# The landscape of polymer quantum cosmology

Lautaro Amadei  Aix Marseille Univ, Université de Toulon, CNRS, CPT,
Marseille, France  Alejandro Perez  Aix Marseille Univ, Université de Toulon,
CNRS, CPT, Marseille, France  Salvatore Ribisi  Aix Marseille Univ, Université
de Toulon, CNRS, CPT, Marseille, France

(September 0 d , 2024)

###### Abstract

We show that the quantization ambiguities of loop quantum cosmology, when
considered in wider generality, can be used to produce discretionary dynamical
behaviour. There is an infinite dimensional space of ambiguities which
parallels the infinite list of higher curvature corrections in perturbative
quantum gravity. There is however an ensemble of qualitative consequences
which are generic in the sense that they are independent of these ambiguities.
Among these, one has well defined fundamental dynamics across the big bang,
and the existence of extra microscopic quantum degrees of freedom that might
be relevant in discussions about unitarity in quantum gravity. We show that
(in addition to the well known bouncing solutions of the effective equations)
there are other generic type of solutions for sufficiently soft initial
conditions in the matter sector (tunnelling solutions) where the scale factor
goes through zero and the spacetime orientation is inverted. We also show that
generically, a contracting semiclassical universe branches off at the big bang
into a quantum superposition of universes with different quantum numbers.
Despite their lack of quantitative predictive power these models offer a
fertile playground for the discussion of qualitative and conceptual issues in
quantum gravity.

##  I Introduction

Quantum field theories with local degrees of freedom generically suffer of
divergences due to uncontrolled UV contributions to amplitudes. At the
mathematical level the latter can be traced to the fact that interactions
involve products of fields (operator valued distributions in quantum field
theory) at the same spacetime point and that such products are ill-defined if
constructed naively. The standard procedure of renormalization eliminates
infinities from the physical amplitudes at the price of introducing counter
terms with free parameters to be fixed by a series of renormalization
conditions taken from physical inputs. In certain simple situations one can
instead take due care in the definition of products of operator valued
distributions and thus completely avoid from the very beginning UV divergences
(see for instance scharf2014finite ). However, such procedure is not unique
and free parameters arise too in the regularization procedure. These
parameters must be fixed (in order to produce physical amplitudes) by the same
number of renormalization conditions of standard textbook treatments. In this
way there is a formal link between the number of counter-terms necessary to
eliminate UV divergences and regularization ambiguities.

A key difficulty of canonical approaches to quantum gravity is that such
intrinsic ambiguity of standard quantization recipes become in appearance out
of control. This is associated to the non-renormalizability of the
gravitational interaction, which in the case of general relativity in metric
variables, is illustrated by the fact that the family of general covariant
functionals of $g_{ab}$ representing a possible action principle describing
the quantum effective action is infinite dimensional and that the parameter
controlling the dimensionality of such free couplings is the quantum gravity
coupling itself. Namely,

$\displaystyle
S[g_{ab}]=\frac{1}{2\kappa}\int\sqrt{|g|}\left(R+\Lambda+\alpha_{1}\ell_{p}^{2}R^{2}+\alpha_{2}\ell_{p}^{4}R^{3}+\cdots+\beta_{1}\ell_{p}^{2}R_{\mu\nu\alpha\sigma}R^{\mu\nu\alpha\sigma}\cdots\right)dx^{4},$
(I.1)

where only some representative terms have been written with dimensionless
couplings $\alpha_{1},\alpha_{2},\cdots$, $\beta_{1},\beta_{2},\cdots$, etc.
Generic radiative corrections produce divergences that need to be cured by
counter terms in correspondence with the infinite number of elements in the
previous general action, requiring infinitely many renormalization conditions,
and compromising the predictive power of the approach111It is possible,
however, that these couplings would flow under the renormalization group in a
non trivial way toward some asymptotic fixed point characterized by a finite
amount of parameters. Such perspective, known as the asymptotic safety
scenario Weinberg:1980gg , is the subject of active investigations
Niedermaier:2006wt . . However, from the previous formal discussion, an
equivalent danger menaces non-perturbative formulations where UV divergences
are avoided via clever choices of variables and or mathematical structures, as
the danger metamorphoses into that of ambiguities.

As a complementary remark one must keep in mind that the previous analysis
sometimes strongly depends on the ‘fundamental variables’ chosen for
quantization. An example of this is the emblematic case of gravity in three
dimensions where a naive metric variable analysis would have led to similar
conclusions as in four dimensions. However, when the most general action is
written in terms of first order variables one discovers that there is only a
finite dimensional set of possibilities. Namely

$\displaystyle S[e,\omega]=\frac{1}{2\kappa}\int e_{I}\wedge
F_{IJ}(\omega)\epsilon^{IJK}+\Lambda\,e_{I}\wedge e_{J}\wedge
e_{K}\epsilon^{IJK}+\alpha S_{\rm CS}(\omega),$ (I.2)

where $e^{I}_{a}$ is a triad field, $\omega_{a}^{IJ}$ is a Lorentz connection,
and $S_{\rm CS}(\omega)$ is the Chern-Simons action. The theory is indeed
integrable, has only global or topological degrees of freedom, and its
quantization is free of (infinite dimensional) ambiguities Witten:1988hc ;
Witten:1988hf . Strikingly, a similar finite dimensionality of the space of
gravity actions is valid in first order variables in four dimensions where one
has that the most general gravitational action is given by

$\displaystyle S[e_{a}^{A},\omega_{a}^{AB}]$ $\displaystyle=$
$\displaystyle\frac{1}{2\kappa}\int\overbrace{\epsilon_{IJKL}e^{I}\wedge
e^{J}\wedge F^{KL}(\omega)}^{\rm
Einstein}+\overbrace{{\Lambda}\,\epsilon_{IJKL}e^{I}\wedge e^{J}\wedge
e^{K}\wedge e^{L}}^{\rm Cosmological\,Constant}+\overbrace{\alpha_{1}\
e_{I}\wedge e_{J}\wedge F^{IJ}(\omega)}^{\rm Holst}$ $\displaystyle+$
$\displaystyle\underbrace{\alpha_{2}\ (d_{\omega}e^{I}\wedge
d_{\omega}e_{I}\,-\ e_{I}\wedge e_{J}\wedge F^{IJ}(\omega))}_{\rm Nieh-
Yan}+\underbrace{\alpha_{3}\ell_{p}^{2}\ F(\omega)_{IJ}\wedge
F^{IJ}(\omega)}_{\rm Pontrjagin}+\underbrace{\alpha_{4}\ell_{p}^{2}\
\epsilon_{IJKL}F(\omega)^{IJ}\wedge F^{KL}(\omega)}_{\rm Euler},$

where $d_{\omega}e^{I}$ is the covariant exterior derivative of $e^{I}$ and
$\alpha_{1}\cdots\alpha_{4}$ are dimensionless coupling constants. For non-
degenerate tetrads Einstein’s field equations follow from the previous action
independently of the values of the $\alpha$’s: the additional terms are called
topological invariants describing global properties of the field
configurations in spacetime. The $\alpha_{1}$-term is called the Holst term
Holst:1995pc , the $\alpha_{2}$-term is the Nieh-Yan invariant, the
$\alpha_{3}$-term is the Pontryagin invariant, and the $\alpha_{4}$-term is
the Euler invariant. Inspite of not changing the equation of motion these
terms can actually be interpreted as producing canonical transformations in
the phase space of gravity 222 In such a context the so-called Immirzi
parameter Immirzi:1996di  corresponds to the combination
$\gamma\equiv\frac{1}{(\alpha_{1}+2\alpha_{2})}$ Rezende:2009sv . This
parameter plays a central role in the spectrum of quantum geometric operators,
and controls, in the presence of Fermions, the strength of an emergent four-
fermion interaction Perez:2005pm ; Freidel:2005sn ; Mercuri:2006um ..

The previous facts motivate the idea of the pertinence of such variables for
the implementation of non-perturbative quantization and thus can be viewed as
natural rational behind the approach of loop quantum gravity Rovelli:2004tv
(although the history of the subject cannot be reduced literally to such
perspective but rather to the discovery of Ashtekar’s new variables
Ashtekar:1986yd ). However, not surprisingly unlike the simple 3d case (which
has no local degrees of freedom) the absence of ambiguities in the quantum
theory remains an open question. Indeed, at early stages of the development of
loop quantum gravity it was found that—thanks to the peculiar Hilbert space of
quantum gravity adapted to diffeomorphism invariance and the Ashtekar-Barbero
connection variables—the quantum gravitational dynamical equations (embodied
by the Hamiltonian constraint that encodes both the gravity and matter
interactions) where free of UV divergences Thiemann:1996aw . Nevertheless, the
quantization of the Hamiltonian constraint suffers of ambiguities of an
infinite dimensional nature suggesting that the renormalizability issue is
still present Perez:2005fn .

There is however an unresolved consistency check concerning the quantization
of the constraints in loop quantum gravity. This is the issue of anomalies.
More precisely, the quantum dynamical equations are represented by a set of
quantum operators that must satisfy a commutation algebra inherited from the
classical algebra of generators of the surface deformation algebra. Checking
the absence of anomalies has shown to be a remarkably hard question suggesting
that such consistency check could reduce the ambiguities in the definition of
the quantum constraints Varadarajan:2018tei ; Tomlin:2012ejk ;
Ashtekar:2020xll . However, one should recognise that such hope is not clear
from the general perspective of our initial discussion as the algebra of
surface deformations is a feature of any diffeomorphism invariant formulation
of gravity. More precisely, in the case of metric variables, the canonical
analysis of the general action (I) would produce the same surface deformation
algebra independently of the values of the undetermined couplings.

The non-triviality of this question has motivated a recent interest in the
application of renomalization group methods to investigate (as in asymptotic
safety scenarios) the possibility that the non-perturbative techniques of loop
quantum gravity could help uncover a non trivial UV completion of the theory
Bahr:2009qc ; Dittrich:2011zh ; Bahr:2014qza ; Bahr:2011aa ; Steinhaus:2020lgb
; Thiemann:2020cuq .

The perspective that we emphasize here is not new Perez:2005fn  but its full
implications in quantum cosmology has been somewhat underestimated. The
problem of quantization ambiguities in the cosmological models inspired by the
full theory has been considered in various works Ashtekar:2007em ; however,
under certain restrictive assumptions that reduce the discussion to finite
dimensional sectors of the space of ambiguities Vandersloot:2005kh ;
BenAchour:2016ajk ; Dapor:2017rwv ; Li:2018opr ; Kowalczyk:2021bwr , it has
been argued not to represent a menace to the predictatbility of the framework.
Our present analysis shows that, as long as the question of ambiguities
remains open in the full theory, polymerized symmetry reduced models cannot
produce accurate quantitative predictions but only qualitative insights. We
will show here that the ambiguities inherited from the full theory have an
important dynamical effect in these models that, naturally, compromises their
predictive power. Nevertheless, due to their simplicity, our analysis does not
reduce in any way the great value of these models for illustrating qualitative
features of quantum gravity. Some of these features are, in these simple
models at least, generic (i.e. independent of the ambiguity issue) suggesting
that they might represent robust features possibly realized in nature.

##  II Unimodular quantum cosmology as a simple testing ground

In order to show the influence of the regularization ambiguities on physical
quantities one first needs to be able to perform explicit calculations within
the framework where these ambiguities appear. Even when quantum cosmology
models correspond to classical systems with finitely many degrees of freedom,
the non-standard representation theory used in the construction of the quantum
theory makes these models sufficiently complicated (in some of its versions)
to prevent explicit calculations. For example, different choices of time
variables (realized by different choices of lapse functions) produce different
quantum constraints which can present supplementary challenges when it comes
to analysing the quantum dynamics.

This is in part the reason why the technique of the so-called effective
dynamics has been developed Taveras:2008ke  where the quantum evolution is
approximated by modified classical equations of motion. These effective
equations are affected by the ambiguities in the definition of the quantum
dynamics. In fact these modifications are supposed to encode the quantum
corrections to general relativity coming from quantum gravity. In this sense
the quantization ambiguities on which we focus here are expected to affect
these quantum corrections. However, showing explicitly the form of the
effective equations can be challenging or (unnecessarily) more involved when
different time variables are chosen. In order to simplify our presentation we
analyse cosmology in the unimodular version of general relativity. Unimodular
gravity is simply equivalent to standard general relativity if the matter
coupling is diffeomorphism invariant Ellis:2010uc . When applied to cosmology,
it has the advantage of resolving the problem of time as the lapse function is
fixed by the unimodular constraint. It is customary in the literature of
quantum cosmology to modify the scalar constraint by assuming different
choices of the lapse function (a choice that it is often referred to as a
‘gauge choice’). Even when unimodular gravity is not a gauge fixing of
standard general relativity, from the previous perspective unimodular
cosmology could be characterized at the technical level by a choice of lapse.
We will see that such choice makes the Hamiltonian evolution particularly
simple in the gravity sector, and thus allows for the most transparent and
simple derivation and resolution of the quantum as well as the effective
dynamical equations. It should be clear that the main point of this work will
not change if one would use a different notion of lapse. The choice we make
has a very natural geometric interpretation that we describe in the following
paragraph.

