#  Trauma lurking in the shadows: A Reddit case study of mental health issues
in online posts about Childhood Sexual Abuse

Orchid Chetia Phukan  Department of CSE
IIIT-Delhi, India
<EMAIL_ADDRESS>Rajesh Sharma  Institute of Computer Science
University of Tartu, Estonia
<EMAIL_ADDRESS>Arun Balaji Buduru  Department of CSE
IIIT-Delhi, India
<EMAIL_ADDRESS>
###### Abstract

Childhood Sexual Abuse (CSA) is a menace to society and has long-lasting
effects on the mental health of the survivors. From time to time CSA survivors
are haunted by various mental health issues in their lifetime. Proper care and
attention towards CSA survivors facing mental health issues can drastically
improve the mental health conditions of CSA survivors. Previous works
leveraging online social media (OSM) data for understanding mental health
issues haven’t focused on mental health issues in individuals with CSA
background. Our work fills this gap by studying Reddit posts related to CSA to
understand their mental health issues. Mental health issues such as
depression, anxiety, and Post-Traumatic Stress Disorder (PTSD) are most
commonly observed in posts with CSA background. Observable differences exist
between posts related to mental health issues with and without CSA background.
Keeping this difference in mind, for identifying mental health issues in posts
with CSA exposure we develop a two-stage framework. The first stage involves
classifying posts with and without CSA background and the second stage
involves recognizing mental health issues in posts that are classified as
belonging to CSA background. The top model in the first stage is able to
achieve accuracy and f1-score (macro) of 96.26% and 96.24%. and in the second
stage, the top model reports hamming score of 67.09%.

Content Warning: Reader discretion is recommended as our study tackles topics
such as child sexual abuse, molestation, etc.

Keywords: Childhood sexual abuse (CSA), mental health, Reddit, Natural
language processing (NLP).

##  I Introduction

Sexually harmful acts against a child (less than 18 years of age) are
characterized as Childhood Sexual Abuse (CSA) and it spans a wide range of
acts including sexual abuse, sexual interaction, rape, sexual grooming, and
sexual exploitation [1]. This form of abuse often entails the perpetrator
using force or making threats [2]. Between ages 7 and 13, children are most
vulnerable to CSA. Perpetrators can be either family members or strangers [3].
It has been reported that survivors of CSA are often accompanied by feelings
of stigma and guilt [4, 5].

CSA is often succeeded by physical and mental difficulties. Physical
difficulties include physical injuries such as genital injuries [6] and
sexually transmitted diseases (STDs) [7]. Individuals with a history of CSA
are more likely to have chronic illnesses such as chronic pain [8] and obesity
[9] in later stages of life. Physical injuries may heal, but psychological
scars from the abuse can last throughout later stages of life [7]. Survivors
of CSA are often likely to be haunted by the recall of the traumatic event
[10]. Risk of occurrence of various mental health issues such as Post-
Traumatic Stress Disorder (PTSD) [11], depression, anxiety [8], and eating
disorders [12] are more in CSA survivors. CSA is also reported to be a key
risk factor for the development of Borderline Personality Disorder (BPD) [13].

For the purpose of understanding mental health issues in CSA survivors,
numerous prior research investigations have been conducted, but they were
primarily restricted to interviews, questionnaires, surveys, and electronic
health records (EHRs) with a limited amount of data [14, 15]. In recent years
research considering OSM for mental health issues is rising [16], as it has
been seen that individuals are resorting to online social media (OSM)
platforms to find shelter for their mental health issues [17] where CSA
survivors are no exception. Especially OSM such as Reddit has turned out to be
a popular medium [18] for such. Reddit is a community-based platform where
each community is called a “subreddit” and is based on a particular topic or
issue [19]. These communities sometimes act as a discussion forum and
sometimes as peer support groups for various issues related to health, mental
complications, etc. Thus, this inspired us to look through various subreddits
and gather posts about mental health issues associated with CSA, and probe the
following three research questions.

RQ1: What are the commonly discussed mental health issues in posts related
with CSA?

For finding out the most commonly discussed mental health issues in posts with
CSA exposure, we analyzed r/adultsurvivors. We found out that depression,
anxiety and PTSD are the most commonly discussed mental health issues.

RQ2: Are there differences between the posts related to mental health issues
with and without CSA background?

Mental health issues can be triggered by reasons other than CSA. So, in order
to identify the specific characteristics of mental health issues related posts
with CSA background which could be different from posts without CSA
background, we make use of NLP techniques such as word-shift, word cloud,
topic analysis, and emotion analysis. We found that posts with CSA background
mention flashbacks of the traumatic event. Trauma-related issues are more
prevalent in posts with CSA background. Emotion “disgust” and “fear” are more
associated with posts with CSA background in contrast to posts without a CSA
background which are more associated with emotion “sadness”.

RQ3: Can fine-tuning domain-specific transformer model help in identifying
mental health issues in CSA survivors from the OSM posts?

The analysis in RQ2 revealed that indeed there are differences between mental
health issues-related posts with and without CSA background. So we developed a
two-stage framework where the first stage is binary classification of posts to
“with CSA” and “without CSA” background. We implemented various classical
machine learning models and transformer-based models for it. Among all the
models, fine-tuned BERT (Bidirectional Encoder Representations from
Transformers) [20] reported the highest accuracy of 96.26% and F1-score
(Macro) of 96.24%. In the second stage, multi-label classification is done on
the posts with CSA background for the identification of different mental
health issues. Here, we fine-tune MentalBERT [21], which is a mental health
domain-specific language model and it reported the highest hamming score of
67.09%. In addition, we provide interpretations of the model’s decision using
the Integrated Gradients method for both stages.

Contributions: Our work is the first, to the best of our knowledge which has
analyzed OSM for the understanding and identification of mental health issues
in CSA survivors. We present the commonly discussed mental health issues
confronted by CSA survivors and the distinctive characteristics of the posts
with CSA background that differ from the posts without CSA background. We
created an anonymized dataset of around 8k posts related to CSA collected from
various subreddits for carrying out our analysis. We highlight the existence
of minor distinctions between the posts with and without CSA background and
with consideration of this, we propose a two-stage framework for detecting
mental health issues in posts with CSA background.

Ethical and Privacy concerns: Although our institutional review board waived
review of our work, we followed preventive measures to protect the privacy of
the users whose data is being used in our study. No sensitive or meta
information related to the users is collected or considered during our
analysis. The examples of posts shown in our study have been paraphrased to
avoid traceability. The predictive model only takes the posts and no user
information is required. In addition, the work does not propose any diagnostic
claims.

##  II Background

In this section, we first discuss existing literature related to mental health
in CSA survivors in Subsection II-A and then works that have used OSM to
understand mental health issues in Subsection II-B.

###  II-A Childhood sexual abuse (CSA) and mental health

A direct relationship between CSA and severe mental health issues has been
observed in previous studies [14, 22]. Studies have discovered that CSA has a
strong association with serious mental health issues in women who were victims
of CSA. Substance addiction and suicidal behavior were also more prevalent
among CSA survivors [23]. Research focused on children exposed to CSA (both
male and female, 16 years of age or younger) reported that children exposed to
CSA are substantially more likely than the general population to need
psychiatric treatment. Schizophrenia was the least common mental health issue
among CSA survivors of all the mental health issues [24]. Men exposed to CSA
are at higher risk of severe mental health issues and also suicidality [25].
Electronic health records (EHRs) were also availed for a better understanding
of the relationship between CSA and serious mental health issues. It was found
that patients who have been exposed to CSA are more likely to develop clinical
depression, PTSD, and personality disorders [15].

###  II-B Mental health issues and online social media (OSM)

Various research studies using OSM data [26, 27] for analyzing and identifying
mental health issues have been conducted during the previous decade and this
has proven to be an effective method [16]. These studies are divided into two
groups: those that focus on a single mental health issue and those that focus
on multiple mental health issues. Reddit posts were leveraged to identify
depressive disorder using multiple predictive modeling approaches such as
support vector machine (SVM) and multilayer perceptron (MLP) [28].
Furthermore, using a multi-modal (text, image, etc) strategy, Chiu et al. [29]
investigated the identification of depressive disorder. Using tweets from
self-proclaimed schizophrenia patients, Mitchell et al. [30] looked into
transmitting linguistic markers for schizophrenia detection. Multiple BERT-
based algorithms were used to identify eating disorders using tweets [31].

Suicidal Intent detection is a crucial task for early detection and prevention
of suicide. For identification of suicidal intent, Facebook posts were used
[32]. Using RoBERTa, investigation was done for the identification and
classification of several mental health issues (depression, anxiety, bipolar
disorder, Attention deficit hyperactivity disorder (ADHD), PTSD) [33]. For
achieving it, Reddit posts were exploited. Previous research also looked into
using a two-stage approach for detecting mental health issues, with the first
stage involving binary classification into mental health or non-mental health,
and the second stage involving multi-class classification of mental health
issues [34, 27]. Recently COVID-19 pandemic took a toll on people’s mental
health which motivated researchers to look into different mental health
subreddits such as r/SuicideWatch, r/depression, etc. to investigate the
impact of different mental health issues on individuals during the COVID-19
pandemic’s early waves [35]. Past research studies involving OSM either
ignored or addressed mental health concerns in CSA survivors as part of a
larger study of mental health issues in the general community, with no
specific emphasis placed on it. Our work takes a step in this direction.

##  III RQ1: What are the commonly discussed mental health issues in posts
related with CSA?

We strive to comprehend the different mental health issues that CSA survivors
face. For achieving it, we explored r/adultsurvivors [36] subreddit defined as
“a peer aid community for adults who experienced sexual abuse as children”,
which provides a platform for CSA survivors to share about their experiences
and seek support for their difficulties. We preferred Reddit over other OSM
platforms as it is a community-focused platform and also provides a
distinctive feature called “throwaway” accounts to its users. This temporary
feature acts as a veil and allows the users to anonymously express their
feelings or subjects that seem inexpressible [19].

