# Phenomenological implications of modified loop cosmologies: an overview

Bao-Fei Li 1,2111E-mail address<EMAIL_ADDRESS>Parampreet Singh1222 E-mail
address<EMAIL_ADDRESS>Anzhong Wang3333E-mail address:
<EMAIL_ADDRESS>1 Department of Physics and Astronomy, Louisiana
State University, Baton Rouge, LA 70803, USA
2 Institute for Theoretical Physics $\&$ Cosmology, Zhejiang University of
Technology, Hangzhou, 310032, China
3 GCAP-CASPER, Department of Physics, Baylor University, Waco, TX, 76798-7316,
USA

###### Abstract

In this paper, we first provide a brief review of the effective dynamics of
two recently well-studied models of modified loop quantum cosmologies (mLQCs),
which arise from different regularizations of the Hamiltonian constraint and
show the robustness of a generic resolution of the big bang singularity,
replaced by a quantum bounce due to non-perturbative Planck scale effects. As
in loop quantum cosmology (LQC), in these modified models the slow-roll
inflation happens generically. We consider the cosmological perturbations
following the dressed and hybrid approaches and clarify some subtle issues
regarding the ambiguity of the extension of the effective potential of the
scalar perturbations across the quantum bounce, and the choice of initial
conditions. Both of the modified regularizations yield primordial power
spectra that are consistent with current observations for the Starobinsky
potential within the framework of either the dressed or the hybrid approach.
But differences in primordial power spectra are also identified among the
mLQCs and LQC. In particular, for mLQC-I, striking differences arise between
the dressed and hybrid approaches in the infrared and oscillatory regimes.
While the differences between the two modified models can be attributed to
differences in the Planck scale physics, the permissible choices of the
initial conditions and the differences between the two perturbation approaches
have been reported for the first time. All these differences, due to either
the different regularizations or the different perturbation approaches , in
principle can be observed in terms of non-Gaussianities.

##  I Introduction

Despite offering a solution to several fundamental and conceptual problems of
the standard big bang cosmology, including the flatness, horizon, and exotic-
relic problems, the cosmic inflation in the early universe also provides a
mechanism to produce density perturbations and primordial gravitational waves
(PGWs) Guth ; inflation ; InfReviews . The latter arise from quantum
fluctuations of spacetimes and produce not only a temperature anisotropy, but
also polarizations in the cosmic microwave background (CMB), a smoking gun of
PGWs. However, the inflationary paradigm is incomplete without the knowledge
of key elements from quantum gravity. First, it is well-known that the cosmic
inflation is sensitive to the ultraviolet (UV) physics, and its successes are
tightly contingent on the understanding of this UV physics trans-planck ; KW14
; stringC . In particular, if the inflationary phase lasts somewhat longer
than the minimal period required to solve the above mentioned problems, the
length scales we observe today will originate from modes that are smaller than
the Planck length during inflation trans-planck . Then, the treatment of the
underlying quantum field theory on a classical spacetime becomes questionable,
as now the quantum geometric effects are expected to be large, and the space
and time cannot be treated classically any more. This is often referred to as
the trans-Planckian problem of cosmological fluctuations 444It has been
conjectured using models in string theory that the trans-Planckian problem
might never arise BV20 , which results on severe constraints on various
cosmological models (See BBLV20 ; Brand20  for more details)..

The second problem of the inflationary paradigm is more severe. It is well
known that inflationary spacetimes are past-incomplete because of the big bang
singularity singularity , with which it is not clear how to impose the initial
conditions. This problem gets aggravated for low energy inflation in
spatially-closed models which are slightly favored by current observations
where the universe encounters a big crunch singularity and lasts only for a
few Planck seconds linde .

Another problematic feature of inflation is that one often ignores the pre-
inflationary dynamics and sets the Bunch-Davies (BD) vacuum at a very early
time. But, it is not clear how such a vacuum state can be realized dynamically
in the framework of quantum cosmology stringC , considering the fact that a
pre-inflationary phase always exists between the Plank and inflation scales,
which are about $10^{3}\sim 10^{12}$ orders of magnitude difference inflation
; stringC . While these problems of inflationary paradigm demand a completion
from quantum theory of spacetimes, they also open an avenue to overcome one of
the main obstacles in the development of quantum gravity, which concerns with
the lack of experimental evidences. Thus, understanding inflation in the
framework of quantum gravity could offer valuable guidances to the
construction of the underlying theory QGs ; string ; LQG .

In particular, when applying the techniques of loop quantum gravity (LQG) to
homogeneous and isotropic universe, namely loop quantum cosmology (LQC) LQC ,
it was shown that, purely due to quantum geometric effects, the big bang
singularity is generically resolved and replaced by a quantum bounce at which
the spacetime curvature becomes Planckian aps1 ; aps2 ; aps3 ; acs2010 . The
robustness of the singularity resolution has been shown for a variety of
isotropic and anisotropic spacetimes giesel-li-singh-inflation .
Interestingly, there exists a reliable effective spacetime description, which
has been used to confirm a generic resolution of all strong curvature
singularities non-singular . Various phenomenological implications have been
studied using this effective spacetime description, whose validity has been
verified for isotropic and anisotropic spacetimes numlsu-1 ; numlsu-2 ;
numlsu-3 ; numlsu-4 ; numlsu-5 . For low energy inflation models with a
positive spatial curvature, the singularity resolution and a successful onset
of inflation for classically inadmissible initial conditions have been
demonstrated dupuy-singh ; gordon-li-singh ; warm-inflation .

In the last couple of years, several approaches have been proposed, in order
to address the impacts of the quantum geometry on the primordial power
spectra. These include the approaches of the deformed algebra bhks2008 ;
cbgv2012 ; cmbg2012 , dressed metric aan2012 ; aan2013 ; aan2013b , and hybrid
mm2012 ; mm2013 ; gmmo2014 ; gbm2015 ; mo2016  (For a recent discussion about
similar ideas in anisotropic Bianchi I LQC spacetimes see Refs. GS12 ; GS13 ;
b1-lett ; b1-long  and references therein.). In particular, the last two
approaches have been widely studied and found that they are all consistent
with current cosmological observations agullo_detailed_2015 ; d1 ; d1b ;
bonga_inflation_2016 ; bonga_phenomenological_2016 ; ZhuA ; ZhuB ; ZWKCS18 ;
WZW18 ; abs2018 ; bo2016 ; gbmo2016 ; nbm2018 ; nbm2018a . In addition, within
the framework of the dressed metric approach recently it has been also shown
that some anomalies from the CMB data Planck2018b ; Planck2019 ; SCHS16  can
be reconciled purely due to the quantum geometric effects AGJS20 ; AKS20 .

In addition to the standard LQC, in which the Lorentzian term of the classical
Hamiltonian constraint is first expressed in terms of the Euclidean term in
the spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) universe, and
then only the quantization of the Euclidean term is considered, the robustness
of the singularity resolution with respect to different quantizations of the
classical Hamiltonian constraint in the symmetry reduced spacetimes have been
extensively studied. Two notable examples are the so-called modified LQC-I
(mLQC-I) and modified LQC-II (mLQC-II) models, which were first proposed by
Yang, Ding and Ma more than a decade ago YDM09 . In a recent study, Dapor and
Liegener (DL) DL17 ; DL18  obtained the expectation values of the Hamiltonian
operator in LQG using complexifier coherent states states , adapted to the
spatially flat FLRW universe. Using the non-graph changing regularization of
the Hamiltonian advocated by Thiemann thiemann , DL obtained an effective
Hamiltonian constraint, which, to the leading order, agrees with the mLQC-I
model first obtained in YDM09 . Sometimes, this model has also been referred
to as the DL model or Thiemann regularized LQC. Strictly speaking, when
constructing loops in DL17  DL treated the edge length $\mu$ as a free
parameter, but in YDM09  it was considered as a specific triad dependent
function, the so-called $\bar{\mu}$ scheme aps3 , which is known to be the
only possible choice in LQC, and results in physics that is independent from
underlying fiducial structures used during quantization, and meanwhile yields
a consistent infrared behavior for all matter obeying the weak energy
condition cs08 . Lately, the studies of DL17  have been extended to the
$\bar{\mu}$ scheme adlp .

In the two modified LQC models, mLQC-I and mLQC-II, since different
regularizations of the Lorentzian term were used, the resulting equations
become the fourth-order and non-singular quantum difference equations, instead
of the second-order difference ones obtained in LQC. In these two models the
big bang singularity is also generically resolved and replaced by a quantum
bounce. In addition, the inflationary phase can naturally take place with a
very high probability lsw2018 ; lsw2018b ; lsw2019 ; SS19a ; SS19b . In
addition, the dynamics in LQC and mLQC-II is qualitatively similar in the
whole evolution of the universe, while the one in mLQC-I becomes significantly
different from LQC (as well as mLQC-II) in the contracting phase, in which an
emergent quasi de Sitter space is present. This implies that the contracting
phase in mLQC-I is purely a quantum regime without any classical limit 555A
similar contracting branch is found in certain anisotropic models in the
standard regularization of LQC (see for eg. dadhich-joe-singh-kantowski-sachs
)..

An important question now is what are the effects of these models and
approaches on the CMB observations. The answer to this question requires the
knowledge of how the quantum fluctuations propagate on a quantum spacetime in
LQC and modified loop cosmological models. In particular, in the framework of
the dressed metric approach the power spectra of the cosmological
perturbations for both mLQC-I and mLQC-II models were investigated lsw2020 .
In the same framework but restricted only to the mLQC-I model, the power
spectra of the cosmological perturbations were studied in IA19 . More
recently, the hybrid approach was applied to mLQC-I qm2019 ; gm20 ; gqm2020 ,
for which the time-dependent mass of the perturbations was studied in detail
qmp2020 . The primordial scalar power spectra obtained in the three models,
LQC, mLQC-I and mLQC-II, were also investigated in the hybrid approach LOSW20
, and found that the relative differences in the amplitudes of the power
spectra among the three models could be as large as $2\%$ in the UV regime of
the spectra, which is relevant to the current observations. Interestingly, in
the above work, differences in primordial power spectra were found between the
hybrid and dressed metric approaches in the infra-red and oscillatory regimes
in mLQC-I.

In this brief review, we shall focus mainly on the states that are sharply
peaked along the classical trajectories, so that the description of the
“effective” dynamics of the universe becomes available LQC , and the questions
raised recently in KKL20  are avoided. This includes the studies of the
“effective” dynamics of the homogeneous and isotropic mLQC-I and mLQC-II
models, and their cosmological perturbations in the framework of the dressed
metric and hybrid approaches. We shall first clarify the issue regarding the
ambiguities in the extension of the effective potential for the scalar
perturbations across the quantum bounce, and then pay particular attention to
the differences among the three models, LQC, mLQC-I and mLQC-II, and possible
observational signals. It is important to note that initial conditions are
another subtle and important issue not only in LQC but also in mLQCs. This
includes two parts: (i) when to impose the initial conditions, and (ii) which
kind of initial conditions one can impose consistently. To clarify this issue,
we discuss it at length by showing the (generalized) comoving Hubble radius in
each model and in each of the dressed and hybrid approaches. From this
analysis, one can see clearly what initial conditions can and cannot be
imposed at a chosen initial time.

The outline of this brief overview is as follows. In Sec. II we consider the
effective dynamics of mLQC-I and mLQC-II, and discuss some universal features
of their dynamics such as the resolution of big bang singularity. In addition,
in this section we also show that for states such that the evolution of the
homogeneous universe was dominated initially at the bounce by the kinetic
energy of the inflaton, that is, $\dot{\phi}_{B}^{2}\gg V(\phi_{B})$, the
post-bounce evolution between the bounce and the reheating can be always
divided universally into three different phases: the bouncing, transition, and
slow-roll inflation [cf. Fig. 1]. During each of these phases the expansion
factor $a(t)$ and the scalar field $\phi(t)$ can be given analytically. In
particular, they are given by Eqs.(II.4)-(2.55) and (II.4)-(2.57) during the
bouncing phase for mLQC-I and mLQC-II, respectively. In this same section, the
probabilities of the slow-roll inflation is considered, and shown that it
occurs generically. This particular consideration is restricted to the
quadratic potential, but is expected to be also true for other cases.

In Sec. III, the cosmological perturbations of mLQC-I and mLQC-II are studied.
We discuss initial conditions and the subtle issue of the ambiguity in the
choice of the variables $\pi_{a}^{-2}$ and $\pi_{a}^{-1}$ (present in the
effective potential), which correspond to the quadratic and linear inverse of
the momentum conjugate to the scale factor. In addition, to understand the
issue of initial conditions properly, we first introduce the comoving Hubble
radius $\lambda_{H}^{2}$ and then state clearly how this is resolved in GR
[cf. Fig. 2], and which are the relevant questions in mLQC-I [cf. Fig. 5] and
mLQC-II [cf. Fig. 6]. From these figures it is clear that the BD vacuum cannot
be consistently imposed at the bounce 666It should be noted that anisotropies
rise during the contracting phase and generically dominate the earliest stages
of the post-bounce of the homogeneous universe GS12 ; GS13 ; b1-lett ; b1-long
. So, cautions must be taken, when imposing initial conditions at the bounce.,
as now some modes are inside the (comoving) Hubble radius while others not.
However, the fourth-order adiabatic vacuum may be imposed at this moment for
both of these two modified LQC models, as that adopted in LQC aan2013b . In
addition, in mLQC-I the de Sitter state given by Eq.(3.7) 777To be
distinguished from the BD vacuum described by Eq.(3.4) we refer to the state
described by Eq.(3.7) as the de Sitter state. The difference between them is
due to the term $i/(k\eta)$, which is not negligible in the deep contracting
phase of the de Sitter background, as now $|k\eta|$ could be very small. For
more details, see the discussions given in Sec. III.A, especially the
paragraph after Eq.(3.9). can be imposed in the contracting phase as long as
$t_{0}$ is sufficiently early, so the universe is well inside the de Sitter
phase. On the other hand, in mLQC-II and LQC, the BD vacuum can be imposed in
the contracting phase as long as $t_{0}$ is sufficiently early, so the
universe becomes so large that the spacetime curvature is very small, and
particle creation is negligible. With these in mind, the power spectra
obtained in the three models, mLQC-I, mLQC-II and LQC, within the framework of
the dressed metric approach were calculated and compared by imposing the
initial conditions in the contracting phase. In particular, the spectra can be
universally divided into three regimes, the infrared, intermediate and UV. In
the infrared and intermediate regimes, the relative differences in the
amplitudes of the spectra can be as large as $100\%$ between mLQC-I and mLQC-
II (the same is also true between mLQC-I and LQC), but in the UV regime such
differences get dramatically reduced, which is no larger than $0.1\%$. Since
the modes in the UV regime are the relevant ones to the current observations
and also their corresponding power spectra are scale-invariant, so these three
models are all consistent with observations.

In Sec. IV, the cosmological perturbations of mLQC-I and mLQC-II are studied
within the hybrid approach, and the subtleties of the initial conditions are
shown in Figs. 8, 9 and 10, where Figs. 8 and 9 are respectively for the
quadratic and Starobinsky potentials in mLQC-I, while Fig. 10 is for the
Starobinsky potential in mLQC-II. The case with the quadratic potential in
mLQC-II is similar to that of mLQC-I, given by Fig. 8. From these figures it
is clear that imposing the initial conditions now becomes a more delicate
issue, and sensitively depends on the potential $V(\phi)$ of the inflaton
field. First, in the cases described by Figs. 8 and 9, all the modes are
oscillating during the time $t_{i}^{p}<t<t_{i}$, so one might intend to impose
the BD vacuum at the bounce. However, for $t<t_{i}^{p}$ the quantity
$\Omega_{\text{tot}}$ defined in Eq.(3.9) experiences a period during which it
is very large and negative. As a result, particle creation is expected not to
be negligible during this period. Then, imposing the BD vacuum at the bounce
will not account for these effects, and the resulting power spectra shall be
quite different from the case, in which in the deep contracting phase ($t\ll
t_{B}$) the BD vacuum is imposed for mLQC-II and LQC, and the de Sitter state
for mLQC-I. On the other hand, in the case described by Fig. 10, even if the
BD vacuum is chosen at the bounce, it may not be quite different from the one
imposed in the deep contracting phase, as now in the whole contracting phase
all the modes are oscillating, and particle creation is not expected to be
important up to the bounce. To compare the results from the three different
models, in this section the second-order adiabatic vacuum conditions are
chosen in the contracting phase, which is expected not to be much different
from the de Sitter state for mLQC-I and the BD vacuum for mLQC-II and LQC, as
long as $t_{0}\ll t_{B}$ in all the cases described by Figs. 8 \- 10. The
ambiguities of the choice of $\pi_{a}^{-2}$ and $\pi_{a}^{-1}$ also occur in
this approach, but as far as the power spectra are concerned, different
choices lead to similar conclusions gqm2020 . So, in this section only the so-
called prescription A is considered. Then, similar conclusions are obtained in
this approach regarding the differences among the amplitudes of the power
spectra in the three different models. In particular, the relative differences
can be as large as $100\%$ between mLQC-I and mLQC-II/LQC, but in the UV
regime such differences are reduced to about $2\%$. A remarkable feature
between the two different approaches is also identified: in the infrared and
oscillatory regimes, the power spectrum in mLQC-I is suppressed as compared
with its counterpart in LQC in the hybrid approach. On the other hand, in the
dressed metric approach, the power spectrum in mLQC-I is largely amplified in
the infrared regime where its magnitude is as large as of the Planck scale
IA19 ; lsw2020 . The main reason for such differences is that the effective
mass in the hybrid approach is strictly positive near the bounce, while it is
strictly negative in the dressed metric approach for states that are initially
dominated by the kinetic energy of the inflaton IA19 ; lsw2020 ; gqm2020 .

