# Metaverse: A Young Gamer’s Perspective

Ivan V. Bajić  Simon Fraser University
Burnaby, BC, Canada
Teo Saeedi-Bajić and Kai Saeedi-Bajić  École Pauline Johnson
West Vancouver, BC, Canada

###### Abstract

When developing technologies for the Metaverse, it is important to understand
the needs and requirements of end users. Relatively little is known about the
specific perspectives on the use of the Metaverse by the youngest audience:
children ten and under. This paper explores the Metaverse from the perspective
of a young gamer. It examines their understanding of the Metaverse in relation
to the physical world and other technologies they may be familiar with, looks
at some of their expectations of the Metaverse, and then relates these to the
specific multimedia signal processing (MMSP) research challenges. The
perspectives presented in the paper may be useful for planning more detailed
subjective experiments involving young gamers, as well as informing the
research on MMSP technologies targeted at these users.

###### Index Terms:

Metaverse, gaming, avatar, Roblox

##  I Introduction

Metaverse is expected to be the next iteration of the Internet. Building on
advances in communications and networking, computer graphics, multimedia
signal processing (MMSP), and other technologies, the Metaverse111We
capitalize Metaverse when used as the noun (e.g., the Metaverse) and show it
lowercase when used as an adjective (e.g., metaverse gaming). will become the
place for interaction, sharing, gaming, and a variety of services operating in
virtual worlds. Several recent reviews of the Metaverse have appeared in the
literature. In [1, 2, 3], the authors provide a general overview of metaverse
technologies with an emphasis on security and privacy. Human-centric
perspectives on metaverse technologies are explored in [4], while [5, 6]
review metaverse technologies related to healthcare, and [7, 8] explore the
role of the Metaverse in education.

The focus of this paper is on the perspectives of young (10-year-old and
younger) persons on metaverse gaming.222User experience in games has been
studied in the past but, to our knowledge, most studies have targeted older
users, e.g., 11+ years of age [9]. This is important for several reasons.
First, young gamers are among the first adopters of the Metaverse. Although
the financial volume of this segment of users may be questioned because they
do not control their own finances, the commercial success of gaming, in
general, means that there are significant commercial opportunities in this
market. Second, the perspectives of young people often differ from those of
adult users, and this is generally underappreciated in the technical
community. Finally, running user studies involving young users is more
complicated than with adult users due to more stringent regulatory
requirements, the need to involve parents/guardians, and their generally lower
attention span that could impact the duration and logistics of such studies.
This means that user studies that involve young users should be planned even
more carefully than those involving adults, and the selection of hypotheses to
be tested should be done judiciously. Our hope is that the present manuscript
will aid in selecting such hypotheses for further studies.

We will use Roblox,333www.roblox.com the most popular metaverse gaming
platform, for concrete illustrations. Roblox is a multi-player gaming platform
where developers can create their own games, and users can select from a
variety of developed games to play on their own or with their friends. Most
Roblox games involve users’ avatars, which themselves represent an important
focal point for young gamers.

This paper was written based on discussions with the last two authors, who are
avid young Roblox gamers and who contributed some of their Roblox creations to
the manuscript. While the discussions were mostly nontechnical, an attempt was
made to map their perspectives and expectations into technical terms and
relate them to relevant MMSP technologies. No new methodologies or algorithms
are presented in this paper; instead, the goal is to illuminate how young
gamers see metaverse gaming, what their expectations are of these
technologies, and how these expectations can be addressed through MMSP
research. Section II presents some of the preliminaries needed for a
subsequent discussion. Section III discusses interaction with the Metaverse,
which is the centerpiece of young gamers’ view of the Metaverse. Section IV
discusses health, safety, security, and privacy, followed by conclusions in
Section V.

##  II Preliminaries

Through discussions with the last two authors, it became clear that their
understanding of the Metaverse is shaped by their prior experience with two
other worlds: _physical_ and _digital_. The physical world is where their
physical body resides, while the digital world is the technology they got used
to prior to taking up metaverse gaming and includes digital images and video,
simpler forms of gaming, as well as simple software. The Metaverse appears as
the third world, which we will refer to as _virtual_. Fig. 1 illustrates the
three worlds.

Figure 1: The three worlds discussed in the paper.

Young people have limited control over their life in the physical world, so
the virtual world provides an environment in which they can exercise greater
control and pursue their interests. Once they learn that many aspects of the
virtual world are not limited by the same constraints as the physical world,
it opens up many possibilities to exercise their imagination and creativity.
One of the young gamers in this study took an interest in designing avatar
clothing (Fig. 1 top), because the opportunities to choose his clothing in the
physical world were limited; the other was unhappy with the layout of his
dwellings so he took up designing houses in the virtual world (Fig. 2). We
will use avatar clothing design to illustrate many of the MMSP research
challenges related to the Metaverse. Table I lists the acronyms used in the
paper.

TABLE I: Acronyms used in the paper. Acronym |  Meaning
---|---
MMSP |  MultiMedia Signal Processing
PNG |  Portable Network Graphics
JPEG |  Joint Photographic Experts Group
UGC |  User-Generated Content
QoE |  Quality of Experience
RGB+D |  Red, Green, Blue plus Depth
LiDAR |  Light Detection and Ranging
PC |  Point Cloud
VAC |  Vergence-Accommodation Conflict
SSO |  Single Sign-On
MFA |  Multi-Factor Authentication

##  III Interactions among the worlds

The current generation of young gamers has grown up surrounded by
touchscreens; for them, interaction with the digital world is fairly seamless,
and the expectation is for such seamlessness to carry over to the virtual
world. Just as moving an icon across a screen is effortless, it should be easy
to move objects into and out of the virtual world. However, for now, this is
far from easy; some of the key research problems stem from such challenges.

###  III-A Interaction between digital and virtual worlds

We use avatar clothing design as a motivating application to illustrate some
of the challenges. Roblox offers many pre-made clothing items for avatars, but
personalizing avatar clothing usually means having to design patterns or logos
outside of Roblox. This can be done in a variety of image-editing software in
the digital world. Fig. 3 shows an example of a logo designed using a simple
image editor transferred to the virtual world. Roblox can import a number of
image formats (PNG, JPEG, etc.) and transferring from the digital to the
virtual world is fairly easy. The key technical question here is related to
how image compression artifacts (for compressed image/video formats) would
affect the quality of rendering the corresponding image/video in the virtual
world. These issues have recently been studied, both in terms of conventional
coding [10] and foveated coding [11].

Figure 2: Top: penthouse; Bottom: living room. Figure 3: Transferring objects
(e.g., a logo) from the digital world to the virtual world is relatively easy,
though not seamless.

There is also a need for moving data the other way: from the virtual to the
digital world. In particular, user-generated content (UGC) videos of games
have been popular on YouTube and other video sharing platforms, and the
quality assessment of these videos has been the subject of recent studies [12,
13]. UGC videos often have both synthetic (the game itself) and natural
content (e.g., the player’s headshot), which highlights the need for codecs
that are able to handle both synthetic and natural components. Screen content
coding has been addressed in the latest video coding standards [14, 15, 16,
17] as well as recent learning-based pre/post-processing [18] and compression
[19]. When encoding game video for video sharing platforms, one is likely to
encounter content that is already encoded in the virtual world. Therefore, an
efficient solution would be to transcode such content into a conventional
video format. However, there appears to be little work on transcoding for such
scenarios.

In terms of digital-virtual world interaction, there is also the question of
playing the game on different devices. To an outsider, it might seem like a
good idea to strive to make the playing experience on different devices (e.g.,
computer and tablet) as similar as possible. However, perhaps surprisingly,
both gamers involved in this study indicated that they _liked the fact that
different devices provided different experiences_ and that they would choose a
device based on the type of experience they preferred on a given occasion.
This point should be further examined in a larger user study.

###  III-B Interaction between physical and virtual worlds

As the virtual world resembles and often tries to mimic the physical world, it
seems natural to want to transfer objects between the two worlds. Indeed, the
increasing sophistication of avatars in metaverse games is an attempt to more
closely mimic physical bodies. However, object transfer between the physical
and the virtual worlds is the source of some of the greatest challenges
related to metaverse technologies.

In the context of our clothing example, Roblox offers advanced options for
virtual clothing design through its Roblox Studio tool. These can incorporate
designs created in other software, as described in the previous section.
However, once a piece of clothing is designed in the virtual world, there is
no easy way to transfer it to the physical world. The necessary steps would
involve choosing the right size and shape (virtual try-on) and then choosing
and cutting the material, sowing, and printing in the physical world. In the
case of a more general object transfer from the virtual to the physical world,
related steps would include sizing, material selection, and manufacturing
(e.g., 3D printing). There has been quite a bit of research on virtual try-on
in recent years [20, 21, 22]. The same goes for 3D printing and advanced
manufacturing, although these areas are not usually considered part of MMSP
research.

