# Deep learning enables accurate sound redistribution via nonlocal
metasurfaces

Hua Ding  Xinsheng Fang  Bin Jia  Nengyin Wang  Institute of Acoustics, School
of Physics Science and Engineering, Tongji University, Shanghai 200092, China
Qian Cheng<EMAIL_ADDRESS>Yong Li<EMAIL_ADDRESS>Institute of
Acoustics, School of Physics Science and Engineering, Tongji University,
Shanghai 200092, China

###### Abstract

Conventional acoustic metasurfaces are constructed with gradiently “local”
phase shift profiles provided by subunits. The local strategy implies the
ignorance of the mutual coupling between subunits, which limits the efficiency
of targeted sound manipulation, especially in complex environments. By taking
into account the “nonlocal” interaction among subunits, nonlocal metasurface
offers an opportunity for accurate control of sound propagation, but the
requirement of the consideration of gathering coupling among all subunits, not
just the nearest-neighbor coupling, greatly increases the complexity of the
system and therefore hinders the explorations of functionalities of nonlocal
metasurfaces. In this work, empowered by deep learning algorithms, the complex
gathering coupling can be learned efficiently from the preset dataset so that
the functionalities of nonlocal metasurfaces can be significantly uncovered.
As an example, we demonstrate that nonlocal metasurfaces, which can redirect
an incident wave into multi-channel reflections with arbitrary energy ratios,
can be accurately predicted by deep learning algorithms. Compared to the
theory, the relative error of the energy ratios is less than 1%. Furthermore,
experiments witness three-channel reflection with three types of energy ratios
of (1, 0, 0), (1/2, 0, 1/2), and (1/3, 1/3, 1/3), proving the validity of the
deep learning enabled nonlocal metasurfaces. Our work might blaze a new trail
in the design of acoustic functional devices, especially for the cases
containing complex wave-matter interactions.

The emergence of acoustic metamaterials, which are artificial materials with
targeted functionalities, have significantly broaden the fields of physical
and material science Liu2000Science ; Fang2006NM ; Yang2008PRL ; Liang2012PRL
; Kaina2015Nature ; Cummer2016NRM . Novel acoustic manipulations based on
metamaterials have been realized, such as acoustic focusing Li2009NM ;
Zhu2011NP ; Chen2018NC , acoustic cloaking Popa2011PRL ; Zhu2011PRL ;
Zhang2011PRL , acoustic holography Marzo2015NC ; Melde2016Nature ; Zhu2018NC ,
and asymmetric acoustic transmission Liang2009PRL ; Liang2010NM ;
Fleury2014Science ; Devaux2015PRL . In the last decade, acoustic metasurfaces,
as 2D acoustic metamaterials, stand out as distinct choices to manipulate
sound for their vanishing thickness Li2013SR ; Zhao2013SR ; Tang2014SR ;
Li2014PRApplied ; Xie2014NC ; Mei2014NJP ; Ma2014NM ; Li2015PRApplied ;
Li2017PRL ; Zhu2017PRX ; Li2018NC ; Assouar2018NRM ; Fu2020SA . By employing
“locally” designed phase shift provided by subunits of metasurfaces,
fascinating acoustic manipulations are extensively explored recently,
including acoustic cloaking Jin2019PRApplied , acoustic holography Zhu2018NC ,
and compact sound absorbers Mei2012NC ; Ma2014NM ; Yang2017MH ; Huang2020SB .
However, in the most of previous designs, the subunits are designed
individually according to the discretized phase shift profile. In this way,
the mutual coupling among the subunits which also contributes the wave-field
forming is neglected. It therefore turns out that the designed metasurfaces
suffer from the lower efficiency, especially in the cases where
incident/reflected/transmitted angles are relatively large. Furthermore, to
reshape wave field with fine resolution, a large number of subunits are
necessary which inevitably introduce undesired loss due to the viscou-thermal
effect induced by small apertures in small subunits. Recently, the concept of
“nonlocal” metasurface is introduced to overcome these limitations Ra2017PRL ;
Asadchy2017PRX ; Hou2019PRApplied ; Ni2019PRB . In these works, the nonlocal
coupling effects among all subunits are taken into accounted to collect the
contribution of nonlocality to the wave-field forming and therefore give rise
to more accurate wave field manipulations. However, it also makes the nonlocal
system quite complex and the relationship between desired wave field and the
geometrical parameters unclear. In this situation, optimization methods are
generally employed to inversely design the functionality of nonlocal
metasurface. However, for different wave manipulation aims, the objective
function needs to be revised correspondingly to search for the best value in
the entire parameter space, resulting in the drawbacks of time-wasting and
computational resource-consuming.

The development of artificial intelligence algorithms provides new insights
into the inverse design of nonlocal metasurfaces. In view of its extreme
success in domains correlated to the detection of diseases Ribli2018Detecting
; Naseer2019Refining , speech recognition Nassif2019Speech , computer vision
Voulodimos2018Deep , and natural language processing Young2018Recent ;
Wu2019Deep , deep learning has aroused wide attention. The excellent learning
and inference capabilities of deep learning enable it to achieve the similar
effect of logical calculations with physical formulas. Many works about
combining deep learning with physics have been attempted, including
optimization of photonic structures Liu2018Training ; Sajedian2019Optimisation
; Asano2018Optimization ; Long2019Inverse ; Tahersima2019Deep ;
Pilozzi2018Machine ; John2018Nanophotonic , high-resolution imaging
Liu2018Learning ; Orazbayev2020Far , and acoustic source imaging
Xu2021Acoustic .

In this work, by harnessing the power of deep learning, we explore the
capabilities of reshaping acoustic fields accurately by nonlocal metasurfaces.
The complex nonlocal coupling can be learned by deep learning efficiently from
the preset datasets. Then for a target wave field, deep learning can predict
the structural parameters of a nonlocal metasurface in a few seconds. In the
following, we will demonstrate that nonlocal metasurfaces predicted by deep
learning can redirect an incident wave into multi-channel reflections with
arbitrary energy ratios. Compared to the direct calculations, the relative
error of the energy ratios is less than 1%. In addition, we also perform
experiments to witness the ability of the deep learning enabled nonlocal
metasurfaces.

Results
Theory of nonlocal metasurface. Our proposed nonlocal metasurface is
schematically shown in Fig. 1, which consists of a suitable engineered
arrangement of grooves. In this work, we set 3 rectangular-shaped grooves in
each period to demonstrate the ability of energy redistribution among three
allowed diffraction beams. We will show that empowered by machine learning
algorithms, the energy proportion of the three diffracted beams can be
redistributed arbitrarily by tuning the depth (${h_{l}}$) and the width
(${t_{l}}$) of each groove and the distance between two adjacent grooves
(${dx_{l}}$).

Figure 1:  The schematic diagram of nonlocal acoustic metasurface. The
metasurface is composed of repeated supercell along $x$ direction with a
period of $a$. Three grooves are etched in each supercell. ${t_{l}}$ and
${h_{l}}$ represent the width and the depth of the $l$th groove, and
${dx_{l}}$ refers to the distance between adjacent grooves ($l=1,\ 2,\ 3$).
The incident wave along the $-y$ direction could be reflected to three allowed
diffracted beams.

When an incident acoustic plane wave along with ${-y}$ direction impinges on
the metasurface (i.e., incident angle ${\theta_{i}}$ = $0^{\circ}$), the
periodicity of the metasurface will give rise to a set of different
diffraction modes. According to the coupled mode theory, the pressure field
$p(x,y)$ and normal velocity field $v(x,y)$ in the region above the structure
(Region I) can be expressed by Ni2019PRB

${p_{\rm
I}}(x,y)=A_{0}^{+}{e^{j{k_{0}}y}}+\sum\limits_{n}{A_{n}^{-}}{e^{-j{k_{x,n}}\cdot
x}}{e^{-j{k_{y,n}}\cdot y}},$ (1) ${v_{\rm
I}}(x,y)=-\frac{{A_{0}^{+}}}{{\rho\omega}}{k_{0}}\cdot{e^{-j{k_{0}}y}}+\sum\limits_{n}{\frac{{A_{n}^{-}}}{{\rho\omega}}}{k_{y,n}}\cdot{e^{-j{k_{x,n}}\cdot
x}}{e^{-j{k_{y,n}}\cdot y}},$ (2)

where ${A_{0}^{+}}$, ${A_{n}^{-}}$ are the amplitude of the incident wave and
the $n$th-order of the reflected diffraction beams.
${k_{x,n}}={k_{0}}\sin{\theta_{i}}+{G_{n}},$ and
${k_{y,n}}=\sqrt{{k_{0}}^{2}-{k_{x,n}^{2}}},$ are the $x$ and $y$ component of
the $n$th-order diffraction wave vector with ${k_{0}}$ being the wave number
of incident waves and ${G_{n}}=n{2\pi/a}$ representing the reciprocal lattice
vector. $\rho$ and $\omega$ refer to the mass density of the medium and the
angular frequency. From the above formulas, the period $a$ of the metasurface
related to $k_{x,n}$ decides the propagation characteristics of different
diffraction modes. It means that by selecting a suitable $a$, the numbers of
the allowed propagating diffraction orders can be tuned.