The review is concluded in Sec. V, in which we summarize the main conclusions
and point out some open questions for future studies.

##  II Effective Quantum Dynamics in Modified LQCs

To facilitate our following discussions, let us first briefly review the
standard process of quantization carried out in LQC, from which one can see
clearly the similarities and differences among the three models, LQC, mLQC-I
and mLQC-II.

###  II.1 Quantum Dynamics of LQC

The key idea of LQC is to use the fundamental variables and quantization
techniques of LQG to cosmological spacetimes, by taking advantage of the
simplifications that arise from the symmetries of these spacetimes. In the
spatially-flat FLRW spacetime,

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-N^{2}(t)dt^{2}+q_{ab}(t)dx^{a}dx^{b}$ (2.1)
$\displaystyle\equiv$
$\displaystyle-N^{2}(t)dt^{2}+a^{2}(t)\delta_{ab}dx^{a}dx^{b},$

there exists only one degree of freedom, the scale factor $a(t)$, where $N(t)$
is the lapse function and can be freely chosen, given the freedom in
reparametrizing $t$, and $q_{ab}(t)$ denotes the 3-dimensional (3D) spatial
metric of the hypersurface $t=$ Constant. In this paper, we shall use the
indices $a,b,c,...$ to denote spatial coordinates and $i,j,k,...$ to denote
the internal su(2) indices. Repeated indices will represent sum, unless
otherwise specified.

In full GR, the gravitational phase space consists of the connection
$A^{i}_{a}$ and density weighted triad $E^{a}_{i}$. In the present case, the
3D spatial space $M$ has a ${\mathbb{R}}^{3}$ topology, from which we can
introduce a fiducial cell $\mathcal{V}$ and restrict all integrations to this
cell, in order to avoid some artificial divergences and have a well-defined
symplectic structure. Within this cell, we introduce a fiducial flat metric
$\mathring{q}_{ab}$ via the relation $q_{ab}(t)=a^{2}(t)\;\mathring{q}_{ab}$,
and then an associated constant orthogonal triad $\mathring{e}^{a}_{i}$ and a
cotriad $\mathring{\omega}^{i}_{a}$. Then, after symmetry reduction
$A^{i}_{a}$ and $E^{a}_{i}$ are given by,

$\displaystyle
A^{i}_{a}=c\;v_{o}^{-1/3}\;\mathring{\omega}^{i}_{a},\;E^{a}_{i}=|p|\;v_{o}^{-2/3}\sqrt{\mathring{q}}\;\mathring{e}^{a}_{i},$
(2.2)

where $|p|=v_{o}^{2/3}a^{2},\;\kappa=8\pi G/c^{4}$, $v_{o}$ denotes the volume
of the fiducial cell measured by $\mathring{q}_{ab}$, $\mathring{q}$ is the
determinant of $\mathring{q}_{ab}$, and $\gamma$ is the Barbero-Immirzi
parameter whose value can be set to $\gamma\approx 0.2375$ using black hole
thermodynamics in LQG Mei04 . For classical solutions, symmetry reduced
connection $c$ is related to time derivative of scale factor as
$c=\gamma\dot{a}$, where an over dot denotes a derivative with respect to $t$
for the choice $N=1$.

The physical triad and cotriad are given by $e_{i}^{a}={\mbox{(sgn
p)}}\;|p|^{-1/2}v_{o}^{1/3}\;\mathring{e}^{a}_{i}$ and
$\omega^{i}_{a}={\mbox{(sgn
p)}}\;|p|^{1/2}v_{o}^{-1/3}\;\mathring{\omega}^{i}_{a}$, where (sgn p) arises
because in connection dynamics the phase space contains triads with both
orientations. In the following we choose this orientation to be positive and
volume of the fiducial cell to be $v_{o}$ = 1. The variables $c$ and $p$
satisfy the communication relation,

$\displaystyle\left\\{c,p\right\\}=\frac{\kappa\gamma}{3}.$ (2.3)

Then, the gravitational part of the Hamiltonian is a sum of the Euclidean and
Lorentzian terms,

$\mathcal{H}_{\mathrm{grav}}={\cal
H}_{\mathrm{grav}}^{(E)}-(1+\gamma^{2}){\cal H}_{\mathrm{grav}}^{(L)},$ (2.4)

where, with the choice $N=1$, these two terms are given, respectively, by

$\displaystyle\mathcal{H}_{\mathrm{grav}}^{(E)}$ $\displaystyle=$
$\displaystyle\frac{1}{2\kappa}\int\mathrm{d}^{3}x\,\epsilon_{ijk}F^{i}_{ab}\frac{E^{aj}E^{bk}}{\sqrt{q}},$
(2.5) $\displaystyle\mathcal{H}_{\mathrm{grav}}^{(L)}$ $\displaystyle=$
$\displaystyle\frac{1}{\kappa}\int\mathrm{d}^{3}x\,K^{j}_{[a}K^{k}_{b]}\frac{E^{aj}E^{bk}}{\sqrt{q}},$
(2.6)

where $q=\mathrm{det}(q_{ab})=a^{6}\;\mathring{q}$, $F_{ab}^{k}$ is the field
strength of the connection $A^{i}_{a}$, and $K^{i}_{a}$ is the extrinsic
curvature, given, respectively, by

$\displaystyle F_{ab}^{k}$ $\displaystyle\equiv$ $\displaystyle
2\partial_{[a}A^{k}_{b]}+\epsilon_{ij}^{\;\;\;k}A^{i}_{a}A^{j}_{b}=c^{2}\epsilon_{ij}^{\;\;\;k}\;\mathring{\omega}^{i}_{a}\;\mathring{\omega}^{j}_{b},$
$\displaystyle K_{a}^{i}$ $\displaystyle\equiv$ $\displaystyle
K_{ab}e^{b}_{i}=\frac{{e^{b}_{i}}}{2N}\left(\dot{q}_{ab}-2D_{(a}N_{b)}\right)=\pm\dot{a}\;\mathring{\omega}^{i}_{a}.~{}~{}~{}~{}$
(2.7)

Upon quantization, ambiguities can arise due to different treatments of the
Euclidean and Lorentzian terms in the Hamiltonian constraint. In particular,
LQC takes the advantage that in the spatially-flat FLRW universe the
Lorentzian part is proportional to the Euclidean part,

${\cal H}_{\mathrm{grav}}^{(L)}=\gamma^{-2}{\cal H}_{\mathrm{grav}}^{(E)},$
(2.8)

so that, when coupled to a massless scalar field, the classical Hamiltonian
can be rewritten as abl ; aps3

$\displaystyle\mathcal{H}$ $\displaystyle\equiv$
$\displaystyle\mathcal{H}_{\mathrm{grav}}+\mathcal{H}_{M}$ (2.9)
$\displaystyle=$
$\displaystyle-\frac{1}{2\kappa\gamma^{2}}\int\mathrm{d}^{3}x\,\epsilon_{ijk}F^{i}_{ab}\frac{E^{aj}E^{bk}}{\sqrt{q}}+\mathcal{H}_{M},$

where $\mathcal{H}_{M}={p_{\phi}^{2}}/\left({2\sqrt{q}}\right)$, with
$p_{\phi}$ being the momentum conjugate of $\phi$.

The elementary operators in the standard LQC are the triads888For a
modification of LQC based on using gauge-covariant fluxes, see liegener-singh
. $p$ and elements of the holonomies given by $\widehat{e^{i\bar{\mu}c/2}}$ of
$c$, where $\bar{\mu}=\sqrt{\Delta l_{pl}^{2}/|p|}$ with $\Delta\equiv
4\sqrt{3}\pi\gamma$, and $\Delta l_{pl}^{2}$ being the minimum non-zero
eigenvalue of the area operator, and the Planck length $l_{pl}$ is defined as
$l_{pl}\equiv\sqrt{\hbar G}$. However, it is found that, instead of using the
eigenket $\left|p\right>$ of the area operator $p$ as the basis, it is more
convenient to use the eigenket $\left|v\right>$ of the volume operator
$\hat{v}\left(\equiv{\mbox{sgn}}(p)|p|^{3/2}\right)$, where

$\hat{v}\left|v\right>=\left(\frac{8\pi\gamma}{6}\right)^{3/2}\frac{|v|}{K}l_{pl}^{3}\left|v\right>,\quad\widehat{e^{i\bar{\mu}c/2}}\left|v\right>=\left|v+1\right>,$
(2.10)

with $K\equiv 2\sqrt{2}\left(3\sqrt{3\sqrt{3}}\right)^{-1}$. Let
$\Psi(v,\phi)$ denote the wavefunction in the kinematical Hilbert space of the
gravitational field coupled with the scalar field $\phi$, we have

$\displaystyle\hat{\phi}\Psi(v,\phi)$ $\displaystyle=$
$\displaystyle\phi\Psi(v,\phi),$ $\displaystyle\hat{p}_{\phi}\Psi(v,\phi)$
$\displaystyle=$
$\displaystyle-i\hbar\frac{\partial}{\partial\phi}\Psi(v,\phi),$
$\displaystyle\widehat{|p|^{-3/2}}\Psi(v,\phi)$ $\displaystyle=$
$\displaystyle{\cal{B}}(v)\Psi(v,\phi),$ (2.11)

where

$\displaystyle{\cal{B}}(v)$ $\displaystyle\equiv$
$\displaystyle\left(\frac{6}{8\pi\gamma l_{pl}^{2}}\right)^{3/2}B(v),$
$\displaystyle B(v)$ $\displaystyle\equiv$
$\displaystyle\left(\frac{3}{2}\right)^{3}K|v|\left||v+1|^{1/3}-|v-1|^{1/3}\right|^{3}.~{}~{}~{}~{}$
(2.12)

Then, the equation satisfied by selecting the physical states
$\hat{\cal{H}}\Psi(v,\phi)=0$ can be cast in the form,

$\displaystyle\partial^{2}_{\phi}\Psi(v,\phi)$ $\displaystyle=$
$\displaystyle\frac{1}{B(v)}\Big{[}C^{+}(v)\Psi(v+4,\phi)-C^{o}(v)\Psi(v,\phi)$
(2.13)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+C^{-}(v)\Psi(v-4,\phi)\Big{]},$

where

$\displaystyle C^{+}(v)$ $\displaystyle\equiv$ $\displaystyle\frac{3\pi
KG}{8}|v+2|\left||v+1|-|v+3|\right|,$ $\displaystyle C^{-}(v)$
$\displaystyle\equiv$ $\displaystyle C^{+}(v-4),\;\;C^{o}(v)\equiv
C^{+}(v)+C^{-}(v).~{}~{}~{}~{}~{}~{}~{}$ (2.14)

This is the main result of LQC aps3 , which shows that: (i) It is a second
order quantum difference equation with uniform discreteness in volume, rather
than a simple differential equation, a direct consequence of the discrete
nature of loop quantum geometry. (ii) It provides the evolution of the quantum
cosmological wavefunction $\Psi(v,\phi)$, in which the scalar field serves as
a clock. Thus, once an initial state $\Psi(v,\phi_{0})$ is given at the
initial moment $\phi_{0}$, the study of the quantum dynamics of LQC can be
carried out. It is found that, instead of a big bang singularity, a quantum
bounce is generically produced, a result confirmed through extensive numerical
simulations numlsu-1 ; numlsu-2 ; numlsu-3 ; numlsu-4 ; numlsu-5  and an
exactly solvable model acs2010 . Using this model, one can compute the
probability for the quantum bounce which turns out to be unity for an
arbitrary superposition of wavefunctions craig-singh .

For the states sharply peaked around a classical solution, we can obtain
“effective” Friedmann and Raychaudhuri (FR) equations, by using the geometric
quantum mechanics in terms of the expectation values of the operators
$(\hat{b},\hat{v},\hat{\phi},\widehat{p_{\phi}})$,

$\displaystyle\dot{b}$ $\displaystyle=$
$\displaystyle\left\\{b,{{\cal{H}}}\right\\},\;\dot{v}=\left\\{v,{{\cal{H}}}\right\\},$
(2.15) $\displaystyle\dot{\phi}$ $\displaystyle=$
$\displaystyle\left\\{\phi,{{\cal{H}}}\right\\},\;\dot{p}_{\phi}=\left\\{p_{\phi},{{\cal{H}}}\right\\},$
(2.16)

which take the same forms as their classical ones, but all the quantities now
represent their expectation values,
$A_{I}\equiv\left<\widehat{{A}_{I}}\right>$. Then, it was found that the
effective Hamiltonian is given by aps3 ,

$\displaystyle{{\cal{H}}}_{\text{eff.}}=-\frac{3}{8\pi\gamma^{2}\bar{\mu}^{2}G}|p|^{1/2}\sin^{2}(\bar{\mu}c)+\frac{1}{2}|p|^{3/2}p_{\phi}^{2},$
(2.17)

which can also be expressed in terms of $v$ and $b$ via the relations
$v=|p|^{3/2}$ and $b=c/\sqrt{|p|}$. Then, from Eqs. (2.15) and (2.16) one can
find that the “effective” FR equations are given by,

$\displaystyle H^{2}$ $\displaystyle=$ $\displaystyle\frac{8\pi
G}{3}\rho\left(1-\frac{\rho}{\rho_{c}}\right),$ (2.18) $\displaystyle\dot{H}$
$\displaystyle=$ $\displaystyle-4\pi
G\left(\rho+P\right)\left(1-\frac{\rho}{\rho_{c}}\right),$ (2.19)

where

$\displaystyle H$ $\displaystyle\equiv$
$\displaystyle\frac{\dot{v}}{3v}=\frac{\dot{a}}{a},\quad\rho_{c}\equiv\frac{3}{8\pi\lambda
a^{2}\gamma^{2}G},$ $\displaystyle\rho$ $\displaystyle\equiv$
$\displaystyle\frac{\mathcal{H}_{M}}{v},\quad
P\equiv-\frac{\partial\mathcal{H}_{M}}{\partial v},$ (2.20)

and $v=v_{o}a^{3}$. Since $H^{2}$ cannot be negative, from Eq.(2.18) we can
see that we must have $\rho\leq\rho_{c}$, and at $\rho=\rho_{c}$ we have
$H^{2}=0$, that is, a quantum bounce occurs at this moment. When
$\rho\ll\rho_{c}$, the quantum gravity effects are negligible, whereby the
classical relativistic limit is obtained.

For a scalar field $\phi$ with its potential $V(\phi)$, we have

${\mathcal{H}_{M}}\equiv{\mathcal{H}_{\phi}}=v\left[\frac{p_{\phi}^{2}}{2v^{2}}+V(\phi)\right].$
(2.21)

Then, from Eq.(2.16) we find that

$\displaystyle\dot{\phi}$ $\displaystyle=$ $\displaystyle\frac{p_{\phi}}{v},$
(2.22) $\displaystyle\dot{p}_{\phi}$ $\displaystyle=$ $\displaystyle-
vV_{,\phi}(\phi).$ (2.23)

In the rest of this review, we shall consider only the states that are sharply
peaked around a classical solution, so the above “effective” descriptions are
valid, and the questions raised recently in KKL20  are avoided.

