# Noncommutative effective LQC: A (pre-)inflationary dynamics investigation

Luis Rey Díaz-Barrón<EMAIL_ADDRESS>Unidad Profesional Interdisciplinaria de
Ingeniería Campus Guanajuato del Instituto Politécnico Nacional.
Av. Mineral de Valenciana No. 200, Col. Fraccionamiento Industrial Puerto
Interior, C.P. 36275, Silao de la Victoria, Guanajuato, México.  Abraham
Espinoza-García<EMAIL_ADDRESS>Unidad Profesional Interdisciplinaria de
Ingeniería Campus Guanajuato del Instituto Politécnico Nacional.
Av. Mineral de Valenciana No. 200, Col. Fraccionamiento Industrial Puerto
Interior, C.P. 36275, Silao de la Victoria, Guanajuato, México.  S. Pérez-
Payán<EMAIL_ADDRESS>Unidad Profesional Interdisciplinaria de Ingeniería
Campus Guanajuato del Instituto Politécnico Nacional.
Av. Mineral de Valenciana No. 200, Col. Fraccionamiento Industrial Puerto
Interior, C.P. 36275, Silao de la Victoria, Guanajuato, México.  J. Socorro
<EMAIL_ADDRESS>Departamento de Física, DCeI, Universidad de
Guanajuato-Campus León, C.P. 37150, León, Guanajuato, México.

###### Abstract

We conduct a (pre-)inflationary dynamics study within the framework of a
simple noncommutative extension of effective loop quantum cosmology—put
forward recently by the authors—which preserves its key features (in
particular, the quantum bounce is maintained). A thorough investigation shows
that the (pre-)inflationary scenario associated to the chaotic quadratic
potential is in the overall the same as the one featured in standard loop
quantum cosmology (which reinforces the conclusion reached by the authors in a
preliminary analysis). Hence, this (pre-)inflationary scenario does not easily
distinguish between standard loop quantum cosmology and the aforementioned
noncommutative scheme. It is argued that a particular tuning of the
noncommutativity parameter could accommodate for subtle effects at the level
of primordial perturbations (the hybrid quantization framework being a
tentative route of analysis).

##  I Introduction

Leading approaches to quantum gravity (loop quantum gravity (LQG) and string/M
theory being at the moment the most prominent candidates) seem to point to a
breakdown of the continuum picture of spacetime, at a fundamental level. In
particular, it is anticipated that an accurate description of the very early
universe be achieved by appending a certain degree of discreteness to the
gravitational field. On one hand, LQG leads to a “quantum geometry” in which
spatial area and volume operators feature a discrete set of eigenvalues whose
associated levels “crowd” exponentially as the eigenvalues grow (the minimum
area eigenvalue being of paramount importance)reviews-lqg ; on the other, low
energy limits of string/M theory lead to noncommutative gauge theories st-ncg
, reviews-ncft . LQG has the appealing of not involving higher dimensions, nor
supersymmetry, in its formulation; furthermore, loop quantum cosmology (LQC)
(the implementation of the ideas and methods of LQG to cosmologically relevant
symmetry-reduced minisuperspace models) has the benefit of leading to a true
initial singularity resolution (particular black hole singularities have also
been shown to be removed as a consequence of the quantum geometry)reviews-lqg
,reviews-lqc . The presumption that the final theory of quantum gravity would
display, at an effective level, relevant features of these two leading
approaches is not too adventurous, given their claimed relative successes. A
rather simple and direct avenue to explore such blending in a single effective
framework is the implementation of a simplified form of noncommutativity at
the effective scheme of LQC (a recent review on implementation of
noncommutativity-related ideas in cosmology is given in ncqc-review ). Of
course, it is most desirable that this construction retains some of the
prominent results of LQC (most notably, the initial singularity resolution and
compatibility with the inflation mechanism). Indeed, the authors of the
present investigation have shown, in a preliminary analysis, that when
considering a momentum space $\theta-$deformation at the effective scheme of
LQC (with a quadratic potential scalar field) there exist solutions that are
singularity free and at the same time display a sufficiently long inflationary
period prd . Such preliminary results make plausible carrying a more careful
study of the (pre-)inflationary dynamics within this noncommutative extension
of effective LQC. The main purpose of this letter is the undertaking of this
more thorough study.

In the following, we review the basics of both, the effective scheme of LQC
and the noncommutativity paradigm as implemented within the minisuperspace
approximation.

###  I.1 Basics of effective LQC

We begin by spelling out some facts pertaining the effective scheme of LQC
(detailed and rather thorough accounts can be found in the standard reviews
reviews-lqc ). As already stated, LQC is the quantization of symmetry-reduced
general relativity, following closely the ideas and methods of LQG.
Succinctly, the usual 3-metric Hamiltonian formulation of general relativity
is rewritten (by means of a canonical transformation) in terms of the SU(2)
Ashtekar-Barbero connection, therefore casting General Relativity (GR) as a
gauge theory. Further, careful implementation of canonical quantization
requires the holonomies of the Ashtekar-Barbero connection to be considered as
the true configuration field variable (instead of the connection itself).
Accordingly, holonomies of a highly symmetric sector (e.g. the homogeneous
sector) of the connection superspace are regarded as the cosmologically
relevant configuration field variable. Canonical quantization is then
implemented on such connection minisuperspace, the resulting quantum framework
is known as loop quantum cosmology. Hence, LQC suffers from the same
minisuperspace approximation drawback as standard quantum cosmology, i.e. LQC
is not a symmetry-reduced LQG. In this respect, it is important to note that,
in particular, the noncommutativity among the conjugate momenta fields
(electric fluxes) in full LQG is completely lost in LQC, signaling possible
significant loss of information when carrying the minisuperspace
approximation.

One of the main results of LQC is the avoidance of the initial singularity via
a quantum bounce. The quantum corrections can be seen as holonomy- and inverse
triad-related, with the former being more relevant. The standard effective
scheme of LQC is achieved by focusing on such holonomy-related quantum
corrections. In the isotropic case, the classical geometro-dynamical variables
which are selected by (effective) LQC are $\beta=\gamma\dot{a}/(aN)$ and
$V=a^{3}$, satisfying

$\\{\beta,V\\}=4\pi G\gamma,\quad\mathsf{P}(y)=4\pi
G\gamma\,\partial_{\beta}\wedge\partial_{V},$ (1)

where $\gamma$ is the so called Barbaro-Immirzi parameter and $\mathsf{P}(y)$
denotes the Poisson structure in variables $y=(\beta,V)$. The (gravitational)
Hamiltonian is written in these variables as ($N=1$)

$\mathcal{H}=-\frac{3}{8\pi G\gamma^{2}}\beta^{2}V.$ (2)

It has been shown (via analytical reviews-lqc  and numerical review-numlqc
investigations) that (in the isotropic case) the quantum corrections
associated to LQC are very well approximated by performing the replacement

$\beta\to\frac{\sin{\lambda\beta}}{\lambda},$ (3)

within the classical Hamiltonian, where
$\lambda^{2}=4\sqrt{3}\pi\gamma\ell^{2}_{p}$ is the lowest eigenvalue
(corresponding to quantum states compatible with the assumed spatially
homogeneous and isotropic geometry) of the area operator in the full loop
quantum gravity—and where $\ell_{p}$ is the Planck lenght. The resulting
effective Hamiltonian is given by

$\mathcal{H}_{\mathrm{eff}}=-\frac{3}{8\pi
G\gamma^{2}\lambda^{2}}\sin^{2}(\lambda\beta)V+\frac{p^{2}_{\phi}}{2V}.$ (4)

Thus, the effective scheme of LQC supports meaningful quantum corrections
within a purely classical setup. This situation has allowed to carry
exploratory investigations of quantum effects on a wide range of cosmological
backgrounds in this rather simplified theoretical framework.

The equations of motion associated to the “holonomized” Hamiltonian (4) with
the standard Poisson structure

$\mathsf{P}(y)=4\pi
G\gamma\,\partial_{\beta}\wedge\partial_{V}+\partial_{\phi}\wedge\partial_{p_{\phi}},\quad
y=(\beta,V,\phi,p_{\phi}),$ (5)

are

$\displaystyle\dot{\beta}=4\pi
G\gamma\frac{\partial\mathcal{H}_{\mathrm{eff}}}{\partial
V}=-\frac{3}{\gamma\lambda^{2}}\sin^{2}(\lambda\beta),$ (6)
$\displaystyle\dot{V}=-4\pi
G\gamma\frac{\partial\mathcal{H}_{\mathrm{eff}}}{\partial\beta}=\frac{3}{\gamma\lambda}V\sin(\lambda\beta)\cos(\lambda\beta),$
(7)
$\displaystyle\dot{\phi}=\frac{\partial\mathcal{H}_{\mathrm{eff}}}{\partial
p_{\phi}}=\frac{p_{\phi}}{V},$ (8)
$\displaystyle\dot{p}_{\phi}=-\frac{\partial\mathcal{H}_{\mathrm{eff}}}{\partial\phi}=0.$
(9)

Since $V=a^{3}$, $\frac{\dot{V}}{3V}=\frac{\dot{a}}{a}=:H$, then, taking into
account the effective Hamiltonian constraint, $\mathcal{H}_{\mathrm{eff}}=0$,
and the equation of motion for $V$, we have,

$H^{2}:=\left(\frac{\dot{V}}{3V}\right)^{2}=\frac{8\pi
G}{3}\rho\left(1-\frac{\rho}{\rho_{\mathrm{c}}}\right),$ (10)

where

$\rho=\frac{\dot{\phi}^{2}}{2}=\frac{p^{2}_{\phi}}{2V^{2}}=\frac{3}{8\pi
G\gamma^{2}\lambda^{2}}\sin^{2}(\lambda\beta),$ (11)

and $\rho_{{c}}$ is the maximum value that $\rho$ can take in view of the
effective Hamiltonian constraint,

$\rho_{{c}}=\frac{3}{8\pi G\gamma^{2}\lambda^{2}}\simeq
0.41\rho_{\mathrm{p}},$ (12)

with $\rho_{\mathrm{p}}$ the Planck density. Equation (10) is the so called
modified Friedmann equation.

