# Knowledge-enriched Two-layered Attention Network for Sentiment Analysis

Abhishek Kumara, Daisuke Kawaharab, Sadao Kurohashib
aIndian Institute of Technology Patna, India
bKyoto University, Japan
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

We propose a novel two-layered attention network based on Bidirectional Long
Short-Term Memory for sentiment analysis. The novel two-layered attention
network takes advantage of the external knowledge bases to improve the
sentiment prediction. It uses the Knowledge Graph Embedding generated using
the WordNet. We build our model by combining the two-layered attention network
with the supervised model based on Support Vector Regression using a
Multilayer Perceptron network for sentiment analysis. We evaluate our model on
the benchmark dataset of SemEval 2017 Task 5. Experimental results show that
the proposed model surpasses the top system of SemEval 2017 Task 5. The model
performs significantly better by improving the state-of-the-art system at
SemEval 2017 Task 5 by 1.7 and 3.7 points for sub-tracks 1 and 2 respectively.

##  1 Introduction

With the rise of microblogging websites, people have access and option to
reach to the large crowd using as few words as possible. Microblog and news
headlines are one of the common ways to dispense information online. The
dynamic nature of these texts can be effectively used in the financial domain
to track and predict the stock prices Goonatilake and Herath (2007). These can
be used by an individual or an organization to make an informed prediction
related to any company or stock Si et al. (2013).

This gives rise to an interesting problem of sentiment analysis in financial
domain. A study indicates that sentiment analysis of public mood derived from
Twitter feeds can be used to eventually forecast movements of individual stock
prices Smailović et al. (2014). An efficient system for sentiment analysis is
a core component of a company involved in financial stock market price
prediction.

Social media texts are prone to word shortening, exaggeration, lack of grammar
and appropriate punctuations. Moreover, the word limit constraint forces a
user to limit their content and squeeze in their opinion about companies.
These inconsistencies make it challenging to solve any natural language
processing tasks including sentiment analysis Khanarian and Alwarez-Melis
(2012).

Bag-of-words and named entities were used by Schumaker and Chen (2009) for
predicting stock market. For predicting the explicit and implicit sentiment in
the financial text, de Kauter et al. (2015) used a fine-grained sentiment
annotation scheme. Kumar et al. (2017) used a classical supervised approach
based on Support Vector Regression for sentiment analysis in financial domain.
Oliveira et al. (2013) relied on multiple regression models. Akhtar et al.
(2017) used an ensemble of four different systems for predicting the
sentiment. It used a combination of Long Short-Term Memory (LSTM) Hochreiter
and Schmidhuber (1997), Gated Recurrent Unit (GRU) Cho et al. (2014),
Convolutional Neural Network (CNN) Kim (2014) and Support Vector Regression
(SVR) Smola and Schölkopf (2004). Yang et al. (2016) used a hierarchical
attention network to build the document representation incrementally for
document classification.

Our model focuses on interpretability and usage of knowledge bases. Knowledge
bases have been recognized important for natural language understanding tasks
Minsky (1986). Our main contribution is a two-layered attention network which
utilizes background knowledge bases to build good word level representation at
the primary level. The secondary attention mechanism works on top of the
primary layer to build meaningful sentence representations. This provides a
good intuitive working insight of the attention network.

##  2 Proposed Methodology

We propose a two-layered attention network which leverages external knowledge
for sentiment analysis. It consists of a bidirectional Long Short-Term Memory
(BiLSTM) Graves et al. (2005) based word encoder, word level attention
mechanism for capturing the background knowledge and a sentence level
attention mechanism aimed at grasping the context and the important words. The
output of the two-layered attention network is then ensembled with the output
of the feature based SVR using the Multilayer perceptron based approach
described in Akhtar et al. (2017). The overall ensembled system is shown in
Figure 2. Each of the components is explained in the following subsections and
an overview of the two-layered attention network is depicted in Figure 1.

Figure 1: Two-layered attention network

###  2.1 Two Layered Attention Model

####  2.1.1 BiLSTM based word encoder

A Long-Short Term Memory (LSTM) is a special kind of Recurrent Neural Network.
It handles the long-term dependencies where the current output is dependent on
many prior inputs. BiLSTM, in essence, is a combination of two different LSTM
- one working in forward and the other working in the backward direction. The
contextual information about both past and future helps in determining the
current output.

The two hidden states $\overrightarrow{h_{t}}$ and $\overleftarrow{h_{t}}$ for
forward and backward LSTM are the information about past and future
respectively at any time step $t$. Their concatenation $h_{t}$ =
[$\overrightarrow{h_{t}}$, $\overleftarrow{h_{t}}$] provides complete
information. Each word of the sentence is fed to the network in form of word
embeddings which are encoded using the BiLSTM.

####  2.1.2 Word Level Attention

External knowledge in form of Knowledge Graph Embedding Yang et al. (2015) or
top-k similar words are captured by using the word level attention mechanism.
This serves the purpose of primary attention which leverages the external
knowledge to get the best representation for each word. At each time step we
get V($x_{t}$) relevant terms of each input $x_{t}$ with $v_{i}$ being the
embedding for each term. (Relevant terms and embeddings are described in next
section). The primary attention mechanism assigns an attention coefficient to
each of relevant term having index $i$ $\in$ V($x_{t}$):

$\alpha_{ti}\propto h_{t}^{T}W_{v}v_{i}$ (1)

where $W_{v}$ is a parameter matrix to be learned.

$m_{t}=\sum_{i\in{V(x_{t})}}\alpha_{ti}v_{i}$ (2)

$\widehat{h_{t}}=m_{t}+h_{t}$ (3)

The knowledge aware vector ($m_{t}$) is calculated as Equation 2, which is
concatenated with the hidden state vector to get the final vector
representation for each word.

####  2.1.3 Sentence Level Attention

The secondary attention mechanism captures important words in a sentence with
the help of context vectors. Each final vector representing the words is
assigned a weight indicating its relative importance with respect to other
words. The attention coefficient $\alpha_{t}$ for each final vector
representation is calculated as:

$\alpha_{t}\propto\widehat{h_{t}^{T}}W_{s}u_{s}$ (4)

$H=\sum_{t}\alpha_{t}\widehat{h_{t}}$ (5)

where $W_{s}$ is a parameter matrix and $u_{s}$ is the context vector to be
learned. $H$ is finally fed to a one layer feed forward neural network.

###  2.2 Relevant Terms and Embeddings

External knowledge can provide explicit information for the model which the
training data lacks. This helps the model to make better predictions. We
relied on Knowledge Graph Embeddings based on WordNet and Distributional
Thesaurus to get relevant terms and their corresponding embeddings for each
word in the text.

####  2.2.1 Knowledge Graph Embedding

WordNet111https://wordnet.princeton.edu is a lexical database which contains
triplets in the form of (subject, relation, object). Both subject and object
are synsets in WordNet. Each word in the text serves as the subject of the
triplet. The relevant terms for the current word are the triplets having the
current word as the subject. We then employ Knowledge Graph Embeddings to
learn the representation of the triplet. A 100-dimensional dense vector
representation for each subject, relation and object were learned using the
DistMult approach Yang et al. (2015) and concatenated. These served as the
relevant embeddings. An example of triplet in WordNet is (bronze_age, part_of,
prehistory).

####  2.2.2 Distributional Thesaurus

Distributional Thesaurus (DT) Biemann and Riedl (2013) is an automatically
computed word list which ranks words according to their semantic similarity.
We use a pre-trained DT to expand a current word. For each current word, top-4
target words are found which are the relevant terms. The relevant embeddings
are obtained by using a 300-dimensional pre-trained Word2Vec Mikolov et al.
(2013) and GloVe Pennington et al. (2014) model. An example of the DT
expansion of the word ’touchpad’ is mouse, trackball, joystick and trackpad.

