Primordial perturbations in kinetically dominated regimes of general relativity and hybrid quantum cosmology

Primordial perturbations in kinetically dominated regimes of general relativity and hybrid quantum cosmology Beatriz Elizaga Navascu´es * JSPS International Research Fellow, Department of Physics, Waseda University, 3-4-1 Okubo, Shinjuku-ku, 169-8555 Tokyo, Japan Rafael Jim´enez-Llamas†and Guillermo A. Mena Marugán ‡ Instituto de Estructura de la Materia, IEM-CSIC, Serrano 121, 28006 Madrid, Spain (Received 10 June 2021; accepted 15 October 2021; published 18 November 2021) Scalar fields with an energy density dominated by its kinetic part may have played a relevant role in the very early stages of the Universe. Compared to the standard inflationary paradigm, they may lead to modifications in observable quantities, e.g., the anisotropies found in the cosmic microwave background. Kinetically dominated regimes arise in classical fast-roll scenarios as well as in quantum bouncing cosmologies. For instance, kinetic dominance is typical in interesting preinflationary phases of loop quantum cosmology. In this work, we analyze the leading-order effects that the presence of a scalar field potential causes on the primordial cosmological perturbations in these kinetically dominated epochs. These effects can be grouped in two sets, namely, those that affect the effective mass of the perturbations and those that affect the choice of their vacuum state. The effective mass is modified directly by terms that include the potential, but also indirectly by the change in the background dynamics and the relation between the parameterization of these dynamics and the conformal time, usually employed to describe the evolution of the perturbations. On the other hand, away from de Sitter inflation, the Bunch-Davies state is no longer the most natural vacuum at all scales. Recent proposals suggest to modify it by carrying out certain Hamiltonian diagonalization with a suitable asymptotic behavior at large wave number scales. Both this diagonalization condition and the imposed asymptotic behavior depend on the effective mass of the perturbations, and therefore the selected vacuum state varies in the presence of the scalar field potential. DOI: 10.1103/PhysRevD.104.103520 I. INTRODUCTION The level of precision that we have reached in cosmological observations has opened a new era in which we can falsify the predictions of the standard cosmological model [1], which includes a sufficiently large epoch of slow-roll inflation and a Gaussian distribution of perturbative inhomogeneities that can be explained by assigning to them a vacuum identified with the Bunch-Davies state [2]. This standard model has proven extremely successful. Nonetheless, some observations have raised the possibility that there may exist tensions with the theory [3–6]. For instance, the observations of the cosmic microwave background (CMB) reported by WMAP and the Planck mission [7–9] indicate a lack of power at large angular scales (for multipoles with number laround and below 30). Another example is the lensing amplitude associated with the gravitational lensing experienced by the CMB radiation in their propagation from the surface of last scattering [10]. Although, individually, the statistical significance of these observational anomalies is small, their combination points toward the exceptionality of the observed Universe unless we alleviate the tension with some new physics, beyond the otherwise successful standard inflationary paradigm. Considerable attention has recently been paid to possible solutions to this tension, in particular to the anomalous lack of power in the CMB, within the framework of classical general relativity for spatially flat cosmological models [11–23]. A solution to this problem could come from the realization that preinflationary cosmological stages can leave a trace in the observable Universe (see, e.g., [24]), for instance if the slow-roll period is not too large, although sufficiently long to avoid conflicts with the observational constraints on the number of e-folds [25–27]. The simplest situation with this behavior is found in models with a single scalar field in which the short slow-roll period is preceded by an epoch in which the energy density of the scalar field is dominated by its kinetic part, experiencing what has been called a fast-roll inflation [11,14,15,23]. The onset of *[email protected] †[email protected]fmac.csic.es ‡[email protected]s PHYSICAL REVIEW D 104, 103520 (2021) 2470-0010=2021=104(10)=103520(15) 103520-1 © 2021 American Physical Society inflation, shortly before slow-roll conditions hold in this class of models, typically introduces an infrared scale from which one may observe traces of fast-roll inflation on the primordial power spectrum [28]. Another possibility to alleviate the tensions comes from quantum cosmology, i.e., the description of the cosmological evolution according to quantum principles [29,30]. In quantum cosmological models, the preinflationary history of the Universe can be modified by quantum gravity phenomena as one approaches the big bang, in some circumstances providing even a bouncing mechanism that avoids this singularity. This is the case, for instance, when one studies certain physical states for the background in loop quantum cosmology (LQC) [31–33]. LQC is a quantum description of cosmological spacetimes [30,34–36] constructed by applying the techniques of nonperturbative loop quantum