To illustrate the nonlocal coupling effect among etched grooves, we need first
get the pressure and velocity field inside each grooves. Considering the fact
that the bottom of the grooves are acoustically hard where the normal velocity
should be zero, the $p(x,y)$ and $v(x,y)$ in grooves (Region II) could be
represented as

${p_{{\rm I}{\rm
I}}}^{l}(x,y)=\sum\limits_{k}{{a_{kl}}}\cos[{\alpha_{kl}}(x-{x_{l}})]({e^{j{\beta_{kl}}\cdot
y}}+{e^{-j{\beta_{kl}}(y+2{h_{l}})}}),$ (3) ${v_{{\rm I}{\rm
I}}}^{l}(x,y)=\sum\limits_{k}{-\frac{{{a_{kl}}}}{{\rho\omega}}}{\beta_{kl}}\cos[{\alpha_{kl}}(x-{x_{l}})]({e^{j{\beta_{kl}}\cdot
y}}-{e^{-j{\beta_{kl}}(y+2{h_{l}})}})$ (4)

where ${a_{kl}}$ represents the amplitude of the $k$th-order of the waveguide
mode, ${x_{l}}$ is the beginning $x$ coordination of the $l$th groove,
${\alpha_{kl}}=k\pi/{t_{l}}$ and
${\beta_{kl}}=\sqrt{{k_{0}}^{2}-{\alpha_{kl}}^{2}}$ are the $x$ and $y$
components of the wave vector for the $k$th-order mode in the groove,
respectively. Conforming to the conditions at the air-metasurface interface,
sound pressure and volume velocity should be continuous at $y$ = 0. By
plugging Eqs. (1)-(4) into continuum conditions and integrating the equations
with orthogonal relationship, we can get the following equations:

$\left({\begin{array}[]{c}{P_{1}^{1}}\\\ {P_{1}^{2}}\\\ {P_{1}^{3}}\\\
\end{array}}\right)A_{0}^{+}+\left({\begin{array}[]{c}{P_{2}^{1}}\\\
{P_{2}^{2}}\\\ {P_{2}^{3}}\\\
\end{array}}\right)A_{n}^{-}=\left({\begin{array}[]{ccc}{P_{3}^{1}}&0&0\\\
0&{P_{3}^{2}}&0\\\ 0&0&{P_{3}^{3}}\\\
\end{array}}\right)\left({\begin{array}[]{c}{a_{kl}^{1}}\\\ {a_{kl}^{2}}\\\
{a_{kl}^{3}}\\\ \end{array}}\right),$ (5)

and

${V_{1}}A_{0}^{+}+{V_{2}}A_{n}^{-}=\left({\begin{array}[]{ccc}{V_{3}^{1}}&{V_{3}^{2}}&{V_{3}^{3}}\\\
\end{array}}\right)\left({\begin{array}[]{c}{a_{kl}^{1}}\\\ {a_{kl}^{2}}\\\
{a_{kl}^{3}}\\\ \end{array}}\right),$ (6)

where

$P_{1}^{l}(k)=\frac{1}{{{t_{l}}}}\int_{{x_{l}}}^{{x_{l}}+{t_{l}}}{\cos{\alpha_{kl}}(x-{x_{l}})}dx,$
(7)
$P_{2}^{l}(k,n)=\frac{1}{{{t_{l}}}}\int_{{x_{l}}}^{{x_{l}}+{t_{l}}}{{e^{-j{k_{x,n}}x}}}\cos{\alpha_{kl}}(x-{x_{l}})dx,$
(8)
$P_{3}^{l}({k_{1}},{k_{2}})=(1+{e^{-j2{\beta_{{k_{1}}l}}{h_{l}}}})\frac{1}{{{t_{l}}}}\int_{{x_{l}}}^{{x_{l}}+{t_{l}}}{\cos{\alpha_{{k_{1}}l}}(x-{x_{l}})\cos{\alpha_{{k_{2}}l}}(x-{x_{l}})}dx,$
(9) ${V_{1}}(m)=-\frac{1}{a}\int_{0}^{a}{{k_{y,0}}{e^{j{k_{{}^{x,m}}}x}}}dx,$
(10)
${V_{2}}(m,n)=\frac{1}{a}\int_{0}^{a}{{k_{y,n}}{e^{-j({k_{{}^{x,n}}}-{k_{{}^{x,m}}})x}}}dx,$
(11)

and

$V_{3}^{l}(m,k)=-(1+{e^{-j2{\beta_{kl}}{h_{l}}}})\frac{{{\beta_{kl}}}}{a}\int_{{x_{l}}}^{{x_{l}}+{t_{l}}}{\cos{\alpha_{kl}}(x-{x_{l}}){e^{j{k_{x,m}}x}}}dx,$
(12)

with $m,n=0,\pm 1,\cdots,\pm(N-1)/2$ and $k,k_{1},k_{2}=1,\cdots,K$.
${P_{1}^{l}}$, ${P_{2}^{l}}$, ${P_{3}^{l}}$ are respectively related to the
the sound pressure of incident wave, reflected wave in Region I and the sound
wave in Region II. And their dimensions are known as $LK$×1, $LK$×$N$ and
$LK$×$LK$. $L$ is the total number of grooves in a period, $K$ is the total
number of modes inside the metasurface, and $N$ is the total number of
diffraction modes. In fact, the superscript and subscript of $V$ have the
similar meaning as those of $P$, but $V$ is related to the volume velocity of
sound waves. The dimensions of ${V_{1}}$, ${V_{2}}$, ${V_{3}^{l}}$ are $N$×1,
$N$×$N$, $N$×$LK$ respectively. We simplify the above equations with matrixes
and rewrite them as:

$\left({\begin{array}[]{c}{{P_{1}}}\\\ {{V_{1}}}\\\
\end{array}}\right)A_{0}^{+}=\left({\begin{array}[]{cc}{-{P_{2}}}&{{P_{3}}}\\\
{-{V_{2}}}&{{V_{3}}}\\\
\end{array}}\right)\left({\begin{array}[]{c}{A_{n}^{-}}\\\ {{a_{kl}}}\\\
\end{array}}\right),$ (13)

From Eq. (13), the reflected energy of each diffraction order could be
calculated. In order to show energy distribution intuitively, we utilize
$I={\left|{\frac{{A_{n}^{-}}}{{A_{0}^{+}}}}\right|^{2}}\cos{\theta_{r}}$ to
denote the power flow ratio of each mode. Here, we choose -1st, 0th and +1st
order diffractive components as reflected channels, and the corresponding
diffraction angles for them are ${\theta_{r}}$ = $-60^{\circ}$, $0^{\circ}$,
$60^{\circ}$. The working frequency is set to $f_{0}=8000$Hz. In real
situations, the harmonic modes of the metasurface and its internal modes are
very complicated, so we choose $K$=20, $N$=51, $L$=3 to be as close to the
actual situations as possible. Obviously, the consideration of non-locality
makes the dimension of the matrixes large which means that the computational
complexity is higher. This brings more troubles to inverse design.

Deep learning model. The special operating mechanism of deep learning will
help to explore the relationship between metasurface and its response to
acoustic waves. Here, we establish a tool that can achieve rapid forward and
inverse design with the support of deep learning. This part of content mainly
explains the process of building the transient reactor through deep learning
algorithms (the specific training mechanism can be found in the Method).

Figure 2:  Schematic representation of data acquisition and the tandem
network. This network is composed of pre-trained forward network ${A_{net}}$
(in the red dotted box) and inverse network ${B_{net}}$ (in the blue dotted
box). The intermediate layer where the blue nodes are located is the output
layer of the inverse design.

To realize the intelligent design of metasurfaces, we firstly should establish
a data set. We introduce a vector
${\bf{S}}={({h_{1}},{h_{2}},{h_{3}},{t_{1}},{t_{2}},{t_{3}},d{x_{1}},d{x_{2}})^{\text{T}}}$
which is composed of the structure parameters of metasurfaces. Meanwhile,
reflected power flow ratio in three channels is expressed as a state vector
${\bf{I}}={({I_{1}},{I_{2}},{I_{3}})^{\text{T}}}$, where ${I_{1}}$, ${I_{2}}$,
${I_{3}}$ respectively correspond to power flow of $-1$st, $0$th and $+1$st
order diffractive components.The dataset based on diffraction theory for
metasurfaces contains many arrays {$\bf{S}$, $\bf{I}$}, including 270,000
training samples and 30000 test samples. Specifically, as shown in Fig. 2, we
randomly generate many different sets of $\bf{S}$ and calculate their
corresponding $\bf{I}$ by MATLAB. Here, we construct two network models
(${A_{net}}$ and ${B_{net}}$), one of which takes $\bf{S}$ as input data and
$\bf{I}$ as label data while the other is the opposite. Obviously, there is a
correspondence between these two vectors, $\bf{I}$ = ${A_{net}}$($\bf{S}$),
$\bf{S}$ = ${B_{net}}$($\bf{I}$), where ${A_{net}}$ is the forward deep
learning network and ${B_{net}}$ is the inverse neural network. It should be
noted that ${B_{net}}$ can hardly converge when it is trained in a similar way
as ${A_{net}}$ since the same power-flow distribution could be accounted for
different arrangement of meta-units.

To tackle this multi-value problem, we exploit tandem neural network
introduced in Ref.Liu2018Training . As Fig. 2 shows, the tandem network model
is composed of ${A_{net}}$ and ${B_{net}}$, which connects the trained
${A_{net}}$ served as a backend behind the ${B_{net}}$. This connected network
is similar to an autoencoder network (consists of two networks: encoder and
decoder) in terms of structures, in which the input data in the first layer
has the same physical meaning as the data in the final layer and structural
parameters exist as the encoded feature data in the intermediate layer. The
entire network trained successfully is actually a transient reactor. In the
construction of it, the forward network with dataset {$\bf{S}$, $\bf{I}$} is
supposed to be trained as the pre-trained network firstly. Then, when training
the connected model, $\bf{I}$ is both the input data and the target value that
needs to be compared with the output data. Different from the general
autoencoder training mechanism which is often used for unsupervised learning,
we have a prepared dataset including label data, so we can train the “decoder”
(${A_{net}}$) first. Training part of the network in advance transforms the
surjective relation between $\bf{S}$ and $\bf{I}$ into bijective relation, and
in this way, the root cause of the failure to converge inverse network is
resolved.

For all training processes, Adaptive Moment Estimation (Adam) optimizer
Kingma2014Adam  is chosen as the global optimizer. Forward training is done by
minimizing the cost function,
${C_{1}}=\frac{1}{N}\sum\limits{|{{\bf{I}}_{true}}}-{{\bf{I}}_{pred}}|$, in
which $N$ is the batch size of training dataset, ${{\bf{I}}_{pred}}$ is the
possible power flow ratio predicted by ${A_{net}}$ and ${{\bf{I}}_{true}}$ is
the target value in dataset. The value of cost function is inclined to be
steady after 1500 epochs of forward training. The successful training of
${A_{net}}$ informs that deep learning model could acquire the relationship
between physical quantities. During the secondary training procedure, every
layer in pre-trained forward model is frozen, that is, weights and bias of
each node are not updated in layers of ${A_{net}}$ but these values in
${B_{net}}$ are optimized all the time. The cost function of this combination
model can be expressed by
${C_{2}}=\frac{1}{N}\sum\limits{|{{\bf{I}}_{true}}-{A_{net}}({\bf{S}})}|$. In
a way, the “decoder” converts the multi-value problem into a single-value
problem without complicated data filtering. As the corresponding relation
between the design $\bf{S}$ and the response $\bf{I}$ changes, the training of
connected network converges easily (nearly after 1500 epochs).

With the entire model trained successfully, the transient reactor for three-
channel metasurfaces is completely constructed. When we take the middle layer
as input and the last layer as the output, it could realize the forward data
prediction. While we use the first layer as the input and the middle layer as
the output, the desired structure parameters would be predicted by the middle
layer, i.e., the output layer of inverse network, as long as we input the
target power flow distribution. Actually, what we are concerned more about is
the function implemented by the latter, because nonlocal metasurfaces do not
have a clear inverse theoretical relationship between structures and their
responses, but the realization of the model allows this theoretical
relationship to be simulated. Moreover, it avoids the troubles of conventional
optimization methods that require to iterate many times to search for the best
value and easily fall into local optimality.