###  2.3 Feature Based Model - SVR

The following hand-crafted features are extracted and used to train a Support
Vector Regression (SVR).

\- Tf-Idf: Term frequency-inverse document frequency (Tf-Idf) reflects the
importance of each word in a document. We use Tf-Idf score as a feature value
for each word.

\- Lexicon Features: Sentiment lexicons are an important resource for
sentiment analysis. We employ the following lexicons: Bing Liu opinion lexicon
Ding et al. (2008) and MPQA subjectivity lexicon Wilson et al. (2005),
SentiWordNet Baccianella et al. (2010) and Vader sentiment Gilbert (2014).
From the above lexicons we extracted the agreement score Rao and Srivastava
(2012) and the count of the number of occurrences of all positive and negative
words in the text.

\- Word embedding: We use the 300-dimensional pre-trained Word2Vec and GloVe
embedding. The sentence embedding was obtained by concatenating the embedding
for all words in the sentence.

Figure 2: Multilayer perceptron based ensemble

##  3 Experiments

###  3.1 Dataset

We evaluate our proposed approach for sentiment analysis on the benchmark
datasets of SemEval-2017 shared task 5. The task ’Fine-Grained Sentiment
Analysis on Financial Microblogs and News’ Keith Cortis and Davis (2017) had
two sub-tracks. Track 1 - ’Microblog Messages’ had 1,700 and 800 train and
test instances respectively. Track 2 - ’News Statements & Headlines’ had 1,142
and 491 train and test instances respectively. The task was to predict a
regression score in between -1 and 1 indicating the sentiment with -1 being
negative and +1 being positive.

###  3.2 Implementation Details

We implement our model using Tensorflow and Scikit-learn on a single GPU. We
use a single layer BiLSTM with the two-layered attention mechanism followed by
a one layer feed forward neural network. The number of units in each LSTM cell
of the BiLSTM was 150. The batch size was 64 and the dropout was 0.3
Srivastava et al. (2014) with the Adam Kingma and Ba (2014) optimizer. The
length of context vector in the secondary attention network was 300. For each
experiment, we report the average of five random runs. Cosine similarity is a
measure of similarity. It represents the degree of agreement between the
predicted and gold values. Cosine similarity was used for evaluation as per
the guideline.

###  3.3 Results

We compare our system with the state-of-the-art systems of SemEval 2017 Task 5
and the system proposed by Akhtar et al. (2017). Table 1 shows evaluation of
our various models. Team ECNU Lan et al. (2017) and Fortia-FBK Mansar et al.
(2017) were the top systems for sub-tracks 1 and 2 respectively. Team ECNU and
Fortia-FBK reported a cosine similarity of 0.777 and 0.745 for sub-tracks 1
and 2 respectively. Team ECNU employed a number of systems - Support Vector
Regression, XGBoost Regressor, AdaBoost Regressor and Bagging Regressor
ensembled together. Team Fortia-FBK used a Convolutional Neural Network for
this task. The system proposed by Akhtar et al. utilizes an ensemble of LSTM,
GRU, CNN and a SVR and reported a cosine similarity of 0.797 and 0.786 for the
two sub-tracks.

Our proposed system has a cosine similarity of 0.794 and 0.782 for sub-tracks
1 and 2 respectively. The proposed system performs significantly better than
top systems of SemEval 2017 Task 5 for both the tasks. Moreover, the system
performs at par with the system proposed by Akhtar et al. with half the number
of subsystems involved in the ensemble. This shows that our proposed system is
not only robust since it performs for both the task equally well but also
powerful as it involves fewer subcomponents while having the same expressive
power.

The two-layered attention network alone performs better than the best system
of SemEval 2017 Task for both the sub-track. It manages to achieve much higher
score than any of the deep learning component utilized by the system proposed
by Akhtar et al. (2017) as shown in Table 2. This shows that the two-layered
attention network helps to reduce overall model complexity without
compromising the performance.

| Models | Microblog | News
---|---|---|---
Layered Attention Network
L1 | Knowledge Graph Embedding | 0.758 | 0.727
L2 | Distributional Thesaurus + GloVe | 0.764 | 0.749
L3 | Distributional Thesaurus + Word2Vec | 0.779 | 0.763
Support Vector Regression
S1 | Tf-Idf + Lexicon | 0.735 | 0.720
S2 | Tf-Idf + Lexicon + GloVe | 0.755 | 0.753
S3 | Tf-Idf + Lexicon + Word2Vec | 0.743 | 0.740
Ensemble
E1 | L3 + S2 | 0.794 | 0.782

Table 1: Cosine similarity score of various models on test dataset.

Models | Microblog | News
---|---|---
Single systems
Mansar et al. (Team Fortia-FBK) | - | 0.745
Akhtar et al. \- LSTM | 0.727 | 0.720
Akhtar et al. \- GRU | 0.721 | 0.721
Akhtar et al. \- CNN | 0.724 | 0.722
L3 (proposed) | 0.779 | 0.763
Ensembled systems
Lan et al. (Team ECNU) | 0.777 | 0.710
Akhtar et al. | 0.797 | 0.786
E1 (proposed) | 0.794 | 0.782

Table 2: Comparison with the state-of-the-art systems.

###  3.4 Error Analysis

We performed error analysis and observed that the proposed system faces
difficulty at times. Following are the situations when the system failed and
incorrectly predicted values of the opposite polarity:

$\bullet$ Sometimes the system fails to identify an intensifier. In the
example below, ’pure’ is used as an intensifier.
Text : Pure garbage stock
Actual: -0.946 Predicted: 0.042

$\bullet$ The system fails when it does not have enough real-world
information. In the example below, a low share price is a good opportunity to
buy for an individual but from a company’s point of view, a low share price
does not indicate a prosperous situation.
Text : Good opportunity to buy
Actual: -0.771 Predicted: 0.260

##  4 Conclusion

In this paper, we proposed an ensemble of a novel two-layered attention
network and a classical supervised Support Vector Regression for sentiment
analysis. The two-layered attention network has an intuitive working. It
builds the representation hierarchically from word to sentence level utilizing
the knowledge bases. The proposed system performed remarkably well on the
benchmark datasets of SemEval 2017 Task 5. It outperformed the existing top
systems for both the sub-tracks comfortably. Experimental results demonstrate
that the system improves the state-of-the-art system of SemEval 2017 Task 5 by
1.7 and 3.7 points for sub-tracks 1 and 2 respectively. This robust system can
be effectively used as a submodule in an end-to-end stock market price
prediction system.

##  5 Acknowledgements

This work was supported by the Kyoto University Cooperative Intelligence
Short-term Internship Program.

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Review Phil. Trans. R. Soc

cosmology, astrophysics

Konstantinos Migkas

# Galaxy clusters as probes of cosmic isotropy

Konstantinos Migkas1,2 1Leiden Observatory, Leiden University, PO Box 9513,
2300 RA Leiden, the Netherlands
2SRON Netherlands Institute for Space Research, Niels Bohrweg 4, NL-2333 CA
Leiden, the Netherlands<EMAIL_ADDRESS>
###### Abstract

Scaling relations of galaxy clusters are a powerful probe of cosmic isotropy
in the late Universe. Owing to their strong cosmological dependence, galaxy
cluster scaling relations can obtain tight constraints on the spatial
variation of cosmological parameters, such as the Hubble constant ($H_{0}$),
and detect large-scale bulk flow motions. Such tests are crucial to scrutinise
the validity of $\Lambda$CDM in the local Universe and determine at what
cosmic scales (if any) extra-galactic objects converge to isotropy within the
Cosmic Microwave Background rest frame. This review describes the methodology
for conducting cosmic isotropy tests with cluster scaling relations and
examines possible systematic biases. We also discuss the results of past
studies that reported statistically significant observed anisotropies in the
local Universe. Finally, we explore the future potential of cluster scaling
relations as a cosmic isotropy probe given the large amount of multi-
wavelength cluster data expected in the near future.