In full LQC it is shown that $\beta$ takes values on the interval
$\left(0,\frac{\pi}{\lambda}\right)$ lqc-status-report , therefore, the volume
function reaches a stationary point at $\beta=\frac{\pi}{2\lambda}$. From the
equation of motion (6) it is established that $\beta$ is a decreasing function
of time ($\dot{\beta}\leq 0$), which in turn implies that such stationary
point is only reached once (and that the maximum of the energy density
function is also attained only once). From this monotonic behavior of $\beta$
it also follows that such stationary point corresponds to a minimum (since
$\rho$ is always bounded, the stationary point associated to a vanishing
volume does not feature in the solutions ofx the equations of motion). Thus,
in the effective scheme of LQC, a minimum of the volume function is reached
precisely once, and this minimum volume corresponds to the maximum of the
energy density function. This state of affairs is summarized by the statement
that singularity resolution is achieved through a (minimum-volume) quantum
bounce. As already stated, this singularity resolution is one of the landmark
features of LQC, and in this effective scheme it is generic: all solutions
undergo a minimum-volume bounce exactly once. One could argue that singularity
resolution is achieved due to the semiclassical assumptions underlying this
effective scheme, but it has been shown that the quantum bounce is likewise
“robust” in the full LQC (of the flat FLRW model with a free scalar field)
robustness .

Note that the modified Friedmann equation (10) incorporates in a rather simple
way the occurrence of the bounce (at $\rho=\rho_{\mathrm{c}}$). In the limit
$\lambda\rightarrow 0$ (no area gap) we recover the ordinary Friedmann
equation ${H}^{2}=\frac{8\pi G}{3}\rho$.

By making use of the Hamiltonian constraint, the equation for $\beta$ can be
decoupled and solved by direct integration. Upon substitution of the solution
$\beta=\beta(t)$ in (7) we find the corresponding solution for $V=V(t)$.
Similarly, we substitute the solution for $V=V(t)$ in (8) to get the solution
$\phi=\phi(t)$. We have,

$\displaystyle\beta(t)=\frac{1}{\lambda}\mathrm{arccot}\left(\frac{3t}{\gamma\lambda}\right),$
(13) $\displaystyle V(t)=C_{1}\sqrt{\gamma^{2}\lambda^{2}+9t^{2}},$ (14)
$\displaystyle\phi(t)=C_{2}+\frac{p_{\phi}}{3C_{1}}\mathrm{log}\left(3t+\sqrt{\gamma^{2}\lambda^{2}+9t^{2}}\right),$
(15) $\displaystyle p_{\phi}(t)=p_{\phi}.$ (16)

From (13) we note that the solution is consistent with
$\beta\in\left(0,\frac{\pi}{\lambda}\right)$ (by considering another “branch”
we get
$\beta\in\left[\left.-\frac{\pi}{2\lambda},\frac{\pi}{2\lambda}\right.\right)$,
which is also employed in the literature of effective LQC (see, for instance,
corichi-tatiana ).

###  I.2 Basics of the noncommutativity paradigm

We now turn to state the main ideas of the noncommutativity paradigm. Let us
first recall some basic facts of the geometrical formulation of classical
mechanics. Remember that the phase space $\Gamma=T^{\star}\mathcal{Q}$ (the
cotangent bundle of $\mathcal{Q}$), where $\mathcal{Q}$ is the configuration
space of a mechanical system, is a symplectic manifold, and that the inverse
$\mathsf{P}$ of the symplectic structure $\omega$
($\omega_{\alpha\beta}\mathsf{P}^{\beta\gamma}=\delta^{\gamma}_{\alpha}$)
defines the Poisson bracket $\\{\cdot,\cdot\\}$ (which endows $\Gamma$ with a
Lie algebra structure). The equations of motion

$\dot{q}^{a}=\partial_{p_{a}}\mathcal{H},\quad\dot{p}_{a}=-\partial_{q^{a}}\mathcal{H}$
(17)

of the corresponding mechanical system $(\Gamma,\mathcal{H})$ ($\mathcal{H}$
being the Hamiltonian of the system) can be written in a purely geometric
coordinate-independent way,

$\dot{x}^{\alpha}=\mathsf{P}^{\alpha\beta}(d\mathcal{H})_{\beta}.$ (18)

In canonical coordinates
$x=(x^{1},...,x^{2n})=(q^{1},q^{2},...,q^{n},p_{1},p_{2},...,p_{n})$ (i.e. in
a Darboux chart), the symplectic structure acquires the usual form
($a=1,...,n$)

$\omega=J_{\alpha\beta}dx^{\alpha}\wedge dx^{\beta}=dx^{n+a}\wedge
dx^{a}=dp_{a}\wedge dq^{a}.$ (19)

We therefore have (in canonical coordinates)

$\mathsf{P}=J^{\alpha\beta}\partial_{\alpha}\wedge\partial_{\beta}=\partial_{a}\wedge\partial_{n+a}=\partial_{q^{a}}\wedge\partial_{p_{a}}.$
(20)

(Notation $J_{\alpha\beta}$, $J^{\alpha\beta}$ is usually reserved for the
components of the symplectic and Poisson structures when expressed in a
Darboux chart.)

Poisson brackets are defined in terms of the Poisson structure by

$\\{f,g\\}:=\mathsf{P}(df,dg)=\mathsf{P}^{\alpha\beta}df_{\alpha}dg_{\beta}.$
(21)

In particular, for the canonical coordinates
$x=(x^{1},...,x^{2n})=(q^{1},q^{2},...,q^{n},p_{1},p_{2},...,p_{n})$, we have
the standard relations

$\\{q^{a},p_{b}\\}=\delta^{a}_{b},\qquad\\{q^{a},q^{b}\\}=\\{p_{a},p_{b}\\}=0.$
(22)

The noncommutativity paradigm we consider is closely related to the so called
quantum mechanics on phase space (QMPS), which provides an alternate way to
consistently formulate quantum mechanics (the recent monograph curtright
provides a comprehensive account on QMPS).

Roughly, in QMPS the canonical commutation relations

$[\hat{q}^{i},\hat{p}_{j}]=i\hbar\delta^{i}_{j},\quad[\hat{q}^{i},\hat{q}^{j}]=[\hat{p}_{i},\hat{p}_{j}]=0$
(23)

are realized by means of a deformation of the usual point-wise multiplication
of phase space functions, $f\cdot g\mapsto f\star g$. This star-product
($\star$-product) can be thought of as a “power series” of
$\frac{i\hbar}{2}\mathsf{P}(df,dg)$ (where $\mathsf{P}$ is the Poisson
structure of classical phase space) in which the “zeroth power” is the usual
commutative product $f\cdot g$. We can therefore obtain standard classical
mechanics by simply taking $\hbar\to 0$. By considering the (classical)
commutator $[f,g]_{\star}=f\star g-g\star f$, the usual canonical commutation
relations (23) are realized in classical phase space $\Gamma$. Central to the
scheme of QMPS is the so called Wigner function, which is a quasi-probability
distribution in phase space whose purpose was to provide quantum corrections
to the classical Liouville measure of classical statistical mechanics. In QMPS
probability amplitudes are calculated employing the Wigner function.

The connection with canonical quantum mechanics is given by working in the
well known Weyl representation (unitarily equivalent to the standard
Schrödinger representation) in which the canonical commutation relations (23)
acquire an “exponentiated form”; this representation is encoded in a
quantization prescription, the so called Weyl map $\mathcal{W}$, which
associates a (symmetric ordered) quantum operator $\hat{f}$ to a phase space
function $f$. The Weyl map $\mathcal{W}$ is invertible
($\mathcal{W}^{-1}(\hat{f})$ is called the Weyl symbol of $\hat{f}$ –it is
also called the Wigner transform of $\hat{f}$–). The Wigner function maps (up
to multiplicative factors of $\hbar$) to the density operator $\hat{\rho}$
under the Weyl transform (i.e. the Wigner function is the Wigner transform of
the density operator). The $\star$-product is the image of the operator
product (in the Weyl representation) under the Wigner transform, $f\star
g=\mathcal{W}^{-1}(\mathcal{W}(f)\mathcal{W}(g))$.

More explicitly, the Weyl-Wigner-Moyal (WWM) correspondence relates a phase
space function $f$ with its operator counterpart $\mathcal{W}(f)=\hat{f}$, via
the following map

$\displaystyle
f\mapsto\mathcal{W}(f)=\hat{f}(\hat{q},\hat{p}),\quad\mathrm{with}$
$\displaystyle\hat{f}(\hat{q},\hat{p})=\int\tilde{f}(\xi,\eta)\exp\left({\frac{i}{\hbar}\left(\hat{p}_{a}\xi^{a}+\hat{q}^{a}\eta_{a}\right)}\right)w(\xi,\eta)d^{l}\xi
d^{l}\eta,$ (24)

where $\tilde{f}$ is the Fourier transform of $f$, $w$ a weight function, and
$\dim(T^{\star}\mathcal{Q})=2l$; with $\hat{q}^{a},\hat{p}_{b}$ satisfying the
canonical Heisenberg algebra (23). We define the $\star$-product with the help
of this correspondence by

$\mathcal{W}(f\star g)=\mathcal{W}(f)\mathcal{W}(g)$ (25)

that is, $f\star g$ is the Weyl symbol of $\hat{f}\hat{g}$. For this case of
the Heisenberg algebra (23) and in the symmetric ordering ($w=1$) we have

$\displaystyle f\star g$
$\displaystyle=\exp\left(\frac{i\hbar}{2}\mathsf{P}(df,dg)\right)$
$\displaystyle=fg+\sum_{r=1}^{\infty}\left(\frac{i\hbar}{2}\right)^{r}\frac{1}{r!}\mathsf{P}^{r}(df,dg)$
$\displaystyle=fg+\frac{i\hbar}{2}\\{f,g\\}+\mathcal{O}(\hbar^{2})$ (26)

where in the expansion, terms of the form
$\mathsf{P}^{r}(df,dg)=J^{\mu_{1}\nu_{1}}\cdot\cdot\cdot
J^{\mu_{r}\nu_{r}}\left(\partial_{\mu_{1}...\mu_{r}}f\right)\left(\partial_{\nu_{1}...\nu_{r}}g\right)$
(the “rth power” of the Poisson bracket of $f$ and $g$) are to be considered.
This is the Moyal $\star$-product moyal , it replaces the ordinary point-wise
multiplication in the algebra of functions defined in phase space. The Moyal
$\star$-bracket $[f,g]_{\star}:=f\star g-g\star f$ is thus responsible for the
realization of the canonical Heisenberg algebra (23). This product, together
with the Wigner function, is the cornerstone of QMPS.