###  II.2 Effective Dynamics of mLQC-I

As mentioned in the introduction, an important open issue in LQC is its
connection with LQG engle . In particular, in LQC the spacetime symmetry is
first imposed (in the classical level), before the quantization process is
carried out. However, it is well-known that this is different from the general
process of LQG LQG , and as a result, different Hamiltonian constraints could
be resulted, hence resulting in different Planck scale physics. Though the
question of ambiguities in obtaining the Hamiltonian in LQG is still open,
based on some rigorous proposals by Thiemann thiemann , various attempts have
been carried out, in order to obtain deeper insights into the question.

One of the first attempts to understand this issue was made in YDM09 , in
which the Euclidean and Lorentzian terms given by Eqs.(2.5) and (2.6) are
treated differently, by closely following the actual construction of LQG. To
be more specific, in the full theory LQG , the extrinsic curvature in the
Lorentzian term (2.6) can be expressed in terms of the connection and the
volume as

$K^{i}_{a}=\frac{1}{\kappa\gamma^{3}}\\{A^{i}_{a},\\{\mathcal{H}_{\mathrm{grav}}^{(E)},V\\}\\},$
(2.24)

which once substituted back into Eq. (2.6) lead to an expression of
$\mathcal{H}_{\mathrm{grav}}^{(L)}$ different from that of
$\mathcal{H}_{\mathrm{grav}}^{(E)}$ in the standard LQC (see YDM09  for more
details). Correspondingly, one is able to obtain the following “effective”
Hamiltonian YDM09 ; lsw2018 ,

$\displaystyle\mathcal{H}^{\scriptscriptstyle{\mathrm{I}}}_{\text{eff.}}$
$\displaystyle=$ $\displaystyle\frac{3v}{8\pi
G\lambda^{2}}\Big{\\{}\sin^{2}(\lambda
b)-\frac{(\gamma^{2}+1)\sin^{2}(2\lambda b)}{4\gamma^{2}}\Big{\\}}$ (2.25)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\mathcal{H}_{M}.$

Hence, the Hamilton’s equations take the form,

$\displaystyle\dot{v}$ $\displaystyle=$ $\displaystyle\frac{3v\sin(2\lambda
b)}{2\gamma\lambda}\Big{\\{}(\gamma^{2}+1)\cos(2\lambda
b)-\gamma^{2}\Big{\\}},~{}~{}~{}$ (2.26) $\displaystyle\dot{b}$
$\displaystyle=$ $\displaystyle\frac{3\sin^{2}(\lambda
b)}{2\gamma\lambda^{2}}\Big{\\{}\gamma^{2}\sin^{2}(\lambda b)-\cos^{2}(\lambda
b)\Big{\\}}$ (2.27) $\displaystyle-4\pi G\gamma P,$

where $P$ represents the pressure defined in Eq.(II.1). Once the matter
Hamiltonian $\mathcal{H}_{M}$ is specified, together with the Hamiltonian
constraint,

$\mathcal{H}\approx 0,$ (2.28)

Eqs.(2.26) and (2.27) uniquely determine the evolution of the universe. Using
the non-graph changing regularization of the Hamiltonian thiemann , the
expectation values of the Hamiltonian operator yield the same “effective”
Hamiltonian of Eq.(2.25) to the leading order DL17 .

It has been shown in detail that the big bang singularity is generically
replaced by a quantum bounce when the energy reaches its maximum
$\rho_{c}^{\scriptscriptstyle{\mathrm{I}}}$ YDM09 ; DL17 ; lsw2018 ; lsw2018b
; lsw2019 , where

$\displaystyle\rho_{c}^{\scriptscriptstyle{\mathrm{I}}}\equiv\frac{\rho_{c}}{4(1+\gamma^{2})},$
(2.29)

and the universe is asymmetric with respect to the bounce, in contrast to LQC.

To write Eqs.(2.26)-(2.28) in terms of $H,\;\rho$ and $P$, it was found that
one must distinguish the pre- and post- bounce phases lsw2018 . In particular,
before the bounce, the modified FR equations take the form lsw2018 ,

$\displaystyle H^{2}$ $\displaystyle=$ $\displaystyle\frac{8\pi
G\alpha\rho_{\Lambda}}{3}\left(1-\frac{\rho}{\rho_{c}^{\scriptscriptstyle{\mathrm{I}}}}\right)\left[1+\left(\frac{1-2\gamma^{2}+\sqrt{1-\rho/\rho_{c}^{\scriptscriptstyle{\mathrm{I}}}}}{4\gamma^{2}\left(1+\sqrt{1-\rho/\rho_{c}^{\scriptscriptstyle{\mathrm{I}}}}\right)}\right)\frac{\rho}{\rho_{c}^{\scriptscriptstyle{\mathrm{I}}}}\right],$
(2.30) $\displaystyle\frac{\ddot{a}}{a}$ $\displaystyle=$
$\displaystyle-\frac{4\pi\alpha G}{3}\left(\rho+3P-2\rho_{\Lambda}\right)+4\pi
G\alpha
P\left(\frac{2-3\gamma^{2}+2\sqrt{1-\rho/\rho^{\scriptscriptstyle{\mathrm{I}}}_{c}}}{(1-5\gamma^{2})\left(1+\sqrt{1-\rho/\rho^{\scriptscriptstyle{\mathrm{I}}}_{c}}\right)}\right)\frac{\rho}{\rho^{\scriptscriptstyle{\mathrm{I}}}_{c}}$
(2.31) $\displaystyle-\frac{4\pi\alpha
G\rho}{3}\left[\frac{2\gamma^{2}+5\gamma^{2}\left(1+\sqrt{1-\rho/\rho^{\scriptscriptstyle{\mathrm{I}}}_{c}}\right)-4\left(1+\sqrt{1-\rho/\rho^{\scriptscriptstyle{\mathrm{I}}}_{c}}\right)^{2}}{(1-5\gamma^{2})\left(1+\sqrt{1-\rho/\rho^{\scriptscriptstyle{\mathrm{I}}}_{c}}\right)^{2}}\right]\frac{\rho}{\rho^{\scriptscriptstyle{\mathrm{I}}}_{c}},$

where

$\displaystyle\alpha\equiv\frac{1-5\gamma^{2}}{\gamma^{2}+1},\quad\rho_{\Lambda}\equiv\frac{\gamma^{2}\rho_{c}}{(1+\gamma^{2})(1-5\gamma^{2})}.$
(2.32)

As $\rho\ll\rho_{c}^{\scriptscriptstyle{\mathrm{I}}}$, Eqs.(2.30) and (2.31)
reduce, respectively, to

$\displaystyle H^{2}$ $\displaystyle\approx$ $\displaystyle\frac{8\pi\alpha
G}{3}\left(\rho+\rho_{\Lambda}\right),$ (2.33)
$\displaystyle\frac{\ddot{a}}{a}$ $\displaystyle\approx$
$\displaystyle-\frac{4\pi\alpha G}{3}\left(\rho+3P-2\rho_{\Lambda}\right).$
(2.34)

These are exactly the FR equations in GR for an ordinary matter field coupled
with a positive cosmological constant $\rho_{\Lambda}$, and a modified
Newton’s constant, $G_{\alpha}\equiv\alpha G$. For $\gamma\approx 0.2375$, we
have $\rho_{\Lambda}\approx 0.03\rho_{pl}$, which is of the same order as the
one deduced conventionally in quantum field theory for the vacuum energy in
our universe. In addition, we also have

$\left|\frac{G_{{\alpha}}}{G}-1\right|_{\gamma\approx 0.2375}\simeq
0.32>\frac{1}{8}.$ (2.35)

Finally, we want to emphasize that the minimal energy density of this branch,
for which the Hubble rate vanishes, turns out to be negative which can be
shown as $\rho_{\mathrm{min}}=-\frac{3}{8\pi G\lambda^{2}}\approx-0.023$. As a
result, the necessary condition to generate a cyclic universe in mLQC-I is the
violation of the weak energy condition which is in contrast to the cyclic
universes in LQC where the energy density is always non-negative ls2021 .

In the post-bounce phase ($t>t_{B}$), from Eqs.(2.26)-(2.28) we find that
lsw2018 ,

$\displaystyle H^{2}$ $\displaystyle=$ $\displaystyle\frac{8\pi
G\rho}{3}\left(1-\frac{\rho}{\rho_{c}^{\scriptscriptstyle{\mathrm{I}}}}\right)\Bigg{[}1+\frac{\gamma^{2}}{\gamma^{2}+1}\left(\frac{\sqrt{\rho/\rho_{c}^{\scriptscriptstyle{\mathrm{I}}}}}{1+\sqrt{1-\rho/\rho_{c}^{\scriptscriptstyle{\mathrm{I}}}}}\right)^{2}\Bigg{]},$
(2.36) $\displaystyle\frac{\ddot{a}}{a}$ $\displaystyle=$
$\displaystyle-\frac{4\pi G}{3}\left(\rho+3P\right)+\frac{4\pi
G\rho}{3}\left[\frac{\left(7\gamma^{2}+8\right)-4\rho/\rho_{c}^{\scriptscriptstyle{\mathrm{I}}}+\left(5\gamma^{2}14+8\right)\sqrt{1-\rho/\rho_{c}^{\scriptscriptstyle{\mathrm{I}}}}}{(\gamma^{2}+1)\left(1+\sqrt{1-\rho/\rho_{c}^{\scriptscriptstyle{\mathrm{I}}}}\right)^{2}}\right]\frac{\rho}{\rho_{c}^{\scriptscriptstyle{\mathrm{I}}}}$
(2.37) $\displaystyle+4\pi
GP\left[\frac{3\gamma^{2}+2+2\sqrt{1-\rho/\rho_{c}^{\scriptscriptstyle{\mathrm{I}}}}}{(\gamma^{2}+1)\left(1+\sqrt{1-\rho/\rho_{c}^{\scriptscriptstyle{\mathrm{I}}}}\right)}\right]\frac{\rho}{\rho_{c}^{\scriptscriptstyle{\mathrm{I}}}},$

from which we obtain

$\displaystyle\dot{H}$ $\displaystyle=$
$\displaystyle\frac{4G\pi(P+\rho)}{(1+\gamma^{2})}\left(2\gamma^{2}+2\frac{\rho}{\rho_{c}^{\scriptscriptstyle{\mathrm{I}}}}-3\gamma^{2}\sqrt{1-\frac{\rho}{\rho_{c}^{\scriptscriptstyle{\mathrm{I}}}}}-1\right).$

Therefore, regardless of the matter content, the super-inflation (starting at
the bounce) will always end at $\rho=\rho_{s}$, where

$\rho_{s}=\frac{\rho_{c}^{\scriptscriptstyle{\mathrm{I}}}}{8}\left(4-8\gamma^{2}-9\gamma^{4}+3\gamma^{2}\sqrt{8+16\gamma^{2}+9\gamma^{4}}\right),$
(2.39)

for which we have $\dot{H}(\rho_{s})=0$.

In the classical limit $\rho/\rho_{c}^{\scriptscriptstyle{\mathrm{I}}}\ll 1$,
Eqs. (2.36) and (2.37) reduce, respectively, to

$\displaystyle H^{2}$ $\displaystyle\approx$ $\displaystyle\frac{8\pi
G}{3}\rho,$ (2.40) $\displaystyle\frac{\ddot{a}}{a}$ $\displaystyle\approx$
$\displaystyle-\frac{4\pi G}{3}\left(\rho+3P\right),$ (2.41)

whereby the standard relativistic cosmology is recovered.

It is remarkable to note that in the pre-bounce phase the limit
$\rho/\rho_{c}^{\scriptscriptstyle{\mathrm{I}}}\ll 1$ leads to Eqs.(2.33) and
(2.34) with a modified Newtonian constant $G_{\alpha}$, while in the post-
bounce the same limits leads to Eqs.(2.40) and (2.41) but now with the precise
Newtonian constant $G$.

###  II.3 Effective Dynamics of mLQC-II

In LQG, the fundamental variables for the gravitational sector are the su(2)
Ashtekar-Barbero connection $A^{i}_{a}$ and the conjugate triad $E^{a}_{i}$.
When the Gauss and spatial diffeomorphism constraints are fixed, in the
homogeneous and isotropic universe the only relevant constraint is the
Hamiltonian constraint, from which we obtain the FR equations, as shown in the
previous section. The Hamiltonian in mLQC-II arises from the substitution

$K^{i}_{a}=\frac{A^{i}_{a}}{\gamma},$ (2.42)

in the Lorentzian term (2.6). Then, the following effective Hamiltonian is
resulted YDM09 ,

$\displaystyle\mathcal{H}^{\scriptscriptstyle{\mathrm{II}}}_{\mathrm{eff.}}$
$\displaystyle=$ $\displaystyle-\frac{3v}{2\pi
G\lambda^{2}\gamma^{2}}\sin^{2}\left(\frac{\lambda
b}{2}\right)\left\\{1+\gamma^{2}\sin^{2}\left(\frac{\lambda
b}{2}\right)\right\\}$ (2.43) $\displaystyle+\mathcal{H}_{M},$

from which we find that the corresponding Hamilton’s equations are given by,

$\displaystyle\dot{v}$ $\displaystyle=$ $\displaystyle\frac{3v\sin(\lambda
b)}{\gamma\lambda}\Big{\\{}1+\gamma^{2}-\gamma^{2}\cos\left(\lambda
b\right)\Big{\\}},$ (2.44) $\displaystyle\dot{b}$ $\displaystyle=$
$\displaystyle-\frac{6\sin^{2}\left(\frac{\lambda
b}{2}\right)}{\gamma\lambda^{2}}\Big{\\{}1+\gamma^{2}\sin^{2}\left(\frac{\lambda
b}{2}\right)\Big{\\}}-4\pi G\gamma P$ (2.45) $\displaystyle=$
$\displaystyle-4\pi G\gamma(\rho+P).$

It can be shown that the corresponding (modified) FR equations now read
lsw2018b ,

$\displaystyle H^{2}$ $\displaystyle=$ $\displaystyle\frac{8\pi
G\rho}{3}\left(1+\gamma^{2}\frac{\rho}{\rho_{c}}\right)\left(1-\frac{(\gamma^{2}+1)\rho}{\Delta^{2}\rho_{c}}\right),~{}~{}~{}$
(2.46) $\displaystyle\frac{\ddot{a}}{a}$ $\displaystyle=$
$\displaystyle-\frac{4\pi G}{3}\left(\rho+3P\right)-\frac{4\pi
GP\rho}{\Delta\rho_{c}}\left[3\left(\gamma^{2}+1\right)-2\Delta\right]$
$\displaystyle-\frac{4\pi
G\rho^{2}}{3\Delta^{2}\rho_{c}}\left[7\gamma^{2}-1+\left(5\gamma^{2}-3\right)\left(\Delta-1\right)-\frac{4\gamma^{2}\rho}{\rho_{c}}\right],$

where $\Delta\equiv 1+\sqrt{1+\gamma^{2}\rho/\rho_{c}}$. From these equations
we can see that now the quantum bounce occurs when
$\rho=\rho_{c}^{\scriptscriptstyle{\mathrm{II}}}$, at which we have $H=0$ and
$\ddot{a}>0$, where

$\rho_{c}^{\scriptscriptstyle{\mathrm{II}}}=4(\gamma^{2}+1)\rho_{c},$ (2.48)

which is different from the critical density $\rho_{c}$ in LQC as well as the
one $\rho_{c}^{{\scriptscriptstyle{\mathrm{I}}}}$ in mLQC-I. Therefore, the
big bang singularity is also resolved in this model, and replaced by a quantum
bounce at $\rho=\rho_{c}^{\scriptscriptstyle{\mathrm{II}}}$, similar to LQC
and mLQC-I, despite the fact that the bounce in each of these models occurs at
a different energy density. However, in contrast to mLQC-I, the evolution of
the universe is symmetric with respect to the bounce, which is quite similar
to the standard LQC model.

In addition, similar to the other two cases, now the bounce is accompanied by
a phase of super-inflation, i.e. $\dot{H}>0$, which ends at
$\dot{H}(\rho_{s})=0$, but now $\rho_{s}$ is given by,

$\rho_{s}=\frac{\rho_{c}}{8\gamma^{2}}\left(3(\gamma^{2}+1)\sqrt{1+2\gamma^{2}+9\gamma^{4}}+9\gamma^{4}+10\gamma^{2}-3\right).$
(2.49)

For $\gamma=0.2375$, we find
$\rho_{s}=0.5132\rho_{c}^{\scriptscriptstyle{\mathrm{II}}}$.