The workaround to the challenges of moving objects from the virtual to the
physical world is to design them in the digital world and then transfer them
to both the virtual and the physical world. An example with the t-shirt logo
is shown in Fig. 4, and utilizes a t-shirt printing service (for digital $\to$
physical transfer) and Roblox Studio (for digital $\to$ virtual transfer).
This workaround is, of course, cumbersome and making this process easier is
among the greatest expectations of the young gamers involved in this study.

Figure 4: Transferring items between virtual and physical worlds (red dashed
line) is challenging. The workaround is to transfer from digital to both
virtual and physical.

The transfer from the physical to the virtual world has been more researched,
although it remains challenging. The main research keywords in this area would
be 3D capture, reconstruction, and compression of 3D models. Great strides
have been made in the last few decades in related technologies. In terms of
sensing, Red-Green-Blue plus Depth (RGB+D), multiview imaging, and Light
Detection And Ranging (LiDAR) have all been popular topics in the MMSP
community. RGB+D sensing, for example, using a Kinect-like sensor, offers a
cost-effective way of obtaining 3D scans [23, 24]. However, this approach
generally offers lower accuracy compared to the alternatives. Multiview 3D
reconstruction [25], LiDAR and industrial laser-based 3D scanners are more
accurate but also more expensive options. For a young gamer who wants to
transfer a 3D object from the physical to the virtual world, an ideal solution
would be to use the same device they play the game on (e.g., a smartphone or
tablet) and a process that takes only a few clicks. Technology is not there
yet, although progress is being made.

Point clouds (PCs) are a popular data format for representing 3D objects. PC
processing, compression, and quality assessment are among the most active
metaverse-related research thrusts in the MMSP community. In terms of PC
processing, sampling [26] and resampling [27], denoising [28], registration
[29], super-resolution [30], and analysis [31, 32] have attracted considerable
research interest. For PC compression, video-based [33] and geometry-based
approaches [34, 35] have been developed. PC quality assessment addresses both
geometry and color [36, 37, 38]. Both full-reference [39] and no-reference
[40] methods have been proposed.

Finally, when discussing interactions between the physical and virtual worlds,
it is worth mentioning latency and game responsiveness. Multiplayer games may
involve hundreds of players distributed across the world, creating enormous
challenges in terms of the need to provide low-latency interaction. Although
dealing with latency is usually the purview of multimedia communications, the
MMSP community has addressed related challenges in game video compression [41,
42] and quality assessment [43, 44].

##  IV Health, safety, security, and privacy

Young gamers have limited understanding of the health, safety, security, and
privacy aspects of metaverse gaming. Hence, this section is mostly written
from the parents’/guardians’ perspective.

###  IV-A Health

When it comes to health, parents/guardians of young gamers are usually
concerned about screen time and its effect on focus and sleep [45]. Another
area of concern is the blue light emitted from screens and its long-term
effect on vision [46]. Although there does not appear to be much MMSP research
on these topics, one health-related area where the MMSP community has made
notable contributions is the study of the vergence-accommodation conflict
(VAC) [47] in stereoscopic 3D displays [48, 49]. This phenomenon, which exists
in 3D display systems that project objects at different depths onto the same
focal plane, has been shown to cause visual fatigue and discomfort. The
proposed solutions include multifocal [50, 51] and time division multiplexed
[52] light field displays. Related quality assessment of light field content
has also been a subject of intense research [53, 54].

###  IV-B Safety

Safety issues for young gamers in the Metaverse center on two main concerns:
(1) Who are they communicating with? and (2) Are they exposed to harmful /
inappropriate content? MMSP research has addressed both concerns. In terms of
identity verification, biometrics are a key technology, and the MMSP community
has a long history of research in this area. Among the many approaches
investigated for person identification are those based on face images [55],
head and mouth dynamics [56], gait [57], audio-visual cues [58, 59], apparel
[60]. The other issue - detection of inappropriate or harmful content - has
been studied in both conventional images [61, 62] and video [63, 64]. However,
there does not yet appear to be any related published work in the specific
context of the Metaverse.

###  IV-C Security

Some young gamers have heard of _hackers_ and _hacking_. Although they lack
technical understanding of these terms, they understand that someone else may
take over their account, thereby also taking away their metaverse creations.
Also, if their friend’s account gets compromised, they may end up
communicating with an unknown person thinking that they are their friend. So,
in this general sense, they appreciate that security in the Metaverse is
important and is not guaranteed.

Account security has been evolving in response to various cyber threats. The
current practice of single sign-on (SSO) federated identity management and
multi-factor authentication (MFA) is likely to need upgrades in the future as
the threats themselves evolve. Biometrics (face and fingerprints) have also
been incorporated into account security, especially on mobile devices. This is
an area where the MMSP community has contributed important research, as
discussed above.

Figure 5: MMSP research related to metaverse gaming. TABLE II: Selected challenges related to metaverse gaming and corresponding MMSP research. Note that the list of challenges, research topics, and especially the list of references are far from exhaustive. Challenge |  MMSP research |  References
---|---|---
Virtual QoE |  Quality assessment in the virtual world |  [10, 11]
UGC QoE |  Quality assessment of UGC video |  [12, 13]
Game video coding |  Screen content coding, transcoding, and pre/post-processing |  [14, 15, 16, 17, 18, 19]
Virtual $\to$ physical |  Virtual try-on |  [20, 21, 22]
Physical $\to$ virtual |  3D capture and reconstruction |  [23, 24, 25]
Point cloud processing |  Point cloud (re)sampling, denoising, reconstruction, and analysis |  [26, 27, 28, 29, 30, 31, 32]
Point cloud coding |  Geometric and video-based point cloud compression |  [33, 34, 35]
Point cloud quality |  Objective and subjective quality assessment of point clouds |  [36, 37, 38, 39, 40]
Multiplayer latency |  Latency-aware game video coding and quality assessment |  [41, 42, 43, 44]
Health |  Addressing VAC, light field display and content assessment |  [48, 49, 50, 51, 52, 53, 54]
Safety and security |  Biometrics, detection of inappropriate or harmful content |  [55, 56, 57, 58, 59, 60, 61, 62, 63, 64]
Privacy |  Player privacy, biometric template protection |  [65, 66, 67, 68]

###  IV-D Privacy

Privacy issues refer to unintended disclosure of private information that may
be used to identify a person or some of their attributes (age, gender, etc.),
which would enable someone to take advantage of them, impersonate them, or
steal their identity. It is encouraging to see that metaverse privacy is on
the research agenda and that solutions such as MetaGuard [65] are beginning to
appear. MetaGuard employs differential privacy to make it more difficult for
others to infer player’s attributes such as age, gender, height, etc.

Since biometrics are an increasing part of metaverse safety and security
measures, privacy issues related to biometric data are also important. In this
context, it is worth mentioning the work on biometric template protection [66,
67] and cancelable biometrics [68], where the MMSP community has made
important contributions.

##  V Summary and conclusions

This paper has provided a young person’s view of metaverse gaming. Related
MMSP research challenges have been summarized in Table II and Fig. 5. It is
evident that MMSP research has an important role to play in the future
development of the Metaverse.

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# The Integrated Sachs-Wolfe Effect in Interacting Dark Matter-Dark Energy
Models

Mina Ghodsi Yengejeh<EMAIL_ADDRESS>PDAT Laboratory, Department of
Physics, K.N. Toosi University of Technology, P.O. Box 15875-4416, Tehran,
Iran  Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran
19839, Iran  Saeed Fakhry<EMAIL_ADDRESS>Department of Physics, Shahid
Beheshti University, G.C., Evin, Tehran 19839, Iran  Javad T. Firouzjaee
<EMAIL_ADDRESS>Department of Physics, K.N. Toosi University of
Technology, P.O. Box 15875-4416, Tehran, Iran  School of Physics, Institute
for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran
PDAT Laboratory, Department of Physics, K.N. Toosi University of Technology,
P.O. Box 15875-4416, Tehran, Iran  Hojatollah Fathi<EMAIL_ADDRESS>PDAT Laboratory, Department of Physics, K.N. Toosi University of Technology,
P.O. Box 15875-4416, Tehran, Iran

###### Abstract

Interacting dark matter-dark energy (IDMDE) models can be taken to account as
one of the present challenges that may affect the cosmic structures. The main
idea of such models is that the mass of dark matter particles can be specified
by their interactions with a scalar field whose energy density is
characterized by dark energy. In this work, we propose to study the integrated
Sachs-Wolfe (ISW) effect in IDMDE models. To this end, we initially introduce
a theoretical framework for IDMDE models. Moreover, we briefly discuss the
stability conditions of IDMDE models and by specifying a simple functional
form for the energy density transfer rate, we calculate the perturbation
equations. In the following, we calculate the amplitude of the matter power
spectrum for the IDMDE model and compare it with the corresponding result
obtained from the $\Lambda$CDM model. Furthermore, we calculate the amplitude
of the ISW auto-power spectrum as a function of multipole order $l$ for the
IDMDE model. The results indicate that the amplitude of the ISW auto-power
spectrum in the IDMDE model for different phantom dark energy equations of
state behaves similar to the one for the $\Lambda$CDM model, whereas, for the
quintessence dark energy equations of state, the amplitude of the ISW-auto
power spectrum for the IDMDE model should be higher than the one for the
$\Lambda$CDM model. Also, it turns out that the corresponding results by
different values of the coupling parameter demonstrate that $\xi$ is inversely
proportional to the amplitude of the ISW-auto power spectrum in the IDMDE
model. Finally, by employing four different surveys, we calculate the
amplitude of the ISW-cross power spectrum as a function of multipole order $l$
for the IDMDE model. The results exhibit that the amplitude of the ISW-cross
power spectrum for the IDMDE model for all values of $\omega_{\rm x}$ is
higher than the one obtained for the $\Lambda$CDM model, while deviations are
still less than $0.1$ order of magnitude. Also, it turns out that the
amplitude of the ISW-cross power spectrum in the IDMDE model changes inversely
with the value of coupling parameter $\xi$.

