# Heart-Darts: Classification of Heartbeats Using Differentiable Architecture
Search

Jindi Lv  Sichuan University
China, Chengdu
<EMAIL_ADDRESS>Qing Ye  Sichuan University
China, Chengdu
<EMAIL_ADDRESS>Yanan Sun  Sichuan University
China, Chengdu
<EMAIL_ADDRESS>Juan Zhao  Sichuan University
China, Chengdu
<EMAIL_ADDRESS>Jiancheng Lv  Sichuan University
China, Chengdu
<EMAIL_ADDRESS>
###### Abstract

Arrhythmia is a cardiovascular disease that manifests irregular heartbeats. In
arrhythmia detection, the electrocardiogram (ECG) signal is an important
diagnostic technique. However, manually evaluating ECG signals is a
complicated and time-consuming task. With the application of convolutional
neural networks (CNNs), the evaluation process has been accelerated and the
performance is improved. It is noteworthy that the performance of CNNs heavily
depends on their architecture design, which is a complex process grounded on
expert experience and trial-and-error. In this paper, we propose a novel
approach, Heart-Darts, to efficiently classify the ECG signals by
automatically designing the CNN model with the differentiable architecture
search (i.e., Darts, a cell-based neural architecture search method).
Specifically, we initially search a cell architecture by Darts and then
customize a novel CNN model for ECG classification based on the obtained
cells. To investigate the efficiency of the proposed method, we evaluate the
constructed model on the MIT-BIH arrhythmia database. Additionally, the
extensibility of the proposed CNN model is validated on two other new
databases. Extensive experimental results demonstrate that the proposed method
outperforms several state-of-the-art CNN models in ECG classification in terms
of both performance and generalization capability.

###### Index Terms:

Arrhythmia, Heartbeat classification, Deep neural networks, Convolutional
neural network, Differentiable architecture search, Skip connection

##  I Introduction

Cardiovascular diseases (CVDs) are the dominant causes of death globally
during the past 20 years [1]. According to the report from the American Heart
Association, CVDs are responsible for 25$\%$ of annual deaths in the United
States. Arrhythmias are abnormal rhythms in heartbeats belonging to the CVDs.
The most common technique to diagnose arrhythmias is to analyze
electrocardiogram (ECG) signals, which consist of three main components: P
wave, QRS complex, and T wave. It supports the detection of heart disease by
measuring the rhythm and electrical activity of the heart. In ECG analysis,
heartbeat classification plays a critical role in determining the
effectiveness of arrhythmias detection [2]. As manually classifying heartbeats
of long-term ECG recordings is a time-consuming task, the application of
computer-aided diagnosis (CAD) algorithms provide essential assistance for
diagnosis.

Many CAD algorithms have been applied to assist the classification of
heartbeats, which require signal preprocessing, waveform detection, and
extraction of hand-crafted features [3, 4, 5, 6]. For instance, Li et al. [4]
utilized the wavelet packet entropy and random forests for ECG classification.
Fatin et al. [5] investigated the representation ability of linear and
nonlinear features for ECG signals and proposed a method to combine these
features together for better ECG classification. Li et al. [6] employed a
multi-domain feature extraction approach and constructed a support vector
machine (SVM) classifier to classify the heartbeats. Generally, these hand-
designed approaches cannot guarantee reliable robustness and easily tend to
over-adapt for specific tasks [7]. In addition, CAD systems typically produce
a large number of false positives, resulting in longer evaluation time and
more biopsies [8]. To address this challenge, deep neural networks (DNNs) were
employed in ECG analysis.

The DNN differs from traditional CAD algorithms in that it integrates feature
extraction and classification together rather than using hand-crafted
features. The two dominant DNN algorithms in ECG analysis are long short-term
memory (LSTM) and convolutional neural networks (CNNs), which have been widely
studied by researchers. For instance, Yildirim et al. [9] adopted the LSTM
algorithm with a new wavelet sequence layer for the classification of ECG
signals. Kiranyaz et al. [10] proposed an adaptive implementation of the 1-D
CNN method to distinguish different heartbeat types. Acharya et al. [11, 12]
designed two CNN models for morphological analysis of ECG signals.

With classical DNNs such as AlexNet [13], GooleNet [14], and ResNet [15],
being extensively applied in computer vision (CV) for their remarkable network
architecture, the interest is shifting from feature design to architecture
design. However, promising architectures in CV do not necessarily have the
same good performance for signal processing. Specifically, the CV task deals
with images with multidimensional spatial information, whereas the ECG
analysis task deals with temporal signals. Applying high-performance networks
from CV directly to the ECG analysis task is a process of model migration,
which may result in domain bias. Additionally, the process of network
architecture design for a specific task relies on trial-and-error experiments
and human experience, which is a time-consuming task. Fortunately, neural
architecture search (NAS), a technique automatically to discover the efficient
architectures of the DNNs for a specific task, is proposed and gaining ground
to effectively address the aforementioned problems. NAS aims to design
efficient and promising DNN architectures based on well-designed search
methods such as reinforce learning method (RL), evolutionary algorithm (EA).
As the EA-based NAS [16] and RL-based NAS have a high demand for calculation
despite the excellent performance, the differentiable architecture search
(Darts) [17] was emerged, which is more efficient and less demanding on
resources. Instead of searching in a discrete architecture candidate set,
Darts searches in a continuous search space and is optimized by gradient
descent algorithm.

Darts has been employed with great success in various fields as a powerful and
resource-friendly automation tool for automated network architecture design.
Inspired by it, we propose a novel approach, name Heart-Darts, to explore
high-performance CNN model for ECG classification based on Darts. To the best
of our knowledge, it is the first work presenting the NAS approach to the ECG
classification. Specifically, we have also customized a CNN model with a novel
architecture for ECG classification. To ensure the model be flexibly applied
in the ECG analysis, we have validated the generalization capability of the
proposed model on two new databases. Extensive experiments have also been
conducted to demonstrate the efficiency of the proposed method. The
contributions of the proposed Heart-Darts method are shown below:

  * •

To explore the impact of CNN network architecture on ECG classification, we
introduce Darts, a cell-based neural architecture search technique, as a
powerful tool for ECG analysis for the first time.

  * •

To further enhance the performance of ECG classification, we design a novel
model architecture, i.e., a special combination method of cells in which the
cellular architecture is obtained by Darts.

  * •

To investigate the generalization capability of the proposed model, we migrate
the raised model to fresh ECG databases and obtain promising results.

The remainder of this paper is organized as follows. The related work and
background are discussed in Section II. Section III elaborates the proposed
Heart-Darts method. Section IV describes the experimental details, including
database information, data preprocessing and parameter settings. The
experimental results are presented and analyzed in Section V. Finally, the
conclusions and future work are drawn in Section VI.

##  II Related Work and Background

In the past decade, a large number of algorithms have been designed for
classifying ECG. These algorithms are broadly grouped into two different
categories. The first type of algorithm performs manual feature extraction and
classification separately. Conversely, the second type of algorithms typically
exploits the end-to-end nature of DNNs [18] to automate feature extraction
followed by the classification.

The statistical method is one of the representative algorithms in the first
category, which relies on manual feature extraction and is prevalent in ECG
analysis. For example, Martis et al. [19] employed principal component
analysis (PCA) for dimensionality reduction. Additionally, they utilized a
four-layer feed-forward neural network and least square-support vector machine
to classify ECG. The method yielded an accurate classification of 93.48$\%$.
Elhaj et al. [5] utilized PCA with the discrete wavelet transform coefficients
followed by SVM classifier and neural network for classification with an
accuracy of 98.91$\%$. Sahoo et al. [3] applied wavelet transform to detect
QRS complexes of ECG signals and extracted information by neural network and
SVM classifier, where the diagnostic accuracy was 99.85$\%$. Although the
statistical methods have made considerable progress in ECG classification,
their scalability is poor.

With the advent of DNNs, which is an end-to-end techinique that automatically
combines feature extraction and classification together, a large number of
works have been built on it for ECG analysis. Particularly, Yildirim et al.
[9] designed two models, i.e., DBLSTM-WS and DULSTM-WS, to classify ECG
signals using the LSTM algorithm and achieved recognition rates of 99.39$\%$
and 99.25$\%$, respectively. In their another work, Yildirim et al. [20]
founded a CAE-LSTM model that significantly improved the time cost of the LSTM
network and obtained an accuracy of 99.23$\%$. The application of DNNs in ECG
classification has achieved promising success, but the design of specific
network architecture has become another major challenge.

Considering the rising demand for network architecture engineering, NAS came
into being. The RL-based NAS [21] is considered a pioneering work in NAS. They
obtained the network architecture using the reinforce learning method and
achieved state-of-the-art performance on image classification. Subsequently,
the EA-based NAS [22] utilized the evolution algorithm to achieve similar
results. However, they both have a high demand for calculation resources.
Therefore, numerous works have arisen on reducing the computation resource and
accelerating the network architecture search. The most notable work is the
Darts approach proposed by Liu et al. [17], which introduced a novel idea of
searching architecture in a continuous search space and achieved state-of-the-
art performance. Given the strengths of Darts, the idea of applying Darts to
address the aforementioned challenges of ECG classification is motivated.

##  III Methodology

Task-specific network architecture design is a challenging task, which has
inspired us to consider using NAS to automate this process for ECG
classification. In view of the limitation of computation resources, Darts, an
approach of NAS with lower computational resource requirements, becomes a
better choice. Additionally, the depth of the DNNs is an important factor in
determining the performance of the network. In general, the learning ability
of DNNs will be improved as the number of network layers increases. But in
fact, simply stacking the number of network layers may cause network
performance degradation, since the deep network model will encounter gradient
vanishing/explosion problem when the number of network layers is deep enough.
To address this issue, ResNet [15] proposes an efficient strategy of skipping
multiple network layers with “shortcut connections”. Based on the above
inspiration, in this paper, we propose a novel method named Heart-Darts for
effective classification of ECG signals.

