Loop Quantum Cosmology: A cosmological theory with a view G A Mena Marug´an1, ∗ 1 Instituto de Estructura de la Materia, CSIC, Serrano 121, 28006 Madrid, Spain. Loop Quantum Gravity is a background independent, nonperturbative approach to the quantization of General Relativity. Its application to models of interest in 1 1 cosmologyandastrophysics,knownasLoopQuantumCosmology,hasledtonewand 0 2 exciting views of the gravitational phenomena that took place in the early universe, n a or that occur in spacetime regions whereEinstein’s theory predicts singularities. We J 0 provideabriefintroductiontothebasesofLoopQuantumCosmologyandsummarize 1 themostimportantresultsobtainedinhomogeneousscenarios. Theseresultsinclude ] c q amechanismtoavoid thecosmological BigBangsingularityandreplaceitwithaBig - r g Bounce, as well as the existence of processes which favor inflation. We also discuss [ the extension of the frame of Loop Quantum Cosmology to inhomogeneous settings. 1 v 8 3 PACS numbers: 04.62.+v,04.60.-m, 98.80.Qc 7 1 . 1 0 1 I. INTRODUCTION 1 : v i X Gravityremains astheonlyfundamental physical interaction which hasnot beensatisfac- r a torily described quantum mechanically. Leaving aside the belief that all interactions should be unified in a single theory, a strong motivation for a quantum theory of gravity comes from the singularity results of General Relativity [1]. In these classical singularities the predictability breaks down, indicating that the regime of applicability of Einstein’s theory has been surpassed and that a new and more fundamental description is needed. Apart from the ability to cure the classical singularities, one expects that a quantum theory of gravity would open a new window to physics, incorporating phenomena which originate in the quantum realm. Any candidate theory should make this compatible with ∗ Electronic address: [email protected] 2 the infrared behavior of General Relativity, so that no quantum effect alters much spacetime regions like those which we observe [2]. Explaining why our Universe is actually so (semi- )classical constitutes a real challenge to quantum gravity. Previous attempts to construct a quantum formalism for the metric fields (in geometrodynamics) employing a Wheeler-De Wittapproachhaveprovenunsuccessful. Inthegeneraltheory, theapproachfindsfunctional analysis andinterpretational obstacles whichprevent further progress. Ontheother hand, in simple models where these obstacles can be circumvented, the classical singularities are not fully resolved (e.g., semiclassical states arepeaked on trajectories where physical observables eventually diverge [3, 4]). Recently, a different approach to quantize General Relativity has been developed: the theory of Loop Quantum Gravity (LQG) [5, 6]. This theory is based on a nonperturbative canonical formalism of gravity and declares diffeomorphism invariance and background independence to be basic guidelines. Its application to simple cosmological models, generally homogeneous ones, gives rise to the new area of gravitational physics known as Loop Quantum Cosmology (LQC) [2, 7]. II. LOOP QUANTUM GRAVITY LQGisa canonical quantizationofGeneral Relativity, constructed starting froma Hamil- tonian formulation of Einstein’s theory. Let us begin by considering a globally hyperbolic spacetime. Introducing then a global foliation, the initial data for the construction of the Lorentzian spacetime metric are contained in the spatial 3-metric and the extrinsic curva- ture on the considered section of the foliation [8]. If one wants to introduce fermions in this framework, the spatial metric must be replaced with a triad, to which fermions couple directly. This coupling occurs in an internal su(2) index, and respects the invariance of the system under internal rotations, realized then as SU(2) gauge transformations. The spatial metric is recovered as the square of the triad, contracted in the internal indices by means of the Euclidean metric [the Killing-Cartan metric on su(2)]. A canonical set of variables to describe the gravitational phase space is formed, up to numerical factors, by the densitized triad (i.e., the triad multiplied by the square root of the determinant of the spatial metric), and the extrinsic curvature expressed in triadic form (i.e., the extrinsic curvature contracted with the triad in one spatial index). At this stage, one can replace the extrinsic curvature by a connection valued 1-form, tak- 3 ing values onthe algebra su(2). Classically, this Ashtekar-Barbero connection is a sum of the spin connection compatible with the co-triad and the triadic extrinsic curvature multiplied by an arbitrary positive number γ, known as the Immirzi parameter. By construction, this connection forms a canonical set with the densitized triad, modulo a factor 8πGγ in their Poisson brackets, where G is the Newton constant (from now on, we set the speed of light equal to the unity). The introduction of the su(2)-connection allows one to incorporate nonperturbative tech- niques in the description of the system, similar to those developed in gauge field theories, like e.g. for Yang-Mills fields. In particular, the gauge invariant information about the