Quantum Cosmology and the Evolution of Inflationary Spectra
Alexander Y. Kamenshchik, Alessandro Tronconi, Giovanni Venturi
QQuantum Cosmology and the Evolution ofInflationary Spectra
Alexander Y. Kamenshchik ∗ , Alessandro Tronconi † andGiovanni Venturi ‡ Dipartimento di Fisica e Astronomia and INFN, Via Irnerio46,40126 Bologna, Italy L.D. Landau Institute for Theoretical Physics of the RussianAcademy of Sciences, Kosygin str. 2, 119334 Moscow, RussiaSeptember 12, 2016
Abstract
We illustrate how it is possible to calculate the quantum gravita-tional effects on the spectra of primordial scalar/tensor perturbationsstarting from the canonical, Wheeler-De Witt, approach to quantumcosmology. The composite matter-gravity system is analysed througha Born-Oppenheimer approach in which gravitation is associated withthe heavy degrees of freedom and matter (here represented by a scalarfield) with the light ones. Once the independent degrees of freedomare identified the system is canonically quantised. The differentialequation governing the dynamics of the primordial spectra with itsquantum-gravitational corrections is then obtained and is applied todiverse inflationary evolutions. Finally, the analytical results are com-pared to observations through a Monte Carlo Markov Chain techniqueand an estimate of the free parameters of our approach is finally pre-sented and the results obtained are compared with previous ones. ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ g r- q c ] S e p Introduction
The paradigm of inflation [1] has led to a beautiful connection between mi-croscopic and macroscopic scales. This occurs since inflation acts as a “mag-nifying glass” insofar as microscopic quantum fluctuations at the beginningof time, when the universe was very small, evolve into inhomogeneous struc-tures [2]. Thus the observed structure of the present-day universe is relatedto the very early time quantum dynamics. As a consequence the formercan be used to test the primordial dynamics and in particular the possibleeffects of quantum gravity at early times corresponding to a very small uni-verse. The reason for this is that because of the huge value of the Planckmass quantum gravity effects are otherwise suppressed (of course one canalso hope to observe quantum gravitational effects in the presence of verystrong gravitational fields, for example in the proximity of black holes).Composite systems which involve two mass (or time) scales such as moleculesare amenable to treatment by a Born-Oppenheimer approach [3]. For moleculesthis is possible because of the different nuclear and electron masses, this al-lows one to suitably factorise the wave-function of the composite systemleading, in a first approximation, to a separate description of the motion ofthe nuclei and the electrons. In particular it is found that the former areinfluenced by the mean hamiltonian of the latter and the latter (electrons)follow the former adiabatically (in the quantum mechanical sense). Similarlyfor the matter gravity system as a consequence of the fact that gravity ischaracterised by the Planck mass, which is much greater than the usual mat-ter mass, the heavy degrees of freedom are associated with gravitation andthe light ones with matter [4]. As a consequence, to lowest order, gravita-tion will be driven by the main matter Hamiltonian and matter will followgravity adiabatically. As mentioned above we shall quantise the compositesystem, by this we mean that we shall perform the canonical quantisation ofEinstein gravity and matter leading to the Wheeler-DeWitt (WDW) equa-tion [5]. This is what we mean by quantum gravity and is quite distinct tothe introduction of so-called trans-Planckian effects (loosely referred to asquantum gravity) through ad hoc modifications of the dispersion relation [6]and/or the initial conditions [7]. Further the equations we shall obtain afterthe BO decomposition will be exact, in the sense that they also include non-adiabatic effects. The above approach has been previously illustrated in amini-superspace model with the aim of studying the semiclassical emergenceof time [4], which is otherwise absent in the quantum system. Conditions were2ound for the usual (unitary) time evolution of quantum matter (Schwinger-Tomonaga or Schrödinger) to emerge, essentially these are that non-adiabatictransitions (fluctuations) be negligible or that the universe be sufficiently farfrom the Planck scale. In a series of papers [8] we have generalized the ap-proach to non-homogeneous cosmology in order to obtain corrections to theusual power spectrum of cosmological fluctuations produced during inflation.These corrections, which essentially amount to the inclusion of the effect ofthe non-adiabatic transitions, affect the infrared part of the spectrum andlead to an amplification or a suppression depending on the background evo-lution. More interestingly they depend on the wavenumber k and scale as k − , in both the scalar and the tensor sectors, when background evolutionis close to de Sitter. That non-adiabatic effects affect the infrared part ofthe spectrum, which is associated with large scales, is not surprising, sinceit is this part of the spectrum which exits the horizon in the early stagesof inflation and is exposed to high energy and curvature effects for a longertime.The latest Planck mission results [9] provide the most accurate constraintsavailable currently to inflationary dynamics [1]. So far the slow roll (SR)mechanism has been confirmed to be a paradigm capable of reproducingthe observed spectrum of cosmological fluctuations and the correct tensorto scalar ratio [2]. Since the inflationary period is the cosmological era de-scribing the transition from the quantum gravitational scale down to the hotbig bang scale, it may, somewhere, exhibit related peculiar features whichcould be associated with quantum gravity effects. Quite interestingly a lossof power, with respect to the expected flatness for the spectrum of cosmo-logical perturbations, can be extrapolated from the data at large scales [10].Since, as mentioned above, it is for such scales that quantum gravity effectsdue to non-adiabaticity may appear, this has motivated us to estimate sucheffects. Unfortunately, such a feature (evident already in the WMAP results)exhibits large errors due to cosmic variance. Nonetheless we feel that it isworth comparing our detailed analytical predictions for the quantum gravityeffects with Planck data through a Monte Carlo Markov Chain (MCMC)based method.The paper is organized as follows. In section 2 the basic equations are re-viewed, the canonical quantization method and the subsequent BO decom-position are illustrated. In the section 3 we calculate the master equationgoverning the dynamics of the two point function of the quantum fluctua-tions when the quantum gravitational effects are taken into account and the3acuum prescription for these fluctuations is briefly discussed. In section4 we review the basic relations for de Sitter, power law and slow-roll (SR)inflation and the quantum corrections to the primordial spectra are explic-itly calculated for these three distinct cases. In section 5 we illustrate howour analytical predictions are compared to observations and we comment ourresults. Finally in section 6 we draw the conclusions. The inflaton-gravity system is described by the following action S = (cid:90) dηd x √− g (cid:20) − M P2 R + 12 ∂ µ φ∂ µ φ − V ( φ ) (cid:21) (1)where M P = (8 πG ) − / is the reduced Planck mass. The above action canbe decomposed into a homogeneous part plus fluctuations around it. Thefluctuations of the metric δg µν ( (cid:126)x, η ) are defined by g µν = g (0) µν + δg µν (2)where g (0) µν = diag [a( η ) (1 , − , − , − and η is the conformal time. Onlythe scalar and the tensor fluctuations “survive” the inflationary expansion: δg = δg ( S ) + δg ( T ) . The scalar fluctuations of the metric can be defined asfollows δg µν = a ( η ) A ( (cid:126)x, η ) − ∂ i B ( (cid:126)x, η ) − ∂ i B ( (cid:126)x, η ) 2 δ ij ψ ( (cid:126)x, η ) − D ij E ( (cid:126)x, η ) (3)with D ij ≡ ∂ i ∂ j − δ ij ∇ . These four degrees of freedom (d.o.f.) mix withthe inflation fluctuation δφ ( (cid:126)x, η ) , defined by φ ( (cid:126)x, η ) ≡ φ ( η ) + δφ ( (cid:126)x, η ) . Thescalar perturbations, defined in (3), are gauge dependent. One can eitherrewrite them in terms of just two Bardeen’s potentials or fix the gauge andset two of them to zero. Finally, on using the equations of motion, thescalar sector can be collectively described by a single field v ( (cid:126)x, η ) which, inthe uniform curvature gauge, is given by v ( (cid:126)x, η ) = a ( η ) δφ ( (cid:126)x, η ) . Its Fouriertransform, v k , can then be decomposed into two parts: v ,k ≡ Re ( v k ) and v ,k ≡ Im ( v k ) . 