# Evolutionary Algorithms and Efficient Data Analytics for Image Processing

Farid Ghareh Mohammadi1, Farzan Shenavarmasouleh1, M. Hadi Amini2, and Hamid
R. Arabnia1 1 Department of Computer Science, University of Georgia, Athens,
GA, USA
2 School of Computing & Information Sciences, Florida International
University, Miami, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Steganography algorithms facilitate communication between a source and a
destination in a secret manner. This is done by embedding messages/text/data
into images without impacting the appearance of the resultant images/videos.
Steganalysis is the science of determining if an image has secret messages
embedded/hidden in it. Because there are numerous steganography algorithms,
and since each one of them requires a different type of steganalysis, the
steganalysis process is extremely challenging. Thus, researchers aim to
develop one universal steganalysis to detect all known and unknown
steganography algorithms, ideally in real-time. Universal steganalysis
extracts a large number of features to distinguish stego images from cover
images. However, the increase in features leads to the problem of the curse of
dimensionality (CoD), which is considered to be an NP-hard problem. This COD
problem additionally makes real-time steganalysis hard. A large number of
features generates large datasets for which machine learning cannot generate
an optimal model. Generating a machine learning based model also takes a long
time which makes real-time processing appear impossible in any optimization
for time-intensive fields such as visual computing. Possible solutions for CoD
are deep learning and evolutionary algorithms that overcome the machine
learning limitations. In this study, we investigate previously developed
evolutionary algorithms for boosting real-time image processing and argue that
they provide the most promising solutions for the CoD problem.

###### Index Terms:

Image steganalysis, Image classification, Feature extraction, Feature
selection, Curse of dimensionality, Dimension reduction, NP-hard problem, Data
science

##  I Introduction

$\diamond$ Motivation Nowadays, researchers seek a way to distinguish between
cover images and stego images focusing on a specific steganographic scheme in
a fraction of a second. However, the problem occurs when we do not know the
scheme; which makes steganalysis challenging, especially in real-time
processing. To solve this problem, scientists utilize a universal steganlaysis
which enables them to recognize if a given image is a stego or cover. This
universal steganalysis has to detect all steganographic schemes to become able
to classify images by extracting important features. The more features we
extract, the higher will be the chance of detecting the steganographic schemes
that lead us to classify images accurately. However, the high number of
features generates a high dimensional problem known as the curse of
dimensionality (CoD). This is an important problem to solve because analyzing
the features in details requires a lot of time and is not easy to do in real-
time. Furthermore, prior works [1] [2] have found the CoD problem to be an NP-
hard problem and have attempted to solve it by reducing the dimensionality of
the data.

In this study, we aim to address this problem and present the potential
solutions. Figure 1 illustrates the overall structure of this paper.
Steganography is an advanced skill and communication method that allows hidden
secret messages to be sent via an innocuous multimedia file [3]. The most
common multimedia communications involve images, audios, videos, even text
files, or internet protocols [4]. The cover multimedia, with secret messages
embedded in, is called stego multimedia. The goal of steganography is to make
the stego multimedia and the associated cover look as identical as possible.
Otherwise, the risk of detection would be high. Steganalysis is a challenging
field because all the different steganography schemes must be recognized using
different steganalysis algorithms. Researchers have proposed steganalysis
algorithms for individual steganography schemes. To this end, first, they
extract important features based on preferred extraction strategies [5]. The
goal of these features is to provide a distinguishing border between stego
images and cover images. Further, researchers propose a universal steganalysis
to detect stego images from cover ones without considering the steganographic
scheme, known or unknown, used to embed the hidden messages. A large number of
research studies have proposed to improve steganalysis performance with a
large number of features like CC-C300 with 48600 features [6], in which the
the authors aimed to identify the goodness of utilizing high-dimensional
features, together with ensemble classifiers by comparing them to selected
existing steganalysis algorithms. Another example is PHARM with 12600 features
[7], in which the authors identified an important discovery that in the pixels
of a decompressed JPEG image are not shift invariant and their statistical
characteristics heavily rely on their position in the filtering (e.x. 8*8
grid). These are examples of the universal (blind) approach. However, to do
universal steganalysis like those mentioned above, we need to extract as many
features as possible, which is also the main disadvantage of it. Because this
leads to the problem of the Curse of dimensionality (CoD). CoD is generated
upon extracting or gathering too much information.

Figure 1: A general overview of this study

$\diamond$ Related Works Deep learning tries to enhance the detection accuracy
while considering high dimensional images. For example, Boroumand _et al_ [3]
proposed a novel convolutional neural network (CNN) architecture called SRNet
for steganalysis. SRNet is independent of many newly introduced design
elements yet it still produces a high performance through deep learning.
Although deep learning enhances the performance of steganalysis significantly,
it still has time complexity issues. Along side of deep learning, evolutionary
algorithms (EA) have been proposed for this task and has shown that they can
improve the performance of steganalysis better than deep learning [8].

It is imperative to develop efficient data analytics algorithms to deal with
emerging engineering problems, e.g., critical infrastructures[9]. Evolutionary
algorithms lend themselves as powerful computational algorithms [8] that can
utilize a large number of techniques to solve large-scale engineering and
science problems . EA provides an environment in which steganalysis yields a
low time complexity. Researchers have adopted evolutionary algorithms for a
wide variety of purposes in steganalysis like feature selection, finding the
most probable sub-images in spatial domain, and much more. In this study, we
focus on evolutionary algorithms that have attempted to improve steganalysis,
such as Artificial Bee Colony (ABC) [10]. These studies [11] utilized feature
selection techniques to solve the COD problem [12].

##  II Image steganalysis using evolutionary algorithms

Recently, researchers have sought to optimize the process of steganalysis,
particularly within image domain. Through their research they developed a
successful algorithms which will be discussed further in detail. The general
overview of the combination of evolutionary algorithms and steganalysis are
shown in Figure 2.

The goal of universal steganlaysis is to find a model that performs optimally
on both known and unknown steganography schemes, however, this leads to the
curse of dimensionality. Researchers have attempted to solve the CoD problem
using evolutionary algorithms. EAs aim to minimize the number of feature
dimensions as much as possible while maintaining or improving performance. EAs
always choose an optimal feature dimension, which will likely yield a
significantly higher performance compared to the traditional machine learning
feature selection algorithms.

Scientists deal with the large-scale engineering and science projects that
seek optimization either by finding global maximum or global minimum in a
continuous or discrete space. However, due to the presence of various local
maximums or minimums, traditional machine learning algorithms run at the risk
of reaching them instead, and rarely finding global maximum or minimum. The
likelihood of being stuck in local maximum or minimum is further exacerbated
by CoD. Therefore, researchers leverage EAs to seek global maximum or minimum
solutions by skipping local ones. Universal steganalysis, specifically,
requires evolutionary algorithms to solve this problem in discrete space. EAs
are used as a feature selection tool to decrease the feature dimensions in
discrete space to reach the global maximum solution, i.e. the most relevant
subsets of features which yield the highest accuracy on detecting stego images
from cover images.

$\diamond$ Contributions The main contribution of this paper is to address the
main problem of big data, COD, and determine the most promising solutions and
trends to tackle this problem. Scientists have conducted research studies to
solve COD and make the machine learning algorithms get trained well and robust
enough to yield high performances. So, our contributions are as follows:

$\diamond$ Recognising one of the most promising solutions to COD problem that
is feature selection using evolutionary algorithms. We discuss state-of-the-
art works that solved this problem with and without evolutionary algorithms.

$\diamond$ Discovering the trends for image classification and image
steganlaysis. We aim to find techniques that have used evolutionary algorithms
to optimize the process of steganalysis.

Figure 2: A general flowchart of image steganalysis using evolutionary
algorithms [12] [11]

##  III Image steganalysis using artificial bee colony

The idea of the artificial bee colony (ABC) algorithm, as developed by
Karaboga [10], is more suitable for continuous and discrete optimization
problems. ABC algorithm is a powerful optimization algorithm, which simulates
the foraging behavior of honeybees to find the best food source. ABC works
based on three types of bees: 1) employed bees, 2) onlooker bees and 3) one
scout bee. ABC starts with employed bees computing the goodness of each
solution, then onlooker bees seek the global maximum solution with respect to
the best goodness using cross over, which generates new offspring out of
selected high solutions to create a better solution. This process repeats
itself until no higher maximum goodness is found or the number of crossover
iterations meet the pre-designed limit. After that, the scout bee goes beyond
of the boundary of onlookers to seek global maximum goodness (solution). This
entire process repeats until the condition is met or again it reaches the
maximum number of iterations. Onlooker bees run until the limit condition is
satisfied. Next, one of employed bees converts into a scout. Note that, in
each iteration, Scout performs mutation, which is the final phase of
evolutionary algorithms.

Researchers have proposed a new customized version of ABC to solve their
discrete engineering problems such as CoD. There are plenty of customized ABC
applications particularly for image processing, including but not limited to
[13] [14]. Owing to its simplified implementation, ABC is a widely-used
algorithm with flexibility, robustness and ability to exploit local solutions.
It is also capable of exploring the whole sample space to find the global
solution. The most important advantages of ABC are its power to solve any
problem and its simplicity, which is duo to the minimum number of parameters
to be tuned. Therefore, ABC seems to be far better than other evolutionary
algorithms like Genetic Algorithm (GA). However, ABC suffers from a low
convergence rate in sequential processing and slow execution processing to
compute precise solutions. These are the reasons that ABC may not obtain the
best solution.[15]

###  III-A Image steganalysis using Feature Selection based on ABC

Mohammadi and Saniee Abadeh in [16] first presented the idea of dimension
reduction using evolutionary algorithms like ABC, then in [13] proposed a new
steganalysis method named IFAB, which helps steganalysis to enhance the
performance of distinguishing stego images from the cover images. Authors
adopted a wrapper-based feature selection technique by customizing the
original ABC into the discrete ABC algorithm. ABC selects the feature subsets,
and a classifier is employed to evaluate every feature subset generated by the
algorithm. According to the IFAB algorithm for feature selection, the process
of feature selection solves the problem of CoD by leveraging employed bees,
onlooker bees and a scout bee.

According to the table I, Mohammadi’s and Saniee’s study [16], and a more
comprehensive study [13] would be considered the firsts to apply the EAs on
steganalysis. IFAB significantly minimizes the feature dimension and selects
80 features out of 686 SPAM (Subtractive Pixel Adjacency Matrix) attributes.
IFAB also shrinks the CC-PEV (Cartesian Calibrated features by PEVny) feature
dimension well by selecting 250 features out of 548. The Final accuracy of
IFAB is 60.98 and 68.22 for SPAM and CC-PEV datasets, respectively.

Table I:  Research study Comparison

Paper | Evol. alg. | Proposed method | FS type | Year
---|---|---|---|---
[13] | ABC | IFAB | Wrapper | 2014
[17] | ABC | IFAB-KNN | Wrapper | 2014
[14] | ABC | RISAB | Wrapper | 2017
[18] | PSO | HYBRID | Wrapper &Filter | 2016
[19] | PSO | APSO | Filter | 2018
[20] | PSO | PSO-AUC | Filter | 2016
[21] | PSO | GLBPSO | Wrapper | 2017
[22] | PSO | MI-APSO | Wrapper∗ | 2018
[23] | FA | DyFA | Wrapper | 2018
[24] | GWO | GRASP-BGWO | Wrapper | 2019
[25] | GWO | LFGWO | Wrapper | 2019
[26] | PGO | GLCM-PGO | Wrapper∗ | 2019

∗ goes to special kind of wrapper that select features by eliminating the non-
informative features. The rest of papers select subsets of features regularly

###  III-B comparative studies

Mohammadi and Saniee Abadeh in [17] presented an improved evolutionary
algorithm approach for image steganalysis to enhance IFAB, named IFAB-KNN.
IFAB-KNN also provided a wrapper based feature selection algorithm. The
authors embedded K-Nearest Neighbor(KNN) into ABC to help it evaluate each
subset of features more carefully. Concretely, KNN plays a significant role as
a fitness function for evaluating subset features in ABC. IFAB-KNN outperforms
IFAB with the updated tuning parameters while keeping the same number of
selected features.