Figure 1:  An overview of Darts: (a) Operations on each edge are initially
unknown. (b) A mixture of candidate operations are placed on each edge. (c)
Optimization with gradient descent. (d) Operation with max probabilities is
selected for each edge to form the final architecture. [17].

###  III-A The proposed Heart-Darts Method

The Heart-Darts method aims to customize a high-performance model for ECG
classification. It mainly consists of two parts: (1) Darts-based cell
architecture search (2) cells-based novel model construction. The specific
algorithm of Heart-Darts is presented in Algorithm 1.

The cell architecture is dominant because of the final model formed by
stacking cells. To obtain the optimal cell architecture for ECG
classification, we employ the Darts as a powerful tool. It introduces a novel
idea of searching for the cell architecture in a continuous search space,
unlike traditional NAS whose search space is discrete such as EA-based NAS
[22] or RL-based NAS [21].

In Darts, a cell is defined as a directed acyclic graph that contains N
ordered nodes, where the cells are of two types: normal cell and reduction
cell. Specifically, each node $x^{(i)}$ is a latent representation of a
feature map computed by Equation (1) and each directed edge $(i,j)$ stands for
some operations $o^{(i,j)}$ on $x^{(i)}$ [17]. Furthermore, the operation
candidate set is denoted as $O$, where each operation represents a function
$o(\cdot)$ with $x^{(i)}$ as input. To make search space continuous, Darts
defines $\alpha=\\{\alpha^{(i,j)}\\}$ as the encoding of architecture, where
$\alpha^{(i,j)}$ is a vector of dimension $|O|$ represents the operation
mixing weights for a pair of nodes $(i,j)$. Then $\bar{o}^{(i,j)}$ can be
obtained from Equation (2), which is a weighted summation of all operations on
each directed edge $(i,j)$. At the end of the search, $\bar{o}^{(i,j)}$ is
replaced with the operation corresponding to the largest $\alpha^{(i,j)}$
value: $o^{(i,j)}=argmax_{o\in O}\alpha_{o}^{(i,j)}$, and the top-2 strongest
operations are selected as inputs for each node. An overview of Darts is shown
in Figure 1. Since the encoding of cell architecture
$\alpha=(\alpha_{normal},\alpha_{reduce})$ is a learnable parameter, the
ultimate goal of Darts is to find $\alpha^{*}$ that minimizes the validation
loss by the gradient descent: $\min\limits_{\alpha}L_{val}(\omega,\alpha)$. It
is noteworthy that due to the time consumption issue, we discard the original
Darts’ bilevel optimization strategy and adopt an alternating update of the
weights and $\alpha$. The whole process of cell architecture search is shown
in Steps 1 to 6 in Algorithm 1.

$x^{(j)}=\sum_{i<j}o^{(i,j)}(x^{(i)})$ (1) $\bar{o}^{(i,j)}(x)=\sum_{o\in
O}\frac{exp(\alpha_{o}^{(i,j)})}{\sum_{o^{{}^{\prime}}\in
O}exp(\alpha_{o^{{}^{\prime}}}^{(i,j)})}o(x)$ (2)

Figure 2: The complete proposed model architecture

After obtaining the optimal cell architecture, a final model is assembled from
the obtained cells, where 1/3 and 2/3 layers are reduction cells follows [23,
24, 22]. Although the final model is based on the stack of obtained cells, it
does not mean that the more cells the better performance, as gradient
vanishing/explosion may occur with the increase of the number of cells. To
prevent this phenomenon, we introduce a shortcut strategy of the ResNet [15]
in the final model architecture. Specifically, we devise a novel model
architecture that incorporates a preprocessing layer and two shortcut
connections as shown in Fig 2. And considering the dimensional consistency,
two skip connections are attached separately before the two reduction cells.
In Algorithm 1, Steps 8 to 19 show the adding of the shortcut connections to
the model constructed in Step 7.

1 Define the operation candidate set $O$.

2 Initialize weights of architecture
$\alpha=(\alpha_{normal},\alpha_{reduce})$.

3 while  _not converged_ do

4

5 Update architecture $\alpha$ by $\nabla_{\alpha}L_{val}(w,\alpha)$;

6

7 Update weights $w$ by $\nabla_{w}L_{train}(w,\alpha)$;

8

9

10Obtain the optimal cell architecture $\alpha^{*}$

11Construct a model without shortcut connections based on the obtained cells.

12 Initalize input nodes $s_{0}$, $s_{1}$.

13 while  _not converged_ do

14

15 for  _$i <layers$_ do

16 compute $s_{0}$, $s_{1}$;

17 if  _$i==0$ or $i==\frac{layers}{3}$ or $i==\frac{2\times layers}{3}$_ then

18 $t=s_{1};$

19 else if  _$i==\frac{layers}{3}-1$ or $i==\frac{2\times layers}{3}-1$_ then

20 $s_{0}=ReLu(s_{0}+t);$

21 $s_{1}=ReLu(s_{1}+t);$

22

23 $output=classifier(s_{1})$

24 Update weights $w$ by descending $\nabla_{w}L_{train}(w)$

return the best model

Algorithm 1 Heart-Darts algorithm

##  IV Experiment Design

In this section, we provide details of the chosen databases, data
preprocessing and parameter settings used in the experiments for investigating
the performance of Heart-Darts. All experiments are performed on a single
1080Ti GPU with PyTorch implementation.

The experiment is divided into three main stages: cell architecture search,
model construction and evaluation, and model generalization validation. The
first part focuses on searching for an optimal cell architecture by Darts.
Then a novel model is constructed and evaluated based on the obtained cells.
In the last part, to validate the generalization capability of the proposed
model, we migrate the model on two new database. It is noteworthy that the
ordinary cross-entropy function is used in all experiments.

Figure 3: The data distribution details for MIT-BIH arrhythmia database

###  IV-A The Details of Database

We used three databases in this study: MIT-BIH arrhythmia database, St
Petersburg Institute of Cardiology Technics(INCART) 12-leads arrhythmia, and
QT database. They were obtained from an open-source PhysioNet database[25].
Among all the data, there are 15 types of heartbeats as shown in Table I.
According to the standards advocated by Advanced Medical Instruments(AAMI)
[26], the types of heartbeat re-classified into five classes: Normal beats(N),
Supraventricular ectopic beats(S), Ventricular ectopic beats(V), Fusion
beats(F), Unclassified beats(Q).

The reclassification data are detailed in Table II. As can be seen from the
table, there is a significant data imbalance problem in each database,
especially in the INCART database. The MIT-BIH arrhythmia database, which has
a total of 94,013 heartbeats, is involved in the cell architecture search and
model evaluation phases. The remaining two databases are availed for
validating model generalization. For each database, we randomly select 80% as
the training set and 20% as the test set. Regarding the cell architecture
search, we need to re-segment the training set of the MIT-BIH arrhythmia
database into two parts, with the first 80$\%$ for training and the last
20$\%$ for validation. The data distribution details of the MIT-BIH arrhythmia
database are shown in Figure 3. The details of the three databases are
described below.

Figure 4: Five types of heartbeats on the MIT-BIH arrhythmia database TABLE I: Fifteen arrhythmia heartbeats in the MIT-BIH database.. Annotation |  Heartbeat type |  Annotation |  Heartbeat type
---|---|---|---
N |  Normal beat |  S |  Supraventricular premature beat
L |  Left bundle branch block beat |  j |  Nodal (junctional) escape beat
R |  Right bundle branch block beat |  f |  Fusion of paced and normal beat
A |  Atrial premature beat |  E |  Ventricular escape beat
V |  Premature ventricular contraction |  J |  Nodal (junctional) premature beat
a |  Aberrated atrial premature beat |  e |  Atrial escape beat
P |  Paced beat |  Q |  Unclassifiable beat
F |  Fusion of ventricular and normal beat |  |
TABLE II: The five classes of division schemes recommended by AAMI. AAMI class |  MIT-BIH classes annotations |  Group description database |  MIT-BIH database |  INCART database |  QT databse
---|---|---|---|---|---
N |  L, N, R, e, j |  Normal beats |  79,508 |  153,350 |  81,022
S |  A, J, S, a |  Supraventricular ectopic beats |  2696 |  1957 |  1485
V |  E, V |  Ventricular ectopic beats |  7129 |  19,973 |  1691
F |  F |  Fusion beats |  793 |  219 |  251
Q |  P, Q, f |  Unclassified beats |  3887 |  - |  2146
Total |  15 |  5 |  94,013 |  175,499 |  86,595
TABLE III: Distribution of lead types and record numbers in MIT-BIH arrhythmia database. Lead A | V5 | V5 | MLII | MLII | MLII | MLII
---|---|---|---|---|---|---
Lead B | V2 | MLII | V2 | V4 | V5 | V1
Records | 102 | 114 | 103 | 124 | 100 123 | 101 105 106 107 108 109 111 112 113 115 116 118 119 121 122 200 201 202 203 205
| 104 |  | 117 |  | 123 | 207 208 209 210 212 213 214 215 217 219 220 221 222 223 228 230 231 232 233 234

  * •

MIT-BIH arrhythmia database contains 48 records obtained from 47 subjects (25
males and 22 females). Each record consists of 30-minutes of dual-channel
(noted as lead A and lead B) ambulatory ECG signals with a sampling rate of
360 Hz. In lead A, modified-lead II (MLII) contains 45 records except for 102,
104, 114 records. In lead B, lead V1 involves most of the records. The details
of the lead shown in Table III. In this study, two datasets are selected. One
is 2-lead dataset consisting of lead V1 and lead MLII with 40 records, and the
other is single-lead dataset consisting of lead V1 with the same records as
2-lead dataset.

  * •

St Petersburg INCART database consists of 75 annotated recordings obtained
from 32 subjects(17 males and 15 females, aged 18-80). Each record contains
30-minutes 12 standard leads ambulatory ECG signals. The sample rate is 257
Hz. In this database, we select lead II and lead V1 to compose a 2-lead
dataset like of the MIT-BIH database described above.

