Satwinder Singh, Feng Hou, Ruili Wang∗

# A Novel Self-training Approach for Low-resource Speech Recognition

###### Abstract

In this paper, we propose a self-training approach for automatic speech
recognition (ASR) for low-resource settings. While self-training approaches
have been extensively developed and evaluated for high-resource languages such
as English, their applications to low-resource languages like Punjabi have
been limited, despite the language being spoken by millions globally. The
scarcity of annotated data has hindered the development of accurate ASR
systems, especially for low-resource languages (e.g., Punjabi and Māori
languages). To address this issue, we propose an effective self-training
approach that generates highly accurate pseudo-labels for unlabeled low-
resource speech. Our experimental analysis demonstrates that our approach
significantly improves word error rate, achieving a relative improvement of
14.94% compared to a baseline model across four real speech datasets. Further,
our proposed approach reports the best results on the Common Voice Punjabi
dataset.

Index Terms: self-training, low-resource, Punjabi ASR

##  1 Introduction

The development of end-to-end (E2E) automatic speech recognition (ASR) systems
for high-resource languages has made significant strides in recent years,
resulting in an excellent performance [1]. The key factors contributing to
this high performance include the presence of large annotated datasets,
advancements in deep learning algorithms, and easy access to high-performance
computational resources. However, the same cannot be said for ASR systems
designed for low-resource languages. These systems still fall short in terms
of performance when compared to their high-resource counterparts. The primary
reason for this discrepancy is the lack of annotated speech datasets and their
availability in the public domain. Without these resources, it is difficult
for ASR systems to be trained effectively and perform at the same level as
those designed for high-resource languages.

The research community has been actively exploring various ways to address the
scarcity of annotated datasets for speech recognition systems [2, 3, 4, 5, 6].
The process of annotating highly accurate speech datasets is a labor-intensive
task that requires a significant amount of time and resources. To overcome
these challenges, many researchers have proposed self-training or pseudo-
labeling (PL) based methods [7]. These methods leverage the vast amount of
available unlabeled data in conjunction with a small labeled dataset to
improve the performance of ASR systems. Self-training has gained increasing
interest in a wide range of research areas, including computer vision [8, 9],
natural language processing [10, 11], and more recently, in the field of
speech recognition [2, 12, 13, 14, 15]. These methods have shown promise in
addressing the shortage of annotated data and improving the performance of ASR
systems.

The fundamental concept behind self-training is to first train an initial seed
model, either from scratch or by fine-tuning an existing pre-trained model,
using a limited amount of labeled data. This seed model is then used to
generate pseudo-labels for unlabeled data. The process can be repeated over
multiple iterations in order to improve the quality of the generated pseudo-
labels. To ensure that only highly accurate pseudo-labels are used, various
filtering methods are applied to filter out any incorrect labels [12].
Additionally, a robust language model can be utilized to achieve accurate
decoding of ASR transcriptions, further improving the quality of the generated
pseudo-labels.

Most of the previous and recent work in self-training/ pseudo-labeling is
based on and demonstrated on high-resource languages like English [2, 12, 13,
14, 16]. However, most of the languages of the world are low-resource and the
ASR system designed for them performs poorly when compared to those with high-
resource languages. With that in mind, in this work, we focus to design a
self-training approach for one of the most widely spoken languages in the
world, i.e., the Punjabi language. Although the Punjabi language is spoken
natively across India and Pakistan region, however, Punjabi speakers are
spread across the globe, especially in countries such as Canada, the United
States, the United Kingdom, Australia, and New Zealand. Despite having more
than 100 million speakers, the Punjabi language still lacks mature ASR systems
due to a lack of annotated datasets.

In the paper, we propose a self-training approach to leverage the available
large unlabeled audio data. We adopt a pre-trained self-supervised
crosslingual wav2vec 2.0 model (XLSR-53) [17] as our seed model to generate
pseudo-labels across multiple iterations. The proposed model first fine-tunes
the XLSR-53 seed model on limited Punjabi datasets in a supervised fashion
with language model decoding. Afterwards, the fine-tuned model is used to
produce pseudo-labels for unlabeled audio data. As we are fine-tuning on
limited data, it is common to get a mixed bag of pseudo-labels. To sieve out
the erroneous pseudo-labels, we propose a length-normalized confidence score.
The pseudo-labels with high confidence scores above the certain cut-off
threshold are selected to be included into the dataset. Finally, the model is
fine-tuned again from scratch using labeled data and highly confident pseudo-
labels. This process is repeated over multiple iterations with gradual
confidence score filtration to refine the pseudo-labels. Our empirical results
demonstrate that our proposed approach is able to achieve significant WER
reduction compared with the baseline and also reports the best results on the
Common Voice Punjabi dataset. To the best of our knowledge, this work is the
first to explore the self-training approach for the Punjabi language on the
mix of datasets (public, non-public, and synthesized datasets).

##  2 Related Work

Semi-supervised approaches such as self-supervision and self-training have
been active areas of research for the past few years. The goal of these
approaches is to leverage the large unlabeled speech data to improve the
overall accuracy of the ASR systems. These approaches showed significant
results for low-resource languages. Self-training is a fairly simple yet
effective semi-supervised approach for utilizing unlabeled audio data. Self-
training has been applied in computer vision and natural language processing
tasks. However, it is recently adopted for E2E speech recognition.

Kahn et al. [7] proposed a self-training approach for E2E ASR model. The
proposed system incorporate a pseudo-label filtering method and language model
decoding to improve the word error rates. Additionally, an ensemble of
multiple models showed further performance improvement. Further, inspired by
noisy student training (NST) from the image processing domain, Park et al.
[12] adopted this method for speech recognition. The proposed approach
exploited the adaptive SpecAugment [18] augmentation method. The NST approach
incorporated filtering, balancing, and mixing techniques to carefully select
highly accurate pseudo-labels. Furthermore, rather than selecting pseudo-
labels over just one iteration, iterative pseudo-labeling (IPL) [13] refines
the acoustic model over multiple iterations. IPL effectively generates the
pseudo-labels for a subset of unlabeled data rather than labeling the entire
unlabeled data. Also, IPL fine-tuned an existing model instead of training an
entirely new model.

Recently, Higuchi et al. [14] presented momentum pseudo-labeling (MPL), which
incorporated online and offline models. At different time steps, weights of
multiple models are averaged using the momentum-based moving average to refine
pseudo-labels. The online model (also called the teacher model), hypothesizes
the pseudo-labels produced by the offline model (also referred to as the
student model) on the go. The MPL approach outperformed standard pseudo-labels
and IPL approaches. The extension to this work is presented in [19], where MPL
is combined with IPL. The resulting approach improved the seed model by
exploiting language model knowledge using the IPL strategy. Additionally, the
Conformer-based ASR architecture further enhanced the overall accuracy of the
system.

While most of the self-training approaches utilized language models for
decoding, the SlimIPL [16] method performed pseudo-labeling without any
language model. The proposed model is built upon the IPL approach, which
produced pseudo-labels over multiple iterations using a single model. SlimIPL
also incorporated a dynamic cache for better and more stable model training.
Furthermore, Lugosch et al. [20] presented a self-training approach to
multilingually generate pseudo-labels for 60 languages. The proposed approach
first pre-trained a single multilingual model followed by fine-tuning it on
the target language using semi-supervision. The fine-tuned model is then used
to produce pseudo-labels for that particular language, which in turn is
utilized to train the final model. Overall, the multilingual pseudo-labeling
demonstrated good performance and better generalization to the data from
different domains.

Xu et al. [21] presented a study, which showed that self-training and pre-
training can effectively be combined and are complementary to each other. Pre-
training based self-supervised approaches like wav2vec [1] and its variants
demonstrated excellent performance for low-resource languages. They leveraged
unlabeled audio data and learn representations using contrastive learning.
When combined with self-training, self-supervision showed a better generation
of pseudo-labels and achieved significant results. Similarly, Jin et al. [22]
exploited a self-supervised pre-trained wav2vec 2.0 model for pseudo-labeling.
The proposed method filtered out the low-probability pseudo-labels and only
selected the pseudo-labels with the highest probability scores. Further, self-
training has also been used to improve representation learning in crosslingual
settings [23]. Furthermore, Khurana et al. [24] proposed the DUST (Dropout-
based Uncertainty-driven Self-Training) approach to alleviate the issues
caused by domain mismatch. The proposed approach efficiently rejected pseudo-
labels that exhibit high levels of uncertainty.

Until recently, limited work has been carried out in the Punjabi language.
Previous efforts relied heavily on statistical and hybrid models, using mostly
non-public datasets [25, 26, 27, 28]. However, in our previous work [6], we
demonstrated that E2E models trained with synthetic speech and real speech
datasets could enhance the effectiveness of E2E ASR models for Punjabi.