gravity [37,38]. In LQC, there exist certain families of Gaussian states that are peaked around trajectories on phase space that never reach the cosmological singularity [30,39,40].When the energy density of the scalar field is large, these trajectories deviate from those of Einsteinian cosmology. In this departure from general relativity, they experience large quantum geometry effects. These quantum preinflationary phenomena may affect the CMB power spectrum [4–6,41–52] if the energy density of the scalar field is kinetically dominated in those epochs [53]. Through the consideration of these families of states, it has been argued that the loop quantization favors a sufficient amount of slow-roll inflation [54–56]. This prediction depends on assumptions about the probability measure on the space of initial data. Actually, with some proposals about this measure, it is claimed that the most probable scenario is that the scalar field is in a kinetically dominated regime at the bounce [55]. For the purposes of our investigations, the viewpoint that we adopt here is a bit more pragmatic, in the sense that we are interested in peak trajectories of phenomenological interest. Namely, they must be compatible with the CMB observations (in what respects the amplitude and the spectral index of the scalar power spectrum) and the contraints on the number of e-folds during inflation, while displaying quantum geometry effects in the window of wavelength scales that can be observed today. For this class of trajectories, indeed, the energy of the scalar field at the bounce happens to be highly dominated by its kinetic contribution [53]. A similar situation, with kinetically dominated regimes that can leave traces in the CMB power spectrum, could be found in other quantum bouncing cosmological models, not necessarily within the context of LQC. A general framework, that allows for more general quantizations of the cosmological background than a loop quantization, is provided by the so-called hybrid quantum cosmology formalism. The hybrid quantization strategy combines such a quantum description of the background with a Fock description of its gauge invariant perturbations. The background quantization should include the construction of a space of states1andaninner product over the background geometry. At least within this hybrid quantization, it is possible to show that the scalar perturbations, eventually responsible for the temperature anisotropies of the CMB and that are usually described in terms of the Mukhanov-Sasaki (MS) field [57–59] (which is perturbatively gauge invariant), satisfy similar evolution equations in conformal time as in general relativity, except for the modification of the effective-mass term that appears in those equations [60]. In hybrid quantum cosmology, this mass is given by the ratio of the expectation values of two geometric operators, on the state that determines the quantum background. These expectation values are defined with the aforementioned inner product over the geometry. The quantum modifications of the effective mass affect the propagation of the primordial perturbations, making it possible that there appear changes in the primordial power spectrum if inflation is short lived. In particular, the quantization of the background cosmology typically introduces a scale of Planck order in the system, from which imprints of quantum cosmology phenomena may appear on this spectrum [28,53]. It is worth commenting that MS equations for the scalar perturbations with an effective mass modified by quantum geometry effects appear as well in other formalisms of quantum cosmology, like another approach to LQC called the dressed-metric formalism [42].2In this case, the background also presents an epoch of kinetic dominance, and the effective mass of the perturbations is given again in terms of a quantum expectation value, but different from the ratio of expectation values of the hybrid approach. A preliminary study of the dependence of the modified MS dynamics on the scalar field potential, when it is small, was carried out in Ref. [67]. Nonetheless, apart from being restricted to the hybrid LQC case, this study is incomplete at least in two aspects. First, any of the considered expectation values is subject to the quantum dynamics of the cosmological background, which is usually described in terms of an internal degree of freedom. On the other hand, the MS equations for the perturbations are parameterized in conformal time. The relation between this time and the internal variable used in the background evolution is also affected by the presence of a potential with respect to the situation with a free scalar field. This important point was not considered in Ref. [67]. Moreover, even if we have specific MS equations at our disposal to dictate the propagation of the scalar perturbations, their evolution cannot be integrated unless we fix 1In general, these states are nonphysical inasmuch as they still have to satisfy quantum constraints. One usually refers to this space of states as the kinematic (Hilbert) space. 