11institutetext: IRAP, Université de Toulouse, CNRS, UPS, CNES, Toulouse,
France
11email<EMAIL_ADDRESS>22institutetext: Institute of Physics,
Laboratory of Astrophysics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Observatoire de Sauverny, 1290 Versoix, Switzerland 33institutetext: Leiden
Observatory, Leiden University, P.O. Box 9513, 2300 RA, Leiden, The
Netherlands

# A case study of early galaxy cluster with the _Athena_ X-IFU

F. Castellani 11 N. Clerc 11 E. Pointecouteau 11 Y.M. Bahé 2233 J. Schaye 33
F. Pajot 11

(Received - ; accepted -)

###### Abstract

Context. Observations of the hot gas in distant clusters of galaxies, though
challenging, are key to understand the role of intense galaxy activity, super-
massive black hole feedback and chemical enrichment in the process of massive
halos assembly.

Aims. We assess the feasibility to retrieve, using X-ray hyperspectral data
only, the thermodymamical hot gas properties and chemical abundances of a
$z=2$ galaxy cluster of mass $M_{500}=7\times 10^{13}$ M⊙, extracted from the
Hydrangea hydrodynamical simulation.

Methods. We create mock X-ray observations of the future X-ray Integral Field
Unit (X-IFU) onboard the _Athena_ mission. By forward-modeling the measured
$0.4-1$ keV surface brightness, the projected gas temperature and abundance
profiles, we reconstruct the three-dimensional distribution for the gas
density, pressure, temperature and entropy.

Results. Thanks to its large field-of-view, high throughput and exquisite
spectral resolution, one X-IFU exposure lasting 100 ks enables reconstructing
density and pressure profiles with $20$% precision out to a characteristic
radius of $R_{500}$, accounting for each quantity’s intrinsic dispersion in
the Hydrangea simulations. Reconstruction of abundance profiles requires both
higher signal-to-noise ratios and specific binning schemes. We assess the
enhancement brought by longer exposures and by observing the same object at
later evolutionary stages (at $z=1$ and $1.5$).

Conclusions. Our analysis highlights the importance of scatter in the radially
binned gas properties, which induces significant effects on the observed
projected quantities. The fidelity of the reconstruction of gas profiles is
sensitive to the degree of gas components mixing along the line-of-sight.
Future analyses should aim at involving dedicated hyper-spectral models and
fitting methods that are able to grasp the complexity of such three-
dimensional, multi-phase, diffuse gas structures.

###### Key Words.:

X-rays: galaxies: clusters – Instrumentation: detectors – Methods: numerical –
Techniques: imaging spectroscopy

##  1 Introduction

The formation epoch of groups and clusters of galaxies ranges from $z\sim 1-3$
when star formation in galaxies and super-massive black holes (SMBH) are at
the peak of their activities. The gas trapped in the forming massive potential
wells of these structures heats up under the dual effect of gravity and
feedback from star formation and SMBH (e.g., Kravtsov & Borgani, 2012;
Vogelsberger et al., 2014; Schaye et al., 2015; Bahé et al., 2017; Schaye et
al., 2023). The questions of how this gas is accreted by galaxies and how it
feeds their star formation and SMBH activity are still open. The same goes for
the timescale within which galaxies evolve into the massive ellipticals that
form the red sequence under the joint influence of their dense environment and
the quenching of star formation by active galactic nucleus (AGN) feedback
(e.g., Behroozi et al., 2013; Leauthaud et al., 2012; Eckert et al., 2021;
Oppenheimer et al., 2021). This phase of violent and intense astrophysical
activity injects large amounts of energy, gas and metals into the forming
intra-cluster medium, shaping its thermal and chemical properties (e.g.,
McNamara & Nulsen, 2007; Biffi et al., 2017; Mernier et al., 2018). As such,
these processes imprint the statistical (scaling and structural) properties of
the population of groups and clusters of galaxies (Lovisari & Maughan, 2022,
for a recent review). Constraining the properties and the evolution of hot gas
in these massive halos out to their epoch of formation is an efficient way to
understand the aforementioned assembly and evolution of the largest
gravitationally bound halos in the Universe.

By construction in a hierarchical scheme of structure formation, lower mass
halos ($M_{500}<10^{14}$ M⊙) constitute the vast majority of the population of
groups of galaxies (e.g., Tinker & Chen, 2008). They are observed in numbers
in large surveys, especially in optical surveys (e.g., Lambert et al., 2020;
Werner et al., 2023), and they dominate the population of simulated halos in
numerical simulations. The self-similar process of structure formation
predicts their properties to be down-scaled versions of massive galaxy
clusters. Though these predictions seem to agree with some of their
statistical properties (e.g., the mass-temperature relation, Babyk & McNamara,
2023), many observations have shown that most actually depart from the scaling
and structure behaviour of their more massive siblings (e.g., Ponman et al.,
2003; Sun et al., 2011; Stott et al., 2012; Sanderson et al., 2013; Lovisari
et al., 2015, 2021). Their shallower potential is more prone to the impact of
AGN feedback in terms of intra-group (IGM) gas heating, but also in the way it
impacts the IGM gas distribution and its depletion from these less deep
gravitational potential wells (e.g., Eckert et al., 2021). The physics
governing the gas content of galaxy groups and clusters is a current true
challenge for numerical simulations, the current predictions of which largely
vary from one work to the other (Oppenheimer et al., 2021).

With their smaller masses and thus shallower potential well with respected to
massive galaxy clusters, the observation of the X-ray emitting hot gas is
harder for groups than for clusters. To date several groups or samples of
groups of galaxies have been studied at X-ray wavelengths, mainly in the local
Universe (e.g., Lovisari & Maughan, 2022). Reaching such objects out to larger
redshifts to investigate their gas content and its evolution remains very
challenging for the current generation of X-ray telescopes. The next
generation of X-ray observatories shall combine a large collective area with
high resolution spectro-imaging capabilities. The X-IFU instrument (Barret et
al., 2023) on board the future European X-ray observatory _Athena_ (Barcons et
al., 2017) should enable studies of groups with masses of a few $10^{13}$ M⊙
out to $z\approx 2$. The understanding of the assembly of structure, and more
specifically of massives halos, is a key objective of the Hot and Energetic
Universe science theme that the _Athena_ mission will implement (Nandra et
al., 2013).

In the wake of other previous feasibility studies addressing the science cases
of chemical enrichment in groups and galaxy clusters (Cucchetti et al., 2018;
Mernier et al., 2020), bulk and turbulent motion in the intra-cluster gas
(Roncarelli et al., 2018; Clerc et al., 2019; Cucchetti et al., 2019), and the
warm hot intergalactic medium (Walsh et al., 2020; Wijers et al., 2020; Wijers
& Schaye, 2022), we address in this work the issue of the observation of
distant galaxy groups and clusters in order to characterise their physical
properties out to the epoch of their formation with the _Athena_ X-IFU
instrument. Following the Athena Mock Observing Plan111Version v4.3 issued on
8th of September 2020 by the ESA Athena Science Study Team, the science
objective concerning the evolution of thermodynamical properties of groups and
clusters of galaxies from $z>0.5$ and out to $z\sim 2$ will be achieved with
X-IFU observations. We therefore focused in this study on this specific
instrument only. We present realistic mock observations with the X-IFU
instrument of one simulated galaxy cluster extracted from the cosmological
hydrodynamic simulations Hydrangea (Bahé et al., 2017). The paper is organised
as follows. We present the input of our simulations in Sec. 2 and the mock
X-IFU observations and processing in Sec. 3. In Sec. 4, we describe our
Bayesian approach analysis. We present our results in Sec. 5 and discuss them
in Sec. 6.