The noncommutativity paradigm employs some of the ideas and methods of QMPS
(most notably, the $\star$-product), and can therefore be viewed from the
perspective of the QMPS scheme: Given more general commutation relations, e.g.
(27), construct the corresponding $\star$-product by accordingly considering
an appropriate modification $\tilde{\mathsf{P}}$ to $\mathsf{P}$. This
$\tilde{\mathsf{P}}$ will be such that the classical Poisson relations
corresponding to the above commutation relations are fulfilled (in the same
dynamical variables). Once the $\star$-product is constructed, the advancement
of a noncommutative quantum mechanics in phase space (NCQMPS) can be pursued.
Instead, we view (27) as an extension to standard quantum mechanics, resulting
in a more noncommutative framework than the original standard one. The
implementation of such additional noncommutativity is achieved by implementing
a $\star$-product on quantum phase space. This point of view results in a
framework known as noncommutative quantum mechanics (NCQM) mezin .

The introduction of the modification $\tilde{\mathsf{P}}$ can, at the
classical level, be interpreted as a manifestation of the phase space
coordinates not being canonical, which in turn would imply that a new phase
space $\tilde{\Gamma}$ is being considered (a “deformation” to the original
one, $\Gamma$). An interesting point of view is the assumption that this new
$\tilde{\Gamma}$ is the true phase space of the mechanical system under study,
and that it was initially overlooked due to the noncommutativity scale being
unnoticed, that is, the mechanical system $(\tilde{\Gamma},\mathcal{H})$ is
considered as the true mechanical system, instead of $(\Gamma,\mathcal{H})$.
This phase space $\tilde{\Gamma}$ would therefore incorporate (noncommutative)
quantum corrections in a purely classical setup. In this interpretation, the
classical equations of motion would not be in canonical form in the original
(now non-canonical) variables $(y)$ (the Hamiltonian function $\mathcal{H}$
being the same). This deviation from the canonical form of the equations of
motion (noncommutative equations of motion) would therefore encode such
quantum corrections. Due to the Darboux theorem, one would be able to find a
new chart with local coordinates $x$ (via a transformation $y=\mathsf{T}(x)$)
in which $\tilde{\mathsf{P}}$ acquires the canonical form (20) (so that the
classical equations of motion would be in canonical standard form), but the
Hamiltonian function would be expressed as
$\mathcal{H}(y)\mapsto\mathcal{H}(x)=\mathcal{H}(\mathsf{T}(x))$; the quantum
corrections from noncommutativity would in this case be encoded in the
Hamiltonian $\mathcal{H}(x)$. In our investigations, when performing
computations, we prefer to work in canonical coordinates $(x)$ than with the
non canonical coordinates $(y)$ (which were canonical in the original
mechanical system $(\Gamma,\mathcal{H})$).

Consider for instance the deformed commutation relations
(“$\theta-$deformation” on momentum sector)

$[\hat{q}^{a},\hat{p}_{b}]=i\hbar\delta^{a}_{b},\quad[\hat{p}_{a},\hat{p}_{b}]=i\theta_{ab},$
(27)

with $\theta_{ab}$ an antisymmetric constant real matrix.

In light of the discussion above, quantum corrections associated to (27) could
be incorporated in a classical setup by considering the associated “deformed”
mechanical system $(\tilde{\Gamma},\mathcal{H})$, for which the Poisson
structure would take the form

$\mathsf{P}(y)=\partial_{y^{a}}\wedge\partial_{y^{n+a}}+\theta_{ab}\,\partial_{y^{n+a}}\wedge\partial_{y^{n+b}}\leavevmode\nobreak\
\leavevmode\nobreak\ {\rm with}\leavevmode\nobreak\ \leavevmode\nobreak\
(a<b)$ (28)

where $y=(q^{1},q^{2},...,q^{n},p_{1},p_{2},...,p_{n})$ are the phase space
coordinates used for the description of the original mechanical system
$(\Gamma,\mathcal{H})$. This concludes our succinct account on the basics of
LQC and $\theta-$deformed (quantum) mechanics.

The remaining of the manuscript is displayed as follows. Section (II)) is
devoted to quickly reviewing the model presented in prd . A careful analysis
of the corresponding post-bounce and pre-inflationary dynamics is presented in
section (III). In section (IV)) we discuss the main results and spell out some
perspectives and undergoing work.

##  II A Noncommutative effective lqc

The contents of this section are based on certain specific parts of Refs.
hindawi , ijmpd  and prd . Its purpose is to make the document somewhat self-
contained, and to highlight certain aspects which will be of relevance in the
development of the main part of the manuscript (Section III).

###  II.1 The free case

Consider the mechanical system $(\Gamma,\mathcal{H})$ defined by (5) and (4).
We would like to consider a new mechanical system
$(\tilde{\Gamma},\mathcal{H})$ which incorporates quantum corrections from an
underlying $\theta-$deformation analogous to (27). The Poisson algebra of this
new $(\tilde{\Gamma},\mathcal{H})$ is therefore given by

$\\{p^{{nc}}_{\phi},V^{{nc}}\\}=\theta,\quad\\{\beta^{{nc}},V^{{nc}}\\}=4\pi
G\gamma,\quad\\{\phi^{{nc}},p^{{nc}}_{\phi}\\}=1,$ (29)

with the remaining brackets zero as usual. The corresponding Poisson structure
$\tilde{\mathsf{P}}$ is therefore expressed as

$\tilde{\mathsf{P}}(y)=4\pi
G\gamma\,\partial_{\beta}\wedge\partial_{V}+\partial_{\phi}\wedge\partial_{p_{\phi}}+\theta\,\partial_{p_{\phi}}\wedge\partial_{V}.$
(30)

Here, variables $y=(\beta^{{nc}},V^{{nc}},\phi^{{nc}},p_{\phi}^{{nc}})$ are
the same variables which were canonical in the original system
$(\Gamma,\mathcal{H})$ defined by (4) and (5) (we append the label “${nc}$” in
order to remind us that they are non canonical variables in the new system
$(\tilde{\Gamma},\mathcal{H})$).

Now, the relations

$V^{{nc}}=V+a\theta\phi,\quad
p_{\phi}^{{nc}}=p_{\phi}+b\theta\beta,\quad\beta^{{nc}}=\beta,\quad\phi^{\mathrm{nc}}=\phi,$
(31)

with $4\pi G\gamma b-a=1$, define a family of phase space coordinate
transformations $x=\mathsf{T}(y)$ to canonical variables
$x=(\beta,\phi,V,p_{\phi})$ for system $(\tilde{\Gamma},\mathcal{H})$. As
already stated, we prefer to work with canonical coordinates so that the
equations of motion take the standard form. Accordingly, the equations of
motion associated to the Hamiltonian,

$\mathcal{H}^{{nc}}_{\mathrm{eff}}=-\frac{3}{8\pi
G\gamma^{2}\lambda^{2}}\sin^{2}(\lambda\beta)(V+a\theta\phi)+\frac{\left(p_{\phi}+b\theta\beta\right)^{2}}{2(V+a\theta\phi)},$
(32)

are to be computed in the standard way for variables
$x=(\beta,\phi,V,p_{\phi})$. Such equations are,

$\displaystyle\dot{\beta}=$ $\displaystyle 4\pi
G\gamma\frac{\partial\mathcal{H}^{{nc}}_{\mathrm{eff}}}{\partial
V}=-\frac{3}{\gamma\lambda^{2}}\sin^{2}(\lambda\beta),$ (33)
$\displaystyle\dot{V}=$ $\displaystyle-4\pi
G\gamma\frac{\partial\mathcal{H}^{{nc}}_{\mathrm{eff}}}{\partial\beta}=\frac{3}{\gamma\lambda}(V+a\theta\phi)\sin(\lambda\beta)\cos(\lambda\beta)$
$\displaystyle-4\pi G\gamma
b\theta\frac{p_{\phi}+b\theta\beta}{V+a\theta\phi},$ (34)
$\displaystyle\dot{\phi}=$
$\displaystyle\frac{\partial\mathcal{H}^{{nc}}_{\mathrm{eff}}}{\partial
p_{\phi}}=\frac{p_{\phi}+b\theta\beta}{V+a\theta\phi},$ (35)
$\displaystyle\dot{p}_{\phi}=$
$\displaystyle-\frac{\partial\mathcal{H}^{{nc}}_{\mathrm{eff}}}{\partial\phi}=-\frac{3a\theta}{8\pi
G\gamma^{2}\lambda^{2}}\sin^{2}(\lambda\beta)$
$\displaystyle-a\theta\frac{\left(p_{\phi}+b\theta\beta\right)^{2}}{2\left(V+a\theta\phi\right)^{2}}.$
(36)

We recover the usual (commutative) equations of motion by taking
$\theta\rightarrow 0$.

We note that the equation for $\beta$ is the same as in the commutative case,
so that
$\beta(t)=\frac{1}{\lambda}\mathrm{arccot}\left(\frac{3t}{\gamma\lambda}\right)$.
We also see that it is the case that $\beta$ is still a decreasing function of
time (taking values on $\left(0,\frac{\pi}{\lambda}\right)$).

From the Hamiltonian constraint and the equation of motion for $\phi$ it
follows that the energy density $\rho$ is exactly the same as in the standard
case:

$\rho=\frac{\dot{\phi}^{2}}{2}=\rho_{\mathrm{c}}\sin^{2}{\lambda\beta},$ (37)

and so it is still bounded and it attains the same maximum value $\rho_{c}$ as
in the commutative case. We emphasize that the energy density function not
only has the same functional form in terms of $\beta$ as in the usual case,
but that it is exactly the same function of time, since the solution
$\beta=\beta(t)$ is the same as in the standard commutative case.

