# Logographic Information Aids Learning Better Representations
for Natural Language Inference

Zijian Jin
Tandon School of Engineering
New York University
<EMAIL_ADDRESS>
Duygu Ataman
Courant Institute of Mathematical Sciences
New York University
<EMAIL_ADDRESS>

###### Abstract

Statistical language models conventionally implement representation learning
based on the contextual distribution of words or other formal units, whereas
any information related to the logographic features of written text are often
ignored, assuming they should be retrieved relying on the cooccurence
statistics. On the other hand, as language models become larger and require
more data to learn reliable representations, such assumptions may start to
fall back, especially under conditions of data sparsity. Many languages,
including Chinese and Vietnamese, use logographic writing systems where
surface forms are represented as a visual organization of smaller graphemic
units, which often contain many semantic cues. In this paper, we present a
novel study which explores the benefits of providing language models with
logographic information in learning better semantic representations. We test
our hypothesis in the natural language inference (NLI) task by evaluating the
benefit of computing multi-modal representations that combine contextual
information with glyph information. Our evaluation results in six languages
with different typology and writing systems suggest significant benefits of
using multi-modal embeddings in languages with logograhic systems, especially
for words with less occurence statistics.

Logographic Information Aids Learning Better Representations
for Natural Language Inference

Zijian Jin Tandon School of Engineering New York University<EMAIL_ADDRESS>Duygu Ataman Courant Institute of Mathematical Sciences New York University
<EMAIL_ADDRESS>

##  1 Introduction

The essential idea in statistical language modeling is to represent the
meaning of a word as a function of its context. The function, modeled via the
conditional probability of observing a word in a given utterance, has most
efficiently been approximated with a neural network based architecture
(Mikolov et al., 2013a, b; Bengio et al., 2003; Mikolov et al., 2010;
Sundermeyer et al., 2012). The outstanding performance of neural methods in
language modeling and their recent development (Peters et al., 2018; Tenney et
al., 2019) have them a preliminary component in various downstream NLP tasks.

One of the main limitations in the formulation of language models lies however
in the choice of ortographic units in calculating the contextual distribution,
which is usually convenient in English and other languages using phonetic
scripts. On the other hand, many languages rely on logographic writing
systems, where surface forms are represented as a visual organization of
smaller graphemic units and the word meaning can be changed through
compositional variations of these units. Although a direct segmentation of
these units has been found quite challenging due to visual compositions in the
final form of the grapheme, previous studies have found potential benefits of
using visual information to aid NLP models in sentence representation Liu et
al. (2017a); Meng et al. (2019); Dai and Cai (2017); Salesky et al. (2021). On
the other hand, none of these studies have focused on isolating the effects of
different linguistic features in relation to their correlation to visual
features.

As shown in Figure 1, logographic information often contain important features
related to the word meaning. In this paper, we perform the first focused
analysis to measure the significance of logographic features specifically to
the semantic information encoded in token or character-level language
representations by evaluating the performance of multi-modal embeddings in the
NLI task. In particular, we aim to answer the following research questions:

  * 1)

How important may logographic information be to for an accurate representation
of semantic information in word or character-level language units

  * 2)

Whether the contribution of logographic information to semantic
representations may depend on the language typology and writing system

Figure 1: Logographic information in Chinese.

In order to answer these questions we implement a multi-modal representation
learning model where each written text segment is representation as a
combination of visual embeddings obtained from prominent convolutional neural
network (CNN) based models Liu et al. (2017a); Meng et al. (2019); Salesky et
al. (2021), and contextual representations obtained from multilingual pre-
trained language models Devlin et al. (2019); Conneau et al. (2020a). We
evaluate the contribution of visual information to the performance in the NLI
task under few-shot learning settings in six languages with varying typology
and writing systems: English, Spanish, Hindi, Urdu, Vietnamese and Chinese. We
also study the optimal representation granularity for semantic information by
comparing word or character-level multi-modal representations in our
experiments.

Figure 2: Method overview.

In conclusion, we find that taking into account the visual information
improves the performance in NLI tasks especially in logographic languages like
Chinese and that the improvements are correlated with the factors that
determine the quality of token representations, such as the occurence of the
tokens in training data as well as language model capacity and
hyperparameters. Our findings suggest multi-modal processing is a promising
direction, especially for processing languages where conditions of data
sparsity may create fall backs in assumptions undertaken in statistical
formulations.

##  2 Computing Visual Glyph Embeddings

Our multi-modal embedding model is composed of two components: (i) the visual
encoder, which computes embeddings based on the input images representing each
text segment, and (ii) the pre-trained language model providing the text-based
embeddings.

#### Image conversion

Text segments consisting of complete sentences are split into words (or
characters) and then converted into images. Sentences are split into 30 x 60
pixel word images using the Jieba111https://github.com/fxsjy/jieba tool. All
graphemes are centralized to the middle of the image.

#### Visual Embeddings

In order to extract the glyph information from text images, we use the CNN
model developed by (Liu et al., 2017b; Sutskever et al., 2014) to generate
visual embeddings. The model consists of a three-layer CNN, augmented with a
two-layer feed-forward network. The full details of the network is given
below.

Layer | Visual encoding model
---|---
1 |  Spatial Conv. (3,3) $\rightarrow$ 32
2 | ReLu
3 |  MaxPool (2,2)
4 |  Spatial Conv. (3,3) $\rightarrow$ 32
5 | ReLu
6 |  MaxPool (2,2)
7 |  Spatial Conv. (3,3) $\rightarrow$ 32
8 | ReLu
9 |  Linear (800,128)
10 | ReLu
11 |  Linear (128,128)
12 | ReLu

The visual features extracted by the CNN are further encoded in a long-short
term memory (LSTM) Hochreiter and Schmidhuber (1997) network to learn the
glyph embeddings.

For a sequence consisting of $t$ tokens $x_{0},x_{1},...,x_{t}$, the visual
embedding $v$ is computed by concatenating (Su et al., 2020; Lu et al., 2019)
the hidden states of the LSTM and averaging them as

$v=\textrm{mean}([h_{0};h_{1};...;h_{t}])$

#### Embedding composition

In order to isolate the learning of representations from two modalities and
measure their effect on the learning task in a controlled setting, we deploy
late fusion in combining the visual embeddings with the text embeddings
obtained by the pre-trained model for prediction in the down-stream task. The
two embeddings are linearly composed through a simple affine projection and
then concatenated. For the down-stream prediction task we use a multi-layer
perceptron classifier.

##  3 Experiments

###  3.1 Character recognition

As an initial verification, we implement the visual encoder and evaluate it
individually in the character recognition task. We use the CASIA Chinese
Handwriting Database Liu et al. (2011) and obtain competitive results (93.23%
accuracy) on this task, confirming the visual encoder works sufficiently in
extracting character features from input images.

###  3.2 NLI

#### Data

We evaluate our model under few-shot learning settings using the XNLI dataset
Conneau et al. (2020b), using only a small portion of the testing data for
training and development, and test the effect of logographic information to
contribute to resolve the high level of semantic ambiguity.

Datasets | Number of Sentences
---|---
Training | 4509
Development | 501
Test | 2490
Table 1: Data statistics for training, development and test sets.

#### Model settings and hyper-parameters

In training the multi-modal models, the learning rates of both XLM-R and mBERT
based pre-trained models are set to 1e-6. The visual encoder is trained on the
images captured from the training sentences, either at word or character-level
resolution, with a learning rate of 4e-6 (for XLM-Roberta) and 1e-6 (for
mBERT). The hidden size of the LSTMs used is 128 and we use dropout of with
0.3 in this layer. All hyper-parameters are tuned with grid-search. For each
task we train 30 epochs and always choose the results with smallest validation
loss.

#### Languages

We pick six languages with varying typology and writing systems, including
English, Spanish, Urdu, Vietnamese, Chinese and Hindi. English and Spanish use
the Latin script; Urdu is written with the Arabic alphabet, whereas Hindi uses
Devanagari, all of which are phonetic writing systems. Chinese uses
logographic writing. Vietnamese, although traditionally have used logographic
writing, recently and in the XNLI data set is written with the Latin script.

#### Contextual representations

We verify the significance of logographic information for contributing to
enrich the language representations by testing our multi-modal approach with
two different pre-trained language models, including the mBert-base and the
XLM-R-base both available from Huggingface222https://huggingface.co. We also
investigate the effects of different segmentation methods for processing
sentence images either at the level of words or characters.

##  4 Results and Discussion

Table 2: Results in the XNLI benchmark. base models represent baseline pre-
trained language model performance in the down-stream task. base-CNN models
represent the multi-modal system performance. (C) denotes character and (W)
denotes word level input representations. Random stands for comparisons to
multi-modal systems where random images were input to the visual encoder to
verify the effect of visual information on the overall performance.

