Towards anisotropic cosmology in group eld theory Andrea Calcinariand Steen Gieleny School of Mathematics and Statistics University of Sheeld

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Towards anisotropic cosmology in group ﬁeld theory

Andrea Calcinari∗and Steﬀen Gielen†

School of Mathematics and Statistics, University of Sheﬃeld,

Hicks Building, Hounsﬁeld Road, Sheﬃeld S3 7RH, United Kingdom

(Dated: March 17, 2023)

In cosmological group ﬁeld theory (GFT) models for quantum gravity coupled to a massless

scalar ﬁeld the total volume, seen as a function of the scalar ﬁeld, follows the classical

Friedmann dynamics of a ﬂat Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) Universe at

low energies while resolving the Big Bang singularity at high energies. An open question

is how to generalise these results to other homogeneous cosmologies. Here we take the

ﬁrst steps towards studying anisotropic Bianchi models in GFT, based on the introduction

of a new anisotropy observable analogous to the βvariables in Misner’s parametrisation.

In a classical Bianchi I spacetime, βbehaves as a massless scalar ﬁeld and can be used

as a (gravitational) relational clock. We construct a GFT model for which in an expanding

Universe βinitially behaves like its classical analogue before “decaying” showing a previously

studied isotropisation. We support numerical results in GFT by analytical approximations

in a toy model. One possible outcome of our work is a deﬁnition of relational dynamics in

GFT that does not require matter.

CONTENTS

I. Introduction 2

II. Short review of GFT cosmology 4

A. Basics of group ﬁeld theory 4

B. Canonical quantisation of GFT 7

C. Emergent FLRW Universe from free theory and single mode 10

III. Relational deﬁnition of cosmological dynamics 14

A. FLRW 15

B. Bianchi I 15

C. Bianchi II 16

IV. Anisotropic GFT model 17

A. Deﬁning β~

j,ı

±: the trisohedral tetrahedron 18

B. Initial conditions 21

C. Three modes with ﬁxed base 23

D. Mean-ﬁeld continuous toy model 26

E. Including more modes into GFT models 29

V. Conclusions 30

A. Classical and quantum geometry of the tetrahedron 33

∗acalcinari1@sheﬃeld.ac.uk

†s.c.gielen@sheﬃeld.ac.uk

arXiv:2210.03149v2 [gr-qc] 24 Apr 2023

I. INTRODUCTION

A major challenge for discrete approaches to quantum gravity is the derivation of an eﬀective

(emergent) continuum description which can be compared with classical general relativity or more

general gravitational theories. The challenge arises on many levels, for instance in recovering the

usual notions of a spacetime manifold from combinatorial structures [1]; recovering an eﬀective

description in terms of coordinates and restoring the continuum notion of diﬀeomorphisms or coor-

dinate changes [2]; and understanding the intricate interplay between a continuum and semiclassical

limit. Deriving such a description, however, is crucial for understanding the phenomenology of such

quantum gravity theories and ensuring their compatibility with observation given that, e.g., new

ﬁfth-force degrees of freedom at low energies would have to be compatible with tight experimental

bounds [3]. A common approach in this situation is to restrict to situations of high symmetry,

in particular spatially homogeneous cosmology or spherically symmetric black holes. While by

assumption they no longer include all degrees of freedom, symmetry-restricted models would be

expected to capture at least some phenomena of the underlying theory (as they do in classical

general relativity), while also connecting directly to phenomenology given the obvious relevance of

cosmological and black hole spacetimes. A prominent example is loop quantum gravity (LQG),

whose cosmological sector has been studied in loop quantum cosmology [4] while there are also a

number of eﬀective black hole models including LQG discretisation eﬀects [5].

