Anisotropic constant-roll inflationary scenario with complex quintessence
field and swampland conjectures
Jafar Sadeghi⋆111Email<EMAIL_ADDRESS>Saeed Noori Gashti⋆222Email:
<EMAIL_ADDRESS>Mehdi Shokri†,‡333Email:
<EMAIL_ADDRESS>
⋆Department of Physics, Faculty of Basic Sciences,
University of Mazandaran P. O. Box 47416-95447, Babolsar, Iran
†Department of Physics, University of Tehran, North Karegar Ave., Tehran
14395-547, Iran
‡School of Physics, Damghan University, P. O. Box 3671641167, Damghan, Iran

###### Abstract

In this paper, with presence of swampland conjecture we use the complex form
of a scalar field and investigate the anisotropic constant-roll of an
inflationary scenario. Recently, the complex quintessence field has been used
to describe the accelerating expansion of the universe which has had
interesting results connecting with various conditions. Therefore, the
Lagrangian density of the quintessence field leads to obtain the equations of
the complex scalar field. Also, we use the anisotropic constant-roll
conditions and the field equation and calculate the exact solutions for some
cosmology parameters such as Hubble parameter and potential. Using the exact
solution of potential and refined swampland conditions, we plot some figures.
In that case, the figures lead us to have challenge between three concepts as
complex quintessence, anisotropic constant-roll conditions and swampland
conjectures. Then we discuss their compatibility and incompatibility with
corresponding results. Finally, we will analyze the complex quintessence in
examining the inflationary scenario.

Keywords: Anisotropic constant-roll, complex quintessence, Refined swampland
conjectures

###### Contents

  1. 1 Introduction
  2. 2 Complex Quintessence theory
  3. 3 Anisotropic constant-roll inflation
    1. 3.1 Condition I
    2. 3.2 Condition II
  4. 4 Refined swampland conjecture
    1. 4.1 discussion and result
  5. 5 Concluding remarks

##  1 Introduction

Recently, reports from various projects such as High-Z Supernova Search and
Supernova Cosmology Project about their observations of SNe Ia have shown that
the universe has an accelerating cosmic expansion [1, 2]. In addition, the CMB
measurements from MAXIMA-1 and BOOMERANG represent the fact that there is
around $l=200$ a sharp zenith, which indicates that the universe is flat [3,
4]. Of course, multiple projects have made various observations so far, each
for a specific mission, to use such cosmic data to identify the large-scale
structures and nature of the universe and how this universe formed. The above
observations suggest that the universe may have a critical density that could
include different structures, representing one-third of ordinary matter and
two-thirds of dark energy with negative pressure[5, 6].
So far, various concepts have been used to explain the universe’s
acceleration. One of the most important candidates is dark energy with unknown
nature. Due to this unknown nature also introduced different types of models
including positive cosmological constant, quintessence, Chaplygin gas models,
interacting dark energy models, ghost condensate, quintom, tachyon models and
braneworld models [7, 8] One of the models we mentioned is the quintessence,
which includes different parts (real and complex). Huterer, Turner, and other
researchers have conducted studies on the real part of the quintessence
field[9, 10, 11, 12]. Also, the complex part of the quintessence field has
recently been considered and discussed in dark energy structures. Sadeghi
Noori Gashti and Azizi studied Tsallis and Kaniadakis holographic dark energy
with Complex Quintessence theory in Brans- Dicke cosmology and Tsallis
holographic dark energy under the Complex form of Quintessence model[13, 14],
also for the further reading you see [15].
Another candidate for examining the universe’s accelerating expansion is the
inflationary scenario, which has been studied from different models and
conditions so far. One of the most important conditions in which much work has
been done is studying the inflation model concerning the constant-roll
condition. Constant- roll condition is as $\ddot{\phi}=\beta H\dot{\phi}$. The
exact solutions calculated in constant-roll conditions can be closer to the
latest observable data constant-roll[16, 17, 18, 19, 20] This condition has
recently been investigated in various theories such as teleparallel $f(t)$
gravity, $f(R)$, $f(T),$ $f(Q)$, etc., and the results have been compared with
the latest observable data. Also, the anisotropic constant-roll condition has
recently been studied by researchers in various inflation models, which has
led to exciting results[21].
Recently, researchers have used a combination of this theory with other
conditions and conjectures, including the condition derived from string
theory, called the swampland program. The swampland program has have been
studied in various cosmological structures, including inflation, dark energy,
and the physics of black holes. Despite the problems and novelty of this idea,
it has also yielded interesting results. For further study of this concept, as
well as the most important conjectures that have been studied in this theory,
such as dS conjecture, weak gravity conjecture, etc., see [22, 23, 24, 25, 26,
27, 28, 29, 30, 31, 32, 33, 34, 35, 36]. In this paper, we want to study the
constant-roll condition of the inflationary scenario in the presence of
refined swampland conjectures considering the complex form of the scalar field
as complex quintessence. We compare the obtained results with the latest
observable data and other studies. This concept has not been studied yet and
can lead to exciting results. According to the concept, as mentioned earlier,
the organization of this article is as follows.
In section 2 we describe the concept of complex quintessence field and basic
equations by introducing the resulting field equations. In section 3, we apply
the anisotropic constant-roll condition concerning the complex form of the
scalar field or the complex quintessence and obtain accurate solutions,
including the Hubble parameter and the potential. In section 4, according to
the exact solutions that we calculated for the potential of the mentioned
model, we apply the refined swampland condition and describe the results by
plotting some figures and challenge the compatibility or incompatibility of
the mentioned concepts. Finally, we describe the results in section 5.

##  2 Complex Quintessence theory

In this section, we first state the basic equations for studying anisotropic
constant-roll conditions. So we will first assume that a flat spatial universe
is dominated by non-relativistic matter and a complex scalar field, which can
be expressed with the following flat metric[39],

$\begin{split}ds^{2}=dt^{2}-a^{2}(t)(dr^{2}+r^{2}d\varphi_{1}^{2}+r^{2}\sin^{2}\varphi_{1}d\varphi_{2}^{2}),\end{split}$
(1)

In the second step, we introduce the corresponding action, which is given by,

$\begin{split}S=\int d^{4}x\sqrt{g}\big{(}-\frac{1}{16\pi
G}\mathcal{R}-\rho_{M}+\mathcal{L}_{\Phi}\big{)},\end{split}$ (2)

where $g$, $G$, $\mathcal{R}$ and $\rho_{M}$ are the absolute value of the
metric tensor $g_{\mu\nu}$, Newton’s constant, and Ricci scalar and matter
density, respectively. According to the last term in equation (2), the
Lagrangian density for this complex scalar field is expressed by,

$\begin{split}\mathcal{L}_{\Phi}=\frac{1}{2}g^{\mu\nu}\partial_{\mu}\Phi^{\ast}\partial_{\nu}\Phi-V(|\Phi|),\end{split}$
(3)

where $\mu,\nu=0,1,2,3$. In the Lagrange equation, we will assume that the
potential term depends only on the exact values of the complex form of the
scalar field. In the following, instead of the two field parameters $\Phi$ and
$\Phi^{\ast}$ we will use alternative field variables such as $\phi$ and
$\theta$ called the amplitude and the phase, respectively. So, we have do
separation $\Phi$ and $\Phi^{\ast}$ in terms of $\phi$ and $\theta$, we will
have following,

$\begin{split}\Phi(x)=\phi(x)\exp(i\theta x),\end{split}$ (4)

Or the more accurate form is as follows.

$\begin{split}\Phi(t)=\phi(t)\exp(i\theta t),\end{split}$ (5)

According to the new definition of variables, the Lagrangian equation (3) is
rewritten as follows,

$\begin{split}\mathcal{L}_{\Phi}=\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi+\frac{1}{2}\phi^{2}g^{\mu\nu}\partial_{\mu}\theta\partial_{\nu}\theta-V(\phi),\end{split}$
(6)

Now with the help of equations (2) and (6), we can calculate Einstein and
field equations of the complex scalar field. Therefore, using the metric
equation (1), the above mentioned equations are obtained in the following
form,

$\begin{split}H^{2}\equiv(\frac{\dot{a}}{a})^{2}=\frac{8\pi
G}{3}\rho=\frac{8\pi
G}{3}\bigg{(}\rho_{M}+\big{(}\frac{1}{2}\dot{\phi}^{2}+\frac{1}{2}\phi^{2}\dot{\theta}^{2}\big{)}+V(\phi)\bigg{)}\end{split}$
(7) $\begin{split}(\frac{\ddot{a}}{a})=-\frac{4\pi G}{3}(\rho+3p)=-\frac{8\pi
G}{3}\bigg{(}\frac{1}{2}\rho_{M}+\big{(}\dot{\phi}^{2}+\phi^{2}\dot{\theta}^{2}\big{)}-V(\phi)\bigg{)},\end{split}$
(8)
$\begin{split}\ddot{\phi}+3H\dot{\phi}-\dot{\theta}^{2}\phi+V^{\prime}(\phi)=0,\end{split}$
(9)
$\begin{split}\ddot{\theta}+\big{(}2\frac{\dot{\phi}}{\phi}+3H\big{)}\dot{\theta}=0,\end{split}$
(10)

The above equations are the fundamental equations that govern the universe’s
evolution and $H$, ′ $dot$ are Hubble parameters, derivatives concerning
$\phi$, and derivatives with respect to $t$, respectively. Also, two
parameters $\rho$ and $p$ are called energy density and pressure. It would be
worth mentioning here that the contribution is non-relativistic matter is
energy density and pressure $\rho_{M}=p_{M}=0$. In that case, energy density
and pressure is expressed in the following form.

$\begin{split}\rho_{\Phi}=\frac{1}{2}\big{(}\dot{\phi}^{2}+\phi^{2}\dot{\theta}^{2}\big{)}+V(\phi)\\\
p_{\Phi}=\frac{1}{2}\big{(}\dot{\phi}^{2}+\phi^{2}\dot{\theta}^{2}\big{)}-V(\phi),\end{split}$
(11)

According to the solution of equation (10), a relation for ”angular velocity”
is calculated in the following form.

$\begin{split}\dot{\theta}=\frac{\omega}{a^{3}\phi^{2}},\end{split}$ (12)

where $\omega$ is an integration constant calculated according to the initial
conditions defined for $\dot{\theta}$. According to the equation derived from
(12), equations (7-10) can be rewritten as follows.