2Certainly, there exist quantum cosmology formalisms in which the quantum corrections to the MS equations cannot be reduced to a modification of the effective mass (see, e.g., the socalled deformed constraint algebra approach [61–66]), which escape from the scheme that we consider in this work. BEATRIZ ELIZAGA NAVASCU ´ ES et al. PHYS. REV. D 104, 103520 (2021) 103520-2 some initial conditions for them. These initial conditions are usually interpreted as the choice of a vacuum state for the perturbations. In a standard slow-roll inflation that lasts long enough, so that the window of observable scales in the CMB exited the cosmological horizon only well inside the slow-roll regime, the relevant part of the evolution can be approximately described as a de Sitter phase (with small corrections), and in this sense it is natural to adopt initial conditions that correspond to the choice of a Bunch-Davies state for the perturbations [2]. Nonetheless, in scenarios of fast-roll inflation, for instance in general relativity or following quantum bounces, or more generically away from genuine slow-roll schemes, the adoption of a BunchDavies vacuum is no longer justified for scales that can be affected by the new physics, compared with the standard model. The vacuum state should be optimally adapted to the new background dynamics, which is not close to de Sitter anymore. Several proposals have been put forward to select a state with those properties [48,68–70]. Among them, it has been recently suggested that the choice of a set of positive frequency solutions, that determines the vacuum state, should come from a process of diagonalization of the contribution of the perturbations to the Hamiltonian of the cosmological system, which involves an extension from the asymptotic sector of modes with infinitely large wave number [71,72]. This diagonalization provides quantum excitations of positive frequency that do not interact (at least asymptotically) when the background dependence of the system is conveniently taken into account, and in this sense it is natural to consider the associated solutions as a natural set to define a vacuum. In addition, it has been shown that this vacuum reproduces the Bunch-Davies one in de Sitter [71] and that, in phenomenologically interesting situations, one can relate it with other proposals to choose a state that minimizes the oscillations in the primordial power spectrum [48,72] (state that is often called the NO-vacuum). The corresponding choice of positive frequencies can be given in terms of the imaginary part of a (mode-dependent) solution to a Riccati equation that includes the effective mass of the perturbations [71].Thedesired solution to this equation must satisfy a specific asymptotic expansion for large wave numbers, expansion that also depends on the effective mass. Therefore, modifications of this effective mass of the perturbations with respect to the free scalar field case caused by a potential affect the choice of the vacuum, both in those preinflationary models in which the background is described classically within general relativity and in quantum cosmology models. This influence of the potential on the choice of vacuum state, and therefore on the resulting power spectrum, was not studied in Ref. [67]. The aim of this work is to discuss the influence of the potential on the perturbations in regimes that are kinetically dominated, analyzing the leading-order corrections around the case of a free scalar field. Thus, the goal is to reduce all the complications that a scalar field potential introduces on the background dynamics to the free case plus manageable corrections. In this way, one can simplify the complicated calculations required to solve the cosmological models with a generic potential, for which there is no analytic solution available in general, and which would typically need the implementation of numerical computation techniques. Furthermore, the evaluation of the effective mass would have to be performed numerically for each of the possible values that the scalar field may take (since the effective mass depends explicitly on this field, both classically and quantum mechanically), with the subsequent aggravation of the computational problems. In addition, numerical simulations would also be needed to invert the relation between the scalar field and the conformal time3in order to determine the effective mass in terms of this time, and these simulations would have to be performed independently for each potential that one wants to consider. All these difficulties would complicate a general investigation of the aforementioned preinflationary regimes to the extreme. On the contrary, such a study should be feasible with the approximations that we develop in this work if one can just handle the case of a free scalar field. The rest of this article is organized as follows. In the next section, we summarize very briefly the hybrid approach to quantum cosmology, the modified MS equations for the scalar perturbations, which can be applied to the case of general relativity in the classical limit, and the basic results of the asymptotic diagonalization criterion for the choice of a vacuum state. Section III deals with the corrections to the effective mass that appears in the (modified) MS equations owing to the consideration of a scalar field potential. In a first subsection we calculate the leading-order correction to the effective mass in terms of the scalar field, interpreted as the internal degree of freedom with respect to which we describe the background evolution. The analysis of that subsection differs from Ref. [67] in several convenient choices of conventions and a neater extraction of the nonfree part of the evolution of the scalar field. In a second subsection we discuss the relation between the conformal time and the scalar field in our leading-order approximation in the potential. Finally, in a