Mohammadi and Saniee Abadeh in [14] proposed a new hybrid approach to
steganalysis named, region based Image Steganalysis using Artificial Bee
colony (RISAB). RISAB enables ABC to search in image space, particularly
spatial domain, to find the most probable sub-image to carry the hidden
messages. As a matter of fact, the likelihood of embedding messages in a sub-
image is high if the amount of intensity and energy is greater than the other
parts of the image. RISAB is a combination of applying IFAB over the whole
image, and the sub-image selected by ABC. The first main step and goal of
their paper is to investigate the whole image to identify such a high energy
sub-image leveraging a customized ABC to go through the spatial domain of the
image. Then, they extract features twice. First, they extract features from
the whole given image. Second, they extract the same features from the sub-
image found in the earlier phase. They apply the same features that IFAB [13]
presents, which are the best subset of features for feature extractors, SPAM
and CC-PEV. Having extracted the features from the whole image and sub-image,
researchers merge the datasets for both feature extractors separately. The
final dataset for SPAM and CC-PEV involves 160 and 500 features, respectively.
In either datasets, features visualize the instances in a way that the
classifier will be able to train and find a high-performance model. Since
authors considered both the whole images and the associated sub-images, their
approach greatly outperforms IFAB.

In short, Mohammadi and Saniee Abadeh applied ABC on two different areas:
Numeric data and Spatial domain. They proposed two unique methods, so called
IFAB and RISAB. Both of them greatly increased the performance of steganalysis
due to their dimension reduction properties while solving the CoD problem. It
is also worth noting that in the state-of-the-art work ABC is being used to
detect Malwares. [27]

##  IV Image steganalysis using particle swarm optimization

Particle Swarm Optimization (PSO) [28] is an optimization method to solve non-
linear optimization problems, presented by Kennedy and Eberhart. PSO is
inspired by the behaviour of a flock of birds or fish swarms. Several
researchers proposed to extend a version of PSO to solve optimization
problems, particularly for steganalysis. [18] [19]

Chhikara _et. al_ [18] proposed a new hybrid approach using PSO for
steganalysis named HYBRID. They proposed a hybrid filter and wrapper based
feature selection approach to deal with the computational complexity in image
steganalysis. The researchers examined their approach for attacking different
steganography algorithms.The customized PSO improved the classification
accuracy of detecting stego images and cover images, while reducing time
complexity of the process as well. The authors enhanced the accuracy of
classification significantly for SPAM and CC-PEV up to 10 percent and 14
percent, respectively while examining the HYBRID algorithm on the different
steganography algorithms.

From another point of view towards steganalysis, a few researchers presented a
novel approach to solve the problem of discovering the message embedded in a
covert multimedia using Area Under Curve (AUC) as a new measure for fitness
function to evaluate the selected features [20] [19]. They considered another
filter-based feature selection algorithm for steganalysis. The authors [19]
introduced an Adaptive inertia weight-based PSO called APSO. APSO is adopted
for steganalysis with two main phases. First, there is a feature selection
process, which decreases the dimensionality of features; second making a final
model to recognize stego images from cover images using the selected subsets
of features. APSO used a novel fitness function which provides Area Under
Curve (AUC) to evaluate the selected feature subset. To evaluate the given
approach, authors used several classifiers such as SVM, DT, NB, and KNN. The
SVM obtained the best result in comparison with the others when the hyper-
plane experienced the largest distance between support vectors of given stego
and cover classes. APSO-AUC decreased the feature dimension and selected top
140 features out of 686 SPAM attributes and 363 features out of 548 features.
The Final accuracy of APSO using SVM yields 82.62 and 87.72 for SPAM and CC-
PEV datasets, respectively. These methods yield a better result using PSO in
comparison with IFAB and RISAB because the time complexity has been reduced on
large data sets.

Chikara and Kumari in [21] proposed another version of PSO for steganalysis.
The authors introduced a new wrapper-based feature selection, named Global
Local PSO (GLBPSO). They adopted neural networks to evaluate the selected
feature subsets by GLBPSO. The GLBPSO algorithm improved the standard PSO by
having the best global and local PSOs simultaneously. In [21], researchers
take Chen’s approach [29], and decrease its dimension by selecting the best
feature subset. The prediction performance of GLBPSO provides no more than 7
percent improvement in comparison with the basic results, where performance is
calculated based on all features. GLBPSO reduced features down to 282 features
out of 486 features. Furthermore, Kaur and Singh [22] proposed a new feature
selection leveraging mutual information and adaptive PSO (MI-APSO) using area
under curve for image steganalysis. MI-APSO was also inspired by IFAB as a
feature selection technique and further improved the performance of image
steganalysis.

##  V Image steganalysis using firefly algorithm

Yang introduced a new evolutionary algorithm in [30] named Firefly algorithm
(FA) which is inspired by the flashing behaviour of the fireflies. FA aims to
attract other objects, namely, their mates, through their light. Yang
presented this algorithm based on the following assumption: all fireflies are
the same gender. It means that the possibility of attacking other mates is the
same. Also the rate of attraction has a direct relation with the light; the
more light, the higher the rate of attraction. Furthermore, if there are no
bright fireflies, mates tend to move towards any of the fireflies randomly.
Researchers developed and tailored FA to work well with steganalysis.

It is worth mentioning that firefly algorithm can improve the convergence rate
problem of ABC. In this section we investigate the application of FA to
steganalysis.

Chikara _et al_[23] proposed a new dynamic algorithm to stegalysis, named
Dynamic firefly (DyFA). They customized firefly algorithm for universal
steganalysis. Feature Selection (FS) is the main role of DyFA, which
alleviates the computational complexity of universal steganalysis. FA provides
two important parameters alpha and gamma, which help FA to converge faster per
each iteration. Basically, tuning these parameters is important for FA to
accelerate the process of feature selection. In addition, the researchers
apply a hybrid FA as DyFA, which combines the filter method (t-test and
regression) and wrapper method for FS. The results reveal that DyFA reduces
features by about 77$\ -\ $93 percent of the original feature dimension, which
improves the accuracy of distinguishing stego images from cover images by
about 2$\ -\ $10 percent. Like other discussed studies, the scientists plug in
CC-PEV and SPAM features to DyFA to decrease their feature dimensions.
Accuracy for SPAM has enhanced by 9$\ -\ $15 percent, and CCPEV shows an
improvement of 10$\ -\ $13 percent. The results reveal that DyFA outperforms
IFAB. However, DyFa has still not been tuned to decrease time complexity.

##  VI Image steganalysis using grey wolf optimizer

Mirjalili _et al_ [31] proposed a new evolutionary algorithm based on the
concept of grey wolf packs named Grey Wolf Optimizer (GWO). GWO outperforms
other evolutionary algorithms in searching for the solution of nonlinear
functions in multidimensional space. The GWO algorithm mimics the leadership
hierarchy and hunting mechanism of grey wolves. The leadership hierarchy is
simulated based on the behavior of different types in pack of wolves, such as
the alpha, beta, delta, and omega. Basically, the GWO, inspired by grey wolf
hunting procedure, involves three main phases. First, seeking for prey, which
is called exploration, encircling prey and lastly, attacking prey, which
provides exploitation. This process ensures that scientists identify the
global optimum to tackle optimization problems.

The GWO is a population based algorithm which employs a collective behavior of
wolves for seeking the optimal solution. GWO starts with the exploration of
search space and exploits gradually to identify the local optimum. GWO
provides the most important parameters for adjusting step size by a parameter
named "A", and controlling convergence by the exploration and exploitation
parameters. The GWO is well-known for its low computational cost. However, it
still has some limitations like slow convergence rate and getting stuck in
local optima at times. It is obvious that controlling the exploration and
exploitation trade-off, defined by "A", plays the main role in GWO.

Pathak _et al_ [25] proposed a new version of GWO for solving steganalysis
using feature selection called levy flight-based grey wolf optimization
(LFGWO). LFGWO overcomes the limitations of GWO and aims to seek the most
prominent features in feature space. The fitness function includes one of the
decision tree classifiers, called random forest. The main advantage of LFGWO
compared with the relevant literature of evolutionary algorithms is its better
convergence precision. They also examined 5 different classifiers - SVM, LDA,
RF, KNN and ZeroR - to analyze the performance of image steganalysis over
selected features. It is worth mentioning that the LDA here is used as a
classifier. LDA also is used as a feature reduction algorithm [12]. The LFGWO
extracted 84 out of 686 and 89 out of 1000 features from SPAM and AlexNet
extracted features, respectively. Although LFGWO obtained better results than
IFAB[13] and and advanced IFAB, called IFAB-KNN [17], IFAB and IFAB-KNN
extracted less features, i.e. 80 out of 686 for SPAM. The results reveal that
the 84 selected features were more relevant than those 80 features,
particularly for SPAM.

Veena _et al_ [24] introduced an optimized method to attack a well-known
steganography algorithm, so called Least Significant Bit (LSB). The authors
aimed to seek optimal features by the proposed hybrid technique of Greedy
Randomized Adaptive Search $\ -\ $ Binary Grey Wolf Optimization (GRASP-BGWO).
They succeeded in enhancing classification accuracy of the ensemble logistic
regression classifier while shrinking the feature size.

Veena _et al_ [24] introduced five different spatial LSB algorithms: LSB
Replacement (LSBR), LSB Matching (LSBM), LSBM Revisited (LSBMR), Two bit LSBR
(LSBR2) and Modulo 5 LSBR (LSBRmod5). The authors applied GRASP-BGWO and
observed that the detection process is highly dependent on three important
properties: training algorithms, payloads and features. According to their
research study, GRASP-BGWO excelled all existing works like SRM [32] (Spatial
Rich Mode), PSRM (Projected SRM) [33] and SPAM, even in low volume payload per
pixel. GRASP-BGWO improved performance up to 13 percent. In addition to the
given GRASP-BGWO and it’s performance on LSBR, Shojae Chae-ikar and Ashmadi
[34] presented a novel ensemble Similarity weight (SW)-based image
steganalysis. The ensemble SW steganalysis comprises three main steps. First
is SW analysis, the second step is to adopt SVM classifier, and the third step
makes a decision of stego vs cover. In SW analysis, the researchers compute
the pixel and channel similarity weights of the given object and generate PSW
and CSW datasets. Then, they compare the datasets with their corresponding
reference profiles. The last step takes the generated datasets from the second
step to make the final decision on whether the image is stego or cover.

##  VII Image steganalysis using Pine Growth Optimization

Pine Growth optimizer (PGO) is inspired by pine tree growth pattern. Pine is a
type of tree that creates two branches at each stage (i.e. time period or
height) while growing. Initially, it has 0 branches (k=0, k denotes the number
of branches on the tree). After that, the tree grows to a particular height,
and it creates branches on two opposite sides (first set of branches as k=1).
A new pair of branches are produced at every stage accordingly (k=2,3,…). PGO
performs the feature selection process based on the pine’s growth pattern. It
is noteworthy that the pine tree growth may be altered based on the problem
requirements. [26].