###### keywords:

isotropy, galaxy clusters, bulk flow, cosmology, Hubble constant

††journal: rsta

##  1 Introduction

The Cosmological Principle (CP) postulates that the Universe is statistically
isotropic and homogeneous over sufficiently large scales. The primary
experimental evidence for this assumption comes from Cosmic Microwave
Background (CMB) observations [1]. CMB data support the existence of a rest
frame (the so-called CMB rest frame) in which cosmic radiation in the very
early Universe was highly isotropic. To reach this rest frame, one needs to
remove the main anisotropic feature of the CMB; the well-known, dominant CMB
dipole. The latter is presumed to arise due to our peculiar motion as
observers within the CMB rest frame and the Doppler effect. Adopting the fully
kinematic nature of the CMB dipole, one concludes our Solar system moves with
$370$ km/s towards the Galactic coordinates
$(l,b)\approx(264^{\circ},+48^{\circ})$ within the CMB rest frame [2].

The adoption of the CP in observational cosmology implies that cosmic matter
in the late, low$-z$ Universe should share the same rest frame as radiation in
the early, high$-z$ Universe. In other words, extra-galactic objects, such as
galaxies, galaxy clusters, and supernovae (SN), should converge to statistical
isotropy within the CMB rest frame when large cosmic volumes are considered.
However, the CP alone does not define the scale at which convergence to
isotropy should be achieved. For this, one needs individual cosmological
models that predict the level of apparent anisotropy as a function of scale,
as different models return different predictions. Hence, when one scrutinises
the apparent (an)isotropy of the local Universe, one essentially tests the
validity of specific cosmological models, and not the CP itself.

If a cosmological model is intrinsically isotropic, apparent anisotropies
might emerge from unaccounted bulk flow motions.111The average peculiar
velocity of a given cosmic volume, estimated by the peculiar velocities of
individual extra-galactic objects within this volume. Such bulk flows can bias
the inference of the Hubble flow $z_{\text{Hubble}}$ (the notation $z\equiv
z_{\text{Hubble}}$ will be used hereafter) from the measured $z_{\text{obs}}$
if the true peculiar velocities are not accurately known or not accounted for.
Subsequently, the inferred cosmological distance from $z$ will also be biased.
The properties of bulk flows strongly depend on the degree of matter
homogeneity at each cosmic scale. Consequently, the bulk flow amplitude as a
function of scale varies for different models.

The standard cosmological model, $\Lambda$CDM, predicts that at $\gtrsim 200$
Mpc scales bulk flows should dissipate to $\lesssim 140$ km/s (e.g., [3]). If
such bulk flows are ignored, they can result in an observed $\lesssim 3\%$
anisotropy in the redshift-distance relation and the Hubble constant ($H_{0}$)
within the CMB rest frame. Subsequently, larger apparent $H_{0}$ anisotropies
or bulk flows at $\gtrsim 200$ Mpc scales would pose a challenge for
$\Lambda$CDM. Interestingly, several studies have reported such observed
anisotropies at $z\lesssim 0.06-0.2$, providing indications that the matter
rest frame might not converge to the CMB rest frame at $\gtrsim 200$ Mpc
(e.g., [4, 5, 6, 7, 8, 9, 10, 3, 11, 12, 13, 14]). On the other hand, only a
few of these findings exhibit sufficiently high statistical significance,
while most are at only mild tension ($\lesssim 3\sigma$) with $\Lambda$CDM. As
a result, the consistency of the local Universe kinematics with $\Lambda$CDM
remains rather ambiguous and a topic of ongoing debate. Thus, it is crucial to
develop novel, powerful techniques to scrutinise the assumption of isotropy in
the late Universe and conclusively test the validity of $\Lambda$CDM at local
scales.

##  2 Galaxy clusters as probes of local Universe isotropy

Galaxy clusters are one of the best probes to trace the isotropy of the
redshift-distance relation at $\sim 200-2000$ Mpc ($z\sim 0.05-0.5$) scales.
Clusters are uniformly distributed across the entire extra-galactic sky while
we have high-quality data for thousands of clusters at these cosmic scales. In
contrast, full-sky galaxy surveys are typically restricted to $z\lesssim 0.1$
scales (e.g., [15]). Moreover, although SNIa samples (e.g., [16]) cover a wide
range of distances ($z\sim 0.01-1.5$), they suffer from strongly inhomogeneous
sky distributions that can bias isotropy studies (e.g., [13]). As a result,
galaxy clusters currently offer the best option for studying bulk flows and
cosmological anisotropies at $z\sim 0.05-0.5$ scales, where, according to
$\Lambda$CDM, extra-galactic objects should behave isotropically within the
CMB rest frame.

###  2.1 Cosmology-dependent cluster properties

A key element of galaxy clusters is that they are observed across most of the
electromagnetic spectrum. Thus, numerous multi-wavelength cluster quantities
can be determined independently. For instance, in X-rays, the hot ionised
cluster gas is detected by its bremsstrahlung continuum and line emission.
This allows one to measure the cluster’s X-ray luminosity ($L_{\text{X}}$),
gas mass ($M_{\text{gas}}$), and X-ray isophotal radius ($R_{50\%}$). In
optical and infrared frequencies, one can examine the galaxy population of
clusters and estimate the luminosity of the brightest central galaxy
($L_{\text{BCG}}$). Employing optical weak lensing data, the total cluster
mass ($M_{\text{tot}}$) can also be estimated. In microwave frequencies, the
total integrated Compton parameter ($Y_{\text{SZ}}$, equivalent to the total
cluster gas pressure) is measured using the thermal Sunyaev-Zeldovich effect.
The latter is caused by the inverse Compton scattering of CMB photons as they
travel through the hot cluster gas.

The above-mentioned cluster properties can only be estimated with knowledge of
the cluster’s distance. However, an absolute calibration of cluster distances
has yet to be available. Hence, assuming a $\Lambda$CDM Universe, one can
infer the cluster’s distance from its $z$ as

$D_{L}(z)=\dfrac{c(1+z)}{H_{0}}\int_{0}^{z}\frac{dz^{\prime}}{E(z^{\prime})}\quad\text{and}\quad
D_{A}(z)=\frac{D_{L}(z)}{(1+z)^{2}},$ (1)

where $D_{L}(z)$ and $D_{A}(z)$ are the luminosity distance and angular
diameter distance respectively, $c$ is the speed of light,
$E(z)=\sqrt{\Omega_{\text{m}}(1+z)^{3}+\Omega_{\Lambda}}$ is the normalised
Hubble parameter, and $\Omega_{\text{m}}$ and $\Omega_{\Lambda}$ are the
normalised matter and dark energy densities respectively. As a result, the
inference of these galaxy cluster properties from observables strongly depends
on the fiducial $H_{0}$ value, making such cluster properties _cosmology-
dependent_. Moreover, the inference of $z$, and consequently $D_{L}(z)$ and
$D_{A}(z)$, depends on the observed $z_{\text{obs}}$ and the peculiar velocity
$z_{\text{pec}}$ as:

$1+z_{\text{obs}}=(1+z)(1+z_{\text{pec}}).$ (2)

Thus, bulk flows can bias the cluster distance if the former are ignored,
i.e., it is assumed that $z_{\text{obs}}\approx z$. This bias then propagates
to all cosmology-dependent cluster properties.

