# Substructure analysis of the RXCJ0232.2-4420 galaxy cluster

Viral Parekh1,2, Tatiana F. Laganá3, and Ruta Kale4
1Department of Physics and Electronics, Rhodes University, Drosty Rd,
Grahamstown, 6139, Republic of South Africa
2South African Radio Astronomy Observatory (SARAO), 2 Fir Street, Black River
Park, Observatory, 7925, South Africa
3Núcleo de Astrofísica, Universidade Cidade de São Paulo, R. Galvão Bueno 868,
Liberdade, São Paulo, SP, 01506-000, Brazil
4National Centre for Radio Astrophysics, Tata Institute of Fundamental
Research, Savitribai Phule Pune University Campus,
Pune 411 007 Maharashtra, INDIA
E-mail<EMAIL_ADDRESS>
(MNRAS Accepted.)

###### Abstract

RXCJ0232.2-4420, at $z$ = 0.28, is a peculiar system hosting a radio halo
source around the cool-core of the cluster. To investigate its formation and
nature, we used archival Chandra and XMM-Newton X-ray data to study the
dynamical state of the cluster and detect possible substructures in the hot
gas. Its X-ray surface brightness distribution shows no clear disruption
except an elongation in the North-East to South-West direction. We perform the
unsharp masking technique and compute morphology parameters (Gini, $M_{20}$
and concentration) to characterise the degree of disturbance in the projected
X-ray emission. Both of these methods revealed a substructure, which is
located at $\sim$ 1′ from the cluster core in the South-West direction.
Previous spectral analysis conducted for RXCJ0232.2-4420 concluded that there
are a short cooling time and low entropy at the cluster centre, indicating
that the cluster has a cool core. Thus, we suggest that RXCJ0232.2-4420 may be
a system where the core of the cluster is not showing any sign of disturbance,
but the South-West substructure could be pumping energy to the detected radio
halo via turbulence.

###### keywords:

Clusters of galaxies; intra-cluster medium, X-ray clusters

††pubyear: 2015††pagerange: Substructure analysis of the RXCJ0232.2-4420
galaxy cluster–LABEL:lastpage

##  1 Introduction

Galaxy clusters are the most massive bound and quasi-relaxed objects in the
Universe. In the mass composition of galaxy clusters, the majority of the
baryonic material is in the form of hot and diffuse gas, namely the intra-
cluster medium (ICM). This hot ICM retains the cluster’s thermodynamic
properties as well as imprints of recent merger history. Galaxy clusters can
be broadly divided into two categories - relaxed and non-relaxed clusters.
Generally, relaxed clusters possess a cool-core (CC) while non-relaxed
clusters (NCC) have either weak cool-core or no cool-core (but see Laganá et
al., 2019). However, there is not an unambiguous procedure that distinguishes
these two classes of clusters with a well-defined separation between NCC and
CC clusters (Lovisari et al., 2017; Parekh et al., 2015; Weißmann et al.,
2013).

The dynamic classification of a cluster has important implications both for
using clusters as cosmological tools and for studying the complex
gravitational and non-gravitational processes acting during large-scale
structure formation and evolution. Furthermore, the measurements of the global
properties of clusters, for example, hydrostatic mass and scaling relations
also depends on the cluster dynamical state (Lovisari et al., 2020; Pratt et
al., 2009).

Galaxy clusters also allow us to study large-scale and non-thermal diffuse
radio sources such as radio halos and relics which are believed to be linked
with substructures and cluster merging process (Cassano et al., 2010; Brunetti
et al., 2009). These diffuse radio sources are generally classified based on
their shapes and locations in the cluster: radio halos which are mostly
circular and situated at the cluster centre, while radio relics are found at
the periphery of the cluster and are elongated perpendicular to the merger
axis (e.g. van Weeren et al., 2019). These diffuse radio sources trace the
cluster magnetic field distribution and reservoir of relativistic electrons.
The radio halos are further divided into two categories, namely giant halos
(of the size of $\sim$ 1 Mpc) and mini-halos (of the size of $\sim$ 500 kpc)
(e.g. Brunetti & Jones, 2014; Feretti et al., 2012). Sometimes the physical
size of the radio halos is ambiguous and depends on the sensitivity and
quality of radio observations. The formation mechanisms of these two classes
of radio halos are still under debate, but from the observational point of
view, they are largely connected with the dynamical state of the host cluster.
Most of the giant radio halos are found in the disturbed or non-relaxed
clusters, while mini-halos are associated with relaxed or cool-core clusters
(Cassano et al., 2010; Buote, 2001). Cluster dynamical states are playing a
vital role to investigate the nature of diffuse radio sources, and recently,
several puzzling large-scale size ($\sim$ Mpc) radio halos are reported in
cool core clusters (Kale et al., 2019; Kale & Parekh, 2016; Sommer & Basu,
2014; Bonafede et al., 2014). But still, the role of the cluster dynamics on
the formation of radio halos is poorly understood.

High-resolution X-ray observations provide the quantitative and robust measure
of the cluster disturbance as well as the identification of subclusters,
bimodality and X-ray centroid shifts in clusters map using morphology
parameters (Parekh et al., 2015; Böhringer et al., 2010; Poole et al., 2006;
Buote & Tsai, 1995a, b). In the literature, there are other imaging and
filtering techniques applied to investigate substructures, for example, the
unsharp masking technique. Some authors have also advocated studying the
coincidence between the brightest cluster galaxy and X-ray peak to evaluate
cluster dynamical states (Lopes et al., 2018; Rossetti et al., 2016; Mann &
Ebeling, 2012). Cluster central cooling time and entropy are also very
important parameters to investigate the cool-core clusters (Hudson et al.,
2010; Cavagnolo et al., 2009). Entropy governs the global property of the
clusters and records the past heating and cooling history of these systems. It
is very crucial to investigate excess heat generation in the ICM due to
substructure or cluster merging.

Some substructures, however, are less evident in the X-ray surface brightness
distribution than the temperature distribution (Andersson et al., 2009; Govoni
et al., 2004). The merging process generates disturbances in the ICM, which
results in the swirling of stripped gas, shock expansion and propagation in
and out of the cluster. Such shock propagation and the resulting turbulence
can create a wide variety of phenomenon in the ICM. These imprints of
substructures or cluster merging process are largely studied by the
computation of the thermodynamic properties (Kale & Parekh, 2016; Ferrari et
al., 2006; Henry et al., 2004).

###  1.1 RXCJ0232.2-4420

RXCJ0232.2-4420 hosts a radio halo whose properties are similar to the mini-
halos (Kale et al., 2019). In the GMRT 606 MHz observation, Kale et al. (2019)
estimated the total linear extent of the halo is 550 $\times$ 800 kpc2 and the
1.4 GHz radio power of the halo is 4.5 $\times$ 1024 W Hz-1. Furthermore, they
argued that the detected radio halo in RXCJ0232.2-4420 is in the transition
phase of mini-halo to giant radio halo. To understand how the radio halo
formed in RXCJ0232.2-4420, it is important to investigate the dynamical state
of the cluster. This also requires to study of how the possible substructure
has triggered the particle re-acceleration process in the ICM.

The dynamical state of RXCJ0232.2-4420 is not clear in the literature. Two
independent parametric substructure analysis was performed for RXCJ0232.2-4420
based on the X-ray observations: Weißmann et al. (2013) have used a sample of
129 clusters observed by XMM-Newton to investigate the substructure occurrence
as a function of redshift. They have used two shape estimators, namely power
ratio ($P3/P0$) and centroid shift ($w$) to quantify the dynamical state of
the clusters. Based on their analysis, they found that the RXCJ0232.2-4420 is
a moderately disturbed cluster. Later on, Lovisari et al. (2017) used eight
morphology parameters for the Planck Early Sunyaev-Zeldovich (ESZ) clusters
that were observed by XMM-Newton telescope. They have shown that the
identification of relaxed clusters using a set of morphology parameters is
relatively easier than identifying disturbed clusters. They also show that
combining the concentration parameter (which is sensitive to the core) with
centroid shift (which is sensitive to substructure) gives better results to
characterise the dynamical state of the system. Based on their analysis, they
found RXCJ0232.2-4420 is a relaxed system.

We show Chandra X-ray and optical superCOSMOS images for RXCJ0232.2-4420 in
Fig. 1(a) and (b), respectively, and we list its X-ray properties in Table 1.
There are two brightest cluster galaxies (BCGs) found in the core of the
cluster (Pierini et al., 2008). In the optical image, we marked their
positions with the blue symbol ‘+’. In Fig. 1(c), diffuse radio halo contours
are shown to be surrounding one of the BCGs at the cluster core, which is a
typical signature of a mini-radio halo. Based on the visual inspection of the
Chandra X-ray image, it looks like a regular and relaxed cluster with
symmetric contours around the X-ray peak. There are no merging activities
visible as compared to other disturbed clusters hosting radio halos. However,
it shows the broad extension in the South-West direction. One of the BCGs is
spatially coincident with the X-ray peak as for a typical relaxed system.

In this paper, we have quantitatively investigated the dynamical state of this
cluster. This will help us to identify any underlying substructure and its
connection with the detected radio halo. This paper is organised as follows;
Section 2 gives X-ray data analysis, Section 3 presents the results, Sections
4 and 5 are the discussion and conclusion, respectively. Throughout this
paper, we assume $H_{0}$ = 70 km s-1 Mpc-1, $\Omega_{M}$ = 0.3 and
$\Omega_{\Lambda}$ = 0.7. At redshift $z$ = 0.2836, 1′′ corresponds to 4.28
kpc and a luminosity distance = 1456 Mpc. In this paper, all confidence
intervals correspond to 1$\sigma$.

