MINOT: Modeling the intracluster medium (non-)thermal content and observable prediction tools
R. Adam, H. Goksu, A. Leingärtner-Goth, S. Ettori, R. Gnatyk, B. Hnatyk, M. Hütten, J. Pérez-Romero, M. A. Sánchez-Conde, O. Sergijenko
AAstronomy & Astrophysics manuscript no. minot˙paper © ESO 2020October 5, 2020
MINOT : Modeling the intracluster medium (non-)thermal contentand observable prediction tools
R. Adam (cid:63) , H. Goksu , A. Leing¨artner-Goth , S. Ettori , , R. Gnatyk , B. Hnatyk , M. H¨utten , J. P´erez-Romero , ,M. A. S´anchez-Conde , , and O. Sergijenko Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS / IN2P3, 91128 Palaiseau, France INAF, Osservatorio di Astrofisica e Scienza dello Spazio, via Pietro Gobetti 93 /
3, 40129 Bologna, Italy INFN, Sezione di Bologna, viale Berti Pichat 6 /
2, I-40127 Bologna, Italy Astronomical Observatory of Taras Shevchenko National University of Kyiv, 3 Observatorna Street, Kyiv, 04053, Ukraine Max-Planck-Institut f¨ur Physik, F¨ohringer Ring 6, 80805 M¨unchen Instituto de F´ısica Te´orica UAM-CSIC, Universidad Aut´onoma de Madrid, C / Nicol´as Cabrera, 13-15, 28049 Madrid, Spain Departamento de F´ısica Te´orica, M-15, Universidad Aut´onoma de Madrid, E-28049 Madrid, SpainLast update: October 5, 2020
Abstract
In the past decade, the observations of di ff use radio synchrotron emission toward galaxy clusters revealed cosmic-ray (CR) electronsand magnetic fields on megaparsec scales. However, their origin remains poorly understood to date, and several models have beendiscussed in the literature. CR protons are also expected to accumulate during the formation of clusters and probably contribute tothe production of these high-energy electrons. In order to understand the physics of CRs in clusters, combining of observations atvarious wavelengths is particularly relevant. The exploitation of such data requires using a self-consistent approach including boththe thermal and the nonthermal components, so that it is capable of predicting observables associated with the multiwavelengthprobes at play, in particular in the radio, millimeter, X-ray, and γ -ray bands. We develop and describe such a self-consistent modelingframework, called MINOT (modeling the intracluster medium (non-)thermal content and observable prediction tools) and make this toolavailable to the community.
MINOT models the intracluster di ff use components of a cluster (thermal and nonthermal) as sphericallysymmetric. It therefore focuses on CRs associated with radio halos. The spectral properties of the cluster CRs are also modeled usingvarious possible approaches. All the thermodynamic properties of a cluster can be computed self-consistently, and the particle physicsinteractions at play are processed using a framework based on the Naima software. The multiwavelength observables (spectra, profiles,flux, and images) are computed based on the relevant physical process, according to the cluster location (sky and redshift), and basedon the sampling defined by the user. With a standard personal computer, the computing time for most cases is far shorter than onesecond and it can reach about one second for the most complex models. This makes
MINOT suitable for instance for Monte Carloanalyses. We describe the implementation of
MINOT and how to use it. We also discuss the di ff erent assumptions and approximationsthat are involved and provide various examples regarding the production of output products at di ff erent wavelengths. As an illustration,we model the clusters Abell 1795, Abell 2142, and Abell 2255 and compare the MINOT predictions to literature data. While
MINOT was originally build to simulate and model data in the γ -ray band, it can be used to model the cluster thermal and nonthermal physicalprocesses for a wide variety of datasets in the radio, millimeter, X-ray, and γ -ray bands, as well as the neutrino emission. Key words.
Galaxies: clusters: intracluster medium – Cosmic rays – Radiation mechanisms: general – Method: numerical
1. Introduction
Galaxy clusters are the largest gravitationally bound structuresthat are decoupled from the expansion of the Universe. Theyform peaks in the matter density field. Their assembly has beendriven by the gravitational collapse of dark matter (Kravtsov &Borgani 2012), which is thought to dominate the matter con-tent of clusters (about 80% in mass). Clusters also consist ofbaryonic matter, essentially in the form of hot ionized thermalplasma, called the intracluster medium (ICM; about 15%), andof galaxies (about 5%). While clusters are used to understandthe formation of large-scale structures and to constrain cosmo-logical models, they are also the place of very rich astrophysicalprocesses and excellent targets for testing fundamental physics(see, e.g., Allen et al. 2011, for a review).Galaxy clusters form through the merging and accretionof other groups and surrounding material (Sarazin 2002). This (cid:63)
Corresponding author: R´emi Adam, [email protected] leads to the propagation of shocks and turbulences in the ICM(Markevitch & Vikhlinin 2007), which can accelerate chargedparticles to very high energies. These cosmic rays (CRs) interactwith the magnetized ICM, generating di ff use radio synchrotronemission (Feretti et al. 2012; van Weeren et al. 2019), and theyare expected to produce a γ -ray signal because of the inverseCompton interaction with background light or the decay of pionsproduced in proton-proton collisions (Brunetti & Jones 2014). Inaddition, clusters also host active galactic nuclei (AGN), whichare known to provide feedback onto the ICM (Fabian 2012). Thisfeedback is only poorly understood, but is expected to have a ma-jor e ff ect on the formation and the evolution of galaxy clusters.Di ff use radio synchrotron emission in galaxy clusters is gen-erally classified as radio halos (including giant and mini-halos),radio relics, and revived AGN fossil plasma source (see vanWeeren et al. 2019, for detailed discussions). While relics arethought to be associated with the shock acceleration of electronsin the periphery of clusters, radio halos might originate from tur- a r X i v : . [ a s t r o - ph . H E ] O c t . Adam, H. Goksu and A. Leing¨artner-Goth: Modeling the ICM (non-)thermal content and observable prediction tools bulent reacceleration of seed electrons and / or secondary elec-trons produced by hadronic interactions. γ -ray emission is alsoexpected as a result of the inverse Compton emission that arisesfrom the scattering of background photon fields onto relativis-tic electrons, or the hadronic interaction from CR protons (CRp)and the ICM (see, e.g., Pinzke & Pfrommer 2010, for the signalexpected based on numerical simulations).The annihilation or the decay of dark matter particlesmight also cause γ -ray emission from galaxy clusters (see, e.g.,Combet 2018), and many searches for this signal have been per-formed (e.g., Ackermann et al. 2010; Aleksi´c et al. 2010; Arlenet al. 2012; Abramowski et al. 2012; Combet et al. 2012; Cadena2017; Acciari et al. 2018). In the case of dark matter decay,galaxy clusters are particularly competitive targets because thesignal scales linearly to the huge dark matter reservoirs in galaxyclusters. In the case of dark matter annihilation, clusters canbe at the same flux level as dwarf galaxies when substructuresare accounted for, and they are thus also highly relevant targets(S´anchez-Conde et al. 2011; Molin´e et al. 2017). However, thelimits that can be set on the properties of dark matter depend onthe uncertainties associated with the modeling of the backgroundemission, so that accurate CR modeling is also essential for darkmatter searches.Many attempts to detect the cluster γ -ray emission have beenmade using ground-based (e.g., Aharonian et al. 2009; Aleksi´cet al. 2012; Arlen et al. 2012; Ahnen et al. 2016, at 50 GeV - 10TeV energies) and space-based observations (e.g., Reimer et al.2003; Huber et al. 2013; Prokhorov & Churazov 2014; Zandanel& Ando 2014; Ackermann et al. 2014, 2015, 2016, at about 30MeV - 300 GeV). While unsuccessful so far, these searches werevery useful to constrain the CR physics and particle accelerationat play in clusters, especially when combined with radio obser-vations (e.g., Vazza et al. 2015; Brunetti et al. 2017). Recently,Xi et al. (2018) claimed the first significant detection of γ -raysignal toward the Coma cluster using data obtained with the Fermi -Large Area Telescope (
Fermi -LAT). However, their re-sults might be confounded by a possible point source because thesignal-to-noise ratio and angular resolution of the observationsare limited. While the
Fermi -LAT satellite is to continue to takedata for several additional years (compared to about the 12 yearsof data collected so far), major discoveries concerning galaxyclusters are unlikely given the modest increase in statistics thatis expected. From the ground, the Cherenkov Telescope Array(CTA, Cherenkov Telescope Array Consortium et al. 2019) isexpected to provide a major improvement in sensitivity in the100 GeV - 100 TeV energy range.In order to address the CR physics in galaxy clusters, mul-tiwavelength observations and analyses are becoming particu-larly relevant with the construction of such new facilities. Whilecluster CRs can essentially be accessed the radio and γ -raybands, their physics is driven by the continuous interaction withthe thermal plasma. When data are compared to modeling, orwhen mock observations are generated, the thermal and thenonthermal components should therefore be modeled together,in a self-consistent way, so that uncertainties and degenera-cies between the two can be accounted for. The thermal emis-sion can be probed in particular in the X-ray and at millimeterwavelengths through thermal Bremsstrahlung emission (Sarazin1986; B¨ohringer & Werner 2010) and the thermal Sunyaev-Zel’dovich (tSZ) e ff ect (Sunyaev & Zeldovich 1970, 1972). Inaddition to the primary components, modeling the particle inter-actions in the ICM relies on particle physics data from acceler-ators or a sophisticated Monte Carlo code (see, e.g., Kafexhiuet al. 2014, for discussions), and they need to be accounted for carefully. Nevertheless, clusters are commonly modeled focus-ing on individual (or just a few) components, and no public self-consistent multiwavelength software exists in the literature. Forinstance, Li et al. (2019) and Br¨uggen & Vazza (2020) recentlymodeled the di ff use radio synchrotron emission of radio halosand radio relics, respectively, and employed the Press-Schechterformalism to estimate the statistical properties of the correspond-ing signal.Here, we present a software dedicated to the self-consistentmodeling of the thermal and nonthermal di ff use componentsof galaxy clusters, for which the main objective is comput-ing accurate and well-characterized multiwavelength predic-tions for the radio, millimeter, X-ray, γ -ray, and neutrino emis-sion. This software is called MINOT, modeling the intraclustermedium (non-)thermal content and observable prediction tools.It is based on the Python language and is available at the fol-lowing url: https://github.com/remi-adam/minot . MINOT includes various parameterizations for the radial profiles andspectral properties of the di ff erent cluster components. The codedoes not aim at computing the CR production rate from micro-physics considerations (e.g., turbulence, shocks, or di ff usion),but instead directly models the spatial and spectral distributionsof the CRs and the thermal gas. The predictions for associatedobservables are available in the radio (synchrotron), millime-ter (tSZ e ff ect), X-ray (thermal Bremsstrahlung), γ -ray (inverseCompton and hadronic processes), and also for neutrino emis-sions (hadronic processes). This includes surface brightness pro-files or maps, spectra, and integrated flux computed with dif-ferent options. For γ -rays, CR electrons (CRe), and neutrinosfrom hadronic origin, MINOT includes the latest description ofthe hadronic interactions in the ICM, based on the
Naima soft-ware (Zabalza 2015). The thermal modeling uses the
XSPEC soft-ware for X-ray predictions (Arnaud 1996), and it includes anaccurate description of the tSZ signal up to high plasma temper-atures.This article is organized as follows. In Section 2 we provide ageneral overview of the code and discuss the di ff erent interfaces.Section 3 discusses the physical modeling of the cluster com-ponents. The physical processes related to particle interactionsare detailed in Section 4. In Section 5 we discuss the predictionof observables in the relevant energy bands. The use of MINOT is illustrated in Section 6 for three nearby massive well-knownclusters for which multiwavelength data are available in the lit-erature. Finally, Section 7 provides a summary and conclusion.Equations are given following the international system of units.
2. General overview and structure of the code
MINOT is a Python-based code available at https://github.com/remi-adam/minot . It essentially depends on standardPython libraries, but some functionalities require specific soft-wares and packages, as discussed below. In this section, we pro-vide a general overview of the working principle of the code,of its structure, and the interactions between the di ff erent mod-ules. The list of the code parameters is also discussed, as wellas the available functional forms for the radial and spectral mod-els. Figure 1 highlights how the input modeling is used to gen-erate observables via the di ff erent plasma processes consideredin MINOT , and the general overview of the code is illustrated inFig. 2. Several Python notebook examples are also available. In particular,the notebook ’demo plot.ipynb’ has been used to generate the figures ofthis paper.2. Adam, H. Goksu and A. Leing¨artner-Goth: Modeling the ICM (non-)thermal content and observable prediction tools
Figure 1.
Overview of the
MINOT input modeling, the considered physical processes at play in the ICM, and the observables that arecomputed. The interdependences are shown by the black arrows.
