Cosmological implications of the hydrodynamical phase of group field theory
CCosmological implications of the hydrodynamical phase ofgroup field theory
Luciano Gabbanelli a, ∗ and Silvia De Bianchi b, † a Institut de F´ısica d’Altes Energies (IFAE), Universitat Aut`onoma de Barcelona,08193 Bellaterra (Barcelona), Spain. b Department of Philosophy, Universitat Aut`onoma de Barcelona,Building B Campus UAB 08193 Bellaterra (Barcelona), Spain.
Abstract
In this review we focus on the main cosmological implications of the Group Field The-ory approach, giving rise to bouncing cosmologies from the Engle–Pereira–Rovelli–LivineGroup Field Theory model. Starting from the kinematics and dynamics, we offer anoverview of the way in which Group Field Theory treats solutions for the homogeneousuniverse and the bounce in a compatible way with Loop Quantum Cosmology results.We conclude with a discussion of the limits and perspectives of the Group Field Theoryapproach.
The goal of the present paper is to review some essential properties of Group Field Theory(GFT) and GFT condensate cosmology in an accessible way from a non-specialist’s perspec-tive, and with a non-GFT audience in mind. For a review with an internalist perspectivesee [1]. Our purpose is to underline the physical consequences of the formalism and to discusssome results that go beyond quantum cosmology and invite conceptual innovation. Strictlyrelated to Loop Quantum Gravity (LQG), tensor models and lattice quantum gravity, GFTcondensate cosmology is able to reach results similar to those of Loop Quantum Cosmology(LQC), namely some solutions of its models of the early universe include a bounce. Symme-try reduced versions of LQG have recently been studied as models of quantum cosmology [2].They possess two main features. The first is a mechanism for avoiding the big bang singu-larity in the framework of mini-superspace models of quantum gravity . In this mechanismthe inverse scale factor is represented by an operator that stays bounded as the universe’sclassical radius shrinks to zero. Other alternatives involve the effective discretization of the ∗ [email protected] † [email protected] It should be noted that ‘singularity avoidance mechanisms’ may exist in more conventional mini-superspace of quantum geometrodynamics. For instance, from simple particle models like [3] to more com-prehensive studies in more realistic situations [4] and recent extension to anisotropic models [5], in whichthe analysis is consistently based on the behaviour of the wave function and not on the bouncing behaviourof quantum-corrected classical equations. For an overview and a comparison between LQC and standardquantum cosmology, see [6]. a r X i v : . [ g r- q c ] A ug amiltonian constraint, which enables the quantum wave function to ‘jump over’ the singu-larity. Whichever the model under consideration, it is not clear how these models can bederived from full-fledge LQG, if it is in fact possible. Hence, there is no common agreementwhether the singularity avoidance a property of the full theory. In fact, calculations regardingthe full LQG theory show that the spectrum of the operator corresponding to the inversevolume is not bounded from above [7] .The second feature implies the possibility that an intrinsically quantum gravitationalmechanism of LQC might trigger inflation, which may eventually be stopped (gracefully) bygravity itself [9]. In this scenario, the inflaton potential characterized by a fine tuning inboth the potential and the initial conditions could become unnecessary in many inflationarymodels. The same key-feature of LQC including a bounce can be found in GFT cosmology.However, the latter offers not only the possibility to build up an Engle–Pereira–Rovelli–Livine(EPRL) GFT model [15], but also to change our definition of cosmology itself. Accordingto GFT, cosmology should be the ‘hydrodynamics’ of quantum gravity describing the macro-scopic universe as a fluid, whose ‘atoms’ are the GFT quanta, and whose main collectivevariable is analogue to a density function, to which a velocity function is added [15]. There-fore, this ‘hydrodynamical’ picture provides a natural link with the usual Wheeler–DeWittapproach to quantum cosmology, but with two main improvements: (i) the theory does notrequire any symmetry reduction, and (ii) includes many-body features of the full Hilbertspace of the microscopic theory.Moreover, GFT condensate cosmology is a tentative realization of geometrogenesis [16],which is also suggested by the phase transitions obtained via GFT renormalization. In otherwords, spacetime and the universe emerge in quantum gravity through a phase transitionfrom some non-geometric phase (or pre-geometric phase) with no notion of locality. Thisis consistent with the aim of (fully or partially) background-independent formulation forquantum gravity theories which is based on the formulation of any physical statement inrelational terms and without referring to any external structure. In this manner, after thephase transition, a more regular and ordered phase is obtained, where geometric data cannow be identified. The ‘appearance’ of space and time in this regime is a requirement forlocal structures to appear and it is also a requirement needed for the emergence of ourmacroscopic universe. This conceptual aspect is shared by LQG and GFT and its implicationsin cosmology give rise to the formalism of the EPRL GFT model. We will particularly focusin the GFT formalism where the macroscopic, homogeneous and isotropic universe emergesdynamically from the collective behaviour of a highly coherent configuration of many discrete“pre-geometric atoms” introduced in Sections 2. This approach suggests the presence ofa ‘hydrodynamical phase’ where a large number of constituents form a condensate phase,starting from which the concepts of space and time are well defined (see Sections 3 and 4). Inthis regime, the classical Friedmann dynamics for a homogeneous, isotropic universe, togetherwith quantum corrections of general relativity, emerge consistently from the fundamentalconstituents. These aspects are presented and discussed in Sections 5 and 6 of our review.We then conclude by discussing the results in recent GFT literature and their cosmologicalimplications, underlying open questions to be further investigated. For further analysis concerning these topics refer to [8]. The relevance of the EPRL model [10, 11], together with its generalized version including the Immirziparameter [12] consists in that it solves a number of problems in the quantization present in the first spinfoam model proposed in [13]. As underlined in [14] the EPRL model implemented linearization of the simplicityconstraints and imposed simplicity constraints in such a way that four triangles described by bivectors thatbelong to the same tetrahedron lie in the same hyperplane. The inclusion of the Immirzi parameter makesthe EPRL model closer to offer a spin foam model consistent with LQG. Quantum gravity with matter reference frames
GFT is a research programme for a non-perturbative quantization of gravity. These theoriesaim at describing the dynamics of quanta of space on background independent grounds andhence are characterized by a lack of any preferred notion of time (there are local relationsbut not global ones). According to GFT, the universe is an ensemble of processes happeningwhere any notion of evolution is purely relational. Nonetheless, there is no reason to considerthat coordinates describing the relational dynamic may be available a priori at the quantumlevel. In fact they are not, as it happens in most quantum gravity approaches. Indeed,discussions focus on how to identify in mathematical terms the available degrees of freedomat the Planck scale.One of the most drastic change of perspective of GFT is certainly how the theory describesthe macroscopic universe starting from the underlying physics and without referring to anyexternal structure. According to this approach, the purely relational dynamics betweenthe elementary degrees of freedom of geometry and matter are interpreted as combinatorialstructures and hence, their quantum dynamics should lead to an effective reconstruction ofcoordinates, i.e. the ‘appearance’ of space and time. Indeed, one might consider the classicaltheory as an emergent phenomenon that agrees with general relativity, together with thediffeomorphism invariance which is one of its most established foundations. Undoubtedly,this viewpoint leaves open the question of how to choose the embedding manifold. As wewill discuss below, the picture is similar to the theory of superfluidity where the fundamentaldisconnected quantum atoms play no individual role at the hydrodynamic level, but thecollective behaviour is what matters. Analogously, in the GFT formalism the macroscopicuniverse emerges dynamically as the collective behaviour of a highly coherent configuration ofmany discrete “pre-geometric atoms”. Thus, the theory enters in a “hydrodynamical” phasewhere a large number of constituents form a condensate structure. It is from the latter thatconcepts of space and time are defined. This approach would suggest that, in this regime, theclassical Friedmann dynamics, together with quantum corrections emerge consistently fromthe fundamental constituents. It seems plausible that, if cosmology and continuum geometryemerge from this hydrodynamic approximation to quantum gravity, the topology of spaceshould also be emergent, rather than determined by microscopic details.Let us first