The equation of motion for $V$ indicates that several stationary points of the
volume function are in principle possible, and therefore the volume function
could attain minima not necessarily all positive, and not necessarily
associated to the maximum of the energy density function. This scenario would
be in striking contrast with the standard commutative case, where all volume
solutions undergo precisely one minimum, and this minimum is always related to
the maximum in the energy density.

In order to be able to simplify the analysis required to establish wether or
not a single minimum-volume bounce is achieved, we restrict to the canonical
coordinates $x=(\beta,\phi,V,p_{\phi})$ given by the particular chart defined
by relations (31) with $b=0$ ($a=-1$). It is important to stress that this
choice merely amounts to fixing a Darboux chart in which further study of the
mechanical system will be carried. Working in this particular chart we observe
that by requiring $V^{{nc}}(t_{c})>0$ a single minimum-volume bounce
(associated to the maximum of the energy density) is guaranteed to take place.
Indeed, if such condition is fulfilled, we have $\dot{V}(t_{c})=0,\
\ddot{V}(t_{c})>0$ at $\rho(t_{c})=\rho_{c}$. Furthermore, given the monotonic
behavior of $\beta$ and the bounded character of $\rho$, this would be the
only stationary point of $V$. In other words, by studying the system in an
appropriate chart, we have shown that the occurrence of a (single) big bounce
at $t=t_{c}$ is assured for all solutions.

The solutions for the remaining functions are,

$\displaystyle V(t)=$ $\displaystyle
A\sqrt{9t^{2}+\gamma^{2}\lambda^{2}}-\frac{\theta(4\pi\gamma
bG-a)}{\lambda\sqrt{12\pi G}}\arctan\left(\frac{3t}{\gamma\lambda}\right)$
$\displaystyle+\frac{a\gamma\theta}{\sqrt{12\pi
G}}\frac{\log(\sqrt{9t^{2}+\gamma^{2}\lambda^{2}}+3t)}{\sqrt{9t^{2}+\gamma^{2}\lambda^{2}}},$
(38) $\displaystyle\phi(t)=$ $\displaystyle\frac{\gamma}{\sqrt{12\pi
G}}\log{(\sqrt{9t^{2}+\gamma^{2}\lambda^{2}}+3t)}.$ (39)

###  II.2 Inclusion of a generic potential term

Consider again the algebra (29), realized by working in canonical variables
defined by the family of transformations (31), and a scalar field $\phi$ with
generic potential term $\mathcal{V}(\phi)$.

The deformed Hamiltonian takes the form

$\mathcal{H}^{{nc}}_{\mathrm{eff}}=-\frac{3}{8\pi
G\gamma^{3}\lambda^{2}}\sin^{2}(\lambda\beta)V^{{nc}}+\frac{(p_{\phi}^{nc})^{2}}{2V^{{nc}}}+\mathcal{V(\phi)}V^{{nc}}.$
(40)

The respective noncommutative equations of motion are (note that both, the
Hamiltonian and the equations are expressions involving the canonical
variables $x=(\beta,\phi,V,p_{\phi})$, we only substitute the non-canonical
variables on the right hand side as a mean to abbreviate the expressions),

$\displaystyle\dot{\beta}=4\pi G\gamma\left[-\frac{3}{4\pi
G\gamma^{2}\lambda^{2}}\sin^{2}(\lambda\beta)+2\mathcal{V(\phi)}\right],$ (41)
$\displaystyle\dot{V}=\frac{3}{\gamma\lambda}V^{{nc}}\sin(\lambda\beta)\cos(\lambda\beta)-\frac{4\pi
G\gamma b\theta p_{\phi}^{{nc}}}{V^{{nc}}},$ (42)
$\displaystyle\dot{\phi}=\frac{p_{\phi}^{{nc}}}{V^{{nc}}},$ (43)
$\displaystyle\dot{p}_{\phi}=\frac{3a\theta}{4\pi
G\gamma^{2}\lambda^{2}}\sin^{2}(\lambda\beta)-2a\theta\mathcal{V(\phi)}-V^{{nc}}\partial_{\phi}\mathcal{V(\phi)}.$
(44)

We observe that in the limit $\theta\rightarrow 0$ the commutative field
equations are recovered. From the equation of motion for $\phi$ and the
constraint $\mathcal{H}_{\mathrm{eff}}^{{nc}}=0$, we can see that the energy
density takes the form
$\rho=\dot{\phi}^{2}/2+V(\phi)=\rho_{c}\sin^{2}(\lambda\beta)$, which is the
same functional form (in terms of $\beta$) as in the standard case, and
therefore attains the same maximum value $\rho_{c}$. However, the functional
form in terms of $t$ would not be in general the same as the standard case,
since the solution of $\beta(t)$ is likely to depend on the noncommutative
parameter $\theta$ (nonetheless, given the monotonic behavior of $\beta$, it
is expected that the behavior of the energy density function is in the overall
the same as in the standard case). It is also noted that several stationary
values of the volume function could in principle be attained, which in turn
could result in several bouncing scenarios (depending on the evolution of all
dynamical variables).

Guided by the free case, we consider again

$b=0,\ \ V^{{nc}}(t_{c})>0$ (45)

(where, as before, $t_{c}$ is such that $\rho(t_{c})=\rho_{c}$). The
fulfillment of (45) is a sufficient condition for a minimum volume bounce to
be reached when the maximum $\rho_{\mathrm{c}}$ of the energy density $\rho$
is attained. That is to say, condition (45) implies that the standard LQC big
bounce is preserved. This will be the only bouncing scenario since $\beta$ is
a decreasing function of time (as (41) indicates), taking values on
$\left(0,\frac{\pi}{\lambda}\right)$, and since $V^{{nc}}=0$ is not allowed
during evolution (in view of the equation of motion for $\phi$ and the energy
density being bounded).

On the other hand, condition

$\dot{\phi}(t_{c})=0,\ b\theta\ddot{\phi}(t_{c})<0$ (46)

which translates into

$\rho(\phi(t_{c}))=\mathcal{V}(\phi(t_{c})),\ b\theta\ddot{\phi}(t_{c})<0,$
(47)

is also a sufficient condition for the occurrence of a minimum volume bounce.
However, it does not precludes the existence of additional bounces (which
could lead to additional minima of volume, not necessarily all positive),
hence, singularity resolution is not guaranteed to always be achieved in this
latter scenario. Condition (46) would allow to fix the value of $\phi$ at this
minimum volume bounce. Furthermore, from the equation of motion for $\phi$ it
is observed that
$p_{\phi}(t_{c})=-b\theta\beta(t_{c})=-\frac{b\theta\pi}{2\lambda}$ must hold.
We are facing the following situation: by demanding (through condition (46))
that a minimum volume bounce takes place when the energy density reaches its
maximum value, initial conditions for all dynamical variables (except for the
volume) get fixed at the bounce. This condition therefore leads to a
dramatically reduced number of solutions which feature a bounce resembling the
(single) big bounce of standard LQC. This state of affairs is obviously not
particularly appealing.

For the scalar field we can construct the modified Klein-Gordon equation using
(41)-(44):

$\ddot{\phi}+\frac{1}{V^{{nc}}}\left[\dot{V}^{{nc}}+a\theta\phi\right]\dot{\phi}+\partial_{\phi}\mathcal{V}(\phi)=0,$
(48)

we can see that in the limit $\theta\to 0$ we recover the commutative Klein-
Gordon equation for the scalar field.

As already stated, in prd  we have also shown that this noncommutative
extension to effective LQC supports universes which (apart from avoiding the
initial singularity by means of a single minimum-volume bounce) exhibit a
sufficiently long period of inflation for $\mathcal{V}(\phi)=1/2m\phi^{2}$.

##  III Inflation in Noncommutative Effective LQC

In this section we will focus on post-bounce and pre-inflationary dynamics by
specifying a potential $\mathcal{V}(\phi)$. Specifically, we consider the
chaotic potential $\mathcal{V}(\phi)=\frac{1}{2}m^{2}\phi$, where
$m=1.26\times 10^{-6}\leavevmode\nobreak\ m_{pl}$. With the specified
potential we solve the field equations (41-44) in the same way as in prd , for
a given set of initial conditions $\phi_{B}=\phi(t_{B})$,
$\dot{\phi}_{B}=\dot{\phi}(t_{B})$, where $t_{B}$ is the time at the bounce.
In the initial investigation prd  a $\theta-$deformation in the momentum
sector of phase-space was shown to support an inflationary period. In this
section we will analyze such inflationary model in more detail.

Before beginning with the analysis, we will introduce some essential
ingredients related to inflation.

  * •

We will denote the ratio of the potential energy to the critical density by

$F_{B}=\frac{\mathcal{V}(\phi_{B})}{\rho_{c}},$ (49)

depending on the initial conditions, $F_{B}$ takes values between $[0,1]$, and
as in sloan , depending on the values that $F_{B}$ takes, we can define
different energy stages after the bounce. If $F_{B}<10^{-4}$, we say that we
are in the extreme kinetic energy domination (EKED) regime;
$10^{-4}<F_{B}<0.5$ characterizes the kinetic energy domination (KED) regime;
the potential energy domination (PED) regime is attained by $0.5<F_{B}<1$.

  * •

The equation of state parameter $\omega_{\phi}$ of the scalar field is given
by

$\omega_{\phi}=\frac{P(t)}{\rho(t)}=\frac{\frac{1}{2}\dot{\phi}-\mathcal{V}(\phi)}{\frac{1}{2}\dot{\phi}+\mathcal{V}(\phi)},$
(50)

it can take values in the interval $[-1,1]$. The values corresponding to
$\omega_{\phi}=-1$ give a PED regime, while for $\omega_{\phi}=1$ a KED is
achieved. The equation of state parameter allows us to divide the evolution of
the scalar field before reheating into three stages, similar to LQC cases
Parampreet ; piulqc , these stages are: the bouncing stage, transitions stage
and slow roll inflation stage.