Eﬀective cosmological models have recently been constructed in the GFT approach to quantum

gravity [6], itself closely related to the spin foam deﬁnition of LQG dynamics [7] and to matrix

and tensor models [8]. Just as LQG, GFT models are fundamentally deﬁned in terms of discrete,

combinatorial structures, interpreted loosely as “quanta of spacetime”. One can associate geometric

notions such as areas, volumes and angles to these degrees of freedom by incorporating concepts

from LQG, but there is no simple way of giving these an eﬀective continuum interpretation, in

particular given that the conceptual status of a given discretisation in GFT is similar to a Feynman

graph in quantum ﬁeld theory, i.e., only one term in an inﬁnite expansion. What can be done

relatively straightforwardly, however, is to derive dynamical equations for global observables such

as the total (spatial) volume of a certain geometry, which can then be contrasted with globally

homogeneous cosmological models. These equations are usually derived for certain GFT states

whose properties make them good candidates for spatially homogeneous geometries. After some

prior groundwork [9] a breakthrough in this line of research came when, in GFT models for quantum

gravity coupled to a massless matter scalar ﬁeld, a “relational” volume observable (corresponding

to the volume of space for a given value of the scalar ﬁeld) was shown to satisfy the Friedmann

dynamics of general relativity at low energies while also replacing the classical Big Bang singularity

by a bounce [10]. Similar results have been obtained using diﬀerent methods and from diﬀerent

starting points [11–14], emphasising their robustness: one can understand the main properties of the

resulting cosmological dynamics from classical solutions for a single ﬁeld mode. While important

for the phenomenology of GFT and for connecting to approaches such as loop quantum cosmology,

these results have so far been restricted to the case of a ﬂat homogeneous, isotropic Universe.1

Some studies have included anisotropies perturbatively [16, 17] showing that they decay leading to

isotropisation, but there is so far no characterisation of, e.g., an anisotropic Bianchi cosmology.

1Inhomogeneities can be included perturbatively as in [15], and match physical expectations at least in a long-

wavelength limit.

Here we take the ﬁrst steps towards the study of anisotropic cosmologies in group ﬁeld theory,

focussing on the simplest possible case of Bianchi I cosmology with local rotational symmetry

so that two out of the three directional scale factors are taken to be equal. There are at least

two, initially quite separate, challenges involved in this extension of past work. The ﬁrst is to

ﬁnd a characterisation of anisotropies in group ﬁeld theory, i.e., to deﬁne an observable that can

distinguish isotropic and anisotropic geometries and quantify the amount of anisotropy. Here

the key idea we use is the Misner parametrisation of Bianchi models (see, e.g., [18]; we will also

review this below) in terms of a volume degree of freedom and two relative anisotropy variables, the

Misner variables β±. In a classical Bianchi I model β±behave as free, massless scalar ﬁelds in a ﬂat

FLRW geometry2, which have already been studied in GFT. On the other hand, the discreteness

of geometry in GFT means that we cannot simply take over a continuum deﬁnition of anisotropy,

so the construction of an analogue β±variable requires careful thought. The second challenge is

to understand which simplifying approximations used in past work need to be relaxed in order to

allow for anisotropies in the eﬀective description. For instance, while the work of [10] only used

“isotropic” states interpreted as describing simplicial building blocks for which all faces have equal

area, it is known (see, e.g., [13]) that this microscopic restriction to isotropy is neither necessary nor

suﬃcient to obtain the correct (ﬂat FLRW) Friedmann dynamics: the more relevant assumption

is to restrict to a single ﬁeld mode in the Peter–Weyl expansion in representation data. Hence, in

order to describe anisotropic geometries, multiple Peter–Weyl modes must be taken into account,

but it is not clear how many (and which) modes are needed to capture physical anisotropies.

In this paper we show how to tackle the ﬁrst challenge; we deﬁne an β±analogue with a clear

geometric interpretation, quantum ambiguities that disappear for large quanta, and correct phys-

ical properties – constant velocity and hence linear evolution – at least for a certain cosmological

period of time, before the isotropisation observed in [16, 17] sets in and the anisotropy disappears.

The second challenge is partially addressed, given that the β±dynamics partially match expecta-

tions from classical relativity, but the observed isotropisation does not correspond to a classically

expected behaviour and, more importantly, anisotropies do not backreact on the eﬀective Fried-

mann equation as expected. This suggests that while our constructions will be useful for future

work, our model needs further reﬁnement to reproduce the correct physics of a classical Bianchi

Universe.

