# Higher-order generalized uncertainty principle applied to gravitational
baryogenesis

Zhong-Wen Feng,11footnotetext: Corresponding author. Xia Zhou  Shi-Qi Zhou

###### Abstract

The gravitational baryogenesis plays an important role in the study of baryon
asymmetry. However, the original mechanism of gravitational baryogenesis in
the radiation-dominated era leads to the asymmetry factor $\eta$ equal to
zero, which indicates this mechanism may not generate a sufficient baryon
asymmetry in the early Universe. In this paper, we investigate the
gravitational baryogenesis for the generation of baryon asymmetry in the early
Universe by using a new higher-order generalized uncertainty principle (GUP).
It is demonstrated that the entropy and the Friedman equation of the Universe
deviate from the original cases due to the effect of the higher-order GUP.
Those modifications break the thermal equilibrium of the Universe, and in turn
produce a non-zero asymmetry factor $\eta$. In particular, our results satisfy
all of Sakharov’s conditions, which indicates that the scheme of explaining
baryon asymmetry in the framework of higher-order GUP is feasible. In
addition, combining our theoretical results with the observational data, we
constraint the GUP parameter $\beta_{0}$, whose bound is between
$8.4\times{10^{10}}\sim 1.1\times{10^{13}}$.

##  1 Introduction

The discovery of antimatter is one of the most important achievements in
modern physics [1]. Initially, it is thought that the number of matter and
antimatter in the Universe was equal [2]. However, this prediction conflicts
with the observational evidence such as the Big Bang Nucleosynthesis (BBN),
the Cosmic Microwave Background (CMB), and Planck observations, which show
that matter exceeds antimatter [3, 4, 5]. This stark contradiction sparks an
open issue of modern physics and cosmology, namely the baryon asymmetry of the
Universe (BAU) [6]. Notably, there are two consensuses on the issue of baryon
asymmetry, one is that baryon asymmetry may generate dynamically in the early
stages of the cosmic expansion, and the other is that the key to the
generation of baryon asymmetry is to satisfy the three Sakharov conditions,
which are [7] (i) existence of reactions violating baryon number; (ii) C
(charge conjugation) and CP (combined charge conjugation and parity
transformation) violation; (iii) departure from thermal equilibrium. Along
this line, various theories have been proposed to explain how the asymmetry
between baryons and antibaryons arose in the early stages of the evolution of
the Universe (see [8, 9, 10] for a review). In particular, Davoudiasl _et al_.
[11] proposed one appealing mechanism called “gravitational baryogenesis”,
which is for generating the baryon number asymmetry in thermal equilibrium
during the expansion of the Universe utilizing a dynamical breaking of CPT.
The gravitational baryogenesis has gained people’s attention as soon as it was
put forward because this mechanism showed that gravitational coupling is a
means to produce material asymmetry. Since then, various scenarios are in
progress in this direction (see e.g., [12, 13, 14, 15, 16]).

Although the gravitational baryogenesis plays an important role in analyzing
BAU, it is found to be flawed in further research, i.e., in the general
relativity regime for flat Friedmann-Robertson-Walker (FRW) metric, one found
that the time changes of the Ricci scalar curvature $\dot{R}$ and the
corresponding baryon asymmetry factor $\eta$ (a factor used to measure the
number of baryonic matter exceeding the antibaryonic matter) are zero, which
indicate the original mechanism cannot generate a sufficient baryon asymmetry
for the radiation dominated universe. To address the problem above, many
effective schemes have been proposed. For example, Li _et al._ [17] improved
the interaction of gravitational baryogenesis to generate the matter-
antimatter asymmetry. In [18, 19, 20, 21, 22, 23, 24, 25], the authors argued
that combining the original mechanism of gravitational baryogenesis with
modified gravity theories can deviate the Friedmann equations from the
classical case and produces a non-vanishing baryon asymmetry factor. Fukushima
_et al._ [26] investigated the gravitational baryogenesis in the anisotropic
spacetime and discussed the BAU at the end of anisotropic inflation. When
considering the loop quantum cosmology effects, Odintsov and Oikonomou [27]
obtained a non-zero baryon-to-entropy ratio from the gravitational
baryogenesis mechanism. Smyth _et al_. [28] discussed that the BAU is created
through Hawking radiation from the primordial black holes via a dynamically-
generated chemical potential.

On the other hand, a distinctive signature of most candidate theories of
quantum gravity (QG) is the prediction of a minimum measurable length near the
Planck scale [29, 30, 31]. In that case, many classical theories should be
modified by the effect of QG. For instance, by incorporating the minimal
measurable length with the Heisenberg algebra, the conventional Heisenberg
uncertainty principle (HUP) can be changed to the so-called generalized
uncertainty principle (GUP). In recent years, the GUP has received wide
attention since it can be used in physical systems with extremely small scale
or high-energy scale [32, 33, 34, 35, 36, 37, 38]. As discussed above, the BAU
may occur at around the radiation domination era, during which the effect of
GUP is also active at that time. In this connection, the combination of
gravitational baryogenesis with GUP allows addressing the BAU. In [39],
according to the GUP that was constructed by Ali, Das, and Vagenas (ADV model)
[40], i.e., $\Delta x\Delta
p\geq\frac{\hbar}{2}\left[{1-{{{\alpha_{0}}{\ell_{p}}\Delta
p}\mathord{\left/{\vphantom{{{\alpha_{0}}{\ell_{p}}\Delta
p}\hbar}}\right.\kern-1.2pt}\hbar}}\right.$
$\left.{+{{{\beta_{0}}\ell_{p}^{2}\Delta{p^{2}}}\mathord{\left/{\vphantom{{{\beta_{0}}\ell_{p}^{2}\Delta{p^{2}}}{{\hbar^{2}}}}}\right.\kern-1.2pt}{{\hbar^{2}}}}}\right]$
with the GUP parameters ${\alpha_{0}}$ and ${\beta_{0}}$,222When ignoring the
${\alpha_{0}}$, the ADV model reduces to $\Delta x\Delta
p\geq{{\hbar\left[{1+{{\beta_{0}\ell_{p}^{2}\Delta{p^{2}}}\mathord{\left/{\vphantom{{\beta_{0}\ell_{p}^{2}\Delta{p^{2}}}{{\hbar^{2}}}}}\right.\kern-1.2pt}{{\hbar^{2}}}}}\right]{\rm{}}}\mathord{\left/{\vphantom{{\hbar\left[{1+{{\beta_{0}\ell_{p}^{2}\Delta{p^{2}}}\mathord{\left/{\vphantom{{\beta_{0}\ell_{p}^{2}\Delta{p^{2}}}{{\hbar^{2}}}}}\right.\kern-1.2pt}{{\hbar^{2}}}}}\right]{\rm{}}}2}}\right.\kern-1.2pt}2}$,
which is consistent with the expression of KMM model. Das _et al_. derived a
non-zero asymmetry factor $\eta$ at the radiation domination era, which
provides a new way to explain BAU. However, it is worth mentioning that the
ADV model has some limitations [41]. Firstly, the perturbative of the ADV
model is only valid for small values of the GUP parameter. Secondly, this GUP
model does not imply noncommutative geometry. In addition, due to the maximal
momentum uncertainty being different from the maximal momentum, it is found
that the ADV model is not appropriate to the doubly special relativity. To
solve those defects, Pedram presented a nonperturbative higher-order GUP that
is in agreement with various proposals of QG and eliminates the objections due
to the doubly special relativity theories [42].

This heuristic work of Pedram is considered as an important step towards
establishing a physically reasonable theory of QG. Therefore, many studies
have converged on the construction and application of high-order GUP [43, 44,
45, 46]. Recently, according to the commutator
$\left[{x,p}\right]=i\hbar\sqrt{1-2\beta{p^{2}}}$, Petruzziello [47]
constructed a new higher-order GUP, viz.

$\displaystyle\Delta x\Delta p$
$\displaystyle\geq\frac{\hbar}{2}\left\langle{\sqrt{1-2\beta{p^{2}}}}\right\rangle\geq\frac{\hbar}{2}\sum\limits_{n=0}^{\infty}{\frac{{{\beta^{n}}\left({2n}\right)!}}{{{2^{n}}\left({1-2n}\right){{\left({n!}\right)}^{2}}}}}\left\langle{{p^{2n}}}\right\rangle$
$\displaystyle\geq\frac{\hbar}{2}\sum\limits_{n=0}^{\infty}{\frac{{{\beta^{n}}\left({2n}\right)!}}{{{2^{n}}\left({1-2n}\right){{\left({n!}\right)}^{2}}}}}{\left\langle{{p^{2}}}\right\rangle^{n}}$
$\displaystyle\geq\frac{\hbar}{2}\sum\limits_{n=0}^{\infty}{\frac{{{\beta^{n}}\left({2n}\right)!}}{{{2^{n}}\left({1-2n}\right){{\left({n!}\right)}^{2}}}}}{\left({\Delta{p^{2}}+{{\langle\hat{p}\rangle}^{2}}}\right)^{n}}$
$\displaystyle\geq\frac{\hbar}{2}\sqrt{1-2\beta\Delta{p^{2}}},$ (1.1)