MINOT was first developed to compute an accurate γ -ray predic-tion for galaxy clusters. As discussed below and shown in Fig. 1,this requires several key ingredients. Because the same ingre-dients also provide diagnoses for other observables at di ff erentwavelengths through various physical processes, MINOT was fur-ther developed to account for them. This allowed us to provideexternal constraints to a given input modeling that is used to gen-erate γ -ray observables, but also to provide further diagnosis ofthe physical state of the cluster.First, the spatial and spectral distributions of primary CRe(CRe ) and protons are crucial. They generate γ -rays through in-verse Compton scattering on the cosmic microwave background(CMB), or through hadronic interactions, respectively. Modelingthe thermal gas is also essential because hadronic processes arisefrom the interaction between CRp and thermal plasma. As weshow in Section 3, the thermal component is based on the ther-mal electron pressure and density. Additionally, the normaliza-tion of the CR distributions is generally given relative to thethermal energy. The hadronic interactions also generate sec-ondary CRe and positrons (CRe ). Because they are a ff ectedby synchrotron losses, they require that the magnetic field isaccounted for as another key ingredient of the input modeling(Section 4). These electrons contribute to the inverse Comptonemission (Section 5). In summary, the necessary input modelingingredients are CRe , magnetic field strength, CRp, and thermalelectron pressure and density.With these ingredients at hand, the radio emission thatarises from CRe (primary and secondary) moving in the mag-netic field can be modeled. Similarly, the thermal pressureand density allow us to compute the tSZ signal and the ther-mal Bremsstrahlung X-ray emission. They also provide a com-plete diagnosis of the thermodynamic properties of the cluster.Neutrinos are also produced during hadronic interactions, andtheir associated observable is thus available. In order to model galaxy clusters and predict observables as-sociated with the di ff use thermal and nonthermal components, MINOT is organized in six main parts, each of which is related tospecific functions and procedures. We list these parts below.1. The main class, called
Cluster , is written in the file model.py and provides an entry point for the user. It allowsdefining the model and solve entanglement between param-eters.2. A subclass called
Admin allows us to handle administrativetasks, in particular, input and output procedures.3. The
Modpar subclass is dedicated to the model parameter-ization. It gathers a library of available radial and spectralmodels.4. The physical modeling of the cluster is performed in the
Physics subclass. It includes many functions for retrievingthe desired physical quantities.5. The
Observable subclass allows us to extract the requestedcluster observables based on the inner physics encoded in themodel.6. Finally, the subclass
Plots is designed for automated plotsto provide a cluster diagnostic based on the current model-ing.In addition to these six main parts,
MINOT also includes a librarycalled
ClusterTools in which numerical tools and astrometrictools are defined. It also includes several classes that are used tocompute various physical processes relevant for
MINOT . In thefollowing subsections, we describe the working principle of thedi ff erent functionalities. As illustrated in Fig. 2, the user can directly define a cluster ob-ject by calling the
Cluster (main) class of
MINOT , cluster = minot.Cluster(optional parameters).
3. Adam, H. Goksu and A. Leing¨artner-Goth: Modeling the ICM (non-)thermal content and observable prediction tools
Figure 2.
Overview of the code structure and interfaces.Optional parameters, such as cluster name, coordinates, or red-shift, can be passed directly to the initialization call. However,any parameter can be modified on the fly, for example, cluster.redshift = 0.1
The entanglement of parameters is solved in the code. For in-stance, changing the cluster redshift will automatically changethe angular diameter distance of the cluster according to the cur-rent cosmological model. Information is provided to the userwhen the ’silent’ parameter is set to ’False’. The list of param-eters that describe the cluster object is available in Table 1. Wenote that whenever possible, the code uses
Astropy units forquantities .We can distinguish four types of parameters, as listed inTable 1. The first type corresponds to administration-like pa-rameters (e.g., output directory used to generate products), thesecond type concerns the global properties of the cluster object(e.g., the redshift), the third type is related to the radial and spec-tral modeling of the physical quantities of interest of the cluster(e.g., CR number density profile), and the last type allows theuser to sample the output observables. In particular, it is possibleto set a map header (e.g., obtained for real data) on which themodel prediction maps are projected, which facilitates a com-parison of data and model. The parameters describing the cluster can be divided into twotypes: global properties that apply to the entire cluster (e.g.,mass, redshift, and coordinates), and properties that vary as afunction of radius or energy. This separation is highlighted inTable 1. Some parameters are assumed to be constant over theentire cluster volume, such as the hydrostatic mass bias or themetal abundances.In addition to the global properties, the primary quantitiesthat are used to define the physical state of the cluster are (seealso Section 3 for further details) the gas pressure of thermalelectrons, the gas number density of thermal electrons, the CRp https://docs.astropy.org/en/stable/units/ number density profile and spectrum, the CRe profile and spec-trum, and the profile of the magnetic field strength. The CR dis-tributions are normalized according to the ratio of CR and ther-mal energy enclosed within a given radius. The physical model-ing of the radial and spectral properties of the cluster relies on alibrary of predefined models in the Modpar subclass of
MINOT .The list of models that are currently available in the code is givenin Tables 2 and 3 for the spectral and spatial component, respec-tively. Figure 3 illustrates the shape of the di ff erent models; theyare further discussed in a more physical context in Section 3.We note that the spatial and spectral parts of the modeling arecurrently decoupled (e.g., the spectrum of CRp does not changewith radius), such that a physical quantity f can be expressed as f ( r , E ) ∝ f ( r ) f ( E ) , (1)where E is the particle energy (only relevant for the CRs), and r is the physical radius in three dimensions. However, functionsthat couple the radius and the energy dependence are ready tobe implemented in the model library because any calculationsrelying on the modeling of f ( r , E ) are made on 2D grids (energyversus radius) that are ignorant of the underlying parameteriza-tion of the distributions. In addition, it is possible to apply somelosses, assuming a given scenario, to the input distribution. Inthis case, f ( r , E ) is considered as an injection rate, and the out-put distribution is a ff ected di ff erently for di ff erent energy andradii. The implementation of the losses is discussed in detail inSection 4.2.A new model is set to a given physical property by passing aPython dictionary, such as cluster.density_gas_model =----- {’name’:’beta’,----- ’n_0’:1e-3*u.cm**-3,----- ’beta’:0.7, ’r_c’:300*u.kpc} or cluster.spectrum_crp_model =----- {’name’:’PowerLaw’,----- ’Index’: 2.5}
4. Adam, H. Goksu and A. Leing¨artner-Goth: Modeling the ICM (non-)thermal content and observable prediction tools
Table 1.
List of the parameters involved in the code.
Parameter Type Default value DescriptionAdministrative parameterssilent Boolean False Allows
MINOT to provide information when runningoutput dir string ’. / minot output’ Full path to the output directory for products savingGlobal physical propertiescosmo cosmology ( a ) Planck15 Cosmological modelname string ’Cluster’ Name of the clustercoord SkyCoord ( b ) [0 ,
0] deg Coordinates of the cluster centerredshift float 0.01 redshift of the clusterD ang quantity ( c ) ( (cid:63) ) Angular diameter distanceD lum quantity ( c ) ( (cid:63) ) Luminosity distanceM500 quantity ( c ) M (cid:12) Characteristic cluster massR500 quantity ( c ) ( (cid:63) ) Characteristic cluster physical radiustheta500 quantity ( c ) ( (cid:63) ) Characteristic cluster angular radiusR truncation quantity ( c ) × R Physical extent (boundary) of the clustertheta truncation quantity ( c ) × θ Angular extent (boundary) of the clusterhelium mass fraction float 0.2735 Helium mass fractionmetallicity sol float 0.0153 Reference metallically of the Sunabundance float 0.3 The metal abundance relative to the solar valueEBL model string ’dominguez’ Name of the extragalactic background light modelhse bias float 0.2 Hydrostatic mass biasEpmin quantity ( c ) ∼ .
22 GeV Minimal energy of the CRpEpmax quantity ( c )
10 PeV Maximal energy of the CRpEemin quantity ( c ) m e c Minimal energy of the CRe Eemax quantity ( c )
10 PeV Maximal energy of the CRe pp interaction model string ’Pythia8’ Name of the proton-proton interaction modelcre1 loss model string ’None’ Loss model to apply to the input primary CRe distributionRadial and spectral modelingpressure gas model dict P13UPP ( d ) Model to be used for the thermal gas pressure profiledensity gas model dict P e ( r )10 keV Model to be used for the thermal gas number density profilemagfield model dict P e ( r ) P e (10 kpc) × µ G Model to be used for the magnetic field profileX crp E dict 1% within R CRp to thermal energy ratio and reference radiusX cre1 E dict 1% within R CRe to thermal energy ratio and reference radiusdensity crp model dict ∝ P e ( r ) Model to be used for the CRp number density profiledensity cre1 model dict ∝ P e ( r ) Model to be used for CRe number density profilespectrum crp model dict Index 2.5 power law Model to be used for the CRp spectrumspectrum cre1 model dict Index 3.0 power law Model to be used for CRe spectrumSampling parametersRmin quantity ( c ) ( b ) [0 ,
0] deg Coordinates of the map centermap reso quantity ( c ) ( c ) list [5 ,
5] deg Map field-of-view size along R.A. and Dec.map header string None Header of the map
Notes. ( a ) From the astropy package. ( b ) From the astropy.coordinates package. ( c ) From the astropy.units pacakge. ( d ) Universal pressureprofile based on mass and redshift, from Planck Collaboration et al. (2013). ( (cid:63) ) Quantities that are computed from other parameters.
It is also possible to automatically set a parameterization of sev-eral quantities to predefined physical states without directly set-ting the model parameters, for instance, forcing the CRp to fol-low the radial distribution of the gas density, cluster.set_density_crp_isodens_scal_param() or to define the thermal electron number density based on thethermal electron pressure in the case of an isothermal clusterwith a given temperature. These functions are written as part ofthe
Modpar subclass.
When the desired physical properties of the cluster are set, func-tions related to the physical description of the cluster, from thesubclass
Physics , can be called to extract the thermodynamic and CR properties of the cluster, or the production rate of non-thermal particles (the physical modeling is further detailed inSection 3), for example, extracting the hydrostatic mass profile,or the neutrino emission rate via r, M = cluster.get_hse_mass_profile()dN_dEdVdt = cluster.get_rate_neutrino().
The user may also generate observables corresponding to theradio synchrotron emission, the tSZ signal in the millimeter, thethermal Bremsstrahlung in the X-ray, the inverse Compton emis-sion, and the hadronic emission in the γ -rays, or the associatedneutrino emission (see also Section 5 for more details). Thisis implemented in the subclass Observable of MINOT , for in-stance,
E, dN_dEdSdt = cluster.get_gamma_spectrum().
5. Adam, H. Goksu and A. Leing¨artner-Goth: Modeling the ICM (non-)thermal content and observable prediction tools
Table 2.
List of spectral models.
Model name Function Dictionary keysPowerLaw f ( E ) = A × (cid:16) EE (cid:17) − α ’name’, ’Index’ExponentialCuto ff PowerLaw f ( E ) = A × (cid:16) EE (cid:17) α × exp (cid:16) − EE cut (cid:17) ’name’, ’Index’, ’Cuto ff EnergyMomentumPowerLaw f ( p ) = A × (cid:16) pp (cid:17) − α , with E = p c + m c = (cid:16) E kin + mc (cid:17) ’name’, ’Index’, ’Mass’InitialInjection f ( E ) = A (cid:16) EE (cid:17) − α (cid:40) (1 − E / E break ) α − E < E break , α (cid:62) E (cid:62) E break ’name’, ’Index’, ’BreakEnergy’ContinuousInjection f ( E ) = A (cid:16) EE (cid:17) − ( α + (cid:40) − (1 − E / E break ) α − E < E break E (cid:62) E break ’name’, ’Index’, ’BreakEnergy’User f ( E ) = anything ’name’, ’User’, ’energy’, ’spectrum’Note: In addition to these models, the parameter cre1 loss model allows applying an energy loss to the given parameterization, thus modifying itsenergy distribution (with a radial dependence). In this case, the parameterizations given here correspond to the injection rate q ( E , r ) and not to theactual CR distribution J CR ≡ dN CR dEdV given in Eq. 15. See Section 4.3 for further details, and in particular Eq. 33 for the steady-state scenario. Table 3.
List of spatial models.
Model name Function Dictionary keysGNFW f ( r ) = P (cid:18) rrp (cid:19) c (cid:18) + (cid:18) rrp (cid:19) a (cid:19) b − ca , with r p = R / c ’name’, ’P 0’, ’c500’ or ’r p’, ’a’, ’b’, ’c’SVM f ( r ) = n (cid:20) + (cid:16) rr c (cid:17) (cid:21) − β/ (cid:16) rr c (cid:17) − α/ (cid:104) + (cid:16) rr s (cid:17) γ (cid:105) − (cid:15)/ γ ’name’, ’n 0’, ’beta’, ’r c’, ’r s’, ’alpha’, ’gamma’, ’epsilon’beta f ( r ) = n (cid:20) + (cid:16) rr c (cid:17) (cid:21) − β/ ’name’, ’n 0’, ’beta’, ’r c’doublebeta f ( r ) = n (cid:20) + (cid:16) rr c (cid:17) (cid:21) − β / + n (cid:20) + (cid:16) rr c (cid:17) (cid:21) − β / ’name’, ’n 02’, ’beta1’, ’r c1’, ’n 02’, ’beta2’, ’r c2’User f ( r ) = anything ’name’, ’User’, ’radius’, ’profile’ Radius (kpc)10 N o r m a li z e d p r o f il e ( a d u ) -modelSVM-modelGNFW-modeldouble -model Energy (GeV)10 N o r m a li z e d s p e c t r u m ( a d u ) PowerLawExponentialCutoffPowerLawPowerLaw changing E min/max MomentumPowerLawInitialInjectionContinuousInjectionPowerLaw + steady state losses
Figure 3. Left : Illustration of the di ff erent radial profiles available in the library. The β -model parameters are ( n , r c , β ) = (1 ,
50 kpc , .