third subsection we combine these results to derive the total leading-order correction to the effective mass of the perturbations, and comment on the particularization of the result to the cases of a classical Einstenian evolution, hybrid LQC, and a hybrid quantum cosmology model based on geometrodynamics [29]. Section IV investigates the leading-order effects of the scalar field potential on the choice of vacuum state if one adopts the asymptotic diagonalization criterion. Finally, Sec. Vcontains the conclusions. We take units such 3In the kinetically dominated regimes that we are considering, the potential remains so small that its presence does not break the monotonicity of the evolution of the scalar field, even when this potential is treated exactly. PRIMORDIAL PERTURBATIONS IN KINETICALLY DOMINATED …PHYS. REV. D 104, 103520 (2021) 103520-3 that the Planck reduced constant, ℏ, and the speed of light, c, are equal to the unit. II. MODIFIED PERTURBATION EQUATIONS AND CHOICE OF A VACUUM In order to motivate the kind of modification to the effective mass of the scalar perturbations that can be due to quantum geometry effects, let us consider the hybrid approach to quantum cosmology [60].We consider a cosmological spacetime of the FriedmmanLemaître-Robertson-Walker (FLRW) type with a homogeneous scalar field as matter content, subject to a potential. We assume compact spatial sections with the compact topology of the three-torus,4with orthogonal coordinates chosen with period equal to 2π.Thisspacetime will play the role of our background. Around it, we introduce perturbations, both in the metric and in the scalar field. We then expand the action in a perturbative series and truncate it at quadratic order, which is the first nontrivial perturbative order. This truncation is essential in all the other steps of the hybrid quantization that we summarize below. For the sake of conciseness, in this work we consider only scalar perturbations. The other physically relevant cosmological perturbations, namely the tensor ones, can be analyzed in a totally parallel way. The perturbative degrees of freedom can be described with a suitable canonical set of variables, similar to those introduced by Langlois [74]. This set contains the Fourier mode coefficients of the MS gauge invariant perturbations, the Fourier mode coefficients of its momentum (that is also a perturbative gauge invariant), the linear perturbative diffeomorphisms constraints of the system in a convenient Abelianized form, and canonical momenta of the latter that correspond to gauge degrees of freedom. This set is canonical as far as the perturbations are concerned, but it does not include variables for the background. The set can be extended to a canonical one for the whole cosmological system composed by the perturbations and the background along the lines first proposed by Pinto-Neto and his collaborators [75,76] and laterdevelopedinRef.[60]. The result is the inclusion of new canonical variables to describe the background, obtained from the Fourier zero-modes of the metric and the scalar field after correcting them with terms that are quadratic in the perturbations and that respect our order of truncation in the action [60]. A quantization of this canonical system leads then to physical states that depend only on (a configuration subset of) these background variables and (e.g.,) on the real mode coefficients of the MS field. We denote these coefficients by v k;ε,where  kis the wave vector of the Fourier mode5and εis a dichotomic label that indicates its parity. The hybrid strategy combines a quantum representation of the background (i.e., the corrected zero-modes) with a Fock representation of the MS gauge invariant field. In principle, the quantization of the background variables can be chosen freely except for basic consistency requirements and the need to include a kinematic space of states with an inner product in the FLRW geometry. For the corrected zero-mode of the scalar field and its momentum, we assume a standard representation, with the configuration variable acting by multiplication and its momentum as a derivative. The hybrid quantum system is subject to a global constraint, not yet satisfied in our states, that arises from the zero-mode of the Hamiltonian constraint of the perturbed cosmological model, written in terms of our canonical set of variables. This constraint is the sum of a contribution of the background and another contribution that is quadratic in the perturbations. If we call ϕthe (perturbatively corrected) zero-mode of the scalar field, πϕits canonical momentum, WðϕÞthe scalar field potential, and Vthe volume of the compact spatial sections, equal to 8π3times the cube of the scale factor, the nonperturbative contribution of the background to the constraint is given in terms of the quantum operators of the chosen representation as ðˆπ2 ϕ− ˆ Hð2Þ 0Þ=2, with ˆ Hð2Þ 0¼½ ˆ HðFÞ 02 −2WðϕÞˆ V2:ð2:1Þ Here, ˆ HðFÞ 0can be viewed as a representation of HðFÞ 0¼ 2ffiffiffiffiffiffiffiffiffi 3πG pjπVjV,whereπVwould be a momentum variable conjugate to the volume Vand Gis the Newton constant. We take it as a positive operator by construction. Its square is the geometric part of the Hamiltonian contribution of the background in the absence of potential and perturbations. On the other hand, for the moment we adopt the operator 2WðϕÞˆ V2 as a natural choice to represent the potential term. Nonetheless, we leave open the possibility of adopting an alternative