$\begin{split}H^{2}\equiv(\frac{\dot{a}}{a})^{2}=\frac{8\pi
G}{3}\rho=\frac{8\pi
G}{3}\bigg{(}\rho_{M}+\frac{1}{2}\dot{\phi}^{2}+\frac{1}{2}\frac{\omega^{2}}{a^{6}\phi^{2}}+V(\phi)\bigg{)}\end{split}$
(13) $\begin{split}(\frac{\ddot{a}}{a})=-\frac{4\pi G}{3}(\rho+3p)=-\frac{8\pi
G}{3}\bigg{(}\frac{1}{2}\rho_{M}+\dot{\phi}^{2}+\frac{\omega^{2}}{a^{6}\phi^{2}}-V(\phi)\bigg{)},\end{split}$
(14)
$\begin{split}\ddot{\phi}+3H\dot{\phi}-\frac{\omega^{2}}{a^{6}\phi^{3}}+V^{\prime}(\phi)=0,\end{split}$
(15)

………..As mentioned earlier, the term $\frac{\omega^{2}}{2a^{6}\phi^{2}}$ is
derived from the angular motion of the complex form of the scalar field, which
can also be referred to as an effective potential. In fact, in a more severe
form, perhaps the centrifugal potential causes a distance $\phi$ from zero by
creating a centrifugal force. Also, since the gravitational effect of dark
energy with negative pressure prevents the ordinary material from forming
configurations, the contributions related to quintessence to pressure and
energy density have been negligible for a short time ago. Also, to maintain
the balance between different theories such as Big Bang Cosmology + Inflation
+ Cold Dark Matter and observations as measurements of CMB, quintessence field
contributions at various approaches such as recombination and primordial
nucleosynthesis should be insignificant. These constraints on quintessence
help distinguish between two types of kinetic energy resulting from the
complex form of the scalar field or (complex quintessence). As shown in the
fundamental equations (13) and (14), the contribution of the angular motion of
the scalar complex field to the energy density and the pressure is
proportional to the expression $a^{-6}\phi^{-2}$. Of course, factor $a^{-6}$
may cause a rapid decrease in these shares, although it is also subject to
conditions; for example, $\phi$ does not decrease with $(a^{-3/2})$. For more
study, you can see [39]. Also, according to equations (13-15), we can
implement the anisotropic constant-roll condition. However, according to the
data $r(z)$ from SNe Ia and some parameters such as $\Omega_{M}$, $H_{0}$, and
$\omega$, the results we can discuss in detail. Of course, the most important
parameters, $z$, $r(z)$, $H_{0}$, and $\Omega_{M}$ are redshift, Robertson-
Walker coordinate distance, Hubble constant, and matter-energy density
fraction. we can also benefit two equations as $1+z=1/a$ where $a$ named the
scale factor and $\rho_{M}=\Omega_{M}\rho_{c}=\frac{3\Omega_{M}H_{0}^{2}}{8\pi
G}(1+z)^{3}$

##  3 Anisotropic constant-roll inflation

This section will discuss the anisotropic constant-roll condition concerning
the complex form of the scalar field, such as complex quintessence. So we can
obtain accurate solutions for parameters such as Hubble and potential, i.e.,
parameters that play an important role in explaining the inflation scenario.
The basic condition that we will assume here is that the first index of slow-
roll condition in the inflation period is very small. Therefore, according to
all concepts mentioned earlier and according to equations (13) and (14), we
will have,

$\begin{split}3\frac{\ddot{a}}{a}-3H^{2}=-\frac{3}{2}\rho_{M}-\frac{3}{2}\dot{\phi}^{2}-\frac{3}{2}\frac{\omega^{2}}{a^{6}\phi^{2}},\end{split}$
(16)

According to the definition of the Hubble parameter $H=\frac{\dot{a}}{a}$ and
equation (16), one can calculate,

$\begin{split}\dot{H}=-\frac{3\Omega_{M}H_{0}^{2}}{2a^{3}}-\frac{1}{2}\dot{\phi}^{2}-\frac{1}{2}\frac{\omega^{2}}{a^{6}\phi^{2}},\end{split}$
(17)

We use the auxiliary equation viz $\dot{H}=\frac{dH}{d\phi}\dot{\phi}$ and
place it in the above equation. so we will have,

$\begin{split}\frac{1}{2}\dot{\phi}^{2}+\frac{dH}{d\phi}\dot{\phi}+\frac{1}{2}\frac{\omega^{2}}{a^{6}\phi^{2}}+\frac{3\Omega_{M}H_{0}^{2}}{2a^{3}},\end{split}$
(18)

We solve the above equation in terms of $\dot{\phi}$, So

$\begin{split}\dot{\phi}=-\frac{dH}{d\phi}\pm\sqrt{\frac{d^{2}H}{d\phi^{2}}-(\frac{\omega^{2}}{a^{6}\phi^{2}}+\frac{3\Omega_{M}H_{0}^{2}}{a^{3}})},\end{split}$
(19)

We take the derivative from the above equation concerning $t$. so we will
have,

$\begin{split}\ddot{\phi}=-\frac{d^{2}H}{d\phi^{2}}\dot{\phi}\pm\frac{\frac{d^{2}H}{d\phi^{2}}\frac{dH}{d\phi}\dot{\phi}-(\frac{-2\omega^{2}\dot{\phi}}{a^{6}\phi^{3}})}{\sqrt{\frac{d^{2}H}{d\phi^{2}}-(\frac{\omega^{2}}{a^{6}\phi^{2}}+\frac{3\Omega_{M}H_{0}^{2}}{a^{3}})}},\end{split}$
(20)

We combine equation (20) with $\ddot{\phi}=-(3+\alpha)H\dot{\phi}$. So we will
have.

$\begin{split}-(3+\alpha)H=-\frac{d^{2}H}{d\phi^{2}}\pm\frac{\frac{d^{2}H}{d\phi^{2}}\frac{dH}{d\phi}-(\frac{-2\omega^{2}}{a^{6}\phi^{3}})}{\sqrt{\frac{d^{2}H}{d\phi^{2}}-(\frac{\omega^{2}}{a^{6}\phi^{2}}+\frac{3\Omega_{M}H_{0}^{2}}{a^{3}})}},\end{split}$
(21)

Assuming $\omega=\Omega_{M}=0$ the above equation becomes as,

$\begin{split}\frac{d^{2}H}{d\phi^{2}}-\frac{3+\alpha}{2}H=0,\end{split}$ (22)

The solution of the above-reduced equation is calculated as follows.

$\begin{split}H=c_{1}\exp(\sqrt{\frac{3+\alpha}{2}}\phi)+c_{2}\exp(-\sqrt{\frac{3+\alpha}{2}}\phi),\end{split}$
(23)

The above equation is a specific solution of the Hubble parameter or the
general solution of the isotropic constant-roll condition. We assumed the
effects of specific parameters in this equation to be zero. According to
equation (23), a particular ansatz can be considered for the general solution
in equation (21) in the following form to specify the effects of the deleted
parameters.

$\begin{split}H=c_{1}\exp(\lambda(\omega,\Omega_{M})\sqrt{\frac{3+\alpha}{2}}\phi)+c_{2}\exp(-\lambda(\omega,\Omega_{M})\sqrt{\frac{3+\alpha}{2}}\phi),\end{split}$
(24)

Here, by placing this ansatz in the equation (21) and with a series of direct
calculations, the parameter $\lambda$ can be specified. We compute the
parameter $\lambda$ and place it in the Hubble equation (24). We then examine
specific solutions using boundary conditions. We will also calculate the exact
solutions for essential parameters, such as the potential for investigating
swampland conjectures.

###  3.1 Condition I

Here we will apply the boundary condition as $(c_{1}=c_{2}=M/2)$ to the Hubble
parameter in equation (24). Since different values are calculated for the
Hubble parameter, we kept only positive solutions. The negative solutions
showed a similar concept in the study of the swampland program. Hence, we only
pursue positive solutions. Negative solutions follow a similar process so that
we will have.

$\begin{split}H=M\cosh\bigg{[}\sqrt{\frac{3+\alpha}{2}}\phi\bigg{(}\pm\frac{2\sqrt{M(3+\alpha)-\frac{2\omega^{2}}{a^{6}\phi^{3}\sqrt{1/2M(3+\alpha)-12H_{0}^{2}\Omega_{M}/a^{3}}}}}{\sqrt{\frac{M\big{\\{}-12H_{0}^{2}(3+\alpha)\Omega_{M}+a^{3}M\big{[}2+(3+\alpha)^{2}+\phi\sqrt{2M(3+\alpha)-12H_{0}^{2}\Omega_{M}/a^{3}}\big{]}\big{\\}}}{a^{3}M(3+\alpha)-6H_{0}^{2}\Omega_{M}/a^{3}}}}\bigg{)}\bigg{]},\end{split}$
(25)

Given the relation obtained for the Hubble parameter in equation (25), we will
have

$\begin{split}&\mathcal{X}_{1}=\sqrt{M(3+\alpha)-\frac{2\omega^{2}}{a^{6}\phi^{3}\sqrt{\frac{1}{2}M(3+\alpha)-3H_{0}^{2}\Omega_{M}/a^{3}}}}\\\
&\mathcal{X}_{2}=\sqrt{\frac{M\big{(}-12H_{0}^{2}(3+\alpha)\Omega_{M}+a^{3}M\big{[}2(3+\alpha)^{2}+\phi\sqrt{2M(3+\alpha)-12H_{0}^{2}\Omega_{M}/a^{3}}\big{]}\big{)}}{a^{3}M(3+\alpha)-6H_{0}^{2}\Omega_{M}/a^{3}}}\\\
&\mathcal{X}_{3}=a^{2}\phi\sqrt{2M(3+\alpha)-12H_{0}^{2}\Omega_{M}/a^{3}}\\\
&V(\phi)=-2M^{2}\bigg{(}-\frac{a^{3}M^{2}\sqrt{3+\alpha}\mathcal{X}_{1}\times\mathcal{X}_{3}}{\sqrt{2}(a^{3}M(3+\alpha)-6H_{0}^{2}\Omega_{M}/a^{3})(\mathcal{X}_{2})^{3}}+\frac{3\sqrt{2}\sqrt{3+\alpha}\omega^{2}}{a^{3}\phi^{2}\mathcal{X}_{1}\times\mathcal{X}_{2}\times\mathcal{X}_{3}}\\\
&+\frac{\sqrt{2}\sqrt{3+\alpha}\mathcal{X}_{1}}{\mathcal{X}_{2}}\bigg{)}^{2}\times\sinh^{2}\big{[}\frac{\sqrt{2}\sqrt{3+\alpha}\phi\mathcal{X}_{1}}{\mathcal{X}_{2}}\big{]}+\cosh^{2}\big{[}\frac{\sqrt{2}\sqrt{3+\alpha}\phi\mathcal{X}_{1}}{\mathcal{X}_{2}}\big{]},\end{split}$
(26)

Also, according to equations (19) and (25), we can calculate $\dot{\phi}$. So
we will have,

$\begin{split}&\mathcal{X}_{1}=\sqrt{M(3+\alpha)-\frac{2\omega^{2}}{a^{6}\phi^{3}\sqrt{\frac{1}{2}M(3+\alpha)-3H_{0}^{2}\Omega_{M}/a^{3}}}}\\\
&\mathcal{X}_{2}=\sqrt{\frac{M\big{(}-12H_{0}^{2}(3+\alpha)\Omega_{M}+a^{3}M\big{[}2(3+\alpha)^{2}+\phi\sqrt{2M(3+\alpha)-12H_{0}^{2}\Omega_{M}/a^{3}}\big{]}\big{)}}{a^{3}M(3+\alpha)-6H_{0}^{2}\Omega_{M}/a^{3}}}\\\
&\mathcal{X}_{3}=a^{2}\phi\sqrt{2M(3+\alpha)-12H_{0}^{2}\Omega_{M}/a^{3}}\\\
&\dot{\phi}=-M\bigg{(}-\frac{a^{3}M^{2}\sqrt{3+\alpha}\mathcal{X}_{3}\mathcal{X}_{1}}{\sqrt{2}(a^{3}M(3+\alpha)-6H_{0}^{2}\Omega_{M}/a^{2})(\mathcal{X}_{2})^{3}}+\frac{3\sqrt{2}\sqrt{3+\alpha}\omega^{2}}{a^{3}\phi^{2}\mathcal{X}_{1}\times\mathcal{X}_{2}\times\mathcal{X}_{3}}\\\
&+\frac{\sqrt{2}\sqrt{3+\alpha}\mathcal{X}_{1}}{\mathcal{X}_{2}}\bigg{)}\sinh\big{[}\frac{\sqrt{2}\sqrt{3+\alpha}\phi\mathcal{X}_{1}}{\mathcal{X}_{2}}\big{]}-\bigg{(}M\bigg{[}-\frac{a^{3}M^{2}\sqrt{3+\alpha}\mathcal{X}_{3}\times\mathcal{X}_{1}}{\sqrt{2}(a^{3}M(3+\alpha)-6H_{0}^{2}\Omega_{M}/a^{3})/(\mathcal{X}_{2})^{3}}\\\
&+\frac{3\sqrt{2}\sqrt{3+\alpha}\omega^{2}}{a^{3}\phi^{2}\mathcal{X}_{1}\times\mathcal{X}_{2}\times\mathcal{X}_{3}}+\frac{\sqrt{2}\sqrt{3+\alpha}\mathcal{X}_{1}}{\mathcal{X}_{2}}\bigg{]}^{2}-\frac{\omega^{2}}{a^{6}\phi^{2}}-3H_{0}^{2}\Omega_{M}/a^{3}\\\
&\times\cosh\big{[}\frac{\sqrt{2}\sqrt{3+\alpha}\phi\mathcal{X}_{1}}{\mathcal{X}_{2}}\big{]}+M\bigg{\\{}\frac{3a^{3}M^{4}\sqrt{3+\alpha}\phi(2M(3+\alpha)-12H_{0}^{2}\Omega_{M}/a^{3})\mathcal{X}_{1}}{2\sqrt{2}(a^{3}M(3+\alpha)-6H_{0}^{2}\Omega_{M}/a^{3}(\mathcal{X}_{2})^{5}}\\\
&-\frac{3\sqrt{2}M^{2}\sqrt{3+\alpha}\omega^{2}\mathcal{X}_{3}}{\phi^{2}(a^{3}M(3+\alpha)-6H_{0}^{2}\Omega_{M}\mathcal{X}_{3}\times\mathcal{X}_{1}\times(\mathcal{X}_{2})^{3}}-\frac{\sqrt{2}a^{3}M^{2}\sqrt{3+\alpha}\mathcal{X}_{3}\times\mathcal{X}_{1}}{(a^{3}M(3+\alpha)-6H_{0}^{2}\Omega_{M}/a^{3})(\mathcal{X}_{2})^{3}}\\\
&-\frac{6\sqrt{2}\sqrt{3+\alpha}\omega^{2}}{a^{3}\phi^{3}\mathcal{X}_{1}\times\mathcal{X}_{2}\times\mathcal{X}_{3}}-\frac{9\sqrt{2}\sqrt{3+\alpha}\omega^{4}}{a^{6}\phi^{6}(1/2M(3+\alpha)-3H_{0}^{2}\Omega_{M}/a^{3})(\mathcal{X}_{1})^{3}\mathcal{X}_{2}}\bigg{\\}}\\\
&\times\sinh\big{[}\frac{\sqrt{2}\sqrt{3+\alpha}\phi\mathcal{X}_{1}}{\mathcal{X}_{2}}\big{]}\bigg{)},\end{split}$
(27)

