Relativistic SZ temperature scaling relations of groups and clusters derived from the BAHAMAS and MACSIS simulations
MMNRAS , 1–21 (2019) Preprint 3 March 2020 Compiled using MNRAS L A TEX style file v3.0
Relativistic SZ temperature scaling relations of groups andclusters derived from the BAHAMAS and MACSISsimulations
Elizabeth Lee, (cid:63) Jens Chluba, Scott T. Kay and David J. Barnes , Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, UK Department of Physics, Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Accepted 2020 February 11. Received 2020 February 11; in original form 2019 December 17
ABSTRACT
The Sunyaev-Zeldovich (SZ) effect has long been recognized as a powerful cosmologicalprobe. Using the BAHAMAS and MACSIS simulations to obtain > , simulatedgalaxy groups and clusters, we compute three temperature measures and quantify thedifferences between them. The first measure is related to the X-ray emission of thecluster, while the second describes the non-relativistic thermal SZ (tSZ) effect. Thethird measure determines the lowest order relativistic correction to the tSZ signal,which is seeing increased observational relevance. Our procedure allows us to accu-rately model the relativistic SZ (rSZ) contribution and we show that a (cid:38) − underestimation of this rSZ cluster temperature is expected when applying standardX-ray relations. The correction also exhibits significant mass and redshift evolution, aswe demonstrate here. We present the mass dependence of each temperature measurealongside their profiles and a short analysis of the temperature dispersion as derivedfrom the aforementioned simulations. We also discuss a new relation connecting thetemperature and Compton- y parameter, which can be directly used for rSZ modelling.Simple fits to the obtained scaling relations and profiles are provided. These shouldbe useful for future studies of the rSZ effect and its relevance to cluster cosmology. Key words: cosmology: Theory - galaxies: clusters: intracluster medium - methods:numerical - cosmic background radiation - galaxies: clusters: general
Galaxy clusters constitute some of the largest structuresin our Universe, forming from the highest overdensities ofthe cosmic web. This makes them excellent probes for cos-mology, sensitive to fundamental cosmological parameters,such as the matter density and power spectrum (e.g., Voit2005; Allen et al. 2011; Kravtsov & Borgani 2012; Wein-berg et al. 2013), as well as interesting in their own right.These clusters, for our purposes, can be considered as giantpockets of hot ionized plasma, which induce X-ray emissionthrough both bremsstrahlung and line-emission processes(see e.g. Sarazin 1986, for a review). They are also observ-able through the Sunyaev-Zeldovich (SZ) effect (Zeldovich& Sunyaev 1969; Sunyaev & Zeldovich 1970), a unique spec-tral signature caused by the upscattering of photons fromthe Cosmic Microwave Background (CMB) by free electronswith temperatures of T e (cid:38) K (i.e., (cid:38) keV). For re- (cid:63)
E-mail: [email protected] In fact, X-rays can be induced already by plasmas at T e (cid:38) K. views of the SZ effect see e.g. Carlstrom et al. (2002) andMroczkowski et al. (2019).Galaxy clusters comprise of giant dark matter haloesin which baryonic plasma is located – while some of thisgas cools to form galaxies, the majority remains as ionizedplasma (Briel et al. 1992), also known as the intraclustermedium (ICM). The ICM is often modelled as an isothermalsphere of electrons, allowing for simple mass-temperaturerelations to be derived. However, both direct measurementsand hydrodynamical simulations indicate that clusters areneither isothermal nor spherical (e.g., Nagai et al. 2003;Vikhlinin et al. 2009). As such, instead of directly obtain-ing the thermodynamic temperature, we obtain volume-averaged temperatures, weighted according to the physicalprocess they derive from. It thus becomes necessary to un-derstand the appropriate weighting of each observable. Inparticular, it has long been established (Pointecouteau et al.1998; Hansen 2004; Kay et al. 2008) that X-ray and SZ mea-surements do give rise to different temperatures once realis-tic cluster atmospheres are being considered.The SZ distortion is dominated by the thermal SZ (tSZ)signal (Zeldovich & Sunyaev 1969), which gives rise to a © a r X i v : . [ a s t r o - ph . C O ] M a r E. Lee et al. redshift and temperature independent spectrum. However,relativistic corrections at typical cluster temperatures leadto both a broadening and drop in magnitude of this signalat fixed y -parameter. The rSZ effect is caused by the factthat for typical cluster temperature kT e (cid:39) (for mass M (cid:39) × M (cid:12) h − ) the electrons move at a fair fraction ofthe speed of light ( (cid:51) / c (cid:39) . − . ). In this case, the classi-cal non-relativistic tSZ formula (Zeldovich & Sunyaev 1969)is no longer sufficient, and higher order temperature correc-tions become relevant (Challinor & Lasenby 1998; Itoh et al.1998; Sazonov & Sunyaev 1998). The rSZ effect can be ef-ficiently modelled using SZpack (Chluba et al. 2012, 2013);however, accurate estimates for the y -weighted temperatureare required. The y -weighted temperature is also relevant toprecise computations of the SZ power spectra and the inter-pretation of SZ data from Planck , as rSZ can cause biasesin cosmological parameters such as σ (Remazeilles et al.2019).In this paper, we examine the differences between threetemperature measures; the first is a proxy for the observedX-ray temperature T sl (the so-called spectroscopic-like tem-perature; Mazzotta et al. 2004); the second is a proxy forthe Compton- y parameter, T m (i.e., the mass-weighted tem-perature), which is closely related to the line-of-sight pres-sure and Compton- y parameter through a cluster; the thirdis a measure that allows us to account for the relativistictemperature correction to the tSZ distortion (rSZ), T y (the y -weighted temperature; Hansen 2004; Remazeilles et al.2019). The latter in particular so far has not been studiedsystematically; the current analyses, both from individualcluster measurements ( ? Hansen 2004; ? ; ? ; Chluba et al.2013) and stacking procedures (Hurier 2016; Erler et al.2018), provide only one measurement of rSZ with signifi-cance greater than 3 σ ( ? ) by relying on the assumption thatkSZ is negligible. However, due to the growing sensitivity ofplanned an ongoing CMB experiments, rSZ is now cominginto reach, and future observations with the Simons Obser-vatory and CCAT-prime ought be able to extract this signalmore accurately.Precise SZ power spectrum calculations furthermoredepend directly on the clusters’ average pressure and y -weighted temperature profiles. Cluster pressure profiles havebeen extensively studied using simulations (e.g., Nagai et al.2003; Battaglia et al. 2010, 2012) and also have been cal-ibrated against X-ray observations (Arnaud et al. 2010;Planck Collaboration et al. 2013). The y -weighted temper-ature profiles again have not been studied directly but willaffect the precise shape of the relativistic temperature powerspectrum (Remazeilles et al. 2019), which could become anovel cluster observable (Basu et al. 2019; Remazeilles &Chluba 2019) for future CMB missions similar to CORE (Melin et al. 2018) and
PICO (Hanany et al. 2019). Here,we carry out a comparative study of various temperatureprofiles with a particular focus on obtaining a new prescrip-tion of the y -weighted temperature profiles.We base our study on BAHAMAS (McCarthy et al.2017, 2018) and MACSIS (Barnes et al. 2017a), two gianthydrodynamical simulations generating over 14,000 haloesof mass (cid:39) M (cid:12) to × M (cid:12) with outputs at redshiftsof z = , 0.5, and 1. These allow us to generate temperature–mass relations for each of our temperature measures as wellas a detailed understanding of their temperature profiles. To concur with the work of Barnes et al. (2017a), we alsoconsider the effects of restricting our analysis to only the hotand relaxed subsets of clusters within our samples. We will,however, find little change in the conclusions for these cases.Moreover, since clusters are not isothermal, further cor-rections to the observed SZ signal must arise. This comesfrom the understanding that the distortions are caused byelectrons of varying temperature along the line of sight, andthus will not be completely modelled by a single tempera-ture. The first corrections to the signal can be found througha temperature moment expansion (Chluba et al. 2012, 2013)and is related to the dispersion of y -weighted temperatureswithin clusters, which we study systematically here. Our re-sults suggest that this dispersion scales at around (cid:39) percent of the cluster temperature, but overall leads to negligi-ble corrections to the rSZ signal (see Sect. 6.1.2). Finally, wewill briefly discuss the relevance of rSZ to determinations of H through SZ measurements (Cavaliere et al. 1979; Birkin-shaw et al. 1991; Hughes & Birkinshaw 1998; Reese et al.2002), showing that it could lead to a systematic shift in thederived H values if rSZ is neglected.The paper is structured as follows: we clarify the math-ematical meaning behind each of the considered tempera-ture measures and their purposes in Sect. 2 and describethe simulations used in Sect. 3. In Section 4, we discuss thecluster-averaged temperature measures and in Sect. 5 theprofiles across the clusters follow. Finally, we discuss the ef-fects of these temperature measures on common observablesin Sect. 6 and conclude in Sect. 7. In this section, we discuss cluster masses and self-similarredshift scaling relations and how they are related to simu-lation quantities. We will then describe the formulations ofour three temperature measures; the spectroscopic-like tem-perature, a proxy for X-ray temperatures, and the mass and y -weighted temperatures, both related to the SZ effect. Themass-weighted temperature will be seen to be a proxy for theintegrated electron pressure, or the Compton- y parameter,within clusters, while the y -weighted temperature character-izes the precise shape of the SZ distortion. Finally, we willdiscuss the higher order y -weighted temperature moments,and their relationship to the observed SZ signals. In general, we can define our dark matter haloes to bespheres colocated with the cluster such that the total mass M ∆ contained within a radius R ∆ is given by M ∆ = R ∆ ∆ ρ crit ( z ) . (1)Here, ρ crit is the critical density for a flat universe. We set ∆ = for our main analysis (although a discussion for ∆ = is presented in Appendix C). In such a halo, for anisothermal sphere of gas, we can find the temperature to be k B T ∆ = GM ∆ µ m p R ∆ . (2)This is equivalent to the virial temperature and can be usedas a reference. As usual, G is the gravitational constant, m p MNRAS000
PICO (Hanany et al. 2019). Here,we carry out a comparative study of various temperatureprofiles with a particular focus on obtaining a new prescrip-tion of the y -weighted temperature profiles.We base our study on BAHAMAS (McCarthy et al.2017, 2018) and MACSIS (Barnes et al. 2017a), two gianthydrodynamical simulations generating over 14,000 haloesof mass (cid:39) M (cid:12) to × M (cid:12) with outputs at redshiftsof z = , 0.5, and 1. These allow us to generate temperature–mass relations for each of our temperature measures as wellas a detailed understanding of their temperature profiles. To concur with the work of Barnes et al. (2017a), we alsoconsider the effects of restricting our analysis to only the hotand relaxed subsets of clusters within our samples. We will,however, find little change in the conclusions for these cases.Moreover, since clusters are not isothermal, further cor-rections to the observed SZ signal must arise. This comesfrom the understanding that the distortions are caused byelectrons of varying temperature along the line of sight, andthus will not be completely modelled by a single tempera-ture. The first corrections to the signal can be found througha temperature moment expansion (Chluba et al. 2012, 2013)and is related to the dispersion of y -weighted temperatureswithin clusters, which we study systematically here. Our re-sults suggest that this dispersion scales at around (cid:39) percent of the cluster temperature, but overall leads to negligi-ble corrections to the rSZ signal (see Sect. 6.1.2). Finally, wewill briefly discuss the relevance of rSZ to determinations of H through SZ measurements (Cavaliere et al. 1979; Birkin-shaw et al. 1991; Hughes & Birkinshaw 1998; Reese et al.2002), showing that it could lead to a systematic shift in thederived H values if rSZ is neglected.The paper is structured as follows: we clarify the math-ematical meaning behind each of the considered tempera-ture measures and their purposes in Sect. 2 and describethe simulations used in Sect. 3. In Section 4, we discuss thecluster-averaged temperature measures and in Sect. 5 theprofiles across the clusters follow. Finally, we discuss the ef-fects of these temperature measures on common observablesin Sect. 6 and conclude in Sect. 7. In this section, we discuss cluster masses and self-similarredshift scaling relations and how they are related to simu-lation quantities. We will then describe the formulations ofour three temperature measures; the spectroscopic-like tem-perature, a proxy for X-ray temperatures, and the mass and y -weighted temperatures, both related to the SZ effect. Themass-weighted temperature will be seen to be a proxy for theintegrated electron pressure, or the Compton- y parameter,within clusters, while the y -weighted temperature character-izes the precise shape of the SZ distortion. Finally, we willdiscuss the higher order y -weighted temperature moments,and their relationship to the observed SZ signals. In general, we can define our dark matter haloes to bespheres colocated with the cluster such that the total mass M ∆ contained within a radius R ∆ is given by M ∆ = R ∆ ∆ ρ crit ( z ) . (1)Here, ρ crit is the critical density for a flat universe. We set ∆ = for our main analysis (although a discussion for ∆ = is presented in Appendix C). In such a halo, for anisothermal sphere of gas, we can find the temperature to be k B T ∆ = GM ∆ µ m p R ∆ . (2)This is equivalent to the virial temperature and can be usedas a reference. As usual, G is the gravitational constant, m p MNRAS000 , 1–21 (2019)
Z temperature scalings the proton mass and µ the mean molecular weight of theplasma . We note that in this work we always use the true(simulation) mass of clusters rather than any proxy for theobserved mass, (e.g., the hydrostatic mass used in Barneset al. 2017a), which may introduce biases. Scaling relations : Assuming self-similarity, cluster tempera-tures are a simple function of their mass and redshift (Kaiser1986). We can recall that the critical density of the Universeis ρ crit ≡ H π G E ( z ) E ( z ) ≡ H ( z ) H = (cid:113) Ω m ( + z ) + Ω Λ . (3)Here, H is the Hubble constant and the exact form of E ( z ) iscosmology dependent. From this, a simple geometrical con-sideration and an assumption of isothermality in the viralsphere gives us that M ∆ ∝ E ( z ) R ∆ T ∆ ∝ E / ( z ) M / ∆ . (4) Temperature measures : Since clusters are not isothermal, wemust instead define weighted averaged temperatures appro-priate to each observable (i.e., X-ray, SZ and rSZ effect).That is, (cid:104) T (cid:105) ≡ ∫ w T d V ∫ w d V , (5)where, as we will discuss in the rest of this section, it hasbeen found that for spectroscopic-like, mass-weighted and y -weighted temperatures we have w = n T − α (Mazzotta et al.2004), n and n T , respectively, and α (cid:39) . . Connection to simulations : To find all aforementioned quan-tities from our simulations we must discretize this process.We first ignore all particles with a temperature lower than . K as they make a negligible contribution to the totalX-ray or SZ emission (cf., Barnes et al. 2017a). We can thenconvert our weighted volume integrals, to weighted sums,recalling that µ m p n d V = d m . With this procedure we cancompute the various temperature measures discussed below. X-ray emission, from hot clusters ( k B T (cid:38) keV), is pri-marily caused by bremsstrahlung radiation within the ICM,and as such has classically been modelled by the emission-weighted temperature. This can be motivated from a simpleconsideration of the X-ray surface brightness, S x = π ( + z ) ∫ n ( l ) Λ ee ( T ( l ) , Z ) d l . (6)Here, n ( l ) , T ( l ) are the electron density and temperaturealong line of sight l and Λ ee ( T , Z ) is the X-ray emissivitymeasured by the instrument within the energy band usedfor the observation; z is the clusters redshift and Z is themetallicity of the ICM. The mean molecular mass is set to µ = . . This cut-off is in large part due to the the dominance of emissionlines rather than bremsstrahlung in the observed X-ray spectrabelow these temperatures.