Languages | English | Chinese | Urdu | Hindi | Vietnamese | Spanish
---|---|---|---|---|---|---
mBERT-base | 65.86 | 55.28 | 51.29 | 56.48 | 57.08 | 62.27
mBERT-base-CNN (W) | 62.87 | 58.88 | 53.49 | 57.68 | 59.88 | 60.47
mBERT-base-CNN (C) | 64.07 | 59.08 | 53.69 | 57.48 | 60.07 | 61.67
mBERT-base-CNN (C) — Random | - | 54.33 | - | - | - | -
XLM-Roberta-base | 69.86 | 64.27 | 59.88 | 63.87 | 63.07 | 65.66
XLM-Roberta-base-CNN (W) | 69.26 | 66.66 | 57.88 | 62.87 | 61.67 | 65.46
XLM-Roberta-base-CNN (C) | 68.26 | 62.07 | 56.28 | 61.67 | 63.07 | 65.26
XLM-Roberta-base-CNN (C) — Random | - | 61.36 | - | - | - | -

Our experiment results are given in Table 2. At a first glance, we observe the
performance of the models are much lower than reported in Conneau et al.
(2020b), since we have significantly less training and development data
available. Under these challenging evaluation settings with high amount of
sparsity, we observe that the logographic information improves the performance
obtained using the mBert-base model in all languages that do not deploy the
Latin script, including Chinese, Urdu, Hindi and Vietnamese. In case of XLM-
Roberta-base, which had better optimization on a larger corpus, the overall
performance are consistently better than mBert-base and the improvements
remain consistent, especially in Chinese and Vietnamese. We hypothesize that
the slightly higher amount of improvements in mBERT-base might be due to
better quality of representations provided with the optimized training regime
of XLM-Roberta-base. Using the mBERT-base model, we find more advantage of
embedding logographic information at the character level in Chinese, Urdu and
Hindi, however, in Hindi, the results are comparable. When using the XLM-
Roberta-base, we observe improvements in Chinese with word-level glyph
embeddings and in Vietnamese using character-level glyph embeddings. While
character-level embeddings might be suitable for a phonetic language like
Vietnamese, the logograhic writing system in Chinese might make word-level
visual embeddings more convenient, since the intra-graphemic dependencies can
be captured at the visual level.

Although the improvements highly correlate with the logograhic nature of the
writing system, the fact that they apply to most languages, even Urdu and
Hindi with phonetic alphabets, point to the suboptimal effects in tokenization
or segmentation and their potential harms to correctly model the contextual
distribution. We also see in high-resource language representations like
English, our fusion method may be harmful to the downstream task, which we
anticipate that could be resolved with higher amount of fine-tuning and
development data. In light of all these considerations, the findings suggest
that multi-modality is a promising direction for overcoming problems related
to data sparsity, and eventually tokenization or segmentation-free language
modeling.

Models | Accuracy | # of UNK
---|---|---
mBERT-base (C) | 45.31 | 128
mBERT-base-CNN (C) | 52.43
Table 3: Results for targeted evaluation, UNK represents unknown tokens.

As an additional analysis investigating the effects of token frequency on the
positive effects of logographic information integrated in the language model,
we sample sentences in the test set that have unknown words in the model
vocabulary and compute the targeted accuracy on this sample of sentences. The
results shows in table 3 further illustrate the boosted performance on the
sample test, suggesting that data sparsity is an important obstacle to
learning high-quality contextual representations, and such conditions can be
the ideal place where logographic information might be useful to improve the
semantic features embedded in representations.

##  5 Conclusion

In this paper, we evaluated the benefits of using logographic information in
language modeling by implementing a multi-modal representation learning model
which combines contextual language representations with visual embeddings. Our
experiments in the NLI task in six languages confirmed the benefits of
logograhic information in obtaining more reliable semantic representations,
especially under sparse learning settings. As future work we hope to
contribute to the development of larger multilingual benchmarks to evaluate
the effect of visual information on more languages and linguistic phenomena.
Our software and the experimental data will be available upon publication.

## Acknowledgements

The authors would like to express deep gratitude to Chinmay Hedge and
Aishwarya Kamath for discussions and their useful feedback. This work was
supported by Samsung Advanced Institute of Technology (under the project Next
Generation Deep Learning: From Pattern Recognition to AI) and NSF Award
1922658 NRT-HDR: FUTURE Foundations, Translation, and Responsibility for Data
Science.

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2021

1]Sardar Vallabhbhai National Institute of Technology, Surat, 395007, Gujarat,
India

2]Delhi Technological University, Delhi, 110042, India

# Analytical Stellar Models of Neutron Stars in Teleparallel Gravity

Jay Solanki<EMAIL_ADDRESS>Rohan Joshi<EMAIL_ADDRESS>Malay
Garg<EMAIL_ADDRESS>[  [

###### Abstract

In this paper, we developed three analytical models and obtained a new class
of solutions describing compact stellar structures using the theory of
teleparallel gravity. We consider the general anisotropic nature of stellar
configurations and solve teleparallel gravity equations. In order to
thoroughly analyze the various parameters of the stars, we developed three
models by choosing various physically acceptable forms of metric potential
$e^{d(r)}$ and radial pressure $p_{r}(r)$. We also analyze the impact of
teleparallel gravity’s parameters $\beta$ and $\beta_{1}$ on the description
of the stellar structures. We calculated model parameters such that models
describing various observed neutron stars obey all physical conditions to be
potentially stable and causal. By analyzing the impact of various parameters
of teleparallel gravity on the description of anisotropic stellar structures,
we found that three models developed in this paper can describe anisotropic
neutron stars ranging from low density to high density. Finally, we obtain a
quadratic Equation of State for each model describing various neutron stars,
which can be utilized to find compositions of the stellar structures. It is
very useful to find models that can exhibit quadratic EOS, since material
compositions of real neutron stars and strange stars are found to exhibit
quadratic EOS by various authors. Non linear $f(T)$ model gives high deviation
of EOS from quadratic behaviour, thus, in this paper we work with linear
$f(T)$ function by using diagonal tetrad to model realistic compact stars.

###### keywords:

teleparallel gravity, stellar structures, neutron stars, EOS

##  1 Introduction

It is well known that general relativity describes the effect of gravity as an
effect of the curvature of space-time. Thus, general relativity is a
geometrical theory of gravitation. General relativity is a very successful
theory describing various astronomical and astrophysical phenomena with great
precision. However, recent observations of galactic dynamics can not be
understood in the tenets of general relativity. Thus, the concepts of dark
matter and dark energy have been introduced to understand the phenomena of
galactic dynamics and the expansion of the universe.COPELAND_2006 ;
sami2009dark ; sami2009primer ; 2011AdAst2011E…8G  However, another way to
account for observations of galactic dynamics like phenomena that can not be
explained by general relativity is to modify general
relativity.RevModPhys.82.451 ; durrer2008dark ; PhysRevD.70.043528 ;
PhysRevD.81.127301 ; Capozziello_2011 ; Nojiri:2010wj ; Nojiri:2017ncd ;
Nojiri:2003ft  The Lagrangian formulation of the general relativity is
constructed from Einstein-Hilbert action written in terms of Ricci scalar and
metric tensor.RevModPhys.82.451  The modified gravity can be constructed by
modifying the Einstein-Hilbert action. Modified field equations can be
obtained by varying modified Einstein-Hilbert action and implementing the
Euler-Lagrangian equation. In recent years there is a huge effort to modify
gravity to solve the issue of non-renormalizability of general
relativity.PhysRevLett.108.031101 ; PhysRevD.16.953

The general relativity is constructed from its dynamical variable - the metric
tensor, defined on a pseudo-Riemannian manifold. In general relativity,
connections are considered to be torsion-less. However, in order to account
for torsion, Einstein introduced Teleparallel Equivalent of General Relativity
known as TEGR.1985FoPh…15..365I ; PhysRevD.19.3524  In this theory, Einstein
introduced absolute parallelism and the tangent space as a frame of reference
to understand the effect of torsion. In TEGR, tetrad fields are used instead
of metric tensor to construct the theory.PhysRevD.19.3524  Because of using
tangent space frame of reference, the curvature at each point becomes zero,
and the only effect that remains is of torsion.doi:10.1142/S0218271808013972
Thus, TEGR describes the gravity as due to the effect of torsion of space-
time. TEGR also can be constructed by Lagrangian formulation by considering
tetrad field as dynamical variables instead of the metric tensor. Thus, we can
also modify this action to construct modified gravity in terms of the torsion
mechanism. This modified theory of gravity is known as $f(T)$ gravity.Cai_2016
; PhysRevD.79.124019 ; Kr_k_2016 . Also, recently the reference
bahamonde2021teleparallel  has provided a detailed review on the topic of
teleparallel gravity and modified $f(T)$ gravity, which readers can consider
for more discussion on the subject.

