# Enhancing Quantised End-to-End ASR Models via Personalisation

###### Abstract

Recent end-to-end automatic speech recognition (ASR) models have become
increasingly larger, making them particularly challenging to be deployed on
resource-constrained devices. Model quantisation is an effective solution that
sometimes causes the word error rate (WER) to increase. In this paper, a novel
strategy of personalisation for a quantised model (PQM) is proposed, which
combines speaker adaptive training (SAT) with model quantisation to improve
the performance of heavily compressed models. Specifically, PQM uses a 4-bit
NormalFloat Quantisation (NF4) approach for model quantisation and low-rank
adaptation (LoRA) for SAT. Experiments have been performed on the LibriSpeech
and the TED-LIUM 3 corpora. Remarkably, with a 7x reduction in model size and
1% additional speaker-specific parameters, 15.1% and 23.3% relative WER
reductions were achieved on quantised Whisper and Conformer-based attention-
based encoder-decoder ASR models respectively, comparing to the original full
precision models.

Index Terms—  speaker adaptive training, quantisation, LoRA, Whisper, end-to-
end ASR

##  1 Introduction

End-to-end neural network models have achieved state-of-the-art results in
various Automatic Speech Recognition (ASR) tasks [1, 2, 3, 4]. However, these
advancements in accuracy usually come at the cost of increasing model size,
which not only substantially increases operational costs on the server but
also presents significant challenges in deploying them on resource-constrained
edge devices. More recently, universal large speech models, such as OpenAI
Whisper [1] have become increasingly popular. These models adopt a very large
amount of model parameters, making the deployment even more challenging and
demanding.

Model quantisation has been widely adopted as an effective way to reduce model
sizes, and has been widely studied and applied in both academia and industry
[5, 6, 7]. Model quantisation replaces floating-point weights with low-
precision values to considerably reduce the model size and inference time
without altering the model architecture. However, model quantisation often
results in a degradation in model performance due to the loss of precision.
Some studies employ quantisation-aware training (QAT) schemes [8, 9, 10] that
consider the effects of quantisation during training, while others utilize
more sophisticated quantisation methods [11, 12, 13]. Although these
approaches have mitigated the performance degradation issue directly from the
generic data perspective, the fact that the edge devices to deploy quantised
models are often personalised is under-explored. For these devices, such as
personalised voice assistants or smart door locks, improving performance for
the target speaker is the critical objective rather than the generic
performance. Consequently, this paper investigates the use of personalisation
to compensate for the degradation due to quantisation.

This paper proposes a novel strategy of personalisation for a quantised model
(PQM) which performs speaker adaptive training on a quantised end-to-end ASR
model. The PQM strategy adopts the block-wise NormalFloat4 (NF4) quantisation
[14] for model compression, which incurs a smaller performance loss compared
to conventional uniform quantisation. The speaker adaptive training is
performed using the Low-Rank Adaptation (LoRA) [15] approach. As the
adaptation data for a particular speaker is often very limited, PQM includes a
LoRA pretraining stage before the speaker adaptive training using existing
similar data to achieve faster and better convergence. Moreover, to further
mitigate the data scarcity issue, semi-supervised training is also explored in
PQM where the quantised model can be trained with labels generated by the
model itself.

The PQM strategy was implemented for the Conformer attention-based encoder-
decoder (AED) model and the Whisper model as two examples of end-to-end ASR
models in this paper. Experiments performed on the LibriSpeech and TED-LIUM 3
datasets demonstrated that, with nearly a 7-fold compression of the model, the
PQM strategy achieves a relative WER reduction of 15.1% and 23.3% on quantised
Whisper and Conformer AED models respectively, compared to the original full-
precision models. The main contribution of this paper can be summarised as
follows.

  * •

The PQM strategy, to the best of our knowledge, is the first work that
investigates personalisation to compensate for quantisation degradation on
edge devices.

  * •

A LoRA pretraining-based initialisation and a semi-supervised training
approach are proposed for speaker adaptive training.

  * •

PQM has been validated on both Conformer AED and Whisper models across two
datasets, with large improvements achieved compared to fully fine-tuned
baselines.

The rest of this paper is organised as follows: Sec. 2 summarises related
work. Sec. 3 explains the three stages and some details of the PQM strategy.
Sec. 4 describes the experimental setup and results are provided in Sec. 5.
The paper concludes in Sec. 6.

Fig. 1: Overview of the PQM strategy.

##  2 Related Work

###  2.1 Quantisation

Quantisation is the process of discretising an input with higher information
content to one with lower information content. Uniform quantisation is a
commonly used quantisation method where the value range is uniformly divided
into multiple intervals. This approach is widely supported in mobile inference
frameworks such as TFLite [16], MNN [17], and NCNN [18]. Recently, Quantile
Quantisation was proposed based on the Lossy Minimum Entropy Encoding [19],
and QLoRA was introduced [14] which employs NF4 quantisation on the weights of
pretrained neural networks to achieve near-lossless quantisation. Unlike
uniform quantisation, Quantile Quantisation determines the quantisation steps
and levels based on the data distribution. In data-dense regions, more
quantisation levels are allocated, ensuring an equal number of quantised
values in each quantisation bin. This allows for a more refined representation
of the data, as well as better handling of outliers and adaptation to non-
uniform distributions.

###  2.2 Speaker Adaptation

The objective of speaker adaptation is to minimize the mismatch between
speakers in training and testing conditions. Current neural network-based
methods for speaker adaptation can be broadly categorized into two types:
Embedding-based and Model-based.

Speaker embeddings map speakers into a continuous space using techniques like
i-vectors [20, 21, 22] or neural network bottlenecks [23, 24, 25]. Model-based
adaptation methods include three primary methods: Structured Transforms,
Speaker Adaptive Training (SAT), and Regularization. Structured Transforms,
such as the Learning Hidden Unit Contributions (LHUC) scheme [26] and
parameterised activation functions [27], modify the model structure or its
activations. SAT methods adjust model parameters to individual speakers using
approaches like SAT-embedding [28] and SAT-LHUC [29]. Regularization
techniques, such as L2 loss or KL divergence [30, 31], work to prevent
overfitting to specific speakers.

##  3 Methodology

###  3.1 Strategy Overview

The PQM strategy is illustrated in Fig. 1, which is divided into three stages.
In stage 1, we apply block-wise NF4 quantisation to the model’s primary weight
parameters. In stage 2, we pretrain the randomly initialised LoRA using data
from a large number of speakers, providing a more robust starting point for
subsequent speaker adaptation. In stage 3, we perform speaker adaptive
training on speaker-specific data, during which the entire model is frozen,
and only the LoRA parameters corresponding to each speaker are updated.

Stage 2 of PQM is particularly beneficial for the application scenario where
reasonable-sized training data of the target domain is available, e.g.
assuming there is some in-house data, with a very limited amount of target
speaker data.

###  3.2 k-bit NormalFloat Quantisation

The block-wise NF4 quantisation is adopted in this paper, which is applied to
the weight matrices that are the primary parameters of the model. While
standard floating point quantisation applies the same set of quantisation bins
to all weight matrices, the dynamic range of parameter values is not taken
into account, resulting in heavily unbalanced quantisation bins. NF4, on the
contrary, ensures each bin has an equal number of values by estimating the
quantile of the input matrices using the empirical cumulative normal
distribution. This leveraged the fact that the parameters of a weight matrix,
in general, follow a normal distribution [14].

Fig. 2: Illustration of the construction of quantiles for NF4 quantisation. It
comprises 16 quantisation bins, where the midpoint of each bin represents the
quantisation level.

Specifically, for a $k$-bit quantisation, $2^{k}+1$ quantiles (i.e. $2^{k}$
quantisation bins) of a theoretical $\mathcal{N}(0,1)$ distribution are first
estimated which equally divides the area under the distribution curve. Then,
these quantisation bins are normalised to be within the range $[-1,1]$, as
illustrated in Fig. 2. Finally, the parameters of a weight matrix are
normalised into the range $[-1,1]$ to find their corresponding quantiles by
dividing the maximum absolute value of those parameters. In this way, a
similar number of quantised values are obtained for each bin, allowing for a
more refined representation of the model parameters. To ensure zero is exactly
quantised to zero which is important for padding, asymmetric quantiles are
used that ensure the mid bin has the quantisation level of zero (see Fig. 2).

To reduce the influence of extreme values in weight matrices (i.e. outliers)
on the maximum absolute value normalisation, block-wise quantisation is
applied which divides the weight matrices into small blocks and quantises each
block with separate normalisation factors. In this way, outliers in the input
tensor are confined to individual blocks, reducing their overall impact on
quantisation. As a result, block-wise quantisation allows for individual
normalisation factors for each block, resulting in a more fine-grained overall
quantisation.

