11institutetext: Innopolis University, Tatarstan, Russian Federation,
22institutetext: Sorbonne Center for Artificial Intelligence, Sorbonne
University, Paris, France 33institutetext: IT University of Copenhagen,
Denmark

# Bridging the Domain Gap for Stance Detection for the Zulu language

Gcinizwe Dlamini 11 Imad Eddine Ibrahim BEKKOUCH 22 Adil Khan 11 Leon
Derczynski 33

###### Abstract

Misinformation has become a major concern in recent last years given its
spread across our information sources. In the past years, many NLP tasks have
been introduced in this area, with some systems reaching good results on
English language datasets. Existing AI based approaches for fighting
misinformation in literature suggest automatic stance detection as an integral
first step to success. Our paper aims at utilizing this progress made for
English to transfers that knowledge into other languages, which is a non-
trivial task due to the domain gap between English and the target languages.
We propose a black-box non-intrusive method that utilizes techniques from
Domain Adaptation to reduce the domain gap, without requiring any human
expertise in the target language, by leveraging low-quality data in both a
supervised and unsupervised manner. This allows us to rapidly achieve similar
results for stance detection for the Zulu language, the target language in
this work, as are found for English. We also provide a stance detection
dataset in the Zulu language. Our experimental results show that by leveraging
English datasets and machine translation we can increase performances on both
English data along with other languages.

###### keywords:

Stance Detection, Domain Adaptation, Less resourced languages, Misinformation,
Disinformation

##  1 Introduction

Social media platforms have become a major source of information and news. At
the same time, the amount of misinformation on them has also become a concern.
Automatic misinformation detection is a challenging task. One was of
categorising rumours is into those that can be grounded and verified against a
knowledge base, and those that cannot (e.g. due to a lack of knowledge base
coverage). This task has required the work of professional fact-checkers. This
work has recently been complemented by fact-checking systems. However, fact
checking only works when there is evidence to refute or confirm a claim.
Automated fact checking relies on databases for this evidence. These databases
often lack the information needed, due to e.g. lag or a lack of notability. In
this case, another information source is needed. Prior work has hypothesised
[30] and shown that the stance, that is the attitude, that people take towards
claims can act as an effective proxy for veracity predication [10, 22].

A primitive solution to analyzing texts written in uncommon languages, i.e.
those which are not supported by the state-of-the-art NLP models, is to
translate the text to a language an appropriate NLP model has been trained on.
However, this solution is not always effective and does not give good
performance. Some of the main difficulties faced by state-of-the-art NLP
models (especially in the task of stance detection using translated LREL) come
from: discrepancies induced by the LREL translation process; LREL data
scarcity [1]; and the noise that exists in the social media data itself [8].

An alternative approach to solving this problem can be ensemble models for
stance detection [19] trained on different languages and combining their
respective results, but we still face the same problem which is the fact that
each model is built from a well-resourced, well-represented language, so if we
introduce another language to it even by translation, its individual
performance is not likely to be good, hence the overall ensemble performance
will probably be the same as the previously suggested solution [27].

Thus, in light of the problem of wide spread of false rumours, mis-
information, and challenges faced by state-of the-art NLP models when it comes
to LREL, we propose an approach to solving these challenges in this paper. We
propose using Domain Adaptation (DA)[17] in the task of Stance Detection on
LREL Twitter text data.

In our approach we use machine translation to translate the model’s training
data, taking into account the amount of noise this introduces. Our intended
stance detection model will be able to learn from variety of low quality data,
whereby the low quality aspect comes from the automated translation. This
comes up as a strength in our approach for two reasons: 1. For most social
media text classification tasks, precise and canonical forms of expression are
not the most important feature for making the decisions. 2. The target data
(The LREL) has an element of being noisy since it’s from social media and is a
direct output of a translator. Our method uses data from multiple languages to
build a stance detection model for a less-privileged language.

In addition to the proposed Domain Adaptation technique for stance detection,
we contribute a new dataset for Stance Detection in a language previously
without resources.

The structure of this paper is as follows: Section 2 outlines the related
works. In Section 3, the model architecture, methodology and datasets used in
detail, while Section 4 presents the obtained results followed by discussion.
In conclusion Section 5 we discuss possible future directions and insights
from the paper results.

##  2 Related Work

Researchers have approached this problem of fake news detection, mis-
information and stance detection from many point of views. Our approach is
about combining concepts from different areas of research, mainly: Domain
Adaptation [3], Domain Generalization[21], Domain Randomization [32] and some
old and new techniques where the concept of using data from one language to
improve performances on another was used or from one task to improve
performances on another [23].

###  2.1 Domain Generalization, Adaptation, Randomization

These three fields of research are very tightly related, the basic idea behind
them is to use multiple simple to collect data sources to improve performance
on a harder to collect (or label) dataset. Domain Adaptation(DA)[3] is the
most widely researched topic and specifically Unsupervised Domain Adaptation
(UDA) where we use both data and labels from one source domain and use only
the data of a target domain without its labels and try to build a model that
gives good performances on both domains. The most common method for performing
UDA is by utilizing Generative Adversarial Networks[12] in multiple ways [16]
, but there are other techniques that use simpler and faster techniques based
only on adversarial losses.

Domain Generalization on the other hand uses multiple source domains and aims
at generalizing to unseen target domains. The current state if the art in DG
is leaning towards improving the ability to learn and merge knowledge both
from supervised and unsupervised learning. the supervised part is by
classifying samples into their corresponding labels whereas the unsupervised
part leverages only data in many ways, one way is by reorganising the images
into a jigsaw puzzle and training a classifier to solve this[6].

Domain Randomization [31] is the extreme case where we only have access to one
domain on training and we want to improve the results on unseen domains. Most
of these techniques agree on the fact that having a lot of messy and non-
perfect data that comes from multiple sources improves accuracy and allows the
model to perform better on real data[32].

###  2.2 Stance Detection

One of the core approaches to automatic fake news assessment is stance
detection which is the extraction of a subject’s reaction to a claim made by a
primary actor [2]. The main approach taken by NLP researchers for the past
years has been shifted towards less hand engineered techniques and into using
Deep Learning by taking two text sequences, encoding them in some form (mainly
by adding a mask that determines which word belongs to which sentence), and
then estimating the type of relationship that joins them.

One way of formulating stance detection was to divide it into two sub-tasks
[25], Task A: fully supervised, classify labeled texts into ”IN-FAVOR”,
”AGAINST”, ”NONE”. The text belongs to 5 topics: ”Atheism”, ”Climate Change is
a Real Concern”, ”Feminist Movement”, ”Hillary Clinton”, and ”Legalization of
Abortion”. Task B on the other hand aims at solving weakly supervised stance
detection by training on data from Task A and unlabeled data from another
topic and measuring the performance on it. The goal of this paper is actually
a next step on top of these two Tasks which is sharing the knowledge from
labeled data into another dataset that comes from a totally different language
not just a different topic. Zhou et al.[33] introduced an approach based on
convolution neural networks(CNN) and Attention to detect stances in tweeter.
Their proposed approach addresses the challenges faced by CNN’s which is
generating and capturing high quality word embedding having global textual
information. Less Resourced and Endangered Languages are not addressed in
their proposed approach for stance detection.

Other major efforts in stance detection include the RumourEval challenge
series [13]. Work on LRELs includes datasets in Czech [14] and Turkish [20].

###  2.3 Explicit & Implicit Transfer Learning for NLP

Adapting Computational Language Processing (CLP) [11] techniques were used in
the early 2000s as a first approach to transfer knowledge from one language to
the other which we can see in the ’MAJO system’ which uses the similarities
between Japanese and Uighur to improve substantially the performance[23]. This
idea although non-generic and a completely human-centric approach it is one of
the first approaches of improving results on a LREL by using a more popular
and widely used language.

Another more recent approach is Bidirectional Encoder Representations from
Transformers (BERT)[9]. BERT is built on the idea of building a general
purpose model for NLP that is trained on large amounts of text data and can be
easily fine tuned to downstream NLP tasks like stance detection for example.
BERT also has a multilingual mode that has learned on data from 104 languages.