4he tensor fluctuations are gauge invariant perturbations of the metric andare defined by ds = a ( η ) (cid:2) dη − ( δ ij + h ij ) dx i dx j (cid:3) (4)with ∂ i h ij = δ ij h ij = 0 . For each direction of propagation of the perturbation k i , the above conditions on h ij , with the requirement g µν = g νµ , give sevenindependent constraint equations for the components of the tensor perturba-tions, leading to only two remaining polarization physical degrees of freedom h (+) and h ( × ) . Then, on defining v ( λ )1 ,k ≡ a M P √ Re (h k ) and v ( λ )2 ,k ≡ a M P √ Im (h k ) ,one can describe the tensor perturbations in a manner similar to the scalarperturbations.In what follows we shall illustrate in detail a point which is often glossedover: namely the fact that on working in a flat 3-space and considering bothhomogeneous and inhomogeneous quantities one must introduce an unspeci-fied length L . Indeed the effective action of the homogeneous inflaton-gravitysystem plus the inhomogeneous perturbations finally is [11] S = (cid:90) dη (cid:40) L (cid:34) − (cid:101) M a (cid:48) + a (cid:0) φ (cid:48) − V ( φ ) a (cid:1)(cid:35) + 12 (cid:88) i =1 , ∞ (cid:88) k (cid:54) =0 (cid:20) v (cid:48) i,k ( η ) + (cid:18) − k + z (cid:48)(cid:48) z (cid:19) v i,k ( η ) (cid:21) + 12 (cid:88) i =1 , (cid:88) λ =+ , × ∞ (cid:88) k (cid:54) =0 (cid:32) v ( λ ) i,k dη (cid:33) + (cid:18) − k + a (cid:48)(cid:48) a (cid:19) (cid:16) v ( λ ) i,k (cid:17) (5)where (cid:101) M P = √ P , z ≡ φ (cid:48) /H , H = a (cid:48) /a is the Hubble parameter and L ≡ (cid:82) d x . Let us note that the action for the perturbations has been con-veniently simplified by means of the homogeneous dynamics.The interval ds has dimension of a length l and one generally may either take [ a ] = l and [ dx ] = [ dη ] = l or [ a ] = l and [ dx ] = [ dη ] = l . Correspond-ingly one then has [ L ] = l or [ L ] = l . One can eliminate the factor L byreplacing a → a/L , η → ηL , v → √ Lv and k → k/L . Such a redefinitionis equivalent to setting L = 1 in the above action (5) (then implicitly as-suming the convention [ a ( η )] = l and [ dx ] = [ dη ] = l ) and then proceedingwith its quantization. Such a choice, although limited to the homogeneouspart, has been previously illustrated [12]. Henceforth we shall use this lattersimplifying choice. Only at the end, in order to compare our results with5bservations, we shall restore all quantities to their original definition andthe dependence on L will become explicit. Let us finally note that the factthat L is infinite does not create a problem. As usual, the transition fromthe Fourier integral w.r.t. the wave number to the Fourier series eliminatesthe corresponding divergence.Once the total action for the matter-gravity system is cast into the form (5),all the dynamical quantities (fields) are expressed through an infinite “tower”of homogeneous variables v i,k . Such an effective description has a simplifyingrole in the quantization procedure, which we shall illustrate in detail in thenext section. The dynamics of each d.o.f. describing the perturbations, is formally anal-ogous to that of a homogeneous scalar field with a time dependent mass.In order to illustrate the quantization procedure and the subsequent Born-Oppenheimer decomposition in detail, without losing generality, we singleout the homogenous part and one real scalar field for the perturbations in(5): S = (cid:90) dη (cid:40)(cid:34) − (cid:101) M a (cid:48) + a (cid:0) φ (cid:48) − V ( φ ) a (cid:1)(cid:35) + 12 ∞ (cid:88) k (cid:54) =0 (cid:2) v (cid:48) k ( η ) − ω k v k ( η ) (cid:3)(cid:41) ≡ (cid:90) dη L tot (6)where ω k = k + m ( η ) is time dependent and L has been set equal to .Let us note that m ( η ) depends on the homogeneous quantities a ( η ) , φ ( η ) and their derivatives. The action describing the evolution of the cosmologicalperturbations, is derived by substituting the homogenous, leading order, so-lutions into the perturbed Lagrangian. Such a derivation does not affect thequantization of the perturbations but may have consequences on the quan-tization of the homogeneous d.o.f.. Let us remember that in obtaining thereduced action (6) we have at most kept terms to quadratic order in the fieldand metric perturbations ( v k ). Therefore, since quantum fluctuations around z (cid:48)(cid:48) /z occur already multiplied by small field perturbations, we shall just re-tain for it its classical homogeneous value. Thus our choice is to consider m ( η ) as a generic function of time and consequently specify it at the end of6he quantization procedure.One can rewrite the above action in terms of an arbitrary time parameter τ with N ( τ ) dτ = a ( η ) dη , where N ( τ ) is the lapse function. The action (6)then becomes S = (cid:90) dτ Na (cid:40)(cid:34) − (cid:101) M a ˙ a N + a (cid:32) ˙ φ N − V ( φ ) (cid:33)(cid:35) + 12 ∞ (cid:88) k (cid:54) =0 (cid:20) a ˙ v k ( η ) N − ω k v k ( η ) (cid:21)(cid:41) ≡ (cid:90) dτ ˜ L tot (7)where the dot indicates the derivative w.r.t. τ . The lapse function plays therole of a Lagrange multiplier in the action. The variation of the action w.r.t N leads to the following equation of motion δ ˜ L tot δN = (cid:101) M a ˙ a N − a ˙ φ N − a V − ∞ (cid:88) k (cid:54) =0 (cid:20) a ˙ v k N + ω k v k a (cid:21) (8)having the form of a constraint equation. The system Hamiltonian is H = − N π a a (cid:101) M + N π φ a + a N V + ∞ (cid:88) k (cid:54) =0 (cid:20) N π k a + N ω k a v k (cid:21) (9)where π N = 0 , π a = − (cid:101) M a ˙ aN , π φ = a ˙ φ N , π k = a ˙ v k N . (10)and is proportional to the above constraint (8): δ ˜ L tot δN = H N (11)which is then called “Hamiltonian constraint”. It is a very particular energyconservation constraint which equates the system’s total energy to zero. Atthe quantum level, when the degrees of freedom are canonically quantized,it plays the role of a time independent Schroedinger equation.The canonical quantization of the action (5) leads to the following Wheeler-De Witt (WDW) equation [5] for the wave function of the universe (matter7lus gravity) (cid:40) (cid:101) M ∂ ∂a − a ∂ ∂φ + V a + ∞ (cid:88) k (cid:54) =0 (cid:20) − ∂ ∂v k + ω k v k (cid:21)(cid:41) Ψ ( a, φ , { v k } ) = 0 . (12)Let us note that the time dependent mass in ω k is m ( η ) = − z (cid:48)(cid:48) z for eachmode of the scalar perturbation and m ( η ) = − a (cid:48)(cid:48) a for each mode of thetensor perturbation, where z ( η ) , a ( η ) are classical expressions. Eq. (12) can be written in the compact form (cid:34) (cid:101) M ∂ ∂a + ˆ H ( M )0 + (cid:88) k ˆ H ( M ) k (cid:35) Ψ ( a, φ , { v k } ) ≡ (cid:34) (cid:101) M ∂ ∂a + ˆ H ( M ) (cid:35) Ψ ( a, φ , { v k } ) = 0 . (13)where ˆ H ( M )0 = − a ∂ ∂φ + V a , (14) ˆ H ( M ) k = − ∂ ∂v k + ω k v k (15)and is formally similar to a time independent Schroedinger equation, exceptfor the sign in front of the kinetic term for