The cosmological constant as an eigenvalue of a Sturm-Liouville problem

Abstract

It is observed that one of Einstein-Friedmann's equations has formally the aspect of a Sturm-Liouville problem, and that the cosmological constant, Λ, plays thereby the role of spectral parameter (what hints to its connection with the Casimir effect). The subsequent formulation of appropriate boundary conditions leads to a set of admissible values for Λ, considered as eigenvalues of the corresponding linear operator. Simplest boundary conditions are assumed, namely that the eigenfunctions belong to L 2 space, with the result that, when all energy conditions are satisfied, they yield a discrete spectrum for Λ > 0 and a continuous one for Λ < 0. A very interesting situation is seen to occur when the discrete spectrum contains only one point: then, there is the possibility to obtain appropriate cosmological conditions without invoking the anthropic principle. This possibility is shown to be realized in cyclic cosmological models, provided the potential of the matter field is similar to the potential of the scalar field. The dynamics of the universe in this case contains a sudden future singularity.Recently Barrow and Shaw [5] suggested a non-anthropic solution to the problem: by making Λ into a field and restricting the variations of the action with respect to it by causality, they managed to obtain an additional Einstein constraint equation. One can say that this approach is based on a different interpretation of the cosmological constant, notably to consider the cosmological constant to be a field variable.In the present work we use a somehow similar procedure to calculate the value of Λ, namely to interpret it as an eigenvalue of a Sturm-Liouville problem, rather than as an integration constant. Our approach is in fact more conservative than the method in [5] since we use the standard Einstein-Friedmann equations without any additional constraint equation. Instead, we consider one of Friedmann's equations as a spectral problem and look for a class of boundary conditions which may allow us to actually calculate the corresponding eigenvalues (in this way, our procedure is closely related to standard investigations of the Casimir effect [6, 1, 2] by spectral methods). A most simple condition is to impose that eigenfunctions be elements of L 2 . As we will show, this choice results in the following interesting consequence: if the universe is filled up with a matter field, φ (except for Λ), such that all the strong energy conditions are satisfied, then one gets a point spectrum for positive values of the cosmological constant, and a continuous spectrum for negative ones.The next step will be to consider models which actually result in a positive discrete spectrum with just one single point. This idea is inspired in the conceptual approach developed in the paper by Linde and Vanchurin [7]: one can fix all the parameters in the landscape, including the value of the cosmological constant, without using any anthropic considerations, what "will give us a chance to return to Einstein's dream of a final theory, which may all...

The cosmological constant of emergent spacetime in the Newtonian approximation

“…To name but a few, in ref. [1] is has appeared as the eigenvalue of a Sturm-Liouville problem, while in ref. [2] it has been regarded as a field.…”

The cosmological constant of emergent spacetime in the Newtonian approximation

“…More modestly, in refs. [5,10] we have obtained S max /k B 10 123 using a finite-dimensional quantum-mechanical model 1 of a Newtonian Universe. This strongly suggests that the cosmological constant Λ must also be obtainable from the same model as in refs.…”

“…To name but a few, in ref. [1] is has appeared as the eigenvalue of a Sturm-Liouville problem, while in ref. [2] it has been regarded as a field.…”