Another very popular approach to Inductive transfer learning in NLP is
Universal Language Model Fine-tuning (ULMFiT)[15], which aims at reducing the
amount of labeled data needed for building any NLP tasks and specially for
classification. It works on leveraging the the existence of big amounts of
unsupervised text that can be used to train a general purpose model which can
later be fine-tuned in two steps, the first one is unsupervised where the
model learns from raw text that is related to the problem and then the second
step is supervised where we train the final layers of the model using a small
amount of labeled data, for example we train a general model on Wikipedia
texts and then fine-tune it for emails by using a large amount of unlabeled
emails in the first step and just a hand-full of labeled emails in the second.

From all the existing state-of-the-art approaches in stance detection,
breakthroughs of transfer learning and domain generalization great performance
in lot of different domains, we have found that there still exists a need to
cover LRELs in the domain of NLP, specifically in stance detection. With our
approach we are hoping to contribute to the existing methods and help the
research community develop methods for fighting mis-information, fake news
detection and understanding social media content better as compared to
existing state-of-the-art methods.

##  3 Architecture, Methodology And Dataset

Our method consists of a two step process. Firstly, the creation of the
training Dataset where training dataset is retrieved and joined from multiple
languages in order to train a more resilient stance detection model. Secondly,
building the training pipeline whereby the chronology for implementing stance
detection for less-privileged languages is outlined.

We use Zulu as a case study for a less-privileged language. The Zulu language
[7] is a good example of a Less Resourced and Endangered Language (LREL) [4]
and uncommon language found in social media. Zulu is spoken by an estimated 10
million speakers, mainly located in the province of KwaZulu-Natal of South
Africa and Swaziland. Zulu is recognised as a one of the 11 official South
African languages and it is understood by over 50% of its population. As
interesting as it is, there are not enough Zulu language resources on the
internet to help build major NLP models; and even though Google has done an
amazing job with providing models that can be fine-tuned for specific tasks on
more than 104 languages using BERT-multilingual [29], Zulu and many other
languages were not part of their research.

###  3.1 Step 1: Build the training Dataset

Our aim is to have a dataset large enough to expose an underlying structure of
a given topic in LREL for any NLP model to capture and generalize. For this
reason we gather our dataset from multiple sources which are chosen randomly
from the languages supported by google translate. The total number of sources
ranges from 2 to 106.

More formally, we will have a training dataset coming from $N$ languages,
where $N_{i}$ is the number of labeled samples in the $i$th source dataset,
such that

$X^{s}_{i}=\\{(x_{i,j}^{s},y_{i,j}^{s})\\}_{j=1}^{N_{i}}$

where $x_{is}^{j}$ denotes the $jth$ text sample from the $ith$ source dataset
and $y_{is}^{j}$ is its label.

In the case where there is just a source dataset $X^{s}_{i}$ that gets
translated into other languages we denote the $kth$ translated version as
$X^{s}_{ik}$.

We denote also $M$ the number of labeled samples in the target dataset and

$X^{t}=\\{(x_{j}^{t},y_{j}^{t})\\}_{j=1}^{M}$

where $x_{j}^{t}$ denotes the $jth$ text sample from the target dataset and
$y_{j}^{t}$ is its label.

In adoption of the ensemble learning principle of using different training
data sets [18], we select data sources with different marginal distributions
(i.e. they have some dissimilarities between them in terms of syntax rules,
language family tree, e.t.c) and from the dataset of stance detection. These
differences in marginal distributions are at the core of the strength of our
stance detection model since the input of the model will be a translator’s
result we will force the model to actually learn from the translator results.
This idea can also be used in the case where your input data isn’t perfect;
for example if your model is being deployed on a task where the target
audience isn’t a language expert they will most likely make many mistakes in
their writing, your model should be familiar with these mistakes and more
resilient to them. Such mistakes can be easily generated from translators
where we can translate from language A to language B and then vice versa,
giving us more variety in the data which can be considered as a data
augmentation technique.

Modeling this domain gap in the training data can be done in one of two ways
which are shown on Figure1. The domain gap in Text datasets can be anything
from the length of sentences, to sentiment changes and even difference in the
grammatical correctness of the sentences. In our case we are mostly focused on
modeling the errors that a translator might make, and putting them in our
training data so that we can correctly classify them on inference time.

(a) Domain Generalization: multiple source languages all converted into
English to use pre-trained models.

(b) Domain Randomization: one source language to which we add multiple
translated versions as data augmentation.

Figure 1: Dataset construction process for Domain Generalization and Domain
Randomization.

Domain Generalization:

For DG we have available for us multiple datasets similar in structure and
purpose but come from different languages, we translate them all to English
and build our classifier. We denote $n_{G}$ as the number of datasets used to
build the model.

Although there are no constraints on which family the language comes from, but
empirical results show that using datasets from different families help to
generalize better to unseen datasets and using datasets from the same family
as the target language helps for generalizing to it better.

Domain Randomization:

In this case we only have one dataset from one language , so we apply some
randomization to increase the size of the dataset and make it more inclusive
to mistakes; We do that by translating the dataset into multiple intermediate
languages and then translate them into English. The results empirically appear
to contain many mistakes but the huge increase in size makes up for it. We
denote $n_{R}$ as the number of intermediate languages used to build the
model.

Same remarks as for DG, using intermediate languages that are quite different
allows the model to learn from an even richer dataset and overall generalizes
better to unseen domains, whereas for a specific target domain, using
languages that are from it’s same family allows the model to perform better on
this target dataset.

###  3.2 Step 2: Build the training pipeline

Now that the data set is ready, the next step will be to build a pipeline that
inputs the dataset, cleans it, tokenizes it, convert it into word embeddings
and train the model on it. This step is the reason why we convert all of our
datasets into English rather than the target language it self or some other
language, given the fact that the tokenization process requires hand crafted,
man-made rules and knowledge about the constructs of the language which is
something that can’t be done automatically for now on all languages and
specially the LREL ones.

For model choice, our method is also model-generic meaning it can be used on
any model given its non-intrusive property. But for the purpose of this
research we will use the same architecture as ULMFiT, where we use a three-
layer LSTM model that was pre-trained on wikitext-103 dataset [24] (which is a
collection of over 100 million tokens extracted from the set of verified Good
and Featured articles on Wikipedia). The later model is then fine-tuned on the
training dataset in an unsupervised fashion and finally we add a 2 layer fully
connected neural network as the classifier which is carefully fine-tuned on
the training dataset using gradual unfreezing of layers[28].

The Final part, is to introduce a new loss which operates on the final layers
of the model. They act as enforcement for the model to dismiss the noise in
the data and only extract features useful for classification. The new loss is
inspired by Linear Discriminant Analysis (LDA) to capture the separability
(based on the assumption that samples which are closer to each other in the
latent space are classified similarly), Ficher defined an optimization
function to maximize the between-class variability and minimize the within-
class variability regardless of the source of the sample. the separability
loss is defined as follows:

$\mathcal{L}_{sep}(W^{E})=\bigg{(}\frac{\sum_{i\in Y}\sum_{z_{ij}\in
Z_{i}}d(z_{ij},\mu_{i})}{\sum_{i\in
Y}d(\mu_{i},\mu)}\bigg{)}\times\lambda_{BF}$

$\lambda_{BF}=\frac{\min_{i}|Y^{t}_{i}|}{\max_{i}|Y^{t}_{i}|}$

where $Z_{i}$ is the set of latent variables that belongs to class $i$ and it
can be expressed as:

$Z_{i}=\cup_{s=1}^{N}Z_{i}^{s}\cup_{t=1}^{M}Z_{i}^{t}$

which is the union of the sets of latent representation of all language
domains that have the same label $i$. $\mu_{i}$ is the mean of the latent
representations that has label $i$, so it can be expressed as
$\mu_{i}=mean(Z_{i})$, while $\mu$ is the mean of all the latent
representations $\mu=mean(Z)$, the calculations of such loss can be hard so
for simplicity we use this loss as batch based and in order to mitigate the
drain of information we used a balancing factor $\lambda_{BF}$ which proved to
be useful in other applications of the separability loss [3]. $d(.,.)$ is the
cosine dissimilarity which is used as a measure of distance between the
samples. This loss allows us to increase the separability of the latent space
according to the labels, regardless of its domain (language), which in return
helps the classifier to generalize and provide more resilient and higher
performances.

