Study of the Inflationary Spectrum in the Presence of Quantum Gravity Corrections

Abstract

After a brief review of the different approaches to predicting the possible quantum gravity corrections to quantum field theory, we discuss in some detail the formulation based on a Gaussian reference frame fixing. Then, we utilize this scenario in the determination of the inflationary spectrum of primordial perturbations. We consider the quantization of an inhomogeneous, free, massless scalar field in a quasi-classical isotropic Universe by developing a WKB expansion of the dynamics of the next order in the Planckian parameter, with respect to the one at which standard QFT emerges. The quantum gravity corrections to the scale-invariant spectrum are discussed in a specific primordial cosmological setting and then in a general minisuperspace formalism, showing that there is no mode-dependent effect, and thus the scale invariant inflationary spectrum is preserved. This result is discussed in connection to the absence of a matter backreaction on the gravitational background in the considered paradigm.

1. Introduction

The quantization of the gravitational field is one of the most challenging open questions in modern theoretical physics [1,2]. In particular, the implementation of the canonical method to quantize geometrodynamics [3,4,5,6] has encountered a number of non-trivial difficulties, among which is the the so-called “problem of time” [7,8,9,10], which is associated the possibility of constructing a viable Hilbert space for the gravitational states. In this respect, significant progress has been achieved by adopting the Ashtekar formulation [11,12], which allowed the authors, via the introduction of non-local variables, to achieve the set-up of a kinematical Hilbert space and to demonstrate the discrete nature of the geometrical operator spectrum [13]. Nonetheless, some important shortcomings also affect this proposal, dubbed loop quantum gravity, such as the impossibility of coherently implementing the system dynamics and the difficulty of defining a proper classical limit [14].

On the contrary, many important achievements have been reached in treating quantum field theory (QFT) on curved spacetime [15,16,17], such as the derivation of the Unruh and Hawking effects [18,19]. Despite the absence of an experimental confirmation, this new physics has been derived by various different approaches and appears as a well-grounded paradigm, although there is not yet a general unique formulation in terms of a generic gravitational field.

Among the two research fields mentioned above, an intermediate point of view should receive more attention, i.e., the description of possible quantum gravity corrections to QFT on curved spacetime. In fact, there are physical settings, both in early cosmology and for the gravitational collapse, in which the background metric cannot be regarded as a purely classical dynamics, but simply an essentially classical dynamics affected by quantum fluctuations of the geometry. This line of research has been till now developed by a relatively limited number of studies [20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36], in which the first problem to be addressed is the emergence of a time variable for QFT from the gravity–matter Wheeler–DeWitt equation.

In a pioneering approach [37], the so-called “Tomonaga time” was introduced to reconstruct QFT on curved spacetime from canonical quantum gravity. However, the most interesting proposal comes from the well-known analysis [20], treating the separation of the total Hamiltonian as a quasi-classical component and a “small” quantum subsystem (see [38] for a physical characterization of the word “small” in this context). A number of interesting implementations of this idea in the cosmological arena can be found in [23,26,27,28,34,36,39,40,41,42,43,44,45,46,47,48,49,50], whose common trait is the description of a “small” quantum subsystem, often identified in the Universe anisotropic degrees of freedom in contrast to a quasi-classical isotropic background. Other approaches aimed at obtaining QFT on curved spacetime as an effective theory limit from quantum gravity can be found in Refs. [51,52,53,54,55].

In [21], the original proposal of [20] was specialized to the problem of quantum gravity corrections to QFT, and overall extended to the next order of approximation, where such a feature affects the QFT functional Schrödinger equation. This study had the merit of outlining the emergence of a non-unitary theory, as the QFT Hamiltonian is amended by quantum gravity contributions, at the first order of expansion in the inverse of a Planckian parameter. This question of the non-unitarity was discussed in later works [29,56,57], and the Born–Oppenheimer character of the adopted scheme was emphasized in [25,28,30]. First in [33] and subsequently in [35], it was argued that, to properly deal with the non-unitarity puzzle, it is necessary to define the physical clock from a different dynamical setting. In fact, in [20,21] (see [33] for a detailed comparison of the two works), the time dependence of the QFT wave functional is essentially recovered from the corresponding dependence on the label time of the quasi-classical metric variables. It is just in this feature that the non-unitarity naturally manifests itself in the perturbation scheme (for a review of the entire line of research, including some minisuperspace applications see [36]). In [33], the presence of the so-called kinematical action was postulated in the quantum gravity–matter dynamics [6,58], and in [35] the whole problem was restated in the framework of [59], i.e., by fixing a Gaussian reference frame which is “materialized” in the dynamics (see also [60]). The idea is that, in the considered Born–Oppenheimer scenario, the non-physical nature of the emerging fluid (i.e., its violation of the so-called “energy conditions”) does not take place, as a result of the perturbative expansion.

In the present study, we re-analyze this idea and then apply it to the natural arena of predicting possible quantum gravity corrections to the inflationary spectrum. The origin of the primordial perturbation spectrum is identified in the quantum fluctuations of the inflaton field during the slow-rolling phase [61,62,63,64], here approximated by an exact de Sitter regime [26]. More specifically, we consider a Robertson–Walker quasi-classical background, described via the conformal time, and we study the resulting spectrum of a free massless scalar field living on a de Sitter phase of the Universe, dominated by the vacuum energy of the transition phase, here represented by a cosmological constant term. By a Fourier decomposition of the scalar mode, we are able to deal with a set of minisuperspace models, one for each value of the wavenumber. The Schrödinger equation for QFT we consider here is the one obtained in [35]—see also [34]—and the aim of the present study is to evaluate how such a correction can affect the spectrum of the inflaton field.

The main result of the considered cosmological scenario is showing how the quantum gravity corrections manifest by a simple phase factor in front of the standard QFT solution on a Robertson–Walker metric, de facto corresponding to the solution of a time-dependent harmonic oscillator. As a natural consequence, the effect of the quantum gravity corrections on the spectrum vanishes at the considered order of expansion. The explanation for such a surprising issue is then discussed in a more general minisuperspace scheme, without a specific reference to the dynamical setting. We remark that the obtained result depends on the possibility of always factorizing the quantum gravity correction to the Universe wave function with respect to the standard QFT state on the considered cosmological background. The physical motivation for such a decoupling of the wave function is finally identified in the absence, up to first order of approximation, of a backreaction of the quantum matter on the quasi-classical background.

Thus, our study has the main merit of clarifying how, in the framework proposed in [35], the phenomenology (here identified in the primordial spectrum) is not affected by quantum gravity modifications, at least up to the considered expansion order, and how this perspective has to be sensitive to the existence of appreciable feedback of the quantum matter dynamics on its background variables.

The paper is structured as follows. In Section 2, we discuss the implementation of the Gaussian frame procedure to define a time parameter, reviewing the original formulation in Section 2.1 and the WKB implementation in Section 2.2. In Section 3, we apply the considered formalism to calculate the possible corrections to the inflationary spectrum by introducing a Fourier decomposition of the inflaton and determining the vacuum expectation values. In Section 4, we discuss the possible physical motivations for dealing with the standard (not modified by quantum gravity) inflationary spectrum discussed in Section 3. The concluding remarks are presented in Section 5.

2. Reference Frame Fixing and Reparametrization

We discuss here the reparametrization procedure illustrated in [59] that allowed us to define a physical clock for the quantum gravity system. The original paradigm was then applied to the case of gravity and matter via a Wentzel–Kramer–Brillouin (WKB) expansion in a Planckian parameter, as shown in [35].

2.1. Kuchar–Torre Gaussian Frame Proposal

A proposal to recover a physical clock for the quantum gravity system was discussed in [59], based on a Gaussian reference frame implementation. The formalism there introduced allows one to fix the Gaussian frame in a quantum field theory, by a reparametrization procedure that preserves the system’s invariance under the coordinate choice.

For this purpose, the following term is adjoined to the action of the system:

$$S^f=\int d^{4}x\left[\frac{\sqrt{-g}}{2}\left(g^{\alpha \beta }{\partial }_{\alpha }T\left(x\right){\partial }_{\beta }T\left(x\right)-1\right)\mathcal{F}+\sqrt{-g}\left(g^{\alpha \beta }{\partial }_{\alpha }T\left(x\right){\partial }_{\beta }{X}^i\left(x\right)\right){\mathcal{F}}_i\right]$$

(1)

where $\mathcal{F},{\mathcal{F}}_i$ are Lagrange multipliers. Here, $T,X^i$ are the Gaussian coordinates associated with the metric ${\gamma }_{\mu \nu }$ satisfying ${\gamma }^{00}=1,{\gamma }^{0i}=0$ (the implemented signature is $(+,-,-,-)$ for coherence with the original paper); the writing $T\left(x\right),X^i\left(x\right)$ in Equation (1) clarifies their dependence as functions of generic coordinates $x^{\alpha }=(t,x^i)$ that instead correspond to the metric $g_{\alpha \beta }$ (${\partial }_{\alpha }$ stands for the derivative with respect to $x^{\alpha }$). Such reparametrization is a necessary tool for recovering a field theory that is diffeomorphism-invariant, as opposed to field equations valid only in the Gaussian frame, simply by providing the map between the Gaussian and the arbitrary desired coordinates. Indeed, the non-reparametrized form of (1) would be

$$S_G^f=\int d^{4}X\left[-\frac{\sqrt{-\gamma }}{2}\left({\gamma }^{00}-1\right)\mathcal{F}+\sqrt{-\gamma }{\gamma }^{0i}{\mathcal{F}}_i\right],$$

(2)

corresponding to a gauge fixing, where $\mathcal{F},{\mathcal{F}}_i$ act as Lagrange multipliers (since their variations give the Gaussian conditions). The reparametrized form (1) is then uniquely obtained by requiring it to be invariant under transformations of the $x^{\alpha }$, and that, for $x^{\alpha }\equiv (T,X^i)$, the expression is equivalent to (2).