When $\rho\ll\rho_{c}^{\scriptscriptstyle{\mathrm{II}}}$, the modified FR
equations (2.46)-(II.3) reduce to,

$\displaystyle H^{2}$ $\displaystyle\approx$ $\displaystyle\frac{8\pi
G}{3}\rho,$ (2.50) $\displaystyle\frac{\ddot{a}}{a}$ $\displaystyle\approx$
$\displaystyle-\frac{4\pi G}{3}\left(\rho+3P\right),$ (2.51)

which are identical to those given in GR. Therefore, in this model, the
classical limit is obtained in both pre- and post-bounce when the energy
density $\rho$ is much lower than the critical one
$\rho_{c}^{\scriptscriptstyle{\mathrm{II}}}$.

###  II.4 Universal Properties of mLQC-I/II Models

To study further the evolution of the universe, it is necessary to specify the
matter content $\mathcal{H}_{M}$. For a single scalar field with its potential
$V(\phi)$, the corresponding Hamiltonian takes the form (2.21). As a result,
the Hamilton’s equations of the matter sector are given by Eqs.(2.22) and
(2.23).

The effective quantum dynamics of LQC, mLQC-I, and mLQC-II were studied in
detail recently in lsw2018b  for a single scalar field with various
potentials, including the chaotic inflation, Starobinsky inflation, fractional
monodromy inflation, non-minimal Higgs inflation, and inflation with an
exponential potential, by using dynamical system analysis. It was found that,
while several features of LQC were shared by the mLQC-I and mLQC-II models,
others belong to particular models. In particular, in the pre-bounce phase,
the qualitative dynamics of LQC and mLQC-II are quite similar, but are
strikingly different from that of mLQC-I. In all the three models, the non-
perturbative quantum gravitational effects always result in a non-singular
post-bounce phase, in which a short period of super-inflation always exists
right after the bounce, and is succeeded by the conventional inflation. The
latter is an attractor in the phase space for all the three models.

Figure 1: The evolution of the scale factor $a(t)$, the scalar field
$\phi(t)$, and the equation $w_{\phi}$ of state of the scalar field (from top
to bottom) in the post-bounce phase are depicted and compared among the three
modes, LQC (red solid curves), mLQC-I (blue dotted curves) and mLQC-II (green
dot-dashed curves), with the Starobinsky potential. In the last panel,
$w_{\phi}$ is defined via ${w}_{\phi}\equiv
P(\phi)/\rho(\phi)=[\dot{\phi}^{2}-2V(\phi)]/[\dot{\phi}^{2}+2V(\phi)]$. The
initial condition for the simulation is chosen at the bounce with
$\phi_{B}=-1.6~{}m_{pl},\dot{\phi}_{B}>0$ lsw2019 .

Moreover, similar to LQC ZhuA ; ZhuB  999In LQC, this universality was first
found for the quadratic and Starobinsky potentials ZhuA ; ZhuB  (see also
BCL19 ), and later was shown that they are also true for other potentials,
including the power-law potentials SSWW17 ; Shahalam18 , $\alpha$-attractor
potentials SSW18 ; SAMW20 , Monodromy potentials SSWW18 , warm inflation XW20
, Tachyonic inflation KX20a  and even in Brans-Dicke LQC SZW19 ; JMZ19 ., for
the initially kinetic energy dominated conditions,

$\frac{1}{2}\dot{\phi}_{B}^{2}\gg V(\phi_{B}),$ (2.52)

it was found that the evolution of the universe before the reheating is
universal. In particular, in the post-bounce phase (between the quantum bounce
and the reheating), the evolution can be uniquely divided into three phases:
bouncing, transition and slow-roll inflation, as shown in Fig. 1 for the
Starobinsky potential,

$V(\phi)=\frac{3m^{2}}{32\pi G}\left(1-e^{-\sqrt{16\pi G/3}\phi}\right)^{2}.$
(2.53)

For other potentials, similar results can be obtained, as long as at the
bounce the evolution of the universe is dominated by the kinetic energy of the
inflaton $w(\phi_{B})\simeq 1$ lsw2019 ; KX20b .

In each of these three phases, the evolutions of $a(t)$ and $\phi(t)$ can be
well approximated by analytical solutions. In particular, during the bouncing
phase, they are given by

$\displaystyle a(t)$ $\displaystyle=$ $\displaystyle\left[1+24\pi
G\rho_{c}^{\scriptscriptstyle{\mathrm{I}}}\left(1+\frac{A\gamma^{2}}{1+Bt}\right)t^{2}\right]^{1/6},$
$\displaystyle\phi(t)$ $\displaystyle=$
$\displaystyle\phi_{\text{B}}\pm\frac{m_{pl}\;\text{arcsinh}{\left(\sqrt{24\pi
G\rho_{c}^{\scriptscriptstyle{\mathrm{I}}}\left(1+\frac{C\gamma^{2}}{1+Dt}\right)}t\right)}}{\sqrt{12\pi
G\left(1+\frac{C\gamma^{2}}{1+Dt}\right)}},$

for mLQC-I model, where the parameters $A$, $B$, $C$ and $D$ are fixed through
numerical simulations. It was found that the best fitting is provided by
lsw2019 ,

$\displaystyle A=C=1.2,\quad B=6,\quad D=2.$ (2.55)

For the mLQC-II model, during the bouncing phase $a(t)$ and $\phi(t)$ are
given by

$\displaystyle a(t)$ $\displaystyle=$ $\displaystyle\left[1+24\pi
G\rho_{c}^{\scriptscriptstyle{\mathrm{II}}}\left(1+\frac{A\gamma^{2}}{1+Bt}\right)t^{2}\right]^{1/6},$
$\displaystyle\phi(t)$ $\displaystyle=$
$\displaystyle\phi_{\text{B}}\pm\frac{m_{pl}\;\text{arcsinh}{\left(\sqrt{24\pi
G\rho_{c}^{\scriptscriptstyle{\mathrm{II}}}\left(1+\frac{C\gamma^{2}}{1+Dt}\right)}t\right)}}{\sqrt{12\pi
G\left(1+\frac{C\gamma^{2}}{1+Dt}\right)}},$

but now with,

$A=2.5,\quad B=7,\quad C=D=2.$ (2.57)

In the transition and slow-roll inflationary phases, the functions $a(t)$ and
$\phi(t)$ were given explicitly in lsw2019 .

For the initially potential energy dominated cases,

$\frac{1}{2}\dot{\phi}_{B}^{2}\ll V(\phi_{B}),$ (2.58)

it was found that such universalities are lost. In particular, for the
Starobinsky potential, the potential energy dominated bounce cannot give rise
to any period of inflation for both mLQC-I and mLQC-II models, quite similar
to what happens in LQC bonga_inflation_2016 ; bonga_phenomenological_2016 .

###  II.5 Probabilities of the Slow-roll Inflation in mLQC-I/II Models

To consider the probability of the slow-roll inflation in the modified LQC
models, let us start with the phase space $\mathbb{S}$ of the modified
Friedmann and Klein-Gordon equations, which now is four-dimensional (4D), and
consists of the four variables, $(v,\;b)$ and $(\phi,\;p_{\phi})$, from the
gravitational and matter sectors, respectively. Using the canonical
commutation relations, the symplectic form on the 4D phase space is given by
SVV06 ; ZL07 ; as2011 ; CK11 ; corichi-sloan ; bedic ; LB13 ; CZ15 ,

$\Omega=dp_{\phi}\wedge d\phi+\frac{dv\wedge db}{4\pi G\gamma}.$ (2.59)

However, after taking the effective Hamiltonian constraint into account,

$\mathcal{C}=16\pi
G\left\\{{\cal{H}}_{grav}(v,b)+\frac{p_{\phi}^{2}}{2v}+vV(\phi)\right\\}\simeq
0,$ (2.60)

where $``\simeq"$ means that the equality holds only on $\bar{\Gamma}$, we can
see that the 4D phase space $\mathbb{S}$ reduces to a three-dimensional (3D)
hypersurface $\bar{\Gamma}$.

On the other hand, the phase space $\mathbb{S}$ is isomorphic to a
2-dimensional (2D) gauge-fixed surface $\hat{\Gamma}$ of $\bar{\Gamma}$, which
is intersected by each dynamical trajectory once and only once as2011 . From
the FR equations, it can be shown that for both mLQC-I and mLQC-II the
variable $b$ satisfies the equation lsw2018 ; lsw2018b ; lsw2019 ,

$\displaystyle\dot{b}=-4\pi G\gamma(\rho+P).$ (2.61)

For any given matter field that satisfies the weak energy condition HE73 , we
have $\rho+P>0$, so the function $b$ is monotonically decreasing. Then, a
natural parameterization of this 2D surface is $b=$ constant, say, $b_{0}$.
Hence, using the Hamiltonian constraint (2.60) we find

$p^{\mathrm{A}}_{\phi}=v\left\\{-2\left[\hat{\cal{H}}_{grav}^{\mathrm{A}}+V(\phi)\right]\right\\}^{1/2},$
(2.62)

where $\mathrm{A}=\mathrm{I},\mathrm{II}$, and

$\hat{\cal{H}}_{grav}^{\mathrm{A}}\equiv
v^{-1}{\cal{H}}_{grav}^{\mathrm{A}}(v,b_{0}).$ (2.63)

On the other hand, from Eqs.(2.25) and (2.43) we find that
$\hat{\cal{H}}_{grav}^{\mathrm{A}}=\hat{\cal{H}}_{grav}^{\mathrm{A}}(b_{0})=$
constant on $\hat{\Gamma}$. Thus, we find

$\displaystyle\left.dp^{\mathrm{A}}_{\phi}\right|_{\hat{\Gamma}}=\frac{p^{\mathrm{A}}_{\phi}}{v}dv-\frac{v^{2}V_{,\phi}}{p_{\phi}}d\phi.$
(2.64)

Inserting this expression into Eq.(2.59), we find that the pulled-back
symplectic structure $\hat{\Omega}$ reads

$\displaystyle\left.\hat{\Omega}^{\mathrm{A}}\right|_{\hat{\Gamma}}=\left\\{-2\left[\hat{\cal{H}}_{grav}^{\mathrm{A}}(b_{0})+V(\phi)\right]\right\\}^{1/2}d\phi\wedge
dv,$ (2.65)

from which we find that the Liouville measure $d\hat{\mu}_{L}$ on
$\hat{\Gamma}$ is given by

$\displaystyle
d\hat{\mu}^{\mathrm{A}}_{L}=\left\\{-2\left[\hat{\cal{H}}_{grav}^{\mathrm{A}}(b_{0})+V(\phi)\right]\right\\}^{1/2}d\phi
dv.$ (2.66)

Note that $d\hat{\mu}^{A}_{L}$ does not depend on $v$, so that the integral
with respect to $dv$ is infinite. However, this divergency shall be cancelled
when calculating the probability, as it will appear in both denominator and
numerator. Therefore, the measure for the space of physically distinct
solutions can be finally taken as

$d\omega^{\scriptscriptstyle{\mathrm{A}}}=\left\\{-2\left[\hat{\cal{H}}_{grav}^{\mathrm{A}}(b_{0})+V(\phi)\right]\right\\}^{1/2}d\phi,$
(2.67)

so that the 2D phase space $\hat{\Gamma}$ is further reduced to an interval
$\phi\in\left(\phi_{\mathrm{min}},\phi_{\mathrm{max}}\right)$. It should be
noted that such a defined measure depends explicitly on $b_{0}$, and its
choice in principle is arbitrary. However, in loop cosmology there exists a
preferred choice, which is its value at the quantum bounce
$b_{0}=b\left(t_{B}\right)$ as2011 . With such a choice, the probability of
the occurrence of an event $E$ becomes

$P(E)=\frac{1}{{\cal{D}}}\int_{\mathcal{I}(E)}{\left\\{-2\left[\hat{\cal{H}}_{grav}^{\mathrm{A}}(b_{B})+V(\phi)\right]\right\\}^{1/2}d\phi},$
(2.68)

where $\mathcal{I}(E)$ is the interval on the $\phi_{B}$-axis, which
corresponds to the physically distinct initial conditions in which the event
$E$ happens, and ${\cal{D}}$ is the total measure

${\cal{D}}\equiv\int_{\phi_{\mathrm{min}}}^{\phi_{\mathrm{max}}}{\left\\{-2\left[\hat{\cal{H}}_{grav}^{\mathrm{A}}(b_{B})+V(\phi)\right]\right\\}^{1/2}d\phi}.$
(2.69)

Once the probability is properly defined, we can calculate it in different
models of the modified LQCs. In LQC as2011 , the calculations were carried out
for the quadratic potential. In order to compare the results obtained in
different models, let us consider the same potential. Then, for the mLQC-I
model it was found that lsw2019

$\displaystyle\sin\left(\lambda
b_{B}^{\scriptscriptstyle{\mathrm{I}}}\right)=\sqrt{\frac{1}{2\gamma^{2}+2}},\quad\sin\left(2\lambda
b_{B}^{\scriptscriptstyle{\mathrm{I}}}\right)=\frac{\sqrt{2\gamma^{2}+1}}{\gamma^{2}+1},$
$\displaystyle
p^{\scriptscriptstyle{\mathrm{I}}}_{\phi}=v\left(2\rho^{\scriptscriptstyle{\mathrm{I}}}_{c}-2V\right)^{\frac{1}{2}},\quad
d\omega^{\scriptscriptstyle{\mathrm{I}}}=\left(2\rho^{\scriptscriptstyle{\mathrm{I}}}_{c}-2V\right)^{\frac{1}{2}}d\phi,~{}~{}~{}~{}~{}~{}~{}~{}$
(2.70)

so that the probability for the desired slow-roll not to happen is,

$P^{\scriptscriptstyle{\mathrm{I}}}(\text{not
realized})\lesssim\frac{\int^{0.917}_{-5.158}d\omega^{\scriptscriptstyle{\mathrm{I}}}}{\int^{\phi^{\scriptscriptstyle{\mathrm{I}}}_{\text{max}}}_{-\phi^{\scriptscriptstyle{\mathrm{I}}}_{\text{max}}}{d\omega^{\scriptscriptstyle{\mathrm{I}}}}}\simeq
1.12\times 10^{-5},$ (2.71)

where $\phi^{\scriptscriptstyle{\mathrm{I}}}_{\text{max}}=3.49\times
10^{5}\;m_{pl}$.

In mLQC-II, following a similar analysis, it can be shown that the probability
for the desired slow-roll not to happen is lsw2019 ,

$P^{\scriptscriptstyle{\mathrm{II}}}(\text{not realized})\lesssim 2.62\times
10^{-6}.$ (2.72)

Note that in LQC the probability for the desired slow roll inflation not to
occur is as2011 ,

$P^{\scriptscriptstyle{\mathrm{LQC}}}(\text{not realized})\lesssim 2.74\times
10^{-6},$ (2.73)

which is smaller than that for mLQC-I and slightly larger than the one for
mLQC-II. However, it is clear that the desired slow-roll inflation is very
likely to occur in all the models, including the two modified LQC ones.

##  III Primordial power spectra of Modified LQCs in dressed metric approach

As mentioned above, in the literature there exist several approaches to
investigate the inhomogeneities of the universe. Such approaches can be
generalized to the modified LQC models, including mLQC-I and mLQC-II. In this
section we shall focus ourselves on cosmological perturbations in the
framework of mLQCs following the dressed metric approach aan2012 ; aan2013 ;
aan2013b , while in the next section we will be following the hybrid approach
mm2012 ; mm2013 ; gmmo2014 ; gbm2015 ; mo2016 . We shall restrict ourselves to
the effective dynamics, as we did with the homogeneous background in the last
section. Such investigations in general include two parts: (a) the initial
conditions; and (b) the dynamical evolutions of perturbations. In the
framework of effective dynamics, the latter is a second-order ordinary
differential equation in the momentum space, so in principle once the initial
conditions are given, it uniquely determines the cosmological (scalar and
tensor) perturbations.

However, the initial conditions are a subtle issue, which is true not only in
LQC but also in mLQCs. This is mainly because that in general there does not
exist a preferred initial time and state for a quantum field in an arbitrarily
curved space-time BD82 . If the universe is sufficiently smooth and its
evolution is sufficiently slow, so that the characteristic scale of
perturbations is much larger than the wavelengths of all the relevant modes, a
well justified initial state can be defined: the BD vacuum. This is precisely
the initial state commonly adopted in GR at the beginning of the slow-roll
inflation, in which all the relevant perturbation modes are well inside the
comoving Hubble radius DB09  [cf. Fig.2].