  * •

QT database contains 100 records of 15-minute dual-lead (noted as lead MLII
and lead V5) ECGs with a sampling rate of 250 Hz. For comparison, we select
lead MLII as a single-lead dataset.

Figure 5: A complete heartbeat observed on an ECG

###  IV-B The Data Preprocessing

As the ECG signals are long-term recordings of cardiac activity, it is
difficult to directly classify the raw data of the ECG signals. Therefore, a
proper data preprocessing should be employed. In this experiment, the data
preprocessing consists of two parts: denoising and heartbeat classification.

ECG signals are waveform data associated with time series. The raw ECG signals
cover a lot of noise affected by the environment, such as power frequency
interference, baseline drift, high-frequency noises caused by muscle
contraction and electrode movement, etc [27]. To restore the real
characteristics of the ECG signals, denoising is a desirable option. In this
study, we use the denoising method based on wavelet transform.

TABLE IV: A confusion matrix for single-lead dataset.

|  | Predicted | +P | Se | Spe | Acc
---|---|---|---|---|---|---
|  | N | S | V | F | Q | ($\%$) | ($\%$) | ($\%$) | ($\%$)
Original | N | 16007 | 28 | 8 | 1 | 0 | 99.63 | 99.77 | 97.83 | 99.35
S | 27 | 472 | 5 | 0 | 0 | 94.21 | 93.65 | 99.84
V | 15 | 1 | 1290 | 13 | 0 | 98.47 | 97.80 | 99.89
F | 14 | 0 | 6 | 120 | 0 | 89.55 | 85.71 | 99.92
Q | 4 | 0 | 1 | 0 | 791 | 100.00 | 99.37 | 100.00

  * 1

Se: Sensitivity, +P: Positive predictivity, Spe: Specificity, Acc: Accuracy.

A complete heartbeat consists of P wave, QRS-complex, T wave, PR segment, and
ST segment as shown in Figure 5. One record contains multiple heartbeats,
which are segmented based on R wave peak position. The standard of
segmentation is to make the heartbeat contains the most pathological
information. A total of three databases are used in this paper, and the
details of the databases are described in Section IV-A. For the MIT-BIH
arrhythmia database, we choose the first 99 sampling points and the last 200
sampling points of the marked R peak position. The five types of heartbeats
with lead MLII and lead V1 in the MIT-BIH arrhythmia database shown in Figure
4. For the INCART database, we choose 99 sampling points before the marked R
peak position and 150 sampling points after it. For the QT database, we take
the first 99 sampling points and the last 120 sampling points of the marked R
peak position.

###  IV-C Parameter Settings

In cell architecture search, we work only with the MIT-BIH arrhythmia 2-lead
dataset (lead MILL and lead V1). According to the previous state-of-the-art
network architectures [28, 11, 12], we define the candidate operation set $O$
that includes the following operations: $3\times 1$ convolution, $5\times 1$
convolution, $9\times 1$ convolution, $13\times 1$ convolution, $17\times 1$
convolution, $27\times 1$ convolution, $3\times 1$ max pooling, $5\times 1$
max pooling, skip connection and zeros. Operation zero($\cdot$) represents
there is no connection between a pair of nodes and the order of the
convolution operation is Conv-BN-ReLU. The preprocessing layer consists of a
1-D convolutional layer (the size of kernels is 5, the strides are 2 and the
number of filters is 32), a batch normalization (BN) layer, a ReLU layer, and
a max-pooling layer (the pooling size is 3 and the strides are 2). And the
convolutional cell consists of 7 nodes, including two input nodes and one
output node. A small network composed of 8 cells is trained by Darts for 100
epochs with a batch size of 100. Additionally, the initial channel of this
small network is 16. The rest of the hyperparameters are the same with [17] on
CIFAR-10.

After obtaining the optimal cell architecture in the first stage, a novel CNN
model is customized with the 15 cells. To investigate the performance of this
model, evaluations are performed on the MIT-BIH arrhythmia single-lead and
2-lead datasets, respectively. We train the model for 100 epochs with a batch
size of 256 from scratch and report the performance on the MIT-BIH arrhythmia
test dataset. The other hyperparameters are identical to the cell architecture
search phase. Additionally, we train three different CNN models, CNN-9 [11],
CNN-11 [12], and ResNet-31 [29], in the same setups to fairly compare the
model performance. The CNN-9 and CNN-11 are generally regarded as the earliest
successful CNN models in ECG classification. The ResNet-31, a block-based
network with a novel shortcut connection architecture similar to ours, is
selected to validate the performance cell architecture.

For the model generalization validation stage, we migrate the proposed model
on two new databases: INCART database and QT database. The model is trained
from scratch with random weights. All hyperparameters are the same as in the
model evaluation phase. Furthermore, to make a fair comparison, we train the
CNN-9 and CNN-11 from scratch with the same settings over the two new
databases.

##  V Results Analysis

TABLE V: A confusion matrix for 2-lead dataset.

|  | Predicted | +P | Se | Spe | Acc
---|---|---|---|---|---|---
|  | N | S | V | F | Q | ($\%$) | ($\%$) | ($\%$) | ($\%$)
Original | N | 16001 | 26 | 7 | 8 | 2 | 99.72 | 99.73 | 98.37 | 99.43
S | 27 | 476 | 1 | 0 | 0 | 94.82 | 94.44 | 99.86
V | 9 | 0 | 1304 | 6 | 0 | 98.56 | 98.86 | 99.89
F | 7 | 0 | 11 | 121 | 1 | 89.63 | 86.43 | 99.92
Q | 2 | 0 | 0 | 0 | 794 | 99.62 | 99.75 | 99.98

  * 1

Se: Sensitivity, +P: Positive Predictivity, Spe: Specificity, Acc: Accuracy.

Table VI presents the results of the model evaluation on the MIT-BIH
arrhythmia test set. A comparison is presented between three different CNN
models, CNN-9 [11], CNN-11 [12] and ResNet-31 [29], and the proposed Heart-
Darts-base model. Additionally, we train an additional Heart-Darts-base model
without shortcut connections to verify the proposed shortcut strategy.

TABLE VI: The detailed results of the architecture evaluation.

Architecture | Lead | Acc | Se | +P | F1
---|---|---|---|---|---
|  | ($\%$) | ($\%$) | ($\%$) | ($\%$)
CNN-9 [11] | Single-lead | 98.89 | 92.11 | 95.92 | 93.98
| 2-Lead | 99.03 | 93.60 | 96.43 | 94.99
CNN-11 [12] | Single-lead | 98.88 | 91.38 | 96.58 | 93.91
| 2-Lead | 99.10 | 93.15 | 97.29 | 95.18
ResNet-31 [29] | Single-lead | 99.00 | 91.11 | 97.05 | 93.98
| 2-Lead | 99.23 | 94.31 | 96.51 | 95.40
Heart-Darts | Single-lead | 99.29 | 94.87 | 95.76 | 95.31
| 2-Lead | 99.39 | 95.57 | 96.28 | 95.92
Heart-Darts + shortcut | Single-lead | 99.35 | 95.26 | 96.37 | 95.81
| 2-Lead | 99.43 | 95.84 | 96.47 | 96.16

  * 1

Acc: Accuracy, Se: Sensitivity, +P: Positive predictivity, F1: F1-score .

TABLE VII: The detailed results of the generalization validation.

Architecture | Database | Lead | Acc | Se | +P | F1
---|---|---|---|---|---|---
|  | ($\%$) | ($\%$) | ($\%$) | ($\%$) | ($\%$)
CNN-9 | QT | single-lead | 99.50 | 89.59 | 96.69 | 93.00
| INCART | 2-lead | 99.45 | 77.25 | 95.39 | 85.37
CNN-11 | QT | single-lead | 99.36 | 88.10 | 97.05 | 92.36
| INCART | 2-lead | 99.54 | 79.42 | 88.63 | 83.77
Heart-Darts | QT | single-lead | 99.69 | 93.76 | 98.38 | 96.01
| INCART | 2-lead | 99.81 | 90.05 | 89.49 | 89.77

  * 1

Acc: Accuracy, Se: Sensitivity, +P: Positive predictivity, F1: F1-score .

Figure 6: The test accuracy of Heart-Darts on single-lead and 2-lead datasets

(a) The normal cell architecture of 2-lead data

(b) The reduction cell architecture of 2-lead data

Figure 7: The cell architectures searched on 2-lead dataset

Results show that our Heart-Darts-base model achieves new state-of-the-art
across all the metrics except precision. Specifically, the Heart-Darts-base
model outperforms other models by approximately 0.4$\%$, 4$\%$, 2$\%$ in terms
of accuracy, sensitivity, and f1-score, respectively, for single-lead dataset.
For the 2-lead dataset, our proposed model achieves approximately 0.3$\%$,
2$\%$, and 1$\%$ gain over other models in terms of accuracy, sensitivity, and
f1-score, respectively. The promising outcomes of sensitivity and f1-score
demonstrated that the Heart-Darts approach is more advantageous in data
imbalance problems. Specifically, model architectures customized by auto-
search methods may have stronger feature extraction capabilities than hand-
crafted models for categories with less data. Additionally, the Heart-Darts-
base model with shortcut connections yielded superior results, which
demonstrates the effectiveness of the proposed shortcut strategy. It is worth
mentioning that the 2-lead dataset further improves the performance of ECG
classification compared with the single-lead as shown in Figure 6.

During the model evaluation phase, the detailed performance metrics of the
Heart-Darts-base model for both datasets are presented in the confusion
matrices, as shown in Tables IV and Tables V. For the single-lead dataset,
Heart-Darts achieved the global accuracy of 99.35$\%$ and the sensitivity of
class F with the least data reach 85.71$\%$. For the 2-lead dataset, Heart-
Darts achieved 99.43$\%$ accuracy and the sensitivity regarding the F class is
86.43$\%$. Figure 7 shows the cell architecture of the Heart-Darts-base model
and the whole process of cell architecture search costs 16 hours.