Spectroscopic-like temperature : It has been shown that, dueto the non-isothermality of the gas, it is more appropriateto use a modified weighting determined by fitting the X-rayspectrum with a thermal emission model (Mazzotta et al.2004; Vikhlinin 2006). This leads to the spectroscopic-liketemperature, T sl ≡ ∫ n T − α d V ∫ n T − α d V (7)where α (cid:39) . . When compared to observational results,this matches well with data from both Chandra and
XMM-Newton , provided the temperatures are all sufficiently high,e.g., kT (cid:38) . keV. Hydrodynamical simulations have beenused to calibrate T sl to the observed ‘X-ray’ temperaturesand confirm the differences between various X-ray derivedtemperatures weightings (e.g., Mathiesen & Evrard 2001;Rasia et al. 2014; Biffi et al. 2016).We also can see that both measures, Eq. (6) and (7),lead to an n dependence in the X-ray temperature measure-ments – and in general a higher weighting of cooler, densergas. This indicates (see Sect. 4.1 and 5) that the X-ray mea-surements are far poorer probes of the outskirts of clusters,where the electron density drops significantly, compared tothe SZ measurements, which we will see has a linear depen-dence on n . This has also been seen observationally since itrequires very long exposures to observe the outskirts of clus-ters through X-ray emission (e.g., Simionescu et al. 2011). Mass-weighted temperature : The classical tSZ effect gives riseto an intensity distortion that can be written in terms of theCompton- y parameter as (Zeldovich & Sunyaev 1969): ∆ I ν ≈ I y x e x ( e x − ) (cid:18) x e x + x − − (cid:19) ≡ I yg ( x ) . (8)The spectral function g ( x ) for the tSZ effect is definedhere implicitly. To characterize the photon energy we use x = h ν / k B T CMB , where T CMB is the temperature of the CMB, k B the Boltzmann constant. The intensity normalizationconstant furthermore is I = ( k B T CMB ) ( hc ) = . (cid:20) T CMB . (cid:21) MJy / sr . (9)The Compton- y parameter, as previously mentioned, is di-rectly related to the integrated electron pressure, P e , alongthe line of sight and is typically written as, y ≡ ∫ k B Tm e c d τ = σ T k B m e c ∫ nT d l = σ T m e c ∫ P e d l . (10)Here, τ is the Thomson scattering optical depth and all theother constants have their usual meaning.The second equality in Eq. (10) leads to an expressionfor the mass-weighted temperature, when we extend this for-malism to a volume-averaged, rather than a line-of-sight,integral: T m ≡ ∫ nT d V ∫ n d V = ∫ T d m ∫ d m . (11)Here m now refers to the mass of the electron gas. We can MNRAS , 1–21 (2019)
E. Lee et al. see that the volume averaged Compton y parameter is Y = σ T k B m e c ∫ nT d V ∝ M T m (12)where M denotes the total gas mass, i.e., Y is the total ther-mal energy of the gas. y -weighted temperature : Including the rSZ corrections, theSZ distortion is no longer temperature/mass independentand the dimensionless signal has to be expressed as S ( ν ) = ∆ I / I = y f ( ν, T e ) . Since the temperature varies within eachcluster, we can then consider a temperature moment ex-pansion about some pivot temperature ¯ T e for each cluster,as detailed in Chluba et al. (2013) and Remazeilles et al.(2019), to obtain S (cid:96) m ( ν ) (cid:39) f ( ν, ¯ T e ) y (cid:96) m + f ( ) ( ν, ¯ T e ) y ( ) (cid:96) m + f ( ) ( ν, ¯ T e ) y ( ) (cid:96) m (13)to second order in ∆ T e = T e − ¯ T e . For more generality, we haveexpressed S ( ν ) using spherical harmonic coefficients, intro-ducing y ( k ) (cid:96) m = [ ∆ T k e y ] (cid:96) m , where [ X ] (cid:96) m denoted the sphericalharmonic expansion of X . These will become relevant for the y -weighted temperature measure introduced below. Here,the derivatives of the SZ signal are f ( k ) = ∂ kT f ( ν, T ) and, ifapplied to isothermal clusters, one has y iso , ( k ) (cid:96) m = ∆ T k e y (cid:96) m .Equation (13) motivates the use of a pivot tempera-ture that eliminates the first-order term in S ( ν ) . We there-fore introduce the y -weighted temperature, by requiring ∫ y ( ) d V = , i.e., T y ≡ ∫ [ T y ] d V ∫ y d V = ∫ y T d V ∫ y d V = ∫ nT d V ∫ nT d V . (14)Thus, setting ¯ T e = T y , removes the first-order correction tothe volume average of S , yielding (cid:104) S ( ν )(cid:105) (cid:39) f ( ν, ¯ T e )(cid:104) y (cid:105) + f ( ) ( ν, ¯ T e )(cid:104) y ( ) (cid:105) . (15)As shown below, even the second-order term can becomerelevant for our simulation clusters. This is consistent withprevious studies (Chluba et al. 2013), but here we deriveexplicit scaling relations.In Planck Collaboration et al. (2016a), the assumptionthat f ( ν, ¯ T e ) (cid:39) f ( ν, ) , or equivalently that the observed sig-nals are well-modelled by the classical tSZ distortion, wasused. However, in Remazeilles et al. (2019), it was shownthat due to rSZ this is insufficient. Relativistic correctionswill lead to a lower amplitude of the SZ signal at fixed y -parameter as well as broadening of the SZ signal, whichcauses a miscalibration and underestimation of the trueCompton- y values for each cluster. In Sect. 6.1, we find thatthis results in a (cid:39) − per cent correction to the derived y -parameters for typical clusters, and thus is worth quanti-fying further. Higher order temperature moments : While using the y -weighted temperature removes the first-order correction tothe SZ signal, higher order terms proportional to y ( k ) (cid:96) m re-main. We thus define the volumetric y -weighted temperaturemoments as T yk = ∫ ∆ T k e y d V ∫ y d V = ∫ y ( T − T y ) k d V ∫ y d V . (16) Table 1.