###  3.3 LoRA for Speaker Adaptation

Compared to full fine-tuning, LoRA adjusts only the low-rank subspace
parameters of the model, thereby achieving higher computational efficiency and
lower costs for computation and storage. In scenarios with limited speaker
data, full fine-tuning methods may be prone to model overfitting, whereas LoRA
can alleviate this issue.

Fig. 3: LoRA. Initially, the pretrained weight parameters $W_{0}$ are frozen.
For $A$, random Gaussian initialisation is employed, whereas $B$ is
initialised with zeros.

For the pretrained ASR model with weight matrix $W_{0}\in\mathbb{R}^{d\times
k}$, its update is expressed through the following equation, where
$B\in\mathbb{R}^{d\times r}$ and $A\in\mathbb{R}^{r\times k}$, with the rank
$r\ll\min(d,k)$.

$W_{0}+\Delta W=W_{0}+BA$ (1)

In speaker adaptive training, only the LoRA parameters corresponding to each
speaker are updated. In this way, effective adaptation to different speakers
can be achieved by updating a minimal set of parameters. Moreover, in cases
where the base model is quantised, full-precision LoRA serves to some extent
to restore full-precision performance to the base model.

Although the target speaker data is always limited, in reality, the target
domain data of other speakers is usually available. Therefore, PQM leverages
those data to find a better initialisation point for LoRA weights before
performing speaker adaptation, referred to as LoRA pretraining. This allows
the LoRA parameters to adapt to the target-domain ASR task, providing a more
robust starting point for subsequent speaker adaptation. When using LoRA, each
speaker corresponds to one set of LoRA parameters. The number of parameters is
given by $|\Theta|=2\times L_{\text{LoRA}}\times d_{\text{model}}\times r$,
where $L_{\text{LoRA}}$ is the number of weight matrices to apply LoRA,
$d_{\text{model}}$ is the attention layer dimension, and $r$ is the rank.

##  4 Experimental Setup

###  4.1 Data

LibriSpeech is an English audiobook dataset. We selected 5 male speakers and 5
female speakers with the largest number of utterances from train-clean-360 as
speaker adaptation data. Each speaker contributes approximately 150
utterances, resulting in a total speech duration of roughly 25 minutes. For
LoRA pre-training, the train-clean-100 set was used which does not have any
speaker overlap with the selected speakers.

TED-LIUM 3 (TL3) is a TED talks dataset. We selected 16 speakers from the test
set as speaker adaptation data. On average, each speaker has 161 utterances
(14 minutes).

Speaker adaptation data for LibriSpeech and TL3 was divided randomly, where
2/5 was divided into the train set, 1/5 was divided into the dev set, and 2/5
was divided into the test set. On average, each speaker has 6-10 minutes of
training data, while the dev and test data remains constant across all
experiments. We denote the partitioned test sets as LibriSpeech-SA and TL3-SA
respectively in the results. Data partition details are provided1.

###  4.2 Model and training specifications

In order to verify the effectiveness of PQM, we use Whisper and Conformer AED
models as two widely used models as examples.

Whisper is a transformer-based AED model released by OpenAI trained on 680k
hours of audio. The base.en model with a full model size of 278MB was used.
The encoder has 6 Transformer blocks with 2048 hidden dimensions, and the
output size is 512. The decoder has 6 Transformer blocks with 2048 hidden
dimensions. The Transformer-related weight matrices are all 512 by 512
dimensional. Feature processing and model training followed [1, 32].

Conformer AED is a hybrid CTC/attention-based encoder-decoder model, whose
FP32 model size is about 131MB. The training follows ESPnet [33] with 0.3 CTC
weight and 80-dim FBank features. The Conformer encoder has 12 blocks with
1024 hidden dimensions. The decoder uses a 6-block transformer architecture
with 2048-dim linear units. The Transformer-related weight matrices are all
256 by 256.

The baseline system is the quantised system without personalisation. When
training LoRA from scratch (rank=4), the LoRA parameter sizes of Whisper and
Conformer are 1.2MB and 0.8MB. When using pretrained LoRA (rank=1), the LoRA
parameter sizes of Whisper and Conformer are 0.3MB and 0.2MB. Identical hyper-
parameter settings were used for all speakers. Models are evaluated using WER
averaged across all utterances from the test set speakers.

##  5 Evaluation Results and Analysis

Table 1: WER on the LibriSpeech-SA using quantised Whisper and Conformer for
the linear, convolution, and embedding layers.

System Quantise WER(%) Model Size (MB) Comp. Ratio linear conv embed Whisper
$\times$ $\times$ $\times$ 10.02 277.8 - $\checkmark$ $\times$ $\times$ 10.46
130.8 2.12 $\checkmark$ $\checkmark$ $\times$ 10.60 127.8 2.17 $\checkmark$
$\times$ $\checkmark$ 11.37 42.2 6.58 $\checkmark$ $\checkmark$ $\checkmark$
11.22 38.3 7.25 Conformer $\times$ $\times$ $\times$ 12.43 130.9 -
$\checkmark$ $\times$ $\times$ 12.51 31.6 4.14 $\checkmark$ $\checkmark$
$\times$ 12.69 23.4 5.59 $\checkmark$ $\times$ $\checkmark$ 12.61 27.3 4.79
$\checkmark$ $\checkmark$ $\checkmark$ 12.77 19.1 6.85

First, WER and model compression ratios of systems after quantising different
parts are shown in Table 1. The majority of parameters in both the Whisper and
Conformer models reside in the linear layers. Notably, applying NF4
quantisation to these layers has a very small impact on the performance.
Furthermore, WER increased by 1.20% for Whisper and only 0.34% for Conformer
upon NF4 quantisation. This suggests that models trained on smaller datasets
are more robust to the quantisation noises under NF4 quantisation. In the
following experiments, models with the highest compression ratios (i.e. the
last row of each model in Table 1) are used.

Table 2: WER on the LibriSpeech-SA and TL3-SA using quantised Whisper models. Whisper-baseline: Whisper after NF4 quantisation without adaptive training. FFT refers to full fine-tuning which trains all model parameters. Scratch refers to initialising LoRA weight randomly, and pretrain refers to the full PQM strategy with LoRA pretraining. System | WER(%)
---|---
LibriSpeech-SA | TL3-SA
Whisper-baseline | 11.22 | 7.71
Whisper-FFT-FP32 | 9.59 | 6.85
Whisper-FFT-NF4 | 10.51 | 7.19
Whisper-LoRA-scratch-NF4 | 9.67 | 6.95
Whisper-LoRA-pretrain-NF4 | 8.51 | 6.72

Table 2 shows the performance of PQM on the Whisper base.en model. Compared to
the baseline, the WER reduction achieved by fine-tuning all model parameters
at full precision on target speaker data was largely reduced after model
quantisation. As a result, the Whsper-FFT-NF4 model only achieved around 6.3%
relative WER reduction after speaker adaptive training. When LoRA was applied
in conjunction with NF4 quantisation (i.e. the PQM strategy), without LoRA
pretraining, the performance was already on par with the Whsper-FFT-FP32 full
precision model and achieved a 13.8% relative WER reduction on LibriSpeech-SA
and a 9.9% relative WER reduction on TL3-SA tasks. Moreover, when LoRA
pretraining was applied in PQM, the improvements were further enlarged, with
24.2% and 12.8% relative WER reductions on LibriSpeech-SA and TL3-SA sets
respectively. Note that the LoRA pretraining for TL3-SA was in fact cross-
data, as the pretraining was done on the LibriSpeech clean-100 training set
while directly applied to the TL3-SA data for adaptive training. This
underscores the effectiveness of pretraining LoRA on data that resembles
speaker-specific data.

Table 3: WER on the LibriSpeech-SA using Conformer AED. Conformer-baseline: Conformer AED after NF4 quantisation without adaptive training. FFT, LoRA, scratch and pretrain follow the same definition as Table 2 System | WER(%)
---|---
Conformer-baseline | 12.77
Conformer-FFT-FP32 | 10.43
Conformer-FFT-NF4 | 11.99
Conformer-LoRA-scratch-NF4 | 10.25
Conformer-LoRA-pretrain-NF4 | 9.54

The same set of experiments was also performed for the Conformer model as
shown in Table 3. Note that as the Conformer AED is trained on train-clean-100
already, we select 250 speakers from LibriSpeech train-clean-360 for LoRA
pretraining. As before, the Conformer-LoRA-scratch-NF4 performed on par with
the Conformer-FFT-FP32 full precision model, which achieved a 19.7% relative
WER reduction compared to the baseline. The Conformer-LoRA-pretrain-NF4 model
achieved a further WER reduction, resulting in a relative 25.3% WER reduction
compared to the baseline.

