Inflationary power spectra with quantum holonomy corrections
PPrepared for submission to JCAP
Inflationary power spectra withquantum holonomy corrections
Jakub Mielczarek
Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Cracow, PolandDepartment of Fundamental Research, National Centre for Nuclear Research,Hoża 69, 00-681 Warsaw, PolandE-mail: [email protected]
Abstract.
In this paper we study slow-roll inflation with holonomy corrections from loopquantum cosmology. Both tensor and scalar power spectra of primordial perturbations arecomputed up to the first order in slow-roll parameters and
V /ρ c , where V is a potential ofthe scalar field and ρ c is a critical energy density (expected to be of the order of the Planckenergy density). Possible normalizations of modes at short scales are discussed. In case thenormalization is performed with use of the Wronskian condition applied to adiabatic vacuum,the tensor and scalar spectral indices are not quantum corrected in the leading order. However,by choosing an alternative method of normalization one can obtain quantum corrections inthe leading order. Furthermore, we show that the holonomy-corrected equation of motion fortensor modes can be derived from an effective background metric. This allows us to provethat the Wronskian normalization condition for the tensor modes preserves the classical form. a r X i v : . [ g r- q c ] N ov ontents Ω − deformed Minkowski vacuum 73.2 Bojowald-Calcagni normalization 83.3 Deformed Wronskian condition 9 Ω = const case 125.2 Ω − deformed Minkowski vacuum 135.3 Bojowald-Calcagni normalization 14 Ω − deformed Minkowski vacuum 166.2 Bojowald-Calcagni normalization 17 Effects of the quantum nature of space at the Planck scale predicted by loop quantum gravity(LQG) [1] can be studied by introducing appropriate modifications at the level of the classicalHamiltonian. This so-called effective approach enables to relate some quantum gravitationalphenomena with the realm of classical physics, which proved to be especially fruitful in thecosmological context, known as loop quantum cosmology (LQC) [2, 3].As discussed in Ref. [4], the effective approach in quantum gravity is conceptuallysimilar to the effective approach in solid state physics. Namely, while calculations based onmany-body Hamiltonian are extremely difficult to execute, there is a whole range of effectivemodels enabling explanation of macroscopic phenomena in terms of atomic-scale physics.As an example, relevant for our further discussion, let us refer to the nature of refractiveindex n . In the effective model of frequency dependence of n , one considers a single atomicdipole interacting with electromagnetic plane wave. By virtue of homogeneity of a sample theformula for n ( ω ) , characterizing macroscopic bulk, can be derived. The formula depends onknown microscopic quantities as electron mass and elementary charge, but also contains somecharacteristic frequencies which cannot be derived from the model. The unknown values canbe either fixed experimentally or derived from quantum mechanical computations.In LQG, there exists an analogue of the many-body Hamiltonian in solid state physics.Action of this so-called Hamiltonian constraint on the spin network states, describing the– 1 –tate gravitational field, is however not fully understood yet. When these difficulties are over-come it will be possible to study some macroscopic or mesoscopic gravitational configurationsnumerically. This is in analogy to computations performed within condensed matter physicsor quantum chemistry. Meanwhile, the effective approach, competitive with the first-principlecomputations, can be utilized.Construction of effective models is facilitated by certain assumptions regarding sym-metries of space. Here, we focus on homogeneous and isotropic background geometry de-scribed by the flat Freidmann-Robertson-Walker (FRW) metric on which inhomogeneitiesare considered perturbatively. Such setup is sufficient to study the generation of primordialperturbations during the phase of slow-roll inflation.Incorporation of LQG effects into cosmological models is performed by taking into ac-count two types of corrections: inverse volume corrections and holonomy corrections. Bothcorrections reflect discrete nature of space at the Planck scale, however in a different man-ner. While strength of inverse volume corrections depends on volume element, holonomycorrections are sensitive to energy density. In case of inverse volume corrections, equations ofmotion for perturbations were derived in Ref. [5]. Based on this, corrections to the inflation-ary power spectra were derived in Ref. [6]. The corrections were shown to be consistent withthe 7-year WMAP data [7, 8].In this paper we focus on derivation of holonomy corrections to inflationary powers spec-trum. As already mentioned, holonomy corrections are sensitive to energy density of matter.The characteristic energy scale, at which holonomy corrections are becoming important is ρ c = 3 m P l πγ L , (1.1)where the Planck mass m P l = 1 . · GeV, γ ∼ O (1) is the so-called Barbero-Immirziparameter and L is a length scale of the order of the Planck scale. Therefore, the criticalenergy density is expected to be of the order of the Planck energy density, ρ c ∼ ρ P l , where ρ P l ≡ m P l .Holonomy corrections are modifying Friedmann equation into the following form [9, 10] H = 8 π m P l ρ (cid:18) − ρρ c (cid:19) , (1.2)where H is a Hubble factor and ρ is energy density of the matter content. Positivity of theleft hand side of this equation implies that energy density of matter is bounded from above, ρ ≤ ρ c . This leads to resolution of singularity problem of homogeneous cosmological modes.The big bang singularity is replaced by non-singular bounce , which merges contracting andexpanding phases [9, 11].Models of inflation are typically constructed with the use of scalar fields. In the simplestcase it can be a single scalar field ϕ , the so-called inflaton field [12]. The single scalar issufficient to construct a reliable model of the inflationary phase. Energy density of this fieldexpresses as ρ = ˙ ϕ V ( ϕ ) , (1.3) The same symmetries, but for crystal lattice, were applied in