  * •

In the bouncing stage dynamics there is a period of super inflation (SI),
which corresponds to an increasing behavior for the Hubble parameter. This
epoch begins at the bounce and ends when the Hubble parameter reaches its
maximum value ($\dot{H}=0$).

  * •

The slow-roll Hubble parameters $\epsilon_{H}$ and $\eta_{H}$, are given by

$\epsilon_{H}=-\frac{\dot{H}}{H^{2}},\qquad\eta_{H}=-\frac{\ddot{H}}{2\dot{H}H}.$
(51)

To guarantee an accelerated expansion and a slow-roll inflation, these
parameters must meet the conditions $\epsilon_{H}\ll 1$, $\eta_{H}\ll 1$. The
number of e-folds corresponding to this period is

$N=\ln{\left(\frac{a_{end}}{a_{i}}\right)},$ (52)

where $a_{end}$ is the value of $a$ at the end of slow-roll inflation and
$a_{i}$ its value at the onset. A period of early inflationary expansion
consistent with observations demands $N\geq 60$ wmap .

  * •

For the scalar field, the continuity equation and Klein-Gordon equation take
the usual form,

$\displaystyle\ddot{\phi}+3H\dot{\phi}+\partial_{\phi}\mathcal{V}(\phi)$
$\displaystyle=0,$ (53) $\displaystyle\dot{\rho}+3H(\rho+P)$
$\displaystyle=0.$ (54)

The modified Klein-Gordon equation for $\phi(t)$ and the corresponding
continuity equation are given by

$\displaystyle\ddot{\phi}+3H^{nc}\dot{\phi}+\partial_{\phi}\mathcal{V}(\phi)$
$\displaystyle=0,$ (55) $\displaystyle\dot{\rho}+3H^{nc}(\rho+p)$
$\displaystyle=0,$ (56)

where we define the the noncommutative Hubble parameter as

$H^{nc}\equiv\frac{\dot{V}^{nc}}{3V^{nc}}.$ (57)

  * •

The modified noncommutative Friedmann equation (ijmpd ) is

$H^{2}=\frac{8\pi
G}{3}\rho\left[1-\frac{\rho+\rho_{\theta}}{\rho_{c}}\right],$ (58)

where $\rho_{\theta}=\rho_{1}(\theta)+\rho_{2}(\theta^{2})$, and these are
given as

$\displaystyle\rho_{1}(\theta)$ $\displaystyle=$
$\displaystyle\frac{2a\theta\phi}{V}(\rho_{c}-\rho)$ (59)
$\displaystyle+\frac{\sqrt{2}b\theta}{\gamma\lambda
V}\sqrt{\frac{\rho_{c}}{\rho}\left(1-\frac{\rho}{\rho_{c}}\right)(\rho-\mathcal{V}(\phi))},$
$\displaystyle\rho_{2}(\theta^{2})$ $\displaystyle=$
$\displaystyle\frac{a^{2}\theta^{2}\phi^{2}}{V^{2}}(\rho_{c}-\rho)$ (60)
$\displaystyle+\frac{\sqrt{2}ab\theta^{2}\phi}{\gamma\lambda
V}\sqrt{\frac{\rho_{c}}{\rho}\left(1-\frac{\rho}{\rho_{c}}\right)(\rho-\mathcal{V}(\phi))}$
$\displaystyle+\frac{4\pi
Gb^{2}\theta^{2}}{3V^{2}}\rho_{c}\left(1-\frac{\mathcal{V}(\phi)}{\rho}\right).$

The complexity of the form of the modified Friedmann equation makes it
difficult for us to obtain an analytical form of the condition (51). In this
case we will carry out a numerical analysis in order to see the effects and
properties of noncommutativity in the inflationary model of LQC.

With these defined quantities we are in a position to perform an analysis of
the pre-inflationary and inflationary dynamics. As we saw above, there are two
initial conditions that guarantee that the noncommutative model has a bounce
with the same characteristics of standard LQC. In the following, we will
discuss how these conditions affect the inflationary dynamics.

###  III.1 $b=0$, $V^{nc}(t_{b})>0$

In this section we will present, with the help of some plots, the behavior of
the evolution of the universe for the different periods EKED, KED, PED, after
the bounce, considering condition $b=0$ (for particular initial conditions).

Taking the Hamiltonian (40) specialized to the quadratic potential term, we
get the equations of motion,

$\mathcal{H}^{nc}_{\mathrm{eff}}=-\frac{3}{8\pi
G\gamma^{2}\lambda^{2}}\sin^{2}(\lambda\beta)V^{nc}+\frac{(p^{nc}_{\phi})^{2}}{2V^{nc}}+\frac{1}{2}m^{2}\phi^{2}V^{nc}$
(61) $\displaystyle\dot{\beta}$ $\displaystyle=$
$\displaystyle-\frac{3}{\gamma\lambda^{2}}\sin^{2}(\lambda\beta)+4\pi G\gamma
m^{2}\phi^{2},$ (62) $\displaystyle\dot{V}$ $\displaystyle=$
$\displaystyle\frac{3}{\gamma\lambda}V^{nc}\sin(\lambda\beta)\cos(\lambda\beta)-4\pi
G\gamma b\theta\frac{p_{\phi}^{nc}}{V^{nc}},$ (63) $\displaystyle\dot{\phi}$
$\displaystyle=$ $\displaystyle\frac{p^{nc}_{\phi}}{V^{nc}},$ (64)
$\displaystyle\dot{p}_{\phi}$ $\displaystyle=$
$\displaystyle\frac{3a\theta}{8\pi
G\gamma^{2}\lambda^{2}}\sin^{2}(\lambda\beta)+a\theta\frac{(p_{\phi}^{nc})^{2}}{2(V^{nc})^{2}}-m^{2}V^{nc}\phi$
(65) $\displaystyle-\frac{1}{2}a\theta m^{2}\phi^{2},$

of course, these field equations reduce to the commutative ones when taking
$\theta\rightarrow 0$. Some numerical solutions for this case are given in prd
.

####  III.1.1 Extreme Kinetic Energy Domination, after the bounce

We will start the analysis with the initial conditions for an EKED after the
bounce, $F_{B}<10^{-4}$, $\phi(B)=0$ and $\dot{\phi}(B)=0.8$. Analytical
solutions to the equations of motion (62-65) could not be found. However, some
curves constructed via numerical solutions are plotted. The main results are
shown in Figs. 1. The scale factor has an exponential growth, whereas at
higher values of the noncommutative parameter, $\theta$, the growth of the
scale factor is lower. On the other hand, the energy density at early times is
not affected by the noncommutativity, but at larger times there is a lag in
the energy density, particularly, noncommutativity accelerates the decay of
the energy density with respect to time, as shown in d) of Fig (1).

Figure 1: Time evolution of several quantities in the EKED period of the
commutative and noncommutative effective model of LQC. The initial conditions
for the numerical results are $\phi_{B}=0$ $\dot{\phi}_{B}=0.8$, $F_{B}=0$ so
we are in an EKED after the bounce, $m=1.26\times 10^{-6}\rm{mpl}$, $b=0$. The
graphs represent the commutative evolution $\theta=0$, and the noncommutative
evolution $\theta=0.05$ and $\theta=0.1$.

Fig. 2 shows the behavior of the total, kinetic and potential energy densities
for the noncommutative and commutative cases, here the domination periods of
kinetic and potential energy are clearly exemplified, given the initial
conditions.

Figure 2: The figure shows the behavior of the energy density $\rho$, the
kinetic energy $T$ and the potential energy $V$, of the commutative and
noncommutative models of LQC for the EKED domain. We take the same conditions
as in Fig 1

As previously pointed out, for the EKED evolution the scalar field takes
values such that $\mathcal{V}(\phi)=0$ and the Hubble parameter $H$ vanishes.
In parts b) and c) of Figure 1 we can see the behavior of the inflaton field.
The Klein-Gordon equation tells us that, for the given the initial condition,
the inflaton will slow down, $\ddot{\phi}<0$, which is due to the friction
term $-3H\dot{\phi}$, causing a loss of kinetic energy and a gain of potential
energy. These behaviors can be observed in Figure 2. By incorporating
noncommutativity into the model, we analyze the form of the modified Klein-
Gordon equation (55). The behavior will be similar due to the form of Eq.
(55), the term that will carry the noncommutativity correction will be the
friction term $-3H_{nc}\dot{\phi}$. These corrections are shown in Figure 2.
For the EKED at the bounce, the dynamics of the universe can be divided for
analysis into stages. In order to have a more detailed picture of the energy
domination periods, we focus on the equation of state parameter,
$\omega_{\phi}$. Figure 3 shows the behavior for the commutative and
noncommutative scenarios. From Figure 3 we identify three stages, the kinetic
energy domination, the bounce period, the transition period, and the
inflationary period, similar to Parampreet , which correspond to the pre-
inflationary and inflationary dynamics of the universe. A first analysis of
Fig. 3 shows that the effect of noncommutativity is to extend the bouncing,
transition, and slow-roll inflation stages, but a more detailed analysis is
necessary. Next will give a more elaborated discussion of these results for
each of these stages.