Fig. 4: WER on the LibriSpeech-SA using Whisper-LoRA-pretrain-NF4 under
different numbers of utterances. Utt-num=0 refers to Whisper-LoRA-pretrain-NF4
without speaker adaptive training.

Next, the influence of the number of training utterances was explored as shown
in Fig. 4. It is evident that LoRA pretraining serves as a favourable starting
point for speaker adaptive training. Moreover, the cost-effectiveness was
maximized when 20 to 30 utterances were used for adaptive training with PQM.

Table 4: WER on the LibriSpeech-SA using Whisper-LoRA-pretrain-NF4 and
Conformer-LoRA-pretrain-NF4 under different label sources.

System | Label Source | WER(%)
---|---|---
Whisper-LoRA-pretrain-NF4 | No adaptive training | 9.52
Ground truth | 8.51
Whisper-large | 9.01
Whisper-base.en | 9.44
Conformer-LoRA-pretrain-NF4 | No adaptive training | 11.81
Ground truth | 9.54
Whisper-large | 10.46
Whisper-base.en | 10.71

Finally, to further alleviate the data scarcity issue, semi-supervised
training for PQM was investigated for the Whisper and Conformer model and
results are shown in Table 4. As shown in the results, both Whisper-large and
Whisper-base generated labels yielded improvements. Remarkably, with the
training guided by Whisper-large labels showing, 5.4% and 11.4% relative WER
reduction were achieved on the Whisper and Conformer model compared to without
doing any speaker adaptive training. Note that this process did not require
any human-annotated labels.

##  6 Conclusions

This paper proposes the PQM strategy to compensate for the performance loss
due to model quantisation via personalisation. PQM adopted the NF4
quantisation approach together with LoRA-based speaker adaptive training and
was applied to both the Conformer-based AED model and the Whisper model.
Experiments on LibriSpeech and TL3 datasets using speaker adaptive training
data partitions showed that personalisation largely improved the performance
of quantised models. Specifically, using PQM, 15.1% and 23.3% relative WER
reductions were achieved on quantised Whisper and Conformer-based AED models
respectively, compared to the full precision models.

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# Study of central light distribution in nearby early-type galaxies hosting
nuclear star clusters

K. Sruthi,1 and C. D. Ravikumar1
1Department of Physics, University of Calicut, Malappuram 673635, India
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present analysis of 63 nearby ($<$ 44 Mpc) early-type galaxies hosting
nuclear star clusters using the recently discovered parameter Central
Intensity Ratio (CIRI) determined from near-infra-red (3.6 $\mu$m)
observations with the Infra-red-array-camera of _Spitzer_ space telescope. The
CIRI, when combined with filters involving age and $B-K$ colour of host
galaxies, helps identify two distinct classes of galaxies hosting nuclear star
clusters. This is independently verified using Gaussian Mixture Model. CIR
shows a positive trend with faint, low mass, and blue galaxies in the sample,
while the opposite is true for bright, high mass, and red galaxies, albeit
with large scatter. The variation of CIRI with central velocity dispersion,
absolute B band magnitude, dynamical mass, and stellar mass of host galaxies
suggests that the mass of nuclear star clusters increases with that of host
galaxies, for faint, low mass, young and blue galaxies in the sample. In
bright, high-mass, old and red galaxies, on the other hand, the evolution of
nuclear star clusters appears complex, with no apparent trends.The analysis
also reveals that redder galaxies ($B-K>3.76$) are more likely to be dominated
by the central black-hole than the nuclear star clusters, while for bluer
galaxies ($B-K<3.76$) in the sample the situation is quite opposite.

###### keywords:

Galaxy: centre – Galaxy: evolution – galaxies: photometry

††pubyear: 2023††pagerange: Study of central light distribution in nearby
early-type galaxies hosting nuclear star clusters–Study of central light
distribution in nearby early-type galaxies hosting nuclear star clusters

##  1 Introduction

The evolution of galaxies is closely connected with interactions and mergers
(Tonnesen & Cen, 2012). The minor and major interactions (e.g. Woods & Geller,
2007; Robaina et al., 2009; Lambas et al., 2012) have significant effects on
the star formation, morphology, kinematics, and physical conditions of gas,
both in the central and outer regions of galaxies. Large quantities of gas can
be funnelled to the central region of galaxies during major mergers,
triggering enormous star formation and rapid fuelling of central black holes
(Dutta et al., 2018). Due to these reasons, the very innermost regions of
galaxies provide examples of extreme astrophysical environments, where
numerous complex mechanisms operate simultaneously that have direct as well as
indirect roles in the formation and evolution of galaxies. The properties of
nuclear regions are thought to be linked to the formation history of galaxies
where the collapse of gas and merging of galaxies can significantly influence
the creation of different types of galaxies (Gullieuszik et al., 2014). As a
result, the very central regions of galaxies host fascinating objects like
supermassive black holes (hereafter SMBHs) and active galactic nuclei (AGN)
whereas nuclei of some galaxies are undergoing extreme star formation at their
centres in the form of central starbursts, and extreme stellar densities
(Pechetti et al., 2020). All these are likely connected to the global
properties of their host galaxies, and the co-evolution of various components
is invoked to account for the observed correlations (Gullieuszik et al.,
2014). For example, masses of SMBHs have been shown to correlate with a range
of host-galaxy properties (e.g. Kormendy & Ho, 2013; Graham, 2016; Saglia et
al., 2016; Davis et al., 2017; Aswathy & Ravikumar, 2018; Sruthi & Ravikumar,
2021).

Over the past two decades, high-resolution observations with the Hubble space
telescope have revealed that, in some galaxies, the central SMBH is surrounded
by a massive, very compact, star cluster composed of stars up to 108 in
number, commonly known as nuclear star cluster (NSC). These massive stellar
clusters reside at the photometric and dynamic centres of most intermediate
and low luminosity galaxies of all Hubble types (Böker et al., 2002; den Brok
et al., 2014; Georgiev & Böker, 2014). NSCs are known to be the densest
stellar systems in the universe with millions of solar masses packed within
the central few parsecs of a galaxy and therefore host unique stellar dynamics
(Cole & Debattista, 2016). These objects are identified as a clear over-
density in stars above the inwardly extrapolated light profile of a galaxy
within the central 50 pc (Neumayer et al., 2020). NSCs are sited at the bottom
of the potential well (Arca-Sedda & Capuzzo-Dolcetta, 2014) and contain old as
well as young stellar populations (e.g. Rossa et al., 2006; Neumayer &
Walcher, 2012). NSCs and SMBHs are proposed to be related to their host
galaxies in similar ways, which activated a new wave of interest in the study
of NSCs and their role in the growth of SMBH (e.g. Rossa et al., 2006; Wehner
& Harris, 2006; Ferrarese et al., 2006). Galaxies and NSCs display a variety
of scaling relations suggesting that, their formation is intricately linked to
that of their host galaxy (e.g. Georgiev & Böker, 2014; Georgiev et al.,
2016). However, unlike SMBHs, the formation history of NSCs is directly
visible through their stellar populations which allows for detailed studies of
the mass accretion history that takes place in galactic nuclei (Kacharov et
al., 2018). Though NSCs are observed in galaxies with every type of the Hubble
sequence, their modes of formation and evolution are still under debate (Arca-
Sedda & Capuzzo-Dolcetta, 2014). Two different mechanisms are proposed for the
formation of NSCs, infall of star clusters that formed elsewhere in the host
galaxy (e.g. Tremaine et al., 1975; Capuzzo-Dolcetta & Miocchi, 2008;
Antonini, 2013) and _in situ_ formation and build-up through star formation
following the accretion of gas in the centre of galaxies (Bekki, 2007). There
is a need to accurately determine scaling relations involving NSCs to improve
our understanding of the possible evolution of NSCs with their host-galaxies
(Pechetti et al., 2020). Studying how NSCs form and how they are related to
the growth of central massive black holes and their host galaxies is crucial
for our understanding of the evolution of galaxies and the physics that have
shaped their central components (Antonini et al., 2015).