the mentioned effective model of refractiveindex. – 2 –here V ( ϕ ) is a potential term. Equation of motion governing evolution of ϕ is not a subjectof holonomy corrections and takes the standard form: ¨ ϕ + 3 H ˙ ϕ + V ,ϕ = 0 . (1.4)The holonomy corrections can be also introduced into equations governing evolution ofcosmological perturbations. In particular, it was found that, while holonomy corrections arepresent, equation of motion for the Mukhanov variable v is[13]: d dτ v − Ω ∇ v − z (cid:48)(cid:48) S z S v = 0 . (1.5)Here τ is a conformal time defined as dτ = dt/a . Moreover, the holonomy correction function Ω = 1 − ρρ c (1.6)and z S = a ˙ ϕH . (1.7)It is clear that, while ρ (cid:28) ρ c the classical expression with Ω = 1 is correctly recovered.Based on the Mukhanov variable v , perturbations of curvature R = vz can be derived.The quantity R is a key object characterizing scalar perturbations, allowing for computationof the scalar power spectrum.The equation (1.5) was originally derived by considering requirements of anomaly free-dom for the scalar perturbations [13]. Later, it was shown that this equation can also beobtained from the lattice loop quantum cosmology [14, 15]. The equation (1.5) can be seenas a result of discretization of space for homogeneous cubic cells with the lattice spacing L .Cosmological consequences of equation (1.5) have not been studied in details yet. As aninteresting application, power spectra from a matter bounce was found [16].For tensor modes (gravitational waves), holonomy corrected version of the equation is[17] d dτ h i + 2 (cid:18) H − d Ω dτ (cid:19) ddτ h i − Ω ∇ h i = 0 , (1.8)where i = ⊗ , ⊕ corresponds to two polarizations of gravitational waves. This equation canbe rewritten into the form ddτ u − Ω∆ u − z (cid:48)(cid:48) T z T u = 0 , (1.9)where z T = a/ √ Ω and u = ah ⊗ , ⊕ √ πG √ Ω . The equation (1.8) differs from the equation of motionfor tensor modes with holonomy corrections originally derived in Ref. [18]. This is becausethe original derivation has not been based on anomaly freedom Hamiltonian. However, tak-ing into account the issue of anomaly freedom became possible thanks to analysis of scalarperturbations with holonomy corrections performed in Ref. [13].So far, the equation (1.8) was applied to study generation of tensor perturbation acrossthe cosmic bounce [16, 19]. Nevertheless, there is a whole aggregation of previous analysesperformed with the use of the original equation for tensor modes with holonomy corrections(See e.g. [20–23]). There were also earlier attempts to study holonomy corrections for scalarperturbations. However, they were not consistent with the requirement of anomaly freedom.– 3 –n particular, the studies for scalar perturbations were performed in Ref. [24]. There is also analternative approach to incorporate loop quantum corrections to cosmological perturbationsdeveloped in Refs. [25–27].It is worth mentioning at this point that because we consider model with the scalarmatter, the vector modes are not activated and identically equal zero [28].For both tensor and scalar perturbations the deformation factor Ω is placed in frontof Laplace operator. Therefore, it can be considered as an effective speed of light squared.Namely, by neglecting the cosmological factor and assuming the plane wave solution v ∝ e i ( k · x − ωτ ) , we find the following dispersion relation ω = Ω k , based on which the phasevelocity v ph = ωk = √ Ω . Therefore, the refractive index n ≡ v ph = 1 √ Ω . (1.10)While ρ → ρ c / ( Ω → ) the refractive index becomes infinite, and speed of propagationtends to zero. As discussed in Ref. [29] this can be associated with the state of asymptoticsilence . At the energy densities ρ ∈ ( ρ c / , ρ c ] the refractive index is purely imaginary. Asdiscussed in Ref. [19] this not necessarily means that space is opaque for the propagationof waves. The waves are not only evanescent in this region, but can be amplified as well.Behavior observed from the numerical computations differs with the intuition gained from e.g analysis of waves in plasma with frequencies lower than the plasma frequency. Furthermore,as discussed in Refs. [30, 31] the region of negative Ω can be associated with the change ofmetric signature from Lorentzian to Euclidean one.However, in our calculations of inflationary power spectra, we restrict ourselves to theregime where Ω > . Therefore, the interesting behavior in vicinity of Ω = 0 and at thenegative values of Ω will not be relevant. We will come back to the issue of evolution ofmodes in the Ω ≤ in our further studies. During the slow-roll roll inflation Universe underwent an almost exponential expansion. Thedeviation from the exponential (de Sitter) growth of the scale factor is parametrized bythe slow-roll parameters, which are much smaller than unity. The slow-roll roll inflation ischaracterized by gradual decreasing of ϕ in a potential V ( ϕ ) . In this regime, energy densityof the scalar field is dominated by its potential energy, therefore ˙ ϕ (cid:28) V ( ϕ ) . Because of that,the modified Friedmann equation (1.2) can be approximated by H (cid:39) π m P l V (cid:18) − Vρ c (cid:19) . (2.1)Furthermore, flatness of the potential implies that ¨ ϕ in equation (1.4) can be neglected, suchthat H ˙ ϕ + V ,ϕ (cid:39) . (2.2)Using (2.2) to eliminate ˙ ϕ from the condition ˙ ϕ (cid:28) V ( ϕ ) and by using (2.1), one can define[24] (cid:15) := m P l π (cid:18) V ,ϕ V (cid:19) − V /ρ c ) = − ˙ HH − V /ρ c ) , (2.3)such that (cid:15) (cid:28) for the slow-roll inflation. – 4 –y differentiating the slow-roll equation ˙ ϕ (cid:39) − V ,ϕ H we find ¨ ϕ = − V ,ϕϕ ˙ ϕ H + V ,ϕ ˙ H H . (2.4)Because | ¨ ϕ | (cid:28) | V ,ϕ | , the absolute value of ¨ ϕV ,ϕ (cid:39) η − (cid:15) (cid:18) − Vρ c (cid:19) (2.5)has to be much smaller than unity. Following Ref. [24] let us introduce the second slow-rollparameter η := m P l π (cid:18) V ,ϕϕ V (cid:19) − V /ρ c ) , (2.6)satisfying | η | (cid:28) for | ¨ ϕ | (cid:28) | V ,ϕ | . Based on (2.5) we can also define δ := η − (cid:15) (cid:18) − Vρ c (cid:19) = − ¨ ϕH ˙ ϕ , (2.7)satisfying δ (cid:28) .While studying cosmological perturbations it is convenient to work with the conformaltime