Collecting posts with CSA background: For finding out commonly discussed
mental health issues among CSA survivors, first we collected posts from
r/adultsurvivors using keywords related to various mental health issues such
as depression (depression, depressed, hopelessness, hopeless, depressive, and
despondent), anxiety (anxiety, panic attack, panic disorder, and phobia), PTSD
(ptsd, post-traumatic stress disorder, posttraumatic stess disroder, trauma,
traumatic, and traumatized), schizophrenia (schizophrenia and psychosis),
eating disorders (eating disorder, anorexia, and bulimia), and personality
disorders (bpd, personality disorder, and ocpd). We got the highest number of
posts related to depression, anxiety, and PTSD. We got few to none posts
related to mental health issues such as schizophrenia, eating disorders, and
personality disorders. Since depression, anxiety, and PTSD are the most
commonly discussed mental health issues so we focus only on these three mental
health issues and discarded other mental health issues for carrying out our
further analysis.

We didn’t collect posts from subreddit such as r/survivorsofabuse, a support
group for abuse survivors [37], as it is a private community. We collected
posts from the subreddits that are publicly accessible. Taking inspiration
from aforementioned, we also collected posts with CSA background from various
mental health subreddits associated with a particular mental health issue such
as r/depression, r/Anxiety, r/ptsd, and r/depression_help [33] related to
commonly discussed mental health issues using keywords such as childhood
sexual abuse and csa. We collected posts related to commonly discussed mental
health issues from r/mentalhealth (most popular health-related subreddit) [38]
using the same keywords used for collecting posts in r/adultsurviors and also
used keywords such as childhood sexual abuse and csa to make sure they are
related to CSA. We collected posts posted by normal accounts as well as
throwaway accounts. Posts from the subreddits are collected using PushShift
API. Posts from the above-mentioned subreddits are from the date these
subreddits started till January 2022. Along with the title and post, author
ID’s for each post is collected. Multiple posts may have the same author ID.
These posts are subjected to a manual inspection to ensure that they are CSA-
related. Table I shows information about posts collected through various
subreddits. “# of posts” column shows the count of posts collected from each
subreddit. We got the highest number of posts from r/adultsurvivors.

We dropped posts tagged as “[deleted]” because these posts are either deleted
or posted by users who have deleted their accounts and also posts tagged as
“[removed]” are dropped as these posts are removed by the moderator of the
subreddit or spam filter. Table I “Final # of posts” column presents the final
count of posts from each subreddit. Subreddits r/depression, r/Anxiety,
r/ptsd, r/depression_help are labelled accordingly to the mental health issue
they are related to. For subreddits r/adultsurvivors and r/mentalhealth, the
posts are labelled using rule-based methods based on the keywords (depression,
panic attack, trauma, etc) used for collection of posts. A manual checkup is
given to the posts for making sure that keywords related to a particular
mental health issue is not found in posts related to a different mental health
issue. Table I “Traits” column shows the label for posts collected from
different subreddits.

TABLE I: Posts collected from different subreddits Subreddits | # of posts | Final # of posts | # of users | Traits
---|---|---|---|---
adultsurvivors | 6619 | 6447 | 4523 | mental health issues
depression | 455 | 419 | 400 | depression
Anxiety | 18 | 16 | 13 | anxiety
ptsd | 738 | 714 | 639 | ptsd
depression_help | 37 | 37 | 37 | depression
mentalhealth | 329 | 324 | 312 | mental health issues
Figure 1: Intersections of different subreddits

After we collected posts from various subreddits, we looked at intersections
(count of common users) (Figure 1) between various mental health subreddits
and r/adultsurvivors using the author IDs of posts collected. It can be viewed
that the arc between r/adultsurvivors and r/ptsd is the most broad with 58
common users. This depicts the higher chances of CSA survivors having trauma-
related issues and it reflects as mentioned in earlier research done by
psychologists [39]. We also notice intersections between different mental
health subreddits and it can be interpreted as the possibility of co-
occurrence of multiple mental health issues simultaneously. Between mental
health subreddits, r/ptsd and r/depression with 10 common users is the
highest. The posts containing multiple mental health issues are labeled
accordingly. Figure 2 presents the distribution of posts among various labels.

Figure 2: Distribution of posts

After labeling the posts we looked at prominent words when different mental
health issues occur. Log-odds ratio [40] is used to score the terms. In
comparison to corpora that is related to different mental health issue, the
log-odds ratio aids in identifying distinctive prevalent words in a corpus of
a certain mental health issue. In the depression (D) corpus, words like
fruitlessly and shortcoming have higher scores. The anxiety (A) corpus
contains words like fogginess, which are associated with brain fog anxiety.
Words connected to memories of the traumatic experience, such as
revictimization and surfacing, are highly rated in the PTSD (P) corpus.
Prominent words when multiple mental health issues co-occur (DA, AP, DP, DAP)
are also provided. Table II and III shows the prominent words when different
mental issues occurs.

TABLE II: Prominent unigrams when different mental health issues occurs. Depression (D), Anxiety (A) and PTSD (P) D | A | P | DA | DP
---|---|---|---|---
Unigram | Score | Unigram | Score | Unigram | Score | Unigram | Score | Unigram | Score
fruitlessly | 4.75 | fogginess | 5.23 | dominance | 4.43 | paroxetine | 5.72 | entrap | 5.79
shortcoming | 4.26 | foam | 5.23 | devastation | 3.62 | diazepam | 5.72 | dreamscape | 5.79
dissatisfaction | 4.26 | hypochondriacs | 4.82 | revictimization | 3.43 | cyclothymia | 5.30 | insensitivity | 4.79
ephebophilia | 4.26 | emojibator | 4.82 | transference | 3.27 | sponge | 5.30 | dissociation | 4.79
exacerbate | 4.26 | submerge | 4.23 | catheter | 3.21 | nervous | 4.72 | leeching | 4.79
absorb | 3.94 | chauvinist | 4.23 | helpless | 3.21 | childtrigger | 4.72 | disheartening | 4.79
suicidally | 3.94 | tainting | 4.23 | cling | 2.94 | colluding | 4.72 | somnophilia | 4.79
subjugation | 3.94 | carelessness | 4.23 | tremble | 2.94 | hypogonadism | 4.72 | blisteringly | 4.79
oxycodone | 3.94 | stuntedness | 4.23 | traumatic | 2.94 | puffing | 4.72 | dysphoric | 4.79
lamotrigine | 3.94 | vaginal | 4.23 | surfacing | 2.94 | anxiousness | 4.72 | retraumatize | 4.79
TABLE III: Prominent unigrams when different mental health issues occurs. Depression (D), Anxiety (A) and PTSD (P) AP | DAP
---|---
Unigram | Score | Unigram | Score
overactivation | 5.41 | incapacity | 5.17
nauseating | 5.41 | limerence | 5.17
pistanthrophobia | 5.41 | taunting | 4.76
sensitive | 4.83 | misperception | 4.76
reanswering | 4.82 | rebound | 4.76
propranolol | 4.83 | jeopardize | 4.17
coddling | 4.83 | sodomize | 4.17
reemerge | 4.83 | sleeplessness | 4.17
hypersexualization | 4.83 | transphobia | 4.17
straining | 4.83 | hypermania | 4.17

##  IV RQ2: Are there differences between the posts related to mental health
issues with and without CSA background?

(a) with CSA

(b) without CSA

Figure 3: Examples of posts with and without CSA background

Mental health issues in users might not be necessarily due to CSA as evident
in some of the posts. For example, comparing the posts shown in Figure 3 with
and without CSA background, it is evident that they are pretty similar though
small differences may exist. Thus, to understand if posts with CSA background
have certain characteristics which are different from posts without CSA
background, we explore various NLP methods. Firstly, we need posts without CSA
background and for that we used an existing dataset [26] annotated for various
mental health issues. The dataset contains posts from various mental health
subreddits, so it may have posts with CSA background. Posts that contains
words closely related with CSA such as childhood sexual abuse and csa are
removed and finally 9136 posts related to depression, anxiety, PTSD are
extracted from the dataset as our work focuses only on these three mental
health issues. These posts are again manually inspected to ensure they are not
CSA-related.

###  IV-A Methodology

We use various NLP approaches such as word-shift, wordcloud, topic analysis,
and emotion analysis [41]. At first we utilised proportion shift, a type of
word-shift [42] graph that ranks words based on scores generated from the
difference between a word’s relative frequency in first textual content and
its relative frequency in the second textual content. For both posts with and
without CSA background, we intended to look at unique unigrams and bigrams. To
score unigrams and bigrams, TF-IDF (Term frequency-Inverse document frequency)
is used and the results are displayed as a wordcloud. Topic analysis is also
done for discovering various topics in both posts with and without CSA
background using BERTopic [43] and words inside each topic are scored using
c-TF-IDF (class-based TF-IDF). Lastly, we use fine-tuned DistilRoBERTa [44]
approach for detecting emotions for both posts with and without CSA
background. This approach labels the posts to the most probable emotion among
the emotions, namely, anger, sadness, fear, disgust, neutral, surprise, and
joy.

###  IV-B Results

As visualized in Figure 4, words that describe CSA (“abuse”, “sexual”, and
“child”) are prominent in posts with CSA background and this helps to paint a
clear picture of how CSA could be the core cause of mental health issues in
CSA survivors. In contrast to posts without CSA background, words that
represent memory or flashbacks of the traumatic event (“remember”, “happened”,
“memories”) are prominent in posts with CSA background. From these words, it
can be deduced that memories or flashbacks of the traumatic event can play a
vital role in the development of mental health issues such as PTSD (“trauma”)
in CSA survivors (Section III). In contrast to posts without CSA background,
words that demonstrate familial relationships (“dad”, “brother”, and “mom”)
stand out in posts with CSA background which may point out the likelihood of
perpetrators being family members [3]. For example, phrases such as
“..cried..over..csa..”, “..memory..being..raped..”, “..remembered..stuff..”
appeared often in the corpus with CSA background.

Words associated with the workplace (“work” and “job”) are more prominent in
posts without CSA background than in posts with CSA background, indicating
that the development of mental health difficulties may be linked to the
workplace. Words “anxiety” and “depression” are more common in posts without
CSA background, indicating that these mental health concerns are more
prevalent. For example, without CSA background, “..depression comes back..”,
“..depression suicidal thoughts..f**king feeling..”, “need..friend..”.