The main advantages of PGO are as follows [26]: 1) It helps any classifier
distinguish stego images from cover images. 2) Selecting the most relevant
features using PGO helps any classifier improve steganlaysis performance. 3)
PGO selects the relevant subsets of features that are sensitive enough to work
with the steganographic scheme. 4) Based on this study, PGO also acts as a
solution for CoD.

##  VIII Generative dataset for steganalysis

In the aforementioned research studies, authors mostly took advantage of well-
known raw image datasets, such as Breaking Out Steganography System (BOSS)
(BOSSbase 1.01), 1000 Pictures, and Photobucket in order to extract features
for a goal of classification, specifically image steganalysis.

$\diamond$ SPAM Researchers have adopted the most common feature extractors.
One of them is proposed in a research study presented by Pevny [35] for
Subtractive Pixel Adjacency Matrix (SPAM) feature extraction from the spatial
domain of the digital images. SPAM has two clusters of features: one group
derived from markov-horizontal and vertical and another extracted from markov-
major and minor diagonal features. Merging these two groups brings the total
number of SPAM features to 686.

$\diamond$ CC-PEV Researchers also propose CC-PEV features [36], from
transforming the domain of the digital images. It worth mentioning that CC-PEV
involves 548 feature vectors which provides 274 features from the original
given image and the second half from the calibrated respective image[5].

Figure 3: Frequency usage of Feature extractors

$\diamond$ AlexNet The third feature extractor found in literature is AlexNet
[37], which is a deep convolutional neural network that extracts the features
directly from images.

$\diamond$ SRM and PSRM There are still other state-of-the-art feature
extractors that researchers take into account: Spatial Rich Model (SRM) [32],
which provides 34,671 features; projected SRM (PSRM), which extracts 12,870
features [33]; and CHEN[29], which is the first common feature extractor that
generates 486 features.

##  IX Algorithms efficiency

The learning process to train a model using training data plays a pivotal role
in machine learning. The trained model helps researchers predict test data
with high precision. When the number of features dimensions are too high, the
learning process takes a long time or the trained model will perform poorly
due to the CoD. Due to mentioned obstacles, data analytics phases take longer
time and do not match well with visual computing solutions. To do real-time
processing, a large number of studies have been explored in this paper to
provide a trend in the world of image processing. All the studies state, and
it goes without saying, that when higher number of resources and processors
are available, the chance of computing in real-time increases [13]. While
reviewing the above-mentioned studies, we found a direct relationship between
the efficiency of the algorithms for data analytics on visual computing and
the richness of systems’ hardware.

##  X Further optimization and efficient data analytic trends

All scientific studies discussed in this paper rely on an optimization
procedure implemented over evolutionary algorithms with imaging applications.
Mohammadi and Saniee abadeh mostly focused on EAs and provided a comprehensive
analysis of image steganalysis in [4]. Mohammadi _et al_ [12] also
investigated a large number of research studies to tackle high dimensional
data using evolutionary algorithms and their applications on real-world
problems. Further, Farahani _et al_ [38] aimed to tackle unsupervised domain
adaptation and worked on groups of the domain adaptation from different
prospectus. Scientists aimed to collect the most important and significant
research studies with a high impact on large-scale engineering and science
problems [39]. Aside from EAs, many researchers proposed feature selection
algorithms based on machine learning and deep learning to tackle high
dimensional data in various fields and all of them reported state-of-the-art
results from them [40] [41] [42] [43] [44].

Additionally, Kumar _et al_ [45] presented a randomised smoothing technique to
deal with CoD. There is also a wide range of distributed optimization
algorithms for complex engineering problems, e.g.image captioning using[46],
distributed methods for smart cities applications [47] and decentralized
artificial intelligence for energy systems [48]. These methods lend themselves
as efficient computing tools to deal with large-scale decision making problems
in a decentralized/distributed fashion.

Figure 4: Feature extractors usage within 2013-2019

##  XI Discussion and conclusion

In recent years, there has been significant development in image steganalysis,
and researchers are proposing a universal steganalysis to attack all
discovered and upcoming steganography algorithms successfully. However, image
steganalysis algorithms always suffer from shortcomings; according to the
papers considered in this study, the common problem with state of the art
image steganalysis algorithms is the curse of dimensionality. This problem may
cause the algorithms to fail to make a proper model and distinguish stego
images from cover ones. Not only that, but this problem will also cause the
time complexity of the learning process to increase.

It is worth mentioning that the majority of researchers use SPAM and CC-PEV as
their feature extractors according to figure 3 and figure 4. Although SPAM is
the most commonly used feature extractor, other feature extractors, like SRM
and PSRM, have shown good results as well.

Figure 5: Evolutionary algorithms usage within last decade for dimension
reduction

Evolutionary algorithms are the most common algorithms associated with high-
dimensional data and complex engineering and science problems. In this study,
we explored those algorithms that were adopted to solve the shortcomings of
image steganalysis with respect to the curse of dimensionality. Five
evolutionary algorithms succeeded in alleviating this problem while improving
the performance of image steganalysis: Artificial Bee Colony, Particle Swarm
Optimization, Firefly Algorithms, Grey Wolf Optimizer and Pine Tree
Optimization. Figure 5 shows the use of EAs for feature selection in image
steganalysis within years 2013-2019.

EAs are used for optimization purposes to ensure real-time image processing
capabilities by tackling issues such as CoD. To that end, they are mostly
adopted for feature selection. Table I illustrates that the majority of EAs
used for steganalysis work for different types of feature selection. The table
shows that most research studies of EAs use wrapper-based feature selection,
followed by filter-based feature selection.

Evolutionary algorithms have been adopted for image steganalysis since 2013
[16]. Researchers started using ABC, a powerful optimization tool tailored for
image processing. A number of research studies with innovative solutions have
been proposed since then to improve the aforementioned traditional
evolutionary algorithms. However, they also had their own shortcomings causing
the algorithm to fail in properly converging to find the global optimum. There
are reasons for failure such as: sensitivity to initial and noisy conditions,
being greedy, having biased assumptions, lacking fine-tuned parameters, and
being undeterministic. Even though the majority of EAs suffer from high
computational costs, they are quite robust in finding the best solution.

The goal of this paper was to provide a comprehensive overview of
evolutionary-based image processing for researchers who are currently working
or will work on real-time image processing, particularly in image
steganalysis. In this research study, we reveal that a limited number of
evolutionary algorithms have been utilized for image steganalysis thus far.
However, a large number of evolutionary algorithms have yet to be considered
to overcome real-time image processing challenges. Therefore, future works
should explore other evolutionary algorithms and try to extend them in order
to make them suitable for real-time image processing.

