Tensor power spectrum with holonomy corrections in LQC
aa r X i v : . [ g r- q c ] A p r Tensor power spectrum with holonomy corrections in LQC
Jakub Mielczarek ∗ Astronomical Observatory, Jagiellonian University, 30-244 Cracow, Orla 171, Poland
In this paper we consider tensor perturbations produced at a bounce phase in presence of theholonomy corrections. Here bounce phase and holonomy corrections originate from Loop QuantumCosmology. We re-derive formulas for the corrections for the model with a scalar field content.Background dynamics with a free scalar field and multi-fluid potential are considered. Since theconsiderations are semi-classical effects of quantum fluctuations of the background dynamics are nottaken into account. Quantum and classical back-reaction effects are also neglected. To find spectrumof the gravitational waves both analytical approximations as well as numerical investigations wereperformed. We have found analytical solutions on super-horizontal and sub-horizontal regimesand derived corresponding tensor power spectra. Also occupation number n k and parameter Ω gw were derived in sub-horizontal limit, leading to its extremely low present value. Final results arenumerical power spectra of the gravitational waves produced in presence of quantum holonomycorrections. The obtained spectrum has two UV and IR branches where P T ∝ k , however withthe different prefactors. Spectrum connecting these regions is in the form of oscillations. We havefound good agreement between numerical spectrum and this obtained from the analytical model.Obtained spectrum can be directly applied as initial conditions for the inflationary modes. Wemention possible resulting observational features of the CMB in particular B-type polarization. I. INTRODUCTION
In the Minkowski background free gravitational wavesfulfil the wave equation ( ∂ t − ∇ ) h µν = 0. Solutions ofthis equation are plane waves h µν = P µν e i ( k · x −| k | t ) , here P µν is polarization tensor. However when cosmologicalexpansion is turned on (we assume flat FRW backgroundhere) additional term appears and the equation of mo-tion is modified to ( ∂ t + 3 H∂ t − ∇ ) h µν = 0, where H isHubble parameter. We see that cosmological term actsas effective friction. When Universe undergoes expansionthen H >
H < ∗ Electronic address: [email protected] heuristic and results obtained have to be verified by thepurely quantum considerations. In particular it has notbeen proved yet whether phase of bounce is generally re-alized for the inhomogeneous loop cosmologies. Howeversome recent studies show that in case of loop quantizedinhomogeneous Gowdy spacetime, singularity is avoided[11]. In our approach inhomogeneities are treated pertur-batively and we neglect their back-reaction on the back-ground dynamics. However in the more detailed studiesthese effects have to be also taken into account. In thesemi-classical approach applied here, quantum gravity ef-fects are introduced by the corrections to the classicalequations of motion. For tensor modes in LQC theseeffects were preliminary studied in Ref. [12, 13]. Laterimproved approach was developed [14] introducing holon-omy corrections. Results of this paper are a backbone ofour investigations. In this paper we assume that thesecorrections are valid during the whole evolution. Somepreliminary studies of influence of the holonomy correc-tions for the gravitational waves production have beendone [15, 16, 17]. However in that papers effects of thecorrections to the source term were neglected. Whilein classical approach this term vanishes (within linearregime) in quantum regime it does contribute. In thepresent paper we improve these studies including a sourceterm.Besides the holonomy corrections also inverse volumecorrections are predicted in the framework of LQC. Ef-fects of inverse volume corrections on gravitational waveswere recently studied in Ref. [18, 19]. However in theflat FRW background inverse volume corrections exhibitfiducial cell dependence. This makes those effects harderto interpret. However, in the curved backgrounds thisproblem disappears. Since holonomy and inverse volumeeffects differ qualitatively, they should be studied sepa-rately. In this paper we follow this line of reasoning. Weconsider consistent model where holonomy corrections in-fluence both background and perturbations parts.The organisation of the text is the following. In sec-tion II we introduce the equation for tensor modes withholonomy corrections. Then in section III we define back-ground dynamics. We consider both the model with freescalar field and with multi-fluid potential. Subsequentlyin section IV and section V we investigate analyticallyand numerically the evolution of the tensor modes. Ef-fects of holonomy corrections are investigated. With useof numerical computations we calculate power spectra ofthe gravitational waves. In section VI we summarize theresults. Finally in Appendix we introduce gravitationalwaves in LQC framework, derive particular form of theholonomy corrections and explain the employed notation.