Figure 3: Evolution of the equation of state $\omega_{\phi}$ for the domain of EKED for the commutative and noncommutative models. We take the same conditions as in Fig 1 $b=0$ | Event | $t$ | $\phi$ | $\dot{\phi}$ | $H$ | $\dot{H}$ | $N$
---|---|---|---|---|---|---|---
| Bounce | 0 | 0 | 0.8 | 0 | 0 | 0
| End SI | 0.333 | 0.235 | 0.566 | 0.5 | $-9.185\times 10^{-17}$ | 0.115
$\theta=0$ | KE=PE | $3.457\times 10^{4}$ | 3.250 | $6.844\times 10^{-6}$ | $1.185\times 10^{-5}$ | $-2.107\times 10^{-10}$ | 3.886
| Onset Slow Roll | $1.075\times 10^{5}$ | 3.444 | $6.947\times 10^{-7}$ | $8.882\times 10^{-6}$ | $-4.643\times 10^{-17}$ | 4.574
| End of Inflation | $1.629\times 10^{7}$ | 0.2013347 | $-1.793801\times 10^{-7}$ | $6.358859\times 10^{-7}$ | $-4.043509\times 10^{-13}$ | 76.594
| Bounce | 0 | 0 | 0.8 | 0 | 0 | 0
| End SI | 0.338 | 0.232 | 0.537 | 0.529 | $-2.687\times 10^{-16}$ | 0.123
$\theta=0.05$ | KE=PE | $3.704\times 10^{4}$ | 3.034 | $5.872\times 10^{-6}$ | $1.106\times 10^{-5}$ | $-1.836\times 10^{-10}$ | 3.955
| Onset Slow Roll | $1.166\times 10^{5}$ | 3.212 | $5.463\times 10^{-7}$ | $8.282\times 10^{-6}$ | $-2.399\times 10^{-16}$ | 4.654
| End of Inflation | $1.514\times 10^{7}$ | 0.2013346 | $-1.793800\times 10^{-7}$ | $6.358856\times 10^{-7}$ | $-4.043505\times 10^{-13}$ | 66.599
| Bounce | 0 | 0 | 0.8 | 0 | 0 | 0
| End SI | 0.340 | 0.228 | 0.515 | 0.557 | $1.862\times 10^{-16}$ | 0.129
$\theta=0.1$ | KE=PE | $3.900\times 10^{4}$ | 2.881 | $5.240\times 10^{-6}$ | $1.051\times 10^{-5}$ | $-1.656\times 10^{-10}$ | 4.010
| Onset Slow Roll | $1.240\times 10^{5}$ | 3.048 | $4.505\times 10^{-7}$ | $7.861\times 10^{-6}$ | $-2.975\times 10^{-16}$ | 4.718
| End of Inflation | $1.434\times 10^{7}$ | 0.2013348 | $-1.793802\times 10^{-7}$ | $6.358861\times 10^{-7}$ | $-4.043512\times 10^{-13}$ | 59.995
| Bounce | 0 | 0.917 | 0.8 | $6.123\times 10^{-17}$ | 0 | 0
| End SI | 0.333 | 1.152 | 0.566 | 0.5 | $-4.249\times 10^{-16}$ | 0.115
$\theta=0$ | KE=PE | $2.744\times 10^{4}$ | 4.105 | $8.609\times 10^{-6}$ | $1.497\times 10^{-5}$ | $-3.362\times 10^{-10}$ | 3.810
| Onset Slow Roll | $9.170\times 10^{4}$ | 4.306 | $7.095\times 10^{-7}$ | $1.110\times 10^{-5}$ | $3.159\times 10^{-17}$ | 4.567
| End of Inflation | $2.044\times 10^{7}$ | 0.2013346 | $-1.793800\times 10^{-7}$ | $6.358856\times 10^{-7}$ | $-4.043506\times 10^{-13}$ | 118.649
| Bounce | 0 | 0.917 | 0.8 | $2.811\times 10^{-17}$ | 0 | 0
| End SI | 0.303 | 1.217 | 0.442 | 0.337 | $1.432\times 10^{-17}$ | 0.133
$\theta=0.05$ | KE=PE | $2.862\times 10^{4}$ | 3.937 | $7.735\times 10^{-6}$ | $1.436\times 10^{-5}$ | $-3.092\times 10^{-10}$ | 3.956
| Onset Slow Roll | $9.680\times 10^{4}$ | 4.125 | $5.901\times 10^{-7}$ | $1.064\times 10^{-5}$ | $-3.392\times 10^{-16}$ | 4.725
| End of Inflation | $1.955\times 10^{7}$ | $0.2013348$ | $-1.793802\times 10^{-7}$ | $6.358862\times 10^{-7}$ | $-4.043512\times 10^{-13}$ | 108.846
| Bounce | 0 | 0.917 | 0.8 | 0 | 0 | 0
| End SI | 0.281 | 1.317 | 0.297 | 0.237 | 0 | 0.145
$\theta=0.1$ | KE=PE | $2.922\times 10^{4}$ | 3.857 | $7.338\times 10^{-6}$ | $1.406\times 10^{-5}$ | $-2.967\times 10^{-10}$ | 4.057
| Onset Slow Roll | $9.945\times 10^{4}$ | 4.039 | $5.363\times 10^{-7}$ | $1.041\times 10^{-5}$ | $-6.402\times 10^{-16}$ | 4.833
| End of Inflation | $1.913\times 10^{7}$ | $0.2013347$ | $-1.793800\times 10^{-7}$ | $6.358858\times 10^{-7}$ | $-4.043508\times 10^{-13}$ | 104.330
Table 1: The table shows the numerical results of the commutative and
noncommutative models of LQC with the potential $\phi^{2}$. The Events:
bounce, the end of SI (superinflation), the equilibrium point KE=PE, the onset
of slow-ll inflation, and the end of inflation, are compared for $\theta=0$,
$\theta=0.05$ and $\theta=0.1$.

  * •

The bouncing stage EKED
The beginning of this stage is at the bounce $t_{B}^{EKED}$ and ends at the
time $t_{eq}^{EKED}$ when the kinetic energy of the scalar field is equal to
the potential energy, regarding the equation of state parameter (50), this
occurs when $\omega_{\phi}=0$. At the end of the bouncing stage, the LQG-
related corrections can be neglected and the classical Friedmann dynamics can
be recovered. For the initial conditions $(\phi_{B}=0,\leavevmode\nobreak\
\dot{\phi}_{B}=0.8)$ the end of the bouncing stage is at
$t_{eq}^{EKED}=3.457\times 10^{4}$ as shown in Table 1. At this stage there is
a period where the Hubble parameter grows very rapidly, which we call super
inflation (SI). This SI period lasts for fractions of seconds. In a) of Fig. 4
the behavior of the Hubble parameter in the SI period is shown. On the other
hand, in b) of Fig. 4 we can see that the SI period comes to an end
($\dot{H}=0$) at $t_{SI}^{EKED}=0.333$, and it lasts for
$N(t_{SI}^{EKED})=0.115$ e-folds, for the commutative case, Table 1. These
results are similar to those reported in sloan  for EKED.

Noncommutativity effects in this stage vary depending on the evolution of the
scalar field. In SI, Fig. 4, the noncommutativity effects cause the duration
of SI to be larger and reach a larger growth of the commutative Hubble
parameter $H(t_{SI}^{EKED})=0.5$, for $\theta=0.05$ and $\theta=0.01$ the end
of SI occurs at $t_{SI_{,0.05}}^{EKED}=0.338$ and $t_{SI_{,0.1}}^{EKED}=0.340$
respectively, while the values of maxima reached by $H$ are
$H(t_{SI_{,0.05}}^{EKED})=0.529$ and $H(t_{SI_{,0.1}}^{EKED})=0.557$. On the
other hand, from the SI term, the noncommutativity effects extend the bouncing
stage, for example the bouncing stage term occurs at
$t_{eq_{,0.05}}^{EKED}=3.704\times 10^{4}$ for $\theta=0.05$, while which for
$\theta=0.1$ $t_{eq_{,0.1}}^{EKED}=3.900\times 10^{4}$. These data are
reported in Table 1.

Figure 4: The figure represents in a) the graph of the Hubble parameter
$H(t)$, in b) the graph of $\dot{H}(t)$ for EKED of the commutative and
noncommutative LQC models. We take the same conditions as in Fig 1

As already pointed out, at this stage, LQG-related quantum corrections are
present, so the primordial power spectra are affected by them. In this
respect, several investigations have been conducted (see, e.g., section 4.2 of
the first review in reviews-lqg ). Two specific frameworks have been mostly
employed, the so called dressed-metric approach (see, for instance, dressed-
metric  and references therein) and the hybrid quantization scheme (a recent
review on the latter is given in Ref. hybrid-quant ; in dressed-metric-hybrid-
quant  a relation between the two approaches at the effective level is
established). As we have seen, noncommutativity effects appear to be present
in a rather simple form at this stage (essentially extending its time
duration). We therefore stress that, within the scope of the present analysis,
the universe predicted by the $\theta-$deformed effective model put forward
presently does not deviates significantly from the universe predicted by
standard (effective) LQC. Put another way, for appropriate values of the
$\theta$ parameter, both models are essentially indistinguishable. This
conclusion had been anticipated in the preliminary investigation prd , and the
present analysis puts such conclusion on rather firmer grounds. However, a
focused study on the primordial perturbations (which are not included here)
may prove to be interesting and seems to be a natural complement to the
present analysis: Noncommutativity may influence the universe in a more subtle
manner at this more fundamental level, and tuning of the $\theta$ parameter
could be exploited with an eye on taming power spectrum-related tensions still
prevalent on standard LQC. Such a study lies outside the scope of the present
investigation but it is currently being pursued by the authors within the
hybrid quantization scheme, and will be reported elsewhere.

  * •

Transition stage EKED
In the transition period the kinetic energy decreases abruptly, which causes
the universe to accelerate, while $\omega_{\phi}^{EKED}$ tends to behave in a
constant manner. This stage starts at equal energies $\omega_{\phi}^{EKED}=0$
and ends when $\omega_{\phi}^{EKED}=-1$ (see Fig. 3). For the commutative
case, the stage starts at $t_{eq}=3.457\times 10^{4}$ and ends at
$t_{i}=1.075\times 10^{5}$, where $t_{i}$ will also be the time at which slow-
roll inflation begins. In terms of e-folds, this period is shorter in duration
than the bouncing and slow-roll inflation stages, this kind of behavior is
also reported in piulqc . The noncommutativity effects in this stage remain
the same as those at the end of the bouncing stage, at larger values of the
noncommutative parameter $\theta$, the duration of the transition stage is
prolonged, for example, in the commutative case,we have $N_{tr}=0.688$, while
for the noncommutative cases $\theta=0.05$ and $\theta=0.1$ are
$N_{tr_{0.05}}=0.699$ and $N_{tr_{0.1}}=0.708$, respectively (Table 1).