For model generalization validation, the results are presented in Table VII.
Note that the Heart-Darts-base model outperforms across all metrics except
precision of the INCART database. Particularly, the proposed model outperforms
the other two models by about 11$\%$ and 6$\%$ for the INCART database and by
approximately 5$\%$ and 3$\%$ for the QT database on sensitivity and f1-score,
respectively, which demonstrated that the Heart-Darts approach gains better
generalization capability.

##  VI Conclusion

The purpose of this paper is to explore a promising CNN model being applied to
ECG classification. To achieve this, we proposed a Heart-Darts approach by
employing the Darts (i.e., an efficient neural architecture search method) to
design an excellent CNN model. The Heart-Darts approach is the first
application of Darts in ECG classification, which also proposes a novel model
architecture to further improve the performance. Specifically, Heart-Darts
proposes a specific Darts for ECG analysis which automatically searches the
CNN cell architecture. After obtaining the cell architecture, Heart-Darts
customizes a CNN model with a novel model architecture for ECG classification.
To validate the performance of the Heart-Darts-base model, we performed the
model evaluation and generalization validation, both of them yielded appealing
results. Despite the promising results we have obtained, the proposed Heart-
Darts approach also suffers from several drawbacks. For example, the Darts
search process relies heavily on the operation candidate set and the problem
of model dimensionality constraints in Darts-based approaches. These potential
improvements are left for future work.

##  VII Acknowledge

This work is supported by the National Key Research and Development Project of
China (No.2017YFB1002201), National Natural Science Fund for Distinguished
Young Scholar (GrantNo. 61625204), the State Key Program of National Science
Foundation of China (GrantNo. 61836006), and the Sichuan Science and
Technology Program (NO. 2020YFG0323).

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# On the Cosmology of Dirac-Born-Infeld dRGT massive gravity

Sobhan Kazempour<EMAIL_ADDRESS>Faculty of Physics, University of
Tabriz, Tabriz 51666-16471, Iran  Amin Rezaei Akbarieh
<EMAIL_ADDRESS>Faculty of Physics, University of Tabriz, Tabriz
51666-16471, Iran  Hossein Motavalli<EMAIL_ADDRESS>Faculty of
Physics, University of Tabriz, Tabriz 51666-16471, Iran  Lijing Shao
<EMAIL_ADDRESS>Kavli Institute for Astronomy and Astrophysics, Peking
University, Beijing 100871, China  National Astronomical Observatories,
Chinese Academy of Sciences, Beijing 100012, China

###### Abstract

We introduce the cosmological analysis of the Dirac-Born-Infeld dRGT massive
gravity theory which is a new extension of de Rham-Gabadadze-Tolley (dRGT)
massive gravity. In this theory, we consider the Dirac-Born-Infeld (DBI)
scalar field which is coupled to the graviton field. Moreover, we perform the
cosmological background equations, and we demonstrate the self-accelerating
background solutions. We show that the theory consists of self-accelerating
solutions with an effective cosmological constant. In the following, we
exhibit tensor perturbations analyses and achieve the dispersion relation of
gravitational waves. We analyze the propagation of gravitational perturbation
in the Friedmann-Lemaître-Robertson-Walker cosmology in the DBI dRGT massive
gravity. Finally, we present the vector and scalar perturbations to show the
stability conditions of the theory.

††preprint: APS/123-QED

##  I Introduction

According to a number of observational evidence such as baryon acoustic
oscillations Beutler:2011hx ; SDSS:2009ocz , CMB Planck:2015fie ; WMAP:2003elm
, and supernovas I-a Phillips:1993ng ; SupernovaSearchTeam:1998fmf , it is
strongly accepted that the Universe is in the accelerated expansion phase. To
explain the origin of the accelerated expansion of the Universe and the
cosmological constant problem, there are some different approaches
Copeland:2006wr ; Sotiriou:2008rp ; Clifton:2011jh ; Nojiri:2006ri ;
Bamba:2012cp . It is clear that general relativity is a unique theory of a
massless Lorentz-invariant spin-2 particle Weinberg:1965rz . However, this
theory can not explain the late-time acceleration of the Universe
Weinberg:1988cp ; Peebles:2002gy . One approach to solving the problems in
cosmology is a massive gravity theory in which gravity is propagated by a
spin-2 nonzero graviton mass deRham:2010ik ; deRham:2010kj ;
Hinterbichler:2011tt ; deRham:2014zqa ; Hassan:2011hr ; Hassan:2011zd .

In 1939, Fierz and Pauli introduced the first massive spin-2 field theory.
They presented the unique Lorentz-invariant linear theory without ghosts in a
flat spacetime Fierz:1939ix . But, there was a discontinuity (van Dam-Veltman-
Zakharov i.e., vDVZ discontinuity) which the theory does not reduce to general
relativity in the limit of $m_{g}\rightarrow 0$ vanDam:1970vg ;
Zakharov:1970cc . While Vainshtein solved this problem by considering the
nonlinear Fierz-Pauli action instead of linear Vainshtein:1972sx , Boulware
and Deser claimed that the nonlinear Fierz-Pauli action has a ghost which is
called the Boulware-Deser ghost Boulware:1972yco . Also, Arkani-Hamed et al.
and Creminelli et al. confirmed this issue which the nonlinear massive gravity
is an unstable theory Arkani-Hamed:2002bjr ; Creminelli:2005qk . However, de
Rham, Gabadadze, and Tolley (dRGT) demonstrated the fully nonlinear massive
gravity without the Boulware-Deser ghost in 2010 deRham:2010ik ; deRham:2010kj
. They constructed a theory with nonlinear interactions which can show the
massive spin-2 field in a flat spacetime.

As the dRGT massive gravity theory is only valid for an open FLRW solution,
and there are not any stable solutions for homogeneous and isotropic Universe
DeFelice:2012mx , alternative theories have been proposed DAmico:2011eto ;
Gumrukcuoglu:2012aa ; DeFelice:2013awa ; Huang:2012pe ; Hassan:2011zd ;
Hinterbichler:2012cn . In addition, in the dRGT massive gravity theory, there
are a strong coupling problem and a nonlinear ghost instability, which is why
the scalar and vector perturbations vanish Gumrukcuoglu:2011zh . It is
noticeable that considering the extra degree of freedom such as an extra
scalar field is one of the great approaches. This way, the quasi-dilaton
massive gravity theory is successful to explain the accelerated expansion of
the Universe DAmico:2012hia . However, because of the perturbations
instability in the quasi-dilaton massive gravity theory, extensions of this
theory have been presented DeFelice:2013tsa ; Mukohyama:2014rca ;
Akbarieh:2021vhv ; Aslmarand:2021qwn . In this paper, we consider an extra
scalar field whose kinetic term has a Dirac-Born-Infeld (DBI) form. Actually,
we introduce the new extension of the dRGT massive gravity theory. Also, we
will show the perturbations analyses for this new extension in order to show
this new extension is free of instability.

As there is an attractor solution in the scalar fields with inverse power-law
potentials, there has been a tendency towards this kind of theory Ratra:1987rm
. In other words, the significance of this issue lies in the fact that the
present-day behavior of the Universe is insensitive to the initial conditions
Martin:2008xw . The Dirac-Born-Infeld model has something to do with inflation
and string theory. Recently, there has been a trend toward finding the
connection between string theory and inflation. It should be mentioned that
the main ideas are that according to the concept of brans, inflation
interprets as the distance between two branes that move in the extra
dimensions along a warped throat Martin:2008xw ; Kachru:2003sx ;
Silverstein:2003hf ; Alishahiha:2004eh ; Lorenz:2007ze .

As several attempts have been done for introducing the new extensions of
massive gravity theories Gannouji:2013rwa ; Mukohyama:2014rca ;
Gumrukcuoglu:2013nza ; Gabadadze:2014kaa ; Motohashi:2014una ;
Kahniashvili:2014wua ; Gumrukcuoglu:2016hic ; Gumrukcuoglu:2020utx ;
Akbarieh:2021vhv ; Aslmarand:2021qwn , we would like to introduce the new
extension of dRGT massive gravity theory which can explain the accelerated
expansion of the Universe in the FLRW cosmology, and we also try to show the
conditions of stability of the system using the perturbations analyses. In the
vector and scalar perturbations, we will exhibit the stability conditions of
the system. Moreover, in the tensor perturbation, the dispersion relation of
gravitational waves which can show the mass of graviton, will be obtained.

It should be noted that in order to impose the constraints on the modified
gravity theories, detecting and analyzing gravitational waves is really
essential. Using the mass of graviton, the speed of gravitational wave
propagation can be determined. So, by comparing the speed of gravitational
waves and their electromagnetic waves, we can find the constraints on the
parameters of the theories. This issue shows the significance of the
investigation of the dispersion relation of gravitational waves.

The outline of this paper is as follows. In Sec II, the Dirac-Born-Infeld dRGT
massive gravity theory are presented. Moreover, we demonstrate the background
equations of motion and self-accelerating solutions. In Sec III, we present
perturbation analysis to determine the dispersion relations of gravitational
waves in this theory. In addition, we elaborate on the stability condition of
the system in the vector and scalar perturbations. In Sec IV, we present the
conclusion and discussion.

In this paper, we use $M_{\rm Pl}^{2}\equiv 8\pi G=1$ where $G$ is Newton’s
gravitational constant. It will be considered natural units ($c=\hslash=1$).

##  II Dirac-Born-Infeld dRGT Massive Gravity

In this section, we begin by introducing a new extension of the dRGT massive
gravity theory. In this new extension, we consider the Dirac-Born-Infeld (DBI)
scalar field which is the response to the dark energy scalar field. We review
the dRGT massive gravity theory which is extended by Dirac-Born-Infeld terms.
Meanwhile, we perform the evolution of a cosmological background.