The Hamiltonian formulation of (1) shows how such a contribution can play the role of a physical clock for quantum gravity, when $S^f$ is adjoined to the Einstein–Hilbert action and the canonical quantization is implemented. Such formulation can be evaluated via the Arnowitt–Deser–Misner (ADM) foliation [65,66], which allows one to write the line element as

$$d{s}^2=N^{2}d{t}^2-h_{ij}d{x}^{i}d{x}^j,$$

(3)

where we label by $h_{ij}$ ($i,j$ are spatial indices) the induced metric on the identified 3d hypersurfaces $\Sigma$, and by N and $N^i$ the lapse function and shift vector describing the separation in the time-like and space-like directions, respectively. The super Hamiltonian and supermomentum contributions are

$$\begin{array}{c}{H}^f=W^{-1}P+W{W}^k{P}_k,\end{array}$$

(4)

$$\begin{array}{c}{H}_i^f=P{\partial }_{i}T+P_k{\partial }_i{X}^k,\end{array}$$

(5)

where P, $P_k$ are the momenta conjugate to $T,X^k$ and

$$\begin{array}{c}W\equiv {(1-h^{jl}{\partial }_{j}T{\partial }_{l}T)}^{-1/2},\end{array}$$

(6)

$$\begin{array}{c}{W}^k\equiv h^{jl}{\partial }_{j}T{\partial }_l{X}^k.\end{array}$$

(7)

The functions (4) and (5), which are linear in the momenta, are added to the analogous functions $H^g,H_i^g$ of the gravitational sector; consequently, the total constraints $H^g+H^f$ and $H_i^g+H_i^f$ must vanish because of diffeomorphism invariance [1,2]. One can thus obtain a functional Schrödinger evolution with the time definition

$$\widehat{\mathcal{H}}\Psi =i\hslash {\partial }_t\Psi =i\hslash {\int }_{\Sigma }{d}^{3}x\frac{\delta \Psi (T,X^k,h^{jl})}{\delta T\left(x\right)}{|}_{T=t},$$

(8)

where $\mathcal{H}={\int }_{\Sigma }{d}^{3}x{\widehat{H}}^g$, i.e., restricting the states to the hypersurfaces where $t\equiv T$, so that $\Psi$ is still a functional of the $X^i$. The choices $x^i\equiv X^i$ and $(t,x^i)\equiv (T,X^i)$ are also examined in the original paper [59].

However, an important characteristic of the Gaussian-frame method emerges at the classical level of the theory. Varying the total action with respect to the metric, it is observed that the corresponding Einsteinian equations are modified by the appearance of a source term:

$$T^{\alpha \beta }=\mathcal{F}{\mathrm{U}}^{\alpha }{\mathrm{U}}^{\beta }+\frac{1}{2}\left({\mathcal{F}}^{\alpha }{\mathrm{U}}^{\beta }+{\mathcal{F}}^{\beta }{\mathrm{U}}^{\alpha }\right),$$

(9)

being that ${\mathrm{U}}^{\alpha }=g^{\alpha \beta }{\partial }_{\beta }T$ and ${\mathcal{F}}_{\alpha }={\mathcal{F}}_i{\partial }_{\alpha }{X}^i$. Thus, the Gaussian-frame terms arise as a fluid component, having four-velocity ${\mathrm{U}}^{\alpha }$, energy density $\mathcal{F}$, and heat flow ${\mathcal{F}}_{\alpha }$. The associated energy conditions give the relation

$$\mathcal{F}\ge 2\sqrt{{\gamma }^{\alpha \beta }{\mathcal{F}}_{\alpha }{\mathcal{F}}_{\beta }},$$

(10)

which, however, is not in general satisfied due to the arbitrariness of the Lagrange multipliers, so the fluid has a non-physical character. Actually, by implementing only the Gaussian time condition with $\mathcal{F}$ (i.e., setting ${\mathcal{F}}_i=0$ in (1)), the fluid reduces to an incoherent dust (no heat flow is present). In this case, the energy conditions are ensured by $\mathcal{F}\ge 0$, which can be cast as an initial condition, since $\sqrt{-g}\mathcal{F}$ is a constant of motion. We stress that this point will be differently addressed in the next subsection.

2.2. WKB Matter Dynamics with the Gaussian Frame Implementation

Here we briefly illustrate the procedure, discussed in [35], by which the kinematical variables associated with the Gaussian reference frame can provide a suitable clock for the matter sector in a quantum gravity–matter system. Indeed, unitary dynamics emerges at the next order of expansion in a Planckian parameter, where quantum gravity corrections arise. This scheme will then be applied for the computation of the modified primordial power spectrum in the next section.

Let us consider a gravity–matter system, where the gravitational Hamiltonian is characterized by a kinetic term and a potential V, and the matter component is a self-interacting scalar field $\varphi$ with potential $U_m\left(\varphi \right)$. Such choice will turn out to be suitable for the cosmological implementation discussed in Section 3. We insert the Gaussian-frame term (1) such that the total action reads:

$$S=\int dt{\int }_{\Sigma }{d}^{3}x\left({\Pi }^{ij}\dot{h_{ij}}+p_{\varphi }\dot{\varphi }-N(H^g+H^m)-N^i(H_i^g+H_i^m)\right)+S^f,$$

(11)

where

$$\begin{array}{c}{H}^g=-\frac{{\hslash }^2}{2M}\left(G_{ijkl}\frac{\partial }{\partial h_{ij}}\frac{\partial }{\partial h_{kl}}+g_{ij}\frac{\partial }{\partial h_{ij}}\right)+MV,\end{array}$$

(12)

$$\begin{array}{c}{H}_i^g=\phantom{\frac{1}{2}}2i\hslash h_{ij}{D}_k\frac{\partial }{\partial h_{kj}},\end{array}$$

(13)

$$\begin{array}{c}{H}^m=-{\hslash }^2\frac{{\partial }^2}{\partial {\varphi }^2}+U_m,\end{array}$$

(14)

$$\begin{array}{c}{H}_i^m=-\left({\partial }_i\varphi \right)\frac{\partial }{\partial \varphi }.\end{array}$$

(15)

In this notation, we will treat functional derivatives as ordinary partial ones. The term $g_{ij}\partial /\partial h_{ij}$ in (12) is inserted to account for a generic factor ordering (see discussion in [21]), and $D_k$ in (13) is the 3d covariant derivative on the hypersurface $\Sigma$. Instead of the Einstein constant $\kappa =8\pi G/c^3$, we have written in (12) and (13) the following Planckian parameter:

$$M:=\frac{c^2}{32\pi G}=\frac{c{m}_{Pl}^2}{4\hslash },$$

(16)

with dimension of mass over length, which will be taken as the order parameter for the expansion. Indeed, the Planckian energy scale, representative of the gravitational sector, is typically larger with respect to the corresponding scale of the matter fields. In principle, it would be possible to construct the WKB expansion via a dimensionless parameter, constructed as the ratio between the present one and the corresponding quantity calculated for a typical energy scale of the quantum matter. In the case we will consider, such an energy scale corresponds to that one of the inflationary process, say, $T\simeq {10}^{15}$ GeV (see Section 3). However, we retain here a dimensional parameter in order to keep contact and comparison with the previous literature, e.g., Refs. [21,26,27,29,33,35].

The previous consideration motivates a Born–Oppenheimer (B-O) separation of the wave function:

$$\Psi \left(h_{ij},\varphi ,X^{\mu }\right)=\psi \left(h_{ij}\right)\chi \left(\varphi ,X^{\mu };h_{ij}\right)$$

(17)

between the gravity and matter components. After performing a WKB expansion [67] in powers of $1/M$, we have

$$\Psi \left(h_{ij},\varphi ,X^{\mu }\right)=e^{\frac{i}{\hslash }\left(M{S}_0+S_1+\frac{1}{M}{S}_2\right)}{e}^{\frac{i}{\hslash }\left(Q_1+\frac{1}{M}{Q}_2\right)}$$

(18)

up to the order $M^{-1}$. Here, the $S_m$ functions (at the m-order) account for the gravitational background, and the $Q_n$ (at the order n) describe the reference fluid and matter components. We stress that the first matter contribution is of the order $M^0$. Similarly to the B-O scheme, we enforce the conditions

$$\begin{array}{c}\frac{\langle {\widehat{H}}^m\rangle }{\langle {\widehat{H}}^g\rangle }=\mathcal{O}\left(M^{-1}\right),\end{array}$$

(19)

$$\begin{array}{c}\frac{\partial Q_n}{\partial h_{ij}}=\mathcal{O}\left(M^{-1}\right),\end{array}$$

(20)

where the expectation values are computed over the corresponding wave functions, due to the “fast” nature of the matter sector with respect to gravity.