In recent years, teleparallel gravity is found to be very successful in
describing various astronomical and astrophysical phenomena. $f(T)$ gravity is
also a widely used theory besides $f(R)$ gravity to explain the expansion of
the universe and dark matter through modification of general
relativity.PhysRevD.84.083518 ; Momeni_2015 ; Momeni_2014a ; Jamil_2014 ;
Daouda_2011 ; El_Hanafy_2016a ; Jamil_2012a ; RODRIGUES_2013 ; HOUNDJO_2012 ;
Jamil_2012b ; El_Hanafy_2016b ; Jamil_2012c ; Jamil_2012d ; Jamil_2013 ;
Jamil_2012e ; Jamil_2012f ; Farooq_2013  The compact astrophysical objects
such as black hole and compact stars have been modeled using TEGRB_hmer_2011 ;
Ulhoa_2013 ; article1 ; Zubair_2015 ; Abbas_2015 ; article2 ; article3 ;
article4  as well as other theories of modified gravityAstashenok:2014nua ;
Capozziello:2015yza ; Astashenok:2020qds ; Astashenok:2021peo ;
Astashenok:2020cqq . Recently Wang et al. studied spherically symmetric static
solution in $f(T)$ gravity models PhysRevD.84.024042 . It is found that only a
limited class of $f(T)$ gravity models can be solved in this frame. Cesmsinnan
Deliduman and Baris Yapiskan investigated neutron stars under modified $f(T)$
gravity. They found that the relativistic neutron star solution in $f(T)$
gravity models is possible only if $f(T)$ is a linear function of the torsion
scalar $T$ for diagonal tetrad. The black hole with a cosmological constant
has been studied in this paperFerraro_2011 . The violation of the first law of
black hole thermodynamics in $f(T)$ gravity due to the lack of local Lorentz
invariance is studied in this work miao2011violation .

Thus, we choose linear function $f(T)$, to be $f(T)=\beta T+\beta_{1}$. This
is essentially known as teleparallel gravity with the extra factors $\beta$
and $\beta_{1}$. Many authors studied the effect of this linear $f(T)$ on the
modeling of anisotropic neutron stars. However, many of the studies are
performed by setting the parameter $\beta_{1}$ equals zero to describe and
analyze graphs of various parameters of neutron stars.Abbas_2015 ; article2 ;
article4  In this paper, we developed three models to understand the role of
various parameters like metric potential, radial pressure, $\beta$,
$\beta_{1}$, etc. on the properties of the neutron stars. We considered
different physically acceptable forms of metric potential $e^{d(r)}$ and
radial pressure $p_{r}(r)$, to solve equations of teleparallel gravity and to
generate models that can describe various types of neutron stars ranging from
low density to high density. In first two models we considered $\beta$ to be a
constant of value $\beta=2$. Then we compute the physically acceptable value
of $\beta_{1}$ along with other parameters of the models. Thus, in the first
two models, we analyze the effect of parameter $\beta_{1}$ on the properties
of observed neutron stars. In third model, we considered $\beta_{1}=0$, and
observed the effect of parameter $\beta$ on the properties of observed neutron
stars. We found that these three models can describe anisotropic neutron stars
ranging from low density to high density.

We organized this paper as follows: In section (2), we formulate the field
equations of teleparallel gravity. In that section, we briefly review the idea
of the tetrad field and write down the elements of the manifold in terms of
tetrad field. By varying the action and considering the Weitzenböck
connection, we write down equations of teleparallel gravity for anisotropic
matter distribution. In sections (3), (4), and (5), we developed the first,
second, and third model describing anisotropic neutron stars in teleparallel
gravity. In each of those sections we developed the model by considering
physically acceptable forms of metric potential $e^{d(r)}$ and radial pressure
$p_{r}(r)$. After that, we impose physical conditions so that model becomes
physically acceptable. After that, we calculated various model parameters for
describing observed neutron stars. We calculated model parameters and neutron
star’s physical parameters for each neutron star mentioned in this paper. We
also plotted graphs of various parameters of neutron stars to study the
general behavior of each model on the description of the stellar structures.
Finally, we obtain a quadratic Equation of State for each model describing
various neutron stars, which can be utilized to find compositions of the
stellar structures. Finally, we summarize the results obtained through
discussion section (6).

##  2 Formulation of the Field Equations in the Teleparallel Gravity

General relativity is constructed from its dynamical variable - the metric
tensor, defined on a pseudo-Riemannian manifold. However, in teleparallel
gravity tetrad field is used as a dynamical variable, which forms an
orthonormal basis in tangent space at each point. Now let us denote Latin
indices ($i$, $j$, …) for coordinates of tangent space and greek indices
($\mu$, $\nu$, …) for space-time coordinates. Thus basis vectors and covectors
for tangent space are given by $e_{i}=\partial_{i}$ and $e^{i}=dx^{i}$. Basis
vectors for space-time coordinates are given by $e_{\mu}=\partial_{\mu}$ and
$e^{\mu}=dx^{\mu}$. The vectors and covectors can be transformed from one base
to another as follow

$\partial_{i}=e\indices{{}_{i}^{\mu}}\partial_{\mu}\quad\mathrm{and}\quad
dx^{i}=e\indices{{}^{i}_{\mu}}dx^{\mu}$ (1)

and conversely

$\partial_{\mu}=e\indices{{}^{i}_{\mu}}\partial_{i}\quad\mathrm{and}\quad
dx^{\mu}=e\indices{{}_{i}^{\mu}}dx^{i}$ (2)

where $e\indices{{}^{i}_{\mu}}$ is known as tetrad field and
$e\indices{{}_{i}^{\mu}}$ is inverse of it, that satisfy the relations
$e\indices{{}^{i}_{\mu}}e\indices{{}_{j}^{\mu}}=\delta^{i}_{j}$ and
$e\indices{{}^{i}_{\mu}}e\indices{{}_{i}^{\nu}}=\delta^{\nu}_{\mu}$. The
relation between the metric of space-time and the tetrad field is given by

$g_{\mu\nu}=\eta_{ij}e\indices{{}^{i}_{\mu}}e\indices{{}^{j}_{\nu}}$ (3)

where $\eta_{ij}=diag[1,-1,-1,-1]$. From equation (1), line element of the
manifold is given in terms of tetrad field as

$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=\eta_{ij}e\indices{{}^{i}_{\mu}}e\indices{{}^{j}_{\nu}}dx^{\mu}dx^{\nu}$
(4)

Now in teleparallel gravity, the curvature-less connection known as
Weitzenböck connection is given by

$\Gamma\indices{{}^{\alpha}_{\mu}{}_{\nu}}=e\indices{{}_{i}^{\alpha}}\partial_{\nu}e\indices{{}^{i}_{\mu}}$
(5)

Now the torsion tensor in terms of Weitzenböck connection is given by

$T\indices{{}^{\alpha}_{\mu}{}_{\nu}}=\Gamma\indices{{}^{\alpha}_{\nu}{}_{\mu}}-\Gamma\indices{{}^{\alpha}_{\mu}{}_{\nu}}$
(6)

Now con-torsion and superpotential tensors have been introduced to calculate
torsion scalar. The con-torsion and superpotential tensors are respectively

$K\indices{{}^{\mu}{}^{\nu}_{\alpha}}=-\frac{1}{2}(T\indices{{}^{\mu}{}^{\nu}_{\alpha}}-T\indices{{}^{\nu}{}^{\mu}_{\alpha}}-T\indices{{}_{\alpha}^{\mu}{}^{\nu}})$
(7)
$S\indices{{}_{\alpha}^{\mu}{}^{\nu}}=\frac{1}{2}(K\indices{{}^{\mu}{}^{\nu}_{\alpha}}+\delta^{\mu}_{\alpha}T\indices{{}^{\rho}{}^{\nu}_{\rho}}-\delta^{\nu}_{\alpha}T\indices{{}^{\rho}{}^{\mu}_{\rho}})$
(8)

Now the torsion scalar can be defined as

$T=S\indices{{}_{\alpha}^{\mu}{}^{\nu}}T\indices{{}^{\alpha}_{\mu}{}_{\nu}}$
(9)

Now in $f(T)$ gravity, modified gravitational action is written in terms of
torsion scalar in the units of (G = c = 1) as

$S=\int d^{4}x\left[\frac{1}{16\pi}f(T)+\mathcal{L}_{m}\right]e$ (10)

Where $f(T)$ is a function of torsion scalar and $\mathcal{L}_{m}$ is
Lagrangian density for matter field. Also $e=\sqrt{-g}=det[e^{i}_{\mu}]$.
Now the variation in action (10) gives the field equations of $f(T)$ gravity.
article2 ; Jamil_2013 ; B_hmer_2012

$S\indices{{}_{\mu}^{\nu}{}^{\gamma}}\partial_{\gamma}Tf_{TT}+e^{-1}e\indices{{}^{i}_{\mu}}\partial_{\gamma}(ee\indices{{}_{i}^{\alpha}}S\indices{{}_{\alpha}^{\nu}{}^{\gamma}})f_{T}+T\indices{{}^{\alpha}_{\rho}{}_{\mu}}S\indices{{}_{\alpha}^{\nu}{}^{\rho}}f_{T}+\frac{1}{4}\delta^{\nu}_{\mu}f=4\pi\mathcal{T}\indices{{}^{\nu}_{\mu}}$
(11)

Where $\mathcal{T}_{\mu\nu}$ is the energy- momentum tensor of matter and
$f_{T}$ and $f_{TT}$ are first and second derivative of $f(T)$ with respect to
T respectively. Now in the case of anisotropic matter distribution, the
energy-momentum tensor is given by

$\mathcal{T}\indices{{}^{\nu}_{\mu}}=(\rho+p_{t})u_{\mu}u^{\nu}-p_{t}\delta^{\nu}_{\mu}+(p_{r}-p_{t})v_{\mu}v^{\nu}$
(12)

Where $\rho$, $p_{r}$ and $p_{t}$ are energy density, radial pressure and
tangential pressure, respectively. $u^{\mu}$ and $v^{\mu}$ are the four
velocity and radial four vectors, respectively. We define anisotropy of the
compact stellar structure as $S=\frac{p_{r}-p_{t}}{\sqrt{3}}$.

