Reﬂected Waves and Quantum Gravity Leonardo ChataignierAlexander Yu. KamenshchikyAlessandro Tronconizand Giovanni Venturix Dipartimento di Fisica e Astronomia Università di Bologna via Irnerio 46 40126 Bologna Italy

2025-04-29 0 0 694.8KB 16 页 10玖币

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Reﬂected Waves and Quantum Gravity

Leonardo Chataignier,∗Alexander Yu. Kamenshchik,†Alessandro Tronconi,‡and Giovanni Venturi§

Dipartimento di Fisica e Astronomia, Università di Bologna, via Irnerio 46, 40126 Bologna, Italy

I.N.F.N., Sezione di Bologna, I.S. FLAG, viale B. Pichat 6/2, 40127 Bologna, Italy

In the context of canonical quantum gravity, we consider the eﬀects of a non-standard expression

for the gravitational wave function on the evolution of inﬂationary perturbations. Such an expression

and its eﬀects may be generated by a sudden variation in the (nearly constant) inﬂaton potential.

The resulting primordial spectra, up to the leading order, are aﬀected in the short and in the long

wavelength regime, where an oscillatory behavior with a non-negligible amplitude is superimposed

on the standard semiclassical result. Moreover, a novel, non-perturbative, approach is used to study

the evolution. Finally, a simpliﬁed application is fully illustrated and commented.

I. INTRODUCTION

In the last few years, the quest for the theory of quantum gravity (QG) has entered a new era. A series of increasingly

precise observations, ranging from cosmic microwave background (CMB) to gravitational wave signals and the direct

observation of the horizon of black holes (BHs), are now in support of the theory and may soon lead us to a consistent

description of gravitational interactions at energy scales which have never been probed before and where quantum

eﬀects may be observable.

Among the several approaches to QG, canonical quantum gravity [1] has an important role. It is obtained from the

canonical quantization of the classical constraints emerging from the spacetime diﬀeomorphism invariance of general

relativity. The resulting Wheeler-DeWitt (WDW) equation for the wave function of the Universe is similar to a time-

independent Schrödinger equation in non-relativistic quantum mechanics. For simple cases, the WDW equation can be

solved and a suitable interpretation of the wave function of the Universe can be given. Matter-gravity systems where

the eﬀective number of relevant degrees of freedom is small, such as BHs and inﬂationary cosmology, are amenable

to the WDW description [2]. Despite several conceptual issues, the canonical quantum gravity framework presents

several advantages with respect to (w.r.t.) other approaches, and we expect that it is a predictive mathematical

description of the QG regime, at least when quantum gravitational eﬀects are small.

In this paper, we shall investigate some possible quantum gravitational eﬀects in the early Universe at the energy

scale of inﬂation [3], which may leave “footprints” in the CMB spectrum. Such eﬀects must be small in order to ﬁt

observations but their magnitude need not be tiny and dependent on the “usual” H/MPratio, i.e., the ratio between the

Hubble parameter and the Planck mass (see for example [4]). As shown in [5], the small (quantum) ﬂuctuations which

seed the structures we observe today can be described within this framework by a set of separate wave functions

obtained through the traditional Born-Oppenheimer (BO) decomposition [6] applied to the entire inﬂaton-gravity

system. The approach leads to a modiﬁed Mukhanov-Sasaki (MS) equation [7] which accounts for diverse quantum

gravitational eﬀects. Non-adiabatic QG eﬀects are obtained as a consequence of the traditional BO treatment and

are tiny, being proportional to (H/MP)2. Further QG eﬀects related to the “BO introduction of time” can also be

present [8].