NSCs and SMBHs are sometimes jointly referred to as central massive objects
(CMO; Ferrarese et al., 2006; Arca-Sedda & Capuzzo-Dolcetta, 2014) whose
presence is connected to the host galaxy mass; galaxies with mass above 1010
M$\odot$ usually host SMBH while lighter galaxies have a well resolved nuclear
star cluster. Moreover, a transition region exists for galaxies with a mass
between 108 M$\odot$ and 1010 M$\odot$ where both the objects co-exist. It
seems that there is a continuous sequence from NSC-dominated galaxies to SMBH-
dominated galaxies as the galaxies grow bigger (Arca-Sedda et al., 2016). The
formation of NSCs may also be connected to the formation of SMBH in the
centre, and these objects could therefore trace the evolutionary history of
the host galaxy formation. For instance, NSCs sit on the low mass
extrapolation of the scaling relation between the SMBH mass and the total mass
of the host galaxy (Gullieuszik et al., 2014). However, interactions with
SMBHs have the potential to destroy NSCs or inhibit NSC growth in most massive
galaxies (Fahrion et al., 2021). Very few NSCs are observed in galaxies
harbouring very massive SMBHs at their centres; this could be an indication of
a physical connection between the presence (or absence) of NSCs and SMBHs in
galaxies (Arca-Sedda et al., 2016). The relationship between NSCs and SMBHs is
not well understood as the size of a sample of galaxies with both NSC and a
central BH is quite limited (Nguyen et al., 2018). Galaxies with masses lower
than 1010 M⊙ show clear evidence for nucleation, but little evidence for an
SMBH. Contrarily, galaxies with masses above $\sim$ 1011 M⊙ are dominated by
SMBH but show no evidence for nucleation (Antonini et al., 2015). Therefore,
it is an open question whether NSCs are an essential ingredient in (or an
intermediate step towards) the formation of a supermassive black hole in the
galaxy nucleus (Georgiev & Böker, 2014).

The co-evolution of nuclear star clusters and central black holes is presently
a very active field of research. In this scenario, we explore the connection
between NSCs, SMBHs and host galaxy properties with the aid of central
intensity ratio (CIR; Aswathy & Ravikumar, 2018, 2020; Sruthi & Ravikumar,
2021). The central intensity ratio (defined in section 2.1) is a photometric
parameter that measures the variation of light intensity at the very centre of
the (projected) galaxy image. Simple Monte Carlo simulations involving CIR
show remarkable stability against variations in distance and orientation of
ellipsoidal systems in the nearby universe (Aswathy & Ravikumar, 2018, 2020).
The straightforward photometric definition of CIR makes it sensitive to any
addition (or subtraction) to the light near the centre of a galaxy. The
addition of light is possible in systems with high star formation near the
centre while quenching via feedback from the central black hole can reduce the
CIR value (Sruthi & Ravikumar, 2021). The various correlations shown by CIR in
field early-type galaxies (hereafter ETGs) (Sruthi & Ravikumar, 2021) explored
the co-evolution scenario in galaxies. The CIR is also found to contain
information about the star formation near the central region of galaxies
(Aswathy & Ravikumar, 2020). Therefore, the CIR can be used as an ideal tool
in studying galaxies hosting NSCs.

This paper is organized as follows. Section 2 describes the sample selection,
observations, data reduction, and calculation of CIRI. Section 3 presents
variations of CIRI with different host galaxy properties. We present a
discussion of results and conclusion in Section 4.

##  2 The Data

We constructed a sample of 80 nearby early-type galaxies hosting nuclear star
clusters based on the availability of _Spitzer_ /IRAC 3.6$\mu$m observations
adopted from literature. We avoided 15 galaxies from the above sample due to
their relatively small size which could affect the determination of CIRI.
Also, we avoided two galaxies NGC 205 and M32 as they are pretty close and
tidally influenced by the nearby galaxy M31. Thus the sample includes 63
nearby ETGs hosting NSCs composed of 26 ellipticals and 37 lenticulars. The
sample along with the references is listed in Table 1.

###  2.1 Central intensity ratio

Central intensity ratio (CIR; Aswathy & Ravikumar, 2018) is defined as,

$CIR=\frac{I_{1}}{I_{2}-I_{1}}=\frac{10^{0.4(m_{2}-m_{1})}}{1-10^{0.4(m_{2}-m_{1})}}$
(1)

where _I 1_ and _I 2_ are the intensities and _m 1_ and _m 2_ are the
corresponding magnitudes within the inner and outer apertures respectively.
Aperture photometry (MAG$\\_$APER) using Source extractor (SEXTRACTOR, Bertin
& Arnouts, 1996) was carried out at the photometric centre of the galaxy
image. CIR measured in the optical band (hereafter CIRV) used the inner and
outer apertures of 1.5 and 3 arcsec respectively. For the present study, we
have chosen the inner and outer radii as 12 and 24 arcsec, respectively for
the calculation of CIRI. The inner radius is selected to contain the effects
of the point spread function (PSF) in the Spitzer images and the outer radius
is chosen to be smaller than the half-light radii of the sample galaxies. This
ensures that the estimation of CIR is reasonably stable against differences in
size and surface brightness profile (Aswathy & Ravikumar, 2018) of and
distance to galaxies. However, a direct comparison of CIRV and CIRI is not
possible as there are large differences in the PSF, depth and aperture sizes
used.

The simple definition of CIR helps avoid any dependence on a form following
central intensity, I(0) (i.e. surface brightness at a radial distance r, I(r)
= I(0) f(r) where f(r) is a function of r). On the other hand, the definition
boosts any addition to (or subtraction from) the central intensity I(0)
(Aswathy & Ravikumar, 2018, 2020; Sruthi & Ravikumar, 2021). NSCs are very
tiny objects near the centre covering around 10 parsecs only. CIR measures the
total projected light along the centre and hence the smaller the apertures the
better for identifying variations in flux due to NSCs.