Cosmological parameters used in the BAHAMAS andMACSIS simulations.Simulation Ω Λ Ω m Ω b σ n s h † BAHAMAS 0.6825 0.3175 0.0490 0.8340 0.9624 0.6711MACSIS 0.6930 0.3070 0.0482 0.8288 0.9611 0.6777 † where h ≡ H /(
100 km s − Mpc − ) From this we see that T y = and T y = . While we couldtheoretically expand to arbitrarily many orders of ∆ T , in thispaper we will consider only the lowest order correction, i.e., T y . We can see that this is closely related to the intrinsicvariance of the electron temperature within the cluster gas.To match the dimensionality of the y -weighted temperature,we will later discuss σ ( T y ) = ( T y ) / instead, which providesa proxy for the standard deviation of temperature variationwithin clusters.The higher order temperature moments further changethe detailed shape of the SZ signal, and thus may causeadditional biases to SZ measurements if omitted (Chlubaet al. 2013). We will see that from simulations this standarddeviation is around (cid:39) per cent of the cluster temperature(Sect. 4.3); however, overall this is likely to only lead to a (cid:46) . per cent correction in y (see Sect. 6.1.2). Compton- y power spectra : As discussed in Remazeilles &Chluba (2019), to correctly calculate the tSZ power spec-trum, we need temperature profiles and in particular the y -weighted temperature profiles. They show that for the tSZpower spectrum one requires a y -weighted or C yy (cid:96) -weightedtemperature as a pivot. This demands that for each multi-pole (cid:96) , (cid:104) y ∗ (cid:96) y ( ) (cid:96) (cid:105) = , for an isotropic homogeneous, sphericalcluster. For an isothermal temperature profile for each clus-ter, this yields k ¯ T yy e ,(cid:96) = (cid:104) kT e ( M , z )| y (cid:96) | (cid:105)(cid:104)| y (cid:96) | (cid:105) = C T e , yy (cid:96) C yy (cid:96) . (17)This assures only second-order terms in ∆ T e remain in thetheoretical tSZ power spectrum, C tSZ (cid:96) ( ν ) ∝ | y (cid:96) m | . With theoutputs from this work we can improve the calculation byusing explicit temperature profiles and their Fourier trans-forms for the computation of the relativistic temperaturepower spectra. We use a combined sample of clusters from the BAHAMASand MACSIS simulations, both of which we explain in moredetail below. From the BAHAMAS project (McCarthy et al.2017), we obtain > , haloes with masses M ≥ M (cid:12) . However, these simulations provide a limited numbers ofhigh mass clusters. These are supplemented by the compat-ible MACSIS project (Barnes et al. 2017a), which generated In the work Chluba et al. (2013), a different definition for the SZtemperature moments is used. First, they take the mass-weightedtemperature moments T mk , so that their moments are weightedby n d V rather than y d V . Furthermore, they use dimensionlessmoments ω ( k ) = T mk + /( T m ) k + . In the limit of many moments,the definitions in terms of T m and T y are equivalent and yieldthe same result. MNRAS000
100 km s − Mpc − ) From this we see that T y = and T y = . While we couldtheoretically expand to arbitrarily many orders of ∆ T , in thispaper we will consider only the lowest order correction, i.e., T y . We can see that this is closely related to the intrinsicvariance of the electron temperature within the cluster gas.To match the dimensionality of the y -weighted temperature,we will later discuss σ ( T y ) = ( T y ) / instead, which providesa proxy for the standard deviation of temperature variationwithin clusters.The higher order temperature moments further changethe detailed shape of the SZ signal, and thus may causeadditional biases to SZ measurements if omitted (Chlubaet al. 2013). We will see that from simulations this standarddeviation is around (cid:39) per cent of the cluster temperature(Sect. 4.3); however, overall this is likely to only lead to a (cid:46) . per cent correction in y (see Sect. 6.1.2). Compton- y power spectra : As discussed in Remazeilles &Chluba (2019), to correctly calculate the tSZ power spec-trum, we need temperature profiles and in particular the y -weighted temperature profiles. They show that for the tSZpower spectrum one requires a y -weighted or C yy (cid:96) -weightedtemperature as a pivot. This demands that for each multi-pole (cid:96) , (cid:104) y ∗ (cid:96) y ( ) (cid:96) (cid:105) = , for an isotropic homogeneous, sphericalcluster. For an isothermal temperature profile for each clus-ter, this yields k ¯ T yy e ,(cid:96) = (cid:104) kT e ( M , z )| y (cid:96) | (cid:105)(cid:104)| y (cid:96) | (cid:105) = C T e , yy (cid:96) C yy (cid:96) . (17)This assures only second-order terms in ∆ T e remain in thetheoretical tSZ power spectrum, C tSZ (cid:96) ( ν ) ∝ | y (cid:96) m | . With theoutputs from this work we can improve the calculation byusing explicit temperature profiles and their Fourier trans-forms for the computation of the relativistic temperaturepower spectra. We use a combined sample of clusters from the BAHAMASand MACSIS simulations, both of which we explain in moredetail below. From the BAHAMAS project (McCarthy et al.2017), we obtain > , haloes with masses M ≥ M (cid:12) . However, these simulations provide a limited numbers ofhigh mass clusters. These are supplemented by the compat-ible MACSIS project (Barnes et al. 2017a), which generated In the work Chluba et al. (2013), a different definition for the SZtemperature moments is used. First, they take the mass-weightedtemperature moments T mk , so that their moments are weightedby n d V rather than y d V . Furthermore, they use dimensionlessmoments ω ( k ) = T mk + /( T m ) k + . In the limit of many moments,the definitions in terms of T m and T y are equivalent and yieldthe same result. MNRAS000 , 1–21 (2019) Z temperature scalings
390 clusters with M > M (cid:12) . The MACSIS simulationswere designed to match the hydrodynamical properties ofthe BAHAMAS simulations and use compatible cosmologies(see Table 1).We note that there is a small redshift discrepancy be-tween the BAHAMAS sample at z = . and the MACSISsample at z = . . However, since the redshift dependenceof our quantities are slight (as we discuss below) this requiresno correction. Further we acknowledge there is a mismatchin cosmological parameters, however, we believe that thishas little effect on our measured values, and again, is leftunadjusted.In this section, we highlight the key properties of thesesimulations and discuss how we combine the samples. Wealso discuss the subsamples used within the work of Barneset al. (2017a) for hot and relaxed clusters, and define theversions we will explore later in this paper. We also explainthe core excision procedure used for X-ray observations andhow it is recreated in simulations. The BAHAMAS simulation (McCarthy et al. 2017, 2018) isa calibrated version of the model used in the cosmo-OWLSsimulations (Le Brun et al. 2014). Following this work, theBAHAMAS simulation consists of a 400 Mpc/ h periodicbox. For the simulations used in this paper, the cosmologi-cal parameters used are consistent with those from Planck × particles, yielding a dark mat-ter mass of m DM = . × M (cid:12) / h and initial baryon particlemass of m gas = . × M (cid:12) / h . The Plummer equivalent grav-itational softening length was fixed to 4 kpc/h in comovingunits for z > and in physical coordinates thereafter. Thesimulations were run with a version of the smoothed parti-cle hydrodynamics code p-gadget3 , which was last publiclydiscussed in Springel (2005) but has since been greatly mod-ified to include new subgrid physics as part of the ambitiousOWLS project (Schaye et al. 2010). The feedback calibrationwas set to match the observed gas mass fraction of groupsand clusters and galaxy stellar mass function at z = (seeMcCarthy et al. 2017, for details). As already mentioned, to extend the BAHAMAS simula-tions to higher mass haloes the MACSIS project (describedin detail in Barnes et al. 2017a) was developed. This entailsa sample of 390 massive clusters. To obtain this numberof massive clusters, with current computational resources,the MACSIS sample was generated using a zoomed simula-tion technique from a very large volume Dark Matter onlysimulation. This ’parent’ simulation was a periodic cubewith a side length of 3.2 Gpc. The cosmological parame-ters were taken from the
Planck M FoF > M (cid:12) , and grouping them into logarithmically spacedbins of width ∆ log ( M FoF ) = . . The bins with masses Table 2.
Selected halo counts with M > M (cid:12) , and with amass cut between the BAHAMAS and MACSIS samples at thegiven values.Redshift BAHAMAS MACSIS M , cut / M (cid:12) . × . / . a . × . × a that is, 0.5 for BAHAMAS and 0.46 for MACSIS. above . M (cid:12) had less than 100 haloes each and all wereselected. The other bins were further subdivided, each into10 logarithmic bins, from each of which 10 haloes were ran-domly selected – this ensured the sample is not biased tolow masses by the steep slope of the mass function.These selected clusters were then re-simulated using thezoomed simulation technique (Katz & White 1993; Tormenet al. 1997) to recreate the chosen sample at an increasedresolution compared to the parent simulation. Both a DMonly and full gas physics resimulation was then carried out.The latter, which we use in this work, had a dark mattermass of m DM = . × M (cid:12) / h and gas particle inital massof m gas = . × M (cid:12) / h . The softening length was fixed asin the BAHAMAS simulation. The simulations were againrun with the same version of the smoothed particle hydro-dynamics code p-gadget3 . The resolution and softening ofthe zoom re-simulations were deliberately chosen to matchthe values of the periodic box simulations of the BAHAMASproject. Barnes et al. (2017a) further shows that the MAC-SIS clusters reproduce the observed mass dependence of thehot gas mass, X-ray luminosity and SZ signal at redshift z = and z = We combine these simulations to allow for clear comparisonwith the work in Barnes et al. (2017a), taking only haloeswith M > M (cid:12) . Further we take a mass cut at each red-shift, as detailed in Table 2, above which we take only MAC-SIS haloes and below which we take only BAHAMAS haloes.The final halo counts at each redshift are detailed there.Haloes are identified in both simulations through the friends-of-friends method described in McCarthy et al. (2017). Thecentre of these haloes is taken to be the minimum of the lo-cal gravitational potential, and any sub-haloes lying outsidea given characteristic radius, R ∆ , are ignored. It is a common technique in X-ray observations to excludethe central regions of clusters to reduce the scatter in X-rayproperties. These core-excised quantities are often consid-ered to be better mass proxies (Pratt et al. 2009). Withinsimulations, this can have an added effect of reducing the po-tential impact of the central (more uncertain) physics insidethe cores. In the work of Barnes et al. (2017a), the excludedregion is that of r < . R .Theoretically it would be possible to core excise all ofour volume averaged quantities, not just the X-ray calcu-lations. However, it can be seen that while T sl has a largecorrection under core-excision – raising the temperatures in-creasingly at higher masses, but undergoing a more complex MNRAS , 1–21 (2019)
E. Lee et al.
Table 3.
Selected halo counts with M > M (cid:12) , and with amass cut between the bahamas and macsis samples at the givenvalues for the Hot and Relaxed samples.Redshift bahamas macsis M , cut / M (cid:12) M , min / M (cid:12) Hot Sample0 271 295 . × . × . / . a
87 263 . × . × . × . × Relaxed Sample0 165 188 . × . × . / . a
50 178 . × . × . × . × a that is, 0.5 for bahamas and 0.46 for macsis . increase across the entire mass range – both T y and T m un-dergo very minimal modifications [the mean corrections are ( T CE − T full )/ T CE = − . ± . and − . ± . for eachmeasure respectively ].In general, SZ measurements are flux and resolution lim-ited and the full volume average is taken (since taking coreexcised values would be difficult in practice). We will thususe the full volume averages for T m and T y , but the coreexcised values for T sl . As previously noted, the models for the X-ray temperatures,all rely on continuum emission, while at low cluster tempera-tures the effects of spectral lines begin to seriously affect theobserved X-ray spectra. Accordingly, following the analysisof Mazzotta et al. (2004), we note that the spectroscopic-like temperature is validated only for higher temperatures.This motivates the use of a Hot Sample, where T sl is a morereliable proxy for the X-ray emission. To avoid biases, we in-troduce a mass cut by finding the minimal mass that fulfills T sl ( M ) ≥ . – this ensures that the maximal tempera-ture at a given mass is T sl ( M ) (cid:38) . . These cuts ariseat log ( M ) = . , 14.32, 14.28 M (cid:12) for z = , 0.5 and 1respectively, with the results summarized in Table 3.The final sample is a relaxed sub-sample of these Hotclusters. Although there are many ways to define a relaxedhalo (see e.g. Neto et al. 2007; Duffy et al. 2008; Klypin et al.2011; Dutton & Macci`o 2014; Klypin et al. 2016; Barneset al. 2017b), in this paper we follow the criteria used inBarnes et al. (2017a), that is X off < . f sub < . λ < . , where X off is the distance offset between the point of min-imum gravitational potential in a cluster and its centre ofmass, divided by its virial radius; f sub is the mass fractionwithin the virial radius that is bound to substructures and λ is the spin parameter for all particles within R . It should These are the values for the volume average over R ; over R instead, arguably a more applicable volume for SZ measurements,these corrections reduce to − . ± . and − . ± . re-spectively. In the work of Barnes et al. (2017a), they take the smallersample of all clusters with T sl > keV be noted that, as in Barnes et al. (2017a), this is not a smallsample of the most relaxed objects, but instead a simplemetric to remove those that are significantly disturbed. To understand the cluster-wide, i.e., volume-averaged tem-peratures, it is instructive to first consider the contributionsto each temperature measure, given by each part of the clus-ter. These lead to variations between the temperature mea-sures calculated over spheres of regions R (as typical forX-ray measurements) and R (a proxy for the viral ra-dius and arguably more applicable for SZ measurements).In this work we will present all our figures with respect tothe R sphere, but tabulate all our fits for both regions inthe Appendix. In this section we will discuss both of theseelements, and present our results for the volume-averagedtemperature measures from the simulations. These allow usto generate both temperature–mass scaling relations as wellas some temperature–temperature scaling relations. Finally,we will discuss the volume-averaged values for σ ( T y ) , thestandard variation of T y within clusters. From an illustrative point of view, we can examine the dif-ferent temperature measures over clusters through the pro-jected temperatures in a selection of clusters. These, as canbe seen in Figure 1, give us an indicative understanding ofvarious features (e.g., shocks, outflows, sub-haloes and fila-mentary behaviours) that might exist within haloes undereach temperature measure.While we generally see that T y > T m > T sl 7 , it is alsothe case that at larger radii, T y is more susceptible to thestructures within the haloes. This can be seen by the in-crease in visibility of features in the haloes from the T sl tothe T y projections. This can be understood fairly simply: T sl depends on the square of the local density, so in regionsof high density – i.e., the core of the cluster or in dramaticsubstructures, this will be clearly visible. On the other hand, T y depends on both the local temperature and density (i.e.,the local pressure), so it is more affected by areas of diffuse,but warm gas, and thus highlights shocks. This particularlyweights the observed temperatures in the outer regions, e.g., R → R which is barely probed by the X-ray temper-atures (as reflected in T sl ). We see that T m typically liesbetween these other two temperature measures. First, we will discuss the difference between the temperaturemeasures averaged over spheres of R and R , and thenquantify the temperature relations to the cluster masses,and the covariance between these values for each tempera-ture measure. We will then discuss both the temperature–temperature fits and the fits for the Hot and Relaxed sub-samples. This can be seen especially in the features, but is generallyevident in the slightly brighter overall colours of the halos fromleft to right. MNRAS000