There is no preferred notion of time in general relativity. This implies that
the dynamics is dictated by constraint equations and leads to the so-called
problem of time in quantum gravity: instead of Schroedinger like evolution
equations one has a timeless dynamics defined by the quantum constraints. This
very complicated technical and conceptual problem can be circumvented in
quantum cosmology by the use of tools that have become customary in the area.
The commonly accepted prescription is the use of some (partial) observable as
clock that allows for the deparametrization of the dynamics that leads to a
Schroedinger like evolution equation and the definition of the so-called
physical Hilbert space of quantum cosmology. Even when such procedure is not
unique and thus might lead to unitarily inequivalent theories, this additional
source of potential ambiguity will not concern us here. The reason why the
problem of time does not seem so serious in quantum cosmology is the fact that
for the study of dynamical questions (sufficiently far form the Planckian
regime) an effective classical description is available. Such classical
description allows for dealing with the problem of time in just the usual way
that is familiar to us in cosmology: via gauge fixing, i.e., particular
choices with some clear interpretation ranging from co-moving time, harmonic
time, conformal time, or (our choice here) unimodular time.

Unimodular time is the time variable that naturally emerges from the
description of cosmology in unimodular gravity Smolin:2010iq . Instead of a
gauge fixing, unimodular gravity can be thought of as a genuine modified
theory of gravity that is (apart from a subtlety in the cosmological constant
sector) is completely equivalent to general relativity. We will use unimodular
quantum cosmology Chiou:2010ne ; Sartini:2020ycs  in what follows just because
in this formulation the gravitational part of the Hamiltonian takes a
particularly simple form; indeed, the geometry degrees of freedom can be
mapped uniquely to those of a non-relativistic free particle. This feature
allows for a very intuitive interpretation of both the effective classical
evolution equations as well as the quantum gravity equations. In most
situations of interest the problem of quantum or classical evolution of
geometry coupled with simple forms of matter can be seen as a regular
scattering problem of a non-relativistic particle in an external potential.
This makes the setting of the dynamical system particularly appealing for its
simplicity; however, it should be clear from our treatment that the
implications drawn are of general validity and should apply (qualitatively
speaking) to any of the customary parametrizations of loop quantum cosmology.

When specializing to (spatially flat) homogeneous and isotropic cosmologies
with metric

$\displaystyle ds^{2}=-N^{2}dt^{2}+a(t)^{2}d\vec{x}^{2}$ (II.1)

the Einstein-Hilbert action supplemented with the unimodular constraint
becomes

$\displaystyle
S=-\kappa^{-1}\int\left(\sqrt{|g|}R+\lambda(\sqrt{|g|}-1)\right)dx^{4}$ (II.2)

where $\kappa=16\pi G$, and $\lambda$ is a Lagrange multiplier imposing the
unimodular constraint $\sqrt{|g|}-1=0$, and we have put an overall minus sign
in front of the action for later convenience. Specializing to the FLRW metric
II.1 one gets

$\displaystyle
S=\kappa^{-1}V_{0}\int\left(6\frac{a\dot{a}^{2}}{N}-\lambda(N|a|^{3}-1)\right)dt,$
(II.3)

where total derivative terms have been eliminated, and the $3$-volume $V_{0}$
of a fiducial cell has been introduced. Resolving the unimodular constraint
fixes $N=|a|^{-3}$ and defines a preferred notion of time; from now on we
denote this new time variable as $s$ and we call it unimodular time. The
action becomes

$\displaystyle S=\kappa^{-1}V_{0}\int{6a^{4}\dot{a}^{2}}ds,$ (II.4)

where $s$ denotes from now on unimodular time. For further reference it is
important to relate unimodular time with the standard comoving time $\tau$,
namely

$\displaystyle ds=-|a|^{3}d\tau.$ (II.5)

At this point we will change variables to more convenient ones that make the
action look like that of a non-relativistic free particle. The new
configuration variable will be chosen to be given by the $3$-volume density

$\displaystyle x=a^{3},$ (II.6)

from which it follows that $\dot{x}=3a^{2}\dot{a}$ and the action then is

$\displaystyle\boxed{S=\int\frac{1}{2}m\dot{x}^{2}ds},$ (II.7)

with

$\displaystyle m\equiv\frac{4V_{0}}{3\kappa},$ (II.8)

and where the dot denotes derivative with respect to the unimodular time $s$.
Note that the minus sign in from of (II.3) was chosen so that the kinetic term
of the particle analog has the usual sign. Also notice that if we use comoving
time $d\tau=-ds/|a|^{3}$ we have

$\displaystyle
p=m\left(\frac{dx}{ds}\right)=-3m\frac{1}{|a|}\frac{da}{d\tau}=-3m\ {{\rm
sign}(a)}H.$ (II.9)

We see that the momentum variable in our parametrization is just proportional
to the Hubble rate $H$ in usual comoving variables! Let us introduce a scalar
field as a matter model. Then the matter action (with the same overall minus
sign convention that we are adopting) is

$\displaystyle S_{M}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int\sqrt{|g|}\left(\nabla_{a}\phi\nabla^{a}\phi+U(\phi)\right)$
(II.10) $\displaystyle=$ $\displaystyle-\frac{1}{2}V_{0}\int
Na^{3}\left(\frac{1}{N^{2}}\left(\frac{d\phi}{ds}\right)^{2}-U(\phi)\right)ds=-\frac{1}{2}V_{0}\int\left(a^{6}\left(\frac{d\phi}{ds}\right)^{2}-U(\phi)\right)ds.$

Therefore, the action including our simple matter model is

$\displaystyle{S(x,\phi)=\int\left(\frac{1}{2}m\left(\frac{dx}{ds}\right)^{2}-\frac{1}{2}V_{0}x^{2}\left(\frac{d\phi}{ds}\right)^{2}-V_{0}U(\phi)\right)ds}$
(II.11)

The previous action can be written in Hamiltonian form as

$\displaystyle{\boxed{S(x,\phi)=\int
p\frac{dx}{ds}+p_{\phi}\frac{d\phi}{ds}-\left(\frac{p^{2}}{2m}-\frac{p_{\phi}^{2}}{2V_{0}x^{2}}-V_{0}U(\phi)\right)ds}},$
(II.12)

where

$\displaystyle p_{\phi}=-V_{0}x^{2}\frac{d\phi}{ds}.$ (II.13)

####  II.0.1 Changing variables to match the standard setup

The previous $(x,p)$ variables in the gravity sector where chosen to emphasize
the simple link between unimodular gravity in the FLRW context and the
dynamics of a point particle. A simple rescaling of these variables leads to
the standard parametrization of the phase space in loop quantum cosmology in
the so-called $\mathrm{bar}\mu$-scheme. The variables customarily used are
called $(b,v)$ and are defined as

$\displaystyle
b\equiv-\frac{\gamma}{3m}p=-\frac{\gamma}{3}\frac{dx}{ds}=-\gamma
a^{2}\frac{da}{ds},$ (II.14)

and

$\displaystyle v\equiv\frac{3m}{\gamma}x=\frac{V_{0}a^{3}}{4\pi G\gamma}.$
(II.15)

Thus one has

$\displaystyle\\{b,v\\}=1.$ (II.16)

With these variables, the Hamiltonian is

$\displaystyle H=\frac{V_{0}}{2\pi
G\gamma^{2}}\left(\frac{3}{4}b^{2}-\frac{p_{\phi}^{2}}{16\pi Gv^{2}}-2\pi
G\gamma^{2}\,U(\phi)\right),$ (II.17)

which is proportional to the scalar constraint $C$ as written in reference
Ashtekar:2011ni  (equation 2.19) simply rescaled by the use of the unimodular
time lapse, namely $H=V_{0}C/(\pi G|v|)$. The advantage of using unimodular
variables resides in the remarkable fact that the gravity part of the
Hamiltonian depends only on the variable $b$ (like a free particle in
classical mechanics). This simple fact simplifies several technical as well as
conceptual discussions of the classical and quantum features of the model.

##  III Regularization ambiguities of the Hamiltonian

There are two aspects of the Hamiltonian that call for a modification of its
classical expression in order to promote it to a well defined self-adjoint
operator in the special Hilbert space of loop quantum cosmology. One of them
is that only quasi-periodic functions of $b$ but not $b$ itself can be
quantized. The second is that inverse volume contributions to the Hamiltonian
(entering through the matter coupling) are also modified by means of the use
of classical expressions that eliminate unboundedness of these at small
volumes. Both modifications are ambiguous by nature and lead to dynamical
effects that we analyze in what follows. Interest in this issue from the
observational perspective has resurfaced recently in Renevey:2021tmh , here we
concentrate on further theoretical implications. There are other approaches
for the definition of the quantum dynamics for cosmology where one starts from
a more fundamental perspective at the quantum level and infers from it the
symmetry reduced model Gielen:2016dss ; Oriti:2016qtz ; Alesci:2013xd , we
note that similar ambiguities are present in these perspectives as well. For
simplicity we concentrate on the loop quantum cosmology formulation where the
problem is embodied in the notion of regularization.

###  III.1 Holonomy corrections

Due to the peculiar choice of representation in the quantization of the model
(inspired by the structure of Loop Quantum Gravity), there is no $b$ operator
in the Hilbert space of loop quantum cosmology but only operators
corresponding to finite $v$ translations Ashtekar:2006wn ; Ashtekar:2011ni ;
from here on referred to as shift operators

$\displaystyle{\exp(i\lambda b)}\triangleright\Psi(v)=\Psi(v-\lambda),$
(III.1)

where $\lambda$ is some arbitrary length scale. As the classical Hamiltonian
explicitly depends on $b$, it needs regularization in order to be promoted to
a self-adjoint operator in the Hilbert space of loop quantum cosmology.
Consequently we replace (II.17) by

$\displaystyle H=\frac{V_{0}}{2\pi G\gamma^{2}}\left(\frac{3f(\lambda
b)^{2}}{4\lambda^{2}}-\frac{p_{\phi}^{2}}{16\pi Gv^{2}}-2\pi
G\gamma^{2}\,U(\phi)\right),$ (III.2)

where

$\displaystyle f(\lambda b)=\sum_{n\in\mathbb{Z}}f_{n}e^{in\lambda b}.$
(III.3)

To be explicit about the regularization choice we write the substitution rule
as

$\displaystyle\boxed{b^{2}\to\frac{f(\lambda b)^{2}}{\lambda^{2}}.}$ (III.4)

This operation is called polymerization in the loop quantum cosmology
literature. The only conditions that consistency with the classical dynamics
imposes are: On the one hand, that $f(x)=\mathrm{bar}f(x)$ which translates
into the condition

$\displaystyle f_{n}=\mathrm{bar}f_{-n}.$ (III.5)

On the other hand one needs that

$\displaystyle\braket{f(\lambda b)}=\lambda b_{0}+{\mathfs{O}}[(\lambda
b_{0})^{2}]$ (III.6)

for $\lambda b_{0}\ll 1$ when expectation values are computed in suitable
semiclassical states peaked at the classical momentum $b_{0}$. This second
condition is necessary to recover the semiclassical dynamics of standard
cosmology at low Hubble rates or the low density regime. This leaves an
infinite dimensional freedom in the choice of the regularized Hamiltonian to
be promoted to an operator in the Hilbert space of our system. The standard
choice in the loop quantum cosmology literature is $f_{n}=i\delta_{n}^{1}/2$,
namely

$\displaystyle b^{2}\to\frac{\sin^{2}(\lambda b)}{\lambda^{2}}.$ (III.7)

A possible justification for this choice is that of simplicity. The link
between such choice in relation to the lowest non-vanishing eigenstate of the
area operator in loop quantum gravity, and the special status given to the
fundamental representation of the gauge group, is sometimes evoked as a
further reason to pick (III.7) (See Ashtekar:2011ni ). This argument connects
the regularization of a certain quantum operator in loop quantum cosmology to
the features of a particular state (the state with minimal area-eigenvalue) in
loop quantum gravity. Even when accepting such possibility it is unclear how
the lowest area eigenstate should play such a central role. Indeed, in quantum
theory the principle of superposition rather suggests that states would
typically be made of arbitrary superpositions of different area eigenvalues.
Consideration of such aspects in full generality brings us back to the
infinite dimensional landscape of polymerizations in (III.4).

When expressed in terms of the $v$-basis the evolution equation (related to
the Hamiltonian constraint) contains a finite difference term, which, with the
so-called traditional choice (III.7) becomes a discrete version of a second
derivative in $v$. For an arbitrary choice (III.4) the finite difference term
can be put in correspondence with a linear combination of the discretization
of higher derivative terms in $v$. If one were looking for eigenstates of the
Hamiltonian (III.2) one would be confronted with a growing multiplicity of
formal solutions of the eigenvalue equation as the order of the corresponding
difference equation grows when considering general functions $f(b)$ with
arbitrarily high Fourier components. This question can be studied in the
simpler context of the pure gravity case which in the analogy with the point
particle system corresponds to the asymptotic large universe regime where
standard matter contributions can be neglected in a logic analogous to that of
scattering theory. In such simpler setting, most of these extra solutions
related to the higher order character of the difference equation for arbitrary
$f(b)$ can be shown not to be normalizable, and hence not to be part of the
spectrum.