Nonlocal metasurfaces predicted by deep leanring. The target structures of
acoustic metasurfaces could be predicted quickly and intelligently by the
transient reactor. We randomly select several groups of power flow
distributions with different ratios to verify the accuracy and efficiency of
this reactor. Part of the prediction results is shown in Table 1. The values
on the left indicate the target ratios fed to the model and the theoretical
values in middle represent the real energy distributions calculated by the
predicted geometric parameters as shown in the right column. All the ratios
agree well with each other, which reveals the accuracy of this reactor.

Table 1: Part of design results by deep learning model. $\bf{Target\quad value}$ | $\bf{Theoretical\quad value}$ | $\bf{Structural\quad parameters(unit:mm)}$
---|---|---
$[0.70,0.00,0.30]$ | $[0.7006,0.0026,0.2967]$ | $[9.01,22.53,10.75,8.46,7.79,8.41,8.40,8.40]$
$[0.80,0.00,0.20]$ | $[0.8005,0.0007,0.1988]$ | $[8.71,23.22,11.32,8.42,7.62,8.25,8.21,8.25]$
$[0.90,0.10,0.00]$ | $[0.9000,0.0996,0.0004]$ | $[8.06,23.66,14.06,8.20,6.40,7.09,7.70,8.36]$
$[0.30,0.30,0.40]$ | $[0.2989,0.3007,0.4004]$ | $[11.80,20.09,11.07,8.39,7.93,8.38,8.47,8.47]$
$[0.20,0.30,0.50]$ | $[0.1990,0.2999,0.5011]$ | $[12.78,20.04,10.39,8.40,7.92,8.40,8.49,8.48]$
$[0.20,0.50,0.30]$ | $[0.2000,0.5005,0.2995]$ | $[12.83,19.46,11.79,8.43,7.56,8.49,8.49,8.50]$
$[0.40,0.20,0.40]$ | $[0.3998,0.2005,0.3997]$ | $[11.06,20.44,11.04,8.47,8.22,8.39,8.49,8.44]$

For further verification, we choose three sets of predicted data for
simulation and experiment. These three target ratios are set to be
${{\bf{I}}_{\rm I}}=[1,0,0]$, ${{\bf{I}}_{{\rm I}{\rm
I}}}=[\frac{{\text{1}}}{{\text{2}}},0,\frac{{\text{1}}}{{\text{2}}}]$,
${{\bf{I}}_{{\rm I}{\rm I}{\rm
I}}}=[\frac{{\text{1}}}{{\text{3}}},\frac{{\text{1}}}{{\text{3}}},\frac{{\text{1}}}{{\text{3}}}]$.
For these cases, the structures designed by the transient reactor are shown as
follows (unit: mm):

${{\bf{S}}_{\rm I}}=[6.80,23.52,11.74,8.19,6.41,7.06,8.09,8.19],$
${{\bf{S}}_{{\rm I}{\rm I}}}=[10.10,23.01,10.12,8.30,7.99,8.46,8.37,8.37],$
${{\bf{S}}_{{\rm I}{\rm I}{\rm
I}}}=[11.53,19.96,11.53,8.44,7.80,8.44,8.46,8.47].$

Corresponding power flow ratios in theory are ${{\bf{I^{\prime}}}_{{\rm
I}}}=[0.9910,0.0026,0.0063]$, ${{\bf{I^{\prime}}}_{{{\rm I}{\rm
I}}}}=[0.4986,\\\ 0.0017,0.4998]$, ${{\bf{I^{\prime}}}_{{{\rm I}{\rm I}{\rm
I}}}}=[0.3341,0.3315,0.3344]$, which fit well with the target values. The
finite element analysis is used here for numerical calculation, which could
reveal intuitive sound field distribution based on various geometric sizes.

Figure 3:  Intensity distribution in the total field. (a)-(c) Power flow value
for ${{\bf{S}}_{\rm I}}$, ${{\bf{S}}_{{\rm I}{\rm I}}}$, and ${{\bf{S}}_{{\rm
I}{\rm I}{\rm I}}}$ along $x$ direction at the surface of metasurface. The
size and direction of white arrows in (d)-(f) respectively indicate the
strength and direction of the power flow here.

As shown in Fig. 1 and each matrix $\bf{S}$, along with these rectangular
pits, which are attached randomly in one period, comes the non-local effect
from metasurfaces. Compared with traditional gradient metasurface, additional
degree of freedom for sound regulation could be utilized in these
metasurfaces, which supports the lateral transport of energy flow. To
demonstrate more details of non-local coupling, the energy flow at $y$ = 0 for
these three metasurfaces in simulation has been extracted in Figs. 3a-c. The
local intensity vector ${I_{x}}$ could be obtained by
${I_{x}}=\frac{1}{2}\operatorname{Re}[p{({v_{x}})^{*}}]$, where $p$ means
pressure and the ${v_{x}}$ is the local velocity in the $x$ direction. The
sign of
${{{I_{x}}}\mathord{\left/{\vphantom{{{I_{x}}}{{I_{0}}}}}\right.\kern-1.2pt}{{I_{0}}}}$
refers to the direction of energy flow in $x$-axis. From Fig. 3a,
${{{I_{x}}}\mathord{\left/{\vphantom{{{I_{x}}}{{I_{0}}}}}\right.\kern-1.2pt}{{I_{0}}}}$
are almost negative, which suggests the sound energy is concentrated and
reflected in the left channel. In contrast, the curves in Figs. 3b-c float
above and below zero, which indicates that the structures ${{\bf{S}}_{{\rm
I}{\rm I}}}$, and ${{\bf{S}}_{{\rm I}{\rm I}{\rm I}}}$ not only allocate
energy to the left channel but also allocate to the others. Figs. 3d-f confirm
this point more intuitively, corresponding to Figs. 3a-c, respectively, which
reveal the strong lateral energy exchange between grooves. In any case,
metasurfaces via non-local effects realize efficient wave manipulation by
using lateral energy transport in the form of etched grooves. Although the
efficiency of non-local metasurfaces could be perfectly exerted in sound
manipulation, the inverse calculation is complicated due to the randomness of
its geometrical parameters. Herein, deep learning vastly simplifies it and
facilitates the mass production for complex meta-structures. Not only that, it
enables designers without theoretical foundations to quickly implement on-
demand design.

Experimental demostrations. When a plane wave fed perpendicularly into the
metagratings, scattering field with different ratios are represented in Figs.
4a-c, corresponding to ${{\bf{S}}_{\rm I}}$-${{\bf{S}}_{{\rm I}{\rm I}{\rm
I}}}$ respectively. The red arrows in the figure indicate the position and
direction of the incident wave whilst white arrows represent reflection. The
reflected sound waves are obviously emitted from three channels we set, which
follow the target energy distribution.

Figure 4:  Finite element simulation and experimental results for scattered
pressure fields. (a)-(c) Scattered fields corresponding to ${{\bf{S}}_{\rm
I}}$, ${{\bf{S}}_{{\rm I}{\rm I}}}$, ${{\bf{S}}_{{\rm I}{\rm I}{\rm I}}}$. The
upper panels are the results of the simulation. To show the reflected fields
more clearly, incident waves are not shown in the figures. Middle panels are
experimental measurement fields, in which each measured area is a square with
a size of $200\times 200$ mm2. The insets in middle panels are 3D-printed
sample photos (one period is shown). The values at black dashed lines in the
middle panels are extracted and plotted in the lower panels, where the
abscissa $R$($\theta$) is the ratio of the real part for ${p_{\theta}}$ and
${p_{max}}$. ${p_{\theta}}$ represents the pressure value at different
positions on the black dotted lines.

The experimental samples with 20 periods are fabricated by 3D-printing
technology using the stereo lithography apparatus with photosensitive resin.
Each measurement area is marked with a red dashed frame in the upper panels of
Figs. 4a-c and the size is $200\times 200$ mm2. The experimental results shown
in the middle of Fig. 4 correspond to the scattering fields with target ratios
(${{\bf{I}}_{\rm I}}$-${{\bf{I}}_{{\rm I}{\rm I}{\rm I}}}$). All the
scattering fields in experiment have been normalized by the incident pressure.
In addition, we select a line in each measurement area and plot the data
extracted from the line in lower panels, defining
$R(\theta)=\operatorname{Re}({{{p_{\theta}}}\mathord{\left/{\vphantom{{{p_{\theta}}}{{p_{\max}}}}}\right.\kern-1.2pt}{{p_{\max}}}})$
as the abscissa. Experimental results agree well with the simulation, which
demonstrates the fantastic functionality of the metasurfaces designed by deep
learning algorithm. In the last experimental field, the pressure field of the
middle channel is weaker than the two others, which may be caused by the
multiple reflection between loudspeaker array and the metasurface.

Discussion

In this work, we investigate the beam splitting function of the grooved
metasurfaces based on nonlocal coupling mechanism and realize the arbitrary
regulation of the energy ratio for the multi-channel metasurfaces. We
successfully conducted model training on $\bf{S}$ and $\bf{I}$ taking into
account the nonlocality among subunits, so that finally the transient reactor
can predict the target structures. Meanwhile, the efficiency and accuracy of
the network model are validated by simulations and experiments. This
intelligent method fairly shortens the optimization time for acoustic device
design and this way could be extended to other forms of device structures. Our
work provides new design solutions for non-local acoustic functional devices
and promotes explorations of more fancy properties of meta-materials.These
results might inspire more combinations of acoustics and artificial
intelligence algorithms to assist in solving acoustic problems.

Method
Neural network training mechanism. Given that one network model has $D$
layers, then the first layer gets the input data $X^{0}$ and the layer $D$
outputs the data $X^{D}$ which has been activated many times by nonlinear
activation functions. $f$ represents the activation function.The complete
neural network training process is mainly composed of two parts. In the former
process, neural network parameters $w$ and $b$ are used to calculate the
output values in conjunction with the activation function. Forward equations
are as follows:

${{z^{d}}={w^{d}}{X^{d-1}}+{b^{d}}}\quad{for}\quad{d=1,2,\cdots,D}\\\ ,$ (14)
${{X^{d}}=f({z^{d}})}\quad{for}\quad{d=1,2,\cdots,D}\\\ ,$ (15)

where $z$ called weighted input will play an important role in the back
propagation. Cost function($C$) evaluates the error between predicted values
and expected values. Actually, the goal of training given data is to seek
suitable model parameters for which cost function return as small value as
possible. Whereas this purpose depends on the later process to complete. In
the back propagation, parameters $w$ and $b$ could be updated by gradient
calculation and the global optimizer to improve the predictive performance of
models. Referring to the back propagation algorithm proposed in
Ref.Rumelhart1986Learning , error gradient of each layer can be expressed as:

${\delta^{D}}=\frac{{\partial C}}{{\partial{X^{D}}}}f^{\prime}({z^{D}}),$ (16)
${{\delta^{d}}=\left[{{{({w^{d+1}})}^{T}}{\delta^{d+1}}}\right]\odot
f^{\prime}({z^{d}})}\quad{for}\quad{d=1,2,\cdots,D-}1,$ (17)

Then the partial differential of the cost function to the internal parameters
of the network can be written as:

$\frac{{\partial C}}{{\partial{w^{d}}}}={a^{d-1}}{\delta^{d}},$ (18)
$\frac{{\partial C}}{{\partial{b^{d}}}}={\delta^{d}}.$ (19)

Note that every neuron will go through these calculation cycles. Considering
that the calculation relationship between two neurons located in adjacent
layers is similar, for the sake of simplicity, we just use the superscript
representing the number of layer for these parameters in above equations
instead of marking the specific position of each neuron in a certain layer.