50 178 . × . × . × . × a that is, 0.5 for bahamas and 0.46 for macsis . increase across the entire mass range – both T y and T m un-dergo very minimal modifications [the mean corrections are ( T CE − T full )/ T CE = − . ± . and − . ± . for eachmeasure respectively ].In general, SZ measurements are flux and resolution lim-ited and the full volume average is taken (since taking coreexcised values would be difficult in practice). We will thususe the full volume averages for T m and T y , but the coreexcised values for T sl . As previously noted, the models for the X-ray temperatures,all rely on continuum emission, while at low cluster tempera-tures the effects of spectral lines begin to seriously affect theobserved X-ray spectra. Accordingly, following the analysisof Mazzotta et al. (2004), we note that the spectroscopic-like temperature is validated only for higher temperatures.This motivates the use of a Hot Sample, where T sl is a morereliable proxy for the X-ray emission. To avoid biases, we in-troduce a mass cut by finding the minimal mass that fulfills T sl ( M ) ≥ . – this ensures that the maximal tempera-ture at a given mass is T sl ( M ) (cid:38) . . These cuts ariseat log ( M ) = . , 14.32, 14.28 M (cid:12) for z = , 0.5 and 1respectively, with the results summarized in Table 3.The final sample is a relaxed sub-sample of these Hotclusters. Although there are many ways to define a relaxedhalo (see e.g. Neto et al. 2007; Duffy et al. 2008; Klypin et al.2011; Dutton & Macci`o 2014; Klypin et al. 2016; Barneset al. 2017b), in this paper we follow the criteria used inBarnes et al. (2017a), that is X off < . f sub < . λ < . , where X off is the distance offset between the point of min-imum gravitational potential in a cluster and its centre ofmass, divided by its virial radius; f sub is the mass fractionwithin the virial radius that is bound to substructures and λ is the spin parameter for all particles within R . It should These are the values for the volume average over R ; over R instead, arguably a more applicable volume for SZ measurements,these corrections reduce to − . ± . and − . ± . re-spectively. In the work of Barnes et al. (2017a), they take the smallersample of all clusters with T sl > keV be noted that, as in Barnes et al. (2017a), this is not a smallsample of the most relaxed objects, but instead a simplemetric to remove those that are significantly disturbed. To understand the cluster-wide, i.e., volume-averaged tem-peratures, it is instructive to first consider the contributionsto each temperature measure, given by each part of the clus-ter. These lead to variations between the temperature mea-sures calculated over spheres of regions R (as typical forX-ray measurements) and R (a proxy for the viral ra-dius and arguably more applicable for SZ measurements).In this work we will present all our figures with respect tothe R sphere, but tabulate all our fits for both regions inthe Appendix. In this section we will discuss both of theseelements, and present our results for the volume-averagedtemperature measures from the simulations. These allow usto generate both temperature–mass scaling relations as wellas some temperature–temperature scaling relations. Finally,we will discuss the volume-averaged values for σ ( T y ) , thestandard variation of T y within clusters. From an illustrative point of view, we can examine the dif-ferent temperature measures over clusters through the pro-jected temperatures in a selection of clusters. These, as canbe seen in Figure 1, give us an indicative understanding ofvarious features (e.g., shocks, outflows, sub-haloes and fila-mentary behaviours) that might exist within haloes undereach temperature measure.While we generally see that T y > T m > T sl 7 , it is alsothe case that at larger radii, T y is more susceptible to thestructures within the haloes. This can be seen by the in-crease in visibility of features in the haloes from the T sl tothe T y projections. This can be understood fairly simply: T sl depends on the square of the local density, so in regionsof high density – i.e., the core of the cluster or in dramaticsubstructures, this will be clearly visible. On the other hand, T y depends on both the local temperature and density (i.e.,the local pressure), so it is more affected by areas of diffuse,but warm gas, and thus highlights shocks. This particularlyweights the observed temperatures in the outer regions, e.g., R → R which is barely probed by the X-ray temper-atures (as reflected in T sl ). We see that T m typically liesbetween these other two temperature measures. First, we will discuss the difference between the temperaturemeasures averaged over spheres of R and R , and thenquantify the temperature relations to the cluster masses,and the covariance between these values for each tempera-ture measure. We will then discuss both the temperature–temperature fits and the fits for the Hot and Relaxed sub-samples. This can be seen especially in the features, but is generallyevident in the slightly brighter overall colours of the halos fromleft to right. MNRAS000 , 1–21 (2019)
Z temperature scalings T sl − M = . × T m T y − M = . × − − M = . × − r/R − . . . . . . . . T /T
Figure 1.
A comparison of the projected temperatures through a range of clusters at z = , relative to T for that cluster. Theseprojections are taken within spheres of radius R about the cluster centre of potential. From left to right we see T sl , T m and T y , andfrom top to bottom clusters of various masses. Since these are just the projections for single clusters, they are subject to variations fromthe median expected behaviours. To guide the eye, on each plot a dotted line at R has been drawn, alongside a hatched region at . × R , which would be the core-excised region. These clusters have been chosen with T sl , > . keV so that it is an appropriateproxy for the X-ray temperature. R or R It is important to determine the difference between averag-ing over spheres of radii R and R . X-ray measurements,in particular, are almost always taken over R , and as such R values are those commonly used in the literature. How-ever, it can be argued that R , as a better proxy for thevirial radius, should also be widely considered. Since R generates a smaller region, it encapsulates only the hottercore with less of the cooler outskirts of the cluster. As such,regardless of temperature measure, it returns a higher tem-perature than that obtained within R .This can be seen graphically in Figure 2. Here, we have plotted the fractional variation between R and R val-ues. These appear to be predominantly redshift independent;while there are variations between each redshift, they are allwithin the scatter. Secondly we see that for all measuresthe differences between the two measures become smaller athigher masses. This may in fact be an averaging effect due tothe the distribution of temperatures in clusters (see Section5), and the mass-dependent changes to the profiles and thusthe fall-off of temperatures nearer the outskirts of clusters.These will lead to the averaged effects that can be seen here.We see in general that the changes to T m are the mostacute, followed by T sl , with T y undergoing the smallest cor-rections. However, this is still a sizeable effect: (cid:39) per centat M = M (cid:12) ( (cid:39) per cent for T m ). This indicates MNRAS , 1–21 (2019)
E. Lee et al. . . . . . . T s l , / T s l , z = 0 z = 0 . z = 11 . . . . . . T m / T m z = 0 z = 0 . z = 110 M [M (cid:12) ]1 . . . . . . T y / T y z = 0 z = 0 . z = 1 Figure 2.
A comparison of the temperature measures dependingon whether they are calculated over a sphere of radius R or R against the mass of the same clusters. We also display theredshift dependence of the same. The vertical dot-dashed, dashedand dotted lines here depict the mass cut offs between the BA-HAMAS and MACSIS samples. The datapoints and errors showthe median, 84 th and 16 th values for various mass bins, while thesolid line and shaded regions demonstrate the best fits (discussedin Section 4.2.2) for the same. that this should potentially be considered in more detail forfuture SZ measurements.For the rest of the paper we will use R to reproducethe results commonly cited in cluster papers – the analysishas also been carried out across a radii of R with fewqualitative variations. The full tabulated numerical resultscan be found in Appendix C. In Figs. 3 and 4, we display the temperature mass scaling re-lationships for our three temperature measures at each red-shift. Figure 3 shows the redshift dependence of each temper- E ( z ) − / T [ k e V ] T sl z = 0 z = 0 . z = 11310 E ( z ) − / T [ k e V ] T m z = 0 z = 0 . z = 1 10 M [M (cid:12) ]1310 E ( z ) − / T [ k e V ] T y z = 0 z = 0 . z = 1 Figure 3.
A comparison of the three temperature measures atthree different redshifts. The plotted points show the medians ofthe binned data, with the error bars demonstrating their 16 th and84 th percentiles. The solid lines show the fits to the data, with theshaded regions showing the 68 per cent confidence region. Thehorizontal lines in the top panel show the 3.5 keV cutoff for thereliability of T sl as a proxy for the X-ray temperature. ature measure individually, relative to self-similar scaling –i.e., scaling out E ( z ) / ; while Figure 4 shows the the resultsdivided through by T m , the mass-weighted temperature, sothat the variations between the three measures are more vis-ible. We see, first, that the spread in the data is far larger for T sl than for T y or T m . This furthers the common observationthat the SZ signal, Y SZ , provides a tighter mass proxy thanthe X-ray signal. In Figure 3, we can see that in general, the redshiftvariation of each temperature measure is similar to the Though, of course, this has many factors, and generally relieson the accurate calibration of the SZ mass relation.MNRAS000
A comparison of the three temperature measures atthree different redshifts. The plotted points show the medians ofthe binned data, with the error bars demonstrating their 16 th and84 th percentiles. The solid lines show the fits to the data, with theshaded regions showing the 68 per cent confidence region. Thehorizontal lines in the top panel show the 3.5 keV cutoff for thereliability of T sl as a proxy for the X-ray temperature. ature measure individually, relative to self-similar scaling –i.e., scaling out E ( z ) / ; while Figure 4 shows the the resultsdivided through by T m , the mass-weighted temperature, sothat the variations between the three measures are more vis-ible. We see, first, that the spread in the data is far larger for T sl than for T y or T m . This furthers the common observationthat the SZ signal, Y SZ , provides a tighter mass proxy thanthe X-ray signal. In Figure 3, we can see that in general, the redshiftvariation of each temperature measure is similar to the Though, of course, this has many factors, and generally relieson the accurate calibration of the SZ mass relation.MNRAS000 , 1–21 (2019)
Z temperature scalings . . . . . T / T m z = 00 . . . . . T / T m z = 0 . M [M (cid:12) ]0 . . . . . T / T m z = 1 T y T m T sl Figure 4.
A comparison of the three temperature measures on acluster by cluster basis. Here we consider the temperature mea-sures with respect to T m . The solid lines indicate the line of bestfit of the data sets, while the dotted lines show the 16 th and 84 th percentiles. The horizontal dot-dashed line lies at T / T m = . toguide the eye. In the case against T y we can see that the mini-mum values of the means lie at 1.11, 1.16 and 1.21 for z = , 0.5and 1 respectively. self-similar relation – i.e., T ∝ E ( z ) / . In particular, whilewith increasing redshift T y falls a little at low massesand has a slightly steeper mass dependence, overall the y -weighted temperature is consistent within the interclus-ter variation with self-similar evolution. The mass-weightedtemperature shows more departures from self-similarity, and T sl shows the greatest departure from this E ( z ) / scaling.The spectroscopic-like temperature both falls in magnitudeand has increasing curvature, indicating that at the highestmasses, the differences under redshift evolution are magni-fied. From a physical point of view, this can be understoodsince at higher redshifts, the haloes have had a shorter cool-ing time, leading to denser cooler gas, and thus a lower T sl .However, the pressure of the gas is largely fixed to matchthe potential wells of the haloes themselves (as they areroughly in hydrostatic equilibrium) and reduces the redshift- dependent T y , which is less affected by the evolution of theclusters themselves.In Figure 4, we can see that T y has a larger magnitudethan T m and T sl , while the latter two are at points consistent,with T m higher at both higher and lower redshifts. Further-more, we see hints of a strong cluster by cluster correlationin the values of T y and T m , from the (cid:38) − per centshift between these two values. This may be a consequenceof the calibration scheme used in defining the spectroscopic-like temperature, which is focused on clusters at low red-shifts with masses M (cid:39) M (cid:12) , but more work wouldhave to be done to fully analyse this effect. In fact, withrespect to T sl we can see that there is a correction for T y of (cid:38) per cent (or (cid:38) percent) at z = ( z = ), increas-ing greatly to both higher and lower masses with equalityaround . × M (cid:12) ( . × M (cid:12) ). We also find thatthe differences between these three temperature measuresincrease strongly with redshift. We see that at z = , for in-stance, T m and T sl lie within each other’s uncertainties, whileby z = they are clearly separated. This means that account-ing for these corrections will become even more importantwhen considering distant clusters, which are typically thosemore easily probed through the SZ signal.We can find in general that our data are well modelledby a 3-parameter fit, which corresponds to a quadratic equa-tion in log–log space. We will express our values as E ( z ) − / T = A (cid:18) MM fid (cid:19) B + C log ( M / M fid ) . keV , (18)where M fid = × h − M (cid:12) . Hence, a self-similar fit around M (cid:39) M fid , would be given by B = / . By simply examin-ing these fit values , as tabulated in Table 4, we can im-mediately see the differences between the three temperaturemeasures. Here, we have also tabulated the scatter aboutthe best fit relation by calculating the root mean squareddispersion across all the haloes according to σ log T = (cid:118)(cid:117)(cid:116) N N (cid:213) i = [ log ( T i / T fit )] , (19)where i indexes all the haloes at a given redshift and T fit isthe value given by the best fit at the mass, M i , associatedwith the halo.In particular, we see, as previously observed in Figs. 3and 4, that T y appears to be systematically higher than T m ,which itself lies above T sl . The gradients of these three tem-perature measures seem to match this same pattern. Finallywe note that T y always have a positive curvature, while T sl has a strong negative curvature and T m seems to developcurvature at higher redshifts. Further, we note that none ofthese are consistent with hydrostatic equilibrium scalings,which would have B = / and C = . While T y has the clos-est gradients to this value for hydrostatic equlibrium, evenat the highest cluster masses the curvature is not sufficientfor T y to match this scaling. MNRAS , 1–21 (2019) E. Lee et al.