Throughout this paper, we made use of the cosmology setup adopted for the
Hydrangea simulations, that is the Planck Collaboration et al. (2014)
cosmology with $h_{100}=0.6777$, $\Omega_{m}=0.307$, $\Omega_{\Lambda}=0.693$.
At a redshift of $z=2$, 1 arcsecond corresponds to a physical size of 8.6
kilo-parsec (kpc).

##  2 A simulated cluster of galaxies at $z=2$

As input for our study, we used a simulated cluster of galaxies from the
cosmological Hydrangea simulations (Bahé et al., 2017).

###  2.1 Hydrangea simulation

Hydrangea is a suite of 24 cosmological zoom-in simulations of massive galaxy
clusters (selected such that $M_{200}=10^{14-15.4}M_{\odot}$ at $z=0$) that is
part of the Cluster-EAGLE (C-EAGLE) project (Barnes et al., 2017). Hydrangea
adopts the EAGLE (Schaye et al., 2015; Crain et al., 2015) galaxy formation
model but for zooms of regions taken from a larger volume (3200 cMpc)3 than
the original EAGLE parent volumes of $\leq$(100 cMpc)3.

Hydrangea uses a modified version of the N-Body Tree-PM smoothed particle
hydrodynamics (SPH) code gadget-3 (Springel, 2005), with hydrodynamical
updates by Schaye et al. (2015) and Schaller et al. (2015). The subgrid
physics of the code is based on that developed for OWLS (Schaye et al., 2010):
it implements for instance radiative cooling and photoheating (Wiersma et al.,
2009a) for 11 chemical elements (H, He, C, N, O, Ne, Mg, Si, S, Ca, and Fe),
hydrogen reionization and ionizing UV/X-ray background (Haardt & Madau, 2001),
star formation rate of gas following Schaye & Dalla Vecchia (2008), stellar
mass loss based on Wiersma et al. (2009b), energy feedback from star formation
uses the thermal implementation of Dalla Vecchia & Schaye (2012) with the
heating of a small number of gas particles by a large increment in temperature
and the feedback from supermassive black holes (“AGN feedback”) is implemented
with a similar method (Booth & Schaye, 2009).

We selected the most massive halo in the simulation at $z=2$. From the three
snapshots at redshifts 1, 1.5 and 2, we extracted all SPH particles within a
comoving sphere of radius 1.5 Mpc centred on the halo center of potential. For
each SPH particle, we retrieved quantities such as the particle position,
velocity, temperature $T$, mass $m$, mass density $\rho$ and chemical
abundance222We made use of the Hydrangea python library:
https://hydrangea.readthedocs.io/en/latest/index.html. The main properties of
the halo at the selected redshifts are gathered in Table 1. The values of the
mass within a radius encompassing 500 times the critical density of the
Universe at the cluster redshift, $M_{500}$, indicate a well-formed cluster of
galaxies at $z=2$ with $M_{500}=7\times 10^{13}$ M⊙, equivalent in mass to
groups of galaxies in the local Universe. It transitioned into the more
massive cluster regime at $z=1$ with $M_{500}=2\times 10^{14}$ M⊙. Such system
will be the progenitor of a massive cluster of galaxy (e.g., such as the
Perseus cluster, at $z=0$). At the redshift of main interest ($z=2$) this
cluster is not in a major merging stage. Consistently with other systems at
this redshift, this cluster is not relaxed.

Table 1: Characteristics of Hydrangea halos

$z$ | $M_{500}$ | $R_{500}$ | $T_{500}$ | $\theta_{500}$
---|---|---|---|---
| ($10^{14}\,M_{\odot}$) | (kpc) | (keV) | (arcmin)
1.016 | 2 | 616 | 4.12 | 1.24
1.493 | 1 | 419 | 3.25 | 0.80
1.993 | 0.7 | 309 | 3.44 | 0.60
333Characteristics of our target halo from the CE-22 simulation at 3 different
redshifts.

The atomic gas content of the simulation is converted into chemical
abundances: for an element $i$, the mass fraction of the element $X_{i}$, with
an atomic mass $\mathcal{M}_{i}$, is converted into the chemical abundance
$Z_{i}$ in solar metallicity units (as a number density ratio), assuming solar
abundance $Z_{\odot,i}$ from Anders & Grevesse (1989):

$Z_{i}=\frac{X_{i}}{X_{H}\times Z_{\odot,i}\times\mathcal{M}_{i}}$ (1)

where $X_{H}$ is the hydrogen mass fraction of the gas particle.

To serve our show-case study, we investigated various lines-of-sight at each
redshift according to the distribution of key physical quantities such as the
temperature, density, abundances. We qualitatively selected the two projected
images presenting (i) the most disturbed and structure-rich (referred to as
‘irregular’), and, (ii) the most regular and smoothed (referred to as
‘regular’) distributions of these quantities. These orientations are used to
bracket the impact of projection effects.

###  2.2 Model of the ICM X-ray emission

To model the X-ray emission from our simulated cluster, we down-selected the
SPH gas particles to those representative of the ICM in the gas temperature-
density plane. We restricted the temperature range to $0.0808<T_{keV}<60$.
These boundaries are forced by the tabulated X-ray emission model used
afterwards (i.e., vapec under XSPEC). We set an upper limit on the gas density
of $n_{e}<1\,cm^{-3}$, to avoid overly dense regions in the simulations. Such
particles are not representative of the ICM properties, but would nonetheless
bias our mock observations. These are likely particles recently affected by
the supernovae and/or AGN feedback implementation. We removed about 100
particles out of a few million.

To model the X-ray emission of the cluster, we followed the procedure
described in Cucchetti et al. (2018). For each selected gas particle, we
assumed the collisionally-ionised diffuse plasma model APEC (Smith et al.,
2001) computed from the AtomDB v3.0.9. atomic database (Foster et al., 2012).
We used its vapec implementation under XSPEC (Arnaud, 1996) to account for the
various chemical abundances traced in the Hydrangea simulations. The
aforementioned reference solar abundances are from Anders & Grevesse (1989)
and the cross section are taken from Verner et al. (1996).

The redshift accounted for each particle reads as follow:

$z=z_{pec}+z_{clus}+z_{clus}\times z_{los}\quad\text{with}\quad
1+z_{pec}=\sqrt{\frac{1+\beta}{1-\beta}}$ (2)

where $z_{pec}$ is linked to the peculiar velocity of the particles along the
line-of-sight, $v_{los}$ within the clusters. $\beta=v_{los}/c$, with $c$
being the speed of light and $z_{clus}$ the cosmological redshift of the
cluster.

The X-ray flux is directly proportional to the integral of the electron times
proton density, $n_{e}\times n_{H}$, over the particle volume, $V$. Following
the implementation in XSPEC the normalisation $\mathcal{N}$ of the vapec model
writes:

$\mathcal{N}=\frac{10^{-14}}{4\pi[D_{A}(1+z)]^{2}}\int_{V}n_{e}n_{H}dV\quad\text{(in
cm${}^{-5}$)}$ (3)

with $D_{A}$ the angular diameter distance to the source (computed within the
cosmological setup of the Hydrangea simulation). We further assume
$n_{e}=1.2\times n_{H}$.