the scale factor. Finding thegeneral solution of the WDW equation, even when the perturbations are setto zero, is a very complicated task due to the interaction between matter andgravity.A set of approximate solutions can be found within a BO approach. The BOapproximation was originally introduced in order to simplify the Schroedingerequation of complex atoms and molecules [3].It consists in factorising the wave function of the Universe into a product Ψ ( a, φ , { v k } ) = ψ ( a ) χ ( a, φ , { v k } ) (16)8here ψ ( a ) is the wave function for the homogeneous gravitational sectorand χ ( a, φ , { v k } ) is that for matter (homogeneous plus perturbations). Asimilar decomposition for atoms consists in factorising the atomic wave func-tion Ψ A ( r, R ) into a nuclear wave function ψ N ( R ) and the electrons’ wavefunctions χ e ( r, R ) , where r and R are the d.o.f. of electrons and nuclei re-spectively. The matter wave function in eq. (16) can be further factorizedas: χ ( a, φ , { v k } ) = χ ( a, φ ) ∞ (cid:89) k (cid:54) =0 χ k ( η, v k ) = ∞ (cid:89) k =0 χ k . (17)Let us note that the wave function of each mode v k depends parametricallyon the conformal time η and, in the semiclassical limit, the evolution of thescale factor a = a ( η ) fixes η as a function of a . The above factorization leadsto the following set of partial differential equations, which are equivalent tothe WDW equation: (cid:34) (cid:101) M ∂ ∂a + (cid:104) ˆ H ( M ) (cid:105) (cid:35) ˜ ψ = − (cid:101) M (cid:104) ∂ ∂a (cid:105) ˜ ψ (18)which is the equation for the gravitational wave function and ˜ ψ ∗ ˜ ψ (cid:104) ˆ H ( M ) − (cid:104) ˆ H ( M ) (cid:105) (cid:105) ˜ χ + 1 (cid:101) M (cid:18) ˜ ψ ∗ ∂∂a ˜ ψ (cid:19) ∂∂a ˜ χ = 12 (cid:101) M ˜ ψ ∗ ˜ ψ (cid:20) (cid:104) ∂ ∂a (cid:105) − ∂ ∂a (cid:21) ˜ χ (19)which is the equation for matter, where ψ = e − i (cid:82) a A da (cid:48) ˜ ψ, χ = e i (cid:82) a A da (cid:48) ˜ χ, A = − i (cid:104) χ | ∂∂a | χ (cid:105) (20)with v = φ , (cid:104) ˆ O (cid:105) = (cid:104) ˜ χ | ˆ O | ˜ χ (cid:105) and each mode is individually normalized by (cid:104) χ k | χ k (cid:105) = (cid:82) dv k χ ∗ k χ k = 1 . The r.h.s. of eqs. (18) and (19) are associatedwith non adiabatic quantum effects. They are generally neglected in theleading order to the BO approximation.On multiplying both sides by ˆ P k = (cid:81) j (cid:54) = k (cid:104) ˜ χ k | eq. (19) can be split into a setof equations, each governing the dynamics of a single mode k of the matter9eld. One is then led to ˜ ψ ∗ ˜ ψ (cid:104) ˆ H ( M ) k − (cid:104) ˜ χ k | ˆ H ( M ) k | ˜ χ k (cid:105) (cid:105) ˜ χ k + 1 (cid:101) M (cid:32) ˜ ψ ∗ ∂ ˜ ψ∂a (cid:33) × ∂ ˜ χ k ∂a = 12 (cid:101) M ˜ ψ ∗ ˜ ψ (cid:20) (cid:104) ˜ χ k | ∂ ∂a | ˜ χ k (cid:105) − ∂ ∂a (cid:21) ˜ χ k . (21)We may now perform the semiclassical limit for the gravitational wave func-tion ψ ( a ) by setting ˜ ψ ( a ) ∼ ( (cid:101) M a (cid:48) ) / exp (cid:18) − i (cid:90) a (cid:101) M a (cid:48) da (cid:19) (22)obtaining the Friedmann equation − (cid:101) M a (cid:48) + (cid:88) k (cid:104) ˆ H ( M ) k (cid:105) = 0 (23)for Eq. (18), to the leading order. In such a way the BO decomposition ofthe wave function of the universe is uniquely determined and a and η arerelated.Now, on defining | χ k (cid:105) s ≡ e − i (cid:82) η (cid:104) ˜ χ k | ˆ H ( M ) k | ˜ χ k (cid:105) dη (cid:48) | ˜ χ k (cid:105) , Eq. (21) becomes i∂ η | χ k (cid:105) s − ˆ H ( M ) k | χ k (cid:105) s = exp (cid:104) i (cid:82) η (cid:104) ˜ χ k | ˆ H ( M ) k | ˜ χ k (cid:105) dη (cid:48) (cid:105) (cid:101) M × (cid:20) ∂ a − a (cid:48)(cid:48) ( a (cid:48) ) ∂ a − (cid:104) ˜ χ k | (cid:18) ∂ a − a (cid:48)(cid:48) ( a (cid:48) ) ∂ a (cid:19) | ˜ χ k (cid:105) (cid:21) | ˜ χ k (cid:105)≡ (cid:15) (cid:104) ˆΩ k − (cid:104) ˆΩ k (cid:105) s (cid:105) | χ k (cid:105) s (24)where (cid:104) ˆ O (cid:105) s ≡ s (cid:104) χ k | ˆ O | χ k (cid:105) s and (cid:15) ≡ (cid:101) M . In Eq. (24) we have retained allterms, in order to consistently include contributions to O (cid:16) (cid:101) M − (cid:17) (differentexpansions have been previously examined and compared for the homoge-neous case [13]). The operator ˆΩ k has the following form: ˆΩ k = 1 a (cid:48) d dη + (cid:34) i (cid:104) ˆ H ( M ) k (cid:105) s a (cid:48) − a (cid:48)(cid:48) a (cid:48) (cid:35) ddη . (25)10he operator on the r.h.s. of Eq. (24) has a nonlinear structure, since itdepends on χ s and χ ∗ s through multiplicative factors of the form (cid:104) ˆ O (cid:105) s . Weimmediately note that ,in the absence of the r.h.s., Eq. (24) becomes theusual matter evolution equation (Schrödinger or Schwinger-Tomonaga). Theterms on the r.h.s. describe the non-adiabatic effects of quantum gravita-tional origin. We are interested in the observable features of the spectrum of the scalar/tensorfluctuations generated during inflation. Such features can be extracted fromthe two-point function p ( η ) ≡ s (cid:104) | ˆ v | (cid:105) s = (cid:104) ˆ v (cid:105) (26)at late times (for the modes well outside the horizon). In (26) the vacuumstate | (cid:105) s satisfies the full equation (24) and, according to standard prescrip-tions, reduces to the Bunch-Davies (BD) vacuum [14] in the short wavelengthregime (more general assumptions may be considered as well). Let us notethat p ( η ) also depends on k but, in order to keep notation compact, wedecided to omit any explicit reference on it. Before tackling the problem of evaluating the evolution of p ( η ) by takinginto account the full dynamics given by (24), in this section we shall brieflyreview the basic formalism for the unperturbed dynamics.For each k mode, on neglecting the quantum gravitational effects, Eq. (24)takes the form of a time dependent Schrödinger equation for a harmonicoscillator with time dependent frequency ˆ H ( M ) k = ˆ π k ω k v k (27)where ω k = ω k ( η ) . The subscript k and the label ( M ) will henceforth beomitted. The following consideration will be valid for both scalar and tensorperturbations. 11t the classical level, v and π satisfy the Hamiltonian equations leading tothe homogeneous classical Klein-Gordon equation (equation of a harmonicoscillator with a time dependent frequency) : v (cid:48)(cid:48) + ω v = 0 . (28)At the quantum level, the solutions of the time dependent Schroedinger equa-tion can be found by introducing a linear invariant operator ˆ I , satisfying thedifferential equation i ddη ˆ I + (cid:104) ˆ I, ˆ H (cid:105) = 0 (29)and building up a complete set of states from the invariant vacuum state | vac (cid:105) , defined by ˆ I | vac (cid:105) = 0 , and then iteratively applying ˆ I † to the vacuum.A linear invariant satisfying (29) is given by I = i (cid:2) ϕ ∗ ˆ π − ( ϕ ∗ ) (cid:48) ˆ v (cid:3) (30)where ϕ ∗ satisfies the classical equation of motion (28). The commutatorsatisfies (cid:104) ˆ I, ˆ I † (cid:105) = 1 , provided the wronskian condition i (cid:2) ϕ ∗ ϕ (cid:48) − ( ϕ ∗ ) (cid:48) ϕ (cid:3) = 1 (31)holds. Then, in the coordinate representation, the properly normalised in-variant vacuum is (cid:104) v | vac (cid:105) = (cid:20) π ( ϕ ∗ ϕ ) (cid:21) / exp (cid:20) i ϕ ∗ ) (cid:48) ϕ ∗ v (cid:21) (32)and a suitable phase is needed in order for | vac (cid:105) to satisfy the Schroedingerequation. One easily finds | (cid:105) s = exp (cid:20) − i (cid:90) η dη (cid:48) ϕ ∗ ϕ (cid:21) | vac (cid:105) . (33)Let us note that the wronskian condition, (31), does not fix the invariantvacuum in a unique way. In general, different linearly independent combina-tions of solutions of eq. (28), satisfying the wronskian condition, are allowed.The BD prescription is only one of the possible choices. Consequently theexpression (32) is a more general vacuum state satisfying the unperturbedquantum dynamics. 