The action includes the Ricci scalar $R$, scalar field $\sigma$, the tension
$T(\sigma)$, the massive gravity term $U({\cal K})$, a dynamical metric
$g_{\mu\nu}$ and it’s determinant $\sqrt{-g}$. The action is given by

$\displaystyle S=\frac{1}{2}\int
d^{4}x\Bigg{\\{}\sqrt{-g}\bigg{[}R+T(\sigma)\bigg{(}1-\sqrt{1-\frac{g^{\mu\nu}\partial_{\mu}\sigma\partial_{\nu}\sigma}{T(\sigma)}}\bigg{)}$
$\displaystyle+2{m}_{g}^{2}U(\mathcal{K})\bigg{]}\Bigg{\\}}.$

It should be pointed out that the mass of graviton $m_{g}$, originates from
the potential $U(\mathcal{K})$ which is

$\displaystyle U(\mathcal{K})$
$\displaystyle=U_{2}+\alpha_{3}U_{3}+\alpha_{4}U_{4},$ $\displaystyle U_{2}$
$\displaystyle=\frac{1}{2}\big{(}[\mathcal{K}]^{2}-[\mathcal{K}^{2}]\big{)},$
$\displaystyle U_{3}$
$\displaystyle=\frac{1}{6}\big{(}[\mathcal{K}]^{3}-3[\mathcal{K}][\mathcal{K}^{2}]+2[\mathcal{K}^{3}]\big{)},$
$\displaystyle U_{4}$
$\displaystyle=\frac{1}{24}\big{(}[\mathcal{K}]^{4}-6[\mathcal{K}]^{2}[\mathcal{K}^{2}]+8[\mathcal{K}][\mathcal{K}^{3}]+3[\mathcal{K}^{2}]^{2}$
(2) $\displaystyle-6[\mathcal{K}^{4}]\big{)},$

where $\alpha_{3}$ and $\alpha_{4}$ are dimensionless free parameters of the
theory. Here the ”$[\cdot]$” is the trace of the tensor inside brackets.
Moreover, it is necessary to note that the building block tensor $\mathcal{K}$
is defined as

$\mathcal{K}^{\mu}_{\nu}=\delta^{\mu}_{\nu}-e^{\sigma}\sqrt{g^{\mu\alpha}f_{\alpha\nu}},$
(3)

where $f_{\alpha\nu}$ is the fiducial metric, which is defined through

$f_{\alpha\nu}=\partial_{\alpha}\phi^{c}\partial_{\nu}\phi^{d}\eta_{cd}.$ (4)

Here $g^{\mu\nu}$ is the physical metric, $\eta_{cd}$ is the Minkowski metric
with $c,d=0,1,2,3$ and $\phi^{c}$ are the Stueckelberg fields which are
introduced to restore general covariance. In addition, the theory is invariant
under a global dilation transformation, $\sigma\rightarrow\sigma+\sigma_{0}$.

The FLRW Universe is considered because of our cosmological motivation. As a
result, we represent the dynamical and fiducial metrics,

$\displaystyle g_{\mu\nu}$ $\displaystyle={\rm
diag}\left[-N^{2},a^{2},a^{2},a^{2}\right],$ (5) $\displaystyle f_{\mu\nu}$
$\displaystyle={\rm diag}\left[-\dot{f}(t)^{2},1,1,1\right].$ (6)

It should be paid attention that $N$ is the lapse function of the dynamical
metric, and it is similar to a gauge function. Furthermore, the $a$ is the
scale factor, and the $\dot{a}$ is the derivative with respect to time. Also,
the lapse function has something to do with the coordinate-time $dt$ and the
proper-time $d\tau$ via $d\tau=Ndt$ Scheel:1994yr ; Christodoulakis:2013xha .
It is worth noting that the function $f(t)$ is the Stueckelberg scalar
function whereas $\phi^{0}=f(t)$ and $\frac{\partial\phi^{0}}{\partial
t}=\dot{f}(t)$ Arkani-Hamed:2002bjr .

So, the point-like Lagrangian of the Dirac-Born-Infeld dRGT massive gravity in
FLRW cosmology is

$\displaystyle\mathcal{L}=\frac{-3a\dot{a}^{2}}{N}-\frac{a^{3}N}{2}T(\sigma)\bigg{[}\sqrt{1-\frac{\dot{\sigma}^{2}}{T(\sigma)N^{2}}}-1\bigg{]}$
$\displaystyle+m_{g}^{2}a^{3}(Y-1)\Bigg{\\{}N\bigg{[}3(Y-2)-(Y-4)(Y-1)\alpha_{3}$
$\displaystyle-(Y-1)^{2}\alpha_{4}\bigg{]}+\dot{f}(t)aY(Y-1)\bigg{[}3-3(Y-1)\alpha_{3}$
$\displaystyle+(Y-1)^{2}\alpha_{4}\bigg{]}\Bigg{\\}},$

where

$Y\equiv\frac{e^{\sigma}}{a}.$ (8)

To simplify expressions later, we define

$H\equiv\frac{\dot{a}}{Na}.$ (9)

###  II.1 Background Equations of Motion

We obtain the cosmological background equations for a FLRW background. This
way, by varying with respect to $f$, and paying attention to the unitary
gauge, i.e. $f(t)=t$, a constraint equation is given by:

$\displaystyle\frac{\delta\mathcal{L}}{\delta
f}=m_{g}^{2}\frac{d}{dt}\bigg{[}a^{4}Y(Y-1)\times\bigg{(}3-3(Y-1)\alpha_{3}$
$\displaystyle+(Y-1)^{2}\alpha_{4}\bigg{)}\Bigg{]}=0.$

On the classical level, the unphysical fields should be eliminated from the
Lagrangian using gauge transformations Grosse-Knetter:1992tbp .

In this step, we calculate the equation of motion which is related to the
lapse function $N$

$\displaystyle\frac{1}{a^{3}}\frac{\delta\mathcal{L}}{\delta
N}=3H^{2}+\frac{T(\sigma)}{2\bigg{(}T(\sigma)N^{2}+\big{(}HN+\frac{\dot{Y}}{Y}\big{)}^{2}\bigg{)}}\bigg{[}\big{(}HN+\frac{\dot{Y}}{Y}\big{)}^{2}-T(\sigma)N^{2}\bigg{(}\sqrt{1+\frac{\big{(}HN+\frac{\dot{Y}}{Y}\big{)}^{2}}{T(\sigma)N^{2}}}-1\bigg{)}\bigg{]}$
$\displaystyle-
m_{g}^{2}(Y-1)\bigg{[}-3(Y-2)+(Y-4)(Y-1)\alpha_{3}+(Y-1)^{2}\alpha_{4}\bigg{]}=0.$

By varying with respect to $\sigma$, we have

$\displaystyle\frac{1}{a^{3}N}\frac{\delta
S}{\delta\sigma}=\frac{1}{4\big{(}H^{2}+T(\sigma)N^{2}\big{)}^{2}}\Bigg{\\{}-2T(\sigma)^{2}N^{4}T^{{}^{\prime}}(\sigma)\bigg{(}\sqrt{1+\frac{H^{2}}{T(\sigma)N^{2}}}-1\bigg{)}+2H^{4}\bigg{(}3T(\sigma)N\sqrt{1+\frac{H^{2}}{T(\sigma)N^{2}}}+T^{{}^{\prime}}(\sigma)\bigg{)}$
$\displaystyle+H^{2}T(\sigma)N^{2}\bigg{(}3\sqrt{1+\frac{H^{2}}{T(\sigma)N^{2}}}\big{(}2T(\sigma)N-T^{{}^{\prime}}(\sigma)\big{)}+4T^{{}^{\prime}}(\sigma)\bigg{)}\Bigg{\\}}+m_{g}^{2}Y\Bigg{\\{}-(3+r)\big{(}3+3\alpha_{3}+\alpha_{4}\big{)}$
$\displaystyle+6(r+1)Y\big{(}\alpha_{4}+2\alpha_{3}+1\big{)}-3(3r+1)(\alpha_{4}+\alpha_{3})Y^{2}+4r\alpha_{4}Y^{3}\Bigg{\\}}=0,$

where

$\displaystyle r\equiv\frac{a}{N}.$ (13)

Notice that the following equations can be obtained using Eq. (8),

$\frac{\dot{\sigma}}{N}=H+\frac{\dot{Y}}{NY},\qquad\ddot{\sigma}=\frac{d}{dt}\Big{(}NH+\frac{\dot{Y}}{Y}\Big{)},$
(14)

According to the time reparametrization invariance introduced by the
Stueckelberg field $f$, there is a Bianchi identity that guarantees that the
equation of motion related to the scale factor $a$ is redundant. The Bianchi
identity is,

$\displaystyle\frac{\delta S}{\delta\sigma}\dot{\sigma}+\frac{\delta S}{\delta
f}\dot{f}-N\frac{d}{dt}\frac{\delta S}{\delta N}+\dot{a}\frac{\delta S}{\delta
a}=0.$ (15)

Therefore, this equation can be ignored.

It should be indicated that in the spacial states, all of the cosmological
background equations reached to those in Refs. DAmico:2012hia ;
Gumrukcuoglu:2013nza .

###  II.2 Self-Accelerating Background Solutions

In this subsection, we indicate the analyses of self-accelerating solutions to
explain the accelerated expansion of the Universe. We now start the discussion
of the solutions by the Stueckelberg constraint in Eq. (II.1). By integrating
that equation we have,

$\displaystyle
Y(Y-1)\bigg{[}3-3(Y-1)\alpha_{3}+(Y-1)^{2}\alpha_{4}\bigg{]}\propto a^{-4}.$
(16)

It is necessary to mention that the constant solutions of $Y$ lead to the
effective energy density and behave similarly to a cosmological constant. In
other words, the massive gravity term affects the expansion like a
cosmological constant. As we would like to consider an expanding universe, and
the right-hand side of the equation (16) will decrease after a long enough
time, $Y$ saturates to a constant value $Y_{\rm SA}$, which is a root of the
left-hand side of Eq. (16).