τ ≡ (cid:82) dta . Here, it is defined such that τ ∈ ( −∞ , . Based on the definition of conformaltime and integrating by parts, we find τ = (cid:90) dta = (cid:90) daa H = − Ha − (cid:90) a ˙ HH dt = − Ha + τ (cid:15) (cid:18) − Vρ c (cid:19) , (2.8)where in the last equality we applied (2.3). This enables us to write expression for the timedependence of the scale factor a = − Hτ (cid:104) − (cid:15) (cid:16) − Vρ c (cid:17)(cid:105) . (2.9)In the slow-roll regime the Ω function, defined in Eq. 1.6, is approximated by Ω (cid:39) − δ H , (2.10)where for the later convenience we introduced parameter δ H := Vρ c . (2.11)This parameter reflects deviation from the classical slow-roll inflation due to holonomy cor-rections. In the classical limit, which corresponds to ρ c → ∞ , δ H goes to zero. In whatfollows we will consider only linear corrections in δ H . This is because δ H is expected to be avery small quantity. The assumption that the slow-roll regime takes place in the Lorentzianregime ( Ω > ) implies that δ H < / . One can however motivate that δ H (cid:28) / unless thecritical energy density ρ c is not much smaller than the Planck energy density. As an example,let us consider model with a massive potential V ( ϕ ) = m ϕ . Taking the inflaton mass m ∼ − m P l and value of the scalar field ϕ ∼ m P l in agreement with cosmological observa-tions one can estimate that V ( ϕ ) ∼ − ρ P l . Therefore, for ρ c ∼ ρ P l one can expect that δ H – 5 –as the extremely small value δ H ∼ − . On the other hand if ρ c ∼ − ρ P l or smaller, theholonomy corrections are becoming observationally relevant and allow to constraint modelswith low critical energy density.Here, we keep terms linear in both slow-roll parameters and δ H as well as the mixedterms O ( (cid:15)δ H ) and O ( ηδ H ) . Contribution from the second order expansion in the slow-roll pa-rameters is not taken into account. However, in case the δ H is extremely small, as estimatedabove, the terms O ( (cid:15) ) will dominate contributions from O ( (cid:15)δ H ) . The derived expressionswill therefore have practical application only to the regime where ρ c ∈ ( ∼ . , ∼ / , wherethe lower limit comes from estimating values of the slow-roll parameters. The estimated rangeoverlaps with the domain which can be probed with use of currently available observationaldata. Furthermore, the theoretical predictions performed here will set a stage for more com-prehensive considerations of the second order expansion in the slow-roll parameters. This willextend a range of testable values of ρ c , of course if it is allowed by observational data. In this section we will present some possible choices of the short scale normalizations forthe perturbations with holonomy corrections. In must be stressed that we do not exploreany representative class of states. However, the considered normalizations seem to be themost reliable and best physically motivated. Because of this ambiguity, the choice of thenormalization is the weakest point of a whole construction of the model of generation ofprimordial perturbations during the inflationary phase. This concerns also the case withoutquantum gravitational corrections. Therefore, here we pay a lot of attention to this issue.Performing the Fourier transform v ( x , τ ) = (cid:82) d x (2 π ) / v k ( τ ) e i k · x of the equations of modesfor tensor and scalar perturbations we find ddτ v k + Ω k v k − z (cid:48)(cid:48) z v k = 0 , (3.1)where k = k · k , and expression on z depends on whether scalar or tensor mods are studied.In both cases z (cid:48)(cid:48) z ≈ H (cid:39) τ . Based on this, one can define super-horizontal limit when √ Ω k (cid:28) H and short scale limit when √ Ω k (cid:29) H . In the super-horizontal limit the Ω k v k factor in Eq. 3.1 can be neglected and an approximate solution v k = c z + c (cid:82) τ dτ (cid:48) z can befound. Because the physical amplitudes of perturbations are proportional to the ratio v k /z ,it is clear that amplitudes are “frozen” at the super-horizontal scales. This process beingswhen √ Ω k ≈ H . In the short scale limit, the factor z (cid:48)(cid:48) z u k in Eq. 3.1 can be neglected andthe equation for modes reduces to d dτ v k + Ω( τ ) k v k ≈ . (3.2) Under assumption that the potential of the inflaton field is quadratic. It is worth noticing that this condition differs from the classical one k ≈ H due to presence of timedependent function Ω . Furthermore, it is worth to stress that z (cid:48)(cid:48) z is a Ω -dependent function leading correctionsin expression z (cid:48)(cid:48) z ≈ H (cid:39) τ . In particular, for the tensor modes z (cid:48)(cid:48) T z T = H [2 − (cid:15) (1 + 5 δ H ) + . . . ] . Thecorrection due to δ H contributes however together with the (cid:15) factor, contrary to the contribution Ω = 1 − δ H + . . . in front of the k factor. – 6 –or the slow-roll inflation, the Ω is only weakly dependent on τ and solution to equation (3.2)can be found by applying the WKB approximation. We find that v k = c (cid:112) k √ Ω e − ik (cid:82) τ √ Ω( τ (cid:48) ) dτ (cid:48) + c (cid:112) k √ Ω e + ik (cid:82) τ √ Ω( τ (cid:48) ) dτ (cid:48) , (3.3)which is superposition of plane waves traveling forward ( e − ik (cid:82) τ √ Ω( τ (cid:48) ) dτ (cid:48) ) and backward( e + ik (cid:82) τ √ Ω( τ (cid:48) ) dτ (cid:48) ) in time. Validity of the WKB approximation requires that (cid:12)(cid:12)(cid:12)(cid:12) Ω (cid:48) Ω (cid:12)(cid:12)(cid:12)(cid:12) k √ Ω (cid:28) . (3.4)Because we are in the short scale limit √ Ω k (cid:29) | τ | and Ω (cid:48) Ω = − (cid:15)τ δ H , (3.5)the condition of validity of the WKB approximation simplifies to (cid:15)δ H < . Because both (cid:15) and δ H are smaller than unity for the considered slow-roll evolution, the WKB approximation(3.3) holds.Canonical commutation relation between quantum field ˆ v and its conjugated momentarequires Wronskian condition W ( v k , v (cid:48) k ) ≡ v k dv ∗ k dτ − v ∗ k dv k dτ = i (3.6)to be satisfied. This is the usual way the modes are normalized and also the place where quan-tum mechanics enters into description of primordial perturbations. The Wronskian conditionapplied to solution (3.3) leads to relation | c | − | c | = 