((a))

((b))

((c))

Figure 1: (a) Exposure corrected and background subtracted Chandra X-ray image of RXCJ0232.2-4420. (b) X-ray contours (red) overlaid on the superCOSMOS optical image. We mark the BCGs positions with the blue ‘+’. (c) GMRT 606 MHz radio contours overlaid on the X-ray Chandra image. The contours levels are the same as in Kale et al. (2019). 1$\sigma$ rms = 0.1 mJy beam-1, resolution of 20′′ $\times$ 20′′. The first contour is drawn at $\pm$3$\sigma$ and increases with a factor of two. White contours are positive and magenta contours are negative. All X-ray images have energy range of 0.5-7 keV and 1.968′′ pixel size. Table 1: RXCJ0232.2-4420 X-ray properties from the MCXC catalogue (Piffaretti et al., 2011). Cluster Name | RA(J2000) | DEC(J2000) | $z$ | $L_{\rm X}[0.1-2.4]\rm\,keV$ | $M_{500}$
---|---|---|---|---|---
| h m s | h m s |  |  ($10^{44}\rm\,erg/s$) |  (1014 $M_{\odot}$)
RXCJ0232.2-4420 | 02 32 16.8 | -44 20 51.0 | 0.2836 | 8.49 | 6.13

##  2 X-ray observation and data analysis

###  2.1 Chandra data analysis

We use Chandra archival data (obsid 4993, date of observation 8th June 2004,
and total exposure time $\sim$ 23.4 ks) to study RXCJ0232.2-4420 for both
imaging and spectral analysis purposes. We followed the standard X-ray data
analysis steps in the data reduction procedure. We processed the Chandra data
with the Chandra Interactive Analysis of Observations (CIAO v 4.11) (Fruscione
et al., 2006) and CALDB 4.8.3 software. In the data reduction steps, in the
beginning, we use the `chandra_repro` task to reprocess the Advanced CCD
Imaging Spectrometer (ACIS) imaging data, followed by removing high background
flares (3$\sigma$ clipping) with the task `lc_sigma_clip`. All filtered event
files (exposure time $\sim$ 22,414 sec after flare removal) are in the energy
band of [0.5–7.0] keV in further analysis. We binned the event files with the
factor of 4, which results in 1.968′′ image pixel size. We detect and subtract
point sources around the cluster emission using the CIAO’s `wavdetect` and
`dmfilth` tasks, respectively. Finally, we divided the count image by the
exposure map and generated the flux image (photon cm-2 s-1). We make the
exposure map using the `fluximage` task of the CIAO software. We also subtract
the background from the flux image using the Chandra blank sky
files111http://cxc.harvard.edu/ciao/threads/acisbackground/.

###  2.2 Chandra spectral analysis

We extract the X-ray spectra using the `specextract` task of the CIAO. Then we
use the package Sherpa (version 1) in order to perform the Chandra spectral
fitting. In the spectral analysis procedure, spectra are grouped such that
each bin has a minimum count of 10. In all Chandra spectral fit procedure, the
redshift of RXCJ0232.2-4420 is fixed at $z=$ 0.2836, metallicity (Z) at 0.3 of
the solar value, and the hydrogen column density value is fixed at $n_{H}$ =
2.48 $\times$ 1020 atoms cm-2 estimated using the colden
toolkit222https://cxc.harvard.edu/toolkit/colden.jsp (NRAO dataset), while the
temperature ($kT$) and the spectrum normalisation (K) were free to vary. A
single temperature fit is used to model the cluster X-ray spectrum with an
absorbed thermal plasma emission model WABS(APEC). Abundances were estimated
assuming the ratios from Asplund et al. (2009). We employed Chandra blank sky
event files, for background subtraction, after taking into account the
background scaling method.

###  2.3 XMM-Newton Data Reduction

We use XMM-Newton archival data of RXCJ0232.2-4420. The cluster was observed
on 2002 July 11th for $\sim$ 13,645 sec for MOS (XMM-Newton,
ObsID:0042340301). In this observation, the prime full-frame mode was employed
for the XMM-Newton’s three cameras (extended mode for pn) with thin filters.
The data reduction was done with the Science Analysis System (SAS) (Gabriel et
al., 2004) version 16.1.0 (July 2017) and calibration files updated to April
2019. For the purpose of filtering background flares, we applied a
2$\sigma$-clipping procedure using the light curves in the [10–14] keV energy
band. The resulting “cleaned” exposure times are 11.3 ks for MOS cameras and
6.4 ks for pn, respectively. Then, to take into account the background
contribution for each detector, we obtained a background spectrum in an outer
annulus of the observation in the [10-12] keV energy band. We matched these
background spectra with the blank sky obtained by Read & Ponman (2003) in the
same energy band and region. Finally, we rescaled the observation background
to the blank sky background to get a normalisation parameter that will be
applied in the spectral fits. Discrete X-ray point sources were detected by
visual inspection, confirmed in the High Energy Catalogue 2XMMi Source, and
excluded from events files.

###  2.4 XMM-Newton spectral analysis

In our work, the XMM-Newton spectral analysis was restricted to the energy
range [0.7–7.0] keV. Furthermore, to avoid any influence from Al and Si
instrumental lines, we also exclude the energy band from 1.2 to 1.9 keV. All
the extracted spectra are re-binned to ensure at least 15 counts in each
energy bin. To model the spectrum, we adopt an absorbed thermal plasma
emission model WABS(APEC) as similarly used for the Chandra spectral analysis.
Abundances were also estimated considering the ratios from Asplund et al.
(2009). In all the fitting procedures performed using the XSPEC version
12.9.1, the redshift and $n_{H}$ were fixed at $z=$ 0.2836 and 2.48 $\times$
1020 atoms cm-2, respectively. The $kT$, Z, K were free to vary.

If we divide the normalization by the number of pixels in each bin (K/$N_{\rm
P}$), we obtain the projected emission measure (EM). EM is proportional to the
square of the electron density $n_{e}^{2}$, integrated along the line of
sight. Thus, with the best-fit $kT$ and K, we estimate the pseudo-entropy
($S$) and pseudo-pressure ($P$) in the following ways:
$S=kT\times[\frac{K}{N_{\rm P}}]^{-1/3}$ and $P=kT\times[\frac{K}{N_{\rm
P}}]^{1/2}$.

##  3 Results

###  3.1 X-ray surface brightness

In this section, we derive the surface brightness profile of RXCJ0232.2-4420.
We adopt the $\beta$-model (Cavaliere & Fusco-Femiano, 1976) to study the 1D
surface brightness profile. This model is described by four parameters as
shown by Eq. 1, where $r_{c}$ is the cluster core radius (in unit of arcmin),
$\beta$ controls the slope of the profile or power law index, $r$ is the
reference point of the profile, and $\eta_{\rm gas}$(0) is the normalisation
or the maximum value of the model, such that

$\eta_{\rm gas}(r)=\eta_{\rm
gas}(0)\left[1+\left(\frac{r}{r_{c}}\right)^{2}\right]^{-3\beta+1/2}+Const.$
(1)

We derive the azimuthal average surface brightness profile over a radius of 4′
($\sim$ 1 Mpc) centred on the peak of the X-ray emission (reference point) as
shown in Fig. 2(a). We fit the surface brightness profile using the PROFFIT
v1.5 software (Eckert et al., 2011), shown in Fig. 2(b), assuming for each bin
an SNR of at least 20. Our best fit values and statistics are listed in Table
2. At the distance of $\sim$ 1′ from the origin, the best fitting profile
deviates from the data points, corresponding to a large fluctuation in
$\chi^{2}$, as can be seen in Fig. 2(b).

To investigate this deviation in the surface brightness profile, we divided
the cluster region into the four 90∘ sectors, as shown in Fig. 3(a). For each
of these sectors, we extracted the surface brightness profiles and plotted
them in Fig. 3(b). As one can see, there is a visible discrepancy in sector
number 4 (270∘ \- 360∘ angle), in the South-West direction of the cluster,
compared to the other three sectors.

((a))

((b))

Figure 2: (a) Annuli centred on the X-ray peak of the cluster RXCJ0232.2-4420, used to obtain the azimuthal surface brightness profile, shown in panel (b) fitted with the $\beta$-model. Table 2:  1D $\beta$ profile and 2D $\beta$-model (see $3.2) best fitting values with statistics. parameter | value
---|---
$\beta$ | 0.43 $\pm$ 0.0014
$r_{c}$ | 0.08 (22 $h_{70}$ kpc)$\pm$ 0.0012 (3 $h_{70}$ kpc)
Norm | 9.6 $\times$ 10-3
Const | 4.4$\times$ 10-6
$\chi^{2}$ | 412
d.o.f | 118
$r_{0}$ | 60${}^{+5}_{-5}$ $h_{70}$ kpc
$\epsilon$ | 0.23${}^{+0.012}_{-0.012}$
$\theta$ | 320${}^{+1.664}_{-1.664}$ deg
$\alpha$ | 0.60${}^{+0.014}_{-0.015}$
$\chi^{2}$ | 6056
d.o.f | 7848

((a))

((b))

Figure 3: (a) Four sectors of 90∘ angle each centered on the X-ray peak. (b)
Surface brightness profiles extracted from the four sectors, that are plotted
with different colors.