12he linear invariants may be alternatively defined in terms of the so-calledPinney variable. In particular ˆ I can be written as: ˆ I = e i Θ √ (cid:20)(cid:18) ρ − iρ (cid:48) (cid:19) ˆ v + iρ ˆ π (cid:21) (34)where ρ is the Pinney variable, a real function satisfying the following nonlinear differential equation (the so-called Ermakov–Pinney equation [15]) ρ (cid:48)(cid:48) + ω ρ = 1 ρ (35)with Θ = (cid:82) η dη (cid:48) ρ . In terms of ρ the commutator (cid:104) ˆ I, ˆ I † (cid:105) = 1 is now triviallysatisfied. The Pinney variable is related to the solution ϕ of the classicalfield equation (28) by ρ = (cid:112) ϕ ∗ ϕ (36)hence it is proportional to its modulus. In the coordinate representation, theproperly normalised vacuum, expressed in terms of the Pinney, variable is (cid:104) v | (cid:105) s = 1( πρ ) / exp (cid:20) − i (cid:90) η dη (cid:48) ρ − v (cid:18) ρ − i ρ (cid:48) ρ (cid:19)(cid:21) . (37)Let us finally note that the two point function is given by p ( η ) = ϕ ∗ ϕ = ρ . (38) When quantum gravitational effects are taken into account, one must solvethe integro-differential equation (24), which is an extremely difficult task.Instead of trying to solve (24) and then calculating the power spectrum, onecan find the differential equation for the spectrum p , by iteratively differenti-ating the two-point function and using the canonical commutation relations.On taking | χ k (cid:105) s = | (cid:105) s in Eq. (24) (we are omitting the subscript k ) oneobtains the evolution equation for the vacuum i ddη | (cid:105) s − ˆ H | (cid:105) s − (cid:104)(cid:16) i (cid:104) ˆ H (cid:105) g ( η ) + g (cid:48) ( η ) (cid:17) × (cid:18) ddη − (cid:104) ddη (cid:105) (cid:19) + g ( η ) (cid:18) d dη − (cid:104) d dη (cid:105) (cid:19)(cid:21) | (cid:105) s (39)13ith (cid:104) ˆ O (cid:105) ≡ s (cid:104) | ˆ O | (cid:105) s and g ( η ) = (cid:101) M a (cid:48) . The evolution of the two-pointfunction can be now calculated by differentiating (26) w.r.t. η and using(39). The first derivative of p w.r.t. the conformal time is i dpdη = (cid:104) (cid:104) ˆ v , ˆ H (cid:105) (cid:105) − (cid:104) ˆ v (cid:105) F ( η ) + G ˆ v ( η ) (40)where F ( η ) = (cid:16) ig (cid:104) ˆ H (cid:105) + g (cid:48) (cid:17) (cid:104) ∂ η (cid:105) + g (cid:104) ∂ η (cid:105) − c.c. , (41) G ˆ v ( η ) = (cid:16) ig (cid:104) ˆ H (cid:105) + g (cid:48) (cid:17) (cid:104) ˆ v ∂ η (cid:105) + g (cid:104) ˆ v ∂ η (cid:105) − c.c.. (42)Let us note that g is a real function and F and G ˆ v are then purely imaginaryfunctions of η by construction. The subscript ˆ v in (42) indicates that thefunction G depends on η and on the operator ˆ v . The commutator in theexpression (40) is (cid:104) ˆ v , ˆ H (cid:105) = i { ˆ v, ˆ π } . In a more compact form Eq. (40) canthen be written as d (cid:104) ˆ v (cid:105) dη = (cid:104){ ˆ v, ˆ π }(cid:105) − iR (ˆ v ) (43)where R contains the quantum gravitational effects and is defined as R ( ˆ O ) = −(cid:104) ˆ O (cid:105) F ( η ) + G ˆ O ( η ) . The above expression can be differentiated once morew.r.t. η and takes the following form d (cid:104) ˆ v (cid:105) dη = d (cid:104){ ˆ v, ˆ π }(cid:105) dη − i dR (ˆ v ) dη . (44)and, in analogy with (40): d (cid:104){ ˆ v, ˆ π }(cid:105) dη = − i (cid:104) (cid:104) { ˆ v, ˆ π } , ˆ H (cid:105) (cid:105) − iR ( { ˆ v, ˆ π } ) . (45)The commutator in the expression above becomes (cid:104) { ˆ v, ˆ π } , ˆ H (cid:105) = 2 i (ˆ π − ω ˆ v ) and (44) can be then rewritten as d (cid:104) ˆ v (cid:105) dη = 2 (cid:0) (cid:104) ˆ π (cid:105) − ω (cid:104) ˆ v (cid:105) (cid:1) − iR ( { ˆ v, ˆ π } ) − i dR (ˆ v ) dη . (46)On then calculating the derivative of Eq. (46) we finally obtain: d (cid:104) ˆ v (cid:105) dη = d (cid:104) ˆ π (cid:105) dη − ωω (cid:48) (cid:104) ˆ v (cid:105) − ω d (cid:104) ˆ v (cid:105) dη − i dR ( { ˆ v, ˆ π } ) dη − i d R (ˆ v ) dη , (47)14here d (cid:104) ˆ π (cid:105) dη + iR (ˆ π ) = − i (cid:104) (cid:104) ˆ π , ˆ H (cid:105) (cid:105) = iω (cid:104) (cid:104) ˆ v , ˆ H (cid:105) (cid:105) (48)and (cid:104) (cid:104) ˆ v , ˆ H (cid:105) (cid:105) = i d (cid:104) ˆ v (cid:105) dη − R (ˆ v ) . (49)Equation (47) finally becomes d (cid:104) ˆ v (cid:105) dη + 4 ω d (cid:104) ˆ v (cid:105) dη + 2 (cid:0) ω (cid:1) (cid:48) (cid:104) ˆ v (cid:105) + 2 iR (ˆ π )+ 2 iω R (ˆ v ) + i dR ( { ˆ v, ˆ π } ) dη + i d R (ˆ v ) dη . (50)Let us note that eq. (50) is exact (no simplifications have been done to obtaineq. (50) starting from (39)). Further eq. (50) has been obtained withoutusing any peculiar property of the vacuum state and is also valid for anystate satisfying the modified Schroedinger equation (24).A perturbative approach is needed in order to solve eq. (50). To the firstorder in (cid:101) M − , one can then evaluate the quantum gravitational correctionson the unperturbed vacuum (37) and then identify ρ → √ p . The differentialmaster equation governing the evolution of the two point function is finally d pdη + 4 ω dpdη + 2 dω dη p + ∆ p = 0 (51)with ∆ p = − P2 (cid:20) d dη h a (cid:48) − d dη p (cid:48) ( h + 2)4 pa (cid:48) − ddη h + 4 p (cid:48) a (cid:48) p + ωω (cid:48) ha (cid:48) (cid:21) (52)where h ≡ p (cid:48) + 4 ω p − . (53)The above equation is valid to the first order in (cid:101) M − and, in the (cid:101) M P → ∞ limit, it must reproduce the standard evolution of the two point function,which is known to satisfy the second order differential equation d pdη − p (cid:18) dpdη (cid:19) + 2 ω p − p = 0 . (54)15s can be easily derived from (35), given the relation (38). Differentiationof eq. (54) leads to the third order equation (51), without quantum gravita-tional effects.The above master equation can be used for the evolution of the vacuum (andnot of a generic quantum state). Let us observe that (51) is a third orderdifferential equation for p and also contains unphysical solutions, which donot satisfy the unperturbed eq. (54) in the (cid:101) M P → ∞ limit. In the short wavelength limit − kη (cid:29) , the classical equation (28) admitsplane wave solutions of the form v ± = √ k exp ( ± ikη ) and an arbitrary linearcombination provides a suitable initial state of the system. In particular,if one retains only positive frequency waves, correspondingly one has p = k . The initial condition p = k corresponds to the so-called BD vacuumprescription. The BD vacuum state is the quantum state which coincideswith the Hamiltonian vacuum, as initial condition. Let us note that, in theshort wavelength limit, h is zero for p = 1 / k and consequently, to the leadingorder, the quantum gravitational corrections calculated in our approach are ∆ p = 0 .The general combination of plane wave solutions is v = 1 √ k ( α exp ( ikη ) + β exp ( − ikη )) (55)corresponding to p = 12 k (cid:2) | α | + | β | + 2Re ( αβ ∗ exp (2 ikη )) (cid:3) . (56)The integration constants α and β are complex numbers, constrained by thewronskian condition (31) which leads to | α | − | β | = 1; (57)the BD vacuum simply corresponds to | β | = 0 . One may rewrite the expres-sion for p , given the condition (57), and find p = 12 k (cid:104) | β | + 2 | β | (cid:112) | β | (cos δ cos 2 kη − sin δ sin 2 kη ) (cid:105) (58)16here δ is the difference between the phases of α and β respectively. Letus note that 2 real parameter ( δ and | β | ) enter the final expression (58),playing the role of the 2 integration constants of the second order differentialequation (54). Let us note that, only for | β | = 0 , h = 0 and the quantumgravitational corrections are negligible in the short wavelength limit.If one solves the third order differential equation (51), even on neglecting thequantum corrections, 3 integrations constants are necessary for the generalsolution. However only a subset of these solutions is physical, i.e. satisfy eq.