##  3 Proposed Approach

We propose a very simple, yet effective self-training method for the Punjabi
language. However, the proposed method could work for any language having a
small amount of annotated data and large unlabeled audio data.

###  3.1 Pre-trained seed model

For the strong baseline seed model, we adopt a self-supervised pre-trained
wav2vec 2.0 based crosslingual XLSR-53 model [17]. The XLSR-53 model is pre-
trained on 53 languages using a contrastive learning approach to learn audio
representations in an unsupervised manner. The crosslingual wav2vec model maps
raw input speech to latent speech representations, which are then transformed
into a set of discrete representations using a quantization module. These
discrete representations are shared across multiple languages (in this case 53
languages excluding the Punjabi language) and used as the target for the self-
supervised objective. Conneau et al. [17] showed that XLSR-53 could greatly
improve the character error rate (CER) and word error rate (WER) in the case
of low-resource languages.

Although pre-trained XLSR-53 does not include Punjabi in the pre-training
setup, the model is still able to learn powerful shared feature representation
across many languages. The crosslingual nature of the model made it easy to
fine-tune the model on any new language with limited labeled data. Following
that, rather than pre-training the model using the Punjabi language, we
leverage the effective feature representations of the pre-trained model and
only fine-tune the model on available Punjabi labeled data. This way, we keep
our self-training approach simple and fast.

###  3.2 Self-training approach

Our proposed self-training approach is fairly simple as shown in Figure 1.
Initially, we fine-tune our pre-trained seed acoustic model on available
limited labeled Punjabi datasets. Once the model is fine-tuned on the Punjabi
language then we use this model to generate the pseudo-labels for unlabeled
Punjabi data. The details of labeled and unlabeled datasets are outlined in
Section 4.1. At this stage, we also employ the 5-gram KenLM language model
[29] for decoding the seed model's outputs.

Since the seed model is only trained on limited Punjabi data, it is fairly
common to produce inaccurate pseudo-labels. To tackle the erroneous pseudo-
labels, we incorporated confidence based scoring strategy [7]. Our language
model produces sentence-level shallow fusion scores while decoding. We
leverage the shallow fusion score as a confidence score (CS) and normalize it
to the length of the sentence. We filter pseudo-labels with different
filtration thresholds of confidence scores in each iteration of self-training,
i.e., [0.5, 0, -0.5, -1, -1.5, -2, -2.5]. We start with a very strict (0.5)
threshold and then with each iteration of self-training, we gradually relax
the filtration threshold. Afterwards, we fine-tune an entirely new model using
a combination of labeled and pseudo-labeled datasets. We repeat this whole
process over multiple iterations until the ASR model is not improving on the
labeled test datasets.

Our approach is similar to the IPL approach [13], but with few distinctions.
Firstly, we compute pseudo-labels for the entire unlabeled dataset instead of
a subset of it. Although it is time-consuming to pseudo-label the entire
unlabeled dataset, it would give us more high-quality filtered pseudo-labels
for fine-tuning. Secondly, since the XLSR-53 model does not show any
improvement over multiple continued fine-tuning phases, we find that fine-
tuning a new model from pre-trained XLSR-53 model produces the best results.
Therefore, rather than continuing fine-tuning like in IPL, in every iteration,
we fine-tune an entirely new model.

Figure 1: Overview of proposed self-training approach.

##  4 Experimental Setup

###  4.1 Datasets

To train our seed model we use a range of labeled Punjabi datasets. We utilize
four low-resource Punjabi real speech datasets and two synthesized datasets.
For real speech datasets, we use the Shrutilipi dataset [30], Common Voice
(CV) [31], Punjabi Speech [6], and
50Languages111https://www.50languages.com/phrasebook/en/pa. Previously, we
have demonstrated the effectiveness of synthetic datasets in improving the
performance of the Punjabi ASR system on real speech datasets [6]. Thus,
following that, we use two synthesized datasets i.e. Google-synth and CMU-
synth [6].

  * •

Shrutilipi: The Shrutilipi corpus is a collection of audio and text pairs
sourced from public resources such as All India Radio news bulletins, and
compiled through data mining techniques. The corpus includes labeled speech
data for a total of 12 Indian languages, including Punjabi.

  * •

Common Voice: The Common Voice dataset is a crowd-sourced dataset available to
the public, featuring speech data in numerous languages. For our purposes, we
utilize the Punjabi data made available in the April 2022 release of Common
Voice. The dataset comprises of a range of speech samples from a total of 39
male and 12 female speakers, providing a diverse array of recordings.

  * •

Punjabi Speech: The Punjabi Speech dataset is a read speech dataset like
Common Voice. The dataset consists of speech data from 2 male speakers.

  * •

50Languages: The dataset is extracted from the 50Languages learning platform1.
The dataset comprises speech utterances from a single male speaker.

  * •

Google-synth: The Google-synth dataset is synthesized labeled speech data
produced using Google's Cloud text-to-speech (TTS)
API222https://cloud.google.com/text-to-speech. The dataset contains speech
samples from 2 male and 2 female speakers.

  * •

CMU-synth: The CMU-synth dataset is synthesized using CMU's Clustergen TTS
model. The dataset contains speech samples from a single female speaker.

  * •

Audiobooks: The Audiobooks dataset consists of speech recordings gathered from
open Punjabi audiobooks featured on various YouTube channels. It is an
unlabeled dataset used primarily for self-training purposes. The dataset
contains audio recordings from multiple speakers and several audiobooks. To
facilitate self-training, we preprocess the audio recordings, chunking them
down into smaller audio segments no longer than 15 seconds.

The synthesized datasets (Google-synth and CMU-synth) are produced using
Punjabi text available in the Old Newspaper
corpus333https://www.kaggle.com/datasets/alvations/old-newspapers. We ensure
that there is no overlap between data used to train the ASR model, self-
training, language model, and synthesized datasets.

Table 1: List of datasets used for our experimentations. Datasets | #Hours | #Utterances
---|---|---
Real speech
Shrutilipi | 94 | 50K
Common Voice | 1 | 1210
Punjabi Speech | 4 | 2431
50Languages | 3 | 3955
Synthesized speech
Google-synth | 38 | 50K
CMU-synth | 170 | 80K
Unlabeled speech
Audiobooks | 450 | 322K
Table 2: Experimental results in terms of WER (%). The best pseudo-label (PL) results are obtained by choosing the best-performing model on labeled datasets over multiple iterations of PL with different confidence thresholds. Model | Datasets
---|---
Shrutilipi | Common Voice | Punjabi Speech | 50Languages | Google-synth | CMU-synth
Seed without LM | 17.23 | 29.88 | 18.88 | 28.08 | 13.39 | 6.82
Seed with LM (baseline) | 16.87 | 14.04 | 9.84 | 24.16 | 2.93 | 4.96
Raw PL (no filter) | 15.23 | 12.30 | 9.10 | 24.00 | 2.75 | 5.12
Best PL | 14.67 | 11.42 | 8.57 | 20.55 | 2.69 | 4.92

###  4.2 Methodology

For our self-training approach, we adopt a pre-trained wav2vec 2.0 based
crosslingual XLSR-53 model. The XLSR-53 model is pre-trained on 53 languages
(50K hours) with 300 million parameters. To save time, we only fine-tune the
pre-trained model on our labeled datasets instead of pre-training it from
scratch. We adopt the pre-trained model by adding a fully connected layer on
top of the Transformer. The fully connected layer outputs the characters using
Connectionist Temporal Classification (CTC) [32]. During the fine-tuning, we
freeze the weights of the feature encoder. We fine-tune a single XLSR-53 model
on combined data from 6 of the labeled datasets as mentioned in Table 1. For
all the datasets except Common Voice, we use 8:1:1 splits for train,
validation, and test sets. For the Common Voice dataset, we use pre-defined
data splits. We fine-tune our model for 10 epochs with a batch size of 32.
Further, we optimize the training process using the Adam optimizer [33] with
an initial learning rate of 3e-4, which is warmed up for the first 10% of the
updates and afterward, linearly decays during the rest of the updates. All the
experiments are conducted on NVIDIA A100 GPUs and results are reported in
terms of word error rate (WER).

Additionally, we decode the ASR model's output with a 5-gram KenLM language
model [29]. We train the Punjabi language model using the IndiCorp dataset
[34]. The dataset contains 29.2 million Punjabi sentences (773 million
tokens). We fine-tune $\alpha$ and $\beta$ parameters and empirically set
$\alpha=0.7$ and $\beta=4.0$. Here $\alpha$ represents the shallow fusion
weight for the language model and $\beta$ refers to the weight to adjust the
score as per the length of the decoded sequence [35]. Further, for selecting
the highly confident pseudo-labels, we adopt a normalized confidence score
filtering.