Table 4.
Best fit values for the medians of each temperaturemeasure at each redshift. The errors are determined through boot-strap methods. The fit parameters correspond to those describedin Equation (18). M A B C (cid:104) σ log T (cid:105) z = . T y . + . − . . + . − . . + . − . . ± . T m . + . − . . + . − . . + . − . . ± . T sl . + . − . . + . − . − . + . − . . ± . z = . T y . + . − . . + . − . . + . − . . ± . T m . + . − . . + . − . − . + . − . . ± . T sl . + . − . . + . − . − . + . − . . ± . z = . T y . + . − . . + . − . . + . − . . ± . T m . + . − . . + . − . − . + . − . . ± . T sl . + . − . . + . − . − . + . − . . ± . − . . . T s l − . . . T m - . . T sl − . . . T y - . . T m - . . T y Figure 5.
A representation of the covariance of log ( T data ( M )/ T fit ( M )) for the three temperature measures at z = . That is, a comparison of the overall distributions aboundthe line of best fit for each temperature measure. The diagonalparts show the overall distributions for each measure, while thelower triangle shows the contours of these covariances. It is now instructive to understand the spread of cluster tem-peratures about the best fits of the temperature measuresas displayed in the previous section. In Fig. 5, we show the These fits are for the median of the distributions, in AppendixC the fits to the 84 th and 16 th percentiles of the data set can befound to clarify the cluster-to-cluster spread in temperatures. covariances at z = of the quantity log ( T data ( M )/ T fit ( M )) foreach of the three temperature measures. Here T fit ( M ) is thetabulated best-fit value, while T data ( M ) refers to the calcu-lated temperature measure for each cluster. We find thatthis behaviour is replicated well for z = . and 1.0.We can immediately see from the diagonal part that,while T m is almost normally distributed in the log–log space(that is, log-normally), the other two temperature measureshave visible skews. This is most apparent for T sl , which skewsto higher temperatures with a long tail to lower tempera-tures, while the y -weighted temperature measure seems onlygently skewed to lower temperatures – thus being almostlog-normally distributed in the log–log space.Furthermore, from the lower triangle we can see thecorrelations between the temperature measures within eachcluster – in particular the strong interdependence between T y and T m . This indicates that on a cluster-by-cluster basisthe difference between the y -weighted and mass-weightedtemperatures are maintained. However, the spectroscopic-like temperature seems to be distributed independently ofthe other two measures.This strong correlation in the values of T m and T y mo-tivates the exploration of temperature–temperature scalingrelations – and moreover, since these two temperatures de-fine the complete SZ signal, they motivate a volume averaged Y − T y scaling relation. This allows for a self-calibration of therelativistic corrections to the SZ signal, from measurementsof the SZ signal itself. Temperature - temperature scaling relations As an alternative to a temperature–mass relations, we canconsider temperature–temperature scaling relations. Theselead to a predominantly mass-independent conversion be-tween temperature measures. We see that a similar fittingformula [to that in equation (18)] can be used, replacing M fid with T fid = , T = A (cid:18) T rel T fid (cid:19) B + C log ( T rel / T fid ) . keV . (20)Since we have already discussed that the cluster tempera-ture is often a good mass proxy we will not discuss thesefits in much detail here as they take a very similar form tothose against the mass, although the full tables fitting thetemperature relations with respect to T rel = T m and T ∆ canbe found in Appendix C. While it is true that, due to thecovariance of T y and T m , we find that the spread in the fitsof T y against T m are smaller than those against M , thiseffect is minimal.A shortened selection of the fits against T can befound in Table 5. First, we see that T y is always the clos-est temperature measure to T , the temperature assumingthe cluster is an isothermal sphere (agreeing with Kay et al.2008). However, we can see that there is significant curvaturein all of these fits alongside the gradient of the temperaturemeasures being significantly lower than that for T , indi-cating further that the assumption of isothermality oftenused in SZ cluster calculations is inaccurate. In fact, we findthat while T is an overestimate of T y for the most mas-sive clusters, it becomes an underestimate for lower mass,cooler, clusters, particularly at higher redshifts. This is likelydue to the increased AGN feedback effects driving gas from MNRAS000
A representation of the covariance of log ( T data ( M )/ T fit ( M )) for the three temperature measures at z = . That is, a comparison of the overall distributions aboundthe line of best fit for each temperature measure. The diagonalparts show the overall distributions for each measure, while thelower triangle shows the contours of these covariances. It is now instructive to understand the spread of cluster tem-peratures about the best fits of the temperature measuresas displayed in the previous section. In Fig. 5, we show the These fits are for the median of the distributions, in AppendixC the fits to the 84 th and 16 th percentiles of the data set can befound to clarify the cluster-to-cluster spread in temperatures. covariances at z = of the quantity log ( T data ( M )/ T fit ( M )) foreach of the three temperature measures. Here T fit ( M ) is thetabulated best-fit value, while T data ( M ) refers to the calcu-lated temperature measure for each cluster. We find thatthis behaviour is replicated well for z = . and 1.0.We can immediately see from the diagonal part that,while T m is almost normally distributed in the log–log space(that is, log-normally), the other two temperature measureshave visible skews. This is most apparent for T sl , which skewsto higher temperatures with a long tail to lower tempera-tures, while the y -weighted temperature measure seems onlygently skewed to lower temperatures – thus being almostlog-normally distributed in the log–log space.Furthermore, from the lower triangle we can see thecorrelations between the temperature measures within eachcluster – in particular the strong interdependence between T y and T m . This indicates that on a cluster-by-cluster basisthe difference between the y -weighted and mass-weightedtemperatures are maintained. However, the spectroscopic-like temperature seems to be distributed independently ofthe other two measures.This strong correlation in the values of T m and T y mo-tivates the exploration of temperature–temperature scalingrelations – and moreover, since these two temperatures de-fine the complete SZ signal, they motivate a volume averaged Y − T y scaling relation. This allows for a self-calibration of therelativistic corrections to the SZ signal, from measurementsof the SZ signal itself. Temperature - temperature scaling relations As an alternative to a temperature–mass relations, we canconsider temperature–temperature scaling relations. Theselead to a predominantly mass-independent conversion be-tween temperature measures. We see that a similar fittingformula [to that in equation (18)] can be used, replacing M fid with T fid = , T = A (cid:18) T rel T fid (cid:19) B + C log ( T rel / T fid ) . keV . (20)Since we have already discussed that the cluster tempera-ture is often a good mass proxy we will not discuss thesefits in much detail here as they take a very similar form tothose against the mass, although the full tables fitting thetemperature relations with respect to T rel = T m and T ∆ canbe found in Appendix C. While it is true that, due to thecovariance of T y and T m , we find that the spread in the fitsof T y against T m are smaller than those against M , thiseffect is minimal.A shortened selection of the fits against T can befound in Table 5. First, we see that T y is always the clos-est temperature measure to T , the temperature assumingthe cluster is an isothermal sphere (agreeing with Kay et al.2008). However, we can see that there is significant curvaturein all of these fits alongside the gradient of the temperaturemeasures being significantly lower than that for T , indi-cating further that the assumption of isothermality oftenused in SZ cluster calculations is inaccurate. In fact, we findthat while T is an overestimate of T y for the most mas-sive clusters, it becomes an underestimate for lower mass,cooler, clusters, particularly at higher redshifts. This is likelydue to the increased AGN feedback effects driving gas from MNRAS000 , 1–21 (2019)
Z temperature scalings Table 5.
Best fit values for the medians of each temperaturemeasure against T at each redshift. The errors are determinedthrough bootstrap methods. The fit parameters correspond tothose described in Equation (20). T rel = T A B C z = . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . z = . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . − . + . − . T sl . + . − . . + . − . − . + . − . z = . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . − . + . − . T sl . + . − . . + . − . − . + . − . Table 6.
Best fit values for the medians, 84 th and 16 th percentilesof T y to Y at each redshift. The errors are determined throughbootstrap methods. The fit parameters correspond to those de-scribed in Equation (21). T Y − Y A B C z = . median . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . z = . median . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . z = . median . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . these lower mass systems. This would lead to a decreased T (which is mass dependent) compared to the y -weightedtemperature. Volume-averaged Y relations : As already noted, T m forms astrong proxy for the volume averaged y -parameter, Y . Sincethis relates to the amplitude of the SZ signal, while the shapeis dependent on T y , it is instructive to consider the scaling of T y with respect to Y . This gives us a self-calibrated scalingrelationship to determine the rSZ signal. We use a fit similarto equations (18) and (20), T y = A (cid:18) YY fid (cid:19) B + C log ( Y / Y fid ) . keV , (21) Table 7.
Best fit values for the medians of each temperaturemeasure for the Hot and Relaxed Samples against M at eachredshift. The errors are determined through bootstrap methods.The fit parameters correspond to those described in Equation(18), taking C = .Hot Sample Relaxed Sample M A B A B z = . T y . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . z = . T y . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . z = . T y . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . where we take Y fid = . . These results are shownin Table 6 – we also tabulate the Y – M relationship in Ap-pendix C. It is interesting to observe that while we haveused 3-parameter fits, there is significantly less curvaturein all of these fits to that seen in our mass-temperatureand temperature–temperature relations. Further, combiningequations (4) and (12), we could expect from self-similarity, T ∝ ( Y E ) / . Although, this proportionality depends on therelationship between T m and T ∆ , we can see that in Table 6, B lies close to the expected value B = / .We note that there is no explicit redshift dependence inthese fits – since we would expect from self-similarity both T y and Y to scale with E ( z ) / . However, we do see dis-tinct redshift evolution in our fit parameters; in particularin the normalisation factor, A , which seems to almost scale ∝ E ( z ) / (similar to, but above, the self-similar prediction),increases dramatically towards higher redshifts. We see asimilar but smaller decrease in the gradient to higher red-shifts. However, overall this dependence shows that at higherredshifts it becomes increasingly important to consider therelativistic corrections to the SZ signal. Finally, it is useful to consider the behaviours of the Hotand Relaxed samples, as defined in Section 3.5, for whichthe median fits are found in Table 7. Here, we have fittedboth the hot and relaxed samples with a simple 2 parame-ter model , or equivalently, we have taken Equation (18),setting C = .While we can see variations in the medians between Since we ultimately find little difference between these valuesand those for the whole combined sample, these 2 parameter fitsallow for comparison with other fits found in previous studies.MNRAS , 1–21 (2019) E. Lee et al. . . . . σ ( T y ) / T y z = 00 . . . . σ ( T y ) / T y z = 0 . M [M (cid:12) ]0 . . . . σ ( T y ) / T y z = 1 Figure 6.