If the average backreaction of the matter degrees of freedom is negligible, both the gravitational and total constraints are satisfied:

$$\begin{array}{c}{\widehat{H}}^g\psi \left(h_{ij}\right)=0,\end{array}$$

(21)

$$\begin{array}{c}{\widehat{H}}_i^g\psi \left(h_{ij}\right)=0,\end{array}$$

(22)

$$\begin{array}{c}({\widehat{H}}^g+{\widehat{H}}^m+{\widehat{H}}^f)\Psi (h_{ij},\varphi ,X^{\mu })=0,\end{array}$$

(23)

$$\begin{array}{c}({\widehat{H}}_i^g+{\widehat{H}}_i^m+{\widehat{H}}_i^f)\Psi (h_{ij},\varphi ,X^{\mu })=0,\end{array}$$

(24)

where we consider also the supermomentum constraints for generality. By substituting the ansatz (18) in the constraints, with the explicit forms (4), (5), and (12)–(15), the dynamics can be analyzed order by order (we refer to the original paper [35] for the explicit computation).

At the Planckian order M, one obtains the classical Hamilton–Jacobi (H-J) equation for the gravitational function $S_0$:

$$\frac{1}{2}{G}_{ijkl}\frac{\partial S_0}{\partial h_{ij}}\frac{\partial S_0}{\partial h_{kl}}+V=0,$$

(25)

together with its diffeomorphism invariance condition. A crucial point must instead be discussed at the order $M^0$: the gravitational constraints (21) and (22) allow one to solve for $S_1$, and after substituting the solutions $S_0$ and $S_1$ into (23) and (24), the remaining equations for the matter sector are

$$\begin{array}{c}\left(-2i\hslash \frac{{\partial }^2{Q}_1}{\partial {\varphi }^2}+U_m-W^{-1}\frac{\partial Q_1}{\partial T}-W{W}^k\frac{\partial Q_1}{\partial X^k}\right)e^{\frac{i}{\hslash }{Q}_1}=0,\end{array}$$

(26)

$$\begin{array}{c}\left(-2h_{ij}{D}_k\frac{\partial S_1}{\partial h_{kj}}-i{\hslash }^{-1}\left({\partial }_i\varphi \right)\frac{\partial Q_1}{\partial \varphi }-\left({\partial }_{i}T\right)\frac{\partial Q_1}{\partial T}-\left({\partial }_i{X}^k\right)\frac{\partial Q_1}{\partial X^k}\right)e^{\frac{i}{\hslash }{Q}_1}=0.\end{array}$$

(27)

Such expressions require further attention for their physical interpretation. Following from the linearity of $H^f$ and $H_i^f$ in the momenta P and $P_k$, a suitable time parameter can be naturally introduced as

$$i\hslash \frac{\partial }{\partial \tau }=i\hslash \int d^{3}x\left[N\left(W^{-1}\frac{\partial }{\partial T}+W{W}^k\frac{\partial }{\partial X^k}\right)+N^i\left(\left({\partial }_{i}T\right)\frac{\partial }{\partial T}+\left({\partial }_i{X}^k\right)\frac{\partial }{\partial X^k}\right)\right].$$

(28)

Then, the linear combination of Equations (26) and (27) with coefficients N and $N^i$, respectively, takes the form

$$i\hslash \frac{\partial {\chi }_0}{\partial \tau }={\widehat{\mathcal{H}}}^m{\chi }_0=\int d^{3}x\left(N{\widehat{H}}^m+N^i{\widehat{H}}_i^m\right){\chi }_0,$$

(29)

where we label ${\chi }_0=e^{\frac{i}{\hslash }{Q}_1}$. In other words, reading the reference fluid clock induced by the definition (28), we observe at this WKB order functional Schrödinger dynamics of the quantum matter field $\varphi$ on the gravitational background. In this limit, the resulting dynamics corresponds to QFT on curved spacetime.

The quantum gravity’s influence on the matter sector emerges at the next order, $M^{-1}$. Proceeding in a similar way, one obtains the following equation for ${\chi }_1=e^{\frac{i}{\hslash }\left(Q_1+\frac{1}{M}{Q}_2\right)}$:

$$i\hslash \frac{\partial {\chi }_1}{\partial \tau }={\widehat{\mathcal{H}}}^m{\chi }_1+\int d^{3}x\left[N{G}_{ijkl}\frac{\partial S_0}{\partial h_{ij}}\left(-i\hslash \frac{\partial }{\partial h_{kl}}\right)-2N^i{h}_{ij}{D}_k\left(-i\hslash \frac{\partial }{\partial h_{kj}}\right)\right]{\chi }_1,$$

(30)

where the additional contributions with respect to the matter Hamiltonian are quantum gravity corrections. These modifications are unitary due to the real nature of the function $S_0$ and the presence of the conjugate momenta with respect to the induced metric, and their smallness is assured by the hypothesis (20) (see [35]). Thus, the clock defined by (28) is a physical clock for the matter sector in the WKB scheme truncated at the order $M^{-1}$.

3. Calculation of the Inflationary Spectrum

Following the model introduced in the previous section, we now turn to the question of how the power spectrum associated with inflationary perturbations is affected by the quantum gravity corrections.

3.1. Perturbations of the Model

Before facing the analysis of the generation of primordial perturbations during the inflationary dynamics of the Universe, and when studying how the quantum gravity corrections can affect the associated power spectrum, it is worth stressing some key differences between the present analysis and other similar approaches, as in [27,68].

In our formulation, apart from the WKB expansion in the Planckian parameter M, we are addressing a B-O separation between the “slow” gravitational component and the “fast” matter contribution, with the latter including also the fluid’s presence. This separation is justified by virtue of a corresponding scale separation between the energy of the quantum matter dynamics, say in the order of the matter Hamiltonian spectrum, and that one of the Planck order, at which the gravity quantization is expected to manifest itself. In view of the adopted B-O approximation we are implementing, the backreaction of the quantum matter on the gravitational background is implicitly negligible. In other words, quantum corrections of the gravitational dynamics are clearly present (as implied by the function $S_1$ in Equation (18), associated with a quantum amplitude for the background metric), but their existence has to be regarded as independent of the matter’s dynamics.

This point of view has been clearly elucidated in [69], where a critical re-analysis of the original formulation [20], and hence, Ref. [21], has been developed. There, limiting the attention up to the zero order in the parameter M, the quantum gravity component has been expressed in terms of gravitons on the vacuum Bianchi I background. This way, the WKB formulation of the gravitational field takes the form of a purely classical background on which a slow quantum graviton field lives, as referred to by independent degrees of freedom. This graviton contribution is independent, due to the B-O separation, from the quantum matter dynamics, thereby reinforcing the previous statement. If implemented to the isotropic Universe we will consider below, this formulation would also imply the presence of scalar perturbations of the metric, represented by independent degrees of freedom, and clearly, not affected by the scalar field fluctuations.

In the following analysis, although developed in the presence of quantum gravity corrections, we will refer to the scalar field only; such case is equivalent to the study of a free massless scalar field fluctuating on a de Sitter background. The classical energy contribution of the scalar field will be identified with the cosmological constant term (i.e., the gap between the false and true vacuum energy density [61,63]). The inhomogeneous fluctuation of this field will be treated as an independent degree of freedom living on the expanding de Sitter space and whose fluctuations are responsible for the emergence of a scalar perturbation spectrum.

3.2. The Inflaton Field

The theory of inflation postulates an early period of exponentially accelerated expansion of the Universe, motivating its primordial inhomogeneities as emerging from the vacuum fluctuations of a scalar field, the so-called inflaton field. One of the most remarkable results of such mechanism is the ability to explain the flatness problem of the Universe [61,62,70,71]. A schematic formulation of this framework is studied by considering as a background a spatially flat universe (any curvature is damped by the exponential expansion), with a scalar field living on top. More specifically, one should consider a Friedmann–Lemaitre–Robertson–Walker (FLRW) model with line element:

$$d{s}^2=-N^2\left(t\right)d{t}^2+a^2\left(t\right)(d{x}^2+d{y}^2+d{z}^2),$$

(31)

a being the cosmic scale factor (here we use the opposite signature with respect to (1) in Section 2 for easier comparison with existing literature), and it inserts the inflaton contribution as a minimally coupled scalar field $\varphi$ with potential $U\left(\varphi \right)$. Then, small perturbations are introduced, in general, both for the metric and for the inflaton, which give rise to scalar and tensor fluctuations (the detailed Hamiltonian formulation of such approach can be found, for example, in [26,27]). Following the discussion presented in Section 3.1, we will now focus on the fluctuations of the scalar field only over the FLRW background, i.e., by variation of the action with respect to those variables.