##  5 Experimental Results and Discussion

###  5.1 Comparative analysis

We report the results in Table 2. To show the effectiveness of our 5-gram
language model (LM), we demonstrate the results on the seed model with and
without the language model. We find that our LM decoding further improves the
seed model, consequently, we choose seed with LM as our baseline model. All of
our experimental results (except seed without LM) are carried out with LM
decoding. Raw PL uses all available pseudo-labels, whereas Best PL selects the
best-performing pseudo-labels over multiple iterations of pseudo-labeling with
different confidence thresholds. The results show that using pseudo-labels
with a filter (Best PL) outperforms Raw PL on all datasets, indicating that
selecting the best pseudo-label is important to improve the performance of the
ASR system. We find that iteration 5 with confidence scores (CS$\geq$-1.5)
produces the best results as shown in Table 2 (Best PL) and Figure 2. As
compared to the baseline, we find our approach achieves relative improvements
of 13.04%, 18.66%, 12.91%, and 14.94% on real speech Shrutilipi, Common Voice,
Punjabi Speech, and 50Languages datasets, respectively. Our PL approach
demonstrates the best results on the Common Voice Punjabi dataset with a
relative improvement of 50.5% over previously reported results (23.07 % vs
11.42% WER)[6].

Further, on synthetic datasets (i.e., Google-synth and CMU-synth), our PL
approach also achieves better results; nevertheless, the improvements are
marginal. We attribute these minimal improvements to the quality of synthetic
data. Following our previous study [6], we only incorporated these datasets to
improve the quality of the overall ASR model.

Figure 2: Performance on real speech datasets against various confidence score
thresholds (left) and % of selected data over the PL iterations (right).

###  5.2 Effectiveness of gradual filtration

Figure 2 illustrates the performance of the model with different CS thresholds
and the amount of PL data selected with filtration over multiple PL
iterations. Initially, we start with aggressive filtering with (CS$\geq$0.5),
which selects only 1.5% of the total unlabeled Audiobooks dataset. With this
threshold, although we select high-quality PL, however, it results in marginal
improvements overall (2.8% of relative improvement). As we gradually relax the
confidence score threshold, our model indeed continues to produce better
results over each iteration of PL. With (CS$\geq$0), selects 9.37% of PL,
which results in 9.5% of overall relative improvement compared to baseline.
Overall, as Figure 2 shows, (CS$\geq$-1.5), which selects 72.66% of PL,
produces the lowest WERs across most datasets. Most consistent improvements
are seen on the Common Voice and Punjabi Speech datasets. Further, our
experimental analysis reveals that PL generated with (CS$\geq$-2 or greater)
selects more than 80% of the total PL data. However, the performance of the
model gradually starts to diminish as compared to the Best PL.

##  6 Conclusions

The paper presents a self-training (pseudo-labeling) approach for automatic
speech recognition for low-resource settings, specifically focusing on the
Punjabi language. The proposed self-training approach generates highly
accurate pseudo-labels for unlabeled Punjabi speech, resulting in a
significant improvement in word error rate (WER) compared to a strong
baseline. The experimental analysis demonstrates the effectiveness of our
approach and highlights the potential to improve the performance of ASR
systems for low-resource languages such as Punjabi.

##  7 Acknowledgement

This work is supported by the 2020 Catalyst: Strategic New Zealand - Singapore
Data Science Research Programme Fund, Ministry of Business, Innovation, and
Employment (MBIE), New Zealand. Ruili Wang is the corresponding author.

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# Singularity formation in asymptotically safe cosmology with inhomogeneous
equation of state

Oem Trivedia<EMAIL_ADDRESS>Maxim Khlopov<EMAIL_ADDRESS>b,c,d

(aSchool of Arts and Sciences, Ahmedabad University,Ahmedabad 380009,India
bInstitute of Physics, Southern Federal University, Stachki 194 Rostov on Don
344090, Russia
cUniversité Paris Cité, CNRS, Astroparticule et Cosmologie F-75013 Paris,
France
dNational Research Nuclear University ”MEPHI” 115409 Moscow, Russia
)

###### Abstract

Interest in cosmological singularities has remarkably grown in recent times,
particularly on future singularities with the discovery of late-time
acceleration of the universe and dark energy. While such explorations have
previously been done in various modified gravity and quantum gravitationally
inspired cosmologies (besides standard general relativistic cosmology), no
such an endeavour has been taken up till now in the realms of renormalization
group approaches to cosmology and we have hence took up on this journey. In
this work, we consider the formation of cosmological singularities in an
asymptotically safe cosmology where the cut off scale is proportional to the
Hubble parameter. We consider a well motivated inhomogeneous form of the
equation of state(EOS) as well. We firstly delve into some basics of this
cosmology and show that such a scenario permits a transition between phantom
and quintessence forms of universal evolution. We then show that one can have
Type I - Type IV singularities in such a cosmology for various version of the
EOS and that the singularities can occur both in finite and infinite time. The
conditions in which these singularities occur is significantly different than
how they occur in the standard cosmology, with the formulations being even
more involved. Interestingly this difference comes about without the need of
any free parameters in the cosmological models, which is usually what one
observes when one goes for the same pursuit via various modified gravity/ QG
inspired cosmological approaches. Furthermore, we showed that usual
singularity removal methods like conformal anomaly effects and f(R) gravity
effects do not offer much hope for singularity removal in this cosmology.

##  1 Introduction

Observations of late time acceleration of the Universe came as a huge surprise
to the cosmological community [1] and ever since then a lot of work has been
done in order to explain this expansion. The cosmological expansion problem
has been addressed from multiple facets till now, which include the standard
approaches of the Cosmological constant [2, 3, 4] alongside more exotic
scenarios like Modified gravity theories[5, 6, 7] and even scalar field driven
late-time cosmic acceleration scenarios [8, 9, 10, 11, 12, 13]. Another very
interesting way to address cosmological concerns which has picked up pace in
recent years is a Renormalization group approach to cosmology, leading to the
so called asymptotically safe cosmology.

While approaches like modified gravity theories affect the whole cosmological
dynamics in an extensive fashion, in asymptotically safe cosmology quantum
corrections only enforce simple modifications of couplings whose scale
dependence gets determined by the renormalization group [14, 15, 16, 17, 18,
19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. Asymptotic Safety, in
principle, belongs in the same space of theories as the corresponding
effective field theory. By adding the additional requirement that the quantum
theory describing our universe is located within a UV critical hypersurface of
any acceptable renormalization group (RG) fixed point, it resolves the
predictability problem observed in effective field theory framework. This
suggests that the fixed point, which makes all dimensionless coupling
constants finite at high energy, controls how the theory behaves at high
energies. The running of the cosmological constant and Newton’s gravitational
constant have been investigated through perturbative RG cosmological
techniques, and their implications for big bang nucleosynthesis and other
phenomena have been examined recently. They have also shown how there are
significant deviations from standard cosmology in such a scenario. Hence
asymptotic safety, based on a non-Gaussian fixed point of the gravitational
renormalization group flow, can provide an elegant mechanism for completing
the gravitational force at sub-Planckian scales.

There has also been an expansive literature in recent times which has been
devoted to the study of various types of singularities that could occur during
the current and far future of the Universe, with the observation of late-time
acceleration having given a significant boost to such works [31, 32, 33, 34,
35, 36, 37, 38, 39, 40, 41]. These cosmological singularities which have been
discussed in recent times can be classified broadly into two types ; strong
and weak (such a classification was initially put forward by [42]). Strong
singularities are those singularities which can distort finite objects and can
mark either the beginning or the end of the universe, with the big bang being
the one for the start of the universe and the so called ”big rip” signalling
the end of the universe. Weak singularities, as the name might suggest, are
those which do not have such far reaching implications and do not distort
finite objects in the same sense as their strong counterparts. We can discuss
these various singularities in more detail as follows :

  * •

Type I (”Big Rip”) : In this case, the scale factor , effective energy density
and effective pressure density diverges. This is a scenario of universal
death, wherein everything which resides in the universe is progressively torn
apart [43].

  * •

Type II (”Sudden/Quiescent singularity”) : In this case, the pressure density
diverges and so does the derivatives of the scalar factor from the second
derivative onwards [44]. The weak and strong energy conditions hold for this
singularity. Also known as quiescent singularities, but this name originally
appeared in contexts related to non-oscillatory singularities [45]. A special
case of this is the big brake singularity [46].