Network model structure and hyperparameter settings. In order to balance the
training speed and prediction accuracy of the network model, we adopt the
batch training method, that is, part of the data is sent for training each
time. Batch size is the number of samples sent for each training. In all our
training process, batch size is set to 256. We use the Leaky Relu as the
activation function for each layer. The initial learning rates of forward
training and tandem training are 0.0014 and 0.0006, respectively. The
architecture of ${A_{net}}$ is selected as 8-512-512-256-256-128-3 and
${B_{net}}$ is 3-256-256-128-128-128-8. The 8 nodes match the number of
structure parameters $\bf{S}$, and the 3 nodes correspond to the number of
elements in the vector $\bf{I}$. Although pre-trained model and connected
model could converge well after training over a thousand times, we terminated
all the model training until epochs reached 5000 times to make the models more
stable.

Finite element simulation. COMSOL MULTIPHYSICS is used as a preliminary
verification tool for preliminary simulation verification of the predicted
results. The pressure acoustic module in the frequency domain is used to
simulate the effect of the metasurface. A fixed-length plane incident wave as
the sound source is incident on metasurfaces.

Experimental setup. The phase shift and amplitude of pressure fields are
collected by a 1/8-inch microphone (Brüel $\&$ Kjær type 2670) and the data-
acquisition hardware (NI PXI-4461 in NI PXIe-1071). The microphone as a probe
scans fields in a step of 5mm. In the automatic scanning platform, we disposed
7 loudspeakers with a total width of 245 mm as the source. Samples are put in
an organic glass plate waveguide about 2 cm high for experimental
measurements. Foams are distributed at the edge of the measurement field to
absorb external sound waves and reduce the influence of internal multiple
reflection waves. We first measured the empty field with no sample placed, and
then measured the total field with the placed sample. In this way, the
difference between the two measurements is the result of the reflected field.

Data availability
The data that support the plots within this paper and other findings of this
study are available from the corresponding author upon reasonable request.

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Acknowledgements
This work is supported by the National Key R & D Program of China under Grant
Nos. 2020YFA0211400 and 2020YFA0211402, the National Natural Science
Foundation of China under Grant No. 12074286, and the Shanghai Science and
Technology Committee under Grant No. 20ZR1460900.

Author contributions
Y.L. conceived the idea. H.D. implemented model construction and training.
X.F., and N.W. provides support for basic physical model. H.D., B.J., and N.W.
performed the experiments. Y.L. and Q.C. supervised the whole project. H.D.,
X.F. and Y.L. wrote the paper with contribution from all authors.

Competing interests
The authors declare no competing interests.


# Non commutative classical and Quantum fractionary Cosmology: FRW case.

J. Socorro1<EMAIL_ADDRESS>J. Juan Rosales 2<EMAIL_ADDRESS>Leonel
Toledo-Sesma3<EMAIL_ADDRESS>1 Departamento de Física, DCI-Campus León,
Universidad de Guanajuato,
C.P. 37150, León, Guanajuato, México
2 Department of Electrical Engineering, Engineering Division Campus Irapuato-
Salamanca, University of Guanajuato, Salamanca 36885, México.
3 Instituto Politécnico Nacional, UPIIH, Carretera Pachuca - Actopan Kilómetro
1+500, 42162. San Agustín Tlaxiaca, Hgo. México

###### Abstract

In this work we shall explore the effects of non commutativity in fractional
classical and quantum schemes using the flat Friedmmann-Robertson-Walker (FRW)
cosmological model coupled to a scalar field in the K-essence formalism. In
previous work we have obtained the commutative solutions in both regimes into
the fractional framework. Here we introduce noncommutative variables,
considering that all minisuperspace variables $\rm q^{i}_{nc}$ do not commute,
so the symplectic structure was modified. In the quantum regime, the
probability density presents new structure in the scalar field corresponding
to the value of the non-commutative parameter, in the sense that this
probability density undergoes a shift back to the direction of the scale
factor, causing classical evolution to arise earlier than in the commutative
world.

Keywords: Fractional derivative, Fractional non-commutative classical and
quantum cosmology, K-essence formalism.

###### pacs:

02.30.Jr; 04.60.Kz; 12.60.Jv; 98.80.Qc.

##  I introduction

The study and applications of fractional calculus (FC) to cosmology is a new
line of research, that was born approximately twenty years ago. We have
recently worked along this line in the theory of K-essence due that in
Socorro1  it is mentioned that by quantifying different epochs of the
K-essence theory, a fractional Wheeler-DeWitt equation (WDW) in the scalar
field component is naturally obtained in different stages of the universe
universe ; fra-fractionary , however we have not found any work in the
literature, where the idea of non-commutativity (NC) is applied to this
formalism, which is why we are interested in studying the effects of NC
variables from the fractional calculus approach, and seeing their effects on
the exact solutions or mathematical structure of the same. It is well known
that there are various ways to introduce non-commutativity in the phase space
and that they produce different dynamical systems from the same Lagrangian
Abreu-2006 , as can be shown for example in reference De-andrade  and
references cited therein. Therefore, distinct choices for the NC algebra among
the brackets render distinct dynamic systems. We will use non-commutativity in
the coordinate space, which is where we have some working practice in the
past, leaving the application of moments space for the future, Sabido ; Aguero
; Guzman ; Ortiz ; Socorro ; Guzman-2011 , where other quantities such as
angular momentum appear between coordinates and momenta sabido2018 ;
sabido2024 .

Usually, K-essence models are restricted to the Lagrangian density of the form
1 ; roland ; chiba ; bose ; arroja ; tejeiro

$S=\int d^{4}x\,\sqrt{-g}\,\left[f(\phi)\,{\cal G}(X)\right],$ (1)

where the canonical kinetic energy is given by ${\cal
G}(X)=X=-\frac{1}{2}\nabla_{\mu}\phi\nabla^{\mu}\phi$, $f(\phi)$ is an
arbitrary function of the scalar field $\phi$, and g is the determinant of the
metric. K-essence was originally proposed as a model for inflation, and then
as a model for dark energy, along with explorations of unifying dark energy
and dark matter. During the development of research in non-commutative
formalism within fractional cosmology in k-essence theory, the presence of
non-commutativity that usually accompanied the term of the scale factor, here
is disrupted, since essentially Non-commutativity is more present in the
scalar field, modifying the mathematical structure that usually occurs in
works in this direction in other non-fractional formalisms.

We construct the Lagrangian and Hamiltonian densities for the plane FLRW
cosmological model, considering a barotropic perfect fluid for the scalar
field in the variable $X$, and present the general case in commutative (II)
and noncommutative formalism (III). We present the quantum version in both
cases, in (IV) and (V), respectively. Finally, Section (VI) is devoted to
discussions.

##  II Commutative fractional classical exact solution

We start with the following classical Lagrangian density that comes from the
flat Friedmann-Robertson-Walker fractionary cosmological model coupled to a
scalar field in the K-essence formalism universe

${\cal
L}=e^{3\Omega}\left[6\frac{\dot{\Omega}^{2}}{N}-\left(\frac{1}{2}\right)^{\alpha}\left(\dot{\phi}\right)^{2\alpha}N^{-2\alpha+1}\right].$
(2)

Using the standard definition of the momenta
$\Pi_{q^{\mu}}=\frac{\partial{\cal L}}{\partial{\dot{q}^{\mu}}}$, where
$q^{\mu}=(\Omega,\phi)$, we obtain

$\displaystyle\Pi_{\Omega}$ $\displaystyle=$
$\displaystyle\frac{12}{N}e^{3\Omega}\dot{\Omega},\quad\rightarrow\quad\dot{\Omega}=\frac{N}{12}e^{-3\Omega}\Pi_{\Omega},$
(3) $\displaystyle\Pi_{\phi}$ $\displaystyle=$
$\displaystyle-\left(\frac{1}{2}\right)^{\alpha}\frac{2\alpha}{N^{2\alpha-1}}e^{3\Omega}{\dot{\phi}}^{2\alpha-1},\quad\rightarrow\quad\dot{\phi}=-N\left[\frac{2^{\alpha-1}}{\alpha}e^{-3\Omega}\Pi_{\phi}\right]^{\frac{1}{2\alpha-1}},$
(4)

and introducing them into the Lagrangian density, we obtain the canonical
Lagrangian ${\cal L}_{canonical}=\Pi_{q^{\mu}}\dot{q}^{\mu}-N{\cal H}$ as

$\displaystyle{\cal L}_{canonical}$ $\displaystyle=$
$\displaystyle\Pi_{q^{\mu}}\dot{q}^{\mu}-\frac{N}{24}e^{-\frac{3}{2\alpha-1}\Omega}\left\\{e^{-\frac{6(\alpha-1)}{2\alpha-1}\Omega}\Pi_{\Omega}^{2}-\frac{12(2\alpha-1)}{\alpha}\,\Pi_{\phi}^{\frac{2\alpha}{2\alpha-1}}\right\\}.$
(5)

Performing the variation with respect to the lapse function $N$,
${\delta{\mathcal{L}}}_{canonical}/\delta N=0$, the Hamiltonian constraint
$\mathcal{H}=0$ is obtained, where the classical density is written as

${\cal
H}=\frac{1}{24}e^{-\frac{3}{2\alpha-1}\Omega}\left\\{e^{-\frac{6(\alpha-1)}{2\alpha-1}\Omega}\Pi_{\Omega}^{2}-\frac{12(2\alpha-1)}{\alpha}\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\,\Pi_{\phi}^{\frac{2\alpha}{2\alpha-1}}\right\\}.$
(6)

For simplicity we work in the gauge $N=24e^{\frac{3}{2\alpha-1}\Omega}$, and
in the follow we use the reduced Hamiltonian density,

${\cal
H}=e^{-\frac{6(\alpha-1)}{2\alpha-1}\Omega}\Pi_{\Omega}^{2}-\frac{12(2\alpha-1)}{\alpha}\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\,\Pi_{\phi}^{\frac{2\alpha}{2\alpha-1}}.$
(7)

In previous work universe , we found that the barotropic parameter in
K-essence theory has the form $\omega_{X}=\frac{1}{2\alpha-1}$, and the
fractional parameter is $\beta=\frac{2\alpha}{2\alpha-1}$, so, when
$\omega_{X}\in[0,1]$, thus $\beta\in[1,2]$, and when $\omega_{X}\in[-1,0)$,
thus $\beta\in(0,1]$. This is relevant because in the quantum regime, the
Laplace transform of a fractional differential equation depends on the
parameter $n=[\beta]$ (integer part of the fractional parameter).