where we have used the property
$\left\langle{{p^{2n}}}\right\rangle\geq{\left\langle{{p^{2}}}\right\rangle^{n}}$
and the mirror-symmetric states $\left\langle p\right\rangle=0$,
$\beta={{{\beta_{0}}\ell_{p}^{2}}\mathord{\left/{\vphantom{{{\beta_{0}}\ell_{p}^{2}}{{\hbar^{2}}}}}\right.\kern-1.2pt}{{\hbar^{2}}}}={{{\beta_{0}}}\mathord{\left/{\vphantom{{{\beta_{0}}}{m_{p}^{2}{c^{2}}}}}\right.\kern-1.2pt}{m_{p}^{2}{c^{2}}}}$
with the dimensionless GUP parameter $\beta_{0}$, the Planck length
${\ell_{p}}$ and the Planck mass ${m_{p}}$. In addition to inheriting the
advantages of the previous higher-order GUP, this model also has the following
characteristics: (i) it has a negative value for the GUP parameter
$-\left|\beta_{0}\right|$, which is considered to be very useful in
astrophysical and cosmology. In [48], it is estimated that the negative GUP
parameter is the only setting meets the Chandrasekhar limit for white dwarfs.
Besides, it is found that the negative GUP parameter may appears in non-
trivial space-time structures and leads to a crystal-like universe [49, 50].
For the sake of simplicity, we omit absolute value of the GUP parameter in
this work; (ii) eq. (1) contains only a maximal observable momentum
$\Delta{p_{\max}}\approx{1\mathord{\left/{\vphantom{1{\sqrt{2\beta}}}}\right.\kern-1.2pt}{\sqrt{2\beta}}}$
and no minimal length uncertainty, which is never occurs in other models. For
$\Delta p=\pm\Delta{p_{\max}}$, the position and momentum operators of above
inequality are interchangeable, which is consistent with the classical case;
(iii) this new higher order GUP reduce to the quadratic form of the GUP
$\Delta x\Delta
p\geq{{\hbar\left[{1+{{\beta_{0}^{{\rm{KMM}}}\ell_{p}^{2}\Delta{p^{2}}}\mathord{\left/{\vphantom{{\beta_{0}^{{\rm{KMM}}}\ell_{p}^{2}\Delta{p^{2}}}{{\hbar^{2}}}}}\right.\kern-1.2pt}{{\hbar^{2}}}}}\right]{\rm{}}}\mathord{\left/{\vphantom{{\hbar\left[{1+{{\beta_{0}^{{\rm{KMM}}}\ell_{p}^{2}\Delta{p^{2}}}\mathord{\left/{\vphantom{{\beta_{0}^{{\rm{KMM}}}\ell_{p}^{2}\Delta{p^{2}}}{{\hbar^{2}}}}}\right.\kern-1.2pt}{{\hbar^{2}}}}}\right]{\rm{}}}2}}\right.\kern-1.2pt}2}$
(hereafter referred to as Kempf-Mangano-Mann model or “KMM model”) with the
negative GUP parameter ${\beta_{0}^{{\rm{KMM}}}}$ for $\sqrt{\beta}\Delta p\ll
1$. To sum up, it is found that the GUP parameters in the two models have has
the relationship ${\beta_{0}}\sim-{\beta_{0}^{{\rm{KMM}}}}$. However, when the
GUP parameters equal to zero, they recover the HUP.

In [51, 52], the authors pointed out that features of QG are imprinted on
evolution of the Universe, which can be decoded through the astronomical
observation (such as CMB [53, 54] and the upcoming Cherenkov Telescope Array
(CTA) [55]). Furthermore, the effect of QG in the higher-order proposal would
greatly change the classical physical system. Therefore, studying the BAU in
the framework of higher-order GUP helps people understand the role of QG in
the early evolution of the Universe. However, to the best of our knowledge,
the higher-order GUP applied to baryogenesis has never been reported. To
address this issue, in this present paper we attempt to employ the new higher-
order GUP to study the BAU. We intend to combine eq. (1) with the
gravitational baryogenesis to derive the modified Friedmann equations, which
in turn produces a non-zero time derivative of the Ricci scaler curvature
$\dot{R}$ as well as a non-zero asymmetry factor $\eta$. Then, the BAU can be
explained due to those modifications. Finally, according to the recent
observation data, it is possible to estimate the bound of the GUP parameter
${\beta_{0}}$. According to the unique properties of the new higher-order GUP,
it is believed our study would give some different results from the previous
works.

This paper is organized as follows. In the next section, we give a brief
review of the gravitational baryogenesis. In Section 3, we derive the modified
entropy and the corresponding Friedmann equations in the framework of the new
higher-order GUP (1). Then, we derive the non-zero Ricci scalar curvature $R$
and the asymmetry factor $\eta$. Finally, the BAU is explained due to those
modifications. In Section 4, according to the recent experiments and
observations, we constrain the bounds of the GUP parameter $\beta_{0}$. The
paper ends with conclusions in Section 5. To simplify the notation, this
research takes the units $\hbar=c={k_{B}}=1$.

##  2 Gravitational baryogenesis for the standard cosmological model

Within supergravity theories, the interaction of the gravitational
baryogenesis between the derivative of the Ricci scalar curvature and the
baryon-number current dynamically breaks CPT in an expanding universe is given
by [11]

$\displaystyle\frac{1}{{M_{*}^{2}}}\int{{{\rm{d}}^{4}}}x\sqrt{-g}\left({{\partial_{\mu}}R}\right){J^{\mu}},$
(2.1)

where $R$ denotes the Ricci scalar curvature, ${J^{\mu}}$ is the baryon
current, ${M_{*}}$ is the cut-off scale of the effective theory. Following the
viewpoint in [18, 19, 20, 21, 22, 23, 24, 25, 26, 56], in the process of the
Universe’s expansion, when the temperature $T$ of the Universe drops below the
critical temperature $T_{D}$ at which the baryon asymmetry generating
interactions occurs, the baryon asymmetry factor (BAF) $\eta$ that measure the
amount of baryon matter exceeds antibaryon matter can be derived from the
equation above

$\displaystyle\eta=\frac{{{n_{B}}}}{s}\simeq-{\left.{\frac{{15{g_{b}}}}{{4{\pi^{2}}{g_{*}}}}\frac{{\dot{R}}}{{M_{*}^{2}T}}}\right|_{{T_{D}}}},$
(2.2)

where ${{{n_{B}}}\mathord{\left/{\vphantom{{{n_{B}}}s}}\right.\kern-1.2pt}s}$
is the baryon to entropy ratio with the number of baryons (antibaryons) per
volume unity ${n_{B}}$ and the entropy density
$s={{2{\pi^{2}}{g_{*}}{T^{3}}}\mathord{\left/{\vphantom{{2{\pi^{2}}{g_{*}}{T^{3}}}{45}}}\right.\kern-1.2pt}{45}}$
for the Universe. $\dot{R}={{\partial R}\mathord{\left/{\vphantom{{\partial
R}{\partial t}}}\right.\kern-1.2pt}{\partial t}}$ is the time derivative of
the Ricci scalar curvature of the Universe, $g_{b}$ and ${g_{*}}$ stand for
the number of intrinsic degrees of freedom of baryons and the degrees of
freedom of particles that contribute to the entropy of universe, respectively.
According to eq. (2.2), one can observe that the BAF is proportional to
$\dot{R}$. In the standard cosmological mode, the Ricci scalar curvature can
be expressed as

$\displaystyle R=-8\pi G\left({\rho-3p}\right),$ (2.3)

with the energy density $\rho$ and pressure $p$ of the Universe. When
considering matter source in the universe as a perfect fluid, one has the
equation of state parameter
$w={p\mathord{\left/{\vphantom{p\rho}}\right.\kern-1.2pt}\rho}$, and the Eq.
(2.3) can be cast as

$\displaystyle R=-8\pi G(1-3w)\rho.$ (2.4)

Notably, when taking into account the radiation-dominated era that is
characterized by $w={1\mathord{\left/{\vphantom{13}}\right.\kern-1.2pt}3}$,
the Ricci scalar curvature $R$ and its derivative $\dot{R}$ vanish. As a
consequence, one has $\eta=0$. However, many astronomical observations
demonstrated the BAF is not equal to zero, which implies the matter in the
Universe is more than the antimatter. For example, the acoustic peaks in CMB
measured by the Wilkinson Microwave Anisotropy Probe (WMAP) give $\eta\leq
6.3\times{10^{-10}}$, BBN exhibits that $3.4\times{10^{-10}}\leq\eta\leq
6.9\times{10^{-10}}$. To fix this contradiction, we investigate the higher-
order GUP corrected gravitational baryogenesis in the next section.

##  3 Gravitational baryogenesis in the framework of higher-order GUP

In [39], the authors pointed out that, the preservation of thermal equilibrium
in the original scenario of gravitational baryogenesis will lead to the Ricci
scalar curvature as well as its derivative equal to zero, and eventually make
$\eta=0$. Therefore, an effective way to generate the baryon asymmetry is to
break the thermal equilibrium of the Universe by deviating its thermodynamic
properties from the classical case. In the following discussion, according to
the holographic principle and the new higher-order GUP (1), we will derive the
modified Bekenstein-Hawking entropy and the modified Friedmann equations,
which break the thermal equilibrium. Finally, according to those
modifications, a non-zero Ricci scalar curvature $R$ and a non-zero $\eta$ can
be obtained.

###  3.1 The modified entropy in the higher-order GUP

According to the holographic principle, when a gravitational system absorbs a
particle, its area and the total energy inside the apparent horizon will
increases, the minimal change of the area $\Delta A$ can be expressed as [57,
58]

$\displaystyle\Delta A\sim Xm,$ (3.1)

where $X$ and $m$ represent the size and mass of the particle, respectively.
In quantum mechanics, the width of the wave packet of a particle is described
as the standard deviation of $X$ distribution (i.e., the position uncertainty
$\Delta x$), hence, one has the relationship $X\sim\Delta x$. Furthermore, in
the process of measuring the position of a particle, the mass of particle
should be larger the momentum uncertainty $\Delta p$ [45]. Hence, eq. (3.1)
can be rewritten as

$\displaystyle\Delta A\geq\Delta x\Delta p.$ (3.2)

The above equation implies that the smallest increase in area in a
gravitational system is restricted by the momentum uncertainty $\Delta p$ and
position uncertainty $\Delta x$ of quantum mechanics.