7) ; the SVM model parameters are ( n , r c , β, r s , γ, (cid:15), α ) = (1 ,
50 kpc , . , , , , .
5) ; the GNFW modelparameters are ( P , r p , a , b , c ) = (1 ,
200 kpc , . , , .
3) ; and the double β -model parameters are ( n , r c , β , ( n , r c , β ) = (1 ,
50 kpc , . , , , Right : Illustration of the di ff erent spectral models available in the library. The index is set to α = . ff or break energy to E break / cut = ClusterTools subdirectory. This alsoincludes many numerical tools.
When it is defined, the cluster object also includes various ad-ministrative functions, gathered in the subclass
Admin . They can be used to display the current values of the parameters, to savethe current status of the cluster object (or load a previously savedmodel), to generate output observable products automatically(maps, profiles, and spectra), or to get the header of the currentmap. The generation of automatic plots corresponding to the var-ious observables included in
MINOT is also available using thesubclass
Plots .
6. Adam, H. Goksu and A. Leing¨artner-Goth: Modeling the ICM (non-)thermal content and observable prediction tools
In the following sections, we use di ff erent cluster models toillustrate the behavior of the MINOT code. First, we define abaseline cluster model using parametric functions in order toshow the e ff ect of changes in the modeling on the observ-ables. The baseline properties of the cluster were set using ageneralized Navarro-Frenk-White profile (GNFW, Nagai et al.2007) thermal electron pressure profile with ( P , c , a , b , c ) = (cid:16) . × − keVcm − , . , , , . , . (cid:17) , and an SVM thermalelectron number density profile with ( n , r c , β, α, r s , γ, (cid:15) ) = (cid:16) × − cm − ,
290 kpc , . , . , , , . (cid:17) . This corre-sponds to a typical massive merging cluster. The redshift wasset to z = .
02 and the mass to M = × M (cid:12) , inspiredby the Coma cluster. The CRp followed an exponential cuto ff power-law spectrum, with spectral index 2.4 and a cuto ff energyof 100 PeV. The normalization was set to have a CRp-to-thermalenergy ratio within R of 10 − , which corresponds to the typ-ical expected values, according to Pinzke & Pfrommer (2010).The CRe followed a continuous injection spectrum, with aninjection spectral index 2.3 and a break energy of 5 GeV. Thenormalization was set to have a CRe -to-thermal energy ratiowithin R of 10 − , that is, about the proton value scaled by theproton-to-electron mass ratio (i.e., a similar Lorentz factor dis-tribution was assumed for the two). The spatial profiles of bothCRp and CRe were set to the same shape as the thermal gasdensity (see Section 3.2). The magnetic field profile was set tofollow the square root of the thermal gas density and was nor-malized to have an amplitude of 5 µ G, assuming similar prop-erties as those measured for the Coma cluster (Bonafede et al.2010). This model, referred to as
Baseline in the following,was varied whenever the e ff ect of relevant quantities to the phys-ical state of the cluster or observable are illustrated.In addition to this baseline model, it is also useful to usereal clusters, with the aim of comparing our model predictionsto measurements available in the literature. To do so, we needclusters whose thermal properties have been measured and areavailable, over a wide range of spatial scales, in order to calibrateour model as well as possible. The sources targeted by the XMMCluster Outskirt project are perfectly suited for this pur-pose because the project allowed for the precise measurement ofthe thermal pressure and density profiles of nearby galaxy clus-ters from about 10 kpc to the cluster outskirts based on XMM-Newton and
Planck data (Tchernin et al. 2016; Eckert et al. 2017;Ghirardini et al. 2019). Because we are interested in the non-thermal component of the ICM, we selected clusters from the12 XCOP for which a di ff use radio halo has been observed, us-ing the GalaxyCluster database , and for which Fermi -LATconstraints have been obtained by Ackermann et al. (2014). Wefound three objects: 1) Abell 1795, a relaxed cool-core sys-tem; 2) Abell 2142, an elongated, dynamically active clusterwith a cool-core; and 3) Abell 2255, a merging cluster with ahighly perturbed core (see also the recent work by Botteon et al.2020). We thus note that these objects also present the advan-tage of sampling di ff erent dynamical states that are generallyobserved in clusters. In order to model these three clusters, weextrapolated the precisely measured density profile of the ther-mal plasma with a high-order polynomial function. The pressureprofiles were fit with a GNFW model, providing a good extrap-olation to the data. The nonthermal properties were set follow-ing what we did for our baseline cluster model, except that the XCOP, see https://dominiqueeckert.wixsite.com/xcop/ https://galaxyclusters.hs.uni-hamburg.de/ magnetic field was normalized to 5 µ G at 100 kpc. The redshift,mass, and coordinates of the cluster were taken from the XCOPdata. Table 4 summarize the properties of these clusters.
3. Physical modeling of the primary components
In this section, we discuss the physical modeling of the clus-ter. First, the global cluster properties are briefly discussed, aswell as several assumptions employed in the modeling. Thenthe properties of the thermal and nonthermal components aredetailed. The cluster modeling relies on primary base physicalquantities from which other cluster properties can be derived, inparticular in the case of the thermal component. The choice ofthe base quantities is discussed. Then, the derivation of the sec-ondary quantities that characterize the cluster are developed bothfor the thermal and nonthermal components.
Before we model the inner structure of the clusters, it is useful tocharacterize the global cluster properties, as listed in the secondblock of parameters in Table 1. The cluster location is definedin terms of redshift and sky coordinates. From the redshift, andgiven a cosmological model, the angular diameter, and luminos-ity distances are computed and used later in the code. The de-fault cosmological model is based on Planck Collaboration et al.(2016b), but can be modified if necessary.Even if it does not play a direct role in the modeling, thecharacteristic mass of the cluster M is part of the global pa-rameters. It can be used to set several internal properties of thecluster, to their universal expectation, according to the fact thatclusters are at first order self-similar objects (in particular for thethermal pressure, see Arnaud et al. 2010). The value of M alsoallows us to set the characteristic radius, R (see also Eq. 14).It is also worth emphasizing one of the global parameters,the truncation radius, which is used in MINOT in order to set aphysical boundary to the cluster, beyond which the density dropsto zero. This is not only useful for numerical issues when thecluster properties are integrated, but might be associated with theaccretion shock radius at which the kinetic energy from accretingstructures is converted into thermal energy (see, e.g., Hurier et al.2019, for the observation of such an accretion shock).In the modeling, the plasma is assumed to be fully ionizedand to follow the ideal gas law. The ions and electrons are as-sumed to be in thermal equilibrium (see Fox & Loeb 1997, fordiscussions of the electron and ion temperatures). While it mightin principle depend on radius, the hydrostatic mass bias, the he-lium mass fraction, and the metallicity of the cluster are assumedto be constant (see, e.g., Leccardi & Molendi 2008; Nelson et al.2014, for measured cluster metallicity profiles and simulationsof the nonthermal radial profile). Some of these parameters, re-lated to the global properties of the cluster, are further discussedin the following subsections.
The base thermal properties are the electron number density andthe electron pressure profiles. This choice is motivated by thefact the X-ray emission is directly sensitive to the electron num-ber density, while the tSZ e ff ect probes the electron pressure, but M is the mass enclosed within a radius R , within whichthe mean cluster density reaches 500 times the critical density of theUniverse at the cluster redshift. 7. Adam, H. Goksu and A. Leing¨artner-Goth: Modeling the ICM (non-)thermal content and observable prediction tools Table 4.
Main physical properties of the clusters used in this work for illustration. The ’Baseline’ case does not correspond to a realcluster.
Name redshift R.A., Dec. M Dynamical state Cool-core Radio emissionBaseline 0.02 – 7 × M (cid:12) Disturbed no –A1795 0.0622 207.21957, 26.589602 deg 4 . × M (cid:12) Relaxed yes Mini-haloA2142 0.0900 239.58615, 27.229434 deg 9 . × M (cid:12) Disturbed / elongated yes Giant haloA2255 0.0809 258.21604, 64.063058 deg 5 . × M (cid:12) Disturbed no Giant halo + relic Pressure (keV cm ) Baseline modelA2255A2142A1795 Density (cm ) 0246810 Temperature (keV)10 Entropy (keV cm ) Voit et al. (2005) Enclosed thermal energy (erg) 10 Gas mass ( M )10 Radius (kpc)10 HSE mass ( M ) 10 Radius (kpc)0.000.050.100.150.200.25 Gas fraction b / m Radius (kpc)10 Overdensity ( ) R Figure 4.
Thermodynamic properties of our cluster sample defined in Table 4, as computed based on the modeling of the thermalelectron pressure and density (i.e., the base quantities used to derive the others). The truncation radius is shown at r = R ) is reached inthe overdensity profile.other choices could have been made (e.g., density and tempera-ture). Generic literature parametric models are available, such asthe β -model (Cavaliere & Fusco-Femiano 1978) or one of its ex-tensions, the simplified Vikhlinin model (SVM, Vikhlinin et al.2006), which is generally used to describe thermal density clus-ter profiles. The GNFW profile is also available, and is generallyused to describe the thermal pressure profile (Arnaud et al. 2010; Planck Collaboration et al. 2013). See Table 3 for the parameter-ization of these models. In all cases, the di ff erent parameters canbe used to control the amplitude, the characteristic or transitionradius, and the slopes at di ff erent radii. In Fig. 3 we show exam-ples of these profiles for a given set of parameters.With the electron pressure, P e ( r ), and electron number den-sity, n e ( r ), profiles at hand, it is possible to compute the total gas
8. Adam, H. Goksu and A. Leing¨artner-Goth: Modeling the ICM (non-)thermal content and observable prediction tools pressure P gas ( r ) = µ e µ gas P e ( r ) , (2)and the thermal proton number density profile as n p ( r ) = µ e µ p n e ( r ) . (3)The mean molecular weights, µ gas , µ e , µ p , and µ He , are computedfrom the helium primordial abundance and the ICM metallicity,as µ gas = − Y − Z ) + Y + Z (cid:39) . µ e = − Y − Z ) (cid:39) . µ p = − Y − Z (cid:39) . µ He = Y (cid:39) . , (4)where Y (cid:39) .
27 is the helium mass fraction and Z (cid:39) .
