diff --git "a/NNE0T4oBgHgl3EQf0QK5/content/tmp_files/2301.02684v1.pdf.txt" "b/NNE0T4oBgHgl3EQf0QK5/content/tmp_files/2301.02684v1.pdf.txt" new file mode 100644--- /dev/null +++ "b/NNE0T4oBgHgl3EQf0QK5/content/tmp_files/2301.02684v1.pdf.txt" @@ -0,0 +1,3479 @@ +MNRAS 000, 1–30 (2022) +Preprint 10 January 2023 +Compiled using MNRAS LATEX style file v3.0 +Rhapsody-C simulations – Anisotropic thermal conduction, black hole +physics, and the robustness of massive galaxy cluster scaling relations +Alisson Pellissier,1,2★ Oliver Hahn,3,4 Chiara Ferrari2 +1AIM, CEA, CNRS, Université Paris-Saclay, Université Paris Diderot, Sorbonne Paris Cité, F-91191 Gif-sur-Yvette, France +2Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, Bd de l’Observatoire, CS 34229, 06304 Nice cedex 4, France +3Department of Astrophysics, University of Vienna, Türkenschanzstraße 17, 1180 Vienna, Austria +4Department of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +We present the Rhapsody-C simulations that extend the Rhapsody-G suite of massive galaxy clusters at the 𝑀vir ∼ 1015M⊙ +scale with cosmological magneto-hydrodynamic zoom-in simulations that include anisotropic thermal conduction, modified +supermassive black hole (SMBH) feedback, new SMBH seeding and SMBH orbital decay model. These modelling improvements +have a dramatic effect on the SMBH growth, star formation and gas depletion in the proto-clusters. We explore the parameter +space of the models and report their effect on both star formation and the thermodynamics of the intra-cluster medium (ICM) +as observed in X-ray and SZ observations. We report that the star formation in proto-clusters is strongly impacted by the choice +of the SMBH seeding as well as the orbital decay of SMBHs. Feedback from AGNs is substantially boosted by the SMBH +decay, its time evolution and impact range differ noticeably depending on the AGN energy injection scheme used. Compared +to a mass-weighted injection whose energy remains confined close to the central SMBHs, a volume-weighted thermal energy +deposition allows to heat the ICM out to large radii which severely quenches the star formation in proto-clusters. By flattening +out temperature gradients in the ICM, anisotropic thermal conduction can reduce star formation early on but weakens and delays +the AGN activity. Despite the dissimilarities found in the stellar and gaseous content of our haloes, the cluster scaling relations +we report are surprisingly insensitive to the subresolution models used and are in good agreement with recent observational and +numerical studies. +Key words: methods: numerical – cosmology: large-scale structure of Universe – galaxies: clusters: intra-cluster medium – +X-rays: galaxies: clusters – conduction +1 INTRODUCTION +Forming the nodes of the cosmic web, clusters of galaxies are the +largest virialised structures in our Universe and their matter content +reflects that of the Universe. Originating from the highest peaks in +the initial cosmic density field (Kaiser 1984; Bardeen et al. 1986), +their spatial distribution and abundance carry the imprints of the +process of structure formation and are heavily sensitive to the un- +derlying cosmology. Therefore, they represent veritable crossroads +of astrophysics and cosmology as they provide valuable information +from the physics driving structure formation to the nature of dark +matter and dark energy (Voit 2005; Allen et al. 2011; Kravtsov & +Borgani 2012; Weinberg et al. 2013). +Counting galaxy clusters (GCs) as a function of their mass and +cosmic time provides an excellent (late Universe) probe of cosmo- +logical parameters (e.g Planck Collaboration et al. 2021; Abbott et al. +2022) including dark energy, the summed neutrino masses (e.g Mad- +★ E-mail: alisson.pellissier@oca.eu +havacheril et al. 2017) and modifications of gravity (e.g Wilcox et al. +2015). +However, the predictive power of GCs as cosmological probes +is limited principally by our ability to accurately measure their +masses using X-ray, Sunyaev-Zeldovich (SZ) or gravitational +lensing analyses. Mass estimations require high-quality data +and rely on various assumptions that are challenged by the +presence of possible biases caused by several factors, such as +deviations from hydrostatic equilibrium, triaxiality, or instrumental +features. Hence, cluster cosmological surveys depend heavily +on well-calibrated scaling relations that relate directly observed +quantities - so-called mass proxies - such as the X-ray luminosity, to +the underlying cluster mass (see Pratt et al. 2019, for a recent review). +GCs grow hierarchically over cosmic time as gravity pulls +baryonic and dark matter to form collapsed structures. They also +grow in mass via major mergers that represent the most energetic +phenomena since the Big Bang. Moreover, the feedback from +supernovae (SN) and active galactic nuclei (AGN) in cluster +galaxies injects a substantial amount of energy into the intra-cluster +© 2022 The Authors +arXiv:2301.02684v1 [astro-ph.CO] 6 Jan 2023 + +2 +A. Pellissier et al. +medium (ICM). Such processes continually shape the baryonic +components of clusters and can inject up to 1064 ergs of gravitational +potential energy during one cluster crossing time (∼ Gyr), primarily +dissipated by shocks into heating of the intra-cluster gas to high +(X-ray emitting) temperatures (Markevitch & Vikhlinin 2007), +but also through large-scale ICM motions generating cluster-wide +turbulence (Hitomi Collaboration et al. 2016, 2018; Li et al. 2020). +A fraction of this energy can also be channeled into non-thermal +plasma components such as cosmic rays (Brunetti & Jones 2014; +Bykov et al. 2019) and magnetic field amplification (Donnert et al. +2018) as revealed by the presence of extended radio emission +(Ferrari et al. 2008; Feretti et al. 2012; van Weeren et al. 2019). +These energetic processes are expected to contribute to the deviation +of cluster properties from self-similar predictions, which only +account for gravitational evolution in scale-free cluster evolution +(Kaiser 1986). Galaxy cluster observables are therefore a complex +interplay of both cosmology and astrophysics. They require detailed +understanding for precisely calibrating cluster scaling relations to +fully exploit the power of galaxy clusters as cosmological probes. +In this context, numerical simulations provide valuable informa- +tion, as they can follow the evolution of galaxy clusters with ex- +actly known properties. They can capture the effects of physical +processes during cluster formation and predict the resulting observ- +ables self-consistently. However, astrophysical processes related to +galaxy formation cannot be resolved in hydrodynamical cosmologi- +cal simulations due to limited computational resources. Astrophys- +ical processes occurring below the typical resolution are accounted +for by so-called sub-grid models. Significant recent advances in the +development and calibration of efficient sub-grid models has led to +cosmological simulations that can reproduce a large number of the +observed galaxy properties (see Vogelsberger et al. 2020, for a re- +view). However, reproducing the galaxy cluster entropy profiles and +the cool-core/non-cool-core dichotomy remains challenging (Rasia +et al. 2015; Hahn et al. 2017; Barnes et al. 2017a). In cosmological +simulations of galaxy clusters, special attention has been dedicated +to the effects of feedback from stars and AGN feedback. These ad- +vances have yielded a range of simulations from several independent +groups that reproduce various cluster observables, such as the X-ray +and SZ scaling relations (e.g. Barnes et al. 2017a,b; Le Brun et al. +2017; Truong et al. 2018; Cui et al. 2018; Henden et al. 2019). +The outcomes of such simulations can rely heavily on the param- +eter choice of the sub-grid AGN feedback models. Le Brun et al. +(2014) showed that the entire baryon gas profile can be varied by +tuning the energy accumulation threshold Δ𝑇 of the feedback model +of Booth & Schaye (2009). In the Rhapsody-G simulations of Hahn +et al. (2017), changes in the AGN feedback model had no significant +impact on the gas outside the cluster core region. This suggests +that the sub-grid modelling of astrophysical processes needs to be +improved or that the addition of new physics is necessary to increase +the realism of such simulations. Indeed, the addition of thermal +conduction in cluster simulations has been shown to significantly +affect the properties of the ICM and provides an additional source +of gas heating (Voit 2011; Voit et al. 2015). Allowing heat transport +in the ICM with the implementation of thermal conduction could +help to decrease the dependence of numerical simulations on the +feedback sub-grid model parameters. While it does not provide +enough heating to offset cooling losses in clusters, recent studies +have shown that thermal conduction has various impacts on AGN +activity and ICM mixing (Yang & Reynolds 2016; Kannan et al. +2017; Barnes et al. 2019; Beckmann et al. 2022). In this context, the +properties of the intra-cluster gas intimately depend on the physical +processes modelled in simulations. Quantifying such dependencies +in simulations is therefore fundamental to providing robust GC +scaling relations that are focused on the (hot) intra-cluster gas. +In this paper, we perform cosmological magneto-hydrodynamic +simulations of massive galaxy clusters (𝑀vir = 1015.0±0.1 M⊙) that +include anisotropic conduction, radiative cooling, stellar and AGN +feedback to study their effects on cluster scaling relations. In Sec- +tion 2, we introduce our Rhapsody-C sample of zoom-in simulations +and the numerical methods and models that we employ for this study. +We focus in Section 3 on our super-massive black hole (SMBH) mod- +elling, presenting a new ‘tidal friction’ model that efficiently controls +their orbits and studying different AGN feedback models. We show +the impact of anisotropic thermal conduction (ATC) on the cluster +stellar and gaseous properties in Sections 4 and 5. In Section 6, we +study the impact of AGN feedback models and ATC on the evolution +of the simulated clusters along several mass-observable scaling rela- +tions relevant to cosmology. Finally, we summarise our results and +conclude in Section 7. Additionally, we show in Appendix A the im- +pact of anisotropic thermal conduction on the ICM in idealised con- +figurations. We also provide in Appendix B and C the bias on cluster +scaling relations induced by the choice of the X-ray temperature es- +timates in the simulations and the impact of core inclusion/exclusion +on X-ray observables, respectively. +2 METHODS +2.1 The Rhapsody-C sample and initial conditions +This paper presents the Rhapsody-C project1, a suite of high- +resolution zoom-in magneto-hydrodynamical (MHD) simulations of +nine haloes in the 𝑀vir = 1015.0±0.1M⊙ range. The haloes were +selected using the cosmICweb database from the Rhapsody-New +simulation2 (Buehlmann et al. 2023, in prep.). We list the properties +of these haloes in Table 1. +Sharing similar masses at 𝑧 = 0, our haloes have different assembly +histories and probe extreme as well as median cluster properties: +two haloes have extreme concentrations (𝑐vir > 8), high and low +number of subhaloes (𝑁sub > 120 and 𝑁sub < 85 respectively). +On average, our haloes share the same number of substructures and +concentrations as the Rhapsody-G sample (Wu et al. 2015; Hahn +et al. 2017; Martizzi et al. 2016). +We use the ΛCDM cosmology of the Rhapsody-New simulation +with density parameters Ωb = 0.049 for baryons, Ωm = 0.309 for +total matter and ΩΛ = 0.691 for the cosmological constant. The pri- +mordial spectral index, the amplitude normalisation and the Hubble +constant are 𝑛𝑠 = 0.9667, 𝜎8 = 0.8159 and 𝐻0 = 67.74 km/s/Mpc +respectively (Planck Collaboration et al. 2016). In this new cosmol- +ogy, we have a lower baryon fraction of 𝑓b = 0.1586 compared to the +Rhapsody-G’s value of 0.18. The updated value is also much closer +to more recent constraints from Planck Collaboration et al. (2021) of +𝑓b = 0.1564. +We generated the initial conditions using Music (Hahn & Abel +2011) for our nine clusters at 𝑧 = 49 from the minimum bound- +ing ellipsoid matrix retrieved from the cosmICweb database using a +traceback-radius of 2𝑅vir centered in a 1 ℎ−1Gpc box with an effective +resolution of 81923 particles3. All initial conditions were performed +1 where the ‘C’ denotes for the inclusion of anisotropic thermal conduction. +2 See https://cosmicweb.astro.univie.ac.at for more details +3 We share the same resolution as the Rhapsody-G 8K run. +MNRAS 000, 1–30 (2022) + +The Rhapsody-C simulations +3 +Table 1. Description of the Rhapsody-C runs presented in this work. In the +top part of the table we list the minimum cell size (Δ𝑥), the initial mass +per hydro cell (𝑚gas), the dark matter (𝑚dm) and minimum stellar particle +mass (𝑚∗,min). The middle part describes the physical models used in the +simulations studied in this work. In the bottom part, we list the properties +of each Rhapsody-C haloes: the internal halo ID in the Rhapsody-New +simulation, the number of substructures (𝑁sub), the virial mass (𝑀vir), radius +(𝑅vir) and concentration (𝑐vir) as well as the radius enclosing 500 times the +critical density of the Universe (𝑅500) and the total mass within (𝑀500). +Summary of the Rhapsody-C simulations +Δ𝑥 [ kpc] +𝑚dm [M⊙] +𝑚gas [M⊙] +𝑚∗,min [M⊙] +2.82 +1.54 × 108 +1.68 × 107 +6.58 × 106 +Sub-grid modelling and baryonic processes +Cooling, +AGN energy +AGN energy +Anisotropic +Label +SF, +deposition +accumulation +thermal +SN +scheme +threshold +conduction +NR +– +– +– +– +VW +✓ +volume-weighted +107 K +– +MW +✓ +mass-weighted +107 K +– +MC +✓ +mass-weighted +107 K +✓ +MW6 +✓ +mass-weighted +106 K +– +MW8 +✓ +mass-weighted +108 K +– +Halo Properties +ID +𝑁sub +𝑐vir +𝑀vir +𝑅vir +𝑀500 +𝑅500 +[1015M⊙] +[Mpc] +[1014M⊙] +[Mpc] +174742934 111 +6.02 +1.22 +2.82 +6.80 +1.37 +176970005 117 +6.69 +1.19 +2.78 +7.55 +1.41 +174824666 124 +6.35 +1.16 +2.77 +7.56 +1.42 +173505201 135 +5.68 +1.12 +2.80 +6.98 +1.38 +176144520 100 +6.80 +1.01 +2.64 +6.32 +1.33 +176061412 105 +9.52 +1.00 +2.63 +6.62 +1.35 +173917492 111 +8.64 +0.99 +2.62 +6.73 +1.36 +174743229 83 +6.50 +0.98 +2.62 +6.24 +1.33 +173587157 84 +5.45 +0.86 +2.51 +5.19 +1.25 +using second-order Lagrangian perturbation theory (LPT) with dark +matter and baryon perturbations at 𝑧 = 49. Compared to the original +Rhapsody-G simulations, we do not use the local Lagrangian ap- +proximation for the construction of the baryon density field. Baryons +and dark matter did not co-move prior to recombination and sub- +percent effects are expected at cluster scales (see e.g. Angulo et al. +2013; Hahn et al. 2021; Khoraminezhad et al. 2021). However, for +the simulations used here, we assume that baryons fully trace cold +dark matter perturbations. +2.2 Numerical approach +For our cluster zoom simulations, we use the Eulerian adaptive mesh +refinement Ramses code (Teyssier 2002) to follow the non-linear +evolution of the initial conditions. Gas dynamics are computed using +a second-order unsplit Godunov scheme for the ideal MHD equations +(Fromang et al. 2006; Teyssier et al. 2006) while collisionless dark +matter particles as well as stars and sink particles are evolved using a +particle-mesh solver. Our simulations use the method introduced by +Dubois & Commerçon (2016) for solving the anisotropic diffusion +of heat using an implicit finite-volume method which is independent +of the Courant time step constraint of the MHD scheme. +We employ a Lagrangian overdensity-based refinement strategy +that splits cells if they reach an overdensity of eight: the refinement +of the base grid by 𝑛 additional levels requires a density of 8𝑛 ¯𝜌. +Our simulation boxes of 1 ℎ−1Gpc on a side, reach a maximum +refinement level by maintaining a smallest cell size of physical +Δ𝑥 = 2.8 kpc. The dark matter 𝑁-body particle mass is 1.54×108M⊙ +and initial mass per hydro cell is 1.68 × 107M⊙. The high-resolution +Lagrangian ellipsoid patch, from which the 2𝑅vir sphere centred on +each cluster will form, is tagged using a passive scalar colour field +that is advected with the gas. Dynamic refinement is restricted to +the regions where this colour field is non-zero and no refinement +is allowed outside the zoom region. We thus focus most of the +computational resources on the forming cluster and its immediate +environment. +The rest of this section details the various physical ingredients +used in our high-resolution zoom-in simulations. See Section 2.2.1 +for gas cooling and heating as well as the star formation and stellar +feedback, Section 2.2.2 for the subgrid modelling of SMBH forma- +tion, evolution and AGN feedback ang finally in Section 2.2.3 we +describe the magnetic field evolution with the anisotropic thermal +conduction. The reader can skip to Section 3 and Section 4 for the +scientific results regarding the impact of the the various BH-related +sub-grid models and anisotropic thermal conduction respectively on +the stellar and gaseous content of a GC, or directly to Sections 5 +and 6 for properties of cluster galaxies and ICM as well as the the +evolution of our clusters along various scaling relations respectively. +2.2.1 Radiative gas cooling, metallicity and stellar evolution +Radiative gas cooling is calculated according to the tabulated rates +of Sutherland & Dopita (1993) for Hydrogen, Helium and metal +line cooling. The total gas metallicity is not evolved separately but +treated as a single species. It is advected with the MHD equations as +a passive scalar and is sourced by the supernovae feedback model. +We consider an UV background radiation according to the Haardt & +Madau (1996) model. An instantaneous reionisation takes place at +𝑧 = 10 to take into account an earlier reionisation in the particularly +overdense proto-cluster regions