focus on gravity only with no mater fields present at all and later, once thegravitational theory is defined, we will show how this approach can embody matter degreesof freedom. In the canonical formulation of GFT, the elementary degrees of freedom ofgeometry are represented by excitations of a quantum (statistical) complex scalar field ϕ defined on an abstract group manifold. As mentioned, this manifold does not carry a priori any notions of spacetime geometry by itself, but stores geometric information –metric orconnection data– beyond its mere combinatorial or topological structure [17]. Then, theelementary excitations occur above a fully degenerate ‘no-space’ vacuum and can be seen asquanta of geometry labelled by data in the domain space of the bosonic GFT field ϕ . Thecoarse-grained microscopic theory is usually depicted as tetrahedra equipped with a discreteSU(2) connection (parallel transports across the four faces), in this manner, the GFT field isdefined as ϕ ( g I ) : SU(2) −→ C . (1)At a quantum level, imposing appropriate conditions, these tetrahedra can be ‘glued’ to oneanother to form extended 4 − dimensional triangulations as quantum states determining thedynamic of the theory.Analogous algebraic data is used to construct the spin network states in LQG (holonomies3f a connection and fluxes of a triad field) [18], and in fact these states can be seen as graphsdual to the triangulations formed by GFT quanta. In this duality, each vertex of the graphis dual to the tetrahedron of the triangulation; the links joining vertex are coloured by SU(2)connections and play the role of the tetrahedron faces where the gluing determines the par-ticular GFT model. Therefore, GFT quantum states are built up from the kinematical dataof LQG and the theory can be understood as a field-theoretic 2 nd quantization formulationof LQG [19]. This correspondence between discrete quantum field theory (QFT) structuresand spin networks can be found also at the dynamical level but treated via standard QFTmethods. In this way, GFT attempts to define a sum over discretised geometries which can beused, once a continuum limit is identified, to obtain a path integral formulation for quantumgravity.However, to extract this effective continuum physics and realistic cosmological modelsrequires more crucial ingredients. Of course, there is plenty of matter in the universe andthe relation of the matter content and its corresponding interaction must be addressed in atheory of Quantum Gravity. In GFT and other background independent and diffeomorphismrelated formalisms, matter fields are the most convenient way to define physical referenceframes. This is a relational approach usually employed to define physical observables indifferent quantum gravity theories (see [20–22]).A standard choice in quantum cosmology is to use free massless scalar fields for definingthe evolution of the theory. This choice ensures diffeomorphism invariance. It is our interesthere to see how these scalar fields can be coupled to a QFT formalism and their implicationsfor the GFT condensate and the cosmological sector.The matter reference frame should be reconstructed from the physical degrees of freedomof the underlying theory. In this manner, in the canonical formulation, the GFT field ϕ in(1) encompasses the new coordinate (scalar) degrees of freedom with real labels, ϕ ( g I , φ J ) : SU(2) × R −→ C . (2)The four parallel transports g I of the gravitational SU(2) connection are four copies of thegroup elements labelled by I = 1 , , ,
4. These SU(2) valued arguments play the role ofparallel transports of a gravitational Ashtekar–Barbero connection across the four faces ofthe tetrahedron or GFT quanta; hence they are associated to the links attached to eachnode of the 4 − valent spin network. Besides, each “chunk of space” is labelled with a φ J ,with J = 0 , , ,
3, specifying the discrete matter (scalar field) degrees of freedom . Thesefields are attached to the vertices corresponding to each tetrahedron and would represent thereadings of all fields: ‘clocks’ and ‘rods’. The coupling of gravity to four scalar fields has beenanalyzed in the LQG context for spin networks given by 4 − valent vertex [19, 23].The 2 nd quantization formalism is favourable for describing quantum many-body systems.The Fock space is built from the Fock vacuum | (cid:105) representing the state with no spin networknodes or no tetrahedra. Therefore, it is a state with no topological nor geometrical infor-mation; a “no-space” vacuum analogous to the Ashtekar–Lewandowski vacuum [24] whereoperators for geometric observables such as volumes and areas from LQG vanish. There-upon, one-particle states can be generated with the creation operators ˆ ϕ † ( g I , φ J ) acting onthe vacuum state. The role of the creation operator ˆ ϕ † and the corresponding annihilationoperator ˆ ϕ is derived directly from the postulated canonical commutation relations for the This procedure is usually generalized to an arbitrary k − number of massless scalar fields, J = 0 , , ..., k .Here we restrict the analysis to only four of them because they will be used for labelling the four spatiotemporaldimensions; i.e. 1 temporal φ and 3 spatial φ i independent components. (cid:104) ˆ ϕ ( g I , φ J ) , ˆ ϕ † ( g (cid:48) I , φ (cid:48) J ) (cid:105) = δ ( φ J − φ (cid:48) J ) (cid:90) SU(2) d h (cid:89) I =1 δ ( g (cid:48) I h g I − ) , (3)while two ˆ ϕ or two ˆ ϕ † operators commute. On the one hand, the Dirac delta functionsbetween the values of φ J that meet at the vertex necessary imply that the interaction islocal in these scalar fields [25, 26]. On the other hand, the integration of the Dirac deltadistribution over SU(2) ensures consistency with the “gauge invariance” under diagonal leftmultiplication of the ϕ field ϕ ( g , . . . , g , φ , . . . , φ ) = ϕ ( hg , . . . , hg , φ , . . . , φ ) ∀ h ∈ SU(2) . (4)Apart from this elementary SU(2) transformation acting on the central vertex, the four copiesof SU(2) associated to the links provides an additional permutation symmetry for the GFTfield ϕ ( g , g , g , g , φ J ) = ϕ ( g , g , g , g , φ J ) = ϕ ( g , g , g , g , φ J ) = ϕ ( g , g , g , g , φ J ) . (5)In this picture, the bosonic excitations are interpreted as geometric tetrahedra or vertices ofthe spin network. This is the one-particle state is depicted asˆ ϕ † ( g I , φ J ) | (cid:105) = (cid:12)(cid:12)(cid:12) g I , φ J (cid:69) =One can create generic states with arbitrary particle number N by acting N times with thecreation operators in the usual way. These N − particle states can be associated to a graph of N disconnected 4 − valent open vertices or disconnected tetrahedra. The GFT Hilbert spaceis the Fock space spanned by all subspaces with different N values. Finally, one can introduceconnections between the atoms of spacetime by gluing the tetrahedra or connecting the openvertices of the graph with links into higher dimensional structures. An important feature ofthe GFT states is that, contrary to LQG states, they are defined with no direct identificationwith the graph used to be constructed; i.e. there is no unambiguous identification between thegraph, the glued tetrahedra or their topology with the N − particle state. Further discussionon the relation between LQG and GFT Hilbert spaces can be found in [23] or in [27] in thecontext of a particular toy model.GFT is fully specified when its dynamics is encoded in the action. Indeed, this theoryis used as a field-theoretic description for the dynamics of LQG. A simple general form forthe latter is the sum of a quadratic kinetic term (that includes a local kinetic operator K that contains derivatives with respect to both variables g I and φ J ) and a generic interactionterm (which is higher order in the field operators and typically has a combinatorial type ofnon-locality due to the simplicial gluing of the QFT building blocks). This is S [ ϕ, ¯ ϕ ] = − (cid:90) SU(2) × R d g d φ ¯ ϕ ( g I , φ J ) K ϕ ( g I , φ J ) + V [ ϕ, ¯ ϕ ] . (6) This also ensures the “gauge invariance” under a diagonal left action of the group on all arguments of thefield. A different convention can be used: the invariance under a diagonal right action. K and V are not needed for deriving a condensate cosmologybecause the latter is derived from a general class of GFT models, the existence of a widenumber of such models in the literature shows that there is no consensus on which are thepreferred candidates for a four dimensional quantum gravity theory. For instance, a trivialkinetic term ( K = 1) is sufficient to define a class of GFT models with Feynman amplitudescorresponding to the amplitudes of a given class of spin foam models for a certain election of V . Furthermore, the perturbative expansion of the partition function of the field theory Z = (cid:90) D ϕ D ¯ ϕ e − S [ ϕ, ¯ ϕ ] (7)generates the Feynman rules for any spin foam model; such perturbative expansion in Feyn-man diagrams equals the sum over discretized path integrals for quantum gravity [28]. Indeed,in [29] it was realized that amplitudes for the Barrett–Crane spin foam model in four dimen-sional quantum gravity [30] could be obtained from a suitable choice of the GFT action.Later, it was shown that any prescription for a spin foam amplitude (within a class of mod-els of interest for quantum gravity) could be obtained directly from GFT [31]. In fact, thegenerality on possible elections of both operators points towards a one-to-one correspondencebetween spin foam models and GFT actions [32]. In this manner, the