## References

  * [1] Nikolaos Kouiroukidis and Georgios Evangelidis.  The effects of dimensionality curse in high dimensional knn search.  In 2011 15th Panhellenic Conference on Informatics, pages 41–45. IEEE, 2011.
  * [2] Stefano Ermon, Carla Gomes, Ashish Sabharwal, and Bart Selman.  Taming the curse of dimensionality: Discrete integration by hashing and optimization.  In International Conference on Machine Learning, pages 334–342, 2013.
  * [3] Mehdi Boroumand, Mo Chen, and Jessica Fridrich.  Deep residual network for steganalysis of digital images.  IEEE Transactions on Information Forensics and Security, 2019.
  * [4] Farid Ghareh Mohammadi and Mohammad Saniee Abadeh.  A survey of data mining techniques for steganalysis.  In Hedieh Sajedi, editor, Recent Advances in Steganography, chapter 1. IntechOpen, Rijeka, 2012.
  * [5] Jan Kodovskỳ and Jessica Fridrich.  Calibration revisited.  In Proceedings of the 11th ACM workshop on Multimedia and security, pages 63–74. ACM, 2009.
  * [6] Jan Kodovskỳ and Jessica Fridrich.  Steganalysis in high dimensions: Fusing classifiers built on random subspaces.  In Media Watermarking, Security, and Forensics III, volume 7880, page 78800L. International Society for Optics and Photonics, 2011.
  * [7] Vojtěch Holub and Jessica Fridrich.  Phase-aware projection model for steganalysis of jpeg images.  In Media Watermarking, Security, and Forensics 2015, volume 9409, page 94090T. International Society for Optics and Photonics, 2015.
  * [8] Thomas Bäck, David B Fogel, and Zbigniew Michalewicz.  Evolutionary computation 1: Basic algorithms and operators.  CRC press, 2018.
  * [9] M Hadi Amini, Ahmed Imteaj, and Panos M Pardalos.  Interdependent networks: A data science perspective.  Patterns, page 100003, 2020.
  * [10] Dervis Karaboga.  An idea based on honey bee swarm for numerical optimization.  Technical report, Technical report-tr06, Erciyes university, engineering faculty, computer …, 2005.
  * [11] F Ghareh Mohammadi, M Hadi Amini, and Hamid R. Arabnia.  Applications of nature-inspired algorithms for dimension Reduction: Enabling efficient data analytics.  In Optimization, Learning, and Control for Interdependent Complex Networks. Springer, 2019.
  * [12] F Ghareh Mohammadi, M Hadi Amini, and Hamid R. Arabnia.  Evolutionary computation, optimization and learning algorithms for data science.  Optimization, Learning, and Control for Interdependent Complex Networks, Springer, 2019.
  * [13] F Ghareh Mohammadi and M Saniee Abadeh.  Image steganalysis using a bee colony based feature selection algorithm.  Engineering Applications of Artificial Intelligence, 31:35–43, 2014\.
  * [14] F Ghareh Mohammadi and H Sajedi.  Region based image steganalysis using artificial bee colony.  Journal of Visual Communication and Image Representation, 44:214–226, 2017.
  * [15] Eduardo Gerhardt and Herbert Martins Gomes.  Artificial bee colony (abc) algorithm for engineering optimization problems.  In International Conference on Engineering Optimization, pages 1–11, 2012.
  * [16] F Ghareh Mohammadi and M Saniee Abadeh.  An improved image steganalysis using a novel feature selection algorithm based on artificial bee colony.  pages 593–598, 2013.
  * [17] Farid Ghareh Mohammadi and Mohammad Saniee Abadeh.  A new metaheuristic feature subset selection approach for image steganalysis.  Journal of Intelligent & Fuzzy Systems, 27(3):1445–1455, 2014\.
  * [18] Rita Rana Chhikara, Prabha Sharma, and Latika Singh.  A hybrid feature selection approach based on improved pso and filter approaches for image steganalysis.  International Journal of Machine Learning and Cybernetics, 7(6):1195–1206, 2016.
  * [19] Ali Adeli and Ali Broumandnia.  Image steganalysis using improved particle swarm optimization based feature selection.  Applied Intelligence, pages 1–14, 2018.
  * [20] Vahid Rostami and Azar Shahmoradi Khiavi.  Particle swarm optimization based feature selection with novel fitness function for image steganalysis.  In 2016 Artificial Intelligence and Robotics (IRANOPEN), pages 109–114. IEEE, 2016.
  * [21] Meena Kumari and Rita Chhikara.  Blind image steganalysis using neural networks and wrapper feature selection.  In 2017 International Conference on Computing, Communication and Automation (ICCCA), pages 1065–1069. IEEE, 2017.
  * [22] Jasmanpreet Kaur and Singara Singh.  Feature selection using mutual information and adaptive particle swarm optimization for image steganalysis.  In 2018 7th International Conference on Reliability, Infocom Technologies and Optimization (Trends and Future Directions)(ICRITO), pages 538–544. IEEE, 2018.
  * [23] Rita Rana Chhikara, Prabha Sharma, and Latika Singh.  An improved dynamic discrete firefly algorithm for blind image steganalysis.  International Journal of Machine Learning and Cybernetics, 9(5):821–835, 2018.
  * [24] ST Veena, S Arivazhagan, and W Sylvia Lilly Jebarani.  Improved detection of steganographic algorithms in spatial lsb stego images using hybrid grasp-bgwo optimisation.  In Nature Inspired Optimization Techniques for Image Processing Applications, pages 89–112. Springer, 2019.
  * [25] Yadunath Pathak, KV Arya, and Shailendra Tiwari.  Feature selection for image steganalysis using levy flight-based grey wolf optimization.  Multimedia Tools and Applications, 78(2):1473–1494, 2019.
  * [26] John Babu Guttikonda and Sridevi R.  A new steganalysis approach with an efficient feature selection and classification algorithms for identifying the stego images.  Multimedia Tools and Applications, 78(15):21113–21131, Aug 2019\.
  * [27] Farid Ghareh Mohammadi, Farzan Shenavarmasouleh, M Hadi Amini, and Hamid R Arabnia.  Malware detection using artificial bee colony algorithm.  In Adjunct Proceedings of the 2020 ACM International Joint Conference on Pervasive and Ubiquitous Computing and Proceedings of the 2020 ACM International Symposium on Wearable Computers, pages 568–572, 2020.
  * [28] JAMES Kennedy.  Eberhart, r.: Particle swarm optimization.  In Proceedings of IEEE international conference on neural networks, volume 4, pages 1942–1948. IEEE Press, 1995.
  * [29] Chunhua Chen and Yun Q Shi.  Jpeg image steganalysis utilizing both intrablock and interblock correlations.  In 2008 IEEE International Symposium on Circuits and Systems, pages 3029–3032. IEEE, 2008.
  * [30] Xin-She Yang.  Nature-Inspired Metaheuristic Algorithms, chap 10,2nd ed.  Luniver Press, 2008.
  * [31] Seyedali Mirjalili, Seyed Mohammad Mirjalili, and Andrew Lewis.  Grey wolf optimizer.  Advances in engineering software, 69:46–61, 2014.
  * [32] Jessica Fridrich and Jan Kodovsky.  Rich models for steganalysis of digital images.  IEEE Transactions on Information Forensics and Security, 7(3):868–882, 2012.
  * [33] Vojtech Holub and Jessica Fridrich.  Random projections of residuals for digital image steganalysis.  IEEE Transactions on Information Forensics and Security, 8(12):1996–2006, 2013.
  * [34] Saman Shojae Chaeikar and Ali Ahmadi.  Ensemble sw image steganalysis: A low dimension method for lsbr detection.  Signal Processing: Image Communication, 70:233–245, 2019.
  * [35] Tomáš Pevny, Patrick Bas, and Jessica Fridrich.  Steganalysis by subtractive pixel adjacency matrix.  IEEE Transactions on information Forensics and Security, 5(2):215–224, 2010.
  * [36] Tomas Pevny and Jessica Fridrich.  Merging markov and dct features for multi-class jpeg steganalysis.  In Security, Steganography, and Watermarking of Multimedia Contents IX, volume 6505, page 650503. International Society for Optics and Photonics, 2007.
  * [37] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton.  Imagenet classification with deep convolutional neural networks.  In Advances in neural information processing systems, pages 1097–1105, 2012.
  * [38] Abolfazl Farahani, Sahar Voghoei, Khaled Rasheed, and Hamid R Arabnia.  A brief review of domain adaptation.  arXiv preprint arXiv:2010.03978, 2020.
  * [39] Farzan Shenavarmasouleh and Hamid R Arabnia.  Drdr: Automatic masking of exudates and microaneurysms caused by diabetic retinopathy using mask r-cnn and transfer learning.  arXiv preprint arXiv:2007.02026, 2020.
  * [40] Yong Zhang, Shi Cheng, Yuhui Shi, Dun-wei Gong, and Xinchao Zhao.  Cost-sensitive feature selection using two-archive multi-objective artificial bee colony algorithm.  Expert Systems with Applications, 137:46–58, 2019.
  * [41] Lucija Brezočnik, Iztok Fister, and Vili Podgorelec.  Swarm intelligence algorithms for feature selection: a review.  Applied Sciences, 8(9):1521, 2018.
  * [42] Ehsan Asali, Farzan Shenavarmasouleh, Farid Ghareh Mohammadi, Prasanth Sengadu Suresh, and Hamid R Arabnia.  Deepmsrf: A novel deep multimodal speaker recognition framework with feature selection.  arXiv preprint arXiv:2007.06809, 2020.
  * [43] Farid Ghareh Mohammadi, Hamid R Arabnia, and M Hadi Amini.  On parameter tuning in meta-learning for computer vision.  In 2019 International Conference on Computational Science and Computational Intelligence (CSCI), pages 300–305. IEEE, 2019.
  * [44] Nihal Soans, Ehsan Asali, Yi Hong, and Prashant Doshi.  Sa-net: Robust state-action recognition for learning from observations.  In 2020 IEEE International Conference on Robotics and Automation (ICRA), pages 2153–2159. IEEE, 2020.
  * [45] Aounon Kumar, Alexander Levine, Tom Goldstein, and Soheil Feizi.  Curse of dimensionality on randomized smoothing for certifiable robustness.  arXiv preprint arXiv:2002.03239, 2020.
  * [46] Soheyla Amirian, Khaled Rasheed, Thiab R Taha, and Hamid R Arabnia.  Image captioning with generative adversarial network.  In 2019 International Conference on Computational Science and Computational Intelligence (CSCI), pages 272–275. IEEE, 2019.
  * [47] M Hadi Amini, Javad Mohammadi, and Soummya Kar.  Distributed holistic framework for smart city infrastructures: Tale of interdependent electrified transportation network and power grid.  IEEE Access, 7:157535–157554, 2019.
  * [48] Ahmed Imteaj, M Hadi Amini, and Javad Mohammadi.  Leveraging decentralized artificial intelligence to enhance resilience of energy networks.  arXiv preprint arXiv:1911.07690, 2019.


# A spherically symmetric gravitational solution of nearly conformally flat
metric measure space

S.Oghbaiee<EMAIL_ADDRESS>Department of Physics, Shahid Beheshti
University, Tehran 1983969411, Iran  N. Rahmanpour<EMAIL_ADDRESS>Department of Computer Science, University of Southern California Los Angeles,
CA, USA  H. Shojaie<EMAIL_ADDRESS>Department of Physics, Shahid
Beheshti University, Tehran 1983969411, Iran

###### Abstract

In this manuscript, we study the nearly flat approximation of a conformally
invariant gravitational theory in metric measure space (MMS). In addition, we
investigate the transformation of the energy-momentum tensor in this context
and obtain the vacuum solution of MMS and its weak field limit in the
spherically symmetric coordinates. We show that while it is already a vacuum
solution, it can simulate dark matter when restricted to the framework of
general relativity, i.e., a symmetry-broken conformal frame. This is done by
employing a density function which is an essential part of MMS. We derive an
equation for the density function for a general profile of a rotation curve
obtained from observations. Specifically, the density function corresponding
to two well-known profiles PSS and NFW are provided.

Keywords

Metric measure space; Conformal gravity; Linearized gravity; Flat rotation
curve.

##  I Introduction

The elegant simplicity in formulation, following a delicate arrangement of
deeply thoughtful principles, together with a robust mathematical framework,
all backed by an ingenious way of reasoning, has made Einstein’s theory of
gravitation, namely general relativity (GR), a relatively special theory of
all time. Although on the way of its construction, the author’s aim was mainly
focused on the generalization of the concept of the observers rather than
justifying little observational evidence available theretofore Einstein
(1915), surprisingly, it has so far outdistanced all of its old and new
competitors not only in successfully passing obtainable tests from weak to
strong regimes but in intriguingly making predictions of gravitational
phenomena unknown at the time Will (2014).

Among all features of GR, observer independence is maybe the most important
aspect of the theory. This generalizes the concept of inertial observers to
all physical observers and even beyond, to formulate underlying physics in any
arbitrary and bias-free framework. This property of the theory, dubbed
diffeomorphism invariance, seems to be a generic property of every geometry-
based theory of physics. However, we are not sure whether the roads to
formulate theories in this way would be paved clear ones. This may be the
origin of unsatisfactory inconsistencies between GR and quantum mechanics
(QM); inconsistencies with models formulated in the latter, such as elementary
particle physics, as well as recipes in this context like Quantum Field
Theories (QFT).

On the other hand, regardless of these inconsistencies, some hints motivate
one to step forward and think of possible improvements to GR. It is not very
hard to mention some deficiencies in physics that GR may be accountable for.
From the observational point of view, for instance, the justification of dark
energy, a clear explanation for dark matter, and a proposal to overcome
$H_{0}$ tension are some topics in this context. Although GR is not credited
for all these problems, it should not be plausibly regarded as entirely
irresponsible; GR has a few exact solutions, and almost always, when applied
to the real world, the approximations come to the stage.

Regardless of this seemingly impenetrable barrier towards unification, GR has
never been historically regarded as an ultimate version of a theory of
gravitation, even not by its inventor. This can be witnessed by first attempts
mainly focused on the constructions of more general geometrized theories to
include Electrodynamics (EM) as well Weyl (1918); Weyl  _et al._ (1929). The
non-stop efforts towards unification have had many ups and downs, led to the
rising of many hypotheses and theories, and have always been in the mainstream
of theoretical physics. The geometrization of other fundamental forces in the
context of gauge theories and the emergence of theories and contexts such as
String Theory and Loop Quantum Gravity (LQG) are just a few examples of these
endeavors.

All these lead to the question of whether GR is a special and symmetry-broken
case of a generalized theory with a larger symmetry group. It is worth noting
that in physics, all equations have global symmetry with respect to unit
invariance, regardless of whether they are dimensionful or not111The unit
change, when physical quantities, parameters, and constants are considered, is
better to be called unit covariance instead of unit invariance, since these
things are not necessarily invariant, but they change covariantly; i.e.,
consistently.. However, like many other global symmetries, this global
symmetry does not generally have physical significance. Nevertheless, in
theories with local unit invariance, where by local units one means units that
are functions of spacetime coordinates, the physical consequences may emerge.
Attempts to construct such theories date back to the introduction of the Weyl
geometry, later revised by Dirac Weyl (1919); Dirac (1973). In particular, in
theories with a dimensionful coupling constant, this symmetry can be
interpreted as gauge invariance, namely scale invariance. This is because this
invariance is responsible for an internal symmetry rather than an external
one. Then, fixing the units, being a kind of gauge fixing, eliminates the
redundant degrees of freedom and leaves only the physical ones. The price one
pays is that one encounters a broken symmetry. However, it is not unusual
throughout physics; for example, when the gravitational waves (GWs) are
considered, full diffeomorphism invariance has not been preserved anymore.