Interacting Dark Sector; Integrated Sachs-Wolfe Effect; Power Spectrum; Cross-
Correlation.

##  I Introduction

The standard model of cosmology, the Lambda cold dark matter ($\Lambda$CDM)
model, gained big success in explaining wide observations in large-scale
physics. This success is due to the great efforts to set up cosmological
surveys like the cosmic microwave background (CMB) WMAP:2003gmp ; WMAP:2006bqn
; Planck:2015fie ; Planck:2018vyg , type Ia supernovae
SupernovaSearchTeam:1998fmf ; SupernovaSearchTeam:1998fmf2  , galaxy surveys
SDSS1 ; SDSS2 , and weak lensing Hildebrandt:2016iqg ; DES:2017myr . All these
surveys have confirmed the $\Lambda$CDM Peebles , which shows the entire
matter budget of the Universe consists primarily of cold dark matter and dark
energy in the present-time Universe.

Recently, some serious tensions in observational data have opened some new
windows to think about the nature of dark energy and dark matter beyond the
$\Lambda$CDM model. The most impressive examples of so-called tensions between
the $\Lambda$CDM predictions can be deduced from the latest analysis of the
Planck collaboration Planck:2018vyg  and independent observational of the
distance ladder measurements. These measurements employ close cepheids as
anchors for measuring the Hubble parameter $H_{0}$ that measures the expansion
rate of the Universe.

Hubble space telescope (HST) has measured this tension through local distance
ladder measurements, and the supernova $H_{0}$ for the equation of state
(SH0ES) collaboration reports $H_{\rm 0}=74.03\pm 1.42\,{\rm
km\,s^{-1}\,Mpc^{-1}}$ Riess:2019cxk , which is at a $4.4\sigma$ difference to
the value measured by the Planck satellite, i.e., $H_{\rm 0}=67.4\pm 0.5\,{\rm
km\,s^{-1}\,Mpc^{-1}}$ Planck:2018vyg . Another tension is the value of
$\sigma_{\rm 8}$, which represents the amount of late-time matter clustering.
To this end, some surveys such as the kilo degree survey (KiDS)
Kohlinger:2017sxk  and the dark energy survey (DES) DES:2017myr ; DES:2017qwj
; DES:2017qwj2  have been utilized, resulting in tensions in $\sigma_{8}$
measurements of the order of $2\sim 3\,\sigma$.

Many different studies have been performed to solve these tensions H0review ,
including models that question the nature of dark matter or dark energy, which
can be studied in mass-temperature relation Naseri:2020uvn  and scaling
relation Naseri:2020dgq  in the galaxy clusters. Moreover, various
cosmological aspects have been discussed, while considering other dark matter
candidates Bernal:2018ins ; Hryczuk:2020jhi ; Egana-Ugrinovic:2021gnu ;
Fakhry:2020plg ; Fakhry:2021tzk . A model that has been studied extensively is
the interaction between dark matter and dark energy (henceforth IDMDE), see
e.g., Wang:2016lxa  for a recent review.

Another interesting approach to constrain dark energy models is to investigate
the integrated Sachs-Wolfe (ISW) effect in the observational data in such a
way that the cross-correlation between the ISW signal and the galaxy
distribution can be specified. The ISW effect is an additional effect
associated with the time-dependence gravitational potential as photons pass
through the observable Universe. It should be noted that gravitational
potentials are constant in time during the matter-dominated era. Hence,
emitted photons from the last scattering surface do not change during the
matter-dominated era. Except at very large scales, the amplitude of the ISW
fluctuations tends to be smaller than that of fluctuations originating at the
epoch of the last scattering. However, the CMB fluctuations were created much
more recently. Therefore, they are expected to be partially correlated with
tracers of the large-scale matter distribution, such as distant galaxies. In
this way, the ISW effect results in a new direction in observational cosmology
to constrain cosmological models with time-varying potentials.

Since the ISW affects the redshift of the measurement as an integrated effect
on large redshift ranges (from the last scattering surface to now) can be more
sensitive to variations in dark energy properties. In this regard, calculation
of the ISW effect can be a convenient criterion for studying dark energy
Crittenden:1995ak  and can probe some modifications of the $\Lambda$CDM model
such as interacting dark sector models and modified gravity Munshi:2014tua ;
Adamek:2019vko ; Kable:2021yws ; Y:2021ybx .

In this work, we calculate the ISW effect within the framework of an IDMDE
model. In this respect, the outline of this work is structured as below. In
Sec. II, we describe the theoretical framework of IDMDE models. In this
regard, by considering a perturbed metric for the inhomogeneous Universe, we
determine density and velocity perturbation equations. Then in Sec. III, we
briefly discuss the stability conditions of IDMDE models and rewrite the
perturbation equations by determining a simple functional form for the energy
density transfer rate. Furthermore, in Sec. V, we calculate the ISW effect
within the context of IDMDE models and compare our results with that of the
$\Lambda$CDM model. Finally, we scrutinize the results and summarize the
findings in Sec. V.

##  II Interacting Dark Matter-Dark Energy Models

Astrophysical and cosmological experiments such as the measurement of
temperature anisotropies from the CMB WMAP:2006bqn , large-scale structure
evaluations 2DFGRS:2001zay ; SDSS:2003eyi ; 2dFGRS:2005yhx ; Springel:2006vs ,
the integrated Sachs-Wolfe effect Fosalba:2003ge ; boughn2004 ; Boughn:2004zm
; boughn2005 , luminosity distances estimation via supernovae type Ia
Boomerang:2000efg ; SupernovaCosmologyProject:2003dcn ;
SupernovaSearchTeam:2003cyd ; SupernovaSearchTeam:2004lze ; SNLS:2005qlf , and
weak lensing predictions Contaldi:2003hi  demonstrate that the late-time
Universe is passing an accelerated expansion phase. In this regard, the
standard model of cosmology suggests an unknown relativistic matter component
with negative pressure, referred to as dark energy, to explain the accelerated
expansion phase in the late-time Universe. It is believed that dark energy is
currently about $75\%$ of the total density of the Universe and the remaining
$25\%$ is specified by the density of nonrelativistic matter component (i.e.,
dark matter and baryon) and very small amounts of relativistic particles
(i.e., neutrinos and photons). Thus, the dynamics of the accelerated expansion
phase may be referred to as a theoretical paradigm called the cosmological
constant. However, providing a cosmological constant to justify the
accelerated expansion phase in the late-time Universe, despite all advantages,
would not be perfect. In other words, the discrepancy between observed values
of cosmological constant and theoretical predictions of zero-point energy
leads to an issue known as the cosmological constant problem wienberg1989a .
Additionally, observational experiments suggest that in the present-time
Universe the ratio of the dark energy density to the matter one is of the
order of unity, which requires special conditions to be satisfied in the early
Universe. The question that may arise is why now $\rho_{\rm DM}/\rho_{\rm
DE}\sim\mathcal{O}(1)$?, which is known as the coincidence problem
Steinhardt1997 .

In order to address these problems, one can consider an interaction between
the components of the dark sectors (i.e., matter and energy) in such a way
that the ratio $\rho_{\rm DM}/\rho_{\rm DE}$ includes an evolution from a
constant and unstable value in the early Universe to a lower constant and
stable value in the late-time Universe and well explain the accelerated
expansion era Zimdahl:2001ar ; Chimento:2003iea . In this regard, dark matter
and dark energy densities do not evolve independently. On the other hand, one
assumes that photon and baryon can not be coupled to dark energy. Hence, they
do not contribute to interactions. Also, radiation can be neglected, as the
focus is on the formation of structures in the late-time Universe Caldera-
Cabral:2009hoy .