We fixed the value of the Galactic hydrogen column density to $0.03\times
10^{22}$ cm-2, a typical value for high galactic latitude. Its associated
absorption of soft X-ray photons is modeled with a wabs model (Morrison &
McCammon, 1983).

From these hypotheses, we computed the flux of photons at the Earth emitted by
each particle. We used each particle’s X-ray spectrum between 0.2-12 keV as a
density probability to draw the appropriate number of photons according to a
given exposure time and collecting area. Stacked together over all cluster
particles, we obtain a photon list at the Earth. The exposure time is
systematically fixed to 10 times this of the mock observation exposure in
order to provide enough statistics for the instrumental simulations (see Sec.
3.) At this stage the collecting area is chosen to be 20,000 cm2, largely
encompassing this of the _Athena_ mirrors over the whole energy band. This
leads to an oversized list of photons providing proper statistics for the
telescope and instrumental simulations and avoid any duplication or under-
sampling biases.

##  3 Simulated X-IFU/_Athena_ observations

###  3.1 The _Athena_ telescope and X-IFU instrument

_Athena_ is the next generation of X-ray telescope from the European Space
Agency (Barcons et al., 2017). It implements the science theme of the Hot and
Energetic Universe (Nandra et al., 2013). With a focal length of 12 m it will
embark a Wolter-I type mirror initially expected to have a collecting area of
14,000 cm2 at 1 keV, and to provide a spatial resolution of 5 arcsec FWHM. Two
instruments will board the payload, the Wide field Imager (WFI – Meidinger et
al., 2017), and the X-ray Integral Field Unit (X-IFU – Barret et al., 2023).

In this study we focus on the capabilities of the X-IFU instrument, whose main
initial high level performance requirements relevant for our study include a
spectral resolution of 2.5 eV over the 0.2-7 keV band. This would represent a
gain of a factor of $\approx\,$50 with respect to XMM-_Newton_. The effective
area (constrained by the mirror collecting area) shall be of 10,000 cm2 at 1
keV, whilst the hexagonal field-of-view shall have an equivalent diameter of 5
arcmin.

We refer the reader to Barret et al. (2023) for an comprehensive description
of the X-IFU instrument.

###  3.2 X-IFU mock observations

To simulate the observations with the X-IFU instrument, we followed the method
described in Cucchetti et al. (2018) and summarised in the following.

We made use of the SImulation of X-ray TElescopes software package (SIXTE,
Wilms et al., 2014; Dauser et al., 2019). SIXTE ingests as input a SIMPUT
(Schmid et al., 2013) file containing either emission spectra of individual
sources or regions, or directly a list of photons at the telescope.

The SIXTE website444https://www.sternwarte.uni-erlangen.de/sixte/ distributes
the baseline setup for the X-IFU/_Athena_ , as described in the above section
and detailed in Barret et al. (2023), formatted for the use of SIXTE. This
includes ressources and configurations such as the focal plane geometry, the
point spread function (PSF), the vignetting, the instrumental background,
crosstalk, the ancillary response file (ARF) and the redistribution matrix
file (RMF), as provided by the X-IFU consortium555http://x-ifu.irap.omp.eu/.
We used this baseline setup in this work.

We took into account the astrophysical emission from foreground and background
emissions, and co-added them to our cluster of galaxies into a total photon
list received at the Earth. We modelled the Galaxy halo and local bubble as
diffuse emission according to the model proposed by McCammon et al. (2002),
that is an absorbed (wabs*apec) and unabsorbed (apec) thermal plasma emission
model, respectively. As per the required _Athena_ spatial resolution, 80% of
the sources constituting the extragalactic background are expected to be
resolved (Moretti et al., 2003). This corresponds to a lower flux limit for
individual simulated sources of $\approx 3\times 10^{-16}$ ergs/s/cm2 for a
100 ksec exposure time (down scaled to $\approx 10^{-16}$ ergs/s/cm2 for 1
Msec). AGN with fluxes below this limit are treated as a diffuse contribution,
and modelled with an absorbed power-law spectrum parameterised as in McCammon
et al. (2002). To avoid double-counting resolved AGN, we set the normalisation
of this model to 20% of its nominal value. The resolved cosmic X-ray
background (CXB) is accounted for by adding the contribution of foreground and
background AGNs according to the procedure described by Clerc et al. (2018).
In short, AGNs are drawn from their luminosity function, $N(L_{0.5-2keV},z)$
(from Hasinger et al., 2005). Each source is assigned an emission spectrum
from Gilli et al. (2007). They are then uniformly distributed over the
simulated solid angle (thus neglecting any clustering effects). For both the
Galactic and non-resolved CXB emission we used the model parameters provided
by Lotti et al. (2014). We fed the total (cluster of galaxies + astrophysical
contamination) photon list to the xifupipeline method of SIXTE in order to
generate a mock event list. All aforementioned instrumental effects and setups
were accounted for, except for the cross-talk, which is irrelevant for our
study. For each three redshifts and two lines-of-sight orientations we
selected for our Hydrangea cluster (see Sec. 2.1 and Table 1), we generated
two mock observations of one single pointing with 100 and 250 ksec,
respectively. For the cluster at $z=2$, we also ran a third deep exposure of 1
Msec.

Figure 1: X-IFU map of a 100 ksec raw number count for our Hydrangea cluster
of galaxies at $z=2$ and for projection ‘regular’. The image includes
astrophysical and instrumental backgrounds. The green circles mark the loci of
the point sources to be excised.

###  3.3 Processing of the mock data

For the purpose of our study, we assumed in our analysis spherical symmetry
for the cluster, and we focus on a radial analysis.

####  3.3.1 Point source masking

We first proceed by generating a raw count map, as shown on Fig. 1 for $z=2$
and projection ‘regular’. As in Cucchetti et al. (2018), we flagged the loci
of the simulated AGNs with fluxes above the limit defined in Sec. 3.2, that is
$\approx 3\times 10^{-16}$ ($\approx 10^{-16}$) ergs/s/cm2 for an exposure
time of 100 ksec (1 Msec) They are masked by excluding all the pixels with
number counts 2$\sigma$ above the mean count number of the pixels neighbouring
their positions. This ensures that the ICM emission is accounted for. This
procedure led to an effective flux threshold asymptotically reaching the value
of $\approx 3\times 10^{-16}$ ($10^{-16}$) ergs/s/cm2 with the decreasing ICM
emission. We performed a visual check and manually masked any remaining
obvious point sources or overdense count regions that could arise from the
Hydrangea simulation itself.

####  3.3.2 Surface brightness profile

We derived the surface brightness (SXB) profile from the raw count image
masked from the point sources in order to account for the geometry of the
X-IFU detector. We thus avoid artificially over sampling the intrinsic spatial
resolution of the instrument (that is the convolution of the mirror PSF and
the pixel size). The focal plane pixels are attributed to the radial annuli
which encompass the position of their centre. Hence the pattern of pixels
populating the various radial annuli slowly tend to actual geometrical annuli
over the detector array with increasing radius. However, the inner annuli
significantly depart from such a regular shape. For instance, the central
annulus contains a single pixel, whilst the second one contains the four
nearest adjacent pixels to the central one in a cross pattern. This specific
pattern is properly accounted for in the fit of the SXB profile (see Sec. 4).