  * •

Type III (”Big Freeze”) : In this case, the derivative of the scale factor
from the first derivative on wards diverges. These were detected in
generalized Chaplygin gas models [47].

  * •

Type IV (”Generalized sudden singularities”): These are finite time
singularities with finite density and pressure instead of diverging pressure.
In this case, the derivative of the scale factor diverges from a derivative
higher than the second [34].

  * •

Type V (”w-singularities”) : In this case, the scale factor, the energy and
pressure densities are all finite but the barotropic index $w=\frac{p}{\rho}$
becomes singular [48].

  * •

Type $\infty$(” Directional singularities ”): Curvature scalars vanish at the
singularity but there are causal geodesics along which the curvature
components diverge [49] and in this sense, the singularity is encountered just
for some observers.

  * •

Inaccessible singularities: These singularities appear in cosmological models
with toral spatial sections, due to infinite winding of trajectories around
the tori. For instance, compactifying spatial sections of the de Sitter model
to cubic tori. However, these singularities cannot be reached by physically
well defined observers and hence this prompts the name inaccessible
singularities [50].

Furthermore, Type I-Type IV singularities were also studied in the context of
a cosmology with inhomogeneous equations of state (EOS), firstly in [32, 33].
The authors of these papers discussed singularity structures in a cosmology
with multiple types of inhomogeneous EOS and showed that Type I-IV
singularities can occur in such a cosmology if certain conditions are
fulfilled. They also showed how there can be a transition between phantom and
quintessence types of evolution for the universes given various inhomogeneous
forms for the EOS. These explorations were, however, done in the standard
general relativistic cosmology. Hence in this paper we would like to discuss
the formation of Type I-IV singularities in an asymptotically safe cosmology
with various inhomogeneous equations of state and the paper will be structured
as follows 333One can raise the point that asymptotically safe cosmology is
only significant in the early universe and not so much for the late universe,
where we have considered it. But we would again like to say this is not
exactly the case as recent literature to supports the consideration of
asymptotically safe cosmology in the very late universe. In particular it was
comprehensively shown in [51] as to how asymptotically safe cosmology can be a
viable late universe expansion alternative.. In the next section, we will
firstly briefly discuss the basics of asymptotically safe cosmology with an
inhomogeneous equation of state after which we will discuss how such a
cosmology can transition between phantom and quintessence forms of evolution.
In section III we will discuss in detail the singularity structure of this
cosmology with three forms of equation of state and we will then conclude our
work in section IV.

##  2 Brief review of asymptotically safe cosmology with the Phantom-to-
Quintessence transition

The capacity to build gravitational RG flow approximations outside of
perturbation theory is necessary for conceptually testing asymptotic safety. a
very strong framework for doing these calculations is the functional
renormalization group equation (FRGE) for the gravitational effective average
action $\Gamma_{k}$

$\partial_{k}\Gamma_{k}[g,\overline{g}]=\frac{1}{2}Tr\left[(\Gamma_{k}^{(2)}+\mathcal{R}_{k})^{-1}\partial_{k}\mathcal{R}_{k}\right]$
(1)

The construction of the FRGE uses the background field formalism, where the
metric $g_{\mu\nu}$ is split into a fixed background $\overline{g}_{\mu\nu}$
and fluctuations $h_{\mu\nu}$. The Hessian $\Gamma_{k}^{(2)}$ is the second
functional derivative of $\Gamma_{k}$ with respect to the fluctuation field in
a fixed background and a scale-dependent mass term for fluctuations with
momenta $p^{2}<<k^{2}$ (where the RG scale k constructed from the background
metric) is provided by $\mathcal{R}_{k}$. As it is a mass term, we can assume
it to be a positive definite in nature. The role that $\mathcal{R}_{k}$ plays
in the numerator and denominator gives us a trace which could be both infrared
and ultraviolet finite and also ensures that the flow of $\Gamma_{k}$ is
actually under the jurisdiction of the fluctuations with momentum
$p^{2}\approx k^{2}$ . A useful approach towards asymptotic safety is the
functional renormalization group equation for the gravitational effective
average action $\Gamma_{k}$.

One could argue that the simplest approximation of the gravitational RG flow
could be obtained from projecting the FRGE onto the Einstein-Hilbert action
approximating $\Gamma_{k}$ by [52]

$\Gamma_{k}=\frac{1}{16\pi G_{k}}\int
d^{4}x\sqrt{-g}\left[-R+2\Lambda_{k}\right]+\text{gauge-fixing and ghost
terms}$ (2)

where R, $\Lambda_{k}$ and $G_{k}$ are the Ricci Scalar, the running
cosmological constant and the running Newton’s gravitational constant. The
scale-dependence of these couplings can be written in terms of their
dimensionless counterparts as

$\Lambda_{k}=k^{2}\lambda_{*}$ (3) $G_{k}=g_{*}/k^{2}$ (4)

where $g_{*}=0.707$ and $\lambda_{*}=0.193$ . There has been a significant
literature which has been invested in considering various forms of the cutoff
scale k [52] . In order to move forward, we consider the background metric to
be that of a flat FLRW cosmology

$ds^{2}=-dt^{2}+{a(t)}^{2}(dx^{2}+dy^{2}+dz^{2})$ (5)

Then, considering a perfect fluid form for the stress-energy tensor ,
$T_{\mu}^{\nu}=\text{diag}[-\rho,p,p,p]$ , one can get the Friedmann equation
and the continuity equation in this scenario to be

$H^{2}=\frac{8\pi G_{k}}{3}+\frac{\Lambda_{k}}{3}$ (6)
$\dot{\rho}+3H(\rho+p)=-\frac{\dot{\Lambda_{k}}+8\pi\dot{G_{k}}}{8\pi G}$ (7)

Where the continuity equation comes about from the Bianchi identity which is
satisfied by Einstein’s equations $D^{\mu}[\lambda(t)g_{\mu\nu}-8\pi
G(t)T_{\mu\nu}]=0$, which has the usual meaning that the divergence $D^{\mu}$
of the Einstein tensor vanishes . The extra terms of the right hand side in
(2.7) can be interpreted as an illustration of the energy transfer between
gravitational degrees of freedom and matter. To proceed further, we need a
form for the cut-off scale k. In the RG improvement scheme, the cutoff scale k
is usually associated with a length scale for the system. In the context of
cosmology, one could have severeal interesting identifications (as discussed
in [52] for example) A popular form for k where one uses k to be proportional
to the Hubble parameter, where the Hubble parameter is a function of time
(which represents a proper distance and hence this could be a viable length
scale identification for the system) and so here we take k as 444One might be
tempted to think that this cut-off identification is arbitrary but it is
hardly so.The cut-off choice can indeed be very different from this as well
Taking the cut-off scale proportional to the Hubble parameter has been a
recurring theme in asymptotically safe cosmology studies and one can see for
example [52] , [53] , [29] to look at discussions about this choice of scale.
In particular one can read the discussion in [52], for instance, which
discusses four different types of cut-off identifications with the one where
the cut-off choice is proportional to the Hubble parameter being one of them.
This choice was firstly introduced in [18] and there has been significant work
on this basis ever since.

$k^{2}=\epsilon{H(t)}^{2}$ (8)

where $\epsilon$ is an a priori undetermined positive parameter of
$\mathcal{O}(1)$. Using this cutoff scale and the values (2.3-2.4), we can
write the Friedmann and continuity equations as

$H^{4}\approx\frac{18\rho}{3\epsilon-0.193\epsilon^{2}}=M\rho$ (9)
$\dot{\rho}\left(\frac{1}{2\rho}-1\right)+3H(\rho+p)\approx 0$ (10)

where $M=\frac{54}{\epsilon-0.064\epsilon^{2}}$ . The form of the equation of
state we will be considering here is given by

$p=-\rho-f(\rho)$ (11)

This sort of equation of state with $f(\rho)=A\rho^{\alpha}$ for $\alpha$
being an arbitrary constant was first proposed in [54] and was investigated in
detail in [55] and there can be diverse physical motivations behind such an
equation of state. This form of EOS can also be equivalent to bulk viscosity
[56]. This type of an equation of state can also come about due to modified
gravity effects [32]. In order to now discuss a model for transition between
phantom and quintessence forms of evolution we consider the following ansatz
for the Hubble parameter [33]

$H=n\left(\frac{1}{t}+\frac{1}{t_{s}-t}\right)$ (12)

where $t_{s}$ refers to the time of the singularity and in this case, it can
also be taken as the age of the universe as when $t=t_{s}$, we see a big rip
scenario 555The ansatz above really comes about by the consideration that the
scale factor is given by $a(t)=a_{o}\left(\frac{t}{t_{s}-t}\right)$ So the
real starting point here is the scale factor but we have proceeded from the
Hubble factor right away and n is a real constant. Using the Friedmann
equation (2.9), the energy density can be written as