GFT partition function Z corresponds to a sum over topologies and spacetime histories (for gravity and matter).Each history itself is discrete and contains a finite number of degrees of freedom. The maintechnical challenges are still the same as for the lower-dimensional matrix models; these areto control the unwieldy sum over Feynman graphs and obtain a continuum limit. However,we are interested here in cosmological implications of GFT, hence we will not go deeper inthe microscopic description of the theory and proceed to the effective picture.Once a structure for the action is chosen, the path integral for the theory can be formallydefined. Subsequently, the complete (although formal) quantum dynamics can be fully spec-ified by deriving the Schwinger–Dyson equations for n − point correlation functions from thepath integral formalism. If in LQG dynamics the principal object of study is the physicalinner product between states projected onto the kernel of the Hamiltonian constraint, thecorresponding object of interest in the GFT formalism consists in the second quantizationof the spin network vertices. On the one hand, the Hamiltonian constraint imposed on spinnetwork vertices can be found in the classical action and their associated equations of motionin the GFT approach. On the other hand, the LQG dynamics which is mainly encoded inthe single evolution history of a spin network state can be found in all its details in the formof a GFT Feynman amplitude. However, from a field theoretic standpoint, this represents atiny piece of the true quantum dynamics. A generic continuum geometry is captured by a(complicated) superposition of spin network states; in a GFT language the analogy is foundin a hugely populated state composed by many-particles.In quantum field theory, the Schwinger–Dyson equations are one way to organize andsum in a natural way the infinitely many diagrams that contribute to n − point functions[33]. In this sense, they automatically contain non-perturbative information and encodethe complete quantum dynamics of the GFT models [34]. The equivalence to other non-perturbative methods such as the Bogoliubov canonical transformation and the Gaussianvariational Ansatz, is well established. Nonetheless, it is clear that the Schwinger–Dysonmethod is much more general than either of these. Therefore, in the continuum limit of GFTmodels it is expected that the Schwinger–Dyson equations would admit an interpretation asHamiltonian and diffeomorphism constraints of the quantum gravity theory, and thus providethe definition of the physical inner product (at least in a regime where topology changes are6uppressed) for a non-perturbative domain.The same quantum dynamics can also be given in an operator form for a GFT modelwith an action S [ ϕ, ¯ ϕ ] given by (6). The quantum equations of motion for a generic state | Ψ (cid:105) can be simply written as δ ˆ S [ ϕ, ¯ ϕ ] δ ¯ ϕ ( g I ) | Ψ (cid:105) = 0 , (8)together with a second equation, obtained from the variation of the action with respect to ϕ ( g I , φ J ). As mentioned above, the kinetic K and interaction V kernels in the generic action(6) can be chosen so as to reproduce the edge and vertex amplitudes of a spin foam model; i.e.a perturbative expansion of the GFT partition function around the Fock vacuum can be madeto coincide with the expansion of the spin foam model. The connection between the previousoperator equations of motion and the path integral formulation lies on the Schwinger–Dysonequations. Nonetheless, when dealing with interacting field theories, solutions to the latterequation cannot be easily obtained; in fact, a general solution is not known without furtherapproximations. Nevertheless, the kinematical Fock space is known to be troublesome fordefining an interacting quantum field theory. According to Haag’s theorem one should notexpect solutions to the quantum dynamics to be defined as elements on the Fock space.The GFT formulation allows to deal with problems that are analogous in condensed mat-ter physics and hence it has allowed to make use of its ideas and methods. It proceeds byseeking for some condensate state that can play the role of a new, nonperturbative vacuumof the theory and that approximates the full solution state | Ψ (cid:105) , at least for a restrictedset of observables. This provides a direct route from Schwinger–Dyson equations to cos-mological observables [35], avoiding the need for discussing an effective Wheeler–DeWittequation. In a systematic treatment, one would have to prove that these solutions to thesimplest Schwinger–Dyson equations approximate the fully dynamical solution (higher-orderequations would then be consistency conditions). In the next section it will be shown thatwhen restricting to states represented by a wave function obeying the homogeneity principle,consistent effective quantum homogeneous cosmologies are obtained [36–38]. The program of the GFT condensate cosmology (see Refs. [1, 15, 39] for reviews on the topicand their application to homogeneous cosmology) is to represent a continuum geometry from aquantum field theory for ϕ , function of four arguments valued in the SU(2) and at least fourscalar matter fields degrees of freedom. The macroscopic and nearly homogeneous descriptionis very well approximated by the condensate phase of a large number of excitations over theFock vacuum of this field. One can think of coherent states such as Bose–Einstein condensatesor as similar to macroscopic electromagnetic fields in quantum optics. Simple models wherea phase transition produces a symmetry breaking providing a condensate phase have beenpresented in Refs. [40], or using Landau–Ginzburg theory in [41].The coherent state | σ (cid:105) is an eigenstate of the field operator ˆ ϕ ; i.e.ˆ ϕ † ( g I , φ J ) | σ (cid:105) = σ ( g I , φ J ) | σ (cid:105) , (9)and as condensate acquires a non-vanishing expectation value σ ( g I , φ J ) := (cid:104) σ | ˆ ϕ ( g I , φ J ) | σ (cid:105) (cid:54) = 0 (10) An election that can be generalized [36, 38]. φ J can macroscopically distinguishbetween different points over the condensate. Therefore, on the emerged spacetime one isable to define a completely relational dynamics.The simplest coherent state is defined by means of the mean-field approximation. Whenany connectivity amongst the spin network nodes is neglected, meaning that tetrahedra arenot glued, the state can be written as a single-particle condensate state | σ (cid:105) ≡ N ( σ ) exp (cid:0) ˆ σ (cid:1) | (cid:105) ; (11)where the normalization factor for a condensate model with an arbitrary large number ofquanta is given N ( σ ) ≡ exp (cid:104) − (cid:90) d φ J (cid:12)(cid:12) σ ( φ J ) (cid:12)(cid:12) (cid:105) (12)and the condensate operator is defined asˆ σ ≡ (cid:90) d g d φ σ ( g I , φ J ) ˆ ϕ † ( g I , φ J ) . (13) σ ( g I , φ J ) is the analogue of the order parameter in condensed matter physics. This definitionis based on the idea that the homogeneity of the wave function for a many-particle state isfully determined by the single-particle wave function. Under this very simple approximation,fluctuations are ignored and σ ( g I , φ J ) represents a condensate wave-function directly as aclassical GFT configuration.In the second-quantized framework, the simplest one-body observable that can be con-structed is the number operatorˆ N ( φ J ) = (cid:90) d g ˆ ϕ † ( g I , φ J ) ˆ ϕ ( g I , φ J ) . (14)The wave function is not normalized; rather its norm determines the number of uncorrelatedquanta in the given state N ( φ J ) = (cid:104) σ | ˆ N ( φ J ) | σ (cid:105) = (cid:90) d g (cid:12)(cid:12) σ ( g I , φ J ) (cid:12)(cid:12) ( < ∞ ) (15)which is the (finite) expectation value of the operator (14) at a value φ J for each of the fourfields. The homogeneity condition for the wave function can be defined over more generalcondensates containing additional topological structure and defined by a sum over connectedgraphs of arbitrary complexity [37].In the remaining part of this section and the next one, the two main testable structuralconsequences of the cosmological principle will be discussed, with emphasis on how they areboth implemented from the GFT point of view. The GFT condensate approach involvescosmological models which reproduce the spatially flat homogeneous and isotropic Fried-mann–Lemaˆıtre–Robertson–Walker (FLRW) spacetime, but in a semiclassical limit wherethe field operators are replaced by classical fields. This approximation seems to be suitableenough for describing the emergence of a macroscopic and nearly homogeneous universe,where small spatial gradients on the effective geometry can take place. In complete anal-ogy with condense matter theory, if interactions get stronger, the mean-field approximationbreaks down. This behaviour is probably expected in the early universe or near the centres of8lack holes where quantum effects are relevant because of high curvatures; a nice discussionon the ranges of validity of the mean-field approximation can be found in [25].A Hartree–Fock mean field approximation (11) is the most simple collective wave functionconstructed under the homogeneity principle. Its dynamics can be looked for in the expecta-tion value of the (normal ordered) operator equations of motion obtained from (8). However,one can appeal to the Schwinger–Dyson equations to be solved approximately (cid:104) σ | (cid:99) δSδ ¯ ϕ | σ (cid:105) = δS [ σ, ¯ σ ] δ ¯ σ ( g I , φ J ) = −K σ ( g I , φ J ) + δ V [ σ, ¯ σ ] δ ¯ σ ( g I , φ J ) = 0 . (16)This expression provides a