connection is captured in the so-called Wilson loops, constructed from holonomies which describe the parallel transport of spinors around loops. We hence replace the connection by SU(2)-holonomies along (piecewise analytic) edges, where we understand that an edge is an embedding of the closed unit interval in the considered spatial manifold. This replacement involves a 1-dimensional smearing of the connections, and renders the information about them gauge invariant except for the effect of transformations at the end points of the edges. By joining a finite number of edges in those vertices to form a graph [5], and combining the holonomies there so that SU(2) invariance is respected everywhere, one obtains what is usually called a spin network. It is worth noticing that the construction of the holonomies, whichcontainalineintegraloftheconnections, ismadewithoutrecurringtoanybackground structure. Since the most relevant field divergences in our formalism come from the appearance of a 3-dimensional delta function on the Poisson brackets between the connection and the densitized triad, and we have already smeared the connection over one dimension, it seems natural to smear the triad similarly, but now over two dimensions. Given that the densitized triad is a (spatial) vector density, this smearing can be carried out again without employing any background structure. For any (piecewise analytic) surface, we can define the flux of the densitized triad through it, obtaining the desired smearing. Holonomies and fluxes form an algebra under Poisson brackets, which we regard from now on as our basic algebra of functions on the gravitational phase space. From this perspective, the quantization of General Relativity amounts to the representation of this algebra as an algebra of operators acting on a Hilbert space. In addition, we must take into account that the system is subject to a series of constraints, which must be imposed quantum mechanically. These are the 4 Gauss [or SU(2)] constraint, the diffeomorphisms constraint, and the Hamiltonian or scalar constraint, which express the invariance of the system under SU(2) transformations, spatial diffeomorphisms, and time reparametrizations [5, 6]. An important result for the robustness of the predictions of LQG is a uniqueness theorem about the admissible representations of the holonomy-flux algebra, known as the LOST theorem (after the initials of its authors [9]). Specifically, this theorem states that there exists a unique cyclic representation of that algebra with a diffeomorphism invariant state (interpretable as a vacuum). In total, the choice of the algebra of elementary variables, motivated by background independence, and the status of spatial diffeomorphisms as a fundamental symmetry pick up a unique quantization, up to unitary equivalence and prior to the imposition of constraints. To gain insight into the kind of quantization adopted in LQG, let us consider the so- called cylindrical functions: complex functions that depend on the connection only via the holonomies along a graph, formed by a finite numbers of edges. Completing this algebra of functions with respect to the norm of the supremum, we obtain a commutative C -algebra ∗ withidentity, wherethe -relationisprovidedbythecomplexconjugation[10,11]. According ∗ to Gel’fand theory, this algebra is then (isomorphic to) the algebra of continuous functions onacertaincompactspace, calledthespectrum. Smoothconnectionsaredenseinthisspace. Besides, the Hilbert space of the representation is just a space of square integrable functions on this Gel’fand spectrum with respect to a certain measure. The LOST theorem guaranties that there exists only one diffeomorphism invariant measure which supports not only a representation of the holonomies, but of the whole holonomy-flux algebra: the Ashtekar- Lewandowski measure, used to construct the representation in LQG. This representation turnsoutnottobeequivalent toastandardone, andthereforeleadstophysical resultswhich are different from those of other, conventional quantizations. In fact, the representation is not continuous; as a consequence, the connection cannot be defined as an operator valued distribution [5]. Finally, the resulting quantum geometry is discrete, with area and volume operators that have a point spectrum [5]. 5 III. LOOP QUANTUM COSMOLOGY: THE FLAT FRW MODEL As a paradigmatic example in LQC, let us now apply this type of loop quantization techniques to Friedmann-Robertson-Walker (FRW) cosmologies (namely homogeneous and isotropic spacetimes) with flat spatial sections of R3 topology and a matter content provided by a homogeneous massless scalar field φ, minimally coupled to the metric [3, 4]. We introduce a fiducial triad and an integration cell, adapted to it, to carry out all integrations and avoid in this way divergences due to the homogeneity and noncompactness of the spatial sections. We call V the fiducial volume of this cell. It is possible to check that all physical 0 results are indeed independent of these choices of fiducial elements [3, 4]. Besides, one can fix the gauge freedom so that both the densitized triad and the connection become diagonal. Given the isotropy, they are then totally specified by one single variable each, which we call p and c, respectively. These variables describe the geometry degrees of freedom, vary only in time, and form a canonical pair: c,p = 8πγG/3. Classically, they are related with the { } scale factor a and its time derivative by the formulas p = V2/3a2 and c = γV1/3a˙. 