Although the scale invariance and conformal invariance are not generally the
same, a theory in the latter class will respect the former Polchinski (1988);
Nakayama (2015). There are different ways to construct a theory that respects
conformal invariance (see, for example, Faci (2013) and references therein).
To generalize the aspect of the conformal invariance to curved spacetime,
however, it seems that the most general symmetry that preserves this is the
Weyl invariance in which metric transforms as $g_{\mu\nu}\rightarrow
e^{2w}g_{\mu\nu}$ Farnsworth  _et al._ (2017). But Weyl symmetry is not
generally the symmetry of physics (especially of GR). To recover the conformal
invariance and eliminate the effect of the Weyl transformation, a scalar
field, usually called ”Weyl compensator” Shaukat and Waldron (2010), is
introduced like what is done in Gover  _et al._ (2009). The joint
transformation is traditionally called conformal transformation. When applied
to GR, it extends the group of symmetry to $Weyl\rtimes Diff$.

Metric Measure Space (MMS), alternatively, as a generalization of
(pseudo-)Riemannian manifolds, is one of such frameworks which lets one build
up conformal invariant objects out of the Riemannian-invariant ones. It
incorporates the Weyl transformation in addition to the transformation of a
compensator scalar field, namely the density function, which relates a more
general measure to the natural volume element by $dm=e^{-f}dvol(g)$. In
Rahmanpour and Shojaie (2015, 2016) following Chang  _et al._ (2006), the
authors provide a conformally invariant theory of gravitation which is
basically consistent with GR, since when the density function $f$ is fixed to
be a constant function, GR is retrieved. In another part of their work, they
investigate similarities and differences between MMS and Weyl integrable space
(WIS). For instance, they show that in contrast to WIS, the metricity and
integrability are maintained in MMS. As a result, Rahmanpour & Shojaie suggest
MMS as an alternative to WIS as a platform to construct conformally invariant
theories of gravitation Rahmanpour and Shojaie (2015). Indeed, despite the
fact that these two mathematical frameworks, namely MMS and WIS, are
constructed based on different assumptions, the authors have shown that some
relations, such as the contracted second Bianchi identity, are entirely
similar in these spaces. It is an interesting result from both mathematical
and physical perspectives.

This manuscript investigates the other aspects of the conformally invariant
theory of gravitation in MMS. To do so, first, in Sect II, MMS is introduced
in more detail and the structure of the theory is reviewed. Since a general
tensor may depend on fields in addition to or rather than metric, recipes for
constructing conformally invariant tensors may not be the same. However, a
vital case in point in a gravitational theory is the energy-momentum tensor,
which includes matter field as well. In Sect. III, as an example of this type,
we study the energy-momentum tensor of a perfect fluid in MMS. In Sect. IV, we
present a linearized version of the conformally invariant theory in the weak
field limit. Indeed, as gravity is a weak force we usually need to use the
full nonlinear equations in the vicinity of massive objects like black holes.
Perturbation methods, based on Taylor series expansion, mainly rely on the
superposition of terms with different orders of magnitude to approximate a
solution to a differential equation. Linearization is a name for considering
perturbations up to first order. GR and its modifications are non-linear
theories and do not have many exact solutions, highlighting the need for
approximation techniques including perturbation methods.

In Sect. V, we obtain the spherically symmetric vacuum solution of the theory.
We intend to use this solution to provide an explanation for the flat rotation
curves (FRC) of galaxies in Sect VI. FRC problem is a discrepancy between the
theoretical and the observational velocities of outer stars in galaxies Oort
(1932); Sofue and Rubin (2001). Indeed, there are different models in the
literature trying to solve this problem Mannheim and Kazanas (1989); Mannheim
(2006); Lobo (2008); Capozziello  _et al._ (2006); Mbelek (2004); McGaugh  _et
al._ (2016); Moffat (2006); Rahvar and Mashhoon (2014). We provide an
alternative method to explain FRC in the context of MMS. To accomplish that,
we need a corresponding density function to justify FRC. To do so, we employ
the Persic–Salucci–Stel velocity profile Persic  _et al._ (1996) and the
Navarro–Frenk–White density profile Navarro (1996), to find a suitable density
function. The last section, i.e., Sect VII, is devoted to some remarks. It is
worth noting that despite using the results of Persic  _et al._ (1996) and
Navarro (1996), no datasets have been generated or analyzed in this
manuscript.

One final point to remember is that the research has been carried out in the
geometrized unit where the gravitational constant and the speed of light are
set to be the unity ($G=c=1$).

##  II Metric Measure Space PRELIMINARIES

On every $n$-dimensional (pseudo-)Riemannian manifold $(M^{n},g)$, the volume
element $dvol(g)$ is defined naturally. To generalize the concept of volume,
one can promote a (pseudo-)Riemannian manifold to a metric measure space
(MMS). In this case, MMS is the triplet $(M^{n},g,m)$ with $(M^{n},g)$ and $m$
being a (pseudo-)Riemannian manifold and the generalized smooth measure,
respectively. The measure $m$ is related to the natural volume element by
$dm=\exp(-f)dvol(g)$ where $f$ is a smooth function on $M^{n}$ called the
density function. Henceforth in this manuscript, as we are concerned with
physical theories, we work with pseudo-Riemannian manifolds, and we use the
name MMS instead of pseudo-Riemann measure space.

In 2005, in an article by Chang et al. Chang  _et al._ (2006), it is shown
that MMS is compatible with conformal transformations. In this space,
conformal transformation is applied by a joint transformation

$\left\\{\begin{array}[]{l}\hat{g}_{\mu\nu}=e^{2\omega}g_{\mu\nu}\\\ \\\
\hat{f}=f+(n-s^{\prime})\omega,\end{array}\right.$ (1)

of metric $g$ and the density function $f$, respectively. In the latter,
$s^{\prime}$ is the conformal weight of measure $dm$. According to their
method, one can construct a conformally invariant tensor of weight $s$, shown
by $\mathcal{I}_{s}(g,f)$, from its Riemannian invariant counterpart $I(g)$.
Indeed, $I(g)$ can be any geometrical tensor in the Riemannian space. It is
natural to expect two essential features for the conformally invariant tensor
$\mathcal{I}_{s}(g,f)$. First, $\mathcal{I}_{s}(g,0)=I(g)$, and second,
$\mathcal{I}_{s}(\hat{g},\hat{f})=\exp(s\omega)\mathcal{I}_{s}(g,f)$. Then it
is easy to show that

$\mathcal{I}_{s}(g,f)=\exp\left(\frac{fs}{n-s^{\prime}}\right)I\left(\exp\left(\frac{-2f}{n-s^{\prime}}\right)g\right),$
(2)

satisfies the desired features. This relation can be rewritten elegantly by
introducing the “canonical metric” $g_{m}$. The canonical metric $g_{m}$ is
defined as $g_{m}=\exp(\frac{-2f}{n-s^{\prime}})g$, and is obtained by setting
$\hat{f}=0$ into the joint transformation. Accordingly, the Eqn. (2) can be
demonstrated by

$\mathcal{I}_{s}(g,f)=\exp\left(\frac{fs}{n-s^{\prime}}\right)I(g_{m}).$ (3)

It is worth noting that there is freedom to choose the conformal weight of the
tensors, including the conformal weight of $dm$, namely $s^{\prime}$. For
instance, Chang et al. in Chang  _et al._ (2006), set $s^{\prime}=0$ for
convenience, while Rahmanpour & Shojaie in Rahmanpour and Shojaie (2016) take
$s^{\prime}=2$ as they were interested to draw an analogy between MMS and WIS.
Hereafter, we also set $s^{\prime}=2$ and confirm that this will not affect
the results. In addition, it should be mentioned that the conformal weights of
some tensors are mutually dependent. For example, the conformal weights
$s_{0}$ and $s_{1}$ of the modified counterparts of the Ricci tensor and the
Ricci scalar, namely $\mathcal{R}_{\mu\nu}$ and $\mathcal{R}$ respectively,
are related via $s_{1}=s_{0}-2$. Therefore within four-dimensional MMS, this
relation leads to

$\displaystyle\mathcal{R}_{\mu\nu}(g,f)$
$\displaystyle=\exp\left(\frac{s_{0}f}{2}\right)\left[R_{\mu\nu}(g)+\nabla_{\mu}\nabla_{\nu}f+\frac{1}{2}\nabla_{\mu}f\nabla_{\nu}f\right.$
(4)
$\displaystyle\left.+\frac{1}{2}\nabla^{\rho}\nabla_{\rho}fg_{\mu\nu}-\frac{1}{2}\nabla^{\rho}f\nabla_{\rho}fg_{\mu\nu}\right],$
$\mathcal{R}(g,f)=\exp\left(\frac{\left(s_{1}+2\right)f}{2}\right)\left[R(g)+3\nabla^{\mu}\nabla_{\mu}f-\frac{3}{2}\nabla^{\mu}f\nabla_{\mu}f\right].$
(5)

The theory introduced in Rahmanpour and Shojaie (2015, 2016), is a
gravitational theory in which the conformal invariance, in addition to
diffeomorphism invariance, holds both at the level of the action and at the
level of the equation of motion. To have a conformally invariant weightless
action in MMS

$\int dm\left(\mathcal{R}(g,f)+8\pi\mathcal{L}_{M}(\psi,g,f)\right),$ (6)

one must set $s_{1}=-s^{\prime}$. Afterward, using the variational method and
introducing a modified energy-momentum tensor $\mathcal{T}_{\mu\nu}$, they
obtain Einstein’s field equation counterpart in MMS as

$\mathcal{G}_{\mu\nu}(g,f)\equiv\mathcal{R}_{\mu\nu}(g,f)-\frac{1}{2}\mathcal{R}(g,f)g_{\mu\nu}=8\pi\mathcal{T}_{\mu\nu}(\psi,g,f).$
(7)

Regarding Eqn. (7), it is worth mentioning that tensor fields, such as the
energy-momentum tensor, which depend on fields in addition to the metric, do
not generally transform via Eqn. (2) by a conformal transformation. In other
words, the suitable transformation of a field is restrictively based on the
other fields on which it depends and the way the tensor is physically defined.
This may lead to observable discrepancies between different conformal frames,
as that observable may be a conformally frame-dependent observation. A more
detailed discussion with an example is provided in Sect III.

Another aspect of MMS, which is directly related to the notion of generalized
measure, is that any quantity or relation which is defined through an integral
over volume may change. In particular, although the covariant derivative in
this space is the same as the covariant derivative in Riemannian geometry, it
is not necessarily the case for the other differential operators. For example,
it can be shown that the divergence operator in MMS is not simply the
contraction of the covariant derivative with the tensor it acts uponRahmanpour
and Shojaie (2015). As a result, the invariance of the geometrical part of the
action under the diffeomorphism leads to a generalized contracted second
Bianchi identity. The invariance of the matter part of the action under the
diffeomorphism on the other hand, yields a generalized identity for the
conservation of energy-momentum tensor. Using this identity, the authors of
Rahmanpour and Shojaie (2015) obtain a conformally invariant geodesic equation
in the context of MMS.

In summary and in the four-dimensional Lorentzian manifold, the equations of
motion (7) are invariant under transformations within the class

$[(g_{m},0)]=\\{(g,f)\lvert g=\exp(2\omega)g_{m},f=2\omega\\}$ (8)

as well as general coordinate transformations, where $g_{m}$ is the canonical
metric with which GR is retrieved; i.e.