quantum cosmology-like equation for the ‘wavefunction’ σ (similarto those obtained in Ref. [42]). The underpinning of the approximation into coherent statesis only valid in regimes where the interaction term, which contributes with a non-linear termof the order of ¯ σ , is subdominant. For instance, in Bose–Einstein condensate theory thebreakdown of the Gross–Pitaevskii equation is signalled by large fluctuation with respect tothe mean field associated to the particular quantum state considered, and nothing has to dowith any singularity of any particular solution. In this context, the microscopic dynamicsof the GFT quanta can be described hydrodynamically in terms of the collective variable σ ( g I , φ J ), the condensate wave function.This hydrodynamic limit is understood as an effective collapse of the Schwinger–Dysontower of equations into the simplest one. The simplicity of the state | σ (cid:105) makes this equationsto become the one-particle correlation function –which is exactly σ ( g I , φ J )–. The continuumnature of the picture arises from the fact that, given the equivalence with the path integralformulation, the coherent state is given by an infinite sum over numbers of disconnectedspin networks nodes (implicitly a sum over ¡¡not yet connected” graphs). Therefore | σ (cid:105) is anon-perturbative state with respect to the Fock vacuum, but now playing the role of a newnon-perturbative vacuum of an effective theory obtained after the hydrodynamical limit.Analogously to the Gross–Pitaevskii equation, the condensate equation (16) is non-linear,as to be expected in a hydrodynamic context. The non-linearities effectively encodes the mi-croscopic interactions between fundamental quanta. These interactions are ultimately respon-sible for developing inhomogeneities at both the microscopic and macroscopic scales. Thisequation is of course a weaker condition than (8); in terms of the truncated Schwinger–Dysonequation, the theoretical error in the resulting effective theory can be estimated by the mag-nitude of the neglected terms. These terms can be reconstructed in terms of the non-Fockrepresentations for describing interacting fields. With this inequivalent representation of thecanonical commutation relations (with respect to the free theory) the interacting theory natu-rally provides fluctuations over the homogeneous background associated with inhomogeneitiesof the condensate.There are various ways of choosing a suitable form for the kinetic and interaction oper-ators. If the aims of GFT is to define the dynamics of LQG, they can be quite genericallychosen in such a way that the Feynman amplitudes of GFT correspond to the amplitudesbetween boundaries of the spin foam models; see for instance [42] for an effective descriptionof inhomogeneities in a non-linear extension of LQC. However, the purpose of this reviewis to show in which degree GFT reproduce in a consistent manner a cosmological scenariowithout making a concrete assumption on a particular model.To get an initial insight on the effective dynamics of GFT condensates, two approximationsare usually imposed. The first one involves all symmetries of the employed free massless scalarfields φ J used as matter to introduce relational cosmological observables [25]. As mentioned,these matter reference frames allow us to define an effective dynamics formulated exclusively9n relational terms, where spacetime points in the emergent spacetime description can nowbe distinguished . Each scalar field satisfies the Klein–Gordon equation which is equivalentlyunderstood as a harmonic coordinate condition for each field ∇ µ ∇ µ φ J = 0 . (17)Concerning the symmetries of the material clocks and rods, the GFT dynamics should beinvariant under1. constant (arbitrary) shifts φ J (cid:55)→ φ J + φ J ,2. the time-reversal or parity transformation φ J → − φ J ,3. rotations φ i → O ij φ j , with O ij ∈ O(3) and i, j = 1 , , K in (even) derivatives with respect to the fields φ J ; the first symme-try forbids any explicit dependence on φ J . In Refs. [25, 43–46] is argued that the low-energyGFT dynamics can be compared with cosmology on large scales when K is truncated up tosecond order derivatives as K = K + (cid:101) K ∂ ∂ ( φ ) + K (cid:88) i =1 ∂ ∂ ( φ i ) + . . . (18)The dots stand for fourth and higher derivatives which are suppressed. Let us note thatthe coefficients K i in this expansion are still differential operators with respect to the SU(2)variables g I and therefore they can contain (even) derivatives with respect to these variables.The second approximation usually made for extracting physical states of the theory is toconsider for the building blocks of geometry to be all in the same microscopic configurationand all in a weakly interacting regime in which the effect of V on the dynamics can beneglected. This implies for the GFT quanta to be uncorrelated; necessary condition fordefining the coherent state | σ (cid:105) in a mean-field treatment given by Eq. (11). This drasticapproximation is not strictly necessary. In fact, it is not suitable for strong coupling regimesand it breaks down with the grown of the particle number intervening in the picture. Thefree approximation is valid only in a mesoscopic regime where the particle number for a givenvolume of the state is not so large. Some studies include the potential V of the effectivedynamics for some particular models of GFT condensates [47]. In some cases, interactionterms become important at late times after a prolonged phase of acceleration. In [48] it ispresented a model with a cyclic evolution leading to a recollapse of the universe, analogousto a negative cosmological constant.Under both approximations, replacing the general expansion for the kinetic kernel (18)in the r.h.s. of Eq. (16), one gets the following equation of motion (cid:18) K + (cid:101) K ∂ ∂ ( φ ) + K (cid:88) i =1 ∂ ∂ ( φ i ) + . . . (cid:19) σ ( g I , φ J ) = 0 . (19)The uncorrelated state solving this equation has interesting cosmological applications to bereviewed in what follows. Being a many-particle state with an analogous hydrodynamicaltreatment, it can contain information about the connection and the metric at many different See Section 5.2 for clocks and Section 6 for rods. φ and relational rods φ i defined over the condensate. Apart from the symmetries 1–3,no assumptions have been made over these fields. One of the main goals of this condensateapproach is to provide an affective cosmological dynamics consistent with general relativityat a low curvature regime, but this description naturally encompasses possible quantumcorrections. The construction of simple coherent states like (11) finds its motivation in their propertiesand their analogies with Bose–Einstein condensates. In fact, the idea that spacetime couldbe treated as a kind of Bose–Einstein condensate of geometric quanta was formulated inother approaches before [49]. As we will see in the following, these states are the buildingblocks for obtaining a cosmological picture with geometries characterized by a nearly spatialhomogeneity and isotropy. However, these states | σ (cid:105) cannot be interpreted as the graphsusually employed in LQG. They are rather “graphs” constituted by a large number of discon-nected constituents (vertex or tetrahedra) understood as GFT quanta. In the dual picture,this is depicted as open spin networks, where four links carrying group theoretical data areattached to each vertex, but these vertices are not connected to each other. These quanta aresupposed to condense into a macroscopically occupied “ground state” described by the meanfield σ ( g I , φ J ) representing the condensate wave function and understood as a superpositionof states with all possible particle numbers. Again, in this approximation, fluctuations overthis condensed phase are ignored and only after a well-established homogeneous configurationis obtained, fluctuations are re-established as a natural feature due to quantum uncertainty.It is worth noticing that it is quite misleading to interpret σ as a quantum mechanicalwave function. Not only due to the normalization linked to the average particle number,but also and more importantly, because the superposition principle does not hold here, andthe probability interpretation is not direct. In the mean-field approximation, self-consistencyimplies the existence of a regime where the field equation (19) is approximately solved onlyconsidering the first kinetic term. The dynamics of σ is typically governed by nonlinear andnonlocal equations (while all dynamical equations on the Hilbert space and on the GFT Fockspace remain linear). As mentioned, at some point interactions should become dominant,since the particle number scales as | σ ( g I , φ J ) | and the potential contains higher powers of σ and ¯ σ . However, in the mean-field regime motivated by the nearly homogeneous character ofour universe, the equation is still linear and explicit solutions can be straightforwardly found.In the following a further restriction on the structure of the wave function of the con-densate will be imposed for capturing the FLRW cosmology. GFT recovers the microscopicconfigurations from an effective continuum dynamics which results a non-linear generaliza-tion of LQC [36, 38]. As it will be shown in this and the following sections, the restrictionto isotropic modes for the microscopic states leads to further simplifications that allow usto reconstruct isotropic quantities like the spatial volume, the cosmological scale factor andthus the Hubble