###  3.3 Dataset

####  3.3.1 Source Domain Dataset

The Source domain dataset which is the training set of our model, used to
predict tweets stances on unseen domain is a dataset for English Tweets. We
use Semantic Evaluation competition train dataset. The data has 4163 samples
with three features being ID, Target, tweet and Stance as the response
variable. Each tweet in the dataset has a target and can classified into
stance class. The are five targets and three stances classes. The summary of
the SemEval-2016 data set is presented in Figure 2

####  3.3.2 Target Domain Dataset

The target domain dataset language is Zulu. The Zulu language is a tonal
language which also features click consonants. In comparison to English, in
these clique consonants’ noises are conveyed in literature with the word
‘tut!’. The Standard Zulu that is taught in schools is also referred to as
‘deep Zulu’, as it uses many older Zulu words and phrases, and is altogether a
purer form of the language than many of the dialects that are used in common
speech. The Zulu language relies heavily on tone to convey meaning; but when
the language itself is written down, often no tones are conveyed in the
writing. This means that the speaker must have a good understanding of spoken
Zulu before being able to read the Zulu language fluently, which is unusual
among languages. All these interesting aspects and facts about our chosen
target language makes it a perfect choice for Domain Generalization tasks
since there exist a wide domain gap between itself and the English language
(which is our source domain). For our approach, we randomly sampled 1343
tweets from Semantic Evaluation competition test dataset (SemEval-2016) [25].
We translated the tweets to Zulu language with the help of the Google
Translate API, together with a native Zulu language speaker, to try to
minimize the grammatical errors from the google translator. Examples of tweets
with Zulu translation:

Atheism : AGAINST

English : The humble trust in God: ’Whoever leans on, trusts in, and is
confident in the Lord happy, blessed, and fortunate is he’

Zulu : Abathobekile bathembela kuNkulunkulu: ’Noma ngubani oncika, athembela
kuye, futhi othembela eNkosini uyajabula, ubusisiwe, futhi unenhlanhla’

Legalization of Abortion : FAVOR

English : Would you rather have women taking dangerous concoctions to induce
abortions or know they are getting a safe & legal one?

Zulu : Ngabe ungathanda ukuthi abesifazane bathathe imiqondo eyingozi
ukukhipha izisu noma wazi ukuthi bathola ephephile futhi esemthethweni ?

Feminist Movement : NONE

English : Some men do not deserve to be called gentlemen

Zulu : Amanye amadoda awakufanele ukubizwa ngokuthi ngamanenekazi

The Stances and Targets distribution in our final target domain data is in
Figure 2

(a) Source Domain: overall balanced w/ less data in Favor of Legal abortion.

(b) Target Domain: not as well balanced as the source dataset, with a big lack
of data for ’against’ in the ’Climate Change’ topic and for both ’not-related’
and ’in-favor’ of the ’Atheism’ topic.

Figure 2: Dataset comparison between the source and the target dataset.

Tested On | English | Zulu
---|---|---
Trained On | F1-score | Accuracy | FAVOR-F1-score | AGAINST-F1-score | F1-score | Accuracy | FAVOR-F1-score | AGAINST-F1-score
Zulu-Only (DLB) | - | - | - | - | 0.3942 | 0.4861 | 0.1929 | 0.4259
English-Only (DUB/DLB) | 0.5792 | / | 0.4476 | 0.6908 | 0.4906 | 0.5258 | 0.3749 | 0.5742
Randomized-English-1 | 0.5686 | / | 0.4591 | 0.6951 | 0.5061 | 0.5386 | 0.3932 | 0.5987
Randomized-English-2 | 0.5993 | / | 0.4626 | 0.6999 | 0.5112 | 0.5443 | 0.3923 | 0.6147
Randomized-English-3 | 0.6070 | / | 0.4658 | 0.7083 | 0.5087 | 0.5512 | 0.3852 | 0.6090
Randomized-English-4 | 0.6125 | / | 0.4656 | 0.7243 | 0.5186 | 0.5635 | 0.4038 | 0.6204
Randomized-English-5 | 0.6296 | / | 0.4710 | 0.7201 | 0.5293 | 0.548 | 0.4014 | 0.6286
Randomized-English-Zulu | 0.5690 | / | 0.4586 | 0.7083 | 0.5493 | 0.5596 | 0.4228 | 0.6423

Table 1: Evaluation of Domain Randomization, We denote a randomized dataset
with index i as the degree of randomization. We use the / symbol to denote
that this result is not reported given that it is not accessible by the
evaluation script of the original dataset. We also use - symbol to denote that
this experiment isn’t feasible. DUB and BLB are used to denote the baselines
described in the baselines description 3.4

###  3.4 Baselines

For the purpose of evaluating our method, we will compare it’s results against
multiple baselines that represent an estimation of a lower-bound and an upper-
bound of the performance.

####  3.4.1 Lower Bound Baselines

Our method should perform better than a model that was directly trained on
English dataset and tested on a Zulu dataset (translated to English) and it
should outperform a model trained only on the Zulu dataset and tested on it,
given that we don’t have enough data to fully train a model. that’s why we
will be using such models as our lower bound. These model are denoted as DLB
(Direct Lower Bound).

####  3.4.2 Upper Bound Baselines

At the same time, Our model should not be able to exceed the performance of a
model trained and tested on the same distribution meaning trained and tested
on English without any translation from other languages. This model is denoted
as DUB (Direct Upper Bound).

##  4 Evaluation and Results

We compare our model with several baselines on for Stance Detection and
transfer learning. using the four metrics used for the stance detection
challenge.

###  4.1 Domain Randomization

We evaluate our stance detection model using Domain Randomization where we use
the English dataset for Tweet Stance Detection, and test it on both its
testing data and our Zulu dataset for Tweets Stance Detection. Here we notice
that the model generalizes to itself even better the more randomization we add
and the same for the Zulu dataset, as the results in Table 1 show. Results are
reported with macro-F1-score and accuracy along with Favor-F1-score and
Against-F1-score because the main challenge reported these metrics.

The different experiments were conducted all on the same dataset by adding
different translated versions on top of it. The results reported are the mean
5 random runs of the models, where we noticed that on some experiments the
results on the English dataset drops because the quality of the data changes
drastically when the target language translation quality is too poor.

We also noticed that by removing @ and # and some other symbols, the accuracy
increases and becomes more stable over several re-runs. It is also worth
mentioning that the best results on the Zulu dataset: F1-score = 0.5634 were
achieved on Domain Randomization towards: English(original dataset), Zulu,
Xhosa, Shona and Afrikaans languages [26], although we have no way of defining
the reason for this increase it is most likely that it was achieved due to the
similarities of the languages since they are all African languages even though
Afrikaans is very different from Zulu, and due to the homogeneous performance
of google Translate on these languages [5].

Some of the draw backs of the Domain Randomization technique is that it takes
up to 100 times more time per epoch to train the model specially in the first
unsupervised part and requires 3 to 5 times more epochs to converge(so that
the accuracy and loss aren’t changing a lot), plus the translation over-head
which can take up to hours and can be faced by blocking IP-address by the
google API. Another potential issue is that after a certain degree of
randomization ($n_{R}=16$ in our case) the models performance drops
drastically even on the training data.

###  4.2 Domain Adaptation

We implemented a supervised domain adaptation scenario where we train on both
English and Zulu datasets both with access to the labels. We used 70% of the
Zulu dataset for training and 30% for testing. This is the only experiment
where we used the Zulu dataset as part of the training set, unlike the
Randomized-English-Zulu experiment where we only used Zulu as an intermediate
language for the randomization process.

| F1-score | Accuracy | FA-F1-score | AG-F1-score
---|---|---|---|---
Eng-Only | 0.4906 | 0.5258 | 0.3749 | 0.5742
Eng-Zulu | 0.5673 | 0.5448 | 0.4434 | 0.7469

Table 2: Domain Adaptation results on the Zulu-30 Test dataset.

The results on table 2 are the output of k-fold cross validation with k set to
10. The Domain Adaptation results show that there is definitely a visible
increase in the performance on the Zulu test dataset (referred to as Zulu-30),
it even outperforms the best domain randomization models evaluated in Table 1.

##  5 Conclusions

We have proposed a non-intrusive method for improving results on Less-
Resourced / Endangered Languages by leveraging low quality data from English
datasets, using Zulu as a demonstration case. We also provide a new dataset
for Stance Detection on Zulu. Our method was able to effectively transferring
knowledge between different languages. In future work, we aim to improve the
technique by adding more intrusive techniques like distribution matching
between Source and Target domains in the latent space by reducing KL
divergence.