representation, taking into account the freedom to add commutators that correspond to different choices of factor orderings (we will actually make use of this freedom in Sec. III A). An interesting ansatz for the search of physical states consists in a separation of variables in their dependence, so that the wave functions factorize in a part that depends on the FLRW geometry and another part that depends on the MS variables. In both partial wave functions, a dependence on the scalar field variable ϕis allowed. The partial state for the background, χ, can then be chosen close to a solution of the unperturbed system. Furthermore, as an ingredient of our ansatz, we assume a unitary evolution of this state χon ϕthat 4The noncompact case is obtained by appropriately taking the limit in which a physical length scale of reference is sent to infinity; see Ref. [73]. 5Zero-modes are excluded in the perturbations previous to adopting the continuous (noncompact) limit. BEATRIZ ELIZAGA NAVASCU ´ ES et al. PHYS. REV. D 104, 103520 (2021) 103520-4 is generated by a positive operator ˆ H0. In consonance with our requirements, and recalling that we are interested in regimes with negligible scalar field potential, we take ˆ H0as the square root of ˆ Hð2Þ 0(or its positive part [67]). Therefore, we have χðV;ϕÞ¼ˆ UðV;ϕÞχ0ðVÞ ¼Pexp iZϕ ϕ0 d ˜ ϕ ˆ H0ðV; ˜ ϕÞχ0ðVÞ;ð2:2Þ where χ0is the initial background state at a given ϕ0, and the symbol Pstands for time ordering with respect to ϕ. Introducing this ansatz, and neglecting backreaction and transitions between background states mediated by perturbations,6one can show that the Hamiltonian constraint leads to a dynamics for the MS variables that can be expressed (as a differential equation for the field solutions in our Fock analysis) in the form [36,60,67]  v k;εþk2þhˆ ϑq eþðˆ ϑo ˆ H0Þsymiχ hˆ ϑeiχv k;ε¼0:ð2:3Þ Here, the dot denotes the derivative with respect to the conformal time, kis the Euclidean norm of the mode wave vector  k, and the subindex sym stands for the symmetrized product. Three new background operators appear in the above formula, that are given by [36] ˆ ϑq e¼1 2πd 1 V1=3 ˆ Hð2Þ 0ð19 −18ðˆ HðFÞ 0Þ−2ˆ Hð2Þ 0Þd 1 V1=3 þ3 8π2G ˆ V4=3W00ðϕÞ− 16πG 3WðϕÞ;ð2:4Þ ˆ ϑo¼3 πffiffiffiffiffiffiffi 3 πG rW0ðϕÞˆ V2=3ðˆ HðFÞ 0Þ−1ˆ ΛðFÞ 0ðˆ HðFÞ 0Þ−1ˆ V2=3; ˆ ϑe¼3 2G ˆ V2=3:ð2:5Þ The prime in the potential denotes the derivative with respect to ϕ, and the operators d ½1=Vand ˆ ΛðFÞ 0have been introduced to take into account some subtleties that arise in the quantization if one follows loop techniques. The operator d ½1=Vis a regularization of the inverse of the physical volume, and as a result its composition with ˆ Vis not the identity operator for volumes of the Planck order (see, e.g., Ref. [60]). On the other hand, the definition of ˆ ΛðFÞ 0is related to that of ˆ HðFÞ 0. It represents the quantity −2ffiffiffiffiffiffiffiffiffi 3πG pVπV, so that jˆ ΛðFÞ 0jcan be viewed as an alternative representation of ˆ HðFÞ 0. In LQC, this alternative representation is obtained using holonomies along edges of the double of the basic coordinate length [36,60], in order to ensure that the action of the operator leaves invariant the superselection sectors on which the unperturbed Hamiltonian constraint of LQC is defined [60]. For other quantum representations of the background, these operators may admit simpler definitions that simplify the above expressions. The operators for the inverse and the square inverse of ˆ HðFÞ 0are well defined because we have assumed in our ansatz that this last operator is positive. As for the conformal time ηthat is used in the modified MS equations (2.3), it turns out to be related with the scalar field ϕby [60,67] hˆ H0iχdη¼hˆ ϑeiχdϕ:ð2:6Þ The modified MS equation (2.3) canbewrittenintheform  v k;εþ½k2þsðηÞv k;ε¼0, with an effective time-dependent mass sequal to a ratio of expectation values. According to the arguments of Ref. [72], that are based in turn on the criterion of asymptotic Hamiltonian diagonalization for the choice of vacuum state put forward in Ref. [71], a preferred set of (normalized) positive frequency solutions fμkgof that equation is μk¼1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −2ImðhkÞ peiRη η0 d˜ ηImðhkÞ;ð2:7Þ where Im is the imaginary part, η0is any fixed reference time, and hkis a solution of a Riccati equation that contains the effective mass, namely _ hk¼k2þsþh2 k:ð2:8Þ More specifically, the desired solution must admit the following asymptotic expansion for infinitely large wave numbers [72]: 1 hk ∼ i k1− 1 2k2X ∞ n¼0−i 2kn γn;ð2:9Þ with γ0¼sand the rest of gamma-coefficients determined by the recurrence relation γnþ1¼−_γnþ4sγn−1þX n−3 m¼0 γmγn−ðmþ3Þ −X n−1 m¼0 γmγn−ðmþ1Þ:ð2:10Þ 6One usually neglects as well a typically small term that would obstruct the reality of the MS equations and that, in any case, can be absorbed with a suitable choice of factor ordering in the perturbative contribution to the constraint [60,67]. PRIMORDIAL PERTURBATIONS IN KINETICALLY DOMINATED …PHYS. REV. D 104, 103520 (2021) 103520-5 Notice that the effective mass fixes γ0and also appears explicitly in this relation. If the time interval where the positive frequency solutions fμkgare valid includes the end of inflation, the power spectrum of the scalar perturbations is given, up to a k-independent multiplicative factor, by [77] PVðk; ηÞ¼ k3 2π2jμkðηÞj2¼− k3 4π2ImðhkÞðηÞ;ð2:11Þ evaluated at a time ηwhen slow-roll inflation has just finished. If the aforementioned