Figure 4: Word shift for posts with CSA background (left) and without CSA
background (right)

(a) with CSA

(b) without CSA

Figure 5: Wordcloud for top unigrams of posts with and without CSA background

(a) with CSA

(b) without CSA

Figure 6: Wordcloud for top bigrams of posts with and without CSA background TABLE IV: Top topics and its top five N-grams for posts with CSA background Topic 1 | Topic 2
---|---
N-gram | Score | N-gram | Score
abuse | 0.0183 | doctor | 0.0924
trauma | 0.0126 | exam | 0.0545
family | 0.0124 | appointment | 0.0256
sex | 0.0111 | pain | 0.0156
parent | 0.0103 | advice | 0.0119
TABLE V: Top topics and its top five N-grams for posts without CSA background Topic 1 | Topic 2
---|---
N-gram | Score | N-gram | Score
anxiety | 0.0096 | bed | 0.0319
depression | 0.0067 | anxiety | 0.0147
work | 0.0067 | week sleep | 0.0078
school | 0.0062 | trouble | 0.0069
problem | 0.0047 | time sleep | 0.0069
Topic 3 | Topic 4
N-gram | Score | N-gram | Score
calorie | 0.0207 | nausea | 0.0502
weight loss | 0.0144 | symptom | 0.0159
fat | 0.0116 | vomit | 0.0150
exercise | 0.0096 | lack appetite | 0.0083
anorexia | 0.0094 | nausea anxiety | 0.0079
Topic 3 | Topic 4
N-gram | Score | N-gram | Score
ugly | 0.0185 | weed | 0.0621
insecure | 0.0148 | cbd | 0.0571
appearance | 0.0140 | cbd oil | 0.0253
body dysmorphia | 0.0103 | anxiety depression | 0.0130
self image | 0.0097 | weed yesterday | 0.0111

In addition to word-shift, we also created wordcloud of the top ten unigrams
and bigrams. Figure 5 presents the top unique unigrams and Figure 6 presents
the top unique bigrams for posts with and without CSA background. These
results converges with those shown in Figure 4 but the method used to score
the unigrams and bigrams is different than the method used in word-shift.
Additionally, bigram such as “self harm” related to suicidal intent is more
prevalent in posts with CSA background.

Next, to understand various topics which are discussed in both with and
without CSA background posts, we perform topic analysis. Two topics are
identified using BERTopic in posts with CSA background based on coherence
score. Topic 1 is associated with CSA, such as cause (“abuse”) and its after-
effect (“trauma”). Topic 2 is concerned with clinical aspects (“doctor”,
“appointment”). Table IV displays the topics and top five N-grams for posts
with CSA background. In posts without CSA background, thirty topics are
discovered based on coherence score, but we only provide the top six topics
from the thirty topics obtained and their top five N-grams in Table V. Various
topics are discovered including mental health concerns relating to the
workplace (“anxiety”, “depression”, “work”), eating disorders (“anorexia”),
and substance use (“cbd”, “cbd oil”, “weed yesterday”).

Finally, we also examined if the emotions being present in posts with CSA
background are different from those without CSA background. Figure 7 shows the
emotions for posts with and without CSA background. Emotion “disgust” and
“fear” are associated more with posts with CSA background than posts without
CSA background as strong correlation is there between these two emotions and
trauma-related issues [45] and the presence of trauma-related issues is more
in posts with CSA background. “sadness” emotion is reported more in posts
without CSA background as “sadness” is associated with depression and it is
one of the most common mental health issue.

Figure 7: Emotions of posts related to mental health issues with and without
CSA background

##  V RQ3: Can fine-tuning domain-specific transformer model help in
identifying mental health issues in CSA survivors from the OSM posts?

For identifying mental health issues in posts related to CSA, differentiating
posts with and without CSA background is crucial as sometimes subtle
differences exist between posts with and without CSA background as reported in
Section IV. To answer it, we develop a two-stage framework that helps in
differentiating posts with and without CSA background and also identification
of mental health issues in posts with CSA background.

###  V-A Methodology

Proposed Framework: The first stage entails binary classification of mental
health issues related posts into “with CSA” and “without CSA” background.
Along with employing classical machine learning algorithms such as Naive
Bayes, Random Forest, and XGBoost, we also use transformer-based models BERT
and MentalBERT. We add fully connected layers on top of BERT and MentalBERT
[21] model’s layers and fine-tune them by retraining all the weights of the
model’s layers on our training data. We design the second stage as multi-label
classification for identifying mental health issues in posts with CSA
background as multiple mental health issues can occur simultaneously (Section
III). We use tree-based classical machine learning algorithms (XGBoost, Random
Forest), Multilayer perceptron (MLP), transformer-based models as used in the
first stage. In this stage, we fine-tune BERT and MentalBERT only on training
set of posts with CSA background. We set each label’s threshold probability as
0.5.

Data split: For the first stage, we combined both posts with CSA (7957 posts)
and without CSA (9136 posts) background. Then we performed a split of 80:20
(13674 and 3419 posts). Train and validation sets contains 12306 and 1368
posts, respectively, out of 13674 posts. Test set consists of 3419 posts. Only
posts with CSA background are considered for the second stage, with an 80:20
split (6365 and 1592 posts). The train and validation sets each have 5728 and
637 posts out of 6365 posts. There are 1592 posts in the test set.

Model Training: BERT and MentalBERT are trained on posts with and without CSA
background for 10 epochs with a batch size of 8 and a learning rate of 5e-5
for the first stage. For multi-label classification, the number of epochs and
learning rate is kept the same for training BERT and MentalBERT as the first
stage. Also, text data is converted into embedding vectors before providing as
input to classical machine learning models.

Model interpretation: When dealing with a sensitive domain like
psychopathology interpreting the model’s decision is critical. Integrated
gradients (IG) [46] method is used to interpret the model’s decision. Both the
stages of the framework are accompanied by interpretation. IG assigns
attribution values to input features, which aids in understanding why the
model makes a particular decision. Alibi [47] library is used to implement
integrated gradients.

TABLE VI: Evaluation results of models for classifying posts to “with CSA” and “without CSA” background Model | Accuracy | F1-score (macro)
---|---|---
Naive Bayes | 0.5607 | 0.3926
Random Forest | 0.6774 | 0.6753
XGBoost | 0.7452 | 0.7429
BERT | 0.9626 | 0.9624
MentalBERT | 0.9509 | 0.9508
TABLE VII: Evaluation results of models for multi-label classification of mental health issues Model | Hamming score
---|---
MLP | 0.4102
XGBoost | 0.5636
Random Forest | 0.5698
BERT | 0.5798
MentalBERT | 0.6709

###  V-B Results

In binary classification, among classical machine learning algorithms XGBoost
outperformed both Naive Bayes and Random Forest. Fine-tuned BERT, on the other
hand, surpassed all the methods, with an accuracy of 96.26% and f1-score
(macro) of 96.24%. This shows using a pretrained model and fine-tuning on our
dataset lead to improved performance. The models and their scores are shown in
Table VI. Multi-label classification models are assessed using hamming score.
With a hamming score of 67.09% fine-tuned MentalBERT surpassed other models
for multi-label classification. This demonstrates that model pre-trained on
mental health data is more effective at identification of mental health issues
in CSA survivors. Table VII illustrates the models and their corresponding
scores.

###  V-C Visualizing interpretation of model’s response

For both stages, we visualize the interpretations of the best model’s
decision. Figure 8 shows the explanations on two instances for fine-tuned
BERT. Figure 8a shows the true positive (correctly classified as “with CSA”
background) and words that are highlighted in green contributed for this
decision and pink represents those that contributed oppositely towards this
decision. Phrases such as “molested by”, “stepfather keeping”, “ptsd crying”
are highlighted in green which explains that fine-tuned BERT is able to
correctly identify phrases that denote the traumatic event, perpetrator, and
trauma related issues commonly related to posts with CSA background. Figure 8b
shows the true negative (correctly classified as “without CSA” background).
Phrases “things like never get better”, “over difficulties” are highlighted in
green which could relate to sadness but are not related to CSA. Figure 9 shows
the explanations for fine-tuned MentalBERT on a true positive instance
(correctly labeled as depression and anxiety). Phrases such as “depression and
anxiety set in” are highlighted in green which shows that fine-tuned
MentalBERT is able to capture information that is related to depression and
anxiety. The phrase “usually triggered” that pertains to anxiety triggers is
also highlighted in green.

(a) with CSA

(b) without CSA

Figure 8: Interpretation of fine-tuned BERT model’s decision on true positive
(top) and true negative (bottom) instances Figure 9: Interpretation of fine-
tuned MentalBERT model’s decision on true positive instance

##  VI Conclusion

As far as we are aware, our work is a pioneer for exploring OSM for the
analysis of mental health issues in CSA survivors as previous studies have not
paid particular attention towards gaining a better understanding of mental
health issues among CSA survivors. We discovered the commonly discussed mental
health issues in posts with CSA background and for achieving this, we created
a dataset by collecting mental health issues related posts with CSA background
from multiple subreddits. Our research also found that minor variations exist
between posts with and without CSA background. We developed a two-stage
framework for identifying mental health issues in CSA survivors, with the
first stage identifying posts with CSA background while taking into
consideration the differences between posts with and without CSA background
and followed by multi-label classification of mental health issues with posts
reported to be related with CSA in second stage. The best model in first stage
reported highest accuracy and f1-score (macro) of 96.26% and 96.24%
respectively and in second stage, the best model reported highest hamming
score of 67.09%. We have multiple plans to extend this work. In our future
work, we would like to include more subreddits for our analysis. One of the
reasons of including more subreddits is to include additional mental health
issues in our analysis.