II. GRAVITATIONAL WAVES WITHHOLONOMY CORRECTIONS
Equation for tensor modes with LQC holonomy cor-rections derived in [14] is given by d dη h ia + 2¯ k ddη h ia − ∇ h ia + T Q h ia = 16 πG Π iQa , (1)where T Q = − (cid:18) ¯ p ¯ µ ∂ ¯ µ∂ ¯ p (cid:19) ¯ µ γ (cid:18) sin ¯ µγ ¯ k ¯ µγ (cid:19) , (2)Π iQa = " V ∂ ¯ H m ∂ ¯ p δE cj δ ja δ ic ¯ p ! cos 2¯ µγ ¯ k + δH m δ ( δE ai ) . (3)For details and explanation of the employed notation wesend to Appendix. To derive specific form of the func-tions T Q and Π iQa , matter content must be defined. Inthis paper we consider models with a scalar field. Weconsider both free and self-interacting fields. In that casematter Hamiltonian is up to the second order H m = ¯ H m + 14 Z Σ d x ¯ N √ ¯ p π φ ¯ p − V ( φ ) ! δ ia δE aj δ jb δE bi . (4)where homogeneous part is given by¯ H m = V ¯ N ¯ p / π φ ¯ p + V ( φ ) ! . (5)Here integration was constrained to fiducial volume V .Further physical results do not depend of this quantity.Energy density can be now defined as ρ := 1 V ¯ p / ∂ ¯ H m ∂ ¯ N . (6)When matter content is defined one can derive partic-ular form of the functions (2) and (3). Expressions for the quantum holonomy corrections simplify to T Q = 8 πG pρ ρ c , (7)Π iQa = Π Q h ia = 12 ¯ p ρρ c (2 V − ρ ) h ia . (8)These expressions were first derived in Ref. [18]. How-ever we have found a discrepancy between the expressionfor Π iQa derived here and this found in Ref. [18]. Toapprove the result presented here we show intermediatesteps of derivation in Appendix. The difference is 1 / µ scheme of quantisation. Namelywe used ¯ µ = p ∆ / ¯ p where ∆ = 2 √ πγl . It is well mo-tivated to use this particular form of the function [20].However, other choices are in principle also permitted.In this paper we consider only ¯ µ scheme, which seems tobe the best motivated.Now equation for the tensor modes (1) simplifies to d dη h ia + 2¯ k ddη h ia − ∇ h ia + ˜ T Q h ia = 0 (9)where we have defined the total holonomy correction˜ T Q = T Q − πG Π Q = 16 πG ¯ p ρρ c (cid:18) ρ − V (cid:19) . (10)Therefore also source term correction has been included.This is in contrast with the analysis performed in [15,16, 17], where this influence was neglected. In the clas-sical theory in fact this term vanish in the linear order.Therefore when fluctuations of vacuum are considered,higher order term can be set to zero. However since, dueto quantum corrections, source term contribute linearly,there is no reason to neglect this term. Therefore in thepresent paper we take it into account.We introduce new common variable u = ah ⊕ √ πG = ah ⊗ √ πG , (11)where h = − h = h ⊕ , h = h = h ⊗ and a = √ ¯ p . Thenperforming the Fourier transform u ( η, x ) = Z d k (2 π ) u ( η, k ) e i k · x , (12)one can rewrite the equation (9) in the form d dη u ( η, k ) + [ k + m ] u ( η, k ) = 0 , (13)where k = k · k and m = ˜ T Q − a ′′ a . (14)In this paper we aim to solve equation (13). Howeverfirst we must specify the background dynamics. III. BACKGROUND DYNAMICS
Background dynamics is governed by the effectiveFriedmann equation (cid:18) p d ¯ pdt (cid:19) = κ ρ (cid:18) − ρρ c (cid:19) , (15)where ρ c = √ π γ l (16)is critical energy density. This equation can be derivedcombining the Hamilton equation ˙¯ p = { ¯ p, ¯ H m + ¯ H phen G } with the scalar constraint ¯ H m + ¯ H phen G = 0.Evolution of the scalar field component is governed bythe Hamilton equations˙ φ = { φ, ¯ H m } = ¯ p − / π φ , (17)˙ π φ = { π φ , ¯ H m } = − ¯ p / dVdφ . (18)Energy density and pressure of the homogeneous scalarfield are expressed as follows ρ φ = 12 ˙ φ + V ( φ ) , (19) p φ = 12 ˙ φ − V ( φ ) . (20) A. Free scalar field
Energy density of the free scalar field has the form ρ = 12 π φ ¯ p (21)and the solution of the effective Friedmann equation (ef-fective background equation) is the following¯ p ( t ) = (cid:0) A + Bt (cid:1) / (22)where A = 16 κπ φ γ ∆ , B = 32 κπ φ . (23)Solution (22) represents non-singular bouncing evolutionand is discussed in Ref. [9].For further applications, it will be useful to relate thecoordinate time with the conformal one dη = dt/a ( t ).Assuming that η ( t = 0) = 0, we obtain η ( t ) = tA / F (cid:20) , ,
32 ; − BA t (cid:21) . (24) B. Scalar field with a multi-fluid potential
One can shown [21] that the restriction p φ = wρ φ ,where w = const in the framework of effective LQC, leadsto the potential in the form V ( φ ) = 12 ρ c (1 − w ) 1cosh hp πG (1 + w ) φ i . (25)Solution of the equations of motion with this potentialhas simple analytic form¯ p ( t ) = ¯ p c (cid:0) πGρ c (1 + w ) t (cid:1) w ) . (26)It is worth to mention that for w = 1 and taking¯ p c = A = 16 κπ φ γ ∆ (27)we recover the solution (22).In analogy with the free field case we derive η ( t ) = t √ ¯ p c F (cid:20) , w ) ,
32 ; − πGρ c (1 + w ) t (cid:21) . (28) IV. ANALYTICAL CONSIDERATIONS