In the Wheeler-DeWitt standard representation of quantum cosmology,
eigenstates of the Hamiltonian are doubly degenerate in correspondence with
the two possible equal ‘energy’ classical solutions corresponding to an
expanding and/or contracting universe for a given cosmological constant: the
two are related to the discrete symmetry $\dot{x}\to-\dot{x}$ involving the
initial conditions of the theory written in the variables (II.11). Therefore,
any additional degeneracy of energy eigenvalues would have no classical
correspondence and its associated conserved quantity would reveal the
existence of new (microscopic) degrees of freedom. We will see that this is
the case in a subtle way but we postpone this discussion until Section IV.

The formal similarities with higher curvature corrections of the Einstein-
Hilbert action due to quantum effects is manifest even when a rigorous
statement in this sense is made difficult by the breaking of explicit
covariance by the Hamiltonian formulation (in the first place), and the
further (possibly explicit) breaking of covariance introduced by the
polymerization itself (see Bojowald:2015zha ). In perturbative quantum
gravity, higher derivative terms arise from higher curvature corrections and
this changes the number of degrees of freedom as seen from a classical
perspective. As higher curvature (higher derivative) terms appear multiplied
by increasing powers of $\ell_{p}^{2}$, all these corrections are taken to be
negligible at energy scales well below the Planck scale. We believe that this
analogy is interesting, thus making polymer models a nice simplified arena
where difficult questions related to renormalization and the definition of the
continuum limit can be explored in the highly simplified context of a model
that (at least classically) starts with a finite number of degrees of freedom.

Finally, it is possible to exhibit the direct relation between the function
$f(\lambda b)$ and the cosmological constant as follows. Standard
considerations in unimodular gravity imply that Amadei:2019wjp ;
Amadei:2019ssp

$\displaystyle\Lambda=8\pi G\frac{E}{V_{0}},$ (III.8)

where $E$ is the eigenvalue of $\eqref{hammm}$ or $\eqref{Hamon}$. The
discussion is simplified if we assume that we are in the massless scalar field
case $U(\phi)=0$. If a non trivial self interaction is present then a more
careful analysis is needed. We restrict to initial conditions given at
$v=\pm\infty$ (large universes) where the contribution of the scalar field to
the Hamiltonian (III.2) vanishes. In this limit the system is the analog of a
non relativistic free particle where eigenvalues of the energy can be labelled
by eigenvalues of momenta333More precisely, in our context these correspond to
the eigenvalues of the shift operators (IV.2), yet the key point is that they
are still labelled by a value of $b$.. Therefore, one obtains from (III.2) the
relation

$\displaystyle\Lambda=\frac{3f^{2}(b_{\infty})}{\gamma^{2}\lambda^{2}},$
(III.9)

where $b_{\infty}$ is the asymmptotic value of $b$ for $v=\pm\infty$.

###  III.2 Inverse volume corrections

Inverse volume terms in the Hamiltonian introduce potential singularities in
the quantum theory. Such potential divergencies are present as well in the
full theory of loop quantum gravity and need regularization when constructing
a well defined quantum scalar constraint operator. Thiemann introduced
Thiemann:1997rt  a natural regularization of such potential UV divergences by
realizing that inverse volume terms can be obtained from the Poisson algebra
between well defined geometric operators and the holonomy of the connection.
In the case of cosmology the idea can be illustrated, for example, by the
following simple classical identity444This is a particular case of a more
general identity leading to additional ambiguities Singh:2013ava . For
simplicity we concentrate on the one given in Ashtekar:2011ni .

$\displaystyle\frac{1}{\sqrt{|v|}}=\frac{2i}{\lambda}{\rm
sgn}(v)\exp{(i\lambda b)}\ \\{\exp{(-i\lambda b)},\sqrt{|v|}\\}$ (III.10)

which suggests a natural regularization of quantities depending on the inverse
volume using ‘holonomies’ and commutators in the quantum theory. Using a
symmetrized factor ordering, for instance

$\displaystyle\widehat{\frac{1}{\sqrt{|v|}}}\to\frac{1}{\hbar\lambda}{\rm
sgn}(v)\left(\exp{(i\lambda b)}\ [\exp{(-i\lambda
b)},\widehat{\sqrt{|v|}}]+[\exp{(-i\lambda b)},\widehat{\sqrt{|v|}}]\
\exp{(i\lambda b)}\right)$ (III.11)

This choice regularizes the singular behaviour of the inverse volume at
$v=0$—where the previous expression vanishes by construction—and produces a
well defined operator in the Hilbert space of loop quantum cosmology. However,
the choice is by no means unique. In fact (in addition to factor ordering and
other sources of ambiguities, such as the choice of the power of $v$ inside
the Poisson bracket in (III.10)) on has an infinite dimensional space of
regularizations that is similar in spirit to the one identified for the
regularization of curvature in (III.2) given by

$\displaystyle\widehat{\frac{1}{\sqrt{|v|}}}\to\frac{{\rm
sgn}(v)}{2\hbar\sum_{m\in\mathbb{Z}}c_{m}}\sum_{n\in\mathbb{Z}}\frac{c_{n}}{\lambda
n}\left(\exp{(i\lambda nb)}\ [\exp{(-i\lambda
nb)},\widehat{\sqrt{|v|}}]+[\exp{(-i\lambda nb)},\widehat{\sqrt{|v|}}]\
\exp{(i\lambda nb)}\right),$ (III.12)

for arbitrary coefficients $c_{n}$. This implies that in addition to the
infinite dimensional family of curvature regularizations one has an (at least)
equally large family of inverse volume regularizations which would generically
enter in the construction of the matter coupling when defining the quantum
Hamiltonian. One can show that the action of the previous operator is simply
given by (see Ashtekar:2011ni )

$\displaystyle\widehat{\frac{1}{\sqrt{|v|}}}\Psi(v)$ $\displaystyle=$
$\displaystyle\frac{\Psi(v)}{\hbar\sum_{m\in\mathbb{Z}}c_{m}}\sum_{n\in\mathbb{Z}}\frac{c_{n}}{\lambda
n}\left(\sqrt{\left|v+\lambda n\right|}-\sqrt{\left|v-\lambda
n\right|}\right)$ (III.13) $\displaystyle\equiv$
$\displaystyle\frac{\Psi(v)}{\sum_{m\in\mathbb{Z}}c_{m}}\sum_{n\in\mathbb{Z}}{c_{n}}\left[\frac{1}{\sqrt{|v|}}\right]_{n},$

where we have introduced the definition

$\displaystyle\left[\frac{1}{\sqrt{|v|}}\right]_{n}\equiv\frac{1}{\hbar\lambda
n}\left(\sqrt{\left|v+\lambda n\right|}-\sqrt{\left|v-\lambda
n\right|}\right).$ (III.14)

One has that

$\displaystyle\sqrt{|v|}\left[\frac{1}{\sqrt{|v|}}\right]_{n}=1+\frac{1}{16}\frac{n^{2}\lambda^{2}}{v^{2}}+\frac{7}{128}\frac{n^{4}\lambda^{4}}{v^{4}}+{\mathfs{O}}\left(\frac{n^{6}\lambda^{6}}{v^{6}}\right),$
(III.15)

which shows that for sufficiently large volume one recovers the classical
expected limit. Notice that the regularization (III.13) vanishes at $v=0$. One
can use the previous series expansion and chose the coefficients $c_{n}$ in
order to improve the convergence to the classical value. For example with the
following choice

$\displaystyle\text{c1}\to 9.42267,\ \text{c2}\to-13.1273,\ \text{c3}\to
6.31659,\ \text{c4}\to-1.93791,\ \text{c5}\to 0.355751,\
\text{c6}\to-0.0297957$ (III.16)

one gets the regularization to coincide with $1/\sqrt{|v|}$ up to order
${\mathfs{O}}\left({n^{10}\lambda^{10}}/{v^{10}}\right)$ (plotted in blue in
Figure 1)! One can continue improving convergence by killing higher order
deviations from $1/\sqrt{v}$, it simply boils down to solving a linear system
of equations with increasing dimension. One might think that such process
would produce a sequence of $c_{n}$ converging pointwise to the Wilson-Ewing
regularization Wilson-Ewing:2012dcf ; Singh:2013ava  of the inverse volume
which is given by $1/\sqrt{|v|}$ for all $v\not=0$ while it vanishes at $v=0$.
By plotting a few members of the above approximating sequence we see that this
will not be the case. In fact, the previous sequence, while it gets better and
better in approximating $1/\sqrt{v}$ for large $v$, it differs more and more
from the function $1/\sqrt{v}$ at around $v=\pm\lambda$ (see Figure 1)

Figure 1: Inverse volume corrections of the classical function $1/\sqrt{|v|}$
(shown in black). The blue curve represents the best approximation for large
volumes which coincides with the classical expression up to order
$(\lambda/v)^{10}$. However, as the approximations get better for large volume
they get worse at Planckian volumes as the plotted sequence illustrates which
deviations becoming the largest at low integer values times $\lambda$.

There are two important features that we would like to emphasize here. The
first one is that the regularization of the inverse volume operator would lead
to a function of $v$ that is not everywhere differentiable due to the presence
of the absolute value in the formulas. This is of course not a problem from
the perspective of the quantum theory where (for the quantum dynamics) only
the evaluation of the regularization on a discrete lattice plays a role.
However, at the non differentiable points the effective equations can simply
not be trusted. The second important feature is that the regularization is a
continuous function and thus bounded. We insist on the point that (even when
some choices might seem natural given some subjective criteria) there is no
well defined rule that would actually eliminate the vast set of possibilities
here either. We will discuss some further consequences of inverse volume
corrections later, in particlar we will see in Section VI.3 their role in the
violation of the null energy condition near the big bang.

##  IV The quantum theory: evolution across the singularity

Let us discuss the main features of the quantum dynamics, before discussing
the validity of the effective dynamical approach that we will use later for
further interpretation—where the quantum dynamics is approximated by looking
at the evolution of semiclassical states. One great advantage of the
unimodular gravity formulation is that (at least in the FLRW context) the
theory has a well identified time evolution (in unimodular time (II.5)) and
the Kinematical Hilbert space of loop quantum cosmology is the physical
Hilbert space. In other words the problem of time is trivialized and the
physical interpretation of the quantum theory becomes closer to that of
standard quantum mechanics 555Closer but not quite exactly the same. Here we
are making reference to the unconfortable questions related to the meaning of
a quantum theory of the universe as a whole. These questions are indeed very
important and remain open to a large extent. In order to concentrate on the
main point of this paper we have to ignore them altogether.. However, an
important difference with standard quantum mechanics is the use of an
unconventional representation of the basic phase space variables which brings
into the system a central property of the full theory of loop quantum gravity:
fundamental discreteness. Concretely, instead of the standard Schroedinger
representation on a gravitational Hilbert space of square integrable wave
functions where $v$ acts by multiplication and $b=-i\partial_{v}$, one
introduces a Hilbert space where only the exponentiated version of $b$, the
shift (or holonomy) operators $\exp{i\lambda b}$ (for arbitrary
$\lambda\in\mathbb{R}$) are well defined. This is called sometimes the
_polymer representation_ and the procedure of using this exotic representation
(well motivated from the full theory) is called polymerization.

Consequently, there is no operator corresponding to $b$ in the loop quantum
cosmology polymer representation but only the operators corresponding to
finite $v$ translations Ashtekar:2006wn ; from here on referred to as shift
operators defined as

$\displaystyle{\exp(i\lambda b)}\triangleright\Psi(v)=\Psi(v-\lambda).$ (IV.1)

There are states diagonalizing the shift operators, denoted
$\ket{b_{0};\Gamma^{\epsilon}_{\lambda}}$, which are labelled by a real value
$b_{0}$ and where $\Gamma^{\epsilon}_{\lambda}$ is a 1d lattice of points, a
graph, in the real line of the form $v=n\lambda+\epsilon$ with
$\epsilon\in[0,\lambda)$ and $n\in\mathbb{N}$. The corresponding wave function
of these eigenstates is given by
$\Psi_{b_{0}}(v)\equiv\braket{v}{b_{0};\Gamma^{\epsilon}_{\lambda}}=\exp{(-i{b_{0}v})}\delta_{\Gamma^{\epsilon}_{\lambda}}$
where the symbol $\delta_{\Gamma^{\epsilon}_{\lambda}}$ evaluates to one when
$v\in\Gamma_{\lambda}^{\epsilon}$ and vanishes otherwise. Assuming that
$k=m\lambda$, it follows from (IV.1) that

$\displaystyle{\exp(ikb)}\triangleright\ket{b_{0};\Gamma^{\epsilon}_{\lambda}}=\exp{(ikb_{0})}\ket{b_{0};\Gamma^{\epsilon}_{\lambda}}.$
(IV.2)

The states $\ket{b;\Gamma^{\epsilon}_{\lambda}}$ are eigenstates of those
shift operators which preserve the lattice $\Gamma^{\epsilon}_{\lambda}$. The
fact that these states are supported on discrete lattices (polymer-like
excitations) is what motivated the name of the representation. Notice that the
eigenvalues are independent of the parameter $\epsilon$, i.e., they are
infinitely degenerate and span a non separable subspace of the quantum
cosmology Hilbert space ${\mathfs{H}}_{lqc}$.