$\rho=\frac{n^{4}{t_{s}}^{4}}{M(t_{s}-t)^{4}t^{4}}$ (13)

And using the continuity equation (2.10) and equation of state (2.11), we can
get

$f(\rho)=\pm\frac{2(1-2\rho)}{3nt_{s}}\sqrt{t_{s}\left(t_{s}\pm\frac{4n}{M^{1/4}\rho^{1/4}}\right)}$
(14)

where $f(\rho)$ turns out to be a 4-valued function instead of a double valued
function as it happens in standard cosmology with the same ansatz [33] and we
start to see the effects of asymptotic safety on cosmological dynamics here.
The equation of state in this scenario is given by

$w=-1\pm\frac{2(1-2\rho)}{3nt_{s}\rho}\sqrt{t_{s}\left(t_{s}\pm\frac{4n}{M^{1/4}\rho^{1/4}}\right)}$
(15)

Although the function $f(\rho)$ is technically four valued, the $\pm$ term in
the square roots is not that important in the context of the equation of
state, as we want to see how w can move away from -1 (also we can just take
the plus part of the square root function if we want, without any loss of
generality). We see that w in this case can transition from phantom region
$w<-1$ to the quintessence region $w>-1$ depending on the parametrization of
$f(\rho)$, again as in [33] . This kind of behaviour of the equation of state
is similar to what happens for first order phase transitions and hence, in an
asymptotically safe cosmology with such inhomogeneous equations of state we
can have transitions between phantom and quintessence evolution of the
universe.

##  3 Singularity formation for various forms of equation of state

In this section, we will discuss singularity formation in the asymptotically
safe cosmology in detail.First up, we now express Friedman equation (2.9) as

$H^{4}=M\rho$ (16)

where $M=N/3$. While this might seem insignificant, we feel that re
parametrizing like this will be more convenient for the forthcoming
formulations. Using the Friedman equation as above and the continuity equation
(2.10) and assuming the equation of state (2.11), we can find the scale factor
to be

$a=a_{o}\exp\left[\int\frac{1-2\rho}{6\rho f(\rho)}d\rho\right]$ (17)

And furthermore, we can write a relation between the cosmological time t and
$\rho$ as

$t-t_{s}=\int\frac{1-2\rho}{6M^{1/4}\rho^{5/4}f(\rho)}d\rho$ (18)

The above two equations will allow us to identify about different types of
cosmic singularities that can occur in this scenario. Broadly, we can write
that

  * •

For Type I singularities $t\to t_{s}$ , $a\to\infty$, $p\to\infty$ and
$\rho\to\infty$ .

  * •

For Type II singularities $t\to t_{s}$ , $a\to a_{s}$, $p\to\infty$ and
$\rho\to\rho_{s}$ or $0$ .

  * •

For Type III singularities $t\to t_{s}$ , $a\to a_{s}$, $p\to\infty$ and
$\rho\to\infty$ .

  * •

For Type IV singularities $t\to t_{s}$ , $a\to a_{s}$, $p\to p_{s}$ and
$\rho\to\rho_{s}$ but derivatives of the Hubble Factor H diverge from the
second derivative onward.

With this cleared up , we would now like to consider two different forms of
$f(\rho)$ and discuss singularity formations for each of these forms.

###  3.1 $f(\rho)=A\rho^{\alpha}$

Here we will consider the case of $f(\rho)=A\rho^{\alpha}$ , for $\alpha$
being an arbitrary constant and A is some real number. For this case (2.10)
becomes

$\dot{\rho}\left(\frac{1}{2\rho}-1\right)=3H(A\rho^{\alpha})$ (19)

Using (3.1), we can further write this as

$\dot{\rho}\left(\frac{1}{2\rho}-1\right)=3M^{1/4}A\rho^{1/4+\alpha}$ (20)

We can now use (2.11) to find the scale factor as a function of the energy
density, which comes out to be

$a(\rho)=a_{o}\exp\left[\frac{\rho^{-\alpha}\left(1+\alpha(2\rho-1)\right)}{6A\alpha(\alpha-1)}\right]$
(21)

We can further write a relation between $\rho$ and cosmic time t (for
$\alpha\neq\frac{3}{4}-\frac{1}{4}$)

$t-t_{o}=\frac{2\rho^{-\alpha-1/4}(3-4\alpha+2\rho+8\alpha\rho)}{3AM^{1/4}(4\alpha-3)(1+4\alpha)}$
(22)

while for $\alpha=\frac{3}{4}$ we have

$t-t_{o}=\frac{1}{6AM^{1/4}}\left[\frac{\rho-\rho_{o}}{2\rho\rho_{o}}-2\ln\left(\frac{\rho}{\rho_{o}}\right)\right]$
(23)

(where $\rho_{o}$ is the energy density at time $t_{o}$) and for
$\alpha=-\frac{1}{4}$ we have

$t-t_{o}=\frac{1}{3M^{1/4}A}\left[\frac{1}{2}\ln\left(\frac{\rho}{\rho_{o}}\right)-(\rho-\rho_{o})\right]$
(24)

Now, we can immediately compare these formulas with their corresponding forms
in the usual standard cosmology with the scale factor being

$a(\rho)=a_{o}\exp\left[\frac{\rho^{1-\alpha}}{3A(1-\alpha)}\right]$ (25)

and the relation between t and $\rho$ for $\alpha\neq\frac{1}{2}$ being

$t-t_{o}=\frac{k\rho^{-\alpha+1/2}}{A(1-2\alpha)}$ (26)

where k is a constant and for $\alpha=1/2$

$t-t_{o}=\frac{k}{2A}\ln\left(\frac{\rho}{\rho_{o}}\right)$ (27)

We immediately see some very significant changes between the formulations in
usual cosmology and the one in asymptotically safe cosmology and hence, there
will be quite a few significant differences in the inferences drawn from
(3.10-3.12), as was the case for (3.6-3.9) in [32].For example, in the usual
cosmology for $\alpha=1/2$, $\rho$ can only diverge in the far future but in
this case, for $\alpha=1/2$ we can have a finite time divergence of $\rho$ as
well. In the usual cosmology the cases $\alpha=3/4$ and $\alpha=-1/4$ did not
have any special solutions but here they have special solutions.
We would now like to discuss the singularity structure in detail. Firstly, the
scale factor here takes a very interesting form and we can draw some useful
conclusions from this.

  * •

When $\alpha>1$ then the scale factor will always be finite for
$\rho\to\infty$ regardless of whether $A>0$ or $A<0$. As $\rho\to\infty$,
$p\to\infty$ because of the EOS (2.11) and so for $\alpha>1$ we will have type
III singularities.

  * •

For $\alpha<1$ and $A>0$ , $a\to 0$ as $\rho\to\infty$ and so in this case, we
will again be dealing with a type III singularity. 666 One might be tempted to
think that for a null scale factor we could be dealing with a big bang
singularity in this case, but that is not true as the equation of state
parameter $w=-1$. This can, however, be considered as a possible grand
bang/rip [57] but we will not be considering the grand bang/rip singularities
in our paper

  * •

For $\alpha<1$ and $A<0$ , $a\to\infty$ as $\rho\to\infty$ and so in this case
as all of a, $\rho$ and p diverge, we will have type I or Big rip singularity.

Furthermore, we can now discuss the special cases of the $\rho-t$ relation
(3.8-3.9). Firstly as $\rho\to\infty$ the right hand side of both (3.8) and
(3.9) diverge towards negative infinity. We can now comment for the cases
$\alpha\neq\frac{3}{4}-\frac{1}{4}$ as follows

  * •

For $\rho\to\infty$ and $A>0$, $t\to-\infty$ and as $\rho$ diverges, so will
the pressure as well. In this case, the scale factor will still stay finite
(3.6). Hence, this points towards the fact that for these values of $\alpha$ ,
we have a Type III singularity in the far past for a positive value of the
parameter A.

  * •

For $\rho\to\infty$ and $A<0$, $t\to\infty$ and the pressure p diverges again,
but in this case the scale factor a will also diverge as $\alpha<1$ as well.
So we see that if $\alpha$ takes these values for $A<0$, then we can have a
big rip (Type I) singularity in the far future of the universe 777A reader
could also be tempted by the possibility of a logarithmic singularity in
(3.8-3.9) for $\rho\to 0$ and we would say that in principle that is a viable
possibility but we have not considered that here as the possibility of
vanishing energy densities in the far future of the universe is something
which is not in the current scope of the work we would want to consider here.
.

