11institutetext:  Univ. Grenoble Alpes, CNRS, LPSC-IN2P3, 53, avenue des
Martyrs, 38000 Grenoble, France 22institutetext: LLR, CNRS, Ãcole
Polytechnique, Institut Polytechnique de Paris, Palaiseau, France
33institutetext: School of Physics and Astronomy, Cardiff University, Queenâs
Buildings, The Parade, Cardiff, CF24 3AA, UK 44institutetext: AIM, CEA, CNRS,
Université Paris-Saclay, Université Paris Diderot, Sorbonne Paris Cité, 91191
Gif-sur-Yvette, France 55institutetext: INAF, IASF-Milano, Via A. Corti 12,
20133 Milano, Italy 66institutetext: Institut d’Astrophysique Spatiale (IAS),
CNRS, Université Paris Sud, Orsay, France 77institutetext: Institut Néel,
CNRS, Université Grenoble Alpes, France 88institutetext: Institut de
RadioAstronomie Millimétrique (IRAM), Grenoble, France 99institutetext: Aix
Marseille Univ, CNRS, CNES, LAM, Marseille, France 1010institutetext:
Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 5,
I-00185 Roma, Italy 1111institutetext: Univ. Grenoble Alpes, CNRS, IPAG, 38000
Grenoble, France 1212institutetext: Centro de Astrobiología (CSIC-INTA),
Torrejón de Ardoz, 28850 Madrid, Spain 1313institutetext: Instituto de
Radioastronomía Milimétrica (IRAM), Granada, Spain 1414institutetext: LERMA,
Observatoire de Paris, PSL Research University, CNRS, Sorbonne Université,
UPMC, 75014 Paris, France 1515institutetext: Univ. de Toulouse, UPS-OMP, CNRS,
IRAP, 31028 Toulouse, France 1616institutetext: Laboratoire de Physique de
lâÉcole Normale Supérieure, ENS, PSL Research University, CNRS, Sorbonne
Université, Université de Paris, 75005 Paris, France 1717institutetext:
Department of Physics and Astronomy, University of Pennsylvania, 209 South
33rd Street, Philadelphia, PA, 19104, USA 1818institutetext: Institut
d’Astrophysique de Paris, CNRS (UMR7095), 98 bis boulevard Arago, 75014 Paris,
France 1919institutetext: Kavli Institute for Astrophysics and Space Research,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA
2020institutetext: School of Earth and Space Exploration and Department of
Physics, Arizona State University, Tempe, AZ 85287, USA 2121institutetext:
Caltech, Pasadena, CA 91125, USA 2222institutetext: Departamento de Física
Teórica and CIAFF, Facultad de Ciencias, Modulo 8, Universidad Autánoma de
Madrid, 28049 Madrid, Spain

# Multi-probe analysis of the galaxy cluster CL J1226.9+3332: hydrostatic mass
and hydrostatic-to-lensing bias

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biasMulti-probe analysis of the galaxy cluster CL J1226.9+3332: hydrostatic
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biasMulti-probe analysis of the galaxy cluster CL J1226.9+3332: hydrostatic
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###### Abstract

We present a multi-probe analysis of the well-known galaxy cluster CL
J1226.9+3332 as a proof of concept for multi-wavelength studies within the
framework of the NIKA2 Sunyaev-Zel’dovich Large Program (LPSZ). CL
J1226.9+3332 is a massive and high redshift (z = 0.888) cluster that has
already been observed at several wavelengths. A joint analysis of the thermal
SZ (tSZ) effect at millimeter wavelength with the NIKA2 camera and in X-ray
with the XMM-Newton satellite permits the reconstruction of the clusterâs
thermodynamical properties and mass assuming hydrostatic equilibrium. We test
the robustness of our mass estimates against different definitions of the data
analysis transfer function. Using convergence maps reconstructed from the data
of the CLASH program we obtain estimates of the lensing mass, which we compare
to the estimated hydrostatic mass. This allows us to measure the hydrostatic-
to-lensing mass bias and the associated systematic effects related to the
NIKA2 measurement. We obtain $M_{500}^{\rm{HSE}}=(7.65\pm 1.03)\times
10^{14}M_{\odot}$ and $M_{500}^{\rm{lens}}=(7.35\pm 0.65)\times
10^{14}M_{\odot}$, which implies a HSE-to-lensing bias consistent with 0
within 20$\%$.