##  3 First Anisotropic Model of Neutron Star in Teleparallel Gravity

We consider the static and spherically symmetric model of the star. Thus
space-time metric describing interior of the star is given by

$ds^{2}=e^{c(r)}dt^{2}-e^{d(r)}dr^{2}-r^{2}(d\theta^{2}+sin^{2}\theta
d\phi^{2})$ (13)

For the metric (13), we calculate the tetrad matrix as

$e^{i}_{\mu}=diag[e^{\frac{c(r)}{2}},e^{\frac{d(r)}{2}},r,rsin(\theta)]$ (14)

From equation (14) we obtain,

$e=det[e^{i}_{\mu}]=e^{\frac{(c+d)}{2}}r^{2}sin(\theta)$ (15)

Now we determine torsion scalar as

$T(r)=\frac{2e^{-d(r)}}{r}\left(c^{\prime}+\frac{1}{r}\right)$ (16)

Now from equation (11), we write equation of $f(T)$ gravity for the metric
(13) as article2 ; Jamil_2013 ; B_hmer_2012

$4\pi\rho=\frac{f}{4}-\frac{f_{T}}{2}\left(T-\frac{e^{-d}}{r}(c^{\prime}+d^{\prime})-\frac{1}{r^{2}}\right)$
(17)

$4\pi p_{r}=\frac{f_{T}}{2}\left(T-\frac{1}{r^{2}}\right)-\frac{1}{4}$ (18)

$4\pi
p_{t}=\left[\frac{T}{2}+e^{-d}\left(\frac{c^{\prime\prime}}{2}+\left(\frac{c^{\prime}}{4}+\frac{1}{2r}\right)\left(c^{\prime}-d^{\prime}\right)\right)\right]\frac{f_{T}}{2}-\frac{f}{4}$
(19)

$\frac{cot(\theta)}{2r^{2}}T^{\prime}f_{TT}=0$ (20)

where single prime and double prime denotes first and second derivative with
respect to r. Now equation (20) leads to the following linear form of $f(T)$

$f(T)=\beta T+\beta_{1}$ (21)

where $\beta$ and $\beta_{1}$ are constants. Thus, the theory becomes
comparable to teleparallel gravity with the extra factors of $\beta$ and
$\beta_{1}$ and we will study the effects of these parameters by generating
different three models. Now in order to solve equations (17) to (19), we
choose following form of metric $e^{d(r)}$,

$e^{d(r)}=1-\frac{ar^{2}}{R^{2}}$ (22)

where a is constant and R is the radius of the neutron star. We consider
$a<1$, so that the metric does not become singular at any point for $r<R$. In
this model we consider $\beta=2$ in equation (21). By substituting equation
(22) in equation (17), we obtain

$4\pi\rho=\frac{-\frac{a}{R^{2}}\left(3-\frac{ar^{2}}{R^{2}}\right)}{\left(1-\frac{ar^{2}}{R^{2}}\right)^{2}}+\frac{\beta_{1}}{4}$
(23)

Now from the equation (18), we write

$c^{\prime}(r)=(4\pi
p_{r})re^{d(r)}+\frac{e^{d(r)}-1}{r}+\frac{\beta_{1}re^{d(r)}}{4}$ (24)

In order to solve equation (24), we consider following form of $4\pi p_{r}$

$4\pi
p_{r}=\frac{\frac{p_{0}}{R^{2}}\left(1-\frac{r^{2}}{R^{2}}\right)}{1-\frac{ar^{2}}{R^{2}}}$
(25)

where $p_{0}$ is another constant, that is the parameter of the model such
that $\frac{p_{0}}{R^{2}}$ denotes the central pressure of the neutron star.
The constraint on the $p_{0}$ is that $p_{0}>1$, so that radial pressure
remains positive for $0<r<R$. The particular form of $p_{r}$ is physically
reasonable because it is monotonically decreasing function of r and satisfy
the condition of vanishing radial pressure at the surface of the neutron star
$r=R$. by substituting values of equations (22) and (25) into (24), we obtain

$c^{\prime}(r)=r\left(\frac{p_{0}-a}{R^{2}}+\frac{\beta_{1}}{4}\right)-r^{3}\left(\frac{p_{0}}{R^{4}}+\frac{\beta_{1}a}{4R^{2}}\right)$
(26)

by integrating above equation, we obtain the metric as

$e^{c(r)}=Ke^{\frac{r^{2}}{2}\left(\frac{p_{0}-a}{R^{2}}+\frac{\beta_{1}}{4}\right)-\frac{r^{4}}{4}\left(\frac{p_{0}}{R^{4}}+\frac{\beta_{1}a}{4R^{2}}\right)}$
(27)

where $K$ is the integration constant. Now interior space-time of the neutron
star in this model is given by

$\displaystyle ds^{2}={}$
$\displaystyle\left(Ke^{\frac{r^{2}}{2}\left(\frac{p_{0}-a}{R^{2}}+\frac{\beta_{1}}{4}\right)-\frac{r^{4}}{4}\left(\frac{p_{0}}{R^{4}}+\frac{\beta_{1}a}{4R^{2}}\right)}\right)dt^{2}-\left(1-\frac{ar^{2}}{R^{2}}\right)dr^{2}$
(28) $\displaystyle-r^{2}(d\theta^{2}+sin^{2}\theta d\phi^{2})$

###  3.1 Constraining parameters for physically well-behaved model

####  3.1.1 Matching exterior space-time with interior space-time

Now the space-time metric should be continuous at the boundary of the star.
Thus, interior metric (28) of the star should be matched to exterior
Schwarzschild space-time metric at $r=R$ in this model.

$ds^{2}=\left(1-\frac{2M}{r}\right)dt^{2}-\left(1-\frac{2M}{r}\right)^{-1}dr^{2}-r^{2}(d\theta^{2}+sin^{2}\theta
d\phi^{2})$ (29)

Now interior metric (28) will be matched to exterior Schwarzschild metric at
the surface of the star if

$1-a=\left(1-\frac{2M}{R}\right)^{-1}$ (30)

and

$Ke^{\left(\frac{p_{0}-2a}{4}+\frac{\beta_{1}R^{2}}{8}\left(1-\frac{a}{2}\right)\right)}=1-\frac{2M}{R}$
(31)

From equation (30) and (31) we obtain the value of $K$ as

$K=\frac{1}{1-a}e^{-\left(\frac{p_{0}-2a}{4}+\frac{\beta_{1}R^{2}}{8}\left(1-\frac{a}{2}\right)\right)}$
(32)

For given value of $a$, $p_{0}$, $\beta_{1}$ and $R$, value of constant $K$ in
equation (28) can be calculated from equation (32).