The redshift evolution of σ ( T y ) = (cid:113) T y with respect to T y , as a function of M . The scattered points show the wholedataset, the error bars show the same data binned and the straightline shows the 2 parameter best fit of the data. the Hot and relaxed samples, we also find that the 16 th and 84 th percentiles are wider for the relaxed sample, sothat these two samples give fits that lie within each other’scluster-to-cluster variance. Further, they agree well withthe 3-parameter combined sample fits for both T m and T y ,though the fit can be found to be less appropriate for thespectroscopic-like temperature due to the strong curvaturein the T sl combined sample fits.We can also find that, while the relaxed fits’ larger 68per cent error region for T y and T m seems to be well cen-tred over the errors predicted by the complete combinedsample fits, for T sl these extend to higher temperatures, in-dicating that Relaxed clusters are more likely to have higherspectroscopic-like temperatures. We can understand this as T sl is largely driven by the denser central region, and sincemore spherical (i.e., more relaxed) clusters are more likelyto have a larger region for the same given mass, they arelikely to lead to higher observed values for T sl . y -weighted Temperature Dispersion As noted in Section 2.3, the second moment of the y -weighted temperature ( T y ) is a measure of the variance ofthe temperature distribution within the cluster . Here wediscuss σ ( T y ) , the standard deviation and its comparisonto T y . We recall that under a temperature moment expan-sion about T y , the leading order correction is proportionalto [ σ ( T y )] [see Eq. (15)].In Figure 6, we explore σ ( T y )/ T y and can see that, whilethere is a small variation of the values across the mass range,they are well approximated by a power law (i.e., straightlines in the log–log space) – which are tabulated in the Ap-pendix (Table C5 ). Generally we can see that, at higherredshifts, σ ( T y )/ T y increases and that at all redshifts in-creases slightly with increased mass, approximately scalingas σ ( T y )/ T y (cid:39) . ( + z ) . [ M / M fid ] . . Since this red-shift evolution closely matches the evolution of T y with re-spect to T , it may be that σ ( T y ) is mainly dependent on T or equivalently the potential well of the cluster ratherthan the specifics of the substructure. That is, the variationin σ ( T y )/ T y with redshift with respect to mass, is domi-nated by the near self-similar redshift evolution of T y . It isalso possible that there is an effect of clusters thermalizingover time, since this would explain the increase in variancefor larger clusters and clusters at higher redshifts. However,since there are no clear differences between the dispersion ofrelaxed sample and the combined sample there is little evi-dence either way. In Section 5.2, we will explore the radialprofiles of these values to see that these clusters see almostconstant values across the whole width of the clusters, sothat the overall dispersion is indicative of the dispersion ateach point in the cluster.Generally we find that the data spread is small, witharound (cid:39) per cent of the values for σ ( T y ) lying at around40 percent of the overall temperature. However, we do see acharacteristic small dip in the values of σ ( T y ) in the middleof our mass range ( (cid:39) − × M (cid:12) at z = ). One possibilityis that as the masses increase from (cid:39) − − M (cid:12) , thesystems become more resilient to AGN feedback due to theincreased potential well. As the masses increase further, thetemperature variance is likely to increase again, due to theclusters still thermalizing (i.e., they are still forming). Theexact details however are not explored in this paper. In this section we discuss various cluster temperature pro-files. To find analytic averages of our temperature profiles(to discern between each in a quantitative manner) we refer Recall that this is different from the distribution between clus-ters at each temperature, and as such is a measure of the intrinsictemperature variation within clusters rather than the variationbetween different clusters. Found in the supplementary online material.MNRAS000
The redshift evolution of σ ( T y ) = (cid:113) T y with respect to T y , as a function of M . The scattered points show the wholedataset, the error bars show the same data binned and the straightline shows the 2 parameter best fit of the data. the Hot and relaxed samples, we also find that the 16 th and 84 th percentiles are wider for the relaxed sample, sothat these two samples give fits that lie within each other’scluster-to-cluster variance. Further, they agree well withthe 3-parameter combined sample fits for both T m and T y ,though the fit can be found to be less appropriate for thespectroscopic-like temperature due to the strong curvaturein the T sl combined sample fits.We can also find that, while the relaxed fits’ larger 68per cent error region for T y and T m seems to be well cen-tred over the errors predicted by the complete combinedsample fits, for T sl these extend to higher temperatures, in-dicating that Relaxed clusters are more likely to have higherspectroscopic-like temperatures. We can understand this as T sl is largely driven by the denser central region, and sincemore spherical (i.e., more relaxed) clusters are more likelyto have a larger region for the same given mass, they arelikely to lead to higher observed values for T sl . y -weighted Temperature Dispersion As noted in Section 2.3, the second moment of the y -weighted temperature ( T y ) is a measure of the variance ofthe temperature distribution within the cluster . Here wediscuss σ ( T y ) , the standard deviation and its comparisonto T y . We recall that under a temperature moment expan-sion about T y , the leading order correction is proportionalto [ σ ( T y )] [see Eq. (15)].In Figure 6, we explore σ ( T y )/ T y and can see that, whilethere is a small variation of the values across the mass range,they are well approximated by a power law (i.e., straightlines in the log–log space) – which are tabulated in the Ap-pendix (Table C5 ). Generally we can see that, at higherredshifts, σ ( T y )/ T y increases and that at all redshifts in-creases slightly with increased mass, approximately scalingas σ ( T y )/ T y (cid:39) . ( + z ) . [ M / M fid ] . . Since this red-shift evolution closely matches the evolution of T y with re-spect to T , it may be that σ ( T y ) is mainly dependent on T or equivalently the potential well of the cluster ratherthan the specifics of the substructure. That is, the variationin σ ( T y )/ T y with redshift with respect to mass, is domi-nated by the near self-similar redshift evolution of T y . It isalso possible that there is an effect of clusters thermalizingover time, since this would explain the increase in variancefor larger clusters and clusters at higher redshifts. However,since there are no clear differences between the dispersion ofrelaxed sample and the combined sample there is little evi-dence either way. In Section 5.2, we will explore the radialprofiles of these values to see that these clusters see almostconstant values across the whole width of the clusters, sothat the overall dispersion is indicative of the dispersion ateach point in the cluster.Generally we find that the data spread is small, witharound (cid:39) per cent of the values for σ ( T y ) lying at around40 percent of the overall temperature. However, we do see acharacteristic small dip in the values of σ ( T y ) in the middleof our mass range ( (cid:39) − × M (cid:12) at z = ). One possibilityis that as the masses increase from (cid:39) − − M (cid:12) , thesystems become more resilient to AGN feedback due to theincreased potential well. As the masses increase further, thetemperature variance is likely to increase again, due to theclusters still thermalizing (i.e., they are still forming). Theexact details however are not explored in this paper. In this section we discuss various cluster temperature pro-files. To find analytic averages of our temperature profiles(to discern between each in a quantitative manner) we refer Recall that this is different from the distribution between clus-ters at each temperature, and as such is a measure of the intrinsictemperature variation within clusters rather than the variationbetween different clusters. Found in the supplementary online material.MNRAS000 , 1–21 (2019)
Z temperature scalings to the fits suggested by Vikhlinin et al. (2006) T tot ( r ) = T t cool ( r ) t ( r ) t cool ( r ) = x cool + T min / T x cool + t ( r ) = x − at [ + x bt ] c / b . (22)Here we defined x t = r / r t and x cool = ( r / r cool ) a cool . T cool ( r ) accounts for the temperature decline of the central regionof most clusters, while t ( r ) acts as a broken power law witha transition region, to model the area outside this centralregion. There are two methods of generating profiles for oursimulation measurement, each intuitive in different manners.From a simulation perspective, it is natural to consider afull radial profile, where we bin the particles in sphericalshells from the centre of the cluster, and volume average theparticles within each bin. However, from an observationalstandpoint, it is perhaps more relevant to consider the line-of-sight profiles, which we will here refer to as cylindricalprofiles. In the next section we will discuss these radial pro-files, as they are those normally discussed of the literature,while an exploration of the cylindrical profiles can be foundin Appendix B.In most observational work, the observed line-of-sightprofiles are deprojected to generate radial profiles, and ra-dial profiles are compared in the literature. However, we cansee from our cylindrical profiles that care must be taken inthis deprojection process, as the different weighting in eachtemperature measure, lead to complicated variations in thebehaviour of the radial and cylindrical profiles. We will thenfinally discuss the profiles derived for the y -weighted tem-perature moments in Section 5.2. In Figure 7 we display the radial profiles at z = wherewe have sorted the clusters into 5 mass bins (three of whichare graphically displayed); the th bin (lowest panel of fig-ure) corresponds to the selection of clusters from the MAC-SIS sample, hence the uneven bin width. In Figure 8, weshow the redshift evolution of the clusters with . ≤ log ( M / M (cid:12) ) ≤ , which are indicative of the variationof all mass bins. The median fits of all of these quantitiescan be found in Appendix C.First, we can see in Figure 7 that T y is once again sys-tematically larger than T m which is in turn larger than T sl .Further we can see that this increase appears systematicallylarger at larger radii. This is in agreement with our previ-ous observations that the y -weighted temperature is moreattuned to the affects of larger radii.We can further see that these differences are enhancedat higher masses (see also Henson et al. 2017; Pearce et al.2019). For instance, we can see that at higher masses T sl developes a defined downwards turn between R and R where the density falls and thus the contribution to the tem-perature drops markedly. We also note that as masses in-crease, the initial peak in the temperature shifts to smaller This model has 8 fit parameters { T , r cool , a cool , T min , r t , a , b , c } ,and requires fitting data within the ’core excised region’ to allowthe fit to access the central cooler region. . . . . . . T / T . < log m < . T y T m T sl . . . . T / T . < log m < . r/R . . . . . . T / T . < log m Figure 7.
The radial profiles of the three different temperaturesacross 3 different mass bins – note here m = M / M (cid:12) . As isstandard the temperatures have all been scaled by T for thesame cluster, and the radii have been scaled by R . The verticaldotted lines indicate the core region (0.15 R ) and virial radius( R ) respectively. The solid lines show the median values at eachradial bin across the clusters and the shaded region the 68 percent confidence region. The dotted lines show the fits using theVikhlinin model. radii; that is that the cooled central region of clusters (whichgenerates the cooling flow) becomes proportionally smallerfor higher mass clusters. This indicates that the highly vari-able inner regions of the clusters will have a smaller effecton the overall temperatures in higher mass clusters thansmaller.Considering the redshift evolution as seen in Figure 8,we see that all of the temperature measures evolve self-similarly in the outskirts of clusters ( r (cid:38) R ) while theinterior appears to heat up comparatively from high to lowredshift. This indicates that there is some true increase intemperature in the centre of clusters not explained by self-similar evolution, as redshift decreases. The differences be-tween the three temperature measures are very small, largely MNRAS , 1–21 (2019) E. Lee et al. . . . T / T T sl z = 0 z = 0 . z = 10 . . . . T / T T m z = 0 z = 0 . z = 10.1 1 r/R . . . T / T T y z = 0 z = 0 . z = 1 Figure 8.
The radial profile evolution across z for the clustersof masses M = . − × M (cid:12) clusters. This is indicativeof the evolution of all of the clusters, through the profile fits forthe others can be found in Appendix C. This figure is otherwisearranged as in Figure 7. Recall also that T is defined to beredshift dependent, so is removing the E / ( z ) dependence. Note,that these are the same clusters traced over the redshift evolution,so they would appear to have lower masses at higher redshifts. dominated by the overall scaling of the three volume aver-aged temperature measures. y -weighted Temperature Moments We find that the radial and cylindrical profiles for σ ( T y ) be-have very similarly across all masses and redshifts, in thatthey are approximately constant with respect to T . Thiscan be seen in Figure 9. This approximate mass indepen-dence matches what we observe in Figure 6 where we see that σ is a roughly constant fraction of T y . Furthermore, we see These are defined, identically to the temperature weightings,as the averaged values over each spherical shell. . . . . . . T / T . < log m < . T y σ ( T y )0 . . . . T / T . < log m < . r/R . . . . . T / T . < log m Figure 9.