However, in LQC/mLQCs, especially near the bounce, the evolution of the
background is far from “slow”, and its geometry is also far from the de
Sitter. In particular, for the perturbations during the bouncing phase, the
wavelengths could be larger, equal, or smaller than the corresponding
characteristic scale, as it can be seen, for example, from Fig. 5. Thus, it is
in general impossible to assume that the universe is in the BD vacuum at the
bounce aan2013b ; d1 ; d1b ; ZhuA ; ZhuB . Therefore, in the following let us
first elaborate in more details about the subtle issues regarding the initial
conditions.

###  III.1 Initial Conditions for Cosmological Perturbations

The initial conditions for cosmological perturbations in fact consists of two
parts: when and which? However, both questions are related to each other. In
LQC literature, for cosmological perturbations, two different moments have
been chosen so far in the dressed and hybrid approaches: (i) the remote past
in the contracting phase lsw2020  and (ii) the bounce aan2013b ; d1 ; d1b . To
see which conditions we need to impose at a given moment, let us first recall
how to impose the initial conditions in GR, in which the scalar perturbations
are governed by the equation,

$\displaystyle
v_{k}^{\prime\prime}+\left(k^{2}-\frac{z^{\prime\prime}}{z}\right)v_{k}=0,$
(3.1)

where $k$ denotes the comoving wave number, and $z\equiv a\dot{\delta\phi}/H$,
with $\delta\phi$ being the scalar field perturbations,
$\phi=\bar{\phi}(t)+\delta\phi(t,x)$. A prime denotes a derivative with
respect to the conformal time $\eta$, while an over dot denotes a derivative
with respect to the cosmic time $t$, where $d\eta=dt/a(t)$. The standard
choice of the initial sate is the Minkowski vacuum of an incoming observer in
the far past, $k\gg aH$ [cf. Fig. 2]. In this limit, Eq.(3.1) becomes
$v_{k}^{\prime\prime}+k^{2}v_{k}=0$, which has the solution,

$v_{k}\simeq\frac{\alpha_{k}}{\sqrt{2k}}e^{-ik\eta}+\frac{\beta_{k}}{\sqrt{2k}}e^{ik\eta},$
(3.2)

where $\alpha_{k}$ and $\beta_{k}$ are two integration constants, and must
satisfy the normalized condition,

$v_{k}^{*}v_{k}^{\prime}-{v_{k}^{*}}^{\prime}v_{k}=-i.$ (3.3)

If we further require the vacuum to be the minimum energy state, a unique
solution exists, which is given by $\alpha_{k}=1,\;\beta_{k}=0$, that is,

$\lim_{k\gg aH}v_{k}\simeq\frac{1}{\sqrt{2k}}e^{-ik\eta},$ (3.4)

which is often referred to as the BD vacuum DB09 .

Figure 2: The evolution of the comoving Hubble radius $\ln(L_{H})$ vs $\ln(a)$
in GR, where $t_{i}$ denotes the moment of the onset of the slow-roll
inflation, $t_{*}$ the horizon crossing time of a mode with the wavenumber
$k$, and $t_{\text{end}}$ the moment that the slow-roll inflation ends.

Consider the de Sitter space as the background, we have $a(\eta)=1/(-\eta H)$,
and $z^{\prime\prime}/z=a^{\prime\prime}/a=2/L_{H}^{2}$, where $L_{H}\equiv
1/(aH)=-\eta$ is the corresponding comoving Hubble radius. Then, Eq.(3.1)
reads,

$\displaystyle
v_{k}^{\prime\prime}+\left(\frac{1}{\lambda^{2}}-\frac{2}{L_{H}^{2}}\right)v_{k}=0,$
(3.5)

where $\lambda\;(\equiv 1/k)$ denotes the comoving wavelength. The above
equation has the following exact solutions,

$v_{k}=\frac{\alpha_{k}}{\sqrt{2k}}e^{-ik\eta}\left(1-\frac{i}{k\eta}\right)+\frac{\beta_{k}}{\sqrt{2k}}e^{ik\eta}\left(1+\frac{i}{k\eta}\right).$
(3.6)

It is clear that on scales much smaller than the comoving Hubble radius
($\lambda\ll L_{H}$), $v_{k}$ is oscillating with frequency $k$ and constant
amplitude, given by Eq.(3.2). Then, setting $(\alpha_{k},\;\beta_{k})=(1,0)$
we find that Eq. (3.6) reduces to

$v_{k}=\frac{1}{\sqrt{2k}}e^{-ik\eta}\left(1-\frac{i}{k\eta}\right).$ (3.7)

Note that if the initial time $t_{i}$ is chosen sufficiently small, i.e.,
$t_{i}\ll t_{\text{end}}$ or $|k\eta|\gg 1$, all the modes are inside the
comoving Hubble radius $L_{H}$ [cf. Fig. 2], and the BD vacuum (3.4) becomes a
natural choice.

However, on the scales much larger than the comoving Hubble radius
($\lambda\gg L_{H}$), the $k^{2}$ term is negligible compared to the squeezing
term, $z^{\prime\prime}/z$, and as a result, the fluctuations will stop
oscillating and the amplitude of $v_{k}$ starts to increase, yielding

$v_{k}\simeq z(\eta).$ (3.8)

As shown in Fig. 2, if the initial time $t_{i}$ is chosen to be sufficiently
early, all the currently observed modes $k_{\text{ph}}\in(0.1,1000)\times
k_{0}^{*}$ will be well inside the comoving Hubble radius at $t=t_{i}$, so the
mode function $v_{k}$ can be well approximately given by Eq.(3.4), which is
the well-known zeroth order adiabatic state, where $k_{0}^{*}=0.002$ Mpc-1 and
$k_{\text{ph}}(t)\equiv k/a(t)$ cosmo .

In modified LQC models, the mode function $v_{k}$ satisfies the following
modified equation,

$\displaystyle v_{k}^{\prime\prime}+\Omega_{\text{tot}}^{2}(\eta,k)v_{k}=0,$
(3.9)

where $\Omega_{\text{tot}}^{2}(\eta,k)$ depends on the homogeneous background
and the inflation potential $V(\phi)$, so it is model-dependent. Therefore,
the choice of the initial conditions will depend on not only the modified LQC
models to be considered but also the moment at which the initial conditions
are imposed.

One of the main reasons to choose the remote past in the contracting phase as
the initial time for perturbations is that at such time either the background
is well described by the de Sitter space (mLQC-I) or the expansion factor
$a(t)$ becomes so large that the curvature of the background is negligible
(mLQC-II and LQC), so imposing the BD vacuum for mLQC-II and LQC and the de
Sitter state given by Eq.(3.7) for mLQC-I at this moment is well justified. It
should be noted that the reason to refer to the state described by Eq.(3.7) as
the de Sitter state is the following: In the slow-roll inflation, the
homogeneous and isotropic universe is almost de Sitter, as the Hubble
parameter $H\equiv\dot{a}/a$ is almost a constant, so we have $a(\eta)\simeq
1/(-H\eta)$. For $t_{i}\ll t_{\text{end}}$ we have $a(\eta)\ll 1$, and $|\eta
k|\simeq|H\eta|\gg 1$, so the choice $(\alpha_{k}\;\beta_{k})=(1,0)$ will lead
Eq.(3.7) directly to Eq.(3.4) at the onset of the slow-roll inflation [cf.
Fig. 2]. However, in the deep contracting phase of the same de Sitter space,
now the universe is very large, that is, $a(\eta)\gg 1$, so we must have
$|H\eta|\ll 1$ and $|\eta k|\simeq|k/H_{\Lambda}|a^{-1}(\eta)$, which can be
very small in sufficiently early times of the contracting phase, so the terms
$i/(k\eta)$ in Eq.(3.6) now cannot be neglected. To distinguish this case from
the one described by Eq.(3.4), In this review we refer the state described by
Eq.(3.7) with the term $i/(k\eta)$ not being negligible as the de Sitter
state, while the state described by Eq.(3.4) is still called the BD vacuum
state, or simply the BD vacuum.

On the other hand, if the initial time is chosen to be at the bounce, cautions
must be taken on what initial conditions can be imposed consistently. In
particular, if at this moment some modes are inside the comoving Hubble radius
and others are not, it is clear that in this case imposing the BD vacuum at
the bounce will lead to inconsistencies. In addition, there also exist the
cases in which particle creation in the contracting phase is not negligible,
then it is unclear how a BD vacuum can be imposed at the bounce, after the
universe is contracting for such a long time before the bounce. Thus, in these
cases other initial conditions need to be considered, such as the fourth-order
adiabatic vacuum aan2013b ; d1 ; d1b ; ZhuA ; ZhuB .

With the above in mind, in the following we turn to consider power spectra of
the cosmological perturbations.

###  III.2 Power Spectra of Cosmological Perturbations

Since the evolutions of the effective dynamics of the homogeneous backgrounds
for mLQC-I and mLQC-II are different, in this subsection let us first consider
the case of mLQC-I and then mLQC-II. To compare the results with those
obtained in LQC, at the end of this subsection, we also discuss the LQC case.

####  III.2.1 mLQC-I

For mLQC-I, the power spectrum of the cosmological scalar perturbations was
first studied in IA19  for the quadratic $\phi^{2}$ potential, and re-examined
in lsw2020 . In the terminology used in IA19 , it was found that the
corresponding mode function $v_{k}(\equiv q_{k}/a)$ satisfies Eq.(3.9) with

$\displaystyle\Omega_{\text{tot}}^{2}$ $\displaystyle=$ $\displaystyle
k^{2}-\frac{a^{\prime\prime}}{a}+\mathfrak{A}_{-},\quad r\equiv\frac{24\pi
G\dot{\phi}^{2}}{\rho},$ $\displaystyle\mathfrak{A}_{-}$ $\displaystyle\equiv$
$\displaystyle
a^{2}\left[V(\phi)r-2V_{,\phi}(\phi)\sqrt{r}+V_{,\phi\phi}(\phi)\right],$
(3.10)

where $r=24\pi G\dot{\phi}^{2}/\rho$ and $V(\phi)$ denotes the scalar field
potential.

It should be noted that, when generalizing the classical expression of the
function $\mathfrak{A}_{-}$ defined in Eq.(III.2.1) to its corresponding
quantum mechanics operator, there exists ambiguities. In fact, classically
$\mathfrak{A}_{-}$ only coincides with $\Omega^{2}_{Q}$ in the expanding
phase. The latter is a function of the phase space variables which is
explicitly given by aan2013b ; abs2018 ; IA19 ; lsw2020 ,

$\Omega^{2}_{Q}=3\kappa\frac{p^{2}_{\phi}}{a^{4}}-18\frac{p^{4}_{\phi}}{a^{6}\pi^{2}_{a}}-12a\frac{p_{\phi}}{\pi_{a}}V_{,\phi}+a^{2}V_{,\phi\phi},$
(3.11)

where $\pi_{a}$ is the moment conjugate to $a$, and given by one of Hamilton’s
dynamical equations,

$\pi_{a}=-\frac{3a^{2}}{4\pi G}H,$ (3.12)

with the choice of the lapse function $N=1$. Therefore,
$\pi_{a}<0\;(\pi_{a}>0)$ corresponds to $H>0\;(H<0)$. At the quantum bounce we
have $H(t_{B})=0$, so that $\pi_{a}(t_{B})=0$. Then, $\Omega^{2}_{Q}$ defined
by Eq.(3.11) diverges at the bounce. Hence, from the Friedmann equation,
$H^{2}=(8\pi G/3)\rho$, we find that

$\frac{1}{\pi_{a}^{2}}=\frac{2\pi
G}{3a^{4}\rho},\quad\frac{1}{\pi_{a}}=\pm\sqrt{\frac{2\pi G}{3a^{4}\rho}},$
(3.13)

where “-” corresponds to $H>0$, and “+” to $H<0$. Then, a direct
generalization leads to lsw2020 ,

$\Omega^{2}_{Q}=\begin{cases}\mathfrak{A}_{-},&H\geq
0,\cr\mathfrak{A}_{+},&H\leq 0,\cr\end{cases}$ (3.14)

where

$\mathfrak{A}_{\pm}\equiv a^{2}\left[V(\phi)r\pm
2V_{,\phi}(\phi)\sqrt{r}+V_{,\phi\phi}(\phi)\right].$ (3.15)

It should be noted that in IA19  only the function $\mathfrak{A}_{-}$ was
chosen over the whole process of the evolution of the universe. The same
choice was also adopted in abs2018 ; AKS20 .

In addition, $\mathfrak{A}$ defined by Eq.(3.13) is not continuous across the
bounce, as the coefficient $2V_{,\phi}(\phi)\sqrt{r}$ in general does not
vanish at the bounce. In ZhuA ; ZhuB ; nbm2018  $\mathfrak{A}_{-}$ appearing
in Eq.(III.2.1) was replaced by $U(\phi)(\equiv\Omega^{2}_{+})$ over the whole
process of the evolution of the homogeneous universe, where

$\Omega^{2}_{\pm}\equiv a^{2}\left[\mathfrak{f}^{2}V(\phi)\pm
2\mathfrak{f}V_{,\phi}(\phi)+V_{,\phi\phi}(\phi)\right],$ (3.16)

and $\mathfrak{f}\equiv(24\pi G/\rho)^{1/2}\;\dot{\phi}$.

Another choice was introduced in lsw2020 , which was motivated from the
following considerations. The functions $\Omega^{2}_{\pm}$ defined above are
not continuous across the bounce, quite similar to $\mathfrak{A}_{\pm}$.
However, if we introduce the quantity $\Omega^{2}$ as,

$\Omega^{2}=a^{2}\left[\mathfrak{f}^{2}V(\phi)+2\Theta(b)\mathfrak{f}V_{,\phi}(\phi)+V_{,\phi\phi}(\phi)\right],$
(3.17)

to replace $\mathfrak{A}_{-}$ in Eq.(III.2.1), it could be continuous across
the bounce by properly choosing $\Theta(b)$. In particular, the variable
$b(t)$ satisfies Eq.(2.61) lsw2018 ; lsw2018b ; lsw2019  101010It is
interesting to note that Eq.(2.61) holds not only in mLQC-I, but also in LQC
and mLQC-II., from which we can see that $b(t)$ is always a monotonically
decreasing function for any matter that satisfies the weak energy condition
HE73 . Moreover, one can construct a step-like function of $b$ with the bounce
as its symmetry axis lsw2018 ; lsw2018b ; lsw2019 . Therefore, if we define
$\Theta(b)$ as

$\displaystyle\Theta(b)=1-2\left(1+\gamma^{2}\right)\sin^{2}\left(\lambda
b\right),$ (3.18)

it behaves precisely as a step function, so that $\Omega^{2}$ smoothly
connects $\Omega_{\pm}^{2}$ across the bounce, as shown in Fig. 3.

Figure 3: The potential terms $\Omega^{2}_{+}$ and $\Omega^{2}_{-}$ are
compared with their smooth extension $\Omega^{2}$ across the bounce in mLQC-I
for the quadratic potential $V(\phi)=\frac{1}{2}m^{2}\phi^{2}$ lsw2020 .

Figure 4: The potential terms $\Omega^{2}$ and $\Omega^{2}_{\mathrm{eff}}$ are
compared with the curvature term $a^{\prime\prime}/a$ in mLQC-I near the
bounce and the preinflationary regime for the quadratic potential
$V(\phi)=\frac{1}{2}m^{2}\phi^{2}$ lsw2020 .

In addition to the above choices, motivated by the hybrid approach mm2012 ;
mm2013 ; gmmo2014 ; gbm2015 ; mo2016 , the following replacements for
$\pi_{a}^{-2}$ and $\pi_{a}^{-1}$ in Eq.(3.11) were also introduced in lsw2020
,

$\displaystyle\frac{1}{\pi^{2}_{a}}$ $\displaystyle\rightarrow$
$\displaystyle\frac{64\pi^{2}G^{2}\lambda^{2}\gamma^{2}}{9a^{4}{\cal{D}}},$
(3.19) $\displaystyle\frac{1}{\pi_{a}}$ $\displaystyle\rightarrow$
$\displaystyle-\frac{8\pi G\lambda\gamma\Theta(b)}{3a^{2}{\cal{D}}^{1/2}},$
(3.20)

where

$\displaystyle{\cal{D}}\equiv\left(1+\gamma^{2}\right)\sin^{2}\left(2\lambda
b\right)-4\gamma^{2}\sin^{2}\left(\lambda b\right).$ (3.21)

Such obtained $\Omega^{2}_{Q}$ was referred to as $\Omega_{\text{eff}}^{2}$ in
lsw2020 , and in Fig. 4, we show the differences among $\Omega^{2}$,
$\Omega_{\text{eff}}^{2}$ and the quantity $a^{\prime\prime}/a$, from which
one can see that the term $a^{\prime\prime}/a$ dominates the other two terms
over the whole range $t/t_{pl}\in\left(-8,10^{4}\right)$.