interval does not cover the inflationary period, one can use our solutions to obtain initial values for the primordial perturbations at the end of the kinetically dominated regime and continue the evolution afterwards with those values. III. LEADING-ORDER CORRECTIONS TO THE EFFECTIVE MASS Since we are interested in studying kinetically dominated regimes, in this section we want to calculate the leadingorder correction of the potential to the effective mass of the perturbations around the free scalar field case. For concreteness, we consider the expression of the effective mass sderived in hybrid quantum cosmology [60], which, according to our discussion in the previous section, is given by the following ratio of expectation values: s¼hˆ ϑq eþðˆ ϑo ˆ H0Þsymiχ hˆ ϑeiχ :ð3:1Þ For cosmological scenarios in general relativity, one can take the classical limit of the above expression, replacing operators by functions on the classical phase space and expectation values by evaluation on classical solutions, after having taken due care of the detailed effects of the potential. We divide our calculations in this section in three parts. First, we consider the influence of the potential at leading order in WðϕÞand its derivatives on the relevant operators and in the evolution of the background state χ. Since this evolution is usually described in terms of the scalar field ϕ, we then discuss the leading-order effect of the potential in the change to conformal time, which is the parameter used in the modified MS equations for the perturbations. Finally, we combine these two classes of corrections to obtain the leading-order effect of the potential in the effective mass regarded as a function of the conformal time, s¼sðηÞ. We also succinctly consider the particularization of our discussion to some classical and quantum cosmology models. A. Corrections to the effective mass in terms of the scalar field In the following, by linear or higher-order dependence in the potential we understand a dependence on WðϕÞor any of its derivatives (or products of them). The leading-order contribution of the potential to the different operators that appear explicitly in the ratio of expectation values that determines the effective mass is easy to calculate. First, we notice that ˆ ϑe, appearing in the denominator, is independent of the potential. On the other hand, ˆ ϑois linear in the potential, without higher-order corrections. Then, its symmetrized product with ˆ H0has a linear contribution of the potential that, with our previous definitions, is simply given by the symmetrized product with ˆ HðFÞ 0. Recall that this last operator is just the generator of the background evolution in the free case, and was introduced in Eq. (2.1). Finally, in the caseof ˆ ϑq e,itisstraightforwardtoseefromourdefinitionsthat ˆ ϑq e¼1 2πd 1 V1=3 ðˆ HðFÞ 0Þ2d 1 V1=3 þ17 πWðϕÞd 1 V1=3 ˆ V2d 1 V1=3 þ3 8π2GW00ðϕÞ− 2 πWðϕÞˆ V4=3þOðW2Þ;ð3:2Þ wherethe firstterm isthefree contribution,the term incurved bracketsistheleading-ordercontributionofthepotential,and the terms OðW2Þare quadratic or higher-order in the potential. The background state χalso introduces a dependence on the potential because its evolution from the initial state χ0ðVÞdepends on it. In this subsection, we calculate the leading-order dependence when the evolution is described in terms of the scalar field ϕ. This dependence can be extracted using an interaction picture for the quantum evolution, along the lines explained in Ref. [67]. However, employing the freedom in the choice of factor ordering of the contribution of the potential that we commented in Eq. (2.1), we use a different approximation to the generator ˆ H0around the free Hamiltonian ˆ HðFÞ 0, which proves to be more convenient for the rest of our calculations. Since most of the arguments parallel the discussion of Ref. [67], we only explain here the most important steps. We first want to approximate ˆ H0around the generator of the free evolution up to quadratic and higher-order terms in the potential. From the definition of ˆ H0as the square root of ˆ Hð2Þ 0, it is not difficult to check that our operator can be represented in the form ˆ H0¼ˆ HðFÞ 0−WðϕÞðˆ HðFÞ 0Þ−1=2ˆ V2ðˆ HðFÞ 0Þ−1=2þOðW2Þ: ð3:3Þ Our statement can be checked by taking its square and realizing that the result gives indeed ˆ Hð2Þ 0at the considered order in the potential, up to double commutators that can be absorbed with a suitable factor ordering [see Ref. [67] and our remarks after Eq. (2.1)]. The second term on the BEATRIZ ELIZAGA NAVASCU ´ ES et al. PHYS. REV. D 104, 103520 (2021) 103520-6 right-hand side of our equation then provides the leadingorder contribution of the potential. Let us call χðFÞðV;ϕÞthe background state that would result from the initial state χ0ðVÞif the evolution in ϕwere generated by the Hamiltonian of the free case. Explicitly, χðFÞðV;ϕÞ¼exp ½i ˆ HðFÞ 0ðϕ−ϕ0Þχ0ðVÞ:ð3:4Þ Employing the interaction picture in a convenient way, it is possible to show that χðV;ϕÞ¼exp½i ˆ HðFÞ 0ðϕ−ϕ0ÞPexpiZϕ ϕ0 d ˜ ϕ ˆ HðIÞ 1ðV; ˜ ϕÞ ×exp½−i ˆ HðFÞ 0ðϕ−ϕ0ÞχðFÞðV;ϕÞ;ð3:5Þ where we have defined ˆ HðIÞ 1¼exp ½−i ˆ HðFÞ 0ðϕ−ϕ0Þð ˆ H0− ˆ HðFÞ 0Þ × exp ½i ˆ HðFÞ 0ðϕ−ϕ0Þ:ð3:6Þ The two last expressions are exact to all orders in the potential. Using them and our approximation (3.3),itis now easy to obtain the leading-order correction of the potential to the freely evolved background state. With the notation χðV;ϕÞ¼ˆ UðWÞðϕÞχðFÞðV;ϕÞ, we get ˆ UðWÞðϕÞ¼1−iZϕ ϕ0 d ˜ ϕ ˆ Kðϕ; ˜ ϕÞWð˜ ϕÞþOðW2Þ;ð3:7Þ ˆ Kðϕ; ˜ ϕÞ¼exp ½i ˆ HðFÞ 0ðϕ− ˜ ϕÞð ˆ HðFÞ 0Þ−1=2ˆ V2 ×ðˆ HðFÞ 0Þ−1=2exp ½−i ˆ HðFÞ 0ðϕ− ˜ ϕÞ:ð3:8Þ Notice that, with our choice of factor ordering in Eq. (3.3), the two operators on the right of ˆ V2commute, as well as the two operators in front of it. Furthermore, the operator ˆ Kðϕ; ˜ ϕÞis symmetric since so are ˆ HðFÞ 0and ˆ V,by construction. The above expressions allow us to pass from the expectation values on the exact state χðV;ϕÞthat determine the effective mass to expectation values on χðFÞðV;ϕÞ, which evolves in the scalar field ϕaccording to the free dynamics. We notice the different approach taken here with respect to the discussion presented in Ref. [67]. Here, the expectation values are finally referred to the case of the free evolution, which is left unspecified, while in Ref. [67] these expectation values were directly rewritten in terms of the initial background state χ0ðVÞ. B. Corrections in the relation between the scalar field and the conformal time In order to obtain the effective mass in terms of the conformal time, we still have to invert the relation between this time and the scalar field used for the parametrization of the quantum background dynamics, according to Eq. (2.6). Again, we are interested in extracting the relation for the free case and the leading-order correction to it produced by the presence of a potential. One can compute them using the expressions for the operator ˆ H0and for the background state χðV;ϕÞat leading order in the potential, provided by Eqs. (3.3) and (3.7) respectively. A simple calculation at this order leads then to the relation 3 2Ghˆ V2=3iχðFÞdϕ−i3 2GhIðFÞ W½ˆ V2=3iχðFÞdϕ ¼hˆ HðFÞ 0iχðFÞdη−WðϕÞhð ˆ HðFÞ 0Þ−1=2ˆ V2ðˆ HðFÞ 0Þ−1=2iχðFÞdη −ihIðFÞ W½ˆ HðFÞ 0iχðFÞdη;ð3:9Þ where we have introduced the notation IðFÞ W½ˆ A¼Zϕ ϕ0 d ˜ ϕWð ˜ ϕÞ½ˆ A; ˆ Kðϕ; ˜ ϕÞ;ð3:10Þ for any operator ˆ A. In the case of ˆ HðFÞ 0, we notice that this operator commutes with all the factors of ˆ Kðϕ; ˜ ϕÞexcept with the central one, ˆ V2. Neglecting in our expressions the contribution of the potential, we obtain the relation for the free evolution, that we call ηðFÞðϕÞ. Explicitly, ηðFÞðϕÞ¼ 3 2GZϕ ϕ0 d ˜ ϕhˆ V2=3iχðFÞ h ˆ HðFÞ 0iχðFÞ ;ð3:11Þ where we have set, without loss of generality, ηðFÞðϕ0Þ¼0, and the expectation values in the integrand are computed on the free evolved state χðFÞðV; ˜ ϕÞ. This is why the integral cannot be calculated trivially and needs the knowledge of (only) the free evolution (but not of the dynamics when the potential is present). This relation can now be substituted in the terms that are linear in the potential in Eq. (3.9) preserving our leadingorder approximation. Thus, in the last two terms of that equation, we can substitute dηby ðηðFÞÞ0dϕ(the prime denoting the derivative with respect to ϕ). Adopting the notation ηðϕÞ¼ηðFÞðϕÞþηðWÞðϕÞþOðW2Þfor the corrections introduced in the dependence of the conformal time by the presence of a potential, we conclude then from Eqs. (3.9) and (3.11) that ηðWÞðϕÞ¼ 3 2GZϕ ϕ0 d ˜ ϕ1 ½h ˆ HðFÞ 0iχðFÞ2 ×hðWð˜ ϕÞhðˆ HðFÞ 0Þ−1=2ˆ V2ðˆ HðFÞ 0Þ−1=2iχðFÞ þihIðFÞ W½ ˆ HðFÞ 0iχðFÞÞhˆ V2=3iχðFÞ −ihˆ HðFÞ 0iχðFÞhIðFÞ W½ˆ V2=3iχðFÞ:ð3:12Þ PRIMORDIAL PERTURBATIONS IN KINETICALLY DOMINATED …PHYS. REV. D 104, 103520 (2021) 103520-7 Actually, in order to obtain the effective mass in terms of the conformal time, we need the inverse of the relation that we have computed, inverse which determines the dependence of the scalar field ϕas a function of η.Letus call ϕðFÞðηÞthe inverse of the functional dependence (3.11) for the free evolution [so that ϕðFÞðηðFÞðϕÞÞ ¼ ϕ and ηðFÞðϕðFÞðηÞÞ ¼ η]. Then, at our leading-order approximation, we can express the relation that we are looking for in the form ϕðηÞ¼ϕðFÞðηÞþϕðWÞðηÞþ OðW2Þwith the following leading-order correction of the potential to the conformal time: ϕðWÞðηÞ¼−ϕðFÞðηðWÞðϕðFÞðηÞÞÞ:ð3:13Þ This is minus the composition of the inverse of the free conformal time function with the correction produced by the potential and composed again with the inverse of the free conformal time function. C. Corrections to the effective mass in terms of the conformal time. Applications We can finally derive the leading-order correction to the effective mass, treated as a function of the conformal time. First, let us define ˜ χðFÞðV;ηÞ¼exp ½i ˆ HðFÞ 0ðϕðFÞðηÞ−ϕ0Þχ0ðVÞ:ð3:14Þ According to our discussion above, we have that χðFÞðV;ϕðηÞÞ ¼ exp ½i ˆ HðFÞ 0ðϕðηÞ−ϕðFÞðηÞÞ˜ χðFÞðV;ηÞ: ð3:15Þ This expression is exact. Approximating it to leading order in the potential, we obtain χðFÞðV;ϕðηÞÞ¼½1þi ˆ HðFÞ 0ϕðWÞðηÞþOðW2Þ˜ χðFÞðV;ηÞ; ð3:16Þ with ϕðWÞðηÞgiven in Eq. (3.13). Combining this with the corrections of the background state already derived in terms of the conformal time [see Eq. (3.7)], we conclude that χðV;ϕðηÞÞ ¼ 1þi ˆ HðFÞ 0ϕðWÞðηÞ−iZϕðFÞðηÞ ϕ0 d ˜ ϕ ˆ KðϕðFÞðηÞ; ˜ ϕÞWð˜ ϕÞþOðW2Þ˜ χðFÞðV;ηÞ:ð3:17Þ We recall that the operator ˆ Kðϕ; ˜ ϕÞwas defined in Eq. (3.8). Other than via their dependence on the state χ, the dependence on ϕðηÞof the expectation values that provide the effective mass appears only in the leading-order correction of the potential, but not on the free contribution independent of it. Therefore, at the dominant order that we are considering, the expression of those leading-order corrections in terms of the conformal time can be obtained by simply evaluating the scalar field on the trajectory of the free case, namely on ϕðFÞðηÞ. With these results, it is then easy to check that the factor corresponding to the