1 . (3.7)The initial four numbers ( c , c ∈ C ) parametrizing solution (3.3) are therefore reduced tothree. Because the total phase is physically irrelevant, the family of normalized solutionsin the short scale limit is characterized by two real numbers. Their values have to be fixedby hand. The obtained solutions are used to normalize general solutions to the equations ofmotion.It is worth stressing at this point that while considering the short scale limit √ Ω k (cid:28) H one has to be cautious about the limit k → ∞ . Such limit can be performed only formallybecause for k > al Pl the classical description of space is expected to be no more valid due toquantum gravitational effects. We do not consider such trans-Planckian modes here. Ω − deformed Minkowski vacuum For the particular choice c = 1 , solution (3.3) contains incoming modes only: v k = 1 (cid:112) k √ Ω e − ik (cid:82) τ √ Ω( τ (cid:48) ) dτ (cid:48) . (3.8)This solution reduces to the so-called Minkowski (Bunch-Davies) vacuum v Mk ≡ e − ikτ √ k (3.9)– 7 –n the classical limit ( Ω → ), which has been extensively used to normalize cosmologicalperturbations.Based on (3.8) we define Ω − deformed Minkowski vacuum to be v Ω Mk ≡ e − ik (cid:82) τ √ Ω( τ (cid:48) ) dτ (cid:48) (cid:112) √ Ω k ≈ v Mk (cid:20) (cid:18)
12 + ikτ (cid:19) δ H + O ( δ H ) (cid:21) , (3.10)where in the second equality we neglected time variation of Ω . Another possibility of normalizing modes was proposed in Ref. [6] for the case of perturbationswith inverse volume corrections. The proposal made by Bojowald and Calcagni was that inthe short scale limit the solution to (3.2) can be written up to the first order in δ H as follows v BCk = v Mk (1 + y ( k, τ ) δ H ) , (3.11)where y ( k, τ ) is some unknown function . By plugin in (3.11) to (3.2) we find the followingequation for the function y : y (cid:48)(cid:48) − H (cid:15) + ik ) y (cid:48) + (4 i H (cid:15)k − (cid:15) H − δ H k ) y − k = 0 , (3.12)where we used relations δ (cid:48) H = − (cid:15) H δ H , (3.13) δ (cid:48)(cid:48) H = − (cid:15) H δ H . (3.14)The equation (3.12) requires certain simplifications. Firstly, because we are interested in thefirst order correction in δ H we can skip the factor − δ H k y in (3.11), which would generatehigher order contribution. Secondly, because we are looking for the short scale solution( √ Ω k (cid:29) H ) the factor − (cid:15) H y can be ignored as well. This second approximation turns outto be beneficial while searching for analytic solution to the equation of motion for y . Thereduced equation (3.12) is now y (cid:48)(cid:48) − H (cid:15) + ik ) y (cid:48) + 4 i H (cid:15)ky − k = 0 . (3.15)For (cid:15) = 0 , solution to this equation is y = ikτ + c + c e ikτ , (3.16)where c and c are constants of integration. Because the e ikτ factor would lead to outgoingmodes we fix c = 0 . Value of the factor c can be determined by considering the case (cid:15) (cid:54) = 0 .Let us now find solution in the form y = a + bx . By applying it to (3.15), where H = − τ , wefind special solution y = a + bx with a = (cid:15) and b = i (cid:15) to equation (3.15). Requirementof analytic continuity of the solution (in respect to (cid:15) ) between the cases (cid:15) = 0 and (cid:15) (cid:54) = 0 fixesthe value of c . In consequence, we obtain v BCk = v Mk (cid:20) (cid:15) (1 + ikτ ) δ H + O ( δ H ) (cid:21) (cid:39) v Mk [1 + (1 + ikτ ) δ H ] . (3.17) In the original paper [6] the expansion was performed not in therms of δ H but δ Pl relevant for inversevolume corrections. Here we use the simplified de Sitter solution instead of H = − τ · − (cid:15) (1 − δ H )]] , which is sufficient withinthe considered order of approximation. – 8 –t is worth noticing a slight difference between this case and predictions of the Ω − deformedMinkowski vacuum (3.10). In contrast to that case, the method presented in this subsectiondoes not utilize the Wronskian condition in order to normalize the mode functions. This mayhave advantages if we have reason to suppose that the Wronskian condition is deformed butthe form of deformation is not known. Let us suppose that indeed the Wronskian condition is deformed due to presence of Ω . Suchdeformation can come from the fact that, because Ω is present in equations of motion, innerproduct must differ from the classical one. We will discuss this issue in more details in thenext section, while here wa assume the classical Wronskian condition (3.6) is deformed to W Ω ( v k , v (cid:48) k ) = v k dv ∗ k dη − v ∗ k dv k dη = if (Ω) , (3.18)where f (Ω) is some function of Ω , defined such that lim Ω → = 1 . In case we have no hintswhat the functional form of f (Ω) is we can investigate a power-low parametrization f (Ω) = Ω n . (3.19)In this case, the counterpart of (3.10) is v ( n ) k ≡ e − i √ Ω kτ √ k Ω / − n = v M (cid:20) (cid:18) − n + ikτ (cid:19) δ H + O ( δ H ) (cid:21) . (3.20)As we see, for n = − , the normalization from the deformed Wronskian condition overlapswith Bojowald-Calcagni normalization up to the first order in δ H : v ( − / k (cid:39) v BCk . (3.21) Let us now address the issue of validity of the Wronskian normalization in presence of holon-omy corrections in more details.For the a pair of fields φ , φ satisfying Klein-Gordon equation ( (cid:3) − m ) φ = 0 , the innerproduct is [32] (cid:104) φ | φ (cid:105) := i (cid:90) Σ d x √ qn µ ( φ ∗ ∂ µ φ − φ ∂ µ φ ∗ ) , (4.1)where n µ is a future-direction unit ( g µν n µ n µ = − ) vector normal to Cauchy surface Σ and q is a determinant of the spatial metric on Σ . To remind, the Cauchy surface is a spatialhypersurface at which the initial conditions are imposed. The inner product (4.1) is definedsuch that it does not depend on the choice of a Cauchy surface: (cid:104) φ | φ (cid:105) Σ = (cid:104) φ | φ (cid:105) Σ . (4.2)The proof is direct and employs a Gauss law, Klein-Gordon equation and vanishing of φ atspatial infinities (See e.g. Ref. [33]). It is also worth noticing, that the inner product (4.1) isnot positive-definite. – 9 –n the case studied in this paper, the Klein-Gordon equations for scalar ad tensor per-turbations are deformed with respect to the classical one. Therefore, in general, one couldexpect that (4.1) is not a good scalar product because the condition (4.2), requiring the Klein-Gordon equation to be satisfied, is not fulfilled. This can imply that the Wronskian condition(3.6), resulting from normalization of modes with use of (4.1), is deformed.However, if we manage to find an effective metric g effµν which leads to holonomy defor-mations of the equations of perturbations, then the proof the condition (4.2) would remain inforce, and the inner product (4.1) can be used. In what follows we show that such constructionis possible for tensor perturbations.For any component φ of the tensor perturbations, the equation of motion is d dτ φ + 2 (cid:18) H − d Ω dτ (cid:19) ddτ φ − Ω ∇ φ = 0 . (4.3)In the coordinate time ( dt = adτ ) this equation can be written as ¨ φ + 3 H ˙ φ − ˙ΩΩ ˙ φ − Ω a ∇ φ = 0 . (4.4)The classical Klein-Gordon equation (cid:3) φ = 0 (the tensor modes are massless) on the FRWbackground is recovered by taking Ω = 1 . It can be proved by direct calculation, that theholonomy corrected equations for tensor modes can be derived from the wave equation (cid:3) φ = 0 at the effective FRW background given by the line element ds eff = g effµν dx µ dx ν = −√ Ω N dt + a √ Ω δ ab da a dx b , (4.5)where N is a lapse function. In particular for the coordinate time ( N = 1 ) we have (cid:3) φ = ∇ µ ∇ µ φ = 1 √− g ∂ µ ( √− gg µν ∂ ν φ )= − √ Ω ¨ φ + √ Ω a ∇ φ − √ Ω (cid:32) H − ˙ΩΩ (cid:33) ˙ φ, (4.6)where we used g µν = g effµν . By equating (4.6) to zero and multiplying by −√ Ω , the equation(4.4) is recovered.It is worth noticing that the effective metric (4.5) is conceptually similar to dressed metric approach to quantum fields on quantum spaces [25, 34]. In our case, quantum gravitationaleffects are “dressing” the FRW metric leading to the effective metric (affected by Ω terms),which is felt by test fields.Now we can check if the Wronskian condition derived based on (4.1) holds the classicalform. We will be interested in the form of the Wronskian condition for the field u = a √ Ω φ, (4.7)which, as can be seen by substituting to (4.3), fulfills equation d dτ u − Ω ∇ u − z (cid:48)(cid:48) z u = 0 , (4.8)– 10 –here z = a/ √ Ω . Because the Cauchy hypersurface is spatial, based on g µν n µ n µ = − , wefind n = N Ω / and n a = 0 , which gives us n µ ∂ µ = N Ω / ∂ t . Then, for the conformal time( N = a ), the inner product of two fileds φ is (cid:104) φ | φ (cid:105) = i (cid:90) Σ d x √ q ( φ ∗ n µ ∂ µ φ − φn µ ∂ µ φ ∗ )= i (cid:90) Σ d x a Ω / a Ω / Ω a (cid:18) u ∗ dudτ − u du ∗ dτ (cid:19) = i (cid:90) Σ d x (cid:18) u ∗ dudτ − u du ∗ dτ (cid:19) = − i (cid:90) Σ d xW ( u, u (cid:48) ) = (cid:90) Σ d x = V = 1 , (4.9)where the classical Wronskian condition (3.6) was used to get the proper normalization (cid:104) φ | φ (cid:105) = 1 . Here, we assumed that the spatial integration is restricted to V , or equivalentlythe spatial topology is compact and has coordinate volume V . This volume can be conven-tionally fixed to one. Alternatively the field φ can be rescaled by φ → √ V φ to compensatethe contribution from the spatial integration over V .In summary, for the tensor modes, the inner product (4.1) is properly defined andnormalization condition (cid:104) φ | φ (cid:105) = 1 leads to the classical Wronskian condition (3.6). The Ω − deformed vacuum seems to therefore be the right choice for the tensor modes. It remainsto show if the similar construction can be performed for the scalar modes as well. In this section we will compute inflationary tensor power spectrum with holonomy corrections.Starting point for our considerations is the equation d dτ u k + Ω k u k − z (cid:48)(cid:48) T z T u k = 0 , (5.1)where k = k · k . Having solutions for u T and z T , tensor power spectrum can be found fromthe definition P T ( k ) = 64 πG k π (cid:12)(cid:12)(cid:12)(cid:12) u k z T (cid:12)(cid:12)(cid:12)(cid:12) . (5.2)With use of z T = a/ √ Ω , expression for the effective mass term can be written as m eff ≡ − z (cid:48)(cid:48) T z T = − a (cid:48)(cid:48) a + a (cid:48) a Ω (cid:48) Ω + 12 Ω (cid:48)(cid:48) Ω − (cid:32) Ω (cid:48) Ω (cid:33) . (5.3)All the factors contributing to m eff can be expressed in terms of conformal time τ as well as (cid:15), η and δ H . With use of the slow-roll conditions, these terms are: a (cid:48) a = − τ [1 + (cid:15) (1 − δ H )] , (5.4) a (cid:48)(cid:48) a = 1 τ [2 + 3 (cid:15) (1 − δ H )] , (5.5) Ω (cid:48)(cid:48) Ω = 4 (cid:15)τ δ H . (5.6)– 11 –he expression for Ω (cid:48) Ω is given in Eq. 3.5. Plugin it into expression for m eff and keepingterms up to the first order in (cid:15) and δ H we obtain m eff = − τ [2 + 3 (cid:15) (1 − δ H )] . (5.7)The equation for tensor modes can be therefore expressed as d u k dτ + (cid:20) Ω k − (cid:18) ν T − (cid:19) τ (cid:21) u k = 0 , (5.8)where | ν T | = (cid:114)
94 + 3 (cid:15) (1 − δ H ) (cid:39)
32 + (cid:15) (1 − δ H ) . (5.9)Equation (5.8) reminds the standard equation for inflationary modes and it is temptingto find its analytic solution in terms of Hankel functions. This however would not be consistentbecause requires assumption of constancy of Ω . The slow variation of Ω cannot be neglected ifwe already included variation of Ω in the expression for m eff . To see it clearly, let us performthe following change of variables: x := − kτ √ Ω , (5.10) f ( x ) := u √ x , (5.11)which transforms (5.8) into (1 − (cid:15)δ H ) x d fdx + x dfdx + ( x − ν T ) f = 0 . (5.12)While the dependence on δ H in m eff is contributing to ν T , the time dependence of Ω infront of k generates factor − (cid:15)δ H . Because of this factor, solutions to equation (5.12) arenot Bessel (or equivalently Hankel) functions, what would be the case if the factor − (cid:15)δ H isabsent. Nevertheless, solutions to equation Ref. (5.8) can be studied numerically.Because analytic solution to equation (5.8) cannot be easily found, we are forced to useanother approach to find tensor power spectrum. Namely we will determine amplitude of theperturbations at the Hubble radius with use of the short scale solutions. However first, inorder to approve consistency of normalization in case on non-vanishing Ω we will considertensor power spectrum for the case with Ω = const.