We can now discuss the singularity formations for the more general case (3.7)
in more detail

  * •

For the case when $\alpha>1$ , when $\rho\to\infty$ then p will also diverge
(2.11) first of all. Secondly, the right hand side of (3.7) will tend towards
zero and hence $t\to t_{o}$ and this would be the result regardless of whether
A is positive or negative. The scale factor does not diverges in this scenario
as $\alpha>1$. Hence, in this case we can have a type III singularity in
finite time.

  * •

For the case when $3/4<\alpha<1$ , when $\rho\to\infty$ then p also diverges.
The right hand side of (3.7) will again tend towards zero and hence $t\to
t_{o}$ and this would again be the result regardless of whether A is positive
or negative. Through (3.6), a can still diverge in this scenario as $\alpha<1$
for $A<0$. Hence, in this scenario we can have a type I singularity in finite
time.

  * •

For the case when $-1/4<\alpha<3/4$ , when $\rho\to\infty$ then p will again
diverge. This is the first case where we the right hand side of (3.7) diverges
and hence the singularities occuring in this case will occur in either the
infinite future or the infinite past. For $A>0$, $t\to-\infty$ and as the
scale factor will not diverge in this case (3.6), we have a Type III
singularity in the far past. For $A<0$, $t\to\infty$ and as the scale factor
diverges in this case, we have a type I singularity in the far future in this
scenario.

  * •

For the case when $\alpha<-1/4$, when $\rho$ diverges the right hand side of
(3.7) diverges as well. In this scenario when $A>0$, $t\to-\infty$ , and as
for positive A we can only have a finite scale factor, in this case the only
the pressure and energy densities diverge in the far past and hence we have a
type III singularity in this scenario. For $A<0$, $t\to\infty$ and as in this
case the scale factor also diverges ( as $A<1$) we have a type I singularity
in the far future in this case.

  * •

Finally, we would also like to comment on the possiblity of Type IV
singularities. We can write Hubble factor H as

$H=\frac{\dot{a}}{a}=\frac{da}{d\rho}\dot{\rho}$ (28)

By using the continuity equation (3.5) and the scale factor (3.6), we can
write the Hubble factor as

$H=\frac{M^{1/4}\rho^{1/4}}{A}$ (29)

Its very interesting to note that the Hubble parameter does not depend on
$\alpha$ in any way and the hubble parameter will diverge for
$\rho\to\infty$.Its second derivative,however, does depend on $\alpha$ and is
given by

$\ddot{H}=\frac{9M^{3/4}\rho^{2\alpha+\frac{3}{4}}(\alpha(2-4\rho)+2\rho+1)}{2A(1-2\rho)^{3}}$
(30)

The second derivative of H remains finite for $\alpha>0$ (regardless of
whether A is positive or negative) when $\rho\to 0$ , $p\to 0$. However, the
second derivative (and the subsequent derivatives ) diverge for $\alpha<0$
(for both positive and negative values of A) but in this case even though
$\rho\to 0$, p diverges. Hence, we cannot have a Type IV singularity with this
form of the EOS for $\rho\to 0$ . However, if $\rho\to 1/2$ then we although
the second derivative of H diverges, neither the scale factor nor the energy
or pressure densities diverge. Hence, we can have a type IV singularity in
this case for $\rho\to 1/2$. Furthermore looking at the expressions (3.7-3.9),
we see that for $\rho\to 1/2$, the cosmic time t does not diverge unless
$\alpha$ values are tuned for it diverge. So we can have type IV singularities
in a finite timeframe in this scenario.

In this way we have described the full singularity structure for the
inhomogenous EOS corresponding to $f(\rho)=\rho^{\alpha}$ in an asymptotically
safe cosmology. Crucially, we have shown that Type I and Type III
singularities can occur both in finite times and also in infinite times while
Type IV singularities can occur in a finite time in such a cosmology. We have
also shown that there is a significant difference between the singularity
structure in standard cosmology for such an EOS and that in our asymptotically
safe cosmology. We have also shown that for this ansatz, a Type IV singularity
is not possible. We take a moment here to look at the status quo of energy
conditions in the scenarios we have discussed so far. In particular, we would
like to discuss about the following energy conditions

$\rho\geq 0\quad\rho\pm p\geq 0\qquad\text{"dominant energy condition"}$ (31)
$\rho+p\geq 0\qquad\text{"null energy condition"}$ (32) $\rho\geq
0\quad\rho+p\geq 0\qquad\text{"weak energy condition"}$ (33) $\rho+3p\geq
0\quad\rho+p\geq 0\qquad\text{"strong energy condition"}$ (34)

We would like to comment about their status in the different scenarios we have
outlined so far as follows

  * •

For $\alpha>0$ , $A>0$ and $\rho\to\infty$ , all of the strong, weak, null and
dominant energy conditions are violated.

  * •

For $\alpha>0$, $A<0$ and $\rho\to\infty$ , all energy conditions are
satisfied except the dominant energy condition.

  * •

For $\alpha>0$ and $\rho\to\infty$ , all energy conditions are satisfied
except the strong energy condition and in this case it does not matter whether
A has a positive or negative value.

  * •

For $A>0$ and $\rho\to 1/2$ , all energy conditions are violated and this
result is independent of whether $\alpha$ has a positive or negative value.

  * •

For $\alpha<0$ , $A<0$ and $\rho\to 1/2$, all energy conditions are satisfied
except the dominant energy condition.

  * •

For For $\alpha>0$ , $A<0$ and $\rho\to 1/2$, all energy conditions are
satisfied and no condition is violated.

###  3.2 $f(\rho)=C(\rho_{o}-\rho)^{-\gamma}$

Let’s now consider the case $f(\rho)=C(\rho_{o}-\rho)^{-\gamma}$, where C is
an arbitrary constant and $\gamma$ is another real constant but it is assumed
to be positive, which was considered in [33] to address type II singularities
in the context of (2.11). For this case, the scale factor and the $\rho$-t
relation in usual cosmology are

$a\sim a_{o}\exp\left[-\frac{(\rho_{o}-\rho)^{\gamma+1}}{3C(\gamma+1)}\right]$
(35) $t\sim t_{o}-\frac{(\rho_{o}-\rho)^{\gamma+1}}{3\kappa
C\sqrt{3\rho_{o}}(\gamma+1)}$ (36)

where $\kappa$ is again a constant. The scale factor and the $\rho$ -t
relation in the asymptotically safe scenario are given by

$a(\rho)=a_{o}\exp\left[\frac{(\rho_{o}-\rho)^{\gamma+1}\left(2\rho_{o}-{{}_{2}F_{1}}(1,1+\gamma;2+\gamma;1-\frac{\rho}{\rho_{o}})\right)}{6(\rho_{o}(1+\gamma))}\right]$
(37)
$t-t_{0}=\frac{1}{9C\rho_{o}N\rho^{1/4}}\left[6(\rho_{o}-\rho)^{\gamma+1}-2\rho^{\gamma+1}(3+2\rho_{o}+4\gamma){{}_{2}F_{1}}\left(\frac{3}{4},-\gamma;\frac{7}{4};\frac{\rho}{\rho_{o}}\right)\right]$
(38)

where ${{}_{2}F_{1}}$ are the hypergeometric functions which are given by the
hypergeometric series

${{}_{2}F_{1}}(a,b;c;z)=\sum_{n=0}^{\infty}\frac{(a)_{n}(b)_{n}}{(c)_{n}}\frac{z^{n}}{n!}=1+\frac{ab}{c}\frac{z}{1!}+\frac{a(a+1)b(b+1)}{c(c+1)}\frac{z^{2}}{2!}+...$
(39)

where $(q)_{n}$ are the Pochhammer symbols defined as,

$(q)_{n}=\left\\{\begin{array}[]{ll}1&\quad n=0\\\ q(q+1)...(q+n-1)&\quad
n>0\end{array}\right.$

Its firstly very interesting to see that the simple forms of the scale factor
and $\rho$-t relation (3.20-3.21) are changed quite substantially in this
scenario, more so with the entry of an exotic hypergeometric function. But
what is eye catching here is that the ansatz
$f(\rho)=C(\rho_{o}-\rho)^{\gamma}$ still provides Type II singularities in
the same way as it does in usual cosmology. In the case of the scale
factor(3.22), as $\rho\to\rho_{o}$ the scale factor still remains finite as
the third argument of the hypergeometric function, $2+\gamma$ will always be
positive and so the function will not blow up. Thereby in this case we can
have diverging pressure with finite scale factor. For (3.23), the second
argument of the Hypergeometric function, $-\gamma$ , will always be negative
and hence the Hypergeometric function will terminate and will be finite.
Hence, even for $\rho\to\rho_{o}$, time t can still be finite. So we can
successfully have type II singularities even in an asymptotically safe
cosmology with this particular ansatz.