005 isthe heavy element mass fraction (defined through the solar refer-ence metallicity multiplied by the metal abundance, see Table 1).Here, we used the approximation that N charge + N nucleon (cid:39) / N charge the number of charge and N nucleon the number ofnucleons.It is also straightforward to compute the temperature assum-ing the ideal gas law as k B T e ( r ) = P e ( r ) / n e ( r ) ≡ k B T gas ( r ) . (5)Similarly, the electron entropy index, which records the thermalhistory of the cluster (Voit 2005), can be defined as K e ( r ) = P e ( r ) n e ( r ) / . (6)Temperature and entropy are useful diagnostics of the ICM.They can show the presence of a cool core (e.g., Cavagnoloet al. 2009), which is itself related to the central AGN activityand possibly to its CR feedback onto the surrounding gas (e.g.,Ruszkowski et al. 2017). They provide information on the dy-namical state and accretion history of the cluster, which are con-nected to its CR content (e.g., radio emission that begins duringmergers, Rossetti et al. 2011).The thermal energy density stored in the gas is given by u th = n gas k B T = P gas (7)and can be integrated over the volume U th ( R ) = π (cid:90) R u th r dr (8)to obtain the total thermal energy up to radius R . This quantity isvery useful when it is compared to the amount of energy storedin the CRs.The cluster total mass within radius r , under the approxima-tion of hydrostatic equilibrium, is given by M HSE ( r ) = − r µ gas m p n e ( r ) G dP e ( r ) dr . (9)The hydrostatic mass is known to be biased with respect to theactual total mass (see, e.g., Pratt et al. 2019, for a review on thecluster mass scale). The two can be related by M tot ( r ) = M HSE ( r )(1 − b HSE ) , (10) where b HSE is the hydrostatic mass bias (see Table 1), which isassumed to be constant (see Planck Collaboration et al. 2014,for detailed discussions of the bias value, which is expected tobe b HSE ∼ . R can be computed as M gas ( R ) = π (cid:90) R µ e m p n e ( r ) r dr , (11)and it provides a measurement of the available target mass forthe interaction with CRp. The gas fraction can be derived using f gas ( r ) = M gas ( r ) M tot ( r ) . (12)The overdensity profile is computed using ρ c ( z ), the critical den-sity of the Universe, by ∆ ( R ) = M tot ( R ) π R ρ c ( z ) , (13)which allows us to extract the value of the characteristic radius, R ∆ , within which the density of the cluster is ∆ times the criticaldensity of the Universe at the cluster redshift. The value of ∆ isgenerally taken to be 500. The enclosed mass within R ∆ is then M ∆ = π ∆ ρ ref ( z ) R ∆ . (14)All the quantities defined here can be extracted as a func-tion of radius from MINOT , according to a given cluster model,using the dedicated functions that are located in the
Physics subclass. In Fig. 4 we illustrate the main thermodynamic prop-erties of our baseline cluster model and the three Abell clustermodels discussed in Section 2.7. We show the thermal electronpressure, electron number density, gas temperature, entropy, en-closed thermal energy, enclosed gas mass, enclosed hydrostaticmass, gas fraction, and overdensity contrast.Depending on the dynamical state and the presence or ab-sence of a cool core, the profiles are di ff erent. For instance,A1795 clearly presents a high-density cool core according to itstemperature and entropy profiles. Its large-scale electron pres-sure and density fall quickly, consistent with a compact, relaxedmorphology. In contrast, A2255 and our Baseline model showdisturbed cores with a high entropy floor, and their pressureand density profiles are much flatter on large scales, consistentwith a redistribution of the thermal energy in a merging event.A2142 is an intermediate case. It presents a peaked density pro-file (showing a compact core), but its pressure profile is relativelyflat on large scales, typical of disturbed clusters. Because parti-cle acceleration is expected to depend on the cluster dynamicalstate, these thermodynamic diagnosis are useful for character-izing individual clusters in the context of understanding clusterCR physics.The enclosed thermal energy is directly related to the pres-sure profile. It is particularly relevant here because it provides anormalization for the number of CRs (see Section 3.3). Based onits high pressure profile and thus thermal energy, and based onits high density (which implies a high gas mass), A2142 wouldthus be the best target to search for γ -rays from proton-proton in-teraction in our sample, assuming the same CR distribution forall clusters.The hydrostatic mass provides a direct way to measure thecluster total mass profile (given a hydrostatic bias, Eq. 10). This
9. Adam, H. Goksu and A. Leing¨artner-Goth: Modeling the ICM (non-)thermal content and observable prediction tools can be particularly relevant for modeling the γ -ray signal asso-ciated with the decay or annihilation of dark matter particles(which is usually done assuming an NFW dark matter densityprofile with a given concentration and normalization). In addi-tion, the gas fraction gives the ratio between the amount of darkmatter and the amount of gas, which is expected to provide aproxy for the dark matter signal to CR background when indi-rect dark matter searches using clusters are performed, and thusthey are an indication for the best targets. The overdensity con-trast allows us to measure the radius R (or R ), and thus thecorresponding mass.For further discussions of the thermodynamic properties ofgalaxy clusters using a similar framework, we refer to the recentwork by Tchernin et al. (2016), Ruppin et al. (2018), Ghirardiniet al. (2019), Ricci et al. (2020). The base nonthermal properties of the cluster are the magneticfield strength, the spectra and profile of the CRp, and the CRe .The CRe di ff er from the CRe because they correspond to apopulation that was accelerated from the plasma microphysics(e.g., shocks or turbulences, as for CRp), while the secondariesare the product of hadronic interactions (see Section 4 for furtherdetails). The radial models available for the nonthermal compo-nent are the same as for the thermal component (see Table 3).The spectral distributions for currently available models arelisted in Table 2. While all these models might be attributed toeither the CRp or the CRe , the initial injection (Ja ff e & Perola1973) and continuous injection (Pacholczyk 1970) models areexpected to account for electron losses and are thus poorly suitedfor CRp (see Turner et al. 2018, for discussions about the param-eterization). We note that the minimum and maximum energy ofthe CRs is part of the parameters, and it is possible to use theseparameters to truncate the spectra. By default, the minimum en-ergy of CRp and CRe corresponds to the energy threshold of theproton-proton interaction and the rest mass of the electrons, re-spectively. Finally, it is also possible to use this parametric func-tion to inject CRe and apply losses (see Section 4.2 for details)to obtain the actual electron population. In Fig. 3, example spec-tra are shown for the CRe . The e ff ect of these spectra on theobservables is shown in Section 5.The radial and spectral distributions of the CRs are currentlydecoupled, and we can express the CR distribution (i.e., the CRnumber density per unit energy) as J CR ( r , E ) = A CR f ( E ) f ( r ) , (15)where A CR is the normalization, and f ( E ) and f ( r ) are the spec-tral and radial distributions, respectively (see Table 2 and 3 foravailable models). In principle, functions f and f could bemerged into f , ( r , E ) to include a radial dependence of the spec-tral component, but this function is not yet implemented. Whenlosses are to be applied to the input distribution, a radial depen-dence a ff ects the spectrum because the losses themselves dependon the radius (see Section 4.2 for details).In order to normalize the CR distribution, we compute theenergy density that is stored between energy E and E , whichcan be expressed by integrating over the energy as u CR ( r ) = P CR ( r ) = (cid:90) E E E CR J CR ( E CR ) dE CR , (16)and it is related to the CR pressure, P CR . Here we assume that theCRs are ultrarelativistic particles, with adiabatic index Γ = /
3. The result is only weakly sensitive to the upper bound, E = E max , CRp / e , because the CR spectrum generally vanishes rapidlyfor a spectral index higher than 2. The default lower bound isset to the minimum proton energy necessary to trigger the pionproduction, E ≡ E th p , for protons and to the electron rest massfor the electrons. The total energy stored in CRs enclosed withinthe radius R can then be computed as U CR ( R ) = π (cid:90) R u CR r dr . (17)The CR-to-thermal energy density ratio is then given by x CR ( r ) = u CR ( r ) u th ( r ) , (18)or similarly, X CR ( R ) = U CR ( R ) U th ( R ) , (19)when integrated over the volume up to the radius R .In practice, the CR distribution, A CR , is normalized by settingthe value of X CR ( R ) at a given radius (e.g., R ). We note thatthis fraction is defined relative to the enclosed energy here, whileit is also common to find this definition in terms of pressure inthe literature. The two di ff er by a factor of 2 because the thermalgas is nonrelativistic, while the CRs are in the relativistic regime.The CR distributions can be integrated over energy as n CR ( r ) ≡ dNdV (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ E , E ] = (cid:90) E E J CR ( r , E ) dE (20)or radius as dNdE ( < R ) = (cid:90) R π r J CR ( r , E ) dr (21)to compute the number density profile within E and E , or thespectrum enclosed within R , respectively.In Fig. 5 we illustrate the integrated CR number density pro-files and spectra for the baseline cluster model. CRe and CRpclearly follow the same profile because they are calibrated to fol-low the thermal electron number density. The number of elec-trons and protons is nearly the same given the chosen normal-ization. The number of CRs drastically decreases when a cutin energy is applied. The spectra show di ff erent shapes for theelectrons and protons, reflecting our baseline choice (power lawfor the protons, and continuous injection with a break at 1 GeVfor the electrons), and the minimum energy is also di ff erent forthe two populations. The number of enclosed CRs naturally in-creases with increasing radius. We note that the figure would bevery similar for the real cluster models because the underlyingCR modeling is the same.In Fig. 6 we show the magnetic field profiles of our clustermodels in the left panel and the ratio between the CRp energyand the thermal energy in the right panel. The magnetic field wascalibrated on the thermal density profile as B ∝ n . e , and nor-malized to 5 µ G (at the peak for the baseline model, and at 100kpc for the real clusters). For the CR-to-thermal energy, we alsovaried the CR number density profiles using di ff erent scaling re-lations with respect to the thermal density and pressure to showthe changes in the resulting profiles. However, we note that theexact shape also depends on the shape of the thermal pressure,which is kept fixed here. Because the pressure profile decreaseswith radius, setting the CR distribution to a flatter profile leads
10. Adam, H. Goksu and A. Leing¨artner-Goth: Modeling the ICM (non-)thermal content and observable prediction tools Radius (kpc)10 d N d V ( c m ) CRe , all energiesCRe , E >100 MeVCRp, all energiesCRp, E >100 GeV Energy (GeV)10 d N d E ( G e V ) CRe , R <10 kpcCRe , R <1000 kpcCRp, R <10 kpcCRp, R <1000 kpc Figure 5.
CR properties of the baseline cluster model.
Left : CR number density profile (i.e., CR distribution integrated between E min and E max , as indicated in the legend) for CRe and CRp. Right : CR spectrum integrated over the radius up to a maximumradius, as given in the legend, for CRe and CRp. Radius (kpc)10 M a g n e t i c f i e l d s t r e n g t h ( G ) Baseline modelA2255A2142A1795 Radius (kpc)10 X C R / X C R ( R n o r m ) Isobaric scaling ( n CR P gas )Isodensity scaling ( n CR n )Flat CRp profile ( n CR = constant)Baseline + XCOP (color), with n CR n gas Figure 6.
Magnetic field and CRp-to-thermal energy properties of the baseline cluster model.
Left : Magnetic field profile obtainedby assuming a fixed magnetic field strength of 5 µ G at 100 kpc, and a scaling relative to the thermal density as B ∝ n . e . Right :CRp-to-thermal energy profile for di ff erent models of the CRp distribution. The color lines show each real cluster, using the samecolors as in the left panel, and with n CRp ∝ n gas . The black lines correspond to variation in the CRp scaling with respect to thethermal gas for the baseline cluster model, as indicated in the legend.to a deficit in the center and an increase in the outskirt for the en-ergy ratio X CR . When the CR number density profile follows thethermal density profile, the CR-to-thermal energy of cool-coreclusters is boosted in the center because the thermal pressure islow relative to the thermal density in the core (see, e.g., the caseof A1795). In all cases, the ratio was set to X CR ( R ) = − .
4. Particle interactions in the ICM
The physical properties of the ICM for its thermal and nonther-mal components have been defined in Section 3. In this section,we model the hadronic interactions that take place in the plasmaand generate secondary particles (see also Fig. 1). We also dis-cuss the loss processes that a ff ect them, in particular, the elec-trons. interactions The collision between high-energy CRp and the thermal am-bient gas produces γ -rays, electrons, positrons, and neutrinos, mainly through the production of pions, following these interac-tion chains: p + p −→ π + π − + π + + others π −→ γπ ± −→ µ ± + ν µ / ¯ ν µ −→ e ± + ν e / ¯ ν e + ¯ ν µ /ν µ. (22)When the thermal plasma is considered to be at rest withrespect to the CRs, the CRp-thermal proton collision rate perunit energy of CRp is given by dN col dE CRp dVdt = σ pp × v CRp × n p × J CRp , (23)where σ pp is the proton-proton interaction cross section, v CRp (cid:39) c is the speed of CRp, n p the number density of thermal pro-tons, and J CRp the number density per unit energy of CRp. Theproduction rate of secondary particles X per unit volume, unitenergy, and unit time can then be expressed as dN X dE X dVdt = (cid:90) + ∞ E X dN col dE CRp dVdt F X (cid:16) E X , E CRp (cid:17) dE CRp . (24)
11. Adam, H. Goksu and A. Leing¨artner-Goth: Modeling the ICM (non-)thermal content and observable prediction tools Energy (GeV)20015010050050100150200 R e l a t i v e d i ff e r e n c e w . r . t . P y t h i a ( % ) Kelner 2006Kelner 2006 (low energy)Kelner 2006 (high energy)Kafexhiu 2014 (SIBYLL)Kafexhiu 2014 (QGSJET)Kafexhiu 2014 (Geant4) Energy (GeV)1.401.421.441.461.481.501.52 d N d E d V d t / d N d E d V d t ( n o H e , n o m e t a l s ) With He, with metalsWith He, no metals Energy (GeV)10 E d N d E d V d t ( G e V c m s ) a t r = k p c e +/ edNdEdVdt E Figure 7.
Production rate of secondary particles from hadronic processes.