Different solutions can be calculated for the parameter $\phi$ by solving the
above equation. Also, we can obtain with straightforward calculation the scale
factor according to equations (25) and $H=\frac{\dot{a}}{a}$. We can examine
another boundary condition and compare the results with each other.

###  3.2 Condition II

In this part, we consider another boundary condition expressed as $c_{1}=M/2$
and $c_{2}=-M/2$. Also, we focus only on positive solutions. Therefore,
according to the computational values for the parameter $\lambda$, we will
calculate the quantities as Hubble, potential, and $\dot{\phi}$ separately.
Hence we will have

$\begin{split}H=M\sinh\bigg{[}\sqrt{\frac{3+\alpha}{2}}\phi\bigg{(}\pm\frac{\sqrt{\frac{M^{2}(3+\alpha)\bigg{\\{}-12H_{0}^{2}(3+\alpha)^{2}\Omega_{M}+\frac{2a^{3}M\sqrt{3+\alpha}\omega^{2}}{(3+\alpha)^{3}\phi+3\sqrt{\frac{\sqrt{2}a^{3}M(3+\alpha)^{3/2}\phi-3H_{0}^{2}\Omega_{M}}{a^{3}}}}\bigg{\\}}}{\sqrt{2}a^{3}M(3+\alpha)^{3/2}\phi-12H_{0}^{2}\Omega_{M}/a^{3}}}}{M(3+\alpha)^{3/2}\phi}\bigg{)}\bigg{]},\end{split}$
(28)

Here, as in the previous part, quantities such as potential and $\dot{\phi}$
will be calculated as follows.

$\begin{split}&\mathcal{Y}_{1}=\bigg{(}\frac{M^{2}(3+\alpha)\bigg{[}-12H_{0}^{2}(3+\alpha)^{2}\Omega_{M}+\frac{2a^{3}M\sqrt{3+\alpha}\omega^{2}}{(3+\alpha)^{3}\phi+3\sqrt{\frac{\sqrt{2}a^{3}M(3+\alpha)^{2}-3H_{0}^{2}\Omega_{M}}{a^{3}}}}\bigg{]}}{\sqrt{2}a^{3}M(3+\alpha)^{3/2}\phi-12H_{0}^{2}\Omega_{M}/a^{3}}\bigg{)}^{\frac{1}{2}}\\\
&\mathcal{Y}_{2}=\big{(}(\sqrt{2}a^{3}M(3+\alpha)^{2}-3H_{0}^{2}\Omega_{M})/a^{3}\big{)}^{1/2}\\\
&\mathcal{Y}_{3}=\sqrt{2}a^{3}M(3+\alpha)^{3/2}\phi-(12H_{0}^{2}\Omega_{M})/a^{3}\\\
&V(\phi)=-2M\bigg{(}\frac{\mathcal{Y}_{1}}{\sqrt{2}M(3+\alpha)}+\frac{\phi\bigg{[}-\frac{2a^{3}M^{3}\omega^{2}(3+\alpha)^{9/2}}{\mathcal{Y}_{3}((3+\alpha)^{3}\phi+3\mathcal{Y}_{2})^{2}-\frac{\sqrt{2}a^{3}M^{3}(3+\alpha)^{5/2}\big{(}-12H_{0}^{2}(3+\alpha)^{2}\Omega_{M}+\frac{2a^{3}M\omega^{2}\sqrt{3+\alpha}}{(3+\alpha)^{3}\phi+3\mathcal{Y}_{2}}\big{)}}{(\mathcal{Y}_{3})^{2}}}\bigg{]}}{2\sqrt{2}M(3+\alpha)\mathcal{Y}_{1}}\bigg{)}^{2}\\\
&\times\cosh^{2}\big{[}\frac{\phi\mathcal{Y}_{1}}{\sqrt{2}M(3+\alpha)}\big{]}+3M^{2}\sinh^{2}\big{[}\frac{\mathcal{Y}_{1}}{\sqrt{2}M(3+\alpha)}\big{]},\end{split}$
(29)

and

$\begin{split}&\mathcal{Y}_{1}=\bigg{(}\frac{M^{2}(3+\alpha)\bigg{[}-12H_{0}^{2}(3+\alpha)^{2}\Omega_{M}+\frac{2a^{3}M\sqrt{3+\alpha}\omega^{2}}{(3+\alpha)^{3}\phi+3\sqrt{\frac{\sqrt{2}a^{3}M(3+\alpha)^{2}-3H_{0}^{2}\Omega_{M}}{a^{3}}}}\bigg{]}}{\sqrt{2}a^{3}M(3+\alpha)^{3/2}\phi-12H_{0}^{2}\Omega_{M}/a^{3}}\bigg{)}^{\frac{1}{2}}\\\
&\mathcal{Y}_{2}=\bigg{(}(\sqrt{2}a^{3}M(3+\alpha)^{2}-3H_{0}^{2}\Omega_{M})/a^{3}\bigg{)}^{1/2}\\\
&\mathcal{Y}_{3}=\sqrt{2}a^{3}M(3+\alpha)^{3/2}\phi-(12H_{0}^{2}\Omega_{M})/a^{3}\\\
&\dot{\phi}=-M\bigg{(}\frac{\mathcal{Y}_{1}}{\sqrt{2}M(3+\alpha)}+\frac{\phi\bigg{[}-\frac{2a^{3}M^{3}\omega^{2}(3+\alpha)^{9/2}}{\mathcal{Y}_{3}((3+\alpha)^{3}\phi+3\mathcal{Y}_{2})^{2}-\frac{\sqrt{2}a^{3}M^{3}(3+\alpha)^{5/2}\big{(}-12H_{0}^{2}(3+\alpha)^{2}\Omega_{M}+\frac{2a^{3}M\omega^{2}\sqrt{3+\alpha}}{(3+\alpha)^{3}\phi+3\mathcal{Y}_{2}}\big{)}}{(\mathcal{Y}_{3})^{2}}}\bigg{]}}{2\sqrt{2}M(3+\alpha)\mathcal{Y}_{1}}\bigg{)}\\\
&\times\cosh\big{[}\frac{\phi\mathcal{Y}_{1}}{\sqrt{2}M(3+\alpha)}\big{]}-\bigg{\\{}-\frac{\omega^{2}}{a^{6}\phi^{2}}-\frac{3H_{0}^{2}\Omega_{M}}{a^{3}}\\\
&+M\bigg{[}\frac{\phi\bigg{(}\frac{4a^{3}M^{3}(3+\alpha)^{15/2}\omega^{2}}{\mathcal{Y}_{3}\big{(}(3+\alpha)^{3}\phi+3\mathcal{Y}_{2}\big{)}^{3}}+\frac{4\sqrt{2}a^{6}M^{4}(3+\alpha)^{6}\omega^{2}}{(\mathcal{Y}_{3})^{2}\big{(}(3+\alpha)^{3}\phi+3\mathcal{Y}_{2}\big{)}^{2}}+\frac{4a^{6}M^{4}(3+\alpha)^{4}\big{[}-12H_{0}^{2}\Omega_{M}(3+\alpha)^{2}+\frac{2a^{3}M\sqrt{3+\alpha}\omega^{2}}{(3+\alpha)^{3}\phi+3\mathcal{Y}_{2}}\big{]}}{(\mathcal{Y}_{3})^{3}}\bigg{)}}{2\sqrt{2}M(3+\alpha)\mathcal{Y}_{1}}\\\
&+\frac{-\frac{2a^{3}M^{3}\omega^{2}(3+\alpha)^{9/2}}{\mathcal{Y}_{2}\big{(}(3+\alpha)^{3}\phi+3\mathcal{Y}_{2}\big{)}^{2}}-\frac{\sqrt{2}a^{3}M^{3}(3+\alpha)^{5/2}\big{(}-12H_{0}^{2}\Omega_{M}(3+\alpha)^{2}+\frac{2a^{3}M\sqrt{3+\alpha}\omega^{2}}{(3+\alpha)^{3}\phi+3\mathcal{Y}_{2}}\big{)}}{(\mathcal{Y}_{2})^{2}}}{\sqrt{2}M(3+\alpha)\mathcal{Y}_{1}}\\\
&-\frac{\phi\bigg{(}-\frac{2a^{3}M^{3}\omega^{2}(3+\alpha)^{9/2}}{\mathcal{Y}_{3}\big{(}(3+\alpha)^{3}\phi+3\mathcal{Y}_{2}\big{)}^{2}}-\frac{\sqrt{2}a^{3}M^{3}(3+\alpha)^{5/2}\big{[}-12H_{0}^{2}\Omega_{M}(3+\alpha)^{2}+\frac{2a^{3}M\sqrt{3+\alpha}\omega^{2}}{(3+\alpha)^{3}\phi+3\mathcal{Y}_{2}}\big{]}}{(\mathcal{Y}_{3})^{2}}\bigg{)}^{2}}{4\sqrt{2}M(\mathcal{Y}_{1})^{3}}\bigg{]}\cosh\big{[}\frac{\phi\mathcal{Y}_{1}}{\sqrt{2}M(3+\alpha)}\big{]}\\\
&+M\bigg{(}\frac{\mathcal{Y}_{1}}{\sqrt{2}M(3+\alpha)}+\frac{\phi\bigg{(}-\frac{2a^{3}M^{3}\omega^{2}(3+\alpha)^{9/2}}{\mathcal{Y}_{3}\big{(}(3+\alpha)^{3}\phi+3\mathcal{Y}_{2}\big{)}^{2}}-\frac{\sqrt{2}a^{3}M^{3}(3+\alpha)^{5/2}\big{[}-12H_{0}^{2}\Omega_{M}(3+\alpha)^{2}+\frac{2a^{3}M\sqrt{3+\alpha}\omega^{2}}{(3+\alpha)^{3}\phi+3\mathcal{Y}_{2}}\big{]}}{(\mathcal{Y}_{3})^{2}}\bigg{)}}{2\sqrt{2}M(3+\alpha)\mathcal{Y}_{1}}\bigg{)}^{2}\\\
&\times\sinh\big{[}\frac{\mathcal{Y}_{1}}{\sqrt{2}M(3+\alpha)}\big{]}\bigg{\\}},\end{split}$
(30)

##  4 Refined swampland conjecture

This section provides a brief description of the swampland program, especially
the refined swampland conjecture. Then, using the exact solutions obtained for
the potential using the anisotropic constant-roll condition concerning complex
quintessence, we combine them with swampland conjectures and explain the
results by plotting some figures. We will also examine the compatibility or
incompatibility of the mentioned concepts. Recently, researchers have
introduced a concept called the swampland program, which has many conjectures.
These conjectures have been studied in various cosmological structures and
string theory, and their results have been compared with the latest observable
data. Although this concept is a developing topic and still faces many
problems, it has also yielded exciting results. Also, many conjectures have
been added to this concept in recent years, and even conjectures that
contradict specific theories have been corrected, including refined swampland
conjecture. We also challenge this conjecture in this article, which is
expressed in the following form.