###  2.2 Cosmology-independent cluster properties

The key cluster property to test cosmic isotropy is the gas temperature
($T_{\text{X}}$). The latter is observationally determined through X-ray
spectroscopy and its measurement does not depend on cosmological parameters or
peculiar velocities.222The only cosmological dependency of the $T_{\text{X}}$
measurement comes from the selection of the cluster spectra extraction region
which usually depends on first estimating the radius of the cluster. However,
the final $T_{\text{X}}$ dependency on $H_{0}$ is $T_{\text{X}}\sim
H_{0}^{-0.07}$ (see Appendix B in [17]). The $T_{\text{X}}$ dependency on bulk
flows is similarly weak (see Appendix E.4 in [17]). Therefore, $T_{\text{X}}$
is _cosmology-independent_. $T_{\text{X}}$ is a measure of the kinetic energy
of the gas and reflects the depth of the cluster’s gravitational well. The
latter is also traced by the kinetic energy of the galaxies, which is
characterised by the galaxy velocity dispersion $\sigma_{\text{vel}}$,
measured in optical/infrared wavelengths. $\sigma_{\text{vel}}$ is also
determined without any cosmological assumptions and can be an alternative to
$T_{\text{X}}$ as a cosmology-independent cluster property to be used for
testing cosmic isotropy. Finally, the optical richness of galaxy clusters
($\lambda$) is another useful cosmology-independent quantity for isotropy
tests.

###  2.3 Galaxy cluster scaling relations

All the multi-wavelength cluster properties mentioned above scale with the
total cluster mass. As a result, they scale with each other, giving rise to
the so-called _galaxy cluster scaling relations_. Most such relations are
theoretically motivated (see e.g., [18]) and have been established
observationally for at least two decades (see [19, 20] for a review on galaxy
cluster scaling relations). They are described by single power laws that in
logarithmic space take the form:

$\log{Y}+C_{YX}\log{E(z)}=A_{YX}+B_{YX}\log{X}\pm\sigma_{\text{intr},YX}$ (3)

where $Y$ and $X$ are the two scaling cluster properties, $A_{YX}$, $B_{YX}$,
and $\sigma_{\text{intr},YX}$ are respectively the intercept, slope, and
intrinsic scatter of the $Y-X$ scaling relation, and $C_{YX}$, together with
the $\log{E(z)}$ term, account for the redshift evolution of the relation.
When one uses a cosmology-dependent and cosmology-independent cluster property
as $Y$ and $X$, the cosmological dependence of such a scaling relation
maximises.

Figure 1: $L_{\text{X}}-T_{\text{X}}$ scaling relation for the 313 clusters
presented in [21] and [17]. The colour of data points corresponds to the
cluster redshift. The blue and grey shaded areas represent the statistical and
total scatter, respectively.

For example, let us consider the $L_{\text{X}}-T_{\text{X}}$ relation (Fig.
1). We know that $L_{\text{X}}=4\pi D_{L}(z)\times
K_{\text{corr}}f_{\text{X}}$, where $f_{\text{X}}$ is the observed X-ray
cluster flux and $K_{\text{corr}}$ is the K-correction that converts the
$L_{\text{X}}$ from the observer’s rest frame to the cluster’s rest frame.
Therefore, Eq. 3 can be written as

$\log{f_{\text{X}}}+2\log{D_{L}(z)}+C_{LT}\log{E(z)}+\log(4\pi
K_{\text{corr}})=A_{LT}+B_{LT}\log{T_{\text{X}}}\pm\sigma_{\text{intr},LT}.$
(4)

All terms that include $E(z)$ and numerical constants can be summarised into a
single term
$G(\Omega_{\text{m}},C_{LT},z)$.333$G(\Omega_{\text{m}},C_{LT},z)=2\log{\left[c(1+z)\int_{0}^{z}\frac{dz^{\prime}}{E(z^{\prime})}\right]}+C_{LT}\log{E(z)}+\log(4\pi
K_{\text{corr}})$ Eq. 4 can be then rewritten separating the cosmology-
dependent and cosmology-independent terms:

$2\log{H_{0}}+A_{LT}+G(\Omega_{\text{m}},C_{LT},z)=\log{f_{\text{X}}}-B_{LT}\log{T_{\text{X}}}\pm\sigma_{\text{intr},LT}.$
(5)

On the right side of Eq. 5, $f_{\text{X}}$ and $T_{\text{X}}$ are observables
and directly measured quantities. On the contrary, $B_{LT}$ and
$\sigma_{\text{intr},LT}$ are nearly completely insensitive to cosmological
changes. On the left side of Eq. 5, there is strong cosmological dependence
due to the $2\log{H_{0}}$ and $G(\Omega_{\text{m}},C_{LT},z)$ terms. The
intercept $A_{LT}$ is not a cosmological quantity but it is completely
degenerated with $H_{0}$ since $A_{LT}$ is not known independently of
cosmological assumptions.

###  2.4 Testing cosmological parameter isotropy and bulk flows

In an isotropic Universe, the best-fit
$2\log{H_{0}}+A_{LT}+G(\Omega_{\text{m}},C_{LT},z)$ sum should not spatially
vary. One can scrutinise the isotropy hypothesis and investigate the
directionality of cosmological parameters (e.g., $H_{0}$) and bulk flows by
constraining Eq. 5 in independent sky patches. To do so, sufficiently large
and uniform cluster samples should be employed. Such an approach reveals how
much each sky direction deviates from the average, all-sky behaviour. This
deviation can then be translated to the necessary variation of cosmological
parameters to reconcile each direction with the rest of the sky. This approach
does not assume the anisotropy shape. Alternatively, one can fit Eq. 5 to the
full, all-sky cluster sample while introducing an anisotropic component with
an assumed shape in the scaling relation model, e.g., a dipole anisotropy. In
this case, the extra free parameters are the direction and amplitude of the
anisotropy. Each approach can be applied to different $z$ bins to characterise
any retrieved anisotropy as a function of cosmic scale. As explained in Sect.
2.4.1, 2.4.2, and 2.4.3, using different $z$ bins one can eventually
distinguish $H_{0}$ and $\Omega_{\text{m}}$ anisotropies, as well as bulk
flows since these cosmological phenomena impact the observed anisotropy of Eq.
5 differently. Thus, the degeneracy between the individual left-side terms of
Eq. 5 can be broken and one can draw conclusions on individual cosmological
parameters.

Every cluster scaling relation can be decomposed to observables and
cosmological parameters as in Eq. 5. Each time, the cosmological sensitivity
of the scaling relation changes according to how the utilised cluster
properties depend on cosmic distance. For example, $M_{\text{gas}}\propto
D_{A}^{\frac{5}{2}}$; as a result, the $M_{\text{gas}}-T_{\text{X}}$ relation
leads to the strongest $H_{0}$ dependence with $2.5\log{H_{0}}$.444The
dependence on $\Omega_{\text{m}}$ and other parameters through $G$ is more
complicated and also depends on the redshift evolution of the relation as
explained in Sect. 2.4.3 Similarly, the $R_{50\%}-T_{\text{X}}$ relation leads
to a $\log{H_{0}}$ dependence. Furthermore, one can study scaling relations
between two cosmology-dependent properties and still see some residual
cosmology dependence. The latter depends on $B_{YX}$ as well. For instance,
the $L_{\text{X}}-M_{\text{gas}}$ relation shows $B_{LM\text{gas}}\sim 1.2$
(e.g., [22, 23]), which results in a $\sim\log{H_{0}}$ dependence. One can
apply the same exercise to any scaling relation.

It is evident from the above that, using just one galaxy cluster sample with
multiple measured quantities per cluster, one can obtain complementary
information on cosmic isotropy. The different isotropy constraints also yield
excellent consistency checks and reveal possible systematic and cluster
physics biases, further reinforcing the robustness of the methodology. The
plethora of (nearly independent) cosmological information given per object
constitutes a substantial advantage of clusters as probes of cosmic isotropy
over other extra-galactic objects.