In our mock observation, the position of the cluster projected centre, as
defined in the Hydrangea simulation, coincides with the instrument optical
axis. In other words, the cluster center is positioned at the centre of the
instrument detector array. The SXB profile is extracted in the 0.4-1 keV band,
where the bremsstrahlung emission of the ICM is still relatively flat as a
function of energy even at $z=2$, hence minimising its dependence on the gas
temperature. The derived SXB profile for $z=2$ and projection ‘regular’ is
presented on Fig. 4, left panel.

####  3.3.3 Spectral analysis

To recover the radial distribution of the gas temperature and chemical
abundances, we binned our mock event list into six concentric annuli. As for
the SXB profile, the centre is chosen to match the position of the projected
centre of the simulated cluster. The six annuli were defined to gather at
least 10,000 counts in order to populate the spectral channels of the X-IFU.
The spectra are extracted using the makespec function of SIXTE and binned
according to the optimal method by Kaastra & Bleeker (2016). A specific
ancillary response is computed per annulus accounting for the mirror
vignetting weighted for each contributing pixel by its number counts.

The local background spectrum is extracted from the area of the field-of-view
beyond $R_{500}$. It is fitted with the model described in Sec. 3, that is
apec+wabs*(apec+powerlaw). All parameters but the three normalisations are
fixed. This best fit background model is then rescaled to the area of each
annulus (i.e., the pixel solid angle times the number of pixels belonging to
the annulus) and fixed as such for the fit of the ICM model. We fitted the
spectra of the galaxy cluster with a wabs*vapec model under XSPEC with Cash
statistics (Cash, 1979), with the gas temperature, the normalisation and the
Fe, Si and Mg abundances as free parameters. We limited our investigation of
the chemical abundances to these three elements, as they present the most
prominent lines with a reasonable probability to be detected out to a redshift
of $z=2$. All other abundances were fixed to the average value of the
projected emission-measure weighted abundances from the Hydrangea halo
particles over a given annulus.

The reconstructed temperature and abundances radial distribution are shown in
Fig. 4, right panel and Fig. 5, respectively.

##  4 Forward-modelling and MCMC analysis

We chose a forward modelling approach to fit the SXB, temperature and
abundance profiles in order to reconstruct the 3D radial distribution of the
thermodynamical and chemical properties of the simulated cluster of galaxies.
We recall that we assume spherical symmetry for our simulated clusters.

###  4.1 The forward-modelling procedure

We considered the gas density and pressure as two independent physical
quantities in our model.

We modelled the density distribution with a ‘simplified’ Vikhlinin functional
(Vikhlinin et al., 2006):

$n_{e}^{2}(x)=\frac{n_{0}^{2}}{(x/r_{s})^{\alpha_{1}}[1+(x/r_{s})^{2}]^{3\beta_{1}-\alpha_{1}/2}}$
(4)

where $x=r/R_{500}$. $n_{0}$ is the central density, $\alpha_{1}$ and
$\beta_{1}$ the inner and outer slopes, respectively, and $r_{s}$ the scale
radius.

The pressure radial distribution is described by a gNFW profile (Nagai et al.,
2007)

$P(x)=\frac{P_{0}}{(c_{500}\leavevmode\nobreak\
x)^{\gamma}[1+(c_{500}\leavevmode\nobreak\
x)^{\alpha_{2}}]^{(\beta_{2}-\gamma)/\alpha_{2}}}$ (5)

Where $P_{0}$ is the central pressure, $c_{500}$ is the concentration at a
radius of $R_{500}$, $\alpha_{2}$ and $\beta_{2}$ the intermediate and outer
slopes, respectively. We do not have enough leverage with our observational
constraints to fit $\gamma$, the inner slope. We thus let it as a free
parameter with a Gaussian prior with mean $0.43$ and standard deviation of
$0.1$, the values derived from the XCOP sample (Ghirardini et al., 2019).

The temperature distribution is simply derived from the gas density and
pressure assuming the ICM to be a perfect gas with $P=n_{e}\times k_{B}T$
($k_{B}$ being the Boltzmann constant). The entropy is reconstructed assuming
the conventionally adopted relation in X-ray astronomy as introduced by Voit
(2005), that is $K=k_{B}T/{n_{e}^{2/3}}$.

For the radial distributions of chemical elements, we adopted a simple power-
law model:

$A(x)=A_{0}\leavevmode\nobreak\ x^{-p}$ (6)

where $A_{0}$ is specific to each element. However we assumed the same slope
$p$, for all elements, considering it a fair assumption according to Mernier
et al. (2017).

The remaining twelve free parameters are listed in Table 2 together with their
initial values and priors adopted in our MCMC fit.

We account for the parameter dispersion in each spherical shell by randomly
drawing 5000 particles in the shell and adopting their values in density and
pressure around the shell fiducial value. We aim to compare our results with
volume-weighted distributions of the thermodynamic quantities (Sect. 5), so as
to minimise the impact of high-density, small-volume particles ; therefore we
weigh the random draws by the volume of each particle. In doing so, we follow
the dispersion of thermodynamic quantities predicted by the Hydrangea
simulation about their fiducial profile. Fig. 2 illustrates this process and
shows how translating the cloud of particles in the $\log n_{e}-\log T$ plane
provides a new distribution of densities and temperatures dispersed around a
newly requested median value. These distributions are clipped at zero and at
the maximal values allowed in our setup (1 cm-3 and 15 keV). This process
leads to 5000 spherically symmetric models, each contributing in equal
proportion to the final model. Taken individually, they do not serve as
realistic descriptions of the cluster ; they serve as intermediate models,
enabling propagation of the scatter in thermodynamic profiles.

We compute the X-ray emission of the ‘toy model’ cluster by averaging that of
the 5000 models together. We apply an Abel transform, using the
PyAbel666https://pyabel.readthedocs.io/en/latest/ python package with the
Hansenlaw transform method (Hansen & Law, 1985), to each of the 5000 models
and average the resulting surface brightness profiles. Assuming circular
symmetry, the profile is distributed into a two-dimensional grid generously
oversampling the X-IFU pixel size. Emission-measure-weighted temperature and
abundance profiles are constructed similarly. We finally convolve these
parameter grids with the PSF kernel and reshape it to the pixelization and
geometry of the X-IFU focal plane. We reproduce the source and pixel masking
applied to the mock data. With this process we ensure faithfully reproducing
the input image characteristics (see Sec. 3.3).