$\exp(-\frac{sf}{2})\mathcal{G}_{s}(g,f)=\mathcal{G}_{s}(g_{m},0)=G(g_{m})$
(9)

with $G$ and $s$ being the traditional Einstein tensor and its conformal
weight, respectively. In other words, each class of type (8) presents a
solution of MMS. By substituting Eqns. (4) and (5) in the LHS of Eqn. (7), it
is reduced to

$\mathcal{G}_{s}(g,f)=\exp(\frac{s_{0}f}{2})\left(G_{\mu\nu}(g)-8\pi\mathcal{H}_{\mu\nu}(g,f)\right).$
(10)

From Eqns. (9) and (10), it can easily be inferred that the conformal weight
of the Einstein tensor is the same as that of the Ricci tensor, i.e.,
$s=s_{0}$. A simple comparison shows that

$\mathcal{H}_{\mu\nu}(g,f)\equiv-\frac{1}{8\pi}\left(\nabla_{\mu}\nabla_{\nu}f+\frac{1}{2}\nabla_{\mu}f\nabla_{\nu}f-\nabla^{\rho}\nabla_{\rho}fg_{\mu\nu}+\frac{1}{4}\nabla^{\rho}f\nabla_{\rho}fg_{\mu\nu}\right)$
(11)

which resembles the effect of some sort of mass distribution. It is evident
that a constant $f$ corresponds to a global Weyl rescaling and is interpreted
as the global change of unit. In this case, $\mathcal{H}=0$, and no physical
consequence occurs. Thus, without loss of generality, we can set $f=0$ if it
is constant. A more detailed discussion on the energy-momentum tensor is
provided in the following section.

##  III Energy–Momentum Tensor in metric measure space

In the MMS theory of gravitation, there is a straightforward recipe to
construct conformally invariant quantities out of Riemannian diffeomorphism
invariant counterparts, i.e. Eqn. (3). However, this instruction is restricted
to tensors that solely depend on metric and its derivatives. Generally,
contributions of other fields (in addition to metric) as sources of
constructing a tensor field, demand a more careful treatment and probably
different instructions to build up a conformally invariant tensor field. In
other words, it is plausible that the most suitable instruction mainly depends
on how a field is defined. In particular, the energy-momentum tensor is built
upon measurements either in labs or via (cosmological) observations. As an
example of this tensor, consider the perfect fluid, which in addition to the
metric, depends on the matter field via the matter density $\rho$ and the
pressure $P$, in the form

$T_{\mu\nu}=\left(\rho+P\right)u_{\mu}u_{\nu}+Pg_{\mu\nu}.$ (12)

It is clear that there is no explicit derivative of any kind in Eqn. (12) that
might cause the derivatives of the density function $f$ to emerge. More
precisely, the Einstein equations in the canonical frame and in another
arbitrary one, are related as

$\displaystyle 8\pi T(\psi_{m},g_{m})=G(g_{m})$
$\displaystyle=\mathcal{G}_{s}(g_{m},0)$ (13)
$\displaystyle=\exp(-\frac{sf}{2})\mathcal{G}_{s}(g,f)=8\pi\exp(-\frac{sf}{2})\mathcal{T}_{s}(\psi,g,f),$

with

$\psi=\exp(\frac{s_{\psi}f}{2})\psi_{m},$ (14)

where $\psi_{m}$, $\psi$, and $s_{\psi}$ are matter fields in the above
conformal frames and the corresponding conformal weight, respectively
Blagojevic (2001).

On the other hand, a straightforward calculation employing Eqns. (9) and (10),
for a perfect fluid, leads to

$8\pi T(\psi_{m},g_{m})=G(g_{m})=G(g)-8\pi\mathcal{H}(g,f)=8\pi T(\psi,g).$
(15)

In fact, when a conformal transformation is applied, the change in metric and
consequently in the proper velocity $u$ is compensated by the change in the
units of the matter density and the pressure; any local changes in scale can
be interpreted as local changes in the units. That is, the difference between
$\psi_{m}$ and $\psi$ is in the different local units used to specify them.
Comparing Eqns. (13) and (15), one can conclude

$\mathcal{T}_{s}(\psi,g,f)=\exp(\frac{sf}{2})T(\psi_{m},g_{m})=\exp(\frac{sf}{2})T(\psi,g).$
(16)

However, recalling the lines after Eqn. (3) for setting $s^{\prime}=2$ in
compatibility with WIS, one deduces that $s=0$ and Eqn. (16) simply becomes

$\mathcal{T}_{s}(\psi,g,f)=T(\psi_{m},g_{m})=T(\psi,g).$ (17)

In the context of large-scale astrophysics, where the energy-momentum tensor
in the form of a perfect fluid is more desirable, the Einstein equation is
usually written as

$G(g)=8\pi\left(T_{b}(\psi_{b},g)+T_{d}(\psi_{d},g)\right),$ (18)

wherein the indices $b$ and $d$ stand for baryonic and dark matter fields,
respectively. However, comparing Eqns. (15) and (18), in the framework of MMS,
can lead one to conclude

$T_{d}(\psi_{d},g)=\mathcal{H}(g,f),$ (19)

and

$T_{b}(\psi_{b},g)=T(\psi_{m},g_{m})=T(\psi,g).$ (20)

In other words, the result of fixing a conformal frame resembles the effect of
an unknown matter field, dubbed dark matter.

##  IV  nearly Conformally flat approximation in metric measure space

GR does not possess many exact solutions, and the perturbation methods are the
practical ways to construct approximate solutions close to the exact ones. In
principle, there are no restrictions on the orders of approximation in
perturbation methods. However, in some cases, a decomposition of metric at the
linear level to $g=g_{0}+h$ seems useful when components of $h$ are small
enough, i.e., with “the clear separation of scales” (according to Maggiore
(2007)) in some coordinate system. This helps one, for instance, achieve $h$
as a field in spacetime with background metric $g_{0}$. Moreover, setting
$g_{0}\equiv\eta$, the Minkowski metric, resembles the weak field limit,
literally known as the linearized theory. Nevertheless, writing down metrics
in the specific form

$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}\hskip 14.22636pt,\hskip
14.22636pt|h_{\mu\nu}|\ll 1$ (21)

in a particular coordinate system is a kind of gauge fixing that breaks the
inherent symmetry of GR, viz., diffeomorphism invariance. It should be noted
that although the whole group of diffeomorphism is no longer the symmetry
group of the linearized theory, a subgroup of it still preserves the symmetry
group for (21) i.e. $h_{\mu\nu}$ and $h^{\prime}_{\mu\nu}$ represent the same
perturbation field if they admit
$h_{\mu\nu}-h^{\prime}_{\mu\nu}=\epsilon\mathcal{L}_{X}\eta_{\mu\nu}$ for a
given vector field $X$ on the background Minkowski spacetime and any arbitrary
infinitesimal constant $\epsilon$ (to guarantee the smallness of $h^{\prime}$
as well). Nonetheless, GWs, interpreted in this gauge, are not an illusion of
gauge fixing and will not fade away by diffeomorphisms. It is worth noting
that in Eqn. (21), one reduces the background to SR but saves the effect of GR
in $h$. In other words, even though SR does not predict GWs, it can still be
used to demonstrate one of the outcomes of GR. From the mathematical point of
view, when (21) is considered, the infinite-dimensional symmetry group of
diffeomorphism has been reduced to a finite-dimensional Poincaré group.

Recalling that MMS has been introduced as a platform for constructing a
gravitational theory with the larger symmetry group, it is useful to derive
the linearized version of MMS, in which the conformal symmetry is also broken.
This is similar in nature to what has been done above for GR and we follow the
same way of reasoning used for the linearized theory except in this case, the
theory is reduced to linearized GR with a perturbation that preserves the
effect of MMS. In other words, we assume

$f=f_{0}+\mathfrak{f}\hskip 14.22636pt,\hskip 14.22636pt|\mathfrak{f}|\ll 1$
(22)

where consequently

$e^{f}\approx 1+\mathfrak{f},$ (23)

in which, for convenience, we set $f_{0}=0$. It is worth noting that there is
a subgroup of the conformal group which still presents the symmetry in (22);
$\mathfrak{f}$ and $\hat{\mathfrak{f}}$ represent the same perturbation in
weights if

$\mathfrak{f}-\hat{\mathfrak{f}}=\mathfrak{w},$ (24)

with $\mathfrak{w}$ being any arbitrary infinitesimal constant to ensure the
smallness of $\hat{\mathfrak{f}}$ as well. Further in this section, we show
that (24) is consistent with the degrees of freedom of a scalar potential.
Therefore, the full linearized conformal transformation in 4D is completed by

$g_{m}=e^{-\mathfrak{f}}g\approx\left(1-\mathfrak{f}\right)\left(\eta+h\right),$
(25)

when a class of linearized solutions is regarded. We call Eqn. (25) the nearly
conformally flat approximation.

The conclusion to be drawn is that when an infinitesimal coordinate
transformation $x^{\mu}\rightarrow x^{\prime\mu}=x^{\mu}+\xi^{\mu}$ where
$\xi^{\mu}\ll 1$, and conformal transformation
$\hat{g}_{\mu\nu}=e^{2\omega}g_{\mu\nu}$ where $\omega\ll 1$ are applied on
metric, some straightforward calculations result in

$h_{\mu\nu}-h^{\prime}_{\mu\nu}=\left(-2\omega\eta_{\mu\nu}+\mathcal{L}_{\xi}\eta_{\mu\nu}\right),$
(26)

up to the first order. For the sake of simplicity, we split the modified
Einstein’s tensor into two parts as shown in Eqn. (10); the ordinary Einstein
tensor follows the common methods by defining $\bar{h}_{\mu\nu}\equiv
h_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}h$ and using the Lorentz gauge
$\partial^{\mu}\bar{h}_{\mu\nu}=0$, and the $\mathcal{H}_{\mu\nu}$ part by
applying Eqn. (21) to Eqn. (11) yields

$\mathcal{H}_{\mu\nu}(g,f)=-\frac{1}{8\pi}\left(\partial_{\mu}\partial_{\nu}\mathfrak{f}-\eta_{\mu\nu}\Box\mathfrak{f}\right)+\mathcal{O}(\epsilon^{2}).$
(27)

Furthermore, to obtain the quasi-Newtonian limit of the theory, we apply slow
motion approximation which is indicated by
$\frac{dx^{i}}{d\tau}=\epsilon\frac{dx^{0}}{d\tau}$. Therefore by adopting
$\epsilon\equiv\frac{v}{c}\ll 1$, the slow-motion condition for any function
$\mathfrak{f}$ can be written as

$\frac{\partial\mathfrak{f}}{\partial
x^{0}}\sim\epsilon\frac{\partial\mathfrak{f}}{\partial x^{i}},$ (28)

demonstrating that as a slowly changing quantity, the changes in the gradient
of $\mathfrak{f}$ are much greater than its time evolution. Thus one can
assume $\nabla_{i}\mathfrak{f}\propto\mathcal{O}(\epsilon)$ that leads to
$\nabla_{0}\mathfrak{f}\propto\mathcal{O}(\epsilon^{2})$.

By applying these assumptions, Eqn. (27) is reduced to

$\mathcal{H}_{00}=-\frac{1}{8\pi}\eta^{pq}\partial_{p}\partial_{q}\mathfrak{f}+\mathcal{O}(\epsilon^{2}),$
(29)

and

$\mathcal{H}_{kl}=-\frac{1}{8\pi}\left(\partial_{k}\partial_{l}\mathfrak{f}-\eta_{kl}\eta^{pq}\partial_{p}\partial_{q}\mathfrak{f}\right)+\mathcal{O}(\epsilon^{2}).$
(30)

Consequently, the components of the weak counterpart of the Einstein tensor in
MMS are

$\mathcal{G}_{00}=-\frac{1}{2}\nabla^{2}\left(\bar{h}_{00}-2\mathfrak{f}\right),$
(31)

and

$\mathcal{G}_{kl}=-\frac{1}{2}\nabla^{2}\left(\bar{h}_{kl}+2\eta_{kl}\mathfrak{f}\right)+\partial_{k}\partial_{l}\mathfrak{f}.$
(32)

up to the first order in $\epsilon$.