According to DAmico:2012hia , there are four constant solutions for $Y$. For
avoiding to encounter strong coupling, we leave this solution DAmico:2012hia .
Thus, we are left with,

$(Y-1)\big{[}3-3(Y-1)\alpha_{3}+(Y-1)^{2}\alpha_{4}\big{]}\bigg{|}_{Y=Y_{\rm
SA}}=0.$ (17)

Furthermore, it is clear that the other solution is $Y=1$. However, this
solution leads to inconsistency and vanishing cosmological constant, and this
is the reason why it is unacceptable DAmico:2012hia .

So, the two remaining solutions of Eq. (16) are

$Y_{\rm
SA}^{\pm}=\frac{3\alpha_{3}+2\alpha_{4}\pm\sqrt{9\alpha_{3}^{2}-12\alpha_{4}}}{2\alpha_{4}}.$
(18)

Here, we obtain the modified Friedmann equation. We represent the Friedmann
equation (II.1) in a different form,

$\displaystyle
3H^{2}+\frac{\tilde{T}}{2\big{(}\tilde{T}N^{2}+H^{2}N^{2}\big{)}}\bigg{[}H^{2}N^{2}-\tilde{T}N^{2}\big{(}\sqrt{1+\frac{H^{2}}{\tilde{T}}}$
$\displaystyle-1\big{)}\bigg{]}=\Lambda_{\rm SA}^{\pm},$

where $\tilde{T}$ is a saturate of $T(\sigma)$. We solve the Eq. (II.2) to
calculate the $H^{2}$, so we have

$\displaystyle H^{2}=\frac{1}{18}\Bigg{\\{}4(\Lambda_{\rm
SA}^{\pm}-2\tilde{T})+\big{(}2\Lambda_{\rm
SA}^{\pm}+5\tilde{T}\big{)}^{2}\bigg{(}9\sqrt{\tilde{T}^{3}\big{(}-16\Lambda_{\rm
SA}^{\pm 3}-120\Lambda_{\rm SA}^{\pm 2}\tilde{T}-300\Lambda_{\rm
SA}^{\pm}\tilde{T}^{2}-169\tilde{T}^{3}\big{)}}-2\big{(}4\Lambda_{\rm SA}^{\pm
3}$ $\displaystyle+30\Lambda_{\rm SA}^{\pm 2}\tilde{T}+75\Lambda_{\rm
SA}^{\pm}\tilde{T}^{2}+22\tilde{T}^{3}\big{)}\bigg{)}^{-\frac{1}{3}}+\bigg{(}-2\big{(}4\Lambda_{\rm
SA}^{\pm 3}+30\Lambda_{\rm SA}^{\pm 2}\tilde{T}+75\Lambda_{\rm
SA}^{\pm}\tilde{T}^{2}+22\tilde{T}^{3}\big{)}$
$\displaystyle+9\sqrt{\tilde{T}^{3}\big{(}-16\Lambda_{\rm SA}^{\pm
3}-120\Lambda_{\rm SA}^{\pm 2}\tilde{T}-300\Lambda_{\rm
SA}^{\pm}\tilde{T}^{2}-169\tilde{T}^{3}\big{)}}\bigg{)}^{\frac{1}{3}}\Bigg{\\}}.$
(20)

Note that the effective cosmological constant from the mass term is

$\displaystyle\Lambda_{\rm SA}^{\pm}\equiv m_{g}^{2}(Y_{\rm
SA}^{\pm}-1)\bigg{[}$ $\displaystyle-3Y_{\rm SA}^{\pm}+6+(Y_{\rm
SA}^{\pm}-4)(Y_{\rm SA}^{\pm}-1)\alpha_{3}$ (21) $\displaystyle+(Y_{\rm
SA}^{\pm}-1)^{2}\alpha_{4}\bigg{]}.$

Using Eq. (18), the above equation can be written as

$\displaystyle\Lambda_{\rm
SA}^{\pm}=\frac{3m^{2}_{g}}{2\alpha^{3}_{4}}\bigg{[}9\alpha^{4}_{3}\pm
3\alpha^{3}_{3}\sqrt{9\alpha^{2}_{3}-12\alpha_{4}}-18\alpha^{2}_{3}\alpha_{4}$
$\displaystyle\mp
4\alpha_{3}\alpha_{4}\sqrt{9\alpha^{2}_{3}-12\alpha_{4}}+6\alpha^{2}_{4}\bigg{]}.$
(22)

Therefore, from Eq. (II.1) we have,

$\displaystyle r_{\rm
SA}=\frac{1}{2\big{(}H^{2}+\tilde{T}\big{)}}\Bigg{\\{}2\tilde{T}+H^{2}\bigg{(}2-\frac{\tilde{T}N\sqrt{1+\frac{H^{2}}{\tilde{T}N^{2}}}}{m_{g}^{2}Y_{\rm
SA}^{\pm\rm 2}\big{(}\alpha_{3}Y_{\rm
SA}^{\pm}-\alpha_{3}-2\big{)}}\bigg{)}\Bigg{\\}}.$

The above equation interprets the self-accelerating universe without any
strong coupling. Thus, we have shown that this theory consists of self-
accelerating solutions with an effective cosmological constant.

##  III PERTURBATIONS ANALYSIS

At this stage, we would like to demonstrate the perturbations analyses. These
analyses are crucial to indicate the stability of solutions. To reach this
goal, we focus on quadratic perturbations. The physical metric $g_{\mu\nu}$
can be expanded in terms of small fluctuations $\delta g_{\mu\nu}$ around a
background solution $g_{\mu\nu}^{(0)}$.

$g_{\mu\nu}=g_{\mu\nu}^{(0)}+\delta g_{\mu\nu}.$ (24)

Note that the metric perturbations can be divided into three parts, namely
scalar, vector, and tensor perturbations. Therefore, we have

$\displaystyle\delta g_{00}=$ $\displaystyle-2N^{2}\Phi,$ $\displaystyle\delta
g_{0i}=$ $\displaystyle Na(B_{i}+\partial_{i}B),$ $\displaystyle\delta
g_{ij}=$ $\displaystyle
a^{2}\bigg{[}h_{ij}+\frac{1}{2}(\partial_{i}E_{j}+\partial_{j}E_{i})+2\delta_{ij}\Psi$
(25)
$\displaystyle+\big{(}\partial_{i}\partial_{j}-\frac{1}{3}\delta_{ij}\partial_{l}\partial^{l}\big{)}E\bigg{]},$

All perturbations agree with the spatial rotation transformations, and they
are functions of time and space. Also, there are these conditions
$\delta^{ij}h_{ij}=\partial^{i}h_{ij}=\partial^{i}E_{i}=\partial^{i}B_{i}=0$
for scalar, vector, and tensor perturbations which means that the tensor
perturbations are transverse and traceless.

The the scalar field $\sigma$ would be perturbed as follows

$\sigma=\sigma^{(0)}+\delta\sigma.$ (26)

In the following procedures, all terms should be kept in quadratic order in
$\delta g_{\mu\nu}$. It is crucial to mention that we have not had any
problems with the form of gauge-invariant combinations because we have
indicated all analyses in the unitary gauge. In addition, the expanded actions
can be written in the Fourier domain with plane waves, i.e.,
$\vec{\nabla}^{2}\rightarrow-k^{2}$, $d^{3}x\rightarrow d^{3}k$. Note that the
spatial indices on perturbations are raised and lowered by $\delta^{ij}$ and
$\delta_{ij}$.

###  III.1 Tensor

The significance of tensor perturbation analysis lies in the fact that it is
the only source of gravitational waves in general relativity. As the
dispersion relation of gravitational waves is different in modified gravity
models, the speed of propagation of the gravitational wave is different from
the speed of light in standard general relativity. In this subsection, we
obtain the dispersion relation of gravitational waves in the Dirac-Born-Infeld
dRGT massive gravity theory. We consider tensor perturbations around the
background,

$\delta g_{ij}=a^{2}h_{ij},$ (27)

where

$\partial^{i}h_{ij}=0\quad{\rm and}\quad g^{ij}h_{ij}=0.$ (28)

Note that we calculate separately the tensor perturbed actions in the second-
order for any part of the main action. The gravity part of the perturbed
action in quadratic order is

$\displaystyle S^{(2)}_{\rm gravity}=$ $\displaystyle\frac{1}{8}\int
d^{3}k\,dt\,a^{3}\bigg{[}\frac{\dot{h}_{ij}\dot{h}^{ij}}{N^{2}}$ (29)
$\displaystyle-\Big{(}\frac{k^{2}}{a^{2}}+4\frac{\dot{H}}{N}+6H^{2}\Big{)}h^{ij}h_{ij}\bigg{]}.$

Furthermore, we obtain the Dirac-Born-Infeld part of the perturbed action in
quadratic order

$\displaystyle S^{(2)}_{\rm DBI}=\frac{1}{8}\int
d^{3}k\,dt\,a^{3}N\bigg{[}T(\sigma)\bigg{(}\sqrt{1-\frac{\dot{\sigma}^{2}}{T(\sigma)N^{2}}}-1\bigg{)}\bigg{]}h^{ij}h_{ij}.$

We derive the massive gravity sector of the perturbed action

$\displaystyle S^{(2)}_{\rm massive}$ $\displaystyle=\frac{1}{8}\int
d^{3}k\,dt\,a^{3}Nm_{g}^{2}\bigg{[}(\alpha_{3}+\alpha_{4})rY^{3}-(1$ (31)
$\displaystyle+2\alpha_{3}+\alpha_{4})(1+3r)Y^{2}+(3+3\alpha_{3}+\alpha_{4})$
$\displaystyle(3+2r)Y-2(6+4\alpha_{3}+\alpha_{4})\bigg{]}h^{ij}h_{ij}.$

Summing up the second-order pieces of the perturbed actions $S^{(2)}_{\rm
gravity}$, $S^{(2)}_{\rm DBI}$, and $S^{(2)}_{\rm massive}$, we calculate the
total action in the second-order for tensor perturbations