To study the effects of the curvature term, let us first introduce the
quantity,

$k_{B}^{\scriptscriptstyle{\mathrm{I}}}=\left.\left(\frac{a^{\prime\prime}}{a}\right)^{1/2}\right|_{t=t_{B}}\approx
1.60,$ (3.22)

which is much larger than other two terms $\Omega^{2}$ and
$\Omega^{2}_{\mathrm{eff}}$, where $\Omega^{2}(t_{B})=1.75\times 10^{-10}$ and
$\Omega^{2}_{\mathrm{eff}}(t_{B})=0.006$. Therefore, the differences between
$\Omega^{2}$ and $\Omega^{2}_{\mathrm{eff}}$ near the bounce are highly
diluted by the background. On the other hand, in the post-bounce phase,
$\Omega^{2}$ and $\Omega^{2}_{\mathrm{eff}}$ coincide after $t/t_{pl}\simeq
10^{4}$, while near the onset of the inflation their amplitudes first become
almost equal to that of the curvature term, and then quickly exceeds it during
the slow-roll inflation, as we can see from Fig. 4.

From Fig. 4 we can also see that the difference between $\Omega^{2}$ and
$\Omega^{2}_{\mathrm{eff}}$ lies mainly in the region near the bounce.
However, as the curvature term $a^{\prime\prime}/a$ overwhelmingly dominates
in this region, it is usually expected that the impact of the different
choices of $\Omega^{2}$ on the power spectrum might not be very large aan2013b
; abs2018 ; IA19 . However, in lsw2020 , it was found that the relative
difference in the magnitude of the power spectrum in the IR and oscillating
regimes could be as large as $10\%$, where the relative difference is defined
as,

${\cal{E}}\equiv
2\left|\frac{{\cal{P}}_{1}-{\cal{P}}_{2}}{{\cal{P}}_{1}+{\cal{P}}_{2}}\right|.$
(3.23)

However, the power spectra obtained from $\Omega^{2}$ and $\Omega^{2}_{\pm}$
are substantially different even in the UV regime due to the (tiny) difference
between $\Omega^{2}_{\pm}$ at the bounce, see Fig. 14 given in lsw2020 . In
fact, the difference is so large that the power spectrum calculated from
$\Omega^{2}_{\pm}$ is essentially already ruled out by current observations.

With the clarification of the ambiguities caused by the quantum mechanical
generalization of the function $\mathfrak{A}_{-}$ defined in Eq.(III.2.1), now
let us turn to the issue of the initial conditions, for which we consider only
two representative potentials, the quadratic and Starobinsky, given explicitly
by

$V=\begin{cases}\frac{1}{2}m^{2}\phi^{2},&\text{quadratic},\cr\frac{3m^{2}}{32\pi
G}\left(1-e^{-\sqrt{16\pi
G/3}\;\phi}\right)^{2},&\text{Starobinsky}.\cr\end{cases}$ (3.24)

In mLQC-I, the evolution of the effective (quantum) homogeneous universe is
asymmetric with respect to the bounce DL17 ; lsw2018 ; lsw2018b ; lsw2019 . In
particular, before the bounce ($t<t_{B}$), the universe is asymptotically de
Sitter, and only very near the bounce (about several Planck seconds), the
Hubble parameter $H$ which is negative in the pre-bounce regime suddenly
increases to zero at the bounce. Then, the universe enters a very short super-
acceleration phase $\dot{H}>0$ (super-inflation) right after the bounce, which
lasts until $\rho\simeq\rho_{c}^{\scriptscriptstyle{\mathrm{I}}}/2$, where
$\dot{H}(t)_{\rho\simeq\frac{1}{2}\rho_{c}^{\scriptscriptstyle{\mathrm{I}}}}=0$.
Afterwards, for a kinetic energy dominated bounce $\dot{\phi}^{2}_{B}\gg
V(\phi_{B})$, it takes about $10^{4}\sim 10^{6}$ Planck seconds before
entering the slow-roll inflation lsw2018 ; lsw2018b ; lsw2019 . Introducing
the quantity,

$\displaystyle\lambda_{H}^{2}\equiv\frac{a}{{a^{\prime\prime}}-a\mathfrak{A}_{-}}=-\frac{1}{m_{\text{eff}}},$
(3.25)

where $m_{\text{eff}}$ is the effective mass of the modes, from Eq.(III.2.1)
we find that

$\Omega_{\text{tot}}^{2}=\frac{1}{\lambda^{2}}-\frac{1}{\lambda_{H}^{2}}=\begin{cases}>0,&\lambda_{H}^{2}>\lambda^{2},\cr<0,&0<\lambda_{H}^{2}<\lambda^{2},\cr>0,&\lambda_{H}^{2}<0,\cr\end{cases}$
(3.26)

where $\lambda\;(\equiv k^{-1})$ denotes the comoving wavelength of the mode
$k$, as mentioned above. Note that such a defined quantity $\lambda_{H}^{2}$
becomes negative when the effective mass is positive. In Fig. 5, we plot
$\lambda_{H}^{2}$ schematically for the quadratic and the Starobinsky
potentials with the mass of the inflaton set to $1.21\times
10^{-6}\;m_{\mathrm{pl}}$ and $2.44\times 10^{-6}\;m_{\mathrm{pl}}$
respectively. The initial conditions for the background evolution are set as
follows: for the quadratic potential, the inflaton starts with a positive
velocity on the right wing of the potential and for the Starobinsky potential
the inflaton is released from the left wing of the potential with a positive
velocity. For both potentials, the initial conditions are set at the bounce
which is dominated by the kinetic energy of the inflaton field. The same mass
parameters and similar initial conditions are also used in the following
figures where the comoving Hubble radius is plotted schematically. In Fig. 5,
the moments $t_{H}$ and $t_{i}$ are defined, respectively, by
$a^{\prime\prime}(t_{H})=a^{\prime\prime}(t_{i})=0$, so $t_{i}$ represents the
beginning of the inflationary phase, and during the slow-roll inflation
(Region III), we have $\lambda_{H}^{2}\approx L_{\text{H}}^{2}/2\simeq
1/(2a^{2}H^{2})$, which is exponentially decreasing, and all the modes
observed today were inside the comoving Hubble radius at $t=t_{i}$. Between
the times $t_{H}$ and $t_{i}$, $\lambda_{H}^{2}$ is negative, and
$\Omega^{2}_{\text{tot}}$ is strictly positive. Therefore, during this period
the mode functions are oscillating, while during the epoch between $t_{B}$ and
$t_{H}$, some modes ($k^{-2}>k_{B}^{-2}$) are inside the comoving Hubble
radius, and others ($k^{-2}<k_{B}^{-2}$) are outside it right after the
bounce, where $k_{B}\equiv\lambda_{B}^{-1}(t_{B})$. In the contracting phase,
when $t\ll t_{B}$, the universe is quasi-de Sitter and $\lambda_{H}^{2}\simeq
1/(2a^{2}H^{2})$ increases exponentially toward the bounce $t\rightarrow
t_{B}$, as now $a(t)$ is decreasing exponentially. However, several Planck
seconds before the bounce, the universe enters a non-de Sitter state, during
which $\lambda_{H}^{2}$ starts to decrease until the bounce, at which a
characteristic Planck scale $k_{B}\;(\equiv 1/\lambda_{H})$ can be well
defined. Therefore, for $t\ll t_{B}$, all the modes are outside the comoving
Hubble radius. Then, following our previous arguments, if the initial moment
is chosen at $t_{0}\ll t_{B}$, the de Sitter state seems not to be viable.
However, when $t_{0}\ll t_{B}$ we have $a(\eta)\simeq 1/(-\eta|H_{\Lambda}|)$,
where $H_{\Lambda}=-[\lambda(1+\gamma^{2})]^{-1}$ and

$\Omega_{\text{tot}}^{2}\simeq k^{2}-\frac{2}{\eta^{2}},$ (3.27)

for which Eq.(3.9) has the exact solutions given by Eq.(3.6). Therefore, at
sufficient early times, choosing $\alpha_{k}=1,\;\beta_{k}=0$ leads us to the
de Sitter state (3.7). From the above analysis it is clear that this is
possible precisely because of the isometry of the de Sitter space, which is
sufficient to single out a preferred state, the de Sitter state IA19 .

With the exact solution (3.7) as the initial conditions imposed at the moment
$t_{0}\;(\ll t_{B})$ in the contracting phase, it was found that the power
spectrum of the cosmological scalar perturbations can be divided into three
different regimes: (i) the ultraviolet (UV) ($k>k_{\text{mLQC-I}}$); (ii)
intermediate ($k_{i}<k<k_{\text{mLQC-I}}$); and (iii) infrared ($k<k_{i}$),
where $k_{\text{mLQC-I}}\equiv a_{B}\sqrt{R_{B}/6}$ and
$k_{i}=a_{i}\sqrt{R_{i}/6}$, and $R_{B}$ and $R_{i}$ are the curvatures given
at the bounce and the beginning of the slow-roll inflation, respectively [cf.
Fig. 5]. During the infrared regime, the power spectrum increases as $k$
increases, while in the intermediate regime it is oscillating very fast and
the averaged amplitude of the power spectrum is decreasing as $k$ increases.
In the UV regime, the spectrum is almost scale-invariant, which is consistent
with the current observations. There exists a narrow band, $0.1\times
k_{0}^{*}<k<k_{\text{mLQC-I}}$, in which the quantum gravitational effects
could be detectable by current or forthcoming cosmological observations IA19 .
Within the dressed metric approach, one of the most distinctive features of
the power spectrum in mLQC-I is that its magnitude in the IR regime is of the
Planck scale IA19 ; lsw2020 . This is because those infrared modes are
originally outside the Hubble horizon in the contracting phase and thus their
magnitudes are frozen as they propagate across the bounce and then into the
inflationary phase. Considering that the contracting phase is a quasi de
Sitter phase with a Plank-scale Hubble rate, the magnitude of the IR modes is
thus also Planckian lsw2020 .

Figure 5: Schematic plot of $\lambda_{H}^{2}$ defined by Eq.(3.25) vs $t$ for
mLQC-I in the dressed metric approach for the quadratic and the Starobinsky
potentials, where $a^{\prime\prime}(t_{H})=0$ and $a^{\prime\prime}(t_{i})=0$
with $t_{i}$ being the starting time of the inflationary phase. During the
slow-roll inflation, we have $\lambda_{H}^{2}\approx L_{\text{H}}^{2}/2$. In
the contracting phase $t<t_{B}$, the universe is initially de Sitter and we
still have $\lambda_{H}^{2}\approx L_{\text{H}}^{2}/2$, but now it increases
exponentially toward bounce $t\rightarrow t_{B}$, as the universe in this
phase is exponentially contracting. However, several Planck seconds before the
bounce, the universe enters a non-de Sitter state, during which
$\lambda_{H}^{2}$ starts to decrease until the bounce. The qualitative
behavior of the comoving Hubble radius is the same for the quadratic and the
Starobinsky potentials. Different potentials will change the the values of
$t_{H}$ and $t_{i}$ correspondingly.

It should be noted that if the initial conditions are imposed at the bounce,
from Fig. 5 we can see clearly that some modes are inside the comoving Hubble
radius, and some are not. In addition, in the neighborhood of the bounce, the
background is far from de Sitter. So, it is impossible to impose either the BD
vacuum or the de Sitter state at the bounce. In this case, one of the choices
of the initial conditions is the fourth-order adiabatic vacuum, similar to
that in LQC aan2013b ; d1 ; d1b ; ZhuA ; ZhuB .

####  III.2.2 mLQC-II

In mLQC-II, the evolution of the effective homogeneous universe is different
from that of mLQC-I. In particular, it is symmetric with respect to the bounce
and in the initially kinetic energy dominated case at the bounce the solutions
can be well approximated by Eq.(II.4) in the bouncing phase lsw2018b ; lsw2019
, similar to that of LQC LQC .

When considering the cosmological perturbations, similar ambiguities in the
choices of $\pi_{a}^{-2}$ and $\pi_{a}^{-1}$ in Eq.(3.11) exist. In
particular, for the choice of Eq.(3.17) now the function $\Theta(b)$ is
replaced by

$\Theta(b)=\cos\left(\frac{\lambda b}{2}\right),$ (3.28)

which behaves also like a step function across the bounce and picks up the
right sign in both contracting and expanding phases, so it smoothly connects
$\Omega_{\pm}$ defined by Eq.(3.16).

On the other hand, $\Omega^{2}_{\mathrm{eff}}$ is obtained from Eq.(3.11) by
the replacements,

$\displaystyle\frac{1}{\pi^{2}_{a}}$ $\displaystyle\rightarrow$
$\displaystyle\frac{4\pi^{2}\gamma^{2}\lambda^{2}}{9a^{4}\sin^{2}\left(\lambda
b/2\right){\cal{D}}},$ (3.29) $\displaystyle\frac{1}{\pi_{a}}$
$\displaystyle\rightarrow$
$\displaystyle\frac{-2\pi\gamma\lambda\cos\left(\lambda
b/2\right)}{3a^{2}\sin\left(\lambda b/2\right){\cal{D}}^{1/2}},$ (3.30)

but now with

${\cal{D}}\equiv 1+\gamma^{2}\sin^{2}\left(\frac{\lambda b}{2}\right).$ (3.31)

Such obtained $\Omega^{2}$, $\Omega^{2}_{\pm}$ and $\Omega^{2}_{\mathrm{eff}}$
are quite similar to those given by Figs. 3 and 4 in mLQC-I. In particular, at
the bounce, we have

$\displaystyle\Omega^{2}(t_{B})=1.59\times
10^{-10},\quad\Omega^{2}_{\mathrm{eff}}(t_{B})=0.265,$ $\displaystyle
k_{B}^{\scriptscriptstyle{\mathrm{II}}}=\left.\left(\frac{a^{\prime\prime}}{a}\right)^{1/2}\right|_{t=t_{B}}\approx
6.84,$ (3.32)

that is, the curvature term $a^{\prime\prime}/a$ still dominates the evolution
near the bounce.

Figure 6:  Schematic plot of $\lambda_{H}^{2}$ defined by Eq.(3.25) vs $t$ for
mLQC-II in the dressed metric approach for the quadratic and the Starobinsky
potentials, where $a^{\prime\prime}(t^{p}_{H})=a^{\prime\prime}(t_{H})=0$ and
$a^{\prime\prime}(t^{p}_{i})=a^{\prime\prime}(t_{i})=0$, and $t=t_{i}$ denotes
the starting time of the inflationary phase, while $t=t^{p}_{i}$ is the end
time of the deflationary phase in the contracting branch. During the slow-roll
inflation, we have $\lambda_{H}^{2}\approx L_{\text{H}}^{2}/2$. In particular,
$\lambda_{H}^{2}$ is decreasing (increasing) exponentially in Region III
(Region III’). The corresponding effective mass near the bounce is always
negative. Similar behavior also happens in LQC in the dressed metric approach
ZhuA . The bounce is dominated by the kinetic energy of the scalar field,
which leads to $t^{p}_{H}\approx-t_{H}$. However, in general we find that
$t^{p}_{i}\neq-t_{i}$ due to the effects of the potential energy of the scalar
field far from the bouncing point. The comoving Hubble radius has the same
qualitative behavior for the quadratic and the Starobinsky potentials, while
the values of $t_{i}$ ($t^{P}_{i}$) and $t_{H}$ ($t^{p}_{H}$) depend on the
type of the potentials and the initial conditions.

To see how to impose the initial conditions, let us introduce the quantity
$\lambda^{2}_{H}$ defined by Eq.(3.25) but now $\mathfrak{A}_{-}$ will be
replaced either by $\Omega^{2}$ or $\Omega^{2}_{\mathrm{eff}}$. The details
here are not important, and $\lambda^{2}_{H}$ is schematically plotted in Fig.
6, from which we can see that if the initial conditions are chosen to be
imposed at the bounce, the BD vacuum (as well as the de Sitter state) is still
not available, and the fourth-order adiabatic vacuum is one of the possible
choices, similar to the LQC case. However, if the initial conditions are
imposed in the contracting phase at $t_{0}\ll t^{p}_{i}$, the universe becomes
very large $a(t)\gg 1$ and can be practically considered as flat, then the BD
vacuum can be chosen.