The radial profiles of T y and the first moment, σ ( T y ) = (cid:113) T y . This figure is arranged as in Figure 7. that under redshift evolution σ ( T y )( r )/ T remains roughlyconstant, suggesting that the variation in σ ( T y ) seen in Sec-tion 4.3 is due to the variation of T y against T rather thanreflective of an increase in temperature dispersions withinclusters at higher redshifts.However, the values are not entirely constant, we can seethat at higher masses σ ( T y ) rises at higher radii, implyingthat as the temperatures fall the variation in the tempera-ture increase. This makes sense if we suppose the outskirtsof clusters to contain clumpy substructure, leading to cooland hot regions at the same radii – this could also be relatedto the cluster asphericities causing similar hot and cool ef-fects in the spherically averaged shells. Similarly we can seethat the variation falls off in the central regions of the clus-ters, implying that the central region (as modelled in thesimulations) are approximately isothermal and we see littlevariation. MNRAS000
The radial profiles of T y and the first moment, σ ( T y ) = (cid:113) T y . This figure is arranged as in Figure 7. that under redshift evolution σ ( T y )( r )/ T remains roughlyconstant, suggesting that the variation in σ ( T y ) seen in Sec-tion 4.3 is due to the variation of T y against T rather thanreflective of an increase in temperature dispersions withinclusters at higher redshifts.However, the values are not entirely constant, we can seethat at higher masses σ ( T y ) rises at higher radii, implyingthat as the temperatures fall the variation in the tempera-ture increase. This makes sense if we suppose the outskirtsof clusters to contain clumpy substructure, leading to cooland hot regions at the same radii – this could also be relatedto the cluster asphericities causing similar hot and cool ef-fects in the spherically averaged shells. Similarly we can seethat the variation falls off in the central regions of the clus-ters, implying that the central region (as modelled in thesimulations) are approximately isothermal and we see littlevariation. MNRAS000 , 1–21 (2019)
Z temperature scalings In this section, we will discuss the effects these different tem-perature measures have on determining Y SZ , and the furthereffects of the higher order moments on the determinationof the y -weighted temperature from examining the spectralshape. Finally, we will discuss the effect of these correctionsand related ‘corrections’ to the radial profiles and their im-pacts on the common method to determine H through theSZ effect – this will give us an indicative view of the magni-tude of the necessary corrections. Y SZ − M relation First we recall that, as mentioned in Section 2.3, to secondorder in ∆ T we can express the SZ signal as S ( ν ) = y f ( ν, T e ) + y ( ) f ( ) ( ν, T e ) + y ( ) f ( ) ( ν, T e ) . (23)By setting the pivot temperature T e = T y , when we take thevolume averages we can find that ∆ I ∝ Y f ( ν, T y ) + Y ( ) T y f ( ) ( ν, T y ) . (24)Here, Y is the volume integrated y -parameter and Y ( ) T y = Y T y = Y [ σ ( T y )] relates to the temperature dispersion. InRemazeilles et al. (2019), it is explained that in the analy-sis of Planck Collaboration et al. (2016a) f ( ν, T e ) (cid:39) f ( ν, ) is implicitly assumed. As mentioned above this leads to anunderestimation of the deduced y -parameter and also bi-ases the tSZ power spectrum amplitude. Remazeilles et al.(2019) characterize the correction to C yy (which is ∝ S )showing that for Planck it scales as C yy (cid:96) ( T e )/ C yy (cid:96) ( ) (cid:39) + . [ k B T e / ] , where the electron temperature should bethe y -weighted temperature. Hence, we could approximatethe correction to the SZ signal around the maximum at ν (cid:39)
353 GHz as f (
353 GHz , T e ) f (
353 GHz , ) (cid:39) − . (cid:20) k B T y (cid:21) , (25)which can also be seen in Fig. 10, where we have plottedthe observed distortions we would expect from our scalingrelations given T e = T y = , 5 and keV. In the presenceof foregrounds, this was found to give a reasonable estimatefor the effect of rSZ on the Planck y -analysis (Remazeilleset al. 2019).When folded into the analysis of Y SZ , for Planck thisleads to a systematic mismatch between the observed rel-ativistic temperature distortions and the magnitude of theintegrated pressure from Y SZ . This leads to the calculationthat: Y ( T y ) Y ( T y = ) (cid:39) + . (cid:20) k B T y (cid:21) . (26)The temperatures here refer to those assumed in the analy-sis of the spectral shape. We have established above that, fora given mass, the spectroscopic-like temperature underesti-mates the y -weighted temperature in a mass-dependent wayby (cid:38) − − per cent. As such, these relativistic correctionslead to even larger errors in the calculations of Y SZ than X-ray measurements alone would suggest, especially for hotterclusters or clusters at higher redshifts. . . . x = hν/k B T CMB − . . . Sp ec tr a l d i s t o rt i o n , ∆ I T y = 0 keV T y = 5 keV T y = 10 keV10 100 1000Frequency [GHz] Figure 10.
A comparison of the different spectral shapes deter-mined by T e = T y given T y = , 5 or 10 keV. The blue verticallines mark the Planck bands, with the black line at 217 GHz toshow the expected mean of the distribution. These plots weremade with
SZpack , taking y = − and τ = . . In Remazeilles et al. (2019), a standard X-ray temper-ature mass relation was used, indicating T e (cid:39) − − to be a typical cluster temperature relevant to tSZ powerspectrum measurements. Using our T y − − M relations, weexpect this typical temperature to increase to (cid:39) − ,which could further help reduce apparent differences in thededuced hydrostatic mass bias seen in various SZ observ-ables (Remazeilles et al. 2019). For refined estimates ournew T y − M (i.e., the true mass) relations should thus bevery useful.As previously discussed, we can also consider the T y −− Y scaling relations to fully calibrate the SZ signal within SZmeasurements. That is, we could consider the SZ signal ex-plicitly as a function of Y , by defining f ( ν, Y ) = f ( ν, T y ( Y )) ,such that ∆ I ∝ Y f ( ν, Y ) . This form of self-calibrated scal-ing allows for an X-ray independent calculation of the rela-tivistically corrected SZ signal, which could theoretically beconfirmed by direct checks of the shape of the signal. In Remazeilles et al. (2019), they use a temperature–massscaling relationship derived from Arnaud et al. (2005) of k B T X e (cid:39) (cid:18) E ( z ) M × h − M (cid:12) (cid:19) / (27)for estimates. Arnaud et al. (2005) used 10 nearby relaxedgalaxy clusters with masses ranging between ( . − − ) × M (cid:12) . This is a form consistent with the results seen inBarnes et al. (2017a), although the latter extends this workto higher masses, which fit the simulated hydrostatic mass tothe simulated observed spectroscopic X-ray temperature us-ing the BAHAMAS and MACSIS simulations. Equation (27)can now be replaced with our T y − M relation from sim-ulations to avoid conversion issues.It is commonly known that there is a hydrostatic massbias between X-ray derived masses and the true total massof clusters (e.g., Rasia et al. 2006, 2012; Nagai et al. 2007; MNRAS , 1–21 (2019) E. Lee et al.
Meneghetti et al. 2010; Nelson et al. 2014; Shi et al. 2015;Biffi et al. 2016; Barnes et al. 2017b; Ansarifard et al. 2020)– which can in particular be seen in comparisons of the X-ray and weak lensing derived masses of clusters. Weak Lens-ing, as a probe of the depth of the gravitational well, gives acloser result to the true mass of clusters than X-ray observa-tions. This underestimate of the hydrostatic model is due tothe limitations of the assumption of hydrostatic equilibriumwithin clusters. In particular, the mass biases calculated tooccur from the MACSIS and BAHAMAS simulations havebeen discussed in e.g., Henson et al. (2017). Generally, thismass bias is considered to be M spec (cid:39) ( − b ) M total with b (cid:39) . ,although in fact, this bias is both mass and redshift depen-dent (e.g., Henson et al. 2017; Pearce et al. 2019; but seealso Ansarifard et al. 2020).However, the temperature–temperature scalings dis-cussed in Section 4.2.4 will hold entirely independently ofthe mass measured of a given cluster. As such, any of thesescaling relationships measured to obtain the X-ray tempera-tures (at high temperatures where T sl is an appropriate proxyfor the spectroscopic X-ray temperature) can be adjusted bythe (cid:38) − per cent conversion discussed before between T sl and T y .We furthermore note that for T y ( M ) we currently canonly rely on numerical simulations, as no accurate directmeasurements of this variable exist. In computation of therSZ effect, the scaling relations given in Table 4 and 6 shouldthus be most useful and directly applicable in computationsof the SZ power spectra, e.g., using Class-SZ (Bolliet et al.2018).
While we have focussed on the leading order rSZ correction,the 2 nd order correction due to the temperature dispersionis also worth discussing. As previously previously noted, thevolume averaged dispersion is significant, scaling with thecluster temperatures, i.e., σ ( T y ) (cid:39) . T y . However, as weargue now, at the current level of precision this rSZ correc-tion remains negligible.Using the asymptotic expansions (e.g., Sazonov & Sun-yaev 1998; Chluba et al. 2012), we can express the fullyrelativistic SZ signal at low temperatures as: f ( ν, θ ) (cid:39) (cid:16) Y ( ν ) + θ Y ( ν ) + θ Y ( ν ) + θ Y ( ν ) + . . . (cid:17) , where we note that these θ = k B T e /( m e c ) , that is, the dimen-sionless temperature. This allows us to directly calculatean approximation for the signal associated with the second-order corrections, f ( ) ( ν, θ ) (cid:39) ( Y ( ν ) + θ Y ( ν ) + . . . ) . As suchwe can express the full signal, with second order correctionsas, S ( ν ) (cid:39) y (cid:32) Y ( ν ) + θ Y ( ν ) + θ (cid:32) + (cid:20) σ ( T y ) T y (cid:21) (cid:33) Y ( ν ) + . . . (cid:33) . Now, Y ( ν ) has an effect on broadening the SZ signal andpushing it to slightly higher frequencies – a full explana-tion of the functions can be found in Chluba et al. (2012). In our range of interest, i.e., temperatures 1–10 keV, θ assumesvalues (cid:39) × − − × − . In particular, at 343 GHz (the frequency most applica-ble for determining the SZ signal magnitude in
Planck ), Y (
343 GHZ )/ Y (
343 GHZ ) (cid:39) ) . Assuming a cluster tem-perature of 5 keV, one has θ (cid:39) . and with σ ( T y )/ T y ≡ . we find a (cid:39) × ( . ) × ( . ) (cid:39) . per cent correction tothe overall SZ signal stemming from the average intraclustertemperature-dispersion.It is worth noting that since the radial σ ( T y ) is constanteven as the temperature changes (see Figure 9), this correc-tion accordingly will be larger proportionally near the out-skirts of clusters. However, these outskirts also correspondto lower temperatures – which would both make the signalitself harder to detect, but also damp further the correctionsfrom the temperature dispersion. More work must be doneto see how different feedback models effect these values of σ ( T y ) – and thus to see if there is any possibility of themgiving detectable results. We also mention that the inter-cluster temperature variations, relating to the shape of themass-function, should also be carefully considered. H It has long been established that H can be determinedthrough a combination of X-ray and SZ measurements (e.g.,Birkinshaw 1979; Reese 2004; Jones et al. 2005; Bonamenteet al. 2006; Kozmanyan et al. 2019). While these are gener-ally less precise than those calculations from the CMB (e.g.,Planck Collaboration et al. 2018) or direct measurements(e.g., Riess et al. 2019), as the systematics in the approachare being accounted for, they are becoming both increasinglycompetitive and complementary.The general approach for this is as follows (see alsoBourdin et al. 2017; Kozmanyan et al. 2019). From the X-ray data, the density and temperature profiles can be con-strained [i.e., n e ( r ) and T sl ( r ) ], and from the SZ data the pres-sure profile, P e ( r ) can be constrained through the measure-ments of y assuming the distortion is wholly non-relativistic.This allows for a second temperature profile to be calculated, T m ( r ) = η T P e ( r )/ n e ( r ) . By assuming these two temperatureprofiles are equal, i.e., T m ( r ) ≡ T sl ( r ) , this allows for a mea-surement of η T , which can be found to depend on (amongother variables) the angular diameter distance, d A . As such, η T ∝ d − / A ∝ H / , which provides a way to obtain H esti-mates.Now, in this consideration, we already have two issues,the first is the estimation of the P e which, as discussed above,will be underestimated due to the omission of relativistic ef-fects [exactly as in Eq. (26)]. The second is the concordanceof T sl ( r ) and T m ( r ) , which, as can be seen in Figure 7, isnot an accurate assumption. We see that, if T m ( r ) > T sl ( r ) ,this method leads to an underestimation of the temperature.As such, these two corrections counteract one another, andwe must determine which one is dominant. The two tem-perature profiles furthermore have slightly different shapes,which will additionally bias the derived value for the H parameter. However, we do not go into more detail here.Overall, we can express the correction due to rSZ as, H , corr H (cid:39) (cid:20) P P corr (cid:21) (cid:20) T m T sl (cid:21) , (28)where P is the pressure calculated assuming there are norelativistic corrections. For instance, to estimate the effect, MNRAS000
343 GHZ ) (cid:39) ) . Assuming a cluster tem-perature of 5 keV, one has θ (cid:39) . and with σ ( T y )/ T y ≡ . we find a (cid:39) × ( . ) × ( . ) (cid:39) . per cent correction tothe overall SZ signal stemming from the average intraclustertemperature-dispersion.It is worth noting that since the radial σ ( T y ) is constanteven as the temperature changes (see Figure 9), this correc-tion accordingly will be larger proportionally near the out-skirts of clusters. However, these outskirts also correspondto lower temperatures – which would both make the signalitself harder to detect, but also damp further the correctionsfrom the temperature dispersion. More work must be doneto see how different feedback models effect these values of σ ( T y ) – and thus to see if there is any possibility of themgiving detectable results. We also mention that the inter-cluster temperature variations, relating to the shape of themass-function, should also be carefully considered. H It has long been established that H can be determinedthrough a combination of X-ray and SZ measurements (e.g.,Birkinshaw 1979; Reese 2004; Jones et al. 2005; Bonamenteet al. 2006; Kozmanyan et al. 2019). While these are gener-ally less precise than those calculations from the CMB (e.g.,Planck Collaboration et al. 2018) or direct measurements(e.g., Riess et al. 2019), as the systematics in the approachare being accounted for, they are becoming both increasinglycompetitive and complementary.The general approach for this is as follows (see alsoBourdin et al. 2017; Kozmanyan et al. 2019). From the X-ray data, the density and temperature profiles can be con-strained [i.e., n e ( r ) and T sl ( r ) ], and from the SZ data the pres-sure profile, P e ( r ) can be constrained through the measure-ments of y assuming the distortion is wholly non-relativistic.This allows for a second temperature profile to be calculated, T m ( r ) = η T P e ( r )/ n e ( r ) . By assuming these two temperatureprofiles are equal, i.e., T m ( r ) ≡ T sl ( r ) , this allows for a mea-surement of η T , which can be found to depend on (amongother variables) the angular diameter distance, d A . As such, η T ∝ d − / A ∝ H / , which provides a way to obtain H esti-mates.Now, in this consideration, we already have two issues,the first is the estimation of the P e which, as discussed above,will be underestimated due to the omission of relativistic ef-fects [exactly as in Eq. (26)]. The second is the concordanceof T sl ( r ) and T m ( r ) , which, as can be seen in Figure 7, isnot an accurate assumption. We see that, if T m ( r ) > T sl ( r ) ,this method leads to an underestimation of the temperature.As such, these two corrections counteract one another, andwe must determine which one is dominant. The two tem-perature profiles furthermore have slightly different shapes,which will additionally bias the derived value for the H parameter. However, we do not go into more detail here.Overall, we can express the correction due to rSZ as, H , corr H (cid:39) (cid:20) P P corr (cid:21) (cid:20) T m T sl (cid:21) , (28)where P is the pressure calculated assuming there are norelativistic corrections. For instance, to estimate the effect, MNRAS000 , 1–21 (2019)
Z temperature scalings M [M (cid:12) ]0 . . . E s t i m a t e d f r a c t i o n a l H c o rr ec t i o n Figure 11.
