from hearing the fundamental tone and all overtones, even though one cannot really see the drum itself. Kac had stated that one most probably cannot hear the shape of a drum, nevertheless he investigated how much about the tambourine’s shape one can infer from the knowledge of all eigenvalues of the wave equation that describes the vibrating object (the membrane).
From the mathematical point of view, the question posed by Kac can be formulated as “How well do we understand the wave equation?”. For ordinary Riemannian geometry, the shape of the Laplacian does not fully determine the metric, so the shape of the drum cannot be heard, implying that we do not have a complete understanding of the (rather simple) wave equation. But what about the product space considered within the Noncommutative Geometry context?
Every space vibrates at certain frequencies, hence one may consider the universe as a vibrating membrane, a tambourine. Let us also consider that space is identified with the product space of the 4-dimensional space-time and the internal space, defined by a mathematical object, called the spectral triple given by the algebra of coordinates, the Hilbert space and the Dirac operator corresponding to the inverse of the Euclidean propagator of fermions. The dynamics of the spectral triple are governed by a spectral action  summing up all frequencies of vibration of the product space. One may thus ask whether he/she can hear the shape of this product space. The spectrum of the Dirac operator together with the unitary equivalence class of its noncommutative spin geometry, fully determine the metric and its spin structure. Hence, one may indeed hear the shape of a spinorial drum.
Considering the spectral action and gravity coupled to matter, it was shown that one can obtain gravity and the Standard Model of elementary particles . Thus, noncommutative spectral geometry offers a purely geometrical explanation for the Standard Model, the most successful particle physics model we still have at hand.
Let us hence consider General Relativity as an effective theory and build a cosmological model based on Noncommutative Geometry along the lines of the spectral action. Given that this model lives by construction at very high energy scales, it provides a natural framework to construct early universe cosmological models. Studying these models one can test the validity of the Noncommutative Geometry proposal based on the spectral action, and in addition address some early universe open issues .
(Submitted on 15 Jul 2014)
Cosmology is currently undergoing one of the most exciting phases of rapid development, with sophisticated theoretical modeling being tested against very accurate observational data for both the Cosmic Microwave Background (CMB) and the matter distribution in the Universe. This is therefore a highly appropriate time for a broad range of mathematical models of particle physics and cosmology to formulate testable predictions that can be confronted with the data.
While model building within the framework of string and brane scenarios and their possible implications for particle physics and cosmology have been widely developed in recent years, less attention has been devoted to other sources of theoretical high energy physics models that are capable of producing a range of predictions, both in the particle physics and cosmology context. It is especially interesting to look for alternative models, which deliver predictions that are distinguishable from those obtained within the framework of string theory. Particle physics models derived within the framework of Noncommutative Geometry recently emerged as a source for new cosmological models, , , , , , , .
Among the most interesting features of these models of particle physics based on noncommutative geometry is the fact that the physical Lagrangian of the model is completely computed from a simple geometric input (the choice of a finite dimensional algebra), so that the physics is very tightly constrained by the underlying geometry.
The features that link the noncommutative geometry models to areas of current interest to theoretical cosmologists are the fact that the action functional of these models, the spectral action, behaves in the large energy asymptotic expansion like a modified gravity model, with additional coupling to matter. Various models of modified gravity have been extensively studied by theoretical cosmologists in recent years. Another feature, which is particular to the noncommutative geometry models we consider here, is the fact that the nonperturbative form of the spectral action determines a slow-roll inflation potential, which shows a coupling of spatial geometry (cosmic topology) and the possible inflation scenarios. The model also exhibits couplings of matter and gravity, which provide early universe models with variable effective gravitational and cosmological constants.
The results described in this paper are mostly based on recent joint work with Elena Pierpaoli , with Elena Pierpaoli and Kevin Teh , and with Daniel Kolodrubetz , as well as on earlier joint work with Ali Chamseddine and Alain Connes .