that we simulate. The unresolved cold +and dense gas that will constitute the inter-stellar medium (ISM) +of galaxies is approximated using a temperature floor given by a +polytropic equation of state, +𝑇floor = 𝑇∗ +� 𝑛H +𝑛∗ +�𝛾∗−1 +, +(1) +with 𝑛H the Hydrogen number density of the gas, 𝑛∗ = 0.1 cm−3 and +𝑇∗ = 104 K being respectively the star formation density threshold +and the ISM polytropic temperature with 𝛾∗ = 5/3 being the ISM +polytropic index. In practice, gas can be heated above the temperature +floor, but cannot cool below it. +Star formation occurs when the gas density exceeds 𝑛∗. A portion +of the gas in a cell is converted into a star particle that decouples from +the gas. We have a minimum stellar particle mass of 5.6 × 106M⊙. +The star particles are randomly drawn from Poisson process (Rasera +& Teyssier 2006) following a Schmidt law +�𝜌∗ = 𝜖∗ 𝜌gas / 𝑡ff, +(2) +with 𝜖∗ = 0.01 and 𝑡ff = �3𝜋/32G𝜌gas +�−1/2, the local free-fall time. +Stellar feedback is included using the model of Dubois & Teyssier +(2008) in which each newly formed star that traces a continuous +stellar mass distribution following the Salpeter (1955) initial mass +function and releases, after 20 Myr, a fraction 𝜂 = 0.1 of its mass +and metals with a yield of 𝑦 = 0.1. Therefore 𝑦𝜂 = 0.01 of the time- +integrated SFR is returned as metals in the ISM. In addition, each +MNRAS 000, 1–30 (2022) + +4 +A. Pellissier et al. +SN feedback event injects a thermal energy of 1051 erg into the sur- +rounding ISM. Compared to the original Rhapsody-G simulations, +we chose to enable the delayed cooling of the SN heated gas with +a dissipation time scale of 20 Myr. This additional sub-grid model +mimics the effect of non-thermal processes, such as turbulence or +CRs (Rodríguez Montero et al. 2022), which can dissipate energy on +longer time scales before being radiated away. The calibration of the +free parameters of the SN feedback listed above is able to reproduce +stellar masses consistent with abundance-matching results at masses +lower than 1012M⊙ for resolved haloes with at least 1000 particles +(see Section 5.1). +2.2.2 Black holes and active galactic nuclei +Ramses uses collisionless sink particles to model black hole growth +and evolution. The SMBH formation and evolution follow the model +of Biernacki et al. (2017), itself based on the precedent models +of Dubois et al. (2010) and Teyssier et al. (2011), and build on +a sink particle implementation developed within the context of +star-forming molecular clouds (Bleuler & Teyssier 2014). +Super-massive black hole seeding. The Phew clump finder +(Bleuler & Teyssier 2014), directly implemented in Ramses, de- +termines potential sites for SMBH sink particle formation by identi- +fying relevant peaks in the density field. We will briefly discuss the +main steps and free parameters of the sink seeding model that we +use. First, all density peaks above a threshold 𝜌peak are identified as +well as their connecting saddle points. To keep only relevant density +peaks, we merge all peaks that have a peak-to-saddle ratio lower than +3 to the neighbouring peak with which it shares the highest density +saddle point. This merging process, or noise removal, is halted when +a saddle density falls below the 𝜌saddle threshold. In short, a noise +removal is performed on the density field to select only the relevant +peaks above a density 𝜌peak which are later divided by the saddle den- +sity threshold 𝜌saddle into clumps to finally yield the sink formation +sites. The gas in the spherical region of radius equal to 4 (highest) +resolution elements Δ𝑥, defining the sink sphere, is investigated to +make sure that the gravitational field is compressive, strong enough +to overcome internal gas support and not only accelerated toward the +sink sphere centre but that this gas is contracting. As a proximity +check, we forbid the gas that is infalling to an already existing sink +to create another sink. While the choice for the initial seed mass is +arbitrary, we set it to be the same as our 𝑁-body dark matter particle +mass with 𝑚BH,seed = 108M⊙. +Gas accretion and black hole dynamics. Once SMBHs are formed, +they grow in mass at the (un-boosted) Bondi-Hoyle accretion rate +(Hoyle & Lyttleton 1939; Bondi & Hoyle 1944; Bondi 1952) capped +by the Eddington rate : +�𝑀acc = min � �𝑀Edd , �𝑀Bondi +� , +(3) +with : +�𝑀Bondi = 4𝜋𝜌∞𝑟2 +Bondi𝑣Bondi, +(4) +�𝑀Edd = 4𝜋G𝑀BH𝑚 𝑝 +𝜖𝑟 𝜎𝑇 𝑐 += 𝑀BH +𝑡𝑆 +, +(5) +where 𝜎𝑇 is the Thomson cross-section, 𝐺 the gravitational constant, +𝑀BH and 𝑚 𝑝, sink and proton mass respectively, 𝜖𝑟 = 0.1 is the +Shakura & Sunyaev (1973) radiative efficiency for a SMBH and +𝑡𝑆 ∼ 45 Myr is the Salpeter time. We also have 𝜌∞ = ¯𝜌/𝛼(𝑥sink) +with 𝛼 is the dimensionless density profile of the Bondi self-similar +solution (see Biernacki et al. 2017), ¯𝜌 the mean density inside the +sink sphere, 𝑥sink = 𝑟sink/𝑟Bondi and the sink radius and velocity +defined as follows : +𝑟Bondi = G𝑀BH +𝑣2 +Bondi +, +(6) +𝑣Bondi = +√︃ +𝑐2𝑠 + 𝑣2 +rel, +(7) +with 𝑣rel the relative velocity of the sink to the average gas velocity +inside the sink sphere and 𝑐𝑠, the local sound speed. While we use +MHD, we generically find high plasma beta values in our simulations. +Therefore, in the Bondi formula, the magneto-sonic speed effectively +reduces to the adiabatic sound speed . In addition to gas accretion, +SMBHs can also grow via mergers. In this work, we do not check +if two sinks form a bound system but directly merge if they are less +than one accretion radius apart, i.e. 4Δ𝑥. +The dynamics of a single SMBH cannot be resolved in cosmolog- +ical simulations. This can lead to spurious oscillations of the SMBH +in the potential well of its host halo, due to external perturbations and +the finite resolution effects, particularly during merger events. Bier- +nacki et al. (2017) implemented in Ramses a physically motivated +model based on Eddington-limited accretion. Their main assump- +tion is that the gas accretion rate onto the accretion disc is set by the +Bondi formula ( �𝑀Bondi) which corresponds to a large scale accretion +flow, while the accretion onto the SMBH is set by the Eddington +rate ( �𝑀Edd). The difference between the two rates therefore gives the +amount of gas not being accreted by the central SMBH. Instead, it +should be pushed away from the accretion disc by the Eddington radi- +ation pressure at a rate �𝑀dec = �𝑀Bondi − �𝑀acc, which we however do +not model explicitly in this work. We also stress that we are not using +radiation hydrodynamics in this work. This process of gas accretion +and ejection leads to an additional momentum exchange between the +gas and the sink particle, hence an additional drag force. This addi- +tional drag force is modelled by requiring a fixed center of mass of +the joint gas+sink system during the accretion and a conserved total +momentum. We implemented in Ramses a further modification to +the model of Biernacki et al. (2017) to move the SMBHs towards the +potential minimun (described in Section 3.2). +Active galactic nucleus feedback The accretion rate of gas onto the +SMBH sink particle is always computed from the cells in the sink +sphere (of radius 4Δ𝑥) using mass-weighting. Following Booth & +Schaye (2009), we do not inject the thermal AGN energy at each +time-step but store the rest-mass energy of the accreted gas until it +would be enough to raise the gas temperature inside the sink sphere by +Δ𝑇 = 107 K (unless specified otherwise, see e.g. studies of Teyssier +et al. 2011 and Le Brun et al. 2014 using this Δ𝑇 threshold strategy). +In other words, we inject this accumulated AGN energy when +𝐸AGN > 3 +2𝑚gaskB Δ𝑇 +(8) +in every gas cell of the sink sphere in a mass- or volume-weighted +way (see Section 3.3) with a maximum allowed temperature of the +AGN feedback set to 𝑇AGN = 1.5 × 1011 K. The rate at which this +thermal energy is released to the ambient gas is given by : +�𝐸AGN = 𝜖𝑐𝜖𝑟 �𝑀accc2, +(9) +where 𝜖𝑐 = 0.15 is the coupling efficiency (Dubois et al. 2012), i.e. +the fraction of radiated energy that couples to the surrounding gas, +and is calibrated on the local 𝑀BH − 𝑀∗ relation. +MNRAS 000, 1–30 (2022) + +The Rhapsody-C simulations +5 +2.2.3 Magnetic fields and anisotropic thermal conduction +To solve the MHD equations, the Ramses code uses the second-order +unsplit Godunov method based on the monotonic upstream-centred +scheme for conservation laws (MUSCL-Hancock method, van Leer +1977; Evans & Hawley 1988). The constrained transport approach is +used to evolve the induction equation +𝜕𝑩 +𝜕𝑡 = ∇ × 𝒖 × 𝑩, +(10) +where 𝒖 is the gas velocity and 𝑩 the magnetic field. The scheme +satisfies the solenoidal constraint ∇ · 𝑩 = 0 to machine precision +(Teyssier et al. 2006). The 2D Riemann problem at the cell edges is +solved using the approximate Harten-Lax-van Leer-Discontinuities +(HLLD) solver from Miyoshi & Kusano (2005) to compute time +averaged electromotive forces. +In the ideal MHD limit, the generation of magnetic fields from +a previously unmagnetised fluid is impossible. Therefore, the +magnetic fields must be seeded in our simulations. For simplicity, we +seed a uniform magnetic field along the box 𝑧 axis with a comoving +magnitude of 𝐵0 = 1.56 × 10−12 G, which ensures a divergence-free +initial field. +In the presence of a magnetic field, the conduction of heat in a +plasma becomes anisotropic since the motion of charged particles +perpendicular to the field lines is restricted. We use the implementa- +tion of Dubois & Commerçon (2016) using an implicit finite-volume +method for solving the anisotropic diffusion of heat through electrons +(Braginskii 1965) +𝜕𝜌𝜖e +𝜕𝑡 += −∇ · Qcond, +(11) +with 𝜖e the specific internal energy of electrons. The conductive heat +flux, Qcond, can saturate once the characteristic length scale of the +electron temperature gradient ℓ𝑇e is comparable to or less than the +mean free path of electron 𝜆e. Hence, following Sarazin (1986), we +introduce an effective conductivity which interpolates between the +unsaturated (Spitzer conductivity) and saturated regime by +Qcond,sat = − 𝑓sat 𝜅Sp∇𝑇e, += − 𝑓sat +� +−𝜅∥b (b · ∇) 𝑇e +� +− 𝑓sat (−𝜅iso∇𝑇e) , +(12) +with 𝑓sat = �1 + 4.2𝜆e/ℓ𝑇e +�−1, b = B/|B| the unit vector in the +direction of the local magnetic field, 𝑇e the electronic temperature, +, 𝜅iso and 𝜅∥ the isotropic and parallel conduction coefficient (with +respect to the magnetic field lines) respectively with 𝜅∥ = 𝜅Sp − +𝜅iso. In many astrophysical cases, 𝜅iso/𝜅∥ ≪ 1 since the Larmor +radius, 𝜆L, is much smaller than the mean-free-path of electrons, 𝜆e. +For instance, in a hot intra-cluster plasma with 𝑇e = 3 keV, 𝑛e = +10−2 cm−3 and 𝐵 = 1𝜇G, we have 𝜆L = 108 cm and 𝜆e = 1021 cm. +Here, we set a perpendicular conductivity coefficient of 1 per cent to +ensure numerical stability (Dubois & Commerçon 2016). +The electron energy is tracked separately from that of the ions as +described in Dubois & Commerçon (2016) and the rate of energy +transfer between the electron and ion temperatures is given by +𝑄e↔i = 𝑇i − 𝑇e +𝜏eq,ei +𝑛e𝑘B +𝛾 − 1, +(13) +with the equilibrium timescale +𝜏eq,ei = +3𝑚e𝑚p +8 +√ +2𝜋𝑛i𝑞4e ln Λ +� 𝑘B𝑇e +𝑚e +� 3 +2 +. +(14) +Both ion and electron adiabatic indexes are equal to 𝛾 = 5/3. +By modelling the anisotropic transport using Braginskii MHD +(equation 11), we follow the Spitzer ansatz that assumes a high +degree of electron-ion collisionality, which is a good assumption +in cluster cores (in which we are particularly interested here), but +would need to be corrected in cluster outskirts. Additionally, we +do not take into account the suppression of thermal conduction by +the ion mirror instability (Komarov et al. 2016) caused by magnetic +trapping of electrons by magnetic field strength fluctuations, or the +Whistler instability (Levinson & Eichler 1992; Pistinner & Eichler +1998; Roberg-Clark et al. 2016, 2018) where electron-whistler scat- +terings can significantly alter conduction at very sharp temperature +gradients such as in cold fronts (Komarov et al. 2018) or at tem- +perature scale lengths below the critical value 𝛽e𝜆e (Drake et al. +2021, with typical values of 𝛽e ∼ 100 and 𝜆e ranging from 0.1 kpc +in cool-cores to 1 kpc in cluster outskirts)4. In the recent idealised +simulations of Berlok et al. (2021) and Beckmann et al. (2022), it +was shown that whistler-based suppression of thermal conduction +has only a small impact on the ICM. In this work, we hence model +the upper limit of anisotropic thermal conduction within the ICM, +which is sufficient for our purposes to study the potential impact on +cosmological observables. +3 THE MODELLING OF SUPER-MASSIVE BLACK HOLES +The key ingredients of our SMBH formation and evolution models +are: (a) the conditions for the formation of the SMBH and the SMBH +seed mass, (b) the SMBH dynamics with a possible inclusion of a +dynamical friction model, (c) the SMBH growth by mass accretion +at the Bondi-Hoyle-Lyttleton rate limited to the Eddington rate and +finally (d) the induced AGN feedback which affects the surrounding +gas which couples back to all the previous model ingredients. Ramses +uses the so-called sink particle technique (Bate et al. 1995) to model +SMBH formation and evolution, which is a point mass which can +move through the fluid accretion and interact with it by the ejection +of mass, energy and momentum. +Motivated by the low efficiency of the AGN feedback model in +the Rhapsody-G simulations, our sub-grid models for the SMBH +formation, evolution and AGN feedback need to be revised. In this +section we test how the free parameters in the model influence the +cluster evolution. Respectively, for (a) we investigate in Section 3.1 +the effect of different SMBH seeding scenarii on the gaseous and +stellar content on one of our proto-clusters. Regarding (b), we will +present in Section 3.2 our new ‘tidal friction’ model which allow +to control SMBH orbits. Lastly, we study for (d) different AGN +feedback models in Section 3.1 which impact completely differently +cluster evolution. These analyses are all carried out on a fiducial halo +(173917492). In the simulations discussed in this section, we do not +implement yet the anisotropic thermal conduction. +3.1 Seeding of SMBHs +The specifics of SMBH seeding in simulations is an important aspect +of controlling the effect of AGN feedback in simulations. Different +models for black hole (BH) seeding are used in cosmological simu- +lations such as placing a BH particle in the centre of every massive +halo (Schaye et al. 2015; Weinberger et al. 2017; McCarthy et al. +2017) or models that use thresholds of local gas properties such as +4 where 𝛽e = 8𝜋𝑛e𝑇e/𝐵2 is the electron plasma beta, the ratio of electron +thermal to magnetic field pressure. +MNRAS 000, 1–30 (2022) + +6 +A. Pellissier et al. +metallicity, density, temperature and velocity (Dubois et al. 2014; +Tremmel et al. 2017; Habouzit et al. 2017; Dubois et al. 2021). +In this work, we generally adopt the same procedure as in +Rhapsody-G for black hole seeding, albeit with modified param- +eters. Following Biernacki et al. (2017), we use the ‘minimal’ Jeans +mass corresponding to the highest refinement level of our simula- +tion to define the initial SMBH sink particle mass 𝑀seed = 108M⊙ +which also correspond to our dark matter particle mass. Sink parti- +cle formation sites are identified on-the-fly using the Phew clump +finder algorithm which identifies density peaks with a given contrast +relative to the next saddle-point (see Bleuler & Teyssier 2014, for a +detailed description) which is directly implemented in the Ramses +code. The Phew parameters adopted for the original Rhapsody-G +simulations favoured the seeding of a sink particle in fewer but larger +patches of gas. Due to the stochastic nature of star formation and +supernova feedback that impact the local gas properties (hence the +SMBH seeding), we observed a large variability in the efficiency of +AGN feedback in this case. For the new suite of simulations dis- +cussed in this paper, we followed a more systematic investigation +into the impact of seeding on the proto-cluster region. In particular +we studied the following scenarios where we varied peak density and +saddle thresholds but kept all other parameters fixed: +• 𝜌peak = 0.5 ¯𝜌, 𝜌saddle = 2 ¯𝜌: Phew parameters as the original +Rhapsody-G setup, with ¯𝜌 = Ω𝑚𝜌𝑐 the mean matter density. +• 𝜌peak = 8 ¯𝜌, 𝜌saddle = 20 ¯𝜌. With a higher 𝜌peak value, only +the highest density regions in the simulation are probed. In that case, +smaller gas patches are selected but spatially more frequent. Thus, +it allows to seed more SMBHs in the simulation compared to the +original Rhapsody-G configuration. +• 𝜌peak = 8 ¯𝜌, 𝜌saddle = 200 ¯𝜌. We increase the saddle density +threshold by a factor of 10. As the result, a much lesser number of +peaks are merged which results in an increased number of SMBH +seeds. +• 𝜌peak = 8 ¯𝜌, 𝜌saddle = 15 ¯𝜌, with a lower saddle threshold which +induce more peak merging hence a lowered number of SMBH seeds +in the simulation. +We show the impact of these choices on the enclosed total stel- +lar mass in the proto-cluster region in Fig. 1 at 𝑧 = 2. At that time, +we find 21, 102, 168, 113 SMBHs of mean masses 3.5 × 108 M⊙, +1.8×108 M⊙, 1.6×108 M⊙ and 2.1×108 M⊙ inside the virial radius +respectively in the above-mentioned simulations. It demonstrates the +tight connection of the 𝜌peak parameter with the total number of +created sinks and the mean mass. The Fig. 1 clearly indicates the +resulting effect on the star formation suppression in the proto-cluster +environment: The simulations hosting a higher number of SMBHs +(being also spatially more frequent), shows a greater amount of AGN +feedback energy injected in haloes. As a result, this more profuse +AGN heating will reduce the gas cooling in haloes which decrease +the accretion of cold gas onto the central SMBH. The resulting mass +accretion rates are seen