####  3.1.2 Imposing physical conditions to the model

Since compact stars are physical objects that exist in nature, their physical
parameters, such as energy density and pressures, must obey the basic laws of
physics incorporated in physical conditions. Thus, any physically acceptable
model of the compact star should obey the following physical conditions for
$0<r<R$
(i) $\rho,p_{r},p_{t}\geq 0$
(ii) $\frac{d\rho}{dr},\frac{dp_{r}}{dr},\frac{dp_{t}}{dr}\leq 0$
(iii) $0\leq\frac{dp_{r}}{d\rho},\frac{dp_{t}}{d\rho}\leq 1$
(iv) $\frac{dp_{t}}{d\rho}-\frac{dp_{r}}{d\rho}<0$
where condition (i) is known as weak energy condition, condition (ii) implies
that the density and pressure of the star must be a decreasing function of
radius. The condition (iii) is known as causality condition, which implies
that radial and transverse speed of sound, $v_{r}=\sqrt{\frac{dp_{r}}{d\rho}}$
and $v_{t}=\sqrt{\frac{dp_{t}}{d\rho}}$ must not exceed the speed of light.
The condition (iv) is known as the stability condition, which implies that the
radial speed of sound must be greater than the transverse speed of sound for
the star to be potentially stable for $0<r<R$.
We have already chosen parameters such that $a<1$ and $p_{0}>0$ for
$\rho,p_{r}\geq 0$. We can get $p_{t}\geq 0$ if the following condition
satisfies

$-64p_{0}+64ap_{0}+R^{4}\beta_{1}^{2}+3a^{2}(4+R^{2}\beta_{1})^{2}\\\
-a^{3}(4+R^{2}\beta_{1})^{2}-aR^{2}\beta_{1}(16+3R^{2}\beta_{1})\geq 0$ (33)

Now in this model we already have $\frac{d\rho}{dr},\frac{dp_{r}}{dr}\leq 0$.
In order to get $\frac{dp_{t}}{dr}\leq 0$ we impose the following condition

$-R^{4}\beta_{1}^{2}-2a^{4}R^{2}\beta_{1}(4+R^{2}\beta_{1})-a^{3}(4+8p_{0}-7R^{2}\beta_{1})(4+R^{2}\beta_{1})+\\\
8p_{0}(16+R^{2}\beta_{1})+a^{2}(-48-32R^{2}\beta_{1}-9R^{4}\beta_{1}^{2}+24p_{0}(4+R^{2}\beta_{1}))+\\\
a(-24p_{0}(8+R^{2}\beta_{1})+R^{2}\beta_{1}(16+5R^{2}\beta_{1}))\geq 0$ (34)

Now conditions (iii) and (iv) satisfy if the model parameters satisfy
following conditions, which we deduce by calculating $\frac{dp_{r}}{d\rho}$,
$\frac{dp_{t}}{d\rho}$ and $\frac{dp_{t}}{d\rho}-\frac{dp_{r}}{d\rho}$ at
$r=0$ and $r=R$.

$0\leq\frac{(1-a)p_{0}}{5a^{2}}\leq 1\quad\mathrm{and}\quad
0\leq\frac{(1-a)^{2}p_{0}}{(5-a)a^{2}}\leq 1$ (35)
$0\leq\frac{-48a^{2}+a(64p_{0}-16R^{2}\beta_{1})+16p_{0}^{2}*+R^{4}\beta_{1}^{2}+8p_{0}(-16+R^{2}\beta_{1})}{320a^{2}}\leq
1$ (36)
$0\leq\frac{1}{64(-5+a)a^{2})}(R^{4}\beta_{1}^{2}+2a^{4}R^{2}\beta_{1}(4+R^{2}\beta_{1})+a^{3}(4+8p_{0}-7R^{2}\beta_{1})(4+R^{2}\beta_{1})\\\
-8p_{0}(16+R^{2}\beta_{1})+a^{2}(48+32R^{2}\beta_{1}+9R^{4}\beta_{1}^{2}-24p_{0}(4+R^{2}\beta_{1}))\\\
+a(24p_{0}(8+R^{2}\beta_{1})-R^{2}\beta_{1}(16+5R^{2}\beta_{1})))\leq 1$ (37)
$48a^{2}+16p_{0}^{2}-16aR^{2}\beta_{1}+R^{4}\beta_{1}^{2}+8p_{0}(-8+R^{2}\beta_{1})>0$
(38)
$R^{4}\beta_{1}^{2}+2a^{4}R^{2}\beta_{1}(4+R^{2}\beta_{1})+a^{3}(4+8p_{0}-7R^{2}\beta_{1})(4+R^{2}\beta_{1})\\\
-8p_{0}(8+R^{2}\beta_{1})+a^{2}(48+32R^{2}\beta_{1}+9R^{4}\beta_{1}^{2}-8p_{0}(4+3R^{2}\beta_{1}))\\\
+a(8p_{0}(8+3R^{2}\beta_{1})-R^{2}\beta_{1}(16+5R^{2}\beta_{1}))>0$ (39)

###  3.2 Physical analysis

In this section, we analyze the behavior of physical parameters such as energy
density and two pressures at the interior of the star for five observed
neutron stars. We calculate parameters for each star such that this first
model describing five neutron stars mentioned in Table 1. satisfy all bound
conditions from (i) to (iv). We obtain Mass(M) and radius(R) of the neutron
stars from referencesarticle4 ; Steiner_2010 ; 2011ApJ…730…25R  and compute
central density($\rho_{c}$), surface density($\rho_{s}$), central
pressure($p_{r}(r=0)=p_{t}(r=0)=p_{c}$) and surface tangential pressure
($p_{t}(r=R)=p_{s}$) of each star, shown in Table 2. We have plotted various
parameters of the star shown in figure 1. (a) to (h) of a neutron star SAX
J1808.4-3658(SS2) (5) shown in Table 1. described by the first model of the
neutron stars in teleparallel gravity. The figures indicate that all the
physical parameters of the star are well-behaved and follow all conditions in
equations (i) to (iv) at all interior points of the star.

Table 1: Values of the model parameters of the first model describing five observed neutron stars with $\beta=2$ Star | R | M | a | $\beta_{1}\times 10^{4}$ | $p_{0}$
---|---|---|---|---|---
| ($km$) | ($M_{\odot}$) |  | ($km^{-2}$) |
HER X-1 | 7.7 | 0.88 | -0.512 | -0.843 | 0.09
RX J1856-3754 | 12.7 | 1.4 | -0.486 | -0.248 | 0.08
SMC X-1 | 8.301 | 1.04 | -0.592 | -1.161 | 0.11
RX J1856.5-3754 | 12 | 1.4 | -0.529 | -0.382 | 0.095
SAX J1808.4-3658(SS2) | 7.951 | 0.9 | -0.506 | -0.759 | 0.089
|  |  |  |  |
Table 2: Values of the physical parameters of the five observed neutron stars described by the first model Star | R | M | $\rho_{c}\times 10^{3}$ | $\rho_{s}\times 10^{3}$ | $p_{c}\times 10^{4}$ | $p_{s}\times 10^{4}$
---|---|---|---|---|---|---
| (km) | ($M_{\odot}$) | ($km^{-2}$) | ($km^{-2}$) | ($km^{-2}$) | ($km^{-2}$)
HER X-1 | 7.7 | 0.88 | 2.06 | 1.05 | 1.21 | 0.55
RX J1856-3754 | 12.7 | 1.4 | 0.72 | 0.38 | 0.39 | 0.19
SMC X-1 | 8.301 | 1.04 | 2.05 | 0.97 | 1.27 | 0.63
RX J1856.5-3754 | 12 | 1.4 | 0.88 | 0.44 | 0.53 | 0.23
SAX J1808.4-3658(SS2) | 7.951 | 0.9 | 1.91 | 0.98 | 1.12 | 0.49
|  |  |  |  |  |

###  3.3 Equation of state

We have derived a physically acceptable model of the neutron star. To predict
the material composition of the stellar configuration, we need to generate an
equation of the state of the stellar configuration. By generating an EOS
governed by the physical laws of the system, one can parametrically relate the
energy density and the radial pressure, which is useful in predicting the
composition of the stellar configuration. By using equations (23) and (25), we
have plotted the variation of the radial pressure against the energy density,
as shown in figure 1.(h). From the plot, we find EOS of the form
$p_{r}=l\rho^{2}+m\rho+n$. where $l$, $m$ and $n$ are constants. By
approximating EOS in figure 1.(h), we obtain $l=-20$, $m=0.18$ and
$n=-0.00015$.

Figure 1: (a) density curve (b) pressure curves (c) anisotropy curve (d)
$\frac{dp}{dr}$ curves (e) $\frac{dp}{d\rho}$ curves (f)
$\frac{dp_{t}}{d\rho}-\frac{dp_{r}}{d\rho}$ curve (g) $\rho-p_{r}-2p_{t}$
curve (h) EOS curve for first model describing neutron star SAX
J1808.4-3658(SS2) mentioned in table 1

.

##  4 Second Anisotropic Model of Neutron Star in Teleparallel Gravity

Now we develop second model for the neutron star. We consider the same metric
potential as given by equation (22), thus we get the same energy density as in
equation (23).