####  2.4.1 $H_{0}$ anisotropies

The dominant cosmological effect in Eq. 5 and other similar scaling relations
comes from the $N_{YX}\log{H_{0}}$ term. $N_{YX}$ varies according to the used
scaling relation, as explained before. Absolute constraints can be put on the
sum $N_{YX}\log{H_{0}}+A_{YX}$ but not on $H_{0}$ alone without assuming the
value of $A_{YX}$. However, $A_{YX}$ is not expected to vary with direction.
$A_{YX}$ depends on inner-cluster (mostly baryonic) physics, such as cluster
gas thermodynamics, and the mechanisms that connect the scaling cluster
properties. Thus, the true, "intrinsic" $A_{YX}$ is independent of the
spacetime between the cluster and the observer. As a result, $A_{YX}$ carries
no information about the redshift-distance relation. Assuming that clusters
are locally described by the same physical laws, $A_{YX}$ should not have any
directional dependence, i.e., it should be statistically isotropic. Thus, when
statistically significant angular variations of $N_{YX}\log{H_{0}}+A_{YX}$ are
observed, the bulk of the effect can be attributed to an $H_{0}$ anisotropy.
In practice, one keeps $A_{YX}$ fixed and treats $H_{0}$ as a free parameter
across the sky. As explained above, the absolute $H_{0}$ values are arbitrary
and depend on the exact choice of $A_{YX}$; for example, $A_{YX}$ can be
chosen so the full-sky cluster sample returns $H_{0}=70$ km/s/Mpc. On the
other hand, relative angular $H_{0}$ variations are directly constrained and
are completely independent of the assumed $A_{YX}$ value.

Finally, an angular variation of $H_{0}$ is degenerate with bulk flows at
$z\lesssim 0.1$ scales and almost entirely independent of bulk flows at
$z\gtrsim 0.2$ (see Sect. 2.4.2). On the other hand, there is strong
degeneracy between $H_{0}$ and $\Omega_{\text{m}}$ anisotropies at $z\gtrsim
0.4$ while there is almost none at $z\lesssim 0.2$ (see Sect. 2.4.3). Thus, at
$z\approx 0.2-0.3$, one can purely trace $H_{0}$ anisotropies with no strong
effects from other possible cosmological anisotropies. At more local or
distant scales, one can independently trace anisotropies on the redshift-
distance relation as a whole. However, to attribute such an apparent
anisotropy to an individual cosmological parameter (or bulk flow), one needs
to assume the other degenerate cosmological parameter to be fixed or vary
within a limited range.

####  2.4.2 Bulk flows

Bulk flows can have a substantial impact on the directional behavior of
$G(\Omega_{\text{m}},C_{YX},z)$ and scaling relations. Bulk flows can affect
the inference of $z$ due to large (unaccounted) cluster peculiar velocities
with non-negligible $z_{\text{pec}}$ compared to $z$ (Eq. 2). Such effects are
mostly relevant for local cluster samples ($z\lesssim 0.15$). For higher$-z$
clusters peculiar velocities become irrelevant compared to the Hubble
recession velocity. In essence, even if strong bulk flows exist at $z\gtrsim
0.2$, they will not cause any detectable anisotropy to scaling relations.

A biased $z$ due to bulk flow presence strongly affects the inferred cluster
distance through the integration limits of
$\int_{0}^{z}\frac{dz^{\prime}}{E(z^{\prime})}$. At the same time, it leaves
$E(z)$ nearly unaffected, constituting the exact redshift evolution term
$C_{YX}\log{E(z)}$ insignificant for bulk flow studies. As an example, a
$+1000$ km/s bulk flow at $z=0.05$ would cause a $\approx 7\%$ overestimation
of $\int_{0}^{z}\frac{dz^{\prime}}{E(z^{\prime})}$ and only a $<0.2\%$ change
in $E(z)$. As previously mentioned, bulk flows at $z\lesssim 0.2$ scales have
similar effects on scaling relations as an $H_{0}$ angular variation. For
example, a $7\%$ $H_{0}$ anisotropy would also cause a $7\%$ change in the
inferred cluster distance, having the same effect with a $+1000$ km/s bulk
flow at $z=0.05$. Consequently, one cannot easily distinguish between the two
phenomena at local scales. Nevertheless, when more distant clusters are
studied, the effect of bulk flows fades out while an $H_{0}$ anisotropy has
the same effect for every cosmic scale. Characteristically, the same $+1000$
km/s bulk flow would only cause a $1.6\%$ change in $D_{L}$ at $z=0.2$.

####  2.4.3 $\Omega_{\text{m}}$, $\Omega_{\Lambda}$, and $w$ anisotropies

An angular variation of the $E(z)$ factor (i.e., $\Omega_{\text{m}}$,
$\Omega_{\Lambda}$, and the dark energy equation-of-state $w$) will cause an
anisotropic effect on $G(\Omega_{\text{m}},C_{YX},z)$ under certain
conditions. For local cluster samples ($z\lesssim 0.3$), $E(z)$ variations
only weakly affect the directional behaviour of most scaling relations,
usually on a lower level than statistical noise. At higher$-z$, directional
variations in $E(z)$ become more impacting on $G(\Omega_{\text{m}},C_{YX},z)$
and can be detected above the statistical noise level. $E(z)$ affects
$G(\Omega_{\text{m}},C_{YX},z)$ through both
$\log{\left[\int_{0}^{z}\frac{dz^{\prime}}{E(z^{\prime})}\right]}$ and
$C_{YX}\log{E(z)}$. However, for scaling relations with $C_{YX}>0$ (e.g.,
$Y_{\text{SZ}}-T_{\text{X}}$ and $M_{\text{gas}}-T_{\text{X}}$), changes in
$E(z)$ tend to partially cancel out between the two terms, suppressing the
impact of an $E(z)$ anisotropy on scaling relations even at high$-z$. On the
other hand, for scaling relations with $C_{YX}<0$ (e.g.,
$L_{\text{X}}-T_{\text{X}}$), the impact of $E(z)$ on
$G(\Omega_{\text{m}},C_{YX},z)$ is enhanced. Consequently, by studying the
anisotropic behaviour of $C_{YX}<0$ and $C_{YX}>0$ scaling relations, it is
trivial to conclusively distinguish between an $E(z)$ anisotropy and a bulk
flow or an $H_{0}$ anisotropy (for which $C_{YX}$ is irrelevant). The value of
$C_{YX}$ can either assumed to be known from the self-similar model [18],
adopted from a past study that constrained it (e.g., [24]), or treat it as a
free parameter. Nonetheless, as mentioned before, the effects of $E(z)$
anisotropies remain small for $z\lesssim 0.3$ samples for any $C_{YX}$. For
now, there is very limited availability of adequately large, all-sky, high$-z$
cluster samples with which $E(z)$ anisotropies could be studied.

Changes in $\Omega_{\text{m}}$ could also cause a directional dependence of
the true $A_{YX}$ and $B_{YX}$ values for scaling relations between properties
that trace the total amount of gas (e.g., $L_{\text{X}}$) and the total
cluster mass (e.g., $M_{\text{tot}}$ or even $T_{\text{X}}$), due to the
different gas fraction per direction. These additional effects might be
challenging to take into account. For all these reasons, the cosmological
parameters included in $E(z)$ are often assumed to be fixed and any observed
anisotropies are attributed to $H_{0}$ variations or bulk flows which have a
much stronger influence at $z\lesssim 0.3$ clusters anyway.