$|\nabla V|\geq\frac{C}{M_{p}}V,\hskip 12.0ptmin\nabla\partial
V\leq-\frac{C^{\prime}}{M_{pl}^{2}}V$ (31)

The above equations for the $V>0$, which is given by

$\sqrt{2\epsilon_{V}}\geq C,\hskip 12.0ptor\hskip
12.0pt\eta_{V}\leq-C^{\prime}$ (32)

###  4.1 discussion and result

We plotted the swampland conjectures for the complex form of the scalar field
concerning the exact solutions calculated for the potential with the
anisotropic constant-roll condition for two different boundary conditions in
figures (1) and (2). Figure (1) shows the first boundary condition, according
to equations (26), (31), and Figure (2) is also related to boundary conditions
2 and is plotted according to equations (29) and (31). In figure (1), we
determine the changes of the swampland conjecture are plotted in terms of the
complex form of the scalar field and the constant parameters mentioned in the
text. The virtual range considered for the first and second components of the
swampland conjecture viz $C$ and $C^{\prime}$ is 0 to 1. So the values of
second component $C^{\prime}$ smaller than the $C$. For this reason, in
figures (1b) and (1d), we limited our figures to acceptable values for these
swampland components. The allowable range of the two components for the first
boundary condition is given according to the constant mentioned parameters and
various values of $\alpha$. For each of the two swampland components (C, C’),
$\alpha=-2$ shows a more acceptable spectrum than the other values. Of course,
by changing the constant parameters, the final results will be almost the
same, and there will be some changes in the range of swampland components that
can be ignored. As mentioned in the text, figures are plotted for positive
potential. However, Negative values of potential had almost similar results,
which we declined. Of course, the computational process is exactly like this
path but with the opposite sign. Figure (2) is also plotted according to the
second boundary condition. According to all the mentioned points, the
allowable range components of the swampland conjectures are specified in
figures (2b) and (2d). As clear in these figures, the acceptable range for the
mentioned model is more volume than the previous case. Similar to the previous
case, We plot the figures according to the mentioned constant parameters[40,
41] and different parameter $\alpha$ . Here, $\alpha=-1,-2$ gives us a more
acceptable range and is therefore similar to the previous case. Of course, the
allowable range for each of the two components $(C,C^{\prime})$ is beyond the
previous case. in general, it can be concluded that compatibility between
different concepts as anisotropic constant-roll condition, swampland
conjectures, and complex quintessence is observed. The calculations of this
section can be expanded and compared with other possible conditions.

Figure 1: The plot of $C$ and $C^{\prime}$ in term of $\phi$ for the case I in
fig (1a, 1b) and fig (1c, 1d) respectively, concerning constant mentioned
parameters $\omega=1000$, $\Omega_{M}=0.3$, $H_{0}=67.9$, $M=1$ and various
values of $\alpha$

Figure 2: The plot of $C$ and $C^{\prime}$ in term of $\phi$ for the case II
in fig (2a, 2b) and fig (2c, 2d) respectively, concerning constant mentioned
parameters $\omega=1000$, $\Omega_{M}=0.3$, $H_{0}=67.9$, $M=1$ and various
values of $\alpha$

##  5 Concluding remarks

Regarding the observations of type Ia supernovae indicating a Universe
accelerating expansion, we considered the complex form of the scalar field,
called the complex quintessence field, to explain this acceleration with a
specific inflationary scenario. We investigated the anisotropic constant-roll
of an inflationary scenario using the complex quintessence in the presence of
swampland conjectures. The recently introduced complex quintessence has
yielded exciting results in the accelerating expansion of the universe
connecting with various conditions. Therefore, using the anisotropic constant-
roll conditions, the Lagrangian density of the quintessence field, which leads
to obtaining the field equations of the complex scalar field, we calculate the
exact solutions for some cosmology parameters such as Hubble parameter,
potential, etc. Using the exact solution of potential and refined swampland
conditions, we plot some figures that challenge the three concepts; complex
quintessence, anisotropic constant-roll conditions, and swampland conjectures.
Then we discuss their compatibility and incompatibility with corresponding
results. Finally, we analyzed the complex quintessence in examining the
inflationary scenario. We analyzed the results of each boundary condition
concerning the exact solutions of the potentials arising from the anisotropic
constant-roll condition relating to the complex form of the scalar field as
complex quintessence and the composition of the swampland conjectures. We
described the results in detail in the previous section. There are also new
questions in this area worth pondering and can be explored in future work.
First, is there a complex form for other fields like phantom and Tachyonic? If
any, one can calculate the field equations, examine them with an anisotropic
constant-roll condition, and compare the results with this paper.
Secondly, the results of this research can be examined in the non-commutative
space, which can lead to exciting results.
It is also possible to study other cosmological concepts of this model and
compare the results with the latest observable data by obtaining other
cosmological parameters.