## Acknowledgments

This research was supported by the Independent Danish Research Fund through
the Verif-AI project grant.

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&=\sum_{\begin{subarray}{c}(x_{1},\ldots,x_{s})\in
S_{n-i,s}^{k_{1}-1}\end{subarray}}\quad\sum_{\begin{subarray}{c}(y_{1},\ldots,y_{s})\in
S_{i,s}^{k_{2}-1}\end{subarray}}q^{y_{1}x_{2}+(y_{1}+y_{2})x_{3}+\cdots+(y_{1}+\cdots+y_{s-1})x_{s}}.\end{split}$

Therefore we can compute the probability of the event $\left\\{L_{n}^{(1)}\geq
k_{1}\ \wedge\ L_{n}^{(0)}\geq k_{2}\right\\}$ as follows

$\begin{split}P_{q,\theta}\Big{(}L_{n}^{(1)}\geq k_{1}\ \wedge\
L_{n}^{(0)}\geq k_{2}\Big{)}=&\sum_{i=k_{2}}^{n-k_{1}}\theta^{n-i}\
{\prod_{j=1}^{i}}\ (1-\theta
q^{j-1})\sum_{s=1}^{i}\Big{[}Q_{q,k_{1},k_{2}}(n-i,\ i,\ s)\\\
&+R_{q,k_{1},k_{2}}(n-i,\ i,\ s)+S_{q,k_{1},k_{2}}(n-i,\ i,\ s+1)\\\
&+T_{q,k_{1},k_{2}}(n-i,\ i,\ s)\Big{]}.\\\ \end{split}$

Thus proof is completed. ∎

It is worth mentioning here that the PMF $P_{q,\theta}\Big{(}L_{n}^{(1)}\geq
k_{1}\ \wedge\ L_{n}^{(0)}\geq k_{2}\Big{)}$ for $n\geq k_{1}+k_{2}$
approaches the probability function of $P_{\theta}\Big{(}L_{n}^{(1)}\geq
k_{1}\ \wedge\ L_{n}^{(0)}\geq k_{2}\Big{)}$ in the limit as $q$ tends to 1 of
IID model. The details are presented in the following remark.

######  Remark 53.

For $q=1$, the PMF $P_{q,\theta}\Big{(}L_{n}^{(1)}\geq k_{1}\ \wedge\
L_{n}^{(0)}\geq k_{2}\Big{)}$ for $n\geq k_{1}+k_{2}$ reduces to the PMF
$P_{\theta}\Big{(}L_{n}^{(1)}\geq k_{1}\ \wedge\ L_{n}^{(0)}\geq k_{2}\Big{)}$
for $n\geq k_{1}+k_{2}$ is given by

$\begin{split}P\Big{(}L_{n}^{(1)}\geq k_{1}\ \wedge\ L_{n}^{(0)}\geq
k_{2}\Big{)}=&\sum_{i=k_{2}}^{n-k_{1}}\theta^{n-i}\ {\prod_{j=1}^{i}}\
(1-\theta q^{j-1})\sum_{s=1}^{i}\Big{[}R(s,n-i,\ k_{1},\ n-i)\\\ &+R(s-1,\
k_{1},\ n-i)+R(s+1,\ k_{1},\ n-i)\\\ &+R(s,\ k_{1},\ n-i)\Big{]}R(s,\ k_{2},\
i).\\\ \end{split}$

where

$\begin{split}R(a,\ b,\ c)=\sum_{j=1}^{min\left(a,\
\left[\frac{c-a}{b-1}\right]\right)}(-1)^{j+1}{a\choose j}{c-j(b-1)-1\choose
a-1}.\end{split}$

See, e.g. Charalambides, (2002).

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1 Department of Physics, Birla Institute of Technology and Science, Hyderabad

# Quasi-stationary sequences of hyper massive neutron stars with exotic
equations of state

Sanika Khadkikar1 Chatrik Singh Mangat1 and Sarmistha Banik1

###### Abstract

In this work, we study the effect of differential rotation, finite temperature
and strangeness on the quasi stationary sequences of hyper massive neutron
stars (HMNS). We generate constant rest mass sequences of differentially
rotating and uniformly rotating stars. The nucleonic matter relevant to the
star interior is described within the framework of relativistic mean field
model with the DD2 parameter set. We also consider the strange $\Lambda$
hyperons using the BHB$\Lambda\phi$ equation of state (EoS). Additionally, we
probe the behaviour of neutron stars (NS) with these compositions at different
temperatures. We report that the addition of hyperons to the EoS produces a
significant boost to the spin-up phenomenon. Moreover, increasing the
temperature can make the spin-up more robust. We also study the impact of
strangeness and thermal effects on the T/W instability. Finally, we analyse
equilibrium sequences of a NS following a stable transition from differential
rotation to uniform rotation. The decrease in frequency relative to angular
momentum loss during this transition, is significantly smaller for EoS
containing hyperons, compared to nucleonic EoS.

###### keywords:

equation of state, neutron stars, strange matter, differential rotation

<EMAIL_ADDRESS>
7 January 20221 April 2022

12.3456/s78910-011-012-3 #### 000 0000 1–11 11

##  1 Introduction

Constraining the Equation of State (EoS) of Neutron Stars (NS) has been one of
the biggest unsolved problems for the Nuclear Astrophysics. The dense core is
inaccessible to any sort of observation, hence its composition is unknown to
us. There has been a lot of speculation on the composition of the NS interior,
based on our knowledge of terrestrial experiments such as nuclear physics and
heavy ion collisions. Various theoretical models and EoSs for the NS are
constructed based on these clues, though the model parameters are extrapolated
to asymmetric nuclear matter, well above the nuclear matter density. These
microscopic EoSs are then used to generate the static structure of the stars,
which on the other hand can be compared with the astrophysical observables -
mass and radius.

With the advancement of observational techniques, we continue to get
increasingly accurate data to constrain the properties of NS. In the recent
past the Neutron Star Interior Composition Explorer (NICER) could determine
the mass and radius of PSR J0030+0451 (Raaijmakers $et~{}al$., 2019)
simultaneously for the first time. Two different analyses of NICER data
estimate the radii of $13.02^{+1.24}_{-1.06}$ km for a
1.44${}^{+0.15}_{-0.14}M_{sol}$ pulsar (Miller $et~{}al$., 2019) and
12.71${}^{+1.14}_{-1.19}$ km for a 1.34${}^{+0.15}_{-0.16}M_{sol}$ pulsar
(Riley $et~{}al$., 2019) respectively. The BNS merger GW170817 event estimates
the radius of 1.4$M_{sol}$ neutron star in the range $11.9^{+1.4}_{-1.4}$ km
(Abbott $et~{}al$., 2018) from its tidal deformability value of
($70\leq{\Lambda_{1.4}}\leq 580$). Mass-observations of galactic pulsars PSR
0348+0432 of $2.01^{+0.04}_{-0.04}M_{sol}$ and PSR 0740+6620 of
$2.08^{+0.07}_{-0.07}M_{sol}$ (Fonseca $et~{}al$., 2021) have made the soft
EoSs redundant. These new observations help us constrain EoS parameters and
improve our understanding of the dynamical processes occuring in these compact
objects.

The maximum possible mass of a non-rotating (static) NS depends heavily on its
EoS. But, all physical NS we expect to observe will be rotating at high
frequencies, since the angular momentum of the progenitor is transferred to
the remnant core in a Type II Supernova. This uniform rotation can allow a NS
to support around 8-20% more mass than a static NS (Khadkikar $et~{}al$.,
2021). A rotating NS which is too massive to be supported without rotation is
called a supramassive NS, while a star with a stable static configuration is
called a normal NS. Moreover, the process of differential rotation (i.e. the
core rotating at a higher frequency than the envelope) can significantly
increase the limit on the maximum mass of a NS (Morrison $et~{}al$., 2004;
Nunna $et~{}al$., 2020). These high mass NS supported by differential rotation
are also known as Hyper Massive Neutron Stars (HMNS). Differential rotation is
also speculated to stabilize Proto Neutron Stars (PNS) during a supernova
event and the high mass remnants created by NS-NS merger events.