Let us start with the Bianchi identity, $\nabla_{\rm\nu}G^{\mu\nu}=0$, which
implies the energy-momentum conservation within the framework of the standard
model of cosmolgy. While it leads to an interacting term in the continuity
equations of dark energy and dark matter fluids in interacting models. In such
a scenario, dark sector energy-momentum tensors cannot be conserved:

$\nabla_{\rm\mu}T^{\rm\mu\nu}_{\rm A}=Q^{\rm\nu}_{\rm A},$ (1)

where $Q^{\rm\nu}$ is the four vector energy-momentum transfer rate, and
subscript “A” is allowed to be “x” (for dark energy) and “c” (for dark
matter). It is essential to remind that the energy-momentum tensor of the
total dark sector is conserved, namely

$\sum_{\rm A}\nabla_{\rm\mu}T^{\rm\mu\nu}_{\rm A}=\sum_{\rm A}Q^{\rm\nu}_{\rm
A}=0,$ (2)

which implies $Q_{\rm c}^{\rm\nu}=-Q_{\rm x}^{\rm\nu}$.

In order to investigate the impact of interacting models at large scales, one
can assume a spatially flat and expanding Universe governed by a perfect
fluid. In light of such considerations, the evolution of the energy-momentum
density is determined by taking $\nu=0$. Therefore, one can have the following
energy balance equations for dark matter and dark energy

$\rho^{\prime}_{\rm c}+3\mathcal{H}\rho_{\rm c}=-a^{2}Q^{0}=-aQ,$ (3)
$\rho^{\prime}_{\rm x}+3\mathcal{H}(1+\omega_{\rm x})\rho_{\rm
x}=a^{2}Q^{0}=aQ,$ (4)

where prime denotes derivative with respect to the conformal time $\tau$,
$a(\tau)$ is the scale factor, $Q\equiv Q^{0}/a$ is a coupling parameter
called the energy density transfer rate, $\mathcal{H}=a^{\prime}/a$ is the
conformal Hubble parameter and $\omega_{\rm x}=p_{\rm x}/\rho_{\rm x}$ is the
dimensionless parameter of the equation of state related to dark energy. In
the framework of IDMDE models one can assume a flat Universe with the
Friedman-Lemaître-Robertson-Walker (FLRW) metric and pressureless dark matter,
i.e., $\omega_{\rm c}=p_{\rm c}/\rho_{\rm c}=0$. Note that Eqs. (3) and (4)
can also be reformed as follows:

$\rho^{\prime}_{\rm c}+3\mathcal{H}(1+\omega^{eff}_{\rm c})=0,$ (5)
$\rho^{\prime}_{\rm x}+3\mathcal{H}(1+\omega^{eff}_{\rm x})=0.$ (6)

In these relations, $\omega^{eff}_{\rm c}$ and $\omega^{eff}_{\rm x}$ are
effective parameters of dark matter and dark energy equations of state that
can be obtained as follows:

$\omega^{eff}_{\rm c}=\dfrac{aQ}{3\mathcal{H}\rho_{c}},$ (7)
$\omega^{eff}_{\rm x}=\omega_{\rm x}-\dfrac{aQ}{3\mathcal{H}\rho_{\rm x}}.$
(8)

As mentioned earlier, $Q$ is the energy density transfer rate, which implies
for $Q>0$ the direction of energy transfer is from dark matter to dark energy,
and vice versa. In addition, the energy density transfer rate leads to the
modification in the redshift evolution of dark sector components. For
instance, if $Q>0$, the redshift evolution of dark matter will be faster.
Thus, there could have been more dark matter at higher redshifts compared to
noninteracting models LopezHonorez:2009tuj . Actually, different forms have
been proposed for the energy density transfer rate Yang:2018xlt ; Yang:2018pej
; Yang:2019vni ; Yang:2019uzo ; Yang:2019uog . Throughout this work, we focus
on $Q\propto H\rho_{\rm x}$.

On the other hand, special attention should be paid to the fact that in order
to provide suitable conditions for the structure formation, interacting models
must be protected from instabilities that may impose to models. To this end,
the stability of interacting models needs to be addressed that will be
discussed in more detail in Sec. III.

###  II.1 The Inhomogeneous Universe

Although on large scales, the Universe seems simple and can be considered as
an isotropic and homogeneous Universe, it contains structures such as galaxies
and galaxy clusters on smaller scales. Therefore, to describe these
structures, the presence of inhomogeneities seems to be necessary on smaller
scale.These structures can be considered as a result of the collapse of
density fluctuations that have exceeded their threshold. During the early
stages of the evolutionary Universe, inhomogeneities in matter density can
arise due to the gravitational instabilities. This leads to the formation of
structures that can be observed in the present-time Universe.

Therefore, to justify the structure formation on small scales, a perturbed
metric can be considered as

$ds^{2}=(\bar{g}_{\rm\mu\nu}+\delta g_{\rm\mu\nu})dx^{\rm\mu}dx^{\rm\nu},$ (9)

where $\bar{g}_{\rm\mu\nu}$ demonstrates the unperturbed background metric and
$|\delta g_{\rm\mu\nu}|\ll 1$ reffers to small perturbations. Given that
vector perturbations decay very quickly and tensor perturbations do not induce
perturbations in the perfect fluid, it makes sence to consider scalar
perturbations for linear fluctuations. Hence, the perturbed FLRW metric within
a homogeneous and isotropic Universe takes the following form Wang:2016lxa

$\displaystyle ds^{2}=a^{2}(\tau)\bigg{[}-$1+2\phi$d\tau^{2}+2\partial_{\rm
i}Bd\tau dx^{\rm i}+$ $\displaystyle$$(1-2\psi)\delta_{\rm ij}+2\partial_{\rm
i}\partial_{\rm j}E$$dx^{\rm i}dx^{\rm j}\bigg{]},$ (10)

where $\tau$ is the conformal time, $\phi$, $\psi$, $B$ and $E$ are the scalar
metric perturbations, and $d\tau=dt/a$.

Moreover, in the rest frame of fluid “A”, the sound speed, $c_{\rm sA}$, is
equal to the propagation velocity of pressure fluctuations Valiviita:2008iv :

$c_{\rm sA}^{2}=\dfrac{\delta P_{\rm A}}{\delta\rho_{A}}\bigg{|}_{\rm rest\
frame}.$ (11)

Also, the adiabatic sound speed can be defined as Valiviita:2008iv

$c_{\rm aA}^{2}\equiv\dfrac{dP_{\rm A}}{d\rho_{\rm A}}=\omega_{\rm
A}+\dfrac{\omega_{\rm A}^{\prime}}{\rho_{\rm A}^{\prime}/\rho_{\rm A}}.$ (12)

If a barotropic equation of state is considered, then both Eqs. (11) and (12)
will be equal. Hence, in the presence of a constant $\omega$, the adiabatic
sound speed will reduce to $c_{\rm a}^{2}=\omega$. Besides, the sound speed
would be imaginary if we consider dark energy as an adiabatic fluid, i.e.,
$c_{\rm sx}^{2}=c_{\rm ax}^{2}=\omega_{\rm x}<0$, which leads to instabilities
in dark energy Valiviita:2008iv . This issue can be addressed by demanding
$c_{\rm sx}^{2}>0$. It should be mentioned that in this work we set $c_{\rm
sx}^{2}=1$ in the calculations.

To calculate the matter power spectrum numerically within the context of
interacting models, a modified version of the Code for Anisotropies in the
Microwave Background (CAMB) Lewis:1999bs  must be employed. For this purpose,
the code has been adopted to solve the IDMDE Friedmann equation. Note that the
CAMB code solves the Friedmann equation of the $\Lambda$CDM model in its
standard form. For the chosen energy density transfer rate as $Q\propto
H\rho_{\rm x}$, analytical solutions for both energy densities, i.e.,
$\rho_{\rm c}$ and $\rho_{\rm x}$, can be specified. The CAMB code requires
the energy densities as functions of the scale factor. To do this, one has to
solve two coupled Eqs. (3) and (4) as functions of scale factor. Furthermore,
compared to the $\Lambda$CDM model, $Q$ modifies linear perturbations that
must be considered in the modified version of the CAMB code.