to be inversely proportional to the number of +SMBHs in our simulations. In consequence, the total stellar mass in +the proto-cluster is consistently reduced with an increasing number +of SMBHs in the simulations. We see that the total stellar mass for +the simulation using 𝜌peak = 8 ¯𝜌, 𝜌saddle = 200 ¯𝜌 is reduced by a +factor of 5 while the number of SMBHs is increased by the same +factor approximately. The total stellar mass in the proto-cluster can +be directly controlled by the number of SMBHs seeded in the simu- +lations. +Galaxy masses at 𝑧 = 2 were found to be in agreement with abun- +dance matching results with the use of 𝜌peak = 8 ¯𝜌, 𝜌saddle = 20 ¯𝜌 +101 +102 +103 +r[kpc] +1011 +1012 +M*( 0 where we see dramatically boosted mass growth. In the +case of a strong decay with 𝑓𝑑 = 1, SMBHs are essentially pinned +to the halo centers at most times. The most massive SMBH accretes +gas at high redshift and experiences, below 𝑧 = 5, frequent mergers +which leads to a very massive central SMBH by 𝑧 = 2. However, +the rapid mass growth of this central SMBH at high redshifts is also +responsible for a very efficient and early AGN feedback (which peaks +at 𝑧 ∼ 4), thus effectively strangulating the growth of the other black +holes. +To reach middle ground between the artificial pinning and the +large swinging of black holes, we found 𝑓𝑑 = 0.1 to yield reasonable +results, but more detailed investigations to tune this parameter might +be helpful in the future. In fact, observations of AGNs in dwarf +galaxies show that BHs are not located at the centers of their host +galaxies with an offset between tens of parsecs and a few kiloparsecs +(e.g. Shen et al. 2019; Reines et al. 2020; Mezcua & Domínguez +Sánchez 2020). A detection of an isolated stellar-mass black hole +MNRAS 000, 1–30 (2022) + +8 +A. Pellissier et al. +100 +fgas(< r) / (Ωb/Ωm) +z = 2 +101 +102 +103 +r [kpc] +1011 +1012 +M∗(< r) [M⊙] +fd = 0.0 +fd = 1.0 +fd = 0.1 +Figure 3. Gas depletion (top) and stellar mass (bottom) radial profile for the +simulations with no (black, 𝑓𝑑 = 0), strong (grey, 𝑓𝑑 = 1) or mild (blue, +𝑓𝑑 = 0.1) tidal descent measured at 𝑧 = 2. We show the universal baryon +fraction, Ωb/Ωm, with the horizontal grey line. We notice that simulations +including the SMBH orbital decay (grey and blue) show lower amount of gas +in the ICM (∼40 per cent). The greater is the SMBH decay, the stronger is the +stellar mass reduction inside the proto-cluster (40 per cent for 𝑓𝑑 = 0.1 and +60 per cent for 𝑓𝑑 = 1.0). +located ∼1.6 kpc away from the galactic center of the Milky Way +has been recently reported by Sahu et al. (2022). Recent simulations +(Pfister et al. 2019; Bellovary et al. 2021, 2019; Boldrini et al. 2020; +Ma et al. 2021) show that BHs in dwarf galaxies are expected to be +wandering around the central regions after the occurrence of mergers +or due to tidal stripping or dynamical friction heating. We observe +in the 𝑓𝑑 = 0.1 case a steadier mass growth of SMBHs which is +mainly driven by accretion of gas down to 𝑧 ∼ 3. As a result the +ICM is heated more gradually by AGN activity, cold gas clumps +can form and be later accreted onto SMBHs. As a consequence, we +observe at 𝑧 ≲ 3 a boosted gas accretion in the less massive black +holes, compared to the simulation with 𝑓𝑑 = 1, by almost an order +of magnitude. +Impact on gas and stars. Finally, in Fig. 3, we show the gas deple- +tion profile as well as the cumulative stellar mass in the proto cluster +region at 𝑧 = 2. At this time, the proto-cluster has a virial radius +of ∼500 kpc. Clearly, in the 𝑓𝑑 = 0 case, the AGN has not heated +the proto-ICM leading to a very high gas fraction at all radii. With +enforced orbital decay, the AGNs become active and we observe as +a consequence a stark reduction of the gas fraction. Thanks to the +tidal descent of SMBHs, AGN feedback is able to deplete the gas +from the central region and efficiently offset radiative losses in the +forming proto-cluster. In the lower panel of Fig. 3, we can see the +resulting reduction of the stellar mass formed. We observe, inside +the virial radius, a reduction by a factor of 40 and 60 per cent for +the simulations with 𝑓𝑑 = 0.1 and 𝑓𝑑 = 1 respectively. This result +is largely consistent with the recent work of Bahé et al. (2021) who +find that the magnitude of the AGN feedback suppression depends +on the ‘drift speed’ towards the center, where a slower SMBH drift +speed toward the halo center in their case also leads to systematically +higher stellar masses. +This new sub-grid model is a first step towards a more physical +solution to the ‘sinking problem’ of SMBH in numerical simulations. +We studied here, the dramatic effect it can have on the SMBH mass +growth, hence AGN activity, which induces amplified gas depletion +and a greater star formation quenching. The dimensionless 𝑓𝑑 pa- +rameter was tuned to reproduce the observed values of stellar masses. +We note here, that in our Rhapsody-C simulation benefiting from +this new sub-grid model, the Booth & Schaye (2009) boost (used +in the previous Rhapsody-G simulations) was dropped. Indeed, the +found SMBH accretion rates are already high enough once the sink +particles are more stably confined to the gas rich centre of haloes. +3.3 Delivering AGN feedback +AGN feedback is believed to proceed in two distinct modes (Best +& Heckman 2012). The quasar mode (or thermal) feedback occurs +when the gas accretion is comparable to the Eddington limit. A large +amount of radiation is emitted from the accretion region which is +able to photoionise and heat the gas in the BH vicinity. In con- +trast, the radio mode (or kinetic) feedback, preferentially triggered +during low-accretion-rate episodes, drives powerful well-collimated +radio-emmiting jets coinciding with cavities in the X-ray emission +(McNamara & Nulsen 2007; Fabian 2012). In some cases both mech- +anisms can be found in the same object (i.e., radio-loud quasar, see +e.g. Bañados et al. 2021). +3.3.1 Implementation +Thermal feedback is usually implemented in astrophysical codes +through the injection of energy or momentum in the surrounding +gas (e.g. Schaye et al. 2015; McCarthy et al. 2017; Tremmel et al. +2017). Radio-mode feedback is often implemented as a second sub- +resolution feedback channel once the accretion rate falls below a +threshold value (e.g. Dubois et al. 2014; Weinberger et al. 2017; +Henden et al. 2018). In this work, we focus purely on thermal feed- +back and will come back to the impact of kinetic AGN feedback in +future work. +Once an AGN event is triggered, in the thermal feedback model, +the released energy is assumed to thermalise within the ‘sink sphere’ +(defined as a sphere of radius 4 high-resolution elements i.e. 4Δ𝑥 +around the SMBH particle) thus effectively leading to an increase in +thermal energy in those cells. Even though these are operations at +the resolution level, multiple ways to distribute this energy among +those few cells are possible, with important consequences. We will +focus on two distinct weighting schemes here. +Mass-weighted (MW) injection. Here the total AGN energy 𝐸AGN +is injected at every fine time step proportionally to the gas mass in a +cell 𝑖 inside the sink sphere as +𝐸AGN,𝑖 = 𝐸AGN +𝜌𝑖Δ𝑥3 +𝑖 +� +𝑖 𝜌𝑖Δ𝑥3 +𝑖 +, +(19) +MNRAS 000, 1–30 (2022) + +The Rhapsody-C simulations +9 +In this case, energy is preferentially injected in denser regions (with +shorter cooling times). The MW scheme predominantly heats the +accretion region fuelling the central SMBH growth and thus reduces +future accretion. +Volume-weighted (VW) injection. Here the AGN energy is injected +at every fine time step proportionally to the volume of the cell 𝑖 inside +the sink sphere as +𝐸AGN,𝑖 = 𝐸AGN +Δ𝑥3 +𝑖 +� +𝑖 Δ𝑥3 +𝑖 +, +(20) +Compared to the MW scheme, here relatively more energy is given +to lower density cells, which for a given energy, leads to a higher +cell temperature. As a result stronger outflows through lower density +regions can be driven, and the immediate accretion gas supply is less +affected. +3.3.2 Validation in simulations +In Fig. 4 we show the effect of the mass-weighted (MW) compared +to the volume-weighted (VW) energy injection on both the ther- +modynamics of the intra-cluster gas and the stellar content of the +cluster over cosmic time. In both simulations, the energy accumu- +lation threshold Δ𝑇 is kept constant (cf. Table 1). We find that the +stellar content in the cluster has been reduced by a factor of ∼ 6−7 at +𝑧 = 0 (𝑅vir ∼ 2𝑅500) for the simulation using the VW AGN feedback +model compared to the MW model (top panel of Fig. 4). +Regarding the ICM, the volume-weighted entropy profiles of the +MW or VW simulations differ at all redshifts and become similar +only at 𝑧 = 0, except in the core (∼ 0.1𝑅500). Outside the core (i.e. +𝑟 > 0.1𝑅500), the entropy profiles of the MW simulation do not +significantly change out to the virial radius (i.e. 1.6𝑅500 and 1.9𝑅500 +at 𝑧 = 3 and 𝑧 = 0 respectively). At the same time, the VW simulation +shows a higher ICM entropy at high redshifts, which drops by 𝑧 = 0.5 +to values similar to the MW simulation. Similarly, the gas fraction +drops below the universal Ω𝑏/Ω𝑚 value for the VW simulation at +𝑧 > 0.5 whereas the MW simulation shows a higher amount of +gas. The higher entropy and lower gas fraction observed for the VW +simulation shows the efficiency of VW AGN deposition at heating +the ICM consistent with a strong star formation quenching. +From 𝑧 = 0.5, we observe an earlier flattening of the entropy profile +in the core of the VW simulation compared to the MW simulation, +but both settle with a similar core entropy of ∼ 100 keV cm2 at +𝑧 = 0. In other words, the VW AGN feedback model can prevent +gas cooling in the core from 𝑧 = 1 in contrast to the MW model. +The reason behind is that the MW model deposits the AGN feedback +energy preferentially in the dense accretion region and therefore has +difficulty to escape from the sink accretion sphere. Meanwhile, the +gas continues to cool outside the accretion region and star formation +proceeds at high rates. Thanks to the large reservoir of cold gas +surrounding the central SMBH, the AGN activity remains energetic +down to 𝑧 = 0 and can therefore gradually heat the cluster core until +it reaches a core entropy comparable to the VW simulation. +Despite the relative similarity of the entropy profiles at 𝑧 = 0, +the evolution of the AGN activity differs significantly. In the MW +simulation, the inefficient AGN feedback at early times fails to +regulate the star formation in the proto-cluster which leads to +over-massive cluster galaxies, whereas in the VW simulation, we see +a strong quenching of the star formation at earlier times as a result +of the VW deposition injecting more energy in the more diffuse gas. +These differences can be seen in the gaseous and stellar maps at +109 +1010 +1011 +1012 +1013 +M∗(< r) [M⊙] +MW +VW +10−1 +100 +K(r)/K500 +10−2 +10−1 +100 +r/R500 +10−1 +100 +fgas(< r) / (Ωb/Ωm) +z = 3.00 +z = 2.00 +z = 1.00 +z = 0.50 +z = 0.25 +z = 0.00 +Figure 4. Total enclosed stellar mass (top), ICM entropy (middle) and en- +closed gas fraction (bottom) radial profiles. Radii have been normalised to +𝑅500 corresponding to the radius enclosing 500 times the critical density at +the indicated redshift (we have 𝑅vir ∼ 1.7 − 2𝑅500). Solid and dotted lines +show the radial profiles of the MW and VW simulations respectively and line +color indicates the redshift at which the radial profile is computed. The en- +tropy profiles has been rescaled by the self-similar value 𝐾500 of Nagai et al. +(2007) to compare profiles at different redshifts more easily. The universal +baryon fraction is shown in the bottom pannel by the horizontal grey line. We +see the greater effect of the AGN with a volume weighted energy deposition +in quenching star formation at all radii. The entropy profiles indicate that the +VW AGN is more efficient in heating the ICM up to large radii at 𝑧 > 1. It +also allows an earlier transition to a NCC cluster by 𝑧 = 0.5, while the MW +simulation still shows in the core low entropy and high 𝑓gas values. +MNRAS 000, 1–30 (2022) + +10 +A. Pellissier et al. +𝑧 = 0 of the VW (left) and MW (center) simulations shown in Fig. 5. +Thanks to the heating at large radii enabled by the VW deposition +model, the pile up of cold gas observed in the MW simulations +has been prevented. Consequently, the stellar content in the cluster +has been greatly reduced in the VW simulation as well as a lower +number of cluster galaxies formed. +In this study, we do not vary the size of the AGN energy injection, +However using a VW deposition scheme, Dubois et al. (2012) showed +that the size of the AGN feedback deposition significantly impacts +the evolution of the SFR, galaxy and SMBH masses. Indeed, a larger +AGN injection region extends to less dense regions, hence far away +from the galaxies, which are more easily affected by AGN feedback +- which is less the case with a MW deposition. +3.3.3 On the energy accumulation threshold +Le Brun et al. (2014) showed that the entire gas profile can be varied +by tuning the energy accumulation threshold of the feedback model +of Booth & Schaye (2009). In contrast, Hahn et al. (2017) found that +none of the AGN models tested on a CC cluster of the Rhapsody-G +sample had a significant effect outside the core. +Similarly, we would like to test the robustness of the ICM +properties to changes in the AGN energy injection threshold, +Δ𝑇, over two orders of magnitude. We emphasise that this pa- +rameter does not control the total energy injected in an AGN +blast, but only its proportions: a higher value of Δ𝑇 results in a +less frequent but more energetic AGN blast and reciprocally, a +lower Δ𝑇 value induces more frequent and less energetic AGN blasts. +We simulate the same halo with the thermal AGN model using a +MW energy deposition and change the energy accumulation thresh- +old Δ𝑇. The simulations presented in this work all use Δ𝑇 = 107 K, +for this section we ran two additional simulations with Δ𝑇 = 106 K +(MW6) and 108 K (MW8) that we compare with the fiducial MW +simulation presented in the previous Section 3.3.2. We show in Fig. 6, +the evolution of the gas fraction as a function of the cluster mass mea- +sured in the 𝑅500 aperture. +We see that that the simulations with higher (MW8) or lower +(MW6) Δ𝑇 value shows a systematically higher gas fraction at all +cluster masses compared to the fiducial MW simulation. Hence, we +observe a non-linear dependence of the gas fraction on the energy +accumulation threshold. This is at odds with the findings of Le Brun +et al. (2014) who report a decreasing 𝑓gas with increasing Δ𝑇 values. +With a higher energy accumulation threshold, the AGN feedback in +the MW8 simulation show the highest amount of gas. Due to energetic +AGN blasts, gas is efficiently depleted from the core region. However, +as the AGN blasts are less frequent, the ICM cools efficiently and gas +condenses towards the cluster core between consecutive AGN blasts. +As a result, a cool core forms which can no longer be impacted by +the AGN activity. With a lower energy accumulation threshold, AGN +feedback events are not energetic enough (albeit more frequent) to +counterbalance the gas cooling outside the core in the MW6 simu- +lation. Consequently, it results in a greater amount of gas within the +𝑅500 region, compared to the MW simulation. +We see that a higher/lower Δ𝑇 does not systematically lead +to smaller/larger gas fractions in such a simulation. Moreover, +it does not significantly affect the overall amount of gas in the +cluster. Changes in the energy accumulation threshold only induce +a variation of 10 per cent at most of the cluster gas content. This is +considerably less compared to the overall 30 per cent gas fraction +reduction observed by Le Brun et al. (2014) when increasing by 0.5 +dex the Δ𝑇 value. This study is consistent with the findings of Hahn +et al. (2017) and shows that the Δ𝑇 parameter indeed has little impact +of the gas content of GCs in our simulations. However, compared +to Hahn et al. (2017), the differences that we observe originate +from the inclusion of our SMBH orbital decay model (presented in +Section 3.2) which keeps SMBHs close to their host halo centre, +resulting in a greater effect on the gas outside the cluster’s core. +3.3.4 Summary on the SMBH modelling +In this section, we discussed the response of the stellar and gaseous +cluster components to small changes of the SMBH and AGN feed- +back modelling. The seeding of SMBHs efficiently regulates the +star formation in the ICM: less massive SMBH seeds trigger more +spatially frequent SMBH formation hence more efficient AGN gas +heating and SF quenching. +Enabled by our new model using the tidal field information, the or- +bital decay of SMBH towards the potential minimum can be robustly +controlled. SMBH stable orbits enable a greater gas accretion. The +resulting enhancement of the AGN activity quenches the SF in the +ICM and deplete gas to large cluster radii. +Regarding AGN feedback, the energy injection scheme appreciably +impacts the ICM gas heating and controls its thermal evolution. When +the AGN feedback energy is delivered proportionally to the local gas +density, it remains confined to the core region but gradually pro- +gresses late to more intermediate radii. With a homogeneous AGN +energy injection, more energy is deposited in diffuse gas region and +thus can escape the cluster core and heat a large radii without pre- +venting the accretion of cold gas fuelling the central SMBH activity. +In that case, the ICM is almost entirely impacted by an early strong +heating which prevents the build up of cold gas and SF later on. +On the other hand, the amount of gas is relatively robust to changes +in the energy accumulation threshold (i.e. the AGN duty cycle and +energetics) of the purely thermal AGN model. +4 ANISOTROPIC