As the operator $b$ does not exist in the Hilbert space one has to construct
approximations in terms of combinations of shift operators which behave like
$b$ in a suitable sense. A procedure that is (as discussed in Section III.1)
intrinsically ambiguous. We would like to understand the influence of
deviating from the standard regularization (III.7) to the quantum dynamics. In
order to do this we will concentrate on the pure gravity case first. Indeed,
the ambiguity (III.4) only affects the gravitational part of the Hamiltonian
and thus this simple case will completely characterize the dynamical influence
of the choice of different regularization functions $f(\lambda b)$ in the
quantum dynamics in the large volume asymptotic regime where matter dilutes
until becoming negligible. Thus we will deal with the special case where,
before quantization, the classical Hamiltonian is regularized as

$H=\frac{b^{2}}{2m}\ \ \ \to\ \ \ \frac{f(\lambda b)^{2}}{2m\lambda^{2}}.$
(IV.3)

Note that this case is non trivial because it admits a non-zero cosmological
constant that is given by the value of the energy in the unimodular framework
(recall (III.8)). In the quantum theory, we are interested in the eigenstates
of the Hamiltonian (the analog of the time independent Schroedinger equation).
Let us first analyze the spectrum of the Hamiltonian in the traditional
polymerization, namely we would like to solve the equation

$\displaystyle\left(\frac{\sin(\lambda
b)^{2}}{2m\lambda^{2}}-E\right)\ket{\Psi_{E}}=0$ (IV.4)

which in the $v$-basis becomes (due to (IV.1)) the difference equation

$\displaystyle\Psi_{E}(v-2\lambda)+\Psi_{E}(v+2\lambda)+(8m\lambda^{2}E-2)\Psi_{E}(v)=0,$
(IV.5)

where the order of the difference equation is directly related to the
polymerization choice. This seems to raise a potential difficulty: if instead
of the traditional choice we take an arbitrary $f(\lambda b)$ the order of the
difference equation will grow arbitrarily. Would this not lead to an
uncontrollable proliferation of spurious solutions? We will see soon that this
is not the case. For the moment we continue the analysis of the present
scenario. As the Hamiltonian is a combination of shift operators (IV.1) of the
kind for which one knows the eigenstates, one can simply express the energy
eigenstates in terms of $\ket{b_{0};\Gamma^{\epsilon}_{2\lambda}}$ (the
eigenstates of the shift operators) and calculate the relationship between
$b_{0}$ and the energy eigenvalues. We could call this the polymerized
dispersion relations. For the standard choice energy eigenstates and
dispersion relations are

$\displaystyle\ket{\Psi_{E(b_{0})}}=\ket{b_{0};\Gamma^{\epsilon}_{2\lambda}},\
\ \ \ \ \ E(b_{0})=\frac{\sin(\lambda b_{0})^{2}}{2m\lambda^{2}}.$ (IV.6)

###  IV.1 The $\epsilon$-sectors

The previous energy eigenstates (eigenstates of the cosmological constant) are
infinitely degenerate due to the $\epsilon$ degeneracy of the shift operators
(IV.1). This over abundance of solutions of the Schroedinger equation is
controlled, in standard accounts, by fixing a volume lattice once and for all
and choising one $\epsilon$-sector. This choice is dynamically consistent
because the Hamiltonian preserves the given lattice; however, in the presence
of matter the choice represents an additional dynamical ambiguity as the
dynamical features will depend on $\epsilon$. This is particularly clear when
we look at the inverse volume corrections of Section III.2. Each different
choice of $\epsilon$ gives a lattice that probes the volume regularization at
different discrete points. As the inverse volume regularization enters the
coupling of gravity with matter (see for instance equation III.2) the details
of the dynamics will depend on this choice. Because the Hamiltonian preserves
$\epsilon$-sectors they are some times called super selection sectors.
However, as there are other (Dirac) observables that do not preserve the
lattice, these sectors are not superselected in any usual sense.

More precisely, in the case of pure gravity observables commuting with the
Hamiltonian and mapping between different values of $\epsilon$ (graph changing
observables) are simply the shift operators introduced in IV.1. In the case of
a non trivial matter coupling other Dirac observables exist, they are
technically hard to characterize explicitly in their full generality because
of the usual difficulty associated to the construction of such conserved
quantities. However, notice that if the matter coupling is such that matter
dilutes as $v\to\infty$ (as expected for regular matter degrees of freedom)
then shift operators remain Dirac asymptotic observables where the universe
becomes large and the Hamiltonian tends to the pure geometry Hamiltonian in
the usual sense of scattering theory. The shift observables (IV.1) define in
this manner a complete set of commuting observables fully characterizing the
positive energy (positive cosmologial constant666Negative $\Lambda$ solutions
exist in the presence of matter and they correspond bound states (in the
analog non-relativistic particle system). These solutions do not reach the
$|v|\to\infty$ asymptotic region and admit no scattering theory interpretation
(as in usual cases).) states. These asymptotic observables are like those
regularly employed in standard situations involving scattering theory. Their
existence shows that $\epsilon$-sectors are not superselected.

There is hence no clear reason to restrict to a single lattice and
superpositions of different lattices can be considered. Some of the
implications of this possibility have beed investigated in Amadei:2019wjp ;
Amadei:2019ssp  where it is shown that these additional degrees of freedom,
which are microscopic or Planckian, can be key in understanding the fate of
information in situations where evolution across would-be-singularities is
relevant like in cosmology and (most importantly) in the context of black hole
formation and evaporation. In order to simplify the following discussion, we
will restrict, from here on, our analysis to the case of states supported a
single lattice.

###  IV.2 Degeneracy of the energy (cosmological constant) eigenstates in the
pure gravity case

In the Schroedinger representation the dispersion relations would have been
the familiar non relativistic particle relation $E(b_{0})=b_{0}^{2}/(2m)$
which is doubly degenerate corresponding to the momentum eigenstates with
$b=\pm b_{0}$. Translating this to unimodular cosmology, these two eigenstates
would correspond to a state of a De Sitter with cosmological constant
$\Lambda=8\pi GE(b_{0})/V_{0}$ that is either contracting or expanding in the
FLRW slicing. With the standard polymerization (III.7) we observe at first
that a new degeneracy has appeared as there are four different shift-operator-
eigenstates that produce the same energy, namely those labelled by the four
roots of the equation on the right of (IV.6) depicted on the left of Figure 2.
We will study the role of these additional solutions below once we have
described this type of degeneracy for an arbitrary regularization.

For an arbitrary polymerization (III.4) the situation is quite similar.
Eigenstates are again given by

$\displaystyle\ket{\Psi_{E(b_{0})}}=\ket{b_{0};\Gamma^{\epsilon}_{2\lambda}}\
\ \ {\rm with}\ \ \ E(b_{0})=\frac{f(\lambda b_{0})^{2}}{2m\lambda^{2}}.$
(IV.7)

However, the degeneracy of the energy eigenvalues is now dependent on the
choice of the function $f(\lambda b)$. An example with 6 different eigenstates
is depicted on the right panel of Figure 4. One can distinguish in this
example two different situations, one where the energy is $E_{1}$ and the 8
solutions correspond to eigenstates of the form (IV.7). In the other case, for
the energy $E_{2}$, the number of solutions of the eigenvalue equation seen as
a difference equation remains 8; however, only the four values of $b_{0}$
explicitly seen in the figure correspond to ‘plane-wave’ eigenstates of the
form (IV.2). It is easy to show777See for instance section 2.3 of
10.5555/230196 . that the other 4 solutions of the difference equation are
diverging in either the $v\to\pm\infty$ limit and thus are not part of the
Hilbert space (this is the analog of non-normalizable solutions of for example
the time independent Schroedinger equation for the Harmonic oscillator). But
the most interesting thing concerning these additional solutions happens when
matter couplings that break $v$-translational invariance of the Hamiltonian
are included.

$\begin{array}[]{c}\includegraphics[width=199.16928pt]{ambi2.pdf}\end{array}\
\ \ \ \ \ \ \ \ \ \ \ \ \
\begin{array}[]{c}\includegraphics[width=256.0748pt]{sca1.pdf}\end{array}$

Figure 2: The polymerization $f(x)^{2}=-2(\cos(x)-1)$ matches the degeneracy
of eigenstates of the Schroedinger representation. On the right panel we show
the different scattering channels in a matter coupling with a massless scalar
that produces (for a given $p_{\phi}$ eigenstate) an effective potential
regularized by inverse volume corrections (shown in black). The universe
bounces into a superposition of transmitted and reflected modes with the same
asymptotic (large $v$) Hubble rates. If we factor by the symmetry $v\to-v$
then we only have a bounce and the superposition disappears. The results of an
analytically solvable model are shown in (IV.3).

$\begin{array}[]{c}\includegraphics[width=199.16928pt]{ambi3.pdf}\end{array}\
\ \ \ \ \ \ \ \ \ \ \ \ \
\begin{array}[]{c}\includegraphics[width=256.0748pt]{sca2.pdf}\end{array}$

Figure 3: In the traditional polymerization $f^{2}(x)=(\sin(x))^{2}$
(dispersion relations on the left) new solutions appear. On the right panel we
show the different scattering channels in a matter coupling with a massless
scalar that produces (for a given $p_{\phi}$ eigenstate) an effective
potential regularized by inverse volume corrections (shown in black). The
universe bounces and tunnels in new channels with different asymptotic Hubble
rates for a given cosmological constant. If the $v\to-v$ symmetry is imposed
(as customarily in the specialized literature) the degeneracy remains and the
universe only bounces yet into the quantum superposition of two semiclassical
solutions. The results of an analytically solvable model are shown in (IV.3).

$\begin{array}[]{c}\includegraphics[width=199.16928pt]{ambi1.pdf}\end{array}\
\ \ \ \ \ \ \ \ \ \ \ \ \
\begin{array}[]{c}\includegraphics[width=256.0748pt]{sca3.pdf}\end{array}$

Figure 4: In a generic polymerization, here
$f(x)^{2}=2/13(2-(\cos[2x]+\cos[3x]))$. On the right panel we show the
different scattering channels in a matter coupling with a massless scalar that
produces (for a given $p_{\phi}$ eigenstate) an effective potential
regularized by inverse volume corrections (shown in black). New channels for
the bounce appear and a given universe evolves through the singularity into a
quantum supperposition of universes with the same cosmological constant (or
expectation value of the cosmological constant for wave packets) but different
Hubble rates. The results of an analytically solvable model are shown in
(IV.3).

###  IV.3 Dynamical consequences when matter couplings are included

Here we show how the inclusion of matter couplings has the generic effect of
producing ‘diffusion’ into the various energy eigensectors which would not be
present in the Schroedinger quantization. The additional energy eigenvalues of
the pure gravity model introduced by the choice of polymerization play an
important dynamical role. We will see that a universe starting in the large
volume limit in one asymptotically De Sitter state—with a given cosmological
constant (energy) and a given Hubble rate $b_{0}$—will ‘scatter’ through the
big bang into a superposition of the various eigenstates of the same
asymptotic energy. In this way, the quantum dynamics of the bounce is way more
complicated than hinted by the effective equation approach that will be
constructed in the later part of the paper. This is a simple instance of the
physical expectation embodied in the statement that anything that can happen
happens in quantum mechanics.  It shows in a crystal clear way that, in a
background independent approach, the most likely result is that an initially
semiclassical state (with a clear spacetime interpretation) will evolve into a
superposition that might not always admit a single spacetime representation.
Forgetting this simple fact about quantum mechanics is one of the most current
errors in setting up important questions such as the ones concerning the fate
of information in black hole evaporation. Here again we see how, the present
models of quantum cosmology, represent a rich and valuable testground for
conceptual ideas in spite of their limited quantitative predictive power as
far as observable effects are concerned.

One of the simplest models of matter coupling is that of a massless scalar
field (i.e. $U(\phi)=0$ in equation (II.10)). In that case the momentum of the
scalar field is conserved and the gravitational dynamics is equivalent to that
of a point particle (with kinetic energy $\propto b^{2}$) moving in a
‘external attractive potential’ that goes like $\propto 1/v^{2}$ (see equation
(II.17)). The divergence in the potential is regularizaed in loop quantum
cosmology using the Thiemann construction that modifies the inverse volume
dependence near the big bang at $v=0$. Such modification is illustrated in the
Figures 2, 3, 4. Such model is already complicated enough to make analytic
statements involved.

However, the qualitative behaviour that one want to illustrate does not depend
on the details of the potential and only on the fact that the coupling with
matter breaks the conservation of the variable $b$. This is simply due to the
fact that matter couplings break translational invariance in the $v$-axes
producing a non trivial dynamics of the Hubble rate (a quite obvious fact from
the standard classical perspective based on the Friedmann equations where only
in pure DeSitter spacetime the Hubble rate remains constant in the FLRW
slicing). Thus, the phenomenon we want to emphasize can be illustrated in a
much simpler model where analytic calculations are trivial. An example of such
model is the one where the regularized $1/v^{2}$ potential produced by the
corresponding contribution to the Hamiltonian (III.2) of the massless scalar
field is replaced by the sum of two Kronecker deltas at $v=0$ and $v=\lambda$
mimicking in some way the two picks in the regularized potential seen in the
previous figures. Notice however, that this example is not meant to
approximate in any precise sense the massless scalar field case. We are only
using it because we can solve it explicitly and because it produces the
phenomenology that will be common (at the qualitative level) to any matter
coupling. The only essential feature here is its breaking of translational
invariance in the $v$-axes.