Within this traditional BO scheme, the emergence of time in QG is usually associated with the “probability current”

of the gravitational wave function. In an expanding universe, one generally assumes that the direction of such a current

follows (and determines) the direction of time. However, since the gravitational wave function obeys a second-order

diﬀerential equation, solutions with opposite probability ﬂuxes always exist, and, in principle, a quantum superposition

of these solutions may be considered. In any case, physical initial conditions must be imposed in order to ﬁx the

form of the gravitational wave function [9], and the possibility of having a small contribution to the gravitational

wave functions evolving in the opposite direction with respect to (w.r.t.) the expanding inﬂationary universe has been

examined [10] with the hypothesis of a bouncing universe. Here, we consider a diﬀerent scenario wherein a (small)

variation of the cosmological constant (inﬂaton potential) generates a reﬂected gravitational wave which inﬂuences

the evolution of inﬂationary perturbations. The amplitude of the reﬂected wave will depend on the variation of the

cosmological constant and its eﬀects may be much larger than the non-adiabatic QG eﬀects which are always present

in the traditional BO approach. On a more technical level, we note that the formalism we shall employ to solve

∗leonardo.chataignier@unibo.it

†kamenshchik@bo.infn.it

‡tronconi@bo.infn.it

§giovanni.venturi@bo.infn.it

arXiv:2210.04927v2 [gr-qc] 6 Jan 2023

the perturbed Schrödinger-like equation, which governs the evolution of inﬂationary perturbations, is novel, and it

consists in solving “exactly” (and numerically) the perturbed equation.

The resulting spectra are aﬀected in the long wavelength region and in the short wavelength interval as well.

In particular, in the short wavelength limit, the QG eﬀects can be analytically well understood, as the equations

can be solved with accurate approximations. In the opposite, long wavelength limit, the eﬀects may be large but

approximations are less precise. It is also important to note that we restrict our study to the case where QG eﬀects

modify the evolution of the “CMB modes” when such modes are still well “inside” the horizon. In such a case, the

evolution of the perturbation modes would be insensitive to the shape of the inﬂaton potential, and, therefore, any

variation of this potential (which is here approximated by a cosmological constant) would have no signiﬁcant eﬀect

on the spectra when the QG eﬀects derived from the WDW equation are neglected.

The article is organized as follows: In Sec. II, the general formalism is introduced; in Sec. III, the traditional BO

approach is illustrated and the gravity equation is solved in the presence of a sudden variation of the cosmological

constant; in Sec. IV, the matter equation in the presence of QG corrections is formally solved, and the primordial

spectrum is calculated in terms of the solution of the so-called Pinney equation; in Sec. V, a simpliﬁed model is

considered and its observational consequences are obtained and analyzed; ﬁnally, in Sec. VI, we draw our conclusions.

II. FORMALISM

We consider the inﬂaton-gravity system described by the following action

S=Zdηd3x√−g−MP2

2R+1

2∂µφ∂µφ−V(φ),(1)

where MP= (8πG)−1/2is the reduced Planck mass. The above action can be decomposed into a homogeneous part

plus ﬂuctuations around it. The ﬂuctuations of the metric δgµν (~x, η)are deﬁned as

gµν =g(0)

µν +δgµν ,(2)

where g(0)

µν = diag a(η)2(1,−1,−1,−1)is a ﬂat Friedmann-Lamaître-Robertson-Walker (FLRW) metric and ηis

conformal time. The scalar and the tensor ﬂuctuations imprint their features in the CMB [11] and are therefore the

relevant perturbations during inﬂation. In particular, the scalar ﬂuctuations, which can be collectively described by

a single ﬁeld (the MS ﬁeld [7]), determine the CMB temperature ﬂuctuations. The homogeneous degrees of freedom

plus the linearized (scalar) perturbations dynamics are described by the following action

S=Zdη (L3"−e

2a02+a2

2φ02

0−2V(φ0)a2#

i=1,2X

k6=0 v0

i,k(η)2+−k2+z00

zvi,k(η)2





≡SG+SI+SMS ,(3)

where the vi,k are Fourier components of the scalar MS ﬁeld and the index iaccounts for the real and imaginary parts

of each component, e

MP=√6MP,z≡φ0

0/H,H=a0/a2is the Hubble parameter, the prime denotes the derivative

w.r.t. conformal time and

L3≡Zd3x. (4)

Notice that we formally split the full action into three contributions: SGand SIare the homogeneous gravity and

inﬂaton actions, respectively, whereas SMS describes the perturbations.

Henceforth, we shall set Lto be equal to 1(see [5] for more details) to keep the notation compact. The Hamiltonian

is ﬁnally

H=−π2

+ π2

2a2+a4V!+∞

k6=0 π2

2+ω2

2v2

k,(5)

where ω2

k=k2−z00/z. For simplicity, we shall limit ourselves to the case of a constant inﬂaton potential V= Λ. One

has

πa=−e

Pa0, πφ=a2φ0

0, πk=v0

k.(6)

The canonical quantization of the Hamiltonian constraint (5) leads to the following WDW equation for the wave

function of the Universe (matter plus gravity)







∂2

∂a2+ˆ

H0+∞

k6=0

Hk



Ψ (a, φ0,{vk})=0,(7)

where we deﬁne

H0:= −1

2a2

∂2

∂φ2

+a4Λ,(8)

Hk:= 1

∂2

∂v2

+ω2

2v2

k,(9)

for k6= 0. Equation (7) will be the starting point in our approach.

III. BO DECOMPOSITION

In this section, we brieﬂy illustrate the traditional BO decomposition for the inﬂaton-gravity system. This decom-

position is one of the approaches to the problem of time in quantum gravity, i.e., to the introduction of an evolution

parameter in the seemingly stationary equation (7), and it has a rather long history (see [12] for a review). In its

traditional version, the BO decomposition follows the formalism described, e.g., in [13], where a factorization of the

wave function leads to the decomposition of the time-independent Schrödinger equation [here, given by (7)] into an

equation for “heavy” variables (here, the scale factor) and one for the “light” variables (here, the matter ﬁelds), and

both equations are, in principle, nonlinear (as they include so-called non-adiabatic eﬀects). The key idea of the

traditional BO decomposition, as applied to quantum cosmology, is that the phase of the gravitational wave function

can be used to deﬁne a time variable that dictates the evolution of matter ﬁelds. The nonlinearities of the matter

equation can be dealt with by suitable “re-phasings” of the gravitational and matter wave functions (see [12, 14, 15])

and by a concrete, iterative procedure that uses perturbation theory (usually in powers of the inverse Planck mass,

see [5, 15–17] and references therein for further details; see also [18] for an application of a BO-inspired approach in

“hybrid quantum cosmology” with techniques used in loop quantum cosmology, and [14] for a comparison of diverse

approaches). In the spirit of this traditional BO approach, we thus start from the ansatz

Ψ (a, φ0,{vk}) = ψ(a)χM(a, φ0,{vk}) = ψ(a)χ0(a, φ)Y

k6=0

χk(a, vk).(10)

If we project out the matter wave function, one is then led to a gravity equation

∂2

a˜

ψ+ 2 e

Phˆ

H0i˜

ψ=h∂a˜χ0|∂a˜χ0i˜

ψ, (11)

where

ψ=˜

ψe−iPkRaAkda0,(12)

χk= ˜χkeiRaAkda0,(13)

Ak:= −ihχk|∂a|χki,(14)

hˆ

Oi=Z+∞

−∞ 

Y

k6=0

dvk

dφ χ∗

Mˆ

OχM,(15)

with k= 0 indicating the homogeneous scalar ﬁeld. Let us note that we neglected the back-reaction originating from

the perturbations vkin (11).