Figure 2: Dispersion of thermodynamic quantities (electronic density $n_{e}$,
units cm-3 and temperature $T$, units keV) in one radial shell with $300$ kpc
$<r<$ 310 kpc of the $z=2$ cluster. The leftmost cloud is a density map of
extracted particles from the Hydrangea-simulated cluster. The leftmost green
cross indicates the position of the median density and temperature. The cloud
of 5000 blue points on the right-hand side illustrates our random generation
of a new model, when requesting a new set of median values (as shown by the
yellow rightmost cross). In this process, the shape of the dispersion in the
$\log-\log$ plane is essentially maintained fixed and translated. Table 2:
Cluster model parameters

Parameter | Initial value | Priors | Unit
---|---|---|---
$n_{0}$ | $10^{-2}$ | $\mathcal{U}(10^{-7}\leavevmode\nobreak\ -\leavevmode\nobreak\ 1)$ | cm-3
$\alpha_{1}$ | $-0.3$ | $\mathcal{U}(-1\leavevmode\nobreak\ -\leavevmode\nobreak\ 3)$ |
$\beta_{1}$ | $0.9$ | $\mathcal{U}(0.1\leavevmode\nobreak\ -\leavevmode\nobreak\ 4)$ |
$r_{s}$ | $0.3$ | $\mathcal{U}(0.1\leavevmode\nobreak\ -\leavevmode\nobreak\ 1)$ | $R/R_{500}$
$P_{0}$ | $5\,10^{-2}$ | $\mathcal{U}(0\leavevmode\nobreak\ -\leavevmode\nobreak\ 0.2)$ | keV cm-3
$\alpha_{2}$ | $3$ | $\mathcal{U}(0\leavevmode\nobreak\ -\leavevmode\nobreak\ 4)$ |
$\beta_{2}$ | $5.17$ | $\mathcal{U}(1\leavevmode\nobreak\ -\leavevmode\nobreak\ 10)$ |
$\gamma$ | $0.43$ | $\mathcal{N}(0.43,0.1)$ |
$c_{500}$ | $2.4$ | $\mathcal{U}(1\leavevmode\nobreak\ -\leavevmode\nobreak\ 4)$ |
$A_{0,Fe}$ | $5\,10^{-2}$ | $\mathcal{U}(0\leavevmode\nobreak\ -\leavevmode\nobreak\ 2)$ | $Z/Z_{\odot}$
$A_{0,Si}$ | $0.13$ | $\mathcal{U}(0\leavevmode\nobreak\ -\leavevmode\nobreak\ 2)$ | $Z/Z_{\odot}$
$A_{0,Mg}$ | $9\,10^{-2}$ | $\mathcal{U}(0\leavevmode\nobreak\ -\leavevmode\nobreak\ 2)$ | $Z/Z_{\odot}$
$p$ | $0.6$ | $\mathcal{U}(-1\leavevmode\nobreak\ -\leavevmode\nobreak\ 3)$ |
$bkg$ | $10^{-2}$ | $\mathcal{U}(0\leavevmode\nobreak\ -\leavevmode\nobreak\ 0.3)$ | cts/s/arcmin2
777Initial values and priors of the 13 free parameters describing our cluster
model in the MCMC fit. The last, fourteenth, parameter is also let free and
accounts for the level of uniform background in the 0.4 – 1 keV image.

###  4.2 The MCMC fit

We simultaneously fitted our SXB, temperature and abundances profiles using a
Bayesian MCMC approach. The modeling procedure described above (including the
assessment and propagation of the parameter dispersion in shell) is reproduced
at each step of the MCMC.

We formulated the associated likelihood as follows:

$\chi^{2}=-2\leavevmode\nobreak\
log\mathcal{L}=\sum_{p_{j}}\sum_{i}\frac{(y_{i,p_{j}}-M_{i,p_{j}})^{2}}{\sigma_{i,p_{j}}^{2}}$
(7)

where $y_{i,p_{J}}$, $\sigma_{i,p_{j}}$ and $M_{i}$ are the mock data,
associated mock error and model for profile $p_{j}$, respectively. $p_{j}$
denotes the set of the SXB, temperature and Mn, Si, Fe abundances profiles.
Such a likelihood implicitly assumes normal-distributed uncertainties. In case
of asymmetric uncertainties in the observable, we take $\sigma_{i,p_{j}}$ as
the arithmetic mean of the upper and lower bounds of the 68% confidence level.

Our MCMC sampling of the parameter space made use of python-based code emcee
(Foreman-Mackey et al., 2019).

###  4.3 Validation

Before proceeding to the analysis of mock Hydrangea data, we first validate
our fitting procedure using a simpler, spherically symmetric model. Its
physical quantities (density, pressure, element abundances) are drawn
following parametric profiles (Eq. 4, 5 and 6). The numerical values are
chosen so as to approximately reproduce the radial behaviour of the $z=2$
cluster that we picked in the Hydrangea simulation (Table 1). In contrast with
the Hydrangea cluster though, we impose that each of these quantities, taking
a single value within a spherical shell at radius $r$. This idealised model is
transformed into a mock 100 ks X-IFU observation (as described in Sect. 3) ;
this step includes the extraction of temperature and surface brightness
profiles. We then fit the X-IFU mock observations with the exact same cluster
model used to fit the actual simulation, letting 14 parameters freely evolve ;
namely: $n_{0}$, $\alpha_{1}$, $\beta_{1}$, $r_{s}$, $P_{0}$, $\alpha_{2}$,
$\beta_{2}$, $\gamma$, $c_{500}$, $A_{0,i}$ ($i$ being one of Fe, Si and Mg),
$p$ and finally a uniform background level in the 0.4–1 keV imaging band.
Priors and initial values for the 14 parameters are listed in Table 2.

Figure 6 shows the results of our validation test. We find overall good
agreement between the input model and the inferred properties. The three-
dimensional density and pressure profiles roughly agree with the input
profiles within the 68% confidence intervals, at least across the radial range
of applicability of our procedure (that is, at radial distances located
between the PSF size, $r\approx 0.1R_{500}$, and the outermost radius,
$R_{500}$). Nevertheless, significant deviations appear, especially when
considering the two-dimensional temperature profiles and the comparison
between the purple and black lines in the top-right panel of Fig. 6. Despite
the simplicity of the input model, projection effects along the line of sight
induce temperature and abundance mixing, which limit the ability of a single
APEC model to account for the observed spectrum. This effect leads to
underestimated uncertainties issued by the XSPEC fit (green error bars). In
addition to this issue comes our (so far unverified) assumption that weighting
the 3-d temperature by the emission-measure provides a fair representation of
the 2-dimensional temperature profile. We have checked that such deviations
are not solely of statistical origin by increasing the exposure time of the
mock observation to 1 Ms and finding similar offsets in the projected
temperature profile (Fig. 7). However, repeating the experiment with a toy-
model cluster placed at $z=1$ provides better agreement in the recovered
profiles (Fig. 8). Indeed, the larger apparent $R_{500}$ as compared to the
$z=2$ system enables defining a finer two-dimensional binning of the
temperature profiles (ultimately limited by the X-IFU pixel and/or PSF size),
hence mitigating the line-of-sight mixing effects.

In summary, our validation tests demonstrate the ability of our procedure to
recover input profiles. However, we highlighted a limitation due to line-of-
sight mixing of temperature (and abundances), attributed to both the finite
X-IFU angular resolution relative to the $z=2$ cluster angular span, and to
the slight inadequacy of the spectral fitting model. These deviations
propagate into the inference of the 3-dimensional thermodynamic profiles and
limit our ability to recover their exact distributions. These limitations are
in fact inherent to any observation of multi-phase diffuse gas, irrespective
of the instrument in use.

