Inflationary spectra and observations in loop quantum cosmology
aa r X i v : . [ g r- q c ] M a y Inflationary spectra and observationsin loop quantum cosmology
Gianluca Calcagni
Max Planck Institute for Gravitational Physics (Albert Einstein Institute)Am M¨uhlenberg 1, D-14476 Golm, GermanyE-mail: [email protected]
Abstract.
We review some recent progress in the extraction of inflationary observables inloop quantum cosmology. Inverse-volume quantum corrections induce a growth of power in thelarge-scale cosmological spectra and are constrained by observations.
Observational implications of quantum gravity present a delicate issue. Corrections tothe general relativistic dynamics are expected to arise in different ways. For instance, loopcorrections are always present in perturbative graviton field theory, which can be captured ineffective actions with higher-curvature corrections to the Einstein–Hilbert action. The additionalterms change the Newton potential as well as the cosmological dynamics. However, in currentlyobservable regimes the curvature scale is very small, and so one expects only tiny corrections of(adimensional) comoving size at most ℓ Pl H , where ℓ Pl is the Planck length and H − = a/a ′ isthe comoving radius of the Hubble region ( a is the scale factor in the flat Friedmann–Lemaˆıtre–Robertson–Walker, FLRW, background and primes denote derivatives with respect to conformaltime τ ). In such cases, tests of quantum gravity are possible at best indirectly, for instance if itprovides concrete and sufficiently constrained models for inflation. So far, however, models donot appear tight enough.In background-independent frameworks such as loop quantum gravity (LQG) [1], strongermodifications of the theory are possible since the usual covariant continuum dynamics isgeneralized, and entirely new effects may be contemplated. In LQG, gauge transformationsas well as the dynamics are generated by constraint equations. Since the latter are modifiedwith respect to the classical constraints, gauge transformations change and new spacetimestructures become apparent. Some of these modifications arise as follows. An inverse ofmetric operators is needed to construct the quantized constraints, but this inverse does notexist because metric operators have discrete spectra containing zero as an eigenvalue. Certainquantization procedures allow one to construct suitable densely-defined operators which giverise to quantum corrections sensitive to the discreteness scale. The modifications are controlledby the requirement that covariance not be broken but at most deformed with respect to itsordinary incarnation, leading to anomaly-free and consistent sets of equations. The resultingdynamics and gauge-invariant observables provide the basis for a cosmological analysis.Inverse-volume corrections constitute the first example that has been consistentlyimplemented, when they are small, in the dynamics of loop quantum cosmology (LQC [2]).Effective dynamical equations with inverse-volume corrections are known for pure FLRW andlinear perturbations in all sectors (scalar, vector and tensor) [3, 4, 5, 6]. The setting is annflationary era driven by a slowly rolling scalar field, the only difference with respect to thestandard case being the presence of the quantum corrections. For instance, the backgroundequations of motion are H = (8 πG/ α [ ϕ ′ / (2 ν ) + a V ( ϕ )] and ϕ ′′ + 2 H (1 − d ln ν/ d ln a ) ϕ ′ + νa V ,ϕ = 0, where V is the scalar field potential and α ≈ α δ Pl , ν ≈ ν δ Pl , δ Pl := ( a ∗ /a ) σ (1)are the inverse-volume quantum corrections. These are parametrized as an inverse power lawof the scale factor. In pure mini-superspace (no inhomogeneities), the parameters α and ν are calculable from the quantization ambiguities of the Hamiltonian constraint and also dependon σ , a constant determined by the way an elementary holonomy cell evolves with the Hubbleexpansion. A natural choice is the so-called ‘improved quantization’ scheme [7], where σ = 6and α ∼ O (0 . ∼ ν . In a flat or open universe, the Hamiltonian formalism is well defined onlyin a finite comoving volume V , which is chosen ad hoc . However, the quantum corrections