Ω = const case
As far as Ω can be considered as a constant, the effective mass term is m eff ≡ − z (cid:48)(cid:48) T z T = − a (cid:48)(cid:48) a = − τ [2 + 3 (cid:15) (1 − δ H )] . (5.13)The equation of motion takes the form (5.8) with Ω = const and | ν T | = (cid:114)
94 + 3 (cid:15) (1 − δ H ) (cid:39)
32 + (cid:15) (1 − δ H ) . (5.14)– 12 –n this case, exact solution to equation (5.8) can be expressed in terms of Hankel functions: u k = √− τ (cid:114) π (cid:104) D H (1) | ν | ( −√ Ω kτ ) + D H (2) | ν | ( −√ Ω kτ ) (cid:105) . (5.15)The constants D and D were normalized such chosen such that the Wronskian condition(3.6) leads to relation | D | − | D | = 1 . (5.16)The Ω − deformed Minkowski vacuum normalization is chosen by taking D = 0 and D = e iπ (2 | ν | +1) / . This can be verified by considering asymptotic behavior of the Hankel function.Namely, for x (cid:28) H (1) | ν | ( x ) ≈ (cid:113) πx exp ( i ( x − | ν | π/ − π/ . With use of this u k = √− τ (cid:114) π e iπ (2 | ν | +1) / H (1) | ν | ( −√ Ω kτ ) ≈ e − i √ Ω kτ (cid:112) √ Ω k = u Ω Mk , (5.17)for −√ Ω kτ (cid:29) .Having the modes correctly normalized we can study the super-horizonal limit −√ Ω kτ (cid:28) . With use of approximation H (1) | ν | ( x ) (cid:39) − iπ Γ( | ν | ) (cid:0) x (cid:1) −| ν | , which holds at x (cid:28) , we obtain | u k | (cid:39)
12 1 aH (cid:32) k √ Ω aH (cid:33) − | ν | , (5.18)where we used − τ (cid:39) aH . Applying it to definition (5.2), the tensor power spectrum from theslow-roll inflation is: P T ( k ) = A T (cid:18) kaH (cid:19) n T , (5.19)where the amplitude A T = 16 π (cid:18) Hm Pl (cid:19) (1 + δ H ) , (5.20)and the tensor spectral index n T = − (cid:15) (1 − δ H ) . (5.21)With use of the modified Friedmann equation (1.2) in the slow-roll regime ( ρ ≈ V ) , onecan rewrite (5.20) into the following form A T = 1283 Vρ P l (1 − δ H ) (1 + δ H ) = 1283 Vρ P l + O ( δ H ) . (5.22)This expression is not a subject of holonomy corrections in the leading order. Ω − deformed Minkowski vacuum Let us now proceed to the proper calculations. The strategy is the following: We will use agiven short scale solution and extrapolate it up to the horizon scale. A mode characterized by k crosses the horizon scale when √ Ω k = aH (cid:39) − τ . Above the horizon scale the modes “freeze– 13 –ut” and the power spectrum remains unchanged. The spectral index can be computed fromthe horizontal spectrum based on the formula n T ≡ d ln P T d ln k . (5.23)Modulus square of the Ω − deformed Minkowski vacuum (3.8) is (cid:12)(cid:12) v Ω Mk (cid:12)(cid:12) = 12 √ Ω k . (5.24)By inserting (5.24) into the definition (5.2) and calculating the value at k √ Ω = aH we find P T ( k ) = 16 π (cid:18) Hm P l (cid:19) √ Ω = 16 π (cid:18) Hm P l (cid:19) (1 + δ H ) + O ( δ H ) , (5.25)which agrees with (5.20).Let us this result and to compute tensor spectral index. By using (5.23), we find n T = 2 kH dHdk − k Ω d Ω dk = − (cid:15) + O ( (cid:15) δ H ) , (5.26)where we used k = − τ √ Ω , (2.3) and (3.5). Here, the correction from δ H contributes togetherwith (cid:15) terms. Therefore, in the leading order the tensor spectral index holds its classicalform. Let us now compute the power spectrum for the Bojowald-Calcagni normalization. While useof condition √ Ω k = aH (cid:39) − τ , the modulus square of v BCk gives (cid:12)(cid:12) v BCk (cid:12)(cid:12) (cid:39) k (1 + 2 δ H ) . (5.27)It is wort mentioning that this value does not depend on the fact that the amplitude iscomputed at the horizon. This is because the ikτ δ H term contributes in the second order,which is neglected. By inserting (5.27) into the definition (5.2) we find P T ( k ) = 16 Gπ k a Ω(1 + 2 δ H ) = 16 π (cid:18) Hm P l (cid:19) (1 + 2 δ H ) + O ( δ H ) . (5.28)Having amplitude of spectrum computed at the horizon scale, the spectral index iscomputed from the relation n T ≡ d ln P T d ln k = 2 kHadk/dτ ˙ HH + 2 dδ H d ln k = − (cid:15) (1 + δ H ) , (5.29)where we have used expression (2.3) and the fact that at the horizon √ Ω kτ = − .Summing up, the tensor power spectrum with the Bojowald-Calcagni normalization canbe expressed as follows P T ( k ) = A T (cid:18) kaH (cid:19) n T , (5.30)– 14 –here the amplitude A T = 16 π (cid:18) Hm P l (cid:19) (1 + 2 δ H ) (5.31)and the tensor spectral index n T = − (cid:15) (1 + δ H ) . (5.32)With use of the modified Friedmann equation (1.2) in the slow-roll regime ( ρ ≈ V ) , onecan rewrite (5.31) into the following form A T = 1283 Vρ P l (1 + δ H ) . (5.33) Here, for the sake of completeness we will derive equation of motion for the scalar modes inthe slow-roll approximation. This equation will not be used to derive spectrum of the scalarinflationary perturbations because of the same reason as in the case of tensor modes.Amplitude of the scalar power spectrum P S ( k ) = k π (cid:12)(cid:12)(cid:12)(cid:12) v k z S (cid:12)(cid:12)(cid:12)(cid:12) (6.1)will be calculated using the short scale solution extrapolated to the horizontal scale. Whilethe spectral scalar is found, the spectral index will be determined by virtue of n S ≡ d ln P S d ln k . (6.2)Similarly as for tensor modes, the Fourier transform of the scalar perturbations fulfillsequation ddτ v k + Ω k v k − z (cid:48)(cid:48) S z S v k = 0 , (6.3)where z S = a ϕ (cid:48) H . The task now is to determine time dependance of z (cid:48)(cid:48) S z S in the slow-rollapproximation.By differentiating z S = a ϕ (cid:48) H with respect to conformal time and by using relation (2.3)we obtain z (cid:48) S z S = (cid:15) (1 − δ H ) H + ϕ (cid:48)(cid:48) ϕ (cid:48) . (6.4)With use this, the expression (2.7) for the parameter δ can be written as δ = 1 − ϕ (cid:48)(cid:48) ϕ (cid:48) H = 1 + (cid:15) (1 − δ H ) − z (cid:48) z H . (6.5)By differentiating this equality with respect to conformal time and neglecting all the non-leading contributions ( i.e. δ (cid:48) , (cid:15) (cid:48) , (cid:15) , η (cid:48) ) we obtain the following equality z (cid:48)(cid:48) S z S = (cid:32) z (cid:48) S z S (cid:33) + H (cid:48) H z (cid:48) S z S . (6.6)– 15 –ombining (6.4) together with (6.5) and (2.7) we find z (cid:48) S z S = [1 − η + 2 (cid:15) (1 − δ H )] H . (6.7)Furthermore H (cid:48) H = H (cid:32) HH (cid:33) = H (1 − (cid:15) (1 − δ H )) , (6.8)where in the second equality we used (2.3). Plugging (6.7) and (6.8) to (6.6) we obtain z (cid:48)(cid:48) S z S = H [2 + 5 (cid:15) (1 − δ H ) − η ] = 1 τ [1 − (cid:15) (1 − δ H )] [2 + 5 (cid:15) (1 − δ H ) − η ]= 1 τ [2 + 9 (cid:15) (1 − δ H ) − η ] . (6.9)Equation for scalar modes with holonomy correction in the (first order) slow-roll approx-imation can be therefore written as d v k dτ + (cid:20) Ω k − (cid:18) ν S − (cid:19) τ (cid:21) v k = 0 , (6.10)where | ν S | = (cid:114)
94 + 9 (cid:15) (1 − δ H ) − η (cid:39)