##  5 Results

We now present the results obtained by fitting the $z=2$ galaxy cluster
extracted from the Hydrangea simulations (Sect. 2) and folded through the
X-IFU instrumental response assuming a 100 ks exposure time. Similarly as for
our validation model (see Sect. 4.3), we fit mock data with a spherically
symmetric model with 14 free parameters. They are listed in Table 2. They
enter Eqs. 4 to 6 and govern the median 3-dimensional model for $n_{e}(r)$,
$P(r)$ and $A(r)$. The thirteenth parameter is an additional background level
in the $0.4-1$ keV imaging band. Accounting for the intrinsic dispersion of
these quantities around their median values requires prior knowledge of its
radial behaviour (or a parameterised model thereof). We simply assumed that
the dispersion in the $(\log n_{e},\log T)$ plane follows that of the
Hydrangea cluster in each radial shell (see Fig. 2 and Sec. 4.1). This
assumption avoids introducing additional model parameters, although one
expects that using the exact dispersion for the forward-model may put our
results on the optimistic side. For computational reasons, we do not assume
any dispersion of the abundance within a spherical shell in our forward model.

Figures 3, 4 and 5 illustrate the outcome of the forward-modelling procedure.
In each figure the uncertainty on the 14 free model parameters is propagated
and provides the envelope indicated with red dashed lines. In contrast to the
validation model (e.g. Fig. 6), our visualization now incorporates the effect
of the intrinsic dispersion in the thermodynamic quantities, which is a key
ingredient in our model. In each figure (but for the chemical abundances) this
intrinsic dispersion is added in quadrature to the parameter uncertainties in
order to provide the red shaded envelope. Profiles built from simulation
particles also incorporate intrinsic dispersion. In each radial shell we
computed the particle-volume weighted histogram of a given quantity (e.g.,
density). Fig. 3 reports the associated 16, 50 and 84 percentiles as a plain
blue line and a blue shaded envelope.

Focusing first on the recovery of physical thermodynamic quantities (electron
density $n_{e}$, pressure $P$, temperature $T$ and entropy $K$ in Fig. 3) we
notice a qualitatively good agreement between the profiles recovered from the
fit and the input profiles. The large amount of intrinsic dispersion within
each spherical shell makes comparison of the median profiles cumbersome, since
there is no one unique density (resp. pressure, temperature, entropy) at a
given radius, rather a distribution of densities (resp. $P$, $T$, $K$). In
order to quantify the agreement between the input and fitted profiles, we
performed a two-sample Kolmogorov-Smirnov (KS) test in thin shells at each
radius $r$, by comparing the distribution of density values $n_{e}$ (resp.
$P$, $T$, $K$) of the Hydrangea simulation to the distribution of densities
(resp. $P$, $T$, $K$) sampled by the MCMC. The KS-statistic is related to the
probability to reject the null hypothesis, being that both distributions
originate from the same (unknown) distribution. The lower the KS value, the
more confident we are that both profiles are in agreement with each other. We
also computed the average value of the KS statistics over the radial range
comprised between the PSF size and $R_{500}$ and we listed the values in Table
3 (in bold characters).

|
---|---
|
Figure 3: Three-dimensional thermodynamic quantities (in blue) for the $z=2$
galaxy cluster with projection ‘regular’ in the Hydrangea sample, and their
best-fit models inferred from an X-IFU 100 ks exposure (in red). Each panel
representing one quantity is made of three plots. The top curves display the
radial profiles ; the blue shaded envelope represents the dispersion of the
gas particles in the hydro-simulation. The red dashed lines indicates the
effect of the variance of the 14 free model parameters ; the shaded envelope
also includes the radial dispersion encapsulated in our model. The middle plot
represents the deviation of the profile relative to the input median profile.
The bottom plot shows the results of a Kolmogorov-Smirnov (KS) test performed
at each radius $r/r_{500}$, related to the probability that the input and
best-fit profiles do not originate from the same distribution (the lower KS,
the closer the agreement between the profiles). Vertical lines indicate the
range of applicability of our modelling procedure.

Folding the MCMC parameter samples through our model provides the distribution
of projected observables, namely the 0.4–1 keV surface brightness (Fig. 4,
left) and the emission-measure-weighted temperature (Fig. 4, right). Most of
the dispersion in surface brightness posterior samples originates from the
intrinsic dispersion of thermodynamic quantities (mostly $n_{e}$), while
statistical uncertainties on the profiles arising from the MCMC have little
impact on the global error budget. The deviation of the reconstructed surface
brightness profile, relative to the statistical error, is at most 1.4. The
variance in the posterior projected temperature is roughly equally shared
between intrinsic dispersion and parameter uncertainties. Similarly as for the
validation case (Sect. 4.3 and Fig. 6), some deviations appear between the
best-fit and input models, although they are contained within the 1-$\sigma$
envelope. A noticeable outlier is the XSPEC-fitted temperature in the fourth
radial bin at $R\simeq 0.45\,R_{500}$. Part of this bias is due to mixing
effects along the line-of-sight, and a spectral model that is unable to
account for such mixed components. The bias is also caused by a relatively
faint CXB point source located within the brighter cluster region. It is
therefore absent from the set of excised points sources (circles in Fig. 1).
This unmasked point source brings an excess of high energy photon in the
fourth radial bin spectrum, that is sufficient to bias high the XSPEC
measurement. The deviation of the reconstructed temperature, relative to the
statistical error, is at most 2.9 and at most 1 if we remove the failed
measurement in the fourth annulus.

|
---|---
Figure 4: Projected observables and their best-fit models for the $z=2$ galaxy
cluster with projection ‘regular’ in the Hydrangea sample, as seen by X-IFU in
a 100 ks exposure. Left: surface brightness profile in the 0.4–1 keV band and
its uncertainties (green points). The posterior mean (‘best-fit model’) and
its 68% confidence envelope are represented in purple colours. The bottom
panel displays the deviation of the measurements relative to the best-fit,
normalised by the errors. Right: two-dimensional temperature profile as
measured by XSPEC in each annular bin (green points and errors). The emission-
measure weighted temperature $T_{EM}$ (known from the hydrodynamic simulation)
is shown as a thick black line. In both panels, the shaded purple envelope
includes both the contribution of the intrinsic dispersion of physical
quantities within the radial shells and the propagated variance of the 14 free
model parameters (dashed lines).

The inference of chemical abundance profiles is depicted in Fig. 5. None of
the profiles is correctly recovered, in other words, the 68% posterior
confidence level (dashed purple envelope) does not reproduce well the EM-
weighted abundance profile (black thick line) known as input from the
hydrodynamical simulation. It is worth noting how XSPEC-fitted abundances
scatter widely around the expected values, denoting both a lack of statistics
and inadequacy in the spectral model due to the mixing of components. The
apparent underestimation of posterior abundance profiles originates from the
low signal-to-noise ratios and to an improper use of a Gaussian likelihood
(Eq. 7) for strictly positive abundance values. We have verified that
increasing the X-IFU exposure to 1 Ms provides a decent recovery of the iron
(Fe) and silicon (Si) profiles, while magnesium (Mg) still suffers from poor
statistics (see Sect. 6 and Fig. 11). Such a result is not surprising due to
the faintness of the cluster emission, and to the large dispersion of
abundances along a single line of sight in the Hydrangea simulation (that is
spatially correlated with the density and temperature) and thus the inability
of our model to adequately capture the 3-dimensional structure of the element
abundances in such a complex object. This issue is exacerbated by our choice
of a crude spatial binning, mostly targeted towards temperature extraction
(see e.g., Cucchetti et al., 2018, presenting an alternative binning scheme
tailored for abundance measurements).