(54), and one then expects some relation holds among the three integrationconstants. On solving the eq. (51), in the short wavelength limit ( − kη (cid:29) )and (cid:101) M P → ∞ , one finds p (cid:39) k [ c + − c − cos (2 kη ) + c sin (2 kη )] . (59)then on comparing with (58) we have c + k = 1 + 2 | β | , c − k = − | β | (cid:112) | β | cos δ, c k = − | β | (cid:112) | β | sin δ (60)or equivalently c − c − − c = k , c + > . (61)The BD vacuum corresponds to c − = c = 0 . In this section we apply our formalism to diverse inflationary backgroundsand calculate the quantum gravitational corrections to primordial spectra.In particular we study a pure de Sitter evolution, power law inflation andfinally SR inflation. Our starting point is the equation d pdη − p (cid:18) dpdη (cid:19) + 2 ω p − p = − p (cid:90) η −∞ dη (cid:48) p ∆ p (62)which is obtained by integrating (51) and imposing the BD initial conditionson p , i.e. p ( −∞ ) = 1 / (2 k ) , p (cid:48) ( −∞ ) = p (cid:48)(cid:48) ( −∞ ) = 0 .17 .1 De Sitter evolution In order to illustrate the main effects of quantum gravity on the spectrum,starting from unperturbed exact expressions, the de Sitter case is first dis-cussed. Such a case can be obtained from realistic inflationary models in thelimit ˙ H → , at least for ∆ p = 0 .When H = const , one has ω = (cid:113) k − η for both scalar and tensor pertur-bations (the equation for the scalar sector must be obtained by starting froma general background evolution and then taking the ˙ H → limit).The BD solution of Eq. (54) is p = 1 + k η k η (63)leading to the following expression for ∆ p : ∆ p = 4 H (cid:101) M k η = − H k (cid:101) M p (cid:48) (64)to the first order in (cid:101) M P . Then eq. (62) can be rewritten as d pdη − p (cid:18) dpdη (cid:19) + 2 ω p − p = 2 H k (cid:101) M p (cid:0) p − p ∞ (cid:1) (65)with p ∞ = 1 / (2 k ) . The latter equation can be recast in the form of theoriginal, unperturbed equation (54), by defining ˜ p = p (cid:113) − H (cid:101) M k p ∞ (66)and ˜ ω = ω − H (cid:101) M k ≡ ˜ k − z (cid:48)(cid:48) z (67)with ˜ k = k (cid:115) − H (cid:101) M k ≡ N k k. (68)The general solution of d ˜ pdη − p (cid:18) d ˜ pdη (cid:19) + 2˜ ω ˜ p − p = 0 (69)18s known and is given by ˜ p = 12˜ k η (cid:26)(cid:113) ˜ k + c + c − (cid:16) k η (cid:17) + cos (cid:16) kη (cid:17) (cid:104) c ˜ kη − c − (cid:16) ˜ k η − (cid:17)(cid:105) + sin (cid:16) kη (cid:17) (cid:104) c (cid:16) ˜ k η − (cid:17) + 2 c − ˜ kη (cid:105)(cid:27) . (70)On setting the oscillatory contribution to zero ( c − = c = 0 ), one finally findsthe perturbed BD vacuum p = 1 + N k k η N k k η . (71)In the long wavelength limit, one finds the observable features of the primor-dial spectra p − kη → −→ k η (cid:16) − H (cid:101) M k (cid:17) (72)and, for H (cid:101) M k (cid:28) , such spectra behave as p − kη → −→ k η (cid:32) H (cid:101) M k (cid:33) = p (cid:32) H (cid:101) M k (cid:33) (73)i.e. quantum gravitational effects lead to a power enhancement w.r.t. thestandard results in the spectrum for large scales.Let us note that the length scale L , defined in the section 2, is hidden in theexpression for the quantum gravitational corrections. On returning to theoriginal physical quantities one has p → p/L , a → La , η → η/L and k → Lk .The scale L ≡ ¯ k − would then appear in the result, as an effect of the initialvolume integration of the homogeneous dynamics. Power-law inflation corresponds to the simplified case in which the Hubbleparameter depends on time, yet still the equations of motions, for both thehomogeneous part and the perturbations, can be solved exactly. In this casethe evolution of the scale factor is given by a ( η ) = a (cid:18) η η (cid:19) qq − . (74)19here q is a constant parameter, which is related to the variation of H by q ≡ (cid:16) − ˙ H/H (cid:17) − and the de Sitter limit is recovered for q → ∞ . Thedynamics of the scalar and the tensor perturbations are governed by thesame equation, which is given by (62) with ω = (cid:113) k − η − (cid:15) (1 − (cid:15) ) . The BDvacuum can be now expressed in term of the Hankel functions: p = − πη (cid:2) H (1) ν ( − kη ) H (2) ν ( − kη ) (cid:3) (75)with ν = + q − . The observable features of the primordial spectra can becalculated by taking the long wavelength limit of (75) , finding p → − η π Γ( ν ) (cid:18) − kη (cid:19) − ν = (cid:20) q +1 q − Γ( ν ) π (cid:21) η − qq − k − ν (cid:18) aa (cid:19) . (76)Alternatively one may solve (54) in the long wavelength regime. In this caseone simply observes that ω → − a (cid:48)(cid:48) /a , p → C a and the normalization, C , can be fixed ,by matching the long wavelength solution with the shortwavelength prescription for the BD vacuum ( p → / (2 k ) ) at the horizoncrossing ( k = a k H k ), namely: C k H k = 12 k ⇒ C = 12 ka k . (77)One then obtains p → k (cid:18) aa k (cid:19) = 12 (cid:20) q ( q − (cid:21) qq − η − qq − k − ν (cid:18) aa (cid:19) (78)On comparing the results (76) and (78), we observe that the normalizationconstant C ,obtained by the matching procedure, is very close to the exactnormalization when q is large (and they coincide in the q → ∞ limit, i.e. forthe de Sitter case).One can also adopt the matching procedure to solve the perturbed equa-tion (51). We already observed that the quantum gravitational correctionsare negligible to leading order, in the short wavelength limit. Conversely theycan be evaluated perturbatively and then the long wavelength limit taken.In such a limit we find that p ∆ p → A (cid:0) p (cid:1) (cid:48) (79)20ith A = − C k (cid:101) M ( q −
1) (2 q + 1) q ( q + 1) (80)where C is the normalization of p .On neglecting ∆ p in the interval ] −∞ , η k ] , one is then led to the followingperturbed equation, valid in the long wavelength regime d pdη − p (cid:18) dpdη (cid:19) + 2 ω p − p = − p (cid:90) ηη k dη (cid:48) p ∆ p (81)where η k is the conformal time at the horizon crossing a k H k = k . Let us notethat (81) is now obtained by integrating (51) and, on imposing the conditions p ( η k ) = 1 / (2 k ) , p (cid:48) ( η k ) = p (cid:48)(cid:48) ( η k ) = 0 .The integral on the r.h.s. of eq. (81) can be easily performed, in the longwavelength regime given (79), and takes the form: d pdη − p (cid:18) dpdη (cid:19) + 2 ω p − p + A p − A k p = 0 . (82)On defining ˜ k = k (cid:113) A k and ˜ p = p/ (cid:16)(cid:113) A k (cid:17) , this latter equationcan be cast in the form of the unperturbed equation , having the followingsolution ˜ p = ˜ C a . (83)The normalization factor ˜ C can here be fixed, by matching the short andthe long wavelength solutions at the horizon crossing, i.e. when a ˜ k H ˜ k = ˜ k .One then finds k (cid:16)(cid:113) A k (cid:17) = ˜ C a k (84)and ˜ C = 12 k (cid:16)(cid:113) A k (cid:17) (cid:18) ν − − η (cid:19) ν − ˜ k − ν +1 a . (85)The perturbed solution in then given by p → (cid:18) qq − (cid:19) qq − ( η ) − qq − k − ν (cid:18) aa (cid:19) (cid:18) A k (cid:19) − qq − (86)21nd the quantum gravitational corrections, which are enconded in the factor (cid:0) A k (cid:1) − qq − , are negligible for large k . Let us note that this behaviouris simply dictated by the dependence of A on k , that is , it is related tothe dependence on k of the unperturbed solution (in the de Sitter limit onecorrectly reproduces the k dependence). The de Sitter and the Power law evolutions are fairly good approximations tothe inflationary dynamics. Furthermore these models permit an almost exacttreatment of the primordial fluctuations and are thus of pedagogical interest.A wider class of more realistic inflationary models is that associated with theslow-roll dynamics. In such a case the evolution of cosmological perturbationsoccurs during a generic inflationary phase having a slowly varying Hubbleparameter and a scalar field. The diverse inflationary models are then treatedwithin the slow-roll (SR) approximation and the features of the spectra ofperturbations, generated during inflation, are accurately estimated in such aframework, with an