Furthermore it is interesting to look at the Hubble parameter in this
scenario, which for the scale factor (3.22) and this ansatz is given by

$H=\frac{6(\gamma+1)C\rho_{o}M^{1/4}\sqrt[4]{\rho}(\rho_{o}-\rho)^{-\gamma-1}}{2\rho_{o}-\,_{2}F_{1}\left(1,\gamma+1;\gamma+2;1-\frac{\rho}{\rho_{o}}\right)}$
(40)

For this form of the Hubble paramter, we can write its second time derivative
as

$\ddot{H}=\frac{27(\gamma+1)C^{3}\rho_{o}M^{3/4}\rho^{3/4}(k-\rho)^{-3(\gamma+1)}\left((k-\rho),_{2}F_{1}\left(1,\gamma+1;\gamma+2;1-\frac{\rho}{\rho_{o}}\right)\left((2\rho_{o}\rho+\rho_{o})\right)\right)}{(1-2\rho)^{3}\left(2\rho_{o}-\,_{2}F_{1}\left(1,\gamma+1;\gamma+2;1-\frac{\rho}{\rho_{o}}\right)\right){}^{3}}\\\
+\frac{2\rho_{o}(3(\gamma+1)-2\rho_{o}(2\rho+1)+2\rho(-8\gamma+10\gamma\rho+8\rho-5))}{(1-2\rho)^{3}\left(2\rho_{o}-\,_{2}F_{1}\left(1,\gamma+1;\gamma+2;1-\frac{\rho}{\rho_{o}}\right)\right){}^{3}}\\\
+\frac{4\rho_{o}^{2}(4(\gamma+1)^{2}+\rho_{o}^{2}(2\rho+1)-3(\gamma+1)k)}{(1-2\rho)^{3}\left(2\rho_{o}-\,_{2}F_{1}\left(1,\gamma+1;\gamma+2;1-\frac{\rho}{\rho_{o}}\right)\right){}^{3}}\\\
+\frac{4\rho_{o}^{2}\left(-2\rho_{o}\rho(-7\gamma+8(\gamma+1)\rho-5)-2(4\gamma+3)^{2}\rho^{3}+\left(48\gamma^{2}+82\gamma+37\right)\rho^{2}-3(\gamma+1)(8\gamma+7)\rho\right)}{(1-2\rho)^{3}\left(2\rho_{o}-\,_{2}F_{1}\left(1,\gamma+1;\gamma+2;1-\frac{\rho}{\rho_{o}}\right)\right){}^{3}}\\\
+\frac{2\rho_{o}(3(\gamma+1)-2\rho_{o}(2\rho+1)+2\rho(-8\gamma+10\gamma\rho+8\rho-5))}{(1-2\rho)^{3}\left(2\rho_{o}-\,_{2}F_{1}\left(1,\gamma+1;\gamma+2;1-\frac{\rho}{\rho_{o}}\right)\right){}^{3}}\\\
-\frac{\rho(2\gamma-4\gamma\rho-2\rho-1)\,_{2}F_{1}\left(1,\gamma+1;\gamma+2;1-\frac{\rho}{\rho_{o}}\right)}{(1-2\rho)^{3}\left(2\rho_{o}-\,_{2}F_{1}\left(1,\gamma+1;\gamma+2;1-\frac{\rho}{\rho_{o}}\right)\right){}^{3}}$
(41)

While the expression above looks humongous, its very easy to see that there is
a finite value for which $\ddot{H}$ diverges. Namely for $\rho\to\frac{1}{2}$
, $\ddot{H}\to\infty$ while the scale factor (3.22) and the pressure given by
the ansatz remain finite. Hence as only the higher derivatives of H diverge
while $\rho$, p and a remain finite, we see that we can have type IV
singularity for this ansatz as well. Furthermore, for $\rho\to\frac{1}{2}$ the
cosmic time t given by (3.23) remains finite as well and so the type IV
singularity can occur in finite time. We would also like to suggest the reader
to have a look at the appendix for a short discussion of other forms of EOS in
this scenario. Finally we make comments on the status of energy conditions for
the various scenarios we have discussed so far as follows

  * •

For $C>0$ and $\rho\to 0$ , the null, weak and dominant energy conditions are
satisfied but the strong energy condition is violated.

  * •

For $C<0$ and $\rho\to 0$, the null, weak and strong energy conditions are
satisfied but the dominant energy condition is violated.

  * •

For $C>0$ and $\rho\to 1/2$, all energy conditions are violated.

  * •

For $C<0$ and $\rho\to 1/2$, all energy conditions are satisfied except the
strong energy condition.

##  4 Possible singularity removal methods

In this section we would like to address some possible methods of singularity
removal which have been previously discussed in the literature as well. In
particular, we would like address singularity removal by the account of
conformal anomaly effects and $f(R)$ gravity realizations. The effect of
quantum backreaction of conformal matter around Type I, Type II and Type III
singularities were taken into consideration in [33, 31, 58]. In these cases,
the curvature of the universe becomes large around the singularity time
$t=t_{s}$, although the scale factor a is finite for type II and III
singularities. Since quantum corrections usually contain the powers of the
curvature or higher derivative terms, such correction terms are important near
the singularity. At this point, it becomes important to add a bit of context
about what conformal anomalies are and how they are usually perceived in high
energy physics. It is fair to assume that there are many matter fields during
inflation in the early universe because the Standard Model of particle physics
has almost 100 fields, and this number may increase by two if the Standard
Model is contained in a supersymmetric theory. Although the behaviour of these
(massless) matter fields—scalars, the Dirac spinors, and vectors in curved
space-time—is conformal invariant, some divergences are observed because of
the presence of the one-loop vacuum contributions. In the renormalized action,
some counterterms are required to break the matter action’s conformal
invariance in order to cancel the poles of the divergence component. From the
classical point of view, the trace of the energy momentum tensor in a
conformally invariant theory is null. But renormalization procedures can lead
to the trace of an anomalous energy momentum tensor, which is the so-called
quantum anomaly or the conformal anomaly(we would recommend the reader [59,
60, 61, 62] for more details on conformal anomaly effects). The conformal
anomaly we have described be considered to have the following form [33]

$T_{A}=b\left(F+\frac{2}{3}\Box R\right)+b^{\prime}G+b^{\prime\prime}\Box R$
(42)

where $T_{A}$ is the trace of the stress-energy tensor, F is the square of the
4d Weyl tensor and G is a Gauss-Bonet curvature invariant, which are given by

$F=(1/3)R^{2}-2R_{ij}R^{ij}+R_{ijkl}R^{ijkl}$ (43)
$G=R^{2}-4R_{ij}R^{ij}+R_{ijkl}R^{ijkl}$ (44)

b and $b^{\prime}$ on the other hand are given by

$b=\frac{N+6N_{1/2}+12N_{1}+611N_{2}-8N_{HD}}{120(4\pi)^{2}}$ (45)
$b^{\prime}=-\frac{N+11N_{1/2}+62N_{1}+1411N_{2}-28N_{HD}}{360(4\pi)^{2}}$
(46)

with N scalar, $N_{1/2}$ spinor , $N_{1}$ vector fields , $N_{2}$ (= 0 or 1 )
gravitons and $N_{HD}$ being higher derivative conformal scalars. For usual
matter, $b>0$ and $b^{\prime}<0$ except for higher derivative conformal
scalars while $b^{\prime\prime}$ can be arbitrary. Quantum effects due to the
conformal anomaly act as a fluid with energy density $\rho_{A}$ and pressure
$p_{A}$. The total energy density is $\rho_{tot}=\rho+\rho_{A}$ . The
conformal anomaly, also known as the trace anomaly, can be given by the trace
of the fluid stress-energy tensor

$T_{A}=-\rho_{A}+3p_{A}$ (47)

The conformal anomaly corrected pressure and energy densities still obey the
continuity equation (2.10) and using that, we can write

$T_{A}=-4\rho_{A}-\frac{\dot{\rho_{A}}}{H}\left(\frac{1}{2\rho_{A}}-1\right)$
(48)

We can immediately contrast this form of the conformal anomaly with the one
obtained for standard cosmology in [33]

$T_{A}=-4\rho_{A}-\frac{\dot{\rho_{A}}}{H}$ (49)

And in standard cosmology using (4.8) one can express $\rho_{A}$ as an
integral in terms of $T_{A}$ as

$\rho_{A}=-\frac{1}{a^{4}}\int a^{4}HT_{A}dt$ (50)