Certainly, one can choose different initial conditions. In particular, the
fourth-order adiabatic vacuum was chosen even in the contracting phase in
lsw2020 . With such a choice, the power spectra from $\Omega^{2}$ and
$\Omega^{2}_{\mathrm{eff}}$ in the region $k\in(5\times 10^{-6},50)$ were
studied and found that the relative difference in the magnitude of the power
spectra is around $30\%$ in the IR regime and less than $10\%$ in the
intermediate regime. In the UV regime, the relative difference can be as small
as $0.1\%$ or even less.

####  III.2.3 LQC

To consider the effects of the ambiguities in the choice of $\pi_{a}^{-2}$ and
$\pi_{a}^{-1}$ in Eq.(3.11) 111111In the framework of LQC, such effects were
also studied in aan2013b ; ZhuA ; ZhuB ; nbm2018 ; lsw2020 . In particular, in
abs2018 ; IA19 ; AKS20  the function $\mathfrak{A}_{-}$ defined in Eq.(3.15)
was chosen over the whole process of the evolution of the universe., power
spectra of the cosmological perturbations were also studied in the framework
of LQC in lsw2020 . In this case, $\Omega^{2}$ is obtained from Eq.(3.17) with

$\Theta(b)=\cos(\lambda b),$ (3.33)

while $\Omega^{2}_{\mathrm{eff}}$ is obtained from Eq.(3.11) by the
replacements,

$\displaystyle\frac{1}{\pi^{2}_{a}}$ $\displaystyle\rightarrow$
$\displaystyle\frac{16\pi^{2}G^{2}\gamma^{2}\lambda^{2}}{9a^{4}\sin^{2}\left(\lambda
b\right)},$ (3.34) $\displaystyle\frac{1}{\pi_{a}}$ $\displaystyle\rightarrow$
$\displaystyle\frac{-4\pi\gamma\lambda\cos\left(\lambda
b\right)}{3a^{2}\sin\left(\lambda b\right)}.$ (3.35)

As shown explicitly, the term $\Omega^{2}_{+}$ is always negligible comparing
with the curvature term $a^{\prime\prime}/a$ in the expression of
$\Omega^{2}_{\text{tot}}$ defined in Eq.(III.2.1) by replacing
$\mathfrak{A}_{-}$ with $\Omega^{2}_{+}$. So, from Eq.(3.25) we find that

$\displaystyle\lambda_{H}^{2}=\frac{1}{a^{\prime\prime}/a-\Omega^{2}_{+}}\simeq\frac{a}{a^{\prime\prime}},$
(3.36)

during the bouncing phase $t\in(t_{B},t_{i})$, and
$\lambda_{H}^{2}\simeq{a}/{a^{\prime\prime}}$ was shown schematically by Fig.
18 in ZhuB , which is quite similar to Fig. 6 given above for mLQC-II.

As a result, the initial states of the linear perturbations can be either
imposed in the contracting phase at a moment $t_{0}\ll t^{p}_{i}$ as the BD
vacuum, or at the bounce as the fourth-order adiabatic vacuum aan2013b .
However, it was shown analytically that such two conditions lead to the same
results ZhuB .

Figure 7: The figure shows the results of the scalar power spectra from three
models presented in lsw2020  when the potential term is given by
$\Omega_{\text{eff}}^{2}$. The inflationary potential is chosen to be the
quadratic potential and the e-foldings of the inflationary phases in all three
models are chosen to be $72.8$. The first panel shows the scalar power
spectrum in mLQC-I which is characterized by its unique infrared regime. In
the second panel, we compare the scalar power spectra from LQC and mLQC-II.

To compare the results obtained from the three different models, in lsw2020
the fourth-order adiabatic vacuum was chosen even in the contracting phase for
LQC. Here, we cite some of the results in Fig. 7. In particular, it was found
that the relative difference in the amplitudes of the power spectra of the
scalar perturbations due to the choice of $\Omega^{2}$ or
$\Omega^{2}_{\mathrm{eff}}$ is about $10\%$ in the infrared regime, about
$100\%$ in the intermediate regime, and about $0.1\%$ in the UV regime. Since
only modes in the UV regime can be observed currently, clearly this difference
is out of the sensitivities of the current and forhcoming observations S4-CMB
.

However, comparing the power spectra obtained from the three different models,
even with the same choice of $\pi_{a}^{-2}$ and $\pi_{a}^{-1}$, it was found
that the relative difference among LQC, mLQC-I and mQLC-II are significant
only in the IR and oscillating regimes, while in the UV regime, all three
models give quite similar results. In particular, with the same regularization
of $\pi_{a}$ the difference can be as large as $100\%$ throughout the IR and
oscillating regimes, while in the UV regime it is about $0.1\%$.

For the tensor perturbations, the potential term $\Omega_{Q}$ vanishes
identically, so no ambiguities related to the choice of $\pi_{a}$ exist. But,
due to different models, the differences of the power spectra of the tensor
perturbations can be still very large in the IR and oscillating regimes among
the three models, although they are very small in the UV regime, see, for
example, Fig. 12 given in lsw2020 .

##  IV Primordial power spectra of Modified LQCs in Hybrid Approach

As in the previous section, in this section we also consider the three
different models, LQC, mLQC-I, and mLQC-II, but now in the hybrid approach,
and pay particular attention to the differences of the power spectra among
these models. Since the scalar perturbations are the most relevant ones in the
current CMB observations, in the following we shall mainly focus on them, and
such studies can be easily extended to the tensor perturbations.

###  IV.1 mLQC-I

Power spectra of the cosmological scalar and tensor perturbations for the
effective Hamilton in mLQC-I were recently studied in the hybrid approach
gqm2020 ; qmp2020 ; LOSW20 . In particular, the mode function $v_{k}$ of the
scalar perturbations satisfies the differential equation LOSW20 ,

$v_{k}^{\prime\prime}+\left(k^{2}+s\right)v_{k}=0,$ (4.1)

where

$\displaystyle s$ $\displaystyle=$ $\displaystyle\frac{4\pi
Gp^{2}_{\phi}}{3v^{4/3}}\left(19-24\pi
G\gamma^{2}\frac{p^{2}_{\phi}}{\pi_{a}^{2}}\right)$ (4.2)
$\displaystyle+v^{2/3}\left(V_{,\phi\phi}+\frac{16\pi G\gamma
p_{\phi}}{\pi_{a}}V_{,\phi}-\frac{16\pi G}{3}V\right)$ $\displaystyle=$
$\displaystyle-\frac{4\pi
G}{3}a^{2}\left(\rho-3P\right)+{\cal{U}},~{}~{}~{}~{}$

which is the effective mass of the scalar mode, with

$\displaystyle{\cal{U}}$ $\displaystyle\equiv$ $\displaystyle
a^{2}\left[V_{,\phi\phi}-12\frac{V_{,\phi}}{\pi_{a}}\right.$ (4.3)
$\displaystyle\left.+\frac{64a^{6}V(\phi)}{\pi
G}\left(\rho-\frac{3V(\phi)}{4\pi G}\right)\frac{1}{\pi_{a}^{2}}\right].$

Note that in LOSW20 , instead of $\pi_{a}$, the symbol $\Omega$ was used. In
addition, the cosmological tensor perturbations are also given by Eqs.(4.1)
and (4.2) but with the vanishing potential ${\cal{U}}=0$. Then, we immediately
realize that in the hybrid approach quantum mechanically there are also
ambiguities in the replacements $\pi_{a}^{-2}$ and $\pi_{a}^{-1}$, as
mentioned in the last section. So far, two possibilities were considered
gqm2020 ; qmp2020 . One is given by the replacements,

$\frac{1}{\pi_{a}^{2}}\rightarrow\frac{1}{\Omega^{2}_{{\scriptscriptstyle{\mathrm{I}}}}},\quad\quad\frac{1}{\pi_{a}}\rightarrow\frac{\Lambda_{{\scriptscriptstyle{\mathrm{I}}}}}{\Omega^{2}_{\scriptscriptstyle{\mathrm{I}}}},$
(4.4)

in Eq.(4.2), where

$\displaystyle\Omega^{2}_{{\scriptscriptstyle{\mathrm{I}}}}$
$\displaystyle\equiv$
$\displaystyle-\frac{v^{2}\gamma^{2}}{\lambda^{2}}\Big{\\{}\sin^{2}\left(\lambda
b\right)-\frac{\gamma^{2}+1}{4\gamma^{2}}\sin^{2}\left(2\lambda
b\right)\Big{\\}},$ $\displaystyle\Lambda_{{\scriptscriptstyle{\mathrm{I}}}}$
$\displaystyle\equiv$ $\displaystyle v\frac{\sin(2\lambda b)}{2\lambda}.$
(4.5)

This is the case referred to as prescription A in qmp2020 .

The other possibility is obtained by the replacement of Eqs.(3.34) and (3.35),
which was referred to as Prescription B qmp2020 , and showed that the two
prescriptions lead to almost the same results. So, in the rest of this section
we restrict ourselves only to prescription A.

Then, for the case in which the evolution of the homogeneous universe was
dominated by kinetic energy at the bounce,

$\dot{\phi}^{2}_{B}\gg 2V(\phi_{B}),$ (4.6)

it was shown that the effective mass is always positive at the bounce qmp2020
. In fact, near the bounce we have WZW18 ,

$\displaystyle s$ $\displaystyle=$ $\displaystyle-\frac{4\pi
G}{3}a^{2}\left(\rho-3P\right)+{\cal{U}}(\eta)$ (4.7) $\displaystyle\simeq$
$\displaystyle\frac{8\pi G}{3}a^{2}\rho>0.$

Note that in writing the above expression, we have used the fact that during
the bouncing phase we have $w_{\phi}\equiv P/\rho\simeq 1$, and
$\left|{\cal{U}}(\eta)\right|\ll 1$. On the other hand, in the pre-bounce
phase, when $t\ll t_{B}$ the background is a contracting de Sitter spacetime,
so we have qmp2020 ,

$\displaystyle s$ $\displaystyle=$ $\displaystyle-\frac{4\pi
G}{3}a^{2}\left(\rho-3P\right)+{\cal{U}}(\eta)\simeq{\cal{U}}(\eta)\simeq
5a^{2}V_{,\phi\phi},$ $\displaystyle a$ $\displaystyle\simeq$ $\displaystyle
a_{B}e^{H_{\Lambda}(t-t_{B})},$ (4.8)

where $H_{\Lambda}\equiv-\sqrt{8\pi\alpha G\rho_{\Lambda}/3}$. Thus, the
effective mass remains positive in the pre-bounce phase, as long as
$V_{,\phi\phi}(t\ll t_{B})>0$. This is the case for both quadratic and
Starobinsky potentials. In fact, from (3.24), we find that

$V_{,\phi\phi}=\begin{cases}m^{2},&\text{quadratic},\cr
m^{2}\left(2-e^{4\sqrt{\pi G/3}\;\phi}\right)e^{-8\sqrt{\pi
G/3}\;\phi},&\text{Starobinsky}.\cr\end{cases}$ (4.9)

For the case that satisfies the condition (4.6) initially at the bounce, we
find that $\phi(t)$ becomes very negative at $t\ll t_{B}$ for the Starobinsky
potential, so $V_{,\phi\phi}(t\ll t_{B})$ is positive even in this case. Then,
the quantity defined by

$\lambda_{H}^{2}\equiv-\frac{1}{s},$ (4.10)

has similar behavior in the post-bounce phases for the case in which the
evolution of the homogeneous universe was dominated by kinetic energy at the
bounce, but has different behaviors in the pre-bounce phases, depending
specifically on the potentials considered.

In Figs. 8 and 9 we show the comoving Hubble radius for the quadratic and
Starobinsky potentials, respectively. From these figures it is clear that for
$t_{i}^{p}<t<t_{i}$, $\lambda_{H}^{2}$ is strictly negative, which implies the
effective mass $s$ is positive in this regime. Hence, all the modes assume the
oscillatory behavior as the modes inside the Hubble horizon, and we may impose
the BD vacuum at the bounce. In addition, when $t\ll t^{p}_{i}$, the
background is well described by the de Sitter space, so the de Sitter state
can be imposed in the deep contracting phase. However, imposing the BD vacuum
at the bounce will clearly lead to different power spectra at the end of the
slow-roll inflation from that obtained by imposing the de Sitter state in the
deep contracting phase. This is because, when the background is contracting to
about the moments $t\simeq t^{p-}_{i}$, the effective mass becomes so large
and negative that the mode function $v_{k}$ will be modified significantly, in
comparison with that given at $t_{0}\;(\ll t^{p}_{i}$), or in other words,
particle creation now becomes not negligible during the contracting phase.
Then, other initial conditions at the bounce may need to be considered.

Figure 8:  Schematic plot of $\lambda_{H}^{2}$ defined by Eq.(4.10) in mLQC-I
for the quadratic potential in the hybrid approach, where
$s(t_{i})=s(t^{p}_{i})=0$, and $t=t_{i}$ is the starting time of the
inflationary phase. During the slow-roll inflation, we have
$\lambda_{H}^{2}\approx L_{\text{H}}^{2}/2$ (Region III). In the contracting
phase, the background is asymptotically de Sitter. The evolution of the
universe is asymmetric with respect to the bounce. In particular,
$\lambda_{H}^{2}$ is strictly negative for $t^{p}_{i}<t<t_{i}$, while for
$t\simeq t^{p-}_{i}$ the “generalized” comoving Hubble radius
$\lambda_{H}^{2}$ becomes positive and large. However, as $t$ decreases,
$\lambda_{H}^{2}$ becomes negative again. Although the values of $t_{i}$ and
$t_{H}$ depend on the initial conditions for the background evolution, for
example, when $\phi_{B}=1.27~{}m_{\mathrm{pl}}$ at the bounce, $t_{i}\approx
7.55\times 10^{4}~{}t_{\mathrm{pl}}$ and
$t_{H}\approx-21.85~{}t_{\mathrm{pl}}$, the qualitative behavior of the
comoving Hubble radius is robust with respect to the choice of the initial
conditions as long as the bounce is dominated by the kinetic energy of the
scalar field.  Figure 9:  Schematic plot of $\lambda_{H}^{2}$ defined by
Eq.(4.10) for the Starobinsky potential and mLQC-I in the hybrid approach,
where $s(t_{i})=s(t^{p}_{i})=s(t^{p}_{H})=0$, and $t=t_{i}$ is the starting
time of the inflationary phase. During the slow-roll inflation, we have
$\lambda_{H}^{2}\approx L_{\text{H}}^{2}/2$ (Region III). In the contracting
phase, the background is asymptotically de Sitter. The evolution of the
universe is asymmetric with respect to the bounce. In particular,
$\lambda_{H}^{2}$ is strictly negative for $t^{p}_{i}<t<t_{i}$, while for
$t\simeq t^{p-}_{i}$ it becomes positive and large. However, as $t$ decreases,
$\lambda_{H}^{2}$ becomes negative again. The qualitative behavior of
$\lambda^{2}_{H}$ does not change with the choice of the initial conditions as
long as the inflaton initially starts from the left wing of the potential at
the kinetic-energy-dominated bounce. However, the exact values of $t_{i}$ and
$t_{H}$ depend on the initial conditions. For example, when
$\phi_{B}=-1.32~{}m_{\mathrm{pl}}$, $t^{p}_{H}=-7.88~{}t_{\mathrm{pl}}$,
$t^{p}_{i}=-4.11~{}t_{\mathrm{pl}}$ and $t_{i}=4.90\times
10^{5}~{}t_{\mathrm{pl}}$.

###  IV.2 mLQC-II

Similar to LQC, the homogeneous universe of mLQC-II is symmetric with respect
to the bounce, and is well described by the analytical solutions given by
Eqs.(II.4) and (2.57) for the states that are dominated by kinetic energy at
the bounce.