The fluctuations $\delta \varphi$ of the inflaton field can be described in a gauge-invariant way via the Mukhanov–Sasaki (M-S) variable v [72,73,74] (see also the discussion in [68]) defined as

$$v:=a\phi =a\delta \varphi .$$

(32)

We here stress that addressing the B-O separation discussed above does not alter the gauge invariance of the perturbation theory. In fact, in the limit in which the backreaction on the metric scalar perturbation is neglected, the M-S variable [72] simply reduces to the inhomogeneous scalar field $\varphi$ of our study times the cosmic scale factor, as in (32), and its gauge invariance is immediately recovered.

The evolution of the inflaton fluctuations is responsible for the formation of primordial structures in the Universe. To analyze their behavior, let us consider modes with physical wavelength ${\lambda }_{phys}\equiv a\left(t\right){\lambda }_0$, ${\lambda }_0$ being the comoving wavelength. It is useful to compare this quantity with the so-called Hubble radius (or micro-physics horizon) ${\mathrm{H}}^{-1}=a/\dot{a}$, that for any given time is the inverse of the Hubble parameter (using $c=1$). This horizon represents the scale separating the gravity-dominated regime from the quantum one: the first happens for modes with physical wavelength such that ${\lambda }_{phys}\gg {\mathrm{H}}^{-1}$, and the second is the case for ${\lambda }_{phys}\ll {\mathrm{H}}^{-1}$. It can be shown that, during the period of accelerated expansion predicted by the theory, the Hubble radius is constant in the physical coordinates, and ${\lambda }_{phys}$ exponentially increases [61,71]. Thus, the quantum fluctuations emerge at early times within the micro-physical scales (i.e., for ${\lambda }_{phys}\ll {\mathrm{H}}^{-1}$), rapidly expand going outside the horizon, and propagate until they re-enter the Hubble radius at later times (when inflation is over, the behavior is opposite, since ${\mathrm{H}}^{-1}$ grows faster than the ${\lambda }_{phys}$) [62,70].

Using the gauge-invariant formalism via the M-S variable (32), it is possible to compute the power spectrum ${\mathcal{P}}_v\left(k\right)$, where k specifies the wavenumber of each Fourier mode associated with the inflaton perturbations (see also [75,76,77,78,79,80] for investigations of such a spectrum in different cosmological settings). However, to investigate the evolution of the primordial Universe, it is more convenient to work with the spectrum associated with the comoving curvature perturbation $\zeta$ (which is the one leaving its fingerprint on the cosmic microwave background radiation) [81]: indeed, $\zeta$ is constant (i.e., it freezes) for all the time in which the perturbations are outside the horizon; therefore, one only needs to compute its spectrum at the end of inflation [61,71]. In the primordial era of our interest, the two quantities $\zeta$ and v are directly related by

$$\zeta =\sqrt{\frac{4\pi G}{ϵ}}\frac{v}{a},$$

(33)

with $ϵ=-\dot{\mathrm{H}}/{\mathrm{H}}^2$ being the first slow-roll parameter. Therefore, in the following, we will focus on the dynamics of the M-S variable v and only at the end use (33) to compute the invariant power spectrum.

Upon decomposition in Fourier modes $v_{\mathbf{k}}$ and assuming Gaussian probability distributions for the quantum amplitudes associated with each $v_{\mathbf{k}}$ [81], all the relevant properties of the inflationary perturbations are contained in the two-point correlation function:

$$\Xi \left(\mathbf{r}\right):=\langle 0\left|\widehat{v}(\eta ,\mathbf{x})\widehat{v}(\eta ,\mathbf{x}+\mathbf{r})\right|0\rangle ,$$

(34)

where $|0\rangle$ is the vacuum state of the inflaton field. In (34), the expectation value implies integration over $\mathbf{k}$-modes, which can be carried out given the expression [81]

$$\Xi \left(\mathbf{r}\right)=\frac{1}{{\left(2\pi \right)}^3}\int d\mathbf{p}{e}^{-i\mathbf{p}·\mathbf{r}}|f_{\mathbf{p}}{|}^2=\frac{1}{2{\pi }^2}{\int }_0^{+\infty }\frac{dp}{p}\frac{sin\left(pr\right)}{pr}{p}^3{\left|f_p\right|}^2.$$

(35)

Here, $f_{\mathbf{p}}$ is the mode function associated with the scalar perturbations, and from (35), the power spectrum is defined as

$${\mathcal{P}}_v\left(k\right)=\frac{k^3}{2{\pi }^2}{\left|f_k\right|}^2,$$

(36)

i.e., the Fourier amplitude of $\Xi \left(0\right)$ per unit logarithmic interval. As mentioned above, this quantity is then evaluated in the super-Hubble limit $k/\left(a\mathrm{H}\right)\ll 1$, when the perturbations essentially freeze. We stress that the vacuum state in (34) must be selected as the one corresponding to the ground level of the scalar field Hamiltonian in the limit $k/\left(a\mathrm{H}\right)\to \infty$ (or equivalently ${\lambda }_{phys}\ll {\mathrm{H}}^{-1}$), also known as the Bunch–Davies vacuum [26,62,81]. We will impose this requirement on the modified wave functional dictated by the model in Section 3.3.

In the following, we will compute (36) in the specific case where the inflaton field follows modified quantum dynamics, as described by Equation (30).

3.3. Perturbation Spectrum in the de Sitter Phase

During the accelerated expansion of inflation, of particular interest is the slow-rolling phase, where the inflaton can be approximately described as a free massless scalar field (the almost constant potential acts as a cosmological term) [61]. In the following, we will consider an exact de Sitter phase; thus, the slow-rolling parameter $ϵ$ is neglected. The analysis of quantum gravity’s effects on the inflationary spectrum is achieved by considering the fluctuations of the scalar field over a quasi-classical background, expressed by a FLRW model with line element (31).

Instead of the general action (11), the considered case can be studied in the minisuperspace formalism (the supermomentum contributions are identically vanishing due to the homogeneity of the background model). The (non-trivial) relevant constraint is thus the superHamiltonian, which takes the form

$$H_{tot}=\frac{{\hslash }^2}{48M{a}^2}{\partial }_a\left(a{\partial }_a\right)+4M\Lambda a^3-i\hslash {\partial }_T+\frac{1}{2a}\sum _{\mathbf{k}}\left(-{\hslash }^2{\partial }_{v_{\mathbf{k}}}^2+{\omega }_k^2{v}_{\mathbf{k}}^2\right).$$

(37)

where we implemented the Laplace–Beltrami factor ordering. Here, the positive cosmological constant $\Lambda$ replaces $U_m$ in (14). The term $-i\hslash {\partial }_T$, that is, the momentum associated with the Gaussian time T, is the only surviving contribution from the insertion of $S^f$ (1) due to homogeneity. The last two terms in (37) are associated with the inflaton field fluctuations, where the $v_{\mathbf{k}}$ correspond to the modes in the Fourier space of the gauge-invariant M-S variable (32); in the considered case of scalar perturbations over a FLRW background, the $v_{\mathbf{k}}$-modes behave as time-dependent harmonic oscillators [26,27,82,83,84], where the frequency depends on the wavenumber modulus only:

$${\omega }_k^2=k^2-\frac{a^2}{N^2}\left(\dot{\mathrm{H}}-\mathrm{H}\frac{\dot{N}}{N}+2{\mathrm{H}}^2\right).$$

(38)

The WDW constraint corresponds to the vanishing of the operator (37) applied to the total system wave function $\Psi (a,T,v_{\mathbf{k}})$. For convenience, we implement the logarithmic scale factor,

$$\alpha :=ln\left(\frac{a}{a_0}\right),$$

(39)

such that the global WDW equation reads

$$i\hslash {\partial }_T\Psi =a_0^{-1}{e}^{-\alpha }\left[\frac{{\hslash }^2}{48M}\frac{1}{a_0^2{e}^{2\alpha }}{\partial }_{\alpha }^2+4a_0^4{e}^{4\alpha }\Lambda M+\frac{1}{2}\sum _{\mathbf{k}}\left(-{\hslash }^2{\partial }_{v_{\mathbf{k}}}^2+{\omega }_k^2{v}_{\mathbf{k}}^2\right)\right]\Psi .$$

(40)

Let us now consider a single Fourier mode identified by a wave number $\mathbf{k}$. Following the scheme discussed above, for each independent mode, the ansatz is taken as

$${\Psi }_{\mathbf{k}}(\alpha ,T,v_{\mathbf{k}})={\psi }_{\mathbf{k}}\left(\alpha \right){\chi }_{\mathbf{k}}(\alpha ,T,v_{\mathbf{k}}),$$

(41)

and then WKB expanded as in (18), obtaining

$$\begin{array}{c}{\psi }_{\mathbf{k}}\left(\alpha \right)=e^{\frac{i}{\hslash }\left[M{S}_0\left(\alpha \right)+S_1\left(\alpha \right)+M^{-1}{S}_2\left(\alpha \right)\right]},\end{array}$$

(42)

$$\begin{array}{c}{\chi }_{\mathbf{k}}(\alpha ,T,v_{\mathbf{k}})=e^{\frac{i}{\hslash }\left[Q_1(\alpha ,T,v_{\mathbf{k}})+M^{-1}{Q}_2(\alpha ,T,v_{\mathbf{k}}))\right]}.\end{array}$$

(43)

Upon substitution into (40), the solutions for the gravitational sector are readily obtained at the three orders:

$$\begin{array}{c}{S}_0\left(\alpha \right)=-8\sqrt{\frac{\Lambda }{3}}{a}_0^3\left(e^{3\alpha }-e^{3{\alpha }_0}\right),\end{array}$$

(44)

$$\begin{array}{c}{S}_1\left(\alpha \right)=i\hslash \frac{3}{2}(\alpha -{\alpha }_0),\end{array}$$

(45)

$$\begin{array}{c}{S}_2\left(\alpha \right)=\frac{{\hslash }^2}{64}\sqrt{\frac{3}{\Lambda }}{a}_0^{-3}\left(e^{-3\alpha }-e^{-3{\alpha }_0}\right).\end{array}$$

(46)

Here, $S_0$ solves the H-J equation and so corresponds to the classical limit of the gravitational component, and the next order functions, $S_1$ and $S_2$, account for quantum gravity effects.