The theory of cosmological creation of particles baseson idea of “freezing” of the vacuum fluctuations. On themathematical level this process can be seen as a squeez-ing and displacement of the vacuum state | i . This isequivalent with the creation of particles. For the non-interacting field theories the wave function is a productof the functions for the particular modes. Therefore thedegree of squeezing and coherence can be different for theparticular modes and is determined by the cosmologicalevolution. The typical scale for which squeezing and dis-placement of the vacuum becomes important is the Hub-ble scale. Modes of quantum fluctuations becomes classi-cal (are described be the coherent states) when crossingthe Hubble radius.To describe process of particles creation quantitativelyone can consider Bogolyubov transformation between ini-tial and final states. Then computing the so-called Bo-golyubov coefficients the number of produced particlescan be obtained. However on the super-horizontal scalesone can in principle obtain ω k = k + m < h | ˆ h ab ( x , η )ˆ h ba ( y , η ) | i = 4 16 πGa Z d k (2 π ) | u ( k, η ) | e − i k · r = Z dkk P T ( k, η ) sin krkr , (29)where we have defined the power spectrum P T ( k, η ) = 64 πGa k π | u ( k, η ) | . (30)The power spectrum can be related later to the ampli-tude of the CMB fluctuations. Therefore it is crucial todeterminate this function.Another way to describe physical properties of thequantum state is the mentioned method of Bogolyubovcoefficients. The relation between annihilation and cre-ation operators for the initial and for the final state isgiven by the Bogolyubov transformationˆ b k = B + ( k )ˆ a k + B − ( k ) ∗ ˆ a †− k , (31)ˆ b † k = B + ( k ) ∗ ˆ a † k + B − ( k )ˆ a − k , (32)where | B + | − | B − | = 1. Since we are working inthe Heisenberg description the vacuum state does notchange during the evolution. It results that ˆ b k | in i = B − ( k ) ∗ ˆ a †− k | in i is different from zero when B − ( k ) ∗ is anonzero function. This means that in the final state thegraviton field considered is no more in the vacuum statewithout particles. The number of produced particles inthe final state is given by n k = 12 h in | h ˆ b † k ˆ b k + ˆ b †− k ˆ b − k i | in i = | B − ( k ) | . (33)The energy density of gravitons is given by dρ gw = 2 · ℏ ω · πω dω (2 πc ) · | B − ( k ) | . (34)where we used definition (33). To describe the spectrumof gravitons it is common to use the parameterΩ gw ( ν ) = νρ ∗ dρ gw dν (35)where ρ gw is the energy density of gravitational wavesand ρ ∗ is the present critical energy density. A. Free scalar field
Based on solution (22) we derive m = κ π φ (cid:0) t + γ ∆ (cid:1) ( A + Bt ) / ≥ . (36)We show this function in Fig. 1. We compare it with theclassical expression m ( ˜ T Q = 0) = κ π φ (cid:0) t − γ ∆ (cid:1) ( A + Bt ) / . (37)The difference is significant since now effective mass isa non-negative function, m ≥
0. One can also com-pare this with the case when the source term corrections
FIG. 1: Evolution of the effective masses m and m ( ˜ T Q =0). In this figure we have assumed π φ = 0 . l Pl were neglected. Then, as can be found in Ref. [16], theeffective mass is negative in some regime and behaveslike the classical one. Here difference is crucial and hasimportant consequences. Namely since m ≥ ω k ≥ ω k ≥ k and a well de-fined vacuum can be found. Otherwise for some k < k x ,the lowest-energy instantaneous vacuum state does notexist.Now we are going to consider the pre-bounce limit.Taking | t | → ∞ , we find m →
14 1 η . (38)The normalised solution of the equation (13) has the form u ( k, η ) = r π e iπ/ √ k p − ηkH (1)0 ( − ηk ) . (39)We have chosen here advanced modes and performed nor-malisation with use of the Wronskian condition. In thesuper-horizontal limit − ηk ≪ H (1)0 ( x ) ≃ i π h ln (cid:16) x (cid:17) + γ E i , (40)where γ E = 0 . . . . is Euler-Mascheroni con-stant. Expression for the power spectrum in the super-horizontal limit is therefore P T ( k ) = A k ( π (cid:20) ln (cid:18) − kη (cid:19) + γ E (cid:21) ) , (41)where A = 4 r π (cid:18) (cid:19) / (cid:18) l Pl π φ (cid:19) . (42)To investigate k dependence in formula (41) we definethe spectral index n T = d ln P T ( k ) d ln k (43)and obtain n T = 3 + 8 π ln (cid:16) − kη (cid:17) + γ E π h ln (cid:16) − kη (cid:17) + γ E i . (44)We show this function for some fixed time in Fig. 2.We find that the resulting spectral index is blue and ap- FIG. 2: Running spectral index on the super-horizontal scales. proaching n T = 3 for k →
0. This blue-tilted spectrumwas predicted earlier in [16]. Recent investigations sug-gest that also for inflationary cosmology with holonomycorrections obtained spectrum is blue-tilted and n T = 3at super-horizontal scales [17]. B. Multi-fluid potential
Now we are going to perform similar analysis for themodel with a multi-fluid potential. We obtain formula m = ¯ p c κ ρ c (1 + w ) (3 w − πGρ c (1 + w ) t ) α ×× (cid:26) t −
43 ∆ γ (1 + w )(3 w − (cid:20) −
23 (1 + 3 w )(1 + w ) (cid:21)(cid:27) , (45)where α = −
23 2 + 3 w w . (46)In the limit | t | → ∞ we obtain m → w − w ) η , (47)where we changed time to conformal. Advanced and nor-malised solution of the equation (13) in the consideredlimit is u ( k, η ) = p − kη r π k e i π ( | ν | + ) H (1) | ν | ( − ηk ) (48) where ν = 94 (1 − w ) (1 + 3 w ) . (49)Power spectrum of the perturbations is then given as P T ( k ) ∝ ( − kη ) − | ν | , (50)where super-horizontal approximation H (1) n ( x ) ≃ − iπ Γ( n ) (cid:16) x (cid:17) − n for x ≪ n = 0 (w=1). In that case anotherexpansion (40) must be applied.It is worth to mention that scale invariant spectrum | ν | = is recovered both for w = − w = 0, asit can be directly seen from (49). This duality was in-vestigated in Ref. [22] in context of the free scalar fieldperturbations. C. Sub-horizontal solutions
Since now we were only concerned with the pre-bouncephase. Now we are going to evolve modes through thebounce. We firstly consider the case of modes which stayunder the Hubble radius before the bounce. For thatmodes the initial vacuum state is given by u in = e − ikη √ k . (52)This can be obtained as a limit of the mode function (39)for − kη ≫
1. To be specific, let us consider the modelwith a free scalar field and π φ = 0 . l Pl .At the Hubble radius we have k H = a | H | (53)which is shown in Fig. 3. We see that for initial time, letus say t = − l Pl , all modes with k > .
