Spectral interaction between universes

Abstract

We derive a perturbative formula for the direct interaction between two four-dimensional geometries. Based on the spectral action principle we give an explicit potential up to the third order perturbation around the flat vacua. We present the leading terms of the interaction as polynomials of the invariants of the two metrics and compare the expansion to the models of bimetric gravity.

Towards modified bimetric theories within non-product spectral geometry

“…We make here some comments on the analysis of doubled geometry models in case when the tensors A µν does not differ sufficiently from the diagonal ones. This is not identical to the situation where the metrics are small perturbation of the Euclidean ones -for the discussion of the latter we refer to [19].…”

“…are polynomial integrals over higher spheres which can be evaluated generalizing the methods from [18] -see also [19] for further discussion. Denoting…”

“…Moreover, by using the generalization of [18, Prop. 2] (see also [19,Prop. A.2] ) one can easily find the recursive formula for c m :…”

“…In particular, the interaction potential for bimetric model is a polynomial one, while for the two-sheeted model it is a rational function. Further similarities and differences for generic metrics were recently analysed in [19], where yet another interpretation of this model in terms of interacting branes was proposed.…”

“…Despite numerous similarities, certain significant differences are also present.In particular, the interaction potential for bimetric model is a polynomial one, while for the two-sheeted model it is a rational function. Further similarities and differences for generic metrics were recently analysed in [19], where yet another interpretation of this model in terms of interacting branes was proposed.In this note we discuss yet another class of models beyond the FLRW framework. We illustrate the generally claimed features on the simplified example -the so-called Hopf model.In this case the interaction potential has nontrivial logarithmic terms but it still possesses bimetric gravity characteristics.…”