## References

  * [1] S. Perlmutter et al, Astrophys. J. 517 565 (1999).
  * [2] A. G. Riess et al, Astron. J. 116 1009(1998).
  * [3] P. de Bernardis et al., Nature 404 955 (2000).
  * [4] S. Hanany et al. Astrophys. J. 545 L5 (2000).
  * [5] P. de Bernardis et al., astro-ph/0011469.
  * [6] A. Balbi et al., Astrophys. J. 545 (2000).
  * [7] P. J. E. Peebles and B. Ratra, Astrophys. J., 325 L17 (1988).; J. P. Ostriker and P. J. Steinhardt, Nature 377 600 (1995).; A. R. Liddle, D. H. Lyth, P. T. Viana and M. White, Mon. Not. Roy. Astron. Soc. 282 281 (1996). C. Armendariz-Picon, V. F. Mukhanov and P. J. Steinhardt, Phys. Rev. D, 63 103510 (2001); R. R. Caldwell, Phys. Lett. B 545 23 (2002); J. A. Frieman, C. T. Hill, A. Stebbins and I. Waga, Phys. Rev. Lett., 75 2077 (1995).; C. Armendariz-Picon, V. F. Mukhanov and P. J. Steinhardt, Phys. Rev. D, 63 103510 (2001); F. Piazza and S. Tsujikawa, JCAP, 0407 004 (2004).; R. S. Nojiri and S. D. Odintsov, Phys. Lett. B 562 147 (2003); B. Ratra and P. J. E. Peebles, Phys. Rev. D, 37 3406 (1988).; M. S. Turner and M. J. White, Phys. Rev. D, 56 4439 (1997).; R. R. Caldwell, R. Dave and P. J. Steinhardt, Phys. Rev. Lett., 80 1582 (1998).; C. Armendariz-Picon, V. F. Mukhanov and P. J. Steinhardt, Phys. Rev. Lett., 85 4438 (2000); C. Wetterich, Nucl. Phys. B, 302 668 (1988).; L. M. Krauss and M. S. Turner, Gen. Rel. Grav. 27 1137 (1995).; M. S. Turner and M. J. White, Phys. Rev. D, 56 4439 (1997).; R. R. Caldwell, Phys. Lett. B 545 23 (2002); R. R. Caldwell, M. Kamionkowski and N. N. Weinberg, Phys. Rev. Lett. 91 071301 (2003); R. S. Nojiri and S. D. Odintsov, Phys. Lett., B 565 1 (2003); A. Sen, JHEP, 0207 (2002) 065.; R. R. Caldwell, Phys. Lett. B 545 23 (2002); N. Arkani-Hamed, H. C. Cheng, M. A. Luty and S. Mukohyama, JHEP 0405 074 (2004).; F. Piazza and S. Tsujikawa, JCAP, 0407 004 (2004).; L. M. Krauss and M. S. Turner, Gen. Rel. Grav. 27 1137 (1995).; M. S. Turner and M. J. White, Phys. Rev. D, 56 4439 (1997).; R. R. Caldwell, R. Dave and P. J. Steinhardt, Phys. Rev. Lett., 80 1582 (1998).; C. Armendariz-Picon, V. F. Mukhanov and P. J. Steinhardt, Phys. Rev. Lett., 85 4438 (2000); L. M. Krauss and M. S. Turner, Gen. Rel. Grav. 27 1137 (1995).; N. Arkani-Hamed, H. C. Cheng, M. A. Luty and S. Mukohyama, JHEP 0405 074 (2004).; F. Piazza and S. Tsujikawa, JCAP, 0407 004 (2004).;
  * [8] R. R. Caldwell, R. Dave and P. J. Steinhardt, Phys. Rev. Lett. 80 1582(1998).; B. Feng, X. L. Wang and X. M. Zhang, Phys. Lett. B, 607 35 (2005); Z. K. Guo, Y. S. Piao, X. M. Zhang and Y. Z. Zhang, Phys. Lett. B, 608 177 (2005); X. Zhang, Commun. Theor. Phys., 44 762 (2005); A. Anisimov, E. Babichev and A. Vikman, JCAP, 0506 006 (2005); E. Elizalde , S. Nojiri, and S. D. Odintsov, Phys. Rev. D, 70 043539 (2004); S. Nojiri, S. D. Odintsov, and S. Tsujikawa, Phys. Rev. D, 71 063004 (2005); C. Deffayet, G. R. Dvali and G. Gabadadze, Phys. Rev. D 65 044023 (2002).; L. Amendola, Phys. Rev. D, 62 043511 (2000).; A. Y. Kamenshchik, U. Moschella and V. Pasquier, Phys. Lett. B 511 265 (2001).
  * [9] I. Zlatev, L. Wang and P. J. Steinhardt, Phys. Rev. Lett. 82 896(1999).; P. J. Steinhardt, L. Wang and I. Zlatev, Phys. Rev. D 59 123504 (1999).
  * [10] D. Huterer and M. S. Turner, Phys. Rev. D 60 081301 (1999).
  * [11] A. A. Starobinsky, JETP Lett. 68 (1998) 757 [Pisma Zh. Eksp. Teor. Fiz. 68 721 (1998).
  * [12] T. Chiba and T. Nakamura, Phys. Rev. D 62 121301 (2000).
  * [13] J. Sadeghi, and S. Noori Gashti, Tsallis holographic dark energy under Complex form of Quintessence model (2022).
  * [14] J. Sadeghi, and S. Noori Gashti, Tsallis and Kaniadakis holographic dark energy with Complex Quintessence theory in Brans–Dicke cosmology (2022).
  * [15] Yang Liu, Eur. Phys. J. C 80 1204 (2020).
  * [16] H. Motohashi, A.A. Starobinsky, J. Yokoyama, JCAP 1509 (2015).
  * [17] H. Motohashi, A.A. Starobinsky, JCAP 11, 025 (2019).
  * [18] H. Motohashi, A.A. Starobinsky, EPL 117 (2017).
  * [19] L. Anguelova, P. Suranyi, L.C.R. Wijewardhana, JCAP 02, 004 (2018).
  * [20] Mehdi Shokri, Jafar Sadeghi, Saeed Noori Gashti, Physics of the Dark Universe, 100923 (2021).
  * [21] T.Q. Do, W.F. Kao, I.C. Lin, Phys. Rev. D 83, 123002 (2011).; K. Murata, J. Soda, JCAP 1106, 037 (2011).; M.A. Watanabe, S. Kanno, J. Soda, Phys. Rev. Lett. 102, 191302 (2009).; J. Soda, Class. Quantum Gravity 29, 083001 (2012) A. Maleknejad, M.M. Sheikh-Jabbari, J. Soda, Phys. Rep. 528, 161 (2013).; A. Ito, J. Soda, Phys. Rev. D 92(12), 123533 (2015).; R. Emami, arXiv:1511.01683 (2015).; J. Sadeghi, S. Noori Gashti, Eur. Phys. J. C 81, 301 (2021).; S. Kanno, J. Soda, M.A. Watanabe, JCAP 12, 024 (2010).; J. Ohashi, J. Soda, S. Tsujikawa, JCAP 1312, 009 (2013).; K. Yamamoto, M.A. Watanabe, J. Soda, Class. Quantum Gravity 29, 145008 (2012).; J. Ohashi, J. Soda, S. Tsujikawa, Phys. Rev. D 87(8), 083520 (2013).; K. Yamamoto, Phys. Rev. D 85, 123504 (2012).; T.Q. Do, W.F. Kao, Phys. Rev. D 84, 123009 (2011).; M. Thorsrud, D.F. Mota, S. Hervik, JHEP 1210, 066 (2012).; J. Ohashi, J. Soda, S. Tsujikawa, Phys. Rev. D 88, 103517 (2013).; S. Hervik, D.F. Mota, M. Thorsrud, JHEP 1111, 146 (2011).; R. Emami, H. Firouzjahi, S.M. SadeghMovahed, M. Zarei, JCAP 1102, 005 (2011).; A. Ito, J. Soda, JCAP 1604(04), 035 (2016).; M. Karciauskas, Mod. Phys. Lett. A 31(21), 1640002 (2016).; S. Lahiri, JCAP 1609(09), 025 (2016).; N. Bartolo, S. Matarrese, M. Peloso, A. Ricciardone, JCAP 1308, 022 (2013).; K.I. Maeda, K. Yamamoto, Phys. Rev. D 87(2), 023528 (2013).; K.I. Maeda, K. Yamamoto, JCAP 1312, 018 (2013).; B. Chen, Z.W. Jin, JCAP 1409(09), 046 (2014).; S. Lahiri, JCAP 1701(01), 022 (2017).; M. Adak, O. Akarsu, T. Dereli, O. Sert, JCAP 11, 026 (2017).; M. Tirandari, K. Saaidi, Nucl. Phys. B 925, 403 (2017).; T.Q. Do, W.F. Kao, Phys. Rev. D 96(2), 023529 (2017).; A.E. Gumrukcuoglu, B. Himmetoglu, M. Peloso, Phys. Rev. D 81, 063528 (2010).; T.Q. Do, S.H.Q. Nguyen, Int. J. Mod. Phys. D 26(07), 1750072 (2017).; T.R. Dulaney, M.I. Gresham, Phys. Rev. D 81, 103532 (2010).;
  * [22] C. Vafa, arXiv:hep-th/0509212 (2005).; Kenji Kadota, Chang Sub Shin, Takahiro Terada and Gansukh Tumurtushaa, JCAP01 008 (2020). ;Oikonomou, V. K. arXiv:2012.01312 (2020).
  * [23] H. Ooguri, C. Vafa, Nucl. Phys. B 766, 21–33 (2007).; Oem Trivedi, arXiv:2101.00638 (2021).; Suratna Das, physics of the dark univers 27, 100432 (2020).
  * [24] N. Arkani-Hamed, L. Motl, A. Nicolis, C. Vafa, JHEP 06, 060 (2007).; Abolhassan Mohammadi, Tayeb Golanbari, Jamil Enayati, arXiv:2012.01512, (2020).
  * [25] J. Sadeghi, S.N. Gashti Pramana 95 (198) (2021).; J. Sadeghi, B. Pourhassan, S.N. Gashti, S. Upadhyay Physica Scripta 96 (12), 125317 (2021).; J. Sadeghi, B. Pourhassan, S.N. Gashti, S. Upadhyay arXiv preprint arXiv:2201.04071 (2022).
  * [26] M. Shokri, J. Sadeghi, R. Herrera, S.N. Gashti arXiv preprint arXiv:2112.12309 (2021).
  * [27] S.N. Gashti, Journal of Holography Applications in Physics 2 (1), 13-24 (2022)
  * [28] M. Orellana, F. Garcia, F. Teppa Pannia, G. Romero, Gen. Relativ. Gravit. 45, 771–783 (2013).; Osses, Constanza and Videla, Nelson and Panotopoulos, Grigoris, arXiv:2101.08882, (2021).;Suddhasattwa Brahma, Phys.Rev.D 101 2, 023526 (2020).; Robert Brandenberger, arXiv:2102.09641 (2021).
  * [29] S. Nojiri, S.D. Odintsov, Phys. Rep. 505, 59 (2011).; William H. Kinney, Phys. Rev. Lett.122, 8 081302 (2019).;William H. Kinney, arXiv:2103.16583 (2021).; Yu, Ten-Yeh and Wen, Wen-Yu, Phys. Lett. B781, 713–718 (2018).; Yang Liu, Eur. Phys. J. C 81, 1122 (2021).
  * [30] S. Capozziello, M. De Laurentis, S.D. Odintsov, A. Stabile, Phys. Rev. D 83, 064004 (2011).
  * [31] S. Capozziello, M. Faizal, M. Hameeda, B. Pourhassan, V. Salzano, S. Upadhyay, Mon. Not. R. Astron. Soc. 474, 2430–2443 (2018).
  * [32] A. Arapoglu, C. Deliduman, K.Y. Eksi, JCAP 1107, 020 (2011).
  * [33] S. Capozziello, R. D’Agostino, O. Luongo, Int. J. Mod. Phys. D 28, 1930016 (2019).
  * [34] S. Capozziello, R. D’Agostino, O. Luongo, JCAP 1805, 008 (2018).
  * [35] S. Capozziello, R. D’Agostino, O. Luongo, Gen. Relativ. Gravit. 51, 2 (2019).
  * [36] P. Channuie, Eur. Phys. J. C 79, 508 (2019).
  * [37] J. Sadeghi, E.N. Mezerji, S.N. Gashti, Modern Physics Letters A 36 (05), 2150027 (2021).
  * [38] R. Myrzakulov, L. Sebastian, S. Vagnozzi, Eur. Phys. J. C 75, 444 (2015).
  * [39] J. Gu, and W. Hwang, Phys. Lett. B, 517 1 (2001).
  * [40] S. Ghaffari, arXiv:2112.05813 (2021).
  * [41] S. Ghaffari, et al., J. C 78, 706 (2018).


# Variants on Digital Covering Maps

Laurence Boxer  Department of Computer and Information Sciences, Niagara
University, Niagara University, NY 14109, USA; and Department of Computer
Science and Engineering, State University of New York at Buffalo. email:
<EMAIL_ADDRESS>
###### Abstract

S-E Han’s paper [11] discusses several variants of digital covering maps. We
show several equivalences among these variants and discuss shortcomings in
Han’s paper.

Key words and phrases: digital topology, digital image, covering map

MSC: 54B20, 54C35

##  1 Introduction

The notion of a covering map has been adapted from classical algebraic
topology to digital topology, where it is an important tool for computing
digital versions of fundamental groups for binary digital images. With varying
success, attempts have been made to modify the notion of a digital covering
map to obtain related results under less restrictive conditions. Among these
attempts are Han’s paper [11], which contains a proof that is murky (see
section 4 for clarification) and citations that are inappropriate. We also
discuss a strangely presented example in Han’s related paper [9] (see Remark
6.6). Further, it turns out that some of Han’s variants on covering maps don’t
really vary from covering maps (see Theorem 6.5). Also, some of material of
[11] is superseded by other papers including [3, 12, 13]. We justify these
claims in the current paper.

##  2 Preliminaries

We use ${\mathbb{N}}$ for the set of natural numbers, ${\mathbb{Z}}$ for the
set of integers, and $\\#X$ for the number of distinct members of $X$.

We typically denote a (binary) digital image as $(X,\kappa)$, where
$X\subset{\mathbb{Z}}^{n}$ for some $n\in{\mathbb{N}}$ and $\kappa$ represents
an adjacency relation of pairs of points in $X$. Thus, $(X,\kappa)$ is a
graph, in which members of $X$ may be thought of as black points, and members
of ${\mathbb{Z}}^{n}\setminus X$ as white points, of a picture of some “real
world” object or scene.

###  2.1 Adjacencies

Let $u,n\in{\mathbb{N}}$, $1\leq u\leq n$. Han’s papers use “$k$-adjacency”
sometimes to mean an arbitrary adjacency, sometimes as an abbreviation for
what he calls “$k(u,n)$-adjacency,” where the digital image $(X,k)$ satisfies
$X\subset{\mathbb{Z}}^{n}$ and
$x=(x_{1},\ldots,x_{n}),~{}y=(y_{1},\ldots,y_{n})\in X$ are $k(u,n)$-adjacent
if and only if

  * •

$x\neq y$, and

  * •

for at most $u$ indices $i$, $\mid x_{i}-y_{i}\mid=1$, and

  * •

for all indices $j$ such that $\mid x_{j}-y_{j}\mid\neq 1$, we have
$x_{j}=y_{j}$.

Other authors refer to this adjacency as $c_{u}$-adjacency. We will prefer the
latter notation in the current paper. The $c_{u}$ adjacencies are the
adjacencies most used in digital topology, especially $c_{1}$ and $c_{n}$.

In low dimensions, it is also common to denote a $c_{u}$ adjacency by the
number of points that can have this adjacency with a given point in
${\mathbb{Z}}^{n}$. E.g.,

  * •

For subsets of ${\mathbb{Z}}^{1}$, $c_{1}$-adjacency is 2-adjacency.

  * •

For subsets of ${\mathbb{Z}}^{2}$, $c_{1}$-adjacency is 4-adjacency and
$c_{2}$-adjacency is 8-adjacency.

  * •

For subsets of ${\mathbb{Z}}^{3}$, $c_{1}$-adjacency is 6-adjacency,
$c_{2}$-adjacency is 18-adjacency, and $c_{3}$-adjacency is 26-adjacency.

We use the notations $y\leftrightarrow_{\kappa}x$, or, when the adjacency
$\kappa$ can be assumed, $y\leftrightarrow x$, to mean $x$ and $y$ are
$\kappa$-adjacent. The notations $y\leftrightarroweq_{\kappa}x$, or, when
$\kappa$ can be assumed, $y\leftrightarroweq x$, mean either $y=x$ or
$y\leftrightarrow_{\kappa}x$. For $x\in X$, let

$N(X,x,\kappa)=\\{\,y\in X\mid x\leftrightarrow_{\kappa}y\,\\}.$

When the image $(X,\kappa)$ under discussion is clear, we will use the
notations $N(x)$ or $N_{\kappa}(x)$ as follows.

$N(x)=\\{\,y\in X\mid
y\leftrightarroweq_{\kappa}x\,\\}=N(X,x,\kappa)\cup\\{x\\}.$

A sequence $P=\\{y_{i}\\}_{i=0}^{m}$ in a digital image $(X,\kappa)$ is a
$\kappa$-path from $a\in X$ to $b\in X$ if $a=y_{0}$, $b=y_{m}$, and
$y_{i}\leftrightarroweq_{\kappa}y_{i+1}$ for $0\leq i<m$.

