Quantum Reduced Loop Gravity
aa r X i v : . [ g r- q c ] J un Quantum Reduced Loop Gravity
Emanuele Alesci * and Francesco Cianfrani *** Instytut Fizyki Teoretycznej, Uniwersytet Warszawski, ul.Pasteura 5, 02-093 Warszawa, Poland, EU, E-mail:[email protected] ** Institute for Theoretical Physics, University of Wrocław, PlacMaksa Borna 9, Pl–50-204 Wrocław, Poland., E-mail:[email protected] 23, 2018
Abstract
Quantum Reduced Loop Gravity provides a promising framework fora consistent characterization of the early Universe dynamics. Inspired byBKL conjecture, a flat Universe is described as a collection of BianchiI homogeneous patches. The resulting quantum dynamics is describedby the scalar constraint operator, whose matrix elements can be analyt-ically computed. The effective semiclassical dynamics is discussed, andthe differences with Loop Quantum Cosmology are emphasized.
Early cosmology is a privileged arena in which our quantum theories ofthe gravitational field can be developed and, eventually, tested. The stan-dard description of the global behavior of our Universe is given in termsof homogeneous and isotropic solutions of Einstein equations in the pres-ence of matter field, i.e.
FRW models. This description is appropriatein the present epoch at scales bigger than 100 Mpc, while at smallerscales the presence of structures (galaxies and cluster of galaxies) cannotbe ignored. Going backward in time, the density contrast of the Uni-verse is smaller and smaller, thus homogeneity and isotropy holds at moreand more scales. The homogeneity and isotropy of the cosmic microwavebackground radiation spectrum provides the most convincing test for thestandard cosmological model at recombination (when the averaged energyof the thermal bath is of the order of hydrogen atom binding energy). Inearlier stages, inflation is expected to occur and it provides a partial ex-planation for homogeneity in later phases. Moreover, the fluctuation ofthe inflaton field during inflation are responsible for the patter of fluctu-ations observed in CMB spectrum. Hence, the inflationary scenario canbe tested. f one goes further back in time and retains the assumptions of ho-mogeneity and isotropy, then the initial Big Bang singularity is reachedand at that time General Relativity is not predictive anymore. A morerealistic scenario is realized skipping both homogeneity and isotropy. Infact, the causal horizon shrinks faster than the cosmological horizon to-wards the singularity, such that soon or later the Universe is a collectionof not-causally-connected sub-Universes. The mathematical treatment ofthis phase is given via the BKL conjecture [1], in which each sub-Universeis a proper homogeneous space, Bianchi type IX. However, each BianchiIX space exhibits chaotic behavior towards the singularity.Therefore, the primordial phase of the Universe cannot be describedin terms of the standard paradigm for cosmology, but it is expected to bethe realm of Quantum Cosmology, which in turn is expected to descendfrom a complete Quantum Gravity theory.Loop Quantum Gravity (LQG) [2] stands among the most promisingquantum theories of gravity, even though a complete dynamic analysis isstill missing. The standard cosmological implementation is Loop Quan-tum Cosmology (LQC) [3], which realizes the quantization of FRW models(minisuperspace quantization) using loop techniques. The main implica-tion of LQC is the replacement of the initial singularity with a bounce.Furthermore, quantum corrections to the Universe dynamics affect thepropagation of inflaton fluctuations during inflation and could providetestable modifications to the CMB spectrum [4].An alternative description of the primordial Universe is given by Quantum-reduced Loop Gravity (QRLG) [5, 6, 7]. Since the homogeneity assump-tion cannot really hold on a quantum level, the idea is to provide a quan-tum description for a flat Universe as a collection of Bianchi I patches,along the same lines as when the BKL conjecture holds. This means re-stricting the metric to the inhomogeneous extension of Bianchi I modeland then neglecting the interaction between different patches in the dy-namic analysis. The latter corresponds to neglecting spatial gradients ofthe metric with respect to time derivatives and in the