Quantum gravitational corrections to the inflationary power spectra …

Quantum gravitational corrections to the inflationary power spectra in scalar-tensor theories

Christian F Steinwachs and Matthijs L van der Wild Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Str. 3, 79104, Freiburg im Breisgau, Germany christian.steinwachs@physik.uni-freiburg.de wild@physik.uni-freiburg.de
Abstract

We derive the first quantum gravitational corrections to the inflationary power spectra for a general single-field scalar-tensor theory which includes a non-minimal coupling to gravity, a non-standard scalar kinetic term and an arbitrary potential of the scalar field. We obtain these corrections from a semiclassical expansion of the Wheeler-DeWitt equation, which, in turn, governs the full quantum dynamics in the canonical approach to quantum gravity. We discuss the magnitude and relevance of these corrections, as well as their characteristic signature in the inflationary spectral observables. We compare our results to similar calculations performed for a minimally coupled scalar field with a canonical kinetic term and discuss the impact of the non-minimal coupling on the quantum gravitational corrections.

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1 Introduction

The inflationary paradigm is an integral part in the dynamics of the early universe and explains the formation of structure out of tiny quantum fluctuations [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. There is a plethora of inflationary models leading to different theoretical predictions that can be tested with observational data [12]. The predictions for the inflationary power spectrum of the perturbations can be tested by analyzing the temperature anisotropy spectrum of the cosmic microwave background radiation (cmb) as measured by satellites such as Planck [13].

In the theoretical description of the inflationary mechanism, the quasi-De Sitter phase of accelerated expansion is realized usually by one or more scalar fields. In the simplest models a single scalar field, the inflaton, which is minimally coupled to gravity with a canonical kinetic term, drives inflation [14]. In this case, the inflationary predictions within the slow-roll approximation are entirely determined by the scalar field potential. However, recent observational data favors inflationary models based on more general scalar-tensor theories [15, 16, 17, 18, 19, 20, 21], as well as geometric modifications of gravity such as theories [22, 1, 23]. Two prominent representatives of these two classes are the model of Higgs inflation, in which the inflaton is non-minimally coupled to gravity and identified with the Standard Model Higgs boson [24], and Starobinsky’s model of inflation in which the inflaton is identified with the scalaron [1], the additional propagating geometrical scalar degree of freedom that arises effectively due to the presence of higher derivatives in the quadratic curvature invariant [25]. Both models lead to almost indistinguishable predictions for the inflationary spectral observables [26, 27, 28, 29]. They are both representatives of a larger class of inflationary attractors [30, 31]. Moreover, these similarities are a manifestation of a more general equivalence between scalar-tensor theories and gravity, see e.g. [20]. Based on the one-loop results [32] obtained within the perturbative covariant approach, this equivalence was shown recently to also hold at the quantum level [33], see also [34] for a similar analysis. These questions are closely related to the question of quantum equivalence between different field parametrizations in scalar-tensor theories [35] for which the one-loop corrections have been obtained in [36, 37, 38].

Recently, it was suggested to combine these two models into a single scalar-tensor theory [39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53], dubbed “scalaron-Higgs inflation” in [49]; see also [54, 55, 56] for a similar analysis within the Palatini formalism. At the classical level, this combined model features an asymptotic scale invariance, which is the theoretical motivation for many interesting models, see e.g. [57, 58, 59, 60, 61, 62, 63].

Quantum corrections in these inflationary models can become important. For example, in the model of Higgs-inflation, the radiative corrections and the renormalization group improvement turned out to be crucial for the consistency with particle physics experiments [26, 64, 65, 66, 67, 68, 28, 69]. While in this case the quantum corrections are dominated by the heavy Standard Model particles, it is in general interesting to also study the effect of quantum gravitational corrections on inflationary predictions. In fact, the strong curvature regime during the inflationary phase make the early universe a natural testing ground for any theory of quantum gravity.

In this paper, we calculate the first quantum gravitational corrections to the inflationary power spectra obtained by a canonical quantization of a general scalar-tensor theory in the framework of quantum geometrodynamics, which is one of the earliest attempts of a non-perturbative quantization of gravity [70]. See [71] for a comprehensive overview about various approaches to quantum gravity.