In this model, the cosmological perturbations are also given by
Eqs.(4.1)-(4.3) but now with the replacement LOSW20 ,

$\frac{1}{\pi_{a}^{2}}\rightarrow\frac{1}{\Omega^{2}_{{\scriptscriptstyle{\mathrm{II}}}}},\quad\quad\frac{1}{\pi_{a}}\rightarrow\frac{\Lambda_{{\scriptscriptstyle{\mathrm{II}}}}}{\Omega^{2}_{\scriptscriptstyle{\mathrm{II}}}},$
(4.11)

where

$\displaystyle\Omega^{2}_{{\scriptscriptstyle{\mathrm{II}}}}$
$\displaystyle\equiv$
$\displaystyle\frac{4v^{2}}{\lambda^{2}}\sin^{2}\left(\frac{\lambda
b}{2}\right)\left\\{1+\gamma^{2}\sin^{2}\left(\frac{\lambda
b}{2}\right)\right\\},$
$\displaystyle\Lambda_{{\scriptscriptstyle{\mathrm{II}}}}$
$\displaystyle\equiv$ $\displaystyle v\frac{\sin\left(\lambda
b\right)}{\lambda}.$ (4.12)

In this case, it can be shown that the effective mass defined by Eq.(4.2) is
always positive in the neighborhood of the bounce, but far from the bounce,
the properties of $\lambda^{2}_{H}$ depend on the potential in the pre-bounce
phase, similar to mLQC-I.

In Fig. 10, we plot $\lambda^{2}_{H}$ for the Starobinsky potential, while for
the quadratic one, it is quite similar to the corresponding one in mLQC-I,
given by Fig. 8. From Fig. 10 we can see that $\lambda^{2}_{H}$ now is
negative not only near the bounce but also in the whole contracting phase, so
that all the modes are oscillating for $t<t_{i}$. Then, one can choose the BD
vacuum at the bounce. It is remarkable that for the quadratic potential, this
is impossible [cf. Fig. 8].

Moreover, as $t\rightarrow-\infty$, the expansion factor becomes very large,
and the corresponding curvature is quite low, so to a good approximation, the
BD vacuum can also be chosen in the distant past, not only for the Starobinsky
potential but also for other potentials. Due to the oscillating behavior of
the mode function over the whole contracting phase, imposing the BD vacuum at
the bounce is expected not to lead to significant difference in the power
spectra from that in which the same condition is imposed in the deep
contracting phase.

Figure 10:  Schematic plot of $\lambda_{H}^{2}$ defined by Eq.(4.10) for the
Starobinsky potential and mLQC-II in the hybrid approach, where $s(t_{i})=0$,
and $t=t_{i}$ is the starting time of the inflationary phase. During the slow-
roll inflation, we have $\lambda_{H}^{2}\approx L_{\text{H}}^{2}/2$ (Region
III) since the contribution from the potential is in general less than
$a^{\prime\prime}/a$. Again the qualitative behavior of $\lambda^{2}_{H}$
remains the same as long as the inflaton starts from the left wing of the
potential with a positive velocity and the bounce is initially dominated by
the kinetic energy of the inflaton field.

###  IV.3 LQC

The evolution of the homogeneous universe of standard LQC model is also
symmetric with respect to the bounce, and is well described by the analytical
solutions given in ZhuA ; ZhuB  for the states that are dominated by kinetic
energy at the bounce.

In this model, the cosmological perturbations are also given by
Eqs.(4.1)-(4.3) but now with the replacement LOSW20 ,

$\frac{1}{\pi_{a}^{2}}\rightarrow\frac{1}{\Omega^{2}_{\text{LQC}}},\quad\quad\frac{1}{\pi_{a}}\rightarrow\frac{\Lambda_{\text{LQC}}}{\Omega^{2}_{LQC}},$
(4.13)

where

$\displaystyle\Omega_{\text{LQC}}$ $\displaystyle\equiv$
$\displaystyle\frac{v\sin\left(\lambda
b\right)}{\lambda},\;\;\;\Lambda_{\text{LQC}}\equiv\frac{v\sin\left(2\lambda
b\right)}{2\lambda}.$ (4.14)

In this case, it can be shown that the effective mass defined by Eq.(4.2) is
always positive for the states that are dominated by kinetic energy at the
bounce nbm2018 ; WZW18 , and the quantity $\lambda^{2}_{H}$ defined by
Eq.(4.10) is negative near the bounce. Again, similar to the mLQC-II case, the
modes are oscillating near the bounce. However, in the contracting phase the
behavior of $\lambda^{2}_{H}$ sensitively depends on the inflation potentials.
For the Starobinsky one, $\lambda^{2}_{H}$ behaves similar to that described
by Fig. 10, so the BD vacuum can be imposed either in the deep contracting
phase or at the bounce, and such resulted power spectra are expected not to be
significantly different from one another. But for the quadratic potential the
situation is quite different, and a preferred choice is to impose the BD
vacuum in the deep contracting phase ($t_{0}\ll t_{B}$).

###  IV.4 Primordial Power Spectra

As it can be seen that one of the preferred moments to impose the initial
conditions for the cosmological perturbations in all these three models is a
moment in the contracting phase $t_{0}<t_{B}$. In this phase, we can impose
the BD vacuum state as long as the moment is sufficiently earlier, $t_{0}\ll
t_{B}$. Certainly, other initial conditions can also be chosen. In particular,
in LOSW20  the second-order adiabatic vacuum conditions were selected, but it
was found that the same results can also be obtained even when the BD vacuum
or the fourth-order adiabatic vacuum is imposed initially.

The nth-order adiabatic vacuum conditions can be obtained as follows: Let us
first consider the solution,

$\nu_{k}=\frac{1}{\sqrt{2W_{k}}}e^{-i\int^{\eta}W_{k}(\bar{\eta})d\bar{\eta}}.$
(4.15)

Then, inserting it into (4.1), one can find an iterative equation for $W_{k}$.
In particular, it can be shown that the zeroth-order solution is given by
$W^{(0)}_{k}=k$, while the second and fourth order adiabatic solutions are
given by,

$W^{(2)}_{k}=\sqrt{k^{2}+s},\quad\quad
W^{(4)}_{k}=\frac{\sqrt{f(s,k)}}{4|k^{2}+s|}.$ (4.16)

Here $f(s,k)=5s^{\prime
2}+16k^{4}(k^{2}+3s)+16s^{2}(3k^{2}+s)-4s^{{}^{\prime\prime}}(s+k^{2})$. It
should be noted that, in order to compare directly with observations, it is
found convenient to calculate the power spectrum of the comoving curvature
perturbation ${\cal R}_{k}$, which is related to the Mukhanov-Sasaki variable
via the relation ${\cal R}_{k}=\nu_{k}/z$, with $z=a\dot{\phi}/H$. Then, its
power spectrum reads

$\mathcal{P}_{{\cal
R}_{k}}=\frac{\mathcal{P}_{\nu_{k}}}{z^{2}}=\frac{k^{3}}{2\pi^{2}}\frac{|\nu_{k}|^{2}}{z^{2}}.$
(4.17)

In addition, the power spectrum is normally evaluated at the end of inflation,
at which all the relevant modes are well outside the Hubble horizon [cf. Fig.
2].

It should be also noted that the above formula is only applicable to the case
where $W^{(2)}_{k}$ and/or $W^{(4)}_{k}$ remains real at the initial time.
This is equivalent to require $k^{2}+s\geq 0$ for $W^{(2)}_{k}$ and
$f(s,k)\geq 0$ for $W^{(4)}_{k}$. Since the effective mass $s$ in general
depends on $t$, it is clear that the validity of (4.16) depends not only on
the initial states but also on the initial times.

Figure 11: The primordial power spectra of the cosmological scalar
perturbations in the hybrid approach with the Starobinsky potential,
respectively, for LQC, mLQC-I, and mLQC-II. The mass of the inflaton field is
set to $2.44\times 10^{-6}m_{\mathrm{pl}}$. The background evolution is chosen
so that the pivot mode is $k_{*}=5.15$ in all three models. The initial states
are the second-order adiabatic states imposed in the contracting phase at the
moment $t_{0}$ with $t_{0}\ll t_{B}$ LOSW20 .

In addition, in the following only the Starobinsky potential given in
Eq.(3.24) will be considered, as it represents one of the most favorable
models by current observations cosmo . Let us turn to consider the power
spectrum of the scalar perturbations in each of the three models. Similar
results can be also obtained for the tensor perturbations. In particular, it
was found that the scalar power spectra in these three models can be still
divided into three distinctive regimes: the infrared, oscillatory and UV, as
shown in Fig. 11.

In the infrared and oscillatory regimes, the relative difference between LQC
and mLQC-I can be as large as $100\%$, while this difference reduces to less
than $1\%$ in the UV regime. This is mainly because LQC and mLQC-I have the
same classical limit in the post-bounce phase, and as shown in Figs. 5 and 8,
the effective masses in both approaches tend to be the same during the
inflationary phase.

However, it is interesting to note that in the infrared and oscillatory
regimes, the power spectrum in mLQC-I is suppressed in comparison with that of
LQC, which has been found only in the hybrid approach. As a matter of fact, in
the dressed metric approach, the power spectrum in mLQC-I is largely amplified
in the infrared regime, and its magnitude is of the Planck scale as depicted
in Fig. 7 IA19 ; lsw2020 . The main reason might root in the distinctive
behavior of the effective masses in the two approaches, as shown explicitly in
Figs. 5 and 8.

On the other hand, the difference of the power spectra between LQC and mLQC-II
is smaller than that between LQC and mLQC-I. In particular, in the infrared
regime, it is about $50\%$. The large relative difference (more than $100\%$)
of the power spectra between mLQC-I and mLQC-II also happens in the infrared
and oscillatory regimes, while in the UV regime it reduces to about $2\%$.

To summarize, in the hybrid approach the maximum relative differences of the
power spectra among these three different models always happen in the infrared
and oscillatory regimes, while in the UV regime, the differences reduce to no
larger than $2\%$, and all the three models predict a scale-invariant power
spectrum, and is consistent with the current CMB observations. However, in the
hybrid approach, the power spectrum in mLQC-I is suppressed in the infrared
and oscillatory regimes. The latter is in a striking contrast to the results
obtained from the dressed metric approach, which might be closely related to
the fact that the effective masses in these two approaches are significantly
different, especially near the bounce and in the prebounce stage.

##  V Conclusions and outlook

In the past two decades, LQC has been studied extensively, and several
remarkable features have been found LQC , including the generic resolution of
the big bang singularity (replaced by a quantum bounce) in the Planckian
scale, the slow-roll inflation as an attractor in the post-bounce evolution of
the universe, and the scale-invariant power spectra of the cosmological
perturbations, which are consistent with the current CMB observations. Even
more interestingly, it was shown recently that some anomalies from the CMB
data Planck2018b ; Planck2019 ; SCHS16  can be reconciled purely due to the
quantum geometric effects in the framework of LQC AGJS20 ; AKS20 .

Despite of all these achievements, LQC is still plagued with some ambiguities
in the quantization procedure. In particular, its connection with LQG is still
not established engle , and the quantization procedure used in LQC owing to
symmetry reduction before quantization can result in different Hamiltonian
constraints than the one of LQG.

Motivated by the above considerations, recently various modified LQC models
have been proposed, see, for example, wilson2017 ; GFT ; QRLG ; DID ; KT ;
GLS20a ; GLS20b ; LSS20 ; HLL20 ; ST20  and references therein. In this brief
review, we have restricted ourselves only to mLQC-I and mLQC-II YDM09 ; DL17 ;
DL18 , as they are the ones that have been extensively studied in the
literature not only the dynamics of the homogeneous universe lsw2018 ;
lsw2018b ; lsw2019 ; SS19a ; SS19b ; qm2019 ; gm20 , but also the cosmological
perturbations IA19 ; lsw2020 ; LOSW20 ; gqm2020 ; qmp2020 .

In these two modified LQC models, it was found that the resolution of the big
bang singularity is also generic lsw2018 ; lsw2018b ; lsw2019 ; SS19a ; SS19b
. This is closely related to the fact that the area operator in LQG has a
minimal but nonzero eigenvalue LQG ; LQC , quite similar to the eigenvalue of
the ground state of the energy operator of a simple harmonic oscillator in
quantum mechanics. This deep connection also shows that the resolution of the
big bang singularity is purely due to the quantum geometric effects. In
addition, similar to LQC, the slow-roll inflation also occurs generically in
both mLQC-I and mLQC-II lsw2019 . In particular, when the inflaton has a
quadratic potential, $V(\phi)=m^{2}\phi^{2}/2$, the probabilities for the
desired slow-roll inflation not to occur are $\lesssim 1.12\times 10^{-5}$,
$\lesssim 2.62\times 10^{-6}$, and $\lesssim 2.74\times 10^{-6}$ for mLQC-I,
mLQC-II and LQC, respectively.

When dealing with perturbations, another ambiguity rises in the replacement of
the momentum conjugate $\pi_{a}$ of the expansion factor $a$ in the effective
potential of the scalar perturbations. This ambiguity occurs not only in the
dressed metric approach [cf. Eq.(3.11)] but also in the hybrid approach [cf.
Eq.(4.3)], as it is closely related to the quantization strategy used in
LQG/LQC, because now only the holonomies (complex exponentials) of $\pi_{a}$
are defined as operators. Several choices have been proposed in the literature
aan2013b ; abs2018 ; MOP11 ; IA19 ; lsw2020 ; LOSW20 ; gqm2020 ; qmp2020 . In
Secs. III and IV, we have shown that for some choices the effects on the power
spectra are non-trivial, while for others the effects are negligible. However,
even with the same choice, the relative differences in the amplitudes of the
power spectra among the three different models can be as large as $100\%$ in
the infrared and intermediate regimes of the spectra, while in the UV regime
the relative differences are no larger than $2\%$, and the corresponding power
spectra are scale-invariant. Since only the modes in the UV regime are
relevant to the current observations, the power spectra obtained in all the
three models are consistent with current observations cosmo .

However, the interactions between the infrared and UV modes appearing in non-
Gaussianities might provide an excellent window to observe such effects. This
was initially done in LQC abs2018 ; ZWKCS18 ; WZW18 , and lately generalized
to bouncing cosmologies AKS20 . It should be noted that in AKS20 , the
expansion factor $a(t)$ near the bounce was assumed to take the form,

$a(t)=a_{B}\left(1+bt^{2}\right)^{n},$

where $b$ and $n$ are two free parameters. For example, for LQC we have
$n=1/6$ and $b=R_{B}/2$, where $R_{B}$ is the Ricci scalar at the bounce ZhuA
; ZhuB . But, it is clear that near the bounce $a(t)$ takes forms different
from the above expression for mLQC-I/II, as one can see from
Eqs.(II.4)-(2.57). Thus, it would be very interesting to study such effects in
mLQC-I/II, and look for some observational signals.

Moreover, initial conditions are another subtle and important issue not only
in LQC but also in mLQCs. As a matter of fact, the initial conditions consist
of two parts: the initial time, and the initial conditions. Different choices
of the initial times lead to different choices of the initial conditions, or
vice versa. To clarify these issues, in Sections III and IV we have discussed
it at length by showing the (generalized) comoving Hubble radius in each model
as well as in each of the two approaches, dressed metric and hybrid. From
these analyses, we have shown clearly which initial conditions can and cannot
be imposed at a given initial time.

In addition, when the universe changes from contraction to expansion at the
bounce, particle and entropy creations are expected to be very large, and it
is crucial to keep such creations under control, so that the basic assumptions
of the models are valid, including the one that the cosmological perturbations
are small and can be treated as test fields propagating on the quantum
homogeneous background, as assumed in both the dressed metric and hybrid
approaches.

Yet, different initial conditions also affect the amplitudes and shapes of the
primordial power spectra, and it would be very interesting to investigate the
consistency of such obtained spectra with current observations, in particular
the possible explanations to the anomalies found in the CMB data Planck2018b ;
Planck2019 ; SCHS16 , and the naturalness of such initial conditions.

On the other hand, bouncing cosmologies, as an alternative to the cosmic
inflation paradigm, have been extensively studied in the literature, see, for
example, BP17  and references therein. However, in such classical bounces,
exotic matter fields are required in order to keep the bounce open. This in
turn raises the question of instabilities of the models. On the other hand,
quantum bounces found in LQC/mLQCs are purely due to the quantum geometric
effects, and the instability problem is automatically out of the question. So,
it would be very interesting to study bouncing cosmologies in the framework of
LQC/mLQCs. The first step in this direction has already been taken LSS20 , and
more detailed and extensive analyses are still needed.

## Acknowledgements

We are grateful to Robert Brandenberger and David Wands for valuable comments
on the manuscript and helpful discussions. We thank Javier Olmedo and Tao Zhu
for various discussions related to works presented in this manuscript. B.-F.L
and P.S. are supported by NSF grant PHY-1454832. B.-F.L is also partially
supported by the National Natural Science Foundation of China (NNSFC) with the
Grants No. 12005186.

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