We consider only the most realistic solutions of differentially rotating stars
that belong to the spheroidal class (Type ‘A’) (Ansorg $et~{}al$., 2009),
which always have a mass-shedding limit. When the degree of differential
rotation is large, the maximum density moves away from the rotation axis,
resulting in a toroidal configuration (Galeazzi $et~{}al$., 2012). But, the
spheroidal class sequence of relativistic solutions is not connected to the
regime of configurations with toroidal shape. Instead, the latter may lead
either to a mass-shedding limit or to a limit of infinite central pressure
(Ansorg $et~{}al$., 2009). We investigate the species of differentially
rotating neutron stars in greater depth, wherein we follow the quasi-static
evolution of a differentially rotating NS down to its static counterpart, if
any. Hence we restrict our calculations to the spheroidal class.

A NS-NS merger can result in a HMNS remnant with densities up to 3-5 times
normal nuclear matter density ($n_{0}$) and temperatures $\sim 50$ MeV (Perego
$et~{}al$., 2019; Bernuzzi, 2020). The high density HMNS core is believed to
contain a few exotic particles, such as hyperons, quarks and/or antikaon
condensate, making the EoS softer (Nunna $et~{}al$., 2020). A strong quark-
hadron phase transition may be detected in the inspiral as well as in post-
merger phase (Chatziioannou & Han, 2020; Bauswein $et~{}al$., 2019).
Appearance of strangeness is more feasible in post-merger remnants due to
their high central density and temperature conditions. Once the HMNS is
formed, large shocks will increase the temperature, and to account for this
additional heating, a finite-temperature EOS is required (Hanauske $et~{}al$.,
2017).

Realistic EoS can support two kinds of equilibrium sequences in an isolated NS
- normal and supramassive. Normal sequences, which contain a stable non-
rotating profile, are considered stable to quasi-radial perturbations. On the
other hand, all supramassive stars have a rest mass which exceeds the maximum
mass of a stable non rotating NS. It has been shown by various authors that
the presence of hyperons, $K^{-}$ condensates and quark matter in an isolated
NS can spin-up its uniform rotation (Zdunik $et~{}al$., 2004; Banik
$et~{}al$., 2004; Spyrou & Stergioulas, 2002; Glendenning, 2000). The spin-up
is accompanied by a loss in angular momentum due to changes in the moment of
inertia as a supramassive star evolves towards the radial instability point.
Beyond this point, both angular momentum and rotational frequency start
increasing, wherein the star crosses into unstable sequence territory. This
behaviour is also known as the back-bending phenomenon for rotating NS.

When dealing with NS evolution, it is very important to look at the stability
with respect to the axi-symmetric perturbations (Zdunik $et~{}al$., 2004). An
important measure of rotation is the T/W parameter i.e. the ratio of
rotational kinetic energy to gravitational binding energy, introduced by
Komatsu $et~{}al$. (1989). The axial symmetry of a rotating NS is
spontaneously broken if T/W exceeds a critical value; this marks the onset of
secular (i.e., viscosity driven) instability. For uniformly rotating,
incompressible ellipsoids, the onset of secular and gravitational wave-
emission instabilities arise at the lowest order for l = 2 modes and $m=\pm 2$
perturbations in Newtonian and general relativistic systems. A limiting value
of the ratio T/W = 0.128 was found for a differentially-rotating NS with a
polytropic EoS (Galeazzi $et~{}al$., 2012). A differentially rotating NS is
reported to be dynamically unstable for T/W > 0.24 (Shibata $et~{}al$., 2000;
Baiotti $et~{}al$., 2007). Relatively low values of T/W $\sim 0.01$ were
calculated to establish the instabilities in NS merger remnants (Shibata
$et~{}al$., 2003; Passamonti & Andersson, 2020). The NS remnant can either
collapse into a black hole or settle as an axisymmetric equilibrium
configuration. While our work uses the barotropic assumption for the EoS where
the thermodynamic properties of the NS are a function of the pressure, there
have been studies of NS with non-barotropic profiles (Camelio $et~{}al$.,
2019). The HMNS is also believed to be surrounded by an accreted disk of mass
$\sim 0.01M_{sol}$ which may affect the long-term stability of the remnant
(Bernuzzi, 2020; Xie $et~{}al$., 2020). However, this is beyond the scope of
this paper, and we leave it for a future study.

The outline of the paper is as follows. In Section 2, we present the
formalism, namely the rotation law for differential rotation, and a brief
description of the different relativistic EoSs employed in this investigation.
We then discuss the numerical scheme employed to obtain the macroscopic NS
structure. In Section 3, we report the main results of this study. We provide
the details of the method used to follow the quasi-static equilibrium
sequences for various differential rotation parameters and EoSs. In Section 4,
we summarise the main findings of this work, its limitations, and the scope
for further study.

##  2 Equation of state and Numerical Scheme of rotating NS

The EoS we work with are realistic, applicable for a charge neutral,
$\beta$-equilibrated NS matter. We consider a NS with (i) pure nucleonic core
(DD2 EoS) consisting of neutrons, protons and electrons (Hempel & Schaffner-
Bielich, 2010), (ii) core with additional $\Lambda$-hyperons (BHB$\Lambda\phi$
EoS) (Banik $et~{}al$., 2014). Two different variants of the EoSs are used,
one corresponding to cold NS (T=0) and the other having constant entropy per
particle (s=2 in units of Boltzmann constant $k_{B}$). Henceforth we consider
$k_{B}=1$ throughout the work. The role of finite temperature on the NS merger
remnant has been explored for s=0 and s=2 cases (Nunna $et~{}al$., 2020; Soma
& Bandyopadhyay, 2020). We use these values to represent cold and hot stars
since studies conclude that the entropy per baryon is $s<2$ at the core of the
remnant just after the merger whereas the bulk of the remnant outside the
unshocked core has the entropy per baryon which is a few times $k_{B}$ (Perego
$et~{}al$., 2019). Although there is a large spread in temperature and entropy
initially, the conditions become homogeneous at a later time.

The hadronic part of both the EoS is computed in the framework of relativistic
mean field model (RMF) (Serot & Walecka, 1986; Walecka, 1995; Hempel &
Schaffner-Bielich, 2010; Glendenning, 2000) where the constituent baryons
interact via $\sigma$, $\omega$ and $\rho$ mesons. Thebaryon-meson couplings
are made density-dependent to take care of the high density behaviour of the
dense matter, whose properties are not known. However, the model parameters
are fitted to the properties of finite nuclei to obtain the bulk properties of
nuclear matter at normal nuclear matter density ($n_{0}$). The saturation
properties of symmetric nuclear matter are $n_{0}=$ 0.149065 fm-3, binding
energy per nucleon 16.02 MeV, incompressibility 242.7 MeV (Typel $et~{}al$.,
2010). The models agree with the values of symmetry energy 31.67 MeV and its
slope coefficient 55.03 MeV, set by experiments. These models can also
reproduce the finite nuclei properties. Further, the pure neutron matter
calculation with the DD2 parameter set is in agreement with that of the chiral
effective theory (Marques $et~{}al$., 2017). The inhomogeneous nuclear matter
at sub-saturation density is composed of light and heavy nuclei along with
unbound nucleons and is described within the extended Nuclear Statistical
Equilibrium (NSE) Model of Hempel & Schaffner-Bielich (2010). This is merged
with the high density part of our EoS by minimisation of free energy, the
thermodynamic consistency is ensured in the process (Banik $et~{}al$., 2014).
The EoSs at sub-saturation density as well as high density, are computed using
the same RMF nucleon-nucleon interaction and can be considered unified.
Moreover, the observational constraint of 2$M_{sol}$ (Demorest $et~{}al$.,
2010; Antoniadis $et~{}al$., 2013) on their mass is satisfied, even in the
presence of strange $\Lambda$ particles.

Figure 1: Gravitational mass vs central energy density of NS with DD2 and
BHB$\Lambda\phi$ EoS for T=0 and s=2 configurations. The solid lines represent
static (black), uniformly rotating (orange) and differentially rotating with
$a$ = 0.2

(blue) stars at Kepler frequencies. The dashed lines are constant rest mass
(in $M_{sol}$) sequences (color online).