####  2.4.4 When cluster physics effects and systematic biases dominate

There are scaling relations that exhibit negligible cosmological dependence
and can be utilised to detect systematic biases or cluster physics effects.
These relations are particularly helpful for disentangling the signal from the
possible cosmological anisotropies from other cluster-related causes that
might influence the conclusions on isotropy. For example, the
$L_{\text{X}}-Y_{\text{SZ}}$ relation has $B_{LY}\sim 0.93$ [17] which results
in a $\lesssim 0.1\log{H_{0}}$ dependency. Consequently, the
$L_{\text{X}}-Y_{\text{SZ}}$ relation is insensitive to cosmological
anisotropies. However, it can effectively trace biases in the X-ray Galactic
absorption correction applied to $L_{\text{X}}$. While the latter is subject
to Galactic absorption effects, $Y_{\text{SZ}}$ is not since it is measured in
microwaves. Thus, a statistically significant angular variation of
$L_{\text{X}}-Y_{\text{SZ}}$ can provide the true X-ray absorption per
direction by revealing a systematic over- or underestimation of the relative
correction factor for different sky patches. Similarly, the
$R_{50\%}-M_{\text{gas}}$ relation yields a rather weak $0.4\log{H_{0}}$
dependency (Migkas et al. in prep.). Hence, its angular variation level is
mainly dictated by the average morphology of the cluster population per
direction. Cluster morphology (e.g., relaxed or disturbed and cool core or
non-cool core) strongly affects $R_{50\%}$ while it does not cause
considerable changes to $M_{\text{gas}}$. Therefore, it is essential to
robustly check if the cosmological sensitivity of a scaling relation is higher
than its sensitivity on different cluster physics effects and systematics
before we conduct isotropy tests.

###  2.5 Properties of cluster samples used for isotropy tests

To carry out an unbiased test of cosmic isotropy using galaxy cluster samples,
the latter should ideally fulfil certain criteria. Firstly, a similar cluster
population across the sky is needed. This ensures that $A_{YX}$ is similar in
every direction. A homogeneous cluster selection, independent of direction,
generally achieves such a cluster population uniformity. It also provides
similar numbers of clusters in every direction, allowing for balanced full-sky
coverage, which is a necessity for an accurate isotropy test. Even if a sample
is not uniformly built, one can apply selection cuts to obtain a uniform
cluster population across different sky patches before conducting isotropy
tests.

If a cluster sample is constructed with an inhomogeneous selection, the sample
might suffer from a directional dependence on cluster properties. For
instance, more relaxed/disturbed clusters or more intrinsically
brighter/fainter clusters may be found in a specific direction than the rest
of the sky. Such an imbalance in the average cluster properties will cause an
anisotropic $A_{YX}$ since distinct cluster populations are described by
somewhat different scaling relations. If one ignores the spatial variation of
$A_{YX}$, the inferred cosmological anisotropy will be overestimated. However,
even in a homogeneously selected cluster sample, $A_{YX}$ can slightly vary
across the sky due to the random variations of the cluster population. That
is, $A_{YX}$ can slightly vary across the sky due to statistical noise. This
effect is small for low-scattered scaling relations, but it should (and can)
be taken into account when inferring the angular variation of cosmological
parameters. If it is assumed to be negligible, it might lead to an
overestimation of the apparent cosmological anisotropies.

Another useful feature of cluster samples used for cosmic isotropy studies is
the similarity of the $z$ distribution across the sky. This serves two
purposes. Firstly, if an inaccurate $C_{YX}$ value is adopted and a rather
high$-z$ sample is studied, a dissimilar $z$ distribution will induce
directional biases in the scaling relation parameters. These could be
wrongfully interpreted as cosmological anisotropies. On the contrary, if the
$z$ distribution is consistent across the sky, a wrong $C_{YX}$ value will
cause the same bias towards every direction, which will eventually cancel out
when comparing independent sky regions; that is, the cosmological anisotropy
constraints will remain unaffected. Nevertheless, for local cluster samples
($z\lesssim 0.2$), $C_{YX}$ biases become insignificant since no strong
cluster evolution is expected in such a local cosmic volume. The second reason
why a similar $z$ is essential is that, when comparing cosmological parameters
across different sky patches, one wishes to trace similar cosmic volumes. Let
us assume a hypothetical anisotropy is present only at, e.g., $z\lesssim 0.1$
due to bulk flows. Then, if clusters in one direction lie at greater distances
on average, they will not be affected by such a bulk flow. On the other hand,
if more local clusters are found in a different direction, they will be
affected by the bulk flow. This will bias the detection of such a cosmological
effect.

##  3 Evidence for local anisotropy from galaxy clusters

###  3.1 First application and preliminary findings

Probing cosmic isotropy with galaxy cluster scaling relations is a relatively
new sub-field of cosmology. The idea was introduced in [25] where they used
the $L_{\text{X}}-T_{\text{X}}$ relation to test cosmic isotropy with two
independent cluster samples; the ASCA Cluster Catalog (ACC, [26]) and the
first data release of the XMM Cluster Survey sample (XCS, [27]). ACC is an
archival sample containing 272 clusters with a rather uniform sky distribution
and median $z\approx 0.09$. XCS was built from archival XMM-Newton data and
contains 364 with $L_{\text{X}}$ and $T_{\text{X}}$ measurements. These
clusters cover the full sky but not uniformly, with some regions having many
more objects than others. XCS is a rather high$-z$ sample with median
$z\approx 0.35$. [25] performed a 1D analysis with respect to the Galactic
longitude $l$ measuring $H_{0}$ and $\Omega_{\text{m}}$ across different $l$
bins. Interestingly, they identified the same anisotropy patterns in both
cluster samples despite the latter being independent while tracing different
cosmic scales. They found $H_{0}$ to vary by $\sim 20\%$ at a mild $\approx
2.5-3\sigma$ level, with the discrepancy being more prominent at $z\lesssim
0.35$. This analysis offered a first glance at the potential of cluster
scaling relations as proxies of cosmic isotropy; nonetheless, the results were
preliminary since both used samples suffered from certain weaknesses. For
example, ACC was compiled by old, low-quality X-ray cluster measurements with
no specific selection. Also, $T_{\text{X}}$ was measured within a non-physical
cluster radius. On the other hand, XCS composed of low-mass galaxy clusters
and groups that generally suffer from larger scaling relation dispersion.
Also, the $T_{\text{X}}$ was measured with large uncertainty in a radius that
strongly depended on the fiducial cosmology (within 300 kpc from the cluster’s
center); several clusters also had photometric $z$ instead of spectroscopic.
Finally, both samples did not have sufficient information to robustly assess
the cluster population characteristics per direction. Thus, there was still
much room for progressing this methodology.

###  3.2 First 2D results and improved cluster catalogues

Results on apparent $H_{0}$ variations from the $L_{\text{X}}-T_{\text{X}}$
scaling relation were significantly improved by [21]. That study presented and
utilised an X-ray flux-limited, homogeneously-selected local cluster sample,
namely, a preliminary version of the extremely expanded HIgh FLUx Galaxy
Cluster Sample (eeHIFLUGCSv0). The latter consists of 313 clusters with self-
consistently measured, high-quality $T_{\text{X}}$ measurements with low
statistical uncertainties as obtained using deep XMM-Newton and Chandra data.
The 313 clusters cover the full sky uniformly and have a median $z\approx
0.075$ while almost all clusters lie at $z\lesssim 0.3$. The
$L_{\text{X}}-T_{\text{X}}$ scatter was shown to be much lower for
eeHIFLUGCSv0 than ACC and XCS. Adequate information to characterise the
cluster population per direction was also provided. Moreover, [21] implemented
cones of varying angular sizes covering all directions in the sky, providing
the first 2D results on apparent $H_{0}$ variations from cluster scaling
relations. For eeHIFLUGCSv0, they identified a $\approx 13\%$ $H_{0}$ maximum
angular variation with a statistical significance of $3.59\sigma$ towards
$(l,b)\sim(281^{\circ},-16^{\circ})$. This finding was consistent with the
directions identified by [25]. The anisotropy was also present when $z<0.03$
clusters were excluded, suggesting that the bulk of the effect continues
beyond $\gtrsim 130$ Mpc. Jointly analysing the 842 independent cluster
samples from eeHIFLUGCSv0, ACC, and XCS, [21] found a combined $\approx 14\%$
apparent $H_{0}$ variation towards $(l,b)\sim(312^{\circ},-21^{\circ})$ at a
$4.55\sigma$ level. However, this finding did not assume the shape of the
anisotropy as they simply compared the two most extreme, independent sky
regions. When a dipole anisotropy was assumed, the $H_{0}$ variation and
statistical significance dropped to $\approx 11\%$ $4\sigma$ level,
respectively. [21] explored an extensive list of possible systematic biases
but failed to identify any bias that would alleviate the tension with the
isotropy hypothesis. Overall, this work confirmed the strong indications for
an anisotropy in the late Universe, as traced by galaxy clusters.
Nevertheless, further confirmation was needed and remaining systematics were
still to be addressed.