THERMAL CONDUCTION +In the simulations of Hahn et al. (2017), the gaseous content of +the Rhapsody-G haloes was found to suffer from the over-cooling +problem with a too gas-rich ICM. None of their AGN feedback +models were able to impact the gas outside the cluster core to +bring it towards more realistic values. This suggested that AGN +feedback was likely not the sole solution to regulating galaxy cluster +thermodynamics. Thanks to improvements in our models, we have +seen in the previous section that AGN feedback is now able to +impact the gas on large scales but its impact remains extremely +dependent to the choice of parameters. +Anisotropic thermal conduction, in conjunction with AGN heating +and radiative cooling, likely plays an important role in setting +the gas properties of clusters (Kannan et al. 2017; Barnes et al. +2019). However, in the presence of thermal conduction, the heat +buoyancy instability (HBI) (Quataert 2008; Parrish et al. 2009) can +reorient the magnetic field lines to tangential configurations leading +to the suppression of conductive heat fluxes (Parrish et al. 2009; +Bogdanović et al. 2009). Simulations of Ruszkowski et al. (2011) +showed that turbulence can counteract the HBI and re-randomise the +magnetic field. The recent work of Beckmann et al. (2022) showed +in idealised massive galaxy cluster simulations that spin-driven +AGN feedback cannot counteract alone the HBI in the cluster center +and suggested that volume-filling turbulence is needed to restore +MNRAS 000, 1–30 (2022) + +The Rhapsody-C simulations +11 +VW +1 Mpc +MW +MC +29 +28 +27 +26 +25 +24 +23 +log10( +gas/gcm 3 ) +VW +1 Mpc +MW +MC +Figure 5. We show the maximal intensity maps of the gas (top panels) and stellar (bottom panels) density in a box of 5.9 Mpc (top) and 1.9 Mpc (bottom) on +the side at 𝑧 = 0 respectively. We have from left to right, the VW, MW and MC simulations. We can see the effect of the VW AGN feedback model at removing +the cold dense gas clumps compared to the MW model. We can also notice a similar effect, albeit lower, of the anisotropic thermal conduction in the MC +simulation at preventing the pile up of gas in dense clumps. As a result, the stellar content in the VW and MC simulations is greatly reduced compared to the +MW simulation. +significant thermal conduction. However, in isolated CC cluster +simulations, Yang & Reynolds (2016) found that magnetic tension +can suppress a significant portion of the HBI-unstable modes which +completely inhibit or significantly impair the HBI for realistic field +strengths on scales smaller than ∼ 50 − 70 kpc. Therefore, if thermal +conduction is not suppressed by the HBI, it can transport heat to +redistribute it in the ICM. It could help alleviating the SMBH/AGN +model parameter dependency found in the previous sections and +help to reach ICM regulation. +The MW simulation shows greater temperature gradients com- +pared to the VW, hence we expect that thermal conduction has a +larger effect in that case. We investigate to what extent anisotropic +thermal conduction (ATC) is able to offset radiative losses in the ICM. +In idealised adiabatic simulations, we have observed that anisotropic +thermal conduction can act as an efficient cooling or heating source +depending on the sign of the ICM temperature gradient (see Ap- +pendix A) where heat is transported from or to the cluster outskirts +in order to flatten out temperature inhomogeneities. In the previous +section, we have seen that a mass-weighted AGN feedback model +deposits its energy predominantly in the gas-rich regions. It causes +the AGN feedback energy to stay confined close to the central SMBH +with difficulty to escape the accretion region. We would like to deter- +mine whether ATC can transport the centrally injected AGN energy +on large distances to regulate the high cooling losses and star for- +mation rates. Ruszkowski et al. (2011) found that ATC was able to +noticeably reduce the effective radiative cooling driven gas accretion +in idealised cool core cluster simulations. In that context, we explore +the effect of ATC on the MW simulation with the shortest cooling +times in the core. We label the simulation with ATC and a mass- +weighted AGN deposition model as MC. The effective conductivity +is given in equation 12 for which we recall the canonical Spitzer +(1962) value, +𝜅Sp = 𝑛ekB𝐷𝑐, +(21) +where 𝐷𝑐 is the thermal diffusivity +𝐷𝑐 = 8 × 1031 +� 𝑘B𝑇e +10 keV +�5/2 � +𝑛e +5 × 10−3 cm−3 +�−1 +cm2 s−1. +(22) +We show in the top panel of Fig. 7 the effect of ATC on star +formation and the distribution of gas within the ICM. We also show +in the right panels of Fig. 5, the impact of ATC on the distribution +of stars and gas compared to the simulation without ATC (middle +panels) at 𝑧 = 0. With ATC, we observe the lower amount of gas +clumpiness in a more diffuse ICM which hosts less massive galaxies. +As seen in the profiles, the amount of stars in the MC simulation +has been reduced by ∼40 per cent already by 𝑧 = 3, before the peak +of the AGN activity. By smoothing out temperature gradients in the +ICM, ATC prevents the formation of cold gas clumps where stars +should form. Therefore, the lower amount of stars in the cluster is +MNRAS 000, 1–30 (2022) + +12 +A. Pellissier et al. +1014 +1015 +Mtot,500 [M⊙] +0.10 +0.11 +0.12 +0.13 +0.14 +0.15 +0.16 +0.17 +fgas,500 +MW6 +MW +MW8 +Eckert+19 - X-COP +Figure 6. Evolution of the cluster gas fraction as a function of its total +mass measured within 𝑅500 for the simulation using a AGN energy injection +threshold of Δ𝑇 = 106 (MW6, orange), 107 (MW, dark red) and 108 K (MW8, +black). The circles show the total gas fraction while the lighter-colored crosses +show only the X-ray emitting gas fraction (i.e. the gas with 𝑇 > 0.5 keV). We +compare our data to the hydrostatic gas fractions and total masses corrected +for the non-thermal pressure of the X-COP sample (Eckert et al. 2019) and +show the universal baryon fraction Ωb/Ωm = 0.1586 by the gray horizontal +line. Variations of the Δ𝑇 parameter by an order of magnitude, compared +to the MW simulation, increase by 10 per cent at most the gas fraction in +the 𝑀500 > 4 × 1914M⊙ range. The trend in gas depletion is observed to be +non-linear with respect to Δ𝑇 . +not due to an enhanced AGN activity but due to a suppression of +cold gas clump formation hence star formation. The entropy profiles +at 𝑧 = 3 shown in the middle panel of Fig. 7, reveal a higher core +entropy in the MC simulation as ATC transports heat to the central +region from the reservoir of hot gas at larger radii. In this case, +thermal conduction contributes in fact to the suppression of cold gas +accretion onto the central SMBH. In consequence, the AGN activity +declines and leads to a build-up of cold gas at later times, as we can +see from the gas fraction profiles at 𝑧 = 0, where the MC simulation +shows a higher amount of gas at all radii with a core contraction. +Kannan et al. (2017) found that ATC isotropises the injected AGN +energy and enhances its coupling with the ICM. They found that +the SFR is hence reduced by an order of magnitude while the overall +amount of AGN feedback energy deposited in the ICM is lower. They +also show that the earlier quenching comes with an earlier transition +to a NCC cluster. From the gas depletion profiles at 𝑧 = 0.25 shown +in the bottom panel of Fig. 7, we also witness an earlier transition to +a NCC cluster where the MC simulation shows a lower amount of +gas (and a higher entropy) in the core compared to the MW simula- +tion which does not include conduction. Additionally, we also found +lower SFRs in the galaxies within the halo by roughly a factor of +two which is however lower than the order of magnitude reduction +found by Kannan et al. (2017). In spite of these similarities, we do +not observe the reported strong quenching induced by a greater AGN +heating efficiency. We find that the simulation with conduction shows +a lower amount of injected AGN feedback energy by almost a factor +of 2. In our simulations, conduction reduces the AGN activity by +109 +1010 +1011 +1012 +1013 +M∗(< r) [M⊙] +MW +MC +10−1 +100 +K(r)/K500 +10−2 +10−1 +100 +r/R500 +10−1 +100 +fgas(< r) / (Ωb/Ωm) +z = 3.00 +z = 2.00 +z = 1.00 +z = 0.50 +z = 0.25 +z = 0.00 +Figure 7. Similarly to Fig. 4, we show the radial profiles for the simulations +with and without anisotropic thermal conduction, labelled MC and MW re- +spectively. Anisotropic thermal conduction allows to reduce the stellar content +in the intra-cluster medium by a factor of ∼2 already by 𝑧 = 3 due to the +transport of heat in the ICM that prevents the formation of cold gas clumps. +However, it also leads to the reduction of the AGN activity which causes +a build-up of cold gas at later times, as we can see from the gas depletion +profiles at 𝑧 = 0. +lowering the amount of cold gas available in the ICM which should +fuel the SMBH accretion. To summarize, heat transport smoothes +temperature inhomogeneities early-on which decreases star forma- +tion in the ICM. As a result, less cold gas clumps are available in the +ICM for SMBH accretion which results in weakening of the AGN +activity. Consequently, this decline in AGN feedback heating leads +to the contraction of the ICM at low redshifts. +MNRAS 000, 1–30 (2022) + +The Rhapsody-C simulations +13 +5 GALAXY PROPERTIES AND ICM PROFILES +5.1 The stellar mass-to-halo mass relation +We next study the properties of the galaxies in and around our +simulated clusters. Since our simulations have a high-resolution +region of twice the virial radius around each cluster at 𝑧 = 0, we +are probing a wide range of haloes mass which enable a statistical +comparison. A rightful test for the realism of our simulations is +to compare the stellar mass of our galaxies (both centrals and +satellites) as a function of their halo mass with results obtained +using the abundance matching technique (Shankar et al. 2017), the +Universe Machine with the semi-empirical model of Behroozi +et al. (2019), with cluster X-ray mass measurements (Kravtsov et al. +2018) and with the IllustrisTNG100 simulation (Nelson et al. +2018; Marinacci et al. 2018; Springel et al. 2018; Pillepich et al. +2018; Naiman et al. 2018). +In Fig. 8 we plot, for all haloes found in the nine Rhapsody-C +clusters at 𝑧 = 0, the total stellar mass 𝑀∗,200 within 𝑅200, the +radius enclosing 200 times the critical density of the Universe, as a +function of the total halo mass 𝑀200. The halo were identified using +the heavily modified version of Rockstar-Galaxies, a special +version of the original version of Rockstar (Behroozi et al. 2013), +designed to work with AMR tree and to compute a wide range of +observables during the sub-halo finding (for more detail see Hahn +et al. 2017) . +We show the different effect of the energy deposition scheme +of our thermal AGN feedback model in the left (MW) and middle +(VW) panels as well as the addition of thermal conduction (MC, right +panel). We observe a strong reduction of the star formation in the VW +simulations by almost an order of magnitude (see Section 3.3.2). As +the stellar masses are systematically lower for the VW simulations +compared to the IllustrisTNG100 simulation or the relations of +Shankar et al. (2017), Kravtsov et al. (2018) and Behroozi et al. +(2019), the stellar masses of the MW simulation are larger at 𝑀200 > +1013M⊙. On the other hand, the stellar masses of the MC simulations +(left panel), which use anisotropic thermal conduction in contrast to +the MW simulation, show stellar masses in good agreement with the +various studies. We see the effect of thermal conduction at regulating +the star formation in haloes (cf. Section 4). However, despite efficient +gas deletion (cf. right panel of Fig. 10), the VW simulations cannot +reproduce realistic galaxy properties since masses are systematically +too low. +5.2 Structure of the ICM +We perform a comparison of electron number density, pressure, +temperature and entropy in Fig. 9 with numerical simulations, SZ +and X-ray observations. We consider stacked profiles over cosmic +time of all our clusters for each type of simulations (NR, MW, +VW and MC, see details in Table 1) separately in order to quantify +the mean profiles. We selected only snapshots in the 0 ⩽ 𝑧 ⩽ 0.5 +redshift range to be consistent with the studies to which we compare +our data. +In the top left panel of Fig. 9, we compare the electron number den- +sity profiles with the low-𝑧 sub-sample of McDonald et al. (2017) +of (X-ray-selected) galaxy clusters at 𝑧 = 0 − 0.1. Additionally, we +compare our simulations to the publicly available data from the AC- +CEPT Chandra archival project described in Cavagnolo et al. (2009). +From the 242 ACCEPT galaxy cluster profiles and using the right +ascensions, declinations and redshifts, we match 141 objects to the +MCXC sample (Piffaretti et al. 2011) which gives access to X-ray ra- +dius and mass estimates. Cavagnolo et al. (2009) found a bi-modality +in the central entropy excess (𝐾0) distribution with two distinct pop- +ulation separated at 𝐾0 ∼ 30 − 50 keVcm2. We therefore classify +the ACCEPTxMCXC clusters as cool-core (CC) and non-cool-core +(NCC) if the core entropy excess 𝐾0 is respectively below or above +50 keVcm2. +The simulations using the MW AGN model (MW and MC) show a +denser core than the CC population of the ACCEPTxMCXC sample. +In contrast, the VW simulations agree almost perfectly with the +NCC ACCEPTxMCXC population within the 1𝜎 scatter shown by +the ribbon. The VW simulations approach the mean radial profile +of McDonald et al. (2017), but have a flatter core electronic density. +The non-radiative simulation shows the flattest core mean density +profile and is consistent with the NCC ACCEPTxMCXC clusters. +On the other hand, outside the core and especially at 𝑟 > 0.7𝑅500, +our simulations consistently show a steeper decrease of the electron +density with radius. +Our VW and NR simulations are consistent with both the CC +and the NCC ACCEPTxMCXC pressure profiles within scatter as +shown in the top right panel of Fig. 9. The VW and NR simulations +are in good agreement with the mean pressure profiles of Planck +Collaboration et al. (2013), the X-COP sample (Ghirardini et al. +2019) and the MUSIC simulated clusters (Gianfagna et al. 2021) out +to large cluster radii. On the other hand, the MW and MC simulations +show a factor 4 higher core pressure, but, meet at 𝑟 ⩾ 0.2𝑅500 the +VW and MC profiles. +In the ICM temperature profiles shown in the bottom left panel, +both ACCEPTxMCXC clusters and Ghirardini et al. (2019) show +large uncertainties. The CC and NCC populations of the ACCEP- +TxMCXC sample show flat temperature profiles with high ICM tem- +peratures out to 𝑅500. Ghirardini et al. (2019) also show, to a lesser +extend, higher temperatures outside the core compared to our simula- +tions. In the core region, we can see that the VW and NR simulations +approach the NCC mean profiles of the ACCEPTxMCXC sample and +the MW and MC simulations tend towards the NCC mean profile. +In the bottom right panel, the entropy slope of all our simulations +at large radii is consistent with the simulations of Voit (2005) and +the observations of the REXCESS sample (Pratt et al. 2009), but is +found to be steeper than the work of Ghirardini et al. (2019). The +ACCEPTxMCXC entropy profiles shows a shallower slope with a +higher normalisation (consistent with the high ACCEPTxMCXC +ICM temperatures). The VW simulations agree well with the +NCC ACCEPTxMCXC population within scatter while the MW +simulations better match the CC sub-sample and the relation of +Ghirardini et al. (2019). The NR simulations are somehow in +between but the MC shows low core entropy still compatible the CC +ACCEPTxMCXC clusters. +Overall, compared to the ACCEPTxMCMC CC and NCC mean +profiles, we can see that the simulations implementing a VW AGN +feedback model tend to produce GCs with NCCs. On the other hand, +the simulations implementing a MW AGN feedback model, tend to +produce more NCC clusters which is even accentuated by the addition +of thermal conduction. Our ICM radial profiles demonstrate that our +simulations are in relatively good agreement with both observations +and state-of-the-art simulations. We note, however, that the MC and +MW mean density profiles show a slight excess of gas in the core +compared to observations. +MNRAS 000, 1–30 (2022) + +14 +A. Pellissier et al. +Figure 8. Comparison of the 𝑀∗ − 𝑀 relations for the MW (right), VW (middle) and MC (right) simulations for all haloes in the nine Rhapsody-C clusters at +𝑧 = 0. Each color represents one of the nine simulations and we show the central galaxies as well as the satellite populations in transparency. We show the total +stellar mass as a function of the total mass within 𝑅200, the radius enclosing 200 times the critical density. Similarly, we also show the total stellar mass inside +𝑅200 for the haloes of the IllustrisTNG100 simulation with the 1𝜎 scatter ribbon. Note that at group and cluster scales (i.e. 