original BKL formu-lation this can be done close to the singularity. Here, the same assumptionis expected to hold both close to the singularity (but a direct proof hasnot been given yet) and in the nearly-homogeneous case, in which spatialgradients are obviously negligible.Such a restriction on the metric is realized in the kinematical Hilbertspace of LQG and, together with a gauge-fixing of the internal SU(2)group, it sets up the kinematical sector of QRLG. The dynamics is de-scribed by neglecting spatial gradients, which implies that the Hamilto-nian is proportional to the euclidean scalar constraint, whose expressioncan be quantized such that the associated matrix elements can be com-puted analytically. The semiclassical limit of the Hamiltonian defines theeffective dynamics, which is studied to infer the phenomenological impli-cations of QRLG [8, 9]. Kinematical Hilbert space
QRLG provides a quantum framework for the investigation of the dy-namics of the inhomogeneous Bianchi I model, which is described by thefollowing line element ds = N ( t ) dt − a ( t, x )( dx ) − a ( t, x )( dx ) − a ( t, x )( dx ) , (1)in the limit in which the spatial gradients of the scale factors a i ( i = 1 , , )are negligible with respect to time derivatives (BKL-like limit).Furthermore, by gauge-fixing internal rotations the triads are set tobe exactly diagonal, such that inverse densitized triads E ai and Ashtekarconnections are diagonal too, i.e. E ai = ( ℓ ) − p i δ ai , | p | = ℓ a a | p | = ℓ a a | p | = ℓ a a (2) A ia = ( ℓ ) − c i δ ia + .. , c i = γℓ N ˙ a i , (3) γ being the Immirzi parameter, while ˙ a i denotes the time derivatives of a i and ℓ is the coordinated length of the considered spatial region. Indeed, E ai can be exactly taken as diagonal, while A ia generically contain some off-diagonal terms (the dots in (3)), related with the spatial spin connectionsand proportional to the spatial gradients of the scale factors, which canbe neglected in the BKL limit.The associated Hilbert space is made of elements based at the cuboidalgraphs Γ whose links l i are parallel to the fiducial vectors ~δ i = δ ai ∂ a . Ateach link we deal with the following basis elements l D j l m l m l ( h l ) = h j l , m l | R − l D j l ( h l ) R l | j l , m l i m l = ± j l , (4) D j l ( h l ) being Wigner matrices associated with the SU(2) group elementbased at l , while R l denotes the rotation mapping the tangent unit vectorto l , ~δ l , in ~δ . For m l = j l , ~δ l is choosen with the same orientation as l ,while for m l = − j l ~δ l has the opposite orientation with respect to l .A basis element based at a generic graph Γ is constructed as follows h h | Γ , m l , x n i = Y n ∈ Γ h j l , x n | m l , ~δ l i Y l l D j l m l m l ( h l ) , m l = ± j l (5)where the products Q n ∈ Γ and Q l extend over all the nodes n ∈ Γ andover all the links l emanating from n , respectively. The intertwiners h j l , x n | m l , ~ u l i are merely coefficients which can be obtained by projectingthe standard SU(2) intertwiner basis elements | j l , x n i in Livine-Speziale[10] coherent states | m l , ~δ l i = Π l | m l , ~δ l i based at the links l .The action of holonomy operators on basis elements is given by usingU(1) recoupling theory at each link, while fluxes operators ˆ E i ( S i ) readthe magnetic indexes at dual links l = l i : ˆ E i ( S i ) l D j l m l m l ( h l ) = 8 πγl P m l l D j l m l m l ( h l ) l i ∩ S i = ⊘ , (6) l P being Planck length.The residual symmetry on a kinematical level is reduced diffeomor-phisms invariance, i.e. the invariance under those transformations actingalong one of the fiducial directions having constant parameters along theothers. It is imposed by constructing the associated s-knots according withthe procedure adopted in LQG to implement background independence. Quantum and effective dynamics