denominator of the effective mass (3.1) in conformal time can be expressed at leading order in the potential as 1 hˆ ϑeiχ¼2G 3hˆ V2=3i˜ χðFÞ1−ih½ˆ V2=3; ˆ JðWÞi˜ χðFÞ hˆ V2=3i˜ χðFÞþOðW2Þ; ð3:18Þ where, to shorten our notation, we have called ˆ JðWÞðηÞ¼ ˆ HðFÞ 0ϕðWÞðηÞ−ZϕðFÞðηÞ ϕ0 d ˜ ϕ ˆ KðϕðFÞðηÞ; ˜ ϕÞWð˜ ϕÞ: ð3:19Þ One can similarly approximate the numerator of the effective mass, expressed in conformal time, employing Eq. (3.2) for the correction of ˆ ϑq eand recalling that the other summand can be replaced at leading order in the potential with ð ˆ ϑo ˆ HðFÞ 0Þsym. Combining this with Eq. (3.18),we obtain that sðηÞ¼sðFÞðηÞþsðWÞðηÞþOðW2Þ;ð3:20Þ sðFÞðηÞ¼G 3πhc ½1 V1=3ð ˆ HðFÞ 0Þ2c ½1 V1=3i˜χðFÞ hˆ V2=3i˜χðFÞ ;ð3:21Þ with the leading-order correction given by BEATRIZ ELIZAGA NAVASCU ´ ES et al. PHYS. REV. D 104, 103520 (2021) 103520-8 sðWÞðηÞ¼ 2G 3πhˆ V2=3i˜ χðFÞ 3 8πGW00 −2Whˆ V4=3i˜χðFÞþ17Wˆ 1 V1=3 ˆ V2d 1 V1=3˜ χðFÞ þ2ffiffiffiffiffiffi 3G p πffiffiffiπ phˆ V2=3i˜ χðFÞ W0hðˆ V2=3ðˆ HðFÞ 0Þ−1ˆ ΛðFÞ 0ðˆ HðFÞ 0Þ−1ˆ V2=3ˆ HðFÞ 0Þsymi˜ χðFÞ þi hˆ V2=3i˜ χðFÞ−sðFÞh½ˆ V2=3; ˆ JðWÞi˜χðFÞþG 3πd 1 V1=3 ð ˆ HðFÞ 0Þ2d 1 V1=3 ; ˆ JðWÞ˜ χðFÞ:ð3:22Þ In the last formula, the potential and its derivatives are evaluated at ϕðFÞðηÞ, and the mass sðFÞ, the state ˜ χðFÞ, and the operator ˆ JðWÞat η. The application of our formulas for the free effective mass and its leading-order correction in the case of classical general relativity can be done as follows. First of all, we notice that the last line of Eq. (3.22) is the implicit correction of the potential to the purely free contribution sðFÞðηÞof the mass. Then, passing from operators to functions on phase space, ignoring the distinction between jˆ ΛF 0jand ˆ HðFÞ 0, identifying the classical analog of the inverse volume operator as 1=V in this passage, translating the commutators of operators in the last line of Eq. (3.22) into itimes the corresponding Poisson brackets f;g, and interpreting the expectation values as the evaluation on free classical solutions, we get sðFÞ GR ¼G 3πðHðFÞ 0Þ2 V4=3¼4G2V2=3π2 V;ð3:23Þ sðWÞ GR ¼2G 3πV2=33 8πGW00 þ15W−3ffiffiffiffiffiffiffi 3 πG rsgnðπVÞW0 −4G2fV2=3π2 V;JðWÞg:ð3:24Þ All of the phase space functions that appear in these two formulas must be evaluated on free classical solutions. The last term in the second equation is the leading-order correction of the potential coming from the evaluation of V2=3π2 Von exact classical solutions instead of free ones, and it is the contribution of the last line in Eq. (3.22).In our calculations, we have used the identification HðFÞ 0¼2ffiffiffiffiffiffiffiffiffi 3πG pjπVjV, the symbol sgn denotes the sign function, and JðWÞis the classical counterpart of the operator ˆ JðWÞ, namely JðWÞ¼2ffiffiffiffiffiffiffiffiffi 3πG pjπVjVϕðWÞ− 1 2ffiffiffiffiffiffiffiffiffi 3πG pjπVjVZϕðFÞ ϕ0 d ˜ ϕWð ˜ ϕÞ ×X ∞ n¼0 1 n!ðϕðFÞ− ˜ ϕÞnfV2;HðFÞ 0gðnÞ;ð3:25Þ where we have obviated the explicit dependence of ϕðFÞand ϕðWÞon the conformal time η,andfV2;HðFÞ 0gðnÞis the nth-order Poisson bracket of V2with the generator HðFÞ 0of the free evolution. These brackets appear because the classical counterpart of Eq. (3.8) is the free evolution of V2=HðFÞ 0from ˜ ϕto ϕðFÞ(and HðFÞ 0remains constant along this free evolution). In particular, we notice that, when the potential is set to zero, the only nonvanishing component sðFÞ GR of the mass turns out to reproduce the standard function −z=z ¼−  a=a in general relativity, with z¼a2_ ϕ=_ a. The operator representation for the case of hybrid LQC is giveninSec.6ofRef.[60] and we do not repeat it here. In that reference, WðϕÞwas particularized to a quadratic potential, m2ϕ2=2. All relevant operators for our calculations can be constructed in terms of the volume operator, which acts by multiplication in the volume representation adopted in Ref. [60], and of an operator ˆ Ω0definedbymeansof holonomies, which shifts the eigenvalues of the volume eigenstates in a way that depends on the so-called Immirzi parameter of loop quantum gravity [78]. For the specific definition that we adopt here for ˆ Ω0and the conventions about the numerical factors in this definition, we use Eq.(28)ofRef.[36]. In particular, in this manner we have ˆ HðFÞ 0¼jˆ Ω0j. The operator ˆ ΛF 0is defined in terms of holonomies exactly in the same way as ˆ Ω0, except for the fact that the coordinate length of the edges of the holonomies are doubled. Finally, the inverse volume operator is obtained by means of a regularization that uses holonomies. It is a standard operator in LQC, and its action on volume eigenstates can be found, e.g., in Ref. [60]. Finally, let us briefly comment on the case of a hybrid quantization based on geometrodynamics [29,30]. We can start with operators ˆ aand ˆπafor the scale factor and its momentum, defined in a geometrodynamic representation in which they respectively act by multiplication, ˆ a¼a, and by differentiation. The representation is provided with an inner product for the background geometry that is given by the integration over the scale factor with a certain continuous measure (not necessarily da). The volume and inverse volume operators can be defined, respectively, as 8π3a3and 1=ð8π3a3Þ. We can also define ˆ ΛðFÞ 0¼−ðaˆπaþˆπaaÞffiffiffiffiffiffiffiffiffiffiffi πG=3 p, and ˆ HðFÞ 0exactly in the same way but replacing −ˆπawith jˆπaj. This simplifies some products of operators in our expressions for the free PRIMORDIAL PERTURBATIONS IN KINETICALLY DOMINATED …PHYS. REV. D 104, 103520 (2021) 103520-9