𝑀200 ⩾ 1013M⊙), the relations +of Shankar et al. (2017), Kravtsov et al. (2018) and Universe Machine (Behroozi et al. 2019) indicate for lower stellar masses as they only take into account +the stellar mass of the central galaxies, while the IllustrisTNG100 relation and our data provide the total stellar mass in haloes i.e. the central galaxies and the +intra-cluster light. +r/R500 +10 5 +10 4 +10 3 +10 2 +10 1 +ne(r)E(z) 2 [cm 3] +r/R500 +10 2 +10 1 +100 +101 +102 +P(r)/P500 +10 2 +10 1 +100 +r/R500 +10 1 +100 +T(r)/T500 +10 2 +10 1 +100 +r/R500 +10 2 +10 1 +100 +K(r)/K500 +MW +VW +MC +NR +McDonald+17 - 0.0 +z + 0.1 +ACCEPTxMCXC [CC] +ACCEPTxMCXC [NCC] +Ghirardini+19 +Planck 2014 +Gianfagna+21 +Pratt+10 +Voit+05 +Figure 9. Mean ICM radial profiles of the electronic number density (top left), dimensionless pressure (top right), temperature (bottom left) and entropy (bottom +right) of our VW, MW, MC and NR simulations for 𝑧 ⩽ 0.5 in comparison with the matched sample of ACCEPT (Cavagnolo et al. 2009) and MCXC (Piffaretti +et al. 2011) and the sample of McDonald et al. (2017). We show the best fit thermodynamic profiles of Ghirardini et al. (2019), the pressure profiles of Planck +Collaboration et al. (2013) and Gianfagna et al. (2021) as well as the outer entropy slopes of Pratt et al. (2009) and Voit (2005) +MNRAS 000, 1–30 (2022) + +14 +MW +13 +12 +200 +11 +*W) +log10 +10 +9 +8 +11 +12 +13 +14 +15 +log10VW +11 +12 +13 +14 +15 +log1n (M200 / Mo )MC +UNIVERSE MACHINE +ILLUSTRISTNG100 +Kravtsov+18 +Shankar+17 +11 +12 +13 +14 +15 +log1n (M200 / M)The Rhapsody-C simulations +15 +6 EVOLUTION ALONG CLUSTER SCALING RELATIONS +Well-calibrated scaling relations between observed (X-ray, SZ or +optical) quantities and the total mass of GCs are not only important +to understand the physical processes that give rise to these relations, +but are a crucial ingredient for cosmology (Giodini et al. 2013). +Hydrodynamical simulations can model the complex processes +of structure formation with the inclusion of baryonic physics in a +cosmological context. Having access to the true cluster mass, such +simulations can be used to explore possible biases in the mass +estimation methods and can help to obtain a definitive measure of the +true cluster mass scale to enhance cosmological parameter analysis +using cluster counts (see Pratt et al. 2019, for a review). However, +it is first important to estimate the degree to which numerical +models impact and potentially bias cluster scaling relations before +directly confronting them with observational results. We studied +independently in Section 3.3 and Section 4 the impact of the AGN +deposition schemes and the addition of ATC, respectively, on the +stellar and gaseous content of a single Rhapsody-C halo. However, +the effect of such numerical schemes on the global properties of the +whole Rhapsody-C sample still needs to be assessed. In this section, +we extend the analysis to the full Rhapsody-C sample and quantify +the changes in the cluster scaling relations with the variation of +sub-grid baryonic models. +We summarise the simulations details in Table 1, where we label +the various simulations NR, VW, MW and MC for convenience. In +short, all simulations share the same resolution and numerical strat- +egy, with the same modelling for gas cooling, star formation, stellar +feedback, black hole seeding and growth (except for the adiabatic run, +NR). The only differences are in the AGN energy injection scheme +and whether ATC is included. +6.1 Synthetic X-ray observables +We chose a different methodology from Hahn et al. (2017) to compute +the X-ray observables from the simulation. Instead of using simple +weighting schemes, we produce a synthetic X-ray spectrum from +which the temperature (and gas density) of the ICM can be estimated +as closely as possible to the observer’s methodology. For each cell +in the range 0.15 ⩽ 𝑟/𝑅500 ⩽ 1,8 we read the gas density, tempera- +ture and metallicity from the simulation and compute the emissivity +𝜖(𝑇, 𝑍) (atomic lines and continuum) using tabulated emission mod- +els from the Astrophysical Plasma Emission Code (APEC, version +3.0.9, Smith et al. 2001). In each spectral bin, we compute the photon +emission rates9 𝜙𝑖 of all the cells 𝑖, +𝜙𝑖 = 𝜖(𝑇𝑖, 𝑍𝑖) 𝑛e,𝑖 𝑛H,𝑖 Δ𝑥3 +𝑖 . +(23) +We sum the individual spectra of all gas cells in the core-excluded +region inside 𝑅500 to produce amockX-ray spectrum(moredetails on +the computation of X-ray observables are given in Appendix B). We +then fit the obtained spectrum, using MCMC via the emcee python +library (Foreman-Mackey et al. 2013), with a single temperature +APEC model generated with PyAtomDB (Foster & Heuer 2020). The +X-ray luminosity is obtained by integrating over the spectra measured +from the simulation (not from the best-fit model) in the soft X-ray +8 We exclude the core (𝑟 < 0.15𝑅500) from the analysis to avoid being biased +by the presence of any cool-core or central AGN activity. See in Appendix C +for more details. +9 We omit any redshift or column density dependence +𝑌 , 𝑋 +𝛽ss +𝛾ss +𝑌0 +𝑋0 +𝑇X, 𝑀 +2/3 +2/3 +5.0 keV +5.0 × 1014M⊙ +𝐿X, 𝑀 +1 +2 +4.0 × 1044 erg/s +5.0 × 1014M⊙ +𝐿X,bol, 𝑀 +4/3 +7/3 +1.0 × 1045 erg/s +5.0 × 1014M⊙ +𝐿X, 𝑇X +3/2 +1 +4.0 × 1044 erg/s +5.0 keV +𝐿X,bol, 𝑇X +2 +1 +1.0 × 1045 erg/s +5.0 keV +𝑀gas,X, 𝑀 +1 +0 +5.0 keV +5.0 × 1014M⊙ +𝑌SZ, 𝑀 +5/3 +2/3 +40 kpc2 +5.0 × 1014M⊙ +𝑌X, 𝑀 +5/3 +2/3 +3.0 × 1014M⊙keV +5.0 × 1014M⊙ +Table 2. Scaling relations in the form 𝑌 ∝ 𝑋 𝛽 +ss 𝐸 (𝑧)𝛾ss expected from the +self-similar theory. We note𝑌 the integrated Comptonization Y parameter and +𝑀 the total cluster mass, 𝑇X, 𝐿X and 𝐿X,bol, the X-ray temperature and the +soft band and bolometric luminosity. The last two columns list the pivot values +used for the fitting the (non-self-similar) scaling relations (equation 24). +(𝐿X,500) and bolometric band (𝐿X,bol,500) i.e. 0.5–2 keV and 0.0– +100.0 keV respectively, with no instrument spectral response. +6.2 X-ray scaling relations +We use the publicly availabe Bayesian regression scheme LIRA of +Sereno (2016b,a) for our cluster scaling relation analysis. We con- +sider a power-law function of the form 𝑌 ∝ 10𝛼𝑋𝛽𝐸(𝑧)𝛾 that de- +scribes the average scaling relation of a given cluster observable 𝑌 +with another cluster observable 𝑋. For the fitting procedure, we fo- +cus on logarithms of the cluster observables of 𝑌 and 𝑋 which are +normalised by their respective pivot values 𝑌0 and 𝑋0 : +log 10 +� 𝑌 +𝑌0 +� += 𝛼 + 𝛽 log 10 +� 𝑋 +𝑋0 +� ++ 𝛾𝐸(𝑧) ± 𝜎𝑌 |𝑋, +(24) +where 𝛼 is the normalisation, 𝛽 is the slope, 𝜎𝑌 |𝑋 is the intrinsic +scatter of 𝑌 at fixed 𝑋 and 𝐸(𝑧) = 𝐻(𝑧)/𝐻0 is the expansion func- +tion, which describes the evolution of the Hubble parameter with +redshift for a given cosmology. We fix its evolution with redshift to +the self-similar expectation, i.e. 𝛾 = 𝛾ss, and we chose pivot values +to be the average sample values which we list in Table 2. We ad- +ditionally fit the intrinsic scatter of 𝑌 at fixed 𝑋. As we are using +‘true’ observables computed directly from the simulation, we do not +assume any selection effects or prior distributions on the regression +parameters. +6.2.1 The 𝑓gas − 𝑀tot relation +We start our study of the cluster scaling relations with the ratio of the +gas mass to the total cluster mass. This quantity is a crucial ingredient +for cosmology because in combination with external information on +the baryon density parameter Ωb, it has provided some of the earliest +and most robust constraints on the cosmic matter density Ωm and +dark energy (Ettori et al. 2009; Allen et al. 2011; Mantz et al. 2022, +e.g. for recent measurements). Moreover, a constant gas fraction is +a key assumption in the self-similar model of Kaiser (1986) from +which the cluster scaling relations ensue but observations indicates +for a mass-dependence (Pratt et al. 2009; Lovisari et al. 2015; Eckert +et al. 2016). +In this section, we discuss how this quantity evolves with mass +when different baryonic physical models are considered. Interest- +ingly, we will see in the following sections that when discussing +the properties of other cluster scaling relations, their outcomes can +already be predicted by the mean of the gas fraction evolution. A +detailed characterisation of this dependence will be investigated in +MNRAS 000, 1–30 (2022) + +16 +A. Pellissier et al. +1014 +1015 +Mtot,500 [M⊙] +0.08 +0.10 +0.12 +0.14 +0.16 +0.18 +0.20 +fgas,X,500 +1014 +1015 +Mtot,500 [M⊙] +fgas,500 +1014 +1015 +Mtot,500 [M⊙] +fb,500 +MW +VW +MC +NR +Lovisari+15 +Eckert+19 - X-COP +Figure 10. Similarly to Fig. 6, we show here the fraction of the X-ray emitting gas (left), gas fraction (middle) and baryonic fraction (right) as a function of the +mass, all measured within 𝑅500. The horizontal grey line show our cosmic baryon fraction of 0.1586. We distinguish the simulation types (MW, VW, MC or +NR) by using different colors (blue, red, purple and black respectively) while individual Rhapsody-C haloes have different symbols. The colored solid lines and +shadding show the fitted relations reported in Table 3 with the 1𝜎 statistical error. The X-ray emitting gas fraction is an increasing function of the total mass +for full-physics simulations except in the non-radiative case (NR), which does not implement physical processes that deplete gas from cluster central regions. In +the middle panel, when we consider all gas and not only the hot gas, the 𝑓gas,500 values show a constant evolution with mass, i.e. time, except in the case of the +VW simulations. The early and strong AGN heating happening at low cluster masses efficiently depletes gas from the 𝑅500 region and quenches star formation. +However, in the VW simulation, due to strong cooling losses at high masses (later times), gas condenses to the cluster centre and the baryonic fraction rises to +values comparable to the other simulations (NR, MW, MC). On the other hand, the origin of the offset between the gas fraction of the simulation without (MW) +and with ATC (MC) comes from the ability of the thermal conduction to prevent the formation of stars in the ICM, leading to a gas rich ICM. +future work. +We show in the left panel of Fig. 10 the X-ray emitting gas fractions +measured inside 𝑅500, 𝑓gas,X,500, of all Rhapsody-C haloes for all +MW (blue), VW (red), MC (purple) and NR (black) simulations as a +function of their total mass enclosing 500 times the critical density. +This X-ray emitting gas fraction is the ratio of the mass of the hot gas, +i.e. with 𝑇gas > 0.5 keV, to the total mass inside 𝑀500. We compare +our data to the hydrostatic gas fractions and total masses corrected +for the non-thermal pressure of the X-COP sample (values are taken +from Table 2 of Eckert et al. 2019) and with the relation of Lovisari +et al. (2015). At the high-mass end, our simulations are in good +agreement with the results of Eckert et al. (2019) but systematically +show higher gas fraction than the result of Lovisari et al. (2015), +which use hydrostatic mass estimates. +Each of the simulation types occupies a different place in the +𝑓gas,X,500 − 𝑀500 plane. The NR simulations systematically show +the highest 𝑓gas,X,500 values, as non-gravitational processes that +could be responsible for any gas depletion are not included. On the +other hand, the radiative runs reveal systematically lower 𝑓gas,X,500 +for lower mass haloes. The VW runs show the steepest increase with +mass to reach comparable values to the NR haloes, but their values +are still lower compared to the NR 𝑓gas,X,500 values. The MW and +MC simulations also show positive slopes (although shallower than +for VW) with increasing mass to reach different (X-ray emitting) +gas fractions values at the highest halo masses, with the MW ones +being the lowest. We list the slopes and normalisations we found +in Table 3 along scaling relations derived in observational studies. +The slopes of our MW and VW simulations are in agreement with +the slopes of Sun et al. (2009); Lovisari et al. (2015); Ettori (2015); +Table 3. Fitted normalisation 𝛼 and slope 𝛽 parameters using LIRA for haloes +with 𝑧 ≲ 1.5 along their standard deviations. In the bottom part of the table, +we give the slopes found by observational studies. +𝑓gas,X,500–𝑀500 +𝛼 +𝛽 +MW +0.873 ± 0.004 +0.145 ± 0.025 +VW +0.973 ± 0.004 +0.221 ± 0.021 +MC +0.951 ± 0.009 +0.103 ± 0.039 +NR +1.030 ± 0.006 +0.015 ± 0.027 +Sun et al. (2009) +0.135 ± 0.030 +Lovisari et al. (2015) +0.16 ± 0.04 +Ettori (2015) +0.198 ± 0.025 +Eckert et al. (2016) +0.21 ± 0.11 +𝑓gas,500–𝑀500 +𝛼 +𝛽 +MW +0.930 ± 0.004 +0.015 ± 0.021 +VW +1.010 ± 0.003 +0.189 ± 0.018 +MC +0.991 ± 0.007 +−0.016 ± 0.031 +NR +1.070 ± 0.006 +0.007 ± 0.026 +𝑓b,500–𝑀500 +𝛼 +𝛽 +MW +1.100 ± 0.002 +0.007 ± 0.014 +VW +1.060 ± 0.003 +0.167 ± 0.018 +MC +1.090 ± 0.006 +−0.047 ± 0.026 +NR +1.070 ± 0.006 +0.007 ± 0.026 +Eckert et al. (2016). On the other hand, the MC simulations using +thermal conduction show shallower slope but is still consistent with +the analysis of Lovisari et al. (2015). As expected, a zero slope +is found for the NR simulation that do not include radiative processes. +In the middle panel of Fig. 10, we now plot the gas fraction 𝑓gas,500 +MNRAS 000, 1–30 (2022) + +The Rhapsody-C simulations +17 +without any cut in the gas temperature. This quantity does not reflect +what X-ray observations measure but can help to understand the +origin of the different slopes found in the 𝑓gas,X,500 − 𝑀500 relations. +We can see that the NR, MW and MC simulations now show a fairly +constant fraction of gas within 𝑅500. MW haloes have ∼10 per cent +lower values compared to their non-radiative counterparts, which is +not the case if we look at the baryonic fraction 𝑓b,500 in the right +panel of Fig. 10. This suggests that the amount of gas ‘missing’ in +the MW simulations, compared to the NR simulations, has been +converted into stars as 𝑓b = 𝑓gas + 𝑓stars. To a lesser extent, we see the +same behaviour for the MC simulations, which indicates that ATC +suppresses star formation at the expense of a denser ICM. +However, haloes in the VW simulations, which benefit from early +and strong AGN activity, show the steepest increase in both 𝑓gas,X,500 +and 𝑓gas,500 with mass (i.e. cosmic time). This demonstrates the +ability of the VW AGN feedback model to efficiently deplete the +gas from the 𝑅500 region in lower mass haloes, where the hot gas +can escape more easily in a shallower potential well. Hence, at earlier +times, the ICM temperatures rose to high values that significantly held +back both the infall of gas (due to a high central gas pressure) and the +formation of cold gas clumps, that can fuel both star formation and +SMBH gas accretion. This explains why at low masses (i.e. earlier +times), VW haloes also show low baryonic fractions. However, at +later times, while this relatively hot gas is slowly radiatively cooling, +it condenses toward the cluster centre to meet 𝑓b,500 values similar to +the other simulations (see the right panel of Fig. 10). This indicates +that VW haloes host cooling flows at late times consequent to early +and efficient AGN activity. +Our radiative simulations suggest a non constant evolution for the +X-ray emitting gas fraction 𝑓gas,X,500 which increase with mass. In +agreement with the self-similar model, we find a roughly constant +evolution of the total gas fractions with mass (i.e. cosmic time), +with the exception of the VW simulations which show efficient gas +depletion due to energetic AGN feedback events that deplete gas from +the central regions of lower mass haloes. +At our resolution, no baryons are expelled beyond 𝑅500 in the MW +and MC simulations in contrast to the VW simulations. Although the +VW model has an effect on the baryon content of our haloes, the +galaxies properties are off (as we can see in Fig. 8) while galaxies of +the MW and MC reproduce more realistic properties. +6.2.2 The 𝑇X − 𝑀tot relation +In Fig. 11, we show a comparison of the X-ray temperatures of the +Rhapsody-C clusters as a function of their mass with various scal- +ing relations from the literature, both simulations and observational +studies, plotted in their respective studied mass ranges10. As ob- +servational mass estimates may be biased with respect to the true +three-dimensional spherical masses in simulations such as ours, we +differentiate them in Fig. 11 by using dash-dotted lines for studies that +use hydrostatic mass estimates. Indeed, we can see that, at a given +temperature, hydrostatic mass based studies systematically indicate +a lower total mass. +In the top panel of Fig. 11, without making any distinction +between the simulation types, we plot the evolution of each +Rhapsody-C halo along published scaling relations as they grow +10 We only show the scaling relation at 𝑧 = 0 for the redshift-dependent +scaling relations of Giles et al. (2016); Lieu et al. (2016); Mantz et al. (2016); +Le Brun et al. (2017); Bulbul et al. (2019); Henden et al. (2019); Lovisari +et al. (2020) +Mtot,500 [M⊙] +100 +101 +E (z)−2/3 T ce +X,500 [keV] +Lieu+16 - XXL [ci] +∗ Le Brun+17 - cosmo-OWLS +∗ Cui+18 - The 300 [mw] +∗ Henden+19 - FABLE +∗ Biffi+14 - MUSIC [ci] +Reichert+11 +∗ Barnes+17a - MACSIS +∗ Barnes+17b - C-EAGLE +Bulbul+19 +Lovisari+20 +1014 +1015 +Mtot,500 [M⊙] +100 +101 +E (z)−2/3 T ce +X,500 [keV] +Figure 11. Core excluded X-ray temperature as a function of the total mass +within 𝑅500. In the top panel we show the evolution of all our Rhapsody- +C haloes along published scaling relations. We compare our data to the +studies using hydrostatic mass estimates or true/weak-lensing masses using +dash-dotted and solid lines, respectively. The scaling relations are plotted +for the same mass considered in each of these works. In the top legend, we +specify in brackets if the measurements include cluster cores (ci) and we +indicate simulation works with asterisks. In the top panel, we distinguish +each Rhapsody-C halo with a unique symbol and color, while in the bottom +panel, the color stands for the type of simulations with black, blue, red and +purple used for the NR, MW, VW and MC simulations, respectively. We plot +in the bottom panel the best fit scaling relations obtained for our haloes and +give the values of the slopes and intercepts found inside the brackets located +in the legend. +in mass. In the high mass end, i.e. for 𝑀500 ⩾ 5 × 1014M⊙, our +results are in good agreement with studies using ‘unbiased’ mass +measurements i.e. weak lensing mass estimates for observational +studies (Lieu et al. 2016) or true total masses measured from +simulations (Biffi et al. 2014; Lieu et al. 2016; Le Brun et al. 2017; +Cui et al. 2018; Henden et al. 2019). Accounting for a hydrostatic +mass bias of ∼20 per cent brings our data into agreement with the +MNRAS 000, 1–30 (2022) + +18 +A. Pellissier et al. +Table 