###  3.2 Cluster residual images

In order to generate the cluster’s residual images, we use two different
techniques: (1) unsharp masking and (2) subtract a 2D beta-model from the
cluster image. In this analysis, we find that both these techniques are
useful; however, the former is important to pinpoint the underlying
substructure in the inner parts of the cluster, while latter is better used
for revealing larger-scale features.

Unsharp mask is a filtering technique that can be used to identify
substructure within a cluster’s image. This is done by smoothing a cluster
emission with a filter of width $\sigma$, so that high-frequency structure can
be washed out from the smoothed image. This map has only high-frequency
components of the galaxy cluster’s flux distribution, representing the
underlying substructures. To make smooth images, we use four different
Gaussian kernels of 5$\sigma$, 10$\sigma$, 20$\sigma$ and 50$\sigma$. Then, we
subtract from original cluster images each of these smoothed images (and take
their absolute values), and the residual images are shown in Fig. 4. There is
an excess of X-ray emission visible in the South-West direction, which is
identified by the green circle as plotted in Fig. 4(a). The distance from the
cluster’s X-ray peak to this region of emission excess is $\sim$ 0.6′, and its
radius is estimated to be $\sim$ 0.35′.

We also use the 2D spherical $\beta$-model to subtract the surface brightness
model from the cluster X-ray image in the [0.5–7 keV] energy band. To take
into account the ellipticity of the plasma emission we use the following
standard coordinates transformation:

$\begin{cases}x^{\prime}=(x-x_{0})\cos\theta-(y-y_{0})\sin\theta\\\
y^{\prime}=(x-x_{0})\sin\theta-(y-y_{0})\cos\theta\\\ r^{2}=x^{\prime
2}+\frac{y^{\prime 2}}{(1-\epsilon)^{2}}&\end{cases}$ (2)

where $(x_{0},y_{0})$ are the coordinates of the X-ray emission peak, $\theta$
is the position angle, and $\epsilon$ is the ellipticity. The $\beta$-model
may now be defined as follows:

$S(r)=S_{0}\bigg{(}1+\frac{r^{2}}{r_{\rm c}^{2}}\bigg{)}^{-3\beta+1/2}+b,$ (3)

where $S_{0}$ is the central brightness, $r_{\rm c}$ the core radius, $\beta$
the shape parameter, and $b$ a residual background emission, assumed to be
constant across the FOV.

In the residual image, shown in Fig. 5, there is an X-ray excess visible in
the South-West direction, identified by the green circle. We mark the position
of the peak of this residual emission excess by a black ‘x’. The projected
distance between the X-ray peak of the cluster core to this emission excess
peak is $\sim$ 1′. We have listed the best-fit parameters in Table 2. In the
residual image, the region of emission excess is larger and more scattered
than in the previous Gaussian subtracted case. The residual emission is
extending toward the North and South-East direction from its peak. This
irregularity makes this region more asymmetric and complex around the emission
excess peak.

Figure 4: Residual images of the unsharp mask filtering. In each image we have
given the width of the Gaussian sigma used for filtering. We also located the
emission excess region inside a green circle in the top-left image. Figure 5:
Residual image after the 2D $\beta$-model subtraction. We marked the emission
excess region and its peak inside the green circle.

###  3.3 Morphology parameters

In this section, we analyse the X-ray morphology of RXCJ0232.2-4420 to
characterise the degree of disturbances in the projected X-ray emission. This
analysis will also allow us to investigate the dynamical state of the cluster
and its comparisons with other X-ray clusters’ morphologies. For this work, we
computed three morphology parameters - Gini, $M_{20}$ and concentration -
using RXCJ0232.2-4420 Chandra X-ray image in the [0.5-7 keV] energy band.
These morphology parameters trace the flux distribution for a given region,
and their computed values depend on the degree of disturbance in a given
cluster image. For more details about the parameters and method, we refer to
the Parekh et al. (2015). We calculate these parameters for a circular region
of 500 $h_{70}$ kpc of radius, centred on the flux weighted centroid (see
Table 3). Fig. (6) shows these three morphology parameters in the parameter-
parameter planes. We represent RXCJ0232.2-4420 along with a sample of relaxed
and merging clusters taken from Vikhlinin et al. (2009), for which their
morphology parameters were computed in our previous work (Parekh et al.,
2015). In the same plot, we have also shown a sample of diffuse radio
clusters, including radio halos, relics and mini-halos (Yuan et al., 2015).

Table 3: RXCJ0232.2-4420 morphology parameter values. | Gini | $M_{20}$ | concentration
---|---|---|---
| 0.55 $\pm$0.005 | -1.58 $\pm$ 0.046 | 1.37 $\pm$ 0.014

((a))

((b))

((c))

Figure 6: Three morphology parameters plotted in the parameter-parameter
planes with RXCJ0232.2-4420 represented by a  $\bigstar$. $\circ$ = strong
relaxed clusters, $\lozenge$ = relaxed clusters,  $+$ = non-relaxed clusters,
and  $\times$ = strong non-relaxed. This sample is taken from Vikhlinin et al.
(2009) and morphology parameters calculated in Parekh et al. (2015). All
clusters with exclusively diffuse radio sources are marked with hexagon. We
took diffuse radio sources sample from (Yuan et al., 2015). Yellow hexagon
correspond to only radio halo, blue hexagon to single relic, cyan hexagon
correspond to double relics, and red hexagon to halo+relic and green hexagon
to mini- halo.

###  3.4 Temperature distribution

In order to understand the temperature distribution, we measure the
temperature for different regions of the RXCJ0232.2-4420 cluster. First of
all, we study the azimuthal mean temperature measurements. Cavagnolo et al.
(2009) have derived the radial temperature of RXCJ0232.2-4420 using the
Chandra data333https://web.pa.msu.edu/astro/MC2/accept/clusters/4993.html.
They used five radial bins (each contains 2500 counts) up to 500 kpc radius
centred on the X-ray peak.

Apart from the azimuthal radial profile, we also study the temperature
distribution; (1) across the North-East to South-West cluster emission by
dividing the cluster into five boxes and (2) within the sector 4 (Fig. 3(a))
which is the 270∘ to 360∘ angle. We show both of these regions in Fig. 7. In
addition to this, we also extract the spectra from the emission excess region
(Fig. 4 green circle). We plot the temperature profiles for both of these
regions (boxes and sector) in Fig. 8. In $3.1, we notice that the surface
brightness profile in sector 4 is deviating from the other profiles. Hence, we
extract the spectra from sector 4 up to 2′ maximum radius from the cluster
core within five bins. This is shown in Fig. 7 and corresponds to the
temperature profile in Fig. 8(b).

Figure 7: Spectra extracted from the various regions of RXCJ0232.2-4420.

((a))

((b))

Figure 8: (a) Temperature plotted for the box regions. (b) Temperature plotted within the sector 4 bins vs the distance from the core. Table 4: Best-fitting spectral values of the of the box regions as shown in Fig. 7. Region | T | Normalisation |  $\chi^{2}$/dof |
---|---|---|---|---
| keV |  $\times$10-3 $cm^{-3}$ |  |
1 | 5.74${}^{+0.74}_{-0.56}$ | 0.840${}^{+0.031}_{-0.031}$ | 80/118 |
2 | 5.09${}^{+0.30}_{-0.29}$ | 2.284${}^{+0.052}_{-0.052}$ | 164/204 |
3 | 7.48${}^{+0.95}_{-0.82}$ | 1.086${}^{+0.033}_{-0.033}$ | 93/152 |
4 | 6.32${}^{+1.01}_{-0.85}$ | 0.747${}^{+0.032}_{-0.031}$ | 66/101 |
5 | 9.52${}^{+2.95}_{-1.96}$ | 0.442${}^{+0.021}_{-0.021}$ | 40/74 |
Table 5: Best-fitting spectral values of the pie regions (sector 4) as shown in Fig. 7. Bins | T | Normalisation |  $\chi^{2}$/dof |
---|---|---|---|---
| keV |  $\times$10-3 $cm^{-3}$ |  |
1 | 6.00${}^{+0.83}_{-0.75}$ | 0.736${}^{+0.029}_{-0.030}$ | 63/110 |
2 | 6.46${}^{+1.01}_{-0.81}$ | 0.771${}^{+0.031}_{-0.031}$ | 65/113 |
3 | 8.61${}^{+1.75}_{-1.32}$ | 0.691${}^{+0.028}_{-0.026}$ | 60/104 |
4 | 6.94${}^{+1.54}_{-0.93}$ | 0.469${}^{+0.022}_{-0.022}$ | 54/79 |
5 | 7.63${}^{+2.49}_{-1.61}$ | 0.345${}^{+0.021}_{-0.021}$ | 29/57 |

###  3.5 Thermodynamic maps

in Fig. 9, we show the 2D temperature ($kT$), pseudo-entropy ($S$) and pseudo-
pressure ($P$) maps for the cluster RXCJ0232.2-4420. We show temperature error
and reduced $\chi^{2}$ maps into Appendix Fig. 1. This was achieved by
dividing the data into small regions from which spectra were extracted (this
procedure was already described in Durret et al., 2010; Durret et al., 2011;
Laganá et al., 2008, 2019). We set a minimum count number of 900 net counts
(that is, after background subtraction) to each pixel and, if necessary, we
increased the region up to a box of 5 $\times$ 5 pixels. Each pixel has a size
of 12.8 $\times$ 12.8 arcsec2, corresponding to 54.8 $\times$ 54.8 $h_{70}$
kpc2. If we still do not have sufficient counts after binning, the pixel is
ignored, and we proceed to the next neighbouring pixel. When we have adequate
counts, the spectra of MOS1, MOS2 and pn are then simultaneously fitted, and
the best-fit $kT$ and $Z$ values are attributed to the central pixel.