The equation for the quantum matter wave function at the first order $M^0$ can be expressed in a clearer form in conformal time $\eta$ (choosing $N=a_0{e}^{\alpha }$), which is related to the Gaussian time constraint via $T^{\prime }\left(\eta \right)=a_{0}exp\left(\alpha \left(\eta \right)\right)$, obtaining

$$i\hslash {\partial }_{\eta }{\chi }_{\mathbf{k}}^{\left(0\right)}=\left(-\frac{{\hslash }^2}{2}{\partial }_{v_{\mathbf{k}}}^2+\frac{1}{2}{\omega }_k^2\left(\eta \right)v_{\mathbf{k}}^2\right){\chi }_{\mathbf{k}}^{\left(0\right)}.$$

(47)

The time-dependent harmonic oscillator system can be exactly solved by implementing the so-called Lewis–Riesenfeld method introduced in [85,86,87,88], which is described in Appendix A. The wave function admits a general representation of the form (A9), where the functions ${\delta }_{n,k}$ and ${\rho }_k$ are defined in (A10) and (A3), respectively. The arbitrary coefficients in those expressions are set by imposing suitable initial conditions. In this specific cosmological setting, we make use of the Bunch–Davies vacuum state requirement [62,81]: the state must correspond to the Minkowskian vacuum in the limit $\eta \to -\infty$ (that is, when the inflaton wavelength is small compared to the curvature of the universe). This condition is satisfied if

$$\begin{array}{c}{\rho }_k\left(\eta \right)\stackrel{\eta \to -\infty }{\to }{k}^{-1/2},\end{array}$$

(48)

$$\begin{array}{c}{c}_{n,k}={\delta }_{0,k}\end{array}$$

(49)

where (49) stems from the observation that the $n=0$ eigenvalue of the invariant (A1) corresponds, for a fixed time, to the lowest-energy state of the oscillator. For the specific ${\rho }_k$ function (A3), its coefficients must be $A=B={\gamma }_1=1$, so that

$${\rho }_k\left(\eta \right)=\sqrt{\frac{1}{k}+\frac{1}{{\eta }^2{k}^3}}$$

(50)

satisfies the required limit. Then, by substituting it into (A10), the ${\delta }_{n,k}$ functions are found to be

$${\delta }_{n,k}=-\left(n+\frac{1}{2}\right)\int d\eta \frac{1}{{\rho }_k^2\left(\eta \right)}=-\left(n+\frac{1}{2}\right)\left(\eta k-arctan\left(\eta k\right)+c\right).$$

(51)

Finally, the solution to Equation (47) satisfying the Bunch–Davies condition is:

$${}^{BD}{\chi }_{\mathbf{k}}^{\left(0\right)}(\eta ,v_{\mathbf{k}})=exp\left[-\frac{i}{2}\left(\eta k-arctan\left(\eta k\right)\right)\right]{\left(\frac{k^3}{\pi \hslash \left(\frac{1}{{\eta }^2}+k^2\right)}\right)}^{\frac{1}{4}}exp\left[\frac{i}{2\hslash }\left(-\frac{1}{{\eta }^3\left(\frac{1}{{\eta }^2}+k^2\right)}+i\frac{k^3}{\frac{1}{{\eta }^2}+k^2}\right)v_{\mathbf{k}}^2\right]$$

(52)

We can now focus on the next order $M^{-1}$, where, due to the quantum gravity corrections, the dynamics is no longer that of a time-dependent oscillator:

$$i\hslash {\partial }_{\eta }{\chi }_{\mathbf{k}}^{\left(1\right)}=\left[\frac{i\hslash }{24}\frac{1}{a_0^2{e}^{2\alpha }}\left({\partial }_{\alpha }{S}_0\right){\partial }_{\alpha }-\frac{{\hslash }^2}{2}{\partial }_{v_{\mathbf{k}}}^2+\frac{1}{2}{\omega }_k^2{v}_{\mathbf{k}}^2\right]{\chi }_{\mathbf{k}}^{\left(1\right)}.$$

(53)

By substituting (44) and the classical background solution $a_0{e}^{\alpha }\left(\eta \right)=-\sqrt{\frac{3}{\Lambda }}\frac{1}{\eta }$, Equation (53) becomes

$$i\hslash {\partial }_{\eta }{\chi }_{\mathbf{k}}^{\left(1\right)}(\alpha ,\eta ,v_{\mathbf{k}})=\left[\frac{i\hslash }{\eta }{\partial }_{\alpha }-\frac{{\hslash }^2}{2}{\partial }_{v_{\mathbf{k}}}^2+\frac{1}{2}{\omega }_k^2\left(\eta \right)v_{\mathbf{k}}^2\right]{\chi }_{\mathbf{k}}^{\left(1\right)}(\alpha ,\eta ,v_{\mathbf{k}}).$$

(54)

We investigate the class of separable solutions of the form

$${\chi }_{\mathbf{k}}^{\left(1\right)}(\alpha ,\eta ,v_{\mathbf{k}})=\theta \left(\alpha \right){\Gamma }_{\mathbf{k}}(\eta ,v_{\mathbf{k}}),$$

(55)

where we remark that the (quantum) degree of freedom $\alpha$ is in principle independent from the chosen conformal time $\eta$, and the classical relation only stands in the appropriate low-energy limit. Then, Equation (54) is solved for

$$\begin{array}{c}-i\hslash {\partial }_{\alpha }\theta \left(\alpha \right)=\lambda \theta \left(\alpha \right),\end{array}$$

(56)

$$\begin{array}{c}i\hslash {\partial }_{\eta }{\Gamma }_{\mathbf{k}}(\eta ,v_{\mathbf{k}})=\left(-\frac{{\hslash }^2}{2}{\partial }_{v_{\mathbf{k}}}^2+\frac{1}{2}{\omega }_k^2\left(\eta \right)v_{\mathbf{k}}^2-\frac{\lambda }{\eta }\right){\Gamma }_{\mathbf{k}}(\eta ,v_{\mathbf{k}}),\end{array}$$

(57)

where the constant $\lambda$ identifies the family of solutions of (56), which gives the eigenvalues of the momentum associated with $\alpha$ and so to the scale factor a. Equation (57) can be solved via another suitable rescaling, ${\Gamma }_{\mathbf{k}}(\eta ,v_{\mathbf{k}})=exp\left[\frac{i}{\hslash }\lambda log(-\eta )\right]{\tilde{\Gamma }}_{\mathbf{k}}(\eta ,v_{\mathbf{k}})$, which absorbs the $\lambda$-factor and maps it into an equation of the form (47) for ${\tilde{\Gamma }}_{\mathbf{k}}$, i.e., the usual time-dependent harmonic oscillator. Therefore, the function ${\tilde{\Gamma }}_{\mathbf{k}}$ coincides with the ${\chi }_{\mathbf{k}}^{\left(0\right)}$ of the previous order, and the ${\Gamma }_{\mathbf{k}}$ is readily obtained from the rescaling above. By putting together the solutions of (56) and (57), we can write the complete matter wave function (55) as

$${\chi }_{\mathbf{k}}^{\left(1\right)}(\alpha ,\eta ,v_{\mathbf{k}})={\theta }_{p_{\alpha }}\left(\alpha \right)e^{\frac{i}{\hslash }{p}_{\alpha }log(-\eta )}{\chi }_{\mathbf{k}}^{\left(0\right)}(\eta ,v_{\mathbf{k}}),$$

(58)

which can then be implemented to analyze the quantum-gravity corrected power spectrum.