32 + 3 (cid:15) (1 − δ H ) − η. (6.11)The classical case is correctly recovered for δ H → . Ω − deformed Minkowski vacuum With use of the WKB approximation (3.8) applied to definition (6.1) we find P S ( k ) = 1 π(cid:15) (cid:18) Hm P l (cid:19) (1 − δ H )Ω / = 1 π(cid:15) (cid:18) Hm P l (cid:19) (1 + 2 δ H ) + O ( δ H ) (6.12)at the horizon √ Ω k = aH .Finally, the inflationary scalar power spectrum: P S ( k ) = A S (cid:18) kaH (cid:19) n S − , (6.13)where amplitude of the scalar perturbations A S = 1 π(cid:15) (cid:18) Hm P l (cid:19) (1 + 2 δ H ) , (6.14)and the spectral power index n S = 1 + 2 η − (cid:15) + O ( (cid:15) δ H , (cid:15)ηδ H ) . (6.15)As in case of the tensor modes, the δ H correction to the spectral index is multiplied by the (cid:15) and (cid:15)η factors which are negligible in the considered order. To see it explicitly let us considerthe case of massive scalar field ( V = m ϕ ) for which (cid:15) = η . Therefore n S = 1 − (cid:15) + O ( (cid:15) δ H ) .– 16 –sing the recent Planck fit n S = 0 . ± . [35], we obtain (cid:15) ≈ (1 − n S ) ≈ . . Thehigher order corrections are therefor of the order O ( (cid:15) δ H , (cid:15)ηδ H ) ∼ − δ H , with | δ H | < .These terms are also typically smaller than contributions from the classical second order slow-roll expansion. With use of the present observational precision is impossible to constrain sucheffects.Moreover, with use of the modified Friedmann equation (1.2) in the slow-roll regime ( ρ ≈ V ) , one can rewrite (6.18) into the following form A S = 83 1 (cid:15) Vρ P l (1 − δ H ) (1 + 2 δ H ) ≈
83 1 (cid:15) Vρ
P l (1 + δ H ) . (6.16) The calculations can be now repeated for the case of Bojowald-Calcagni normalization. Theobtained inflationary scalar power spectrum is P S ( k ) = A S (cid:18) kaH (cid:19) n S − , (6.17)where amplitude of the scalar perturbations A S = 1 π(cid:15) (cid:18) Hm P l (cid:19) (1 + 3 δ H ) (6.18)and the spectral index n S = 1 + 2 η − (cid:15) (cid:18) δ H (cid:19) . (6.19)In contrary to the previous case, the spectral index is holonomy-corrected in the leading orderfor the Bojowald-Calcagni normalizationFor completeness, with use of the modified Friedmann equation (1.2) in the slow-rollregime ( ρ ≈ V ) , one can rewrite (6.18) into the following form A S = 83 1 (cid:15) Vρ P l (1 − δ H ) (1 + 3 δ H ) ≈
83 1 (cid:15) Vρ
P l (1 + 2 δ H ) . (6.20) In theoretical studies of inflation as well in confronting theoretical predictions with obser-vations it is often useful to work with tensor-to-scalar ratio r . This dimensionless quantity,defined as r := A T A S , (7.1)measures ratio between amplitudes of tensor and scalar perturbations. There is at present ahuge effort to detect B-type polarization of the CMB radiation which would make determi-nation of the amplitude of the tensor perturbations A T possible . At present, knowing thevalue of A S and having observational constraint on A T , upper bound on the value of r canbe found. The strongest constraint comes from observations of the Planck satellite: r < . To be precise, the B-type polarization of the primordial origin was not detected yet. The B-type po-larization due to gravitational lensing was recently observed for the first time by the SPTpol observatory[36]. – 17 –95% CL) [35]. The theoretically predicted values of r can be confronted with this boundallowing for elimination of some possible inflationary scenarios. In particular, the massivemodel of inflation is no more preferred in the light of the new Planck constraint [35, 37].Let us calculate the tensor-to-scalar ratio r for the models studied in this paper. Forthe case with Ω − deformed Minkowski vacuum normalization we obtain r = 16 (cid:15) (1 + δ H )(1 + 2 δ H ) = 16 (cid:15) (1 − δ H ) + O ( δ H ) . (7.2)Based on this and equation (5.26) the expression for the tensor spectral index r ≈ − n T (1 − δ H ) . (7.3)As we have shown in the previous section, for the massive scalar field (cid:15) ≈ . . For theclassical case ( δ H = 0 ) this would give us r = 16 (cid:15) = 0 . , which is in contradiction with thePlanck constraint r < . . This reflects the mentioned disagreement between the massivescalar field model of inflation and the new Planck data. In the past, when the observationalbound on the value of r was weaker, the massive scalar field model of inflation was favoredby the data. It is worth noticing that, by applying (cid:15) ≈ . to (7.2), together with thePlanck constraint on r , we find that δ H (cid:38) . . Therefore, presence of the quantum holonomycorrections helps to fulfill the observational bound. However, this would require the criticalenergy density ρ c to be much smaller than the Planck energy density.For the Bojowald-Calcagni normalization we obtain r = 16 (cid:15) (1 + 2 δ H )(1 + 3 δ H ) = 16 (cid:15) (1 − δ H ) + O ( δ H ) , (7.4)which is the same as for the Ω − deformed Minkowski vacuum normalization. Finally, basedon this and (5.32) the expression for the tensor spectral index r ≈ − n T (1 − δ H ) . (7.5) In this paper we found holonomy corrections to inflationary power spectra. Such correctionsreflect a discrete nature of space at the Planck scale predicted by loop quantum gravity.Calculations were performed for the slow-roll type inflation driven by a single self-interactingscalar field. The derivations were done up to the first order in the slow-roll parameters (cid:15) and η as well as in the leading order in the parameter δ H , characterizing holonomy corrections.An important issue while considering quantum fields on expanding backgrounds is aproper normalization of the modes. In our calculations we assumed that only ingoing modesare present. Short scale normalization of these modes is a subject of ambiguity due to presenceof the quantum holonomy effects. We considered two, best motivated, types of normalization.The first one was based on adiabatic vacuum (WKB) approximation, while the second onewas based on the method proposed by Bojowald and Calcagni in Ref. [6].For the first type of normalization, spectral indices are not quantum corrected in theleading order. To be precise, linear corrections in δ H are expected. However, they are mul-tiplied not by (cid:15) or η but (cid:15) or η terms. These higher order contributions were not studiedsystematically in this paper. Nevertheless, calculation of the holonomy-corrected inflation-ary spectrum including O ( (cid:15) , η(cid:15), η ) terms is a natural generalization of the results presented– 18 –ere. This would help constraining δ H if sufficiently accurate observational data are available.Investigation of the higher order corrections in the light of the present CMB data is, however,not possible.As we have shown, equation of motion for tensor modes with holonomy correctionscan be derived from the wave equation defined on effective metric , which encodes quantumgravitational effects. This observation allowed us to define a proper inner product for thetensor modes and to show that normalization of tensor modes is obtained by satisfying theclassical Wronskian condition.In this paper we focused on the region where Ω > . Much more interesting is behaviorof modes in the vicinity of Ω = 0 and for Ω < where the equations of modes become elliptic.The issue of imposing initial conditions at Ω = 0 will be a subject of the forthcoming paper[38]. Evolution of tensor modes across the region with negative Ω was addressed in Ref. [19].As it was shown there, tensor power spectrum is enormously amplified in the UV regime.This new behavior certainly deserves more detailed studies. Furthermore, investigation ofsimultaneous effects of holonomy and inverse volume corrections is now possible thanks tonew results presented in Ref. [39]. Acknowledgments
I would like to thank Gianluca Calcagni for useful discussion.