((a))

((b))

((c))

Figure 9: Thermodynamic maps of RXCJ0232.2-4420. (a) Temperature ($kT$) map.
(b) Pseudo-entropy ($S$) map. (c) Pseudo-pressure ($P$) map. In all maps,
Chandra smoothed image contours are overlaid.

##  4 Discussion

###  4.1 Radio halo, morphology and substructure

RXCJ0232.2-4420 is a peculiar system hosting a radio halo of the size 550
$\times$ 800 kpc2 (Kale et al., 2019). The size of the halo is intermediate
between large radio halos and mini halos. In line with the characteristic of
mini-halos to occur around BCGs, and unlike typical radio halos in merging
clusters where the BCG is often displaced from the centre, this radio halo
surrounds the BCG in the cluster. From a visual inspection, its X-ray image or
surface brightness distribution does not show an obvious departure from the
thermodynamic equilibrium, unlike other merging systems. Thus, it was
classified as a system in transition from mini-halos to radio halos. The
cluster is elongated in the North-East to the South-West direction, similar to
the A1995 merging system (Boschin et al., 2012) in which a diffuse radio halo
has been detected. For the A1995 merging cluster, the authors used X-ray and
optical data and identified two main optical substructures causing the NE-SW
elongation of the galaxy distribution what suggests a bimodal merger caught in
a phase of post core-core passage.

RXCJ0232.2-4420 cluster was part of the Archive of Chandra Cluster Entropy
Profile Tables (ACCEPT) sample (Cavagnolo et al., 2009). They estimated the
entropy and cooling time of the central (30 kpc) region to be $\sim$ 50 keV
cm2 and 1 Gyr, respectively. Santos et al. (2008) classified as strong cool-
core clusters those having cooling time $<$ 3 Gyr, which puts RXCJ0232.2-4420
in this category. Further, Hudson et al. (2010) have measured the central
entropy for HIFLUGCS (HIghest X-ray FLUx Galaxy Cluster Sample) cluster sample
and subdivide clusters into (i) strong cool-core, (ii) weak cool-core or (iii)
no cool-core clusters. Based on their analysis, they found the strong cool-
core clusters have central entropy $<$ 30 keV cm2, and no cool-core clusters
have central entropy $>$ 110 keV cm2. According to this classification, the
RXCJ0232.2-4420 cluster falls into the weak cool-core cluster.

We study the azimuthal average surface brightness profile over 4′ region
centred on the peak of the X-ray emission. We could see the departure from the
$\beta$-model at $\sim$ 1′ from the fitting (Fig. 2), and it suggests
disturbances in surface brightness distribution. To investigate this
disturbance, we divided the cluster emission into four sectors and studied the
profiles in each of the sectors (Fig. 3). Out of the four sectors, there is
one sector that clearly shows the disturbances in the South-West direction,
which is the elongation axis of the cluster. This sector spans the 270-360∘
area and hosts the possible substructure. To further investigate this
substructure, we use two different filtering techniques. One is simple unsharp
masking which can be used to amplify the high-frequency components of the
image. To do so, we adopt four different Gaussian kernels (as shown in Fig.
4), and in all of the residual images, we detect an X-ray emission excess in
the South-West direction. We consider this to be an underlying substructure
associated with RXCJ0232.2-4420. The distance from the peak (or BCG) to this
substructure is $\sim$ 1′. We measure the size of this substructure to be
0.35′ or 90 $h_{70}$ kpc of the radius. Similarly, we also modelled the
cluster emission with the 2D $\beta$-model and subtracted it from the original
image. This residual image (Fig. 5) clearly depicts the substructure, much
extended than what is seen in the unsharp masked image. Both techniques reveal
the presence of substructure in the South-West direction where the cluster is
elongated.

To add to our analysis, we study the three morphology parameters, namely,
Gini, $M_{20}$ and concentration, to compare the morphology of RXCJ0232.2-4420
with other samples of clusters with different dynamical states. Following the
classification criteria provided by Parekh et al. (2015) for the dynamical
state based on these three parameters, the strong relaxed (strong cool-core)
clusters have Gini $>$ 0.65, concentration $>$ 1.55 and $M_{20}$ $<$ -2.0,
while the disturbed clusters have Gini $<$ 0.40, concentration $<$ 1.0 and
$M_{20}$ $>$ -1.4. All other clusters falling in the intermediate dynamical
states (or weak cool core) have the parameters between these ranges. The
spatial parameter values obtained for RXCJ0232.2-4420 do not place this
cluster clearly in any particular category, besides displaying some level of
the substructure. We showed the position of RXCJ0232.2-4420 along with other
samples of clusters in Fig. 6. In this figure, the position of RXCJ0232.2-4420
also matches the single radio halo clusters. Overall X-ray morphology
indicates that the RXCJ0232.2-4420 is a slightly disturbed cluster that is
elongated in one particular direction and has a bright core. The flux weighted
centroid does not coincide with the X-ray peak, falling on the elongated
South-West axis below the X-ray peak (17′′ away).

The low cooling time of the core of RXCJ0232.2-4420 and spatial coincidence
between X-ray peak and BCG may suggest that the core of the cluster might not
be disturbed by a cluster merger yet. Rossetti et al. (2007) analysed the
Chandra and XMM observation of A3558 and found that the cool-core of the A3558
had survived in a merger. Chatzikos et al. (2006) showed that the A2065 is an
ongoing merging cluster that might be a possible reason behind the survival of
one cool-core in this unequal mass merger. Several observations and
simulations show that the elongation of a cluster is the result of the
accretion of groups along filaments (Durret et al., 1998; Roettiger et al.,
1997). RXCJ0232.2-4420 is showing that the possible interaction with other
groups or clusters has been in the elongated direction, but the merger shocks
have not reached the core of the cluster. Hence, the core presents low
temperature (and entropy) and cooling time values as its cool-core has still
survived in the merging process.

###  4.2 Temperature distributions

RXCJ0232.2-4420 is a hot and luminous cluster. Cavagnolo et al. (2009)
estimated the temperature over a 5.94′ region (which corresponds to a linear
size of $\sim$ 1.5 Mpc) to be $kT$ $\sim$ 7.8 keV and its corresponding
bolometric luminosity is $L_{\rm X}=18.30\times 10^{44}$ erg/s. Generally,
clusters with radio halos are hotter and luminous (e.g. Feretti et al., 2012).
It was observed that about 30% of the high X-ray luminosity ($L_{\rm X}$ $>$
5$\times$1044 erg s-1) galaxy clusters host radio halos. Furthermore, we note
that the simulations and observations of merging galaxy clusters have shown
that mergers can temporarily boost the X-ray temperature and luminosity of the
systems (Sarazin, 2004; Randall et al., 2002). The underlying substructure
could cause the high temperature of RXCJ0232.2-4420. Its radial temperature
profile (Cavagnolo et al., 2009) suggests a lower temperature of $\sim$ 5-6
keV in the inner core ($<$ 50 kpc) region, rising to $\sim$ 8-9 keV at the
distance of 200 kpc from the centre where the identified substructure is
located. Going towards the outskirts, the derived temperature has large error
bars and thus becomes unreliable. We also study the temperature distribution
across the North-East to South-West direction, dividing it into five boxes of
size 354 $\times$ 97 $h_{70}$ kpc2, as shown in Fig. 7. As can be seen in the
temperature profile of the box regions (Fig. 8(a)), the temperature is low,
$\sim$ 5-5.7 kev, in boxes 1 and 2, corresponding to the inner regions of
RXCJ0232.2-4420, close to the BCGs. Then temperature rises up from $\sim$ 6.3
to 9.5 keV into the boxes numbered 4 and 5. It is important to state that the
errors associated with temperature measurements are large (but temperatures
from region 4 and 5 still disagree within 1-$\sigma$), hence the statistical
significance of results should be evaluated carefully. These boxes are
spatially coincident with the position of the substructure. We also study the
temperature profile in the sector between 270-360∘ angle centred on the X-ray
peak. Similarly to the box profile, the temperature is low in the inner bins
close to the BCGs. At the distance of $\sim$ 1′ temperature rises to $\sim$
8.5 keV, and then it decreases again. This suggests the substructure is
surrounded by low-temperature gas. We measured the temperature of the
substructure region, indicated by the green circle in Fig. 7, and it is
$kT=8.82\pm 2.16$ keV. This analysis suggests that the presence of this
substructure could affect the global X-ray properties of the cluster.

###  4.3 Thermodynamic properties

We also perform the analysis of 2D thermodynamic maps of RXCJ0232.2-4420 using
XMM-Newton data. As shown in Fig. 9, the pseudo-pressure map shows the high-
pressure fluctuations in the South-West direction near the position of the
substructure as detected in the Chandra residual map. We marked the position
of the substructure in Fig. 9(a) with the green circle. We note that in this
(XMM-Newton) pressure map, the position of the substructure as revealed by the
Chandra image does not coincide with the South-West high-pressure region. We
found there is an offset of $\sim$ 30′′ between the substructure regions in
Chandra and XMM-Newton. The possible reason behind the disagreement between
substructure position could be the lower spatial resolution of XMM-Newton
telescope. XMM-Newton temperature map shows the high temperature ($\sim$ 8-9
keV) fluctuations (at the location of high pressure) which is surrounded by
the low-temperature gas of $\sim$ 6-7 keV. This is supported by the
temperature profile in sector 4 (Fig. 8(b)), where the temperature is dropping
outside the substructure region. The pseudo-entropy map also shows the
fluctuations at the location of the high-pressure region. There is a low
entropy gas near the cluster core, but high entropy gas extends towards the
substructure position. The fluctuation in the temperature and entropy maps
could be the result of a mixing of hot gas between RXCJ0232.2-4420 and its
substructure.