rate, all from σ ( g I , φ J ). Given the equivalence between the representationas spin network nodes and tetrahedra, in the GFT context it is argued that the natural wayto require isotropy is to impose the most ‘isotropic’ condensate configuration. In classical ge-ometry one would think of equilateral tetrahedra whose four faces are equal and the resultingvolume is maximized. This ideas have been translated to the quantum picture in [25, 43].At a quantum level, one approaches Eq. (19) using Peter–Weyl theorem [50] to decomposethe wave function into SU(2) representations. The left “gauge symmetry” in (4) implies for11he GFT condensate to obey the identity σ ( g I , φ J ) = σ ( hg I , φ J ) for all h ∈ SU(2). It isdesirable to give a precise geometric interpretation to the condensate as a continuous andhomogeneous spatial geometry. In this picture, the condensate wave function is interpretedas a probability distribution on the space of such homogeneous geometries. However thisinterpretation requires a right invariance under the diagonal group action for the condensate;i.e. σ ( g I , φ J ) = σ ( g I k, φ J ) for all k ∈ SU(2). This yields for the state to only contain thegauge-invariant degrees of freedom of a tetrahedron and consequently σ becomes a functionon SU(2) \ SU(2) /SU(2), which is isomorphic to the space of connection degrees of freedomof a homogeneous universe in LQC [51]. However, this symmetry is not a symmetry of theGFT field, as it is the left invariance , but an imposed property on certain states with thepurpose of reducing the number of dynamical degrees of freedom for obtaining the previousinterpretation. This approximation captures well enough the diluted or weakly-interactingphysics of Bose–Einstein condensates [52].As GFT makes use of the kinematical space of LQG, the picture of the fundamentalbuilding blocks of the theory coincide. Given the analogy between graphs and simplices, eachnode represents a “chunk” of space carrying a quantum of volume which depends on thecolours of the node. These volumes can be associated to the volumes of tetrahedra, each ofthem with its four faces taken as “elementary surfaces” dual to the links, which carry quantaof area [53]. However, the novelty of GFT concerns the dynamics of these quanta. Althoughboth theories share the underling building blocks, the way GFT accounts for the effectivesemiclassical description is very different with respect to the continuum and classical limit ofLQG. In this sense, in both theories the quantized space does not reside “somewhere”, butit itself defines the “where”; yet the hydrodynamical limit of GFT defines this “where” asan effective theory that is not useful for describing the dynamics of the quanta, but only thecollective behaviour of a big number of them. In this manner, the SU(2) − invariant subspacecarries a unitary representation of Diff Σ and reduces the wave function to a simple formif written in terms of linear combinations of a pair of suitable intertwiners: ¯ I j,ı l mn and I j,ı r mn ,one associated to the left gauge invariance and one to the right closure condition. Theseintertwiners define invariant mappings between SU(2) representations and they are elementsof the Hilbert space of states of a single tetrahedron. Given the volume maximization analogywith a classical tetrahedron, the intertwiners should be chosen so that they are eigenstateof the LQG volume operator with an associated eigenvalue being the largest possible for thegiven j .As mentioned, each of the four links g I (or faces) are coloured with spin − j irreduciblerepresentation of SU(2). The analogy between the equal areas of the faces of the tetrahedronand the restriction to an expansion over isotropic modes only for the underlying GFT quantais understood as expanding over the same four j I , i.e. j = j = j = j , one for each (ofthe same) coloured link associated to each node of the spin network. In this approximation,the spin network vertices are said to be monochromatic and, together with the homogeneousrestriction, all of them are exactly equal.With the aim of extracting a homogeneous, isotropic and spatially flat cosmological sectorfrom quantum gravity, let us decompose the wave function, solution for the equation ofmotion (19), into a basis of orthonormal functions given by the Wigner D jmn ( g I ) − matrices.With this procedure, we group all dependence on g I in fixed functions D j ( g I ), which are anappropriate convolution of four Wigner D − matrices with intertwiners , such that quantum This election is a convention and can be exchanged; on the other way around, one can start with a right“gauge symmetry” on the ϕ field and then impose the left invariance over σ obtaining equivalent results. In the convolution of Wigner D − matrices with SU(2) intertwiners, the usual range of values for the σ j I , ı l ı r ≡ σ j ( φ J ) thatnow only depend on the volume of the tetrahedron (or equivalently, on the surface area of oneof its faces), as well as on the scalar field φ J . Therefore, the restricted mean field expandedin irreducible SU(2) representations is written as σ ( g I , φ J ) = (cid:88) j ∈ N σ j ( φ J ) D j ( g I ) (20)where the coarse-grained degrees of freedom are now captured by each of the wave functions σ j ( φ J ). Refs. [54, 55] discuss whether the restriction to expansions in only a single spin j la-belling the irreducible representations of SU(2) can be relaxed, together with their consequenteffective dynamics in the large-scale limit. Recall that lifting the isotropic restriction allowsto investigate anisotropic GFT condensate configurations. According to [55], anisotropiesplay an important role only at small values of the relational clock φ (i.e. at small volumes),whereas at late times the isotropic mode become dominant.For the usual GFT actions, the kinetic operator K only contains derivatives, but noexplicit dependence on g I . In the common situation where all terms in the expansion of K inEq. (19) are general functions of the Laplace–Beltrami operators with respect to the SU(2)variables g I , we can define the following coefficients K D j ( g I ) := − B j D j ( g I ) , (cid:101) K D j ( g I ) := A j D j ( g I ) , K D j ( g I ) := C j D j ( g i ) ; (21) A j , B j and C j are j − dependent couplings depending on the original GFT kinetic terms andwith no further derivatives. Each Laplacian acting on each g I contributes with an eigenvalue − j ( j + 1). Recall that D j ( g I ) encode the monochromatic (equilateral) character of the spinnetwork nodes (tetrahedra). The Wigner matrices are eigenfunctions of the SU(2) Laplacian,then the Peter–Weyl decomposition leads to a decoupling of (19) into independent equationsfor each j , written as (cid:18) − B j + A j ∂ φ + C j (cid:88) i =1 ∂ φ i (cid:19) σ j ( φ J ) = 0 . (22)In homogeneous configurations, the handling of “rods” φ i loses meaning. Therefore, whenderiving the global aspects of a FLRW cosmology only the first two terms matter. In the nextsection, it will be shown that the two corresponding coefficients, A j and B j , can be constrainedwhen requiring the theory to be compatible with Friedmann equations. However, condensatefluctuations are expected to break the homogeneity; analogously as when in the cosmologicalmodel one considers deviations from the cosmological principle. In such a case, “rods” mustbe reintroduced to locate these deviations, whose power spectrum is probably expected to beassociated to classical inhomogeneities observed in the cosmic microwave background (CMB)spectrum. At this point, the third term becomes meaningful. We will return to discuss thistopic in Section 6.Interestingly, if one expands σ j in Fourier modes with respect to the spatial coordinatespictured as the scalar fields φ i , a complete set of solutions to (22) can be obtained; this is σ K i j ( φ J ) = e iK i φ i α + j exp (cid:32)(cid:114) B j + C j K A j φ (cid:33) + α − j exp (cid:32) − (cid:114) B j + C j K A j φ (cid:33) , (23) magnetic indices is taken: − j ≤ m , n ≤ j . The indices ı labels the possible intertwiners elements in a basis ofthe Hilbert space; ı l and ı r points to the imposition of the left and right invariance to the field. To a detailedconstruction of the wave function see for instance [25]; here we just sum up the main steps for deriving acosmological sector from the full theory. α + j and α − j as arbitrary constants. The coupled scalar fields do not only act as sourcesof matter fields driving the expansion of the universe and the corresponding cosmologicalperturbations, but also as ‘tools’ to define local coordinates. GFT models are constructeddemanding background independence; hence any coordinate system constructed of physicaldegrees of freedom must be relational. In the limit in where they are turned off, we obtaina homogeneous solution. Being the condensate wave function at a given time φ entirelydetermined by just one geometric quantity, the spin j , the only geometric quantities that canbe extracted from this condensate wave function are isotropic quantities like the total spatialvolume, the Hubble rate, etc. Let us now consider proper GFT cosmological models. In this section it will be shown howto obtain a FLRW universe from the condensate wave function (23). However, we first givethe main ingredients of cosmological implications derived from General Relativity to showexplicitly how the previous approximations to GFT quantum gravity formalism lead us to aquantum picture consistent with classical results in the continuum and semiclassical limit.