For a static and spherically gravitational system, the position uncertainty is
approximately equal to the radius of the apparent horizon, that is $\Delta
x\approx 2r$ [59]. Substituting this relationship into the higher-order GUP
(1), one has

$\displaystyle\frac{{\sqrt{1-2\Delta{p^{2}}\beta}}}{{2\Delta p}}\leq\Delta
x\leq 2r,$ (3.3)

and the momentum uncertainty is given by

$\displaystyle-\frac{1}{{\sqrt{4\Delta{x^{2}}+2\beta}}}\leq\Delta
p\leq\frac{1}{{\sqrt{4\Delta{x^{2}}+2\beta}}}.$ (3.4)

Inserting eq. (3.3) and eq. (3.4) into eq. (3.2), the minimal change of the
area can be rewritten as follows

$\displaystyle\Delta A\geq\chi\tilde{\hbar}\left(\beta\right),$ (3.5)

with the effective Planck constant
${\tilde{\hbar}}\left(\beta\right)={{2r}\mathord{\left/{\vphantom{{2r}{\sqrt{16{r^{2}}+2\beta}}}}\right.\kern-1.2pt}{\sqrt{16{r^{2}}+2\beta}}}$
and the calibration factor $\chi=4\ln 2$ [60]. Notedly, in the limit
$\beta\rightarrow 0$, one yields ${\tilde{\hbar}}\left(\beta\right)=1/2$.
Based on the information theory, the minimal increase of entropy is
conjectured related to the value of the area

$\displaystyle\frac{{{\rm{d}}S}}{{{\rm{d}}A}}\simeq\frac{{\Delta{S_{\min}}}}{{\Delta{A_{\min}}}}=\frac{1}{{8{\tilde{\hbar}}\left(\beta\right)}},$
(3.6)

where ${\Delta{S_{\min}}}=\ln 2$ represents the minimal increase in entropy,
which implies to one bit of information. In the classical limit, the original
entropy of a gravitational system can be expressed as
${S_{0}}={A\mathord{\left/{\vphantom{A4}}\right.\kern-1.2pt}4}$, which
indicates the entropy is related to the horizon area $A$. However, when
considering the effect of GUP, the general expression of entropy should be
modified as
$S={{f\left(A\right)}\mathord{\left/{\vphantom{{f\left(A\right)}4}}\right.\kern-1.2pt}4}$
with the function of area ${f\left(A\right)}$. Correspondingly, the entropy-
area relation can be expressed as follows [61]

$\displaystyle\frac{{{\rm{d}}S}}{{{\rm{d}}A}}{\rm{=}}\frac{{f^{\prime}\left(A\right)}}{4},$
(3.7)

where
$f^{\prime}\left(A\right)={{{\rm{d}}f\left(A\right)}\mathord{\left/{\vphantom{{{\rm{d}}f\left(A\right)}{{\rm{d}}A}}}\right.\kern-1.2pt}{{\rm{d}}A}}$.
By comparing eq. (3.7) with eq. (3.6), one has

$\displaystyle
f^{\prime}\left(A\right)=\frac{1}{{2{\tilde{\hbar}}\left(\beta\right)}}=\sqrt{1+\frac{{\pi\beta}}{{2A}}}.$
(3.8)

For $\beta\to 0$, one has $f^{\prime}\left(A\right)=1$, consistently with the
standard result in classical limit. Then, by integrating eq. (3.7), the GUP
corrected entropy is given by

$\displaystyle{S_{{\rm{GUP}}}}$
$\displaystyle=\int{\frac{{{\rm{d}}S}}{{{\rm{d}}A}}{\rm{d}}A}\simeq\int{\frac{{\Delta{S_{\min}}}}{{\Delta{A_{\min}}}}{\rm{d}}A}\simeq\int{\frac{{{\rm{d}}A}}{{8{\tilde{\hbar}}\left(\beta\right)}}}$
$\displaystyle=\frac{{\sqrt{2\left({2A+\pi\beta}\right)A}}}{8}+\frac{\pi}{8}\beta\ln\left({2\sqrt{A}+\sqrt{4A+2\pi\beta}}\right).$
(3.9)

Obviously, the effect of GUP leads to a logarithmic term with the deformation
parameter $\beta$ in the parentheses of eq. (3.1), which meet the requirements
of QG [57, 62, 63]. Besides, in the limit where $\beta\to 0$, it agrees with
the original entropy ${S_{0}}$.

###  3.2 GUP corrected Friedmann equations

In [64, 65, 66, 67], it is found that the Friedmann equations can be derived
from the Bekenstein-Hawking entropy and the first law of thermodynamics.
Therefore, according to the modified entropy (3.1), we will derive the GUP
corrected Friedmann equations that deviates from the thermal equilibrium. In
the homogeneous and isotropic spacetime, the FRW universe is described by the
line element:

$\displaystyle{\rm{d}}{s^{2}}={h_{\mu\nu}}{\rm{d}}{x^{\mu}}{\rm{d}}{x^{\nu}}+{\tilde{r}^{2}}\left({\rm{d}{\theta^{2}}+{{\sin}^{2}}\theta\rm{d}{\varphi^{2}}}\right),$
(3.10)

where ${x^{\mu}}=\left({t,r}\right)$, $\tilde{r}=ra\left(t\right)$ with the
scale factor $a\left(t\right)$, ${h_{\mu\nu}}={\mathop{\rm
diag}\nolimits}\left[{-1,{{{a^{2}}}\mathord{\left/{\vphantom{{{a^{2}}}{\left({1-k{r^{2}}}\right)}}}\right.\kern-1.2pt}{\left({1-k{r^{2}}}\right)}}}\right]$
is the two-dimensional metric with the spatial curvature constant $k$, and
$\mu=\nu=0,1$, respectively. By using the relation
${h^{\mu\nu}}{\partial_{u}}\tilde{r}{\partial_{\nu}}\tilde{r}=0$, the
dynamical apparent horizon of the FRW universe reads
$\tilde{r}=ar={\left({{H^{2}}+{k\mathord{\left/{\vphantom{k{{a^{2}}}}}\right.\kern-1.2pt}{{a^{2}}}}}\right)^{-\frac{1}{2}}}$
with the Hubble parameter
$H={{\dot{a}}\mathord{\left/{\vphantom{{\dot{a}}a}}\right.\kern-1.2pt}a}$. By
supposing the matter of the FRW universe is a perfect fluid, the energy-
momentum tensor can be expressed as

$\displaystyle{T_{\mu\nu}}=\left({\rho+p}\right){u_{\mu}}{u_{\nu}}+p{g_{\mu\nu}},$
(3.11)

where ${u_{\mu}}$ and ${g_{\mu\nu}}$ are the four velocity of the fluid and
the space-time metric of FRW universe, respectively. $\rho_{0}$ is the energy
density, and $p$ denotes the pressure. Due to the conservation law of energy-
momentum $T_{;\nu}^{\mu\nu}=0$, one obtains the continuity equation, i.e.,

$\displaystyle\dot{\rho}+3H(\rho+p)=0.$ (3.12)

Now, following the viewpoint in [68], we will use the first law of
thermodynamics for the matter content within the apparent horizon to derive
the Friedmann equations. The first law of thermodynamics can be expressed as

$\displaystyle{\rm{d}}E=T{\rm{d}}S+W{\rm{d}}V,$ (3.13)

where $E=\rho V$ represents the total energy of matter contained in the
apparent horizon,
$V=\left({{4\mathord{\left/{\vphantom{43}}\right.\kern-1.2pt}3}}\right)\pi{\tilde{r}^{3}}$
is the volume of 3-dimensional sphere, and
$W={{\left({\rho-p}\right)}\mathord{\left/{\vphantom{{\left({\rho-p}\right)}2}}\right.\kern-1.2pt}2}$
is the work density, respectively. Based on the continuity equation (3.12),
the energy differential turns out to be

$\displaystyle{\rm{d}}E=\rho{\rm{d}}V+V{\rm{d}}\rho=4\pi{\tilde{r}^{2}}\rho{\rm{d}}\tilde{r}+\frac{{4\pi{{\tilde{r}}^{3}}}}{3}{\rm{d}}\rho.$
(3.14)

Then, considering the temperature of the apparent horizon
$T={\kappa\mathord{\left/{\vphantom{\kappa{2\pi}}}\right.\kern-1.2pt}{2\pi}}$
with the surface gravity for the metric FRW universe
$\kappa={{-\left[{1-{{\left({{{\partial\tilde{r}}\mathord{\left/{\vphantom{{\partial\tilde{r}}{\partial
t}}}\right.\kern-1.2pt}{\partial
t}}}\right)}\mathord{\left/{\vphantom{{\left({{{\partial\tilde{r}}\mathord{\left/{\vphantom{{\partial\tilde{r}}{\partial
t}}}\right.\kern-1.2pt}{\partial
t}}}\right)}{2H\tilde{r}}}}\right.\kern-1.2pt}{2H\tilde{r}}}}\right]}\mathord{\left/{\vphantom{{-\left[{1-{{\left({{{\partial\tilde{r}}\mathord{\left/{\vphantom{{\partial\tilde{r}}{\partial
t}}}\right.\kern-1.2pt}{\partial
t}}}\right)}\mathord{\left/{\vphantom{{\left({{{\partial\tilde{r}}\mathord{\left/{\vphantom{{\partial\tilde{r}}{\partial
t}}}\right.\kern-1.2pt}{\partial
t}}}\right)}{2H\tilde{r}}}}\right.\kern-1.2pt}{2H\tilde{r}}}}\right]}{\tilde{r}}}}\right.\kern-1.2pt}{\tilde{r}}}$
and the expression of entropy (3.1), the first term on the right hand side of
eq. (3.13) can be rewritten as