##  5 Conclusion

RXCJ0232.2-4420 is a hot (of $kT$ $\sim$ 7.8 keV) and luminous ($L_{\rm
X}=18.30\times 10^{44}$ erg/s) cluster with a small scale (of 180$h_{70}$ kpc
diameter) substructure. One of the characteristics of the system is the
transition phase of its radio halo from mini-halo to giant radio halo (Kale et
al., 2019). In a typical mini-halo system, the core of the host cluster shows
disturbances in the form of cold fronts, swirling of the hot gas or AGN
activity (Gitti et al., 2015; ZuHone et al., 2011). These disturbances may
provide turbulent re-acceleration to the seed relativistic particles. While in
the RXCJ0232.2-4420 case, there are no disturbances detected at the core in
X-ray data, there is a faint substructure located $\sim$ 1′ away from the core
that could have survived in a past merging process. Furthermore, this
substructure could be in the process of injecting turbulence in the
RXCJ0232.2-4420 system and hence slowly power the radio halo due to the
particle re-acceleration. This indicates the radio halo is still growing
around the core of the RXCJ0232.2-4420 cluster. Our future MeerKAT and uGMRT
wideband and more sensitive radio observations will shed light on the nature
of the radio halo of RXCJ0232.2-4420, as well as how substructure affects the
magnetic field distributions.

## Acknowledgements

The financial assistance of the South African Radio Astronomy Observatory
(SARAO) towards this research is hereby acknowledged (www.ska.ac.za). This
work is based upon research supported by the South African Research Chairs
Initiative of the Department of Science and Technology and National Research
Foundation. TFL acknowledges financial support from the Brazilian agencies
FAPESP and CNPq through grants 2018/02626-8 and 306163/2019-5, respectively.
RK acknowledges the support of the Department of Atomic Energy, Government of
India, under project no. 12-R&D-TFR-5.02-0700. We thank the staff of the GMRT
who have made these observations possible. The GMRT is run by the National
Centre for Radio Astrophysics of the Tata Institute of Fundamental Research.

## Data Availability

The data underlying this paper are available in Chandra and XMM-Newton
telescopes archive systems, at https://cda.harvard.edu/chaser/ and
https://nxsa.esac.esa.int/nxsa-web/.

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## Appendix

((a))

((b))

Figure 10: (a) $kT$ error and (b) reduced $\chi^{2}$ maps corresponding to
XMM-Newton 2D $kT$ map. In all maps, Chandra smoothed image contours are
overlaid.


Negative single-trace $T\bar{T}$ holography
versus
de Sitter holography

Amit Giveon

Racah Institute of Physics, The Hebrew University

Jerusalem, 91904, Israel

###### Abstract

In single-trace $T\bar{T}$ holography, for the negative sign of the
deformation, the geometries dual to high energy states consist of a singular
UV wall that separates black strings in the IR from an asymptotically linear
dilaton spacetime. In the hologram, the black strings amount to states in a
symmetric product of $T\bar{T}$ deformed $CFT_{2}$, whose energy and momentum
are split equally among the different blocks. The properties of this
holographic duality have intriguing similarities with the proposed principles
of de Sitter holography. In this note, the analogy between the two is
presented. In particular, as the horizon location approaches the UV wall, one
obtains a state of maximum finite entropy and infinite temperature. On the
other hand, the bulk temperature of the geometry obtained in the limit is
small. These properties are identical to those proposed in empty de Sitter
holography.

Consider supersting theory on the deformed BTZ black holes background, formed
in the near $k$ NS5 branes on $S^{1}\times T^{4}$ with $p$ (negative)
fundamental strings wrapping $S^{1}$, whose radius we denote by $R$, and which
carry momentum number $n$ on this circle. There are two possible signs of the
deformation parameter, denoted by $\lambda\equiv\pm\alpha^{\prime}/R^{2}$ in
[1, 2, 3]; here we consider the $\lambda<0$ case (which amounts to $p$
negative strings, [1]):

$\lambda\equiv-{\alpha^{\prime}\over R^{2}}<0~{}.$ (1)

The background thus obtained describes a rotating black string 111Which we may
regard (upon KK reduction) as a two-dimensional black hole with winding and
momentum fundamental string charges, $p$ and $n$, respectively, as in [4] with
$(q_{L},q_{R})=\left({n\over R}+{pR\over\alpha^{\prime}},{n\over
R}-{pR\over\alpha^{\prime}}\right)$ (note that a subset of the backgrounds in
appendix C of [4] are identical to those below; we shall not present the
coordinate transformation between the two here). in a three-dimensional
spacetime, denoted by ${\cal M}_{3}^{-}$ in [1]; 222Empty ${\cal M}_{3}^{-}$
is a current-current deformation of massless BTZ in a negative direction, [5];
from the point of view of the boundary at infinity, this background can be
thought of as a vacuum of Little String Theory which contains a large number
of negative strings, [1]. its metric, $B$-field and dilaton appear in
equations (3.10) with (3.11)) of [3]:

$ds^{2}=-{N^{2}\over 1-{\rho^{2}\over R^{2}}}d\tau^{2}+{d\rho^{2}\over
N^{2}}+{\rho^{2}\over 1-{\rho^{2}\over R^{2}}}\left(d\varphi-
N_{\varphi}d\tau\right)^{2}~{},$ (2) $B_{\tau\varphi}={\rho^{2}\over
r_{5}}\sqrt{\left(1-{\rho_{-}^{2}\over R^{2}}\right)\left(1-{\rho_{+}^{2}\over
R^{2}}\right)}{1\over 1-{\rho^{2}\over R^{2}}}~{},$ (3) $e^{2\Phi}={kv\over
p}\sqrt{\left(1-{\rho_{-}^{2}\over R^{2}}\right)\left(1-{\rho_{+}^{2}\over
R^{2}}\right)}{1\over 1-{\rho^{2}\over R^{2}}}~{},$ (4)

where

$N^{2}={(\rho^{2}-\rho_{+}^{2})(\rho^{2}-\rho_{-}^{2})\over
r_{5}^{2}\rho^{2}}~{},\qquad N_{\varphi}={\rho_{+}\rho_{-}\over
r_{5}\rho^{2}}~{},$ (5) $r_{5}\equiv\sqrt{k\alpha^{\prime}}~{},\qquad
v\equiv{\rm Volume}(T^{4})/\left(2\pi\sqrt{\alpha^{\prime}}\right)^{4}~{},$
(6)

and

$\rho_{-}\leq\rho_{+}\leq R$ (7)

are the locations of the inner horizon and outer horizon of the black string
and the singular wall in ${\cal M}_{3}^{-}$, respectively. 333See [5, 1, 3]
for more details.

The angular momentum, $n$, of the black string, (2)–(6), 444Which originates
from the momentum $n$ carried by the $p$ negative strings that form it. is,
[3],

$n={p\over\alpha^{\prime}}{\rho_{-}\rho_{+}\over\sqrt{\left(1-{\rho_{-}^{2}\over
R^{2}}\right)\left(1-{\rho_{+}^{2}\over R^{2}}\right)}}~{}.$ (8)

Below, we inspect the properties of the one-parameter family of the
backgrounds above for certain $R,k,p$ and $n$. It can be parameterized e.g. by
the location of the event horizon, $\rho_{+}$. Along this one-parameter
family, the entropy and temperature are, [3],

$S={2\pi kp\over r_{5}}{\rho_{+}\over\sqrt{1-{\rho_{-}^{2}\over R^{2}}}}~{},$
(9)

and

$T={\rho_{+}\over 2\pi
r_{5}R}{1-{\rho_{-}^{2}\over\rho_{+}^{2}}\over\sqrt{1-{\rho_{+}^{2}\over
R^{2}}}}~{},$ (10)

respectively, where $\rho_{-}$ is related to $\rho_{+}$ via equation (8). The
energy of the black hole is, [3],

$E={p\over\lambda R}\left\\{-1+\left[1-\left(\lambda n\over
p\right)^{2}\right]\sqrt{\left(1-{\rho_{-}^{2}\over
R^{2}}\right)\left(1-{\rho_{+}^{2}\over R^{2}}\right)}\right\\}~{},$ (11)

where $\lambda$ is given in (1).

A few comments are in order:

  1. 1.

As usual, the interior of the black hole, $\rho_{-}<\rho<\rho_{+}$, is time
dependent (with time being $\rho$), while the exterior of the black hole,
$\rho>\rho_{+}$, is time independent, with $\tau$ ($\varphi$) being a timelike
direction when $\rho<R$ ($\rho>R$), instead of $\rho$. We shall refer to the
regime between the event horizon and the UV wall, $\rho_{+}<\rho<R$, as the
static patch. 555The other static regime, $\rho>R$, a.k.a. the asymptotically
linear dilaton regime beyond the UV wall, plays an important role in single-
trace $T\bar{T}$ holography, [1, 3]; we shall return to this point below.

  2. 2.