Table 1: Table 1 lists the properties of the sample galaxies. Name of the galaxy with its environment adopted from NED (column 1) where ‘*’(star) symbol denotes galaxy from dense environment whereas ‘+’(plus) symbol denotes galaxy from rare environment, References from which the sample galaxies are adopted (2) where N20 - Neumayer et al. (2020), SG13 - Scott & Graham (2013), F21 - Fahrion et al. (2021) and P20 - Pechetti et al. (2020), Morphological type (3) taken from Scott & Graham (2013) and NED, Distance of the galaxy (4) adopted from NED, for the galaxy NGC 5389, distance is adopted from Erroz-Ferrer et al. (2013), for NGC 4552,the distance is taken from Xu et al. (2005), CIR computed in the near-infra-red band (5), uncertainty in the measurement of CIRI (6), Mass of central black hole (7) with references where a -Neumayer et al. (2020), b - Graham & Soria (2019), c - Kormendy & Ho (2013), d - Rusli et al. (2011), e - de Nicola et al. (2019) , f - Krajnović et al. (2009), g - Ding et al. (2020), h - Shapiro et al. (2006), i - Smith et al. (2021), j - Nguyen et al. (2022), k -Walsh et al. (2012), l - Mould et al. (2015), m - Zhu et al. (2021), Age of the galaxy (8) adopted from Chilingarian (2009), Amblard et al. (2014) and Dabringhausen & Fellhauer (2016), Dynamical mass (9) of the galaxy taken from Scott & Graham (2013) and Dabringhausen & Fellhauer (2016), Central velocity dispersion of the galaxy (10) and absolute B band magnitude of the galaxy (11) adopted from HyperLEDA, Stellar mass of the galaxy (12) taken from references listed in column 2, $B-K$ colour of the galaxy (13) calculated using apparent total B and K magnitudes obtained from HyperLEDA, Mass of the nuclear star cluster (14) of the galaxy adopted from references listed in column 2. Galaxy | Ref. | Type | Dist. | CIRI | $\bigtriangleup_{CIR_{I}}$ | MBH | Age | Mdyn. | $\sigma$ | MB | Mstr | $B-K$ | MNSC
---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  |  | (Mpc) |  |  | (log M$\odot$) | (Gyr) | (log M$\odot$) | (kms-1) | (mag) | (log M$\odot$) | (mag) | (log M$\odot$)
IC1459+ | N20 | E | 28.70 | 0.93 | 0.01 | 9.39a | - | 11.67 | 296.11 | -21.40 | 11.10 | 4.12 | 7.69
IC3032∗ | SG13 | dE | 15.00 | 0.52 | 0.09 | 4.82b | 8.30 | 9.31 | 20.80 | -15.79 | 8.70 | 2.97 | 5.66
IC3065∗ | SG13 | S0 | 17.15 | 0.63 | 0.05 | 5.21b | 5.30 | 9.30 | - | -16.84 | 9.30 | 3.37 | 5.70
IC3292+ | SG13 | dS0 | 15.59 | 0.80 | 0.05 | 4.78b | 10.60 | 9.70 | 32.10 | -16.09 | 8.90 | 3.27 | 6.13
IC3381∗ | SG13 | dE | 17.16 | 0.55 | 0.06 | 4.94b | 5.70 | 9.40 | 40.35 | -17.27 | 9.30 | 2.72 | 6.73
IC3468∗ | SG13 | E | 16.27 | 0.62 | 0.05 | 4.60b | 4.80 | 9.50 | 32.95 | -17.18 | 9.40 | 3.14 | 6.66
IC3470∗ | SG13 | dE | 17.50 | 0.66 | 0.05 | 5.43b | 6.20 | 9.50 | 52.40 | -16.70 | 9.20 | 3.39 | 6.80
IC3586∗ | SG13 | dS0 | 20.04 | 0.69 | 0.07 | 5.35b | 6.50 | 9.30 | 24.40 | -16.77 | 8.80 | 2.33 | 5.81
IC3735∗ | SG13 | dE | 15.44 | 0.56 | 0.08 | 4.83b | 7.70 | 9.60 | 37.00 | -16.69 | 9.10 | 3.01 | 6.54
IC3773+ | SG13 | dS0 | 17.42 | 0.71 | 0.05 | 5.80b | 4.90 | - | 59.49 | -17.34 | 9.30 | 3.06 | 6.10
IC798∗ | SG13 | E | 18.90 | 0.44 | 0.07 | 4.76b | 9.94 | 9.03 | - | -16.11 | 9.00 | 3.19 | 6.93
IC809∗ | SG13 | dE | 15.77 | 0.62 | 0.05 | 5.15b | 5.00 | 9.70 | 37.00 | -16.68 | 9.40 | 3.78 | 6.79
NGC0404+ | N20 | S0 | 3.13 | 0.93 | 0.02 | 5.74a | - | 10.04 | 34.59 | -16.43 | 8.86 | 3.66 | 7.08
NGC1023+ | SG13 | SB0 | 10.84 | 0.81 | 0.01 | 7.62c | 12.30 | 11.00 | 197.10 | -20.61 | 10.80 | 3.31 | 6.64
NGC1332+ | N20 | S0 | 21.90 | 0.82 | 0.01 | 9.18d | - | - | 294.62 | -20.74 | 11.69 | 4.13 | 7.14
NGC1336+ | F21 | S0 | 17.30 | 1.21 | 0.03 | - | - | - | 97.67 | -18.21 | 9.97 | 3.28 | 8.74
NGC1375+ | F21 | S0 | 16.20 | 0.99 | 0.03 | - | - | 9.76 | 64.56 | -18.39 | 9.76 | 3.54 | 8.37
NGC1381+ | F21 | S0 | 17.50 | 0.96 | 0.01 | - | - | 9.75 | 147.14 | -19.34 | 10.35 | 3.97 | 8.45
NGC1389+ | F21 | S0 | 16.80 | 0.95 | 0.02 | - | - | - | 124.57 | -19.25 | 10.52 | 3.77 | 8.15
NGC1399+ | SG13 | E | 17.71 | 0.64 | 0.01 | 8.95e | - | 11.70 | 332.23 | -21.22 | 11.30 | 4.10 | 6.81
NGC2549+ | SG13 | SB0 | 16.78 | 1.04 | 0.01 | 7.15f | 4.80 | 10.40 | 141.11 | -18.75 | 10.20 | 4.02 | 7.04
NGC2778+ | SG13 | SB0 | 42.72 | 1.39 | 0.02 | 7.15g | 13.40 | 10.70 | 154.03 | -18.74 | 10.10 | 3.79 | 6.83
NGC2787+ | N20 | S0 | 7.41 | 0.83 | 0.01 | 7.85a | - | - | 191.92 | -18.45 | 10.14 | 4.30 | 6.28
NGC3115+ | SG13 | S0 | 10.28 | 0.92 | 0.01 | 8.95e | - | 11.00 | 260.23 | -20.05 | 10.90 | 4.19 | 7.18
NGC3115B+ | P20 | S0 | 11.00 | 0.56 | 0.07 | - | - | - | 26.0 | -16.97 | 8.95 | 3.12 | 6.86
NGC3379∗ | N20 | E | 6.80 | 0.77 | 0.01 | 8.08h | 13.40 | 10.61 | 202.47 | -20.16 | 10.83 | 3.94 | 4.15
NGC3384+ | SG13 | SB0 | 11.26 | 0.52 | 0.01 | 7.03i | 7.70 | 10.50 | 144.11 | -19.11 | 10.60 | 4.13 | 7.34
NGC3585+ | SG13 | S0 | 17.29 | 0.56 | 0.01 | 8.52c | - | 11.20 | 214.20 | -20.97 | 11.10 | 4.10 | 6.60
NGC3593+ | P20 | S0 | 5.50 | 0.68 | 0.01 | 6.38j | - | - | 73.91 | -18.36 | 9.78 | 4.42 | 8.20
NGC3998+ | N20 | S0 | 14.10 | 1.25 | 0.01 | 8.38k | 11.30 | 10.64 | 265.09 | -19.40 | 10.66 | 4.00 | 5.92
NGC4026+ | SG13 | S0 | 17.83 | 0.74 | 0.01 | 8.26e | - | 10.60 | 173.49 | -19.11 | 10.40 | 4.10 | 7.11
NGC4261+ | N20 | E | 24.00 | 0.81 | 0.01 | 8.70a | 16.20 | 11.42 | 296.73 | -21.44 | 11.56 | 3.90 | 6.32
NGC4291+ | N20 | E | 24.30 | 1.26 | 0.01 | 8.51a | - | 11.35 | 292.66 | -20.35 | 10.80 | 3.93 | 6.70
NGC4352∗ | SG13 | S0 | 19.00 | 0.97 | 0.03 | 6.09b | 4.50 | 10.30 | 67.58 | -18.04 | 9.80 | 3.52 | 6.84
NGC4374∗ | N20 | E | 10.10 | 0.70 | 0.01 | 9.18a | 14.90 | 11.28 | 277.59 | -21.73 | 11.56 | 3.76 | 7.80
NGC4379∗ | SG13 | S0 | 16.27 | 0.93 | 0.02 | 7.39b | 2.60 | 10.40 | 110.46 | -18.55 | 10.10 | 3.74 | 7.66
NGC4387∗ | SG13 | E | 17.04 | 0.80 | 0.02 | 6.89b | 2.70 | 10.20 | 99.95 | -18.47 | 10.10 | 3.75 | 7.54
NGC4452∗ | SG13 | S0 | 16.04 | 0.70 | 0.03 | 6.89b | 4.20 | 9.80 | 99.89 | - | 10.00 | 3.66 | 6.38
NGC4458∗ | SG13 | E | 18.39 | 1.12 | 0.02 | 6.83b | 3.80 | 10.40 | 97.35 | -18.27 | 9.90 | 3.57 | -
NGC4460∗ | P20 | S0 | 9.20 | 0.58 | 0.03 | - | - | - | 39.67 | -17.57 | 9.44 | 3.33 | -
NGC4476∗ | SG13 | S0 | 17.69 | 0.78 | 0.02 | 5.93b | 2.90 | 10.00 | 62.81 | -18.25 | 9.90 | 3.61 | 7.05
NGC4479∗ | SG13 | S0 | 18.33 | 0.70 | 0.03 | 6.43b | 4.30 | 10.00 | 80.48 | -17.83 | 9.80 | 3.70 | 6.70
NGC4482∗ | SG13 | dE | 18.91 | 0.68 | 0.05 | 4.99b | 4.30 | 10.70 | 41.83 | -17.73 | 9.50 | 3.06 | 6.95
NGC4486∗ | N20 | E | 9.20 | 0.53 | 0.01 | 9.80a | 17.70 | 11.43 | 323.00 | -21.59 | 11.78 | 3.83 | 8.30
NGC4550∗ | SG13 | E/S0 | 16.10 | 0.73 | 0.02 | 6.80b | 2.20 | 10.20 | 96.25 | -18.65 | 10.10 | 3.72 | -
NGC4551∗ | SG13 | E | 17.01 | 0.84 | 0.02 | 6.93b | 3.70 | 10.30 | 102.20 | -18.37 | 10.10 | 3.96 | -
NGC4552∗ | N20 | E | 17.10 | 0.86 | 0.01 | 8.70a | 12.60 | 10.90 | 250.31 | -20.48 | 11.42 | 3.93 | 6.96
NGC4578∗ | SG13 | S0 | 17.03 | 1.01 | 0.02 | 7.28b | 0.53 | 10.80 | 111.92 | -18.90 | 10.30 | 3.84 | 7.59
NGC4612∗ | SG13 | S0 | 17.53 | 1.19 | 0.02 | 6.58b | 2.30 | 10.40 | 85.76 | -19.11 | 10.30 | 3.51 | 7.18
NGC4623∗ | SG13 | E | 16.83 | 0.92 | 0.03 | 6.35b | 3.90 | 10.00 | 76.99 | -18.15 | 9.90 | 3.66 | -
NGC4649∗ | N20 | E | 10.20 | 0.54 | 0.01 | 9.32a | 17.70 | 11.42 | 330.50 | -21.50 | 11.69 | 4.04 | 6.30
NGC4697+ | SG13 | E | 12.17 | 0.88 | 0.01 | 8.13e | 11.30 | 11.10 | 165.22 | -20.33 | 11.10 | 3.90 | 7.45