Left : Relative comparison of the production rate of γ -raysin the case of no helium and zero metallicity in the Kafexhiu et al. (2014) and the Kelner et al. (2006) parameterization, using thePythia8 parameterization from Kafexhiu et al. (2014) as a reference. For Kelner et al. (2006), we also show the low- and high-energy( δ -approximation) limits as dashed lines, which were combined to compute the rate over the total energy range. Middle : E ff ect ofincluding helium and metals in the model. This was done with Pythia8 using the Kafexhiu et al. (2014) parameterization. Right :Comparison of the production rate for γ -rays, electrons, positrons, and neutrinos, computed for a radius r =
100 kpc. We alsoinclude the power-law function proportional to E − . for comparison because it corresponds to the injected CRp distribution.Two ingredients are thus necessary: 1) the total inelastic crosssection of the proton-proton interaction and its evolution as afunction of energy, σ pp ( E CRp ) ; and 2) the number of secondaryparticles produced in a collision per unit energy of the pro-duced particle as a function of the initial energy of the CRp,namely F X ( E X , E CRp ). These ingredients are usually obtainedby fitting parametric functions to accelerator data, together withMonte Carlo simulations performed with sophisticated codes(e.g., Kelner et al. 2006; Kamae et al. 2006; Kafexhiu et al.2014).The Kafexhiu et al. (2014) parameterization was imple-mented in
Naima (Zabalza 2015), a publicly available Pythonpackage dedicated to the computation of nonthermal radiationfrom relativistic particle populations. The work presented hereis based on
Naima , to which the radial dimension was added,and thus, we also use the work by Kafexhiu et al. (2014) as ourbaseline.As we show below, heavy elements also contribute signifi-cantly to the particle production rate. This contribution is onlyavailable in the work by Kafexhiu et al. (2014), which is alsoexpected to have the most current data, in particular at the high-est energies. However, Kafexhiu et al. (2014) only focused onthe γ -ray production rate, while Kelner et al. (2006) also pro-vided a parameterization for the leptons (electrons, positrons,and neutrinos), but did not include heavy elements. Thereforewe employed a hybrid approach. We used the parameterizationby Kafexhiu et al. (2014) for the γ -ray production and that fromKelner et al. (2006) for the leptons. To account for heavy ele-ments in the case of leptons, we applied a rescaling of the pro-duction rate given by Kelner et al. (2006). To do so, we assumedthat the ratio between the production rate of leptons and that of γ -rays does not depend on whether heavy elements are included.This is motivated by the fact that Kafexhiu et al. (2014) ac-counted for heavy elements using a multiplicative correction tothe cross section. The production rate of leptons is finally givenby dN e ± ,ν µ, e dEdVdt = dN e ± ,ν µ, e dEdVdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Kelner2006 × dN γ dEdVdt (cid:12)(cid:12)(cid:12)(cid:12) Kafexhiu2014 dN γ dEdVdt (cid:12)(cid:12)(cid:12)(cid:12) Kelner2006 . (25)While this approach allowed us to compute the electron,positron, and neutrino production rate in the presence of he- lium and nonzero metallicity of the ICM, it uses the so-called δ -approximation for the ratio of leptons to γ -rays in the high-energy regime (see Kelner et al. 2006), which is expected tobe a relatively crude approximation (see Kafexhiu et al. 2014).Nevertheless, the accuracy of the lepton-to- γ -ray ratio is ex-pected to be a much better fit because biases in the spectra areexpected to cancel each other out.In Fig. 7 we illustrate the computation of the secondary par-ticle production rate in the case of our baseline cluster model(see Section 3), with a power-law model with index 2.4 for theCRp. The left panel shows the relative di ff erence between theKafexhiu et al. (2014) and Kelner et al. (2006) parameterizationsfor the γ -ray production rate when the helium and metal abun-dances were set to zero. We used the Pythia8 parametrizationfrom Kafexhiu et al. (2014) as a reference. In practice, the Kelneret al. (2006) parameterization is the combination of a calculationat low energy and the use of the δ -approximation at high energy,which are both shown as dashed lines. The agreement betweenPythia8 and Kelner et al. (2006) is relatively good over mostof the energy range (lower than 25% for most of it, but a peakreaches more than 100% around 100 MeV). The di ff erent high-energy parameterizations available in the work by Kafexhiu et al.(2014), namely using the Monte Carlo codes Pythia8, SIBYLL,QGSJET, or Geant4, are also shown. As expected, the di ff erencewith respect to Pythia8 is only large at high energy. It remainsbelow 25% for energies below 1 TeV, and increases to more than50% above 100 TeV (see Kafexhiu et al. 2014, for further dis-cussions). Based on the comparison of the top panels of Fig. 7,systematic uncertainties in the modeling are expected to be about30% over most of the energy range probed here.The middle panel quantifies the e ff ect of accounting for he-lium and metals in the ICM. The helium mass fraction was cho-sen to be 0.27, and the metallicity was set to the the solar value.The helium can clearly boost the signal by more than 50%, espe-cially at low energies, but has a strong e ff ect over the full energyrange where its contribution remains higher than 40%. The met-als, on the other hand, only account for percent-level changes inthe spectrum. We note that the ratio µ e /µ p depends on the ICMcomposition, which a ff ects the value of n p for fixed n e , and ex-plains why the γ -ray production rate can become lower whenmetals are included compared to the helium-only case; this isvisible around 100 MeV. Based on these results, we expect that
12. Adam, H. Goksu and A. Leing¨artner-Goth: Modeling the ICM (non-)thermal content and observable prediction tools Energy (GeV)10 d E d t ( G e V s ) Synchrotron (10 kpc)Inverse Compton (10 kpc)Bremsstrahlung (10 kpc)Coulomb (10 kpc)Synchrotron (1 Mpc)Inverse Compton (1 Mpc)Bremsstrahlung (1 Mpc)Coulomb (1 Mpc)
Figure 8.
Contributions to the energy loss rate as a function ofenergy for a radius r =
10 kpc and r = ff ect in the model amplitude that underestimates the signal byabout 40-50%.Finally, the right panel of Fig. 7 provides the particle injec-tion rate for γ -rays, electrons, and positrons, and both muonicand electronic neutrinos. The high-energy cuto ff is due to themaximum energy of the CRp, which was set to 10 PeV, while thedecrease below 1 GeV is due to the kinematic production thresh-old of the proton-proton interaction of about 1.2 GeV. Betweenthese energies, the slope of the secondary particle productionrate nearly follows that of the injected CRp, as shown by thedotted black line. While the neutrinos and the γ -rays can escape the cluster andbe detected by ground- and space-based instruments, the elec-trons evolve in the ICM and are a ff ected by several sources ofenergy loss. We considered the main sources of energy loss:synchrotron radiation, inverse Compton interaction, Coulomblosses, and Bremsstrahlung radiation.We first define the Lorentz factor of the electrons γ = Em e c ,and the reduced speed of the electrons, β = (cid:112) − /γ . The syn-chrotron radiation loss is given (in S.I. units) by Longair (2011), dEdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sync = − σ T c β γ B µ , (26)with σ T the Thomson cross section and µ the vacuum perme-ability. It is proportional to the amplitude of the magnetic fieldsquared, B . We therefore expect it to be most e ffi cient in thecentral regions of the cluster. Because of the γ dependency, thesynchrotron loss will be higher at high energy.Inverse Compton losses can be expressed in a very similarway (Longair 2011) as dEdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) IC = − σ T c β γ u CMB , (27)where the dependence on magnetic field is replaced by the ambi-ent photon field, assumed to be dominated by the CMB, whoseenergy density is given by u CMB = π ( k B T CMB (1 + z )) hc ) . (28) The inverse Compton energy losses do not depend on the clusterlocation, but increase with redshift because of the CMB depen-dence.The Coulomb losses are computed as (Gould 1972) dEdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Coulomb = − σ T n e m e c β ln m e c β (cid:112) γ − h ω p − ln (2) (cid:32) β + γ (cid:33) + (cid:32) γ − γ (cid:33) + , (29)where ω p = (cid:113) e n e m e (cid:15) is the plasma frequency. The Coulomb lossesare proportional to the thermal electron number density and aretherefore more e ff ective in the cluster core. They are almost in-dependent of energy.Finally, the Bremsstrahlung losses are computed as(Blumenthal & Gould 1970) dEdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Brem . = − α cr E ( n p + n He ) (cid:32) ln (2 γ ) + (cid:33) , (30)where α = e π(cid:15) ¯ hc is the fine-structure constant and r = e π(cid:15) m e c is the classical electron radius. We neglected elements heavierthan helium and used the completely unscreened limit that is ap-propriate for low-density plasma. As for the Coulomb losses, theBremsstrahlung losses depend on the number density of thermalnuclei, but they increase with energy.In Fig. 8 we provide the loss function for the synchrotron,inverse Compton, Bremsstrahlung, and Coulomb contributionsfor two di ff erent radii from the center, 10 and 1000 kpc, for ourbaseline model. At low energy, the Coulomb losses are expectedto dominate, while at high energy, the synchrotron and inverseCompton losses dominate, depending on the relative value ofthe magnetic field and the CMB photon field. The contributionby Bremsstrahlung is always subdominant. When CRs are injected into the ICM, their evolution is expectedto follow the di ff usion-loss equation (Berezinskii et al. 1990), ∂ n ( E , r , t ) ∂ t = − (cid:126) ∇ (cid:0) n ( E , r , t ) (cid:126) v (cid:1) + ∇ [ D ( E ) ∇ n ( E , r , t )] + ∂∂ E [ (cid:96) ( E , r ) n ( E , r )] + q ( E , r , t ) , (31)where n ( E , r , t ) ≡ dN CRe dEdV is the number of CRe per unit volumeand energy, q ( E , r , t ) is the injection rate, (cid:126) v is the ICM velocity,and D ( E ) is the di ff usion coe ffi cient, and where the loss function, (cid:96) ( E , r ) , is given by (cid:96) ( E , r ) = − (cid:32) dEdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sync + dEdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) IC + dEdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Coulomb + dEdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Brem . (cid:33) . (32)Assuming that the CRe do not significantly di ff use, and assum-ing steady-state condition, we write the number density of CReat equilibrium as dN CRe dEdV ( E , r ) = (cid:96) ( E , r ) (cid:90) ∞ E q ( (cid:15), r ) d (cid:15), (33)where q ( (cid:15), r ) is computed as the output of Eq. 25.We note that we did not account explicitly for the possi-ble reacceleration of seed electrons by ICM turbulences (e.g.,Brunetti et al. 2017, and references therein). However, becausewe did not model the details of the microphysics here, such a
13. Adam, H. Goksu and A. Leing¨artner-Goth: Modeling the ICM (non-)thermal content and observable prediction tools Radius (kpc)10 d N d E d V ( G e V c m )
10 MeV10 TeV ( ×10 ) n gas ×10, n CRp ×1/10 B ×10 Energy (GeV)10 E d N d E d V ( G e V c m )
100 kpc1000 kpc n gas ×10, n CRp ×1/10 B ×10 Figure 9. Left : Profile of the CRe , taken at 10 MeV and 10 TeV (after applying a multiplicative factor of 10 ). Right : Spectrumof the secondary electron at 100 and 1000 kpc.population might be included in the CRe , which are modeled in-dependently, as discussed in Section 3.3. By doing so, we wouldimplicitly assume that the reacceleration process, which woulde ff ectively contribute to a loss for our secondary electron pop-ulation (i.e., a population transfer) is subdominant with respectto the losses. In this case the physical consistency between thedi ff erent CR populations will not necessarily be verified. Whilereacceleration models are beyond the scope of the current work,we leave room for implementing reacceleration options in the MINOT code in the future.In Fig. 9 we present the radial profile and the spectra ofthe CRe in the steady-state approximation, with no di ff usion.The profile becomes steeper at higher energy because inverseCompton and synchrotron losses become more important in thisregime relative to the Coulomb loss, which is more e ffi cientin the core. Moreover, electrons accumulate around 100 MeVbecause lower energy electrons quickly disappear because ofCoulomb losses and higher energy electrons are more a ff ectedby inverse Compton and synchrotron losses. We also provide thesame profiles and spectra when the magnetic field and thermaldensity are higher by a factor of 10. In the latter case, we alsodecreased the number of CRp by a factor of 10 so that the rate ofproton-proton collisions was conserved. An increase in thermalplasma density leads to a much flatter profile because Coulomblosses are far higher in the core where the density is high. Anincreased magnetic field also flattens the profile, but less dras-tically because the magnetic field profile itself is flatter. In thespectrum, the magnetic field has a stronger e ff ect at high en-ergy, while the increase in thermal plasma density leads to morelosses at low energy. The peak of the secondary electron spec-trum therefore depends on the competition between the energylosses in the magnetic field and in thermal plasma.