It is well known how to introduce physical reference frames and how to define relationaldynamics in general relativity. Let us consider a massless free scalar field that plays the roleof a relational clock in a flat FLRW metric of the formd s = − N ( t ) d t + a ( t ) d x . (24)The structure of the metric entails a foliation for the universe on isotropic and homogeneoushypersurfaces d x with flat intrinsic geometry in R parametrized, relative to one another,by a scale factor a ( t ).The action for the scalar field, considering limiting cases in which the backreaction ofreference matter on the geometry can be neglected, is S φ = − (cid:90) d x √− g g µν ∂ µ φ ∂ ν φ ; (25)hence the matter clock obeys the Klein–Gordon equation, which reduces to ∇ µ ∇ µ φ = 0 ⇒ dd t (cid:32) a N d φ d t (cid:33) = 0 ⇒ a N d φ d t = constant . (26)The assumption of a nonnull constant provides a good characteristic for the clock φ ; as itscorresponding momentum π φ is conserved, it has a monotonic evolution; hence it can bewritten as φ = φ T . If for instance φ has dimensions of mass, the ‘temporal’ scalar field T becomes dimensionless.The other equation to solve is (cid:18) d a d T (cid:19) = 4 πG φ a , (27)14hich is a Friedmann equation giving two independent solutions: a ( T ) = a exp (cid:32) ± (cid:114) πG φ (cid:33) , (28)each of them corresponds to an expanding or contracting universe, respectively. Accordingto the classical solution, a singularity V → φ , corresponds to infinity. However, this singularity is reached in a finite propertime if written in the propitious coordinates.A quantum theory of gravity coupled to a massless scalar field should reproduce at somepoint the last equation but avoiding the singularity behaviour. This is the basic idea sincethe early days of quantum cosmology [56] that later on also informed the foundations ofLQC. The requirement for the temporal coordinate to satisfy the harmonic condition avoidsquantization ambiguities when choosing the lapse function [57]. At this point we have a condensate with an isotropic structure imposed from the quantumsymmetries. If spatial homogeneity is also desired, this would correspond to demand theFourier mode (cid:126)K = 0 for the mean field solution in the general solution (23). This requirementmakes the rods φ i , with i = 1 , ,
3, meaningless, as there is no need to refer to locations overan exactly homogeneous state. Then, the mean field would have the form σ j ( φ J ) ≡ σ j ( φ ) , (29)being only a function of one scalar “time field” φ , playing the role of a relational clock. Thisimplies for the condensate wave function to become σ j ( φ ) = α + j exp (cid:115) B j A j φ + α − j exp − (cid:115) B j A j φ . (30)If we assume that the condensate mean field takes its homogeneous form, the associated(background) universe of course would result homogeneous.Once the mean field solution is found, it is of interest to define the relational 3 − volumefor this state. This can be done by means of the second-quantized vertex volume operator.This one body operator generically would define for the mean field wave function (23) alocal volume element at the spacetime point specified by values of the reference fields, this isˆ V ( ϕ J ). In the particular case of the homogeneous universe under consideration, the constrain(29) would then define the element volume only at a given relational “time” φ ˆ V ( φ ) = (cid:90) SU(2) × SU(2) d g d g (cid:48) ˆ ϕ † ( g I , φ ) V LQG ( g I , g (cid:48) I ) ˆ ϕ ( g (cid:48) I , φ ) . (31)The matrix elements V LQG ( g, g (cid:48) ) ≡ (cid:104) g I | V LQG (cid:12)(cid:12) g (cid:48) I (cid:11) are the matrix element of the volumeoperator between single-vertex spin networks states in LQG. Although there are several dif-ferent definitions of the volume operator in the theory [58], all of them agree when 4 − valentvertex are considered [59]. In fact, it is helpful to choose a basis of intertwiners I thatdiagonalizes the action of the LQG volume operator on a spin-network node. Hence, the Recall the discussion in Section 4 concerning the functions D ( g I ) ϕ will carry a definitevolume given by the corresponding eigenvalue of the LQG volume operator.Let us now go back to the homogeneous GFT state σ j ( φ ), the expectation value for thevolume operator (31) can be evaluated immediately when coherent states of the form (11)are considered (cid:104) ˆ V ( φ ) (cid:105) = (cid:90) d g d g (cid:48) ¯ σ ( g I , φ ) V ( g I , g (cid:48) I ) σ ( g (cid:48) I , φ ) . (32)This result corresponds to the total 3 − volume at a relational time φ , associated to suchcondensate state. This procedure is not a novelty of GFT; for instance, the total volume ofthe universe at a fixed value of the scalar field is one of the main relational observables ofinterest in LQC [57, 60]If we also impose the isotropic wave function constraint discussed in (20), since the volumeoperator is diagonal when written in terms of SU(2) representations, the volume expectationvalue of the condensate in such a state reduces simply to (cid:104) ˆ V ( φ ) (cid:105) = ∞ (cid:88) j =0 V j (cid:12)(cid:12) σ j ( φ ) (cid:12)(cid:12) . (33)The latter expression is written in terms of the local particle number density for each quantaof spin j . The approximate eigenvalue of the first quantized volume operator acting upon anode, although depending on the intertwiner used to define D j ( g I ), is very well approximatedfor each j by V j ∼ (cid:96) P l j / .The evolution of the local volume elements then provides the macroscopic behaviour of thestate (30), which will depend on the choice of the initial parameters α + j and α − j . Some generalstatements can be sketched out for some GFT models: if the ratio B j /A j is positive anddevelops a maximum for a given j = j , except for the fine-tuned cases with α + j = 0 or α − j = 0,the spin j will dominate over all others. Hence for almost any condensate homogeneous wavefunction of the form (29), its associated volume will asymptotically behaves as (cid:104) ˆ V ( φ ) (cid:105) φ →−∞ −−−−−−→ (cid:12)(cid:12) σ − j (cid:12)(cid:12) exp − (cid:115) B j A j φ , (cid:104) ˆ V ( φ ) (cid:105) φ → + ∞ −−−−−−→ (cid:12)(cid:12) σ + j (cid:12)(cid:12) exp +2 (cid:115) B j A j φ ; (34)where the global constants are related to the volume eigenvalue assigned to the spin j by | σ ± j | = V j | α ± j | . In such a situation an exponentially small number of quanta are char-acterised by a single spin j excitation, implying mainly a constant volume per quantum.This domination of a single and small spin component in the cosmological dynamics of thehomogeneous and isotropic background can be shown to take place at later times [61]; how-ever it is also achieved exponentially fast and hence it can be expected to be an acceptableapproximation also at earlier times [54]. In this manner, the evolution of the total volumeonly depends on the growth of the number of particles with spin j given by the exponentialfactor in Eq. (34).These GFT states closely match the heuristic relation between LQG and LQC, wherethis type of quantum states are usually assumed [62]. Despite the exact relation betweenboth theories remains open, some proposals analyse a cosmological sector of LQG built up16n states with large number of spin network nodes, all labelled by the same quantum num-bers. The nodes are considered to be disconnected and all links are dressed with the sameSU(2) representation label. Commonly, the spins are taken to be j = 1 /
2, and homogene-ity considerations justify the same number of links per node, typically chosen as 4 − valentnodes. Shortly, LQG also suggests to consider quantum geometry condensates where all itsconstituents are quanta in the same state [63]. All these features are naturally encoded in thecosmological results obtained from the GFT formalism; hence the latter can be considered asa field theory reformulation of LQG and spin foam models. However, it is worth mentioningthat a derivation of LQC from Hamiltonian formulations of LQG is a largely outstandingchallenge [64].Interestingly, for large (positive or negative) φ , the coefficients A j and B j are identifiedwith the low energy (emergent) Newton constant G as follows: B j /A j = 3 πG , Eq. (34)reproduces the classical Friedmann equation of general relativity in (28). Therefore, theresulting expression point towards a regime where the universe expands to a macroscopic size(if σ ± j (cid:54) = 0). This compatibility was one of the main results of [25, 43] and is obtained for arange of parameters of the microscopic dynamics in a suitable semiclassical regime and, asmentioned before, for generic initial conditions [61].It is also worth mentioning some properties characterizing this range of GFT models withthe desired asymptotic behaviour. First, at small volumes, in the Planck regime, the theoryinterpolates between the classical expanding and contracting solutions (28) of the classicalFriedmann dynamics. This implies that the universe undergoes a bounce, i.e. the volumeelements never go through zero avoiding or ‘resolving’ the classical big bang singularity. Infact, it is possible to show that a singularity where (cid:104) ˆ V ( φ ) (cid:105) strictly vanishes for some value ofthe clock field φ is only possible for special (hence fine-tuned) initial conditions. Therefore,instead of a singularity, there is just a very dense region where an effective quantum forceappears like a repulsion that prevents the collapse. Secondly, the asymptotic behaviour inEq. (34) shows an exponentially growing phase in both temporal directions: to the far pastand the future (contrary to the classical solution that, as mentioned in Section 5.1, presentsa singularity in the far past). Third, properties of interest are the corrections to the classicalFriedmann dynamics. Indeed, some mean field solutions provide corrections that can bematched to the “improved dynamics” of LQC [65]. For instance, similar derivations to LQCdynamics but from GFT condensates can be found in [66].More recently, a different analysis of GFT using Hamiltonian methods has been developedin Ref. [67]; this topic is further