Table 1
(continued) Galaxy Ref. Type Dist. CIRI $\bigtriangleup_{CIR_{I}}$ MBH Age
Mdyn. $\sigma$ MB Mstr $B-K$ MNSC (Mpc) (log M$\odot$) (Gyr) (log M$\odot$)
(kms-1) (mag) (log M$\odot$) (mag) (log M$\odot$) NGC5102+ N20 S0 3.05 1.08
0.01 5.96a - - 61.10 -17.91 9.84 3.29 7.87 NGC5389+ SG13 S0 32.20 1.04 0.02 -
- 10.60 116.13 -19.77 10.60 4.25 - NGC5422∗ SG13 S0 28.05 1.16 0.02 - 9.70
10.65 160.29 -19.75 10.50 4.02 7.13 NGC5587+ SG13 S0 35.60 1.00 0.03 - - -
89.12 -19.30 10.30 4.09 - NGC5689+ SG13 S0 43.30 1.06 0.01 - - - 150.02 -20.51
10.80 4.31 - NGC5838+ SG13 E/S0 19.72 1.02 0.01 8.00l 11.30 11.20 273.56
-20.04 10.70 4.17 7.67 NGC5854+ SG13 S0 14.62 0.93 0.02 - 4.30 10.30 102.36
-19.81 10.40 3.81 7.98 NGC6010+ SG13 S0 21.10 1.24 0.02 - 12.00 10.52 147.84
-20.00 10.50 4.21 7.53 NGC7457+ SG13 S0 36.03 0.87 0.02 6.95m - 9.89 67.97
-19.14 10.20 3.65 7.50 UGC7399A+ SG13 dE 16.90 0.53 0.07 5.33b 7.10 9.80 43.06
-17.22 9.10 2.50 6.59 UGC7436∗ SG13 dE 16.04 0.56 0.07 4.40b 5.90 9.81 32.17
-16.60 9.10 2.81 5.81

##  3 Variation of CIRI with host galaxy properties

CIR is found to be an important parameter in galaxy evolution studies
manifested through different correlations and trends reported in the
literature (e.g. Aswathy & Ravikumar, 2018, 2020; Sruthi & Ravikumar, 2021).
These studies suggest the use of CIR as a tool to explore the co-evolution
between various structural and dynamical properties of different Hubble types
of galaxies. This paper discusses the variation of CIRI with different host
galaxy properties for a sample of 63 nearby ETGs hosting NSCs.

Figure 1: Variation of CIRI with central velocity dispersion, $\sigma$, with
colouring schemes based on $B-K$ (panel a), age (panel b), environment (panel
c) and $B-K$ and age together (panel d) of the sample galaxies. In panel a,
blue galaxies ($B-K$ $<$ 3.76) are denoted with blue filled circles whereas
red galaxies ($B-K$ $>$ 3.76) are given in red filled circles. In panel b,
young galaxies (with age less than 10 Gyr) are shown in blue filled circles
while old galaxies (age greater than 10 Gyr) are marked with red filled
circles. In panel c, galaxies from the dense environment (DE) are denoted with
blue filled circles and galaxies from the rare environment (RE) are
represented as red filled circles, respectively. Please see the text for the
criteria used for defining the environment. In panel d, blue ($B-K$ $<$ 3.76)
as well as young (with age less than 10 Gyr) galaxies are represented in blue
filled circles whereas red ($B-K$ $>$ 3.76) and old (age greater than 10 Gyr)
galaxies are denoted with red filled circles. The bottom panel shows the
distribution of two clusters statistically identified using GMM. GMM cluster 1
galaxies are denoted with green filled circles whereas GMM cluster 2 galaxies
are given in purple filled circles. In the upper panel, the best fit linear
relations for the sub-samples separately are shown as dotted lines where
correlations are significant.

###  3.1 Variation of CIRI with central velocity dispersion

In Fig.1, variation of CIRI with central velocity dispersion is shown. Central
stellar velocity dispersion is a tracer of galaxy mass validated by various
studies (e.g. Auger et al., 2010). It is well known that, the luminosity as
well as Mg2 index of a galaxy tightly correlates with velocity dispersion
(e.g. Faber & Jackson, 1976; Colless et al., 1999). Likewise, a tight
correlation exists between the masses of supermassive black holes and the
velocity dispersions of the bulges that host them (Gebhardt et al., 2000). It
is also reported to exhibit a significantly strong correlation with the total
gravitating mass of a galaxy making it a very fundamental parameter in galaxy
evolution studies (Bogdán & Goulding, 2015). Even though no apparent
connection between CIR and velocity dispersion is seen, there appears the
possibility of at least two populations showing different trends. This is not
completely surprising. Owing to the simple definition of CIR, higher will be
its value when the (projected) intensity in the inner aperture is increased
(by virtue of, say, increased star formation near the centre) or when light
contribution in the outer shell ($r_{2}$ \- $r_{1}$) is decreased (as can
happen when obscured by dust). Hence it’s quite natural to expect the CIR to
be less effective when a sample contains sets with different characteristics
that can affect the value of CIR differently. In order to explore this, in
Fig. 1(a), we split the sample into blue and red ones using the median $B-K$
colour (3.76) of the sample galaxies. The median value was used for minimising
uncertainties associated with the low number statistics in a group. $B-K$
colour is a good indicator of the amount of dust extinction (Peletier et al.,
1995). Galaxies bluer than the median $B-K$ colour (3.76), hereafter blue
galaxies, are represented as blue filled circles and those redder than the
median, hereafter red galaxies, are shown as red filled circles. Blue galaxies
show a mild positive trend (linear correlation coefficient, $r=0.64$ with a
significance, $S=99.95$ per cent, see Table 2) in the plot suggesting an
increase in galaxy mass as CIRI increases while red galaxies in our sample do
not show any apparent association. The nature of clustering in the sample is
independently verified using Gaussian Mixture Model analysis (GMM; Bovy et
al., 2011; Wagenveld et al., 2022). A basic assumption of GMM is that all the
data points are drawn from a mixture of a finite number of Gaussian components
with unknown distribution parameters (Gao, 2018). The GMM can assign a
membership score (posterior probability) to each data point that describes how
similar each point is to each cluster’s centre; which is an automated process
and does not require manual intervention. GMM is being used in various fields,
including astrophysics (Chattopadhyay & Maitra, 2017; Kaplan et al., 2018), as
an unsupervised clustering technique. The representation of regularly
distributed subpopulations within a population can be done probabilistically
using GMM. This statistical model uses an Expectation Maximization (EM)
algorithm (Dempster et al., 1977) to optimize the model’s likelihood. In Fig.
1(e) the different groups identified by GMM are given in green and purple
filled circles. The result of GMM also suggests the presence of two groups
within the sample which are statistically different.

In Fig. 1(b), we use age as a parameter to explore the possible subsets. Here
SSP age reported in literature (Chilingarian, 2009; Amblard et al., 2014;
Dabringhausen & Fellhauer, 2016) is used. Galaxies with an age less than 10
Gyr (hereafter young galaxies) are denoted as blue filled circles and those
with an age more than 10 Gyr (hereafter old galaxies) are represented as red
filled circles. Here young galaxies follow a positive trend ($r=0.79$,
$S>99.99$ per cent), while the old ones appear to possess a negative trend
($r=-0.77$, $S=99.45$ per cent) though not reliable due to the limited number
of galaxies. Here also the GMM analysis identifies clusters similar to the
ones we constructed using age (Fig. 1(f)).

Next, we explore the influence of environment of galaxies. In order to do that
a galaxy is considered in the rare environment if it possesses a surrounding
galaxy density below 36 Mpc-3 within 0.5 Mpc in the plane of the sky and if no
nearest neighbour falls within 250 km s-1 in recession velocity, otherwise it
is treated from dense environment111Determined using the link
https://ned.ipac.caltech.edu/forms/denv.html.. In Fig. 1(c), galaxies from the
dense environment (hereafter DE) are given as blue filled circles whereas
those from the rare environment (hereafter RE) are shown as red filled
circles. No apparent trends are visible here. However, the statistical
clusters identified by GMM, shown in Fig. 1(g), are not comparable here
suggesting the environment of a galaxy does not appear to be a good
differentiator in this sample.

Application of both age and colour criteria together, however, reveals the
presence of two distinct sets of galaxies present in the sample and is shown
in Fig. 1(d). It is evident from the Figure that, young and dusty systems seem
to follow a positive trend ($r=0.73$, $S=99.94$ per cent). On the other hand,
old and less dusty systems tend to follow a near-perfect negative correlation
($r=-0.86$, $S=99.73$ per cent). As shown in Fig. 1(h), the application of GMM
also reveals two statistical sets of galaxies. The central velocity dispersion
of galaxies enjoys a very strong and positive correlation with the mass of
central SMBH. The mass of SMBH of most of our sample galaxies is estimated
using their $\sigma$ values. Hence all the correlations/trends with $\sigma$
mentioned here can stand as a proxy to those involving the mass of SMBH as
well.