5. Multiwavelength observables
In this section, the physical properties of the cluster (Section 3)and the production of secondary particles in the ICM (Section 4)are used to compute the observables of galaxy clusters relatedto the di ff use gas component. This includes the tSZ e ff ect, thethermal X-ray emission, the radio synchrotron emission, the in-verse Compton emission, and the γ and neutrino emission fromhadronic processes. We focus here on showing the e ff ect ofmodel changes on the observables using our baseline clustermodel. In general, the cluster observables are associated with physicalprocesses at play in the ICM, which can be described in termsof production rate (this does not strictly apply to the tSZ signalbecause it is a spectral distortion, as we discuss in Section 5.3).We define Q ( r , E ) ≡ dNdEdVdt , the emission rate associated with thephysical process considered. For instance, in the case of X-rayemission, Q would be the number of X-ray photons emitted perunit volume, per unit of time, and per unit energy in the ICM.The surface brightness (or flux per solid angle) at a pro-jected distance R from the center is therefore given by integrating Q ( r , E ) over the line of sight as dNdEdS dtd Ω ( R , E ) = D A π D L (cid:90) + ∞−∞ Q ( r ) d (cid:96) = D A π D L (cid:90) R max R rQ ( r ) √ r − R dr , (34)where the factor D A accounts for the conversion from physicalarea into solid angle, and the normalization by 4 π D L assumesthat the emission is isotropic. We note that Eq. 34 is valid in thesmall-angle approximation (i.e., assumes the cluster size to besmall against the distance to the observer), which is expected tobe accurate for our purpose because the extent of clusters neverexceeds a few degrees. It also neglects the redshift extent of thecluster.Based on Eq. 34, we now wish to compute several quantitiesaccessible from observations: 1) the surface brightness profile;2) the spectrum, by integrating the signal over the cluster volumeor the solid angle; 3) the total flux, by integrating over both thevolume and the energy (or at fixed energy); and 4) the map of thesignal, which in our case is equivalent to the surface brightnessprofile because of azimuthal symmetry, but allows us to generatespatial templates for dedicated analysis.The surface brightness profiles (and maps), are computed bylog-log integration of the quantity dNdEdS dtd Ω ( R , E ) over the re-quested energy range, or simply by fixing the energy to the onerequired by the observation (as is done, e.g., in the radio and mil-limeter domain). In the case of the map, the signal is projectedon a grid corresponding to the header (or sampling properties)set by the user. An option allows the user to normalize the mapto the total flux so that the map only accounts for the spatial de-pendence of the signal in units proportional to the inverse solid
14. Adam, H. Goksu and A. Leing¨artner-Goth: Modeling the ICM (non-)thermal content and observable prediction tools angle. This proves useful for a γ -ray analysis, for example, inwhich image templates are needed.The spectra are computed in a similar way, by log-log inte-gration over the volume. In this case, two possibilities are avail-able. 1) Integration over the solid angle within a circle of radius R max , so that the total integration volume is a cylinder. 2) Theemission rate Q ( r , E ) can be integrated spherically up to R max before normalization by 4 π D L . The two quantities only di ff erby the definition of the integration volume, and should convergewhen all the cluster emission is accounted for with increasing R max . The cylindrical integration resembles more what would beaccessible directly from observations, while the spherical inte-gration is a more natural from a physical point of view becauseit returns a quantity that is computed in a single physical (3D)radius.In order to compute the flux, we integrated over the clus-ter volume and the energy. Alternatively, the energy might befixed, as discussed above. By definition, the luminosity of agiven source within the energy band ∆ E ≡ [ E , E ] is given by L ∆ E = π D L F ∆ E . (35)We also applied the redshift stretching to the energy of thephotons. This function can be switched o ff by the user. Because of the gas temperature (a few keV) and the density(typically 10 − − − cm − ), the leading emission process atX-ray energies is thermal Bremsstrahlung (see Sarazin 1986;B¨ohringer & Werner 2010, for reviews). The X-ray emissionis thus a direct probe for the thermal gas density. It presents acharacteristic exponential cuto ff at high energies, determined bythe gas temperature. Heavy elements also induce a large num-ber of spectral lines. The X-ray surface brightness is generallyexpressed as S X = π (1 + z ) (cid:90) n e Λ ( T e , Z ) dl , (36)where Λ ( T e , Z ) is the cooling function, which varies with tem-perature.In practice, MINOT uses the
XSPEC software (Arnaud 1996) todirectly compute the counts using either the MEKAL or APECX-ray plasma spectral models. These models require the ICMabundance, the redshift, the temperature, the energy range, anda normalization defined asnorm = − π (cid:16)(cid:16) D A (cid:17) (1 + z ) (cid:17) (cid:90) (cid:18) n e (cid:19) (cid:18) n H (cid:19) dV . (37) MINOT also accounts for the foreground photoelectric absorptionusing the value of the hydrogen column density at the clusterlocation. The
XSPEC outputs are then normalized to compute theemission rate (counts or energy) per unit volume and time. Itis also possible to account for the response function of X-raysatellites, so that the outputs are normalized by the e ff ective areaof the observation. The spectrum, surface brightness profile andmaps, and flux are then extracted as discussed in Section 5.1.In Fig. 10 we illustrate these X-ray observables in the case ofour baseline cluster. The two models MEKAL and APEC agreewell. The dashed blue spectrum shows the e ff ect of the photo-electric absorption from the foreground, leading to low energycuts. At high energy, the exponential cuto ff is clearly visible, andthe spectral lines are also visible below 10 keV. The raw signal associated with the inverse Compton emission is also shown; itis discussed in Section 5.6. We note that it is well below the ther-mal X-ray emission, but might become significant with increas-ing energy. The surface brightness profile, computed between0.1 keV and 2.4 keV, drops very rapidly in the outskirt becausethe signal is proportional to the density squared. The integratedflux reaches about 1 ph cm − s − at large radii. The tSZ e ff ect distorts the CMB blackbody spectrum as a resultof inverse Compton scattering onto energetic thermal electrons(see Birkinshaw 1999; Mroczkowski et al. 2019, for reviews).Because it is a spectral distortion, it does not su ff er from redshiftdimming, and the general considerations of Section 5.1 do notstrictly apply here. The change in surface brightness is expressedas ∆ I tSZ I = y f ( x , T e ) (38)with respect to the CMB, I = k B T CMB ) ( hc ) (cid:39) . − . Theparameter y is the so-called Compton parameter, which gives thenormalization of the tSZ e ff ect. It provides a measurement of thethermal electron pressure integrated along the line of sight as y = σ T m e c (cid:90) P e d (cid:96). (39)The frequency dependence of the tSZ e ff ect is given by f ( x , T e ) = x e x ( e x − (cid:18) x coth (cid:18) x (cid:19) − (cid:19) (1 + δ tSZ ( x , T e )) , (40)where x = h ν k B T CMB . The term δ tSZ ( x , T e ) is a relativistic correctionthat introduces a small temperature dependence to the tSZ e ff ect,and becomes important when the temperature becomes higherthan about 10 keV, depending on the frequency. The relativisticcorrection was implemented following Itoh & Nozawa (2003),who are expected to be accurate at the percent level up to 50 keV.When relativistic corrections are neglected, the tSZ spectrum isnull at 217 GHz, negative below (with a minimum around 150GHz), and positive above (peaking at about 350 GHz).The tSZ integrated flux, often used to track the cluster mass,can be expressed as Y cyl ( R ) = (cid:90) R π rydr (41)in the case of cylindrical integration, or as Y sph ( R ) = σ T m e c (cid:90) R π r P e dr (42)for the spherically integrated flux. The integrated flux, Y cyl , sph ,can be expressed in units of surface or normalized by D A to behomogeneous to solid angle, as is commonly done in the litera-ture.In Fig. 11 we illustrate the spectrum and profile (shown interms of the Compton parameter) of our reference cluster. ThetSZ flux can be significant down to a few GHz, and could thus af-fect radio synchrotron observations. On the other hand, the syn-chrotron emission is not shown here, but is far lower than the tSZsignal in the considered frequency range. Given the linear sen-sitivity of the tSZ signal to the pressure, the profile is relativelyflat (e.g., compared to the X-ray surface brightness).
15. Adam, H. Goksu and A. Leing¨artner-Goth: Modeling the ICM (non-)thermal content and observable prediction tools Energy (keV)10 E d N d E dSd t ( k e V c m s k e V ) APECAPEC + Galactic abs. ( N H =10 cm )MEKALInverse Compton, no Galactic abs. Radius (kpc)10 d N dSd t d ( c m s s r ) Thermal emissionInverse Compton
Figure 10.
Observables associated with the X-ray emission.
Left : X-ray spectrum within R shown for both MEKAL and APECmodels, as well as with and without galactic photoelectric absorption. The MEKAL model is di ffi cult to distinguish because itcoincides very well with the APEC model. Right : Surface brightness profile in the band 0.1-2.4 keV. The dynamical range of theprofile amplitude has been set to the same value for all observables. Frequency (GHz)7.55.02.50.02.55.07.510.012.5 d E dSd t d ( J y ) with relativistic correctionswithout relativistic corrections Radius (kpc)10 y Figure 11.
Observables associated with the tSZ e ff ect. Left : tSZ spectrum within R . We also provide the spectrum in the casewhen relativistic corrections are neglected for illustration. Right : Compton parameter profile. The dynamical range of the profileamplitude has been set to the same value for all observables. γ -ray hadronic emission In the case of high-energy photons, the absorption by the ex-tragalactic background light (EBL, Dwek & Krennrich 2013)needs to be accounted for, which is thought to have been pro-duced by the sum of all light contributions (e.g., starlight ordust reemission) at all epochs in the Universe. While travel-ing from the cluster to the Earth, γ -rays may interact with theEBL through electron-positron pair production, and thus be ef-fectively absorbed along the way as dNdEdS dt −→ dNdEdS dt × exp ( − τ ( E )) , (43)with τ ( E ) the optical depth. EBL absorption depends on redshiftand on the energy of the γ -rays. To account for EBL absorption,we used the ebltable Python package , which reads in andinterpolates tables for the photon density of the EBL and theresulting opacity for high-energy γ -rays. This package providesdi ff erent models for the EBL based on Franceschini et al. (2008),Kneiske & Dole (2010), Finke et al. (2010), Dom´ınguez et al.(2011), and Gilmore et al. (2012). We illustrate the e ff ect of the https://github.com/me-manu/ebltable/ EBL for various models available in Fig. 12 for a cluster redshift z = .
02 . While it is crucial to account for the EBL, especiallyat very high energies, the uncertainties associated with the EBLmodels are expected to be small.The intergalactic magnetic fields probably also a ff ect the pre-dictions for the γ -ray observables (e.g., Neronov & Semikoz2009). However, because of the current uncertainties on theproperties of the magnetic field, its e ff ect remains uncertain.While this e ff ect is not yet implemented, it is being consideredfor the future improvement of the MINOT code.The γ -ray production rate resulting from hadronic interac-tion is computed following Kafexhiu et al. (2014), as describedin Section 4.1. We compute the spectrum (within R , usingspherical integration) and profile as detailed in Section 5.1.These quantities are displayed in Fig. 13 for our baseline clustermodel. The spectrum peaks at GeV energies, quickly vanishes atlower energies, and is a ff ected by a cuto ff at high energy due tothe EBL. We also show the e ff ect of the change in CRp slope onthe γ -ray spectrum for a fixed-normalization X CRp . In the caseof a flatter CRp profile, the amplitude of the spectrum is reducedbecause the number of proton-proton collisions is reduced bythe lower spatial coincidence of thermal and CRp. The inverseCompton signal computed for our baseline model is also shown
16. Adam, H. Goksu and A. Leing¨artner-Goth: Modeling the ICM (non-)thermal content and observable prediction tools Energy (GeV)10 E d N dSd E d t ( a d u ) NoneFranceschiniKneiskeFinkeDominguezDominguez-upperDominguez-lowerGilmoreGilmore-fixed
Figure 12. E ff ect of the EBL on the normalized spectra of abaseline model cluster at z = .
02. All the available models areshown as indicated in the legend. Because of the redshift stretch-ing, the shape of the γ -ray spectrum slightly changes even in thecase without EBL absorption (’none’).for comparison. It is below the hadronic emission except at lowenergy. The profile presents a compact signal because it arisesfrom the product of the thermal electron number density and theCR density. When the CR number density profile is flattened, the γ -ray signal itself becomes flatter. Nevertheless, it still decreaseswith radius even for a completely flat CR number density profilebecause the thermal density profile remains peaked. The inverseCompton signal is expected to be flatter than the signal of thehadronic emission, but it is also lower in amplitude in our base-line model. The integrated spectrum between 1 GeV and 1 TeValmost reaches 10 − photon cm − s − in the case of this baselinemodel. In contrast to the γ -rays, the neutrinos are not a ff ected by theEBL absorption. Except for the EBL, their observables are com-puted in the same way as the γ -rays associated with hadronicinteractions (Section 5.4); the production rate is computed fol-lowing a combination of Kelner et al. (2006) and Kafexhiu et al.(2014), as described in Section 4.1.Figure 14 illustrates the neutrino observable both for themuonic and electronic neutrinos in the case of the spectrum andthe profile, and for the sum of the two for the flux. Because of theneutrino oscillation, we would in practice expect a mix betweenthe ratio of neutrinos of di ff erent flavors. Because the processesassociated with the neutrino is the same as that of the γ -rays,their observables are very similar to one another, except for asmall di ff erence in the normalization. The inverse Compton emission is also a ff ected by the EBL ab-sorption in the high-energy limit. We refer to Section 5.4 andFig. 13 for this e ff ect.We use the analytical approximation for the treatment of in-verse Compton scattering of relativistic electrons in the CMBblackbody radiation field given by Khangulyan et al. (2014),which is expected to be accurate within 1% uncertainty through-out the application range. In particular, their Eq. 14 gives us thenumber of inverse Compton photons produced per unit energy and time per CRe as a function of the CRe energy, dN IC dEdt , whichwe express as dN IC dE IC dt = r m e c π ¯ h (cid:32) k B T CMB (1 + z ) E CRe (cid:33) G IC ( E IC , E CRe ) , (44)with G IC ( E IC , E CRe ) an analytical function computed followingKhangulyan et al. (2014), using the approximation given by theirEq. 24. We thus integrate this quantity over the electron energy,accounting for the amount of CRe in the ICM. The emissivity isexpressed as dN IC dE IC dVdt = (cid:90) J e ( E CRe ) dN IC dE IC dt dE CRe , (45)where J e ( E CRe ) ≡ dN CRe dE CRe dV is the CRe number density, summingthe contributions from primary and secondary electrons.In Fig. 15 we illustrate the observables associated with in-verse Compton emission for our baseline cluster model. Thespectrum shape reflects the complex processing of secondaryelectrons by their production rate in hadronic interaction, theirlosses in the ICM, and the production of inverse Compton af-ter having also summed the CRe . In particular, changing thedistribution of CRe to an initial injection model removes rela-tivistic electrons with energy higher than E break , and thus doesnot remove the inverse Compton emission at energies above ∼ E break / ff ect offlattening the profiles of either the CRe or the CRp. The inte-grated flux reaches more than 10 − ph cm − s − between 1 GeVand 100 TeV in our baseline model. The
MINOT code focuses on the di ff use galaxy cluster emissionassociated with the bulk of the X-ray emitting ICM. In the caseof the di ff use synchrotron emission, we therefore focus on theemission associated with radio halos and leave radio shocks (orrelics, see van Weeren et al. 2019) aside. Because the orienta-tion of the magnetic field is expected to be chaotic in the bulkICM regions of a galaxy cluster, we need to average the standardenergy distribution of the synchrotron emission over the direc-tions of the field orientation. To do so, we follow the results ofAharonian et al. (2010), Appendix D, in which the orientationof the magnetic field is assumed to be randomized. This pro-vides a convenient approximation with an accuracy better than0.2% over the entire energy range (see Aharonian et al. 2010, formore details on the approximation and its accuracy).As in the case of inverse Compton emission, we express dN sync dE sync dVdt = (cid:90) J e ( E CRe ) dN sync dE sync dt dE CRe , (46)
17. Adam, H. Goksu and A. Leing¨artner-Goth: Modeling the ICM (non-)thermal content and observable prediction tools Energy (GeV)10 E d N d E dSd t ( G e V c m s ) Hadronic emission, baselineCRp slope to =2.1
CRp slope to =2.8
Isodensity scaling n CRp n Inverse Compton Radius (kpc)10 d N dSd t d ( c m s s r ) Hadronic emission, baselineIsodensity scaling: n CRp n n CRp = constantInverse Compton Figure 13.