$\displaystyle S^{(2)}_{\rm total}=\frac{1}{8}\int
d^{3}k\,dt\,a^{3}N\bigg{\\{}\frac{\dot{h}_{ij}\dot{h}^{ij}}{N^{2}}-\Big{(}\frac{k^{2}}{a^{2}}+M_{\rm
GW}^{2}\Big{)}h^{ij}h_{ij}\bigg{\\}}.$

At this stage, using Eq. (18), we calculate $\alpha_{3}$ and substitute it. As
a result, the dispersion relation of gravitational waves is obtained as

$\displaystyle M^{2}_{\rm
GW}=4\frac{\dot{H}}{N}+6H^{2}-T(\sigma)\bigg{(}\sqrt{1-\frac{\dot{\sigma}^{2}}{T(\sigma)N^{2}}}-1\bigg{)}-\Delta,$

where

$\displaystyle\Delta=\frac{m_{g}^{2}}{(Y_{\rm SA}^{\pm}-1)}\Bigg{\\{}Y_{\rm
SA}^{\pm}\bigg{[}18+8\alpha_{3}+Y_{\rm SA}^{\pm}\bigg{(}2\alpha_{3}Y_{\rm
SA}^{\pm}+Y_{\rm SA}^{\pm}-r_{\rm SA}\big{(}3(\alpha_{3}+2)$
$\displaystyle+Y_{\rm SA}^{\pm}(\alpha_{3}Y_{\rm
SA}^{\pm}-4\alpha_{3}-3)\big{)}-8\alpha_{3}-10\bigg{)}\bigg{]}-2(\alpha_{3}+3)\Bigg{\\}},$

where $Y_{\rm SA}^{\pm}$ can not be equal to 1, because the role of the
massive gravity term vanishes.

It is noticeable that if the mass square of gravitational waves would be
positive, the stability of long-wavelength gravitational waves is guaranteed.
However, if it would be negative, it is tachyonic. So, the instability can
take the age of the Universe to develop if the mass of the tachyon would be
the order of the Hubble scale.

In fact, we exhibit the modified dispersion relation of gravitational waves.
Actually, the propagation of gravitational perturbations in the FLRW cosmology
in the Dirac-Born-Infeld dRGT massive gravity is presented.

###  III.2 Vector

Here the vector perturbation analyses are shown for the Dirac-Born-Infeld dRGT
massive gravity theory. This part’s main goal is to show the conditions in
which there would not be any instabilities.

We consider the vector perturbations,

$\displaystyle
B_{i}=\frac{a(r^{2}-1)k^{2}}{\bigg{[}2k^{2}(r-1)+4T(\sigma)\big{(}\sqrt{\frac{H^{2}a^{2}}{T(\sigma)N^{2}}}-1\big{)}\bigg{]}}\frac{\dot{E}_{i}}{N}.$
(35)

As the field $B_{i}$ is nondynamical, it can be entered into the action as an
auxiliary field. So, a single propagating vector is given

$\displaystyle S_{\rm vector}^{(2)}=\frac{1}{8}\int
d^{3}k\,dt\,a^{3}N\bigg{(}\frac{\beta}{N^{2}}|\dot{E}_{i}|^{2}-\frac{k^{2}}{2}M_{\rm
GW}^{2}|E_{i}|^{2}\bigg{)},$

where

$\displaystyle\beta=\frac{k^{2}}{2}\bigg{(}1+\frac{k^{2}(r^{2}-1)}{2T(\sigma)(\sqrt{\frac{H^{2}a^{2}}{T(\sigma)N^{2}}}-1)}\bigg{)}^{-1}.$
(37)

It can be found that there are two cases, in the first one, if we have
$\frac{(r^{2}-1)}{2T(\sigma)\big{(}\sqrt{\frac{H^{2}a^{2}}{T(\sigma)N^{2}}}-1\big{)}}\geq
0$, we do not need the critical momentum scale. However, in the second one, if
we have
$\frac{(r^{2}-1)}{2T(\sigma)\big{(}\sqrt{\frac{H^{2}a^{2}}{T(\sigma)N^{2}}}-1\big{)}}<0$,
we should have a critical momentum scale $k_{c}$ to avoid a ghost, which is

$\displaystyle
k_{c}=\sqrt{\frac{2T(\sigma)\big{[}\sqrt{\frac{H^{2}a^{2}}{T(\sigma)N^{2}}}-1\big{]}}{(1-r^{2})}}\quad{\rm
if}\quad\frac{(r^{2}-1)}{2T(\sigma)\big{(}\sqrt{\frac{H^{2}a^{2}}{T(\sigma)N^{2}}}-1\big{)}}<0,$

in other words, to have stability in the system we require the $k$ in Eq. (37)
to be smaller than a critical momentum scale $k_{c}$.

In the following, we consider the canonically normalized fields for
determining the other instabilities in the vector modes as below

$\displaystyle\zeta_{i}=\frac{\beta E_{i}}{2}.$ (39)

We have substituted Eq. (39) to the Eq. (III.2),

$\displaystyle S=\frac{1}{2}\int
d^{3}k\,dt\,a^{3}N\bigg{(}\frac{|\dot{\zeta_{i}}|^{2}}{N^{2}}-c_{V}^{2}|\zeta_{i}|^{2}\bigg{)}.$
(40)

The sound speed for vector modes is given by

$\displaystyle c_{V}^{2}=M_{\rm
GW}^{2}(1+u^{2})-\frac{H^{2}u^{2}(1+4u^{2})}{(1+u^{2})^{2}},$ (41)

where the dimensionless quantity is considered as below

$\displaystyle
u^{2}\equiv\frac{k^{2}(r^{2}-1)}{2T(\sigma)\big{(}\sqrt{\frac{H^{2}a^{2}}{T(\sigma)N^{2}}}-1\big{)}}.$
(42)

In fact, we evaluate the conditions which can cause instability in the system.
Notice that to avoid tachyonic instability which can be originated from the
first part of Eq. (41), if we have $M_{\rm GW}^{2}<0$ and $u^{2}>0$, and to
avoid instability the below conditions must be there.

$\displaystyle
k_{c}^{2}\lesssim\frac{2T(\sigma)\big{(}\sqrt{\frac{H^{2}a^{2}}{T(\sigma)N^{2}}}-1\big{)}}{(r^{2}-1)},$
$\displaystyle{\rm
if}\quad\frac{(r^{2}-1)}{2T(\sigma)\big{(}\sqrt{\frac{H^{2}a^{2}}{T(\sigma)N^{2}}}-1\big{)}}>0\quad{\rm
and}\quad M_{\rm GW}^{2}<0.$ (43)

If we consider all physical momenta below a critical momentum scale $k_{c}$,
thus, a growth rate of instability should be lower than the cosmological
scale.

Furthermore, if we pay attention to the second part of Eq. (41), two cases can
be considered. In the first case, if we have $u^{2}>0$, there is not any
instability faster than the Hubble expansion. In the second case which is
$u^{2}<0$, as we have the no-ghost condition Eq. (III.2), to avoid instability
we should have $|u^{2}|\lesssim\frac{k^{2}}{k_{c}^{2}}$. This way, there is
not any instability in the second part of Eq. (41).

In the end, in order to guarantee the stability for vector modes, we should
consider $c_{V}^{2}>0$. In addition, for avoiding the instability the mass
square of the dispersion relation of gravitational waves should be positive
$M_{\rm GW}^{2}>0$, as we pointed out in the Sec. III.1.

###  III.3 Scalar

In this subsection, we exhibit the scalar perturbations analyses in the Dirac-
Born-Infeld dRGT massive gravity theory to elaborate on the stability of the
system.

The process would be started by the action quadratic in scalar perturbations

$\displaystyle\delta g_{00}=$ $\displaystyle-2N^{2}\Phi,$ $\displaystyle\delta
g_{0i}=$ $\displaystyle N\,a\,\partial_{i}B,$ $\displaystyle\delta g_{ij}=$
$\displaystyle
a^{2}\bigg{[}2\delta_{ij}\Psi+\big{(}\partial_{i}\partial_{j}-\frac{1}{3}\delta_{ij}\partial_{l}\partial^{l}\big{)}E\bigg{]},$
(44) $\sigma=\sigma^{(0)}+\delta\sigma.$ (45)

According to the fact that the perturbation $\Phi$ and $B$ are free of time
derivatives, they can be used as auxiliary fields using their equations of
motion.

$\displaystyle
B=\frac{(r^{2}-1)a}{3T(\sigma)\bigg{[}1-\sqrt{\frac{H^{2}a^{2}}{T(\sigma)N^{2}}}\bigg{]}}\Bigg{\\{}6H\Phi-\frac{1}{N}(k^{2}\dot{E}+6\dot{\Psi})$
$\displaystyle+\frac{3T(\sigma)}{Ha^{2}}\bigg{[}1-\sqrt{\frac{H^{2}a^{2}}{T(\sigma)N^{2}}}\bigg{]}\delta\sigma\Bigg{\\}},$

$\displaystyle\Phi=\frac{1}{3H^{4}\bigg{[}4k^{2}\big{(}r^{2}-1\big{)}+\frac{1}{H^{2}a^{2}}\big{(}2QT(\sigma)-T(\sigma)^{2}\big{)}+6Q-7T(\sigma)\bigg{]}}\Bigg{\\{}\frac{k^{4}}{a^{4}}\bigg{(}2k^{2}\big{(}r^{2}-1\big{)}$
$\displaystyle-\frac{3}{(r-1)}\big{(}Q-T(\sigma)\big{)}\bigg{)}\bigg{(}T(\sigma)-Q\bigg{)}^{2}\big{(}E+3\big{)}\delta\sigma+\frac{3T(\sigma)H^{2}}{(r-1)a^{2}}\bigg{[}\bigg{(}2k^{2}(r-1)$
$\displaystyle+3\big{(}Q-T(\sigma)\big{)}\bigg{)}\bigg{]}\bigg{(}\frac{Q}{T(\sigma)}-1\bigg{)}\Psi+\frac{2H^{3}k^{2}}{N}\bigg{(}k^{2}\dot{E}+6\dot{\Psi}\bigg{)}\big{(}r^{2}-1\big{)}$
$\displaystyle-\frac{3HT(\sigma)}{Na^{2}}\bigg{(}\big{(}Q-T(\sigma)\big{)}\delta\dot{\sigma}-6H^{2}a^{2}\dot{\Psi}\bigg{)}\big{(}\frac{Q}{T(\sigma)}-1\big{)}\Bigg{\\}},$

where

$\displaystyle Q\equiv\sqrt{\frac{T(\sigma)H^{2}a^{2}}{N^{2}}}.$ (48)