By taking the perturbed part of Eq. (1) in terms of the Fourier space,
evolution equations of the dimensionless density perturbation $\delta_{\rm
A}=\delta\rho_{\rm A}/\rho_{\rm A}$ (the continuity equation) and of the
velocity perturbation equation $\theta_{\rm A}\equiv\partial_{i}v^{i}_{\rm A}$
(the Euler equation) for derk sector components can be calculated as follows:

$\displaystyle\delta^{\prime}_{\rm x}=-$1+\omega_{\rm x}$\bigg{(}\theta_{\rm
x}+\dfrac{h^{\prime}}{2}\bigg{)}-3\mathcal{H}$c_{\rm sx}^{2}-\omega_{\rm x}$$
$\displaystyle\times\left[\delta_{\rm x}+3\mathcal{H}$1+\omega_{\rm
x}$\dfrac{\theta_{\rm x}}{k^{2}}\right]-3\mathcal{H}\omega_{\rm
x}^{\prime}\dfrac{\theta_{\rm x}}{k^{2}}$ $\displaystyle+\dfrac{aQ}{\rho_{\rm
x}}\left[-\delta{\rm x}+\dfrac{\delta Q}{Q}+3\mathcal{H}(c^{2}_{\rm
sx}-\omega_{\rm x})\dfrac{\theta_{\rm x}}{k^{2}}\right],$ (13)
$\displaystyle\theta_{\rm x}^{\prime}=-\mathcal{H}$1-3c^{2}_{\rm
sx}$\theta_{\rm x}+\dfrac{c^{2}_{\rm sx}}{(1+\omega_{\rm x})}k^{2}\delta_{\rm
x}$ $\displaystyle+\dfrac{aQ}{\rho_{\rm x}}\left[\dfrac{\theta_{\rm
c}-$1+c^{2}_{\rm sx}$\theta_{\rm x}}{1+\omega_{\rm x}}\right],$ (14)
$\delta_{\rm c}^{\prime}=-\bigg{(}\theta_{\rm
c}+\dfrac{h^{\prime}}{2}\bigg{)}+\dfrac{aQ}{\rho_{\rm c}}\bigg{(}\delta_{\rm
c}-\dfrac{\delta Q}{Q}\bigg{)},$ (15) $\theta_{\rm
c}^{\prime}=-\mathcal{H}\theta_{\rm c}.$ (16)

In these relations, $k$ is the wavenumber of Fourier mode and
$h(\tau,\vec{k})$ is the synchronous field in the $k\mbox{-}$space Ma:1995ey .
As can be seen, the dark matter velocity perturbation equation is the same as
the uncoupled one, since there is no momentum transfer in the dark matter
frame. Thus, one can consistently set $\theta_{\rm c}=0$ in a synchronous
gauge Valiviita:2008iv .

In the next section, we will briefly discuss the stability conditions within
the framework of IDMDE models.

##  III Stability and Instability

As mentioned earlier, the large-scale stability of IDMDE models is very
important, because the Universe would not yield structure formations without
stable perturbations Yang:2018xlt . But it should be noted that IDMDE models
may initially suffer from non-adiabatic instabilities on large scales. These
instabilities take place due to the presence of the coupling terms in the
propagation of dark energy pressure waves Gavela:2009cy ; LopezHonorez:2009tuj
; LopezHonorez:2009hpu . It should be noted that Eqs. (13)-(16) are obtained
in the rest frame of dark matter, while the sound speed of dark energy,
$c^{2}_{\rm sx}$, should be evaluated in the rest frame of dark energy. As a
result, one can specify the pressure perturbation in the rest frame of sound
speed as follow Bean:2003fb :

$\delta p_{\rm x}=c^{2}_{\rm sx}\delta\rho_{\rm x}-(c^{2}_{\rm sx}-c^{2}_{\rm
ax})\rho_{\rm x}^{\prime}\dfrac{\theta_{\rm x}}{k^{2}}.$ (17)

If there is an interaction between dark sector components (i.e., in the
presence of a nonzero $Q$), the dark energy pressure perturbation in terms of
the density perturbation can be deduced from Eqs. (4) and (17) as

$\displaystyle\dfrac{\delta p_{\rm x}}{\delta\rho_{\rm x}}=c^{2}_{\rm
sx}+3$c^{2}_{\rm sx}-c^{2}_{\rm ax}$$1+\omega^{\rm eff}_{\rm
x}$\dfrac{\mathcal{H}\theta_{\rm x}}{k^{2}\delta_{\rm x}}\hskip 29.87547pt$
$\displaystyle=c^{2}_{\rm sx}+3$c^{2}_{\rm sx}-c^{2}_{\rm ax}$$1+\omega_{\rm
x}$$1+d$\dfrac{\mathcal{H}\theta_{\rm x}}{k^{2}\delta_{\rm x}},$ (18)

where $d$ represents the doom factor and is defined as:

$d\equiv-\dfrac{aQ}{3\mathcal{H}\rho_{\rm x}(1+\omega_{\rm x})}.$ (19)

In Ref. Gavela:2009cy , it has been stated that the stability or instability
of interacting models depends on the doom factor in such a way that for $d>1$,
i.e., strong coupling regime, an exponential growth of the dark energy
perturbations leads to the large-scale instability. In the mentioned reference
the CAMB and Cosmological MonteCarlo (CosmoMC) codes have been employed with
type Ia supernova and CMB data. As a result, it has been shown that the best
fit case can be obtained when $d<0$, which is stable and consistent with
observational data. In other words, the negative doom factor can guarantee the
stability of IDMDE models.

In this work, we choose a specific functional form of the energy density
transfer rate as

$Q=3H\xi(1+\omega_{\rm x})\rho_{\rm x},$ (20)

where $\xi$ is the coupling parameter. According to Ref. Yang:2017zjs , the
novelty of the selected form for the energy density transfer rate is it
includes the factor $(1+\omega_{\rm x})$. This factor can be simplified by the
expression in the denominator of Eq. (19) and only a simple constraint on the
coupling parameter remains. In other words, the model can alleviate large-
scale perturbation instability if and only if the coupling parameter adheres
to $\xi\geq 0$. Under these situations, once the energy density transfer rate
is determined, Eqs. (13)-(16) can be reformed as follows:

$\displaystyle\delta^{\prime}_{\rm x}=-$1+\omega_{\rm x}$\bigg{(}\theta_{\rm
x}+\dfrac{h^{\prime}}{2}\bigg{)}-3\mathcal{H}$c_{\rm sx}^{2}-\omega_{\rm x}$$
$\displaystyle\times\bigg{[}\delta_{\rm x}+3\mathcal{H}$1+\omega_{\rm
x}$\dfrac{\theta_{\rm x}}{k^{2}}\bigg{]}+3\mathcal{H}\xi$1+\omega_{\rm x}$$
$\displaystyle\times\bigg{[}\dfrac{\theta+h^{\prime}/2}{3\mathcal{H}}+3\mathcal{H}(c^{2}_{\rm
sx}-\omega_{\rm x})\dfrac{\theta_{\rm x}}{k^{2}}\bigg{]},$ (21)
$\displaystyle\theta_{\rm x}^{\prime}=-\mathcal{H}$1-3c^{2}_{\rm
sx}$\theta_{\rm x}+\dfrac{c^{2}_{\rm sx}}{(1+\omega_{\rm x})}k^{2}\delta_{\rm
x}$ $\displaystyle+3\mathcal{H}\xi$$\theta_{\rm c}-$1+c^{2}_{\rm
sx}$\theta_{\rm x}$$,$ (22) $\displaystyle\delta_{\rm
c}^{\prime}=-\bigg{(}\theta_{\rm
c}+\dfrac{h^{\prime}}{2}\bigg{)}+3\mathcal{H}\xi$1+\omega_{\rm
x}$\dfrac{\rho_{\rm x}}{\rho_{\rm c}}$ $\displaystyle\times\left(\delta_{\rm
c}-\delta_{\rm x}-\dfrac{\theta+\frac{h^{\prime}}{2}}{3\mathcal{H}}\right),$
(23) $\theta_{\rm c}^{\prime}=-\mathcal{H}\theta_{\rm c}.$ (24)

Up to here, we have described the theoretical framework and stability
conditions of IDMDE models. In the following, we will calculate the ISW effect
within the context of such models.

##  IV The Integrated Sachs-Wolfe Effect

From the last scattering surface to us, CMB photons move through the
gravitational potential of the matter, like the galaxy clusters. When photons
move into a gravitational potential well, they become blueshifted. On the
other hand, they will be redshifted as they move out of the gravitational
potential. During about the $13.8$ billion years it takes the first free
photons to reach us, they pass through different gravitational potential
wells. Therefore, the shifts we see will be accumulated along the observer’s
line of sight, which can potentially reveal some phenomenological aspects of
the late-time Universe.

To derive the ISW power spectrum, one can start with the matter power
spectrum. To do this, we modify the publicly available CAMB package
Lewis:1999bs , and employ mean values of Table 1. The second column of this
table shows the cosmological parameters of the $\Lambda$CDM model, derived
from the CMB temperature fluctuations measured by Planck with the addition of
external data sets. Error bars are given at $68\%$ confidence level
Planck:2015fie , and the third column reveals the mean values of the free and
derived cosmological parameters with their errors at $68.3\%$ and $95.4\%$
confidence regions for the IDMDE model Yang:2017zjs .