With this, the Hamiltonian density is rewritten as

${\cal
H}=e^{-3(2-\beta)\Omega}\Pi_{\Omega}^{2}-\frac{24}{\beta}\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\,\Pi_{\phi}^{\beta},$
(8)

then, the Hamilton equations are

$\displaystyle\dot{\Omega}$ $\displaystyle=$ $\displaystyle
2e^{-3(2-\beta)\Omega}\Pi_{\Omega},$ (9) $\displaystyle\dot{\phi}$
$\displaystyle=$
$\displaystyle-24\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\,\Pi_{\phi}^{\beta-1},$
(10) $\displaystyle\dot{\Pi}_{\Omega}$ $\displaystyle=$ $\displaystyle
3(2-\beta)e^{-3(2-\beta)\Omega}\Pi_{\Omega}^{2},$ (11)
$\displaystyle\dot{\Pi}_{\phi}$ $\displaystyle=$ $\displaystyle
0,\qquad\Pi_{\phi}=p_{\phi}=constant.$ (12)

substituting these results in the Hamiltonian constraint, we have that

$\dot{\Pi}_{\Omega}=p_{\beta},\qquad\Pi_{\Omega}(t)=p_{0}+p_{\beta}(t-t_{0}),$
(13)

where $p_{0}$ is an integration constant and
$p_{\beta}=\frac{72(2-\beta)}{\beta}\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\,p_{\phi}^{\beta}$.
With this and using the equation (9), the solution for the scale factor
$A(t)=e^{\Omega}$ becomes,

$A(t)=\left[A_{0}+6(2-\beta)p_{0}(t-t_{0})+3(2-\beta)p_{\beta}(t-t_{0})^{2}\right]^{\frac{1}{3(2-\beta)}},$
(14)

and the solution for the scalar field $\phi$ is

$\phi(t)=\phi_{0}-24\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\,p_{\phi}^{\beta-1}(t-t_{0}).$
(15)

##  III Noncommutative fractional classical exact solution

We start with the following classical hamiltonian that comes from the flat
Friedmann-Robertson-Walker fractionary cosmological model coupled to a scalar
field in the K-essence formalism (8), written in term of the fractional
parameter $\beta=\frac{2\alpha}{2\alpha-1}$ and in particular gauge, where in
order to find the commutative equation of motion, we use the classical phase
space variables $\rm q^{\mu}=(\Omega,\phi)$, where the Poisson algebra for
these minisuperspace variables are

$\left\\{q^{\mu},q^{\nu}\right\\}=0\qquad\left\\{\Pi_{q^{\mu}},\Pi_{q^{\nu}}\right\\}=0,\qquad\left\\{q^{\mu},\Pi_{q^{\nu}}\right\\}=\delta^{\mu}_{\nu},$
(16)

${\cal
H}=\frac{1}{24}e^{3(1-\beta)\Omega}\left\\{e^{-3(2-\beta)\Omega}\Pi_{\Omega}^{2}-\frac{24}{\beta}\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\,\Pi_{\phi}^{\beta}\right\\}.$
(17)

In the commutative model the solutions to the Hamiltonian equations are the
same as in General Relativity, modified only by the fractional parameter. Now
the natural extension is to consider the noncommutative version of our model,
with the idea of non-commutative between the two variables
$\rm(\Omega_{nc},\phi_{nc})$, so we apply a deformation of the Poisson
algebra. For this, we start with the usual Hamiltonian (8), but the symplectic
structure is modify as follow

$\left\\{\Pi_{\Omega},\Pi_{\phi}\right\\}_{\star}=0,\qquad\left\\{q^{\mu},\Pi_{q^{\mu}}\right\\}_{\star}=1,\qquad\left\\{\Omega,\phi\right\\}_{\star}=\theta,$
(18)

where the $\star$ is the Moyal product Szabo2 , and the resulting Hamiltonian
density is

${\cal
H}_{nc}=e^{-3(2-\beta)\Omega_{nc}}\Pi_{\Omega}^{2}-\frac{24}{\beta}\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\,\Pi_{\phi}^{\beta},$
(19)

but the symplectic structure is the one that we know, the commutative one
(16). It is well known that, there are two formalisms to study the non-
commutative equations of motion, for the first formalism that we exposed has
the original variables, but with the modified symplectic structure,

$\displaystyle\dot{q^{\mu}_{nc}}$ $\displaystyle=$
$\displaystyle\rm\\{q^{\mu},{\cal H}\\}_{\star},$
$\displaystyle\dot{\Pi^{\mu}_{nc}}$ $\displaystyle=$
$\displaystyle\rm\\{P^{\mu},{\cal H}\\}_{\star},$ (20)

and for the second formalism we use the shifted variables (Bopp shift
approach) but with the original (commutative) symplectic structure

$\displaystyle\dot{q^{\mu}_{nc}}$ $\displaystyle=$
$\displaystyle\rm\\{q^{\mu}_{nc},{\cal H}_{nc}\\},$
$\displaystyle\dot{\Pi^{\mu}_{nc}}$ $\displaystyle=$
$\displaystyle\rm\\{\Pi^{\mu}_{nc},{\cal H}_{nc}\\},$ (21)

in both approaches, we have the same result.

The commutation relations (16) can be implemented in terms of the commuting
coordinates of the standard quantum mechanics ( Bopp shift) and it results in
a modification of the potential like term of the Hamiltonian density and one
possibility is, for example,

$\rm\Omega_{nc}=\Omega+\frac{\theta}{2}\Pi_{\phi},\qquad\phi_{nc}=\phi-\frac{\theta}{2}\Pi_{\Omega}$
(22)

These transformations are not the most general possible to define non-
commutative fields. With this in mind, our hamiltonian density has the form

${\cal
H}_{nc}=e^{-3(2-\beta)\left[\Omega+\frac{\theta}{2}\Pi_{\phi}\right]}\Pi_{\Omega}^{2}-\frac{24}{\beta}\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\,\Pi_{\phi}^{\beta},$
(23)

the Hamilton equations are

$\displaystyle\dot{\Omega}$ $\displaystyle=$ $\displaystyle
2e^{-3(2-\beta)[\Omega+\frac{\theta}{2}\Pi_{\phi}]}\Pi_{\Omega},$ (24)
$\displaystyle\dot{\phi}$ $\displaystyle=$
$\displaystyle-\frac{3\theta(2-\beta)}{2}e^{-3(2-\beta)[\Omega+\frac{\theta}{2}\Pi_{\phi}]}\Pi_{\Omega}^{2}-24\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\,\Pi_{\phi}^{\beta-1},$
(25) $\displaystyle\dot{\Pi}_{\phi}$ $\displaystyle=$ $\displaystyle
0,\qquad\dot{\Pi}_{\Omega}=3(2-\beta)e^{-3(2-\beta)[\Omega+\frac{\theta}{2}\Pi_{\phi}]}\Pi^{2}_{\Omega}.$
(26)

with these equations, the solution for $\Pi_{\Omega}$ is the same as in the
commutative case, so the solution for the scale factor becomes

$A_{\theta}(t)=e^{-\frac{\theta}{2}p_{\phi}}A(t),$ (27)

where $A(t)$ is the solution presented in equation (14). The solution for the
scalar field is related with the $\Pi_{\Omega}$ as

$\dot{\phi}=-\frac{\theta}{2}\dot{\Pi}_{\Omega}-24\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\,p_{\phi}^{\beta-1},$

then

$\phi(t)=\phi_{0}-24\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\,p_{\phi}^{\beta-1}(t-t_{0})-\frac{\theta}{2}\Pi_{\Omega},$
(28)

for both commutative solutions, the scale factor y scalar field $\phi$ are
obtained when the non-commutative parameter $\theta$, goes to zero.

##  IV Commutative fractional quantum exact solution

The Wheeler-DeWitt (WDW) equation for this model is obtained by making the
usual substitution $\Pi_{q^{\mu}}=-i\hbar\frac{\partial}{\partial q^{\mu}}$
into (8) and promoting the classical Hamiltonian density in the differential
operator, applied to the wave function $\Psi(\Omega,\phi)$, $\hat{\cal
H}\Psi=0$. Then, we have

$-\hbar^{2}e^{-3(2-\beta)\Omega}\frac{\partial^{2}\Psi}{\partial\Omega^{2}}-\frac{24}{\beta}\,\hbar^{\beta}\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\frac{\partial^{\beta}\Psi}{\partial\phi^{\beta}}\,=0.$
(29)

For simplicity, the factor $e^{-3(2-\beta)\Omega}$ may be the factor ordered
with $\hat{\Pi}_{\Omega}$ in many ways. Hartle and Hawking HH  suggested what
might be called semi-general factor ordering, which, in this case, would order
the terms $e^{-3(2-\beta)\Omega}\hat{\Pi}^{2}_{\Omega}$ as
$-\hbar^{2}e^{-[3(2-\beta)-Q]\Omega}\,\partial_{\Omega}e^{-Q\Omega}\partial_{\Omega}=-\hbar^{2}\,e^{-3(2-\beta)\Omega}\,\partial^{2}_{\Omega}+\hbar^{2}\,Q\,e^{-3(2-\beta)\Omega}\partial_{\Omega}$,
where $Q$ is any real constant that measures the ambiguity in the factor
ordering in the variables $\Omega$ and its corresponding momenta.