It is worth pointing out that if we substitute these equations into the
action, we obtain the action which contains three fields, $E$, $\Psi$, and
$\delta\sigma$. Meanwhile, we determine another nondynamical combination to
remove the sixth degree of freedom, which is

$\displaystyle\tilde{\Psi}=\frac{1}{\sqrt{2}}(\Psi+\delta\sigma).$ (49)

Moreover, an orthogonal combination can be defined,

$\displaystyle\tilde{\delta\sigma}=\frac{1}{\sqrt{2}k^{2}}(\Psi-\delta\sigma).$
(50)

Note that using above field redefinitions, we write the action in terms of
$\tilde{\Psi}$, $\tilde{\delta\sigma}$, and $E$, with no time derivatives on
$\tilde{\Psi}$. Using the following equation the auxiliary of $\tilde{\Psi}$
can be eliminated,

$\displaystyle\tilde{\Psi}=\frac{1}{3r(r-1)\bigg{[}\big{(}T(\sigma)^{2}-2QT(\sigma)\big{)}+H^{2}a^{2}\bigg{(}4k^{2}+\big{(}7T(\sigma)-6Q\big{)}\bigg{)}\bigg{]}}\Bigg{\\{}\bigg{[}-3k^{2}r\big{(}r-1\big{)}\bigg{(}2QT(\sigma)$
$\displaystyle-T(\sigma)^{2}\bigg{)}+72H^{4}a^{4}\bigg{(}4k^{2}\big{(}r^{2}-1\big{)}+\big{(}6Q-7T(\sigma)\big{)}\bigg{)}-3H^{2}a^{2}\bigg{(}4k^{4}r\big{(}r-1\big{)}+k^{2}r\big{(}6Q$
$\displaystyle+6rQ-T(\sigma)(7r-5)\big{)}+24\big{(}T(\sigma)^{2}-2QT(\sigma)\big{)}\bigg{)}\bigg{]}\tilde{\delta\sigma}-2\sqrt{2}H^{2}a^{2}k^{4}r\big{(}r-1\big{)}E$
$\displaystyle+\bigg{[}18H^{3}a^{4}\bigg{(}4k^{2}\big{(}r^{2}-1\big{)}+6Q-7T(\sigma)\bigg{)}+6Ha^{2}\bigg{(}-2k^{2}r(r-1)\big{(}Q-T(\sigma)\big{)}$
$\displaystyle-3T(\sigma)^{2}+6QT(\sigma)\bigg{)}\bigg{]}\frac{\dot{\tilde{\delta\sigma}}}{N}+\sqrt{2}Ha^{2}k^{2}r\big{(}r-1\big{)}\bigg{[}6H^{2}a^{2}+T(\sigma)-Q\bigg{]}\frac{\dot{E}}{N}\Bigg{\\}}.$

We substitute this solution in the action, and by considering the notation
$A\equiv(\tilde{\delta\sigma},E)$, the scalar action is given by,

$\displaystyle S=\frac{1}{2}\int
d^{3}k\,dt\,a^{3}N\Bigg{\\{}\frac{\dot{A}^{\dagger}}{N}\mathcal{F}\frac{\dot{A}}{N}+\frac{\dot{A}^{\dagger}}{N}\mathcal{D}A$
$\displaystyle+A^{\dagger}D^{T}\frac{\dot{A}}{N}-A^{T}\varpi^{2}A\Bigg{\\}},$
(52)

here, we should pay attention that $D$ is a real anti-symmetric $2\times 2$
matrix, and $\mathcal{F}$ and $\varpi^{2}$ are real symmetric $2\times 2$
matrices.

This way, we demonstrate the components of the matrix $\mathcal{F}$ as below

$\displaystyle\mathcal{F}_{11}=\frac{2k^{4}\big{(}Q-1\big{)}}{H^{2}a^{2}}\Bigg{[}1+\frac{9H^{2}a^{2}}{k^{2}\big{(}r-1\big{)}^{2}}-\frac{\bigg{(}\big{(}Q-T(\sigma)\big{)}(r-1)-6H^{2}a^{2}r\bigg{)}^{2}}{\bigg{[}\big{(}T(\sigma)^{2}-2QT(\sigma)\big{)}+H^{2}a^{2}\bigg{(}4k^{2}+\big{(}7T(\sigma)-6Q\big{)}\bigg{)}\bigg{]}\big{(}r-1\big{)}^{2}}\Bigg{]},$
(53)
$\displaystyle\mathcal{F}_{12}=\frac{\sqrt{2}k^{4}\big{(}Q-T(\sigma)\big{)}\bigg{[}2k^{2}\big{(}r-1\big{)}-3r\big{(}6H^{2}a^{2}+T(\sigma)-Q\big{)}\bigg{]}}{3\big{(}r-1\big{)}\bigg{[}\big{(}T(\sigma)^{2}-2T(\sigma)Q\big{)}+H^{2}a^{2}\big{(}4k^{2}+7T(\sigma)-6Q\big{)}\bigg{]}},$
(54)
$\displaystyle\mathcal{F}_{22}=\frac{k^{4}\big{(}Q-T(\sigma)\big{)}\Bigg{[}2k^{2}-18H^{2}a^{2}+3\big{(}Q-T(\sigma)\big{)}\Bigg{]}}{18\big{(}T(\sigma)^{2}-2QT(\sigma)\big{)}+18H^{2}a^{2}\bigg{[}4k^{2}+7T(\sigma)-6Q\bigg{]}}.$
(55)

For determining the sign of the eigenvalues, we elaborate the determinant of
the kinetic matrix $\mathcal{F}$. Therefore, we have

$\displaystyle{\rm det}\,\mathcal{F}\equiv$
$\displaystyle\mathcal{F}_{11}\mathcal{F}_{22}-\mathcal{F}_{12}^{2}=$ (56)
$\displaystyle\frac{3k^{6}\bigg{(}T(\sigma)-Q\bigg{)}^{2}}{(r-1)^{2}\bigg{(}Q-T(\sigma)-\frac{4k^{2}H^{2}a^{2}}{6H^{2}a^{2}+T(\sigma)-Q}\bigg{)}}.$

To avoid appearing the ghosts in the scalar sector, the following condition
should be taken into consideration,

$\displaystyle
k<\frac{1}{2aH}\sqrt{\bigg{(}6H^{2}a^{2}+T(\sigma)-\sqrt{\frac{H^{2}a^{2}T(\sigma)}{N^{2}}}\bigg{)}\bigg{(}\sqrt{\frac{H^{2}a^{2}T(\sigma)}{N^{2}}}-T(\sigma)\bigg{)}}.$

Thus, if the determinant is positive, we do not have a ghost degree of
freedom. In other words, it should be mentioned that the stability of the
scalar sector could be guaranteed with the use of the determinant of the
kinetic matrix. We have shown the conditions under which the system can
maintain its stability.

##  IV Conclusion

In conclusion, we should like to point out that introducing the new extended
dRGT massive gravity theory is so important due to this fact that can give us
the opportunity to understand how the extended theory can behave around their
cosmological backgrounds. In this present work, we have presented the Dirac-
Born-Infeld dRGT massive gravity which is constructed by considering coupled
DBI scalar field to the graviton field.

Initially, we have obtained the details of the new action and total
Lagrangian. We also performed the full set of equations of motion for a FLRW
background. In order to explain the late-time acceleration of the Universe in
the context of the Dirac-Born-Infeld dRGT massive gravity, we exhibited the
self-accelerating background solutions. In fact, we have shown that the theory
consists of self-accelerating solutions with an effective cosmological
constant which has something to do with massive gravity term.

Finally, we have presented the cosmological perturbations analyses, which
consist of tensor, vector, and scalar perturbations. In tensor perturbation,
we have demonstrated the tensor perturbation calculation to investigate the
mass of graviton for the Dirac-Born-Infeld dRGT massive gravity theory. In
other words, we have presented the modified dispersion relation of
gravitational waves. Furthermore, we have indicated that to guarantee the
stability of long-wavelength, the square of gravitational waves should be
positive i.e., $M_{\rm GW}^{2}>0$. Presenting the propagation of gravitational
perturbation in the FLRW cosmology in the Dirac-Born-Infeld dRGT massive
gravity theory is really vital in the era of gravitational waves. Empirically,
the mass of graviton can be constrained by many means deRham:2016nuf ,
including the propagation of gravitational waves Will:1997bb ;
LIGOScientific:2021sio , the dynamics of Solar system Will:2018gku , the
timing of binary pulsars Finn:2001qi ; Miao:2019nhf ; Shao:2020fka , and so
on. These methods probe different aspects of massive gravity theories
deRham:2016nuf . At the end, we have evaluated the vector and tensor
perturbations to determine the stability conditions of the system.

## Acknowledgements

We are really grateful to Nishant Agarwal for helpful notes and codes which
are related to tensor perturbations. Also, S.K and A.R.A would like to thank
A. Emir Gumrukcuolu for his useful comments. L.S was supported by the National
Natural Science Foundation of China (Nos. 11991053, 11975027, 11721303), the
National SKA Program of China (No. 2020SKA0120300), and the Max Planck Partner
Group Program funded by the Max Planck Society.

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