In this new model we consider different form of radial pressure as following

$4\pi
p_{r}=\frac{\frac{p_{0}}{R^{2}}\left(1-\frac{r^{2}}{R^{2}}\right)}{\left(1-\frac{ar^{2}}{R^{2}}\right)^{2}}$
(40)

where $p_{0}$ is also another constant, that is the parameter of this model
such that $\frac{p_{0}}{R^{2}}$ denotes the central pressure of the neutron
star. The constraint on the $p_{0}$ is that $p_{0}>1$, so that radial pressure
remains positive for $0<r<R$. The particular form of $p_{r}$ is also
physically reasonable because it is a monotonically decreasing function of r
and satisfies the condition of vanishing radial pressure at the surface of the
neutron star $r=R$. by substituting values of equations (22) and (40) into
(24), we obtain

$c^{\prime}(r)=\frac{\frac{p_{0}r}{R^{2}}\left(1-\frac{r^{2}}{R^{2}}\right)}{1-\frac{ar^{2}}{R^{2}}}+r\left(\frac{\beta_{1}}{4}-\frac{a}{R^{2}}\right)-r^{3}\frac{\beta_{1}a}{4R^{2}}$
(41)

By integrating above equation, we obtain the metric as

$e^{c(r)}=K\left(1-\frac{ar^{2}}{R^{2}}\right)^{\frac{p_{0}(1-a)}{2a^{2}}}e^{\frac{r^{2}}{2}\left(\frac{p_{0}}{aR^{2}}+\frac{\beta_{1}}{4}-\frac{a}{R^{2}}\right)-\frac{r^{4}\beta_{1}a}{16R^{2}}}$
(42)

where $K$ is the integration constant. Now, interior space-time of the neutron
star in this model is given by

$\displaystyle ds^{2}={}$
$\displaystyle\left(K\left(1-\frac{ar^{2}}{R^{2}}\right)^{\frac{p_{0}(1-a)}{2a^{2}}}e^{\frac{r^{2}}{2}\left(\frac{p_{0}}{aR^{2}}+\frac{\beta_{1}}{4}-\frac{a}{R^{2}}\right)-\frac{r^{4}\beta_{1}a}{16R^{2}}}\right)dt^{2}-\left(1-\frac{ar^{2}}{R^{2}}\right)dr^{2}$
(43) $\displaystyle-r^{2}(d\theta^{2}+sin^{2}\theta d\phi^{2})$

###  4.1 Constraining parameters for physically well-behaved model

####  4.1.1 Matching exterior space-time with interior space-time

Now the space-time metric should be continuous at the boundary of the star.
Thus, the star’s interior metric (43) should be matched to the exterior
Schwarzschild space-time metric (29) at $r=R$ for this second model. Now
interior metric (43) will be matched to exterior Schwarzschild metric at the
surface of the star if

$1-a=\left(1-\frac{2M}{R}\right)^{-1}$ (44)

and

$K(1-a)^{\frac{p_{0}(1-a)}{2a^{2}}}e^{\left(\frac{p_{0}-a^{2}}{2a}+\frac{\beta_{1}R^{2}}{8}\left(1-\frac{a}{2}\right)\right)}=1-\frac{2M}{R}$
(45)

From equation (44) and (45) we obtain the value of $K$ for the second model as

$K=\frac{1}{(1-a)^{\frac{p_{0}(1-a)}{2a^{2}}+1}}e^{-\left(\frac{p_{0}-a^{2}}{2a}+\frac{\beta_{1}R^{2}}{8}\left(1-\frac{a}{2}\right)\right)}$
(46)

####  4.1.2 Imposing physical conditions to the second model

Any physically acceptable model of the compact star should obey physical
conditions (i) to (iv) as described in section (3.1.2). We satisfy condition
(i) by imposing the following inequality computed from $p_{t}\geq 0$ in the
second model.

$-64p_{0}+R^{4}\beta_{1}^{2}+3a^{2}(4+R^{2}\beta_{1})^{2}-a^{3}(4+R^{2}\beta_{1})^{2}-aR^{2}\beta_{1}(16+3R^{2}\beta_{1})\geq
0$ (47)

Now in the second model we already have
$\frac{d\rho}{dr},\frac{dp_{r}}{dr}\leq 0$. In order to get
$\frac{dp_{t}}{dr}\leq 0$ we impose following condition to the second model

$-R^{4}\beta_{1}^{2}-2a^{4}R^{2}\beta_{1}(4+R^{2}\beta_{1})+8p_{0}(16+R^{2}\beta_{1})+a^{3}(-16+24R^{2}\beta_{1}\\\
+7R^{4}\beta_{1}^{2})+a^{2}(-48-32R^{2}\beta_{1}-9R^{4}\beta_{1}^{2}+8p_{0}(4+R^{2}\beta_{1}))\\\
+a(-16p_{0}(-4+R^{2}\beta_{1})+R^{2}\beta_{1}*(16+5R^{2}\beta_{1}))\geq 0$
(48)

Now conditions (iii) and (iv) satisfy if the model parameters of the second
satisfy following conditions, which we deduce by calculating
$\frac{dp_{r}}{d\rho}$, $\frac{dp_{t}}{d\rho}$ and
$\frac{dp_{t}}{d\rho}-\frac{dp_{r}}{d\rho}$ at $r=0$ and $r=R$.

$0\leq\frac{p_{0}(1-2a)}{5a^{2}}\leq 1\quad\mathrm{and}\quad
0\leq\frac{p_{0}(1-a)}{a^{2}(a-5)}\leq 1$ (49)
$0\leq-\frac{(48a^{2}-128p_{0}+192ap_{0}+16p_{0}^{2}-16aR^{2}\beta_{1}+8p_{0}R^{2}\beta_{1}+R^{4}\beta_{1}^{2})}{320a^{2}}\leq
1$ (50)
$0\leq\frac{1}{64(-5+a)a^{2}}(R^{4}\beta_{1}^{2}+2a^{4}R^{2}\beta_{1}(4+R^{2}\beta_{1})-8p_{0}(16+R^{2}\beta_{1})\\\
+a^{3}(16-24R^{2}\beta_{1}-7R^{4}\beta_{1}^{2})+a^{2}(48+32R^{2}\beta_{1}+9R^{4}\beta_{1}^{2}\\\
-8p_{0}(4+R^{2}\beta_{1}))+a(16p_{0}(-4+R^{2}\beta_{1})-R^{2}\beta_{1}(16+5R^{2}\beta_{1})))\leq
1$ (51)
$48a^{2}-64p_{0}+64ap_{0}+16p_{0}^{2}-16aR^{2}\beta_{1}+8p_{0}R^{2}\beta_{1}+R^{4}\beta_{1}^{2}>0$
(52)
$R^{4}\beta_{1}^{2}+2a^{4}R^{2}\beta_{1}(4+R^{2}\beta_{1})-8p_{0}(8+R^{2}\beta_{1})+a^{3}(16-24R^{2}\beta_{1}-7R^{4}\beta_{1}^{2})\\\
+a^{2}(48+32R^{2}\beta_{1}+9R^{4}\beta_{1}^{2}-8p_{0}(4+R^{2}\beta_{1}))\\\
+a(16p_{0}(-8+R^{2}\beta_{1})-R^{2}\beta_{1}(16+5R^{2}\beta_{1}))>0$ (53)

###  4.2 Physical analysis

This section analyzes the behavior of physical parameters such as energy
density and two pressures at the interior of the star of the second model
describing four observed neutron stars. We calculate parameters for each star
such that this second model describing four neutron stars mentioned in Table
3. satisfy all bound conditions from (i) to (iv). We obtain Mass(M) and
radius(R) of the neutron stars from referencesLattimer_2014 ; Arzoumanian_2018
and calculate central density($\rho_{c}$), surface density($\rho_{s}$),
central pressure($p_{r}(r=0)=p_{t}(r=0)=p_{c}$) and surface tangential
pressure ($p_{t}(r=R)=p_{s}$) of each star, shown in Table 4. It can be
observed that this second model describe potentially stable neutron stars of
higher central density and higher central pressure than neutron stars
described by first model. The modification in radial pressure for second model
leads to description of stable neutron stars of comparable higher density and
higher pressure profile. We have plotted various parameters of the star shown
in figure 2. (a) to (h) of a neutron star RX J 1856-37 (1) shown in Table 3.
described by this second model of the neutron stars in teleparallel gravity.
The figures indicate that all the star’s physical parameters are well-behaved
and follow all conditions in equations (i) to (iv) at all interior points of
the star.

Table 3: Values of the model parameters of the second model describing four observed neutron stars with $\beta=2$ Star | R | M | a | $\beta_{1}\times 10^{4}$ | $p_{0}$
---|---|---|---|---|---
| (km) | ($M_{\odot}$) |  | ($km^{-2}$) |
RX J 1856-37 | 6 | 0.9031 | -0.805 | -11.944 | 0.21
Cen X-3 | 9.51 | 1.49 | -0.869 | -49.757 | 0.2
EXO 1785-248 | 8.99 | 1.3 | -0.751 | -37.119 | 0.2
PSR J1614-2230 | 13 | 1.908 | -0.77 | -18.343 | 0.21
|  |  |  |  |
Table 4: Values of the physical parameters of the four observed neutron stars described by the second model. Star | R | M | $\rho_{c}\times 10^{3}$ | $\rho_{s}\times 10^{3}$ | $p_{c}\times 10^{4}$ | $p_{s}\times 10^{4}$
---|---|---|---|---|---|---
| (km) | ($M_{\odot}$) | ($km^{-2}$) | ($km^{-2}$) | ($km^{-2}$) | ($km^{-2}$)
RX J 1856-37 | 6 | 0.9031 | 5.32 | 2.06 | 4.64 | 2.61
Cen X-3 | 9.51 | 1.49 | 2.19 | 0.75 | 1.76 | 0.73
EXO 1785-248 | 8.99 | 1.3 | 2.15 | 0.83 | 1.97 | 0.65
PSR J1614-2230 | 13 | 1.908 | 1.05 | 0.40 | 0.98 | 0.32
|  |  |  |  |  |

###  4.3 Equation of state

We have derived a second physically acceptable model of the neutron star. To
predict the material composition of the stellar configuration, we need to
generate an equation of the state of the stellar configuration for this model.
By using equations (23) and (40), we have plotted the variation of the radial
pressure against the energy density for the second model, as shown in figure 2
(h). From the plot, we find EOS of the form $p_{r}=l\rho^{2}+m\rho+n$. where
$l$, $m$ and $n$ are constants. By approximating EOS in figure 2.(h), we
obtain $l=24$, $m=0.077$ and $n=-7\times 10^{-5}$.