Figure 5: Projected abundance profiles and their best-fit models for the $z=2$
galaxy cluster with projection ‘regular’ in the Hydrangea sample, as seen by
X-IFU in a 100 ks exposure. Each panel corresponds to one chemical element of
Fe, Si and Mg. The measurement output by XSPEC is shown as green points (and
errors). The emission-measure weighted abundance profile (known as input) is
displayed with a thick black line. The best-fit model is displayed in purple
and the dashed lines represent the error of the free model parameters.  Table
3: Kolmogorov-Smirnov test results

|  | [ksec] | $n_{e}$ | $T$ | $P$ | $K$
---|---|---|---|---|---|---
$z=1$ | ‘regular’ | 100 | 0.152 | 0.157 | 0.121 | 0.168
250 | 0.107 | 0.158 | 0.117 | 0.127
‘irregular’ | 100 | 0.072 | 0.084 | 0.102 | 0.065
250 | 0.079 | 0.152 | 0.150 | 0.122
$z=1.5$ | ‘regular’ | 100 | 0.119 | 0.167 | 0.142 | 0.188
250 | 0.105 | 0.116 | 0.100 | 0.159
‘irregular’ | 100 | 0.088 | 0.243 | 0.149 | 0.235
250 | 0.060 | 0.113 | 0.088 | 0.112
$z=2$ | ‘regular’ | 100 | 0.136 | 0.234 | 0.080 | 0.241
250 | 0.431 | 0.368 | 0.614 | 0.387
1000 | 0.161 | 0.332 | 0.140 | 0.321
‘irregular’ | 100 | 0.113 | 0.327 | 0.220 | 0.291
250 | 0.097 | 0.309 | 0.305 | 0.217
1000 | 0.048 | 0.269 | 0.229 | 0.207
888Radial average of Kolmogorov-Smirnov test statistics, for each of the four
thermodynamic profiles recovered by our fitting procedure. The KS tests are
computed with respect to the input profiles from the Hydrangea simulation
(lower KS values indicate closer agreement). The radial average is evaluated
between the size of the PSF and $R_{500}$. Bold characters refer to the
configuration specifically discussed in Sect. 5 and shown in Fig. 3.

##  6 Discussion

###  6.1 On inferring the properties of a $z=2$ cluster of galaxies

Our analysis demonstrates the capability to infer the (volume-weighted)
thermodynamic properties of a realistic cluster of galaxies located at $z=2$,
using a moderate exposure budget of 100 ks with X-IFU onboard _Athena_.
Despite the compact faint appearance of this low-mass object at such large
cosmological distances (Fig. 1) the effective radial range accessible to X-IFU
spans almost one decade between $\approx 0.1-1$ $R_{500}$. This enables
recovery of the shape, amplitude and characteristic slopes of the profiles.
The finite instrumental angular resolution prevents accessing smaller scales
and resolving core properties.

The density and pressure profiles are the quantities best reconstructed in
comparison to the input data, with deviations of the median profile reaching
at maximum 20%. Temperature and entropy display more noticeable deviations, up
to 50% when considering the median profiles. The explanation for this
difference relates to the fact that temperature and entropy are deduced from
the other two. Therefore, their uncertainties propagate the errors of both
density and pressure profiles. Moreover, this difference is also related to
our primary observables used for inference. Since surface brightness profiles
are measured with much less uncertainty than projected temperature (see e.g.
Fig. 4), quantities heavily dependent on temperature (temperature itself, and
entropy) are much more strongly affected by measurement systematics than
density and pressure.

Our $z=2$ study highlights a key point, being that the ICM is not spherically
symmetric, nor is it homogeneous within a given radial shell, making a mere
comparison of median profiles incapable of grasping the complete reality of
the scientific problem. For this reason we have also compared distributions of
thermodynamic quantities at fixed radius. Our radially-dependent KS-statistic
test indicates again that the recovered density and pressure relate well to
the Hydrangea simulation, with KS statistics spanning values between $0$ and
$0.2$. Entropy and pressure may display KS-values up to $0.4-0.5$, notably in
the centre where PSF blurring is significant. The KS indicator is also
elevated at radial locations affected by a faulty XSPEC measurement (Fig. 4,
right, at $R\simeq R_{500}/2$ in this example). Such a systematic error is not
solely due to poor statistics, nor to inhomogeneity in the cluster gas
distribution. Indeed, we have shown that this error acts as a floor
uncertainty, inherent to our analysis setup. First, the inability of a single
APEC model to account for a multi-temperature plasma projected along the line-
of-sight induces discrepancies that are not well captured by the XSPEC error
bars. Development of multi-temperature spectral models with arbitrary
distributions of emission measure (e.g., generalizing the class of gadem
models) would certainly benefit high-resolution spectroscopy of diffuse
astrophysical plasmas. Moreover, identification of point sources contaminating
the spectral measurements and buried in the cluster emission should also
enhance the quality of spectral fits. Second, we have worked under the
assumption that emission-measure weighting fairly represents the measured
X-IFU spectroscopic temperature. Previous studies focusing on _XMM_ -Newton
and _Chandra_ have instead proposed ‘spectroscopic-like’ weightings in order
to alleviate this concern (e.g. Mazzotta et al., 2004; Vikhlinin, 2006). We
have verified that spectroscopic-like temperature profiles are even more
discrepant with measurements than emission-measure weighted profiles. Such a
work is pending realization in the case of high-resolution instruments like
the X-IFU. More generally, our study calls for further development of new
analysis tools dedicated to the analysis of hyperspectral imaging of extended
structures (e.g. Picquenot et al., 2019), with an ability to handle the regime
of low number of counts.

###  6.2 Impact of deeper observations and of targeting a cluster in a more
mature evolutionary stage

Up to now, our results and discussion have focused on a single galaxy cluster
extracted from the $z=2$ simulation snapshot. Observing this cluster of
galaxies at later times (i.e., at lower redshifts, $z=1$ and $1.5$) brings a
supplementary amount of information to our study. At later epochs, this
cluster is more massive, more extended, hotter and intrinsically more luminous
(Table 1). This leads to an increase of the signal-to-noise ratios in
observables, both the surface brightness and spectra. Being closer to the
observer, the surface brightness is also less faint, hence another increase in
signal-to-noise ratios. We note however that the angular diameter distance
hardly changes over this range of redshifts, by a few percent at most.
Therefore, little gain is expected from angular resolution effects. We have
replicated the analysis shown previously for all 14 configurations displayed
in Table 3. In each case we inferred the four thermodynamic profiles and the
three abundance profiles, accounting for the intrinsic dispersion within a
radial shell. We inspected the results in light of deviations from the known
input profiles. Although the fidelity of the profile reconstruction has a
strong radial dependence, we summarise here our results with a single
quantity, namely the average of the KS tests performed over the whole range of
radii comprised between the PSF size and $R_{500}$ (Table 3).

In general, moving the cluster closer to the observer provides an enhancement
in the accuracy of the reconstructed thermodynamic profiles and abundance
profiles, as does increasing the exposure time. This is especially visible for
the density profiles, whose inference relies primarily on surface brightness
profiles. However, we found significant outliers to this overall trend, due to
the systematic effects already discussed in Sect. 5 and App. B. Indeed, a
single faulty temperature measurement in one radial bin (e.g., due to mixing
components by projection along the line-of-sight) has a negative impact on all
reconstructed quantities. Surprisingly, such situations may occur even at low
redshifts and/or for large exposure times. One such example is the $z=1$,
‘irregular’ configuration, which seems better characterised at $100$ ks than
at $250$ ks. A second example is the $z=2$, $250$ ks, ‘regular’ configuration,
which comprises a catastrophic temperature measurement, hence shifting the
reconstructed profiles considerably away from the true one. We also find that
observing the cluster in an orientation that minimises the projection of gas
phases along the line-of-sight and maximises the asymmetries in the plane of
the image (i.e., the so-called ‘irregular’ case) often slightly improves the
reconstruction of profiles, consistent with our expectations.