Observables associated with the γ -ray hadronic emission. The signal coming from inverse Compton interactions is alsoshown for comparison. Left : γ -ray spectrum within R . Right : γ -ray profile integrated between 1 GeV and 1 TeV. The dynamicalrange of the profile amplitude has been set to the same value for all observables. Energy (GeV)10 E d N d E dSd t ( G e V c m s ) e + e Radius (kpc)10 d N dSd t d ( c m s s r ) e + e Figure 14.
Observables associated with neutrino hadronic emission.
Left : Neutrino spectrum within R . Right : Neutrino profileintegrated between 1 GeV and 1 TeV. The dynamical range of the profile amplitude has been set to the same value for all observables. Energy (GeV)10 E d N d E dSd t ( G e V c m s ) Primary CRe onlySecondary CRe onlyTotalInitialInjectionPowerLaw ( X CR to )CRp slope to 2.8CRp slope to 2.1Isodensity scaling n CRp n Radius (kpc)10 d N dSd t d ( c m s s r ) Primary CRe onlySecondary CRe onlyTotalIsodensity scaling: n CRe1 n n CRe1 = constantIsodensity scaling: n CRp n n CRp = constant Figure 15.
Observables associated with inverse Compton emission.
Left : Inverse Compton spectrum within R , with the con-tribution from both primary and secondary electrons. Right : Inverse Compton profile integrated between 1 GeV and 1 TeV. Thedynamical range of the profile amplitude has been set to the same value for all observables.where dN sync dE sync dt = e B π (cid:15) m e c ¯ hE sync ˜ G ( E sync / E c ) . (47) The quantity E c = eB ¯ h γ m e is the synchrotron characteristic en-ergy and ˜ G ( x ) the emissivity function of synchrotron radiation,which quickly increases from x = x (cid:39) .
23 and smoothlyvanishes for increasing x (see Aharonian et al. 2010).
18. Adam, H. Goksu and A. Leing¨artner-Goth: Modeling the ICM (non-)thermal content and observable prediction tools Frequency (MHz)10 d E dSd t d ( J y ) Primary CRe onlySecondary CRe onlyTotalInitialInjectionPowerLaw ( X CR to )CRp slope to 2.8CRp slope to 2.1Isodensity scaling n CRp n tSZ absolute value Radius (kpc)10 d E dSd t dd ( J y s r ) Primary CRe onlySecondary CRe onlyTotalIsodensity scaling: n CRe1 n n CRe1 = constantIsodensity scaling: n CRp n n CRp = constant Figure 16.
Observables associated with the synchrotron emission.
Left : Synchrotron spectrum within R , with the contributionfrom both primary and secondary electrons. Right : Synchrotron emission profile at 100 MHz. The dynamical range of the profileamplitude has been set to the same value for all observables.In Fig. 16 we illustrate the observables associated with syn-chrotron emission, including the contribution from primary andsecondary electrons. As for the inverse Compton case, the emis-sion reflects the complex processing of secondary electrons,while it is more direct for the primary electrons. In particular,the curvature in the synchrotron emission is due to the lossesof CRe at high energies. When the CRe population model ischanged to an initial injection scenario, the high-frequency cur-vature is significantly enhanced by the lack of very high energyelectrons. In the case of a power-law CRe population, which ex-tends to a lower energy, the spectrum is enhanced at low fre-quency. The slope of the CRp for secondary electrons is directlyreflected in the synchrotron spectrum slope. Similarly as in thecase of hadronic γ -ray emission, flattening the CRp profile de-creases the amount of synchrotron emission because fewer sec-ondary electrons are produced. As highlighted in the figure, thetSZ contribution might be high at high frequencies, and it mightmimic a curved synchrotron spectrum if not accounted for. Thesynchrotron profile is more compact than that of the inverseCompton emission because it also depends on the magnetic field,which decreases with radius. For the inverse Compton emission,the profile is more compact for secondary electrons because itdepends on the product of the thermal density and the CRp num-ber density. We also show that the flattening of the CR popula-tion is reflected in the synchrotron profile. The flux at 100 MHzreaches typical values of 10 Jy in our baseline model.
6. Comparison to the literature
In order to validate the modeling and to further illustrate the useof the
MINOT code, we compared the model predictions to re-sults obtained in the literature and existing data. To do so, weused the models of the XCOP clusters defined in Section 2.7 andTable 4. We focused on millimeter and X-ray data for the thermalpart and on γ -rays and radio data for the nonthermal component.Because the sensitivity of current neutrino telescopes is limitedand because of the typical predicted fluxes, we leave the neutrinoemission model predictions aside. First we compared the measured X-ray luminosity given byEckert et al. (2017) to the luminosity we recover, integrated
Table 5.
Comparison of the rest frame X-ray luminosity in the[0 . −
2] keV band, computed with
MINOT and with the codefrom Eckert et al. (2017).
Cluster L X , , MINOT L X , , Eckert et al. (2017)(10 erg s − )A1795 4.10 4 . ± . . ± . . ± . within R . We used the same values of R and set the MINOT cosmological model to the one used in Eckert et al. (2017) tomitigate di ff erences. The comparison of the obtained rest frameluminosity are given in Table 5. We obtain comparable luminosi-ties, with di ff erences of a few percent for A1795 and A2142,but di ff erences of up to 19% for A2255. Because our model isbased on the interpolation of the results by the XCOP project,we expect consistency between the two. However, our modelhas been defined by extrapolating the thermal plasma densityprofiles down to small radii. While these profile were measureddown to about 10 kpc or even less for A1795 and A2142, it wasonly measured down to about 30 kpc for A2255. Thus, uncer-tainties coming from the extrapolation are likely to be larger forthis cluster. The di ff erences that we observe in Table 5 are thuslikely due to extrapolation uncertainties.We also compared the tSZ flux computed using MINOT tothe flux given in the
Planck
PSZ2 catalog (Planck Collaborationet al. 2016a). Again, our model is based on the XCOP out-puts, which are themselves obtained using
Planck data, so thatwe expect consistency. Table 6 shows that the total integratedCompton parameter (i.e., computed within 5 R ) agrees well forA1795 and A2142, and it di ff ers by 2.3 σ for A2255. BecauseA2255 is a strongly merging cluster, the flux arising from thecluster outskirt is likely to be higher than for A1795 and A2142,and the truncation radius involved in MINOT (set at 5 Mpc) maynot be enough to account for all the tSZ flux. When we increasethe truncation radius to 10 R , the flux rises to nearly 18 × − arcmin , which better agrees with the PSZ2 value. As in the caseof the X-ray luminosity, the di ff erences that we observe are thuslikely due to the interpolation and assumption that we have madein defining the clusters.
19. Adam, H. Goksu and A. Leing¨artner-Goth: Modeling the ICM (non-)thermal content and observable prediction tools
A2255
Data Model Residual
A2142A179520 arcmin X - r a y s u r f a c e b r i g h t n e ss ( s d e g ) A2255
Data Model Residual
A2142A179520 arcmin C o m p t o n p a r a m e t e r m a p ( × ) Figure 17.
Comparison between the thermal gas observables and the
MINOT model prediction.
Left : Comparison between
ROSAT
X-ray images and the
MINOT model prediction. The scale is linear from -10 to 10 s − deg − and logarithmic above. Contours areas ± × N s − deg − , with N the contour index starting at 0. Right : Comparison between the
Planck
Compton parameter MILCAimage and the
MINOT model prediction. The contours are multiples of 3 σ , where σ is the rms of the residual map. The Planck × Table 6.
Comparison of the total tSZ flux, Y R , obtained with MINOT and with the PSZ2 catalog (Planck Collaboration et al.2016a).
Cluster Y R , MINOT Y R , PSZ2(10 − arcmin )A1795 13.62 11 . ± . . ± . . ± . In addition to the fluxes, we also directly compared the X-rayand tSZ images to existing data. We used the publicly available
ROSAT (Truemper 1993) X-ray pointed data obtained for ourthree targets to produce maps in the [0 . , .
4] keV energy band.The maps were subtracted from the background and normalizedby the exposure. The
ROSAT
PSPC response matrices are ac-counted for in
MINOT through the use of
XSPEC , as well as thehydrogen column density taken at the location of each cluster, asobtained by the LAB survey (Kalberla et al. 2005). The model isprojected on the same header as the original
ROSAT data and ac-counts for the
ROSAT e ff ective area in units of counts per unit oftime and solid angle. We accounted for the point spread function(PSF) by smoothing our model with a mean e ff ective Gaussianfunction with a full width at half maximum (FWHM) of 30 arc-sec, which is the typical number expected for ROSAT pointedobservations. While the detailed analysis of the X-ray data in-cluding all instrumental e ff ects is beyond the scope of this work,this comparison already provides a useful qualitative compari-son to our modeling. The data, model, and residual images aredisplayed in the left panel of Fig. 17. The data and the modelare shown on a log scale, and the residual is shown on a linear We used ObsID rp800105n00, rp800096n00, and rp800512n00for A1795, A2142, and A2255, respectively, see https://heasarc.gsfc.nasa.gov/docs/rosat/rhp_archive.html scale. While the overall agreement is good for all three clusters,several features are evident. First, many point sources that arenot accounted for here a ff ect the residual. Then, the real clustersare not perfectly azimuthally symmetric, which is shown for theresidual with a positive and negative butterfly shape in the coreof all targets. Finally, the model of A2255 slightly overpredictsthe signal, which agrees with the prediction of the model lumi-nosity, which is too high, as discussed above.We also used the MILCA (Hurier et al. 2013) Compton pa-rameter map obtained from Planck (Planck Collaboration et al.2016c) to compare our tSZ model to real data. We extracted a1 degree × MINOT
Compton parameter map on thesame grid for comparison. We smoothed the model with a 10arcmin FWHM Gaussian beam to account for the angular reso-lution of
Planck . In the right panel of Fig. 17 we show the data,the model, and the residual. The data and the model are shownon a log scale, and the residual is shown on a linear scale. Themodel and the data agree well for all three clusters. Nevertheless,a low excess in the central part of the A2142 model is evident,which can be explained by the fact that this cluster is slightlyelongated, as is also seen in the
ROSAT image.In conclusion, we have compared the prediction from
MINOT to the X-ray luminosity and tSZ fluxes, finding an overall goodconsistency between the two. The di ff erences are likely ex-plained by uncertainties in the model extrapolation. Similarly, MINOT is able with dedicated functions to predict the X-ray andtSZ images associated with a cluster model. The comparison to
ROSAT and
Planck maps has shown an overall good consistencyfor the targets tested here. In the context of modeling the non-thermal component of galaxy clusters, it is therefore possible touse
MINOT together with X-ray or tSZ data to calibrate the ther-mal part of the model.
20. Adam, H. Goksu and A. Leing¨artner-Goth: Modeling the ICM (non-)thermal content and observable prediction tools
Table 7.
Comparison of the γ -ray (hadronic) flux prediction obtained by Ackermann et al. (2014) and this work, using the samevalue for the CR-to-thermal energy ratio. We have converted the pressure ratio into an energy ratio as defined in this work. Fluxesare given for energies E >
500 MeV, in units of s − cm − . The integration radius was set to the truncation radius (total volume). U th was rescaled using Eq. 48. Cluster X CRp ( R ) Ackermann et al. (2014) Reference model Applying U th rescaling Gas density to β -model– (10 − cm − s − )A1795 0.022 3.01 0.81 3.71 3.46A2142 0.028 3.45 1.60 4.11 4.14A2255 0.022 0.85 0.37 0.83 0.88 Energy (GeV)10 F l u x a b o v e E ( s c m ) A2255Hadronic emissionInverse ComptonA2142A1795A2255 upper limitA2142 upper limitA1795 upper limit
Figure 18.