## References

  * [1] L. K. Murray, A. Nguyen, and J. A. Cohen, “Child sexual abuse,” _Child and Adolescent Psychiatric Clinics_ , vol. 23, no. 2, pp. 321–337, 2014.
  * [2] A. Browne and D. Finkelhor, “Impact of child sexual abuse: A review of the research.” _Psychological bulletin_ , vol. 99, no. 1, p. 66, 1986.
  * [3] D. Finkelhor, “Current information on the scope and nature of child sexual abuse,” _Future Child._ , vol. 4, no. 2, pp. 31–53, 1994.
  * [4] P. Coffey, H. Leitenberg, K. Henning, T. Turner, and R. T. Bennett, “Mediators of the long-term impact of child sexual abuse: Perceived stigma, betrayal, powerlessness, and self-blame,” _Child Abuse & Neglect_, vol. 20, no. 5, pp. 447–455, 1996.
  * [5] A. C. Kennedy and K. A. Prock, ““i still feel like i am not normal”: A review of the role of stigma and stigmatization among female survivors of child sexual abuse, sexual assault, and intimate partner violence,” _Trauma, Violence, & Abuse_, vol. 19, no. 5, pp. 512–527, 2018.
  * [6] J. McCann, J. Voris, and M. Simon, “Genital injuries resulting from sexual abuse: a longitudinal study,” _Pediatrics_ , vol. 89, no. 2, pp. 307–317, 1992.
  * [7] C. F. Johnson, “Child sexual abuse,” _The Lancet_ , vol. 364, no. 9432, pp. 462–470, 2004.
  * [8] Y. Kamiya, V. Timonen, and R. A. Kenny, “The impact of childhood sexual abuse on the mental and physical health, and healthcare utilization of older adults,” _International psychogeriatrics_ , vol. 28, no. 3, pp. 415–422, 2016.
  * [9] T. Gustafson and D. Sarwer, “Childhood sexual abuse and obesity,” _Obesity reviews_ , vol. 5, no. 3, pp. 129–135, 2004.
  * [10] E. F. Loftus, S. Polonsky, and M. T. Fullilove, “Memories of childhood sexual abuse: Remembering and repressing,” _Psychology of women quarterly_ , vol. 18, no. 1, pp. 67–84, 1994.
  * [11] J. Briere and M. Runtz, “Post sexual abuse trauma: Data and implications for clinical practice,” _Journal of interpersonal violence_ , vol. 2, no. 4, pp. 367–379, 1987.
  * [12] S. A. Wonderlich, T. D. Brewerton, Z. Jocic, B. S. Dansky, and D. W. Abbott, “Relationship of childhood sexual abuse and eating disorders,” _Journal of the American Academy of Child & Adolescent Psychiatry_, vol. 36, no. 8, pp. 1107–1115, 1997.
  * [13] L. F. de Aquino Ferreira, F. H. Q. Pereira, A. M. L. N. Benevides, and M. C. A. Melo, “Borderline personality disorder and sexual abuse: a systematic review,” _Psychiatry research_ , vol. 262, pp. 70–77, 2018.
  * [14] V. L. Banyard, L. M. Williams, and J. A. Siegel, “The long-term mental health consequences of child sexual abuse: An exploratory study of the impact of multiple traumas in a sample of women,” _Journal of traumatic stress_ , vol. 14, no. 4, pp. 697–715, 2001.
  * [15] N. Werbeloff, J. Hilge Thygesen, J. F. Hayes, E. M. Viding, S. Johnson, and D. P. Osborn, “Childhood sexual abuse in patients with severe mental illness: Demographic, clinical and functional correlates,” _Acta Psychiatrica Scandinavica_ , vol. 143, no. 6, pp. 495–502, 2021.
  * [16] S. Chancellor and M. De Choudhury, “Methods in predictive techniques for mental health status on social media: a critical review,” _NPJ digital medicine_ , vol. 3, no. 1, pp. 1–11, 2020.
  * [17] S. Bucci, M. Schwannauer, and N. Berry, “The digital revolution and its impact on mental health care,” _Psychology and Psychotherapy: Theory, Research and Practice_ , vol. 92, no. 2, pp. 277–297, 2019.
  * [18] B. S. Fraga, A. P. C. da Silva, and F. Murai, “Online social networks in health care: a study of mental disorders on reddit,” in _2018 IEEE/WIC/ACM International Conference on Web Intelligence (WI)_. IEEE, 2018, pp. 568–573.
  * [19] M. De Choudhury and S. De, “Mental health discourse on reddit: Self-disclosure, social support, and anonymity,” in _Eighth international AAAI conference on weblogs and social media_ , 2014.
  * [20] J. Devlin, M.-W. Chang, K. Lee, and K. Toutanova, “Bert: Pre-training of deep bidirectional transformers for language understanding,” _arXiv preprint arXiv:1810.04805_ , 2018.
  * [21] S. Ji, T. Zhang, L. Ansari, J. Fu, P. Tiwari, and E. Cambria, “Mentalbert: Publicly available pretrained language models for mental healthcare,” _arXiv preprint arXiv:2110.15621_ , 2021.
  * [22] E. C. Nelson, A. C. Heath, P. A. Madden, M. L. Cooper, S. H. Dinwiddie, K. K. Bucholz, A. Glowinski, T. McLaughlin, M. P. Dunne, D. J. Statham _et al._ , “Association between self-reported childhood sexual abuse and adverse psychosocial outcomes: results from a twin study,” _Archives of general psychiatry_ , vol. 59, no. 2, pp. 139–145, 2002.
  * [23] P. E. Mullen, J. L. Martin, J. C. Anderson, S. E. Romans, and G. P. Herbison, “Childhood sexual abuse and mental health in adult life,” _The British Journal of Psychiatry_ , vol. 163, no. 6, pp. 721–732, 1993.
  * [24] J. Spataro, P. E. Mullen, P. M. Burgess, D. L. Wells, and S. A. Moss, “Impact of child sexual abuse on mental health: prospective study in males and females,” _The British Journal of Psychiatry_ , vol. 184, no. 5, pp. 416–421, 2004.
  * [25] S. D. Easton, L. M. Renner, and P. O’Leary, “Suicide attempts among men with histories of child sexual abuse: Examining abuse severity, mental health, and masculine norms,” _Child Abuse & Neglect_, vol. 37, no. 6, pp. 380–387, 2013.
  * [26] J. Kim, J. Lee, E. Park, and J. Han, “A deep learning model for detecting mental illness from user content on social media,” _Scientific Reports_ , vol. 10, no. 1, p. 11846, Jul 2020. [Online]. Available: https://doi.org/10.1038/s41598-020-68764-y
  * [27] V. Vajre, M. Naylor, U. Kamath, and A. Shehu, “Psychbert: A mental health language model for social media mental health behavioral analysis,” in _2021 IEEE International Conference on Bioinformatics and Biomedicine (BIBM)_. IEEE, 2021, pp. 1077–1082.
  * [28] M. M. Tadesse, H. Lin, B. Xu, and L. Yang, “Detection of depression-related posts in reddit social media forum,” _IEEE Access_ , vol. 7, pp. 44 883–44 893, 2019.
  * [29] C. Y. Chiu, H. Y. Lane, J. L. Koh, and A. L. Chen, “Multimodal depression detection on instagram considering time interval of posts,” _Journal of Intelligent Information Systems_ , vol. 56, no. 1, pp. 25–47, 2021.
  * [30] M. Mitchell, K. Hollingshead, and G. Coppersmith, “Quantifying the language of schizophrenia in social media,” in _Proceedings of the 2nd workshop on Computational linguistics and clinical psychology: From linguistic signal to clinical reality_ , 2015, pp. 11–20.
  * [31] J. A. Benítez-Andrades, J. M. Alija-Pérez, I. García-Rodríguez, C. Benavides, H. Alaiz-Moretón, R. P. Vargas, and M. T. García-Ordás, “Bert model-based approach for detecting categories of tweets in the field of eating disorders (ed),” in _2021 IEEE 34th International Symposium on Computer-Based Medical Systems (CBMS)_. IEEE, 2021, pp. 586–590.
  * [32] Y. Ophir, R. Tikochinski, C. S. Asterhan, I. Sisso, and R. Reichart, “Deep neural networks detect suicide risk from textual facebook posts,” _Scientific reports_ , vol. 10, no. 1, pp. 1–10, 2020.
  * [33] A. Murarka, B. Radhakrishnan, and S. Ravichandran, “Detection and classification of mental illnesses on social media using roberta,” _arXiv preprint arXiv:2011.11226_ , 2020.
  * [34] G. Gkotsis, A. Oellrich, S. Velupillai, M. Liakata, T. J. Hubbard, R. J. Dobson, and R. Dutta, “Characterisation of mental health conditions in social media using informed deep learning,” _Scientific reports_ , vol. 7, no. 1, pp. 1–11, 2017.
  * [35] D. M. Low, L. Rumker, T. Talkar, J. Torous, G. Cecchi, and S. S. Ghosh, “Natural language processing reveals vulnerable mental health support groups and heightened health anxiety on reddit during covid-19: Observational study,” _Journal of medical Internet research_ , vol. 22, no. 10, p. e22635, 2020.
  * [36] A. M. Stanton, C. M. Meston, and R. L. Boyd, “Sexual self-schemas in the real world: Investigating the ecological validity of language-based markers of childhood sexual abuse,” _Cyberpsychol. Behav. Soc. Netw._ , vol. 20, no. 6, pp. 382–388, Jun. 2017.
  * [37] N. TeBlunthuis and B. M. Hill, “Identifying competition and mutualism between online groups,” in _Proceedings of the International AAAI Conference on Web and Social Media_ , vol. 16, 2022, pp. 993–1004.
  * [38] M. Gaur, U. Kursuncu, A. Alambo, A. Sheth, R. Daniulaityte, K. Thirunarayan, and J. Pathak, “” let me tell you about your mental health!” contextualized classification of reddit posts to dsm-5 for web-based intervention,” in _Proceedings of the 27th ACM International Conference on Information and Knowledge Management_ , 2018, pp. 753–762.
  * [39] J. E. Aspelmeier, A. N. Elliott, and C. H. Smith, “Childhood sexual abuse, attachment, and trauma symptoms in college females: The moderating role of attachment,” _Child abuse & neglect_, vol. 31, no. 5, pp. 549–566, 2007\.
  * [40] B. Dickinson, “Generate Meaningful Word Clouds in Python — towardsdatascience.com,” https://towardsdatascience.com/generate-meaningful-word-clouds-in-python-5b85f5668eeb, [Accessed 12-Oct-2022].
  * [41] K. Neha, V. Agrawal, V. Kumar, T. Mohan, A. Chopra, A. B. Buduru, R. Sharma, and P. Kumaraguru, “A tale of two sides: Study of protesters and counter-protesters on# citizenshipamendmentact campaign on twitter,” in _14th ACM Web Science Conference 2022_ , 2022, pp. 279–289.
  * [42] R. J. Gallagher, M. R. Frank, L. Mitchell, A. J. Schwartz, A. J. Reagan, C. M. Danforth, and P. S. Dodds, “Generalized word shift graphs: a method for visualizing and explaining pairwise comparisons between texts,” _EPJ Data Science_ , vol. 10, no. 1, p. 4, Jan 2021. [Online]. Available: https://doi.org/10.1140/epjds/s13688-021-00260-3
  * [43] M. Grootendorst, “Bertopic: Neural topic modeling with a class-based tf-idf procedure,” _arXiv preprint arXiv:2203.05794_ , 2022.
  * [44] J. Hartmann, “Emotion english distilroberta-base,” https://huggingface.co/j-hartmann/emotion-english-distilroberta-base/, 2022\.
  * [45] E. Coyle, T. Karatzias, A. Summers, and M. Power, “Emotions and emotion regulation in survivors of childhood sexual abuse: the importance of “disgust” in traumatic stress and psychopathology,” _European journal of psychotraumatology_ , vol. 5, no. 1, p. 23306, 2014.
  * [46] M. Sundararajan, A. Taly, and Q. Yan, “Axiomatic attribution for deep networks,” in _International conference on machine learning_. PMLR, 2017, pp. 3319–3328.
  * [47] J. Klaise, A. V. Looveren, G. Vacanti, and A. Coca, “Alibi explain: Algorithms for explaining machine learning models,” _Journal of Machine Learning Research_ , vol. 22, no. 181, pp. 1–7, 2021. [Online]. Available: http://jmlr.org/papers/v22/21-0017.html