areof the form δ Pl ∼ ( ℓ / V ) σ/ a − σ , and they depend explicitly on the unphysical fiducial volume(or, in other words, on the constant a ∗ which can be rescaled arbitrarily).Switching on inhomogeneities, the interpretation of inverse-volume corrections and thefiducial-volume problem change considerably. Once a closed constraint algebra and the full set oflinearized perturbed equations are obtained, one can study the role of different parametrizationschemes and a number of other issues. We mention the conservation law for the curvatureperturbation, the construction of the inflationary spectra and other cosmological observablesand, finally, the extraction of observational constraints from large-scale data. These aspects,analyzed in [6, 8, 9] will be presently reviewed here. Another characteristic effect of LQCquantization, holonomy corrections, should also enter the picture, but so far the closure of theconstraint algebra has been obtained only in the tensor and vector sector [3, 10]. Consequently,the phenomenology of holonomy effects has been limited to the structure of the primordialbounce and to the inflationary tensor signal [11].The first step is to implement first-order perturbation theory in the classical constraints. Thefundamental canonical variables, the densitized triad and the Ashtekar–Barbero connection aregiven by a background contribution plus a linear inhomogeneous correction: E ai = pδ ai + δE ai , A ia = cδ ia + (cid:0) δ Γ ia + γδK ia (cid:1) , where indices i = 1 , , a = 1 , , p = | a | , c = γ ˙ a classically, γ is the Barbero–Immirzi parameter, Γ isthe spin connection and K is the extrinsic curvature. Perturbations obey the canonical Poissonrelation { δK ia ( x ) , δE bj ( y ) } = 8 πGδ ba δ ij δ ( x , y ). The smeared effective Hamiltonian constraint withinverse-volume correction function reads H [ N ] ∼ R d xN [ α ( E ) H g + ν ( E ) H π + ρ ( E ) H ∇ + H V ],where N is the lapse function and different contributions H g,π, ∇ ,V pertain the gravitationalsector and the scalar field kinetic, gradient, and potential terms, respectively. Similarly, oneconsiders the perturbed Gauss and diffeomorphism constraints, and imposes closure of theeffective constraint algebra, { C α , C β } = f γαβ ( A, E ) C γ . The resulting perturbed equationscontain counterterms which fix the functions α , ν and ρ and guarantee anomaly cancellationin the constraint algebra [3, 5]. These counterterms only depend on the three parameters α , ν and σ , but a consistency condition further reduces the parameter space to two dimensions.Notably, inflationary and de Sitter background solutions exist for 0 . σ . σ = 6. This problem is bypassed in the latticerefinement framework, where the proper fiducial volume V = V a is replaced by the volume L = V / N of a microscopic cell of size L , determined by the number of vertices (or ‘patches’)of an underlying discrete state. The number of cells can be parametrized as N = N a n , where0 ≤ n ≤ / N and L must not decrease with the volume in a discrete geometricalsetting). Then, δ Pl := ( ℓ Pl /L ) m = ( ℓ N / V ) m/ a − (1 − n ) m =: ( a ∗ /a ) σ , where m is a positiveparameter. Eventually, one can argue that the parameter range is σ ≥ α ≥ ν ≥ δϕ in the metric and in the scalar field generate the gauge-invariantcurvature perturbation on comoving hypersurfaces R = Ψ + ( H /ϕ ′ )(1 − σν δ Pl / δϕ . Atlarge scales, this quantity is conserved thanks to a delicate cancellation of counterterms [6]: R ′ = [1 + ( α / ν ) δ Pl ][ H / (4 πGϕ ′ )]∆Ψ, where ∆ is the Laplacian. Because of this property,one can argue (and also rigorously show) that the Mukhanov scalar variable u = z R , where z := ( aϕ ′ / H )[1 + ( α / − ν ) δ Pl ], obeys the simple dynamical equation u ′′ − ( s ∆ + z ′′ /z ) u = 0,where s ( α , ν , σ ) is the propagation speed of the perturbation. According to the inflationaryparadigm, observables are expanded in terms of small slow-roll parameters ǫ := 1 − H ′ / H , η := 1 − ϕ ′′ / ( H ϕ ′ ), ξ := H − ( ϕ ′′ /ϕ ′ ) ′ + ǫ + η −