In the canonical approach spacetime is foliated into leaves of spatial hypersurfaces and the spatial metric as well as its conjugated momentum are the natural variables to be quantized. Making use of the Arnowitt-Deser-Misner (adm) formalism, the covariant action is cast into a constrained Hamiltonian system [72]. The Dirac quantization of the Hamiltonian constraint, which governs the dynamics of this system, leads to the Wheeler-DeWitt equation [73, 74]. Although the canonical approach to quantum gravity does not come without difficulties, both at the conceptual and technical level, the Wheeler-DeWitt equation can be considered as a natural starting point for the analysis of quantum gravitational effects, as its semiclassical expansion reproduces the classical theory and the functional Schrödinger equation for quantized matter fields on a curved background at the lowest orders of the expansion [75, 76]. Therefore, higher order terms in the expansion can be clearly attributed to the first quantum gravitational corrections. When applied to the inflationary universe, these quantum gravitational corrections leave observational signatures in the primordial power spectrum. This has been investigated for a minimally coupled scalar field with a canonical kinetic term [77, 78, 79, 80, 81, 82, 83, 84]. In this work we generalize these analyses to a general scalar-tensor theory of a single scalar field with an arbitrary non-minimal coupling to gravity, a non-standard kinetic term, and an arbitrary scalar potential.

The paper is structured as follows: in Sec. 2, we introduce the model, derive the equations of motion, perform the reduction to a homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker (flrw) universe and discuss the inflationary dynamics of the background and the cosmological perturbations. In Sec. 3, we derive the classical Hamiltonian constraint, perform the Dirac quantization, and obtain the Wheeler-DeWitt equation for the background and perturbation variables. In Sec 4, we perform a semiclassical expansion based on a combined Born-Oppenheimer and Wentzel-Kramers-Brillouin (wkb)-type approximation. At the lowest orders we recover the dynamical background equations and the notion of a semiclassical time. At the next order, we obtain a Schrödinger equation for the perturbations. Finally, the subsequent order yields the first quantum gravitational corrections to the Schrödinger equation. In Sec. 5, we derive the connection between the results found from the semiclassical expansion of the Wheeler-DeWitt equation and the inflationary power spectra. In Sec. 6, we discuss the impact of the leading quantum gravitational corrections on the inflationary power spectra and their observational consequences. Finally, we summarize our main results and conclude in Sec. 7. The results for the subleading slow-roll contributions to the quantum gravitational corrections are provided in A.

2 Scalar-tensor theories of inflation

Almost all models of inflation driven by a single scalar field can be covered by the general scalar-tensor theory

(1)

Here, , , and are three arbitrary functions of the scalar inflaton field . They parametrize the non-minimal coupling to gravity, the non-canonical kinetic term, and the scalar field potential, respectively. We work in four dimensional spacetime with metric of mostly plus signature. The scalar curvature is denoted by . Spacetime coordinates are labeled by , , with capital letters.

2.1 Field equations and energy-momentum tensor

The field equations for the metric and the Klein-Gordon equation of the inflaton are obtained by varying (1) with respect to and , respectively:

(2)
(3)

where the effective energy-momentum tensor is defined as

(4)

Here denotes the covariant d’Alembert operator.

2.2 Spacetime foliation

It is useful to reformulate the action in terms of the adm formalism [72], where the four dimensional metric is expressed in terms of the lapse function , the spatial shift vector , and the spatial metric ,

(5)

Here, the spatial coordinates , are denoted by small letters. In terms of the adm variables, the action (1) can be compactly represented as

(6)

with the dynamical configuration space variables collectively denoted by and the reparametrization invariant covariant time derivative , with the Lie derivative along the spatial shift vector . The bilinear form corresponds to the inverse of the configuration space metric derived in [85],

(7)

with the DeWitt metric . 111Note that the configuration space metric (7) was defined with additional inverse factors of the lapse function in [85]. The potential , which includes the spatial gradient terms of the scalar field, is defined as

(8)

Here, is the positive definite spatial Laplacian, the spatial covariant derivative compatible with and is the three-dimensional spatial curvature. In addition, we have defined the suppression function [65, 66, 38, 85]:

(9)

The subscript is a shorthand for a derivative with respect to the argument, i.e. we denote the derivative of a general field-dependent scalar function with respect to as

(10)

2.3 Cosmological background evolution

The background spacetime is described by a spatially flat () flrw line element

(11)

Comparing with the adm line element (5), spatial flatness, homogeneity and isotropy imply and , where the lapse function and the scale factor are functions of time only. Similarly, homogeneity implies that the scalar field is a function of time only . Moreover, for the isotropic line element (11) the reparametrization invariant time derivative reduces to . In addition, it is convenient to introduce the conformal time , related to the coordinate time by . This choice corresponds to the gauge , which we will adopt in what follows. In terms of , the flrw metric (11) acquires the manifestly conformally flat structure , and the reparametrization invariant covariant time derivative reduces to a partial derivative with respect to conformal time . The conformal Hubble parameter is defined as