The classical Hamiltonian describing the dynamics of the metric (1) withinthe BKL approximation scheme is the sum over all points of that ofBianchi I model, for which the Euclidean scalar constraint is proportionalto the Lorentzian one. Hence, the full Hamiltonian is proportional to thesum over all points of the Euclidean scalar constraint, written in terms ofholonomies and fluxes of QRLG. The resulting expression can be quan-tized by considering a graph-dependent cubulation of the spatial manifoldand the associated operators reads ˆ H [ N ] := 2 i πγ l P X N ( n ) ǫ ijk Tr h R ˆ h α ij R ˆ h − s k (cid:2) R ˆ h s k , R ˆ V (cid:3)i C ( m ) = 2 i πγl P , (7) N ( n ) being the lapse function at the node n . The sum in the expres-sion above extends over all the nodes and over all the cubulations. Theholonomies h are in the fundamental representation and they are based atlinks, which belong to the graph at which the states are based. In partic-ular, s k and α ij denote a link and a square of the graph emanating fromthe considered node. The operator ˆ V is the volume operator of the regiondual to the node and from (6) one can see how it just reads the square rootof the magnetic indexes along the fiducial directions. Therefore, the ma-trix elements of the Hamiltonian operator among basis elements (5) canbe analytically computed and the whole dynamic problem can be solvedin QRLG.The effective dynamics has been analyzed by discussing the dynamicsgenerated by the expectation value of ˆ H over semiclassical states. Thesesemiclassical states have been constructed starting from coherent statesat each link l as follows ψ α Γ H ′ = X m l Y n ∈ Γ h j l , x n | m l , ~ u l i ∗ Y l ∈ Γ ψ αH ′ l ( m l ) h h | Γ , m l , x n i , (8)in which the functions ψ αH ′ l ( m l ) are labeled by H ′ l = h l e α πγl P E ′ l τ l andthey are peaked around the classical values h l and E l for the holonomyand the dual flux E l along the links l of the graph Γ .The semiclassical analysis provides the following expectation value forthe Hamiltonian operator (we neglected the correction due to the finitespread of the semiclassical wave packet) h ˆ H i N ≈ γ N (cid:18) N N p p p p p + p − p p − p p sin c N sin c N ++ N N p p p p p + p − p p − p p sin c N sin c N ++ N N p p p p p + p − p p − p p sin c N sin c N (cid:19) . (9)where p i = 4 πγ NN i l P , (10) i being the total number of nodes along the fiducial direction i withineach homogeneous patch. It is worth noting how for p i ≫ p i and c i ≪ N i the leading order term coincides with the Hamiltonian for the inhomo-geneous extension of the Bianchi I model, thus reconciling our quantumformulation with the classical limit.We see how generically there are two kind of corrections. Holonomycorrections are due the terms N i sin c i N i . They retain the same form asthe analogous ones in LQC as soon as one identifies the regulator µ i with the inverse numbers of nodes along the direction i , ı.e. µ i = 1 /N i .Since in LQC such corrections are responsible for the bounce, our effectivesemiclassical dynamics predicts a bouncing scenario too.Inverse volume corrections comes from the next-to-the-leading-orderterms in the expansion of the functions √ p i + p i − √ p i − p i p i for p i ≫ p i . Inparticular, one gets p p i + p i − p p i − p i p i = 1 p p i (cid:18) p i p i (cid:19) + O (cid:18) p i p i (cid:19) ! , (11)from which one sees how the corrections are enhanced with respect tothose of LQC, because of the factor N in the right-hand side of (10). QRLG provides a framework for the quantum description of the earlyUniverse. The quantum scalar constraint within the adopted approxima-tion scheme can be analytically defined. The effective dynamics resemblesthat of LQC. In particular, holonomy corrections coincide once the reg-ulators are identified with the numbers of nodes, as in lattice refinement[11], while inverse volume corrections are enhanced. Therefore, the re-placement of the initial sigularity with a bounce is still predicted, whilestronger corrections are expected to occur when analyzing the behaviorof perturbations.Furthermore, since in QRLG one has a complete quantum descrip-tion of the Universe, not restricted to minisuperspace, one can use looptechniques in the quantization of the matter sector. As a consequence, ad-ditional modifications with respect to LQC are to be found in the presenceof matter fields. These will be the subject of forthcoming investigations.The work of FC was supported by funds provided by the NationalScience Center under the agreement DEC12 2011/02/A/ST2/00294. Thework of E.A. was supported by the grant of Polish Narodowe CentrumNauki nr 2011/02/A/ST2/00300.
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