As one varies the location of the horizon $\rho_{+}$ (while keeping $R$ and
$n$ fixed, (8)), one interpolates between empty $\mathcal{M}_{3}^{-}$ (when
$\rho_{\pm}=0$) and the case of a maximal black hole size, $\rho_{+}=R$ (in
which case $\rho_{-}=0$, (8)) in $\mathcal{M}_{3}^{-}$. 666We shall discuss
the $\rho_{+}\to R$ limit in detail below.

  3. 3.

The energy is minimal, $E_{min}=0$, for empty $\mathcal{M}_{3}^{-}$ (for any
deformation parameter, (1), of the massless BTZ black hole), and is maximal,

$E_{max}={pR\over\alpha^{\prime}}~{},$ (12)

when the black hole size is maximal.

  4. 4.

The entropy is minimal, $S_{min}=0$, for empty $\mathcal{M}_{3}^{-}$, and is
maximal,

$S_{max}=\beta_{bh}E_{max}~{},\qquad\beta_{bh}\equiv 2\pi r_{5}~{},$ (13)

when $\rho_{+}=R$. 777Note that $\beta_{bh}$ in (13) is not the inverse $T$ in
(10); instead, it is the inverse temperature of an $SL(2)_{k}/U(1)$ black
hole, that will show up in the $\rho_{+}\to R$ limit (see below).

  5. 5.

The temperature diverges when the black hole size is maximal,

$T\to\infty\quad{\rm when}\quad\rho_{+}\to R~{}.$ (14)

The $\rho_{+}\to R$ limit is intriguing; we pause to inspect it next.

When $\rho_{+}\to R$, $\rho_{-}\to 0$ for any finite $n$, (8), and for
$\rho\neq 0,R$, the metric, $B$-field and dilaton, (2), (3) and (4),
respectively, approach

$ds^{2}\to{R^{2}\over r_{5}^{2}}d\tau^{2}-{r_{5}^{2}\over
R^{2}-\rho^{2}}d\rho^{2}+{R^{2}\rho^{2}\over
R^{2}-\rho^{2}}d\varphi^{2}~{},\quad B\to
0~{},\quad\Phi\to-\infty-\log\sqrt{R^{2}-\rho^{2}}~{},$ (15)

in the limit. Note that $B=0$ for any $\rho$, in the limit, and $e^{2\Phi}\to
0$ for any $\rho\neq R$. Moreover, any point in the static patch between $R$
and $\rho_{+}$ has shrunk to the singularity, a.k.a. $\rho\to R$ for any
$\rho_{+}<\rho<R$, and $N_{\varphi}\to 0$ for any $\rho\neq 0$, (5), in the
limit.

In the time-dependent regime $0<\rho<R$ of (15), a change of coordinate, 888We
take $\alpha^{\prime}\equiv l_{s}^{2}=1$ here and below, when it ain’t
present.

$\rho=R\cos\phi~{},$ (16)

gives

$ds^{2}={R^{2}\over r_{5}^{2}}d\tau^{2}-r_{5}^{2}d\phi^{2}+R^{2}\cot^{2}\phi
d\varphi^{2}~{},\quad\Phi=-\infty-\log(\sin\phi)~{},$ (17)

which is a $2d$ cosmology with time $0<\phi<\pi/2$ and compact space
$\varphi$, times a non-compact specelike direction $\tau$, and a vanishing
string coupling for any $\phi>0$; it is singular at $\phi=0$ (a.k.a. $\rho=R$)
and has a horizon at $\phi=\pi/2$ (a.k.a. $\rho=0$).

In the regime $\rho>R$ of (15), a change of coordinate,

$\rho=R\cosh\phi~{},$ (18)

gives

$ds^{2}={R^{2}\over r_{5}^{2}}d\tau^{2}+r_{5}^{2}d\phi^{2}-R^{2}\coth^{2}\phi
d\varphi^{2}~{},\quad\Phi=-\infty\pm{i\pi\over 2}-\log(\sinh\phi)~{},$ (19)

which is the geometry of the regime beyond the singularity of an
$SL(2)_{k}/U(1)$ black hole, 999For a review, see e.g. [6]; the relation of
the level $k$ of $SL(2)_{k}$ and the $r_{5}$ in (19) is that in (6). with
radial direction $\phi>0$ and compact time $\varphi$, times a spacelike
direction $\tau$. The $2d$ cosmological geometry above is thus the interior of
an $SL(2)_{k}/U(1)$ black hole with compact time $\varphi\sim\varphi+2\pi$.

The exterior of the $SL(2)_{k}/U(1)$ black hole amounts to a continuation of
the deformed BTZ geometry (2)–(6) beyond its singularity (at $\rho=0$).
Concretely, in the limit (15), the continuation

$\rho=iR\sinh\phi$ (20)

gives

$ds^{2}={R^{2}\over r_{5}^{2}}d\tau^{2}+r_{5}^{2}d\phi^{2}-R^{2}\tanh^{2}\phi
d\varphi^{2}~{},\quad\Phi=-\infty-\log(\cosh\phi)~{}.$ (21)

The horizon of the $SL(2)_{k}/U(1)$ black hole with compact time in the
$(\varphi,\phi)$ directions is at $\rho=0$, a.k.a. at $\phi=0$ in (21), and
its inverse temperature $\beta_{bh}$ is the one presented in (13); we shall
refer to this temperature as the bulk temperature of the maximally deformed
BTZ theory. 101010In a Schwarzschild-like radial coordinate,
$r\equiv(R^{2}-\rho^{2})/R$, the maximally deformed BTZ metric, (15), reads
$ds^{2}={R^{2}\over
r_{5}^{2}}d\tau^{2}-f(r)R^{2}d\varphi^{2}+{r_{5}^{2}dr^{2}\over 4r^{2}f(r)}$
with $f(r)\equiv 1-{R\over r}$, and the string coupling vanishes in the limit,
$e^{\Phi(r)}\to 0$, for any $r\neq 0$. Note that while the bulk temperature of
the maximally deformed BTZ geometry is finite, $\beta_{bh}=2\pi r_{5}$, the
bulk mass, $M_{bh}$, and bulk entropy, $S_{bh}$, a.k.a. the mass and entropy
of the $2d$ black hole with compact time in $(\varphi,r)$, are infinite,
$S_{bh}=\beta_{bh}M_{bh}=2\pi e^{-2\Phi(\rm horizon)}=\infty$. Needless to say
that exposing the significance of these properties in negative single-trace
$T\bar{T}$ holography is desired.

Finally, the energy, entropy and temperature of the deformed BTZ theory in
(2)–(6), $E$, $S$ and $T$ in (11), (9) and (10), respectively, approach their
maximal value, (12), (13) and (14), respectively, in the limit.

Now, consider de Sitter black holes.

The Reissner-Nordstrom de Sitter metric is given by (see e.g. section 2 of
[7]) 111111The metric is supported by an electric (or magnetic) field,
$A={Q\over r}dt$.

$ds^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+r^{2}d\Omega^{2}~{},\qquad
f(r)=1-{r^{2}\over\ell^{2}}-{2GM\over r}+{Q^{2}\over r^{2}}~{}.$ (22)

The three positive roots,

$r_{c}\geq r_{o}\geq r_{i}~{},$ (23)

are the locations of the cosmological, outer and inner horizons of the black
hole (BH) in de Sitter ($dS_{4}$), respectively.

For simplicity, 121212And since the location of the inner horizon $r_{i}$ will
not play a significant role below. we will focus on Schwarzschild de Sitter
black holes (whose geometry is (22) with $Q=0$), and extremal de Sitter black
holes (whose geometry is (22) with $Q=GM$), in turn.

Schwarzschild de Sitter black holes have

$r_{i}=0\qquad(Q=0)~{}.$ (24)

In this case (see e.g. [8]), the location $r_{c,o}$ of the cosmic and event
horizons and the de Sitter size $\ell$ satisfy the relation

$\ell^{2}-r_{c}^{2}-r_{o}^{2}={1\over
3}\left(\ell^{2}-(r_{c}-r_{o})^{2}\right)~{},\qquad 0\leq
r_{c}-r_{o}\leq\ell~{}.$ (25)

The total entropy $S$ of a black hole in de Sitter is the sum of the black
hole and cosmic entropies,

$S={\pi\over G}\left(r_{c}^{2}+r_{o}^{2}\right),$ (26)

hence, (25),

$S=S_{dS}\left[1-{1\over
3}\left(1-\left({r_{c}-r_{o}\over\ell}\right)^{2}\right)\right]~{},$ (27)

where

$S_{dS}={\pi\ell^{2}\over G}$ (28)

is the entropy of empty de Sitter.

Extremal de Sitter black holes have

$r_{i}=r_{o}={\ell\over 2}\left(1-\sqrt{1-{4Q\over\ell}}\right)~{},\quad
r_{c}={\ell\over 2}\left(1+\sqrt{1-{4Q\over\ell}}\right)\qquad(Q=GM)~{},$ (29)

and hence entropy, (26),

$S=S_{dS}\left(1-{2Q\over\ell}\right)=S_{dS}\left[1-{1\over
2}\left(1-\left({r_{c}-r_{o}\over\ell}\right)^{2}\right)\right]~{},$ (30)

instead.

A few of comments are in order:

  1. 1.

The regime between the event horizon and cosmological horizon,
$r_{o}<r<r_{c}$, is referred to as the static patch. 131313The static patch
plays an important role in de Sitter holography; we shall return to this
below. The regime beyond the cosmological horizon, $r>r_{c}$, is time
dependent (with time being $r$ instead of $t$ in the static patch). 141414The
interior of the black hole, $r_{i}<r<r_{o}$, will not play a role below.