Concretely, we concentrate on the difference equation

$\displaystyle\Psi_{e}(v-2\lambda)+\Psi_{e}(v+2\lambda)+(e-2)\Psi_{e}(v)-\alpha\delta\left(\frac{v}{\lambda},1\right)\Psi_{e}(v)-\alpha\delta\left(\frac{v}{\lambda},0\right)\Psi_{e}(v)=0,$
(IV.8)

where $e\equiv 8m\lambda^{2}E$ and the delta functions are Kronecker deltas on
the lattice $v=\lambda n$ with $n\in\mathbb{Z}$, and $\alpha$ is a coupling
constant. This is a simple scattering problem which is resolved via the ansatz

$\Psi_{b_{1}}(v)=\begin{cases}e^{-i{b_{1}(e)}v}+R_{1}(e)\,e^{i{b_{1}}v}+R_{2}(e)\,e^{i{b_{2}}v}&\text{($v\geq
0$)}\\\ T_{1}(e)\,e^{-i{b_{1}(e)}v}+T_{2}(e)\,e^{-i{b_{2}(e)}v}&\text{($v\leq
0$)},\end{cases},$ (IV.9)

where $b_{1}(e)>b_{2}(e)>0$ are the two positive solutions of the dispersion
relation—plotted in Figure 3—for the given value of $e$. The previous discrete
Schroedinger equation boils down to 4 independent linear equations from which
one determines that reflexion and transmission amplitudes. They are given by

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!R_{1}(e)=\frac{\alpha\left(i\alpha\sqrt{-(e-1)e}+\alpha(1-e)-(1-e)\left(e-i\sqrt{-(e-1)e}\right)\right)}{(e-1)\left(\alpha^{2}+e\right)}\
\ \ \ \ \
R_{2}(e)=\frac{i\alpha^{2}\sqrt{e}}{\sqrt{1-e}\left(\alpha^{2}+e\right)}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!T_{1}(e)=\frac{e}{\alpha^{2}+e}-\frac{i\alpha\sqrt{e}}{\sqrt{1-e}\left(\alpha^{2}+e\right)}\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \
T_{2}(e)=\frac{\alpha\left(1+\frac{i\sqrt{e}}{\sqrt{1-e}}\right)e}{\alpha^{2}+e}.$
(IV.10)

There are two interesting limits of the previous result that will be relevant
for the discussion in Section VI.2: the reflection amplitudes vanish in the
limit $\alpha\to 0$ where the bounce is completely suppressed, while the
transmission amplitudes vanish in the hard-scattering limit $\alpha\to\infty$.
Here we have used a simplistic model where explicit calculations can be done.
As mentioned before, the qualitative features present in this model remain in
the realistic case (this is confirmed by numerical simulations that we have
omitted form the paper for simplicity).

##  V The modified cosmological effective equations

Let us start from the unimodular Hamiltonian constraint where (III.2) is
equated to some energy value that plays the role of the cosmological constant,
namely

$\displaystyle C\equiv{\cal H}-V_{0}\frac{\Lambda}{8\pi G}=\frac{V_{0}}{2\pi
G\gamma^{2}}\left(\frac{3f(\lambda
b)^{2}}{4\lambda^{2}}-\frac{p_{\phi}^{2}}{16\pi Gv^{2}}-2\pi
G\gamma^{2}\,U(\phi)-\frac{1}{4}\Lambda\right)\approx 0,$ (V.1)

Let us study the evolution of the volume variable

$\displaystyle v=\frac{V_{0}a^{3}}{4\pi G\gamma}$ (V.2)
$\frac{d\braket{v}}{ds}=-i\braket{\left[v,{\cal
H}\right]}\approx\frac{\partial\braket{{\cal H}}}{\partial
b}=\frac{3V_{0}}{4\pi G\gamma^{2}\lambda}f^{\prime}(\lambda b)f(\lambda b)$
(V.3)

where we have used the results of Appendix A in the derivation of the
effective equations, and prime denotes derivatives with respect to $\lambda
b$, and s denotes unimodular time given in terms of co-moving (cosmic) time
$\tau$ by

$\displaystyle ds$ $\displaystyle=$ $\displaystyle-|a|^{3}d\tau$ (V.4)
$\displaystyle=$ $\displaystyle-\frac{4\pi G\gamma}{V_{0}}|v|d\tau.$

Indeed, the previous equation gives us an expression for $\dot{a}/a$, namely

$\displaystyle\frac{\dot{a}}{a}=-\frac{1}{\gamma\lambda}f^{\prime}(\lambda
b)f(\lambda b)$ (V.5)

From now on we denote $\braket{v}$ simply $v$. Using the standard definition
of the Hubble rate $H\equiv\dot{a}/a$ we can write

$H^{2}=\frac{1}{\gamma^{2}\lambda^{2}}f^{\prime}(\lambda b)^{2}f(\lambda
b)^{2}.$ (V.6)

The constraint (V.1) can be rewritten as

$\displaystyle\boxed{C=\frac{3V_{0}}{8\pi
G\gamma^{2}\lambda^{2}}\left(f(\lambda
b)^{2}-\frac{\rho+\rho_{\Lambda}}{\overline{\rho}}\right)\approx 0}$ (V.7)

where $\rho_{\Lambda}\equiv\Lambda/(8\pi G)$

$\displaystyle\mathrm{bar}{\rho}=\frac{3}{8\pi G\gamma^{2}\lambda^{2}},$ (V.8)

and

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{p_{\phi}^{2}}{32\pi^{2}G^{2}\gamma^{2}v^{2}}+U(\phi)+\frac{\Lambda}{8\pi
G}$ (V.9) $\displaystyle=$
$\displaystyle\frac{\dot{\phi}^{2}}{2}+U(\phi)+\frac{\Lambda}{8\pi G}$

is the standard energy density. This implies

$\displaystyle f(\lambda b)=\sqrt{\left(\rho/\mathrm{bar}{\rho}\right)}.$
(V.10)

It is also convenient to introduce the pressure

$\displaystyle P$ $\displaystyle=$
$\displaystyle\frac{p_{\phi}^{2}}{32\pi^{2}G^{2}\gamma^{2}v^{2}}-U(\phi)-\frac{\Lambda}{8\pi
G}$ (V.11) $\displaystyle=$
$\displaystyle\frac{\dot{\phi}^{2}}{2}-U(\phi)-\frac{\Lambda}{8\pi G}.$

We arrive thus to the modified Friedman equation

$\boxed{H^{2}=\frac{8\pi
G\rho}{3}\left[f^{\prime}\left(f^{-1}\left(\sqrt{\rho/\mathrm{bar}{\rho}}\right)\right)\right]^{2}.}$
(V.12)

We can now look at the evolution equation of the energy density (V.9)

$\displaystyle\frac{d\braket{\rho}}{ds}$ $\displaystyle=$
$\displaystyle-i\braket{[\rho,{\cal H}]}$ (V.13) $\displaystyle=$
$\displaystyle-i\frac{3V_{0}}{8\pi
G\gamma^{2}\lambda^{2}}\braket{[\rho,f(\lambda b)^{2}]}=\frac{3}{8\pi
G\gamma^{2}\lambda^{2}}\frac{d\braket{f(\lambda b)^{2}}}{ds}$

where we have used that the Hamiltonian is ${\cal H}=3V_{0}f(\lambda
b)^{2}/(8\pi G\gamma^{2}\lambda^{2})-V_{0}\rho$. Now using Remark 4 we get

$\displaystyle\frac{d\braket{\rho}}{ds}$ $\displaystyle=$
$\displaystyle-\frac{3}{16\pi
G\gamma^{2}\lambda^{2}}\frac{\braket{p_{\phi}^{2}}}{(4\pi
G\gamma^{2})v^{3}}4\lambda f^{\prime}(\lambda b)f(\lambda b)$ (V.14)
$\displaystyle=$ $\displaystyle\frac{3}{16\pi
G\gamma^{2}\lambda^{2}}\frac{4\braket{p_{\phi}^{2}}}{(4\pi
G\gamma^{2})v^{3}}\lambda^{2}\gamma H.$

An important corollary of the previous algebra (or simply from Remark 4) is
that

$\displaystyle\dot{b}=-4\pi G\gamma(\rho+P)\frac{|v|}{v},$ (V.15)

which follows directly from Remark 4, the definition of $\rho$ and $P$, and
the sign comes from the relationship (V.4) between comoving time $\tau$ and
unimodular time $s$. Now using (V.4) we get the continuity equation

$\displaystyle\boxed{\dot{\rho}+3H(\rho+P)=0}$ (V.16)

where the quantities in the equation are to be taken as expectation values. It
is now a simple exercise to show that from (V.16) and (V.12) the following
modified Raychaudhuri equation follows

$\dot{H}=-4\pi
G\left(\rho+P\right)\left[f^{\prime}\left(f^{-1}\left(\sqrt{\rho/\mathrm{bar}{\rho}}\right)\right)^{2}+f^{\prime\prime}\left(f^{-1}\left(\sqrt{\rho/\mathrm{bar}{\rho}}\right)\right)f\left(f^{-1}\left(\sqrt{\rho/\mathrm{bar}{\rho}}\right)\right)\right].$
(V.17)

This concludes the derivation of the effective cosmological equations for
arbitrary regularizations of the Hamiltonian encoded in the arbitrary function
$f(\lambda b)$. We see that in regions where the latter behaves linearly as in
(III.6), one recovers the standard classical Eintein’s equations in the
cosmological context. However, and this is the key point of this paper,
deviations from Einstein’s equations can be introduced by ‘tuning’ the
function $f(\lambda b)$. Such modifications, as we will see, do have important
physical consequences and thus make the large number of Fourier coefficients
in $f(\lambda b)$ relevant ambiguity parameters compromising the use of these
models for physical predictions.

##  VI The landscape of polymerized models of quantum cosmology

In this section we analyse the generic implications of the effective dynamical
equations. We will assume their validity through the region corresponding to
the would-be-singularity of classical cosmology where the scale factor $a$
approaches zero. Even when for certain suitable initial semiclassical states
for contracting universes this approximation might hold true in some cases
(for instance for suitable bouncing solutions) we know from our analysis of
Section IV that the state of the universe branches off into other solutions
that go right through the $a=0$ regime. In these other branches the effective
dynamical equations break down unless one considers a rather artificial
regularization of the inverse volume corrections. This is why such solutions
can only be understood in full generality using the quantum theory. Due to
this behaviour we call these solutions _tunneling solutions_.

###  VI.1 Bouncing branches

We will first study the bouncing solutions of the effective equations that are
usually described in the LQC literature, analysing the generic effect of the
choice of the polymerization function $f(\lambda b)$. We recall equation (V.5)

$\displaystyle\frac{\dot{a}}{a}=-\frac{1}{\kappa\gamma}f^{\prime}(\lambda
b)f(\lambda b)\quad,$ (VI.1)

where, $f^{\prime}(\lambda b)$ denotes the derivative with respect to $\lambda
b$. We also need the modified Raychaudhuri equation (V.17) which can be
written in the form

$3\frac{\ddot{a}}{a}=-{4\pi G}\left((\rho+3P)f^{\prime
2}+3(\rho+P)f^{\prime\prime}f\right)+\Lambda f^{\prime 2}\quad.$ (VI.2)

From (V.15) and assuming the validity of the null energy condition (NEC),
$\rho+P\geq 0$, we can determine the direction of change of $b$ depending on
the sign of the volume of the universe. This greatly simplifies the analysis
of the landscape dynamics. NEC are violated due to quantum gravity effects
when inverse volume corrections in the matter coupling are taken into account.
However, this is not relevant for the bouncing branches for states such that
the effective equations are valid as the bounce prevents $v$ from reaching the
regions where $NEC$ are violated (see below Section VI.3).