  * •

Slow-roll inflation EKED
When $\omega_{\phi}=-1$ the slow-roll inflation stage begins. At this point
the energy density is totally dominated by the potential $\phi^{2}$, and a
time $t_{i}=1.075\times 10^{5}$ has elapsed , the corresponding value of the
field is $\phi(t_{i})=3.444$. These results are similar to those reported in
sloan ; Parampreet . Figure 5 shows the regions in which inflationary periods
are found, $\epsilon_{H}\ll 1$. This stage continues until the value of
$\epsilon_{H}=1$ is reached, which occurs at $t_{end}=1.629\times 10^{7}$ and
$N=76.594$. A period of early inflationary expansion consistent with
observations demands $N\geq 60$ wmap , the results are shown in Figure 6 and
Table 1.These results indicate that for EKED after the bounce, there is an
inflationary period with more than 60 e-folds for the given initial
conditions.

Figure 5: Plot of the commutative and noncommutative slow-roll parameter
$\epsilon(t)$ for the EKED period. We take the same conditions as in Fig 1

In the noncommutative scenario, slow-roll inflationary dynamics behave in a
similar way to its commutative counterpart. However, the effects of
noncommutativity imply that the duration of the inflationary period is shorter
when the parameter $\theta$ increases in value. We can see this by calculating
the number of e-folds as a function of $t$,

$N(t)=\int_{0}^{t}H(\tau)d\tau,$ (66)

where $H(\tau)$ is given by (58). Fig. 6 shows the behavior of $N(t)$. As
shown in Table 1, the corresponding number of e-folds for $\theta=0.05,0.1$
are $N_{.05}=66.599$ and $N_{.1}=59.995$, respectively, being less in number
than those obtained in the commutative counterpart. Given this behavior, large
values of $\theta$ imply that the duration of the inflationary period does not
reach the minimum number of e-folds required for the universe to evolve up to
the present epoch, this is one of the reasons why we do not notice the
noncommutative effects, for the universe to evolve in the proper way it is
necessary that $\theta\ll 1$.

Figure 6: Plot of the commutative and noncommutative e-folds $N(t)$ for the
EKED period. We take the same conditions as in Fig 1

The behavior described above for $N(t)$ holds for the initial conditions
$(\phi_{B}>0,\leavevmode\nobreak\ \dot{\phi}_{B}>=0.5)$, Fig. 7 shows some
examples with initial conditions in this range. A peculiar behavior of the
noncommutative model is that for initial conditions
$(\phi_{B}>0,\leavevmode\nobreak\ \dot{\phi}_{B}<0.5)$, the number of e-folds,
$N(t)$, increases as the parameter $\theta$ increases. Therefore, in this
scenario, it could very well be the case that commutative standard models with
initial conditions not consistent with long enough inflationary periods could,
upon introduction of noncommutativity, transit to models that can accommodate
for a sufficiently large number of e-folds consistent with cosmological
observations (see Fig. 8). In standard LQC, theoretical studies on global
aspects of the space of solutions suggest that these initial conditions not
consistent with a sufficiently long inflationary period are otherwise rare
sloan . Assuming, nonetheless, that the universe evolved from such uncommon,
from the point of view of LQC, initial conditions could help to find an upper
bound for the noncommutative parameter.

Figure 7: The Figure shows the commutative and noncommutative $N(t)$, for
different initial conditions $(\phi_{B},\dot{\phi}_{B})$. Each of the columns
keeps $\dot{\phi}_{B}$ constant and for each row $\phi_{B}$ vary. We take
values of $\dot{\phi}_{\geq}0.5$. Figure 8: The Figure shows the commutative
and noncommutative $N(t)$, for different initial conditions
$(\phi_{B},\dot{\phi}_{B})$. Each of the columns keeps $\dot{\phi}_{B}$
constant and for each row $\phi_{B}$ vary. We take values of
$\dot{\phi}_{<}0.5$.

####  III.1.2 Kinetic Energy Domination, after the bounce

For an KED after the bounce $10^{-4}<F_{B}<0.5$, we take $\phi(B)=5\times
10^{4}$ and $\dot{\phi}(B)=0.8$, $F_{B}=1.662\times 10^{-2}$. The commutative
behavior of the solutions for the initial conditions is similar to the case of
EKED, there is a bounce at $t_{B}=0$ and an exponential growth in the scale
factor (see Fig. 9). In contrast to the EKED case, the larger the values taken
by the noncommutative parameter, $\theta$, the faster $a(t)$ grows.

Figure 9: Numerical solution for the scale factor evolution $a(t)$,
considering the commutative and noncommutative model with the potential
$\phi^{2}$, for the initial conditions $\phi_{B}=5\times 10^{4}$, $\dot{\
}phi_{B}=0.8$, $m=1.26^{-6}$, $b=0$, in KED after the bounce.

In this KED period, the bouncing stage, the transition stage and the slow-roll
inflation stage are still featured, although the duration of these stages is
considerably modified. In Fig. 10 we can see this behavior, as well as the
variation between the commutative and noncommutative parts, in the
$\omega_{\phi}$, is minimal.

Figure 10: Numerical solution for the state equation evolution
$\omega_{\phi}$,in KED after the bounce. We take the same initial condition as
the Fig 9.

  * •

The bouncing stage KED
Again we find an SI stage, in which we have the growth of $H(t)$, where it
reaches a maximum value of $H_{max}=0.5$. The duration of this period is
$t=0.341$ with $N=0.118$, similar to that of EKED ($t=0.333$,$N=0.115$). The
noncommutative model of SI presents a much shorter duration than its
commutative counterpart, $t=3.503\times 10^{-3}$ for $\theta=0.05$, while for
$\theta=0.1$ , SI ends at $t=2.477\times 10^{-3}$. Another characteristic that
differentiates the EKED and KED periods from their commutative counterparts
are the maximums reached by the Hubble parameter $H(t)$, while for the
commutative case the difference is minimal, for the noncommutative case there
are big differences. For example, in EKED $H_{0.05}^{\rm{\small EKED}}=0.529$,
$H_{0.1}^{\tiny EKED}=0.557$, while in KED $H_{0.05}^{KED}=95.137$,
$H_{0.1}^{KED}=134.544$. From the point of view of the evolution of $H(t)$,
the commutative case KED behaves similarly to EKED for the maximum value of
$H(t)$. Analyzing the behavior of $H(t)$, for the noncommutative case, the
increase of $\theta$ causes the maximum value of $H(t)$ to increase, so the
existence of $\theta$ implies that the SI stage has a larger amplitude
(regarding the Hubble parameter) compared to its commutative counterpart, see
Fig. 11. The end of the bouncing stage occurs when KE = PE, at
$\omega_{\phi}^{KED}=0$. In the case of KED, the commutative and
noncommutative behaviors at this time are similar up to $10^{-6}$, and there
are fluctuations of up to $10^{-7}$.

Figure 11: Noncommutative and commutative numerical solution for the Hubble
parameter evolution $H(t)$,in KED after the bounce. We take the same initial
condition as the Fig 9. In part a) it is shown how the noncommutative
parameter $\theta$ causes a considerable increase in the maximum value of
$H(t)$ for the SI stage. b) It is shown how $H(t)$ for the commutative case
has a SI stage.

  * •

Transition stage KED
In the KED period there is a much shorter transition stage than in EKED. For
the commutative case, the EKED period has a duration of $\Delta
t^{EKED}_{tr}=72930$, while the KED period lasts $\Delta
t^{KED}_{tr}=21.3664$. For the noncommutative case there is a similar
behavior. As in EKED, the kinetic energy decreases rapidly, the end of the
transition stage happens when $\omega_{\phi}^{KED}=-1$, for the commutative
case this happens at $t_{tr}^{KED}=23.6330$, while for the noncommutative
cases $\theta=0.05$ and $\theta=0.1$ it happens at $t_{tr,0.05}^{KED}=23.6336$
and $t_{tr,0.1}^{KED}=23.6329$ respectively, which is consistent with Fig. 10.

  * •

Slow-roll inflation KED
Fig. 12 shows that there is a region in which $\epsilon^{KED}<1$, which
implies that the commutative and noncommutative models experience an
accelerated expansion. In Fig. 12 there are two regions in which
$\epsilon^{KED}<1$, the first region consists of the super inflation stage,
while the second region starts at $t_{i}^{KED}=2.919$ for the commutative
case, while for the noncommutative model it starts at
$t_{i,0.05}^{KED}=3.22962$ and $t_{i,0.1}^{KED}=3.22963$. At
$t_{i}^{KED},t_{i,\theta}^{KED}$ times, the universe begins to undergo
acceleration, but also $\omega_{\phi}^{KED}\neq-1$, so there are still
contributions from kinetic energy. To ensure that we are in slow-roll
inflation it is necessary to review the second parameter of slow-roll.

Figure 12: Noncommutative and commutative numerical solution for the slow-roll
parameter evolution $\epsilon(t)$,in KED after the bounce. We take the same
initial condition as the Fig. 9.

Fig. 13 show us the behavior of the second slow-roll parameter $|\eta|$. As in
Parampreet , the beginning of the slow-roll inflation stage is when
$|\eta|<0.03$. Fig. 13 shows us that $|\eta|$ has large oscillations in its
behavior, but on average its value is decreasing and tends to zero. This
happens (at an average value) when $t=111.0405$, $t_{0.05}=86.6205$,
$t_{0.1}=72.5000$. From here we can see that by increasing the noncommutative
parameter, $\theta$, the start of the slow-roll stage occurs at earlier times.

Figure 13: Noncommutative and commutative numerical solution for the slow-roll
parameter evolution $\eta(t)$,in KED after the bounce. We take the same
initial condition as the Fig. 9.

Similar to the EKED and KED, we can compute the number of e-folds as a
function of time $N(t)$. Since the PED has a larger range than the EKED and
KED, the number of e-folds grows much faster in time compared to the both EKED
and KED. Fig. 14 shows that a $t=1000$ the amount of e-folds is $128.467$ for
the commutative case while $N_{0.05}=131.741$ and $N_{0.1}=131.972$ for the
noncommutative counterparts. The effect, on the number of e-folds, of
including noncommutativity is that the greater the value of the parameter
$\theta$, the greater the number of e-folds, which contrasts with the EKED
associated to condition $\dot{\phi_{B}}=0$.