However, before that computation, we stress one important remark of this approach. The requirement (20) imposed in Section 2.2 due to the B-O approximation scheme translates, in this specific minisuperspace setting, to $|p_{\alpha }|<1/M$. Therefore, one must consider for (58) a convolution over the suitable values of the momentum $p_{\alpha }$ 

$${\chi }_{\mathbf{k}}^{\left(1\right)}(\alpha ,\eta ,v_{\mathbf{k}})={\chi }_{\mathbf{k}}^{\left(0\right)}(\eta ,v_{\mathbf{k}})\int d{p}_{\alpha }g\left(p_{\alpha }\right){\theta }_{p_{\alpha }}\left(\alpha \right)e^{\frac{i}{\hslash }log(-\eta )p_{\alpha }},$$

(59)

with $g\left(p_{\alpha }\right)$ being a generic distribution. More specifically, choosing a Gaussian weight with deviation $\sigma$ and zero mean value

$$g\left(p_{\alpha }\right)=\frac{1}{{\left(\sqrt{2\pi }\sigma \right)}^{1/2}}{e}^{-\frac{p_{\alpha }^2}{4{\sigma }^2}},$$

(60)

the matter wave function modified by quantum gravity corrections ends up as

$${\chi }_{\mathbf{k},Gauss}^{\left(1\right)}(\alpha ,\eta ,v_{\mathbf{k}})={\chi }_{\mathbf{k}}^{\left(0\right)}(\eta ,v_{\mathbf{k}})\left\{{\left(8\pi {\sigma }^2\right)}^{1/4}exp\left[-\frac{{\sigma }^2}{{\hslash }^2}{\left(\alpha +log(-\eta )\right)}^2\right]\right\}.$$

(61)

We observe that the effect of the quantum gravity corrections has clearly factorized, an aspect which will deeply impact the result of the power spectrum analysis. Indeed, the obtained wave function shall be considered as the “new” vacuum state in order to derive the primordial power spectrum for the order $M^{-1}$ of the prescribed theory, i.e., modified by quantum gravity effects. However, since the modification affecting the wave function (61) takes the form of a time factor only, such a spectrum will coincide with the previous order result, which is computed with the wave function (52) in the absence of quantum gravitational corrections.

At this stage, the wave function ${\chi }^{\left(1\right)}$ retains remarkable dependence on the quantum variable $\alpha$ in the proposed paradigm, a property which has to be carefully addressed when studying phenomenological implications. Following the considerations in [35], we consider an “averaged” wave function in the form of

$$\overline{\chi }(\eta ,v_{\mathbf{k}})=\int d\alpha {\left|A\right|}^2\left(\alpha \right)\chi (\alpha ,\eta ,v_{\mathbf{k}})$$

(62)

where $A=e^{i{S}_1/\hslash }$ is the (quantum) amplitude coming from the lowest-order quantum gravitational component. This choice corresponds to averaging on the quasi-classical gravitational probability density, which in the selected minisuperspace is associated with the logarithmic scale factor $\alpha$ only. It is worth stressing that weighting the matter wave function on the WKB amplitude of the gravitational field is, on the present level, a purely phenomenological procedure. In fact, it is clear that such a wave function can in principle no longer satisfy the Schrödinger equation (53). Nonetheless, the applicability of the analysis in [69] is reliable, where it has been shown that such a calibrated wave function is actually a solution of the Schrödinger equation when suitable gauge invariance of the B-O procedure is taken into account.

Upon substitution of (61) and (45) into Equation (62), the averaged wave function for each mode becomes

$${\overline{\chi }}_{\mathbf{k},Gauss}^{\left(1\right)}(\eta ,v_{\mathbf{k}})={\chi }_{\mathbf{k}}^{\left(0\right)}(\eta ,v_{\mathbf{k}})\left[\hslash {\left(\frac{8{\pi }^3}{{\sigma }^2}\right)}^{\frac{1}{4}}{(-\eta )}^{3}exp\left(\frac{9{\hslash }^2}{4{\sigma }^2}\right)\right].$$

(63)

Requiring normalization over the possible $v_{\mathbf{k}}$ values, i.e., dividing by the wave function integrated on such variables, the term in squared brackets (which depends only on time and on the specific form of the weight (60)) clearly factors out of the integration. Therefore, we have for the averaged and normalized wave function

$${\overline{\chi }}_{\mathbf{k},Gauss}^{\left(1\right)}\stackrel{\mathit{integration}\mathit{over}\alpha }{\to }{\chi }_{\mathbf{k}}^{\left(0\right)}(\eta ,v_{\mathbf{k}}),$$

(64)

namely, we recover the previous order state.

Therefore, we now proceed to the computation of the inflationary power spectrum in the described setting, by computing the two-point correlation function of the M-S variable on the the Bunch–Davies state (52). For convenience, we rewrite ${}^{BD}{\chi }_{\mathbf{k}}^{\left(0\right)}$ in the following way:

$${}^{BD}{\chi }_{\mathbf{k}}^{\left(0\right)}(\eta ,v_{\mathbf{k}})=N_k\left(\eta \right)exp\left(i{\delta }_{0,k}\left(\eta \right)-{\Omega }_k\left(\eta \right)v_{\mathbf{k}}^2\right),$$

(65)

where

$$\begin{array}{c}{\Omega }_k\left(\eta \right):=\frac{1}{2\hslash }\left(\frac{i}{{\eta }^3\left(\frac{1}{{\eta }^2}+k^2\right)}+\frac{k^3}{\frac{1}{{\eta }^2}+k^2}\right),\end{array}$$

(66)

$$\begin{array}{c}{N}_k\left(\eta \right):={\left(\frac{2}{\pi }\Re \left({\Omega }_k\right)\right)}^{1/4}={\left(\frac{k^3}{\pi \hslash \left(\frac{1}{{\eta }^2}+k^2\right)}\right)}^{\frac{1}{4}},\end{array}$$

(67)

and $\Re (·)$ isolates the real part. In the following, we also isolate the real and imaginary parts of the (complex) variable $v_{\mathbf{k}}$ as

$$v_{\mathbf{k}}=\frac{1}{\sqrt{2}}(v_{\mathbf{k}}^R+i{v}_{\mathbf{k}}^I)$$

(68)

for the computation of the correlation function. Then, the two-point correlation function of the complex M-S variable computed on the Bunch–Davies vacuum state corresponds to (see [81], we are here dropping the prefix in ${}^{BD}{\chi }_{\mathbf{k}}^{\left(0\right)}$ for readability):

$$\begin{array}{cc}\hfill \Xi \left(\mathbf{r}\right)& =\langle 0\left|v(\eta ,\mathbf{x})v(\eta ,\mathbf{x}+\mathbf{r})\right|0\rangle =\int \prod _{\mathbf{k}}d{v}_{\mathbf{k}}^{R}d{v}_{\mathbf{k}}^I\left(\prod _{{\mathbf{k}}^{\prime }}{\chi }_{{\mathbf{k}}^{\prime }}^{\left(0\right)*}(\eta ,v_{{\mathbf{k}}^{\prime }})\right)v(\eta ,\mathbf{x})v(\eta ,\mathbf{x}+\mathbf{r})\left(\prod _{{\mathbf{k}}^{″}}{\chi }_{{\mathbf{k}}^{″}}^{\left(0\right)}(\eta ,v_{{\mathbf{k}}^{″}})\right)\hfill \\ \hfill & =\left(\prod _{\mathbf{l}}{\left|N_l\left(\eta \right)\right|}^4\right)\int \prod _{\mathbf{k}}d{v}_{\mathbf{k}}^{R}d{v}_{\mathbf{k}}^I\left(\prod _{{\mathbf{k}}^{\prime }}{e}^{-2\Re \left({\Omega }_{k^{\prime }}\right)\left[{\left(v_{{\mathbf{k}}^{\prime }}^R\right)}^2+{\left(v_{{\mathbf{k}}^{\prime }}^I\right)}^2\right]}\right)v(\eta ,\mathbf{x})v(\eta ,\mathbf{x}+\mathbf{r})\hfill \\ \hfill & =\left(\prod _{\mathbf{l}}\frac{2\Re \left({\Omega }_l\right)}{\pi }\right)\int \frac{d\mathbf{p}}{{\left(2\pi \right)}^{3/2}}\int \frac{d\mathbf{q}}{{\left(2\pi \right)}^{3/2}}{e}^{i\mathbf{p}·\mathbf{x}}{e}^{i\mathbf{q}·(\mathbf{x}+\mathbf{r})}\int \prod _{\mathbf{k}}d{v}_{\mathbf{k}}^{R}d{v}_{\mathbf{k}}^I\left[v_{\mathbf{p}}{v}_{\mathbf{q}}{e}^{-2{\sum }_{{\mathbf{k}}^{\prime }}\Re \left({\Omega }_{k^{\prime }}\right)\left({\left(v_{{\mathbf{k}}^{\prime }}^R\right)}^2+{\left(v_{{\mathbf{k}}^{\prime }}^I\right)}^2\right)}\right]\hfill \end{array}$$