$X$ is $\kappa$-connected [14], or connected when $\kappa$ is understood, if
for every pair of points $a,b\in X$ there exists a $\kappa$-path in $X$ from
$a$ to $b$.

A (digital) $\kappa$-closed curve is a path $S=\\{s_{i}\\}_{i=0}^{m}$ such
that $s_{0}=s_{m}$, and $0<|i-j|<m$ implies $s_{i}\neq s_{j}$. If, also,
$0\leq i<m$ implies

$N(S,x_{i},\kappa)=\\{x_{(i-1)\mod n},~{}x_{(i+1)\mod m}\\}$

then $S$ is a (digital) $\kappa$-simple closed curve.

###  2.2 Digitally continuous functions

Digital continuity is defined to preserve connectedness, as at Definition 2.1
below. By using adjacency as our standard of “closeness,” we get Theorem 2.2
below.

######  Definition 2.1.

[2] (generalizing a definition of [14]) Let $(X,\kappa)$ and $(Y,\lambda)$ be
digital images. A function $f:X\rightarrow Y$ is $(\kappa,\lambda)$-continuous
if for every $\kappa$-connected $A\subset X$ we have that $f(A)$ is a
$\lambda$-connected subset of $Y$.

If either of $X$ or $Y$ is a subset of the other, we use the abbreviation
$\kappa$-continuous for $(\kappa,\kappa)$-continuous.

When the adjacency relations are understood, we will simply say that $f$ is
_continuous_. Continuity can be expressed in terms of adjacency of points:

######  Theorem 2.2.

[14, 2] A function $f:X\to Y$ is continuous if and only if $x\leftrightarrow
x^{\prime}$ in $X$ implies $f(x)\leftrightarroweq f(x^{\prime})$.

Han’s papers generally use the equivalent formulation that $f$ is continuous
if and only for every $x\in X$, $f(N_{\kappa}(x))\subset N_{\lambda}(f(x))$.

See also [5, 6], where similar notions are referred to as immersions,
gradually varied operators, and gradually varied mappings.

A digital isomorphism (called homeomorphism in [1]) is a
$(\kappa,\lambda)$-continuous surjection $f:X\to Y$ such that $f^{-1}:Y\to X$
is $(\lambda,\kappa)$-continuous.

The literature uses path polymorphically: a $(c_{1},\kappa)$-continuous
function $f:[0,m]_{{\mathbb{Z}}}\to X$ is a $\kappa$-path if
$f([0,m]_{{\mathbb{Z}}})$ is a $\kappa$-path as described above from $f(0)$ to
$f(m)$.

##  3 Han’s variants on local isomorphisms

The definition [8] of a digital covering map was simplified to the following.

######  Definition 3.1.

[4] Let $p:(E,\kappa)\to(B,\lambda)$ be a continuous surjection of digital
images. The map $p$ is a $(\kappa,\lambda)$ covering map if and only if

  * •

for every $b\in B$, there is an index set $M$ such that

$p^{-1}(N_{\lambda}(b))=\bigcup_{i\in M}N_{\kappa}(e_{i}),\mbox{ where
}e_{i}\in p^{-1}(b);$

  * •

if $i,j\in M$, $i\neq j$, then $N_{\kappa}(e_{i})\cap
N_{\kappa}(e_{j})=\emptyset$; and

  * •

$p|_{N_{\kappa}(e_{i})}:N_{\kappa}(e_{i})\to N_{\lambda}(b)$ is a
$(\kappa,\lambda)$-isomorphism.

We find the following definition in Han’s paper [10] (not in [7] despite the
claims to the contrary in [10, 11]).

######  Definition 3.2.

A digitally continuous map $h:(X,\kappa)\to(Y,\lambda)$ is a pseudo-local (PL)
isomorphism if for every $x\in X$, $h(N_{\kappa}(x))\subset Y$ is
$\lambda$-isomorphic to $N_{\lambda}(h(x))\subset Y$.

In his paper [7], Han gives the following.

######  Definition 3.3.

A digitally continuous map $h:(X,\kappa)\to(Y,\lambda)$ is a local
homeomorphism [in more recent terminology, a local isomorphism] if for all
$x\in X$, $h|_{N_{\kappa}(x)}$ is a $(\kappa,\lambda)$-homeomorphism
[$(\kappa,\lambda)$-isomorphism] onto $N_{\lambda}(h(x))$.

We have the following.

######  Proposition 3.4.

Let $h:(X,\kappa)\to(Y,\lambda)$ be a digitally continuous map. If $h$ is a
local isomorphism then $h$ is a PL isomorphism.

###### Proof.

Elementary and left to the reader. ∎

######  Theorem 3.5.

([13], correcting an error of [7]) Let $f:(X,\kappa)\to(Y,\lambda)$ be a
continuous surjection. Then $f$ is a digital covering map if and only if $f$
is a local isomorphism.

We will also discuss the following notion.

######  Definition 3.6.

[9] A function $h:(X,\kappa)\to(Y,\lambda)$ is a weakly local (WL) isomorphism
if for all $x\in X$, $h|_{N(x,1)}$ is an isomorphism onto $h(N(x,1))$.

##  4 Theorem 3.15(3) of [11]

Part (3) of Theorem 3.15 of [11] states that

> Neither of a PL-$(k_{0},k_{1})$-isomorphism and a
> WL-$(k_{0},k_{1})$-isomorphism implies the other.

The assertion is correct, but Han’s argument for the existence of $(X,k_{0})$,
$(Y,k_{1})$, and a WL-$(k_{0},k_{1})$-isomorphism $h:X\to Y$ that is not a
PL-$(k_{0},k_{1})$-isomorphism, is not as clear as it could be. In the
following, we clarify Han’s argument.

In his example, Han makes use of an unstated assumption, namely that
$(Y,k_{1})$ is connected. He also assumes $k_{0}=k_{1}=\kappa$, that

$X\subset Y\mbox{ but }X\neq Y$ (1)

and that $h$ is the inclusion map, trivially a
WL-$(\kappa,\kappa)$-isomorphism.

Note that since $(Y,\kappa)$ is connected, (1) implies there is a
$\kappa$-path $\\{y,y^{\prime}\\}\subset Y$ such that $y\in X$, $y^{\prime}\in
Y\setminus X$. Therefore,

$h(N(X,y,\kappa))=N(X,y,\kappa)\mbox{ is a proper subset of
}N(X,y,\kappa)\cup\\{y^{\prime}\\}\subset N(Y,h(y),\kappa).$

Hence $h$ is not a PL-$(\kappa,\kappa)$-isomorphism.

##  5 Theorem 3.20 of [11]

Let $(X,\kappa)$ and $(Y,\lambda)$ be digital simple closed curves of
$\ell_{1}$ and $\ell_{2}$ points, respectively. Theorem 3.20 of [11] states
that $(X,\kappa)$ embeds into $(Y,\lambda)$ if and only if
$\ell_{1}=\ell_{2}$. Since a connected nonempty subset of $(Y,\lambda)$ is
either $(Y,\lambda)$ itself or is isomorphic to a digital interval - which is
not even of the same digital homotopy type as $(X,\kappa)$ \- Han’s assertion
is an easy consequence of the much older Theorem 5.1 of [3], which states that
$(X,\kappa)$ and $(Y,\lambda)$ have the same digital homotopy type if and only
if $\ell_{1}=\ell_{2}$.

##  6 Han’s pseudo-covering maps in [11]

Han defines a digital pseudo-covering as follows.

######  Definition 6.1.

[9] Let $p:(E,\kappa)\to(B,\lambda)$ be a surjection such that for every $b\in
B$,

  1. 1.

there is an index set $M$ such that $p^{-1}(N_{\kappa}(b,1))=\bigcup_{i\in
M}N_{\kappa}(e_{i},1)$, where $e_{i}\in p^{-1}(b)$;

  2. 2.

if $i,j\in M$ and $i\neq j$, then $N_{\kappa}(e_{i},1)\cap
N_{\kappa}(e_{j},1)=\emptyset$; and

  3. 3.

$p|_{N_{\kappa}(e_{i},1)}:N_{\kappa}(e_{i},1)\to N_{\lambda}(b,1)$ is a WL-
isomorphism for all $i\in M$.

Then $p$ is a pseudo-covering map.

However, A. Pakdaman shows in [12] that Han’s definition does not effectively
give us a new object of study. In particular, Pakdaman shows the following.

######  Theorem 6.2.

A digital pseudo-covering map as defined in Definition 6.1 is in fact a
digital covering map.

######  Definition 6.3.

[8] Let $p:(E,\kappa)\to(B,\lambda)$ be $(\kappa,\lambda)$-continuous. Let
$f:[0,m]_{{\mathbb{Z}}}\to B$ be $(c_{1},\lambda)$-continuous. A
$(c_{1},\kappa)$-continuous function $\tilde{f}:[0,m]_{{\mathbb{Z}}}\to E$
such that $p\circ\tilde{f}=f$ is a (digital) path lifting of $f$. If for every
$b_{0}\in B$, every $e_{0}\in p^{-1}(b_{0})$, and every path $f$ such that
$f(0)=b_{0}$,

there is a unique lifting $\tilde{f}$ such that $\tilde{f}(0)=e_{0}$,

then $p$ has the unique path lifting property.

######  Theorem 6.4.

[8] A digital covering map has the unique path lifting property.

Next, we show that several of the variants of covering maps that we have
discussed are equivalent.

######  Theorem 6.5.

Let $p:(X,\kappa)\to(Y,\lambda)$ be a continuous surjection. Then the
following are equivalent.

  1. 1.

$p$ is a digital covering map.

  2. 2.

$p$ is a local isomorphism.

  3. 3.

$p$ is a pseudo-covering in the sense of Definition 6.1.

  4. 4.

$p$ is a WL-isomorphism with the unique path lifting property.

###### Proof.

That 1) and 2) are equivalent is stated in Theorem 3.5.

That 1) implies 3) follows from Definitions 3.1 and 6.1.

That 3) implies 1) is stated in Theorem 6.2.

It follows from Theorem 6.4 that 1) implies 4).

To show 4) implies 2): suppose $p$ is a WL-isomorphism with the unique path
lifting property. Let $x\in X$, $p(x)=y\in Y$, $y^{\prime}\in
N_{\lambda}(y)\setminus\\{y\\}$. Then $\\{y,y^{\prime}\\}$ is a path in
$(Y,\lambda)$, hence lifts to a unique path $\\{x,x^{\prime}\\}$ in
$(X,\kappa)$ with $p(x)=y$, $p(x^{\prime})=y^{\prime}$. Thus
$N_{\lambda}(y)\subset p(N_{\kappa}(x))$. Since continuity implies
$p(N_{\kappa}(x))\subset N_{\lambda}(y)$, we have
$p(N_{\kappa}(x))=N_{\lambda}(y)$. Since $p$ is a WL-isomorphism, we have that
$p$ is a local isomorphism. ∎

######  Remark 6.6.

Han’s Example 4.3(4) of [9] considers (please note here “$c_{1}$” is the label
of a point, so we will avoid using this notation for 4-adjacency)
$C=\\{\,c_{i}\,\\}_{i=0}^{3}$, $D=\\{\,d_{i}\,\\}_{i=0}^{3}$, where

$(0,0)=c_{0}=d_{0},~{}~{}~{}(1,0)=c_{1}=d_{1},~{}~{}~{}(1,1)=c_{2}=d_{3},~{}~{}~{}(0,1)=c_{3}=d_{2},$

See Figure 1. Han’s claim, that $p$ is not a pseudo-4-covering (a pseudo-
covering when $A$ and $B$ both use 4-adjacency), is correct, but this example
should not have been considered since $p$ is not 4-continuous:

$c_{0}\leftrightarrow_{4}c_{3}~{}~{}~{}\mbox{ but
}~{}~{}~{}p(c_{0})=d_{0}\not\leftrightarrow_{4}d_{3}=p(c_{3}).$

Since $p$ is (4,8)-continuous, perhaps Han intended to show that $p$ is not a
(4,8)-pseudocovering as defined at Definition 6.1. This can be done by
observing that

$\\#N_{4}(c_{0})=3\neq 4=\\#N_{8}(d_{0})=\\#N_{8}(p(c_{0})).$

Therefore, $p$ is not a (4,8)-local isomorphism, so by Theorem 6.5 is not a
(4,8)-pseudocovering as defined at Definition 6.1.