Thus, the equation (29) is rewritten as

$\hbar^{2}e^{-3(2-\beta)\Omega}\left[-\frac{\partial^{2}\Psi}{\partial\Omega^{2}}+Q\frac{\partial\Psi}{\partial\Omega}\right]-\frac{24}{\beta}\,\hbar^{\beta}\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\frac{\partial^{\beta}\Psi}{\partial\phi^{\beta}}\,=0.$
(30)

By employing the separation variables method for the wave function $\Psi={\cal
A}(\Omega)\,{\cal B}(\phi)$, we have the following two differential equations
for $(\Omega,\phi)$

$\displaystyle\frac{d^{2}{\cal A}}{d\Omega^{2}}\,-Q\frac{d{\cal
A}}{d\Omega}\mp\frac{\mu^{2}}{\hbar^{2}}e^{3(2-\beta)\Omega}{\cal A}$
$\displaystyle=$ $\displaystyle 0,$ (31) $\displaystyle\frac{d^{\beta}{\cal
B_{\pm}}}{d\phi^{\beta}}\pm\left(\frac{\alpha}{2^{\alpha-1}}\right)^{\frac{1}{2\alpha-1}}\frac{\mu^{2}\,\beta}{24\hbar^{\beta}}{\cal
B}_{\pm}$ $\displaystyle=$ $\displaystyle 0,$ (32)

where ${\cal B}_{\pm}$ considers the sign in the differential equation. The
fractional differential Equation (32) can be given in the fractional
frameworks, following Rosales1 ; Rosales2  and identifying
$\gamma=\frac{\beta}{2}=\frac{\alpha}{2\alpha-1},$ where now, $\gamma$ is the
order of the fractional derivative taking values in $0<\gamma\leq 1$; then, we
can write

$\frac{d^{2\gamma}{\cal
B}_{\pm}}{d\phi^{2\gamma}}\pm\left(\frac{\alpha}{2^{\alpha-1}}\right)^{\frac{1}{2\alpha-1}}\frac{\gamma\mu^{2}}{12\hbar^{2\gamma}}{\cal
B}_{\pm}=0,\qquad 0<\gamma\leq 1,$ (33)

the solution of the Equation (33) with a positive sign may be obtained by
applying direct and inverse Laplace transforms Rosales2 , providing

${\cal B}_{+}(\phi,\gamma)=\mathbb{E}_{2\gamma}\left(-z^{2}\right),\qquad
z=\left(\frac{\alpha}{2^{\alpha-1}}\right)^{\frac{1}{2(2\alpha-1)}}\frac{\sqrt{\gamma}\mu}{2\sqrt{3}\hbar^{\gamma}}\phi^{\gamma},\qquad
0<\gamma\leq 1.$ (34)

In the ordinary case, $\gamma=1$; then, the solution is Rosales2 ,

${\cal
B}_{+}(\phi,1)=\mathbb{E}_{2}\left[-\left(\frac{\mu}{2\sqrt{3}\hbar}(\phi-\phi_{0})\right)^{2}\right]=cos\left(\frac{\mu}{2\sqrt{3}\hbar}(\phi-\phi_{0})\right),$
(35)

Following the book of Polyanin polyanin  (page 179.10), we discovered the
solution for the first equation for $\gamma\not=1$, considering different
values in the factor ordering parameter and both signs in the constant
$\mu^{2}$,

${\cal
A}=A_{0}\,e^{\frac{Q\Omega}{2}}\,Z_{\nu}\left[\frac{2\sqrt{\pm\mu^{2}}}{3\hbar(2-\beta)}e^{\frac{3(2-\beta)}{2}\Omega}\right]=\left\\{\begin{tabular}[]{ll}$A_{0}\,e^{\frac{Q\Omega}{2}}\,K_{\nu}\left[\frac{\mu}{3\hbar(1-\gamma)}e^{3(1-\gamma)\Omega}\right],$&\quad
to\,\,$-\mu^{2}$,\\\
$A_{0}\,e^{\frac{Q\Omega}{2}}\,J_{\nu}\left[\frac{\mu}{3\hbar(1-\gamma)}e^{3(1-\gamma)\Omega}\right],$&\quad
to\,\, $+\mu^{2}$\end{tabular}\right.$ (36)

with order $\nu=\pm\frac{Q}{6(1-\gamma)}$, where we had written the second
expression in terms of the fractional order $\gamma=\frac{\beta}{2}$, and the
solutions which become dependent on the sign of its argument; when
$\sqrt{\mu^{2}}$ (for ${\cal B}_{-}$), the Bessel function $Z_{\nu}$ becomes
the ordinary Bessel function $J_{\nu}$. When $\sqrt{-1\mu^{2}}$ (for ${\cal
B}_{+}$), this becomes the modified Bessel function $K_{\nu}$. For the
particular values $\beta=2$ ($\gamma=1$), it will be necessary to solve the
original differential equation for this variable.

Then, we have the probability density $|\Psi|^{2}$ by considering only ${\cal
B}_{+}$, $\gamma\not=1$,

$|\Psi|^{2}=\psi_{0}^{2}\,e^{Q\Omega}\,\mathbb{E}^{2}_{2\gamma}\left(-z^{2}\right)\,\,K_{\nu}\left[\frac{\mu}{3\hbar(1-\gamma)}e^{3(1-\gamma)\Omega}\right],\quad
z=\left(\frac{\alpha}{2^{\alpha-1}}\right)^{\frac{1}{2(2\alpha-1)}}\frac{\sqrt{\gamma}\mu}{2\sqrt{3}\hbar^{\gamma}}\phi^{\gamma}.$
(37)

We will now report the solution for the $\beta=2,\to\gamma=1$ case, which we
have not reported before, considering the minus sign in the constant
$\mu^{2}$, the general solution for the function ${\cal A}$ becomes

${\cal
A}=e^{\frac{Q}{2}\Omega}\left[A_{0}e^{r_{1}\Omega}+B_{0}e^{-r_{1}\Omega}\right],\qquad
r_{1}=\frac{1}{2}\sqrt{Q^{2}+4\frac{\mu^{2}}{\hbar^{2}}},$ (38)

and for the other sign $+\mu^{2}$, becomes

${\cal
A}=e^{\frac{Q}{2}\Omega}\left[A_{0}e^{r_{2}\Omega}+B_{0}e^{-r_{2}\Omega}\right],\qquad
r_{2}=\frac{1}{2}\sqrt{Q^{2}-4\frac{\mu^{2}}{\hbar^{2}}},$ (39)

and the corresponding solutions to (33) for both signs are

${\cal
B}_{\pm}=\left\\{\begin{tabular}[]{l}$Cos\left(\frac{\mu}{2\hbar\sqrt{3}}(\phi-\phi_{0})\right)$\\\
$Cosh\left(\frac{\mu}{2\hbar\sqrt{3}}(\phi+\phi_{0})\right)$\end{tabular}\right.$
(40)

so, the probability density becomes

$|\Psi|^{2}=\left\\{\begin{tabular}[]{ll}$Cos^{2}\left(\frac{\mu}{2\hbar\sqrt{3}}(\phi-\phi_{0})\right)e^{Q\Omega}\left[A_{0}e^{r_{1}\Omega}+B_{0}\,e^{-r_{1}\Omega}\right]^{2},$&\quad
for $-\mu^{2}$\\\
$Cosh^{2}\left(\frac{\mu}{2\hbar\sqrt{3}}(\phi+\phi_{0})\right)e^{Q\Omega}\left[A_{0}e^{r_{2}\Omega}+B_{0}e^{-r_{2}\Omega}\right]^{2},$&\quad
for $+\mu^{2}$\end{tabular}\right.$ (41)

On the other hand, it is well-known that in the standard quantum cosmology,
the wave function is unnormalized. There is no systematic method to do this,
as the Hamiltonian density is not Hermitian. In particular cases, wave packets
can be constructed, and from these wave packets we can construct a normalized
wave function. In this work, we could not construct these wave packets. We
hope to be able to do it in future studies.

##  V Non-commutative fractional quantum exact solution

As already mentioned, we are looking for the non-commutative deformation of
the flat FRW quantum cosmological model. In order to find the non-commutative
generalization, we need to solve the non-commutative Einstein equation, this
is a formidable task due to the highly non linear character of the theory,
fortunately we can circumvent these difficulties by following Ref. ncqc .

Now we can proceed to the non-commutative model, we will consider, that the
minisuperspace variables $\rm q^{i}=(\Omega,\phi)$ do not commute, so that the
symplectic structure is modified as follows

$\rm[q^{i},q^{j}]=i\theta^{ij},\quad[\hat{\Pi}_{i},\hat{\Pi}_{j}]=0,\quad[q^{i},\hat{\Pi}_{j}]=i\delta^{i}_{j},$
(42)

in particular, we choose the following representation

$\displaystyle\rm[\Omega,\phi]$ $\displaystyle=$ $\displaystyle\rm i\theta,$
(43)

where the $\rm\theta$ parameters are a measure of the non commutativity
between the minisuperspace variables. The commutation relations (42) or (43)
are not the most general ones to define a non-commutative field.

We consider the non-commutative hamiltonian density in a simple way, as

$\hbar^{2}e^{-3(2-\beta)\Omega_{nc}}\left[-\frac{\partial^{2}\Psi}{\partial\Omega^{2}}+Q\frac{\partial\Psi}{\partial\Omega}\right]-\frac{24}{\beta}\,\hbar^{\beta}\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\frac{\partial^{\beta}\Psi}{\partial\phi^{\beta}}\,=0.$
(44)

It is well known that this non-commutativity can be formulated in term of non-
commutative minisuperspace functions with the Moyal star product $\star$ of
functions. The commutation relations (42) can be implemented in terms of the
commuting coordinates of the standard quantum mechanics ( Bopp shift) and it
results in a modification of the potential like term of the WDW equation ncqc
; pimentel , and one possibility is, for example,

$\Omega_{nc}\to\Omega+\frac{\theta}{2}\hat{\Pi}_{\phi},\qquad\phi_{nc}\to\phi-\frac{\theta}{2}\hat{\Pi}_{\Omega}$
(45)

These transformation are not the most general possible to define
noncommutative fields,

However, these shifts modify the potential term in the following way

$\hbar^{2}e^{-3(2-\beta)[\Omega+\frac{\theta}{2}\hat{\Pi}_{\phi}]}\left[-\frac{\partial^{2}\Psi}{\partial\Omega^{2}}+Q\frac{\partial\Psi}{\partial\Omega}\right]-\frac{24}{\beta}\,\hbar^{\beta}\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\frac{\partial^{\beta}\Psi}{\partial\phi^{\beta}}\,=0.$
(46)

As in the commutative case, we choose the wave function to be separable,
$\Psi(\Omega,\phi)={\cal A}(\Omega)\,{\cal C}(\phi)$, getting

$e^{-i\hbar\frac{\theta}{2}\frac{d}{d\phi}}{\cal
C}\left[\hbar^{2}e^{-3(2-\beta)\Omega}\left(-\frac{d^{2}{\cal
A}}{d\Omega^{2}}+Q\,\frac{d{\cal A}}{d\Omega}\right)\right]-{\cal
A}\frac{24}{\beta}\,\hbar^{\beta}\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\frac{d^{\beta}{\cal
C}}{d\phi^{\beta}}$ (47)

which can be rewritten as

$e^{-i\hbar\frac{\theta}{2}\frac{d}{d\phi}}{\cal
C}\left[\hbar^{2}e^{-3(2-\beta)\Omega}\frac{\left(-\frac{d^{2}{\cal
A}}{d\Omega^{2}}+Q\,\frac{d{\cal A}}{d\Omega}\right)}{{\cal
A}}\right]-\frac{24}{\beta}\,\hbar^{\beta}\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\frac{d^{\beta}{\cal
C}}{d\phi^{\beta}},$ (48)

if we want this equation to be separable, we must choose to make the term
within the square parentheses [ ] a constant, in particular $\mp\mu^{2}$, with
this choice, we retrieve the commutative quantum equation for the function
${\cal A}$, (31), with the same quantum solution (36).