$\displaystyle
T{\rm{d}}S=-\frac{1}{G}\left({1-\frac{{\dot{\tilde{r}}}}{{2H\tilde{r}}}}\right)f^{\prime}\left(A\right),$
(3.15)

where we denoted
$\dot{\tilde{r}}={{\partial\tilde{r}}\mathord{\left/{\vphantom{{\partial\tilde{r}}{\partial
t}}}\right.\kern-1.2pt}{\partial t}}$. Furthermore, the second term on the
right hand side of eq. (3.13) is evaluated by

$\displaystyle
W{\rm{d}}V=2\pi{\tilde{r}^{2}}\left({\rho-p}\right){\rm{d}}\tilde{r}.$ (3.16)

Applying eq. (3.14)-eq. (3.16) and the dynamical apparent horizon of the FRW
universe $\tilde{r}$ to eq. (3.14), one gets the first Friedmann equation

$\displaystyle-4\pi
G\left({\rho+p}\right)=\left({\dot{H}-\frac{k}{{{a^{2}}}}}\right)f^{\prime}(A),$
(3.17)

where we set $\dot{\tilde{r}}=0$ since the apparent horizon radius is fixed in
an infinitesimal time interval. Next, substituting the continuity equation
(3.12) into eq. (3.17), and then integrating, the result reads

$\displaystyle\frac{8}{3}\pi
G\rho=-4\pi\int{f^{\prime}\left(A\right)\frac{{{\rm{d}}A}}{{{A^{2}}}}},$
(3.18)

which is the second Friedmann equation. Now, by using eq. (3.8), the GUP
corrected Friedmann equations evolves as

$\displaystyle-4\pi
G\left({\rho+p}\right)=\left({\dot{H}-\frac{k}{{{a^{2}}}}}\right)\sqrt{1+\frac{{\pi\beta}}{{2A}}},$
(3.19)

and

$\displaystyle\frac{8}{3}\pi
G\rho=\frac{2}{{3\beta}}{\left({4+\frac{{2\pi\beta}}{A}}\right)^{3/2}}+\mathcal{C},$
(3.20)

where $\mathcal{C}$ refers to an integration constant, which specific
expression is determined by the boundary conditions in the vacuum energy
dominated era [39]. When $A$ goes to infinity, the energy density becomes
cosmological constant (i.e., $\rho=\Lambda$), which leads to
$\mathcal{C}={{8\pi G\Lambda}\mathord{\left/{\vphantom{{8\pi
G\Lambda}3}}\right.\kern-1.2pt}3}-{{16}\mathord{\left/{\vphantom{{16}{3\beta}}}\right.\kern-1.2pt}{3\beta}}$.
Then, inserting this expression of integration constant into eq. (3.20) and
considering
$A=4\pi{\tilde{r}^{2}}={{4\pi}\mathord{\left/{\vphantom{{4\pi}{\left({{H^{2}}+{k\mathord{\left/{\vphantom{k{{a^{2}}}}}\right.\kern-1.2pt}{{a^{2}}}}}\right)}}}\right.\kern-1.2pt}{\left({{H^{2}}+{k\mathord{\left/{\vphantom{k{{a^{2}}}}}\right.\kern-1.2pt}{{a^{2}}}}}\right)}}$,
one yields

$\displaystyle-4\pi G\left({\rho+p}\right)$
$\displaystyle=\left({\dot{H}-\frac{k}{{{a^{2}}}}}\right)\sqrt{1+\frac{1}{8}\left({{H^{2}}+\frac{k}{{{a^{2}}}}}\right)\beta},$
(3.21a) $\displaystyle\frac{8}{3}\pi G\left({\rho+\Lambda}\right)$
$\displaystyle=\frac{2}{{3\beta}}{\left[{4+\frac{1}{2}\left({{H^{2}}+\frac{k}{{{a^{2}}}}}\right)\beta}\right]^{3/2}}-\frac{{16}}{{3\beta}}.$
(3.21b)

According to the research goal, we should study a flat universe that dominated
by radiation. Hence, the tiny observed cosmological constant $\Lambda$ and the
spatial curvature constant $k$ can be ignored [39]. This argument implies that

$\displaystyle-4\pi
G\left({\rho+p}\right)=\dot{H}\sqrt{1+\frac{{{H^{2}}}}{8}\beta},$ (3.22a)
$\displaystyle\frac{8}{3}\pi
G\rho=\frac{2}{{3\beta}}{\left({4+\frac{{{H^{2}}}}{2}\beta}\right)^{3/2}}-\frac{{16}}{{3\beta}}.$
(3.22b)

Clearly, the above two equations are the GUP corrected Friedmann equations
which show the changes of energy density and pressure in the early Universe.
When $\beta\to 0$ the original Friedmann equations are recovered.

###  3.3 GUP corrected baryon asymmetry factor

For investigating the deviation of thermal equilibrium of the Universe, it is
necessary to re-express the energy density and pressure of Friedmann equations
as $\rho={\rho_{0}}+\delta\rho$ and $p={p_{0}}+\delta p$, where $\rho_{0}$ and
$p_{0}$ represent the pressure and density at thermal equilibrium,
respectively. Substituting $\rho$ and $p$ into GUP corrected Friedmann
equations (3.22), one has

$\displaystyle{\rho_{{\rm{GUP}}}}$
$\displaystyle=-\frac{2}{{G\pi\beta}}+\left({\frac{2}{{G\pi\beta}}+\frac{2}{3}{\rho_{0}}}\right)\sqrt{1+\frac{1}{3}G\pi\beta{\rho_{0}}}$
$\displaystyle={\rho_{0}}+\frac{{G\pi{\rho^{2}}\beta}}{{12}}-\frac{{{G^{2}}{\pi^{2}}{\rho^{3}}{\beta^{2}}}}{{216}}+\mathcal{O}\left({{\beta^{3}}}\right),$
(3.23a) $\displaystyle{p_{{\rm{GUP}}}}$
$\displaystyle=\frac{2}{{G\pi\beta}}+\left[{\frac{{{\rho_{0}}}}{9}\left({1+3w}\right)-\frac{2}{{3G\pi\beta}}}\right]\sqrt{9+3G\pi\beta{\rho_{0}}}.$
$\displaystyle=w{\rho_{0}}+\frac{1}{{12}}G\pi\left({1+2w}\right)\beta\rho_{0}^{2}-\frac{1}{{216}}{G^{2}}{\pi^{2}}\left({2+3w}\right)\rho_{0}^{3}{\beta^{2}}+\mathcal{O}\left({{\beta^{3}}}\right).$
(3.23b)

It is noteworthy that, in the early stages of the Universe, the effect of QG
made the ${\rho_{{\rm{GUP}}}}$ and ${p_{{\rm{GUP}}}}$ deviate from the cases
at thermal equilibrium. As increases of the scale of the Universe, the effect
of QG gradually weakens. So far, the QG effect in our universe is so weak that
it can be ignored. Next, substituting eq. (3.23) into the trace of the
Einstein equation (2.4), it follows that

$\displaystyle{R_{{\rm{GUP}}}}=$ $\displaystyle-8\pi
G\left({{\rho_{{\rm{GUP}}}}-3{p_{{\rm{GUP}}}}}\right)$ $\displaystyle=$
$\displaystyle\frac{8}{3}G\pi\left({1+9w}\right){\rho_{0}}\sqrt{1+\frac{1}{3}G\pi\beta{\rho_{0}}}+\frac{{64}}{{3\beta}}\left({3-\sqrt{9+3G\pi\beta{\rho_{0}}}}\right).$
(3.24)

Applying the continuity equation (3.12) and the second Friedmann equation in
thermal equilibrium ${H^{2}}={{8\pi
G{\rho_{0}}}\mathord{\left/{\vphantom{{8\pi
G{\rho_{0}}}3}}\right.\kern-1.2pt}3}$ to eq. (3.3), the modified time
derivative of the Ricci scalar takes the following form

$\displaystyle{\dot{R}_{{\rm{GUP}}}}=-\frac{{8\sqrt{2}G{\pi^{3/2}}\left({1+w}\right){\rho_{0}}\left[{G\pi\beta{\rho_{0}}-6+9w\left({2+G\pi\beta{\rho_{0}}}\right)}\right]}}{{\sqrt{\pi\beta+{3\mathord{\left/{\vphantom{3{G{\rho_{0}}}}}\right.\kern-1.2pt}{G{\rho_{0}}}}}}}.$
(3.25)

Here we set $w={1\mathord{\left/{\vphantom{13}}\right.\kern-1.2pt}3}$ since
the universe is in an era dominated by radiation. With this setting, eq.
(3.25) can be rewritten as follows

$\displaystyle{\dot{R}_{{\rm{GUP}}}}=-\frac{{128\sqrt{2}{G^{2}}{\pi^{5/2}}\beta\rho_{0}^{2}}}{{3\sqrt{\pi\beta+{3\mathord{\left/{\vphantom{3{G{\rho_{0}}}}}\right.\kern-1.2pt}{G{\rho_{0}}}}}}},$
(3.26)

where the original case ${\dot{R}}=0$ is obtained for $\beta\rightarrow 0$.
Finally, substituting eq. (3.26) into eq. (2.2), the expression for BAF is
modified as follows

$\displaystyle{\eta_{{\rm{GUP}}}}={\left.{\frac{{160{g_{b}}{G^{5/2}}\beta\rho_{0}^{5/2}}}{{{g_{*}}M_{*}^{2}T}}\sqrt{\frac{{2\pi}}{{3+G\pi\beta{\rho_{0}}}}}}\right|_{{T_{D}}}}.$
(3.27)

Obviously, the modified BAF is not only related to the parameters from the
standard result (2.2), but also to the deformation parameter $\beta$.
According to the previous discussion, the effect of GUP is active in the
radiation domination era, so our result can effectively avoid the occurrence
of non-zero BAF. Besides, it is found that the original result (2.2)
osatisfies only the first two Sakharov conditions [11]. However, by using the
modified corrected entropy (3.1) and Friedmann equations (3.22), we broke the
thermodynamic equilibrium and met the third Sakharov condition. Therefore, our
result satisfies all three Sakharov conditions, which indicates the GUP can
generate a sufficient baryon asymmetry for at the radiation domination era.