In the presence of matter with the stress-energy tensor
$T_{\mu\nu}=(\rho+P)U_{\mu}U_{\nu}+Pg_{\mu\nu}$, recalling Eqns. (7) and (17)
and adopting slow motion condition, i.e., $|v|\ll 1$, one can write
$|T_{00}|\gg|T_{0i}|$ and $|T_{00}|\gg|T_{ij}|$ Misner  _et al._ (1973). The
leading order terms of the weak field equation in MMS theory then is

$-\frac{1}{2}\nabla^{2}\left(\bar{h}_{00}-2\mathfrak{f}\right)=8\pi\rho.$ (33)

In comparison to the equation of motion of the Newtonian limit (gravity)
$\nabla^{2}\Phi=4\pi\rho$, where $\Phi$ refers to the gravitational potential
of the matter distribution, one can write $\bar{h}_{00}=2\mathfrak{f}-4\Phi$.
This simply leads to

$h_{00}=\mathfrak{f}-2\Phi.$ (34)

Accordingly, some straightforward calculations, using $\bar{h}_{\mu\nu}\equiv
h_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}h$ result in
$h_{kl}=(\mathfrak{f}-2\Phi)\delta_{kl}$. The Eqn. (34) is consistent with the
results presented in Rahmanpour and Shojaie (2016). Finally, the weak metric
corresponding to MMS theory is obtained as

${ds_{m}}^{2}=-(1+2\Phi-\mathfrak{f})dt^{2}+(1-2\Phi+\mathfrak{f})\left(dx^{2}+dy^{2}+dz^{2}\right).$
(35)

##  V the vacuum solution

In this section, we find a vacuum solution of gravitational theory in MMS,
which corresponds to the spherically symmetric vacuum solution of the 4D
spacetime of GR. By this, we mean there is a canonical metric $g_{m}$ in MMS
with

$\mathcal{G}_{s}(g_{m},0)=0.$ (36)

Using two essential features of constructing the conformally invariant tensors
in MMS, namely, $\mathcal{I}_{s}(g,0)=I(g)$, and,
$\mathcal{I}_{s}(\hat{g},\hat{f})=\exp(s\omega)\mathcal{I}_{s}(g,f)$, one has

$\mathcal{G}_{s}(g_{m},0)=G(g_{m}),$ (37)

which by applying Eqns. (36), results in

$G(g_{m})=0,$ (38)

where $g_{m}$ is the Schwarzschild metric

${ds_{m}}^{2}=-\frac{\left(1-\frac{C_{0}}{4r}\right)^{2}}{\left(1+\frac{C_{0}}{4r}\right)^{2}}dt^{2}+\left(1+\frac{C_{0}}{4r}\right)^{4}\left({dr}^{2}+{r}^{2}d\Omega^{2}\right),$
(39)

in the isotropic coordinates. In this equation $C_{0}$ is a constant and the
index $m$ denotes the canonical metric. As mentioned before and regarding the
coincidence principle, GR should be retrieved by setting $f=0$. Then according
to GR, Eqns. (38) and (39) together lead to $C_{0}=2M$.

On the other hand Eqn. (36), for any other metric which is conformally related
to $g_{m}$ via $g=e^{f}g_{m}$ yields

$G_{\mu\nu}(g)=8\pi\mathcal{H}_{\mu\nu}(g,f).$ (40)

For the sake of simplicity, if we restrict $f$ to be a function of radial
coordinate only, the spherical symmetry of the vacuum solution in MMS is
retained. Therefore, one can assume $g$ to be of the form

$ds^{2}=-e^{2\alpha(r)}dt^{2}+e^{2\beta(r)}\left(dr^{2}+r^{2}d\Omega^{2}\right),$
(41)

which is related to $g_{m}$ by

${ds_{m}}^{2}=-e^{2\alpha(r)-f(r)}dt^{2}+e^{2\beta(r)-f(r)}\left(dr^{2}+r^{2}d\Omega^{2}\right).$
(42)

By comparing Eqns. (39) and (42) of this gravitational theory, one can obtain
components of the metric

$2\alpha(r)=f+2\ln\left(\left(1-\frac{2M}{4r}\right)-\left(1+\frac{2M}{4r}\right)\right),$
(43) $2\beta(r)=f+4\ln\left(1+\frac{2M}{4r}\right).$ (44)

It is worth noting that, while in a conformal frame $(g_{m},0)$, Eqn. (38)
reminds one of vacuum Einstein’s field equation, in another conformal frame
$(g,f)$, Eqn. (40) simulates an illusion of matter in GR. We employ this idea
in its weak field limit to explain the FRC of galaxies, as we shall
investigate in the next section. For further convenience, we note that the
weak field approximation of Eqns. (43) and (44) are

$2\alpha(r)=\mathfrak{f}-\frac{2M}{r}$ (45)

and

$2\beta(r)=\mathfrak{f}+\frac{2M}{r}.$ (46)

##  VI  resolution for FRC problem

Briefly, the FRC problem is a disagreement between the prediction of Newtonian
gravity and observational data; although the former predicts the rates of
tangential velocities of rotating stars to be decreasing by increasing radius
in the outer regions of the galactic bulge, the latter shows nearly constant
values (see for example Bertone and Hooper (2018) and references therein). In
other words, considering the amount of visible mass in galaxies only, the
stars should rotate slower than what is observed Oort (1932). Therefore, a
kind of unknown mass distribution, namely dark matter, may be responsible for
it. However, there are several conceivable scenarios to explain this
discrepancy. In this section, we use the idea discussed in the previous
sections that a vacuum solution of gravitational theory in MMS, when
restricted to the context of GR (in a conformal symmetry broken frame), can
simulate an illusion of a matter field, dubbed as $\mathcal{H}$ in this
manuscript, to justify FRC. For an appropriate density function, $\mathcal{H}$
can play the role of dark matter and explain FRC. It is customary to use the
spherically symmetric metric in order to model FRC in the simplest form. To do
this, the Newtonian gravitational potential $\Phi(r)$, appeared in metric

${ds}^{2}=-e^{2\Phi(r)}dt^{2}+e^{2\gamma(r)}\left(dr^{2}+r^{2}d\Omega^{2}\right),$
(47)

is tuned to explain FRC and consequently is supposed to be produced by dark
matter distribution. Instead, here we seek a density function $f(r)$ that
conformally transforms this metric into the vacuum solution of GR, recalling
that these two apparently different solutions are the same in the framework of
MMS. As mentioned before, to respect the spherical symmetric nature of the
vacuum solution in MMS, we consider the density function to be a function of
radial coordinate only. This is consistent with the observed more or less
spherically symmetric nature of FRC.

Moreover, since FRC can be explained in the Newtonian regime, we use the
benefit of the weak field approximation Eqns. (45) and (46) in consistency
with (35), and find

$\frac{-2M}{r}=2\Phi(r)-\mathfrak{f}(r)$ (48)

and

$\frac{2M}{r}=2\gamma(r)-\mathfrak{f}(r),$ (49)

wherein $\alpha(r)$ and $\beta(r)$ are replaced by $\Phi(r)$ and $\gamma(r)$,
respectively. Using

${v_{t}}^{2}=\frac{r\Phi^{\prime}(r)}{1+r\gamma^{\prime}(r)},$ (50)

derived in appendix A, and where $v_{t}(r)$ is the tangential velocity of
stars rotating around the galactic center coming from observations, Eqn. (48)
eventually leads to the general form of the density function

$\mathfrak{f}(r)=\int_{r_{0}}^{r}dr^{\prime}\frac{2}{1-{v_{t}}^{2}(r^{\prime})}\left[\frac{{v_{t}}^{2}(r^{\prime})}{r^{\prime}}-(1+{v_{t}}^{2}(r^{\prime}))\frac{M}{r^{\prime
2}}\right].$ (51)

in which $r_{0}$ is an arbitrary radius used to denote the origin of the
density function. The choice of $r_{0}$, which globally shifts
$\mathfrak{f}(r)$ by a constant, corresponds to a global rescaling and is a
matter of convenience. According to Eqn. (51), for any tangential velocity
profile, one can obtain the corresponding density function that accounts for
the flattening of the rotation curve. For the plausible assumption $v_{t}\ll
1$ which is consistent with the slow-motion condition, Eqn. (51) reduces to

$\mathfrak{f}(r)=\frac{2M}{r}-\frac{2M}{r_{0}}+2\int_{r_{0}}^{r}dr^{\prime}\frac{{v_{t}}^{2}(r^{\prime})}{r^{\prime}}.$
(52)

In the following subsections, we try to find the form of the corresponding
density functions, beginning from a naive constant tangential velocity model
and proceeding to two well-known PSS and NFW profiles of velocity and mass
density, respectively.

###  VI.1 constant velocity profile

By way of explanation, we start with the simplest case in which the tangential
velocity does not depend on coordinates. Then Eqn. (52) simply yields

$\mathfrak{f}_{c}(r)=\frac{2M}{r}-\frac{2M}{r_{0}}+2{v_{t}}^{2}\ln(\frac{r}{r_{0}}).$
(53)

For this spherically symmetric toy model, the entire baryonic mass of a
galaxy, including the stellar and gas, is considered to be clustered in a
sphere with the radius $R_{b}$ and mass $M_{b}$, which are called the radius
and mass of concentrated baryonic matter, respectively. Additionally, we
expect the flattening of the galactic rotation curve to occur outside of this
radius. Supposing the start point of FRC is where
$v_{t}(r)=(\frac{M_{b}}{R_{b}})^{1/2}$, The density function (52) can be
reduced to

$\mathfrak{f}_{c}(r)=\frac{2M_{b}}{r}+\frac{2M_{b}}{R_{b}}\ln\left(\frac{r}{R_{b}}\right).$
(54)

where the constant term $-\frac{2M_{b}}{R_{b}}$, responsible for a global
rescaling, is ignored. It is worth noting that for galaxies whose most of the
mass is accumulated in their spherical bulge, the above parameters might be
the mass and the radius of their central bulges. In other words, having the
parameters $M_{b}$ and $R_{b}$ for a galaxy, Eqn. (54) can describe FRC as a
part of the geometry that can eliminate the need for dark matter.