Comparison between the cluster model flux predic-tions and the
Fermi -LAT upper limit from Ackermann et al.(2014). γ -ray constraints have been obtained by Ackermann et al. (2014)for A1795, A2142 and A2255 as part of a larger sample using Fermi -LAT data (Atwood et al. 2009). For all clusters, they pre-dicted the expected fraction of CRp pressure over the thermalpressure based on Pinzke & Pfrommer (2010) and Pinzke et al.(2011), given the mass of the clusters. They used these predic-tions together with a model for the spatial distribution of the CRsto compute flux prediction for these clusters. In order to compareour predictions to theirs, we first set the value of X CRp ( R ) tothat of Ackermann et al. (2014), taking into account the fact thatthe pressure ratio is twice the energy ratio (which we used forour parametrization). Our baseline CRp density model matchestheir baseline well (based on simulations, Pinzke & Pfrommer2010) because the CRs are tied to the gas, as they neglect CRtransport.Table 7 shows that our predictions are lower than those ofAckermann et al. (2014) by a factor of about 3. However, wenote that the masses of our selected clusters used in Ackermannet al. (2014) that were taken from Chen et al. (2007), are higherby a factor of 1.5-2.1 than ours. In order to account for this dif-ferences, we therefore rescaled our thermal energy according toself-similarity expectations as U th → U th (cid:32) M , this work M , Ackermann 2014 (cid:33) − / . (48)After we applied this rescaling, our fluxes agreed far better,within 20%. The di ff erences may arise from the γ -ray produc-tion rate modeling, scatter in the thermal pressure, or di ff erencesin the thermal gas distribution that are not necessarily the same. To determine the e ff ect of the latter, we also computed our fluxesby changing our density profiles to the best-fit β -model of thetrue density. We find that this change leads to di ff erences in the γ -ray flux of up to 7% in the case of these clusters, which issignificant.In Fig. 18 we compute the energy-integrated flux as a func-tion of energy in the case of our reference model (Section 2.7)and compare it to the upper limit set by Ackermann et al. (2014).While the predictions are relatively close to the upper limit, theyremain below it for all three clusters.In conclusion, we have shown that our model gives compara-ble predictions for the γ -ray flux compared to what is used in theliterature. However, significant di ff erences may arise as a resultof the inner structure modeling of the clusters, which is generallyignored when large samples are used. In this section, we quantitatively compare the radio predictionsof our model to measurements available in the literature. Todo so, we used the database at https://galaxyclusters.hs.uni-hamburg.de/ , in which available radio data for manyclusters are listed. All of our target clusters present di ff use ra-dio emission: a radio mini-halo for A1795 (Giacintucci et al.2014), a giant radio halo for A2142 (Giovannini & Feretti 2000;Venturi et al. 2017), and a giant radio halo plus a relic for A2255(Kempner & Sarazin 2001; Govoni et al. 2005). In the caseof A2142, we note that two components were distinguished inVenturi et al. (2017), and they likely arise from di ff erent physi-cal processes. However, in this qualitative comparison, we onlyconsidered the global emission and sum the contribution fromthe two components.In Fig. 19 we compare our models (defined in Section 2.7)to the flux measured in the literature. In order to compute themodel flux emission, we used an aperture radius that matchedthe signal from the respective articles (100, 500, and 930 kpc, forA1795, A2142, and A2255, respectively) and performed cylin-drical integration of the synchrotron emission. For each cluster,we illustrate how it is possible to qualitatively change our modelparameters to match the radio data, and also show how thesechanges translate into the γ -ray prediction and its comparison tothe upper limits of Ackermann et al. (2014).In the case of A1795, our default model underpredicts theradio flux by a factor of about 50%. Increasing the magnetic fieldby a factor of 1.4 would solve the di ff erence without a ff ectingthe γ -ray prediction. Alternatively, the number of CRp might beinceased by a factor of about 1.5, but a the cost of increasingthe γ -ray prediction by a similar amount and approaching the Fermi -LAT limit. Finally, increasing the number of CRe by afactor of 3 would also solve the di ff erence, with changes in the
21. Adam, H. Goksu and A. Leing¨artner-Goth: Modeling the ICM (non-)thermal content and observable prediction tools Frequency (MHz)10 F l u x ( m J y ) A1795 - Radio
CRe2 onlyCRe1 onlyTotal B ×1.4 X CRp ×1.5 X CRe1 ×3 Giacintucci (2014) Energy (GeV)10 F l u x a b o v e E ( s c m ) A1795 - Gamma-ray X CRp ×1.5 X CRe1 ×3 Fermi upper limit Frequency (MHz)10 F l u x ( m J y ) A2142 - Radio
CRe2 onlyCRe1 onlytSZ contributionTotalCRe2 only, CRp index + 0.5, X CRp ×0.5
CRe1 only, CRe1 index + 1, X CRe ×500 Venturi (2017) Energy (GeV)10 F l u x a b o v e E ( s c m ) A2142 - Gamma-ray
CRe2 only, CRp index + 0.5, X CRp ×0.5
CRe1 only, CRe1 index + 1, X CRe ×500 Fermi upper limit Frequency (MHz)10 F l u x ( m J y ) A2255 - Radio
CRe2 onlyCRe1 onlyTotalCRe2 only, CRp index + 0.5, X CRp ×10
CRe1 only, CRe1 index + 0.8, X CRe ×500 Kempner (2001)Govoni (2005) Energy (GeV)10 F l u x a b o v e E ( s c m ) A2255 - Gamma-ray
No CRe1, CRp index + 0.5, X CRp ×10
No CRp, CRe1 index + 0.8, X CRe ×500 Fermi upper limit
Figure 19.
Comparison of our model prediction and the radio flux observed in our three clusters (left), and the e ff ect of the radiomodels on the γ -ray constraints (right). Top : A1795.
Middle : A2142.
Bottom : A2255. γ -ray prediction only at energies below 0.1 GeV, which is barelyaccessible for Fermi -LAT.In the case of A2142, our default model overpredicts the ra-dio emission by almost an order of magnitude, and our spec-trum appears too flat compared to the data. First, we considered apurely hadronic scenario (i.e., no CRe ), for which a slope of theCRp spectrum needs to be set to about 2.9 and the amplitude re-duced by a factor of two in order to match the data. Another op-tion is to consider only CRe , in which case the spectrum slopeshould be set to about 3.3 and the normalization increased by afactor of 500 to match the data. A combination of the two sce- narii might also be used, especially because the radio emissionpresents two distinct components (see Venturi et al. 2017, fordiscussions). In both scenarii, the predicted γ -ray emission alsodecreases, and thus remains in agreement with the Fermi -LATlimit.In the case of A2255, we observe a disagreement betweenthe 1400 GHz flux from Kempner & Sarazin (2001) and Govoniet al. (2005). For our purpose, we chose to use the value ofGovoni et al. (2005) as a reference (flux of the halo alone, with-out the relic). The amplitude of our default model broadly agreeswith the observation, but our spectral index is too low. In the
22. Adam, H. Goksu and A. Leing¨artner-Goth: Modeling the ICM (non-)thermal content and observable prediction tools purely hadronic scenario, increasing the slope of the CRp spec-trum to 2.9 together with increasing its normalization by a factorof 10 would bring agreement between the model and the data.Alternatively, in a model with only CRe , we would need to in-crease the CR spectrum slope to 3.1 and multiply the normaliza-tion by 500 in order to reach agreement. In the two cases, the γ -ray flux remains below the Fermi -LAT upper limit.In this section, we showed that we can change the modelparameters to match the radio data in a qualitative way. We fo-cused on the slope and normalization of the CR content of thecluster, but opening the parameter space to spatial distributionsor a functional form of the spectra might also play an impor-tant role. In all the considered cases, it is not possible to rule outany model using γ -ray data because the limits remain too high.However, purely hadronic model predictions are just a factor ofa few below the Fermi -LAT limits in the case of these clusters.
7. Conclusions and summary
We have provided an exhaustive description of
MINOT , a newsoftware dedicated to the modeling the nonthermal componentsof galaxy clusters and predicting associated observables. Whilethe software was originally developed to describe the γ -ray emis-sion from galaxy clusters, MINOT also accounts for most of theemission associated with the di ff use ICM component: X-raysfrom thermal bremsstrahlung, tSZ signal in the millimeter band, γ -rays and neutrino emission form hadronic processes, γ -raysfrom inverse Compton emission, and radio synchrotron emis-sion. Because the γ -ray emission is connected to the same un-derlying cluster physical properties as these other observables, MNOS provides a useful self-consistent modeling of the signal,and these additional observables can be used, for example, tocalibrate a γ -ray model. However, MINOT can also be used toindependently model observables in the di ff erent bands.The software is made publicly available at https://github.com/remi-adam/minot , and this paper aims at pro-viding a reference for any user of the code. To this aim, we havediscussed the structure of the code and the interdependencies ofthe di ff erent modules in Section 2, while Sections 3, 4, and 5provided details about the physical processes considered, howthey are accounted for, and the way observables are computed.The di ff erent functions were illustrated with the use of a refer-ence cluster model. It allowed us to show the dependence of eachwavelength on the physical properties of the cluster. In Section 6we also compared the predictions from MINOT to data availablein the literature in order to show how the code can be used tomodel real data. We used
Planck and
ROSAT data for the ther-mal component, and
Fermi -LAT plus various radio data for thenonthermal component. Finally, we note that the
MINOT code iswell documented, and many examples are provided in the publicrepository.The di ff erent assumptions made in the code were discussed.In particular, the modeling relies on primary base quantities thatare used to derive secondary properties of the cluster and gen-erate observables under the assumption of spherical symmetry.The primary quantities are the thermal electron pressure anddensity profiles, the CRe and CRp profiles and spectra, and theprofile of the magnetic field strength. Regarding the CRe , theyare processed assuming no di ff usion in a steady-state scenario.However, other electron populations can be accounted for usingthe CRe . In order to set the base physical properties of the clus-ter, several predefined models are available, but it is also possibleto provide any user-defined quantity. The accuracy of the modeling was addressed. Regarding thethermal component, modeling uncertainties associated with X-rays are below the percent level, and the uncertainties of thetSZ signal are at the percent level when a relativistic correctionat high temperature is considered and much smaller otherwise.The hadronic processes ( γ -rays, neutrinos, and secondary elec-trons), on the other hand, present uncertainties at a level of typ-ically 25% above the considered energy range when the latestmodels available in the literature are considered. In addition, westress that the e ff ect of helium is about 50% of the signal andshould be accounted for (as done in MINOT ). The computationof inverse Compton and synchrotron emission is based on an-alytical approximations whose precision is expected to remainwithin 1% and 0.2%, respectively. Nevertheless, we stress thatthe main limitations of the modeling is not the accuracy of thecomputation, but the underlying assumptions discussed above.In particular the use of spherical symmetry and the assumptionof stationarity to compute the distribution of secondary electronsare likely to be the dominant sources of mismodeling.The
MINOT software can be used for a wide variety of opera-tions, and we list just a few examples here: – The joint modeling of the nonthermal emission in galaxyclusters for which detailed multi-wavelength data are avail-able. The parameters of the model can be fit jointly to suchdata to constrain di ff erent scenarios while accounting for un-certainties in the di ff erent components. – The prediction of the expected signal, based on ancillarydata, for observation proposals. For instance, it is possibleto predict the tSZ emission associated with that of an X-rayobserved cluster, assuming that the pressure follows a uni-versal profile. – The prediction of the background CR induced γ -rays in thecontext of dark matter searches. MINOT provides an easy wayto model the CR background, which needs to be marginal-ized over to obtain constraints on the nature of dark matter. – The simulation of sky maps associated with the observablesconsidered here, given a halo catalog. – Pedagogical purposes. Because it includes most of the ICMassociated processes,
MINOT can be used to understand thee ff ect of some given parameters on the observable.Historically, the understanding of the physical properties ofgalaxy clusters has strongly benefited from multiwavelength ob-servations and analysis. With the current and upcoming facilitiesaiming at exploring the nonthermal component of galaxy clus-ters, in particular in the radio and γ -ray bands, it has becomevery useful to have an easy-to-use self-consistent modeling soft-ware that allows us to predict the expected signal based on someassumptions. The MINOT software provides such a tool.
Acknowledgements. We are thankful to the anonymous referee for useful com-ments that helped improve the quality of the paper. We would like to thank R.Alves Batista for useful comments. The work of JPR and MASC was supportedby the Spanish Agencia Estatal de Investigaci´on through the grants PGC2018-095161-B-I00 and IFT Centro de Excelencia Severo Ochoa SEV-2016-0597,and the
Atracci´on de Talento contract no. 2016-T1 / TIC-1542 granted by theComunidad de Madrid in Spain. We acknowledge the use of HEALPix (G´orskiet al. 2005). This research made use of Astropy, a community-developed corePython package for Astronomy (Astropy Collaboration et al. 2013), in additionto NumPy (van der Walt et al. 2011), SciPy (Jones et al. 2001) and Ipython (P´erez& Granger 2007). Figures were generated using Matplotlib (Hunter 2007).Several modules of
MINOT are based on the
Naima software (Zabalza 2015).
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