##  4 Constraints for the GUP parameter $\beta_{0}$

Theoretically, the GUP parameters are always assumed as 1 so that the effect
of GUP is valid when the energy is close to the Planck scale. However, when
ignoring this assumption, the bound of the GUP parameter can be estimated by
experiment data and observation results. The research on the bounds of GUP
parameters has always been one of the important focuses, which could help to
understand the effect of QG in the theoretical realm [75, 76, 77, 78].
Therefore, in this section, we will confront the theoretical result (3.27)
with the observational results to obtain the bound of the parameter of the
high-order GUP. In order to achieve this goal, one needs to replace the
gravitational constant to Planck mass (i.e.
$G={1\mathord{\left/{\vphantom{1{m_{p}^{2}}}}\right.\kern-1.2pt}{m_{p}^{2}}}$
with ${m_{p}}\sim 1.22\times{10^{19}}{\rm{GeV}}$) and express the density at
thermal equilibrium as
${\rho_{0}}={{\pi{g_{*}}{T^{4}}}\mathord{\left/{\vphantom{{\pi{g_{*}}{T^{4}}}{30}}}\right.\kern-1.2pt}{30}}$
[6], then the GUP corrected BAF can be rewritten as

$\displaystyle{\eta_{{\rm{GUP}}}}=\frac{{8{g_{b}}g_{*}^{3/2}{\pi^{3}}T_{D}^{9}\beta}}{{45m_{p}^{5}M_{*}^{2}\sqrt{45+{{{g_{*}}{\pi^{2}}T_{D}^{4}\beta}\mathord{\left/{\vphantom{{{g_{*}}{\pi^{2}}T_{D}^{4}\beta}{2m_{p}^{2}}}}\right.\kern-1.2pt}{2m_{p}^{2}}}}}}.$
(4.1)

By employing
$\beta={{{\beta_{0}}}\mathord{\left/{\vphantom{{{\beta_{0}}}{m_{p}^{2}}}}\right.\kern-1.2pt}{m_{p}^{2}}}$
and the values of ${g_{b}}\sim\mathcal{O}\left(1\right)$, ${g_{*}}\sim 106$,
${M_{*}}={{{m_{p}}}\mathord{\left/{\vphantom{{{m_{p}}}{\sqrt{8\pi}}}}\right.\kern-1.2pt}{\sqrt{8\pi}}}$
and ${T_{D}}\sim 2\times{10^{16}}{\rm{GeV}}$ [69], then solving eq. (4.1), one
has

$\displaystyle{\beta_{0}}=4.8\times{10^{-82}}\left({2.3\times{{10}^{112}}\eta_{{\rm{GUP}}}^{2}}{+\sqrt{1.2\times{{10}^{204}}\eta_{{\rm{GUP}}}^{2}+5.5\times{{10}^{224}}\eta_{{\rm{GUP}}}^{4}}}\right).$
(4.2)

Clearly, the above equation implies that the bound of $\beta_{0}$ is
determined by the observed value of ${\eta_{{\rm{GUP}}}}$. In the past forty
years, many astronomical observations have given the range of $\eta$, which
can be used to constraint the region of the higher-order GUP. Now, according
to the data in [5, 70, 71, 72, 73], the bounds on $\beta_{0}$ are shown in
table 1.

Data source | $\eta$ | $\beta_{0}$
---|---|---
Planck observations [5] | $\eta\leq 6.2\times{10^{-10}}$ | ${\beta_{0}}\leq 8.7\times{10^{12}}$
Deuterium and Hydrogenium abundance [70] | $5.9\times{10^{-10}}\leq\eta\leq 6.3\times{10^{-10}}$ | $7.9\times{10^{12}}\leq{\beta_{0}}\leq 9.0\times{10^{12}}$
Deuterium and 3He abundances [71] | $5.7\times{10^{-11}}\leq\eta\leq 9.9\times{10^{-11}}$ | $8.4\times{10^{10}}\leq{\beta_{0}}\leq 2.3\times{10^{11}}$
BBN [72] | $3.4\times{10^{-10}}\leq\eta\leq 6.9\times{10^{-10}}$ | $2.6\times{10^{12}}\leq{\beta_{0}}\leq 1.1\times{10^{13}}$
Acoustic peaks in CMB measured by WMAP [73] | $\eta\leq 6.3\times{10^{-10}}$ | ${\beta_{0}}\leq 9.0\times{10^{12}}$
Particle Data Group [74] | $\eta\leq 8.6\times{10^{-11}}$ | ${\beta_{0}}\leq 1.8\times{10^{11}}$

Table 1:  The data source, the ranges of $\eta$, and the bounds of
$\beta_{0}$.

From table 1, by employing data from different astronomical observations, one
can see the bound of $\beta_{0}$ is between $8.4\times{10^{10}}\sim
1.1\times{10^{13}}$. Besides, the parameter of the higher-order GUP has a
special relationship with that of the KMM model, that is,
${\beta_{0}}\sim-\beta_{0}^{{\rm{KMM}}}$. Therefore, comparing the results
$\beta_{0}^{{\rm{KMM}}}={{-\eta}\mathord{\left/{\vphantom{{-\eta}{2.16\times{{10}^{-19}}}}}\right.\kern-1.2pt}{2.16\times{{10}^{-19}}}}$
from [39], which shows
$-4.58\times{10^{8}}\leq\beta_{0}^{{\rm{KMM}}}\leq-2.64\times{10^{8}}$
(Deuterium and 3He abundances), it is easy to find that our results
$\beta_{0}$ are 2-5 orders of magnitude weaker than the absolute value of
those of the KMM model $\left|{\beta_{0}^{{\rm{KMM}}}}\right|$. In order to
further study the influence of the two models on the “$\eta-{{\beta_{0}}}$"
relationship, we plot fig. 1. One can see that the rad dashed line illustrates
the KMM model case increases monotonically with
$\left|{\beta_{0}^{{\rm{KMM}}}}\right|$ with a constant slope, whereas the
higher-order GUP case presents a blue parabola with an increasing slope. As we
stated previously, the higher-order GUP would reduces to the KMM model for
$\sqrt{\beta}\Delta p\ll 1$. Therefore, we conclude that the above differences
are caused by higher-order terms ${\cal O}\left({{\beta_{0}}}\right)$ of GUP.

Furthermore, with the continuous improvement of astronomical observation
equipment, one can now use gravitational wave signals to constrain the effect
of QG at cosmological scales. For example, Nishizawa [79] tested effect of QG
with the gravitational-wave propagation and the modified photonic dispersion
relation. Inspired by that work, we can constrain the parameter of the higher-
order GUP models via the primordial gravitational wave experiments with space
interferometers (such as LIGO, VIRGO, and KAGRA), those works would lay the
foundation for understanding quantum gravity from a multi-messenger era
perspective [80].

Figure 1:  The effect of different GUP models on the $\eta-{{\beta_{0}}}$
relationship.

##  5 Conclusion

In the current work, the baryon asymmetry in the early Universe is studied
from the perspective of thermodynamics and the effect of higher-order GUP. We
started our investigation by analyzing the original gravitational
baryogenesis. It is found that, although the original mechanism has coupled
the spacetime and the baryon current, it still cannot explain the baryon
asymmetry well since the Universe is in thermal equilibrium, which leads to a
zero BAF and makes the third Sakharov condition unsatisfied. For fixing this
problem, we derived the modified Bekenstein-Hawking entropy by employing a new
higher-order GUP together with the holographic principle. Subsequently,
according to this modification and the first law of thermodynamics, we
obtained the GUP corrected Friedmann equations. It is found that the modified
Friedmann equations are in non-thermal equilibrium, which satisfies the third
Sakharov condition and leads to $\eta\neq 0$ in the radiation-dominated era.
Those results showed that the effect of GUP can break thermal equilibrium and
provide a feasible scheme for explaining baryon asymmetry in a radiation
dominated Universe. Finally, using the different astronomical observations
data, we constrained the bounds of the GUP parameter $\beta_{0}$, whose bound
between $8.4\times{10^{10}}\sim 1.1\times{10^{13}}$. It is found that our
results the parameter of High-order GUP $\beta_{0}$ are 2-5 orders of
magnitude weaker than the absolute value of that of KMM model
$\left|{\beta_{0}^{{\rm{KMM}}}}\right|$. This indicates the bound of GUP
parameter is depended on the its model, and the high-order terms ${\cal
O}\left({{\beta_{0}}}\right)$ of GUP have an important impact on the baryon
asymmetry. In a word, our study provides the possibility to analyze the QG
effect and its role in the early evolution of the Universe.