Furthermore $T_{A}$ can be expressed in terms of the Hubble parameter as

$T_{A}=-12b\dot{H}^{2}+24b^{\prime}(-\dot{H}^{2}+H^{2}\dot{H}+H^{4})-(4b+6b^{\prime\prime})(H^{(3)}+7H\ddot{H}+4\dot{H}^{2}+12H^{2}\dot{H})$
(51)

And using this, one can have an expression for $\rho_{A}$ taking into account
conformal anomaly effects near the singularity. But obtaining a corresponding
integral for $\rho_{A}$ for the equation (4.7) is not possible in the same way
as in (4.10). Hence its not feasible to address a possible removal of Type I-
Type III singularities using conformal anomaly effects in this asymptotically
safe cosmology. Furthermore, it was shown in [63] that $f(R)$ gravity effects
can also facilitate in singularity removal. To understand whether $f(R)$
effects can do similarly for this cosmology, we modify the action (2.2) for
the $f(R)$ gravity case as

$\Gamma_{k}=\frac{1}{16\pi G_{k}}\int
d^{4}x\sqrt{-g}\left[f(R)+2\Lambda_{k}\right]$ (52)

Although the continuity equation remains the same as (2.10) for the $f(R)$
case, the Friedmann equation changes substantially. While we can consider many
complex $f(R)$ forms, for now lets consider the simplest form
$f(R)=R+f_{2}R^{2}$. In this scenario, the Friedmann equation, takes the form
[24]

$3H^{2}+18f_{2}(2H\ddot{H}-\dot{H}^{2}+6H^{2}\dot{H})=8\pi
G_{k}\rho+\Lambda_{k}$ (53)

The simplest $f(R)$ modification for this asymptotically safe cosmology, as
shown above, is still very complex.In fact, trying to ascertain the
singularity structure for various inhomogeneous EOS as we have done in section
3 for the usual action (2.2) is not looking feasible in this case and hence
considering any potential singularity removals through $f(R)$ gravity as well
does not look like a very optimistic prospect. So we conclude that the chances
singularity removals in an inhomogenous EOS asymptotically safe cosmology
through conformal anomaly and $f(R)$ gravity effects look very bleak.

##  5 Concluding remarks

In this paper we have considered the formation of cosmological singularities
in an asymptotically safe cosmology with inhomogenous equations of state(EOS).
While such explorations were previously done in various modified gravity and
quantum gravitationally inspired cosmologies (besides standard general
relativistic cosmology), no such an endeavour has been taken up till now in
the realms of renormalization group approaches to cosmology and we have hence
took up on this journey. We briefly discussed about the basics of asymptotic
safety and how it could be applied to gravity and cosmology. We also showed
that such a scenario permits a transition between phantom and quintessence
forms of universal evolution After this we discussed a particular form of
asymptotically safe cosmology where the cut off scale is proportional to the
Hubble parameter and we presented the cosmological basics like the Friedmann
equation of such a cosmology. We then considered an inhomogeneous form of EOS,
namely $p=-\rho-f(\rho)$, which has been well motivated physically from
various cosmological approaches. We then considered two different ansatz for
$f(\rho)$ and showed that one can have Type I - Type IV singularities in such
a cosmology, where we demonstrated plenty of cases where the singularities can
occur both in finite and infinite time. The conditions in which these
singularities occur is significantly different than how they occur in the
standard cosmology, with the formulations being even more involved.
Interestingly this difference comes about without the need of any free
parameters in the cosmological models, which is usually what one observes when
one goes for the same pursuit via various modified gravity/ QG inspired
cosmological approaches. Furthermore, we showed that usual singularity removal
methods like conformal anomaly effects and $f(R)$ gravity effects do not offer
much hope for singularity removal in this cosmology. In conclusion we have
discussed in detail, for the first time, how cosmological singularities can
form in asymptotically safe cosmologies with various inhomogeneous EOS and we
hope that our work will propel more investigations towards singularities in
such cosmologies. We would also like to mention something with regards to the
novelty of the work here. One may raise the point that in recent years a lot
of work has been focused on singularity formation in various cosmological
settings and this work is yet another one in that fray. But what separates our
work from others and provides its novelty is the paradigm we have considered.
While other approaches since the initial work of Nojiri and Odintsov have
considered such formations in modified gravity / quantum gravitational
cosmologies which are paradigms where the background dynamics as a whole
changes with a lot of free parameters ( like the brane tension for RS-II
cosmologies etc. ) , the asymptotically safe cosmology we have considered does
not completely changes the setup at the background level and nor does it rely
on additional free parameters in order for it to give out a singularity
formation status quo adequately different from that in usual GR for a similar
scenario. It is interesting in the sense that while the background cosmology
remains essentially like GR, the runnings of both the Newtonian coupling and
the cosmological constant, alongside a modified continuity equation, give out
a very fascinating singularity structure with the conditions for singularity
formation vastly different than what it is for a usual GR based cosmology.

##  6 Acknowledgements

The work by MK has been supported by the grant of the Russian Science
Foundation No-18-12-00213-P https://rscf.ru/project/18-12-00213/ and performed
in Southern Federal University. The authors would also like to wholeheartedly
thank Sergei Odintsov for helpful discussions with him on various aspects
related to the work in this paper. We would also like to thank the reviewer of
the paper for constructive comments on the work.

##  7 Appendix A : Possible alternative EOS forms

In passing, we would also like to entertain the possibility of other forms of
inhomogenous EOS. More exotic equations of states have also been discussed
previously [32] , like

$p=-\rho+f(\rho)+G(H)$ (54)

Let’s consider a first example of this EOS as

$p=w_{o}\rho+w_{1}H^{4}$ (55)

where $w_{o}$ is the EOS parameter in the simple homogeneous case. Using the
Friedmann equation (9), one can write

$p=\left(w_{o}+w_{1}N\right)\rho$ (56)

Hence, we see that the EOS parameter w has shifted effectively as

$w\to w_{eff}=w_{o}+Nw_{1}$ (57)

So we see that even if $w_{o}<-1$, as long as $w_{eff}>-1$ a big rip
singularity does not occur. From the other side we can start with quintessence
value of $w_{o}$ and the inhomogeneous EOS (56) with sufficiently negative
$w_{1}$ will brings the cosmological evolution towards a phantom era. Here we
see that the transition behaviour of EOS in asymptotically safe cosmology
mimics of what it does in the standard cosmology as well [32].

Furthermore,we can also discuss an EOS of this form

$p=-\rho-A\rho^{\alpha}-BH^{4\beta}$ (58)

where $f(\rho)=A\rho^{\alpha}$ and $G(H)=BH^{4\beta}$ . This EOS can reduced
to the form of (2.11) as well by considering the Friedmann equation (2.9).
thereafter which it becomes

$p=-\rho-A\rho^{\alpha}-B^{\prime}\rho^{\beta}$ (59)

We can write the scale factor and the $\rho-t$ relation in this case as

$a=a_{o}\exp\left(\frac{\rho^{-\beta}\left((1-\beta)\,_{2}F_{1}\left(1,-\frac{\beta}{\alpha-\beta};\frac{a}{\beta-\alpha}+2;-\frac{A\rho^{\alpha-\beta}}{B}\right)+2\beta\rho\,_{2}F_{1}\left(1,\frac{\beta-1}{\beta-\alpha};\frac{\alpha-1}{\beta-\alpha}+2;-\frac{A\rho^{\alpha-\beta}}{B}\right)\right)}{6(\beta-1)\beta
B}\right)$ (60)
$t-t_{o}=\frac{\rho^{-\beta-\frac{1}{4}}\left(8(4\beta\rho+\rho)\,_{2}F_{1}\left(1,\frac{3-4\beta}{4a-4\beta};\frac{\alpha-2\beta+\frac{3}{4}}{\alpha-\beta};-\frac{A\rho^{\alpha-\beta}}{B}\right)\right)}{6(4\beta-3)(4\beta+1)BM^{1/4}}+\\\
\frac{4(3-4\beta)\,_{2}F_{1}\left(1,-\frac{\beta+\frac{1}{4}}{\alpha-\beta};\frac{\alpha-2\beta-\frac{1}{4}}{\alpha-\beta};-\frac{A\rho^{\alpha-\beta}}{B}\right)}{6(4\beta-3)(4\beta+1)BM^{1/4}}$
(61)

While the expressions above look very interesting and would make up for a
worthwhile analysis, we would not be pursuing this endeavour here as we have
already discussed the existence of Type I-IV singularities in this scenario
for two different ansatz both in finite and infinite time. Hence this
exploration will be suitable for another time.

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