In the first paragraph of page 5104 of [11], Han attributes the definition of
a digital pointed continuous function to his paper [8]. The definition should
be attributed to the earlier paper [2].

Figure 1: The function $p(c_{i})=d_{i}$ of Han s Example 4.3(4) of [9];
discussed here in Remark 6.6

Pakdaman modifies Han’s Definition 6.1 as follows.

######  Definition 6.7.

[12] Let $p:(E,\kappa)\to(B,\lambda)$ be a surjection of digital images.
Suppose for all $b\in B$ we have the following.

  1. 1.

for some index set $M$, $\bigcup_{i\in M}N_{\kappa}(e_{i},1)\subset
p^{-1}(N_{\lambda}(b,1))$ where $e_{i}\in p^{-1}(b)$;

  2. 2.

if $i,j\in M$ and $i\neq j$ then $N_{\kappa}(e_{i},1)\cap
N_{\kappa}(e_{i},1)=\emptyset$; and

  3. 3.

for all $i\in M$, $p|{N_{\kappa}(e_{i},1)}:N_{\kappa}(e_{i},1)\to
p(N_{\kappa}(e_{i},1))$ is a $(\kappa,\lambda)$-isomorphism.

Then $p$ is a $(\kappa,\lambda)$-pseudocovering map.

Pakdaman proceeds to compare unique path lifting results for pseudocovering
maps based on Definition 6.7 with those asserted by Han in [11] based on
Definition 6.1. He showed that Definition 6.7 gives something not equivalent
to a covering map, since such a pseudocovering need not have the unique path
lifting property.

##  7 Further remarks

We have discussed various flaws in Han’s paper [11]. We have shown that
several variants of digital covering maps that were presented in [11] are in
fact equivalent.

Corrections and suggestions of an anonymous referee are acknowledged with
gratitude.

## References

  * [1] L. Boxer, Digitally continuous functions, Pattern Recognition Letters 15 (1994), 833-839.
  * [2] L. Boxer, A classical construction for the digital fundamental group, Journal of Mathematical Imaging and Vision 10 (1999), 51-62
  * [3] L. Boxer, Properties of digital homotopy, Journal of Mathematical Imaging and Vision 22 (2005), 19-26.
  * [4] L. Boxer, Digital products, wedges, and covering spaces, Journal of Mathematical Imaging and Vision 25 (2006), 159-171
  * [5] L. Chen, Gradually varied surfaces and its optimal uniform approximation, SPIE Proceedings 2182 (1994), 300-307.
  * [6] L. Chen, Discrete Surfaces and Manifolds, Scientific Practical Computing, Rockville, MD, 2004
  * [7] S.-E. Han, Digital $(k_{0},k_{1})$-covering map and its properties, Honam Math. J. 26 (2004) 107-117
  * [8] S-E Han, Non-product property of the digital fundamental group, Information Sciences 171 (2005), 73-91
  * [9] S-E Han, Unique pseudolifting property in digital topology, Filomat 26 (4) (2012), 739-746
  * [10] S-E Han, The most refined axiom for a digital covering space and its utilities. Mathematics 8 (11) (2020),
https://www.mdpi.com/2227-7390/8/11/1868

  * [11] S-E Han, An equivalent condition for a pseudo $(k_{0},k_{1})$-Covering Space, Filomat 36 (15) (2022), 5093-5105
  * [12] A. Pakdaman, Is there any digital pseudocovering map?, Caspian Journal of Mathematical Sciences 11 (1) (2022), 210-216
  * [13] A. Pakdaman and M. Zakki, Equivalent conditions for digital covering maps, Filomat 34 (12) (2020), 4005-4014
  * [14] A. Rosenfeld, ‘Continuous’ functions on digital images, Pattern Recognition Letters 4 (1987), 177-184


# Analyzing Multimodal Objectives Through the Lens of Generative Diffusion
Guidance

Chaerin Kong, Nojun Kwak
Seoul National University
<EMAIL_ADDRESS>
###### Abstract

Recent years have witnessed astonishing advances in the field of multimodal
representation learning, with contrastive learning being the cornerstone for
major breakthroughs. Latest works delivered further improvements by
incorporating different objectives such as masked modeling and captioning into
the frameworks, but our understanding on how these objectives facilitate
learning remains vastly incomplete. In this paper, we leverage the fact that
classifier-guided diffusion models generate images that reflect the semantic
signals provided by the classifier to study the characteristics of multimodal
learning objectives. Specifically, we compare contrastive, matching and
captioning loss in terms of their semantic signals, and introduce a simple
baseline that not only supports our analyses but also improves the quality of
generative guidance in a straightforward manner.

##  1 Introduction

Vision-Language Pretraining (VLP) has attracted great attention from the
community for its wide and robust applications in different downstream tasks.
The seminal work of CLIP (Radford et al., 2021) employs the image-text
contrastive objective to successfully embed images and text descriptions in a
common feature space, inspiring numerous subsequent works that explore
different objectives (Li et al., 2022a; Yu et al., 2022; Yang et al., 2022;
Jang et al., 2023) and architectures (Li et al., 2021; Jang et al., 2022; Wang
et al., 2022). Recently, cross-modal generative models (Ramesh et al., 2021;
Saharia et al., 2022; Rombach et al., 2022; Kong et al., 2022) are also
gaining wide popularity thanks to the powerful capacity of diffusion models
(Sohl-Dickstein et al., 2015; Ho et al., 2020) and readily available guidance
of vision-language foundation models. These models aim to synthesize or edit
images so that the outputs are both realistic and condition-aligned.

Conditional diffusion models embody the conditioning information in two ways:
classifier guidance (Dhariwal & Nichol, 2021) and classifier-free guidance (Ho
& Salimans, 2022). While classifier-guided models (Nichol et al., 2021;
Dhariwal & Nichol, 2021; Kong et al., 2023) typically leverage multimodal
embeddings of a joint embedding network (e.g., CLIP) using cosine
similarities, it is unclear whether this approach is optimal. For example,
recent works (Zhong et al., 2022; Li et al., 2022b) have pointed out that
global representations of CLIP learned by the contrastive objective are not
suitable for handling fine-grained correspondences between image and text as
they represent the image and the text as a whole. This indicates that features
learned from sequence level contrastive learning may have certain blind spots
that can be made visually apparent when directly applied to the diffusion
process as the guidance signal.

In this paper, we aim to obtain a better understanding of different multimodal
learning objectives (e.g., image-text contrastive, image-text matching, image
captioning) by analyzing their semantic signals as generative guidance. That
is, we utilize the classifier-guided diffusion process to study the
characteristics of different objectives by carefully inspecting the samples
they produce, and further present a straightforward modification to the
previous method that both supports our findings and improves the generation
quality. We note that the aim of this paper is not to present a high-
performing generative model. Rather, we are simply leveraging the diffusion
process to visually analyze different objectives and hypothesize about the
properties of the accordingly learned representations.

##  2 Multimodal Objective as Generative Guidance

###  2.1 Classifier-guided Diffusion

Dhariwal & Nichol (2021) has introduced classifier guidance as a means to
steer the generative diffusion process towards the conditioning class. This
requires a noise-aware classifier whose gradient can be used to guide the
diffusion process. Formally, denoting the predicted parameters of timestep $t$
as $\mu_{\theta}(x_{t})$, $\Sigma_{\theta}(x_{t})$, the next step diffusion
sampling becomes

$x_{t-1}\sim\mathcal{N}(\mu_{\theta}(x_{t})+s\Sigma_{\theta}(x_{t})\nabla_{x_{t}}\log
p_{\phi}(y|x_{t}),\Sigma_{\theta}(x_{t})),$ (1)

where $s$ is the step size, $y$ indicates the class label and $p_{\phi}$
refers to the classifier. This formulation was altered by Nichol et al. (2021)
to suit text-to-image generation as follows:

$x_{t-1}\sim\mathcal{N}(\mu_{\theta}(x_{t})+s\Sigma_{\theta}(x_{t})\nabla_{x_{t}}\langle
f(x_{t}),g(c)\rangle,\Sigma_{\theta}(x_{t})),$ (2)

where $\langle\cdot,\cdot\rangle$ indicates the inner product and $f$, $g$,
$c$ are the image encoder, text encoder, and the text condition, respectively.

Overall, the classifier guidance encourages the model to generate samples that
are well-aligned with the condition according to a predefined metric.
Obviously, the choice of this metric affects the final output sample,
revealing how each metric (objective) connects the two modalities (image and
text, in our case) in multiple levels.

###  2.2 Pretrained Models

As our goal is to study the semantic signals encoded in different objectives,
we employ a pretrained diffusion backbone and a pretrained vision-language
guidance model for our analysis.

Generative Model For the image generator, we use a 256$\times$256
unconditional diffusion model pretrained on
Imagenet111https://github.com/openai/guided-diffusion (Russakovsky et al.,
2015). We note that this is not a state-of-the-art text-to-image diffusion
model. The unconditional nature of this model renders it well-suited for our
purpose, as it solely relies on the classifier signal for condition-aware
synthesis, making it possible to analyze the encoded semantic information in a
disentangled manner. Employing an excessively powerful generator can similarly
obfuscate our analysis, as its generative capacity can compensate for the
weakness in the guidance signal and mask its blind spots.

Guidance Model Among many available candidates, we choose BLIP (Li et al.,
2022a) as our main guidance model222https://github.com/salesforce/BLIP. This
model is trained on 129M image-text pairs simultaneously optimizing for three
objectives: image-text contrastive (ITC), image-text matching (ITM) and image
captioning (CAP). The fact that a single model can evaluate these three scores
makes it an excellent guidance model for our analysis, as we can safely
minimize the compounding effects coming from using different models trained
with different dataset, architecture and optimization scheme. We adopt the
idea from Avrahami et al. (2022) to first predict the denoised version for
guidance signal computation.

###  2.3 Objectives and Benchmarks

We analyze three commonly used multimodal objectives: ITC, ITM and CAP. We
simply replace the classifier logit in eq. 1 with the corresponding loss terms
of BLIP (with the sign reversed). For details about the loss computation,
please refer to Li et al. (2022a). For systematic evaluation, we mainly follow
the benchmark of Saharia et al. (2022), namely DrawBench. We further add
prompts from COCO (Lin et al., 2014) captions. Qualitative evaluations as well
as quantitative comparisons based on user study are performed to draw
insights.

##  3 Analyses

In this section, we first present our empirical findings and propose a
straightforward modification called SHIFT that reflects the insights by
employing the coarse-to-fine guidance of CAP and ITC. We acknowledge that
using a more powerful model would relieve some of the issues demonstrated in
this section as the data scale can compensate for the blind spots, but we
believe the findings we present here nevertheless hold true and will come into
play in an increasingly challenging problem setting.

###  3.1 Findings

Figure 1: Text-to-image samples using each objective as the classifier-
guidance. We only present preliminary examples for illustrative purpose.
Please refer to the Appendix for more cases.

1\. While ITC focuses on the fine details of the salient object, CAP tends to
reason about the global scene composition.