##  1 Introduction

Galaxy clusters are excellent probes for cosmology, in particular for the
understanding of large-scale structure formation processes boquet ; planck1 .
For this purpose, a precise knowledge of cluster masses is needed. The NIKA2
SZ Large Program (LPSZ) LPSZ consists in 300 hours of Guaranteed Time
dedicated to the observation and detailed analysis of 45 high redshift (z in
0.5-0.9) galaxy clusters taking advantage of the capabilites of the NIKA2
camera: a large field of view combined with high angular resolution at 150 and
260 GHz adam1 ; calvo ; NIKA2-electronics ; perotto . These characteristics
allow us to accurately map galaxy clusters via the thermal Sunyaev-Zel’dovich
effect sz and, together with high quality X-ray observations from XMM-Newton,
will enable the investigation of the SZâmass scaling relation at high redshift
for cosmological exploitation.

After a first analysis of a LPSZ cluster for a science verification study
ruppin1 and a second one for proving the quality of NIKA2 in the most
challenging case keruzore , we present here a study on the robustness of the
hydrostatic mass reconstruction combining SZ and X-ray data. We test the
impact of systematic effects from the estimation of the filtering induced by
the data processing pipeline and from the pressure model fitting. Systematic
effects that may arise from other effects (e.g. density estimates at large
radii) are not studied in this work.

##  2 The galaxy cluster CL J1226.9+3332

The object of this analysis is the well-known galaxy cluster CL J1226.9+3332,
also known as PSZ2-G160.83+81.66. It is located at (R.A., Dec.)J2000 =
(12h26m58.37s, +33d32m47.4s) according to the X-ray peak from cavagnolo with
mass $M_{500}^{\rm{HSE}}\sim 5.70\times 10^{14}M_{\odot}$ and redshift 0.888
psz2 , which makes it the highest-redshift cluster of the LPSZ sample.
Discovered by ROSAT ebeling , it has already been observed at several
wavelengths adam2 ; maughan1 ; maughan2 ; jeetyson ; joy ; bonamente ; allen ;
postman ; zitrin1 ; romero , appearing, from the first SZ observations with
BIMA joy as a very spherical cluster.

X-ray analyses using XMM-Newton and Chandra data maughan1 found that the
temperature of the cluster in a region at $\sim$ 40" to the south-west of the
X-ray peak is much higher than the average temperature of the intra-cluster
medium (ICM). Posterior SZ analyses with MUSTANG korngut and NIKA adam2 ;
adam3 , as well as lensing zitrin1 and galaxy distribution studies jeetyson ,
showed also the presence of such a substructure. Various explanations have
been proposed jeetyson ; maughan2 , suggesting mainly that the cluster
consists of a two-halo system, which is being observed after the less massive
cluster passed through the central one and the gas followed it. In short, CL
J1226.9+3332 shows a relaxed morphology at large scales and evidence of
disturbance in the core.

##  3 Map-making and data processing filtering

The cluster was observed for 3.6 hours during the 15th NIKA2 science-purpose
observation campaign (13-20 February 2018) as part of the NIKA2 Guaranteed
Time at the IRAM 30-m telescope. The data were calibrated and reduced using
the standard NIKA2 collaboration LPSZ pipeline described in perotto ; keruzore
; rigby . We present in Figure 1 the obtained maps at 150 and 260 GHz. The map
at 150 GHz shows the cluster as a decrement of flux, contaminated by the
positive signal from point sources. These point sources are also detected in
the 260 GHz map.

The filtering induced by the data processing in the 150 GHz map is computed
using simulations via a transfer function (TF) that relates, in the Fourier
space, the real astrophysical signal (the cluster) with the filtered one. The
standard LPSZ analysis ruppin1 ; keruzore uses one-dimensional transfer
functions which are obtained by averaging the filtering in Fourier-domain
annuli at a fixed angular scale, supposing therefore isotropy. The black line
in the left panel in Figure 2 shows the 1D TF for the 150 GHz map in Figure 1.
In this work we test a new approach: a two-dimensional transfer function (2D
TF) presented in Figure 2. That allows us to deal with anisotropy effects
coming for example from the effect of scanning angles and the deviation from
sphericity of clusters. Color lines in the left panel in Figure 2 show the
one-dimensional TFs for the different directions represented in the 2D TF and
illustrate the impact of anisotropies in the angular frequency space.