Table 1: Mean values of the free cosmological parameters for the $\Lambda$CDM and IDMDE models. Parameter |  $\Lambda$CDM model | IDMDE model
---|---|---
$\Omega_{\rm b}h^{2}$ | $0.02230\pm 0.00014$ | $0.0223^{+0.0001+0.0003}_{-0.0001-0.0003}$
$\Omega_{\rm c}h^{2}$ | $0.1188\pm 0.0010$ | $0.1183^{+0.0014+0.0030}_{-0.0014-0.0029}$
$100\,\theta_{\rm MC}$ | $1.04093\pm 0.00030$ | $1.0406^{+0.0003+0.0006}_{-0.0003-0.0006}$
$\tau$ | $0.066\pm 0.012$ | $0.0663^{+0.0161+0.0315}_{-0.0162-0.0319}$
$n_{\rm s}$ | $0.9667\pm 0.0040$ | $0.9760^{+0.0036+0.0071}_{-0.0038-0.0070}$
$\ln(10^{10}A_{\rm s})$ | $3.064\pm 0.023$ | $3.0722^{+0.0311+0.0605}_{-0.0288-0.0616}$
$H_{\rm 0}$ | $67.74\pm 0.46$ | $68.4646^{+0.8199+1.3348}_{-0.7380-1.3616}$
$\omega_{\rm x}$ | $-1.000$ | $-1.0230^{+0.0329+0.0527}_{-0.0257-0.0603}$
$\xi$ | - | $0.0360^{+0.0091+0.0507}_{-0.0360-0.0360}$

In Fig. 1, we have shown the matter power spectrum in the present-time
Universe, while considering the IDMDE model, and have compared it with the
corresponding result obtained from $\Lambda$CDM model. The top panel of this
figure is plotted for different values of the dimensionless parameter of the
dark energy equation of state. As can be seen from this figure, the peak
position of the matter power spectrum is shifted toward larger scales (i.e.,
toward smaller $k$) in $\omega_{\rm x}=-1.5$ and increasing in the matter
power spectrum. Also, the peak of the matter power spectrum is shifted toward
smaller scales (i.e., toward larger $k$) in $\omega_{\rm x}=-0.6$ and
decreasing in the matter power spectrum. In addition, in the bottom panel of
this figure, similar calculations are performed for different values of the
coupling parameter $\xi$. As it turns out, the coupling parameter $\xi$ does
not yield meaningful changes in the matter power spectrum of the IDMDE model,
and only a very slight deviation from the $\Lambda$CDM model is
distinguishable on small scales.

Figure 1: The matter power spectrum as a function of comoving wavenumber $k$
for the IDMDE and $\Lambda$CDM models.Top: For different values of the
dimensionless parameter of dark energy equation of state $\omega_{\rm x}$.
Bottom: For different values of the coupling parameter $\xi$. For the upper
panel, the coupling paramete is kept fixed as $\xi=0.0360$, and for the lower
panel, the dimensionless parameter of dark energy equation of state is set to
be $\omega_{\rm x}=-1.0230$.

Moreover, it is known that the gravitational potential is constant for a
matter-dominated Universe and therefore does not yield ISW signal. Hence, just
as we are looking for it to be, the ISW effect can serve as an indicator of
something different from ordinary pressureless matter. Moreover, the late-time
ISW effect is referred to the integrated differential gravitational redshift
caused by the evolution of gravitational potentials along the path traveled by
photons. In this regard, on large angular scales, the ISW effect produces a
weak positive correlation for all frequencies SDSS:2003lnz .

On the other hand, the ISW effect is the main secondary anisotropy at large
angles (i.e., at low $l$) that reflects the decay of the gravitational
potential. A direct way to specify the late-time ISW effect is to cross-
correlate the large-angle CMB anisotropies with large-scale structures at low
redshifts such as angular galaxy correlations. Therefore, the temperature
anisotropy due to the ISW effect is specified by taking an integration on the
time-varing quantity, ($\phi-\psi$), along the line-of-sight Schaefer:2008qs ;
Y:2021ybx , i.e.,

$\displaystyle\Theta_{\rm ISW}=\dfrac{\Delta T}{T_{\rm CMB}}\bigg{|}_{\rm
ISW}=\dfrac{2}{c^{2}}\int^{1}_{\rm a_{\rm dec}}\dfrac{\partial\phi}{\partial
a}da\hskip 45.52458pt$ $\displaystyle=-\dfrac{2}{c^{3}}\int_{0}^{\rm\chi_{\rm
H}}a^{2}H(a)\dfrac{\partial\phi}{\partial a}d\chi,$ (25)

where $T_{\rm CMB}=2.725\ K$ is the CMB temperature, $a_{\rm dec}$ is the time
at which photons decouple, $c$ is the velocity of light in vacuum. Also,
$\chi$ is the comoving distance between the CMB and us that can be calculated
as a function of scale factor, namely

$\chi(a)=\int_{0}^{1}\dfrac{c\ da}{a^{2}H(a)}.$ (26)

Moreover, if the perturbations are sufficiently within the horizon, the
Poisson’s equation would allow us to relate the matter fluctuation field,
$\delta_{\rm m}$, to the potential at late times Manzotti:2014kta ;
Stolzner:2017ged . Hence, one can have

$\phi(k,a)=-\dfrac{3}{2}\ a^{2}\ H^{2}(a)\ \Omega_{\rm m}(a)\
\dfrac{\delta_{\rm m}(k,a)}{k^{2}},$ (27)

where $\Omega_{\rm m}(a)$ is the dimensionless matter density parameter. In
this regard, by using Eqs. (IV) and (27), the temperature anisotropy due to
the ISW effect can take the following form:

$\Theta_{\rm ISW}=\dfrac{3}{c^{3}}\int^{\rm\chi_{\rm
H}}_{0}a^{2}H(a)\dfrac{\partial\mathcal{F}(a)}{\partial a}\dfrac{\delta_{\rm
m}(k,a=1)}{k^{2}}d\chi.$ (28)

In this relation, $\mathcal{F}(a)$ is defined as

$\mathcal{F}(a)\equiv a^{2}H^{2}(a)\Omega_{\rm m}(a)D_{\rm+}(a),$ (29)

where $D_{\rm+}(a)=\delta_{\rm m}(k,a)/\delta_{\rm m}(k,a=1)$ is the linear
growth factor.

In Fig. 2, we have depicted the time evolution of the ISW source term,
$\mathcal{F}(a)$, and its derivative with respect to the scale factor,
$d\mathcal{F}(a)/da$, as functions of scale factor. It is clear from Eq. (29)
that in the present-time Universe (i.e., at $a=1$), both models converge at
$\mathcal{F}(a)=\Omega_{\rm m,0}$.

It can be turned out form the top panel of this figure that for the case of
phantom dark energy (i.e., for $\omega_{\rm x}<-1$), compared to the result
obtained for the $\Lambda$CDM model, lower values of $\mathcal{F}(a)$ can be
achieved at higher redshifts within the framework of the IDMDE model. Also,
the quintessence dark energy (i.e., $\omega_{\rm x}>-1$) in the IDMDE model
leads to higher values of $\mathcal{F}(a)$ than that obtained from the
$\Lambda$CDM one. Additionally, $d\mathcal{F}/da$ in the IDMDE model is
similar to the corresponding result obtained from the $\Lambda$CDM model at
high redshifts , while it deviates from the $\Lambda$CDM model at low
redshifts. On the other hand, the bottom panel of this figure demonstrates
that at higher redshifts for all nonvanishing values of the coupling
parameter, $\mathcal{F}(a)$ in the IDMDE model takes lower values than the
corresponding result obtained from the $\Lambda$CDM model.

Figure 2: The time evolution $\mathcal{F}(a)$ of the ISW source term and its
derivative $d\mathcal{F}(a)/da$ as functions of scale factor $a$, for the
IDMDE and $\Lambda$CDM models. Top: For different values of the dark energy
equation of state $\omega_{\rm x}$. Bottom: For different values of the
coupling parameter $\xi$.