$\Lambda$’s are populated at the cost of neutrons, when their chemical
potentials attain equilibrium. We use the BHB $\Lambda\phi$ EoS for strange
matter, that is an extension of DD2 EoS. The additional $\phi$ mesons take
care of the $\Lambda-\Lambda$ interaction. The density-dependent
$\Lambda$-vector meson hyperon vertices are obtained from the SU(6) symmetry
of the quark model (Dover & Gal, 1984; Schaffner & Mishustin, 1996). On the
other hand, the $\gamma$-ray spectroscopy for $\Lambda$-hypernuclei indicates
a potential depth of around 30 MeV. The $\Lambda$ hyperon - scalar meson
couplings are obtained from the potential of $\Lambda$ hyperons in nuclear
normal matter and the vector meson couplings. The ratio of $g_{\sigma\Lambda}$
to $g_{\sigma N}$ is 0.62008 (Banik $et~{}al$., 2014). However, the nature of
hyperon-nucleon interaction cannot be ascertained without ambiguity because of
limited scattering data and measured properties of hyper-nuclei, for other
hyperons like $\Sigma$ and $\Xi$. Hence, we keep our calculations restricted
to the lightest hyperon of the octet.

We use codes based on the LORENE (Gourgoulhon $et~{}al$., 2016) library to
study rapidly rotating NS. There are already existing numerical schemes that
compute equilibrium solutions of uniform and differentially rotating cold NSs.
It can also deal with isentropic EoS (Goussard $et~{}al$., 1997, 1998). The
algorithm to construct rotating equilibrium models numerically in a self-
consistent-field method is explicitly described in Bonazzola $et~{}al$.
(1993); Nunna $et~{}al$. (2020). We employ the simple KEH (Komatsu $et~{}al$.,
1989) or j-constant rotation law defined by the velocity profile to compute
various properties of the differentially rotating HMNS:

$F(\Omega)=R_{0}^{2}\left(\Omega_{c}-\Omega\right)$

where $\Omega_{c}$ is the central angular frequency. The angular velocity
decreases monotonically from core to the surface of the star and the degree of
differential rotation is encapsulated in the parameter $a=R_{e}/R_{0}$ where
$R_{e}$ is the equatorial radius of the NS. Our results apply particularly to
the class of solutions of ‘type A’, as classified by Ansorg $et~{}al$. (2009).
These solutions possess a mass-shedding limit, and their rest-mass density is
the maximum at the stellar centre. Higher values of $a$ will lead to higher
angular momenta $J$, which will shift the mass-shedding limit to higher masses
(Weih $et~{}al$., 2018), and might move the maximum density away from the
rotational axis. Moreover, an overall mass-shedding limit may not exist for
higher degrees of differential rotation. Hence, throughout this work, we have
taken $a=0.2$ unless specified otherwise. We observed that the numerical
fluctuations in the differential Kepler frequency curves could be reduced by
giving a high frequency as input with a $10^{-6}$ order convergence threshold.
Incidentally, the differential rotation law with the specific angular momentum
is not constant in general, but only in a particular limit and in Newtonian
gravity (Camelio $et~{}al$., 2021). A number of binary NS merger simulations
suggest the angular velocity profile of the HMNS is characterized by a slowly
rotating core and an envelope that rotates at frequencies scaling like
$r^{-3/2}$ (Hanauske $et~{}al$., 2017) unlike the j-constant law. Uryu et al.
(2017) formulated new models with 3 and 4 parameters for differential rotation
of compact stars (Uryū $et~{}al$., 2017). Several authors have worked on more
realistic rotation profiles (Bozzola $et~{}al$., 2018; Iosif & Stergioulas,
2021; Camelio $et~{}al$., 2021) to construct equilibrium models of
differentially rotating BNS remnants. Iosif and Stergioulas have compared the
Uryu differential law and KEH law and concluded that the gravitational mass
does not have any difference for type A, B, C sequences of (Ansorg $et~{}al$.,
2009), but the choice of rotation law has larger effect on the radius (Iosif &
Stergioulas, 2021). Passamonti and Andersson studied the impact of
differential rotation on the low T/W instability using the 3-parameter
rotation law of Uryu and highlighted the similarity with the widely used
j-constant law (Passamonti & Andersson, 2020). Hence, we stick to the commonly
used j-constant law in this work to first observe the effects of finite
temperature and strangeness on differentially rotating NS. We will explore the
effect of modified rotation laws in this regime in future work.

[ DD2, $T=0$ ]
---
Configuration | $M_{b}(M_{sol})$ | $M_{g}(M_{sol})$ | Radius($km$) | $\rho_{c}^{max}(\rho_{nuc}c^{2})$ | $f_{c}(Hz)$ | $J(G{M_{sol}^{2}}/c)$ | T/W
Differential Kepler | 3.60 | 3.05 | 11.32 | 9.99 | 1783.25 | 7.16 | 0.16
Uniform Kepler | 3.43 | 2.90 | 10.83 | 9.82 | 1535.34 | 5.92 | 0.14
Static | 2.88 | 2.41 | 8.01 | 11.96 | 0.0 | 0.0 | 0.0
[ DD2, s=2 ]
Configuration | $M_{b}(M_{sol})$ | $M_{g}(M_{sol})$ | Radius($km$) | $\rho_{c}^{max}(\rho_{nuc}c^{2})$ | $f_{c}(Hz)$ | $J(G{M_{sol}^{2}}/c)$ | T/W
Differential Kepler | 3.28 | 2.87 | 11.95 | 10.24 | 1537.87 | 5.55 | 0.12
Uniform Kepler | 3.18 | 2.78 | 11.65 | 9.81 | 1343.29 | 4.81 | 0.10
Static | 2.79 | 2.42 | 8.65 | 11.18 | 0.0 | 0.0 | 0.0
[ BHB$\Lambda\phi$, $T=0$ ]
Configuration | $M_{b}(M_{sol})$ | $M_{g}(M_{sol})$ | Radius($km$) | $\rho_{c}^{max}(\rho_{nuc}c^{2})$ | $f_{c}(Hz)$ | $J(G{M_{sol}^{2}}/c)$ | T/W
Differential Kepler | 3.04 | 2.63 | 11.41 | 10.37 | 1573.40 | 4.98 | 0.14
Uniform Kepler | 2.91 | 2.52 | 10.92 | 10.48 | 1422.22 | 4.25 | 0.12
Static | 2.43 | 2.10 | 7.84 | 12.58 | 0.0 | 0.0 | 0.0
[ BHB$\Lambda\phi$, s=2 ]
Configuration | $M_{b}(M_{sol})$ | $M_{g}(M_{sol})$ | Radius($km$) | $\rho_{c}^{max}(\rho_{nuc}c^{2})$ | $f_{c}(Hz)$ | $J(G{M_{sol}^{2}}/c)$ | T/W
Differential Kepler | 2.77 | 2.48 | 11.98 | 10.89 | 1384.66 | 3.78 | 0.10
Uniform Kepler | 2.71 | 2.42 | 11.63 | 10.90 | 1265.43 | 3.39 | 0.09
Static | 2.39 | 2.13 | 8.49 | 12.23 | 0.0 | 0.0 | 0.0

Table 1: Properties of the maximum mass NS spinning at mass-shedding limits,
rotating differentially with $a$ = 0.2

and uniformly. The static case corresponds to properties of the maximum mass
non-rotating NS for each EoS.

##  3 Results

We now present the results of our investigation. We compare the properties of
NS generated using the DD2 EoS and BHB$\Lambda\phi$ EoS at zero temperature
and finite temperature (s=2). We also explore the effect of varying degree of
differential rotation on the evolution of the NS. In Fig. 1, we plot the
gravitational mass (in solar masses) versus central energy density (in units
of $\rho_{nuc}c^{2}$, where $\rho_{nuc}=1.66\times 10^{17}kg/m^{3}$). The
top(bottom) panels are for cold(hot) stars, whereas the left(right) panels are
for DD2(BHB$\Lambda\phi$) EoS. The equilibrium sequences for non-rotating
(black solid curve) and mass-shedding or “Kepler” limits of uniformly rotating
NSs (orange solid curve) and differentially rotating NSs with $a=0.2$ (blue
solid curve) are plotted. We see that differential rotation allows a NS to
support more mass than a non-rotating or a uniformly rotating NS.