# Decay of baryon inhomogeneities in an expanding universe

Pratik K. Das, Sovan Sau, Abhisek Saha and Soma Sanyal  University of
Hyderabad, Prof. C.R. Rao Road, Hyderabad 500046, India

###### Abstract

Baryon inhomogeneities can be generated very early in the universe. These
inhomogeneities then decay by particle diffusion in an expanding universe. We
study the decay of these baryon inhomogeneities in the early universe using
the diffusion equation in the Friedmann–Lemaître–Robertson–Walker (FLRW)
metric. We have studied the decay starting from the electroweak phase
transition. We calculate the interaction cross section of the quarks with the
neutrinos, the electrons and the muons and obtain the diffusion coefficients.
The diffusion coefficients are temperature dependent. We find that the
expansion of the universe causes the inhomogeneities to decay at a faster
rate. We find that the baryon inhomogeneities generated at the electroweak
epoch have very low amplitudes at the time of the quark hadron phase
transition. So unless inhomogeneities are generated with a very high amplitude
(greater than $10^{5}$ times the background density), they will have no effect
on the quark hadron phase transition. After the quark hadron phase transition,
we include the interaction of the muons with the neutrons and the protons till
100 MeV. We also find that large density inhomogeneities generated during the
quark hadron transition with sizes of the order of 1 km must have amplitudes
greater than $10^{5}$ times the background density to survive upto the
nucleosynthesis epoch in an expanding universe.

Inhomogeneities, diffusion, Expanding universe

##  I Introduction

Primordial cosmological fluctuations are an important part of modern cosmology
as they link the current Cosmic Microwave Background Radiation (CMBR) data to
the early universe. Initially the theory related to fluctuations was developed
by Lifshitz Lifshitz . Later on significant work has been done by Hawkings
Hawkings  and Bardeen Bardeen . In this work, we are interested in the decay
of baryon density fluctuations or inhomogeneities generated in the early
universe. Baryon density fluctuations can be generated in the electroweak
phase transition heckler ; megevand ; sanchez  as well as the QCD phase
transition fuller ; layek1 . There are various defect mediated mechanisms
which generate these inhomogeneities at the electroweak scales electroweak .
Though baryon inhomogeneities have other roles to play in the evolution of the
early universe, the primary method of constraining these inhomogeneities is by
studying their effects on the Big Bang Nucleosynthesis calculations. The
inhomogeneities generated at the QCD scales have a higher chance of surviving
to the nucleosynthesis epoch, hence they have been studied more extensively.
The baryon inhomogeneities generated in the electroweak epoch, have less
chance of surviving till the nucleosynthesis epoch, hence they are often
ignored. However it has been shown that baryon over densities generated in the
electroweak epoch can definitely affect the quark hadron phase transition
sanyal  and may also survive till the nucleosynthesis epoch jedamzik . One of
the effect of these baryon over-densities is to delay the quark hadron phase
transition in regions of the inhomogeneities. This, along with the fact that
large scale inhomogeneities from the electroweak epoch may survive till the
nucleosynthesis epoch and affect the abundances of the light elements makes it
important to study the diffusion of particles in an inhomogeneity at the
electroweak epoch.

The only detailed study of the evolution of non-linear sub horizon entropy
fluctuations between $100$ GeV and $1$ MeV was done in ref. jedamzik . The
entropy fluctuations that had been considered in that particular study did not
necessarily come from baryon inhomogeneities only, the authors were not
interested in the source of the entropy fluctuations. They had assumed certain
amplitudes and length scales and evolved them with time. The evolution of the
inhomogeneity was performed as a succession of pressure equilibrium states,
that dissipate or expand due to neutrino heat transport. The neutrino
contribution to the heat transport equation was used and after the quark
hadron phase transition the neutron and proton diffusion were taken into
account. They concluded that large scale entropy fluctuations at $100$ GeV
could survive until the nucleosynthesis epoch and affect the nucleosynthesis
calculations. The main difference between their study and our current study is
that we look at baryon inhomogeneities generated at the electroweak scale in
the diffusive limit taking into consideration both the quark and the lepton
diffusion. Studies have already shown that baryon inhomogeneities can be
generated in the electroweak scale. Generally, they have an amplitude of
around $10^{4}$ times the background baryon density. Since, the baryon number
is carried by the quarks at these high temperatures, the baryon
inhomogeneities at these scales will primarily mean an excess of the number
density of quarks in a certain region. These quarks diffuse out by colliding
with the leptons, which are the electrons, muons and the neutrinos. In this
study we focus on obtaining the diffusion coefficient of these quarks as they
move through the plasma. The diffusion coefficient is then used in the
particle diffusion equation to study the decay of baryon inhomogeneities in an
expanding universe.

The diffusion equation has been studied in the early universe mostly in the
context of cosmic rays cosmicrays . For the case of baryon diffusion it has
been studied mostly in the hadronic phase sasaki ; scherrer ; sau  when the
baryon number is carried by the neutrons and the protons. Here, for the first
time, we use it to study, baryon diffusion in the quark gluon plasma phase.
Since we are using the diffusion equation pertaining to an expanding universe,
we also extend our study to the QCD scales. We do find that the expansion term
makes a significant difference in the decay of the inhomogeneities. We find
that in a static universe the decay rate is slow, while in an expanding
universe, the inhomogeneities decay much faster. In the electroweak case, the
particles that we consider are the quarks, muon, electrons and the neutrinos.
For the QCD epoch, we have the hadrons which carry the baryon number.
Generally, the inhomogeneities which are generated at the QCD epoch have
higher amplitude and sizes than those generated in the electroweak epoch
layek2 . Hence we consider these two cases separately.

In this work, we only look at sub horizon fluctuations. We also assume that
the size of the fluctuations are larger than the mean free path of the
relevant particles. The diffusion equation, is re-written in the
Friedmann–Lemaître–Robertson–Walker (FLRW) metric. We neglect baryon number
violating processes at the temperatures around $200$ GeV and assume that the
total baryon number density is conserved throughout our calculations. All the
baryon density fluctuations we consider are assumed to be Gaussian
fluctuations.

The chief motivation of understanding the decay of the baryon inhomogeneities
is the different signatures of Inhomogeneous Big Bang Nucleosynthesis (IBBN)
that have been predicted arbey . An inhomogeneous BBN will result from patches
of region where there is a baryon inhomogeneity applegate . However, all the
study so far has concentrated only on inhomogeneities generated in the quark
hadron phase transition. The only study for entropy fluctuations generated in
the electroweak epoch has indicated that there will be no significant decay
during the period prior to the quark hadron phase transition. We were
motivated to see whether this result will hold for baryon inhomogeneities
also. We however, find that this result does not hold for baryon
inhomogeneities. The inhomogeneities decay rapidly during this period and they
decrease by about three orders of magnitude. Since some of the inhomogeneities
generated in the electroweak epoch have an amplitude of only $10^{4}$, these
will decay and have no effect on the quark hadron phase transition or the
nucleosynthesis epoch. We also study the decay of the inhomogeneities after
the quark hadron phase transition. Though these will undergo significant
decay, some of them might survive till the nucleosynthesis epoch. The BBN can
then be used to constrain the models that generate these fluctuations
constraints .

In section II, we discuss the amplitude and size of the baryon inhomogeneities
that are of interest to us. In section III we discuss the diffusion equation
in the FLRW metric, in section IV, we obtain the diffusion coefficient in the
quark gluon plasma phase. Since we take into account the scattering of the
quarks with electrons, muons and neutrinos, we have divided this section into
three subsections. In section V we present the numerical results of the decay
of the baryon overdensities between the temperature $200$ GeV and $200$ MeV.
This is the period after the electroweak phase transition and before the quark
hadron transition. In section VI, we discuss the decay of the inhomogeneities
after the quark hadron transition. Inhomogeneities which survive upto the
nucleosynthesis period will affect the light element abundances. In section
VII, we summarize our work and present our conclusions.

##  II Baryon inhomogeneities in the early universe

There are several ways in which baryon over densities can be generated in the
early universe. Topological defects such as electroweak strings are unstable
and generate baryon number when they decay Dziarmaga ; barriola . These give
rise to local baryon density fluctuations. The scale of these fluctuations
will be given approximately by the bubble nucleation distance at the
electroweak scale. Inhomogeneities may thus be generated by these strings over
small length scales ($~{}10^{-10}$ cms). These inhomogeneities have a large
amplitude and consequently diffuse out. Other than the electroweak strings,
superconducting strings are also capable of generating baryon inhomogeneities
over small lengthscales brandenburger . The most detailed study of baryon
inhomogeneity generation, their amplitude and size was done in ref. megevand .
The lengthscales of the inhomogeneities generated were about $10^{-3}$ cm and
they had an amplitude of $10^{4}$ over the background baryon density. We will
not be considering any specific model during this epoch, our focus would be to
see if these baryon inhomogeneities can survive at least upto the QCD epoch.
We have taken Gaussian fluctuations to represent the baryon inhomogeneities.
These are characterised by the amplitude ($A$) and the full width at half
maxima ($\sigma$). We have kept the $A\sim 10^{3}$ as it is expected that the
baryon inhomogeneities at the electroweak scale are not too high. The $\sigma$
is varied over the horizon range.