Critical points in the function $f(\lambda b)$ correspond to two
possibilities: bounces (minimum volume configurations where the universe stops
contracting and starts expanding) and turning points (maximum volume
configurations where the volume of the universe stops increasing and starts
decreasing). Such situations are identified by the condition $\dot{a}=0$,
which, from (VI.1), arises when $f=0$ and $f^{\prime}=0$. We will study first
the case $f^{\prime}=0$. In order to understand if we are at the presence of a
bounce or a turning point, we have to study the sign of the second derivative
of the volume (a bounce occurs for $\ddot{v}>0$, a turning point for
$\ddot{v}<0$). Evaluating (VI.2) at points where $f^{\prime}=0$ we get

$\left.\frac{\ddot{a}}{a}\right|_{f^{\prime}=0}=-4\pi
G(\rho+P)f^{\prime\prime}f\quad.$ (VI.3)

Assuming that the NEC is valid, local maxima of the function $f$ represent a
bounce when $f>0$ as well as for local minima of $f$ when $f<0$. Conversely,
we have turning points at local minima of $f$ for $f>0$, and at local maxima
of $f$ for $f<0$. What about those points where $f=0$? If the cosmological
constant is positive, then the from of the Hamiltonian imposes that

$\displaystyle f^{2}(\lambda b)\geq\sqrt{\frac{\rho_{\Lambda}}{\rho_{c}}},$
(VI.4)

so that these points are inside a ‘classically forbidden’ region but they can
be reached by setting $\Lambda=0$. In this case they become a special case of
points where $f(\lambda b)^{2}\rho_{c}=\rho_{\Lambda}$. In general there are
De Sitter asymptotic configurations (Minkowski being a limiting case) where
the contribution of other forms of matter to $\rho$ and $P$ vanish. All this
is illustrated in Figure 5. Assuming that the universe is in the $v>0$ branch,
then equation (V.15) implies that $\dot{b}\leq 0$ as long as the NEC hold with
$\dot{b}=0$ at the De Sitter configurations where $\rho+P=0$. These
configurations are fixed points of the flow of $b$ (recall that $b$ is simply
related to the Hubble rate according to the classical analysis from where we
started).

With all this information we can interpret the situation described in Figure 5
qualitatively by observing that it represents two distinct possible histories
for the universe. The first starts on the classically allowed region on the
right in an (asymptotically) De Sitter state defined by the furthest
intersection of $f(\lambda b)$ with $\sqrt{\rho_{\Lambda}/\rho_{c}}$ to the
right. The universe contracts and goes out of the purely De Sitter state
entering a phase where other forms of matter start playing a dynamical role.
At the first minimum the universe bounces for the first time and starts
expanding. The expansion continues until the universe gets to the first
maximum (from right to left) where it starts contracting again until the
second minimum is reached and a last bounce leads to an expanding universe
that expands forever towards a final asymptotic De Sitter state. A second
sequence of events can be described in a similar fashion for the evolution
along the classically allowed region on the left of Figure 5.

Note that the initial and final asymptotic De Sitter phases are described by
different Hubble rates are the former is modulated by the value of the
$f^{\prime 2}$ at the asymptotic points according to (V.12). One could
introduce an effective cosmological constant at such De Sitter fixed points

$\displaystyle\Lambda_{\rm ds-fp}\equiv\Lambda f^{\prime 2}.$ (VI.5)

Notice that these fixed point correspond to low energy regions where the
density of matter (other than the cosmological constant) goes to zero. More
generally, one can expand the modified Friedman (V.12) around an arbitrary
density $\rho_{0}$ and write

$\displaystyle\frac{\dot{a}^{2}}{a^{2}}=\frac{8\pi G_{\rm
eff}}{3}\rho+\frac{\Lambda_{\rm
eff}}{3}+{\mathfs{O}}\left(\frac{(\rho-\rho_{0})^{2}}{\rho^{2}_{c}}\right)$
(VI.6)

A simple calculation gives

$\displaystyle G_{\rm eff}=G(f^{\prime 2}+f^{\prime\prime}f),$ (VI.7)

and

$\displaystyle\Lambda_{\rm eff}=(f^{\prime 2}+f^{\prime\prime}f)\Lambda-8\pi
G\rho_{0}ff^{\prime\prime}$ (VI.8)

For $\rho_{0}=\Lambda/(8\pi G)$ we recover the De Sitter fixed point value
(VI.5). Notice, that as we approach a bouncing point $G_{\rm eff}$ becomes
negative turning gravity repulsive. The same functional dependence of $G_{\rm
eff}$ is responsible for the effect interpreted as a change of signature in
Bojowald:2015gra .

Figure 5: Illustration of the dynamical features brought by the use of a
generic $f(\lambda b)$ when assuming that $\Lambda\geq 0$. If we require
$\rho\geq 0$, we have a bound on $f$ given by
$f^{2}\geq\rho_{\Lambda}/\mathrm{bar}\rho$. Close to $\lambda b_{0}$ with
$f^{2}(\lambda b_{0})=\rho_{\Lambda}/\mathrm{bar}\rho$ the function must be
well approximated by $f(\lambda b)\approx\pm\lambda(b-b_{0})$ in order to
recover general relativity at low regular matter densities. We denote bounce
points with B, turning points with $\rm t.p.$, and fixed points where the
universe becomes asymptotically De Sitter with $\rm d.s.$

Finally, in the case of $\Lambda<0$ there is a non trivial lower bound for
$\rho$ given by $\rho_{\text{min}}=-\rho_{\Lambda}$ corresponding to points
where $f=0$. In the point particle analogy these are to turning points of a
bound state where the kinetic energy vanishes, here $f(\lambda b)=0$. For the
universe these are turning points where the universe achieves minimal regular
matter density before recollapsing in to a denser regime. The other
qualitative features at critical points remain the same as in the previous
discussion.

###  VI.2 Tunnelling branches (in the case of a massless scalar field)

In Section IV we have shown that the quantum theory predicts that, in addition
to the traditional bounce evoked in the previous discussion and advocated in
the LQC literature at large, the universe can also tunnel across the
singularity into an expanding phase. The bounce is something that is clearly
captured by the effective equations. Can tunnelling also become apparent from
these equations? It is possible to see this if one considers for a moment
inverse volume corrections in the matter coupling described in Section III.2.
This perspective will turn out not to be important by the end of the
discussion in this section; however, it gives a concrete classical classical
picture of the process we have in mind. We will see that for this classical
picture to hold one has to push to the extreme non-Planckain region the
inverse volume corrections in the inverse volume regularization. However, the
dynamical channel remains open in the fundamental quantum theory where
semiclassical descriptions are only a good interpretational tool away from the
big bang.

For simplicity we concentrate on the case of a massless scalar field matter
model. Assuming that we are in an eigenstate of the momentum $\pi_{\phi}$
(conserved in this case) we have already observed that its contribution to the
Hamiltonian III.2 can be seen as an effective potential in the analogy with a
non relativistic particle parametrization of Section II. However, this
contribution is now everywhere finite as the $1/v^{2}$ classical behaviour is
regularized by the Thiemann trick. This means that there must exist solutions
of the effective equations where the the universe evolves right through $v=0$
(or scale factor $a=0$) into the $v<0$ without experiencing the bounce
produced by the kinetic term when the variable $b$ reaches the suitable
critical points of $f(\lambda b)$ described above. For this to happen the
universe must scatter thought the big bang singularity ‘softly’ in the sense
that the variable $b$ must not grow up to one of the critical points of
$f(\lambda b)$. This happens when the universe—interpreted as the point
particle—rolls down the potential

$\displaystyle V(v)\equiv-p_{\phi}^{2}\left[\frac{1}{v^{2}}\right]_{\rm reg}$
(VI.9)

in a way such that its ‘kinetic energy’ does not grow beyond the bound

$\displaystyle\frac{p_{\phi}^{2}}{4\pi G\gamma^{2}}{\rm
max}\left[\frac{1}{v^{2}}\right]_{\rm
reg}\leq\frac{3}{\gamma^{2}\lambda^{2}}f^{2}(\lambda
b_{c})-\Lambda=\frac{3}{\gamma^{2}\lambda^{2}}(f^{2}(\lambda
b_{c})-f^{2}(\lambda b_{\infty})).$ (VI.10)

We observe that, as the regularized potential is bounded, for $|p_{\phi}|$
sufficiently low the universe experiences (according to the effective
equations) a soft-transition from $v>0$ to $v<0$ in finite unimodular time
$\Delta s$ instead of a bounce. The scale factor crosses $a=0$; however, there
is no singularity as one can easily check from the effective equations (VI.1)
and (VI.2) and the fact that both $P$ and $\rho$ vanish there. Indeed, the
universe goes through a De Sitter phase where $\Lambda$ dominates. Even when
$a=0$ is reached in finite unimodular time $s$, the would-be-singularity $a=0$
is reached at infinite comoving time $\tau$. If valid, the effective equations
predict an infinite number of e-folds of inflation at around the soft-
transmission from $v>0$ to $v<0$. Even when here there is only one such
transitions, the scenario resembles in spirit the eon-transition of conformal
cyclic cosmology proposed by Penrose Penrose:2010zz . However, these
conclusions are not really correct for inverse volume regularizations that
modify the matter coupling from the expected classical behaviour at around the
Planck scale only.

Concretely, if we take the standard inverse volume correction based on
$[1/\sqrt{v}]_{1}$ (recall equation III.14) for which

$\displaystyle{\rm max}\left[\frac{1}{v^{2}}\right]_{\rm
reg}=\frac{4}{\lambda^{2}\hbar}$ (VI.11) $\displaystyle{p_{\phi}^{2}}\leq
3\pi\ell_{p}^{2}\left(f^{2}(\lambda b_{c})-f^{2}(\lambda b_{\infty})\right).$
(VI.12)

However, we see that in order to trust the previous conclusions the effective
equations would have to be valid in the description of the universe from
$v=\lambda$ to $v=-\lambda$. If $\lambda$ is taken to be of the order of the
Planck scale then it is clear that the details of the dynamics evoked above
(De Sitter inflation for an unlimited number of e-folds) does not survive in
the fundamental description where the variable $v$ jumps on discrete values of
the order of $\lambda$. However, the conclusion that the transmission channel
exists in addition to the well known bouncing channel remains. A precise
analysis of such transitions would require using a fully quantum treatment
which is of course possible.

In this respect it is interesting to revisit the results of Section IV under
the light of the present discussion. Notice that, qualitatively speaking, the
parameter $\alpha$ regulating the strength of the toy-model potential in
(IV.3) is the analog of $p_{\phi}^{2}$ here. We observe that, even when always
non vanishing, the transmission amplitudes go to zero in the limit
$\alpha\to\infty$ and only the bouncing channels remain available.
Consideration of the quantum theory uncovers a feature that we have evoked
previously. Indeed, the criterion for soft bounce (VI.10) looses its
quantitative relevance and we realize that even if one considers un-bounded
regularizations such as the one proposed in Wilson-Ewing:2012dcf ;
Singh:2013ava  there will be a component of wave function in the transmission
sector in addition to the bouncing sector for suitable initial states that
probe the potential on sufficiently soft points of the potential. More
precisely, consider the regularization where

$\displaystyle\left[\frac{1}{v^{2}}\right]_{\rm reg}=\frac{1}{v^{2}}\ \ \
\forall\ \ \ v\not=0,\ \ \ {\rm while}\ \ \ \left[\frac{1}{0^{2}}\right]_{\rm
reg}=0.$ (VI.13)

Consider a semiclassical state defined on a lattice of $v=n\lambda$ with
$n\in\mathbb{Z}$, i.e. a superposition of volume eigen-states that will evolve
on this lattice in such a way that the potential will be probed only on such
lattice points. The criterion (VI.10) can be written in this case as

$\displaystyle{p_{\phi}^{2}}\leq p^{2}_{\rm C}\equiv
12\pi\ell_{p}^{2}\left(f^{2}(\lambda b_{c})-f^{2}(\lambda b_{\infty})\right).$
(VI.14)

Now this cannot be a sharp bound because its construction relies on the
effective dynamics. However, it remains an order of magnitude criterion in the
sense that as $p_{\phi}^{2}$ becomes smaller and $p_{\phi}^{2}\ll p^{2}_{\rm
C}$ the transmission probability is expected to dominate while the bouncing
probability will become smaller and vice versa. Indeed a more direct
dimensional analysis argument is perhaps clearer. Assuming the change in the
function $f(\lambda b)$ in the region of interest is order unity (which is
about right for a continuous function unless one would dramatically tune
$f(\lambda b)$) then the criterion of softness is very simple and boils down
to the condition that888It is interesting to notice that if we assume that the
Universe is described by an massless scalar field before the onset of
inflation one can estimate $p_{\phi}$ as follows. Using equation (V.9) we have
$\rho\sim\frac{p_{\phi}^{2}}{V_{0}^{2}a^{6}}.$ (VI.15) If we assume that the
density of the onset of inflation is $\sim 10^{-5}m_{\mathrm{p}}^{4}$—as is
the case, for instance, in power law inflationary model Weinberg:2008zzc —and
we take the physical volume of the fiducial set at the onset of inflation to
be $\sim 10^{2}m_{\mathrm{p}}^{-3}$ we obtain that $p_{\phi}^{2}\sim
10^{-1}m_{\mathrm{p}}^{-2}\sim 10^{-1}\ell_{\mathrm{p}}^{2}$. Note that that a
fiducial cell with physical volume of $10^{2}$ Planck volumes will inflate to
a size much larger than the observable universe today. For a discussion of the
role of $V_{0}$ in quantum fluctuations see Rovelli:2013zaa , this of course
adds and additional dimension to the ambiguity discussion.