003 are welldescribed by the function (52). This solutions howeverdo not hold during the phase of bounce. Close to thebounce one can approximate m ≈ m ( t = 0) = 1(54) / κ ( π φ ρ c ) / ≡ k . (54)For the considered conditions we obtain k ≃ .
12. Inthis approximation the solutions during the bounce phaseare u bounce = A k √ e − i Ω η + B k √ e i Ω η , (55)where Ω = p k + k . Finally, in the post-bounce phasewe have a superposition of advanced and retarded modes u out = α k √ k e − ikη + β k √ k e ikη . (56) FIG. 3: Evolution of the Hubble wave number k H = a | H | .Here π φ = 0 . l Pl . Here the relation | α k | − | β k | = 1 holds, as a conse-quence of the normalisation condition. Now we have tomatch solutions from the three considered regions to de-terminate coefficients α k and β k . In order to do thatwe must specify a time when the matching is performed.We choose it in the mirror points − t − = t + where H reaches its maximal value. Then − t − = t + = t where t = 1 √ πGρ c . (57)With use of equation (24) we obtain η = η ( t ) = F (cid:2) , , ; − (cid:3) √ κρ / c (cid:16) π φ / (cid:17) / . (58)For the considered setup we obtain η ≃ . α k and β k , we define the matrices M = e − ikη − √ k e ikη − √ k − i q k e − ikη − i q k e ikη − , (59) M = e − i Ω η − √ e i Ω η − √ − i q Ω2 e − i Ω η − i q Ω2 e i Ω η − , (60) M = e − i Ω η + √ e i Ω η + √ − i q Ω2 e − i Ω η + i q Ω2 e i Ω η + , (61) M = e − ikη + √ k e ikη + √ k − i q k e − ikη + i q k e ikη + . (62)Then matching conditions can be economically writtenas (cid:18) α k β k (cid:19) = M − M M − M (cid:18) (cid:19) . (63)Multiplying these matrices we obtain α k = − i cos(2 η k ) + sin(2 η k )2 k Ω ×× (cid:2) ik Ω cos(2 η Ω) + (cid:0) k + Ω (cid:1) sin(2 η Ω) (cid:3) (64) β k = − i (cid:0) k − Ω (cid:1) sin(2 η Ω)2 k Ω (65)The resulting square of the amplitude for the out statemodes is | u out | = (cid:0) k + Ω (cid:1) − k cos[4 η Ω] − k sin[2 η Ω] (cid:0) ( k + Ω) sin[2 k ( η − η ) + 2Ω η ] − ( k − Ω) sin[2 k ( η − η ) − η ] (cid:1) k ( k + k ) . (66)Based on this result one can calculate power spectrumof perturbations. We show this spectrum in Fig. 4.Obtained spectrum exhibits sub-horizontal oscillations.This effect can be intuitively understood when anal-ogy with Schr¨odinger equation is employed. Namelythe mode equations are equivalent to a one dimensionalSchr¨odinger equation with potential V = − m . Herespatial variable is replaced by the conformal time η . Inthe employed approximation potential is square well ofwidth 2 η and depth m ( t = 0). Therefore the evolutionof the given mode can be seen as transition of a parti-cle over the potential well. Amplifications of the ampli-tude of transmission correspond to resonances between the width of the potential and the phase shift.It can be shown that the obtained coefficients ( α k , β k )are in fact the Bogolyubov coefficients α k = B + and β k = B − . Therefore the number of produced gravitonsis given by n k = | β k | = k sin (cid:16) η p k + k (cid:17) k ( k + k ) . (67)We show this dependence in Fig. 5. Now it is straight-forward to calculate the parameter Ω gw . We show thisfunction in Fig. 6. We compare it with the obtainedlow energy approximation Ω gw ∝ ν − . Obtained values k P T FIG. 4: Oscillating tensor power spectrum of the sub-horizontal modes at t = 50 l Pl . Straight line represents spec-trum of reference P T ∝ k . k - - n k FIG. 5: Occupation number of the gravitons in the post-bounce state. - - - - - Ν ë Hz ì - - - - - W gw FIG. 6: Parameter Ω gw (bottom curve). Straight line repre-sents low energy approximation Ω gw ∝ ν − . of Ω gw are many orders of magnitude below the presentthreshold for detection. The results obtained were per-formed for some simplified model and for the fixed valueof π φ . In particular in the low energy limit Ω gw ∼ π / φ ,therefore effect of varying π φ is considerable. However,we do not expect significant changes due to the approxi- mations performed. This statement will be confirmed bythe numerical simulations in Section V. D. Super-horizontal solutions