For the inhomogeneities at the QCD scale we have taken a higher amplitude.
This is because there is a greater probability of generating large over
densities at the QCD scale. Inhomogeneities can be generated by moving cosmic
strings layek2 , collapsing Z(3) domain walls atreya  and inhomogeneous
nucleation of bubbles in a first order phase transition madsen .
Inhomogeneities can be spread over an area of one meter and can have
amplitudes of the order of $10^{12}-10^{13}$ through these various mechanisms.
Larger inhomogeneities may also be generated but they will have lower
amplitudes. Again we do not focus on the mechanism that will lead to the
generation of these baryon inhomogeneities, we only study the inhomogeneities
over different ranges for an approximate amplitude of $10^{4}$ times the
background baryon density. Diffusion and decay of baryon inhomogeneities in
the QCD epoch has been studied before. In a previous work sau  we have worked
out the decay of baryon inhomogeneities using the diffusion equation but
ignoring the expansion of the universe. In this work, we redo the
multiparticle diffusion that we had done previously in sau  for the case of an
expanding universe. We find that the expanding universe has a significant
contribution to the decay of the large scale inhomogeneities. Whereas in the
previous work we had considered inhomogeneities with $\sigma$ values much
smaller than the horizon size, in this work we consider large scale baryon
inhomogeneities and see how they decay in the expanding universe.

##  III The diffusion equation in the FLRW metric

The diffusion equation in an expanding universe has been obtained in the
context of cosmic ray propagation cosmicrays . We briefly describe it here in
terms of baryon diffusion in the early universe. The FLRW metric for the flat
universe is defined by,

$ds^{2}=c^{2}dt^{2}-a^{2}(t)d\vec{r}^{2}$ (1)

Here $a(t)$ is the scale factor of the expanding universe and $\vec{r}$ is the
spatial coordinate. This is the comoving distance in an expanding universe.
Consider a region of the universe with an inhomogeneity given by
$n(\vec{r},t)$. As time evolves, the particles in the over dense region tend
to move towards the lesser dense region to restore equilibrium and a particle
flux is generated. In this case, we consider the diffusion to be isotropic.
The local observer then sees the particle flux as,

$j_{k}=-D(t)\frac{\partial}{\partial x^{k}}n(\vec{r},t)$ (2)

The diffusion coefficient $D(t)$ depends on the scattering cross section and
the velocity of the particles. Since there are different kinds of particles in
this plasma, we are dealing with multi-particle diffusion here. The
conservation of current gives us,

$\frac{\partial}{\partial x^{\mu}}(\sqrt{g}j^{\mu})=0$ (3)

Here $\sqrt{g}=a^{3}(t)$ and using the definition of the Hubble parameter as
$H(t)=\frac{\dot{a}(t)}{a(t)}$, the diffusion equation can be written as,

$\frac{\partial}{\partial
t}n(\vec{r},t)+3H(t)n(\vec{r},t)-\frac{D(t)}{a^{2}}\nabla^{2}n(\vec{r},t)=0$
(4)

This is the diffusion equation that we will solve numerically for a time
dependent diffusion coefficient. As mentioned before, the diffusion
coefficient depends on the scattering cross section of the particles. Hence it
is different at different temperatures. Since the scattering cross sections
are obtained in terms of temperatures, we use the time - temperature relation
in the radiation dominated universe to convert our time to temperature.

$t=\frac{(0.95\times 10^{10})^{2}}{T^{2}}$ (5)

Here $t$ is in secs and $T$ is in Kelvin. We then solve the diffusion equation
in the FLRW metric numerically over the entire range of temperature from $200$
GeV - $200$ MeV. In the next section, we first present the details of
calculating the diffusion coefficient in the quark phase.

##  IV Diffusion in the electroweak scale

We start by studying particle diffusion in the electroweak scale. The
inhomogeneities are formed at $200$ GeV during the electroweak phase
transition. So the diffusion of particle will start around the same time.
During this epoch the most abundant particles are the quarks, electrons, muons
and the neutrinos. Out of all these, it is the quarks which carry the baryon
number. So the baryon over-densities would predominantly have a larger density
of quarks as compared to the background number density. As the quarks diffuse
out of the inhomogeneities trying to reach an equilibrium state they will
collide with the electrons, muons and the neutrinos. Here we take two cases
depending on the mass of the particles. This is because the quarks are lighter
than the muons but heavier than the electrons. So as the quarks move through
the muons, we have a lighter gas diffusing into a gas of heavier particles but
as the quarks diffuse through the electrons and neutrinos, we have a heavier
particle diffusing through a lighter gaslandaufm . Since we are not going into
the detailed transport equation of the particles, we choose a distribution
function for the particles of the light gas.

In the first case, for the quarks moving through the muons, the diffusion
coefficient is given by,

$D=\frac{1}{3N}\left\langle\frac{v}{\sigma_{t}}\right\rangle=\Bigg{(}\frac{T}{\pi
m}\Bigg{)}^{1/2}\frac{2^{3/2}}{3\sigma_{t}}$ (6)

In the second case, for a heavier particle moving through a lighter fluid, to
obtain the diffusion coefficient, we have to first compute the mobility of the
particle in the background fluid. If the velocity of the particle is
$\vec{v}$, then the mobility $b$ is related to the external force ($\vec{f}$)
by, $\vec{v}=b\vec{f}$ and the diffusion coefficient is given by,

$D=bT$ (7)

We assume the distribution of particles to be Maxwellian, then the mobility of
the particles is given by,

$b^{-1}=\frac{16\pi}{T}\int\frac{p^{2}dp}{3h}vp^{2}\sigma_{t}e^{-E/T}=\frac{16{\sigma_{t}}m^{2}t^{2}}{3{\pi}^{2}}.$
(8)

Here $\sigma_{t}$ is the scattering cross-section, $m$ is the mass of the
particle. Once the scattering cross section is known, substituting it in the
expression for $b$ would enable us to obtain the diffusion coefficient $D$.
Since the scattering cross-sections are temperature dependent, the diffusion
coefficient too would be temperature dependent. To find the diffusion
coefficient at these temperatures we therefore, obtain the scattering cross
section of the quarks with the leptons. In the next subsections, we will
calculate the different scattering cross section for the different
interactions.

###  IV.1 Quark-electron scattering

We start with the motion of quarks through the electron gas. For this we need
to find the scattering cross section for the $e^{-}e^{+}\longrightarrow
q\bar{q}$ interaction. The differential cross section is given by,

$\frac{d\sigma}{d\Omega}=\frac{Q_{f}^{2}\alpha^{2}}{2s}\left(\frac{u^{2}+t^{2}}{s^{2}}\right)$
(9)

Here $\alpha\sim 10^{-2}$ is the fine structure constant and $Q_{f}$ is the
momentum transfer in this interaction. The variables $u$, $t$ and $s$ are the
Mandelstam variables. This gives,

$\sigma_{t}=\frac{Q_{f}^{2}\alpha^{2}}{2s}\int\left(\frac{u^{2}+t^{2}}{s^{2}}\right)(1-cos\theta)d\Omega$
(10)

The total scattering cross section can be obtained after integrating over the
solid angle. The numerical value can be obtained once the energy scale of the
colliding particles is known. Since we are working around the electroweak
scale, the colliding energy of the particles are also in the GeV range. The
mobility factor is thus given by,

$b^{-1}=\frac{2\sigma_{t}m^{2}}{3\pi^{2}}[8T^{2}(1-e^{-E/T})-2E(2E+4T)e^{-E/T}]$
(11)

The diffusion coefficient can be calculated numerically after obtaining the
mobility at various temperatures.

###  IV.2 Quark-neutrino scattering

Neutrinos do not have any charges, they have weak interactions. Though there
are different flavors of quarks as well as neutrinos, since we only need order
of magnitude estimations, we just consider,

$\sigma_{t}=\frac{G_{F}^{2}\hat{s}}{\pi}$ (12)

Here $G_{F}$ is the Fermi constant given by, $G_{F}=1.166\times
10^{-5}GeV^{-2}$. Numerically, the cross section turns out to be
$\sigma_{t}=17.2\times 10^{-42}cm^{2}\times\frac{E_{\nu}}{GeV}$ farland .
Though we are working at very high temperatures in the GeV scale, the value of
the diffusion coefficient is difficult to handle numerically with this value
of $\sigma_{t}$. For the numerical calculation we therefore rescale the
variables suitably to obtain a stable numerical solution.

###  IV.3 Quark-muon scattering

In both the previous cases we had a heavier particle moving through a lighter
gas of particles, however the scenario changes considerably when we consider
the quarks moving through a gas of muons. For the
$\mu^{-}\mu^{+}\longrightarrow q\bar{q}$, though the expression for the
interaction cross sections are similar to the electrons, but here the quark is
the lighter particle which is moving through a heavier gas of particles (the
muons). This means that, the diffusion coefficient is given by eqn. 6,

$D=\left(\frac{1}{2\pi}\right)^{\frac{3}{2}}\left(\frac{s^{\frac{1}{2}}}{Q_{f}\alpha}\right)^{2}\left(\frac{1}{mT}\right)^{\frac{1}{2}}[2T(1-e^{-E/T})-2Ee^{-E/T}]$
(13)

We have the total cross section given by,

$\sigma_{t}=\frac{4\pi Q_{f}^{2}\alpha^{2}}{3s}$ (14)

In all these cases, we determine the diffusion coefficient at different
temperatures numerically. Similar to the quark neutrino cross section, even at
such high temperatures, the diffusion coefficient are numerically very large
quantities hence for all the different cases we need to do some scaling to
obtain numerically stable solutions. This we have done by scaling the energy
appropriately so that the value of the diffusion coefficient is of reasonable
orders of magnitude. While plotting we have plotted only the amplitude at the
different length scales depending on the temperature. The amplitude is
dimensionless as it is the ratio of the enhanced density to the background
density ($\frac{\Delta n_{B}}{n_{B}}$). Thus the rescaling does not affect the
change in the overdensity that we are interested in. Only, the length scales
are changed appropriately to reflect the decrease in the temperature.

##  V Decay of inhomogeneities in the quark gluon plasma phase

We now look at baryon inhomogeneities generated during the electroweak phase
transition. Since the maximum amplitude of these inhomogeneities is of the
order of $10^{4}$ times the background density, for our simulations we take a
low amplitude Gaussian fluctuation. The amplitude is taken to be of the order
of $10^{3}$. We consider large inhomogeneities whose decay will be affected by
the expanding universe. The horizon at these temperatures is of the order of
$10mm$ schwarz . Though the diffusion coefficients can be obtained
numerically, the problem is that they vary considerably in their numerical
values. This indicates that the particle content in the inhomogeneity would
ultimate define how they decay. Though there are multi particles present in
the plasma, we do not go for multi particle diffusion as it becomes
numerically quite challenging. We look at each of these interactions
separately and see how much each of them contributes to the decay of the
baryon inhomogeneity. We believe this will give us some idea of how the baryon
inhomogeneity decays in this temperature range.