Looking at Figure 7 row 1, we can clearly see that contrastive loss is more
effective in forming fine details of the main object while it often leaves out
less important objects or attributes in the given prompt. We hypothesize the
former is due to the core dynamic of contrastive learning that aims to learn
relative distances by distinguishing objects. At the same time, as the
objective only compares fully abstracted representations (i.e., [cls] tokens)
that dominate each entity as a whole, it fails to densely parse the scene. On
the other hand, captioning objective forces to understand the scene structure
in a deeper level, making captioning-guided samples more faithful to complex
texts.

2\. ITC commonly fuses visual semantics together to forcefully form a global
semantic.

2

in Figure 7 illustrates a more extreme case where ITC not only omits semantic
components but arbitrarily mixes them. From these examples, we can diagnose
that the contrastive objective does reflect semantic attributes (e.g., blue)
but fails to relate them to the correct object (e.g., dog). This can be
another side effect of simplified distance learning. Captioning, in contrast,
requires the model to reason about both objects and their relations, deepening
the scene-level understanding.

3\. Patch-token cross-attention plays a key role in fine-grained visual
understanding.

We now widen the scope of our analysis by further looking at ITM. In contrary
to ITC that simply compares two [cls] tokens, ITM involves lower-level cross-
attention between image patch tokens and text tokens (as in CAP) to output a
matching score between 0 and 1. We discovered that this operation plays an
important role in fine-grained visual understanding and representation
robustness. To our surprise, ITM, more or less an auxiliary loss to polish
multimodal representations, encodes strong semantic signals that involves
dense scene understanding. Looking at 1, 2, 3, cross-attention-based
objectives (ITM, CAP and SHIFT) demonstrate capacity for fine-grained visual
reasoning, and ITM signal successfully materializes objects with corresponding
attributes and relations, though at a lower visual quality.

4\. Dense supervision makes the representations more robust to noise
perturbations.

Last two rows of Figure 7 show the impact of noise in the text prompt. 4
depicts that as opposed to ITC that generates random objects under mild typo,
other losses render relatively consistent outputs. 5 delivers a similar
insight, where the phrase ‘camel in the dessert’ not ‘desert’ is likely
mistaken by the text provider. These cases are very probable in the typical
setting where massive noisy image-text data are crawled from the web, and we
observe that dense supervision that involves low-level patch-token cross-
attention shows better robustness against textual perturbations, as perturbed
text inputs attend to not only themselves but also visual information to form
a more robust representation.

5\. CAP is a more indirect if not challenging form of supervision than ITC or
ITM.

Figure 2: Generated samples for each objective and the number of diffusion
sampling steps.

Lastly, we inspect the optimization complexity of each objective by differing
diffusion sampling steps. As each diffusion step corresponds to an update
using the loss gradient, we regard an objective that generates reasonable
sample with fewer steps to have lower optimization complexity. Referring to
Figure 2, we see that ITC and ITM clearly take less steps to output realistic
samples compared to caption-based losses. This observation coincides with
Radford et al. (2021) and Yu et al. (2022), where the former explicitly chose
contrastive loss for training efficiency and the latter has been reported to
take much more resources to converge due to captioning. We conclude that as
captioning demands a more semantic visual understanding, learning becomes
trickier compared to the simple distance learning.

###  3.2 New Baseline: Guidance Shift

Based on the above findings, we propose a simple yet effective baseline that
takes advantage from both ends, i.e., contrastive learning and captioning. To
leverage the strengths from both, we introduce guidance shift, where we start
with captioning loss and gradually shift to contrastive loss for the
generative guidance. Formally, our SHIFT loss can be written as:

$\mathcal{L}_{SHIFT}=t\mathcal{L}_{ITC}+(1-t)\mathcal{L}_{CAP},$ (3)

where $t$ is the normalized time step, progressing from 0 to 1. The idea is to
first outline the overall structure with CAP and then refine the details with
ITC. To study its effectiveness, we conduct quantitative user study as well as
qualitative evaluations presented in the Appendix.

Figure 3: Human evaluation for photo-realism and condition-alignment.

Fig.3 delivers the result. Compared to simple ITC baseline, SHIFT outperforms
in both fidelity and alignment. Although CAP-only shows better condition
alignment, SHIFT clearly outputs better quality samples, which is apparent
from qualitative results as well. BLEND, a naive baseline that simply mixes
CAP and ITC without gradual transition, performs significantly worse as these
two signals can often be conflicting and difficult to optimize simultaneously.

##  4 Conclusion

In this paper, we have studied the semantic information encoded in different
multimodal objectives by visually analyzing their properties as generative
diffusion guidance. We hope it provides useful insights for ensuing works and
sparks further advances in the field.

## References

  * Avrahami et al. (2022) Omri Avrahami, Dani Lischinski, and Ohad Fried.  Blended diffusion for text-driven editing of natural images.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pp. 18208–18218, 2022.
  * Dhariwal & Nichol (2021) Prafulla Dhariwal and Alexander Nichol.  Diffusion models beat gans on image synthesis.  _Advances in Neural Information Processing Systems_ , 34:8780–8794, 2021.
  * Ho & Salimans (2022) Jonathan Ho and Tim Salimans.  Classifier-free diffusion guidance.  _arXiv preprint arXiv:2207.12598_ , 2022.
  * Ho et al. (2020) Jonathan Ho, Ajay Jain, and Pieter Abbeel.  Denoising diffusion probabilistic models.  _Advances in Neural Information Processing Systems_ , 33:6840–6851, 2020.
  * Jang et al. (2022) Jiho Jang, Chaerin Kong, Donghyeon Jeon, Seonhoon Kim, and Nojun Kwak.  Unifying vision-language representation space with single-tower transformer.  _arXiv preprint arXiv:2211.11153_ , 2022.
  * Jang et al. (2023) Jiho Jang, Seonhoon Kim, Kiyoon Yoo, Chaerin Kong, Jangho Kim, and Nojun Kwak.  Self-distilled self-supervised representation learning.  In _Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision_ , pp. 2829–2839, 2023.
  * Kong et al. (2022) Chaerin Kong, Jeesoo Kim, Donghoon Han, and Nojun Kwak.  Few-shot image generation with mixup-based distance learning.  In _Computer Vision–ECCV 2022: 17th European Conference, Tel Aviv, Israel, October 23–27, 2022, Proceedings, Part XV_ , pp. 563–580. Springer, 2022.
  * Kong et al. (2023) Chaerin Kong, DongHyeon Jeon, Ohjoon Kwon, and Nojun Kwak.  Leveraging off-the-shelf diffusion model for multi-attribute fashion image manipulation.  In _Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision_ , pp. 848–857, 2023.
  * Li et al. (2021) Junnan Li, Ramprasaath Selvaraju, Akhilesh Gotmare, Shafiq Joty, Caiming Xiong, and Steven Chu Hong Hoi.  Align before fuse: Vision and language representation learning with momentum distillation.  _Advances in neural information processing systems_ , 34:9694–9705, 2021.
  * Li et al. (2022a) Junnan Li, Dongxu Li, Caiming Xiong, and Steven Hoi.  Blip: Bootstrapping language-image pre-training for unified vision-language understanding and generation.  _arXiv preprint arXiv:2201.12086_ , 2022a.
  * Li et al. (2022b) Liunian Harold Li, Pengchuan Zhang, Haotian Zhang, Jianwei Yang, Chunyuan Li, Yiwu Zhong, Lijuan Wang, Lu Yuan, Lei Zhang, Jenq-Neng Hwang, et al.  Grounded language-image pre-training.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pp. 10965–10975, 2022b.
  * Lin et al. (2014) Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollár, and C Lawrence Zitnick.  Microsoft coco: Common objects in context.  In _Computer Vision–ECCV 2014: 13th European Conference, Zurich, Switzerland, September 6-12, 2014, Proceedings, Part V 13_ , pp. 740–755. Springer, 2014.
  * Nichol et al. (2021) Alex Nichol, Prafulla Dhariwal, Aditya Ramesh, Pranav Shyam, Pamela Mishkin, Bob McGrew, Ilya Sutskever, and Mark Chen.  Glide: Towards photorealistic image generation and editing with text-guided diffusion models.  _arXiv preprint arXiv:2112.10741_ , 2021.
  * Radford et al. (2021) Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al.  Learning transferable visual models from natural language supervision.  In _International Conference on Machine Learning_ , pp. 8748–8763. PMLR, 2021.
  * Ramesh et al. (2021) Aditya Ramesh, Mikhail Pavlov, Gabriel Goh, Scott Gray, Chelsea Voss, Alec Radford, Mark Chen, and Ilya Sutskever.  Zero-shot text-to-image generation.  In _International Conference on Machine Learning_ , pp. 8821–8831. PMLR, 2021.
  * Rombach et al. (2022) Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Björn Ommer.  High-resolution image synthesis with latent diffusion models.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pp. 10684–10695, 2022.
  * Russakovsky et al. (2015) Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, et al.  Imagenet large scale visual recognition challenge.  _International journal of computer vision_ , 115(3):211–252, 2015.
  * Saharia et al. (2022) Chitwan Saharia, William Chan, Saurabh Saxena, Lala Li, Jay Whang, Emily Denton, Seyed Kamyar Seyed Ghasemipour, Burcu Karagol Ayan, S Sara Mahdavi, Rapha Gontijo Lopes, et al.  Photorealistic text-to-image diffusion models with deep language understanding.  _arXiv preprint arXiv:2205.11487_ , 2022.
  * Sohl-Dickstein et al. (2015) Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli.  Deep unsupervised learning using nonequilibrium thermodynamics.  In _International Conference on Machine Learning_ , pp. 2256–2265. PMLR, 2015.
  * Wang et al. (2022) Wenhui Wang, Hangbo Bao, Li Dong, Johan Bjorck, Zhiliang Peng, Qiang Liu, Kriti Aggarwal, Owais Khan Mohammed, Saksham Singhal, Subhojit Som, et al.  Image as a foreign language: Beit pretraining for all vision and vision-language tasks.  _arXiv preprint arXiv:2208.10442_ , 2022.
  * Yang et al. (2022) Jinyu Yang, Jiali Duan, Son Tran, Yi Xu, Sampath Chanda, Liqun Chen, Belinda Zeng, Trishul Chilimbi, and Junzhou Huang.  Vision-language pre-training with triple contrastive learning.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pp. 15671–15680, 2022.
  * Yu et al. (2022) Jiahui Yu, Zirui Wang, Vijay Vasudevan, Legg Yeung, Mojtaba Seyedhosseini, and Yonghui Wu.  Coca: Contrastive captioners are image-text foundation models.  _arXiv preprint arXiv:2205.01917_ , 2022.
  * Zhong et al. (2022) Yiwu Zhong, Jianwei Yang, Pengchuan Zhang, Chunyuan Li, Noel Codella, Liunian Harold Li, Luowei Zhou, Xiyang Dai, Lu Yuan, Yin Li, et al.  Regionclip: Region-based language-image pretraining.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pp. 16793–16803, 2022.

##  Appendix A Appendix

Figure 4: Additional text-to-image generation examples. We can consistently
observe that while ITC focuses on detailed formulation of the salient object,
CAP and its variant SHIFT understand the prompt in a finer level and output
more faithful visualizations. It is also apparent that ITC alone often leaves
out certain objects or mixes different visual semantics (e.g., colorful
entrance). Figure 5: Additional experiments for noise robustness. Although ITC
produces realistic images with clean prompts, minor typos can completely ruin
their semantic signals. In contrast, losses that provide denser supervisions
generally output consistent results despite textual noise, showing better
robustness. Figure 6: Additional results for optimization complexity.
Captioning-based losses require more diffusion steps to generate realistic
images, while ITC and ITM quickly forms reasonable shapes and appearances.
Figure 7: Qualitative comparison between baselines that combine multiple
objectives. BLEND mixes CAP and ITC with no transition. We observe that
gradually shifting from CAP to ITC enjoys advantages from both sides, i.e.,
faithful scene composition and realistic details.