Table 2: The Table lists the linear correlation coefficients (r) along with
significance (S) for various relations given in Fig.1 and Fig.2. N denotes the
number of galaxies. For completeness, the table also includes relations that
show little correlation. The sub-samples are defined in section 3.

Correlation Sub-sample r S N CIR - log $\sigma$ Blue 0.64 99.95 28 Red -0.30
87.78 30 Young 0.79 >99.99 26 Old -0.77 99.45 13 DE 0.20 67.68 28 RE 0.23
79.53 33 Blue & Young 0.73 99.94 20 Red & Old -0.86 99.73 11 CIR - MB Blue
-0.65 99.98 25 Red 0.68 99.97 27 CIR - Mdyn Blue 0.63 99.91 26 Red -0.33 88.42
26 CIR - Mstr Blue 0.64 99.98 30 Red -0.30 89.1 32 CIR - Mnsc Blue 0.71 99.99
27 Red 0.01 3 28

###  3.2 Variation of CIRI with MB

Variation of absolute B band magnitude with CIRI is shown in Fig. 2(a).
Galaxies bluer than the median $B-K$ colour are denoted with blue filled
circles whereas the redder galaxies are represented using red filled circles.
Blue galaxies in the sample show a mild positive trend ($r=-0.65$, $S=99.98$
per cent) whereas red galaxies show a weak negative trend ($r=0.68$, $S=99.97$
per cent). Optical B band is particularly sensitive to star-forming components
in low redshift galaxies (Jahnke & Wisotzki, 2003). Most of the blue galaxies
are faint (with M${}_{B}<-19$), while red galaxies are aligned with bright
galaxies in the sample. As the blue (and faint) galaxies may contain more dust
(in the central region) than their red counterparts, an increased infra-red
emission from their centres is possible (da Cunha et al., 2010) which can rise
their CIRI value. On the other hand, a black hole’s ability to remove dust
from its sphere of influence (Ricci et al., 2017) and associated quenching may
be the primary physical driver of the negative trend shown by red galaxies. A
very similar clustering is obtained using GMM and is shown in Fig. 2(b) where
the green filled circles represent the galaxies in GMM cluster 1, while the
purple filled circles represent the galaxies in GMM cluster 2.

Figure 2: Left panel shows the variation of CIRI with absolute B band
magnitude of the host galaxy (panel a), dynamical mass of the galaxy Mdyn
(panel c), stellar mass of the host galaxy, Mstr(panel e) and mass of NSC of
the host galaxies, Mnsc (panel g). Galaxies bluer than the median of $B-K$
colour are denoted with blue filled circles whereas galaxies redder than the
median of $B-K$ colour are represented as red filled circles. The right panel
shows the distribution of two clusters identified statistically using GMM.
Galaxies in GMM cluster 1 are marked with green filled circles, whereas
galaxies in GMM cluster 2 are marked with purple filled circles. In the left
panel, the best fit linear relations for the sub-samples separately are shown
as dotted lines where correlations are significant.

###  3.3 Variation of CIRI with Mdyn

The dynamical mass (Mdyn) is a galaxy parameter that depends on velocity
dispersion and size of the galaxy (Auger et al., 2010). For the sample
galaxies, Mdyn estimations are adopted from Scott & Graham (2013) or
Dabringhausen & Fellhauer (2016). However, they have used different methods to
estimate it. Scott & Graham (2013) used virial estimator Mdyn =
$\alpha\sigma_{e}^{2}$Re/G, where Re is the effective half-light radius and
$\sigma_{e}$, is the luminosity-weighted velocity dispersion measured within a
1Re aperture. Dabringhausen & Fellhauer (2016) calculated Mdyn using the
following equation;

$M_{dyn}=\frac{K_{V}}{G}R_{e}\sigma_{0}^{2},$ (2)

where _K V_ is a factor that depends on the shape of the density profile of
the ETG, _G_ is the gravitational constant, _R e_ is the effective radius of
the galaxy and _$\sigma_{0}$_ is the velocity dispersion at the centre of the
galaxy. The variation of Mdyn is plotted against CIRI in Fig. 2(c) and
clusters identified using GMM are shown in Fig. 2(d). Albeit the heterogeneity
in the method of determination of Mdyn, here also we can observe that the blue
galaxies show a weak positive trend ($r=0.63$, $S=99.91$ per cent) with CIRI.
For the red systems in the plot, no trend is visible. Dust content in galaxies
is reported to possess good correlations with stellar mass (Santini et al.,
2014) and star formation rate (da Cunha et al., 2010; Santini et al., 2014).
In a galaxy with a high dust content, the star formation rate can increase,
causing the CIRI to increase with its dynamical mass.

###  3.4 Variation of CIRI with Mstr

Stellar mass of the sample galaxies, Mstr, were compiled from Scott & Graham
(2013), Neumayer et al. (2020), Pechetti et al. (2020) and Fahrion et al.
(2021). Pechetti et al. (2020) used _Spitzer_ 3.6$\mu$m luminosities and a
mass-to-light ratio (M/L) of 0.5 at 3.6$\mu$m to estimate the galaxy stellar
masses. Scott & Graham (2013) obtained stellar mass from the total galaxy
luminosity in $K$ band, MK (using 2MASS observations), assuming a standard
mass-to-light ratio, M/LK = 0.8 (e.g. Balcells et al., 2007; Bell & de Jong,
2001). Fig. 2(e) shows the variation of Mstr with CIRI and Fig. 2(f) shows the
same using GMM clustering. Here also the blue galaxies show a systematic
increase in CIRI with increasing stellar mass ($r=0.64$, $S=99.98$ per cent)
which is not completely surprising as dusty galaxies can harbour high star
formation (e.g. Cortese et al., 2012; Bourne et al., 2012). For the red
systems in the plot, however, no apparent association can be found.

###  3.5 Variation of CIRI with Mnsc

The variation of CIRI with Mnsc is shown in Fig. 2(g). Blue galaxies show a
slight positive trend ($r=0.71$, $S=99.99$ per cent) whereas red galaxies seem
to be scattered. The statistical cluster association obtained using GMM, shown
in Fig. 2(h), does not show any similarity with that obtained using $B-K$
colour. The masses of NSCs obey correlations with properties of the host
galaxies such as velocity dispersion and bulge mass (Ferrarese et al., 2006).
Early-type galaxies exhibit a low contrast between NSC and host-galaxy body
making the spectroscopic studies of NSCs in ETGs extremely difficult (Böker,
2008). Most of the existing mass estimates of NSCs are photometric, based
generally on SED fitting or colour-mass-to-light ratios of galaxies. These may
not be the best indicators of mass due to age-metallicity degeneracy and
because NSCs contain complex star formation history (Pechetti et al., 2020).
Despite these, CIRI seems to vary with Mnsc for dusty systems in the sample.
The CIRI, estimated using _Spitzer_ space telescope, appears to trace many
properties of galaxies for dusty systems in the sample.

##  4 DISCUSSION AND CONCLUSION

In this paper, we determine the central intensity ratio of early-type galaxies
hosting nuclear star clusters using near-infrared light and present its
variations with different host galaxy properties such as central velocity
dispersion ($\sigma$), absolute B band magnitude (MB), dynamical mass (Mdyn),
stellar mass (Mstr) and mass of nuclear star cluster (Mnsc) of the host galaxy
and reflect the possible connection between the evolution of galaxies and
their central light distribution.

The values of CIR measured from nearby bands are known not to disturb
correlations or trends, involving CIR (Sruthi & Ravikumar, 2021). However, CIR
measured from different broadbands can vary differently. In the optical band,
the dust present in the (projected) central region of galaxies can limit the
light visible to observers. On the other hand, in near-infrared, one gets
dust-penetrated light from the centres of galaxies which may originate from
small dust grains that are heated by energetic photons produced by young stars
(Magdis et al., 2013). Dusty galaxies in the sample where CIRI shows a
positive trend with Mnsc, thus suggesting a mutual growth of NSCs with CIRI.
It is likely that NSCs are part of star formation near the centre in low-mass
galaxies. The growth of NSCs in the centers of these galaxies may add light to
their centers. However, high-mass galaxies in the sample do not show such a
trend. Variation of Mnsc with the host galaxy parameters such as central
velocity dispersion, absolute B band magnitude, dynamical mass, and stellar
mass of the sample galaxies seem to be scattered. Moreover, CIRI shows a
positive trend with all the above-mentioned parameters for low mass, blue,
young, and faint galaxies in the sample. However, in high-mass galaxies in the
sample the increased mass of SMBH might have reduced the NSC domination in the
centre, resulting in the lack of any correlation with CIRI.