It should be noted that many new higher-order GUP [43, 44, 45, 46, 47] and
extended uncertainty principle models [81, 82] have been proposed recently. By
combining them with baryon asymmetry could help us have a deeper understanding
of the Universe. In addition, the CTA will start to reveal whether quantum
gravity eventually plays a role in the Universe’s dynamics, via photon
dispersion relations [55], the data released by this observations will be
important for our analysis of QG effects. We hope to address these issues in
the future works.

## Acknowledgments

The authors thank Guansheng He and the the anonymous referees for helpful
suggestions and enlightening comments, which helped to improve the quality of
this paper. This work was supported by the National Natural Science Foundation
of China (Grant Nos. 12105231 and 11847048), the Guiding Local Science and
Technology Development Projects by the Central Government of China (Grant No.
2021ZYD0031), the Sichuan Youth Science and Technology Innovation Research
Team(Grant No. 21CXTD0038), and the Fundamental Research Funds of China West
Normal University (Grant No. 20B009).

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# Hybrid Reconfigurable Intelligent Surface Enabled Over-the-Air Index-
Modulation

1st Emre Arslan  Emre Arslan13, Ali Tugberk Dogukan23, Fatih Kilinc23,
Ertugrul Basar3  16GEN Laboratory, Next-Generation R&D, Network Technologies,
Turkcell, İstanbul, Turkiye  2Department of Research and Development, ULAK
Communications Inc., İstanbul, Turkiye  3Department of Electrical and
Electronics Engineering, Koç University, İstanbul, Turkiye
Emails:
emre.arslan@turkcell.com.tr,{fatih.kilinc,tugberk.dogukan}<EMAIL_ADDRESS>Next Generation Research and Development
6GEN. Lab, Turkcell Iletisim Hizmetleri Inc.
Istanbul, Turkey
<EMAIL_ADDRESS>2nd Ali Tugberk Dogukan  3rd Fatih Kilinc
Department of Research and Development
ULAK Communications Inc.
Istanbul, Turkey
<EMAIL_ADDRESS>Department of Research and Development
ULAK Communications Inc.
Istanbul, Turkey
<EMAIL_ADDRESS>4th Ertugrul Basar  Department of
Electrical and Electronics
Engineering, Koç University
Istanbul, Turkey
<EMAIL_ADDRESS>
###### Abstract

Users’ desire for enhanced performance drive the inevitable technological
progress on vertical applications in wireless communication systems. To meet
these demands, researchers vigorously investigate potential 6G and beyond
technologies and solutions. Reconfigurable intelligent surfaces (RISs) have
risen in popularity and attracted the attention of academia as well as
industry and seem to be a promising candidate for 6G technology. RISs are
reflective metamaterials with many configurable elements consisting of pin
diodes that have the ability to manipulate the impinging signals’ properties,
hence, enabling some sort or virtual control over the wireless channel. In
this paper, we propose a novel over-the-air index modulation (IM) scheme
through the use of a hybrid RIS with active and passive partitions to convey
additional IM bits over-the-air. In addition we propose another modified
scheme that allows the transmission for even more IM bits but at the expense
of certain trade-offs. First, we present the system model for both of the
proposed schemes. Furthermore, comprehensive computer simulation results are
provided and discussed presenting the superior bit error rate (BER)
performance and additional benefits of the proposed systems compared to
similar benchmarks.

###### Index Terms:

reconfigurable intelligent surface (RIS), hybrid RIS, index modulation,
partitioning, 6G, BER, spectral efficiency, data rate

##  I Introduction

Figure 1:  System model of OTA-RIS-IM.

Wireless communication technologies are rapidly developing as researchers
migrate from one generation (5G) to another (6G), striving to meet users’
stringent demands as a consequence of imminent vertical applications. Numerous
candidate solutions and novel technologies are proposed in the literature each
manifesting high potential for future wireless communication systems. While
some methods are adopted, most of these are dismissed and are not favored due
to distinct drawbacks and critical issues which may be in terms of
practicality, performance, complexity and more. However, as new hot topics
arise, researchers must revert to previous solutions and consider them and
their integration with recently developed and considered technologies which
may provide new insights and together provide superior performance. Hence,
previously dismissed technologies may be rekindled as once again significantly
promising solutions.

Index modulation (IM) is a simple, spectrum and energy efficient scheme to
embed and transmit additional information bits through the indices of resource
blocks in communication systems [1, 2, 3, 4]. These resource blocks on which
information is embedded can be subcarriers, antennas, time-slots,
constellation, activation orders and much more [5, 3]. It is a very popular
concept with a significant amount of research and studies conducted on it and
has proven itself to be a worthy tool and is considered for future
communication networks. IM shows benefits such as the ability to improve error
performance, provide enhanced energy and spectral efficiency, and all while
providing low complexity [6]. One of the most important characteristic of IM
is its flexibility and compatibility attribute since it can be employed and
coexist in a wireless communication system in numerous methods [5, 7]. One
example is orthogonal frequency division multiplexing-IM (OFDM-IM) where IM is
incorporated in the subcarriers of an OFDM waveform as an addition layer of
modulation by activating certain subcarriers to transmit additional bits over
OFDM subblock [8, 9]. In [10] and [11], IM has been integrated with non-
orthogonal multiple access and joint radar and communication technologies
introducing novel system designs for future technologies; proposing a flexible
OFDM-IM subcarrier activation and power allocation method according to users’
needs and a unique waveform to serve both communication and radar
functionalities, respectively. Furthermore, additional interesting studies
have been conducted such as [12] where a novel technique is proposed by
dividing the subcarriers into two groups and modulating them with
distinguishable mode constellations and [13] where bits are transmitted
through the indexing of multiple codebooks to be used in sparse vector coding
for ultra-reliable low-latency communication networks.

Reconfigurable intelligent surfaces (RISs) have recently risen in popularity
in the wireless communication community and exhibit high potential as a
promising candidate for future 6G and beyond wireless communication systems.
RISs are large reflective surfaces with numerous small and low-cost meta-
material elements that have the ability to manipulate the characteristics of
impinging signals. Hence, RISs are able to reflect, refract, absorb and even
amplify impinging signals without the need for additional buffering or
processing, providing some sort of virtual control over the wireless channel.
Generally, RISs are envisioned to be employed for coverage enhancement,
increased signal quality, elevated capacity and other system performance
improvements [14, 15, 16, 17, 18, 19]. One significant advantage of the RIS
technology is its flexibility in the integration with conventional
technologies. There are many studies that present substantial benefits in the
integration of RIS with previously considered technologies such as non-
orthogonal multiple accessing [20], machine learning [21], localization [22],
OFDM [23], mutiple-input and multiple-output [24], and much more [25, 26, 27].

Considering both the advantages and flexibility of RIS and IM, the integration
of these technologies are being investigated to further improve wireless
communication system performances. In [28], the authors present a greedy
detector for an RIS space-shift keying system where the RIS phases are aligned
to maximize the instantaneous signal-to-noise ratio (SNR) at the target
receive antenna which is selected through IM bits. Furthermore, an intelligent
RIS grouping-based IM (RGB-IM) is proposed to enhance the spectral efficiency
and bit error rate (BER) performance of conventional RIS systems [29]. In this
study the RIS is divided in to multiple groups each with the same number of
elements, and according to the incoming IM bits a group of RIS elements are
activated while the other RIS elements are inactive and introduce zero phase.
However, completely inactivating remaining groups and elements are not optimal
and could be used in a more efficient manner to improve the reliability and
spectral efficiency of the system.

In this study, we propose a novel over-the-air RIS-IM (OTA-RIS-IM) technique
exploiting the entire RIS enhancing the spectral efficiency and BER
performance. We present a hybrid RIS that has multiple groups and each group
can be active and passive according to the IM bits. Furthermore, we further
expand this system design and propose an enhanced OTA-RIS-IM (E-OTA-RIS-IM)
system by introducing an absorption mode at the RIS to send additional IM bits
at the expense of slightly reduced BER performance improvement compared to our
benchmark.

The rest of the paper can be summarized as follows. In Section II, the system
framework of the proposed OTA-RIS-IM and E-OTA-RIS-IM scheme is presented. In
addition, Section III consists of the computer simulation results and
discussions while Section IV concludes the paper.

Figure 2:  System model of E-OTA-RIS-IM.

##  II System Model

In this section, we provide the system model for both of the proposed schemes.
First we will present the initial OTA-RIS-IM system design. Furthermore, we
will build on top of it to propose the E-OTA-RIS-IM system. As seen from Fig.
1, RIS includes $N$ elements and is partitioned into $G$ groups, each
including $\bar{N}$ elements, $N=G\bar{N}$. Here, the reflection coefficient
of the $i$th element of the $g$th RIS group is
$\theta_{gi}=e^{j\vartheta_{gi}}$ with unit amplitude, where
$\vartheta_{gi}\in\left[0,2\pi\right)$ is the phase shift adjusted by the RIS
controller, $i=1,2,\cdots,\bar{N}$, and $g=1,2,\cdots,G$.