discussed in Section ?? . There are many results that can serveas a starting point for GFT phenomenology: the generic quantum bounce can be followed bya subsequent acceleration replacing the inflaton theory [68], the inclusion of interactions inthe GFT [55, 69] and their subsequent effects which become dominant away from the bounce. We have seen how GFT offers a consistent proposal for condensate states in a low curvatureregime where they describe an effective macroscopic spacetime that emerges dynamically.As repeatedly mentioned, GFT models are connected to previous work done in the LQCcontext; both provide analogous mechanism for the singularity resolution. However, GFTmight explain the emergence of a semiclassical universe as the simplest approximation andmore complex situations are expected to be derived. The extension of this framework be-yond the spatial homogeneity by including, for instance, cosmological perturbations into theformalism, allows more realistic cosmological scenarios where inhomogeneities are present.17ne would be interested in computing the non-vanishing power spectrum of cosmologicalperturbations over this mean field state. Let us approach this issue by bearing in mind theanalogy with Bose–Einstein condensates. Perturbations can be added to (30) in an analogousmanner as phonons appear as deviations from condensates with exact homogeneity. Hence,perturbations around the mean field solution are considered as vacuum fluctuations that areto be converted into classical inhomogeneities in a later stage of the universe [45]. The mainpoint when introducing first corrections to the previously found homogeneous state, is tore-establish the rods φ i eliminated in Eq. (29).Different approaches have been considered to include perturbations. For instance, one canassociate a constant mean field solution but only to ‘local’ patches, labelling them by makinguse of the four scalar fields coordinates. The inhomogeneity relies in the fact that differentpatches do not necessarily have the same constant mean field solution, thus the effectivehomogeneous geometry does not necessary coincide among different patches. This picturefor incorporating inhomogeneities is based on the so-called ‘separate universe approach’ [70],whose main characteristics are considered in the GFT condensate cosmology [71].In the remaining of this section, we will focus instead on quantum fluctuations of the local3 − volume around the exactly homogeneous background condensate derived in Eq. (30). TheGFT models discussed above have enough degrees of freedom for describing inhomogeneousquantum geometries and their effective dynamics which is expected to be a realistic pictureof fundamental cosmology. Quantum fluctuations would then represent the quantum grav-itational mechanism for explaining the origin of these inhomogeneities. This procedure isanalogous to the usual treatment in inflation where the power spectrum of quantum fluc-tuations over a homogeneous quantum state is computed (instead of a quantum state ona classically perturbed geometry). These fluctuations are generically expected because ofquantum uncertainty and they would freeze out producing the classical pattern on inhomo-geneities currently observed in the CMB [72]. Besides, these homogeneities would provide alower bound on deviations from exact homogeneity in GFT [73] .The procedure for introducing vacuum fluctuations into the picture is represented as inSection 5.2 where an effective cosmological dynamics under cosmological principle can beextracted from the volume operator. The main obstacle of this treatment is the absence of anotion of “wavenumber”, which is usually defined through the spectrum of a suitable Laplaceoperator. Therefore the results in Refs. [73, 74] only describe homogeneous perturbations orglobal quantities. Hence one needs to generalize the operator (31) to encompass notions ofspatiality to be capable of describing the fluctuations over the condensate. The procedure isdirect: we extend the formalism attaching a part from the clock φ , the three other masslessscalar fields φ i , that is, by replacing φ → φ J so as to assign a clock and rods to each point ofthe condensate by coupling a four reference matter frame to a GFT for gravity. The volumeoperator is then written asˆ V ( φ J ) = (cid:90) SU(2) × SU(2) d g d g (cid:48) ˆ ϕ † ( g I , φ J ) V ( g I , g (cid:48) I ) ˆ ϕ ( g (cid:48) I , φ J ) . (35)The meaning of the latter expression differs from (31); here ˆ V ( φ J ) refers to the local vol-ume element at the spacetime location specified by the components of the φ J field. Scalarperturbations in cosmology are then obtained from perturbations in these local volume ele-ments. Strictly speaking, ˆ V ( φ J ) corresponds to a density. The infinitesimal local volume is Further work has to be done in the GFT framework to derive the CMB power spectrum and that to ourknowledge there is no evidence for it yet. V ( φ J ) δ φ J and the total 3 − volume (31) is still interpreted as the total volume of theuniverse modelled as a condensate state. At a given moment of the relational time φ , itsvalue is obtained by integrating over the rods φ i ,ˆ V ( φ ) ≡ (cid:90) d φ i ˆ V ( φ , φ i ) . (36)The main idea is that cosmic structures are expected to be formed from early localvolume fluctuations. In the GFT cosmology approach, this pattern is expected to be encodedin the correlation functions for the geometric observables. These correlation functions encodethe ‘true’ fundamental quantum dynamics and are understood as sum over graphs, dual todiscretisations of manifolds given the aforementioned duality between tetrahedra and vertices.For the ongoing discussion, let us compute correlations in local volume fluctuations over thestate (11). It is defined the local volume fluctuation operator as δ ˆ V ( φ J ) = ˆ V ( φ J ) − (cid:104) ˆ V ( φ J ) (cid:105) (37)with respect to the generalized volume operator (35). Then, it is of interest to compute thefollowing two-points function (cid:104) δ ˆ V ( φ J ) δ ˆ V ( φ (cid:48) J ) (cid:105) . (38)The idea of characterizing perturbations employing matrix elements of the one-body squaredvolume operator V ( g I , g (cid:48) I ) is not new. This procedure has been first presented in [73] butwithout referring to any notion of rods. Consequently, the results that can be derived fromthis formalism can only achieve global properties. Later on, in [46] the formalism has beengeneralized to include rods. This modification enables us to extract local information re-garding perturbations; for instance, the Fourier transformation from φ i to their momenta k i provides notions of wave length with respect to the reference frame fields. As discussedbelow in a minute, the transformation to momentum representation allows to write the powerspectrum of inhomogeneities into the usual Fourier space form.Let us now consider perturbations around exact homogeneity. These are written as σ j ( φ J ) = σ j ( φ ) (cid:2) (cid:15) ψ j ( φ J ) (cid:3) ; (39)where the field ψ j ( φ J ) represents condensate perturbations ‘located’ by means of the fourscalar fields. As already mentioned, this procedure is analogous to the propagation of phononsover a condensate. These GFT phonons were firstly proposed in [36], however their interpre-tation was not clear until rod matter fields were included. This inclusion is the conventionalmanner to consider quantum fluctuations in the local volume as seeds of cosmological inho-mogeneities.As shown in [46], the power spectrum for the volume perturbations can be derived forthe GFT condensate state but now taking into account quantum corrections. Computing thequantum fluctuations of the volume expressed as the two point function (38) for the state(39) one obtain the following expression: (cid:104) ˆ V ( φ , k i ) ˆ V ( φ (cid:48) , k (cid:48) i ) (cid:105) − (cid:104) ˆ V ( φ , k i ) (cid:105)(cid:104) ˆ V ( φ (cid:48) , k (cid:48) i ) (cid:105) = δ ( φ − φ (cid:48) ) (cid:88) j V j (cid:12)(cid:12) σ j ( φ ) (cid:12)(cid:12) × (cid:110) (2 π ) δ ( k i + k (cid:48) i ) + (cid:15) (cid:104) ψ j ( φ , k i + k (cid:48) i ) + ψ j ( φ , − k i − k (cid:48) i ) (cid:105)(cid:111) . (40)The delta function in φ is obtained because ˆ V ( φ J ) is as said before a density on the scalarfield space. In agreement with the results obtained when considering an exact homogeneous19ackground, the first term is naturally scale invariant with respect to the rod wavenumbers k i , and its scale depends only on the reference matter through the matter clock φ . Besides,in this very same term, the delta function in the momentum implies a deep connectionbetween scale invariance and translational invariance. The second term corresponds to firstdeviations from the scale invariant homogeneous mean-field, associated to inhomogeneousfluctuations, which naturally have small relative amplitude. In agreement with the usualcosmological perturbations, their shape must solve the condensate dynamics and they arefully determined in a two-fold manner by the coupling with the background on the one side,and by their own dynamics on the other one.A magnitude of particular interest can be defined by the amplitude of the volume fluc-tuations relative to the background; this is the quotient between (37) and the backgroundvolume (cid:104) V ( φ ) (cid:105) , both at a given ‘time’ φ . It is worth to note that the background volume isnot exactly the one given in Eq. (33) because, as mentioned in (36), it should be regularizedby the integral over the added matter rods φ i . Following this procedure, it is written as (cid:104) V ( φ ) (cid:105) = (cid:90) d φ i (cid:88) j V j | σ j ( φ ) | . (41)It is useful to remind that this magnitude is directly associated with the number of quantathat make up the condensate.To encompass first corrections, we should use at least the two correlation functions com-puted in (40). If this is so, this magnitude should be divided by the square of the backgroundvolume written above. Let us recall the previous analysis regarding states with a dominanceof a single spin j . Keeping the dominant part of the power spectrum of such perturbations,we have P δV ( k ) = V j (cid:12)(cid:12) σ j ( φ ) (cid:12)(cid:12) (cid:0)(cid:82) d φ i V j (cid:12)(cid:12) σ j ( φ ) (cid:12)(cid:12) (cid:1) = V j ( (cid:82) d φ i ) V ( φ ) . (42)In the particular cases where in the equation of motion (22) we have C j /B j <