4. Similarly to Table 3, we show the fitted normalisation (𝛼) and slope +(𝛽) for the 𝑇 ce +𝑋,500–𝑀500 scaling relation for haloes with 𝑧 ≲ 1.5. In the +second and last part of the table, we show the slopes found by the analyses +based on numerical simulations and observations respectively, for which we +compare our data to in the upper panels of Figs. 11. We highlight both in +Figs. 11 and this table, the simulation works with asterisks. +𝑇 ce +𝑋,500–𝑀500 +𝛼 +𝛽 +MW +−0.059 ± 0.003 +0.683 ± 0.016 +VW +−0.037 ± 0.003 +0.621 ± 0.014 +MC +−0.075 ± 0.005 +0.699 ± 0.021 +NR +−0.060 ± 0.004 +0.724 ± 0.018 +∗ Barnes et al. (2017a) +0.58 ± 0.01 +∗ Barnes et al. (2017b) +0.47 ± 0.07 +∗ Biffi et al. (2014) +0.56 ± 0.03 +∗ Cui et al. (2018) +0.627 ± 0.007 +∗ Henden et al. (2019) +0.64 ± 0.02 +∗ Le Brun et al. (2017) +0.577 ± 0.006 +Bulbul et al. (2019) +0.83 ± 0.10 +Lieu et al. (2016) +0.56 ± 0.12 +Lovisari et al. (2020) +0.66 ± 0.06 +Reichert et al. (2011) +0.57 ± 0.03 +observational studies using hydrostatic mass estimates (Reichert +et al. 2011; Bulbul et al. 2019; Lovisari et al. 2020) or simulations +estimating mass with a mock X-ray analysis (Barnes et al. 2017a,b). +However, a more precise characterisation of the hydrostatic mass +bias measured in our simulations will be the subject of a future paper. +Comparison with observations Compared to the literature, our +simulations indicate slightly steeper slopes than most of the stud- +ies with the exception of the observations by Bulbul et al. (2019). +While being consistent in the high mass range (𝑀500 ⩾ 5×1014M⊙), +Lieu et al. (2016) indicates for higher X-ray temperatures in the lower +mass range compared to our results, which might be induced by the +inclusion of the cluster core in the X-ray temperature measurements +(see Appendix C). +Comparison with simulations Similarly, while being in agreement +with Cui et al. (2018) and Henden et al. (2019), we report system- +atically slightly steeper slopes compared to the other simulation +works but our data agree well within scatter with these studies in the +high mass range. The shallower slope of Biffi et al. (2014) could be +induced by the inclusion of the core in the temperature estimation +(see Appendix C). However, we are aware that our spectral fits for the +temperature estimation in the low mass range can be slightly biased +low (as discussed in Appendix B), which could explain the steeper +slopes we find. As discussed more extensively in Appendix B, we +show that the mass-weighted temperature estimates are a factor of +∼2 lower than the ones resulting from our spectral fit. However, +assuming such a factor of 2 lower temperatures shifts the scaling +relation of Cui et al. (2018) to even lower temperatures. +We quantitatively compare our scaling relation slopes with the +above-mentioned studies in Table 4. We observe that the inferred +slopes and normalisations are rather insensitive to the physical mod- +els used for our simulations as the temperature reflect the depth of +the cluster’s potential well. The core-excluded X-ray temperatures are +similar for simulations with or without ATC (MC and MW respec- +tively). Therefore, thermal conduction does not play an important +role in offsetting the X-ray temperatures outside the core in our sim- +ulations. While the VW simulations indicate a shallower slope with a +higher normalisation, they converge to the same core-excluded X-ray +temperature values in the high mass range. We see again the effi- +ciency of the VW AGN heating in raising the ICM temperature to +higher values, especially in lower mass haloes where the potential +well is shallowest. Most importantly, we see that all simulations con- +verge to the same temperatures with similar scatter for masses above +5 × 1014M⊙. On average, the slopes agree with a ∼2 per cent steeper +value than the self-similar expectation and no significant effect of the +AGN models or ATC on our 𝑇X − 𝑀tot scaling relation is seen. +Surprisingly, the non-radiative simulations are able to reproduce the +same core-excised temperatures as the full-physics simulations. Be- +sides the fact that the temperature is less affected by feedback pro- +cesses as it reflects more the cluster potential well, this results also +indicates that non-gravitational processes mostly affect the core. In +the radial range 0.15 ⩽ 𝑅/𝑅500 ⩽ 1, radiative cooling, thermal con- +duction, AGN and SF feedback do not play a major role in offsetting +the ICM core-excluded X-ray temperatures. We note that only the +VW AGN feedback, being the most effective, is able to heat the gas +at these radii in the lower mass regime. +From Table 4, we see that different slopes and normalisations for +the 𝑇X − 𝑀tot scaling relation are found in the literature. These nor- +malisation differences can be attributed to the method used to infer +cluster masses, which might be biased compared to the true mass (e.g. +due to the hydrostatic bias or biased weak lensing estimates). The +observational studies of Sun et al. (2009) and Lovisari et al. (2020) +showed that the slopes remain consistent for low mass groups to mas- +sive galaxy clusters. This consistency implies that non-gravitational +processes are not affecting the 𝑇X − 𝑀 scaling relation in a different +manner in distinct mass (or temperature) ranges. Bulbul et al. (2019) +actually found the steepest slope in their observations. They explain +this apparent tension by the fact that they simultaneously fit the mass +and redshift trend of the scaling relation, in contrast to the assumed +self-similar redshift evolution in other studies. Lovisari et al. (2020) +claimed that it could also be explained if their SPT-SZ masses suffer +from a mass-dependent bias (similar to the Planck mass estimates). +We observe slightly steeper slopes compared to the simulation +works of Biffi et al. (2014); Barnes et al. (2017a,b) and Le Brun +et al. (2017) but our results agree with the studies of Cui et al. (2018) +and Henden et al. (2019). The discrepancy could originate from the +higher temperature found in lower mass haloes in those works. It can +be attributed to the method used to estimates X-ray temperatures (see +discussion in Appendix B and C) but also from the efficiency of the +feedback model to deplete and heat the gas in lower mass halo. +6.2.3 The 𝐿X − 𝑀tot relation +The X-ray luminosity mass scaling relation is important as it can +relate one of the ‘cheapest’ X-ray observables to the total cluster +mass. To fully exploit the data from large galaxy cluster samples +provided by X-ray surveys such as e-ROSITA (Liu et al. 2021), which +collects too few photons to infer any spectra or construct any mass +profiles, it is of great use to have a well calibrated 𝐿X − 𝑀tot scaling +relation and an accurate determination of its scatter. However, the +X-ray luminosity measurement depends on the energy band and +the aperture from which it is derived as well as the flux extraction +method. As a consequence, among all the X-ray scaling relations, +it is the one that shows the largest scatter. Moreover, due to its +density squared dependence, the X-ray luminosity can be easily +biased by the presence of gas-rich substructures, a cool core and +non-gravitational processes (Reichert et al. 2011) which motivates +MNRAS 000, 1–30 (2022) + +The Rhapsody-C simulations +19 +Mtot,500 [M⊙] +1043 +1044 +1045 +E (z)−2 Lce +X,500 +� +erg s−1� +Mantz+16 - WtG +∗ Barnes+17b - C-EAGLE +Bulbul+19 +1014 +1015 +Mtot,500 [M⊙] +1043 +1044 +1045 +E (z)−2 Lce +X,500 +� +erg s−1� +Mtot,500 [M⊙] +1044 +1045 +E (z)−7/3 Lce +X,bol,500 +� +erg s−1� +∗ Biffi+14 - MUSIC [ci] +∗ Henden+19 - FABLE +Reichert+11 +∗ Barnes+17a - MACSIS +Lovisari+20 +1014 +1015 +Mtot,500 [M⊙] +1044 +1045 +E (z)−7/3 Lce +X,bol,500 +� +erg s−1� +Figure 12. As Fig. 11, but for the core-excluded X-ray soft band (left) and bolometric (right) luminosities measured in the radial range 0.15 ⩽ 𝑅/𝑅500 ⩽ 1 as a +function of halo mass inside 𝑅500. We notice the large differences in slope and intercept between the published scaling relations. +the exclusion of the core from analyses. +In figure 12, we show the core excluded soft-band (0.5 − 2 keV) +and bolometric (0.01 − 100 keV) X-ray luminosities as a function of +the cluster mass for all our haloes. The X-ray luminosity is rather sen- +sitive to non-gravitational processes and to the ICM clumpiness, and +this can explain why such a diversity of slopes and normalisations +is observed. As we can see in Table 5, published studies show on +average a pronounced deviation, which is on average 50 and 30 per +cent greater than the expected self-similar scaling for soft-band and +bolometric luminosities respectively. In the bottom panels of Fig. 12, +we make a distinction between simulations that incorporate galaxy +formation physics (MW, VW, MC) and those without (NR). The X- +ray luminosity is more sensitive to the physical models used in the +simulations, hence on radiative processes, compared to the temper- +ature that do not show such large discrepancy between the different +simulations. Therefore, the calibration of scaling relations using the +Table 5. Similarly to Table 4 for the scaling of 𝐿ce +X,500 and 𝐿ce +X,bol,500 with +𝑀500. The scaling relations listed here are shown in Fig. 12. In constrast to +observations, we denote numerical works with an asterisk. +𝐿ce +X,500–𝑀500 +𝐿ce +X,bol,500–𝑀500 +𝛼 +𝛽 +𝛼 +𝛽 +MW +−0.186 ± 0.007 +1.230 ± 0.038 +−0.165 ± 0.009 +1.255 ± 0.049 +VW +−0.182 ± 0.004 +1.440 ± 0.023 +−0.203 ± 0.005 +1.497 ± 0.028 +MC +−0.090 ± 0.009 +1.150 ± 0.043 +−0.095 ± 0.014 +1.112 ± 0.062 +NR +0.048 ± 0.007 +0.920 ± 0.033 +−0.010 ± 0.007 +1.146 ± 0.033 +∗ Barnes et al. (2017b) +1.33 ± 0.13 +∗ Biffi et al. (2014) +1.45 ± 0.05 +∗ Barnes et al. (2017a) +1.88 ± 0.05 +Mantz et al. (2016) +1.65 ± 0.14 +∗ Henden et al. (2019) +1.97 ± 0.10 +Bulbul et al. (2019) +1.60 ± 0.17 +Reichert et al. (2011) +1.52 ± 0.04 +Lovisari et al. (2020) +1.82 ± 0.25 +X-ray luminosity is more complex than relations using the X-ray tem- +perature or the X-ray analogue of the Sunyaev-Zeldovich𝑌 parameter +(see later in Section 6.3). +MNRAS 000, 1–30 (2022) + +20 +A. Pellissier et al. +Comparison with observations For the soft-band luminosity, we +systematically find shallower slopes compared to the relations of +Mantz et al. (2016) and Bulbul et al. (2019). With the exception of +the NR simulations which show a 25 per cent higher value, our slopes +for the bolometric luminosity do not significantly change from the +ones derived using the soft-band luminosities. With the exception +of the VW simulations which agree within scatter with the slope +of Reichert et al. (2011), we find even greater discrepancy with +observation for the bolometric luminosity. When setting the redshift +evolution of the scaling relation to the self-similar value (which is +the choice we have made for this work), Lovisari et al. (2020) find a +slope of 1.45 ± 0.10 which agrees only the VW simulations. +If we account for a mass bias of 20 percent (which is also shown to +be a good fit for the 𝑇X −𝑀 scaling relation), we can bring our data in +agreement with the study Reichert et al. (2011) which use hydrostatic +mass estimates but widen the gap with the scaling relation of Bulbul +et al. (2019) and Lovisari et al. (2020) which show lower luminosities +(higher mass) at fixed mass (X-ray luminosity). We observe a higher +normalisation than Mantz et al. (2015). +Comparison with simulations Our radiative simulations have +slopes that are consistent with the simulations of Barnes et al. +(2017b) for the soft-band luminosity but only the VW simulations +agree with the slope of Biffi et al. (2014) for the bolometric +luminosity. Barnes et al. (2017a) and Henden et al. (2019) find +50 per cent steeper slopes compared to our radiative simulations. +The Macsis simulations (Barnes et al. 2017a) indicate a 50 per +cent steeper slope on average compared to our radiative simulations. +Some of this discrepancy can be attributed the ability of their +AGN feedback model to efficiently heat gas in lower mass haloes, +lowering their X-ray luminosity. Regarding normalisation, assuming +a hydrostatic mass bias of 20 percent increases the offset with +the C-Eagle and the Macsis simulations. This discrepancy could +originate from the difference in the energy injection of the thermal +AGN feedback model (which use different Δ𝑇 values and the number +of heated neighbour particles). The Music simulations (Biffi et al. +2014) show, at fixed total mass, lower X-ray luminosities compared +to our data. +By looking at the differences in slope and normalisation between +our simulation types, we observe that the X-ray luminosity follows +the same trend with mass as the X-ray emitting gas fraction (see +the left panel in Fig. 10) with the steepest slope being for the VW +simulations, then in descending order, MW, MC and finally NR. +This illustrates once more the relation between the X-ray luminosity +and the halo gas content, which is driven by the different physical +models (AGN and ATC) that the simulations use. The shallower +slope of the NR compared to both the self-similar scaling and the +other simulations originates from the absence of galaxy formation +physics or radiative gas cooling, which could boost the X-ray +luminosity. The higher NR normalisation can be explained by the +higher gas content in haloes as no star formation (which should turn +cold and dense gas into stars) or feedback processes (which could +deplete the gas in haloes) occur. +6.2.4 The 𝐿X − 𝑇X relation +We now look at the X-ray scaling relations in Fig. 13, which relates +the core-excluded X-ray temperatures and luminosities, and collect +their parameters (slope and intersect) in Table 6. As for the 𝐿X − 𝑀 +Table 6. Similarly to Table 5 for the scaling of 𝐿ce +X,500 and 𝐿ce +X,bol,500 with +𝑇 ce +X,500. +𝐿ce +X,500–𝑇 ce +X,500 +𝐿ce +X,bol,500–𝑇 ce +X,500 +𝛼 +𝛽 +𝛼 +𝛽 +MW +−0.096 ± 0.009 +1.569 ± 0.096 +−0.066 ± 0.011 +1.215 ± 0.125 +VW +−0.126 ± 0.006 +2.339 ± 0.076 +−0.132 ± 0.007 +2.202 ± 0.084 +MC +0.014 ± 0.014 +1.545 ± 0.112 +0.012 ± 0.020 +1.123 ± 0.155 +NR +0.132 ± 0.008 +1.173 ± 0.067 +0.099 ± 0.008 +1.383 ± 0.065 +Giles et al. (2016) +2.63 ± 0.15 +∗ Biffi et al. (2014) +2.29 ± 0.07 +Pratt et al. (2009) +2.34 ± 0.13 +∗ Barnes et al. (2017a) +3.01 ± 0.04 +∗ Henden et al. (2019) +3.02 ± 0.15 +scaling relations, we observe a large offset between recent numerical +and observational works both in normalisation and slope. +Comparison with observations The relations of Giles et al. (2016) +and Pratt et al. (2009), from observations of the XXL and REXCESS +clusters respectively, are rather shifted to higher temperatures at fixed +soft-band X-ray luminosity. We note however that Giles et al. (2016) +use core-included measurements, unlike Pratt et al. (2009), which +can significantly bias the X-ray luminosity (see Appendix C). While +the VW simulations agrees with the slopes of the Giles et al. (2016) +and Pratt et al. (2009), our haloes follows on average 40 per cent +shallower scaling relations as we can see in Table 6. +Comparison with simulations All haloes agree, within scatter, +with the values found by the Music and Fable simulations while +the Macsis simulations indicate a slightly lower normalisation +i.e. higher temperatures at fixed bolometric luminosity. When +comparing quantitatively the slopes for the 𝐿X,bol − 𝑀 scaling +relations with these simulations, we found shallower slopes by more +than a factor 2 on average. Only our VW simulations agree with the +slope of Biffi et al. (2014) which however consider core-included +bolometric luminosities. +In more detail, we systematically find significantly shallower +slopes compared to the literature as we can see in Fig. 13 and Table 6. +The VW simulations, however, show the steepest slopes compatible +with the studies of Pratt et al. (2009) and Biffi et al. (2014) for +the soft-band and bolometric luminosities respectively. The steeper +evolution of the X-ray luminosity with temperature of the VW +simulations is the result of both the more effective AGN feedback +at lower halo masses and the gas enrichment at high halo masses +(see Fig. 10), which significantly boosts the X-ray luminosities. The +effect the different radiative models explored in this work is the most +visibly seen for the 𝐿X −𝑇X relations. Although this scaling relation +might be the most straightforward to derive from observations, +its calibration remains challenging as it demonstrates the most +significant sensitivity on the physical models used in simulations. +6.2.5 The 𝑀gas,X − 𝑀tot relation +We show in Fig. 14 the evolution of the X-ray emitting gas mass (i.e. +the gas with 𝑇 > 0.5 keV) with the cluster total mass enclosed within +𝑅500. We see that the ICM mass correlates well with the total cluster +mass with a relatively small scatter. The study of observed relaxed and +disturbed clusters by Lovisari et al. (2020) showed that this relation +is quite insensitive to the dynamical state of the clusters, however +with a higher scatter for their disturbed sample. In the upper panel of +Fig. 14, we can see that our haloes show a relatively higher gas mass +for haloes with 𝑀500 < 5 × 1014M⊙. As we can see in Table 7, our +MNRAS 000, 1–30 (2022) + +The Rhapsody-C simulations +21 +T ce +X,500 [keVM] +1043 +1044 +1045 +E (z)−1 Lce +X,500 +� +erg s−1� +Giles+16 [ci] - XXL +Pratt+09 - REXCESS +100 +101 +T ce +X,500 [keVM] +1043 +1044 +1045 +E (z)−1 Lce +X,500 +� +erg s−1� +T ce +X,500 [keV] +1044 +1045 +E (z)−1 Lce +X,bol,500 +� +erg s−1� +∗ Biffi+14 [ci] - MUSIC +∗ Henden+19 - FABLE +∗ Barnes+17a - MACSIS +100 +101 +T ce +X,500 [keV] +1044 +1045 +E (z)−1 Lce +X,bol,500 +� +erg s−1� +Figure 13. A pure X-ray scaling relation – the core excluded X-ray soft band (left) and bolometric (right) luminosities as a function of the core excluded X-ray +temperature. We keep the same figure properties as in Fig. 11. +simulations indicate slopes