accuracy comparable with the magnitude of the so calledSR parameters. It is then worth generalizing our procedure to such a case.In the GR framework it is quite common to introduce the SR parameters (cid:15) SR ≡ − ˙ HH and η SR ≡ − ¨ φ H ˙ φ (87)and calculating the spectra just in terms of these two. The SR approximationconsists of neglecting their derivatives (that is treating them as constants)or, equivalently, to only keeping first order contributions in the SR variables. To first order in the SR approximations, the scale factor evolution satisfiesthe equation aH (cid:39) − (cid:15) η (88)and its solution is then given by a = a (cid:18) η η (cid:19) (cid:15) . (89)In terms of the above quantities one finds ω = k − z (cid:48)(cid:48) z = k − (cid:0) (cid:15) SR − η SR (cid:1) η (90)22or the scalar perturbation and ω = k − a (cid:48)(cid:48) a = k − (cid:0) (cid:15) SR (cid:1) η (91)for the tensor perturbations. In contrast with the de Sitter and power-lawcases, the equations for the scalar and the tensor perturbations are now differ-ent. However, because of the forms of (90) and (91), it is possible to recoverthe equation/solution for the tensor perturbations starting from the equa-tion/solution for the scalar perturbations and taking the limit η SR → (cid:15) SR .We shall then focus on the scalar case and finally extract the tensor caseresults in the above limit.We proceed in a fashion analogous to the power-law case. In the short wave-length regime, the quantum gravitational corrections evaluated perturba-tively are absent at the leading and next to leading order. We thus neglecttheir contribution in such a limit. Conversely, in the long wavelength regime,the quantum gravitational correction should be taken into account and canbe evaluated perturbatively. Finally the matching at the horizon crossing isperformed.In the long wavelength regime, the quantum corrections may be rewrittenas ∆ p = a H k (cid:101) M (cid:18) (cid:15) SR − η SR ) − k a H (cid:19) ≡ ∆ + ∆ (92)where the first term ∆ ≡ a H k (cid:101) M ( (cid:15) SR − η SR ) , (93)is peculiar for the scalar sector in the SR case and the second term ∆ ≡ − a H k (cid:101) M (94)is common for de Sitter and Power Law cases. To the leading order, p = C a (cid:15) SR with c = H k / (2 k (cid:15) SR ) , p (cid:48) /p = 2 aH and p (cid:48)(cid:48) /p = 6 a H . Theperturbed second order equation for p is (81), where the integration on ther.h.s. is taken from η k = − /k to η .On integrating by parts one then finds p (cid:90) η − /k dη (cid:48) p ∆ = A p (cid:90) η − /k dη (cid:48) a H p (cid:48) = A p (cid:18) a H p − k H k (cid:19) (95)23ith A = (cid:15) SR − η SR )2 k (cid:101) M and p (cid:90) η − /k dη (cid:48) p ∆ = − B p (cid:90) η − /k dη (cid:48) a H p (cid:48) = − B p (cid:18) a H p − kH k (cid:19) (96)with B = k (cid:101) M .The equation for p then takes the following form: (cid:34) (cid:15) SR − η SR ) H H k k (cid:101) M (cid:35) p (cid:48)(cid:48) − ( p (cid:48) ) p + 2 (cid:32) k − H kH k (cid:101) M − z (cid:48)(cid:48) z (cid:33) p = 12 p (cid:32) − H k k (cid:101) M (cid:33) (97)and can be rewritten as (cid:32) δ k (cid:101) M (cid:33) ˜ p (cid:48)(cid:48) − (˜ p (cid:48) ) p + 2 (cid:18) ˜ k − z (cid:48)(cid:48) z (cid:19) ˜ p = 12˜ p (98)with δ k ≡
718 ( (cid:15) SR − η SR ) H k k , (99) ˜ k ≡ k (cid:115) − H k k (cid:101) M , (100) ˜ p ≡ (cid:32) − H k k (cid:101) M (cid:33) − / p (101)where, on replacing H → H k , we neglected, to the leading order in SR, thetime dependence of H . The equation for ˜ p is very similar to (69), exceptfor the contribution proportional to δ k / (cid:101) M . If δ k / (cid:101) M (cid:28) , which is consis-tent with our perturbative approach, one finds the following long wavelengthsolution for ˜ p ˜ p (cid:39) ˜ C z (cid:18) − δk (cid:101) M2P (cid:19) (102)and consequently one has p = ˜ C (cid:115) − H k k M P2 z (cid:18) − δk (cid:101) M2P (cid:19) . (103)24he integration constant ˜ C is fixed by connecting the long wavelength solu-tion to p = 1 / k , when each mode ˜ k crosses the horizon ( ˜ k = a ˜ k H ˜ k ). Finallyone has p = 12 k a H k k (cid:16) − H k k (cid:101) M (cid:17) − ( (cid:15) SR − η SR ) H kk (cid:101) M2P (104)in the long wavelength regime and, given the smallness of the quantum grav-itational corrections ( H k / M P (cid:28) ), one finally finds the expression p (cid:39) C a (cid:15) SR (cid:34) H k (cid:101) M (cid:18) −
718 ( (cid:15) SR − η SR ) ln a H k (cid:19)(cid:35) (105)valid for the scalar sector. In the tensor sector one easily obtains the correc-tions in the limit η SR → (cid:15) SR . For such a case p = a H k (cid:16) − H k (cid:101) M (cid:17) (106)and p (cid:39) C a (cid:15) SR (cid:32) H k (cid:101) M (cid:33) . (107) The effect of ∆ p on the evolution of the two-point function p is that of addingto the standard, unperturbed, BD solution p BD a contribution of order (cid:101) M − .When realistic inflationary models are considered, these modified spectra arederived from (105) and (107) by replacing k → (cid:0) k/ ¯ k (cid:1) , where ¯ k is an unspec-ified reference wave number. The appearance of ¯ k = L − ,in the quantumcorrections, can be traced back to the three volume integral in the originalaction for the homogeneous inflaton-gravity system plus perturbations (seethe action (5)). Such a volume, on a spatially flat homogeneous space-time,is formally infinite and consequently the value of ¯ k remains undetermined.Naively one may argue that ¯ k is related to an infrared problem (divergence)and indeed, in the literature, its value is taken to be the infrared cut-off forthe perturbations, namely the largest observable scale in the CMB. Alterna-tively one may consider it to be the scale at which new effects or physics set25n. We shall briefly return to this in the conclusions.In the previous section we calculated the form of the quantum gravitationalmodifications to the primordial scalar spectrum, in the case of SR inflation Q k = 1 + H ¯ k (cid:101) M k (cid:18) −
718 ( (cid:15) SR − η SR ) ln a H k (cid:19) . (108)In such an expression, the wavenumber k necessarily refers to the scales,around the pivot scale k ∗ , which are probed by the CMB and exited from thehorizon N ∗ ∼ e-folds before inflation ends. Its contribution to (108) is (cid:18) kaH (cid:19) − (cid:15) SR − η SR ) (cid:39) (cid:18) k ∗ a ∗ e N ∗ H k ∗ (cid:19) − (cid:15) SR − η SR ) (cid:39) e N ∗ ( (cid:15) SR − η SR ) (109)and may well lead to a contribution of O (1) for reasonable values of the SRparameters of the order of 1 per cent. Let us note that the first equality,in (109), is strictly valid for the modes very close to the pivot scale k ∼ k ∗ = a ∗ H k ∗ . Away from the pivot scale, small deviations proportional to theSR parameters, − (cid:15) SR − η SR ) ln (cid:16) kk ∗ (cid:17) , are neglected. Depending on the SRparameters and on N ∗ , the quantum corrections Q k may lead to a power lossor a power increase for large scales which can be generically parametrized inthe following form: p ( L ) (cid:39) p ( L )0 (cid:34) ± q (cid:18) k ∗ k (cid:19) (cid:35) (110)where p ( L )0 is p without quantum corrections and evaluated in the long wave-length regime. The quantity inside the square brackets is Q k . An analogousparametrization holds for the tensor sector with a different q . The parametrization of the primordial spectra by (110) is still not suitable forcomparison with observations. In the k (cid:28) k ∗ limit the quantum gravitationalcorrections are either negative or very large (infinite in the k → limit).Such an apparently pathological behavior is simply a consequence of theperturbative technique employed to evaluate the corrections. One may hopethat resummation to all orders leads to a finite result. In any case we are not26able 1: Range of parameters varied τ ln (10 A s ) n s r α s q q [0 . , .
8] [2 . , .