In the previous subsection we have shown solutions ofthe mode equation (13) in the sub-horizontal limit. Nowwe are going to study the super-horizontal k → f = √ au, (68)and change the conformal time to the coordinate one dt = adη . Then equation (13) can be rewritten in the form d fdt + Ω ( k, t ) f = 0 . (69)Here the parameter Ω ( k, t ) is defined as followsΩ ( k, t ) = (cid:18) ka (cid:19) + ǫ ˜ T Q a − (cid:18) ¨ aa + 12 H (cid:19) . (70)We have introduced here parameter the ǫ to trace effectsof the holonomy corrections in the later equations. Inthe classical limit we should take ǫ = 0 while in presenceof the holonomy corrections ǫ = 1. Taking k = 0 andintroducing new complex variable z ∈ C , z = 12 + i p πGρ c (1 + w ) t (71)we can rewrite equation (69) in the following form d fdz + Q ( z ) f = 0 (72)where Q ( z ) = α z + α z + α z ( z − . (73)The coefficients are α = 9(1 + 2 w ) − ǫ (1 + 3 w )36(1 + w ) , (74) α = − w (1 + w ) , (75) α = w (1 + w ) (76)and it will be useful later to remember that α + α = 0.Now introducing the new variable f ( z ) = z L ( z − K g ( z ) (77)with L = c K = a + b + 1 − c z (1 − z ) d gdz + [ c − ( a + b + 1) z ] dgdz − abg = 0 . (80)Solution of this equation is given by the hypergeometricfunctions g ( z ) = C F ( a, b, c ; z ) . (81)Furthermore, we have a system of equations for the co-efficients α + L ( L −
1) = 0 , (82) α + ab − KL − L ( L −
1) = 0 , (83) α − ab + 2 KL + L ( L −
1) + K ( K −
1) = 0 . (84)One can find that, since α + α = 0, we have either K = L or K = 1 − L , where L = (cid:0) ± √ − α (cid:1) . For K = L we find a = 12 (cid:0) c − ± √ α (cid:1) , (85) b = 2 c − a − , (86) c = 2 L = 1 ± √ − α , (87)and while K = 1 − L we have a = 12 (cid:0) ± √ α (cid:1) , (88) b = 1 − a, (89) c = 2 L = 1 ± √ − α . (90)As an exemplary solution we consider w = 1 case bothclassically ( ǫ = 0) and with holonomy corrections to themode equation ( ǫ = 1). Then since α = − α = − / a = b = c −
12 (91)where c ± ( ǫ = 1) = 1 ± √
476 and c ± ( ǫ = 0) = 1 ± √
72 (92)In Fig. 7 we show solutions for the real components of h variable.In Fig. 8 we show solutions for the imaginary compo-nents of the h variable.In Fig. 9 we show solutions for the absolute value ofthe h variable.As it can be seen, solutions with and without the quan-tum holonomy corrections are qualitatively similar. An-other observations is that for the times t ≫ k →
0. Then one can find approximate solution in theform h ≃ A k + B k Z η dη ′ a ( η ′ ) (93) - - t - - Re h FIG. 7: Real components of h variable. Dashed (blue) linerepresents solution with c + ( ǫ = 1) while straight (red) linerepresents solution with c + ( ǫ = 0). - - t - - Im h FIG. 8: Imaginary components of h variable. Dashed (blue)line represents solution with c + ( ǫ = 1) while straight (red)line represents solution with c + ( ǫ = 0). - -
50 0 50 100 t h ¤ FIG. 9: Absolute values of h variable. Dashed (blue) linerepresents solution with c + ( ǫ = 1) while straight (red) linerepresents solution with c + ( ǫ = 0). where A k and B k are some constants. For the modelsconsidered in the present paper we have a ∝ ( ± η ) / (1+3 w ) and ± η ∝ ( ± t ) w w ) . Here we have + sign for the ex-panding phase and − sign for the contracting one. There-fore for the considered w = 1 case we find h ≃ ˜ A k + ˜ B k ln( ± t ) for | t | ≫ V. NUMERICAL INVESTIGATIONS
As it was shown in the previous section, analytic so-lutions of the mode equation are available only in somelimits. Namely for both t → ∞ and k →
0. Also for k → ∞ an approximate solution was found. It is howevernot sufficient to describe whole spectrum of the gravita-tional waves produced on the bounce phase since the in-teresting intermediate regimes are unexplored. Thereforenumerical analysis is required.In the numerical computations we are going to solvethe autonomous system of equations dudη = π u , (95) dπ u dη = − (cid:2) k + m ( t ) (cid:3) u, (96) dtdη = a ( t ) , (97)where a ( t ) and m ( t ) are defined for particular back-ground dynamics. In the considered models with freescalar field and multi-fluid potential these functions aregiven by analytical expressions. Since canonical variables u, π u ∈ C we decompose u = u + iu , (98) π u = π u + iπ u . (99)Now it is crucial to define proper initial conditions for( u , u , π u , π u ) for some time η . It is always unam-biguous how to choose a proper vacuum defined on thecosmological background. However on the sub-horizontalscales, when Minkowski space approximation holds, wecan set u ( η ) = 1 √ k cos( kη ) , (100) u ( η ) = − √ k sin( kη ) (101)and π u ( η ) = − r k kη ) , (102) π u ( η ) = − r k kη ) (103)at some time η . Here we set initial values like in themodel of sub-horizontal modes studied in the previous section. Therefore analysis is correct for the modes withwith k > . k = 0 . FIG. 10: Evolution of the modes with k = 0 .