1. Asymptotic solutions to the Mukhanovequation at large scales, evaluated at horizon crossing, eventually yield the scalar spectrum, thescalar index and its running: P s := k π z (cid:10) | u k ≪H | (cid:11) (cid:12)(cid:12)(cid:12) k | τ | =1 = Gπ H a ǫ (1 + γ s δ Pl ) , (2) n s − P s d ln k = 2 η − ǫ + σγ n s δ Pl , (3) α s := d n s d ln k = 2(5 ǫη − ǫ − ξ ) + σ (4˜ ǫ − σγ n s ) δ Pl = O ( ǫ ) + O ( σδ Pl ) , (4)where γ s and γ n s depend on the parameters α , ν , σ in a precise way [6]. One can notice alarge-scale enhancement of power via the term δ Pl ∼ a − σ ∼ (1 / | τ | ) − σ ∼ k − σ . If large enough,quantum corrections dominate and α s = σf s ( α , σ ) δ Pl , where f s is a specific function of theparameters. Bounds on the scalar running, in fact, turn out to be the main constraint on theparameters. The power spectrum can be expressed in terms of the comoving wavenumber k ofperturbations and of a pivot scale k , which must be chosen within the scale range probed bythe given experiment. Letting x := ln( k/k ), one has P s ( k ) P s ( k ) = exp (cid:26) [ n s ( k ) − x + α s ( k )2 x + f s δ Pl ( k ) (cid:20) x (cid:18) − σx (cid:19) + 1 σ ( e − σx − (cid:21)(cid:27) . (5)Due to cosmic variance, there is an intrinsic uncertainty in the determination of the spectrumat large scales (small multipoles ℓ ), which should be compared with the strength of the typicalsignal from quantum corrections. Tensor observables can be calculated analogously and displaythe same type of corrections. From those, one can extract a consistency relation between scalarand tensor perturbations: r = − { n t + [ n t ( γ t − γ s ) + σγ t ] δ Pl } .In order to compare with observations, one can choose an inflationary potential and recastall observables as its functions. Then, for any given choice of σ , one can find an upper boundfor the quantum parameter δ := α δ Pl (we recall that ν is not independent). An example oflikelihood contours and power-spectrum enhancement is illustrated in figure 1, for a quadraticpotential. We use the 7-year WMAP data combined with large-scale structure, the Hubbleconstant measurement from the Hubble Space Telescope, Supernovae type Ia and Big BangNucleosynthesis. For a range of e-folds between 45 and 65, the probability distribution of ǫ is consistent with the theoretical range 0 . < ǫ < . not the case for σ . σ >
2, even tighterbounds are obtained, corresponding to practically unobservable inverse-volume effects. The 95%-confidence-level upper limits of δ constrained by observations for the potential V ( ϕ ) = V ϕ ,with k = 0 .
002 Mpc − and different values of σ , are [9] δ ( σ = 0 .
5) = 0 . , δ ( σ = 1) = 3 . × − , δ ( σ = 1 .
5) = 1 . × − ,δ ( σ = 2) = 6 . × − , δ ( σ = 3) = 4 . × − . (6) igure 1. Left: Combined marginalized distribution for the quantum-gravity parameter δ ( k ) = α δ Pl ( k ) and the slow-roll parameter ǫ ( k ) with the pivot k = 0 .
002 Mpc − for V ∝ ϕ and σ = 2. Internal and external solid lines correspond to the 68% and 95% confidencelevels, respectively. Right: Primordial scalar power spectrum P s ( ℓ ) for the same potential andpivot wavenumber (corresponding to ℓ = 29), with three different values of δ ( k ): 0 (dottedline), 7 × − (experimental upper bound, solid line), 4 . × − (from the a-priori upper bound δ Pl ( ℓ = 2) < .
1, dashed line). The shaded region is affected by cosmic variance [8, 9]. ε V (k ) δ ( k ) −4 { P s H { L (cid:144) P s H { L The likelihood analysis has not been performed for σ = 6 since the signal is below the cosmicvariance threshold already when σ = 2. For σ = 3, the parameter δ = ν δ Pl has been usedinstead. The analysis of [9] includes also different potentials and values of the pivot scale.To conclude, observations highlight a tension between mini-superspace parametrizations ofFLRW LQC ( σ >
1) and lattice refinement parametrizations. The latters are the only onecompatible (at least in this model) with anomaly cancellation in inhomogeneous LQC and power-law inflationary solutions. Tight upper bounds can be obtained for inverse-volume quantumcorrections; their improvement with future missions such as Planck will further constrain theparameter space of the theory and, hopefully, stimulate our understanding of the semi-classicallimit of loop quantum cosmology.
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