(12)

where the prime denotes a derivative with respect to conformal time . In the flrw universe takes on the form of the energy-momentum tensor of a perfect fluid:

(13)

Here, is the fluid’s four-velocity with norm , is its energy density, and is its pressure. Comparison with (4) leads to the identifications

(14)
(15)

The symmetry reduced flrw action expressed in terms of the compact notation has a form similar to (6) with and

(16)

The explicit expression for the background action is given by

(17)
(18)

In particular, the derivative coupling between the gravitational and scalar field degrees of freedom induced by the non-minimal coupling becomes manifest. The Friedmann equations and the Klein-Gordon equation are obtained from varying (17) with respect to , , and , or directly from symmetry reducing the equations of motion (2) and (3):

(19)
(20)
(21)

The dimensionless ratio , related to the Einstein frame potential [85], is defined as

(22)

In a flat spatially homogeneous flrw universe, the spatial integral in the action (17) is formally divergent, corresponding to an infinite spatial volume . In order to regularize the spatial integral, we need to introduce some large but finite reference length scale such that . The reference volume can be removed from the formalism by absorbing it into a redefinition of the time variable and the scale factor, such that the action (17) is independent of [80, 81]:

(23)

While in this way any dependence on the reference scale has been eliminated form the formalism, the restriction of the spatial volume to a compact subregion nevertheless has observational consequences, which we discuss in Sec. 6.

2.4 Inflationary background dynamics in the slow-roll approximation

During inflation the universe undergoes a quasi-De Sitter stage in which the energy density is approximately constant and effectively dominated by the potential of the slowly rolling scalar field. In scalar-tensor theories, the slow-roll conditions can be generalized for any function of the scalar field as [86]:

(24)

In particular, for the scalar-tensor theory (1), this encompasses the generalized potentials . Making use of (14) and (15), within the slow-roll regime (24) the background equations (19)-(21) lead to

(25)

The slow-roll conditions (24) motivate the definitions of the following four slow-roll parameters [87], which quantify small deviations from De Sitter space:

(26)

The slow-roll parameters and are the same as for a minimally coupled canonical scalar field, while the slow-roll parameters , contain information about the non-minimal coupling and the generalized kinetic term via the function defined in (9). We work to linear order in the slow-roll approximation, where the , are treated as constant. In addition to (26), we define another set of “potential” slow-roll parameters , which are expressed directly in terms of the generalized potentials , and and their derivatives,

(27)

Within the slow-roll approximation . Therefore, in what follows we simply write for both sets of slow-roll parameters (26) and (27). During the slow-roll regime, a sufficiently long quasi-De Sitter phase of inflation is realized for .

2.5 Cosmological perturbations

We split the fields and into background , and perturbation and ,

(28)

In the following, we omit the bar. In addition, we decompose the metric perturbation into its irreducible components. The cosmological line element including linear perturbations can be parametrized as

(29)

Here, is the scalar perturbation of the lapse function. The perturbation of the shift vector , as well as the spatial metric , are further decomposed as

(30)

with the scalar perturbations , , , the transverse vector perturbations , and the transverse traceless tensor perturbation . Since vector modes decay during inflation, we neglect them in what follows and focus on the scalar and tensor perturbations. The gauge invariant transverse traceless part of the metric perturbation can be associated with primordial gravitational waves

(31)

where denotes the polarization tensor. Since the scalar perturbations are gauge dependent, it is convenient to work with the single scalar gauge invariant combination, the Mukhanov-Sasaki variable [88]:

(32)

Finally, we introduce the canonical field variables for the scalar perturbation and the transverse-traceless tensor perturbation :

(33)

The corresponding factors and defined as [87]:

(34)

The action quadratic in the perturbations and reads [87]:

(35)
(36)
(37)

In the derivation of (36) and (37), total derivative terms are neglected and it is assumed that the background fields satisfy their equations of motion (19)-(21). Since we consider only linear perturbations, the expansion stops at second order and the total combined action of background plus perturbations reads

(38)

Next, we perform a Fourier transformation of the inhomogeneous perturbations:

(39)

Since the position space perturbations are real, we have and . The restriction of the spatial volume to a compact subregion makes it necessary to perform the discrete Fourier transform (39) with the volume factor , introduced to regularize the spatial integral in (17). Moreover, due to the isotropy of the flrw background, the mode components can only depend on the magnitude of the wave vector , rather than its direction. The Fourier transformed action (38) then acquires the form of a sum of harmonic oscillators

(40)
(41)
(42)

with time-dependent frequencies

(43)

In a similar fashion as in (23), it is possible to eliminate any explicit occurrence of the reference volume in the Fourier transformed action (40) by the rescalings [80, 81],

(44)

The Fourier transformed version of the total action (38) is the starting point for the Hamiltonian formulation carried out in the next section.