1. Dotted (blue)curve represent solution of mode equations with holonomycorrections. Straight (red) curve represent solution of modeequations without holonomy effects. tion of modes with and without holonomy corrections tothe modes equation. We see that close to the turningpoint the effects of the holonomy corrections become sig-nificant. However the further oscillating evolution doesnot change qualitatively. The difference is some suppres-sion of the amplitude of perturbations due to the quan-tum corrections. This feature can be also seen in Fig.11 where classical and quantum corrected tensor powerspectra are shown. We find, comparing with the clas- classical with holonomy corrections t = @ l Pl D k P T FIG. 11: Comparison between sub-horizontal spectra ob-tained with and without holonomy corrections to mode equa-tion. Dashed (blue) line represents rescaled initial vacuumpower spectrum. sical case, that quantum holonomy effects amplify lowenergy modes. Therefore tensor power spectra increasesby about one order of magnitude. For the high energiesclassical spectra starts to dominate slightly. It is alsoworth to notice that oscillations do not overlap.0To impose initial conditions on the super-horizontalscales one can use instantaneous vacuum. This is how-ever possible only for values of k fulfilling ω k ≥
0. As wehave found earlier this condition is fulfilled for all k in themodel with the free scalar field. Therefore initial instan-taneous vacuum state can be defined on all length scales.It can be shown that Hamiltonian of perturbations attime η is minimised for u ( η ) = 1 √ ω k (104) π u ( η ) = − i r ω k ω k = p k + m .In Fig. 12 we show tensor power spectrum at post-bounce stage ( t = 50 l Pl ) with imposed instantaneousvacuum initial conditions at t = − l Pl . The charac- IR UV t = @ l Pl D k - P T FIG. 12: Whole post-bounce tensor spectra in presence of theholonomy corrections. Green points comes from the numericalsimulations. Black line is the analytical spectrum from themodel given by Eq. 66. Dashed (red and blue) lines representsUV and IR behaviours, in both cases P T ∝ k . teristic feature of the spectrum are oscillations. Moreoverboth UV and IR behaviours are in the form P T ∝ k .We see that analytical model given by Eq. 66 fairly goodoverlap with the numerical results. Especially structureof oscillations is exactly recovered. Also the asymptoticbehaviours are consistent. The evident discrepancy is thedifference in the total amplitude. In fact this differencecan be suitably adjusted varying parameters of the model η and k . Then low energy behaviour can be exactly re-covered. However it introduces additional phase shift andstructures of oscillation no longer overlap. It is also im-portant to note that effect of the imposed instantaneousvacuum initial conditions is negligible in the range stud-ied. Therefore Minkowski vacuum approximation is stillvalid.The obtained power spectrum can be now applied asan initial condition for the inflationary modes. We ex-pect that super-horizontal part of the spectrum does notchange during the inflationary phase. However the UV part becomes nearly flat (depending on the model of in-flation). It is possible that the oscillating features willalso survive giving the footprints of the bouncing phase.However the further analysis has to be performed to ap-prove these speculations. In particular inflationary powerspectrum with the obtained bouncing initial conditionsmust be calculated. Then it will be possible to computethe B-type polarization spectra of CMB. Therefore a wayto relate the quantum cosmological effects with the lowenergy physics becomes potentially available. VI. SUMMARY
In this paper we have considered influence of LoopQuantum Gravity effects on the gravitational waves prop-agation in the flat FRW cosmological background. Theconsiderations presented based on the semi-classical ap-proach where quantum effects are introduced by correc-tions to the classical equations of motion. This approachwas successfully applied to the homogeneous models. Inthis case good agreement between results of fully quan-tum and semi-classical analyses was found. Here we haveapplied semi-classical approach to the inhomogeneousmodel where inhomogeneity is treated perturbatively.Therefore perturbations had no influence on background.In general both classical and quantum back-reaction ef-fects can be important close to the phase of bounce. Herewe assumed that they can be neglected. We have also notconsidered effects of the quantum fluctuations of back-ground on the inhomogeneities. Quantum effects wereintroduced by the so called holonomy corrections. In thehomogeneous models these corrections lead to absenceof the initial singularity and emergence of the bouncephase. Effects of the other known type of LQG correc-tions, the inverse volume ones, were studied earlier in nu-merous papers. Here we considered self-consistent modelwhere holonomy effects influence both background andperturbations (gravitational waves). In the earlier stud-ies effects on background and perturbations were studiedindependently. In particular in Ref. [16] a model of gravi-tational waves production during the holonomy-inductedbounce phase was investigated. In