0 0 To retain all the gauge invariant information about the su(2)-connection, taking into account the homogeneity, it suffices to consider holonomies along (fiducial) straight edges. Similarly, triads are now smeared across (fiducial) squares. The fluxes are then totally determined by the variable p. Returning to the holonomies, it is easy to check that, for an 1/3 edge of coordinate length λV in any fiducial direction, the matrix elements of the SU(2)- 0 holonomy are linear combinations of exponentials of the form e iλc/2. The corresponding ± configuration algebra is then the linear space of continuous and bounded complex functions in the real line (c ∈ R), with elements of the form f(c) = Pjfjeiλjc/2. It is well known that the completion of this algebra with the supremum norm is just the Bohr C -algebra ∗ of almost periodic functions [11]. Its Gel’fand spectrum is the Bohr compactification of the real line, R . This compactification can be seen as the set of group homomorphisms from Bohr the group of real numbers (with the sum) to the multiplicative group C of complex numbers of unit norm. Indeed, for every real number c we have a homomorphism x : R C of c → this kind, namely x (λ) = eiλc/2. Moreover, it is possible to see that the real line is actually c dense in R , using the fact that our initial configuration algebra separates points c R Bohr ∈ [10, 11]. The operation xx˜(λ) = x(λ)x˜(λ) provides a commutative group structure in R . Since Bohr 6 thegroupR iscompact, ithasa(unique)invariantHaarmeasure. ThefunctionsonR Bohr Bohr consisting in the evaluation at a real point µ form an orthonormal basis in the corresponding Hilbert space of square integrable functions with the norm defined by that measure [11]. We designate each element in this basis with a ket µ . This basis allows us to pass from our | i configuration representation, in which holonomies act by multiplication, to a “momentum” representation in which the triad has a multiplicative action [10, 11]. Calling N = eiλc/2, λ this “momentum” representation is given by pˆµ = (4πγG/3)µ µ and Nˆ µ = µ + λ λ | i | i | i | i (we set ~ = 1). Clearly, the basis µ ; µ R is uncountable, and therefore the Hilbert {| i ∈ } space is nonseparable. Nevertheless, normalizable states can get nonvanishing contributions only from a countable subset of states µ ; otherwise their norm would not be finite. This | i “momentum” representation is the one usually employed in LQC. It is worth remarking that the representation fails to be continuous, owing to the discrete norm on the Hilbert space, µ µ˜ = δµ˜. As a result, a connection operator does not truly exist, and the representation h | i µ is inequivalent to the Wheeler-De Witt one, in total parallelism with the situation found in LQG for the general case. Although homogeneity ensures that the diffeomorphisms constraint is satisfied, and the SU(2) constraint has been removed by gauge fixing, the system is still subject to a Hamilto- nian constraint, which must be imposed now quantum mechanically. In order to introduce a Hamiltonian constraint operator, there are essentially two building blocks which must ˆ be defined in terms of our elementary operators pˆ and N . First, we need an operator to λ represent the phase space function t(p) = sign(p)/p p . This function contains all the p- | | dependence of the (nondensitized) triad of the model. Note that this triad diverges at the Big Bang, where the variable p vanishes. Correspondingly, the operator representing t(p) cannot be defined exclusively from pˆ using the spectral theorem: pˆ has a point spectrum which contains the zero, and hence its inverse operator is not well defined. But it is possible to construct a regularized triad operator using commutators with holonomies, in addition to pˆ[12]: t(p) = 3(Nˆ pˆ1/2Nˆ Nˆ pˆ1/2Nˆ )/(4πγGµ¯). In principle, µ¯ may take any real d −µ¯| | µ¯ − µ¯| | −µ¯ value. It will be fixed later on in our discussion. The resulting operator is diagonal in the considered basis of µ-states. Furthermore, it turns out to be bounded from above, so that, in particular, the classical divergence disappears. Actually, our regularized operator annihilates the kernel of pˆ. The other block that we need is the SU(2)-curvature operator. Recall that the connection 7 operator is not well defined, therefore we cannot use it to construct the curvature. Nonethe- less, it is possible to determine it using a square loop of holonomies. We use again edges of 1/3 fiducial length µ¯V . Classically, the expression of the curvature would be recovered exactly 0 in the limit of zero area, when µ¯ tends to zero. However, this limit is not well defined in LQC. The idea is to shrunk the square up to the minimum physical area ∆ allowed in LQG, where the spectrum of the area operator is discrete [4]. This introduces a certain nonlocal- ity