3 Quantum Geometrodynamics

3.1 Hamiltonian formalism

The canonical quantization of gravity is based on its Hamiltonian formulation. We perform a Legendre transformation of with the generalized momenta

(45)

which leads to the Hamiltonian constraint

(46)

The individual Hamiltonians of the background and perturbation variables read

(47)
(48)
(49)
(50)

3.2 Quantum Geometrodynamics and Wheeler-DeWitt equation

In the canonical quantization procedure, the configuration space variables , , , , and momenta , , , , are promoted to operators that act on states and obey the canonical commutation relations (in units ) 222Formally, for a consistent quantization, the configuration space variables associated with the perturbations should be doubled by decomposing the complex Fourier modes and into real and imaginary parts [89]. For the sake of a compact formulation, we proceed with quantizing terms such as by simply treating them as – the final results are not affected by this.,

(51)

with all other commutators equal to zero. In the Schrödinger representation, the position space operators act multiplicatively and the momentum space operators act as differential operators with the explicit form

(52)

The Wheeler-DeWitt equation is obtained by promoting (46) to an operator equation, acting on a wave function . Following the prescription for the quantization of constrained systems, introduced by Dirac [73], the implementation of the classical constraint equation (46) at the quantum level corresponds to selecting only those states which are annihilated by ,

(53)

The Wheeler-DeWitt equation (53) is defined only up to operator ordering. The results for the semi-classical expansion performed in the subsequent sections are however independent of the factor ordering, see e.g. [75, 85] for details.

4 Semiclassical expansion of the Wheeler-DeWitt equation

For almost all cases, the full Wheeler-DeWitt equation cannot be solved exactly. Since we are interested only in the first quantum gravitational corrections, we do not need to find exact solutions but instead perform a systematic semiclassical expansion of the Wheeler-DeWitt equation. This semiclassical expansion is based on the combined use of a Born-Oppenheimer and wkb-type approximation scheme. The former relies on a clear distinction between the “heavy” and “light” degrees of freedom. In the original Born-Oppenheimer approach to molecular physics, this distinction is based on the presence of a mass hierarchy between different degrees of freedom. For a scalar field minimally coupled to Einstein gravity, such a mass hierarchy could be related to the ratio , with the effective scalar field mass . In this context, the gravitational degrees of freedom are the heavy or “slow” ones, while the scalar field degrees of freedom are the light or “fast” ones [90, 91, 92, 93, 94, 75, 95, 79, 80]. Such a scenario would correspond to a slowly varying background geometry on which the quantum matter (scalar field) degrees of freedom propagate. For a scalar field non-minimally coupled to gravity, the identification of light and heavy degrees of freedom becomes more subtle [85]. In the case of the Hamiltonian (46), the heavy degrees of freedom are identified with the homogeneous background variables and , while the light degrees of freedom are associated with the infinitely many degrees of freedom corresponding to the Fourier components of the inhomogeneous perturbations and . In the cosmological framework, this distinction follows naturally from the observed temperature anisotropies in the cmb.

4.1 Implementation of the semiclassical expansion

In the following we use a condensed notation and collectively denote the heavy degrees of freedom by and the light degrees of freedom by . The index labels both the Fourier modes as well as the different types of perturbations. At a technical level, the distinction between heavy and light degrees of freedom can be implemented by introducing a formal weighting parameter in the Hamiltonian for the heavy degrees of freedom, which can be set to one after the expansion[85],

(54)

Here, in correspondence with the notation in (6), collectively denotes the operators and , and and denote the operator versions of (16). Combining (46) with (52) and the weighting of the background Hamiltonian (54), the Wheeler-DeWitt equation has the form

(55)

The Hamiltonian of the perturbation has the explicit form333We are implicitly treating the background variables , in the frequencies and as classical, i.e. not subjected to the canonical quantization procedure. Within the semiclassical expansion, this means that the variables enter the frequencies only parametrically via . This procedure might be justified a posteriori, by showing that a full quantum treatment of these variables would only affect terms at higher order in the semiclassical expansion [81].,