Figure 14: Noncommutative and commutative numerical solution for $N(t)$, in
KED after the bounce. We take the same initial condition as the Fig 9.

Due to the behavior of the slow-roll parameters, the universe will remain in
accelerated expansion and eternal inflation, such as those presented in Guth .

####  III.1.3 Potential Energy Domination, after the bounce

In this stage the potential energy dominates the evolution of the universe
after the bounce, $F_{B}>0.5$. For the analysis of this stage we will take the
initial conditions $(\phi_{B}=3.5\times 10^{5},\leavevmode\nobreak\
\dot{\phi}_{B}=0.1)$, which implies that $F_{B}=0.9$. As the potential energy
dominates, the universe is in an inflationary period (the bouncing and
transition phases are not present in this stage). As in Refs. sloan ;
Parampreet , we find that the universe has an exponential growth.
Noncommutativity has the effect of increasing the rate to which the volume of
the universe grows (with increasing values of $\theta$) see Fig. 15.

Figure 15: Numerical solution for volume evolution $V(t)$, considering the
commutative and noncommutative model with the potential $\phi^{2}$, for the
initial conditions $\phi_{B}=3.8\times 10^{5}$, $\dot{\phi}_{B}=0.1$,
$m=1.26^{-6}$, $b=0$, in PED after the bounce.

The commutative Hubble parameter in the PED grows and then remains constant as
the universe evolves Parampreet . On the other hand, incorporating
noncommutativity in this stage produces the appearance of a very brief SI
stage, as the noncommutative parameter, $\theta$, increases. The increase of
the Hubble parameter, $H$, is greater, Fig. 16. After a short period of time
it tends to the same constant value featured in the commutative counterpart.

Figure 16: Noncommutative and commutative numerical solution for $H(t)$, in
PED after the bounce. We take the same initial condition as the Fig 15.

The duration of the SI induced by noncommutativity one is very short, for
$\theta=0.05$ we have that the end of SI is $t_{0.05}=6.32\times 10^{-3}$,
while for $\theta=0.1$ the end of SI occurs in less time $t_{0.1}=4.47\times
10^{-3}$, as shown in Fig. 17.

Figure 17: Noncommutative and commutative numerical solution for $\dot{H}(t)$,
in PED after the bounce. We take the same initial condition as the Fig 15.

From Fig. 18 we can clearly visualize that the kinetic energy after the bounce
is almost zero and that the potential energy dominates after the bounce, the
bouncing and transition stages do not appear because $\omega_{\phi}\sim-1$,
that is to say that we are in the slow-roll stage, this behavior is shared by
both, commuative and noncommutative schemes.

Figure 18: Noncommutative and commutative numerical solution for
$\omega_{\phi}$, in PED after the bounce. We take the same initial condition
as the Fig 15. Figure 19: Noncommutative and commutative numerical solution
for $\epsilon_{H}(t)$, in PED after the bounce. We take the same initial
condition as the Fig 15.

##  IV Discussion

In this letter we examined the pre-inflationary and inflationary epochs of the
noncommutative effective LQC (with a massive scalar field with quadratic
potential) presented in prd . As in sloan , we analyzed three periods of
energy domination after the bounce, the EKED, KED and PED, which are specified
by their corresponding values of $F_{B}$, Eq. 49. In prd  there are two sets
of conditions that ensure the existence of a bounce, for this investigation
only the condition $b=0$ was taken into account (since the other one does not
precludes existence of additional minima of volume, and dramatically reduce
the number of compatible initial conditions).

The noncommutativity effects are expected to have its main contribution in
EKED and KED after the bounce, more precisely, at the bouncing stage of these
scenarios.

###  IV.1 Regarding the EKED

The initial conditions compatible with the EKED period are $\phi_{B}$,
$F_{B}<10^{-4}$. For this analysis we used $\phi_{B}=0,\leavevmode\nobreak\
\dot{\phi}_{B}=0.8$. By solving the field equations (62-65) the numerical
solutions for the commutative and noncommutative evolution of the effective
LQC model with the potential $\phi^{2}$ were found. Using the modified
Friedmann equation 57 and the $\omega_{\phi}$ 50 equation of state for the
scalar field, the post bounce evolution can be divided into the bouncing
stage, transition stage and the slow-roll inflation. For the commutative case
$\theta=0$, the results obtained in these three stages are similar to the
results reported in sloan ; Parampreet ; piulqc , as expected. By taking
$\theta\neq 0$, the equation of state $\omega_{\phi}$ undergoes modifications
for each value of $\theta$, which causes a displacement at the beginning of
the pre-inflationary and inflationary stages. Taking increments in $\theta$,
the stages occur at later times than their commutative counterparts
($\theta=0$). In the bouncing stage, the SI period is modified by an increase
in the length of the period and by the maximum value reached by $H(t)$, see
Tab. 1. There is slow-roll inflation, which occurs when $\omega_{\phi}=-1$,
Fig. 6 shows the interval in which $\epsilon_{H}<1$. Finding the number of
e-folds $N(t)$ we observe that the minimum bound $N\geq 60$ wmap  is reached
for $\theta=0$. For values of $\theta>0$, $N(t)$ decreases with respect to its
commutative counterpart, that is, for $\theta\gg 1$ the minimum number of
e-folds is not reached. Therefore, to guarantee the condition $N\geq 60$, the
value of the noncommutative parameter must necessarily be close to zero, for
the given initial conditions (this, of course, is consistent with the idea
that the noncommutative parameter must be “small”). In addition to this
initial condition, several other initial conditions $\phi_{B},\dot{\phi}_{B}$
were analyzed to see if there is a similar behavior for $\theta\neq 0$. It was
found that this is indeed the case for initial conditions where
$(\phi_{B},\leavevmode\nobreak\ \dot{\phi}_{B}>0.5)$. But for conditions where
$(\phi_{B},0\leq\dot{\phi}_{B}<0.5)$, $N\geq 60$ is not reached for
$\theta\neq 0$, (not even for $\theta=0$). For these conditions the
noncommutative behavior is inverted, as $\theta$ increases, now the number of
noncommutative e-folds is greater than the number of e-folds reached in the
commutative case, see Figs. 7, 8.

###  IV.2 Regarding the KED

In the KED, $10^{-4}<F_{B}<0.5$, the pre-inflationary and inflationary stages
prevail, since the potential energy density has a greater contribution
compared to the EKED, both the bouncing and transition stages are reduced
temporarily, while slow-roll inflation exists in earlier times. The
noncommutative effects do not significantly affect the duration of the stages,
see Fig. 10, the fluctuations due to noncommutative effects are in the order
of $10^{-7}$. The main effect of noncommutativity in KED is in the SI period,
as in EKED, the effect of noncommutativity increases the duration and the
maximum value of $H(t)$ when $\theta$ increases. In KED, there is a much
larger maximum increase of $H(t)$ than for EKED, see Fig. 11, which implies
that in the SI period noncommutativity increases the range of growth of the
scale factor in such a way that the universe grows and expands at a faster
rate. For the end of the transition stage $\omega_{\phi}=-1$, but in KED the
field $\ddot{\phi}$ still contributes to the dynamics of the system. Fig. 14
shows the range in which $\epsilon_{H}<1$, while Fig. 13 indicates the start
of the slow-roll inflation stage at $t_{i}^{KED}\approx 111.0405$. By taking
$\theta\neq 0$ the period in which $\epsilon_{H}<1$ occurs at later times than
its commutative counterpart, the opposite effect appears for the condition
$\eta_{H}\ll 1$, which for $\theta>0$, occurs earlier than the $\theta=0$
case. The condition $N\geq 60$ is fulfilled for both commutative and
noncommutative cases ($N(t)$ increases with $\theta$), see Fig 14.

###  IV.3 Regarding the PED

For PED, $F_{B}>0.5$, the potential energy density completely dominates after
the bounce, the bouncing and transition stages disappear in PED. For the
commutative case there is an SI period, which is partially spliced with the
slow-roll inflation stage, just like in Parampreet . In the noncommutative
case, in the SI period, a behavior similar to KED occurs, in which $H_{max}$
increases considerably, to later have a decrease and join the commutative
behavior, see Fig. 16.

In summary, we can assert that, within the scope of the present investigation,
the early universe predicted by the noncommutative scheme considered here does
not deviates in an essential way from the one predicted by the standard
(effective) LQC. That is, i) the singularity is resolved via a single minimum-
volume quantum bounce; ii) solutions exist that feature a sufficiently long
inflationary period; iii) the different evolution stages pertaining to the
pre-inflationary dynamics are in essential agreement with their standard LQC
counterparts (the only meaningful effect being that of extending the time
duration of the different stages). This opens the possibility of seriously
considering a study related to the more fundamental level of primordial
fluctuations. It could very well be the case that noncommutativity enters in a
non-trivial way at this level, prompting meaningful deviations which could
help temper tensions currently featured (at this level) within the standard
LQC paradigm. As pointed out in Section I, an additional motivation (besides
the more phenomenological-related ones) for considering the inclusion of a
momentum-sector $\theta-$deformation at the effective scheme of LQC is that
the noncommutativity among electric fluxes (momenta fields) is completely lost
due to the minisuperspace approximation underlying LQC. The noncommutative
scheme considered here therefore restores (some) of the non-canonical nature
of the fundamental field variables of standard LQG acz .

## Acknowledgments

This work was partially supported by the following grants: J.S.was partially
supported by PROMEP UGTO-CA-3 and SNI-CONAHCyT. S. P. P. was partially
supported by SNI-CONAHCyT and Secretaria de Investigación y Posgrado del
Instituto Politécnico Nacional, grant SIP20231773. A. E. G. was partially
supported by SNI-CONAHCyT and Secretaria de Investigación y Posgrado del
Instituto Politécnico Nacional, grant SIP20231739. L. R. D. B. was partially
supported by SNI-CONAHCyT and Secretaria de Investigación y Posgrado del
Instituto Politécnico Nacional, grant SIP20230114.

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