Assume that $L_{h}=\zeta_{0}d_{h}^{-\varkappa_{h}}$ and
$L_{f}=\zeta_{0}d_{f}^{-\varkappa_{f}}$ represent the large scale path loss of
the BS-to-RIS and RIS-to-UE, respectively, where $\zeta_{0}$, is the path loss
at the reference distance of 1 meter, and $d_{h}$, $d_{f}$ and
$\varkappa_{h}$, $\varkappa_{f}$ denote the distance from BS to RIS and RIS to
UE, the path loss exponents at BS to RIS and RIS to UE paths, respectively.
Then, we define the channel coefficients from the BS to the
$\left(g,i\right)$th RIS element and from the $\left(g,i\right)$th RIS element
to the UE as
$h_{gi}=\left|h_{gi}\right|e^{j\phi_{gi}}=\sqrt{L_{h}}\bar{h}_{gi}$ and
$f_{gi}=\left|f_{gi}\right|e^{j\varphi_{gi}}=\sqrt{L_{f}}\bar{f}_{gi}$,
respectively. Moreover, $\bar{h}_{gi}$ and $\bar{f}_{gi}$ are independent
Rician fading channel coefficients as follows:

$\displaystyle\bar{h}_{gi}$
$\displaystyle=\sqrt{\frac{K_{h}}{K_{h}+1}}\bar{h}_{gi}^{\mathrm{LoS}}+\sqrt{\frac{1}{K_{h}+1}}\bar{h}_{gi}^{\mathrm{NLoS}},$
(1) $\displaystyle\bar{f}_{gi}$
$\displaystyle=\sqrt{\frac{K_{f}}{K_{f}+1}}\bar{f}_{gi}^{\mathrm{LoS}}+\sqrt{\frac{1}{K_{f}+1}}\bar{f}_{gi}^{\mathrm{NLoS}},$
(2)

where $\bar{h}_{gi}^{\mathrm{LoS}}$ and $\bar{f}_{gi}^{\mathrm{LoS}}$
represent the line-of-sight (LoS) components and
$\bar{h}_{gi}^{\mathrm{NLoS}}$ and $\bar{f}_{gi}^{\mathrm{NLoS}}$ represent
the non-LoS (NLoS) components of $\bar{h}_{gi}$ and $\bar{f}_{gi}$,
respectively, whereas $K_{h}$ and $K_{f}$ denote the Rician $K$ factor of the
paths between the BS-RIS and the RIS-UE, respectively. Here,
$\bar{h}_{gi}^{\mathrm{LoS}}$, $\bar{f}_{gi}^{\mathrm{LoS}}$,
$\bar{h}_{gi}^{\mathrm{NLoS}}$, and $\bar{f}_{gi}^{\mathrm{NLoS}}$ are
independent and identically distributed complex Gaussian random variables with
$\mathcal{CN}(0,1)$.

###  II-A OTA-RIS-IM

The system model of OTA-RIS-IM is demonstrated in Fig. 1. In OTA-RIS-IM,
information bits are transmitted both through the conventional $M$-ary PSK/QAM
signal constellation and also the indices of the active/passive states of RIS
groups. Firstly,
$\mathbf{b}_{1}=\left[b_{11},b_{12},\cdots,b_{1\kappa}\right]^{\mathrm{T}}$
bit vector is mapped into the modulated symbol $x$, $x\in\mathcal{M}$, where
$\mathcal{M}$ is the $M$-ary PSK/QAM signal constellation and
$\kappa=\log_{2}(M)$. Then,
$\mathbf{b}_{2}=\left[b_{21},b_{22},\cdots,b_{2G}\right]^{\mathrm{T}}$ bit
vector determines the indices of active/passive states of RIS groups
$\mathbf{p}=[p_{1},p_{2},\cdots,p_{G}]^{\mathrm{T}}$, $p_{g}\in\\{0,1\\}$,
$g=1,2,\dots,G$. Here, the $g$th element of $\mathbf{p}$ is equal to the $g$th
element of $\mathbf{b}_{2}$, i.e., $p_{g}=b_{2g}$. If $p_{g}$ is equal to $0$
($\bar{p}_{g}=1$), then the $g$th RIS group is in passive state, and if
$p_{g}$ is equal to $1$ ($\bar{p}_{g}=0$), then the $g$th RIS group is in
active state, where $\bar{p}_{g}$ represents the logical NOT of $p_{g}$. Here,
the number of ones and zeros in $\mathbf{p}$ is equal to the number of active
($G_{A}$) and passive ($G_{P}$) RIS groups, respectively, $G_{A}+G_{P}=G$. In
Table I, an example of bit mapping is given for $G=2$. The spectral efficiency
of OTA-RIS-IM in terms of bit per channel use (bpcu) can be given as:

$\eta=\log_{2}\left(M\right)+G.$ (3)

In OTA-RIS-IM, the $(g,i)$th RIS element ($\theta_{gi}$) is selected in such a
way that the phase terms in $h_{gi}$ and $f_{gi}$ are eliminated, i.e.,
$\vartheta_{gi}=-(\phi_{gi}+\varphi_{gi})$. Hence, the received signal at UE
can be given as:

$\displaystyle
y=\sqrt{P_{t}}x\left(\sum_{g=1}^{G}\left[\mathcal{H}_{g}\left(\sqrt{\alpha}p_{g}+\bar{p}_{g}\right)+{p}_{g}w^{a}_{g}\right]\right)+w,$
(4)
$\displaystyle\mathcal{H}_{g}=\sum_{i=1}^{\bar{N}}h_{gi}\theta_{gi}f_{gi}=\sum_{i=1}^{\bar{N}}\left|h_{gi}\right|\left|f_{gi}\right|,$
$\displaystyle
w^{a}_{g}=\sqrt{\alpha}\sum_{i=1}^{\bar{N}}\theta_{gi}f_{gi}v_{gi},$

where $P_{t}$, $\alpha$, $w$, and $v_{gi}$ are the transmit power,
amplification factor at the active RIS groups, additive white Gaussian noise
(AWGN) sample with distribution $\mathcal{CN}(0,N_{0})$, and amplifier noise
term at the $(g,i)$th RIS element with distribution $\mathcal{CN}(0,V_{0})$,
respectively. Finally, at the UE, the maximum-likelihood (ML) detection rule
can be exploited to decode the received signal as follows:

$\left(\hat{x},\hat{\mathbf{p}}\right)=\underset{x\in\mathcal{M},\mathbf{p}\in\mathcal{P}}{\arg\min}\left\|y-\sqrt{P_{t}}x\left[\sum_{g=1}^{G}\mathcal{H}_{g}\left(\sqrt{\alpha}p_{g}+\bar{p}_{g}\right)\right]\right\|^{2},$
(5)

where $\hat{x}$ and $\hat{\mathbf{p}}$ are the estimates of $x$ and
$\mathbf{p}$, respectively, and $\mathcal{P}$ is a set including all possible
$\mathbf{p}$ vectors. Here, $M2^{G}$ number of iterations are performed in the
search space to decode the received signal.

The signal-to-noise ratio (SNR) of the received signal can be given as:

$\gamma=\frac{P_{t}\left|\sum_{g=1}^{G}\mathcal{H}_{g}\left(\sqrt{\alpha}p_{g}+\bar{p}_{g}\right)\right|^{2}}{\alpha
V_{0}\sum_{g=1}^{G}p_{g}\sum_{i=1}^{\bar{N}}\left|\theta_{gi}f_{gi}\right|^{2}+N_{0}}.$
(6)

TABLE I: OTA-RIS-IM Example of Bit Mapping For $G=2$

$[b_{21},b_{22}]$ | $G_{A}$ | $G_{P}$ | $[p_{1},p_{2}]$ | $[\bar{p}_{1},\bar{p}_{2}]$ |  | First
---
RIS Group
State
| Second
---
RIS Group
State
$[0,0]$ | $0$ | $2$ | $[0,0]$ | $[1,1]$ | Passive | Passive
$[0,1]$ | $1$ | $1$ | $[0,1]$ | $[1,0]$ | Passive | Active
$[1,0]$ | $1$ | $1$ | $[1,0]$ | $[0,1]$ | Active | Passive
$[1,1]$ | $2$ | $0$ | $[1,1]$ | $[1,1]$ | Active | Active

###  II-B E-OTA-RIS-IM

In this subsection, we present an enhanced OTA-RIS-IM scheme (E-OTA-RIS-IM) to
further improve the number of bits transmitted in the system by introducing an
absorption state at the hybrid RIS. The system model of E-OTA-RIS-IM is
illustrated in Fig. 2. Here, according to the incoming bits from the backhaul
link, the hybrid RIS will determine whether the partitions will behave in the
passive, active or absorption state. As in OTA-RIS-IM, firstly,
$\mathbf{b}_{1}=\left[b_{11},b_{12},\cdots,b_{1\kappa}\right]^{\mathrm{T}}$
bit vector is used to obtain the $M$-ary modulated symbol $x$,
$\kappa=\log_{2}(M)$. Let us define a codebook $\mathcal{S}$ which includes
all permutations with repetition of active/passive/absorption states of RIS
groups,
$\mathcal{S}=\left\\{\mathbf{s}_{1},\mathbf{s}_{2},\cdots,\mathbf{s}_{2^{\varpi}}\right\\}$,
$\varpi=\left\lfloor\log_{2}\left(3^{G}\right)\right\rfloor$, where
$\mathbf{s}_{\varrho}=\left[s_{\varrho 1},s_{\varrho 2},\cdots,s_{\varrho
G}\right]^{\mathrm{T}}$ is the $\varrho$th codeword of set $\mathcal{S}$ and
$s_{\varrho\mu}\in\\{\sqrt{\alpha},1,0\\}$, $\varrho=1,2,\cdots,2^{\varpi}$,
$\mu=1,2,\cdots,G$. Here, $s_{\varrho\mu}$ being $\alpha$, $1$, and $0$
represent active, passive, and absorption states, respectively. Hence,
$\mathbf{b}_{2}=\left[b_{21},b_{22},\cdots,b_{2\varpi}\right]^{\mathrm{T}}$
bit vector is converted into its decimal value ($\xi$), and $\xi$th codeword
($\mathbf{s}_{\xi}$) of $\mathcal{S}$ is selected. As seen in Table II, an
example of states of RIS groups selection is provided for $G=2$. Here, a total
number of $3^{G}$ permutations can be generated; however, only $2^{\varpi}$
can be used to transmit information. As can be seen in the table, for the same
number of RIS partitions $G$, we can send an additional bit compared to the
previous method. Hence, the spectral efficiency of E-OTA-RIS-IM in terms of
bpcu can be given as:

$\eta=\log_{2}\left(M\right)+\left\lfloor\log_{2}\left(3^{G}\right)\right\rfloor.$
(7)

As in OTA-RIS-IM, the $(g,i)$th element of RIS is coherently aligned with the
corresponding channel coefficients $h_{gi}$ and $f_{gi}$, i.e.,
$\vartheta_{gi}=-(\phi_{gi}+\varphi_{gi})$. Hence, the received signal at UE
can be written as:

$\displaystyle y=\sqrt{P_{t}}x\left(\sum_{g=1}^{G}\left[\mathcal{H}_{g}s_{\xi
g}+\sigma\left(s_{\xi g}\right)w^{a}_{g}\right]\right)+w,$ (8)

where $\sigma\left(\beta\right)$ is a function which provides output as $1$ if
$\beta$ equals to $\sqrt{\alpha}$, and $0$ otherwise.

In order to decode the received signal, the ML detector can be used as
follows:

$\left(\hat{x},\hat{\varrho}\right)=\underset{x\in\mathcal{M},\varrho=1,\cdots,2^{\varpi}}{\arg\min}\left\|y-\sqrt{P_{t}}x\left[\sum_{g=1}^{G}\mathcal{H}_{g}s_{\varrho
g}\right]\right\|^{2}.$ (9)

Moreover, the SNR of the received signal can be obtained as:

$\gamma=\frac{P_{t}\left|\sum_{g=1}^{G}\mathcal{H}_{g}s_{\xi
g}\right|^{2}}{\alpha V_{0}\sum_{g=1}^{G}\sigma\left(s_{\xi
g}\right)\sum_{i=1}^{\bar{N}}\left|\theta_{gi}f_{gi}\right|^{2}+N_{0}}.$ (10)

TABLE II: E-OTA-RIS-IM Example of Bit Mapping For $G=2$

| Index Bits
---
$[b_{21},b_{22},b_{23}]$
$(\xi)$ | Codeword |  | First RIS
---
Group State
| Second RIS
---
Group State
$[0,0,0]$ | $0$ | $\mathbf{s}_{1}=\left[\sqrt{\alpha},\sqrt{\alpha}\right]^{\mathrm{T}}$ | Active | Active
$[0,0,1]$ | $1$ | $\mathbf{s}_{2}=\left[\sqrt{\alpha},1\right]^{\mathrm{T}}$ | Active | Passive
$[0,1,0]$ | $2$ | $\mathbf{s}_{3}=\left[\sqrt{\alpha},0\right]^{\mathrm{T}}$ | Active | Absorption
$[0,1,1]$ | $3$ | $\mathbf{s}_{4}=\left[1,\sqrt{\alpha}\right]^{\mathrm{T}}$ | Passive | Active
$[1,0,0]$ | $4$ | $\mathbf{s}_{5}=\left[1,1\right]^{\mathrm{T}}$ | Passive | Passive
$[1,0,1]$ | $5$ | $\mathbf{s}_{6}=\left[1,0\right]^{\mathrm{T}}$ | Passive | Absorption
$[1,1,0]$ | $6$ | $\mathbf{s}_{7}=\left[0,1\right]^{\mathrm{T}}$ | Absorption | Active
$[1,1,1]$ | $7$ | $\mathbf{s}_{8}=\left[0,1\right]^{\mathrm{T}}$ | Absorption | Passive
NaN | NaN | NaN | Absorption | Absorption

##  III Simulation Results and Comparisons

Figure 3: Error performance comparison of OTA-RIS-IM, E-OTA-RIS-IM, and RGB-IM
for varying $\alpha$, $N=256$, and $4$ bpcu. Figure 4: Error performance
comparison of OTA-RIS-IM, E-OTA-RIS-IM, and RGB-IM for varying $N$,
$\alpha=30$ dB, and $4$ bpcu. Figure 5: Error performance comparison of OTA-
RIS-IM, E-OTA-RIS-IM, and RGB-IM varying $G$, $M=2$, $\alpha=30$ dB, $N=480$,
and $5$ bpcu. Figure 6:  Spectrall efficiency comparison of all schemes for
varying $G$ and $M=2$.

In this section, we will discuss the simulation results of the proposed OTA-
RIS-IM and E-OTA-RIS-IM schemes in terms of BER performance compared to the
RGB-IM benchmark [29] which only activates one group of the RIS. For a
different number of RIS groups $G$, various $M$, $N$ and other parameters, we
show the superior performance of the proposed scheme over the RGB-IM. Unless
otherwise indicated, the following system parameters are assumed in all
simulations: the distances $d_{h}=20$ and $d_{f}=50$ meters, the path loss
exponents $\varkappa_{h}=2.2$ and $\varkappa_{f}=2.8$, the Rician shape
parameters $K_{h}=K_{f}=0$, the noise variances and the reference path loss
value of $\zeta_{0}=-30$ dB. For the BER analyses, the noise power $N_{0}$ and
$V_{0}$ are fixed to $-130$ dBm and the transmit power $P_{t}$ is varied.

Fig. 3 presents the error performance for all schemes with $G=2$, $N=256$, and
$4$ bpcu for varying $\alpha$. It is observed that E-OTA-RIS-IM requires the
least modulation order of $M=2$ since it can transmit additional bits through
IM and is the most spectral efficient. On the other hand, OTA-RIS-IM needs
$M=4$ and RGB-IM benchmark operates at $M=8$ modulation orders to keep the
same bpcu of $4$. It is seen that both proposed schemes outperform the
benchmark and OTA-RIS-IM provides the best performance of all schemes since it
uses the entire RIS at all times due to not having absorption state as in
E-OTA-RIS-IM and RGB-IM. Furthermore, the error performance of all schemes
improves with respect to $\alpha$.

In Fig. 4, we can see the BER performance of the proposed schemes compared to
the benchmark with $G=2$ for varying $N$. To obtain same bpcu values in each
scheme, the modulation order varies. It is observed that while E-OTA-RIS-IM
needs only $M=2$, OTA-RIS-IM needs to operate at $M=4$ modulation order.
Despite a higher modulation order, OTA-RIS-IM outperforms E-OTA-RIS-IM in
terms of error performance since it has all RIS elements on at all
combinations. On the other hand, RGB-IM requires $M=8$ modulation order to
keep up the same data rate. It is shown that the RGB-IM at the same bpcu
performs significantly worse compared to both of the proposed schemes.

Fig. 5 presents the BER performance of the proposed systems compared to the
benchmark when all of the systems are at the same bpcu and $M=2$. As we can
see, to reach the same bpcu RGB-IM requires a large amount of grouping $G$ to
send additional bits through IM. On the other hand, OTA-RIS-IM needs only
$G=4$ RIS groups to reach the same bpcu and E-OTA-IM needs even a fewer number
$G=3$ since it has an additional state the hybrid RIS can take. It can also be
observed that since there is no absorption state, even though the bpcu is less
compared to E-OTA-RIS-IM, the BER performance is significantly superior
because all RIS elements are exploited at all times.

Fig. 6 shows the increase in bpcu as the number of groups $G$ increases for
each scheme. It can be observed, that as the number of groups increases the
benchmark’s bpcu does not increase much since only one group is activated
according to the $\left\lfloor\log_{2}(G)\right\rfloor$ IM bits each time and
the rest of the RIS is inactive. On the other hand, OTA-RIS-IM uses the entire
hybrid-RIS where each group may take an active and passive state according to
$G$ IM bits and provides a significantly higher bpcu compared to the
benchmark. Furthermore, E-OTA-RIS-IM introduces an absorption state at the RIS
to further enhance the bpcu for a trade-off from reliability as we have seen
from the previous BER figures. However, we can observe that the bpcu increases
drastically for the E-OTA-RIS-IM scheme since the IM bits are increased to
$\left\lfloor\log_{2}(3^{G})\right\rfloor$.

##  IV Conclusion

In this study, a novel OTA-RIS-IM and E-OTA-RIS-IM scheme has been proposed,
exploiting the benefits of IM and RIS technology efficiently to enhance the
spectral efficiency and BER performance of RIS-aided wireless communication
systems. Through partitioning and varying the states of the groups at the
hybrid-RIS, additional IM bits are conveyed over-the-air. Furthermore,
extensive computer simulation results demonstrate and discuss the performance
benefits of the proposed system. Future directions may include applications of
machine learning, various methods to improve the transmission of IM bits,
considering a multiple antenna system design, and other methods to further
improve the performance and complexity of the proposed system design.

## Acknowledgment

This study has been supported by the 1515 Frontier Research and Development
Laboratories Support Program of TÜBİTAK under Project 5229901 - 6GEN. Lab: 6G
and Artificial Intelligence Laboratory.

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