$\displaystyle p_{\phi}^{2}\lesssim\ell_{p}^{2}.$ (VI.16)

###  VI.3 Violation of the NEC due to inverse volume corrections

The NEC requires that $T_{ab}k^{a}k^{b}\geq 0$ for any future directed null
vector. In our cosmological setting any matter coupling can be considered as a
perfect fluid as demanded by isotropy, and thus the NEC reduces to the
statement that $\rho+P\geq 0$. For a scalar field model of the type considered
here (and independently of the self-interaction potential) this condition is
classically given by

$\displaystyle\rho+P=\frac{p_{\phi}^{2}}{16\pi^{2}G^{2}\gamma^{2}v^{2}},$
(VI.17)

which satisfies the NEC trivially. In the quantum theory the NEC can be
violated by the inverse volume corrections introduced by a regularization, for
instance of the class (III.14). As an example we plot the regularization

$\displaystyle\left[\frac{1}{v^{2}}\right]_{\rm
reg}=\left(2\left[\frac{1}{\sqrt{v}}\right]_{20}-\left[\frac{1}{\sqrt{v}}\right]_{2}\right)\left(\left[\frac{1}{\sqrt{v}}\right]_{2}\right)^{3}$
(VI.18)

in Figure 6, where one observes that the NEC are violated near the big bang.
Such possibility (which is again directly related to the ambiguities of the
polymer quantization) has a strong dynamical effect. In the special case of a
massless scalar field the previous effect implies also a violation of the weak
and the strong energy conditions in the matter sector. When translated into
the non relativistic particle analogy of Section II, one observes that the
effective potential in the Hamiltonian (III.2) is no longer negative definite.
This implies that for sufficiently low cosmological constant—and under
suitable conditions where the function $f(\lambda b)$ will play a role—the
universe might bounce through yet another different channel due to the
repulsive potential produced by the negative energy brought about by the
regularization before reaching one of the critical points of $f^{\prime}=0$.
Once again this is possible if the initial conditions for the matter fields
are sufficiently soft so that the probability of this new channel is activated
before the standard kinetic bounce described in Section VI.1 takes place. A
simple analysis that evaluates the amount of ‘kinetic’ energy acquired by the
universe (in the non-relativistic particle analogy) as it evolves toward the
would-be-singuarity shows that the condition is

$\displaystyle\left|\frac{p_{\phi}}{\ell_{p}}\right|\lesssim\left|\frac{v_{c}}{\lambda}\right|,$
(VI.19)

where $v_{c}$ is the value of $v$ that maximizes the regularization
$v^{-2}|_{\rm reg}$ before the NEC are violated (explicitly seen around $v=20$
in Figure 6 in this particular example). As $v_{c}$ can be made large by
tuning the inverse volume regularization, the present criterion of softness is
weaker than the one for tunnelling (VI.16).

$\begin{array}[]{c}\includegraphics[width=227.62204pt]{NEC1.pdf}\end{array}\ \
\ \ \ \ \ \ \ \ \ \ \ \
\begin{array}[]{c}\includegraphics[width=199.16928pt]{NEC2.pdf}\end{array}$

Figure 6: Regularization of the function $1/v^{2}$: the classical function
corresponds to the dotted line. The regularization shown here violates the
positivity in a range around $v=0$ and coincides in the IR with the classical
function as the plot on the right illustrates. This leads from equation
(VI.17) to violations of the NEC that when considered in the Hamiltonian
produce a different type of bounce for suitable initial conditions. The plot
represents the regularization given in (VI.18).

###  VI.4 Illustrating examples

####  VI.4.1 Inflation induced by ambiguity parameters

In order to illustrate how the ambiguities of loop quantum cosmology can
actually affect the physics in a non trivial manner in this section we show
that the modifications introduced by the function $f(\lambda b)$ can for
instance drive inflation for a large number of e-folds in a way that is weakly
dependent of the matter content and dynamics and basically due to the
dynamical modifications brought by the ‘holonomy corrections’ in $f(\lambda
b)$. We will illustrate this with two simple models: first with a model of a
universe filled with thermal radiation, second in the case of a model of
inflation with a scalar field with potential $U(\phi)=\lambda\phi^{4}/2$. The
first example shows that one can get many e-folds of inflation without an
inflaton, the second contains an scalar field but the inflation will be driven
by effects brought by $f(\lambda b)$, as a consequence, the phenomenology
observable in the CMB fluctuations can be tuned, as we will show, by
judiciously choosing the later function. Thus, let us consider a function
$f(\lambda b)$ such that

$f^{\prime}(\lambda b)^{2}=\begin{cases}1\ \ \ \ \ \ \ \ \ \,\quad b<b_{c}\\\
\frac{3H_{0}^{2}}{8\pi G}\frac{1}{\rho[\lambda b]}\quad b>b_{c}.\end{cases}$
(VI.20)

where from (V.1) we have

$\displaystyle{\rho}[\lambda b]=\mathrm{bar}{\rho}f(\lambda b)^{2},$ (VI.21)

and continuity requires

$\displaystyle\rho[\lambda b_{c}]=\frac{8\pi G}{3H_{0}^{2}}\equiv\rho_{c},$
(VI.22)

with $\rho_{c}$ a critical density depending on the choice of $H_{0}$. A
solution of the differential equation (VI.23) is given by

$f(\lambda b)=\begin{cases}\ \lambda b\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \,b<b_{c}\\\
\sqrt{2\gamma\lambda^{2}H_{0}(b-b_{c})+\lambda^{2}b^{2}_{c}}\ \ \ \ \ \ \
b>b_{c}.\end{cases}$ (VI.23)

This choice of $f(\lambda b)$ produces the standard Frieman equation for
$\rho\leq\rho_{c}$ while it produces a Friedman equation with constant Hubble
rate $H_{0}$ (De Sitter inflation) for $\rho\geq\rho_{c}$ regardless of the
equations of state of matter! The only thing that the matter equations of
state will control (via (V.16)) is for how long the universe will remain in
the inflationary phase. We will construct two explicit examples in what
follows.

####  VI.4.2 A pure radiation inflationary model

The first model consists of a universe filled with radiation $\rho=3P$. In
this case one has from (V.16) that

$\displaystyle\rho=\rho_{\rm in}\frac{a_{\rm in}^{4}}{a^{4}}.$ (VI.24)

Assuming that the initial value of $\rho_{\rm in}=m_{p}^{4}$ (Planck density)
and setting $\rho_{c}=10^{-68}\rho_{\rm in}$ in (VI.22) to the electro-weak
transition density999The electro-weak transition energy scale is $E_{\rm
ew}\approx 100$ GeV which corresponds to $E_{\rm ew}\approx 10^{-17}m_{p}$.
we see that inflation can be sustained as long as

$\displaystyle\frac{a_{\rm in}^{4}}{a^{4}}\geq 10^{-68},$ (VI.25)

in other works for a number of e-folds of about

$\displaystyle{\mathfs{N}}_{\rm rad}=17\log(10)\approx 39.$ (VI.26)

Using a massless scalar field (which is an often used example) with equation
of state $P=\rho$ one has

$\displaystyle\rho=\rho_{\rm in}\frac{a_{\rm in}^{6}}{a^{6}},$ (VI.27)

and the same $\rho_{c}$ we get ${\mathfs{N}}_{\rm\phi}=\log[10]68/6\approx 26$
which is still a considerable number.

###  VI.5 Inflation with a scalar field

It is also possible to design a model without an inflaton just using the
matter content of the standard model of particle physics where inflation is
driven by polymer corrections. The model is consistent with the observed
fluctuations in the CMB if the usual paradigm where quantum fluctuations of
the Higgs (with potential ${U}(\phi)=\frac{\alpha}{2}\phi^{4}$) are
responsible for the generation of inhomogeneities is used. More precisely,
starting from the Klein Gordon equation for the Higgs zero mode

$\ddot{\phi}+3H\dot{\phi}+2\alpha\phi^{3}=0,$ (VI.28)

and using the standard terminal velocity approximation
($\frac{\ddot{\phi}}{H\dot{\phi}}\ll 1$), the solution of (VI.28) is given by

$\phi(t)=\frac{\phi_{0}}{\sqrt{1+\frac{4}{3}\alpha\frac{\phi_{0}^{2}}{H_{0}^{2}}H_{0}t}}\quad\text{
or
}\quad\phi({\mathfs{N}})=\frac{\phi_{0}}{\sqrt{1+\frac{4}{3}\alpha\frac{\phi_{0}^{2}}{H_{0}^{2}}{\mathfs{N}}}},$
(VI.29)

where we introduced the number of e-folds ${\mathfs{N}}=\log(a)\approx
H_{0}t$. Now using that
$\rho=\frac{\dot{\phi}^{2}}{2}+\frac{\alpha}{2}\phi^{4}$ and equation (VI.29)
we obtain

$\rho({\mathfs{N}})=\frac{\alpha}{2}\frac{\phi_{0}^{4}}{\left[1+\frac{4}{3}\alpha\frac{\phi_{0}^{2}}{H_{0}^{2}}{\mathfs{N}}\right]^{2}}\left(\frac{2\alpha}{3}\frac{\frac{\phi_{0}^{2}}{H_{0}^{2}}}{\left[1+\frac{4}{3}\alpha\frac{\phi_{0}^{2}}{H_{0}^{2}}{\mathfs{N}}\right]}+1\right),$
(VI.30)

where $\phi_{0}$ is the initial value of the scalar field. Let us assume that
we want

$\displaystyle{\mathfs{N}}=50$ (VI.31)

then previous expression must satisfy the condition (VI.22). Introducing the
variables $y\equiv m_{p}/\phi_{0}$ and $x\equiv\phi_{0}/H_{0}$ we can write it
as

$\displaystyle\frac{3y^{2}}{8\pi}=\frac{\alpha}{2}\frac{x^{2}}{\left[1+\frac{200\alpha}{3}x^{2}\right]^{2}}\left(\frac{2\alpha}{3}\frac{x^{2}}{\left[1+\frac{200\alpha}{3}x^{2}\right]}+1\right),$
(VI.32)

which imposes some algebraic restrictions on the initial value $\phi_{0}$ and
the Hubble rate $H_{0}$. We can solve this numerically. For instance we find a
solution $\phi_{0}\approx 10\,m_{p}$ and $H_{0}\approx m_{p}$ if we choose
$\alpha=10^{-3}$ (for such small coupling we could solve the previous
constraint analytically neglecting the subleading corrections in $\alpha$ but
this is not really important here as we only want to exhibit an example). At
the end of inflation $\phi_{\rm end}\approx 9.7\,m_{p}$ and the reheating
phase could start as in usual approaches such as those of chaotic inflation.
This example is perhaps more suitable for our point as here the densities
remain Planckian all the way during the inflationary phase so that our
deviations from general relativity can be more safely attributed to ‘quantum
gravity’ effects. In contrast with the previous example where densities went
down to almost standard particle physics densities during the anomalous
expansion era. One could investigate the mechanism of structure formation. The
point of our example is to show the intrinsic discretional nature of these
models which precludes the possibility to actually use them for such
predictions.

##  VII Conclusions

We have investigated regularization ambiguities associated with the so-called
polymerization process imposed upon us when quantizing cosmological models
using the loop quantum cosmology framework. We showed that quantitative
predictability is compromised by the strong dependence on free parameters.
However, some qualitative features are robust and independent of the
polymerization choice. Among these one finds the well defined quantum
evolution across of the big bang which can also be recover at the level of
effective dynamical equations valid for suitable semiclassical states. Thanks
to the fact that our quantum dynamics could be explicitly solved, we were able
to exhibit the existence of new channels (tunneling) for the transition across
the big bang that are not apparent at the level of the effective dynamical
equations. This was possible due to the use of unimodular quantum cosmology;
however, it is easy to understand that these features hold true in the
standard formulation.

The richness of these models should be relevant for the discussion of
conceptual and qualitative issues in quantum gravity. For instance in the
discussion of questions of unitarity in the context of the black hole
information puzzle (where some initial studies have been performed
Amadei:2019wjp ; Amadei:2019ssp ), or in the context of gravitational collapse
where the new tuneling modes discussed here could be simplified toy models
relevant to investigate the black to white hole transition paradigm of
Haggard:2015iya ; Rignon-Bret:2021jch ; Martin-Dussaud:2019wqc ;
Rovelli:2018okm ; Bianchi:2018mml .

##  VIII Acknowledgements

We thank useful discussions with A. Barrau, K. Martineau, K. Noui, C. Renevey,
and E. Wilson-Ewing. We acknowledge support of the ID61466 grant from the John
Templeton Foundation, as part of the “Quantum Information Structure of
Spacetime (QISS)” project (qiss.fr), and I$\varphi$U of Aix-Marseille
University.

##  Appendix A Some properties of Gaussian states in LQC

The work of Willis and Taveras Taveras:2008ke ; Willis:2004br  shows that one
can approximate the quantum dynamics of loop quantum cosmology by effective
classical equations when using suitable semiclassical states defined in terms
of gaussian states. However, their analysis does not include the type of
generalized regularizations studied in this paper. In this section we show
that the effective dynamics approximation continues to make sense for
arbitrary regularizations of the quantum Hamiltonian as defined in (III.2).