###  VI.2 Persic–Salucci–Stel (PSS) velocity profile

The Persic–Salucci–Stel (PSS) is a velocity profile to describe FRC proposed
by Persic et al. Persic  _et al._ (1996). They claim their universal profile
strongly depends on the luminosity of the galaxies, which shows a slight
discrepancy for the bright galaxies while in good accordance with low luminous
ones. The velocity profile is

${v_{PSS}}(r)=\left(\frac{{v_{0}}^{2}r^{2}}{r^{2}+{R^{2}}}\right)^{1/2},$ (55)

with two fitting parameters, $v_{0}$ and $R$, viz, the ’terminal velocity’ and
’the velocity core radius’, respectively Barranco  _et al._ (2015); Persic
_et al._ (1996). The density function related to the PSS velocity profile,
utilizing Eqn. (52), is

$\mathfrak{f}_{PSS}(r)=\frac{2M}{r}-\frac{2M}{r_{0}}+v_{0}^{2}\ln\left(\frac{r^{2}+R^{2}}{r_{0}^{2}+R^{2}}\right).$
(56)

To lower the number of parameters, we set $r_{0}=R$ then remove the constant
terms, similar to what has been done in subsection VI.1, and find

$\mathfrak{f}_{PSS}(r)=\frac{2M}{r}+v_{0}^{2}\ln\left(1+\frac{r^{2}}{R^{2}}\right).$
(57)

###  VI.3 Navarro–Frenk–white (NFW) density profile

One of the most commonly used mass density profiles to describe dark matter
halo is the NFW profile Navarro (1996). This universal profile is obtained
from N-body simulations and describes the dark halo density as a function of
radial coordinate, which varies from the inner to the outer parts of the
galaxy Łokas and Mamon (2001). The profile is

$\rho_{NFW}(r)=\frac{\sigma}{r(r+R_{s})^{2}},$ (58)

with $\sigma\equiv\rho_{cri}\delta_{c}{R_{s}}^{3}$, where $\rho_{cri}$ and
$R_{s}$ are called the critical density of the universe and the characteristic
radius, respectively and $\delta_{c}$ is a dimensionless parameter. Solving
the corresponding Poisson’s equation and substituting the gravitational
potential to Eqn. (52), the density function according to this profile becomes

$\mathfrak{f}_{NFW}=\frac{2M}{r}+\frac{8\pi\sigma}{r}\ln\left(1+\frac{r}{R_{s}}\right),$
(59)

following the similar procedure for Eqns. (54) and (57).

##  VII Conclusion and Remarks

In this manuscript, by gravitational theory in MMS we mean a gravitational
theory that, in addition to diffeomorphism invariance, respects the conformal
invariance. To compensate for the side-effect of Weyl rescaling of metric, the
volume element is subject to change consistently by the introduction of a
density function. These joint transformations, simply called conformal
transformations, enable us to promote the pseudo-Riemannian invariants of GR
to conformal invariant ones.

Although the density function plays the role of a gauge and appears as a
degree of freedom, one is not restricted to assuming it as a pure gauge. For
instance, the platform of MMS may be a fruitful framework for theories that
can fix the density function for other reasons; non-local gravity theories
could be the case. It is worth noting that in MMS, the matter field is no
longer forced to be traceless.

Although there are considerable debates in the literature about the
perforation of either larger or smaller symmetry groups of a theory to be, the
aspect of conformal invariance, when added to a gravitational theory like GR,
seems to widen the strength and the realm of the theory as well. For example,
it may equip the theory in a way that becomes more suitable for quantization
requirements such as renormalization. Also, it is deemed to be more suitable
as a theory to unify physics from small to large scales.

In this work, we have concentrated on the vacuum solution of MMS. Not
surprisingly, this solution when regarded in different conformal frames is
generally equivalent to different solutions of GR. Following this idea, we
suggest how an appropriate conformal frame can resemble the illusion of a dark
matter field and deviation of the rotation curve of galaxies from what is
expected in the Newtonian regime. In other words, this phenomenon is a
(conformally) frame-dependent observation. This encourages one to seek a
similar explanation for dark energy in further studies.

## Appendix A

In this appendix, following Chandrasekhar and Chandrasekhar (1998); Matos  _et
al._ (2002), we try to derive the velocity of a test particle rotating in a
circular orbit in the spherically symmetric and static spacetime represented
by an isotropic line element

$ds^{2}=-e^{2\Phi(r)}dt^{2}+e^{2\gamma(r)}\left(dr^{2}+r^{2}d\Omega^{2}\right).$
(60)

We aim to find the velocity and consequently the corresponding Newtonian
potential field in the context of GR. Hence the Lagrangian

$\mathcal{L}=\frac{1}{2}g_{\mu\nu}\frac{\mathrm{d}x^{\mu}}{\mathrm{d}\tau}\frac{\mathrm{d}x^{\nu}}{\mathrm{d}\tau},$
(61)

reads as

$\mathcal{L}=\frac{1}{2}\left[-e^{2\Phi(r)}\dot{t}^{2}+e^{2\gamma(r)}\left(\dot{r}^{2}+r^{2}\left(\dot{\theta}^{2}+\sin^{2}\theta\dot{\phi}^{2}\right)\right)\right],$
(62)

where the dot stands for the derivative with respect to $\tau$, which is the
proper time of a particle moving along the geodesic. By defining $E\equiv
e^{2\Phi(r)}\dot{t}$, $L_{\theta}\equiv e^{\gamma(r)}r^{2}\dot{\theta}$ and
$L_{\phi}\equiv e^{\gamma(r)}r^{2}\sin^{2}\theta\dot{\phi}$, and setting
$2\mathcal{L}=-1$ for a timelike geodesic, one can write the radial equation
of motion

$\dot{r}^{2}+V(r)=0,$ (63)

with the potential $V(r)$

$V(r)=-e^{-2\gamma(r)}\left(e^{-2\Phi(r)}E^{2}-r^{-2}e^{-2\gamma(r)}L^{2}-1\right),$
(64)

wherein $E$ and
$L^{2}={L_{\theta}}^{2}+\left(\frac{L_{\phi}}{\sin\theta}\right)^{2}$ are
considered as the total energy and angular momentum, respectively.

Employing the conditions of stability for a test particle rotating in a
circular orbit, the total energy and angular momentum in terms of the
coefficients of metric are

$\displaystyle{E^{2}}=\frac{e^{2\Phi(r)}\left(1+r\gamma^{\prime}(r)\right)}{1-r\Phi^{\prime}(r)+r\gamma^{\prime}(r)},$
(65) $\displaystyle
L^{2}=\frac{e^{2\gamma(r)}r^{3}\Phi^{\prime}(r)}{1-r\Phi^{\prime}(r)+r\gamma^{\prime}(r)},$

with prime denotes the derivative with respect to $r$. To find the four-
velocity of a particle moving along its time-like worldline, we set
$d\tau^{2}=-ds^{2}$ in Eqn. (60)

$d\tau^{2}=e^{2\Phi(r)}dt^{2}\left[1-e^{2\gamma(r)-2\Phi(r)}\left(\left(\frac{dr}{dt}\right)^{2}+r^{2}\left(\frac{d\Omega}{dt}\right)^{2}\right)\right],$
(66)

wherein the tangential velocity
${v_{t}}^{2}=e^{2\gamma(r)-2\Phi(r)}r^{2}\left(\frac{d\Omega}{dt}\right)^{2}$
with a straightforward calculation, can be written as

${v_{t}}^{2}=r^{-2}e^{2\Phi(r)-2\gamma(r)}L^{2}E^{-2}=\frac{r\Phi^{\prime}(r)}{1+r\gamma^{\prime}(r)},$
(67)

in terms of the coefficients of metric.

## References

  * Einstein (1915) A. Einstein, Sitzungsberichte, Berlin , 778 (1915).
  * Will (2014) C. M. Will, Living reviews in relativity 17, 1 (2014).
  * Weyl (1918) H. Weyl, Sitzungsber. Königl. Preuss. Akad. Wiss. 26, 465 (1918).
  * Weyl  _et al._ (1929) H. Weyl _et al._ , z. Phys 56, 330 (1929).
  * Weyl (1919) H. Weyl, Annalen der Physik 364, 101 (1919).
  * Dirac (1973) P. A. M. Dirac, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 333, 403 (1973).
  * Polchinski (1988) J. Polchinski, Nuclear Physics B 303, 226 (1988).
  * Nakayama (2015) Y. Nakayama, Physics Reports 569, 1 (2015).
  * Faci (2013) S. Faci, Classical and Quantum Gravity 30, 115005 (2013).
  * Farnsworth  _et al._ (2017) K. Farnsworth, M. A. Luty, and V. Prilepina, Journal of High Energy Physics 2017, 1 (2017).
  * Shaukat and Waldron (2010) A. Shaukat and A. Waldron, Nuclear physics B 829, 28 (2010).
  * Gover  _et al._ (2009) A. Gover, A. Shaukat, and A. Waldron, Physics Letters B 675, 93 (2009).
  * Rahmanpour and Shojaie (2015) N. Rahmanpour and H. Shojaie, General Relativity and Gravitation 47, 123 (2015).
  * Rahmanpour and Shojaie (2016) N. Rahmanpour and H. Shojaie, General Relativity and Gravitation 48, 122 (2016).
  * Chang  _et al._ (2006) S.-Y. A. Chang, M. J. Gursky, and P. Yang, Proceedings of the National Academy of Sciences 103, 2535 (2006).
  * Oort (1932) J. H. Oort, Bulletin of the Astronomical Institutes of the Netherlands 6, 249 (1932).
  * Sofue and Rubin (2001) Y. Sofue and V. Rubin, Annual Review of Astronomy and Astrophysics 39, 137 (2001).
  * Mannheim and Kazanas (1989) P. D. Mannheim and D. Kazanas, The Astrophysical Journal 342, 635 (1989).
  * Mannheim (2006) P. D. Mannheim, Progress in Particle and Nuclear Physics 56, 340 (2006).
  * Lobo (2008) F. S. Lobo, arXiv preprint arXiv:0807.1640 (2008).
  * Capozziello  _et al._ (2006) S. Capozziello, V. Cardone, and A. Troisi, Journal of Cosmology and Astroparticle Physics 2006, 001 (2006).
  * Mbelek (2004) J. Mbelek, Astronomy & Astrophysics 424, 761 (2004).
  * McGaugh  _et al._ (2016) S. S. McGaugh, F. Lelli, and J. M. Schombert, Physical Review Letters 117, 201101 (2016).
  * Moffat (2006) J. W. Moffat, Journal of Cosmology and Astroparticle Physics 2006, 004 (2006).
  * Rahvar and Mashhoon (2014) S. Rahvar and B. Mashhoon, Physical Review D 89, 104011 (2014).
  * Persic  _et al._ (1996) M. Persic, P. Salucci, and F. Stel, Monthly Notices of the Royal Astronomical Society 281, 27 (1996).
  * Navarro (1996) J. F. Navarro, in  _Symposium-international astronomical union_ , Vol. 171 (Cambridge University Press, 1996) pp. 255–258.
  * Blagojevic (2001) M. Blagojevic,  _Gravitation and gauge symmetries_ (CRC Press, 2001).
  * Maggiore (2007) M. Maggiore,  _Gravitational waves: Volume 1: Theory and experiments_ (OUP Oxford, 2007).
  * Misner  _et al._ (1973) C. Misner, K. Thorne, and J. Wheeler,  _Gravitation_ (W. Freeman, 1973).
  * Bertone and Hooper (2018) G. Bertone and D. Hooper, Reviews of Modern Physics 90, 045002 (2018).
  * Barranco  _et al._ (2015) J. Barranco, A. Bernal, and D. Nunez, Monthly Notices of the Royal Astronomical Society 449, 403 (2015).
  * Łokas and Mamon (2001) E. L. Łokas and G. A. Mamon, Monthly Notices of the Royal Astronomical Society 321, 155 (2001).
  * Chandrasekhar and Chandrasekhar (1998) S. Chandrasekhar and S. Chandrasekhar,  _The mathematical theory of black holes_ , Vol. 69 (Oxford university press, 1998).
  * Matos  _et al._ (2002) T. Matos, D. Nunez, F. S. Guzmán, and E. Ramirez, General Relativity and Gravitation 34, 283 (2002).