consistent with the observations of Mantz +et al. (2016) (1.04) and the C-Eagle simulations (1.07, Barnes et al. +2017b), albeit shallower compared to the observations of Bulbul et al. +(2019) and Lovisari et al. (2020) but also the cosmo-OWLS (Le Brun +et al. 2017) Macsis (Barnes et al. 2017a) and Fable (Henden et al. +2019) simulations. +The dependence of the gas fractions on the physical models of our +simulations obviously translates in this scaling relation. Therefore, +similarly to the findings of Section 6.2.1, we see a ∼10 per cent steeper +evolution of the gas mass with the total cluster mass for the VW +simulation while the NR, MW and MC show relatively similar slopes. +6.3 Sunyaev–Zeldovich scaling relations +The Sunyaev-Zel’dovich (SZ) effect (Zeldovich & Sunyaev 1969; +Sunyaev & Zeldovich 1970) - which is the distortion of the cosmic +microwave background (CMB) spectrum by the inverse Compton +scattering of the low-energy CMB photons with free electrons in the +ICM - provides an unique view of the ICM baryons. By probing the +line-of-sight integral of the ICM thermal pressure support, it yields +an ideal proxy for the gas mass in a galaxy cluster and therefore the +total mass. +We compute 𝑌SZ,500, the integrated Comptonization parameter 𝑌 +within 𝑅500, directly from the simulation using the cell gas temper- +ature, 𝑇𝑖, and electronic density, 𝑛e,𝑖 = 𝜌gas,𝑖/(𝜇emp), as +𝑌SZ,500 = +𝜎𝑇 +mec2 +𝑟𝑖⩽𝑅500 +∑︁ +𝑖 +kB𝑇𝑖𝑛e,𝑖d𝑉𝑖, +(25) +where 𝜎𝑇 , me, c and d𝑉𝑖 are respectively the Compton cross +section, the electron mass, the speed of light and the volume of the +considered gas cell. +The 𝑌SZ,500 quantity does not show any particular scatter as the +ICM pressure profiles in clusters tend to be universal within 𝑅500 +(Arnaud et al. 2010). It is less sensitive to the gas density than X-ray +MNRAS 000, 1–30 (2022) + +22 +A. Pellissier et al. +Mtot,500 [M⊙] +1013 +1014 +Mgas,X,500 [M⊙] +∗ Henden+19 - FABLE +Mantz+16 - WtG +∗ Le Brun+17 - cosmo-OWLS +∗ Barnes+17a - MACSIS +∗ Barnes+17b - C-EAGLE +Bulbul+19 +Lovisari+20 +1014 +1015 +Mtot,500 [M⊙] +1013 +1014 +Mgas,X,500 [M⊙] +Figure 14. As Fig. 11, but for the evolution of the (X-ray emitting) gas +mass to the total cluster mass inside 𝑅500 compared to both numerical and +observational studies. +observables due to its linear dependence, and hence core exclusion +is not necessary. Therefore measure 𝑌SZ,500 in the 𝑟 ⩽ 𝑅500 range. +In Fig. 15 we show the 𝑌SZ,500 − 𝑀tot,500 scaling relations. We +see that the 𝑌SZ,500 parameter is tightly connected to the cluster +mass, where we observe the lowest scatter compared to the X-ray +scaling relations. Indeed, this parameter probes the mass-weighted +temperature, which is much less sensitive to the gas clumpiness (as +opposed to the emission measure weighted temperature of X-ray +quantities). +Our Rhapsody-C haloes agree well with all previously published +scaling relations from both numerical simulations (Cui et al. 2018; +Henden et al. 2019; Barnes et al. 2017a; Le Brun et al. 2017) and +observational works (Planck Collaboration et al. 2014; Nagarajan +et al. 2019). Due to the very low scatter in the 𝑌SZ − 𝑀 scaling +relation, its use seems well suited for studies that aim to constrain +cosmological parameters. In a more quantitative comparison, we can +see in Table 7 that Planck Collaboration et al. (2014); Nagarajan et al. +Mtot,500 [M⊙] +100 +101 +102 +103 +E (z)−2/3 YSZ,500 +� +kpc2� +Nagarajan+18 +∗ Cui+18 - The 300 +∗ Henden+19 - FABLE +∗ Le Brun+17 - cosmo-OWLS +∗ Barnes+17a - MACSIS +Planck14 Baseline +1014 +1015 +Mtot,500 [M⊙] +100 +101 +102 +103 +E (z)−2/3 YSZ,500 +� +kpc2� +Figure 15. Evolution of the integrated Compton 𝑌SZ,500 parameter of the +Sunyaev-Zel’dovich effect as a function of the total halo mass computed +within 𝑅500. We keep the same properties as of Fig. 11. We see in the upper +panel the tight evolution of the Rhapsody-C haloes along published scaling +relations irrespective of the physical models used in our simulations. In the +bottom panel, we see that the models used in our simulations induce a very +slight change in the slope of our scaling relations. +(2019); Cui et al. (2018); Nagarajan et al. (2019) have slope values in +agreement with our simulations (within errors), but the latter shows +a 17 per cent lower slope value on average. On the other hand, the +simulations of Le Brun et al. (2017) and Henden et al. (2019) indicate +slightly steeper slopes. For observational studies, this difference can +be understood by 𝑌SZ,500 being a mass-dependent observable and +therefore less constrained at lower masses, which can explain the +difference in the slopes of Nagarajan et al. (2019) and the Planck +Collaboration et al. (2014) baseline that probe different cluster mass +ranges. +Between our simulations, the slopes of the radiative simulations are +in very good agreement and the non-radiative simulations. We find +similar slopes for all simulations with a difference being at most 4 per +MNRAS 000, 1–30 (2022) + +The Rhapsody-C simulations +23 +cent between the MW and VW simulations. Although the difference +in the AGN feedback model between the MW and VW simulations +yields disparate X-ray scaling relations, the difference here weakens +for the 𝑌SZ − 𝑀 scaling relation. We can understand this slight differ- +ence as the VW simulations produce higher ICM temperatures (and +hence higher pressures) at lower halo masses due to a more efficient +AGN gas heating at early times. The slightly shallower slope observed +for simulations with conduction (MC) compared to the simulations +without (MW) is a consequence of the higher fraction of cooling gas +at high halo masses which results in a lower pressure support, hence +lower 𝑌SZ,500 values at higher halo mass (cf. Section 6.2.1). There- +fore, we see here that the physical models used in our simulations +do not play a particular role in offsetting the 𝑌SZ,500 − 𝑀tot scaling +relation. +Again, the slight normalisation and slope changes can be under- +stood in the same way as the conclusions of Section 6.2.1: the simu- +lations with ATC show a shallower evolution with mass than simu- +lations without, as the ICM is more quiescent with a suppressed star +formation, and the higher normalisation of VW simulations can be +ascribed to their higher ICM pressure support caused by their AGN +feedback model. +We also measure the X-ray analogue of 𝑌SZ,500 by taking the +product of the mass of the X-ray emitting gas (𝑀gas,X,500) and +the X-ray temperature from our spectral fit (𝑇ce +X,500). In Fig. 16 +we show our data along other 𝑌X − 𝑀 scaling relations and we +observe a increased scatter than from the SZ scaling relation, as +this 𝑌X parameter is more sensitive to the internal structure of the +ICM11. Our slope values remain rather constant and very similar +to the slopes found for the 𝑌SZ,500 − 𝑀tot,500 relation (see Table 7) +and overall, our data is consistent with the results of the studies +shown in Fig. 16, but our slopes indicate a ∼6 per cent lower value +on average. In the lower mass range, where differences between +published scaling relations become more obvious, our haloes follow +the relations of the numerical studies of Le Brun et al. (2017); +Barnes et al. (2017a) and Henden et al. (2019) while the former +indicates somewhat lower 𝑌X,500 values. On the other hand, the +C-EAGLE simulations (Barnes et al. 2017b) indicate a 10 per cent +shallower slope with higher 𝑌X,500 values, especially at low halo +masses. +To summarize, we have seen in this section that the evolution +of our haloes along cluster scaling relations are not significantly +affected by changes in the AGN feedback model or the inclusion +of ATC. Yet, in this study we focused on massive systems whereas +the effect of feedback processes might be more pronounced in the +group regime. Despite their profound impact on the cluster gaseous +and stellar components, the global properties of our haloes and their +evolution with mass are not significantly affected by the baryonic +processes in the ICM (with the exception of the X-ray luminosity, +which is relatively sensitive to the ICM thermodynamical state). +This finding is good news for cluster cosmology, which relies on +scaling relations to derive cluster total masses. Numerical simulations +are proven to be a reliable and suitable tool for such calibrations. +11 For instance, a completely smooth ICM will give equality between 𝑌X,500 +and 𝑌SZ,500, as in this case we have ⟨𝑛2⟩ = ⟨𝑛⟩2 (which shows the respective +dependence of the 𝑌X,500 and 𝑌SZ,500 on the gas number density). +Mtot,500 [M⊙] +1013 +1014 +1015 +E (z)−2/3 Y ce +X,500 [M⊙ keV] +∗ Henden+19 - FABLE +∗ Le Brun+17 - cosmo-OWLS +∗ Barnes+17a - MACSIS +∗ Barnes+17b - C-EAGLE +Bulbul+19 +Lovisari+20 +1014 +1015 +Mtot,500 [M⊙] +1013 +1014 +1015 +E (z)−2/3 Y ce +X,500 [M⊙ keV] +Figure 16. Similarly to Fig. 15, we show the X-ray analogue of the core- +excluded integrated SZ signal, i.e. 𝑌X = 𝑀gas,X × 𝑇 ce +X , as a function of the +total halo mass. Compared to 𝑌SZ − 𝑀, we observe a greater scatter, expected +from the sensitivity of X-ray observables. However, it shows the lowest scatter +compared to the other X-ray scaling relations (𝑇X − 𝑀, 𝐿X − 𝑀 or 𝐿X −𝑇X). +Our data is in fair agreement with published studies even at lower halo masses +where the scatter is the greatest. Similarly to Fig. 15, the difference in the used +physical models only induce a slight difference in the inferred slopes. +7 SUMMARY AND CONCLUSIONS +We presented the Rhapsody-C suite, a series of zoom-in magneto- +hydrodynamical simulations of massive galaxy clusters (𝑀vir ∼ +1015M⊙) with a physical resolution of 2.8 kpc. The simulations in- +clude radiative gas cooling, star formation, feedback from supernovae +(SN) and active galactic nuclei (AGN) as well as anisotropic thermal +conduction (ATC). This work was motivated by the Rhapsody-G +suite (Wu et al. 2015; Hahn et al. 2017; Martizzi et al. 2016), which +suggested shortcomings in the thermal AGN model and the need for +additional sources of energy injection. Hence, in this paper we re- +visit thoroughly the seeding of super massive black holes (SMBHs), +introduce a new model for their dynamical evolution, and consider +different AGN energy deposition schemes as well as the anisotropic +transport of heat within the intra-cluster medium (ICM). We study +MNRAS 000, 1–30 (2022) + +24 +A. Pellissier et al. +Table 7. Same as Table 4 for the X-ray emitting gas mass (𝑀gas,X,500) integrated 𝑌SZ parameter and its X-ray analogue (𝑌X). +𝑀gas,X,500–𝑀500 +𝑌SZ,500–𝑀500 +𝑌X,500–𝑀500 +𝛼 +𝛽 +𝛼 +𝛽 +𝛼 +𝛽 +MW +−0.030 ± 0.002 +1.075 ± 0.009 +−0.034 ± 0.003 +1.812 ± 0.014 +−0.022 ± 0.004 +1.757 ± 0.022 +VW +0.018 ± 0.001 +1.105 ± 0.007 +0.029 ± 0.003 +1.737 ± 0.014 +0.047 ± 0.003 +1.725 ± 0.017 +MC +0.008 ± 0.003 +1.050 ± 0.013 +−0.031 ± 0.004 +1.752 ± 0.019 +−0.000 ± 0.006 +1.748 ± 0.027 +NR +0.043 ± 0.002 +1.006 ± 0.008 +0.027 ± 0.005 +1.784 ± 0.023 +0.050 ± 0.004 +1.730 ± 0.021 +∗ Barnes et al. (2017a) +1.25 ± 0.03 +∗ Barnes et al. (2017b) +1.69 ± 0.07 +∗ Barnes et al. (2017a) +1.84 ± 0.05 +∗ Barnes et al. (2017b) +1.07 ± 0.05 +∗ Cui et al. (2018) +1.62 ± 0.31 +∗ Barnes et al. (2017b) +1.57 ± 0.07 +∗Le Brun et al. (2017) +1.32 ± 0.01 +∗ Henden et al. (2019) +1.88 ± 0.05 +∗ Henden et al. (2019) +1.88 ± 0.05 +∗ Henden et al. (2019) +1.25 ± 0.04 +∗Le Brun et al. (2017) +1.948 ± 0.018 +∗Le Brun et al. (2017) +1.948 ± 0.018 +Mantz et al. (2016) +1.04 ± 0.05 +Nagarajan et al. (2019) +1.51 ± 0.31 +Bulbul et al. (2019) +2.01 ± 0.20 +Bulbul et al. (2019) +1.26 ± 0.10 +Planck Collaboration et al. (2014) +1.79 ± 0.065 +Lovisari et al. (2020) +1.85 ± 0.10 +Lovisari et al. (2020) +1.25 ± 0.05 +the impact of each of the models on the cluster stellar component and +examine how they shape the intra-cluster gas. We next investigate the +evolution of our simulated clusters over cosmic time with a range of +cosmological observables that serve as mass proxies when the above- +mentioned sub-grid models are changed. We focus in this analysis on +the total cluster mass versus X-ray temperature, luminosity, gas mass +and the integrated Comptonization parameter. The main findings of +our analysis are as follows : +• The star formation in the proto-cluster can be efficiently +controlled by the seeding of the SMBHs in the ICM. With a low +number of SMBH seeds, the AGN heating cannot prevent the +gas from over-cooling in the proto-cluster. Seeding less massive +but more numerous SMBHs enables an efficient, more fre- +quent AGN heating in time and space. Our simulations indicate that +this might be one of the most crucial parameters to regulate feedback. +• We develop a new model for the SMBH dynamics, which +is made publicly available for the Ramses code. It consists of +decaying the SMBHs towards the local potential minimum along +the steepest gradient with a magnitude that depends on the tidal +forces experienced by the SMBH during its evolution. This new +model makes the accretion of gas onto SMBHs easily tunable by +keeping the SMBHs relatively close to the potential minimum. +Consequently, the amount of AGN feedback energy injected into the +ICM can be controlled and it is shown to have a signi���cant effect on +reducing the gas content in the proto-cluster as well as quenching +star formation. The abundance (see point above) and the locations +of the SMBHs therefore appear critical to regulate AGN feedback. +• By changing only the AGN energy injection scheme (i.e. +volume- or mass-weighted energy deposition), we observe a +dramatic change in both the distribution of gas and in star formation. +Mass-weighted deposition preferentially injects the AGN feedback +energy into the dense accretion regions, which then has difficulty +escaping and is thermalised by the cold gas reservoir surrounding +the central SMBH. As a result, a build-up of cold gas occurs in +the ICM and a high star formation rate follows. At late times, this +larger cold gas reservoir fuels AGN activity that can increase the gas +entropy in the core to produce a non cool-core (NCC) state at 𝑧 = 0. +On the other hand, volume-weighted deposition injects more +energy in more diffuse regions, which allows the AGN feedback +energy to escape the accretion region early to heat the gas over +large distances. Star formation is dramatically quenched, the gas +in the core is efficiently depleted and the transition to an NCC +proceeds from 𝑧 = 1. When volume-weighted deposition is used, +the more diffuse ICM leads to the cessation of AGN activity at +lower redshifts because of a low cold gas supply. Thus, with the +decline of the AGN activity, the ICM gradually cools to settle into a +similar ICM thermodynamical state as the simulation implementing +mass-weighted AGN energy deposition at 𝑧 = 0. +In spite of this relative similarity between the two simulations at +𝑧 = 0, the star formation and AGN activity histories greatly differ, +as do the galaxy masses and the ICM clumpiness. +• Anisotropic thermal conduction appears to reduce star forma- +tion in the ICM by almost a factor of 2 by flattening out temperature +gradients in the ICM. ATC leads to an earlier transition to an NCC +cluster thanks to the transport of heat within the ICM. However, in +our simulations, we do not observe enhanced AGN activity but the +opposite: ATC delays the AGN activity by preventing the formation +of cold gas in the ICM that would otherwise fuel the SMBH cold +gas accretion. +• The cluster gas fractions are not appreciably altered by changes +in the energy accumulation threshold of the thermal AGN feedback. +The observed slight gas depletion does not linearly scale with this +threshold. It suggests that purely thermal feedback cannot shape the +ICM on large scales. +• The evolution of our simulated cluster observables over cosmic +time is in relatively good agreement with both observational and +numerical studies. Among the X-ray scaling relations, the 𝑇X − 𝑀 +relation is rather insensitive, especially in the high halo mass regime +(𝑀500 ⩾ 5 × 1014M⊙), to the use of the different galaxy formation +models (radiative gas cooling, ATC, mass- or volume-weighted AGN +energy deposition) as they show no noticeable difference with the +scalings derived for adiabatic simulations. The same conclusions +hold for the mean Sunyaev-Zeldovich flux scaling relations (and its +X-ray analogue) with much lower scatter. This suggests that galaxy +formation physics does not play a particular role in significantly +offsetting the global cluster observables, especially at the high mass +end. +To aid the astrophysical and cosmological interpretation of current +and future galaxy cluster surveys, we have increased the complexity +of the Rhapsody-C high-resolution cosmological simulations by +including anisotropic thermal conduction and various SMBH/AGN +models, as well as generating more sophisticated X-ray observables +compared to the Rhapsody-G simulations. Despite the apparent +sensitivity of the ICM and cluster galaxies to the numerical models +used, the evolution of the simulated cluster observables over cosmic +MNRAS 000, 1–30 (2022) + +The Rhapsody-C simulations +25 +time is remarkably insensitive to changes in our astrophysical +models. A notable exception of the X-ray luminosity, which is +sensitive to clumping. The power of the Rhapsody-C simulations is +to have a sample of clusters sharing a similar mass at 𝑧 = 0 but with +diverse assembly histories, shapes and richness. We are therefore in +a position to study the scatter around the mean scaling relations and +relate it to the astrophysical processes that shape each cluster. This +will be investigated in future research. +In this work, by looking at the 𝑇X − 𝑀tot scaling relation, we +observed that accounting for a ∼20 per cent mass bias can make our +data consistent with studies based on hydrostatic mass estimates. A +detailed study of the energy budget in our simulated galaxy cluster +is needed to quantify the level of non-gravitational energy and non- +thermal pressure support. Moreover, thanks to the implementation +of the thermal conduction of Dubois & Commerçon (2016) which +includes a separate treatment of electrons and ions, we are able +to resolve differences in their distribution and energetics. We will +investigate these aspects in future work. +ACKNOWLEDGEMENTS +We are grateful to Pawel Biernacki for helpful discussions about +the modelling of super-massive black holes in Ramses, to Yohan +Dubois regarding the thermal conduction scheme, as well as Chris- +tian Garrel for his invaluable help with the LIRA code. We are in- +debted to Lorenzo Lovisari and Yohan Dubois for thorough feed- +back on an early version of the manuscript, and we thank Ricarda +Beckmann, Sandrine Codis, Frédéric Bournaud, Aoife Boyle, and +Sunayana Bhargava for useful discussion and comments. +AP and OH acknowledges funding from the European Research +Council (ERC) under the European Union’s Horizon 2020 research +and innovation programme (grant agreement No. 679145, project +‘COSMO-SIMS’). This work was granted access to the HPC re- +sources of TGCC under the allocation A0040410487 made by +GENCI. +This work made use of the following open source software: Matplotlib (Hunter +2007), NumPy (Harris et al. 2020), SciPy (Virtanen et al. 2020), emcee +(Foreman-Mackey et al. 2013), PyAtomDB (Foster & Heuer 2020), APEC +(Smith et al. 2001), Astropy (Price-Whelan et al. 2018). +DATA AVAILABILITY +The simulation data and post-processed data can be made available +per reasonable request to the authors on an individual basis. +REFERENCES +Abbott T. M. C., et al., 2022, Phys. Rev. D, 105, 023520 +Allen S. W., Evrard A. E., Mantz A. B., 2011, ARA&A, 49, 409 +Anders E., Grevesse N., 1989, Geochimica Cosmochimica Acta, 53, 197 +Angulo R. E., Hahn O., Abel T., 2013, MNRAS, 434, 1756 +Arnaud M., Pratt G. W., Piffaretti R., Böhringer H., Croston J. H., Pointe- +couteau E., 2010, A&A, 517, A92 +Bañados E., et al., 2021, ApJ, 909, 80 +Bahé Y. 