The other challenge in studying the diffusion of particles over such a large
temperature range, is the fact that the horizon will also change considerably.
From $200$ GeV where the horizon is of the order of $1$ cm to $200$ MeV where
the horizon is of the order of $10$ kms. We divide it into two parts. We
evolve the inhomogeneity from $200$ GeV to about $1$ GeV and then again from
$1$ GeV to $200$ MeV. Interestingly, we find that the inhomogeneity decays
considerably during this period depending on the particle interactions being
considered. However, we find that on the log scale, the amplitude decays quite
rapidly irrespective of the interactions considered. This means that low
amplitude inhomogeneities will be completely wiped out before the quark hadron
phase transition.

Let us first look at the decay of the inhomogeneities between $200$ GeV and
$1$ GeV. Here the diffusion coefficients differ considerably as they are
dependent on the temperature and this is quite a large temperature range. So
we look at the decay of the inhomogeneities for different interactions
separately. For all the different figures, we have the baryon inhomogeneity
($\frac{\Delta n_{B}}{n_{B}}$), on the y axis and the length scale on the x-
axis. In figure 1, we see the decay due to the quarks moving through the
electrons. The initial fluctuation is taken at $200$ GeV and the final is
taken at $17$ GeV.

Figure 1: The decay of the fluctuation is shown in logscale between $200$ GeV
- $17$ GeV as the quarks moves through a sea of electrons.  Figure 2: The
decay of the fluctuation is shown in logscale between $200$ GeV - $17$ GeV as
the quarks are predominantly surrounded by neutrinos.

As we see the peak of the inhomogeneities goes down by more than three orders
of magnitude. The inhomogeneity also spreads out. The decay is similar in the
case when the surrounding particles are muons and neutrinos. We have used a
different scaling for the neutrinos but as mentioned before the scaling will
not affect the relative decay of the amplitude. Though the final amplitude is
lower in the case of the neutrinos, the order of magnitude is similar. Since
all the graphs have similar decay in orders of magnitude, we have only shown
selected graphs. Fig 1, shows the decay due to the motion of the quarks
through the electrons while fig 2 shows the decay due to the motion of the
quarks through the neutrinos. Since the plasma at those high temperatures is
predominantly dominated by electrons, our results clearly show that the baryon
inhomogeneities decay by about three orders of magnitude in the high
temperature GeV range. This is true, even if there are a large number of muons
and neutrinos in the plasma.

We now look at the decay of the inhomogeneities between the temperatures $1$
GeV to $200$ MeV. We find that the order of magnitude decay is again quite
large for the three diffusion coefficients. The individual numbers vary but we
plot only the order of magnitude estimates as before. In figure 3, we find
that the inhomogeneity has decreased by three orders of magnitude. In the case
of fig. 3, the initial fluctuation is at $1$ GeV while the final is plotted at
$236$ MeV.

Figure 3: The decay of the fluctuation is shown in log scale between $1$ GeV -
$236$ MeV for a plasma where a quark is moving through a sea of electrons.

For the case of the quarks moving through a large number of muons, the
amplitude decay is less than the decay in the case when the particles
surrounding the quarks are the electrons. However, the decay is still quite
significant. For the neutrinos, again we have the inhomogeneity decaying by
three orders of magnitude. So independent of the particle distribution in the
plasma, the amplitude of the inhomogeneities goes down significantly. In fig
4, we have the quarks moving in a region of muons while in fig 5, the quarks
move through the neutrinos.

Figure 4: The decay of the fluctuation is shown in log scale between $1$ GeV -
$236$ MeV for a plasma where a quark is moving through muons.  Figure 5: The
decay of the fluctuation is shown in log scale between $1$ GeV - $236$ MeV for
a plasma where a quark is moving through neutrinos.

We thus find that the inhomogeneity decays by three orders of magnitude
between $200$ GeV to $1$ GeV, and again decays by at least two orders of
magnitude between $1$ GeV to $200$ MeV. This means that any inhomogeneity
generated at the electroweak epoch needs to have an amplitude greater than
$10^{5}$ times the background density to survive till the quark hadron
transition. Thus if we had an inhomogeneity at the electroweak scale with an
amplitude less than $10^{5}$, it would be completely wiped out before the
quark hadron transition.

##  VI Decay of inhomogeneities in the hadronic phase

We have studied the decay of inhomogeneities in the hadronic phase previously
sau , however in that case we were interested in specific inhomogeneities
generated by Z(3) domain walls whose size was much smaller compared to the
horizon size. The expansion of the universe was ignored in those cases. We are
now interested to see the decay of larger inhomogeneities in the hadronic
phase. As mentioned before, inhomogeneities generated during or after the
quark hadron phase transition are not only larger in amplitude but they may
also be larger in size. Consequently, the decay of these inhomogeneities would
be affected by the expanding universe. The plasma during this period consists
of the muons, neutrons, protons. electrons and neutrinos. In our previous
work, we had shown that the presence of muons enhances the diffusion
coefficient of the neutrons/protons, however we had not factored in the
expansion of the universe in the previous work. This time we use the diffusion
equation for the expanding universe and find that the expansion of the
universe causes the inhomogeneities to decay much faster. In the current
section, we briefly describe the diffusion coefficient in the hadronic phase
and then proceed to present the results of the decay of the inhomogeneities in
the hadronic plasma.

As we have mentioned, the calculation of the diffusion coefficient will depend
on which particle is moving through the plasma. The baryon number is carried
by the neutrons and the protons, hence here we will be considering the motion
of a heavier particle through a lighter gas. The heavier particle is the
neutron or the proton, while the lighter gas is a gas of electrons and
neutrinos. The muons only play a role till 100 MeV. We thus have to use eqn.7
and the scattering cross-section of the neutrons with the electron-positron
gas to obtain the diffusion coefficient of the neutrons in the electron
positron gas. The scattering cross section is given by,

$\frac{d\sigma}{d\Omega}=\frac{\alpha^{2}\kappa^{2}q^{2}}{16M^{2}E^{2}sin^{4}(\theta/2)}\frac{E^{\prime}}{E}\times\bigg{[}1+sin^{2}(\theta/2)\bigg{]}$
(15)

Here $\theta$ is the scattering angle, while $E$ is the electron energy before
the scattering and $E^{\prime}$ is the electron energy after the scattering.
The values of $E$ and $E^{\prime}$ depend on the temperature of the
surrounding plasma. The transport cross section $\sigma_{t}$, is given by

$\sigma_{t}=\int\frac{d\sigma}{d\Omega}(1-cos\theta)d\Omega$ (16)

We can then substitute the scattering cross-section in the transport cross
section to obtain,

$\sigma_{t}=3\pi\bigg{[}\frac{\alpha\kappa}{M}\bigg{]}^{2}$ (17)

The diffusion coefficient is obtained by substituting the expression for the
transport cross-section and we get,

$D_{ne}=\frac{M^{2}}{32m^{3}}\frac{1}{\alpha\kappa^{2}}\frac{e^{1/T}}{Tf(T)}.$
(18)

Here $M$, is neutron mass, $m$ is electron mass and $\kappa=-1.91$ is the
anomalous magnetic moment. The temperature in this case is dimensionless as it
is scaled by a factor of $m_{e}c^{2}$. Finally, the function $f(T)$ is given
by, $f(T)=1+3T+3T^{2}$.

Similar to the neutron - electron cross section, we can obtain the nucleon-
muon scattering cross-section too. We have assumed that the heavy neutron
particle is moving through a muon-antimuon gas. The mobility of the neutron is
then given by the force on the neutron due to the gas. This force is given by
the interaction cross section. The differential scattering cross-section is
given by,

$\frac{d\sigma}{d\Omega}=\frac{\alpha^{2}\kappa^{2}q^{2}}{8M^{2}E^{2}sin^{4}(\theta/2)}\frac{1}{1+{2Esin^{2}(\theta/2)}/M}\times\\\
\bigg{[}\frac{cos^{2}(\theta/2)}{1-q^{2}/{4M^{2}}}\bigg{(}\frac{q^{2}}{4M^{2}}-1\bigg{)}-2sin^{2}(\theta/2)\bigg{]}\\\
$ (19)

Here we have assumed that the muon energy and mass are less than the neutron
mass. The cross-section calculation is simplified by this assumption and we
can write the cross-section as,

$\frac{d\sigma}{d\Omega}\approx
K\frac{\alpha^{2}\kappa^{2}}{4M^{2}}[1+cosec^{2}(\theta/2)]$ (20)

All the constants are replaced by a single constant $K=\frac{1}{2}$. We then
substitute the cross section in the diffusion constant. Finally, the diffusion
coefficient is given by,

$D_{n\mu}=\frac{M^{2}}{32m_{\mu}^{3}}\frac{1}{\alpha\kappa^{2}}\frac{e^{1/T^{\prime}}}{T^{\prime}f(T^{\prime})}$
(21)

Here $T^{\prime}=\frac{T}{m_{\mu}c^{2}}$. After obtaining both $D_{ne}$ and
$D_{n\mu}$, we calculate the total diffusion coefficient of the neutron moving
through the plasma.