###  3.3 More precise results from multi-wavelength cluster scaling relations

The most robust evidence for a local Universe anisotropy using galaxy clusters
was presented by [17]. In that study, they used the eeHIFLUGCSv0 sample as
presented in [21] and additionally measured the microwave $Y_{\text{SZ}}$,
infrared $L_{\text{BCG}}$, and X-ray $R_{50\%}$ for each cluster using Planck,
2MASS, and ROSAT data respectively. They also measured $Y_{\text{SZ}}$ for the
ACC sample. These new measurements allowed the construction of ten multi-
wavelength scaling relations. The statistical methodology was also noticeably
improved compared to past studies. Combining all the available information by
properly accounting for all scatter and measurement covariance, [17] detected
an apparent $9\%\ H_{0}$ dipole anisotropy towards
$(l,b)\sim(280^{\circ},-15^{\circ})$ at an unprecedented $5.4\sigma$ level
(displayed in Fig. 2). They assessed the statistical significance of the
$H_{0}$ dipole by employing isotropic Monte Carlo simulations. Furthermore,
[17] fully exploited 14 cosmology-independent cluster properties (e.g.,
relaxation state, Galactic absorption, metallicity, exposure time, and cluster
core $T_{\text{X}}$) to predict the expected $H_{0}$ behaviour of a cluster
sub-sample purely based on cluster population criteria. This sophisticated
approach confirmed that the emerging $H_{0}$ dipole is not significantly
affected by cluster population differences across the sky. Overall, numerous
tests were performed to ensure the validity of their results.

Figure 2: $H_{0}$ anisotropy map as derived by [17] from the joint analysis of
different multiwavelength scaling relations and cluster samples. A $9\%$
$H_{0}$ dipole was detected with a statistical significance of $5.4\sigma$
(credit: Migkas et al., 2021, A&A, 649, A151).

Since eeHIFLUGCSv0 is a local cluster sample, one can attribute the apparent
$H_{0}$ anisotropy to large cluster bulk flow. As explained in Sect. 2.4.2,
the two cosmological effects are nearly indistinguishable at $z\lesssim 0.2$.
[17] explored this scenario and provided the first bulk flow constraints from
galaxy cluster scaling relations. They found that the apparent $H_{0}$ dipole
could emerge due to a $\approx 900\pm 200$ km/s bulk flow extending beyond 500
Mpc ($z\gtrsim 0.12$). As displayed in Fig. 3, they constrained the cluster
bulk flow as a function of different cosmic volumes and shells. Different
scaling relations and independent cluster samples returned consistent results
with the best-fit bulk flow amplitude ranging from $\sim 600-1200$ km/s. They
found that the bulk flow direction did not significantly deviate within $z\sim
0.05-0.15$ scales. However, they noted that the results beyond $z\gtrsim 0.1$
are likely overestimated as they are dominated by lower$-z$ objects. However,
there is no available comparison with another independent study yet since
galaxy clusters are the only probes for now that cover these cosmic scales
uniformly across the sky. Overall, this study provided the strongest evidence
until now that the cluster rest frame does not coincide with the CMB rest
frame at $z\lesssim 0.15$, which is in strong contrast with $\Lambda$CDM.

Figure 3: Left: Bulk flow amplitude and its $1\sigma$ uncertainty as a
function of the redshift (or comoving distance) radius as determined by [17].
The results from different scaling relations, cluster samples, and statistical
approaches are displayed in different shapes and colours. Right: Same as in
left panel but as a function of redshift (or comoving distance) shells instead
of spheres (credit: Migkas et al., 2021, A&A, 649, A151).

It is important to note that the $5.4\sigma$ statistical significance of the
apparent $H_{0}$ dipole detection in [17] does not directly correspond to the
tension with $\Lambda$CDM. The latter still allows for limited residual
anisotropy due to bulk flow motions at these scales. For $\sim 500$ Mpc
volumes ($z\lesssim 0.12$), $\Lambda$CDM predicts an average $\approx 80$ km/s
bulk flow which roughly corresponds to a $\sim 1\%$ dipole $H_{0}$ anisotropy.
Consequently, the findings of [17] are at a $\sim 4.5\sigma$ tension with
$\Lambda$CDM. Finally, they fixed $A_{YX}$ for all scaling relations; this
might underestimate the $H_{0}$ dipole uncertainty. However, the final
statistical significance is not expected to be overestimated since they follow
the same procedure using isotropic Monte Carlo simulations.

To conclude, past studies using various multi-wavelength scaling relations and
independent cluster samples strongly and consistently indicate an apparent
$H_{0}$ angular variation within $\lesssim 1$ Gpc. However, the apparent
anisotropy can be alleviated by a considerable cluster bulk flow motion,
although the latter is in notable tension with $\Lambda$CDM as well. Future
studies with new data and further improved methodologies will reveal if the
observed dipole anisotropy truly constitutes a remarkable failure of the
standard cosmological model in the local Universe or if its statistical
significance has been overestimated.

##  4 Future Outlook

We live in the era of next-generation, state-of-the-art cosmological surveys
that remarkably increase the amount of available data and aim at precise
cosmological constraints. Furthermore, recent hydrodynamical cosmological
simulations offer a unique tool to help us better understand observational
findings and uncover potential systematic biases. As a result, there is vast
potential for scrutinising cosmic isotropy with galaxy clusters using various
approaches. Moreover, new advances in the statistical and modelling
methodology might offer new valuable insights. Here we present a non-
exhaustive list of expected future advances in this sub-field that will yield
significant results within the next years.

###  4.1 New cluster data from multi-wavelength surveys

There are numerous cosmological experiments that will soon provide (or have
already done so) valuable new cluster data. First and foremost, the extended
ROentgen Survey with an Imaging Telescope Array (eROSITA, [28]) and its
planned 4-year eROSITA All-Sky Survey (eRASS) will substantially progress
cluster science owing to the largest X-ray cluster catalogues ever compiled.
Already from the first all-sky scan (eRASS1), eROSITA has detected $>12,000$
X-ray clusters, with $>8,000$ of them as new detections [29]. Almost all of
these clusters have $L_{\text{X}}$ and $M_{\text{gas}}$ measurements, while
for $\sim 450$ of them, precise $T_{\text{X}}$ measurements are already
available.555eROSITA $T_{\text{X}}$ measurements were recently found to
deviate from $T_{\text{X}}$ measured by other major X-ray telescopes, such as
Chandra and XMM-Newton [30]. However, this deviation is accurately known and
one can correct for it, not jeopardising isotropy tests with eROSITA
$T_{\text{X}}$ measurements. These numbers will significantly increase as
deeper eRASS data are analysed, reaching $\gtrsim 50,000$ massive clusters for
eRASS:8 [31]. Additionally, $1499$ eRASS1 clusters have $\sigma_{\text{vel}}$
measurements [32] that will allow the study of X-ray-optical scaling
relations, such as $L_{\text{X}}-\sigma_{\text{vel}}$. Due to the large amount
of new X-ray cluster information eROSITA will return, a significant
improvement in cosmic isotropy tests with X-ray clusters is expected.

