Identifying cosmological perturbations in group field theory condensates

Steffen Gielen

doi:10.1007/jhep08(2015)010

Abstract

AbstractOne proposal for deriving effective cosmological models from theories of quantum gravity is to view the former as a mean-field (hydrodynamic) description of the latter, which describes a universe formed by a ‘condensate’ of quanta of geometry. This idea has been successfully applied within the setting of group field theory (GFT), a quantum field theory of ‘atoms of space’ which can form such a condensate. We further clarify the interpretation of this mean-field approximation, and show how it can be used to obtain a semiclassical description of the GFT, in which the mean field encodes a classical statistical distribution of geometric data. In this sense, GFT condensates are quantum homogeneous geometries that also contain statistical information about cosmological inhomogeneities. We show in the isotropic case how this information can be extracted from geometric GFT observables and mapped to quantities of observational interest. Basic uncertainty relations of (non-commutative) Fourier transforms imply that this statistical description can only be compatible with the observed near-homogeneity of the Universe if the typical length scale associated to the distribution is much larger than the fundamental ‘Planck’ scale. As an example of effective cosmological equations derived from the GFT dynamics, we then use a simple approximation in one class of GFT models to derive the ‘improved dynamics’ prescription of holonomy corrections in loop quantum cosmology.

Highlights

JHEP08(2015)010 to the separate-universe approach in cosmology [16]
As an example of effective cosmological equations derived from the group field theory (GFT) dynamics, we use a simple approximation in one class of GFT models to derive the ‘improved dynamics’ prescription of holonomy corrections in loop quantum cosmology
The viewpoint that a cosmological universe arises from the condensation of many ‘atoms of space’ is most naturally investigated in the setting of group field theory (GFT), which provides a quantum field theory language for discrete geometry in which the concept of a condensate can be made sense of: a GFT condensate defines a nonperturbative ground state, describing a phase away from the Fock vacuum around which perturbative physics is defined in LQG

Summary

Effective classical geometry from global GFT observables

Part of the geometric information contained in any GFT Fock state, e.g. a condensate state of the form (2.6), can be expressed in terms of expectation values of suitable secondquantised operators. Note again the difference between first and second quantisation; the analogue of χC in quantum mechanics would correspond to a projective measurement with eigenvalues 0 and 1, whereas the set of eigenvalues of χC is N0 In this interpretation, the mean field σ(gI ) is used to give a classical statistical description of the hydrodynamic approximation, in which a generic condensate state does not describe a perfectly homogeneous universe, but rather a distribution of patches with different values for geometric quantities such as e.g. curvature invariants constructed from the gI. The precise connection between this GFT formalism and the setting of linear perturbations in cosmology, and the physical interpretation of a scale factor a obtained from such an average, will presumably only be clear for condensates for which ψ 1 and one can treat ψ as a linear perturbation In this approximation in which the statistical distribution over geometric data is approximated by N classical quantities AiI , the expectation value of the operator αI defined in (3.11) is αI = (AiI )2 =. Viewing quantum cosmology as the hydrodynamics of quantum gravity suggests a similar mechanism for quantum geometry [42]

Homogeneity and classicality conditions for GFT condensates

Example

Discussion