in the formalism, and turns the parameter µ¯ into a state dependent quantity, since the physical area depends on the particular state under consideration. Explicitly, the relation that must be satisfied for each state is µ¯2 p = ∆. At this stage, it is convenient to relabel | | the basis of µ-states as a basis of volume eigenstates, introducing an affine parameter v for the translation generated by µ¯c/2 [4]. By construction, we then get that Nˆ v = v + 1 . µ¯ | i | i The parameter v is related with the physical volume of the fiducial cell, V = p3/2, by the formula v = sign(p)V/(2πγG√∆). Finally, for the quantization of the matter field φ, we use the standard Schro¨dinger representation. So, the kinematical Hilbert space is the tensor product of the gravitational space of LQC and the standard one for matter. With a suitable factor ordering and choice of densitization [13], one then gets a Hamiltonian constraint of the form Hˆ = 6Ωˆ2 +Pˆ2, − φ where Pˆ is the momentum operator of the matter field, and acts by differentiation (namely, φ Pˆ = i∂ ). ThegravitationalpartoftheconstraintisgivenbytheoperatorΩˆ2. Remarkably, φ − this constraint leaves invariant the zero-volume state v = 0 , as well as its orthogonal | i complement. Therefore, the analog of the classical singularity is removed in practice from the Hilbert space, and we can restrict all physical considerations to its complement. In this sense, the singularity gets resolved at a kinematical level. Moreover, the operator Ωˆ2 has an action of the following type: Ωˆ2 v = f (v) v +4 +f(v) v +f (v) v 4 . Here, the real + | i | i | i − | − i functions f (v) and f (v) have the outstanding property that they vanish in the respective + − intervals [-4,0] and [0,4] [13]. Thus, the action of Ωˆ2 preserves each of the subspaces of the gravitational Hilbert space obtained by restricting the label of the v-states to any of the semilattices = v = (ε+4n); n N , where ε (0,4]. Then, each of these semilattices L±ε { ± ∈ } ∈ provides a superselection sector. In each sector, the orientation of the triad is definite (v does not change sign) and v has an strictly positive minimum, equal to ε. For concreteness, | | we choose sectors with v > 0 from now on. On the other hand, it is possible to show that, on each sector, the operator Ωˆ2 has 8 a nondegenerate absolutely continuous spectrum equal to the positive real line [13, 14]. Recalling its action, this gravitational constraint operator might be understood as a second- order difference operator. But its eigenfunctions are entirely determined by their value at ε, point from which they can be constructed by solving the eigenvalue equation. In this sense, thegravitational constraint operatorleadsto aNo-Boundarydescription, inwhich the eigenstates which encode the information about the quantum geometry are all determined without the need to introduce any boundary condition in the region around the origin. We also notice that, up to a global phase, these eigenfunctions eε(v) are real, since so is the δ gravitational constraint operator. With such eigenfunctions, one easily finds the solutions to the Hamiltonian constraint, which have the form ψ(v,φ) = R0∞dδeεδ(v)[ψ+(δ)ei√6δφ + ψ (δ)e−i√6δφ]. The scalar field φ − plays the role of an emergent time. Then, physical states can be identified (e.g.) with the positive frequency solutions ψ (δ) that are square integrable over the spectral parameter + δ R+ [13]. A complete set of Dirac observables is formed by Pˆ and vˆ , the latter being ∈ φ | |φ0 defined by the action of the volume operator when the field equals φ . On the Hilbert space 0 of physical states specified above, these observables are self-adjoint operators. IV. THE BIG BOUNCE In the previous section we have completed the quantization of the flat FRW with a homogeneous massless scalar field. In order to analyze the physical predictions of this quantum theory, we will consider now the evolution of (positive frequency) quantum states with a semiclassical behavior. We study Gaussian-like states which, at an instant φ = φ in 0 the region of large emergent times φ 1, are peaked on certain values of the elementary 0 ≫ observables of themodel, namely, the matter momentum, P = P0, andthephysical volume, φ φ v = v0. We restrict our attention to states with large values of P0 and v0 [3, 4]. The φ numerical analysis of the quantum evolution unveils an outstanding phenomenon in these states: the Big Bang singularity is resolved dynamically and is replaced by a bounce that connects the universe with another branch of the evolution, dictated again by the equations of General Relativity. This mechanism to elude the cosmological singularity is known as the Big Bounce [3, 4]. The numerical studies show that the considered semiclassical states remain peaked on a 9 well defined trajectory during the whole evolution. On these states, the Big Bounce does not occur in a genuinely quantum region where one were to loose an effective notion of geometry and spacetime. The trajectory deviates from the one predicted by General Relativity only when the matter energy density ρ becomes of the order of one percent of a critical density, ρ . This scale for the onset of