Figure 1: NIKA2 maps obtained with the LPSZ pipeline at 150 GHz (left) and 260
GHz (right) in Jy/beam units. Contours show S/N levels multiples of $\pm
3\sigma$. Both maps have been smoothed with a 10â FWHM Gaussian kernel.

Figure 2: 1D (left) and 2D (right) transfer functions representing the data
reduction-induced filtering for the 150 GHz map in Figure 1. Color lines in
the left panel show the value of the transfer function for different
directions in the 2D TF, represented with the same colors in the right panel.

##  4 SZ analysis: pressure profile determination

The CL J1226.9+3332 surface brightness map is proportional to the integrated
electron pressure along the line of sight. To recover the latter, we perform a
Monte Carlo Markov chain (MCMC) fit of a model to the 150 GHz map considering
the Compton parameter signal associated with the cluster, the point sources
and a zero level. The contamination from point sources at 150 GHz is estimated
to account for their positive flux that may compensate the tSZ signal of the
cluster. The details are given in keruzore ; keruzore2 . In this work we
performed the fit by adjusting two different pressure profile models: 1) a
generalized Navarro-Frenk-White (gNFW, nagai ) and 2) a radially binned model.
For the first one we fit for the 5 parameters of the model. The radially
binned profile is defined by
$P_{e}(r)=P_{i}\left(\frac{r}{r_{i}}\right)^{-\alpha_{i}}$, where for each
radial bin $r_{i}$ a value of the pressure $P_{i}$ and the slope $\alpha_{i}$
are fitted. Moreover, we repeated the fit considering either 1D or 2D transfer
functions. In the following we take as reference the radially binned pressure
model with the 1D transfer function.

##  5 Hydrostatic mass and robustness tests

Combining the spherical pressure profile resulting from the fit with the
electron density profile obtained from X-ray data, we can compute, under the
hydrostatic equilibrium (HSE) hypothesis ruppin1 ; keruzore ; adam2 , the
hydrostatic mass profile for CL J1226.9+3332 and with it the probability
distribution for $R_{500}^{\rm{HSE,}}$111Radius at which the mean mass (HSE
mass in this case) density of the cluster is 500 times the critical density of
the universe at its redshift.. We note here that before deriving the pressure,
we fitted a gNFW model to the radially binned profiles. This allowed us to
have smooth pressure profiles to calculate the derivative of the profile for
the mass estimate, but still testing the use of a different model when
reconstructing the pressure from the maps.

We show in Figure 3 (left) the obtained probability distributions for
$R_{500}^{\rm{HSE}}$, as well as the tSZ flux $Y_{500}$ and
$M_{500}^{\rm{HSE}}$. These results are compared to previous HSE mass
estimates obtained from NIKA maps adam2 . Our results give higher
$R_{500}^{\rm{HSE}}$ and $M_{500}^{\rm{HSE}}$ values than expected from
previous studies. Nevertheless, we observe a correlation between
$R_{500}^{\rm{HSE}}$ and $M_{500}^{\rm{HSE}}$. This is expected since
$R_{500}^{\rm{HSE}}$ and $M_{500}^{\rm{HSE}}$ are simultaneously obtained from
the over-density definition giving directly proportional
$(R_{500}^{\rm{HSE}})^{3}$ and $M_{500}^{\rm{HSE}}$. Therefore, the comparison
of probability distributions seems more representative when comparing
different hydrostatic mass estimates. In the left panel in Figure 3 we also
compare the results of the robustness tests we have performed starting from
the 150 GHz map in Figure 1. We compare the probability distributions obtained
with the reference model to the results using 2D transfer function instead. In
the latter case, slightly lower values of $R_{500}^{\rm{HSE}}$ and
$M_{500}^{\rm{HSE}}$ are favored. We also show results for the two pressure
profile models. We find tighter constraints for the gNFW model when it is
directly fitted to the map than when the radially binned profile is fitted and
then gNFWs are adjusted to the resulting radially binned profiles. However,
computing the pressure derivative in a consistent way (here always deriving
from a gNFW), our mass estimate is robust. From the combination of the
posterior probability distributions obtained for the four cases that we have
considered, we measure an hydrostatic mass for CL J1226.9+3332 of
$M_{500}^{\rm{HSE}}=(7.65\pm 1.03)\times 10^{14}M_{\odot}$, where $\sim$ 5-10
$\%$ of the uncertainty comes from the contribution of the tested systematic
effects.