(56)

In what follows we suppress hats on operators and resort to the abbreviated notation

(57)

The additive structure of the Wheeler-DeWitt equation (55) suggests the product ansatz

(58)
(59)

Inserting the ansatz (59) into the Wheeler-DeWitt equation (55), dividing the result by , and separating those terms which only depend on the background variables from those that depend additionally on the perturbations , leads to a family of separate equations [96, 95]:

(60)
(61)

Here, is arbitrary function corresponding to the backreaction of the perturbations on the background. In addition, we assume the random phase approximation

(62)

Under these assumptions, we can write and obtain from (61) a family of separate equations for each . In the following we neglect these backreaction terms by choosing , such that the background wave function satisfies the background part of the Wheeler-DeWitt equation and (55) reduces to the following family of equations

(63)
(64)

In order to proceed, we perform a wkb-type approximation and assume that the depend only adiabatically on the background variables , i.e. that a change of background variables causes the to change much slower than ,

(65)

This motivates the following ansatz for and , where the expansion in starts at order rather than ,

(66)
(67)

Inserting (66) and (67) into (63) and (64), and collecting terms of equal powers in leads to two families of equations for the background functions and the perturbation functions at each order in . Once this set of equations has been obtained, the formal expansion parameter is set to one. The resulting equations are then solved order by order, by first solving the equations for the background as their solutions enter the equations for the perturbations. In order to extract the first quantum gravitational corrections, it is sufficient to consider the expansions (66) and (67) up to .

4.2 Hierarchy of background equations

By successively solving for , and , we reconstruct the wave function up to .

4.2.1 :

At this order, we obtain a Hamilton-Jacobi type equation for ,

(68)

The semiclassical time arises from the expansion of the timeless Wheeler-DeWitt equation (53) as the projection along the gradient of the background geometry ,

(69)

The consistency of the semiclassical expansion requires that the classical theory is recovered at the lowest order. Indeed, by identifying the semi-classical time (69) with the conformal time and the gradient of with the background momenta,

(70)

the Hamilton-Jacobi equation (68) yields the equations of motion (19)-(21), upon using (70). The Hamilton-Jacobi equation therefore implies the classical equations of motion for the background variables .

4.2.2 :

Equipped with the semi-classical notion of time (69), at the next order of the semiclassical expansion we obtain

(71)

Using the definition of the semiclassical time (69), we find

(72)

Here, is a function satisfying the transport equation

(73)

This is consistent with the first order corrections to the wkb prefactor, where is associated with the Van Vleck determinant.

4.2.3 :

At this level of the expansion we obtain

(74)

Substituting (72) into (74), we find that satisfies the differential equation

(75)

This shows that corresponds to the second order correction to the wkb prefactor.

4.3 Hierarchy of perturbation equations

Next, we consider the expansion of (64). Using the equations (68)-(74) of the background equations, we reconstruct the up to first order in the expansion parameter .

4.3.1 :

At this order in the expansion, using (69), we obtain

(76)

This equation is equivalent to the Schrödinger equation for the states ,

(77)

4.3.2 :

The first quantum gravitational corrections arise from the semiclassical expansion at order . Making use of (69), we obtain

(78)

This equation can be written in the form of a corrected Schrödinger equation for the state , with satisfying (77)444In the transition from (78) to (79), terms of order are neglected when converting derivatives of in terms of derivatives of .,

(79)

The terms proportional to are identified as the first quantum gravitational corrections. We follow the strategy introduced in [75] and project these terms along the direction normal to the hypersurfaces of constant . By using (68), (71) and (77), the quantum gravitational correction terms can be represented in the form

(80)

We follow the treatment of [77, 81, 82] and only take the real part of the corrections (80) in order to preserve unitarity defined with respect to the Schrödinger inner product on the Hilbert space of the perturbations. The question of unitarity in the context of the canonical approach to quantum gravity and the semiclassical expansion is controversially discussed and an interesting topic on its own, see e.g. [75, 97, 95, 98, 99, 85, 100, 101]. The term in (80) might be viewed as a contribution to the effective potential

(81)

The iterative scheme of the semiclassical expansion implies that equations obtained at lower orders in are used to derive equations arising at higher order in the expansion. In order to solve the corrected Schrödinger equation (81) for , in addition knowledge about the solution of the uncorrected Schrödinger equation (77), which enters (80), is required.

5 Cosmological power spectra in the Schrödinger picture

In order to extract physical information from the semiclassical expansion we need to relate observations to the wkb states