In the optical and infrared regime, the Euclid telescope and its Euclid Wide
Survey will scan $\approx 15,000^{\circ}$ in the extra-galactic sky,
discovering $\gtrsim 10^{5}$ massive clusters ($M_{\text{tot}}\gtrsim 2\times
10^{14}\ M_{\odot}$) at $z>0.2$ [33]. Additionally, $\gtrsim 10^{4}$ of these
clusters will be at $z>1.3$. For a large fraction of them, Euclid will measure
$\sigma_{\text{vel}}$ and $\lambda$ values, as well as $M_{\text{tot}}$
through weak lensing data. This will allow for constructing optical cluster
scaling relations with vast numbers of used objects. As a result, Euclid will
allow for an unprecedented characterisation of isotropy at large cosmic
scales, insensitive to local effects, such as bulk flows. Other optical
surveys that (will) cover large areas of the sky, such as the ones from Rubin
Observatory and DESI, will also detect tens and hundreds of thousands of
clusters, respectively, measuring useful optical properties for many of them
[34, 35, 36]. It is evident that the abundance of optical galaxy cluster
information to become available in the next years is striking and will
immensely advance our understanding of cosmic isotropy at large scales.

In microwave, the Atacama Cosmology Telescope (ACT) has already provided
$>4000$ SZ-detected clusters with $Y_{\text{SZ}}$ measurements and a median
$z\approx 0.5$ [37]. ACTsignificantly increased the availability of
$Y_{\text{SZ}}$ values compared to past cluster samples. The efficiency of
X-ray-SZ scaling relations for testing cosmic isotropy was demonstrated in
[17]. Finally, this list is non-exhaustive since many more cosmological
experiments will provide valuable cluster data in the near future.

###  4.2 Low-scattered scaling relations and improved modelling

There are two primary approaches to improve constraints on cosmic isotropy;
firstly, employ new, larger samples as outlined in Sect. 4.1, and secondly,
reduce the scaling relation scatter of current samples. The constraining power
roughly scales with $\sim\delta/\sqrt{N}$, where $\delta$ is a relation’s
dispersion (or scatter) and $N$ is the number of used objects. Hence, lowering
the scatter of a scaling relation to half has the same constraining effect as
increasing the sample size by a factor of four. Consequently, it is evident
that suppressing the scatter of current scaling relations with improved
modelling is highly important.

Certain scaling relations with lower scatter than those utilised in past
studies are expected to return valuable insights in the near future. For
example, the $M_{\text{gas}}-T_{\text{X}}$ scaling relation is particularly
useful for cosmic isotropy tests due to its low $H_{0}$ dispersion
($\delta_{H0}\approx 11\%$). Moreover, the $L_{\text{X}}-T_{\text{X}}$
relation (and other X-ray relations) exhibits significantly reduced scatter
when the core-excised $L_{\text{X}}$ is used instead of the total
$L_{\text{X}}$ [38]. Hence, core-excised cluster measurements can be
implemented to improve existing isotropy results. Furthermore, cluster
morphology parameters, such as the X-ray concentration and centroid shift,
correlate with the scatter of some scaling relations [39, 40, 41]. A similar
scatter correlation has been found with the cluster dynamical state [42, 43,
17, 44]. If one measures the cluster morphology and includes it in the scaling
relation model, this could potentially reduce the scatter and enhance the
cosmological constraints.

###  4.3 Cosmological simulations

The validity of methodologies that search for apparent, statistically
significant cosmic anisotropies should be cross-checked using cosmological
simulations. Recently, the state-of-the-art hydrodynamical FLAMINGO
simulations were introduced [45]. FLAMINGO recreate different realisations of
isotropic $\Lambda$CDM universes with $\sim 10^{9}\ M_{\odot}$ mass resolution
and implements full baryonic physics effects. The simulations also provide
lightcones for multiple different observers reaching out to 2.8 Gpc ($z\sim
0.8$) and for different cosmologies. Numerous properties for each simulated
halo are available for each lightcone, including all the cluster properties
discussed in this review. As a result, different methods for detecting cosmic
anisotropies can be applied to the FLAMINGO data to investigate the robustness
of each method. Such a test will also assess the true rarity of the observed
anisotropies within a $\Lambda$CDM universe.

##  5 Discussion and Summary

Galaxy clusters and their scaling relations offer an extraordinary tool for
probing cosmic isotropy at a wide range of cosmic scales. For now, galaxy
clusters are the only isotropy probe for which we have uniform, all-sky data
at $z\sim 0.05-0.5$. Moreover, clusters have a unique advantage over other
isotropy probes; they can provide information on the angular variation of
cosmological parameters across many different wavelengths. This enables the
construction of numerous multi-wavelength scaling relations that probe cosmic
isotropy almost independently, with only a single cluster sample. Low-
scattered scaling relations between cosmology-dependent (e.g., $L_{\text{X}}$,
$Y_{\text{SZ}}$, and $M_{\text{gas}}$) and cosmology-independent (e.g.,
$T_{\text{X}}$ and $\sigma_{\text{vel}}$) cluster properties are exceptionally
useful for such tests. Nonetheless, every scaling relation can be decomposed
into observables and cosmological parameters. Thus, by determining the angular
variation of each scaling relation, robust conclusions on the anisotropy of
cosmological parameters and the existence of large-scale bulk flows can be
drawn. Angular $H_{0}$ variations identically affect all cosmic scales, while
bulk flows and $\Omega_{\text{m}}$ anisotropies mainly affect $z\lesssim 0.2$
and $z\gtrsim 0.3$ scales, respectively. Scaling relations that are
insensitive to cosmological anisotropies (e.g., $L_{\text{X}}-Y_{\text{SZ}}$
and $R_{50\%}-M_{\text{gas}}$) can reveal multiple directional cluster physics
effects and help us isolate the cosmological signal from such potential
biases.

There are, however, certain pitfalls in this method. The most critical is that
the used sample should show a similar cluster population throughout the sky.
Varying cluster populations (e.g., relaxed or disturbed, low or high mass, and
centrally peaked or diffuse) are often (but not always) described by slightly
different scaling relations. A uniform cluster selection that results in
similar cluster populations is essential to ensure that the underlying scaling
relations of the independent cluster sub-samples are statistically equivalent.
Thus, any observed difference can be then attributed to cosmological effects.
If there are variations in the cluster population between compared regions,
the inference of any cosmological effect can be distorted. Correspondingly,
compared cluster sub-samples must also have comparable $z$ distributions to
make certain we trace the same cosmic scales per direction.

Recent studies used several multi-wavelength cluster scaling relations to
probe the isotropy of the local ($z\lesssim 0.3$) Universe. Specifically, [17]
reported the detection of a $9.0\pm 1.7\%$ $H_{0}$ dipole using X-ray, SZ, and
infrared cluster data, substantially challenging the convergence to isotropy
hypothesis at these cosmic scales. They argued that the apparent $H_{0}$
variation might be caused by a $\sim 900$ km/s cluster bulk flow extending to
$500$ Mpc, at $\gtrsim 4\sigma$ tension with $\Lambda$CDM.

New cluster samples of unprecedented sizes from current and future surveys
will considerably boost the cosmic isotropy constraints from scaling
relations. Such surveys include eROSITA and Euclid that will eventually
measure X-ray and optical/infrared cluster quantities for $\gtrsim 50,000$ and
$\gtrsim 10^{5}$ clusters, respectively. Moreover, improvements in scaling
relations can be achieved by considering the cluster morphology and dynamical
state. Such refined modelling will result in lower scatter and more precise
cosmological constraints.

Finally, utilising large, cosmological, hydrodynamical simulations will enable
testing the robustness of different anisotropy detection methods using cluster
scaling relations. This is crucial in order to understand the importance of
the observational findings, eliminating possible systematic biases in the
applied methodology. Overall, multi-wavelength galaxy cluster scaling
relations will play a pivotal role in testing the isotropy of the late
Universe in the near future.

I acknowledge support in the form of the X-ray Oort Fellowship at Leiden
Observatory. I would also like to thank the Royal Society and the organisers
for inviting me as a speaker to the "Challenging the standard cosmological
model" meeting, which sparked the writing of this review. Finally, I thank my
wife for taking several vacation days off her job to care for our son so I can
finish writing this review on time.

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