Ref. [15, 17] effects ofholonomy corrections on the gravitational waves in infla-tionary phase were studied. However quantum effects onthe background dynamics were neglected there. More-over quantum-corrected source term was not taken intoaccount in those studies. Linear part of this term van-ish in the classical limit. However, its contribute whileholonomy corrections are present. Therefore source termhas to be taken into account in the full treatment. In thepresent paper we have included effects of this term.We have considered models with both free scalar fieldand self-interacting field with multi-fluid potential. Inboth cases scalar field is a monotonic function and canbe treated as a internal time variable.We have shown that in the model with the free field,effective mass term m for gravitational waves is a non-1negative function. This is not the case for the modelswith multi-fluid potential. We have found solutions ofthe mode function in the pre-bounce phase and deter-mined the power spectra of the obtained perturbations.Then we have considered sub-horizontal solutions dur-ing the bounce phase. We matched the solutions frompre-bounce, bounce and post-bounce phases. Based onthis we have found power spectrum of gravitational wavesand determined Bogolyubov coefficients. Then numberof produced gravitons n k and the parameter Ω gw werecalculated. We have found that Ω gw reaches 10 − whichis far below any observational bounds. These results wereobtained for fixed parameter π φ = 0 . l Pl .Based on analytical considerations we have found thatpower spectrum exhibits oscillations on sub-horizontalscales. An intuitive explanation of this effect was given.We have also solved the model analytically in the super-horizontal limit. These results indicate that quantumcorrections do not introduce qualitative difference in thepower spectrum on these scales. Therefore the obtainedlack of power on the large scales is a feature of the bounc-ing evolution and not of the quantum corrections to themode equation.Subsequently we have investigated the model numer-ically. We have approved presence of the oscillationsemerged from the simplified analytical considerations.Both numerical and analytical results were compared.We have found good qualitative and quantitative agree-ment. We have also approved earlier observation thatquantum corrections does not introduce qualitative dif-ference in the power spectrum. The only differences ob-served were in total amplitude and phase of oscillations.Imposing initial instantaneous vacuum state we havealso studied the low energy part of the power spectrum.Therefore we have finally found the full shape of the ten-sor power spectrum. This spectrum can be used to studyfurther phenomenological consequences. In particular, itcan be applied as an initial condition for the inflationarymodes. Then we expect that the sub-horizontal part ofthe spectrum becomes flat while super-horizontal formsurvive. It is also possible that sub-horizontal oscilla-tions survive as features of the dominant nearly flat in-flationary spectrum. Therefore two observational effectsof the bouncing phase can be distinguished: oscillationsand lack of power on the super-horizontal scales. Theseeffects can potentially be tested with the future CMBmissions like Planck [23] or proposed CMBPol [24]. Es-pecially promising are observations of the CMB polar-ization. Here bounce can lead to the low multipoles sup-pression in the B-type spectrum. At present projectslike Clover [25], QUaD [26] or QUIET [27] are aimingto detect this spectrum and first results are expected inthe near future. Therefore it is the next step to derivequantitative predictions of the CMB features from thepresented model. Acknowledgments
Author is grateful to Francesca Vidotto for discussionduring the conference ”Quantum Gravity in Cracow ”19-21 XII 2008, Poland, where part of these results werepresented. APPENDIX A: LOOP QUANTUM GRAVITYWITH GRAVITATIONAL WAVES
Loop Quantum Gravity (LQG) describes the gravita-tional field as SU (2) non-Abelian gauge field using back-ground independent methods. The canonical fields areso called Ashtekar variables ( A = A ia τ i dx a , E = E ai τ i ∂ a )which take value in su (2) and su (2) ∗ algebras respectivelyand they fulfil the Poisson bracket { A ia ( x ) , E bj ( y ) } = γκδ ba δ ij δ (3) ( x − y ) (A1)where κ = 8 πG and γ is the Barbero-Immirzi parame-ter. These variables are analogues of the vector potentialand the electric field in electrodynamics. The Ashtekarvariables are related with triad representation. In LQGgauge fields describe only spatial part Σ when time istreated separately.In cosmological applications we perturb basic variablesaround a background E ai = ¯ E ai + δE ai , (A2) A ia = ¯ A ia + δA ia . (A3)For the spatially flat FRW background components havethe following form ¯ E ai = ¯ pδ ai , (A4)¯ A ia = γ ¯ kδ ia , (A5)where ¯ p = a and ¯ k = ˙¯ p/ p . Perturbations can be splitfor the scalar, vector and tensor parts. For the purpose ofthis paper we consider here only the gravitational waves(tensor part). Tensor perturbations of the flat FRW met-ric are introduced as follows g = − N + q ab N a N b = − ¯ N = − a ,g a = q ab N b = 0 ,g ab = q ab = a [ δ ab + h ab ] , with the conditions h aa = ∂ a h ab = 0 and | h ab | ≪