Apart from the neutron, we need to find the diffusion coefficient of the
proton moving through the electron positron gas too. For proton-electron
scattering, the Coulomb force has to be taken into consideration. The
scattering cross section for the proton and electron is then given by,

$\frac{d\sigma}{d\Omega}=\frac{\alpha^{2}m_{e}^{2}}{4k^{4}sin^{4}(\theta/2}\bigg{[}1+\frac{k^{2}}{m_{e}^{2}}cos^{2}(\theta/2)\bigg{]}$
(22)

We can obtain the transport cross section from these equations,

$\sigma_{t}=4\pi\alpha^{2}\bigg{[}\frac{E_{e}h}{2\pi
k^{2}}\bigg{]}^{2}ln(\frac{2}{\theta_{0}})$ (23)

where $\theta_{0}$ is the minimum scattering angle. Substituting all the
previous equations, we get the diffusion coefficient as,

$D_{pe}=\frac{3\pi}{8\alpha^{2}ln(\frac{2}{\theta_{0}})}\bigg{[}\frac{h}{2\pi
m_{e}}\bigg{]}\frac{Te^{1/T}}{f(T)}.$ (24)

Since the muons also constitute a significant part of the plasma till $100$
MeV, we calculate the proton muon cross section too. The differential cross
section is given by,

$\frac{d\sigma}{d\Omega}=\frac{\alpha^{2}}{4E^{2}sin^{4}(\theta/2)}\frac{1}{1+{2Esin^{2}(\theta/2)}/M}\times\\\
\bigg{[}\bigg{(}1-\frac{\kappa^{2}q^{2}}{4M^{2}}\bigg{)}cos^{2}(\theta/2)-\frac{q^{2}}{2M^{2}}(1+\kappa)^{2}sin^{2}(\theta/2)\bigg{]}$
(25)

We obtain the numerical value of this diffusion coefficient by substituting
the constants in the transport cross section. Once we have the diffusion
coefficients, we numerically solve the diffusion equation in the FLRW metric.

As mentioned before we are considering inhomogeneities whose sizes are in the
range of $1$ km. Since the horizon size is around $10$ kms in the hadronic
phase, these are large inhomogeneities. We have considered high amplitudes of
the order of $10^{14}$ as well as smaller amplitudes, we find that the decay
rate does not depend significantly on the amplitudes. However, we find that in
an expanding universe the overdensity falls far more rapidly than in an non-
expanding universe. We have shown both the cases in figure 6. for comparison.
We have checked for the decay separately in the range $200$ MeV - $100$ MeV as
the muon is still present in the plasma at these temperatures. At lower
temperatures the muon density in the plasma becomes negligible.

Figure 6: The decay of the initial fluctuation is shown in logscale between
$200$ MeV - $100$ MeV.

It seems that large inhomogeneities do decay significantly in an expanding
universe but as long as they have very large amplitude, they may still survive
upto the nucleosynthesis epoch. So an inhomogeneity whose amplitude is of the
order of $10^{8}$ will be decreased to an amplitude of the order of $10^{7}$.
Hence inhomogeneities with low amplitudes of the order of $10$ will be wiped
out. Finally, we look at the temperature range from $100$ MeV - $1$ MeV. The
muons will be negligible in this epoch but the diffusion coefficients will not
change. Figure 7 shows the decay of the inhomogeneities in this epoch.

Figure 7: The decay of the initial fluctuation is shown in logscale between
$100$ MeV - $1$ MeV.

We find that the amplitude of the inhomogeneity decreases by an order of
$10^{4}$ in this period. This means that any inhomogeneity with an amplitude
less that $10^{4}$ will be wiped out before the nucleosynthesis epoch. So
large baryon inhomogeneities generated during the quark hadron transition must
have amplitudes greater than $10^{5}$ times the background density to survive
till the nucleosynthesis epoch.

##  VII Summary

In summary, we have done a detailed study of the decay of the baryon
inhomogeneities generated at the electroweak scale. Baryon inhomogeneities
have important consequences in the early universe. If they survive till the
quark hadron phase transition they will affect the phase transition dynamics.
The quark hadron phase transition is very important in the thermal history of
the universe. Moreover, baryon inhomogeneities can also be generated during
the quark hadron phase transition. These will have an effect on the Big Bang
Nucleosynthesis calculations. Thus the decay of baryon inhomogeneities are
important in the early universe. There has been no previous studies of the
decay of baryon inhomogeneities in the early universe during the electroweak
scale. We have studied the decay of these inhomogeneities in the presence of
electrons, muons and neutrinos. The baryon number is carried by the quarks at
these high temperatures, so as the inhomogeneity decays, the quarks diffuse
through the electrons, muons and neutrinos. The diffusion coefficients for the
different particle interactions are calculated. We then use these diffusion
coefficients to study the diffusion of the baryon inhomogeneity using the
diffusion equation in the FLRW metric.

We have found that baryon inhomogeneities generated in the electroweak epoch
should have an amplitude greater than $10^{5}$ for them to survive till the
quark hadron phase transition. This makes it difficult for the baryon
inhomogeneities generated in a first order electroweak phase transition to
have any effect on the quark hadron epoch. The baryon inhomogeneities have to
have a very high amplitude, they decay substantially during the period between
$200$ GeV - $400$ MeV. This is because the diffusion coefficient is
temperature dependent. The quark hadron transition occurs around $200$ MeV. We
have found that the amplitude of the baryon inhomogeneity decreases to about
five orders of magnitude during this period. This means any inhomogeneity with
an amplitude of $10^{5}$ (or less) will be wiped away before the quark hadron
phase transition. We therefore conclude that any model which generates
inhomogeneities with less than $10^{5}$ amplitude in the electroweak epoch
cannot affect the quark hadron phase transition. They will therefore not
contribute to inhomogeneous BBN either.

Finally in a previous work, we had looked at the decay of baryon
inhomogeneities in the QCD epoch for a stationary universe. This would work
only for small scale inhomogeneities for which the expansion of the horizon
does not matter. We have extended that work for an expanding universe where we
can work with large baryon inhomogeneities. So we look at the decay of large
baryon inhomogeneities in the QCD epoch. We find that the baryon
inhomogeneities decrease by $5$-$6$ orders of magnitude. This means that if
large baryon inhomogeneities are generated by collapsing domain walls and
other topological defects during the quark hadron transition they will survive
till the nucleosynthesis epoch. We conclude that the big bang nucleosynthesis,
can thus be used to constrain models which generate large amplitude
inhomogeneities in the QCD epoch only.

Acknowledgments

The authors acknowledge discussions with Soumen Nayak and Salil Joshi. A.S is
supported by the INSPIRE Fellowship of the Department of Science and
Technology (DST) Govt. of India, through Grant no: IF170627.

## References

  * (1) E. M. Lifschitz, Phys. (Moscow) 10, 116 (1946)
  * (2) S. W. Hawking, Astrophys. J. 145, 544 (1966)
  * (3) J. M. Bardeen, Phys. Rev. D, 22, 1882 (1980)
  * (4) A. F. Heckler, Phys. Rev. D 51, 405 (1995).
  * (5) A. Megevand and F. Astorga, Phys. Rev. D 71, 023502 (2005).
  * (6) A. Megevand and A. D. Sanchez, Phys.Rev. D 77, 063519 (2008).
  * (7) G.M. Fuller, G.J. Mathews, C.R. Alcock Phys. Rev. D, 37, 1380 (1988); H. Kurki-Suonio, Nuclear Physics B - Proceedings Supplements, Volume 24, Issue 2, 67-73 (1991).
  * (8) B. Layek, S. Sanyal and A.M. Srivastava, Phys. Rev. D 67, 083508, (2003).
  * (9) A. Cohen, D. Kaplan, and A. Nelson, Phys. Lett. B 245, 561 (1990); Nucl. Phys. B349, 727 (1991); Phys. Lett. B 263, 86 (1991); L. D. McLerran, M. E. Shaposhnikov, N. Turok, and M. B. Voloshin, ibid. 256, 451 (1991); L. D. McLerran, Phys.Rev. Lett. 62, 1075 (1989); M. E. Shaposhnikov, JETP Lett. 44, 465 (1986); Nucl. Phys. B287, 757 (1987); B299, 797 (1988); N. Turok and P. Zadrozny, Phys. Rev. Lett. 65, 2331 (1990); Nucl. Phys. B358, 471 (1991).
  * (10) S. Sanyal, Phys. Rev. D 67, 074009, (2003).
  * (11) K. Jedamzik and G. M. Fuller, Astrophys. J. 423, 33-49 (1994).
  * (12) V. Berezinsky and A. Z. Gazizov, The Astrophysical Journal, 643:8–13, (2006).
  * (13) N. Sasaki, O. Miyamura, S. Muroya, and C. Nonaka, Phys. Rev. C, 62(1), 011901. (2000).
  * (14) R.J. Scherrer and M.S. Turner, Phys. Rev. D 100.4, 043545 (2019).
  * (15) S. Sau, S. Bhattacharya and S. Sanyal, The European Physical Journal C 79(5) 439 (2019).
  * (16) B. Layek, S. Sanyal and A.M. Srivastava, Phys.Rev.D 63, 083512 (2001).
  * (17) Cherubini, S., Figuera, P., Musumarra, A. et al., Eur. Phys. J. A 20, 355–358 (2004); A. Arbey, J. Auffinger and J. Silk, Phys. Rev. D 102, 023503 (2020).
  * (18) J.H. Applegate, C.J. Hogan, R.J. Scherrer, Phys. Rev. D35 1151-1160, (1987);In-Saeng Suh and G.J. Mathews, Phys.Rev.D 58 123002 (1998).
  * (19) R. Nakamura, M. Hashimoto, R. Ichimasa and K. Arai, International Journal of Modern Physics E Vol. 26, No. 08, 1741003 (2017); K. Inomata, M. Kawasaki, A. Kusenko and L. Yang, Journal of Cosmology and Astrophysics 12, 003, (2018).
  * (20) J. Dziarmaga, Phys. Rev. D 52, R569 (1995); M. Nagasawa, Astropart. Phys. 5, 231 (1996); M. Sato, Phys. Lett. B 376, 41 (1996); M. Nagasawa and J. Yokoyama, Phys. Rev. Lett. 77, 2166 (1996); H. K. Lo, Phys. Rev. D 51, 7152 (1995).
  * (21) M. Barriola, Phys. Rev. D 51, R300 (1995).
  * (22) R. H. Brandenberger and A. Riotto, Phys. Lett. B 445, 323 (1999); T. Matsuda, Phys. Rev. D 64, 083512 (2001).
  * (23) A. Atreya, A. Sarkar and A. M. Srivastava, J. Phys.: Conf. Ser. 484 012053,(2012); A. Atreya, A. Sarkar and A. M. Srivastava Phys.Rev.D 90 4, 045010 (2014).
  * (24) Michael B. Christiansen and Jes Madsen,Phys.Rev.D 53, 5446-5454, (1996).
  * (25) L. D. Landau and E. M. Lifshitz, Physical Kinetics. Pergamon Press Ltd.,(1981).
  * (26) K. S. McFarland, Nuclear Physics. B, Proceedings Supplements 235, 143-148 (2013).
  * (27) D. J. Schwarz, Annalen Phys.12:220-270, (2003).