When comparing stars with similar compositions, a hot static star, owing to
thermal pressure, can support more mass than the cold one. However, we observe
a different trend for both uniform and differentially rotating stars. The
Kepler frequency (rotation rate corresponding to the mass-shedding limit) for
hot stars is lower than that of cold stars. This is because the additional
thermal energy in a hot star compensates for a part of the rotational energy,
making the hot star start shedding mass at a lower rotation rate than a cold
star. Hence, the maximum mass corresponding to a cold star rotating at its
Kepler frequency is found to be more massive than that of a hot star. This is
evident from the properties listed in Table 1.

Figure 2: Central rotation frequency vs angular momentum of NS with DD2 and
BHB$\Lambda\phi$ EoS for T=0 and s=2 configurations. The solid lines represent
uniformly rotating (orange) and differentially rotating with $a$ = 0.2

(blue) stars at Kepler frequencies. The gray dash-dotted line indicates the
quasi-stationary stability limit of the dashed constant-rest mass sequences
(color online).

Figure 3: Rotational frequency as a function angular momentum for
differentially rotating NS with the DD2 EoS at T=0 for different degrees of
differential rotation $a$. The solid curves represent NS rotating at the mass-
shedding limit uniformly (orange) and differentially (blue). The dashed curves
represent constant rest mass sequences of differentially rotating NS (color
online). Figure 4: Central rotation frequency vs T/W of NS with DD2 and
BHB$\Lambda\phi$ EoS for T=0 and s=2 configurations. The format of the graph
is the same as Figure 2, with the dash-dotted line indicating the quasi-
stationary stability limit (color online).

Figure 5: A comparison of spin-up in supramassive stars with
$M_{b}=3.11M_{sol}$ for cold (top) and hot (bottom) NS with DD2 EoS. The stars
represented by the purple dash-dotted curves rotate differentially with $a$ =
0.2

throughout their evolution. The red dashed curves represent stars where
differential rotation ($a$ = 0.2) decays to uniform rotation ($a$ = 0.0) as
the star evolves. We also plot the Kepler sequences for uniform rotation
(orange) and differential rotation (blue) for reference.

Comparing the left and right panels of Fig. 1 reveals that the NS can support
a smaller maximum mass in the presence of hyperons, other qualitative features
remaining same as DD2 EoS. This reduction in the NS mass is consistent with
the softening of the EoS due to the addition of hyperons.

The maximum mass $M_{g}^{max}$ allowed for each case of Fig. 1 is tabulated
along with the central density $\rho_{c}^{max}$ in Table 1. The static NS
masses are well within the observational constraint (Antoniadis $et~{}al$.,
2013; Cromartie $et~{}al$., 2020). We also observe that the relative increase
in $M_{g}^{max}$ due to differential rotation is higher for cold stars than
hot stars, for both the compositions. The condition,
$\frac{dM_{g}}{d\rho_{c}}<0$, marks the onset of the secular instability for
isentropic (or isothermal), rigidly rotating (Goussard $et~{}al$., 1997;
Marques $et~{}al$., 2017) as well as for differentially rotating stars
(Bozzola $et~{}al$., 2018; Weih $et~{}al$., 2018) . The configurations with
$\rho_{c}<\rho_{c}^{max}$ are stable whereas those with
$\rho_{c}>\rho_{c}^{max}$ with respect to small radial perturbations.

We then plot constant rest mass sequences for differentially rotating neutron
stars. Each sequence represents an isolated NS of fixed rest mass, that
rotates differentially and traces a curve of decreasing $M_{g}$ initially. The
lower mass stars slowly lose energy and eventually meet the static curve when
they stop rotating, these are called normal sequences. On the other hand, the
higher mass sequences, also called supramassive sequences, never meet the
static curve. The higher mass sequences trace a curve of decreasing mass to
reach a minimum $M_{g}$ before rising again to meet the differential Kepler
curve once again. The turn around point in these sequences is called the quasi
stationary stability limit. The minimum $M_{g}$ configurations in these
sequences correspond to the quasi stationary stability limit. These
supramassive NS have masses larger than the maximum mass non-rotating NS. The
behaviour is similar to that of uniformly rotating stars in Cook $et~{}al$.
(1994). For DD2 EoS, a 2.89 $M_{sol}$ baryon mass sequence barely touches the
static curve for the T=0 case, whereas it does not have any static counterpart
for the s=2 case. For BHB$\Lambda\phi$ EoS, this happens at $M_{b}\sim
2.45M_{sol}$. Importantly, we note that the constant rest mass sequences, when
plotted in Fig.1, do not vary much for (i) stars that rotate differentially
for a brief period and then continue to rotate uniformly and (ii) stars that
rotate differentially throughout their evolution.

In Fig. 2, we plot the rotational frequency against angular momentum to
investigate the spin-up phenomenon. The solid curves mark the mass-shedding
limits; we use blue for the differentially rotating stars with $a$ = 0.2 and
orange for the uniformly rotating ones. The dashed lines are constant baryon
mass sequences. Along normal sequences, the NS slows down as it evolves by
radiating energy and angular momentum to move towards the origin. However,
along supramassive sequences, the NS is observed to exhibit spin-up at some
point in its evolution. This can be seen as back-bending, where the angular
momentum decreases with increasing frequency for a short time, for all the
cases considered here. The spin-up or back-bending rate is visibly more
prominent for higher entropy stars. If we compare the massive NS with DD2 EoS
of a particular baryon mass ($3.2M_{sol}$ for example), a star with entropy
s=2 is observed to spin-up by $120Hz$, which is more than a cold star which
spins up by $40Hz$. Inversely, the angular momentum loss for the hot star is
$0.28\text{ }GM^{2}_{sol}/c$, which is smaller compared to $0.49\text{
}GM^{2}_{sol}/c$ for the cold star. The gray dashed-dotted line represents the
quasi-stationary stability line, above which the supramassive sequences are
unstable. We find that the quasi-stationary line can be fit with a polynomial
function of the form

$\dfrac{f_{c}}{f_{c}^{\text{M}}}=c_{1}\bigg{(}\dfrac{J}{J^{\text{M}}}\bigg{)}^{2}+c_{2}\bigg{(}\dfrac{J}{J^{\text{M}}}\bigg{)}$

where the rotational frequency $f_{c}$ and the angular momentum $J$ are
normalised using respective properties ($f_{c}^{\text{M}}$ and $J^{\text{M}}$)
for the maximum mass differentially rotating NS for each EoS from Table 1. The
coefficients for the fitted polynomial for each EoS are listed in Table 2. We
note that this fit also depends on the differential rotation profile adopted
for the study, we will look at the effects of other differential rotation
profiles in our future work.

Configuration | $c_{1}$ | $c_{2}$
---|---|---
DD2, T=0 | -0.995 | 1.982
DD2, s=2 | -0.794 | 1.803
BHB$\Lambda\phi$, T=0 | -1.093 | 2.089
BHB$\Lambda\phi$, s=2 | -0.729 | 1.732

Table 2: The values of the fitting parameters for the polynomial function
relating rotational frequency to angular momentum of differentially rotating
($a=0.2$) HMNS at the quasi-stationary stability limit.

Configuration | $M_{b}(M_{sol})$ | $\Delta f_{c}(Hz)$ |  $\Delta$J($GM_{sol}^{2}/c$)
---|---|---|---
DD2, T=0 | 3.11 | 79.58 | 0.45
DD2, s=2 | 3.11 | 43.61 | 0.33
DD2, T=0 | 2.89 | 71.72 | 0.37
BHB$\Lambda\phi$, T=0 | 2.89 | 7.73 | 0.42

Table 3: Change in frequency and angular momentum in the transition from
differential rotation ($a=0.2$) to uniform rotation (Red sequences plotted in
Fig. 5). The first two rows compare NS of the same composition and $M_{b}$ at
different temperatures. The last two rows compare cold NS having the same
$M_{b}$ but different compositions.

In the presence of $\Lambda$ hyperons, the spin-up is more robust. If we
follow the spin evolution of a cold NS of baryon mass $2.89M_{sol}$ with DD2
and BHB$\Lambda\phi$ EoSs, this becomes more evident. Here, the NS with
hyperons spins up to $1530Hz$, whereas a NS with only nucleons, spins down
from $1290Hz$. The spin-up rate against angular momentum is further enhanced
by increasing entropy. For example a strange, hot NS with baryon mass
$2.75M_{sol}$ spins up from $1280Hz$ to $1370Hz$ for a small angular momentum
loss of 0.1 $GM^{2}_{sol}/c$. For its cold counterpart, the corresponding rise
in frequency is $130Hz$, over a larger angular momentum loss of 1.07
$GM^{2}_{sol}/c$. A similar trend can be seen for the DD2 EoS at s=2.