If the dark sectors are not coupled, matter density will be proportional to
$\rho_{\rm m}(a)\propto a^{-3}$. Hence, one can have the following relation
for the matter density

$\dfrac{\Omega_{\rm m}(a)}{\Omega_{\rm
m,0}}=\dfrac{H_{0}^{2}}{a^{3}H^{2}(a)}.$ (30)

However, in the framework of IDMDE models the above equation cannot be used.
Therefore, one has to solve the modified conservation equations for the IDMDE
model (i.e., Eq. (2)). Due to this consideration, the matter density for the
IDMDE model can be represented as follows:

$\dfrac{\Omega_{\rm m}(a)}{\Omega_{\rm
m}}=\dfrac{H_{0}^{2}}{H^{2}(a)}\dfrac{\bigg{[}\rho_{\rm c,0}+\rho_{\rm
b,0}-\dfrac{-1+a^{3(\omega(-1+\xi)+\xi)}}{\omega(-1+\xi)+\xi}\rho_{\rm
x,0}\bigg{]}}{\big{(}\rho_{\rm c,0}+\rho_{\rm b,0}\big{)}a^{3}}.$ (31)

It is known that on small scales, the ISW effect should be small.
Nevertheless, the most significant ISW effects usually occur in large-scale
potentials, where cosmic variance is most problematic. In other words, the ISW
effect is severely limited when it comes to providing information about dark
energy due to cosmic variance at large angular scales. Although it is not
ideal, galaxies can be the convenient option for tracer of large-scale
structures in the late-time Universe. An analysis of the correlation between
ISW temperature and galaxy density can reveal potential wells and hills that
may lead to anisotropies boughn2004 . In addition, in order to better
understand galaxy clusters, large-scale structure surveys should be employed
that calculate the abundance of galaxy clusters as a function of redshift.
Correspondingly, the density of galaxies can be calculated using the following
integral

$\displaystyle\delta_{\rm g}=\int\dfrac{H}{c}f(z)\delta_{\rm m}(k,a)d\chi,$
(32)

where $f(z)=b(z)dN/dz$ is the redshift distribution function of the observed
samples, $b(z)$ is the galaxy bias that relates the visible matter
distribution to dark matter, and $dN/dz$ is the redshift distribution function
of the survey in the comoving distance. Moreover, by using of a CMB map and a
survey of galaxies, one can define the angular auto-correlation and cross-
correlation as

$\displaystyle C^{\rm TT}\equiv\big{<}\Theta_{\rm ISW}\ \Theta_{\rm
ISW}\big{>},$ $\displaystyle C^{\rm Tg}\equiv\big{<}\Theta_{\rm ISW}\
\delta_{\rm g}\big{>}.$ (33)

Note that in the absence of massive neutrinos, the evolution of density
fluctuations is separable in wavenumber $k$ and redshift $z$ on linear scales.
Hence, by employing Eqs. (28), (32), and (33), angular auto-correlation and
cross-correlation can be obtained as follows:

$\begin{array}[]{c}\vspace{0.5cm}C^{\rm TT}_{\rm
ISW}(\mathit{l})=\bigintss_{0}^{\rm\chi_{\rm H}}\dfrac{W_{\rm
T}^{2}(\chi)}{\chi^{2}}\dfrac{H_{0}^{4}}{k^{4}}P(k=\dfrac{\mathit{l}+1/2}{\chi})d\chi,\\\
C^{\rm Tg}(\mathit{l})=\bigintss_{0}^{\rm\chi_{\rm H}}\dfrac{W_{\rm
T}(\chi)W_{\rm
g}(\chi)}{\chi^{2}}\dfrac{H_{0}^{2}}{k^{2}}P(k=\dfrac{\mathit{l}+1/2}{\chi})d\chi,\end{array}$
(34)

where $P(k)$ is the present-time matter power spectrum, $W_{\rm\mathit{l}}$ is
the weight function, and $k=(l+1/2)/\chi$ is obtained through the Limber
approximation of the small angle (i.e., large $\mathit{l}$) Stolzner:2017ged ;
Afshordi2003-ku .

###  IV.1 Redshift Distribution

The redshift distribution of galaxies in galaxy surveys is crucial for
cosmological analyses. In this regard, a spectroscopic and/or multi-band
photometric calibration survey of a small patch of sky typically yields this
information Sanchez2020-tj . As mentioned earlier, to compute the cross-
correlation between CMB and large-scale structure, the redshift distribution
function of the observed galaxies seems to be necessary. It is important to
note that each survey has a unique photometric redshift distribution and
histogram function. In this work, four surveys are considered:

Figure 3: Photometric redshift distributions for the four catalogs used for
the cross-correlation ($dN/dz$ curves are normalised to unity).

(i) Dark Universe Explorer (DUNE) survey is designed to detect both visible
light and three near-infrared bands. The main mission of this survey is to
probe weak gravitational lensing candidates and the ISW effect. The redshift
distribution function of this survey is defined as follow:

$f_{\rm DUNE}(z)=b_{\rm
eff}\left[\dfrac{z_{*}}{\alpha_{*}}\Gamma(\dfrac{3}{\alpha_{*}})\right]^{-1}\bigg{(}\dfrac{z}{z_{*}}\bigg{)}^{2}\exp\left[-\left(\dfrac{z}{z_{*}}\right)^{\alpha_{*}}\right],$
(35)

where $b_{\rm eff}$, $z_{*}$ and $\alpha_{*}$ are free parameters to be
determined, and $\Gamma(x)$ is the Gamma funcion.

(ii) The National Radio Astronomy Observatory (NRAO) Very Large Array (VLA)
Sky Survey (NVSS) is a continuum survey that is covering about $82\%$ of the
sky. The redshift distribution function of this survey has the following form:

$f_{\rm NVSS}(z)=b_{\rm
eff}\,\dfrac{\alpha_{*}^{\alpha_{*}+1}}{\Gamma(\alpha_{*})}\frac{z^{\alpha_{*}}}{z_{*}^{\alpha_{*}+1}}\exp\left(-\dfrac{\alpha_{*}z}{z_{*}}\right).$
(36)

(iii) The Sloan Digital Sky Survey (SDSS) is an imaging and spectroscopic
survey that collect galaxies, quasars, and stars. The redshift distribution of
the main photometric SDSS galaxy sample (SDSS-MphG) is specified as fllow:

$f_{\rm SDSS}(z)=b_{\rm
eff}\,\dfrac{\alpha_{*}}{\Gamma(\frac{m+1}{\alpha_{*}})}\frac{z^{m}}{z_{*}^{m+1}}\exp\left[-\left(\dfrac{z}{z_{*}}\right)^{\alpha_{*}}\right].$
(37)

(iv) The Euclid-like mission will investigate the distance-redshift relations
of galaxies out to redshift $z\sim 2$. It is currently scheduled to launch in
2023. For this survey, the redshift distribution takes the following form
Weaverdyck2017-nw ; Martinet:2015wza

$f_{\rm Euclid-like}(z)=b_{\rm eff}\
\dfrac{3}{2z_{\rm*}^{3}}z^{2}\exp\left[-\left(\dfrac{z}{z_{\rm*}}\right)^{3/2}\right].$
(38)

Table 2: Redshift distribution parameters for DUNE, NVSS, SDSS and Euclid-like surveys. Survey |  $b_{\rm eff}$ |  $z_{*}$ |  $\alpha_{*}$ |  $m$
---|---|---|---|---
DUNE | $1.00$ | $0.640$ | $1.500$ | -
NVSS | $1.98$ | $0.790$ | $1.180$ | -
SDSS | $1.00$ | $0.113$ | $1.197$ | $3.457$
Euclid-like | $1.00$ | $0.700$ | - | -

As can be seen from Eqs. (35)-(38), there are some free parameters that should
be specified. To this end, the best-fit values of these parameters have been
presented in Table 2.

In Fig. 3, we have also shown the photometric normalized redshift distribution
for the mentioned surveys as a function of redshift. Obviously, the NVSS
survey yields the widest redshift coverage, whereas the SDSS survey leads to
the narrowest redshift coverage. Also, DUNE and Euclid-like surveys provide an
intermediate range of redshifts coverage. For more information such as sky
coverage, number of sources, and the analytical approximation, see
Stolzner:2017ged ; Martinet:2015wza ; Weaverdyck2017-nw ; Manzotti:2014kta ;
Condon-NRAO ; Wang:2010zzj ; Douspis:2008xv ; Schiavon:2012fc .

###  IV.2 The ISW Power Spectrum

In Fig. 4, we have shown the amplitude of the ISW-auto power spectrum $C_{\rm
TT}^{\rm ISW}$ as a function of multipole order $l=(10-100)$ for the IDMDE and
$\Lambda$CDM models. The black lines show the theoretical prediction for the
$\Lambda$CDM model, while the red lines indicate the theoretical prediction
for the IDMDE model with the best fit values. As can be seen from the figure,
the IDMDE model behaves similar to the $\Lambda$CDM model for different
phantom dark energy equations of state, whereas, for the quintessence dark
energy equations of state, the amplitude of the ISW-auto power spectrum for
the IDMDE model is higher than that obtained from the $\Lambda$CDM model.
Additionally, we have demonstrated the results for different values of the
coupling parameter $\xi$. The results indicate that the value of the coupling
parameter $\xi$ is inversely proportional to the amplitude of the ISW-auto
power spectrum in the IDMDE model.

Moreover, in Fig. 5, we have depicted the amplitude of the ISW-cross power
spectrum as a function of multipole order $l=(10-100)$ for the DUNE survey,
while considering the IDMDE model, and have compared it with the corresponding
result obtained from the $\Lambda$CDM model. As can be seen from the figure,
the amplitude of the ISW-cross power spectrum for the IDMDE model for all
values of $\omega_{\rm x}$ is higher than that obtained from the $\Lambda$CDM
model. The results exhibit that for the case of quintessence dark energy,
e.g., for $\omega_{\rm x}=-0.6$, the amplitude of the ISW-cross power spectrum
in the IDMDE model can be in the maximum state. Also, it can be inferred from
the figure that the amplitude of the ISW-cross power spectrum in the IDMDE
model changes inversely with the value of coupling parameter $\xi$.