At this point we want to note that in commutative quantum cosmology, the
prefactor that accompanies its moments is not contemplated when we use a
particular gauge, and usually the non-commutative parameter enters the
solution of the $\Omega$ function, not that of scalar field $\phi$. In this
case, the appearance of the prefactor in fractional cosmology makes the
solution in $\Omega$ remain the same, but not the part of the scalar field,
where the non-commutative parameter appears and the mathematical structure is
completely different.

That said, the expression (48) becomes

$\mp\mu^{2}e^{-i\hbar\frac{\theta}{2}\frac{d}{d\phi}}{\cal
C}-\frac{24}{\beta}\,\hbar^{\beta}\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\frac{d^{\beta}{\cal
C}}{d\phi^{\beta}},$ (49)

since the non-commutative parameter $\theta$ is very small, we can stay until
the first term in this one, obtaining

$\frac{d^{\beta}{\cal
C_{\pm}}}{d\phi^{\beta}}\pm\left(\frac{\alpha}{2^{\alpha-1}}\right)^{\frac{1}{2\alpha-1}}\frac{\mu^{2}\,\beta}{24\hbar^{\beta}}{\cal
C}_{\pm}\mp\hbar\theta\left(\frac{\alpha}{2^{\alpha-1}}\right)^{\frac{1}{2\alpha-1}}\frac{i\mu^{2}\,\beta}{48\hbar^{\beta}}\frac{d{\cal
C}_{\pm}}{d\phi}=0,$ (50)

in this fractional differential equation, when $\theta=0$ we recover the
commutative equation for the quantum function ${\cal C}={\cal B}$ (32). Now we
solve the equation (50) written as follows

$\frac{d^{\beta}{\cal
C_{\pm}}}{d\phi^{\beta}}\mp\frac{i\,\hbar\theta}{2}q_{\alpha,\beta}\frac{d{\cal
C}_{\pm}}{d\phi}\pm q_{\alpha,\beta}{\cal C}_{\pm}=0,$ (51)

where
$q_{\alpha,\beta}=\left(\frac{\alpha}{2^{\alpha-1}}\right)^{\frac{1}{2\alpha-1}}\frac{\mu^{2}\,\beta}{24\hbar^{\beta}}$.
For the particular value $\beta=2$, we can observe that the equation (44),
becomes the ordinary commutative quantum equation, then the quantum solutions,
commutative and non-commutative, are the same in this approach to k-essence
theory.

However, in the dust scenario
$(\beta=1,\alpha\to\infty),q_{\infty,1}=\frac{\sqrt{2}\mu^{2}}{48\hbar}$.
Equation (51) takes the form

$z_{0}\frac{d\mathcal{C}_{\pm}}{d\phi}\pm\mathcal{C}_{\pm}=0,\qquad
z_{0}=\frac{48\hbar}{\sqrt{2}\mu^{2}}\mp i\frac{\hbar\theta}{2},$ (52)

whose solution is given by

$\mathcal{C}_{\pm}=\eta_{\pm}\,e^{\mp\,\varepsilon\Delta\phi}\,e^{-i\,\theta\psi\Delta\phi},\qquad\varepsilon=\frac{96\sqrt{2}\hbar\mu^{2}}{\mu^{4}\hbar^{2}\theta^{2}+4608\hbar^{2}},\quad\psi=\frac{2\hbar\mu^{4}}{\mu^{4}\hbar^{2}\theta^{2}+4608\hbar^{2}}.$
(53)

Thus, the probability density becomes (considering only the real part of the
complex exponential in $\theta$)

$\Psi^{2}=\left\\{\begin{tabular}[]{ll}$A_{0}\,e^{-2\varepsilon\Delta\phi}\,Cos^{2}\left[\theta\psi\Delta\phi\right]\,e^{Q\Omega}\,K_{\nu}^{2}\left[\frac{2\mu}{3\hbar}e^{\frac{3}{2}\Omega}\right],$&\quad
to\,\,$-\mu^{2}$,\\\
$A_{0}\,e^{2\varepsilon\Delta\phi}\,Cos^{2}\left[\theta\psi\Delta\phi\right]\,e^{Q\Omega}\,J_{\nu}^{2}\left[\frac{2\mu}{3\hbar}e^{\frac{3}{2}\Omega}\right],$&\quad
to\,\, $+\mu^{2}$\end{tabular}\right.$ (54)

To make the figure 1, we use the ordinary Bessel function. We can see the
effect of the combination of the parameters $\theta$ and $\mu$, where the
probability density undergoes a shift in the behavior of the scalar field, at
the beginning and at the end, that is, modifying the structure. As we can see,
at $\theta=0$, a crack appears, at $\theta=0.1$, it separates and a peak
appears, at $\theta=0.5$, the peak decreases and disappears at $\theta=1$,
when $\mu=$5\. However, the fact that some peaks no longer appear does not
mean that they have been cancelled, but rather that, due to the change in
probability density, the scales of these peaks are no longer on the graph.

Figure 1: In the following plots, we show the behavior of the probability
density of equation (54), considering the sign in $+\mu^{2}$, taking the
values $\mu=5$, $Q=0$ and $\theta=0,0.1,0.5,1$, respectively.

In the figure 2, the previously mentioned pattern is repeated when the factor
ordering parameter is $Q=1$, but more noticeably in the shift towards the
origin of the $\Omega$ variable. In the figure 3 for the factor ordering
$Q=-1$, the shift is slower, but persists.

Figure 2: In the following plots we show the behavior of the probability
density of equation (54), considering the sign in $+\mu^{2}$, taking the
values $\mu=5$, $Q=1$ and $\theta=0,0.1,0.5,1$, respectively.

Figure 3: In the following plots we show the behavior of the probability
density of equation (54), considering the sign in $+\mu^{2}$, taking the
values $\mu=5$, $Q=-1$ and $\theta=0,0.1,0.5,1$, respectively.

For the other scenarios, employing the modified Bessel function, the behavior
is very different, as shown in the figures 4, 5 and 6, when the combination of
the parameters $\mu=15$ and $\theta=0,0.5,0.8,1$, having decaying behavior in
the direction of evolution of the scale factor like ($\Omega$) and oscillatory
behavior in the direction of the scalar field, or making the scalar field
relevant in quantum evolution and remaining in classical evolution, as has
been found in other alternative models to Einstein’s theory omar2017 ;
omar2018 ; omar2020 ; abraham2021 ; barron2021 ; abraham2022 ; abraham2023 .

Since, we do not know the initial conditions of the universe in the dust
epoch, we have graphed both probability densities, where it is observed that
the scalar field persists in the evolution of both densities, remaining as a
remnant towards the classical evolution of the universe, being a cosmic
background currently.

Figure 4: In the following plots, we show the behavior of the probability
density of equation (54), considering the sign in $-\mu^{2}$, taking the
values $\mu=15$, $Q=0$ and $\theta=0,0.5,0.8,1$, respectively.

Figure 5: In the following plots, we show the behavior of the probability
density of equation (54), considering the sign in $-\mu^{2}$, taking the
values $\mu=15$, $Q=1$ and $\theta=0,0.5,0.8,1$, respectively.

Figure 6: In the following plots, we show the behavior of the probability
density of equation (54), considering the sign in $-\mu^{2}$, taking the
values $\mu=15$, $Q=-1$ and $\theta=0,0.5,0.8,1$, respectively.

The global effect of the non-commutativity between the field coordinates of
the system in fractional quantum cosmology scheme causes the probability
density to shift or shrink in the opposite direction to the scale factor,
causing the classical universe to emerge sooner, which would mean that the
current universe should have more time than is usually mentioned, as in
mentioned in the reference Jalalzadeh2023 , employing the fractional
framework.

On the other hand, if the order of the differential equation (51) is a
rational, then solutions have two cases

###  V.1 $\omega_{x}\in[0,1]$, $\beta\in[1,2]$, $n=\lceil\beta\rceil=2$

Taking into account the Laplace transform in Machado , considering that ${\cal
L}[_{c}D^{\beta}f(t)]=s^{\beta}F(s)-s^{\beta-1}f(0)-s^{\beta-2}f^{\prime}(0)$,
${\cal L}[\frac{df(t)}{dt}]=sF(s)-f(0)$, and ${\cal L}[f(t)]=F(s)$. Then, let
the fractional differential equation

$\frac{d^{\beta}{\cal C}}{d\phi^{\beta}}+A\frac{d{\cal C}}{d\phi}+B{\cal
C}=0,$ (55)

where

$A=\mp\frac{i\theta}{2}\Big{(}\frac{\alpha}{2^{\alpha-1}}\Big{)}^{\frac{1}{2\alpha-1}}\frac{\mu^{2}\beta}{24\hbar^{2\gamma}}\qquad
B=\pm\Big{(}\frac{\alpha}{2^{\alpha-1}}\Big{)}^{\frac{1}{2\alpha-1}}\frac{\mu^{2}\beta}{24\hbar^{2\gamma}}.$
(56)

Applying the Laplace transform to all the terms in (55), we have

$s^{\beta}C(s)-s^{\beta-1}C(0)-s^{\beta-2}C^{\prime}(0)+As\,C(s)-A\,C(0)+B\,C(s)=0$
(57)

Solving with respect to $C(s)$, we get

$C(s)=\frac{C(0)s^{\beta-1}}{s^{\beta}+As+B}+\frac{C^{\prime}(0)s^{\beta-2}}{s^{\beta}+As+B}+\frac{AC(0)}{s^{\beta}+As+B}$
(58)

for the particular value $\beta=2$, the two last terms can be consider as one,
making that $C^{\prime}(0)+AC(0)=\kappa=constant$, and for $\beta=1$, the
first and last term can be simplify to $C(0)+AC(0)=\kappa_{1}=constant$

From the formula in Machado  (page 40, equation (3.11) with a correction), we
have

${\cal
L}^{-1}\Big{[}\frac{s^{\gamma}}{s^{\alpha}+as^{\beta}+b}\Big{]}=t^{\alpha-\gamma-1}\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\frac{(-b)^{n}(-a)^{k}\binom{n+1+k}{k}}{\Gamma[k(\alpha-\beta)+(n+1)\alpha-\gamma]}t^{k(\alpha-\beta)+n\alpha},$
(59)

adapting our parameters to the master equation (59), we have the following
three cases