corrections is of the Planck order and universal: it is the crit same for all the semiclassical states which suffer the bounce. Explicitly, the critical density is ρ = (√3/32π2γ3G2) 0.41ρ , where ρ is the Planck density. For densities crit Planck Planck ≈ close to the critical one, i.e., in the regime close to the bounce, gravity behaves as a repulsive force owing to the effects of quantum geometry [2]. In addition, the trajectory followed by the peak of these states matches an effective dynamics, which has been deduced in detail (under certain assumptions on the family of states under consideration) using techniques of geometric quantum mechanics [15]. The agreement between the numerical simulations and the predictions of this effective dynamics is remarkable. In particular, the effective dynamics predicts a bounce precisely when the matter density reaches the critical value ρ . Further support to the role played by this crit critical density comes from the study of the operator which represents the matter density in the quantum theory. It is possible to prove that it has a bounded spectrum, the bound being given again by ρ . Then, the overview picture is that the emergence of important crit quantum geometry effects in this model is controlled by the value of the matter energy density. When this density approaches the Planck scale, quantum geometry phenomena enter thescene, preventing thatitkeeps onincreasing andconsequently avoiding thecollapse into a cosmological singularity. It is worth pointing out that these quantum phenomena can be relevant even in regions which one would not consider to belong to the deep Planck regime. For instance, the volume v at the bounce is proportional to the value of the matter field momentum, which is conserved in the evolution. Hence, when the bounce occurs, the volume can be as large as desired. The presence of a quantum bounce is actually generic in this quantum model, with implications that exceed the restriction to the discussed class of semiclassical states. We have already commented that the eigenfunctions of the gravitational constraint operator are real (up to a global phase). Studying the Wheeler-De Witt limit of this operator, one can prove that its eigenfunctions lead in fact to positive frequency solutions with ingoing and outgoing components of equal amplitude in this limit [13] (see [16] for the case of a specific 10 superselection sector). Furthermore, the Big Bounce mechanism is not restricted just to the flat FRW model with a massless scalar field, but is rather general. On the one hand, assuming the validity of the effective dynamical equations for other matter contents (assumption that is supported by the numerical analyses carried out so far), one can show that all strong singularities (´a la Kr´olak) are resolved in flat FRW for any kind of matter [17]. Only Type II and Type IV singularities may remain [17], but these singularities can be considered physically harmless, since geodesics can be extended beyond them (then, sufficiently strong in-falling detectors can survive these singularities). On the other hand, similar conclusions about the occurrence of the Big Bounce have been reached in other FRW models quantized in LQC. These include the flat model with negative cosmological constant [18], the closed model [19], the open model [20] (some problems of the treatment presented in that reference can be solved with the techniques of [21]), and the flat model with positive cosmological constant (recently studied by Ashtekar and Pawl owski), all of them with a homogenous scalar matter field present as well. For the flat FRW universe with negative cosmological constant and the closed FRW model, the classical evolution leads to a Big Crunch (i.e., the universe recollapses into a cosmological singularity). In the quantum theory, this Big Crunch is also resolved via a Big Bounce, like in the case of the Big Bang. In all cases, there exists an upper bound for the matter energy density, which is given again exactly by ρ , and the crit infrared regime shows an outstanding agreement with General Relativity. In spite of some statements that have appeared in the literature of LQC [17], the effective equations for flat FRW in the presence of generic matter do not necessarily lead to an asymptotically de Sitter behavior if a vanishing or a divergent value of the scale factor a were to be approached in the evolution, without further assumptions. The confusion comes a from the consideration of the identity ln(ρ/ρ ) = R [1 + w(a˜)]da˜/a˜, deduced from the 0 a0 conservation equation for the matter energy density, and where a is a reference value for 0 the scale factor, ρ = ρ(a ), and w(a) is the ratio between the pressure and the energy 0 0 density of matter. In fact, the convergence of the above integral when a does not → ∞ need that w(a) tend to minus the unity [22] (the value that would correspond to a de Sitter regime). Besides, even if the energy density is required to be positive and bounded from above by the critical density, one may have a vanishing limit for it. Then, the considered integral would diverge in that limit, allowing for values of w different from minus one [22].