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Depending on the temperature gradient in the core (⩽ 200 kpc), +anisotropic thermal conduction can act as a cooling or heating source and +endeavours to smooth out substructures in the ICM. +Wu H.-Y., Evrard A. E., Hahn O., Martizzi D., Teyssier R., Wechsler R. H., +2015, MNRAS, 452, 1982 +Yang H. Y. K., Reynolds C. S., 2016, ApJ, 818, 181 +Zeldovich Y. B., Sunyaev R. A., 1969, Ap&SS, 4, 301 +APPENDIX A: ANISOTROPIC THERMAL CONDUCTION +AS A HEATING OR COOLING SOURCE +To understand the effect of anisotropic thermal conduction (ATC) on +the ICM before adding physics related to galaxy formation. We run +our simulation at a lower resolution corresponding to an effective +resolution of 40963 cells for the full simulation box, yielding a DM +particle mass of 8.22 × 108 ℎ−1Mpc. +We show in Fig. A1, the gas depletion profiles for an adiabatic sim- +ulation (grey line) and where only ATC was added (blue). Similarly, +we compare a simulation where the gas is allowed to radiatively cool +(black) to the same simulation when ATC is switched on (red). +In the adiabatic case, as the cluster forms, the gas in the center +is adiabatically heated and the higher pressure support prevents the +gravitation collapse of gas from the cluster outskirts. +However when thermal conduction is added, we observe that while +the cluster forms and the more gas collapse towards the center, heat +generated during this gas compression is now transported outwards +were the gas temperature is lower. As the result more low-entropy +gas fall inwards and the cluster centre gets denser. With thermal +conduction, the heat generated by the gas compression during the +cluster evolution is transported to the colder outskirts which lowers +the pressure support in the cluster center. Consequently, gas flows +inwards, the cluster core contracts and a higher gas fraction can +be observed at all radii in Fig. A1. In that case, thermal diffusion +is actually behaving like a gas cooling source driving a cooler and +denser cluster core. +We now allow the gas to radiatively cool. Because this process is +quadratically sensitive to the gas density, as the gas in the center is +getting denser, it also cools at faster rates, which leads to a runaway +instability: the cooling catastrophe. To prevent the overcooling of +the gas, we set an artificial limit by imposing a pressure floor below +which no gas can cool but can only increase its density along the +adiabat 𝑃gas ∝ 𝜌𝛾 +gas. In this configuration, we observe the formation +of a much lower entropy and higher density core compared to the +adiabatic case which translates into a high fraction in the central +100 kpc. By comparing the gas depletion profiles of the radiative +simulations in Fig. A1, we can see by comparing the black and red +line that thermal conduction can deplete gas from the core to the +outskirts. In the cooling-only simulation (black), we see the fraction +of gas peaking at 60 kpc, defining the extent of the core, which drops +to reach the universal baryon faction at 700 kpc. With the addition +of thermal conduction (red), the core extends now to 100 kpc with a +lower amount of gas. The gas fraction decreases less steeply towards +the Ω𝑏/Ω𝑚 value at a greater distance of 2 Mpc. In that config- +uration, thermal conduction drives the transport of heat from the +outskirts toward the overcooling core: it now acts as a heating source. +Interestingly, we see that thermal conduction can act as a cooling +source by transporting heat from the core to larger radii, or, as a +heating source by transporting heat inwards. This behaviour is dic- +tated by the sign of the temperature gradient in the inner cluster +region. More generally, thermal conduction is effective at flattening +out temperature substructures in the ICM. +APPENDIX B: ESTIMATING THE X-RAY TEMPERATURE +Having access to the true density and pressure of the gas inside a +simulation cell, we can estimate the true gas temperature from our +simulations. However, the comparison between real and simulated +data is complicated by different problems like sky background, pro- +jection effects and instrumental noise. Additionally there is a possible +mismatch between the spectroscopic temperature estimated from X- +ray observations and the temperatures usually defined in numerical +studies. The former is a mean projected temperature obtained by fit- +ting a single (or multitemperature) thermal model to the observed +photon spectrum while the later fully exploits the three-dimensional +thermal information of gas cells/particles. The average gas temper- +ature in simulated cluster can be obtained by a weighted sum of all +cell/particle temperatures 𝑇𝑖 as +𝑇𝑤 = +� +𝑖 𝑤𝑖𝑇𝑖 +𝑤𝑖 +, +(B1) +where 𝑤𝑖,vw = d𝑉𝑖 in case of a volume-weighted temperature +(𝑇vw), with d𝑉𝑖 is the AMR cell volume determined by the +local resolution of the simulation, or with 𝑤𝑖,mw = 𝑛𝑖d𝑉𝑖 for a +mass-weighted temperature (𝑇mw) with 𝑛𝑖 the gas cell density. This +mass-weighted temperature gives a more ‘physical’ average as it +emphasizes dense regions that participate more to the X-ray emission. +However, X-ray astronomy suffers from well-known biases +due as its intensity depends quadratically on the density since +both Bremsstrahlung and the collisional excitation responsible +for the metal line emissions results from two-body processes. +For this reason, X-ray observations are especially biased by the +dense regions such as the cluster core or the presence of gas-rich +substructures which motivate the definition of a ‘emission-weighted’ +temperature (𝑇emw) where the weighting function is proportional +to the emissivity of each gas element 𝑤𝑖,emw = 𝑛2 +𝑖 Λ(𝑇𝑖) with the +cooling function Λ(𝑇) mainly being the so-called bolometric cooling +MNRAS 000, 1–30 (2022) + +28 +A. Pellissier et al. +function Λ(𝑇) ∝ √𝑇𝑖 which implicitly assume that bremsstrahlung +(free-free) emission is the dominant mechanism at high X-ray +temperatures (> 3 keV)12. Mazzotta et al. (2004) found that the +above-defined emission-weighted temperature over-estimates the +projected spectroscopic X-ray temperatures of thermally complex +clusters and propose another spectroscopic-like temperature 𝑇sl +with 𝑤𝑖,sl = 𝑛2 +𝑖 𝑇3/4 +𝑖 +which better approximates the spectroscopic +temperature in Chandra and XMM-Newton observations. This +weighting function, beside being biased toward the densest regions +of the clusters such as in the emission-weighted case, will also be +biased toward the coolest regions. +To circumvent the shortcomings of simple weighting schemes, we +instead produce an X-ray spectrum from which we derive the gas +temperature and density using a single thermal as close as possible +to the observers’ methodology. The simulated ICM spectrum is ob- +tained by computing the continuum and line emission of a gas cell +with 𝑇 ⩾ 0.5 keV and a metallicity 𝑍, with the publicly available +AtomDB atomic database13. As such, from the continuum and line +emissivity 𝜖𝑖(𝑇𝑖, 𝑍𝑖), we compute the rate of emitted photon Φ𝑖 of +the cell 𝑖 +𝜙𝑖 = 𝜖𝑖(𝑇e,𝑖, 𝑍𝑖) +���︁ +𝑖 +𝑛e,𝑖 𝑛H,𝑖d𝑉𝑖, +(B2) +which allow us to produce a mock X-ray spectrum by summing of +the individual rest frame spectra of each gas cell. +In X-ray observation, the most common method to obtain the ICM +temperature and density is to fit the observed spectra by a single +temperature APEC model. Therefore, we follow a similar method- +ology and chose to perform fits using a Monte Carlo Markov Chain +(MCMC) sampling method thanks to the emcee python library +(Foreman-Mackey et al. 2013). We fit our data to a single temperature +spectra generated using the PyAtomDB library. We show the result of +a such fits in the upper panel Fig. B1 . +We note that the presence of cold gas at high redshifts can produce +a low-energy bump in the spectrum which complicate the fitting +procedure. As the result, it could induce the fit to converge faster +to high values of the density (i.e. higher spectrum normalisation). +In order to maximize the likelihood, the MCMC chain will later try +converging to lower temperatures (i.e. steeper cutoff) to compensate +for the overestimated gas density. Consequently, the spectral fit can +slightly underestimate the temperature of haloes in the case of a +high fraction of cold the gas, which is predominantly the case at +high redshifts (𝑧 ⩾ 1). +To overcome the overestimation of the gas density (and a un- +derestimation of the temperature), we first fit the gas density +0.20–2.00 keV band first as the X-ray flux is not very sensitive on the +temperature and metallicity in this band (as long as the metallicity +is low, i.e. 𝑍 ≲ 0.5). We use the posterior distribution of this first +MCMC sampling as the prior distribution of the gas density for a +second MCMC while using flat priors for the gas temperature and +metallicity. The fits in Fig. B1 result from this two-step MCMC. +However, the low energy bump cannot be constrained with a single +temperature model and a double (or multi) temperature model would +be more suited. +12 Nevertheless, at lower temperatures, metal lines participate significantly +to the X-ray emission which becomes temperature and metallicity dependent. +13 We use the abundances of Anders & Grevesse (1989) as well as APEC +equilibrium line and continuum fits files from the 3.0.9 version of AtomDB - +https://atomdb.org/ +100 +101 +E [keV] +1048 +1049 +1050 +1051 +1052 +1053 +Φ [ph s−1] +Data +Best fit ; z = 0 +Best fit ; z = 1 +1014 +1015 +Mtot,500 [M⊙] +100 +101 +E (z)−2/3 T ce +X,500 [keV] +SL +SF +MW +VW +Figure B1. Top panel: ICM photon emission rate directly computed from +the simulation in the core-excised 𝑅500 sphere (black) along the best fit +APEC model resulting from our MCMC sampling for the same halo at 𝑧 = 0 +(orange) and 𝑧 = 1 (dark orange). At low redshifts, our fitting procedure +performs well. We note that the gas metallicity responsible for the emission +lines is not well constrained but is not relevant for our analysis as the lines +does not significantly participate to the X-ray flux. For the 𝑧 = 1 spectrum, +the high fraction of cold gas (i.e. 𝐸 ⩽ 1 keV) lead to a slight overestimation +of the density (i.e. higher normalisation) and an underestimation of the gas +temperature (i.e. steeper spectrum). +Bottom panel: Scaling of the core-excluded temperature with the total mass +inside the 𝑅500 using different temperature estimates. 𝑇vw (blue) and 𝑇mw +(red) are a weighted average using respectively the cell volume and the cell +density. 𝑇sl (green) uses the cell emissivity and the Mazzotta et al. (2004) +weights of hot (𝐸 ⩾ 0.5 keV) gas cells only. We show the temperatures +resulting from the spectral fits 𝑇sf in orange. We can see that our 𝑇sf estimates +approach well the spectroscopic-like temperature. +MNRAS 000, 1–30 (2022) + +The Rhapsody-C simulations +29 +1014 +1015 +Mtot,500 [M⊙] +1043 +1044 +1045 +1046 +E (z)−2 Lce +X,500 +� +erg s−2� +ci +ce +Figure C1. Mass versus X-ray luminosity for all haloes with 𝑧 ⩽ 1.5 for the +all simulations type combined (NR, MW, VW and MC, see details in Table 1. +The blue symbols show the core-included (ci) X-ray luminosity within 𝑅500 +and the black symbols shows the luminosity in the core-excluded region +corresponding to the [0.15 − 1] 𝑅500 aperture. The see that the exclusion +of the core reduces dramatically the scatter and indicates for 40 per cent the +X-ray luminosities on average. +We show in the bottom panel of Fig. B1, the differences between +the different gas temperature estimates (𝑇vw, 𝑇mw, 𝑇sl and 𝑇sf). To +compute the average within 𝑅500 of 𝑇sl and 𝑇sf, only the hot X-ray +emitting gas (𝐸 ⩾ 0.5 keV) is considered while 𝑇vw and 𝑇mw use the +information of all cells with no cut in minimum gas temperature. +We see that the 𝑇sl and 𝑇sf are similar and shows that the formula +of Mazzotta et al. (2004) gives a good estimate of the spectroscopic +temperature, especially in the lower mass range. However, these two +X-ray’ estimates are, on average, 10 and 20 per cent higher than 𝑇mw +and 𝑇vw respectively. This shows that accounting for the bias induced +by the X-ray emitting gas offsets to higher temperatures the simple +mass- or volume-weighted averages, which also do not have any cut +in minimum gas temperature. +𝑇mw is 10 per cent higher compared to 𝑇vw at higher halo masses but +shows the steepest slope as we can see in Fig. B1 (we have for haloes +with 𝑧 ⩽ 1.5 slopes of 0.700 ± 0.019, 0.723 ± 0.018, 0.726 ± 0.013 +and 0.689 ± 0.014 when using 𝑇sl, 𝑇sf, 𝑇mw and 𝑇vw respectively). +APPENDIX C: ON THE CORE INCLUSION +The presence of AGN activity in the core of GCs can introduce +a strong variability of the X-ray luminosity as the central density +fluctuates and unrealistically high luminosities can be obtained. We +compare in Fig. C1 the core-included (ci) and core-excluded (ce) +X-ray luminosities computed for two apertures: the entire cluster +emission interior to 𝑅500 and in the [0.15 − 1] 𝑅500 aperture respec- +tively. +The inclusion of the core typically boost the X-ray luminosity with a +much greater scatter as it is greatly sensitive to the thermodynamic +state of the cluster core (high gas density and possible AGN heat- +ing). On average, the exclusion of the core decreases by 40 per cent +the X-ray luminosity and shows a steeper slope of (1.169 ± 0.033, +1014 +1015 +Mtot,500 [M⊙] +0.85 +0.90 +0.95 +1.00 +1.05 +1.10 +1.15 +1.20 +1.25 +T ce +w,500 / T ci +w,500 +Tsl +Tmw +Tvw +Figure C2. Ratio of the core-excluded to the core-included temperature versus +mass for all haloes (NR, MW, VW and MC combined) with 𝑧 ⩽ 1.5. We +show the difference induced by the core exclusion on different temperature +estimates: 𝑇vw (blue) and 𝑇mw (red) are a weighted average using respectively +the cell volume and the cell density while 𝑇sl (green) uses the cell emissivity +and the Mazzotta et al. (2004) weights of hot (𝐸 ⩾ 0.5 keV) gas cells only. +compared to 0.681 ± 0.070 for the ci). +It also significantly reduces the scatter and hence is more suited for +galaxy cluster sample studies where clusters can have very different +central states, consistent with the finding of Pratt et al. (2009) and +Mantz et al. (2018). +The inclusion of the core does not significantly impacts the 𝑇 − 𝑀 +scaling relation.14 For instance, we find slopes of 0.717 ± 0.020 for +the core-included and 0.700 ± 0.019 for the core-excluded 𝑇sl − 𝑀 +scaling relation which are consistent.15 In Fig. C2, we see that the VW +temperatures are widely insensitive to the core inclusion/exclusion +because the volume inside 0.15× 𝑅500 represents only a tiny fraction +of the total 𝑅500 sphere. +We can see that, in halo masses lower than ∼ 2 × 1014M⊙, the +exclusion of the core tends to increase the MW temperatures as +the measurement is biased by the presence of dense and cold gas +in lower mass haloes (i.e. higher redshifts). On the other hand, for +𝑀500 > 3 × 1014M⊙ the MW temperature is on average 2 per cent +lower when the core is excluded from the measurement . While +showing a larger scatter, the ratio of the ce to ci 𝑇sl can be as low as +0.85 with a median value of 0.94. +On average, we see that the ce temperature estimates are more +biased low at higher halo masses which can help to explain why very +slightly steeper slopes are found in the ci 𝑇 − 𝑀 scaling relation. +14 As the measurement of the temperature from a MCMC spectral fit is +expensive, we only have measurement of 𝑇sf in the core-excluded region. +Therefore, we only discuss here the 3 other estimates (𝑇vw, 𝑇mw and 𝑇sl) for +which we have both ci and ce measurements. +15 Similarly, we find for the 𝑇mw − 𝑀 relation slopes of 0.745 ± 0.014 and +0.726 ± 0.013for the core-included and core-excluded estimates respectively +and for the 𝑇vw − 𝑀 relation, 0.706 ± 0.014 and 0.689 ± 0.014 respectively. +MNRAS 000, 1–30 (2022) + +30 +A. Pellissier et al. +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–30 (2022) +