References [1] A. Ashtekar and J. Lewandowski, Class. Quant. Grav. (2004) R53 [gr-qc/0404018].[2] M. Bojowald, Living Rev. Rel. (2005) 11 [gr-qc/0601085].[3] A. Ashtekar and P. Singh, Class. Quant. Grav. (2011) 213001 [arXiv:1108.0893 [gr-qc]].[4] M. Bojowald, Phys. Today (2012) 3, 35.[5] M. Bojowald, G. M. Hossain, M. Kagan and S. Shankaranarayanan, Phys. Rev. D (2009)043505 [Erratum-ibid. D (2010) 109903] [arXiv:0811.1572 [gr-qc]].[6] M. Bojowald and G. Calcagni, JCAP (2011) 032 [arXiv:1011.2779 [gr-qc]].[7] M. Bojowald, G. Calcagni and S. Tsujikawa, Phys. Rev. Lett. (2011) 211302[arXiv:1101.5391 [astro-ph.CO]].[8] M. Bojowald, G. Calcagni and S. Tsujikawa, JCAP (2011) 046 [arXiv:1107.1540 [gr-qc]].[9] A. Ashtekar, T. Pawlowski and P. Singh, Phys. Rev. Lett. (2006) 141301 [gr-qc/0602086].[10] A. Ashtekar, T. Pawlowski and P. Singh, Phys. Rev. D (2006) 084003 [gr-qc/0607039].[11] M. Bojowald, Nature Phys. (2007) 523.[12] A. D. Linde, Phys. Lett. B (1983) 177.[13] T. Cailleteau, J. Mielczarek, A. Barrau and J. Grain, Class. Quant. Grav. (2012) 095010[arXiv:1111.3535 [gr-qc]].[14] E. Wilson-Ewing, Class. Quant. Grav. (2012) 085005 [arXiv:1108.6265 [gr-qc]].[15] E. Wilson-Ewing, Class. Quant. Grav. (2012) 215013 [arXiv:1205.3370 [gr-qc]].[16] E. Wilson-Ewing, JCAP (2013) 026 [arXiv:1211.6269 [gr-qc]].[17] T. Cailleteau, A. Barrau, J. Grain and F. Vidotto, Phys. Rev. D (2012) 087301[arXiv:1206.6736 [gr-qc]]. – 19 –
18] M. Bojowald and G. M. Hossain, Phys. Rev. D (2008) 023508 [arXiv:0709.2365 [gr-qc]].[19] L. Linsefors, T. Cailleteau, A. Barrau and J. Grain, arXiv:1212.2852 [gr-qc].[20] E. J. Copeland, D. J. Mulryne, N. J. Nunes and M. Shaeri, Phys. Rev. D (2009) 023508[arXiv:0810.0104 [astro-ph]].[21] J. Grain and A. Barrau, Phys. Rev. Lett. (2009) 081301 [arXiv:0902.0145 [gr-qc]].[22] J. Mielczarek, T. Cailleteau, J. Grain and A. Barrau, Phys. Rev. D (2010) 104049[arXiv:1003.4660 [gr-qc]].[23] J. Mielczarek, JCAP (2008) 011 [arXiv:0807.0712 [gr-qc]].[24] M. Artymowski, Z. Lalak and L. Szulc, JCAP (2009) 004 [arXiv:0807.0160 [gr-qc]].[25] I. Agullo, A. Ashtekar and W. Nelson, Phys. Rev. Lett. (2012) 251301 [arXiv:1209.1609[gr-qc]].[26] I. Agullo, A. Ashtekar and W. Nelson, Phys. Rev. D (2013) 043507 [arXiv:1211.1354 [gr-qc]].[27] I. Agullo, A. Ashtekar and W. Nelson, Class. Quant. Grav. (2013) 085014 [arXiv:1302.0254[gr-qc]].[28] J. Mielczarek, T. Cailleteau, A. Barrau and J. Grain, Class. Quant. Grav. (2012) 085009[arXiv:1106.3744 [gr-qc]].[29] J. Mielczarek, AIP Conf. Proc. (2012) 81 [arXiv:1212.3527].[30] M. Bojowald and G. M. Paily, Phys. Rev. D (2012) 104018 [arXiv:1112.1899 [gr-qc]].[31] J. Mielczarek, arXiv:1207.4657 [gr-qc].[32] R. M. Wald, “Quantum field theory in curved spacetime and black hole dynamics," TheUniversity of Chicago Press, Chicago 1994.[33] L. H. Ford, “Quantum field theory in curved space-time,” In *Campos do Jordao 1997,Particles and fields* 345-388 [gr-qc/9707062].[34] A. Ashtekar, W. Kaminski and J. Lewandowski, Phys. Rev. D (2009) 064030[arXiv:0901.0933 [gr-qc]].[35] P. A. R. Ade et al. [Planck Collaboration], arXiv:1303.5082 [astro-ph.CO].[36] D. Hanson et al. [SPTpol Collaboration], Phys. Rev. Lett. (2013) 141301 [arXiv:1307.5830[astro-ph.CO]].[37] A. Ijjas, P. J. Steinhardt and A. Loeb, Phys. Lett. B (2013) 261 [arXiv:1304.2785[astro-ph.CO]].[38] J. Mielczarek, L. Linsefors and A. Barrau, (in preparation).[39] T. Cailleteau, L. Linsefors and A. Barrau, arXiv:1307.5238 [gr-qc].(2013) 261 [arXiv:1304.2785[astro-ph.CO]].[38] J. Mielczarek, L. Linsefors and A. Barrau, (in preparation).[39] T. Cailleteau, L. Linsefors and A. Barrau, arXiv:1307.5238 [gr-qc].