(69)

where we are considering each Fourier mode of the vacuum state, substituting the expression (65) in the second equality, and expanding both variables in Fourier modes in the third. We observe that the last integral, due to its form, vanishes for $\mathbf{p}\ne \pm \mathbf{q}$, and the same happens for $\mathbf{p}=\mathbf{q}$, since we obtain exponents of the form $\left[{\left(v_{\mathbf{p}}^R\right)}^2-{\left(v_{\mathbf{p}}^I\right)}^2\right]/2$, and the real and imaginary parts contribute the same amounts. Therefore, the surviving contribution is in the case $\mathbf{p}=-\mathbf{q}$, that is,

$$\begin{array}{cc}\hfill \Xi \left(\mathbf{r}\right)& =\left(\prod _{\mathbf{l}}\frac{2\Re \left({\Omega }_l\right)}{\pi }\right)\int \frac{d\mathbf{p}}{{\left(2\pi \right)}^3}{e}^{-i\mathbf{p}·\mathbf{r}}2\int \prod _{\mathbf{k}}d{v}_{\mathbf{k}}^{R}d{v}_{\mathbf{k}}^I\left[{\left(v_{\mathbf{p}}^R\right)}^2{e}^{-2{\sum }_{{\mathbf{k}}^{\prime }}\Re \left({\Omega }_{k^{\prime }}\right)\left({\left(v_{{\mathbf{k}}^{\prime }}^R\right)}^2+{\left(v_{{\mathbf{k}}^{\prime }}^I\right)}^2\right)}\right]\hfill \\ \hfill & =\int \frac{d\mathbf{p}}{{\left(2\pi \right)}^3}{e}^{-i\mathbf{p}·\mathbf{r}}\frac{1}{2\Re \left({\Omega }_p\right)}\hfill \end{array}$$

(70)

where we remind that ${\Omega }_p={\Omega }_p\left(\eta \right)$ as from the definition (66). This corresponds, from (35) and the definition (36), to a power spectrum of the form

$${\mathcal{P}}_v\left(k\right)=\frac{k^3}{4{\pi }^2}\frac{1}{\Re \left({\Omega }_k\right)}.$$

(71)

Therefore, the invariant power spectrum associated with the curvature perturbation $\zeta$ (33) is given by

$${\mathcal{P}}_{\zeta }\left(k\right)=\frac{4\pi G}{ϵa_0^2{e}^{2\alpha }}{\mathcal{P}}_v\left(k\right)=\frac{G}{\pi ϵ}\frac{k^3}{a_0^2{e}^{2\alpha }}\frac{1}{\Re \left({\Omega }_k\right)}.$$

(72)

We now evaluate this quantity in the super-Hubble limit, which in conformal time corresponds to modes for which $k\eta \to 0^{-}$. In this case, we note from the definition (66) that the function $\Re \left({\Omega }_k\right)$ becomes

$$\Re \left({\Omega }_k\left(\eta \right)\right)\approx k^3{\eta }^2$$

(73)

(we are using $\hslash =1$ for easier comparison with the literature). When implementing this limit and substituting the classical solution $\alpha \left(\eta \right)$, we arrive at the following result for the primordial power spectrum in the de Sitter phase:

$${\mathcal{P}}_{\zeta }\left(k\right)=\frac{G{\mathrm{H}}_{\Lambda }^2}{\pi ϵ}{|}_{k=a{\mathrm{H}}_{\Lambda }},$$

(74)

where ${\mathrm{H}}_{\Lambda }=\sqrt{8\pi G\Lambda /3}$ and the slow-roll parameter $ϵ$ is evaluated at the horizon crossing. Recent satellite missions, such as WMAP [89] and PLANCK [90,91], provided an accurate detection of the fluctuation spectrum in the cosmic microwave background temperature. These observations, and in particular, the Gaussian profile of the fluctuations, properly fulfill the prediction of the inflation paradigm, and in this respect, a significant constraint for the spectral index $n_s$ 

$$n_s-1:=\frac{dln{\mathcal{P}}_{\zeta }}{dlnk}$$

(75)

is now available [90]. Nonetheless, some recent data analyses suggest the possibility of some anomaly in the Gaussianity of the fluctuations [92] and called attention to the possibility to be interpreted via a multifield inflationary scenario [93].

Clearly, the quantum gravity corrections we are searching for are extremely small with respect to the accuracy of the current fluctuation measurements, since they are in the order of the square ratio of the inflationary energy scale to the corresponding Planckian one, namely, about ${10}^{-8}$. Despite the possibility of detecting such quantum gravity modifications of the spectrum in current or near-future experiments appearing unlikely, nonetheless, their prediction looks to be a fundamental conceptual challenge.

We recall that we have here recovered the standard QFT spectrum for the primordial fluctuations via a functional approach, implementing the Gaussian fluid as a time parameter (28). It is evident that the quantum gravity corrections in (54) do not modify, but preserve the inflationary power spectrum up to this expansion order; an analogous result derived in a different context is present in [94]. Such result is clearly to be attributed to the form of the modified Schrödinger equation (53), which presents no coupling between the quantum gravitational degree of freedom $\alpha$ and the perturbation variables $v_{\mathbf{k}}$. It then follows that the correction to the “fast” wave function $\chi$ (59) factorizes, and due its time-dependent form, does not influence the evolution of the perturbation modes in the considered setting.

4. Towards the General Case

The result presented in Section 3 suggests that the quantum gravity-induced corrections on the matter evolution, obtained in the WKB expansion and via the time parameter introduced in (28), give as a net effect a time-dependent factor. Such term could be considered a posteriori a phase rescaling acting on the matter wave function, as we show here in the general case.

Let us start from the modified dynamics (30) analyzed for a generic minisuperspace model (the supermomentum is identically vanishing); for this purpose we work with the (homogeneous) generalized variable $h_a$ (i.e., the degrees of freedom associated with the 3-geometries) and the corresponding minisupermetric $G_{ab}$, instead of the spatial metric $h_{ij}$ [20]. We adopt for convenience the synchronous time $N=1$ such that the definition (28) coincides with the derivative with respect to T, up to a fiducial volume set to unit, but the result here discussed stands for a generic lapse function N. Explicitly, the dynamics up to the order $M^{-1}$ are described by

$$i\hslash \frac{\partial \chi }{\partial T}={\widehat{\mathcal{H}}}^m\chi -i\hslash G_{ab}\frac{\partial S_0}{\partial h_a}\frac{\partial }{\partial h_b}\chi ,$$

(76)

and we write the matter wave functional as

$$\chi (h_a,T,\varphi )={\xi }_g\left(h_a\right){\Theta }_m(T,\varphi ).$$

(77)

We remark that this is a stronger requirement and is inherently different from the Born–Oppenheimer separation (18), since ${\Theta }_m$ is now assumed to be independent of the generalized coordinate $h_a$. Such separation is backed by the observation that, since there is no quantum matter back-reaction in the present model, we can consider the two sets of degrees of freedom as independent. By substituting (77) into (76), and dividing by the non-trivial functional ${\xi }_g$, we obtain

$$i\hslash \frac{\partial {\Theta }_m}{\partial T}={\widehat{\mathcal{H}}}^m{\Theta }_m-\frac{i\hslash }{{\xi }_g}{G}_{ab}\frac{\partial S_0}{\partial h_a}\frac{\partial {\xi }_g}{\partial h_b}{\Theta }_m.$$

(78)

Here, $S_0$ belongs to the classical solution (see Equation (25)); thus, the corresponding factor is a function of time only: ${\partial }_{h_a}{S}_0=f\left(T\right)$, where the form of f depends on the specific cosmological model. Additionally, the modified dynamics cannot induce dependence of $\Theta$ on the $h_a$, since that was separated in (77). Then, we can express the factor containing ${\xi }_g$ as a constant, whose value can depend on the quantum number associated with $h_a$; i.e., its value is fixed during the dynamics once a specific foliation is selected:

$$\frac{1}{{\xi }_g}\frac{\partial {\xi }_g}{\partial h_a}=i{k}_{\left(h_a\right)}$$

(79)

where for convenience, we have inverted the couple of indices a and b in (76), making use of the symmetry of the minisupermetric $G_{ab}$. The writing $k_{\left(h_a\right)}$ is to be understood as a function of the gravitational variable $h_a$. The solution to (79) has a plane wave structure

$${\xi }_g\left(h_a\right)=e^{i{k}_{\left(h_a\right)}·h_a}.$$

(80)

The functions (80) constitute a complete basis that can be adopted to construct wave packets, which will describe the quantum gravitational contribution to $\chi$. In what follows, we limit our attention to the plane wave (80) associated with a specific value $k_{\left(h_a\right)}$; in this case, the modified dynamics take the form

$$i\hslash \frac{\partial {\Theta }_m}{\partial T}={\widehat{\mathcal{H}}}^m{\Theta }_m+\hslash f\left(T\right)k_{\left(h_a\right)}{\Theta }_m$$

(81)

We now rewrite the function ${\Theta }_m$, which is useful for the computation of the corrective effects, as:

$${\Theta }_m(T,\varphi )=e^{i\Lambda \left(T\right)}\varrho (T,\varphi ),$$

(82)

where $\varrho$ has the same degrees of freedom with respect to ${\Theta }_m$, and a (complex) time-dependent phase $\Lambda$ has been separated. In the general case, such a phase can acquire different forms depending on the wave number $k_{\left(h_a\right)}$ present in (79) and (80) (or, as we will discuss later, depending on the considered wave packet). It is exactly the phase factor $\Lambda \left(T\right)$ that will account for the quantum gravity corrections, since we will see that $\varrho$ exactly solves the unperturbed matter dynamics at such order. Indeed, by substituting (82) into (81) and requiring that

$$\frac{\partial \Lambda }{\partial T}=f\left(T\right)k_{\left(h_a\right)},$$

(83)

the additional contribution on the right-hand side of (81) cancels out via the phase rescaling, and the function $\varrho$ satisfies the unperturbed Schrödinger evolution:

$$i\hslash \frac{\partial \varrho (T,\varphi )}{\partial T}={\widehat{\mathcal{H}}}_m\varrho (T,\varphi ).$$

(84)

Here, the matter Hamiltonian ${\mathcal{H}}_m$ is left as a generic expression; for the purpose of the cosmological implementation above, it took the form of a time-dependent harmonic oscillator in Section 3.3.