Figure 2: (a) density curve (b) pressure curves (c) anisotropy curve (d)
$\frac{dp}{dr}$ curves (e) $\frac{dp}{d\rho}$ curves (f)
$\frac{dp_{t}}{d\rho}-\frac{dp_{r}}{d\rho}$ curve (g) $\rho-p_{r}-2p_{t}$
curve (h) EOS curve for second model describing neutron star RX J 1856-37
mentioned in table 3

.

##  5 Third Anisotropic Model of Neutron Star in Teleparallel Gravity

Now, we develop the third model for the neutron star. We consider the
following form of metric potential:

$e^{d(r)}=\Big{(}1-\frac{ar^{2}}{R^{2}}\Big{)}^{2}$ (54)

where a is constant and R is the radius of the neutron star. We consider
$a<1$, so that the metric does not become singular at any point for $r<R$. By
substituting equation (54) in equation (17), we obtain

$4\pi\rho=\frac{\beta_{1}}{4}+\frac{\beta\Big{(}\frac{a^{2}r^{2}}{R^{4}}-\frac{2a}{R^{2}}\Big{)}}{\Big{(}1-\frac{ar^{2}}{R^{2}}\Big{)}^{2}}-\frac{2a\beta}{\Big{(}1-\frac{ar^{2}}{R^{2}}\Big{)}^{3}}$
(55)

Now from the equation (18), we write

$c^{\prime}(r)=(4\pi
p_{r})\frac{re^{d(r)}}{\beta}+\frac{e^{d(r)}-1}{r}+\frac{\beta_{1}re^{d(r)}}{2\beta}$
(56)

In order to solve the equation, we consider following form of $4\pi p_{r}$

$4\pi
p_{r}=\frac{\frac{p_{0}}{R^{2}}\left(1-\frac{r^{2}}{R^{2}}\right)}{1-\frac{ar^{2}}{R^{2}}}$
(57)

where $p_{0}$ is constant in this model also, that is the parameter of the
model such that $\frac{p_{0}}{R^{2}}$ denotes the central pressure of the
neutron star. The constraint on the $p_{0}$ is that $p_{0}>1$, so that radial
pressure remains positive for $0<r<R$. The particular form of $p_{r}$ is
physically reasonable because it is a monotonically decreasing function of r
and satisfies the condition of vanishing radial pressure at the surface of the
neutron star $r=R$. By substituting values of equations (54) and (57) into
(56), we obtain

$c^{\prime}(r)=\frac{2p_{0}r}{\beta
R^{2}}\Bigg{(}1-\frac{r^{2}}{R^{2}}\Bigg{)}\Bigg{(}1-\frac{ar^{2}}{R^{2}}\Bigg{)}+\frac{a^{2}r^{3}}{R^{4}}-\frac{2ar}{R^{2}}+\frac{r\beta_{1}}{2\beta}\Bigg{(}1-\frac{r^{2}}{R^{2}}\Bigg{)}$
(58)

By integrating the above equation, we obtain the metric as

$e^{c(r)}=Ke^{\frac{r^{6}}{6}\big{(}\frac{2ap_{0}}{\beta
R^{6}}+\frac{a^{2}\beta_{1}}{2\beta
R^{4}}\big{)}-\frac{r^{4}}{4}\big{(}-\frac{a^{2}}{R^{4}}+\frac{2(1+a)p_{0}}{\beta
R^{4}}+\frac{a\beta_{1}}{\beta
R^{2}}\big{)}+\frac{r^{2}}{2}\big{(}\frac{\beta_{1}}{2\beta}-\frac{2a}{R^{2}}+\frac{2p_{0}}{\beta
R^{2}}\big{)}}$ (59)

Now interior space-time of the neutron star in this model is given by

$\displaystyle ds^{2}={}$
$\displaystyle\Bigg{(}Ke^{\frac{r^{6}}{6}\big{(}\frac{2ap_{0}}{\beta
R^{6}}+\frac{a^{2}\beta_{1}}{2\beta
R^{4}}\big{)}-\frac{r^{4}}{4}\big{(}-\frac{a^{2}}{R^{4}}+\frac{2(1+a)p_{0}}{\beta
R^{4}}+\frac{a\beta_{1}}{\beta
R^{2}}\big{)}+\frac{r^{2}}{2}\big{(}\frac{\beta_{1}}{2\beta}-\frac{2a}{R^{2}}+\frac{2p_{0}}{\beta
R^{2}}\big{)}}\Bigg{)}dt^{2}$ (60)
$\displaystyle-\left(1-\frac{ar^{2}}{R^{2}}\right)^{2}dr^{2}-r^{2}(d\theta^{2}+sin^{2}\theta
d\phi^{2})$

###  5.1 Constraining parameters for physically well-behaved model

####  5.1.1 Matching exterior space-time with interior space-time

Now the space-time metric should be continuous at the boundary of the star.
Thus, the star’s interior metric (60) should be matched to the exterior
Schwarzschild space-time metric (29) at $r=R$ for this third model. Now
interior metric (60) will be matched to exterior Schwarzschild metric at the
surface of the star if

$(1-a)^{2}=\left(1-\frac{2M}{R}\right)^{-1}$ (61)

and

$Ke^{\frac{\beta_{1}R^{2}}{12\beta}(a^{2}+3a+3)+\frac{a^{2}}{4}-a\big{(}1+\frac{p}{6\beta}\big{)}}=1-\frac{2M}{R}$
(62)

From equation (61) and (62) we obtain the value of $K$ for the second model as

$K=\frac{1}{(1-a)^{2}}e^{-\frac{\beta_{1}R^{2}}{12\beta}(a^{2}+3a+3)+\frac{a^{2}}{4}-a\big{(}1+\frac{p}{6\beta}\big{)}}$
(63)

For given value of $a$, $p_{0}$, $\beta$, $\beta_{1}$ and $R$, value of
constant $K$ in equation (60) can be calculated from equation (63).

####  5.1.2 Imposing physical conditions to the third model

Any physically acceptable model of the compact star should obey physical
conditions (i) to (iv) as described in section (3.1.2). We satisfy condition
(i) by imposing the following inequality computed from $p_{t}\geq 0$ in the
third model.

$32\beta
p_{0}-\beta_{1}^{2}R^{4}+5a\beta_{1}^{2}R^{4}-5a^{4}(2\beta+\beta_{1}R^{2})^{2}+a^{5}(2\beta+\beta_{1}R^{2})^{2}\\\
+16a\beta(-4p_{0}+\beta_{1}R^{2})+2a^{3}(24\beta^{2}+22\beta\beta_{1}R^{2}+5\beta_{1}^{2}R^{4})\\\
-2a^{2}(24\beta^{2}-16\beta p_{0}+22\beta\beta_{1}^{2}R^{4})\leq 0$ (64)

Now in the third model we already have $\frac{d\rho}{dr},\frac{dp_{r}}{dr}\leq
0$. In order to get $\frac{dp_{t}}{dr}\leq 0$ we impose following condition to
the third model

$-64\beta
p_{0}+\beta_{1}R^{2}(-8p_{0}+\beta_{1}R^{2})-8a^{5}(2\beta+\beta_{1}R^{2})(\beta-
p_{0}+2\beta_{1}R^{2})+\\\
8a^{3}(2p_{0}-\beta_{1}R^{2})(11\beta+5\beta_{1}R^{2})+8a(24\beta
p_{0}-2\beta\beta_{1}R^{2}+5\beta_{1}p_{0}R^{2}-\beta_{1}^{2}R^{4})\\\
+a^{6}(4\beta^{2}+8\beta\beta_{1}R^{2}+3\beta_{1}^{2}R^{4})+a^{2}(48\beta^{2}-240\beta
p_{0}+56\beta\beta_{1}R^{2}-80\beta_{1}p_{0}R^{2}\\\
+25\beta_{1}^{2}R^{4})+a^{4}(12\beta^{2}-80\beta(p_{0}-\beta_{1}R^{2})+5\beta_{1}R^{2}(-8p_{0}+7\beta_{1}R^{2}))\geq
0$ (65)

Now conditions (iii) and (iv) satisfy if the model parameters of the third
satisfy following conditions, which we deduce by calculating
$\frac{dp_{r}}{d\rho}$, $\frac{dp_{t}}{d\rho}$ and
$\frac{dp_{t}}{d\rho}-\frac{dp_{r}}{d\rho}$ at $r=0$ and $r=R$.