0] [0 . , . [0,0.8] [-0.1,0.1] [0,21] [0,0.5]allowed to extend the validity of the perturbative corrections up to O (1) .Thus, instead of introducing a sharp cut-off on the NLO expressions for themodified spectra by multiplying q by an ad hoc step function which keepsthe correction small but leads to a discontinuous spectrum, we interpolateour expression through a well defined function, with a finite and reasonablebehavior in the k → limit. Such a function, which must reproduce (110)when q ( k ∗ /k ) (cid:28) , may be regarded as a resummation of the perturbativeseries.In order to restrict the number of parameters which will be fitted by thecomparison with the data and still allow for different limits when k → ,weconsider the following parametrization: p ( L ) (cid:39) p ( L )0 q (cid:0) k ∗ k (cid:1) q (cid:0) k ∗ k (cid:1) ∼ p ( L )0 (cid:34) q − ˜ q ) (cid:18) k ∗ k (cid:19) (cid:35) . (111)where one more parameter w.r.t. (110) has been added, in order to obtain aregular expression for k small. Let us note that the above modifications aresubstantially different from considering a running spectral index α s , such as p ( L ) (cid:39) p ( L )0 (cid:18) k ∗ k (cid:19) − αs ln ( kk ∗ ) . (112)Indeed for the latter case, the standard power law dependence is affected atboth large and small scales and, in particular, a negative running would leadto a zero amplitude in the k → limit and a smaller amplitude w.r.t. simplepower law when k (cid:29) k ∗ . On the other hand the modified spectrum (111)reduces to the power law case when k (cid:29) k ∗ and may lead to a non zeroamplitude when k → , depending on the choice of the parameters ˜ q , . In this section we report the comparison between the theoretical predictionsgiven by (111) and Planck 2015 [16] dataset. The analysis is performed using27he Markov Chain Monte Carlo (MCMC) code
COSMOMC [17], which hasbeen properly modified to take into account the estimated quantum gravita-tional effects.Let us note that BD vacuum in the tensor sector gives a power increasefor large scales in the tensor spectrum. Such an increase would be coun-terbalanced by a loss of power in the scalar sector, as far as temperaturecorrelations are concerned. One may parametrize such a power increase ina suitable way, just as we did for the scalar sector ,in order to eliminate thedivergence for small k and fit the corresponding parameter with the data atour disposal. Since our main source of data comes from temperature correla-tions, which do not discriminate between scalar and tensor fluctuations, weneglect a priori quantum gravitational corrections in the tensor spectrum.Such a choice is a simplifying assumption done in order not to have to dis-entangle possible degenerate parameters. Let us note, however, that sucha choice can be realized physically either by an appropriate vacuum choice,differing from a pure BD, or by a very long cutoff scale associated with tensordynamics. Thus we limit our analysis to a subset of the more general case,for which the quantum gravitational corrections affect the tensor sector in anon negligible way, thus minimizing the power loss in the scalar sector. Thetensor spectrum is then given by the unperturbed power law expression p t = A t (cid:18) kk ∗ (cid:19) n t . (113)and we assume that the LO spectra are generated by the conventional SRmechanism and single field inflation. The consistency condition, relatingscalar and tensor spectral indices and the tensor to scalar ration, is validwhen quantum gravitational corrections are neglected. Indeed throughoutthe analysis we assume that the consistency relation (already implementedin COSMOMC ) between the spectral indices and the tensor to scalar ratio n t = − r (cid:16) − n s − r (cid:17) (114)holds to the second order in the SR approximation and the amplitude of thespectrum of tensor perturbations is given by A t = rA s , to the leading orderin M P − , i.e. on neglecting the quantum gravitational corrections. We thenconsider a primordial scalar spectrum p ( L ) parametrized by p s (cid:39) p ( L )0 − q ) (cid:0) k ∗ e q k (cid:1) (cid:0) k ∗ e q k (cid:1) (115)28able 2: List of ModelsModel A s , n s , r A s , n s , r , α s A s , n s , r , q , q − q simply fixes the limit of p s when k → . The parameter q isrelated to the scale ¯ k , i.e. that at which the quantum gravitational modifi-cations of the spectrum become important. In the limit q → ∞ ( ¯ k → ),the quantum gravitational corrections are suppressed and for q = 0 one has ¯ k = k ∗ . Let us note that q = 0 ,or q → ∞ , correspond to the standardpower-law case with no loss of power ( p s = p ( L )0 ) and q = 0 . corresponds tozero power at k = 0 . The expression (115) is a parametrization equivalentto (111), with ˜ q = exp ( − q ) (1 − q ) and ˜ q = exp ( − q ) , which we havefound to be more convenient to be used in COSMOMC .Our analysis is based on the Planck datasets released in 2015 and in-cludes the Planck TT data with polarization at low l (PL), and the dataof the BICEP2/ Keck Array -Planck joint analysis (BK) [18]. In particular weuse plik_dx11dr2_HM_v18_TT , lowTEB and BKPlanck publicly availablePlanck likelihoods. We find the best fit for our model with and without BKdata and compare it with standard power law predictions, and with thoseassuming a non negligible running of the spectral index (112).For simplicity we obtained the best-fits for the parameters of the primordialspectra shown in Table 1 and the parameters are taken to vary with uniformpriors in the intervals indicated in the same table. The priors for τ , A s , n s , r and α s are those used by the Planck 2015 analysis. The remaining cos-mological parameters are fixed to the Planck best-fit and in particular wechose θ MC = 1 . , Ω b h = 0 . , Ω c h = 0 . (116)Let us note that the pivot scale k ∗ is .
05 Mpc − and is the same forboth the scalar and the tensor sector. The additional parameters q and q are chosen to vary in the largest possible interval leading to a power loss29or large scales (compared with the pivot scale), with the parametrizationchosen. At present our theoretical predictions are not able to constraint thevalue of such parameters, or estimate possible allowed intervals where to letthem vary (see [19] for an attempt to estimate priors from quantum gravity),thus the choice of broad enough priors seems reasonable.In particular the prior for q is chosen to let it vary between q = 0 , wherethe quantum gravitational corrections cancel out independently of q , and q = 1 / . The values for q with q > / , lead to an increase of power,those with q < , lead to a physically unacceptable negative spectrum andare thus excluded from the analysis.The choice of the prior for q is rather delicate with the parametrization cho-sen (115). On expanding (115) to the first order in the quantum gravitationalcorrections and comparing the result with the theoretical predictions (108),one finds, after some algebra, the following relation among the parametersof our model exp (3 q ) = 24 q π r · A s · Q ( n s , r, N ∗ ) (cid:18) k ∗ ¯ k (cid:19) (117)with Q ( n s , r, N ∗ ) ≡ (cid:16) − n s − r (cid:17) N ∗ − (118)where we have used the following standard SR relations for single field infla-tion: H ∗ M P2 (cid:39) π A s · r (119)and r = 16 (cid:15) SR , n s = 1 + 2 η SR − (cid:15) SR . (120)The prior for q then depends on some other priors of observables quan-tities (such as r and n s ) and on a few, related, physical assumptions.Let us first note that, with the priors considered for the quantities on ther.h.s. of (117), such expression may vary from −∞ to + ∞ . The case ofpower loss, which we are investigating, is only reproduced by positive val-ues of Q and ,correspondingly, the r.h.s. of (117) then varies in the interval [0 , + ∞ ] (let us note that the relation (117) is otherwise undefined). Such apositivity requirement can be fulfilled only by particular inflationary models,as figure (1) shows, with larger values of N ∗ generically favoured comparedto smaller ones. For example consider the case of chaotic inflation, driven by30igure 1: The figure plots the region (yellow area) compatible with a loss ofpower in the ( n s , r ) plane for N ∗ = 60 . The dotted lines are the contours ofthe N ∗ = 50 (small dots) and N ∗ = 70 (large dots) areas. These contoursare superimposed on the Planck 2015 analysis of various inflationary models.Hilltop quartic models and Natural inflation models lead to a loss in power,conversely chaotic inflation is not compatible with such a loss. a power-law potential V ∝ φ n . For such a case ( n s , r ) = (cid:18) − n + 2)4 N ∗ + n , n N ∗ + n (cid:19) N ∗ (cid:29) n −→ (cid:18) − n + 22 N ∗ , nN ∗ (cid:19) , (121) Q ∼ − (122)and (117) is undefined.Conversely for the Hilltop inflationary models, one has ( n s , r ) = (cid:32) − n − N ∗ ( n − , N ∗ ) − n − n − (cid:33) , (123)where n is defined by the shape of the potential V hilltop = V (cid:20) − (cid:18) φµ (cid:19) n (cid:21) (124)31nd Q ∼ − − n − n (125)leading to a loss in power for < n < / .Let us note that, with the form obtained for the quantum gravitational cor-rections, our model leads to severe constraints on the shape of the inflationarypotential. As shown in the figure (1), only a small subset of the inflationarymodels, satisfying the observed values of n s and r , lead to a loss of powerfor large scales. The remaining models would give a power increase, whichmay be a distinguishing feature, unless ¯ k is too small to be observed in theCMB. More generally, on referring to the classification in [20], power loss isassociated only with a sub set of class I models, with n s (cid:39) b/N ∗ , r ∝ /N − b ∗ and < b < . for < N ∗ < .For the models which lead to a loss in power we assume − Q ( n s , r, N ∗ ) ∼ O (10) (126)which is the order of magnitude of (125), far from the boundaries , / .The ratio k ∗ / ¯ k , where ¯ k − is the scale at which the power loss begins tobe observable, is taken in the interval [10 − , ] , where − is the order ofmagnitude of the shortest scales probed by Planck and corresponds tolargest scale one can observe (in units of the pivot scale). The tensor to scalarratio r , appearing at the denominator, in principle can be , as we vary itin the interval [0 , . . However, on attempting to provide a finite prior for q (and only in this context), we observe that most single field inflationarymodels generate a non zero tensor to scalar ratio and, therefore, we shallassume it varies in the interval [10 − , . where − ∼ O (1 /N ∗ ) . Similarly q is taken in the interval [10 − , / , where − corresponds to a powerloss, as smaller values of q would be indistinguishable from zero and leadto an infinite prior for q . Finally the amplitude A s varies in the interval [1 . · − , . · − ] . Given all such assumptions ,the prior for q can beestimated to be [0 , .The different combinations of primordial spectra and datasets considered,are listed in Table 2 with an index specifying the model number. The bestfits found, for the parameters we varied ,are presented in Table 3 and thecorresponding effective χ , defined as − L where L is the likelihood, arelisted in Tables 4 and 5. The differences between the total χ for the different32able 3: Monte Carlo Best-fits τ ln (10 A s ) n s r α s q q . · − . · − - - -2 . · − . · − - - -3 . · − . · − − . · − - -4 . · − . · − − . · − - -5 . · − . · − - .
48 1 . · − . · − . · − - .