We have also shown the corresponding results of other surveys, i.e., the NVSS,
SDSS, and Euclid-like, in Figs. 6, 7, and 8. Similar to what is obtained for
the DUNE survey, the results demonstrate that the amplitude of the ISW-cross
power spectrum, while considering the IDMDE model, for all values of
$\omega_{\rm x}$ is higher than the one obtained from the $\Lambda$CDM model.
In addition, for the NVSS, SDSS, and Euclid-like surveys, the results confirm
the inverse proportion of the amplitude of the ISW-cross power spectrum with
the value of the coupling parameter $\xi$. As a consequence, the IDMDE model
with larger coupling parameters leads to smaller amplitudes than the one
obtained from the $\Lambda$CDM model.

Figure 4: The amplitude of the ISW-auto power spectrum as a function of
multipole order $l$ for the IDMDE and $\Lambda$CDM models. Top: For different
values of the dark energy equation of state $\omega_{\rm x}$. Bottom: For
different values of the coupling parameter $\xi$.

Figure 5: The amplitude of the ISW cross power spectrum as a function of
multipole order $l$ for the IDMDE and $\Lambda$CDM models. The DUNE survey has
been considered. Top: For different values of the dark energy equation of
state $\omega_{\rm x}$. Bottom: For different values of the coupling parameter
$\xi$.

Figure 6: The amplitude of the ISW cross power spectrum as a function of
multipole order $l$ for the IDMDE and $\Lambda$CDM models. The NVSS survey has
been considered. Top: For different values of the dark energy equation of
state $\omega_{\rm x}$. Bottom: For different values of the coupling parameter
$\xi$.

Figure 7: The amplitude of the ISW cross power spectrum as a function of
multipole order $l$ for the IDMDE and $\Lambda$CDM models. The SDSS survey has
been considered. Top: For different values of the dark energy equation of
state $\omega_{\rm x}$. Bottom: For different values of the coupling parameter
$\xi$.

Figure 8: The amplitude of the ISW cross power spectrum as a function of
multipole order $l$ for the IDMDE and $\Lambda$CDM models. The Euclid-like
survey has been considered. Top: For different values of the dark energy
equation of state $\omega_{\rm x}$. Bottom: For different values of the
coupling parameter $\xi$.

##  V Conclusions

In this work, we have calculated the ISW effect in the framework of the IDMDE
model, and have compared it with the corresponding result obtained from the
$\Lambda$CDM model. The ISW effect is sensitive to the time evolution of the
gravitational potentials sourced by large-scale structures. Thus, such an
effect seems to be significant in the late-time Universe, which is driven via
an accelerated expansion phase. The ISW signal is caused by a stretching
effect that results from an imbalance between the structure growth and the
cosmic expansion. The rise and fall of large-scale gravitational potentials
along the paths of CMB photons in late-time Universe leaves tiny secondary CMB
temperature anisotropies on the primary CMB temperature fluctuations.
Therefore, CMB photons can be studied to determine the cosmic acceleration and
the structure growth rate. In other words, the ISW effect can be used to probe
the physics governing the late-time evolution of the Universe and distinguish
between possible mechanisms generating cosmic acceleration, including dark
energy and modified gravity.

In this regard, we have initially introduced a theoretical framework of IDMDE
models. Additionally, we have determined density and velocity perturbation
equations, while considering a perturbed metric for the inhomogeneous
Universe. Moreover, we have discussed the stability condition of the IDMDE
model. Due to the coupling term in dark energy pressure perturbations, early
time instabilities at large scales play a crucial role in the IDMDE model.
Hence, the non-adiabatic part of the pressure perturbation might grow quickly
at earlier times due to the energy transfer of the IDMDE model, leading to the
rapid growth of curvature perturbation on large scales. The results exhibit
that by restricting the coupling parameter ($\xi>0$), large-scale perturbation
instabilities can be mitigated for a given energy density transfer rate.

In Fig. 1, we have shown the amplitude of the matter power spectrum for the
IDMDE model, and have compared it with the one obtained from the $\Lambda$CDM
model. The results illustrate that the amplitude of the matter power spectrum
in the IDMDE model deviates slightly from the one obtained from the
$\Lambda$CDM model. In other words, deviations from the $\Lambda$CDM model
become larger at lower values of $k$, though they are still much smaller than
a half order of magnitude.

Moreover, in Fig. 2, we have indicated the evolution of the ISW source term,
$\mathcal{F}(a)$, and its derivative with respect to the scale factor in the
late-time Universe for the IDMDE model, and have compared it with that of the
$\Lambda$CDM model. It is known that the ISW signals passing through the
matter-dominated era would be negligible, since the gravitational potential is
static during this era. On the other hand, observational data indicate that
the Universe appears to be very close to the flat geometry. Thus, as
illustrated in Fig. 2, the ISW effect can only be explained by the transition
to a dark energy-dominated era in the late-time Universe. Also, the results
indicate that for the phantom dark energy (i.e., for $\omega_{\rm x}<-1$), the
values of $\mathcal{F}(a)$ in IDMDE model is lower than that of the
$\Lambda$CDM model at at higher redshifts, whereas the quintessence dark
energy (i.e., $\omega_{\rm x}>-1$) in the IDMDE model leads to higher values
of $\mathcal{F}(a)$ than the one obtained from the $\Lambda$CDM model. In
addition, the derivative of the ISW source term in the IDMDE model is similar
to the corresponding result obtained from the $\Lambda$CDM model at higher
redshifts, while it deviates from the $\Lambda$CDM model at lower redshifts.
Additionally, in the corresponding calculation according to coupling parameter
$\xi$, it has been confirmed that for all values of $\xi$, the value of the
ISW source term in the IDMDE model is lower than the result obtained for the
$\Lambda$CDM model.

In Fig. 3, we have shown the normalized redshift distributions of the DUNE,
NVSS, SDSS, and Euclid-like surveys as a function of redshift. This
distributions span a wide range of redshift up to $z\sim 4$. The results
indicate that the NVSS survey leads to the widest redshift coverage, while the
SDSS survey yields to the narrowest redshift coverage. Furthermore, the DUNE
and Euclid-like surveys provide an intermediate range of redshifts coverage.

In the following, we have calculated the ISW power spectrum within the context
of the IDMDE model, and have comapred it with the one obtained from the
$\Lambda$CDM model. It is known that the auto-correlation power spectrum and
the cross-correlation power spectrum between the CMB temperature and large-
scale structures such as galaxies and quasars can be represented in the ISW
effect. In this regard, the auto-correlation contribution is subdominant
compared to the primordial contributions. Therefore, it cannot be detected via
observational data. However, the cross-correlation effect is large enough in a
way that it can be detected by multiple studies Enander:2015vja .

In Fig. 4, we have indicated the amplitude of the ISW auto-power spectrum as a
function of multipole order $l$ for the IDMDE model, and have comapred it with
that obtained from the $\Lambda$CDM model. The results show that for different
phantom dark energy equations of state, the amplitude of the ISW auto-power
spectrum in the IDMDE model behaves similar to the one for the $\Lambda$CDM
model, while, for the quintessence dark energy equations of state, the
amplitude of the ISW-auto power spectrum for the IDMDE model is higher than
that obtained from the $\Lambda$CDM model. Also, the corresponding results by
different values of the coupling parameter demonstrate that $\xi$ is inversely
proportional to the amplitude of the ISW-auto power spectrum in the framework
of the IDMDE model.

Finally, by employing four different surveys, i.e., the DUNE, NVSS, SDSS, and
Euclid-like surveys, we have calculated the amplitude of the ISW-cross power
spectrum as a function of multipole order $l$ for the IDMDE model, and have
compared them with the corresponding result obtained from the $\Lambda$CDM
model. We have also shown the results of these surveys in Figs. 5, 6, 7, and
8. The results indicate that the amplitude of the ISW-cross power spectrum for
the IDMDE model for all values of $\omega_{\rm x}$ is higher than the one
obtained for the $\Lambda$CDM model, while deviations are still less than
$0.1$ order of magnitude. Moreover, for the case of quintessence dark energy,
the amplitude of the ISW-cross power spectrum in the IDMDE model can be in the
maximum state. Also, it turns out that the amplitude of the ISW-cross power
spectrum in the IDMDE model changes inversely with the value of coupling
parameter $\xi$. In addition, it is confirmed that the amplitude of the ISW-
cross power spectrum in the IDMDE model for large values of the coupling
parameter is lower than that obtained from the $\Lambda$CDM model.

## Acknowledgment

M.G. and H.F. gratefully thank Abdolali Banihashemi for the useful
discussions. Also, M.G. would like to appreciate Bjoern Malte Schaefer and
Andras Kovacs for their constructive comments.

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