CIRI is showing a positive trend with $\sigma$ for young and dusty galaxies in
the sample suggesting an increase in dust and light content corresponding to
the emission of 3.6$\mu$m band as the galaxy grows, while their older
counterparts show no significant trend. Environment seems insignificant in the
CIRI \- $\sigma$ plot though $\sigma$ is coupled with the local galaxy density
(Spindler & Wake, 2017). Dust abundance is directly connected with galaxy
growth through the formation of new stars (Santini et al., 2014). Thus, it is
possible to attribute the growth in central velocity dispersion to that in
galaxy mass.

An increase in $\sigma$ also indicates the possibility of growth in the size
of the galaxy’s central black hole, its central potential, and its total mass
validated by various studies (e.g. Auger et al., 2010; Gebhardt et al., 2000).
Aswathy & Ravikumar (2018), hereafter AR18, reported that CIR calculated in
the optical band (CIRV) shows a strong negative correlation with the mass of
the supermassive black hole and central velocity dispersion for classical
bulges, whereas that of spirals and lenticulars with pseudo-bulges remained
outliers. 8 ellipticals and 6 lenticulars in AR18 are present in our sample
also. The pseudo-bulges, which behaved as outliers in AR18, seem to obey a
positive trend here along with the blue galaxies whereas classical bulges in
their correlation seem to be a part of the red galaxies in our sample. Sruthi
& Ravikumar (2021) found that, CIRV follows negative correlations with MBH and
$\sigma$ for field ETGs also. These studies discussed the possible quenching
that happened in these systems as the black hole grows. In our sample, CIRV
does not show any significant correlation with MBH or $\sigma$. From Fig.1(a),
it is evident that the majority of our sample galaxies are carrying dust in
them. The presence of dust can disturb the relation in the optical band. The
correlations/trends described by CIR are context-dependent. In the studies of
Aswathy & Ravikumar (2018) and Sruthi & Ravikumar (2021), the correlations
involving CIRV were possible as most of their sample galaxies contained low
amounts of dust. In the current study, on the contrary, most of the NSC host
galaxies are reported to contain high amounts of dust, making observations in
the near-infra-red band well suited to study their central light distribution.
Interstellar dust in galaxies absorbs energy from starlight; this absorbed
energy is then re-radiated at infra-red and far-infra-red wavelengths (Draine
& Li, 2007). The contribution of thermal dust to the observed emission at
3.6$\mu m$ can be powered by AGN and/or star formation (Sani et al., 2011).
Graham & Spitler (2009) discuss two possible ways of BH growth in galaxies
hosting NSCs, (i) some massive BHs may grow through runaway collision of NSC
stars, (ii) inward gas flow may result in star formation at the galactic
centre, that itself may be fuelling the growth of BHs. It is likely that the
same gas that is feeding star formation also feeds the black hole (Hopkins &
Quataert, 2010). In that sense, the presence of an NSC may, in fact, enhance
the growth rate of the central black hole (Neumayer et al., 2020). Nuclear
star formation is more solidly coupled to the growth of central black holes
than the global star formation rate (Hopkins & Quataert, 2010). Dusty galaxies
in our sample reveal that their central light distribution is coupled to NSC
growth. If NSC growth can boost central BH growth, it is natural to assume
that there may be a growth in the central potential also. Thus our analysis
supports the co-evolution of NSCs with their host galaxies reported in the
literature, at least for dusty galaxies.

Figure 3: The variation of the ratio of the masses of the central massive
black hole and the NSC with their host galaxy stellar mass. The top panel
(panel a) gives $B-K$ clustering and the bottom panel (panel b) gives GMM
clustering results of the sample galaxies. The dashed line indicates equal-
mass black holes and NSCs in galaxies, while objects above the line have more
massive black holes than NSCs. Symbols of galaxies are the same as shown in
Fig.1.

Ferrarese et al. (2006) reported that, in low-mass galaxies (Galaxy mass <
1010 M⊙), which may not have supermassive black holes (searches are ongoing
for low-mass central BHs ($\leq$ 106 M⊙), the masses of nuclear star clusters
correlated with the one-dimensional velocity dispersion ($\sigma_{bulge}$).
However, the connection between NSCs and central BHs is not well understood.
The sample of galaxies with both a detected NSC and a BH is quite limited.
Even from this data, it is clear that, low-mass (109 \- 1010 M⊙) galactic
nuclei are dominated by NSCs while high-mass (Galaxy mass > 1010 M⊙) galaxy
nuclei are dominated by SMBHs (Nguyen et al., 2018). Increasing evidence also
points out that, massive star clusters are sites of inter-mediate mass black
hole formation (Seth et al., 2006). But direct observational pieces of
evidence for such BHs are less since such small BHs are extremely difficult
targets for dynamical searches (Neumayer & Walcher, 2012). Nuclear star
clusters are common in those spheroids with stellar mass Mstr $\sim$ 108-11
M$\odot$ harbouring a MBH < 107 M$\odot$ and may contribute significantly to
the mass of central blackhole. A larger value of MBH can prevent the co-
existence with NSC because the very central stars should be either disrupted
by the BH gravity or kicked away by tidal forces (Sani et al., 2011). These
interactions with NSCs and BHs possibly explain the negligible nucleation
fraction for galaxies with Mstr > 1011 M$\odot$ (Fahrion et al., 2021). The
reason for massive galaxies showing deviation from the positive trend
exhibited in CIRI \- $\sigma$ plot can be explained using the above-proposed
interactions. Neumayer & Walcher (2012) discuss two possible scenarios that
can happen in high mass galaxies, (i) either the galaxies with massive BHs
never had a sizeable NSC, possibly because their central BHs grew early in the
age of the universe; thus stopping NSC growth, or, (ii) massive BHs destroy
their host NSCs. The positive CIRI-$\sigma$ trend in Fig. 1 possibly suggests
the simultaneous growth of NSC and BH. However, we can not make a strong
statement on red galaxies in the sample as no such trends are visible in
CIRI-$\sigma$ variation.

A positive correlation is reported between Mstr and $\log$ (MBH/Mnsc) for a
sample of galaxies hosting NSC (Neumayer et al., 2020). They applied very
strict conditions to construct their limited sample, such as dynamically
estimated masses for BHs and NSCs. However, no such correlation is observed
for this study, as can be seen in Fig. 3, where we used all early-type
galaxies hosting NSCs. In the figure, a comparison of $B-K$ clustering and GMM
clustering is given. In panel a, blue galaxies are shown in blue filled
circles whereas red galaxies are denoted as red filled circles. In panel b,
galaxies in GMM cluster 1 are denoted by green filled circles, and galaxies in
GMM cluster 2 are represented by purple filled circles. The dashed line
represents the equal mass NSCs and BHs in galaxies. Galaxies above the dashed
line host more massive BHs than NSCs while galaxies below the dashed line host
BHs less massive than NSCs. It can be seen that in the red galaxies in the
sample, the central region is dominated by the (central) blackhole, where as
it is the nuclear star cluster that dominates the centre of blue galaxies. The
GMM clustering also reproduces the result.

Using simple photometric techniques, we analyzed centers of nearby ETGs
hosting NSCs. In these galaxies, NSCs appear to be involved in central star
formation. Improved mass determinations of NSCs in the future may allow us to
study galaxies hosting NSCs using CIRI. CIRI can be used as a proxy to measure
central velocity dispersion in dusty galaxies where AGN/heavy star formation
prevents its measurement. In this context, analysis involving the simple
photometric quantity CIR seems promising to shed more light on the galaxy
evolution scenarios.

## Acknowledgements

We thank the anonymous reviewer for his/her comments that greatly improved the
content of this paper. KS would like to acknowledge the financial support from
INSPIRE program conducted by the Department of Science and Technology (DST),
Government of India. KS wishes to thank UGC-SAP and FIST 2
(SR/FIST/PS1159/2010) (DST, Government of India) for the research facilities
in the Department of Physics, University of Calicut. This work is based on
archival data obtained with the _Spitzer_ Space Telescope, which was operated
by the Jet Propulsion Laboratory, California Institute of Technology under a
contract with NASA. Support for this work was provided by NASA through an
award issued by JPL/Caltech. We acknowledge the use of the HyperLeda database
(http://leda.univ-lyon1.fr) and the NED, NASA/IPAC Extragalactic Database
(http://ned.ipac.caltech.edu), which is operated by the Jet Propulsion
Laboratory, California Institute of Technology, under contract with the
National Aeronautics and Space Administration. This research has made use of
NASA’s Astrophysics Data System Bibliographic Services.

## Data Availability

The near-infra-red observations used in this work are freely accessible on the
Spitzer Heritage Archive. The data presented in each figure will be shared on
reasonable request to the corresponding author.

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