Let us specialize to a natural choice of semiclassical states (customarily
used in the literature studying the issues involved here Taveras:2008ke ;
Willis:2004br ). For that we chose a state
$\ket{\Psi}\in{\mathfs{H}}_{lqc}\otimes{\mathfs{H}}_{\phi}$ as a Gaussian
state given by

$\ket{\Psi}=\sum_{v_{n}\in\Gamma_{\lambda}}\int_{\mathbb{R}}dp\
\Psi_{v_{0},b_{0}}(v_{n})\Phi_{p_{0},\phi_{0}}(p)\ \ket{v_{n},p}$ (A.1)

where one is using the basis eigenstates of $v$ and $p$ respectively, with the
(physical) inner product

$\displaystyle\braket{{v_{m},p^{\prime}}}{{v_{n},p}}=\delta_{n,m}\delta(p,p^{\prime}),$
(A.2)

and where the wave function

$\displaystyle\Psi_{v_{0},b_{0}}(v)=\sqrt{\frac{\lambda\sigma_{b}}{\sqrt{\pi}}}e^{-\frac{\sigma_{b}^{2}}{2}\left(v-v_{0}\right)^{2}}e^{ib_{0}\left(v-v_{0}\right)},$
(A.3)

is peaked at the geometry phase space point $(v_{0},b_{0})$, and the wave
function

$\displaystyle\Phi_{p_{0},\phi_{0}}(p)=\sqrt{\frac{\sigma_{\phi}}{\sqrt{\pi}}}e^{-\frac{\sigma^{2}_{\phi}}{2}\left(p-p_{0}\right)^{2}}e^{i\phi_{0}\left(p-p_{0}\right)},$
(A.4)

representing a semiclassical state peaked at the point $(\phi_{0},p_{0})$ of
the scalar field phase space. In the previous expressions $\Gamma_{\lambda}$
denotes a regular lattice with lattice spacing $\lambda$ that identifies of
the the so-called super selected sectors of the quantum geometry Hilbert space
(for a discussion of the nature of such choice see Amadei:2019wjp ;
Amadei:2019ssp  and references there in).

###  A.1 On the equivalence between calculations using the discrete or the
continuum inner products

The following Lemma gives the means to translating expressions involving
discrete sums in the loop quantum cosmology inner product to the more familiar
continuous integrals of the Schrodinger representation.

######  Remark 1.

For any operator $O(b,p)=\sum_{k}o_{k}(p)e^{ib\lambda k}$, and gaussian
semiclassical states as in (A.1), one has that

$\displaystyle\boxed{\braket{O(b,p)}\equiv{\braket{\Psi}{O(b,p)}{\Psi}}.}$
(A.5)

Proof: By linearity it is enough to prove the previous statement for the
operator $e^{ikb\lambda}$ for arbitrary $k$. One has

$\displaystyle{\braket{\Psi}{e^{ikb\lambda}}{\Psi}}$ $\displaystyle=$
$\displaystyle{\frac{\lambda\sigma_{b}}{\sqrt{\pi}}}\sum_{n,m}e^{-\frac{\sigma_{b}^{2}}{2}\left(\lambda
n-v_{0}\right)^{2}}e^{-ib_{0}\left(\lambda
n-v_{0}\right)}e^{-\frac{\sigma_{b}^{2}}{2}\left(\lambda
m-v_{0}\right)^{2}}e^{ib_{0}\left(\lambda m-v_{0}\right)}\braket{n}{{m-k}}$
(A.6) $\displaystyle=$ $\displaystyle{\frac{\lambda\sigma_{b}e^{i2b_{0}\lambda
k}}{\sqrt{\pi}}}\sum_{m}e^{-\frac{\sigma_{b}^{2}}{2}\left(\lambda m-\lambda
k-v_{0}\right)^{2}}e^{-\frac{\sigma_{b}^{2}}{2}\left(\lambda
m-v_{0}\right)^{2}}$ $\displaystyle=$
$\displaystyle{\frac{\lambda\sigma_{b}e^{ib_{0}\lambda
k}}{\sqrt{\pi}}}e^{-\frac{1}{4}\sigma_{b}^{2}{\lambda^{2}k^{2}}}\sum_{m}e^{-\sigma_{b}^{2}{\left(\lambda
m-v_{0}-\lambda\frac{k}{2}\right)^{2}}}={\frac{\sigma_{b}e^{ib_{0}\lambda
k}}{\sqrt{\pi}}}e^{-\frac{1}{4}\sigma_{b}^{2}{\lambda^{2}k^{2}}}\vartheta_{3}\left[-\frac{\pi}{2}\left(k+2\frac{v_{0}}{\lambda}\right);e^{-\frac{\pi^{2}}{\lambda^{2}\sigma_{b}^{2}}}\right]$
$\displaystyle=$ $\displaystyle e^{ib_{0}\lambda
k}e^{-\frac{1}{4}\sigma_{b}^{2}\lambda^{2}k^{2}}\left(1+{\mathfs{O}}\left(e^{-\frac{\pi^{2}}{\lambda^{2}\sigma_{b}^{2}}}\right)\right),$

where

$\displaystyle\vartheta_{3}[u;q]\equiv
1+2\sum_{n=1}^{\infty}q^{n^{2}}\cos[2nu]$ (A.7)

in the first line we used the definition of the state (A.1) and (IV.1) and
then we just rearranged the sums completing squares to arrive at the final
result.

######  Corolary 1.

For any operator $O(\lambda b,p)=\sum_{k}o_{k}(p)e^{ikb\lambda}$, and gaussian
semiclassical states as in (A.1), one has that

$\displaystyle\frac{d\braket{\Psi}{O(\lambda b,p)\Psi}}{d(\lambda
b_{0})}=\braket{\frac{dO(\lambda b,p)}{d(\lambda b)}}.$ (A.8)

The proof of the previous statement follows directly from (A.6) 2.

######  Corolary 2.

For any operator $f(\lambda b)=\sum_{k}f_{k}e^{ikb\lambda}$, and gaussian
semiclassical states as in (A.1), one has that

$\displaystyle\braket{f(b)^{2}}-\braket{f(b)}^{2}=2f^{\prime}(\lambda
b_{0})^{2}\
\lambda^{2}\sigma_{b}^{2}+{\mathfs{O}}(\lambda^{4}\sigma_{b}^{4}).$ (A.9)

Proof: From Remark 1 we have that

$\displaystyle\braket{f(\lambda b)}=f(\lambda
b_{0})+\frac{1}{4}f^{\prime\prime}(\lambda b_{0})\
\lambda^{2}\sigma_{b}^{2}+{\mathfs{O}}(\lambda^{4}\sigma_{b}^{4}).$ (A.10)

The present statement follows from the previous equation when applied to
$O(b)=f^{2}(b)$ and $O(b)=f(b)$ respectively and replacing the result in the
expression of the fluctuation 2.

######  Corolary 3.

The previous two results imply that

$\displaystyle\frac{d\braket{f(\lambda
b)^{2}}}{db_{0}}=2\frac{d\braket{f(\lambda b)}}{db_{0}}\braket{f(\lambda
b)}+{\mathfs{O}}(\lambda^{2}\sigma^{2}).$ (A.11)

###  A.2 The generating function and the expectation value of operators
depending on the volume

######  Remark 2.

For any operator $O(b,p)=\sum_{k}o_{k}(p)e^{ib\lambda k}$, and gaussian
semiclassical states as in (A.1), one has the generating function on the left

$\displaystyle\boxed{\frac{\braket{\Psi}{e^{\mathcal{j}(v-v_{0})}O(b,p)}{\Psi}}{\braket{\Psi}{\Psi}}=\sum_{k}o_{k}(p)e^{ib_{0}k}e^{-\frac{1}{4}\sigma_{b}^{2}\lambda^{2}k^{2}+\frac{\mathcal{j}^{2}}{4\sigma_{b}^{2}}-\mathcal{j}\lambda\frac{k}{2}}+\left(1+{\mathfs{O}}\left(e^{-\frac{\pi^{2}}{\lambda^{2}\sigma_{b}^{2}}}\right)\right),}$
(A.12)

and the generating function on the right

$\displaystyle\boxed{\frac{\braket{\Psi}{O(b,p)e^{\mathcal{j}(v-v_{0})}}{\Psi}}{\braket{\Psi}{\Psi}}=\sum_{k}o_{k}(p)e^{ib_{0}k}e^{-\frac{1}{4}\sigma_{b}^{2}\lambda^{2}k^{2}+\frac{\mathcal{j}^{2}}{4\sigma_{b}^{2}}+\mathcal{j}\lambda\frac{k}{2}}+\left(1+{\mathfs{O}}\left(e^{-\frac{\pi^{2}}{\lambda^{2}\sigma_{b}^{2}}}\right)\right).}$
(A.13)

Proof: Consider

(A.14)

where in the second line we completed the square and performed the gaussian
integration. Equation (A.13) follows from the last line. A similar
manipulation gives (A.12) 2.

###  A.3 Some statements about the truncation of the Fourier expansion

Given a bounded square integrable function $f(\lambda b)$ of period $2\pi$ we
can write it as

$f(\lambda b)=\sum_{n\in\mathbb{Z}}a_{n}e^{in\lambda b}\quad,$ (A.15)

with the coefficients $a_{n}$ given by

$a_{n}=\frac{1}{2\pi}\int_{0}^{2\pi}f(\lambda b)e^{-in\lambda b}d(\lambda
b)\quad.$ (A.16)

which can be bounded

$|a_{n}|\leq f_{max}\quad.$ (A.17)

where $f_{\rm max}\equiv\max{|f(\lambda b)|}$. Let us define the truncated
function as

$f_{N}(\lambda b)=\sum_{n=-N}^{+N}a_{n}e^{in\lambda b}\quad.$ (A.18)

######  Remark 3.

When evaluated on gaussian states (A.1) one has that

$\displaystyle\left|\braket{f(\lambda b)}-\braket{f_{N}(\lambda b)}\right|\leq
2f_{\rm max}\
e^{-\sigma_{b}^{2}\lambda^{2}N^{2}}(1+{\mathfs{O}}(e^{-\sigma_{b}^{2}\lambda^{2}}))\quad.$
(A.19)

Therefore, the expectation value of the full series and the truncation agree
extremely quickly as $N$ grows. We can say the the difference between the two
will be negligible as long as

$\boxed{\lambda^{2}\sigma^{2}_{b}N^{2}>1}\quad.$ (A.20)

###### Proof.

It follows from Proposition 1 that

$\displaystyle\left|\braket{f(\lambda b)}-\braket{f_{N}(\lambda b)}\right|$
$\displaystyle=\left|\sum_{|n|>N}a_{n}e^{in\lambda
b_{0}}e^{-\sigma_{b}^{2}\lambda^{2}n^{2}}\right|$ $\displaystyle\leq
2\sum_{n=N+1}^{+\infty}|a_{n}|\ e^{-\sigma_{b}^{2}\lambda^{2}n^{2}}=2\
e^{-\sigma_{b}^{2}\lambda^{2}N^{2}}\sum_{n=N+1}^{+\infty}|a_{n}|\
e^{-\sigma_{b}^{2}\lambda^{2}(n^{2}-N^{2})}$ $\displaystyle\leq 2\
e^{-\sigma_{b}^{2}\lambda^{2}N^{2}}\sum_{n=N+1}^{+\infty}|a_{n}|\
e^{-\sigma_{b}^{2}\lambda^{2}(n-N)^{2}}$ $\displaystyle=2\
e^{-\sigma_{b}^{2}\lambda^{2}N^{2}}\sum_{m=1}^{+\infty}|a_{m+N}|\
e^{-\sigma_{b}^{2}\lambda^{2}m^{2}}$ $\displaystyle\leq 2f_{\rm max}\
e^{-\sigma_{b}^{2}\lambda^{2}N^{2}}(1+{\mathfs{O}}(e^{-\sigma_{b}^{2}\lambda^{2}}))\quad\mathrm{{}^{2}}.$
(A.21)

######  Corolary 4.

For a given function $f(\lambda b)$, and Gaussian states (A.1) we have that

$\displaystyle\braket{f_{N}(\lambda b)}\approx f(\lambda b_{0})$ (A.22)

as long as

$\displaystyle\boxed{\frac{1}{N^{2}}<\lambda^{2}\sigma_{b}^{2}<\left|\frac{4f(\lambda
b_{0})}{f^{\prime\prime}(\lambda b_{0})}\right|}$ (A.23)

which can be achieved for sufficiently large $N$.

We therefore arrive at the conclusion that for any arbitrary function
$f(\lambda b)$ we can find $N$ and $\sigma_{p}$ such that the Gaussian
expectation value would agree with the desired accuracy with the function of
our choice satisfying the minimal requirement (III.3).

######  Remark 4.

For arbitrary operators $O(\beta b)$ following time evolution rule holds

$\displaystyle\frac{d\braket{O(\beta b)}}{ds}$ $\displaystyle\equiv$
$\displaystyle-i\braket{[O(\beta b),{\cal H}]}$ (A.24) $\displaystyle=$
$\displaystyle\frac{V_{0}}{4\pi G\gamma}\frac{\braket{p_{\phi}^{2}}}{\pi
G\gamma v_{0}^{3}}\frac{d\braket{O(\beta
b)}}{db_{0}}+{\mathfs{O}}\left(\frac{\beta^{2}}{v_{0}^{4}}\right)$

where one needs to assume that $v_{0}\gg\sigma_{b}$ and $v_{0}\gg N$.

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