1. In theTT gauge h = − h = h ⊕ and h = h = h ⊗ .Now we are going to perturb the Hamiltonian of thetheory. The full Hamiltonian is composed of the gravi-tational and H G and matter H m parts. Hamiltonian H G takes the form of a liner combination of the constraints H G = Z Σ d x ( N i G i + N a C a + N S ) . Spatial diffeomorphisms constraint: C a = E bi F iab − (1 − γ ) K ia G i . G i = D a E ai = ∂ a E ai + ǫ ijk A ja E ak . Scalar constraint: S = E ai E bj p | det E | h ε ijk F kab − γ ) K i [ a K jb ] i where F = dA + [ A, A ]. However thanks to the quan-tum gravity effect this Hamiltonian undergoes modifi-cations. These modifications can be introduced on thephenomenological level by the replacement¯ k → sin n ¯ µγ ¯ kn ¯ µγ (A6)in the classical expressions. Here¯ µ = s ∆¯ p where ∆ = 2 √ πγl . This kind of corrections we call holonomy ones. Factor n can be fixed from requirement of the anomaly cancel-lation [14, 28]. Effective second order Hamiltonian withholonomy corrections takes the form H phen G = 116 πG Z Σ d x ¯ N " − √ ¯ p (cid:18) sin ¯ µγ ¯ k ¯ µγ (cid:19) − p / (cid:18) sin ¯ µγ ¯ k ¯ µγ (cid:19) ( δE cj δE dk δ kc δ jd )+ √ ¯ p ( δK jc δK kd δ ck δ dj ) − √ ¯ p (cid:18) sin 2¯ µγ ¯ k µγ (cid:19) ( δE cj δK jc )+ 1¯ p / ( δ cd δ jk δ ef ∂ e E cj ∂ f E dk ) (cid:21) (A7)where for tensor modes δE ai = −
12 ¯ ph ai (A8) δK ia = 12 (cid:20) ˙ h ia + (cid:18) sin 2¯ µγ ¯ k µγ (cid:19) h ia (cid:21) (A9)Based on the Hamilton equations δ ˙ E ai = n δE ai , H phen G + H m o , (A10) δ ˙ K ia = n δK ia , H phen G + H m o , (A11)we obtain equation¨ h ia + 2¯ k ˙ h ia − ∇ h ia + T Q h ia = 16 πG Π iQa (A12)where T Q = − (cid:18) ¯ p ¯ µ ∂ ¯ µ∂ ¯ p (cid:19) ¯ µ γ (cid:18) sin ¯ µγ ¯ k ¯ µγ (cid:19) , (A13)Π iQa = " V ∂ ¯ H m ∂ ¯ p δE cj δ ja δ ic ¯ p ! cos 2¯ µγ ¯ k + δH m δ ( δE ai ) (A14) are quantum holonomy corrections.We consider homogeneous scalar field with the Hamil-tonian H m = Z Σ d x ¯ N π φ p | det E | + p | det E | V ( φ ) ! , (A15)where up to the second order √ det E = ¯ p (cid:20) p δ ia δE ai − p δ ia δE aj δ jb δE bi + 18¯ p δ ia δE ai δ jb δE bj (cid:21) , (A16)1 √ det E = 1¯ p (cid:20) − p δ ia δE ai + 14¯ p δ ia δE aj δ jb δE bi + 18¯ p δ ia δE ai δ jb δE bj (cid:21) . (A17)However, since δ ab h ab = 0 ⇒ δ ia δE ai = 0 the above ex-pansion simplifies. Then H m = ¯ H m + 14 Z Σ d x ¯ N √ ¯ p π φ ¯ p − V ( φ ) ! δ ia δE aj δ jb δE bi + O ( E ) . Now we can derive variation δH m δ ( δE ai ) = 12 ¯ N √ ¯ p π φ ¯ p − V ( φ ) ! δ ib δ ka δE bk (A18)and derivative ∂ ¯ H m ∂ ¯ p = 32 V ¯ N √ ¯ p − π φ ¯ p + V ( φ ) ! . (A19)One can now easily find that in the classical limit,when we set cos(2¯ µγ ¯ k ) = 1 in expression (A14), thesource term vanish. This is due to the opposite signs ofthe bracketed expression in equations (A18) and (A19).When quantum holonomy corrections are present we havecos(2¯ µγ ¯ k ) = 1 − ρρ c , (A20)which can be found from background equations of mo-tion. Therefore the form of the quantum corrections sim-plifies to T Q = 8 πG pρ ρ c , (A21)Π iQa = Π Q h ia = 12 ¯ p ρρ c (2 V − ρ ) h ia , (A22)where we have chosen ¯ N = √ ¯ p and adopted the expres-sion (A8).3 [1] M. Bojowald, Living Rev. Rel. (2008) 4.[2] Y. Shtanov and V. Sahni, Phys. Lett. B (2003) 1[arXiv:gr-qc/0208047].[3] M. Gasperini and G. Veneziano, Phys. Rept. (2003)1 [arXiv:hep-th/0207130].[4] A. Ashtekar, T. Pawlowski and P. Singh, Phys. Rev. Lett. (2006) 141301 [arXiv:gr-qc/0602086].[5] D. Wands, arXiv:0809.4556 [astro-ph].[6] Y. F. Cai and X. Zhang, arXiv:0808.2551 [astro-ph].[7] Y. F. Cai, T. Qiu, R. Brandenberger, Y. S. Piao andX. Zhang, JCAP (2008) 013 [arXiv:0711.2187 [hep-th]].[8] P. Singh, K. Vandersloot and G. V. Vereshchagin, Phys.Rev. D (2006) 043510 [arXiv:gr-qc/0606032].[9] J. Mielczarek, T. Stachowiak and M. Szydlowski, Phys.Rev. D (2008) 123506 [arXiv:0801.0502 [gr-qc]].[10] M. Bojowald, H. H. Hernandez, M. Kagan, P. Singhand A. Skirzewski, Phys. Rev. D (2006) 123512[arXiv:gr-qc/0609057].[11] D. Brizuela, G. A. D. Mena Marugan and T. Pawlowski,arXiv:0902.0697 [gr-qc].[12] J. Mielczarek and M. Szydlowski, Phys. Lett. B (2007) 20 [arXiv:0705.4449 [gr-qc]].[13] J. Mielczarek and M. Szydlowski, arXiv:0710.2742 [gr-qc]. [14] M. Bojowald and G. M. Hossain, Phys. Rev. D (2008)023508 [arXiv:0709.2365 [gr-qc]].[15] A. Barrau and J. Grain, arXiv:0805.0356 [gr-qc].[16] J. Mielczarek, JCAP (2008) 011 [arXiv:0807.0712[gr-qc]].[17] J. Grain and A. Barrau, arXiv:0902.0145 [gr-qc].[18] E. J. Copeland, D. J. Mulryne, N. J. Nunes andM. Shaeri, arXiv:0810.0104 [astro-ph].[19] G. Calcagni and G. M. Hossain, arXiv:0810.4330 [gr-qc].[20] A. Corichi and P. Singh, Phys. Rev. D (2008) 024034[arXiv:0805.0136 [gr-qc]].[21] J. Mielczarek, arXiv:0809.2469 [gr-qc].[22] D. Wands, Phys. Rev. D (1999) 023507[arXiv:gr-qc/9809062].[23] [Planck Collaboration], arXiv:astro-ph/0604069.[24] D. Baumann et al. [CMBPol Study Team Collaboration],arXiv:0811.3919 [astro-ph].[25] C. E. North et al. , arXiv:0805.3690 [astro-ph].[26] :. P. G. Castro et al. [QUaD collaboration],arXiv:0901.0810 [astro-ph.CO].[27] D. Samtleben and f. t. Q. collaboration, arXiv:0806.4334[astro-ph].[28] M. Bojowald and G. M. Hossain, Class. Quant. Grav.24