$0\leq-\frac{2p_{0}(a-1)}{15\beta a^{2}}\leq 1\quad\mathrm{and}\quad
0\leq-\frac{2p_{0}(a-1)^{3}}{\beta a^{2}(a^{2}-4a+15)}\leq 1$ (66)
$0\leq-\frac{48a^{2}\beta^{2}-16a\beta\beta_{1}R^{2}+16p_{0}^{2}+\beta_{1}^{2}R^{4}+8p_{0}(-8\beta+\beta_{1}R^{2})}{240a^{2}\beta^{2}}\leq
1$ (67) $0\leq\frac{1}{16a^{2}(15-4a+a^{2})\beta^{2}}(64\beta
p_{0}+\beta_{1}R^{2}(8p_{0}-\beta_{1}R^{2})+8a^{5}(2\beta+\beta_{1}R^{2})\\\
\times(\beta-
p_{0}+2\beta_{1}R^{2})-8a^{3}(2p_{0}-\beta_{1}R^{2})(11\beta+5\beta_{1}R^{2})-8a(24\beta
p_{0}\\\
-2\beta\beta_{1}R^{2}+5\beta_{1}p_{0}R^{2}-\beta_{1}^{2}R^{4})-a^{6}(4\beta^{2}+8\beta\beta_{1}R^{2}+3\beta_{1}^{2}R^{4})\\\
+a^{2}(-48\beta^{2}+8\beta(30p_{0}-7\beta_{1}R^{2})+5\beta_{1}R^{2}(16p_{0}-5\beta_{1}R^{2}))\\\
+a^{4}(-12\beta^{2}+5\beta_{1}R^{2}(8p_{0}-7\beta_{1}R^{2})+80\beta(p_{0}-\beta_{1}R^{2})))\leq
1$ (68) $48a^{2}\beta^{2}-32\beta
p-16a\beta(2p+\beta_{1}R^{2})+(4p+\beta_{1}R^{2})^{2}<0$ (69)
$\frac{1}{16a^{2}(a^{2}-4a+15)\beta^{2}}(32\beta
p+\beta_{1}R^{2}(8p-\beta_{1}R^{2})+8a^{5}(2\beta+\beta_{1}R^{2})\\\
\times(\beta-p+2\beta_{1}R^{2})-8a^{3}(18\beta
p-11\beta\beta_{1}R^{2}+10\beta_{1}pR^{2}-5\beta_{1}^{2}R^{4})\\\ -8a(12\beta
p-2\beta\beta_{1}R^{2}+5\beta_{1}pR^{2}-\beta_{1}^{2}R^{4})-a^{6}(4\beta^{2}+8\beta\beta_{1}R^{2}+3\beta_{1}^{2}R^{4})\\\
+a^{2}(-48\beta^{2}+8\beta(18p-7\beta_{1}R^{2})+5\beta_{1}R^{2}(16p-5\beta_{1}R^{2}))\\\
+a^{4}(-12\beta^{2}+5\beta_{1}R^{2}(8p-7\beta_{1}R^{2})+80\beta(p-\beta_{1}R^{2})))>0$
(70)

###  5.2 Physical analysis

This section analyzes the behavior of physical parameters such as energy
density and two pressures at the interior of the star of the third model
describing three observed neutron stars. We calculate parameters for each star
such that this third model describing three neutron stars mentioned in Table
5. satisfy all bound conditions from (i) to (iv). We obtain Mass(M) and
radius(R) of the neutron stars from referencesSteiner_2015 ; _zel_2016  and
calculate central density($\rho_{c}$), surface density($\rho_{s}$), central
pressure($p_{r}(r=0)=p_{t}(r=0)=p_{c}$) and surface tangential pressure
($p_{t}(r=R)=p_{s}$) of each star, shown in Table 6. From Table 6. we can
observe that this third model describes stable neutron stars of low mass and
higher outer radius comparable to neutron stars described by first two models.
Neutron stars described by the third model has low density and low pressure
profile comparable to neutron stars described by first two models. The
particular modification in metric potential and chosen model parameters for
third model leads to description of stable neutron stars of lower density and
lower pressure profile comparable to both the first two models. We have
plotted various parameters of the star shown in figure 3. (a) to (h) of a
neutron star PSR J0737-3039A (2) shown in Table 5. described by this third
model of the neutron stars in teleparallel gravity. The figures indicate that
all the star’s physical parameters are well-behaved and follow all conditions
in equations (i) to (iv) at all interior points of the star.

Table 5: Values of the model parameters of the third model describing three neutron stars with $\beta_{1}=0$. Star | R | M | a | $\beta$ | $p_{0}$
---|---|---|---|---|---
| (km) | ($M_{\odot}$) |  |  |
HER X-1 | 10.7655 | 0.85 | -0.2 | 3 | 0.101
PSR J0737-3039A | 14.68 | 1.4 | -0.181 | 3.2 | 0.09
PSR J0437-4715 | 13.8 | 1.4 | -0.194 | 3 | 0.095
|  |  |  |  |
Table 6: Values of the model parameters of the first model describing five observed neutron stars with $\beta=2$ Star | R | M | $\rho_{c}\times 10^{3}$ | $\rho_{s}\times 10^{3}$ | $p_{c}\times 10^{4}$ | $p_{s}\times 10^{4}$
---|---|---|---|---|---|---
| (km) | ($M_{\odot}$) | ($km^{-2}$) | ($km^{-2}$) | ($km^{-2}$) | ($km^{-2}$)
HER X-1 | 10.7655 | 0.85 | 1.24 | 0.79 | 0.70 | 0.30
PSR J0737-3039A | 14.68 | 1.4 | 0.64 | 0.42 | 0.33 | 0.14
PSR J0437-4715 | 13.8 | 1.4 | 0.73 | 0.47 | 0.40 | 0.17
|  |  |  |  |  |

###  5.3 Equation of state

We have derived a third physically acceptable model of the neutron star. To
predict the material composition of the stellar configuration, we need to
generate an equation of the state of the stellar configuration for this model.
By using equations (55) and (57), we have plotted the variation of the radial
pressure against the energy density for the third model, as shown in figure 3
(h). From the plot, we find EOS of the form $p_{r}=l\rho^{2}+m\rho+n$. where
$l$, $m$ and $n$ are constants. By approximating EOS in figure 3.(h), we
obtain $l=-99$, $m=0.26$ and $n=-9.3\times 10^{-5}$.

Figure 3: (a) density curve (b) pressure curves (c) anisotropy curve (d)
$\frac{dp}{dr}$ curves (e) $\frac{dp}{d\rho}$ curves (f)
$\frac{dp_{t}}{d\rho}-\frac{dp_{r}}{d\rho}$ curve (g) $\rho-p_{r}-2p_{t}$
curve (h) EOS curve for third model describing neutron star J0737-3039A
mentioned in table 5

.

##  6 Discussion

A new class of solutions for three analytical models describing anisotropic
stellar structures for various observed neutron stars has been developed using
the theory of teleparallel gravity. We have briefly reviewed the idea of
tetrad field and formulation of field equations of teleparallel gravity. Then
we solved equations of teleparallel gravity for spherically symmetric
anisotropic matter distribution for some physically reliable metric potentials
and radial pressures. Thus, we developed three analytical models describing
anisotropic stellar structures of neutron stars. We calculated various model
parameters by imposing physical bound conditions on them so that models become
physically acceptable. We calculated model parameters and star’s physical
parameters for various observed neutron stars described by three models
developed in this paper. We observed that particular modifications in metric
potential and radial pressure leads to description of stable neutron stars
ranging from low density and pressure to high density and pressure as observed
in physical analysis of each model. We have shown that models with very good
approximations, admits quadratic equations of state, which is very useful in
determining compositions of stellar structures of particular neutron stars as
well as strange stars and also very useful in modelling such realistic neutron
stars. It is also worth to note that, linear form of $f(T)$ is responsible for
quadratic behaviour of EOS. The non linear forms of $f(T)$ leads to high
deviation of EOS from quadratic behaviour. Thus, the authors choose to work
with linear EOS. We have also plotted graphs of various parameters of observed
neutron stars described by each model developed in this paper. Graphs shows
that each model is physically reliable and realistic in all aspects. The
research work presented in this paper provides understanding of how
modification in model parameters can lead to description of various neutron
stars described by teleparallel gravity. This work can be also utilized to
investigate the reason for anisotropy in neutron stars due to pion
condensation, hyperon interaction and electromagnetic forces in teleparallel
gravity theory.

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