0, the inho-mogeneous term is hence further suppressed. Therefore, for large volumes, inhomogeneousperturbations decay relative to the homogeneous background. Besides, if interacting GFTare considered [48], the obtained long-lasting accelerated expansion (after the bounce) isaccompanied by a further suppression of the deviations from scale invariance.Considering that in Eq. (42) V ( φ ) = N ( φ ) V j , the relative amplitude of these scalarperturbations decreases as ∼ /N ; i.e. they decrease with the growth of the number of quanta N while φ evolves and the universe expands. These scaling results are in agreement withthe typical relative size of fluctuations in a condensate. Analogously, these fluctuations arisenaturally in the GFT condensate approach, but within a quantum gravity theory for gravityand matter, which has a properly defined ultraviolet completion. The consistency of the GFT condensate cosmology has grown in the last couple of yearsbecause it could rely not only on the convergent results with models derived from LQG, butalso from Asymptotic Safety. An interesting consequence of Asymptotic Safety is that theoriginal four-dimensional spacetime undergoes a dimensional reduction in the short-distanceregime. Starting from the classical four-dimensional spacetime, the spectral dimension of the20emergent” effective spacetime varies with the energy scale and reaches the value d eff = 2in the ultraviolet limit [75].The same result has been obtained in other approaches to Quantum Gravity, such asHoˇrava–Lifshitz gravity [76], causal dynamical triangulation [77], LQG [78] and, quite re-cently, double special relativity [79]. All these approaches could describe different facets ofthe same quantum theory and GFT is not excluded from the list. For instance, a candidateultraviolet fixed point has been found for GFT [80]. This fixed point suggests the presenceof two distinct infrared phases and is associated to results supporting the existence of GFTphases of the condensate type.Another aspect that we would like to underline is the important role played by the EPRLGFT model for describing the effective cosmological dynamics in hydrodynamical terms fora specific class of condensate wave functions which encode the microscopic dynamics of theEPRL model [15].The key property of the GFT based on the EPRL spin foam model is that (i) the kineticterm contains a Kronecker delta between the j , m and intertwiner labels ı , and (ii) theinteraction term contains a Kronecker delta for the j labels colouring the links that meetin the interaction. The input from the EPRL model is in the combinatoric form of the j and m arguments in the field variables which is due to the presence of Kronecker deltafunctions in the interaction term. The functions encode a specific relation between the SU(2)representation labels and SL(2, C ) representations, as well as a condition of invariance of thesame functions under SL(2, C ). This specific relation between SU(2) and SL(2, C ) data isthe end result of the EPRL prescription for imposing the constraints reducing topologicalBF theory to gravity, and these are the conditions enforcing geometricity of the simplicialstructures of the model, see [25]. These are all aspects that are fundamental to obtain theeffective cosmological dynamics that is extracted from the microscopic quantum dynamicsof the model. Let us remark that for the time being, although these general argumentsprovide some restrictions on the possible terms that could appear in the GFT action (6), animportant open problem is to determine exactly which choices of K and V are required forgiving a good quantum gravity theory that recovers general relativity in the classical limit.Anyway, in order to translate the theory into a set of equations for cosmological observables,the addition of a scalar field variable is crucial, since it allows us to define within the fulltheory a set of relational observables with a clear physical meaning.Of course, the ways of including a scalar field as matter content is an open problemnot only in GFT but in any discrete quantum gravity model per se . The use of matterreference frames is not new; it dates back at least to DeWitt proposal [81] where coordinatesare proposed to be constructed with convenient matter scalar variables (in [82] an extendeddiscussion can be found). More contemporary advances have been obtained by using dustmatter to account for this effect. First insights have been proposed by Brown and Kuchaˇr [22]and generalizations to LQG have been developed in Ref. [83]. Relevant advances have beendone also by Gielen [46,84]. With regard to models constructed from the theories discussed inthis review, the employment of a massless scalar field as a relational clock defining relationaldynamics also appears in canonical LQG [85] and LQC [86, 87].To conclude, it is worth mentioning two different research lines studying the effectivemacroscopic dynamics that builds up the cosmological model. On the one hand, within theGFT condensate cosmology it is possible to realize an early era of geometric inflation. Thisis a period of accelerated expansion in absence of an inflaton field and its associated ad hoc potential. A detailed study of the condition for inflation can be found in [15]; however,in [88] it is argued that the number of e-folds computed for the free theory – V = 0 in the21ction (6)– suggests that such a geometric inflation cannot last sufficiently long to accomplishobservational data. This implies that GFT cosmology in absence of interactions betweenbuilding blocks cannot replace the standard inflationary scenario. The authors explore theimplications of including these interactions, which is indeed a more natural and consistentscenario, as the quanta of geometry should be somehow ‘glued’ with each other insteadof being in a sort of diluted gas regime of tetrahedra. Therefore, the results obtained in[48] may be able to give an alternative prescription on how to build a GFT model withspecific type of interactions, such that in the semiclassical limit the desired properties of ourhomogeneous and isotropic universe are obtained as an emergent 3–geometry. Interestingly,in the interacting case, one can find a range of the parameter space for which the inflationaryera last for sufficiently long. However, to obtain a successful scenario one needs to verify thatthere is no intermediate stage of deceleration between the bounce and the end of inflation.According to [55] a real-valued condensate field has solutions avoiding the singularity andalso growing exponentially after a bounce, if and only if the GFT energy is negative. Adiscussion on the possible values of the parameter space of the interacting potential can befound in [88], together with the stability properties of the evolving isotropic system, givingrise to effective continuous and homogeneous 3–geometries built from many smallest andalmost flat building blocks of quantum geometry. On the other hand, although the successfuluse of relational observables to extract an effective dynamics in the cosmological sector ofGFT, recent work argues that it may be possible to use a more general relational framework inGFT [89]. Indeed, particularly interesting consequences are derived when defining a relationalHamiltonian ˆ H generating the evolution with respect to the massless ‘clock’ φ [27]. The ideais to define a deparametrized setting in which some degrees of freedom serve as coordinatesparametrizing the remaining ones. For certain choices of ˆ H , the Fock vacuum state | (cid:105) is notan eigenstate and cannot be the vacuum of the relational Hamiltonian. Then, the ‘no-space’state is unstable and, given arbitrary initial conditions, quantum fluctuations will push thesystem away from | (cid:105) . As each quantum of geometry contributes with some spatial volume,the resulting state can be understood as a sort of expanding universe. According to thisreasoning, the instability of the ‘no-space’ state | (cid:105) provides a realization of geometrogenesis.More precisely, in the bounce picture, the state goes through the zero-volume state sincethe evolution in the other direction of the relational time is identical. Hence, the quantumdynamics does not break down at the zero-volume state but simply evolves over it. Moreover,the afore-mentioned instability in the Hamiltonian leads to a further cosmological implication:it provides a way to allow an unending expansion of the universe.Different directions can be taken starting from the state of the art. For instance, itwould be interesting to contrast the latter deparametrized framework for a single clock witha covariant setting in which one can choose different clocks, following the ideas of [90]. An-other potentially important generalization would be to examine whether non-vanishing scalarfield potentials can be taken into account when constructing this relational Hamiltonian. Asuccessful result would provide a GFT action depending explicitly on the relational timevariable [44]. Therefore it is in the cosmological sector of GFT where one should find whichchoice for ˆ H is the correct one. Nevertheless, there is a last open question that GFT leavesopen and that has further implications not just for physics. The ad hoc introduction of mass-less scalar fields as a standard of time works well at an effective level, but from a conceptualand fundamental standpoint this move can appear as an arbitrary one and implies more workalso from the philosophical perspective. 22 cknowledgements We would like to thank Marco De Cesare, Claus Kiefer and Daniele Oriti for fruitful dis-cussions and suggestions. The research leading to this paper has received funding from theEuropean Union’s Horizon 2020 research and innovation programme under grant agreementNo 758145.
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