###  6.3 Perspectives

This work presents results on one single test case only. This object may
display peculiarities that are not representative of the entire population of
groups and clusters. A complete assessment of the scientific feasibility
related to thermodynamic profiles inference with X-IFU would involve a larger
sample of objects. On the one hand, singularities associated to a single test
case would average out ; on the other hand, this would more closely match the
approach that observers take in studying intra-cluster and intra-group
physics.

The study presented in this paper was conducted with the current public
science requirements for the _Athena_ missions and the X-IFU instruments. The
_Athena_ mission is currently undergoing a complete reformulation of its
science case and consequently of the specifications of its instruments. The
outcomes of our study might be modulated by the outcome of this reformulation.

As stressed in the introduction, we limited our study to an investigation with
the X-IFU instrument following the specifications from the Athena Mock
Observing Plan. However, we note that a natural extension of the presented
work would be to investigate distant groups of galaxies with deep pointed
observations with the second Athena instrument, the Wide Field Imager (WFI,
Rau et al., 2017), and in combination with X-IFU observations. This would
optimise the physical characterisation of these objects, at the expense of
exposure time as the two instruments will not be observing simultaneously.

Accounting for the current Athena mock observing plan specifications and the
ongoing reformulation process for the Athena mission, we will implement this
dual combination in a forthcoming investigation. The upcoming XRISM mission
will soon provide the community with unprecedented X-ray observations with
high spectral resolution of nearby bright objects. More distant objects such
as the first groups of galaxies will have to wait for the advent of
observations by the next generation of X-ray integral field unit such as the
X-IFU instrument on board the _Athena_ mission or the LEM mission concept
(Kraft et al., 2022).

###### Acknowledgements.

The authors would like to thank Dominique Eckert for refereeing this paper and
for providing insightful comments on the study. EP, FC and NC acknowledge the
support of CNRS/INSU and CNES. JS acknowledges the support of The Netherlands
Organisation for Scientific Research (NWO) through research programme Athena
184.034.002. YMB gratefully acknowledges funding from the Netherlands Research
Organisation (NWO) through Veni grant number 639.041.751, and financial
support from the Swiss National Science Foundation (SNSF) under project
200021_213076.

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##  Appendix A Results of our validation tests

We show here the results of our validation tests on a simpler, spherically
symmetric model. These tests are described in Sect. 4.3. Figures 6 and 7
differ only by the net exposure time of the mock X-IFU observation,
respectively 100 ks and 1 Ms.

Fig. 6 is related to our baseline validation at 100 ks X-IFU exposure for a
$z=2$ cluster. Fig. 7 refers to the same cluster, but for a 1 Ms exposure and
we highlight residual uncertainties not solved by increasing the photon
statistics. Finally, Fig. 8 is for a 100 ks exposure of a much nearer $z=1$
prototypical cluster, whose physical and angular sizes ($R_{500}$ in kpc and
in arcminutes) are about twice that of the $z=2$ cluster. In this case the
cluster emission is better resolved by the X-IFU instrument, and mixing
effects in the spectral fits are less of a concern.

|
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Figure 6: Validation of the forward-modelling procedure. The input model is a
spherically symmetric cluster at $z=2$, transformed into a 100 ks mock X-IFU
observation. Top left: surface brightness profile from the mock 0.4–1 keV
image (green points and errors). The posterior mean (‘best-fit model’) and 68%
confidence range appear as a purple solid line and dashed envelope. The sub-
panel below represents the difference between measurements and the best-fit
model, normalised by the uncertainties. Top right: two-dimensional temperature
profile measured from X-IFU mock events using XSPEC (green points and errors).
The emission-measure weighted temperature $T_{EM}$ known as input is displayed
with a thick black line. The posterior mean (‘best-fit model’) and 68%
envelope are shown in purple with solid and dashed lines respectively. Bottom
panels: the inferred electron density (left) and pressure (right) profiles are
represented with the red line and dashed envelope (68% confidence level). The
input profiles are displayed in blue and follow Eq. 4 and 5 respectively. Both
sub-panels below represent the deviation of the best-fit models relative to
the true input profiles. The range of fidelity of our fit is located between
the PSF size (indicated with the leftmost vertical dashed line) and $R_{500}$
(rightmost line).  Figure 7: Similar figure as Fig. 6 (top-right), but for a 1
Ms X-IFU exposure. This represents the outcome of our validation test in a
regime of high photon statistics and it exacerbates the issue due to mixing
gas temperatures along the line of sight of this $z=2$ idealised cluster
model.  Figure 8: Similar figure as Fig. 6 (top-right), but for a validation
cluster placed at $z=1$ (still at a 100 ks X-IFU exposure). Because each
resolution element in the X-IFU image sees a smaller region (in units of
$R_{500}$), mixing effects appear less prominent than in the $z=2$ case ; the
reconstructed and XSPEC-fitted temperature profiles are therefore closer to
the input EM-weighted profile.

##  Appendix B Results for a 1 Ms exposure on the $z=2$ cluster

In order to gauge the ultimate capabilities of X-IFU in precisely determining
the gas content of the $z=2$ cluster, we reproduced the experiment of Sect. 5
using a 1 Ms exposure instead of 100 ks. We illustrate two salient results
obtained by increasing the exposure time by a factor 10. On the one hand,
XSPEC temperature measurements show reduced error bars due to higher signal-
to-noise ratios in the fitted spectra (Figs. 9), however the disagreement with
the EM-weighted input model still persists. As hinted in our validation
experiment (Sect. 4.3), this may be attributed to projection effects mixing
components in the resulting observables, which are not solved solely by
increasing the statistics. We demonstrate this by running our analysis on the
same cluster, albeit observed from an alternative orientation selected so that
it minimises the projection effects along the line-of-sight. Fig. 10 shows the
result obtained for that configuration ; clearly the green, black and purple
curves appear closer to each other, despite some residual deviations due to
remaining mixing effects.

Figure 9: Similar figure as Fig. 4 (right), but for a 1 Ms mock X-IFU
exposure. Despite the 10-fold increase in photon statistics, the complex
structure of the $z=2$ Hydrangea cluster prevents us from achieving a perfect
reconstruction of the projected temperature profile, mainly because of faulty
XSPEC single-temperature fits (green points and errors). Figure 10: Similar
figure as Fig. 9, but the cluster is observed along a line-of-sight that
minimises projection effects. Mixing issues are mitigated by selecting this
cluster orientation, enabling a more faithful reconstruction of the cluster
profiles.

On the other hand, the reconstruction of abundance profiles at a 1 Ms exposure
(Fig. 11) appears more satisfactory when compared to the 100 ks case (Fig. 5),
with both iron (Fe) and silicon (Si) profiles well recovered by the forward
model. Only magnesium still presents diverging results despite the 10-fold
increase in photon statistics.

Figure 11: Similar figure as Fig. 5. but for a 1 Ms mock X-IFU exposure.