Any monotonically evolving quantity in a cosmological model can be used as a relational clock:

all other dynamical variables can be written, at least in principle, as functions of this “clock”. In

a vacuum Bianchi I model, the anisotropy variables β±have this property and hence, in contrast

to what is often done in quantum cosmology and in particular in GFT, no coupling to matter

would be needed to be able to express the dynamics in relational terms. The fact that we have

deﬁned a new quantity with monotonic evolution in GFT cosmological models hence raises the

possibility of deﬁning relational evolution in GFT without adding matter ﬁelds, which might help

in understanding the “problem of time”, or possible dependence of dynamics on the choice of clock,

in GFT (see [20] for some work on this issue in models with multiple possible clocks).

The remaining parts of the paper are structured as follows. In section II we review basic

ideas of the GFT formalism, possible deﬁnitions of a canonical Hilbert space quantisation, and

application to cosmology: we show how one can derive eﬀective Friedmann equations by restricting

2This property makes the Misner parametrisation particularly natural in classical general relativity. For comparison

we should mention that in loop quantum cosmology the situation is diﬀerent, as one quantises an LQG-corrected

Hamiltonian constraint and the particular type of corrections makes Misner variables less convenient [19].

to a single ﬁeld mode and neglecting interactions. Readers familiar with GFT cosmology may skip

this review section. Similarly, section III is a review of classical FLRW and Bianchi cosmologies

written in relational terms, using a scalar ﬁeld clock; this is the classical theory that any eﬀective

description of GFT can be compared to. Section IV includes the main new results: we motivate

the introduction of an eﬀective β±variable used to characterise anisotropies in GFT. We then

propose models based on a few Peter–Weyl modes and study the eﬀective dynamics of both the

anisotropies and the spatial volume, comparing both with the dynamics of general relativity. We

also propose a simpliﬁed “toy model” in which some of our main results, in particular the linear

growth in anisotropy which matches classical expectations, can be derived analytically rather than

numerically as in the main part. We conclude in section V. An appendix gives details on the

classical and quantum geometry of tetrahedra as used in LQG and GFT.

II. SHORT REVIEW OF GFT COSMOLOGY

In this section we brieﬂy summarise past work on deriving eﬀective cosmological dynamics from

group ﬁeld theory. In this past work, eﬀective Friedmann equations were obtained after truncating

the full dynamics and choosing simple GFT states, following two diﬀerent approaches. The two

approaches, which we will call algebraic and deparametrised, will be introduced in section II B.

A. Basics of group ﬁeld theory

We are interested in GFT models for simplicial gravity coupled to a (free, massless) scalar ﬁeld

χ. In such models one deﬁnes a group ﬁeld ϕwhose arguments are delements of a Lie group G

(hence the name “group ﬁeld”) and a real variable corresponding to the scalar matter ﬁeld χ∈R,

ϕ:Gd×R→K,(1)

where Kcan be Ror C. When applied to four-dimensional quantum gravity, d= 4 and one usually

takes Gto be the local gauge group of general relativity: Gis typically SO(3,1) or SL(2,C) in

the Lorentzian case, SO(4) or Spin(4) in the Euclidean case, or their rotation subgroup SU(2)

which is the gauge group of loop gravity (or the Ashtekar–Barbero formulation of classical general

relativity [21]). The last choice is the one we will use here. To implement a notion of discrete

gauge invariance in the resulting simplicial gravity description, one usually requires invariance of

the ﬁeld under the right diagonal group action,

ϕ(gI, χ)≡ϕ(g1, . . . , gd, χ) = ϕ(g1h, . . . , gdh, χ),∀h∈G . (2)

The action has the general form

S[ϕ, ¯ϕ] = Zdgdg0dχ¯ϕ(gI, χ)K(gI, g0

I)ϕ(g0

I, χ) + V[ϕ, ¯ϕ],(3)

where for a real ﬁeld ¯ϕ=ϕ. Here Rdgstands for an integration over dcopies of the group, using

the Haar measure normalised to unity. The action is therefore split into a quadratic part and an

interaction part V. The kernel Kis assumed to respect the symmetries associated to a minimally

coupled massless scalar ﬁeld on a curved background, namely shift (χ→χ+c) and sign reversal