The projection of the WDW along Qp6=khχp|leads to a set of equations, one for each k-mode of the inﬂaton, of the

following form

∂a˜

ψ∂a˜χk+ˆ

Hk− h ˆ

Hkik˜χk+1

P∂2

a− h˜

∂2

aik˜χk= 0 ,(16)

where

hˆ

Oik=Z+∞

−∞

dvk˜χ∗

kˆ

O˜χk.(17)

We note that (16) plays a central role in our approach to the fully quantized inﬂaton-gravity system. Its structure

is a consequence of the BO decomposition performed and it contains: a ﬁrst term with the ﬁrst derivative of the

matter wave function and also dependent on gravitational wave function (this term is usually associated with the

introduction of time), the Hamiltonian contribution given by the matter Hamiltonian minus its expectation value,

and a third term given by the second derivative of matter wave function minus its expectation value. This latter term

is associated with the non-adiabatic eﬀects in the traditional BO decomposition (when such a BO decomposition is

applied to molecules it describes the eﬀects of the “slowly” moving nucleus on the electron cloud) and in this context

it is usually associated with quantum gravitational eﬀects. Let us note that also the ﬁrst contribution may give rise to

quantum gravitational eﬀects associated with the introduction/deﬁnition of time and we then keep these letter eﬀects

distinct w.r.t. to the former ones. Let us further note, as discussed in detail in [14], that, the structure of (16), apart

from the ﬁrst term, has the form hˆ

Oik−ˆ

O, where ˆ

Ois not necessarily Hermitian. Thus, since e

M−2

P∂a˜

ψ/ ˜

ψappears as

a c-number, one has

h˜χk|∂a|˜χki= 0 ,(18)

and

∂ah˜χk|˜χki=h∂a˜χk|˜χki+h˜χk|∂a˜χki=e

P˜

ψ∗

∂a˜

ψ∗h˜χk|ˆ

O†− h ˆ

O†ik|˜χki+e

P˜

∂a˜

ψh˜χk|ˆ

O− h ˆ

Oik|˜χki= 0 (19)

which means h˜χk|˜χkinormalization is conserved w.r.t. the variation of aor of any function of it (and in particular

the semiclassical time).

If k= 0, the equation (16) is that of the homogeneous inﬂaton. When the inﬂaton potential is constant, such an

equation simpliﬁes to

∂a˜

ψ∂a˜χ0−1

2a2∂2

∂φ2

0− h ∂2

∂φ2

0i0˜χ0+1

P∂2

a− h˜

∂2

ai0˜χ0= 0 ,(20)

and it is easily solved by (scale-factor independent) eigenstates of the operator −i∂/∂φ0(i.e., plane waves ˜χ0∼

eiP φ0/e

MP). Correspondingly, the gravity equation (11) becomes

∂2

a˜

ψ+ 2 e

P P2

2a2e

+a4Λ!˜

ψ= 0 ,(21)

and it can be solved as well. In particular, its solutions have a simple form in the large alimit. Let y=e

Pa3and

Λ≡e

Pλ, then (21) becomes

∂2

y˜

ψ+2

3y∂y˜

ψ+P2

9y2+2

9λ˜

ψ= 0,(22)

and for y1its solution is a (quantum and coherent) superposition of plane waves moving forward and backward

in “time”, respectively

ψ=A1e−i√2λy/3+A2ei√2λy/3.(23)

Notice that the expression becomes increasingly accurate in the y1limit provided P/y →0in the same limit. In

particular, for (23), the second term of Eq. (22) is ∼√λ/y ˜

ψand is much smaller than the remaining two if

√λ

Pa3λ⇒a1

MPλ1/6.(24)

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ReectedWavesandQuantumGravityLeonardoChataignier,AlexanderYu.Kamenshchik,yAlessandroTronconi,zandGiovanniVenturixDipartimentodiFisicaeAstronomia,UniversitàdiBologna,viaIrnerio46,40126Bologna,ItalyI.N.F.N.,SezionediBologna,I.S.FLAG,vialeB.Pichat6/2,40127Bologna,ItalyInthecontextofcanonicalquantumgr...

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Reﬂected Waves and Quantum Gravity Leonardo ChataignierAlexander Yu. KamenshchikyAlessandro Tronconizand Giovanni Venturix Dipartimento di Fisica e Astronomia Università di Bologna via Irnerio 46 40126 Bologna Italy

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