We also investigate the role of differential rotation and confirm that a star
with a higher degree of differential rotation can support a higher maximum
mass. In Fig. 3, the rotational frequency of differentially rotating sequences
for $a=0.1$ and $a=0.3$ is plotted as a function of angular momentum for the
DD2 EoS at T=0. The Kepler frequency corresponding to the maximum mass NS is
$1600Hz$ for $a=0.1$, and $2100Hz$ for $a=0.3$. When two supramassive constant
rest mass sequences are compared for different degrees of differential
rotation, the back-bending is more pronounced for the case with higher value
of $a$.

In Fig. 4, we can see the relationship between the frequency and the ratio of
rotational kinetic energy to gravitational binding energy (T/W) for the same
sets of cold and hot EoSs. The solid curves correspond to Kepler limits for NS
rotating uniformly (orange) and differentially (blue). We note that a
differentially rotating star can reach a much larger T/W ratio compared to a
uniformly rotating star. The triaxial instability could play a more important
role as the T/W value can be as high as 0.164 for a differentially rotating NS
with the DD2 T=0 configuration, which comes down to 0.136 for the uniformly
rotating one. Moreover, the T/W values are lesser for higher entropy stars
than cold stars. We have seen that a hotter star is larger in size (Batra
$et~{}al$., 2018; Nunna $et~{}al$., 2020), or in other words, thermal pressure
results in a less tightly bound star. This was also reported by (Camelio
$et~{}al$., 2021) for a different rotation law. It is further lowered for a
softer EoS, in the presence of $\Lambda$ hyperons. Hence the NS with strange
matter and/or higher entropy are stable with respect to axisymmetric
perturbation. The maximum values of T/W for different cases considered here
are listed in Table 1. The bar-modes in a rapidly rotating, gravitationally
bound fluid body may be excited only for cold NS (T/W $\geq 0.14$). For
general relativistic stars, the instability may occur at lower values of
$\text{T/W}\sim 0.08$, with gravitational wave-emission (Cook $et~{}al$.,
1994). For all the cases, instability may occur before the NS can reach the
mass-shedding limits. However, a stable configuration may exist with smaller
T/W as the stars spin-up. The constant rest mass sequences are plotted with
dashed lines. Here, all the normal sequences spin down as they can evolve from
a differentially rotating state to a non-rotating/static state while being
stable under quasi radial perturbations. One can observe the back-bending
phenomenon in this graph, wherein, the constant $M_{b}$ curves for
supramassive stars turn up to intersect the quasi stationary stability curve
(dash-dotted) and eventually reach the Kepler curve, instead of hitting the
origin like the normal sequences do. This was also observed by (Cook
$et~{}al$., 1994) for uniform rotation. The qualitative features for the NS
with nucleons-only or strange matter are more or less the same. Both the
nucleonic and strange models at T=0 with higher masses may undergo an
instability starting at some frequency, but spin-up and attain stable
configurations.

Finally, we compare two evolution scenarios for NS with the same rest mass. In
case 1, the star continues to rotate differentially throughout the evolution
with no change in $a$. In case 2, the differential rotation gradually decays
to uniform rotation as the star evolves. For the two cases, the constant rest
mass sequences in the $M_{g}\text{ vs }\rho_{c}$ space overlap within
systematic error. However, the back-bending patterns change with angular
momentum and T/W loss. The constant rest mass ($M_{b}=3.11M_{sol}$) sequences
for the two cases are plotted in Fig. 5 case the DD2 EoS with different
entropies. At any given angular momentum, a case 1 star rotates at a higher
frequency than a case 2 star.

We tabulate the change in frequency and angular momentum for the transition
from differential rotation ($a=0.2$) to uniform rotation for different
profiles in Table. 3. In the first two rows, we see that increasing the
temperature for a fixed composition leads to a smaller loss in both frequency
and angular momentum during the transition. Additionally, in the last two
rows, we observe that the change in frequency is significantly smaller for a
star which contains hyperons when compared to a star with a nucleonic EoS. On
the other hand, the change in angular momentum is slightly larger for the star
containing hyperons compared to the star with the nucleonic EoS.

##  4 Summary and Discussions

We have generated constant rest mass sequences for isolated differentially
rotating neutron stars and analysed the effect of strangeness and finite
temperature on their equilibrium sequences. We have found that our results for
the maximum mass and frequency vary significantly based on the choice of EoS
and the degree of differential rotation. Additionally, we see that both
increasing temperature and addition of hyperons tend to boost the spin-up in
differentially rotating NS. While these general trends are more robust, we go
into more detailed discussions of our study one by one.

First, we report the effect of differential rotation on the maximal mass and
rotational frequency of a NS. It is observed that differential rotation
enables a NS to support a significantly higher mass than a non-rotating NS.
This relative increase in maximum supported mass is higher in cold stars than
hot stars. Differential rotation also allows the star to attain higher central
rotational frequency than a uniformly rotating NS. Further increase in the
degree of differential rotation enables the star to support even higher mass
and rotational frequency while staying within the Kepler mass-shedding limit.
For the j-constant law, the degree of differential rotation has no effect on a
constant rest mass sequence being normal or supramassive. This classification
depends on the mass of the NS and the choice of the EoS.

We also obtain constant rest mass sequences which do not cross the Kepler
curve for uniform rotation (for example: $M_{b}$ = 3.20$M_{sol}$ curve for DD2
s=2 EoS and $M_{b}$ = 2.75$M_{sol}$ curve for BHB$\Lambda\phi$ EoS). These
curves, theoretically, represent stars that cannot be stabilised by uniform
rotation. Stars born with this configuration will be highly unstable, as
reduction in the degree of differential rotation through processes such as
magnetic braking and viscosity would inevitably lead to a collapse. This
collapse will occur because a uniformly rotating counterpart to this
configuration does not exist. This avenue is not well explored in our study
and may be a subject for future work.

When investigating the spin-up phenomenon, we observe that the presence of
hyperons in the EoS makes the spin-up more robust. The NSs following
supramassive sequences with a cold nucleonic EoS have very high masses, which
makes them unlikely candidates for observation. But, the presence of hyperons
can make the EoS softer, which allows relatively lower mass stars to follow
supramassive sequences. Moreover, we observe that a higher temperature can
help boost the spin-up rate further. Therefore, a hot star with hyperons can
be a viable candidate for modelling observations of spin-up in massive NS.

We find that the maximum T/W attained by the NS decreases with higher entropy.
Thermal pressure or the presence of strangeness is observed to increase the
stability of the NS by reducing the maximal T/W values. However, high mass
stars with nucleonic or strange models at T=0 may undergo an instability at
some frequency to attain stable configurations upon spin-up.

We also compare the supramassive sequences of a star that rotates
differentially throughout its evolution and a star that transitions from
differential rotation to uniform rotation. The latter case is more plausible
as differential rotation can be damped by physical processes on a timescale
which is shorter than the star’s age. In the equilibrium sequences for a
transition from differential to uniform rotation, the spin-up starts only
after the transition to uniform rotation is complete. This means that a stable
transition might require the frequency of the star to decrease with the degree
of differential rotation, to keep the star within the Kepler mass-shedding
limit. For two stars with the same baryon mass and composition, increasing the
temperature decreases the rate of spin-down during the transition. When we
compare two stars with the same baryon mass but different compositions, we see
that the presence of hyperons drastically reduces the rate of spin down during
the transition from differential to uniform rotation.

We find that differentially rotating hot stars with hyperons can be a viable
option to model the HMNS observed as remnants of binary NS mergers. Our
results also provide evidence for the enhancement of spin-up in the presence
of hyperons in the EoS. For our choice of EoS, spin-up is also supported by an
increase in the temperature of the HMNS. This work opens up avenues for
further investigation of differentially rotating HMNS with different rotation
laws, compositions and temperatures. We plan to explore these avenues in
detail in subsequent papers.

## Acknowledgements

Authors are grateful to Micaela Oertel and Prasanta Char for insightful
discussions.

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