(χ→ −χ) symmetries. For this reason, Kcannot depend explicitly on χ, but it is in general a

diﬀerential operator in χ, which does not involve odd powers [10, 22]. χusually plays the role of a

relational time variable, so that other observables are deﬁned relative to χ, as is common in many

approaches to quantum cosmology, in particular in loop quantum cosmology. The kinetic term can

be written as an expansion in derivatives with respect to χ,

K(gI, g0

I) = ∞

n=0

K(2n)(gI, g0

I)∂2n

∂χ2n=K(0)(gI, g0

I) + K(2)(gI, g0

I)∂2

χ+. . . , (4)

which is usually truncated after the second term: starting from the simplest Kthat only includes

a constant “mass term” suggested by the relation to spin foam models [23], a Laplacian term is

generated by radiative corrections [24]. One could stop here, given that no higher derivative terms

are required, or make the weaker assumption that higher derivatives are present but suppressed,

i.e., |K(2n)/K(0)|  |K(2)/K(0)|nfor n > 1. For concreteness, we will follow previous work (see,

e.g., [9, 25]) and assume that Khas the minimal form

K=m2−∂2

χ+M2

I=1

∆gI,(5)

where mand Mare coupling constants and ∆gIis the Laplace–Beltrami operator acting on the

I-th group argument. Evidently, this corresponds to K(0) =m2+M2PI∆gI,K(2) =−1 in (4).

To construct an interaction term in simplicial gravity models, one can think of the group ﬁeld ϕ

as representing a (d−1)-simplex. The interaction term then describes the gluing of such simplices

to form d-dimensional structures, here a d-simplex, similarly to what happens in tensor models

[8, 26]. This means that an appropriate interaction term consists of products of ﬁelds that are

paired according to a pattern which encodes the combinatorics of a d-simplex. In four dimensions,

we could for example choose (here for a real ﬁeld)

V[ϕ] = λ

5Z(dg)5V(g1

I, ..., g5

a=1

ϕ(ga

I, χ),(6)

where λis a coupling constant, R(dg)5means integration over G20, and V(g1

I, ..., g5

I) is a product of

ten Dirac delta distributions on the group, imposing appropriate matching between group elements

appearing as arguments of the ﬁelds ϕ(gj

I, χ), in order to encode the pattern of gluings needed to

form a four-simplex out of ﬁve tetrahedra (see right panel of ﬁgure 1). Such an interaction should

allow the structure of four-dimensional spacetime to emerge from the dynamics of the theory.

We will now ﬁx Gd=SU (2)4. Then the ﬁeld ϕ(gI, χ) can also be represented as a four-valent

spin network node. The gIare associated to the links dual to the faces of a tetrahedron (see left

panel of ﬁgure 1), and the matter ﬁeld χ“sits” on the node dual to the tetrahedron.

It is often convenient to use the Peter–Weyl theorem to express the group ﬁeld as

ϕ(gI, χ) = X

j, ~m,~n,ı

ϕ~

j,ı

~m (χ)I~

j,ı

I=1 p2jI+ 1 D(jI)

mI,nI(gI),(7)

where the D(jI)

mI,nI(gI) are Wigner matrices and ϕ~

j,ı

~m (χ)≡ϕj1,...,j4,ı

m1,...,m4(χ) are complex functions3.

The intertwiners I~

j,ı

~n ≡ Ij1,...,j4,ı

n1,...,n4arise because of the invariance under group multiplication from

3If the group ﬁeld is real, the complex Peter-Weyl coeﬃcients satisfy the reality condition [12, 27]

ϕ~

j,ı

~m (χ) = (−1)PI(jI−mI)ϕ~

j,ı

−~m(χ).(8)

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TowardsanisotropiccosmologyingroupeldtheoryAndreaCalcinariandSteenGielenySchoolofMathematicsandStatistics,UniversityofSheeld,HicksBuilding,HounseldRoad,SheeldS37RH,UnitedKingdom(Dated:March17,2023)Incosmologicalgroupeldtheory(GFT)modelsforquantumgravitycoupledtoamasslessscalareldthetotalvolu...

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Towards anisotropic cosmology in group eld theory Andrea Calcinariand Steen Gieleny School of Mathematics and Statistics University of Sheeld

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