It is then possible to discuss any effects of such quantum gravity contributions to the scalar field’s power spectrum. As previously stated, the net effect is encased in the time-dependent phase $\Lambda \left(T\right)$ solution of (83), which is actually real-valued, since $f\left(T\right)$ follows from the classical solution $S_0$. The complete matter wave function at $\mathcal{O}\left(M^{-1}\right)$ thus reads

$$\chi (h_a,T,\varphi )=e^{i{k}_{\left(h_a\right)}·h_a}{e}^{i{k}_{\left(h_a\right)}\int d{T}^{\prime }f\left(T^{\prime }\right)}\varrho (T,\varphi )$$

(85)

where the integral in the second term $\int d{T}^{\prime }f\left(T^{\prime }\right)$ is intended to be between values $T_0$ and T, for which the WKB approximation holds. We observe that the solution (85) has the same shape of the result discussed in Section 3.3. Due to the peculiar morphology of the quantum gravity factors, arising from (79) and (83) (which originally stem from the requirement (82)), the effect on the matter spectrum is canceled once the matter wave function is properly normalized. This is the reason for which, as shown in Section 3.3, the quantum gravity corrections preserve the primordial inflationary spectrum.

Clearly, the fact that, at the order $M^{-1}$, no corrections emerge for the inflationary spectrum from quantum gravity effects, does not mean that a possible deformation of the scale invariance property cannot come out at the next orders of approximation. However, here is a peculiar point that deserves specific attention: the absence of a spectral modification is a consequence of the phase form that the quantum gravity corrections take in the matter wave function, and in turn, this feature is induced by the possibility of factorizing such a wave function into a gravitational and a matter component. The physical meaning of this assumption must be searched in the absence of a quantum matter backreaction on the classical gravitational background.

4.1. On the Role of the Matter Backreaction

We observe that the $S_0$ solution for the gravitational field, and in particular, the classical momentum term appearing in the quantum gravity corrections in Equation (76), do not depend, by the considered WKB perturbation scheme, on the quantum matter degrees of freedom. It is exactly this point which enters the possibility of factorizing the matter wave function into two independent components (77). On the contrary, if the H-J equation, Equation (25), contains the expectation value of the quantum matter Hamiltonian, then also the classical momentum would be, on average, affected by the quantum degrees of freedom. Then, the choice of a factorized form for the matter wave function, even if still possible, would no longer appear as a natural solution to the perturbed dynamics.

To elucidate this point of view, we here discuss in more detail the role played by the matter backreaction. In fact, when implementing a standard B-O scheme (see the original formulation [95]) in the WKB approximation order by order in $1/M$ [25], it is immediately recognizable that the quantum matter expectation value enters both the right-hand side of the H-J equation (25), and the Schrödinger equation (29) (see also [96] and for a review [97]). As stated in [98], this contribution can be easily removed from the Schrödinger dynamics by phase rescaling, where the phase contains the matter backreaction term, though as a function of the gravitational degrees of freedom only. This operation is allowed by a natural gauge invariance of the total B-O wave function. However, as shown in [33], this redefinition of the matter wave function induces an opposite change of phase to the gravitational one, with the net effect that the backreaction term is also removed from the H-J equation.

These considerations suggest that such a contribution could always be neglected in view of the gauge invariance analyzed above, and hence, that the emergence of a quantum gravity correction originating from the matter backreaction cannot be inferred. Nonetheless, we question here the correctness of doing such a phase redefinition via the matter expectation value. Actually, in the B-O procedure, the gauge invariance is used to eliminate the Berry phase [99,100,101], but not to cancel the (fast) electronic eigenvalue contribution in the (slow) nuclear dynamics [102]. From this point of view, it is more natural to maintain the expectation value contribution both in the H-J equation and in the Schrödinger one. This would lead to a non-trivial coupled integro-partial differential system which could be treated with a self-consistent method (for a related treatment of the backreaction in a different context, see [79]).

The discussion above was thought to refer to order $M^1$ and $M^0$ of the WKB approximation, but it naturally extends to the $M^{-1}$ order. Thus, if we include the matter Hamiltonian term in Equation (76), we arrive to a coupled system that only at the lowest order of approximation in a Hartree self-consistent approach can be reduced to the form discussed in Section 2.2. The complete problem naturally introduces dependence of the H-J function $S_0$ on the matter one (via an integral of the matter’s degrees of freedom); this point clarifies the technical content of the discussion above on the role of the matter backreaction in the separability of Equation (76) to some order of approximation. Therefore, we are led to conclude that the proposed WKB expansion in the quantity $1/M$ must carefully take into account the evaluation of the matter (average) backreaction on the gravitational quasi-classical background.

5. Concluding Remarks

Here, we reviewed the analysis presented in [35], aiming to calculate the quantum gravity corrections to QFT, in the theoretical framework of fixing a Gaussian reference frame, as discussed in [59]. The motivations for addressing such a revised scheme came from the search for a formulation which is not affected by the non-unitarity questions faced in [21,29] when reconstructing a time variable for the matter wave function from the classical limit of the background gravitational field. The physical clock in [35] is provided by the materialization of the Gaussian frame as a dust fluid. It is important to recall here that such an emerging dust-like contribution no longer has, in the WKB expansion, the shortcoming of a non-positive defined energy density.

The present study implemented the procedure mentioned above to calculate the possible quantum gravity corrections to the primordial inflationary spectrum. We considered the quasi-classical background corresponding to a Robertson–Walker geometry in the presence of a cosmological constant term, mimicking the vacuum energy of an inflationary phase transition, as viewed in the resulting de Sitter evolution. The matter field we quantized in the proposed scheme was clearly the inflaton scalar degree of freedom, for which we applied a Fourier decomposition and introduced the gauge-invariant M-S formulation [72,73,74]. The field would have been in principle associated with a wave functional, describing its dynamics, but the Fourier decomposition of its Hamiltonian allowed us to deal with a minisuperspace formulation for each independent wavenumber modulus (we recall that the inflaton can be regarded, with a very good approximation, as a free massless scalar field during the slow-rolling phase of the inflation process). Clearly, we solved the wave equation amended for the quantum gravity corrections, and we ended up showing that the solution of the time-dependent harmonic oscillator associated with each k-mode is rescaled by a phase factor, due to the additional contribution to the Schrödinger equation. As an immediate consequence, we could conclude that, at the considered approximation in the WKB scheme, no modification of the inflationary spectrum of the Universe can be determined.

It would be worth analyzing the present formulation in the case in which the background gravitational field is described by a modified theory of gravity. For a discussion of how modified gravity affects the inflationary spectrum, see [103,104,105,106,107,108] but it calls attention for further investigation how these results would appear in the present framework, i.e., including quantum gravity corrections of the extended formulation (for approaches which quantize the modified metric $f\left(R\right)$ gravity, see [109,110,111]).

We then discussed this result in the scheme of a generic minisuperspace model, in order to outline the real physical explanation for such surprising preservation of the scale-invariant spectrum in the theory proposed in [35]. From a mathematical point of view, we recognized that the emergence of a phase term in the matter wave function, depending on the scale factor, is a consequence of the possibility of factorizing such a wave function ab initio. Clearly, by considering higher-order contributions to the inflaton Schrödinger equation in the expansion with respect to $1/M$, a modification of the scale invariant spectrum could arise. However, in the considered theoretical framework, the factorization of the matter wave function came from the absence of an average matter backreaction in the classical Robertson–Walker dynamics, i.e., in the H-J equation, as discussed in detail in Section 4.1. A study of the modifications to the power spectrum when the backreaction is taken into account is beyond the analysis here presented and could be investigated in future works.

The analysis above suggests that the scheme in [35] could require further restatement in order to better separate the classical and quantum degrees of freedom, which is beyond the scope of this paper. Particular attention has to be focused on the procedure by which the gravitational degrees of freedom are treated—i.e., their classical and quantum components would have to be described via independent variables; see, for example, the proposal [69]. Only after such a reformulation of the gravitational background could the question concerning the matter backreaction be properly addressed in the B-O WKB picture proposed here. This perspective should call attention for future developments calculating the quantum gravity corrections to the inflationary spectrum.