  2. 2.

As one varies the location of the black hole horizon $r_{o}$ (while keeping
the de Sitter radius $\ell$ fixed), one interpolates between empty de Sitter
(when $r_{o}=0$) and Nariai geometry, when the black hole horizon and the
cosmic horizon coincide, $r_{o}=r_{c}$ (a.k.a. when the de Sitter black hole
size is maximal).

  3. 3.

The entropy is maximal,

$S_{max}=S_{dS}~{},$ (31)

for empty de Sitter ($r_{o}=0$, in which case $r_{c}=\ell$, (25), (29)), and
is minimal when the de Sitter black hole size is maximal ($r_{o}=r_{c}$).
151515The minimal value of the entropy is different in the different branches:
$S_{min}={2\over 3}S_{dS}$ in the Schwarzschild branch, (27), while
$S_{min}={1\over 2}S_{dS}$ in the extremal branch, (30), instead.

  4. 4.

The inverse Hawking temperature of empty de Sitter is 161616In de Sitter
holography, it is assumed though that the formal temperature in the Boltzmann
distribution is infinite in empty de Sitter, $T_{Boltzmann}=\infty$, while the
finite Hawking temperature is a derived or emergent quantity, [9]; we shall
return to this point below.

$\beta_{dS}=2\pi\ell~{}.$ (32)

The above suggests an analogy between the properties of de Sitter black holes
and the negatively deformed BTZ black holes, a.k.a. black holes in
$\mathcal{M}_{3}^{-}$; it is summarized in table 1. 171717The regimes that do
not appear in table 1, $r<r_{o}$ and $\rho<\rho_{-}$, do not have a manifest
analogy. This is due to the coordinate $\varphi$ of $\mathcal{M}_{3}^{-}$, on
top of $(\tau,\rho)$, playing an important role beyond the UV wall, at
$\rho>R$, [1, 10], on the one side, and $dS_{4}$ ending at $r\to\infty$, on
the other side; correspondingly, the regime in the interior of the Reissner-
Nordstrom de Sitter black hole (a.k.a. $r_{i}\leq r<r_{o}$) and the regime
beyond the inner horizon of the black hole in ${\cal M}_{3}^{-}$ (a.k.a.
$0\leq\rho<\rho_{-}$), respectively, do not participate in the analogy table.

Black hole (BH) in $\mathcal{M}_{3}^{-}$ | Black hole (BH) in de Sitter ($dS_{4}$)
---|---
Singular UV wall of $\mathcal{M}_{3}^{-}$ at $\rho=R$ | Outer horizon of BH in $dS_{4}$ at $r=r_{o}$
Outer horizon of BH in $\mathcal{M}_{3}^{-}$ at $\rho=\rho_{+}$ | Cosmological horizon of $dS_{4}$ at $r=r_{c}$
Inner horizon of BH in $\mathcal{M}_{3}^{-}$ at $\rho=\rho_{-}$ | The end of the de Sitter world at $r\to\infty$
Static patch of BH in $\mathcal{M}_{3}^{-}$ at $R>\rho>\rho_{+}$ | Static patch of BH in $dS_{4}$ at $r_{o}<r<r_{c}$
Time-dependent regime at $\rho_{+}>\rho>\rho_{-}$ | Time-dependent regime at $r_{c}<r<\infty$
Empty $\mathcal{M}_{3}^{-}$ (a.k.a. $\rho_{+}=0)$ | Largest BH in $dS_{4}$ (a.k.a. $r_{o}=r_{c}$)
$S_{min}$ when $\rho_{+}=0$ (a.k.a. empty $\mathcal{M}_{3}^{-}$) |  $S_{min}$ when $r_{o}=r_{c}$ (a.k.a. largest BH)
Largest BH in $\mathcal{M}_{3}^{-}$ (a.k.a. $\rho_{+}=R)$ | Empty de Sitter (a.k.a. $r_{o}=0$)
$S_{max}$ when $\rho_{+}=R$ (a.k.a. largest BH) |  $S_{max}$ when $r_{o}=0$ (a.k.a. empty $dS_{4}$)
$T\to\infty$ when $\rho_{+}\to R$ |  $T_{Boltzmann}\to\infty$ when $r_{o}\to 0$
$\beta_{bh}=2\pi r_{5}$ for largest BH in $\mathcal{M}_{3}^{-}$ |  $\beta_{dS}=2\pi\ell$ in empty de Sitter
Table 1: The analogy table.

Consequently, there is an analogy between negative single-trace $T\bar{T}$
holography (see e.g. [5, 2, 3] and references therein) and de Sitter
holography (see e.g. [9, 11, 12] and references therein), which we discuss
below.

In both de Sitter and negatively deformed BTZ geometries, the holographic
principle may be applied to the static patch in a similar 181818Though
opposite (e.g. the state of maximum finite entropy and infinite temperature
amounts on the one hand to empty de Sitter, whose static patch size is
maximal, $r_{c}-r_{o}\to\ell$, and on the other hand to a maximally filled
$\mathcal{M}_{3}^{-}$, whose static patch size is minimal, $R-\rho_{+}\to 0$;
see more details below). spirit. Next, we present the proposed principles, in
turn.

In de Sitter holography, the proposal is the following (see [11] and
references therein for details). Assume that the dimension of the Hilbert
space that describes all possible states of the static patch, including any
matter and/or black holes it may contain and also those associated with its
cosmological horizon, is measured roughly by the area of the cosmological
horizon of empty de Sitter, a.k.a. $e^{\pi\ell^{2}\over G}\equiv e^{S_{dS}}$,
(28). Now, place a particle with energy $E$ at the center of the static patch.
For e.g. a black hole with mass $M=E$, (22), this reduces the size of the
cosmological horizon to $r_{c}=\ell-GE+{\cal O}(E^{2},Q^{2})$, and so

${\pi r_{c}^{2}\over G}=S_{dS}-\beta_{dS}E~{},$ (33)

to leading order in $E,Q$, where $\beta_{dS}$ is given in (32). Since the
total number of microstates is $e^{S_{dS}}$, and the number of microstates
corresponding to a particle with energy $E$ is $e^{A_{c}/4G}$, where
$A_{c}=4\pi r_{c}^{2}$ is the area of the cosmological horizon, then assuming
that there is an equal probability to observe any particle with the same $E$
at the center of the static patch, the probability to observe such a particle
is

$P(E)=e^{-\beta_{dS}E}~{}.$ (34)

Consequently, the density matrix of the static patch of empty de Sitter space
is proportional to the identity, a.k.a. empty de Sitter space is described by
the maximally mixed state of the finite dimensional Hilbert space proposed
above.

Equivalently, the formal temperature in the Boltzmann distribution is
infinite, while the finite Hawking temperature is a derived or emergent
quantity. Concretely, it is argued that a bulk can emerge at a finite
effective temperature from a hologram at infinite temperature (see [9, 12] and
references therein for details).

To recapitulate, the proposal is that empty de Sitter space maximizes the
entropy of any state of the static patch, $S_{max}=S_{dS}$, (31), and
consequently its horizon temperature is infinite, $T_{Boltzman}=\infty$, while
its bulk is in an effective low temperature, $T_{dS}={1\over 2\pi\ell}$, (32).

In negative single-trace $T\bar{T}$ holography, the proposal is the following
(see e.g. [5, 2, 3] and references therein).

First, recall (see [3] for details), that eqs. (8), (9), and (11), can be
rewritten as

$n={p\over\alpha^{\prime}}\tilde{\rho}_{-}\tilde{\rho}_{+}~{},$ (35) $S={2\pi
kp\over r_{5}}\tilde{\rho}_{+}~{},$ (36) $E={p\over\lambda
R}\left[-1+\sqrt{\left(1-{\tilde{\rho}_{-}^{2}\over
R^{2}}\right)\left(1-{\tilde{\rho}_{+}^{2}\over R^{2}}\right)}\right]~{},$
(37)

respectively, where $\lambda<0$ is the one in (1), and

$\tilde{\rho}_{-}\equiv{\rho_{-}\over\sqrt{1-{\rho_{+}^{2}\over
R^{2}}}}~{},\qquad\tilde{\rho}_{+}\equiv{\rho_{+}\over\sqrt{1-{\rho_{-}^{2}\over
R^{2}}}}~{}.$ (38)

Consequently, on the $\lambda$-deformed BTZ backgrounds (2)–(6) with fixed
angular momentum $n$ and entropy $S$, a.k.a. with fixed $\tilde{\rho}_{\pm}$,
the $p$’th fraction of the energy $E(\lambda)$ takes the form

${1\over p}E(\lambda)={1\over\lambda R}\left(-1+\sqrt{1+2\lambda R{E(0)\over
p}+\left({\lambda RP\over p}\right)^{2}}\right)~{},$ (39)

where

$E(0)\equiv{r_{5}\over R}M_{btz}~{},\qquad P\equiv{n\over R}$ (40)

are the energy and momentum of the states in the $CFT_{2}$ hologram of
$AdS_{3}$ that amount to a BTZ black hole with mass $M_{btz}$ and angular
momentum $n$. Finally, in terms of the hologram quantities, the $p$’th
fraction of the entropy reads

${1\over p}S=2\pi\sqrt{c\over 6}\left(\sqrt{E_{L}(1+\lambda
E_{R})}+\sqrt{E_{R}(1+\lambda E_{L})}\right)~{},\qquad c\equiv 6k~{},$ (41)

where

$E_{L,R}\equiv{R\over 2p}\left(E(\lambda)\pm P\right)$ (42)

are the $p$’th fractions of the left and right handed dimensionless energies,
${R\over 2}(E\pm P)$.