64 5 . · − cases are reported, using our model as reference. In particular the cases 1and 3 are compared with 5 and the cases 2 and 4 are compared with 6.Table 4: Monte Carlo Comparison (PL) χ T ot ∆ χ ≡ χ − χ χ T ot ∆ χ ≡ χ − χ The MCMC results (see Tables 4 and 5) show that the quantum gravita-tional modification of the standard power law form for the primordial scalarspectrum, improves the fit to the data. Such improvements are much moresignificant w.r.t the standard modifications of the primordial spectra obtainedon considering a running spectral index. Let us note that the 2015 Planck33
Figure 2:
The figure shows the and confidence level constraints on r and n s . data give constraints on the running, which are quite different from thosecoming from the 2013 data. In particular the fit to the 2015 data does notimprove much if one considers a running spectral index in the scalar sector.The comparison of the marginalized 1-D likelihoods for the parameters q and q in Fig. (4) show that the two datasets lead to close predictions. Inparticular their marginalized maxima are q (cid:39) . , q (cid:39) . (127)when Planck data alone are considered and q (cid:39) . , q (cid:39) . (128)when BK data are added to the analysis. Correspondingly n s and A s alsotake very similar values for the best fit.The value of q indicates a ∼ − loss in power when k approacheszero. Let us note that the tensor to scalar ratio r is weakly constrained.From tables (4-5) we observe that cases 3 and 4, with a running spectralindex, are disfavoured w.r.t cases 1 and 2 respectively, since they almost thesame, effective, χ , but with one more independent d.o.f. to fit the data.Conversely the cases 5 and 6 ( ∆ χ > ) are favoured w.r.t. the cases 1-4, as34 lanck TT Planck TT + BK2.5 3.0 3.5 4.0 4.5 5.02.53.03.54.04.55.05.56.0 n
Log ( k M p c ) Figure 3:
Constraints on ¯ k for hilltop inflation as a function of n without(blue region) and with (red region) BK data. The region spans different N ∗ ∈ [50 , . an improvement greater than 2 for the effective χ is obtained, through theaddition of 2 independent parameters.In figure 4 we finally plot the marginalized likelihoods for r , q and q . Thecorresponding marginalized , and confidence intervals are listedin Tables 6-7. The marginalized likelihoods for q and q show a σ devia-tion from standard power law for both cases 5 and 6. Let us note that, oncomparing the results with those obtained from the Planck 2013 data, theconstraints on q and q are now weaker [8].Finally let us discuss the constraint on ¯ k . On assuming, for example,Hilltop inflation (123), one can invert the relation (117) obtaining ¯ kk ∗ (cid:39) exp ( − q ) q N n − n − ∗ π A s n − − n / . (129)Given that the amplitude A s is quite constrained by observations and,on using < N ∗ < , n = 4 , we obtain the corresponding values for ¯ k , which are very large compared to the wave number associated with thelargest observable scale in the CMB namely k min (cid:39) . · − Mpc − . Thesevalues are illustrated in figure (3) for the cases n (defined by (124)). Let us note that the existence of such a (relatively)small fundamental length may have relevant consequences on astrophysicalobservation. Indeed it is associated with distances which are comparable35igure 4: Marginalized 1-D likelihoods for r , q and q without (black line)and with (red line) BK data. with the diameter of a large galaxy or a galaxy cluster. We further observethat a 3 order of magnitude variation of the value of ¯ k can be obtained on“re-tuning” the parameters used for its estimate. Let us note that the es-timate for ¯ k , although illustrated for a specific inflationary model, is quitegeneral and can also be found for other diverse power loss compatible models.Table 6: Marginalized confidence intervals - Case 5
68% 95% 99% r [0 . , . · − ] [0 . , . · − ] [0 . , . · − ] q [1 . , . · ] [2 . , . · ] [2 . . . · ] q [0 . , . · − ] [0 . , . · − ] [0 . , . · − ] Table 7: Marginalized confidence intervals - Case 6
68% 95% 99% r [0 . , . · − ] [0 . , . · − ] [0 . , . · − ] q [2 . , . · ] [2 . , . · ] [2 . , . · ] q [0 . , . · − ] [0 . , . · − ] [0 . , . · − ] Conclusions
As we mentioned in the introduction the matter-gravity system is amenableto a Born-Oppenheimer treatment, wherein gravitation is associated with theheavy (slow) degrees of freedom and matter with the light (fast) degrees offreedom. Once the system is canonically quantised and the associated wavefunction suitably decomposed, one obtains that, on neglecting terms due tofluctuations (non-adiabatic effects), in the semiclassical limit gravitation isdriven by the mean matter Hamiltonian and matter follows gravitation adi-abatically, while evolving according to the usual Schwinger-Tomonaga (orSchrödinger) equation. Our scope in this paper has been to study perturba-tively the effect of the non-adiabatic contributions, for different inflationarybackgrounds. In particular we wished to see such effects on the observablefeatures of the scalar/tensor fluctuations generated during inflation. In orderto do this we obtained a master equation for the two-point function for suchfluctuations, which includes the lowest order quantum gravitational correc-tions. These corrections manifest themselves on the largest scales, since theassociated perturbations are more effected by quantum gravitational effects,as they exit the horizon at the early stages of inflation and are exposed tohigh energy and curvature effects for a longer period of time. Interestinglythe very short wavelength part of the spectrum remains unaffected and onemay consistently assume the BD vacuum as an initial condition for the evo-lution of the quantum fluctuations. Computationally this feature is relevantas it allows one to find the long wavelength part of the spectrum of the fluc-tuation through a matching procedure (similar to the standard case withoutquantum gravitational corrections).In particular one finds, for a de Sitter evolution, a power enhancement w.r.t.the standard results for the spectrum at large scales, with corrections be-having as k − . Such a k − was also found with similar approaches [21] andmay appear to be a peculiarity of such quantum gravity models. Howeverthe case of power law inflation is different: while power enhancement is alsotrue for power-law inflation, of interest for this case is that one finds thatthe k dependence of the quantum gravitational corrections differs from k − and is, perhaps not surprisingly, directly related to the k dependence of theunperturbed spectra.Finally it is the slow roll case that is more realistic and of greatest interest.The quantum gravitational corrections for the SR case have peculiar featuresand are very different from the de Sitter case. In particular, for the case of37he scalar fluctuations, their form is not simply a deformation of the de Sitterresult proportional to the SR parameters. New contributions arise due to SRand their effect is comparable with the de Sitter-like contributions for verylarge wavelengths. The new contributions are proportional to (cid:15) SR − η SR andare zero for the de Sitter and power-law cases. They can lead to a power-lossterm for low k in the spectrum of the scalar curvature perturbations at theend of inflation, providing the difference (cid:15) SR − η SR > . The evolution ofthe primordial gravitational waves has also been addressed. The quantumgravitational corrections also affect the dynamics of tensor perturbations anddetermine a deviation from the standard results in the low multipole region,which always leads to a power enhancement. In performing the analysis, forsimplicity, we restricted ourselves to the particular case of negligible quan-tum gravitational contributions to the spectrum of primordial gravitationalwaves. Further, since our corrections are perturbative,in order to keep themso for all values of k, we have suitably extrapolated our predictions for thescalar sector beyond the leading order, describing this in terms of two pa-rameters, and examined them down to k → . Other parametrizations havealso been considered, however the one we presented is the simplest and leadsto the best results.It is found that, given the form obtained for the quantum gravitational correc-tions, our model imposes severe constraints on the shape of the inflationarypotential ,as a loss in power at large scales is compatible with observations,whereas a power enhancement must be zero or extremely small to fit thedata. Only a small subset of the inflationary models, satisfying the observedvalues for n s and r , lead to a loss of power at large scales. The remainingmodels give a power increase which may be a distinguishable feature, unless ¯ k is too small to be observed in the CMB.Finally the analysis performed was based on Planck datasets released in 2015, include the Planck TT data with polarization at low l (PL) and the data ofthe BICEP2/ Keck Array -Planck joint analysis (BK) [18]. In our precedingpaper [8] our model predictions were tested through Planck 2013 and BI-CEP2 earlier data and the results were different. The MCMC results (seeTables 4 and 5) show that the quantum gravitational modification of thestandard power law form for the primordial scalar spectrum, improves the fitto the data. Such improvements are much more significant w.r.t the standardmodifications of the primordial spectra, obtained by considering a runningspectral index. Let us note that the 2015 Planck data give constraints onthe running, which are quite different from those coming from 2013 data. In38articular the fit to the 2015 data does not improve much if one considers arunning spectral index in the scalar sector. On including the BK data in ouranalysis, we find that the results take vey similar values for the best fit. Fur-thermore comparison with the data predicts, for our model, a loss in power ofabout − w.r.t. the standard power law as k approaches zero. and fixesthe scale ¯ k , which necessarily appears in the theoretical model. One findsvalues for ¯ k which are very large, compared to the wave number associatedwith the largest observable scale in the CMB (namely k min (cid:39) . · − Mpc ).Let us note that the existence of such a small fundamental length may haverelevant consequences on astrophysical observation. Indeed it is associatedwith distances which are comparable with the diameter of a large galaxy ora galaxy cluster. We further observe that a 3 order of magnitude variationof the value of ¯ k can be obtained on "re-tuning" the parameters used for itsestimate. Further we observe that the value of ¯ k , although illustrated for aspecific inflationary model, is quite general and is found for diverse powerloss compatible models. This is rather surprising and of course,assumingour proposed mechanism is correct, indicates the possible presence of newphysics at such scales. Actually such a result is not new. Indications for thishave been seen both from a study of the stability of clusters of galaxies or isassociated with the running of Newton’s constant ([22]). Acknowledgments
The work of A. K. was partially supported by the RFBR grant 14-02-00894.
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