Figure 3: Left: $R_{500}^{\rm{HSE}}$, $Y_{500}$ and $M_{500}^{\rm{HSE}}$
probability distributions for the radially binned pressure model with 1D TF in
blue (reference), radially binned 2D TF in purple, gNFW 1D TF in red and gNFW
2D TF in orange. The grey star, cyan circle and green diamond correspond to
the results from adam2 for the tested PPC, FPC and NNN models, respectively.
Right: Hydrostatic mass with respect to lensing mass at $R_{500}$. In dark
shades of blue, 1$\sigma$ and 2$\sigma$ contours for $M_{500}^{\rm{HSE}}$ from
the reference model and in light shades of blue $M_{500}^{\rm{HSE}}$ as the
combination of the four probability distributions in the left panel.

##  6 Hydrostatic-to-lensing mass bias

We have performed an analysis of the convergence maps reconstructed from the
CLASH data zitrin1 to obtain lensing mass estimates for CL J1226.9+3332.
Combining them we obtain $M_{500}^{\rm{lens}}=(7.35\pm 0.65)\times
10^{14}M_{\odot}$. Details on this analysis will be given in munoz . The
hydrostatic-to-lensing mass bias
$b_{\rm{HSE/lens}}=1-M_{500}^{\rm{HSE}}/M_{500}^{\rm{lens}}$, can be an
interesting indicator of effects coming from the non-thermal contributions to
the pressure of the cluster and systematic effects in mass estimations
lovisari ; pratt . Supposing that they are uncorrelated estimates, we have
combined the probability distributions obtained for $M_{500}^{\rm{HSE}}$ and
$M_{500}^{\rm{lens}}$ to get a distribution of the ratio, which can be
translated into a probability distribution of the hydrostatic-to-lensing mass
bias for CL J1226.9+3332. In the right panel in Figure 3 we present the
probability distributions for $M_{500}^{\rm{HSE}}/M_{500}^{\rm{lens}}$,
$M_{500}^{\rm{HSE}}$ and $M_{500}^{\rm{lens}}$. We have computed the ratio
using the $M_{500}^{\rm{HSE}}$ from the reference model, as well as using the
combination of the four probability distributions derived for
$M_{500}^{\rm{HSE}}$, therefore accounting for all the considered effects. The
figure shows that for both cases the bias is consistent with 0 within $\sim$
20$\%$.

##  7 Conclusions

Combined tSZ and X-ray analysis allow us to estimate the hydrostatic mass of
this high redshift galaxy cluster, CL J1226.9+3332. We have demonstrated that
our hydrostatic mass estimates are robust against the application of the 1D or
2D transfer function. The reconstruction of the pressure profile derivative is
a key element for the determination of the hydrostatic mass profiles, and
needs to be dealt with carefully. This analysis has also been an important
step towards developing a standard pipeline to compare and combine hydrostatic
and lensing mass estimates. For CL J1226.9+3332 we find a hydrostatic-to-
lensing mass bias consistent with 0.

##  8 Acknowledgements

We would like to thank the IRAM staff for their support during the campaigns.
The NIKA2 dilution cryostat has been designed and built at the Institut Néel.
In particular, we acknowledge the crucial contribution of the Cryogenics
Group, and in particular Gregory Garde, Henri Rodenas, Jean Paul Leggeri,
Philippe Camus. This work has been partially funded by the Foundation
Nanoscience Grenoble and the LabEx FOCUS ANR-11-LABX-0013. This work is
supported by the French National Research Agency under the contracts "MKIDS",
"NIKA" and ANR-15-CE31-0017 and in the framework of the "Investissements
dâavenirâ program (ANR-15-IDEX-02). This work has benefited from the support
of the European Research Council Advanced Grant ORISTARS under the European
Union’s Seventh Framework Programme (Grant Agreement no. 291294). F.R.
acknowledges financial supports provided by NASA through SAO Award Number
SV2-82023 issued by the Chandra X-Ray Observatory Center, which is operated by
the Smithsonian Astrophysical Observatory for and on behalf of NASA under
contract NAS8-03060.

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