Equations (39)–(42) are the same as what one would obtain by studying a
symmetric product $CFT_{2}$, ${\cal M}^{p}/S_{p}$. 191919On a circle with
radius $R$, and with the central charge of the block ${\cal M}$ being $c=6k$,
as in (41). The generic state with large energy and momentum in the undeformed
symmetric product CFT has (at large $E,P,p$) its energy and momentum equally
split among the different copies of ${\cal M}$. Equation (39) gives the
deformed energy of such states under a $T\bar{T}$ deformation of the seed CFT,
${\cal M}$, with a dimensionless deformation parameter $\lambda$, (1), and eq.
(41) is the corresponding entropy in the block.

To recapitulate, in single-trace $T\bar{T}$ holography, the proposal is thus
that the physics of deformed BTZ black holes is captured by the properties of
a symmetric product theory whose block is a $T\bar{T}$ deformed CFT (with the
energy and momentum distributed equally in each of the blocks). 202020The
status of the proposed symmetric product hologram is summarized in [2] and
references therein; we shall not present it here. Indeed, one can check that
there is a detailed match between the properties of the geometries (2)–(6) and
the corresponding energies and entropies, (39)–(42), in the proposed
holographic dual. In particular, since the energy (12) is a maximal energy in
negatively deformed BTZ, for all $n$, it is natural to conjecture that an
analogous statement is true for standard $T\bar{T}$ deformed CFT with negative
$\lambda$ (see [3] and references therein for more details). Consequently, the
negatively deformed single-trace $T\bar{T}$ theory has a state of maximum
finite entropy, (13), and this state is maximally mixed, (14).

In (15)–(21), it was shown that the geometry which corresponds to the maximal
energy and entropy, (12) and (13), respectively, and infinite temperature, is
universal (in the sense that it does not depend on the particular value of
$n$); the static patch of the deformed BTZ geometries is degenerated in these
cases, $\rho_{+}\to R$, and the resulting background is an $SL(2)_{k}/U(1)$
black hole with compact time (times a non-compact spatial direction). In
addition to $T=\infty$, (10), (14), there is thus a bulk temperature, a.k.a.
that of the superstring theory on $SL(2)_{k}/U(1)$, $T_{bh}={1\over 2\pi
r_{5}}\left(={1\over 2\pi\sqrt{\alpha^{\prime}k}}\right)$ in (13) (with (6)).
This appears in harmony with the proposed properties of de Sitter holography,
presented above; see the analogy table 1. There are however intriguing
differences. In the following, we shall discuss both the similarities and
differences.

In both negative single-trace $T\bar{T}$ holography and de Sitter holography,
the fact that there is a state of maximum finite entropy can be understood
from the geometry. In the former, it corresponds to the largest black hole in
negatively deformed BTZ. In the latter, it corresponds to the largest
cosmological horizon size in de Sitter space, a.k.a. to empty de Sitter.

In de Sitter holography, it was argued [9] that for the empty de Sitter state,
the horizon system, at $r=r_{c}=\ell$, is all that there is, a.k.a. the bulk,
(22) with $M=Q=0$, is not a second system; it is a holographic construct made
of the horizon degrees of freedom. In negative single-trace $T\bar{T}$
holography, for the maximal black hole state, the static patch shrinks to the
location of the event horizon, at $\rho=\rho_{+}=R$, and one may say
analogously that the horizon system is all that there is, a.k.a. the bulk (15)
isn’t a 2nd system; it’s a holographic construct made of the horizon d.o.f.
212121This is similar to what is argued [13] in the case of a geometric cutoff
that removes the asymptotic region of $AdS_{3}$ and places the QFT on a
Dirichlet wall at finite radial distance $r=R$ in the bulk; in our maximally
deformed BTZ geometry, the UV wall is the $\rho=\rho_{+}=R$ one..

In negative single-trace $T\bar{T}$ holography, we have a concrete proposal
for the description of the microstates via holography 222222Apart for a
measure zero amount of states (see [2] and references therein). – those of
single-trace $T\bar{T}$ deformed symmetric product $CFT_{2}$ with maximal
energy. On the other hand, it seems fair to say that the description of the
microstates that constitute the entropy associated with cosmological de Sitter
horizons is more mysterious.

In both negative single-trace $T\bar{T}$ holography and de Sitter holography,
it is proposed that the state with maximum entropy is maximally mixed.
Equivalently, the temperature parameter in the Boltzmann distribution must be
infinite, $T=\infty$. In the former, such an infinite temperature is supported
by explicit properties of both the geometry and the proposed holographic
theory. On the other hand, there is no concrete way to obtain this property
from the de Sitter geometry, and it seems fair to say that an explicit
holographic candidate of de Sitter gravity with these properties is still a
mystery.

As to the bulk inverse temperatures, $\beta_{dS}=2\pi\ell$ and
$\beta_{bh}=2\pi r_{5}$ in empty de Sitter, (32), and maximally deformed BTZ,
(13), respectively, while they can be obtained from the surface gravity both
in empty de Sitter and maximally deformed BTZ geometries, there is an
educational difference that we discuss next.

The deformed BTZ geometry, (2)–(5), has an asymptotically linear dilaton
regime beyond the naked singularity at $\rho=R$, which plays an important role
in negative single-trace $T\bar{T}$ holography (see [1, 10] for details). In
particular, the signature of $\tau$ and $\varphi$ beyond the UV wall,
$\rho>R$, is flipped relative to the static patch, $\rho_{+}<\rho<R$. While
the time in the holographic single-trace $T\bar{T}$ deformed symmetric product
is that in the static patch of (2,5), $\tau$, the timelike direction in the
asymptotically linear dilaton regime is $\varphi$, instead, and the bulk
$T_{bh}$ above is w.r.t. to $\varphi$ being time, (19), (21), instead of
$\tau$. This allowed us to obtain both $T\to\infty$ and $T_{bh}=1/2\pi r_{5}$
from the $\rho_{+}\to R$ limit of the geometry (2,5). 232323Recall [1, 3] that
the temperature in eq. (10) is obtained from the geometry e.g. by Wick
rotating the canonically normalized $t\equiv R\tau/r_{5}$, $t\to i\tau_{E}$,
compactifying $\tau_{E}$ on a circle with circumference $\beta=1/T$, and
demanding regularity of the metric (2,5) at the event horizon,
$\rho=\rho_{+}$. In particular, the $T\to\infty$ result is obtained from the
$\rho_{+}\to R$ limit of the geometry in this way. On the other hand, the bulk
temperature, $T_{bh}=1/2\pi r_{5}$, is obtained from the $\rho_{+}\to R$ limit
of the geometry e.g. by Wick rotating the canonically normalized bulk time
$y\equiv R\varphi$ in eq. (21), $y\to i\varphi_{E}$, compactifying
$\varphi_{E}$ on a circle with circumference $\beta_{bh}=1/T_{bh}$, and
demanding regularity of the metric in (21) at the horizon, $\phi=0$, of the
$SL(2)_{k}/U(1)$ black hole.

It is important to recall though that obtaining the bulk inverse temperature
in maximally deformed BTZ, from the surface gravity at the horizon of an
$SL(2)_{k}/U(1)$ black hole, required us to extend the deformed BTZ geometry
beyond its black hole singularity, (20), (21). This is not the case in de
Sitter space, whose bulk temperature is obtained from the surface gravity at
its cosmological horizon.

On the other hand, the de Sitter geometry does not have the analog of the
asymptotically linear dilaton regime of (2). Consequently, the empty de Sitter
geometry reveals only a single temperature – the bulk $T_{dS}=1/2\pi\ell$
above, while the proposed infinite Boltzmann temperature in the hologram is
hidden from the geometry.

To recapitulate, the analogy and difference between negative single-trace
$T\bar{T}$ holography and de Sitter holography is hinting towards the
temptation to seek an extension of the de Sitter geometry, in a way that will
give rise to an analog of the $\rho>R$ regime in the negatively deformed BTZ
geometry (2).

## Acknowledgments

I thank Soumangsu Chakraborty and David Kutasov for discussions. This work was
supported in part by the ISF (grant number 256/22) and the BSF (grant number
2018068).

## Appendix – A comment on $2$pf

According to [12], there is a contradiction between the semiclassical fixed-
background approximation and the holographic description in which the density
matrix is maximally mixed, since in the former, the two point functions
($2$pf) have a non-zero imaginary part, but in de Sitter holography it must
vanish. This paradox and its proposed resolution within the principles of de
Sitter holography are discussed in [12].

This is analogous to the fact that in ${\cal M}_{3}$ with an $R^{1,1}$
boundary, while the $2$pf in the zero winding sector, $w=0$, which amounts to
the semiclassical fixed-background approximation (up to corrections in $1/k$,
(6)), has a non-zero imaginary part, [14], in the sectors with non-zero
winding, $w\neq 0$, a.k.a those that amount to stringy properties of the
theory, it doesn’t, [15, 16].

It is possible that in a full fledged consistent string theory on the
negatively deformed BTZ black holes in (2)–(6), the states in the $w=0$
sector, with a non-zero imaginary part, are projected out, leaving behind only
real two point functions. The investigation of this issue is left for future
work. 242424For that purpose, one should be able to extend the calculations in
[14, 15, 16] to the superstring theory on (2)–(6), in particular, for the
$\rho_{+}\to R$ case, (15).

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