  1. 1.

first term in (58), we use $\gamma=\beta-1$, $\alpha=\beta$, $\beta=1$

${\cal
L}^{-1}\Big{[}\frac{s^{\beta-1}}{s^{\beta}+as+b}\Big{]}=\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\frac{(-B)^{n}(-A)^{k}\binom{n+1+k}{k}}{\Gamma[k(\beta-1)+(n+1)\beta-(\beta-1)]}\phi^{k(\beta-1)+n\beta}$
(60)

  2. 2.

second term in (58), we use $\gamma=\beta-2$, $\alpha=\beta$, $\beta=1$

${\cal
L}^{-1}\Big{[}\frac{s^{\beta-2}}{s^{\beta}+as+b}\Big{]}=\phi\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\frac{(-B)^{n}(-A)^{k}\binom{n+1+k}{k}}{\Gamma[k(\beta-1)+(n+1)\beta-(\beta-2)]}\phi^{k(\beta-1)+n\beta}$
(61)

  3. 3.

third term in (58), we use $\gamma=0$, $\alpha=\beta$, $\beta=1$

${\cal
L}^{-1}\Big{[}\frac{1}{s^{\beta}+as+b}\Big{]}=\phi^{\beta-1}\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\frac{(-B)^{n}(-A)^{k}\binom{n+1+k}{k}}{\Gamma[k(\beta-1)+(n+1)\beta]}\phi^{k(\beta-1)+n\beta}$
(62)

Then, the inverse Laplace transform of (58), is Then, the inverse Laplace
transform of (58), is

$\displaystyle{\cal C}(\phi)$ $\displaystyle=$ $\displaystyle{\cal
L}^{-1}\Big{[}\frac{C(0)s^{\beta-1}}{s^{\beta}+As+B}+\frac{C^{\prime}(0)s^{\beta-2}}{s^{\beta}+As+B}+Ac_{0}\frac{1}{s^{2\gamma}+As+B}\Big{]}=$
(63) $\displaystyle=$ $\displaystyle
C(0)\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\frac{(-B)^{n}(-A)^{k}\binom{n+1+k}{k}}{\Gamma[k(\beta-1)+(n+1)\beta-(\beta-1)]}\phi^{k(\beta-1)+n\beta}+$
$\displaystyle+$ $\displaystyle
C^{\prime}(0)\phi\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\frac{(-B)^{n}(-A)^{k}\binom{n+1+k}{k}}{\Gamma[k(\beta-1)+(n+1)\beta-(\beta-2)]}\phi^{k(\beta-1)+n\beta}+$
$\displaystyle+$ $\displaystyle
A\,C(0)\phi^{\beta-1}\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\frac{(-B)^{n}(-A)^{k}\binom{n+1+k}{k}}{\Gamma[k(\beta-1)+(n+1)\beta]}\phi^{k(\beta-1)+n\beta}$

For the case when $A=\mp i\frac{\theta}{2}q_{\alpha,\gamma}$ and $B=\pm
q_{\alpha,\gamma}$,
$q_{\alpha,\gamma}=\Big{(}\frac{\alpha}{2^{\alpha-1}}\Big{)}^{\frac{1}{2\alpha-1}}\frac{\mu^{2}\beta}{24\hbar^{2\gamma}}$,
we see that the complex solution can be read as

$\displaystyle\mathcal{C}$
$\displaystyle=a\Biggl{\\{}1-\frac{\theta^{2}q^{2}}{4\,\Gamma(2\beta-1)}\phi^{2\beta-2}\mp\frac{q}{\Gamma(\beta+1)}\phi^{\beta}\pm\frac{3\theta^{2}q^{3}}{4\,\Gamma(3\beta-1)}\phi^{3\beta-2}+\frac{q^{2}}{\Gamma(2\beta+1)}\phi^{2\beta}$
$\displaystyle-\frac{3\theta^{2}q^{4}}{2\,\Gamma(4\beta-1)}\phi^{4\beta-2}+\ldots\Biggr{\\}}+b\Biggl{\\{}\phi-\frac{3q^{2}\theta^{2}}{4\,\Gamma(2\beta)}\phi^{2\beta-1}\mp\frac{q}{\Gamma(\beta+2)}\phi^{\beta+1}\pm\frac{3\theta^{2}q^{3}}{2\,\Gamma(3\beta)}\phi^{3\beta-1}$
$\displaystyle+\frac{q^{2}}{\Gamma(2\beta+2)}\phi^{2\beta+1}-\frac{5\theta^{2}q^{4}}{2\,\Gamma(4\beta)}\phi^{4\beta-1}+\ldots\Biggr{\\}}\pm
i\Biggl{\\{}a\Biggr{[}\frac{q\theta}{2\,\Gamma(\beta)}\phi^{\beta-1}+\frac{3\theta^{3}q^{3}}{8\,\Gamma(3\beta-2)}\phi^{3\beta-3}$
$\displaystyle+\frac{\theta
q^{2}}{\Gamma(2\beta)}\phi^{2\beta-1}+\ldots\Biggr{]}+b\Biggr{[}\frac{q\theta}{\Gamma(\beta+1)}\phi^{\beta}+\frac{3q^{2}\theta}{2\,\Gamma(2\beta+1)}\phi^{2\beta}+\frac{2q^{3}\theta}{\Gamma(3\beta+1)}\phi^{3\beta}+\ldots\Biggl{]}\Biggl{\\}},$
(64)

where $C(0)=a$ and $C^{\prime}(0)=b$.

###  V.2 $\omega_{x}\in(0,-1]$, $\beta\in[0,1]$, $n=\lceil\beta\rceil=1$

For this case, the equation to solve is

$C(s)=\frac{C(0)s^{\beta-1}}{s^{\beta}+As+B}+\frac{AC(0)}{s^{\beta}+As+B},$
(65)

similarly, as in the previous case, we have

$\displaystyle{\cal C}(\phi)$ $\displaystyle=$ $\displaystyle{\cal
L}^{-1}\Big{[}\frac{C(0)s^{\beta-1}}{s^{\beta}+As+B}+\frac{AC(0)}{s^{\beta}+As+B}\Big{]}=$
(66) $\displaystyle=$ $\displaystyle
C(0)\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\frac{(-B)^{n}(-A)^{k}\binom{n+1+k}{k}}{\Gamma[k(\beta-1)+(n+1)\beta-(\beta-1)]}\phi^{k(\beta-1)+n\beta}+$
$\displaystyle+$ $\displaystyle
A\,C(0)\phi^{\beta-1}\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\frac{(-B)^{n}(-A)^{k}\binom{n+1+k}{k}}{\Gamma[k(\beta-1)+(n+1)\beta]}\phi^{k(\beta-1)+n\beta}.$

which can be rewritten as

$\displaystyle\mathcal{C}(\phi)$
$\displaystyle=a\Biggl{\\{}1-\frac{q^{2}\theta^{2}}{4\,\Gamma(2\beta-1)}\phi^{2\beta-2}\mp\frac{q}{\Gamma(\beta+1)}\phi^{\beta}\pm\frac{3q^{3}\theta^{2}}{4\,\Gamma(3\beta-1)}\phi^{3\beta-2}+\frac{q^{2}}{\Gamma(2\beta+1)}\phi^{2\beta}$
$\displaystyle-\frac{3q^{4}\theta^{2}}{2\,\Gamma(4\beta-1)}\phi^{4\beta-2}+\ldots\Biggr{\\}}\pm
ia\Biggl{\\{}\frac{q\theta}{2\,\Gamma(\beta)}\phi^{\beta-1}+\frac{3q^{3}\theta^{3}}{8\,\Gamma(3\beta-2)}\phi^{3\beta-3}-\frac{q^{2}\theta}{\Gamma(2\beta)}\phi^{2\beta-1}$
$\displaystyle-\frac{3q^{4}\theta^{3}}{4\,\Gamma(4\beta-2)}\phi^{4\beta-3}+\frac{3q^{3}\theta}{2\,\Gamma(3\beta)}\phi^{3\beta-1}-\frac{5q^{5}\theta^{3}}{4\,\Gamma(5\beta-2)}\phi^{5\beta-3}+\ldots\Biggr{\\}},$
(67)

##  VI Conclusions

During the development of research in the non-commutative formalism within
fractional cosmology in $k$-essence theory, the presence of non-commutativity
that usually accompanied the term of the scale factor, here is disrupted,
since essentially Non-commutativity is more present in the scalar field,
modifying the mathematical structure that usually occurs in works in this
direction in other non-fractional formalisms.

In our non-commutative quantum development, the method of separation of
variables does not appear in a traditional way as the sum of the operators in
their variables, now it is produced as factors, thanks to this it can be
separated, in addition, now complex fractional differential equations arise,
even in cases with derivatives of integer order, which means that these
solutions in the scalar field have a real part and an imaginary part.

In previous non-commutative quantum works Sabido ; Aguero ; Socorro , the term
is usually modified with the scale factor, but in fractional cosmology in
essence K, this term remains unchanged, only the scalar field term undergoes
important modifications, in the sense that the probability density undergoes a
shift back to the direction of the scale factor, causing classical evolution
to arise earlier than in the commutative world. This effect is due to the non-
commutativity between the field coordinates in this formalism, which is
related to some crucial effects due to the fact of having a fractional
equation, such that the age of the universe is greater, of the order of
13.8196 Gyr. , or more Jalalzadeh2023 . These results on fractal K-essence
theory add to the fact that this formalism without considering ordinary matter
is falsified with this approach according to the classical solutions that are
identical using the FRW model universe , but it is found that this is a more
general result mentioned in the reference sasaki2010 .

Since the prefactor that is usually linked to the ordering of the factors
under a certain gauge does not appear in the standard quantum Hamiltonian, the
important contribution of noncommutativity appears in the wave function linked
to the scale factors, which is why this term continues to persist. This causes
the momentum associated with the scalar field to produce an additional total
derivative term to the non-commutative fractional equation due to the Bopp
shift in the scale factor term, producing in this case a significant
contribution of the non-commutative parameter in the wave function, see
equation (51).

###### Acknowledgements.

J.S. was partially supported by PROMEP grants UGTO-CA-3. Both authors were
partially supported by SNI-CONACyT. J.J. Rosales is supported by the UGTO-
CA-20 nonlinear photonics and Department of Electrical Engineering. L.T.S. is
supported by Secretaria de Investigación y Posgrado del Instituto Politécnico
Nacional, grant SIP20211444. This work is part of the collaboration within the
Instituto Avanzado de Cosmología. Many calculations were done by Symbolic
Program REDUCE 3.8.

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