An indicative plot of the potential magnitudes of thecorrections to H . The dashed line is merely to guide the eye. Itshould be noted that this is not a full or complete accounting ofthe corrections, merely a indication of the necessity of carrying outthese two opposing corrections. These corrections are measured as H , corr / H , . Note: No errors are quoted as this is a fast calculationand a true representation of the errors would require an in depthstudy of the various interlocking factors. at T y = keV, we have already determined that P corr (cid:39) . P . We can also use our previous profile fits to estimatethe mismatch in the T m ( r ) and T sl ( r ) profiles. Since T y = keV corresponds to a M (cid:39) . × M (cid:12) , we can seethis correction as T sl ( r ) (cid:39) . T m ( r ) . In this specific case,the two corrections match well and cancel each other, butwe can expect that generally not to hold.In Figure 11, we ran a calculation of the indicative cor-rection over. While this is not a full or complete account-ing of the rSZ corrections, this exercise indicates that thesecorrections have the potential to swing by (cid:39) per centin either direction, tending to higher values of H for lowermasses and smaller values for higher masses. In, for instance,Kozmanyan et al. (2019) the median of the observed sampleof clusters lies at M = . × M (cid:12) , which would indi-cate a potential overestimation of (cid:39) per cent (i.e., naivelyshifting the derived value of H to (cid:39) ± ). This indicates apotentially sizeable correction in the deduced values of H ;however, it is not clear which way this correction will ulti-mately fall, and a more careful analysis of the effect shouldbe undertaken, in particular focusing on the assessment ofthe error budget.At lower masses, we note that this effect will be domi-nated by the profiles of the spectroscopic-like temperature –which, below masses of (cid:39) . × M (cid:12) is no longer a goodprobe of the observed X-ray signal. Furthermore, these cal-culation are all at z = , while at higher redshifts E ( z ) − / T y will remain almost constant with mass and the higher ordercorrections may increase; however, the behaviours of the pro-files are harder to predict. The exact details of this correc-tion should be studied more carefully, including an in depthcomparison of the different radial profiles from T m and T sl . The importance of rSZ corrections is increasing with grow-ing sensitivity of future CMB experiments. To incorporate the expected effects on SZ observables reliable temperature–mass scaling relations and temperature profiles are required.Here, we have greatly extended the works of Pointecouteauet al. (1998); Hansen (2004); Kay et al. (2008) to classify, indetail, the three temperature measure T sl , T m , and T y acrossthe mass ranges allowed through the combined BAHAMASand MACSIS simulations. We find differences (cid:39) − percent between the three temperature measures, with a gen-eral trend that T sl < T m < T y . The differences increase toboth higher redshifts, and when the temperature measuresare determined over the virial radius (i.e., R ), as opposedto the more commonly (and less applicable for SZ measure-ments) used radius, R (i.e., Figures 4 and 3). We find that T y scales almost self-similarly, i.e., ∝ E ( z ) / , out to z = ,while T sl and T m both undergo significant evolution rela-tive to this ‘expected’ scaling. Hence, for higher mass clus-ters, and clusters at higher redshifts (e.g., those detected in Planck ), T sl is an increasingly poor proxy for T y , or equiva-lently, the rSZ signal will be larger than X-ray measurementswould imply. Our analysis also suggests that the y -weightedtemperature is a better proxy for cluster mass, a possibilitythat could be used for self-calibration of cluster masses usingrSZ measurements.We find a strong correlation between T y and T m , with T y (cid:38) . T m at z = . While this correction is more complexfor T sl , we none-the-less find that T y (cid:38) . T sl at z = ,with similarity around M ∼ . × M (cid:12) ( T sl (cid:39) . keV)and these values diverging increasingly to both higher andlower masses, or equivalently temperatures (see Figure 4).We find, moreover, that these corrections depend very littleof the nature of the cluster, i.e., whether they are relaxed ornot. This strong correlation leads to tight scaling relationsbetween Y , the volume averaged compton- y parameter, and T y [see Eq. (21)]. This relationship can be used to calibratethe relativistic corrections to the SZ signal, from the signalitself. This allows for an estimate of the rSZ signal in, forinstance, the Planck
SZ whole sky maps and in computationsof the SZ power spectra, e.g., using
Class-SZ (Bolliet et al.2018).On average our findings suggest that X-ray derived tem-peratures underestimate the level of the rSZ by (cid:39) − per cent. For instance, we can estimate a correction for theaveraged temperature of clusters in the Planck maps cal-culated in Hurier (2016); Remazeilles et al. (2019). Thesepapers determined them to be T X = . keV or T X (cid:38) keVrespectively, which would naively lead to T y = . keV or T y (cid:38) . keV, a correction (cid:38) per cent in both cases. Thesedifferences will also affect the expected value for the sky-averaged SZ contribution, as calculated in, e.g., Hill et al.(2015). There a X-ray temperature–mass scaling relationwas used to determine the size of the relativistic corrections,finding a value of kT e (cid:39) . . This value could increase ifour T y − − M relation is used. Given that in particular low-mass haloes ( M (cid:46) M (cid:12) ) contribute to the average SZsignal, the differences in this prediction are further ampli-fied by redshift-evolution, likely leading to another increaseof the expected value, although they may be mediated bythe true spectroscopic temperature in such regimes beingpoorly modelled by the spectroscopic-like temperature. Mea-surements of the sky-averaged rSZ effect with future CMBspectrometers (Chluba et al. 2019; Kogut et al. 2019) could MNRAS , 1–21 (2019) E. Lee et al. lead to interesting constraints to feedback models and thusdeserves more attention.The profiles of these three radial temperature measuresshow similar trends (see Figure 7). These differences willbe very important when interpreting and combining futureX-ray and high-resolution SZ profile measurements (e.g.,Ameglio et al. 2009; Morandi et al. 2013). From these pro-jected profiles, it will also be possible (see Remazeilles et al.2019) to calculate a corrected power spectrum for the tSZeffect, which could play a role in reducing the tension be-tween σ found with Planck and the SZ measurements. Anunderstanding of the differences between the three profilescould also be useful for quantifying conversions between theobserved X-ray and SZ signals – in particular an understand-ing of the different behaviour of T sl ( r ) and T m ( r ) , which arecommonly taken to be identical. These differences can leadto various miscalculations where these are used interchange-ably, for instance in the SZ-derived H as discussed in Sec-tion 6.2.The intracluster temperature dispersion is found to bealmost mass independent (at around σ ( T y ) (cid:39) . T y , see Fig-ure 6), but increases slightly towards higher redshifts as aresult of cluster evolution. However, we find that this addslittle modification (cid:46) . per cent to the SZ signal. Largereffects due to temperature dispersion could arise from inter-cluster temperature variation, which directly relate to theshape of the halo mass function; however, an estimation ofthis correction is beyond the scope of this paper.While we have presented a classification of all three tem-perature measures and the y -weighted temperature disper-sion, further work must be done to establish the indepen-dence of these results from the simulations (i.e., BAHAMASand MACSIS) used. Through comparisons to other simula-tions it will be possible to assess the robustness of theseresults with respect to feedback models and other aspectsof the gas physics used to generate these clusters. In partic-ular, it would be interesting to understand how variationsof the microphysics between simulations may lead to differ-ences in the calculated intracluster temperature dispersion, σ ( T y ) and T y − Y or T y − M relations. All these could po-tentially be used to learn about the dynamical state of thecluster.Extracting the rSZ signals with future CMB experi-ments still presents a challenge (Basu et al. 2019; Chlubaet al. 2019). However, there is work to be done to estab-lish the utility of rSZ quantities across a variety of clustermodels and simulations. Further, the significant tempera-ture differences from using the more appropriate tempera-ture measures ( T y rather than T X ), compounded with cor-rections from the temperature dispersion effects (and higherorder terms to be considered in future works), will lead toimprovements in the ability to interpret the rSZ signal. ACKNOWLEDGEMENTS
The authors would like to thank the referee for their helpful com-ments, Ian McCarthy for use of the BAHAMAS simulations andFrancesca Pearce for useful discussions regarding the MACSISand BAHAMAS samples. EL was supported by the Royal So-ciety on grant No RGF/EA/180053. JC was supported by theRoyal Society as a Royal Society University Research Fellow at the University of Manchester, UK. This work was also supportedby the ERC Consolidator Grant
CMBSPEC (No. 725456) as partof the European Union’s Horizon 2020 research and innovationprogram.
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A graphical depiction of the spread of data aroundthe T – M fits at z = . The shaded regions here are the per-centile regions associated with the σ values, were the data spreadnormally about the mean, that is at the 0.2–99.8, 2.5–97.5, 16–84and 31–69 percentiles for the 3, 2, 1, and 0.5 σ regions labelled.The medians of the data are plotted in yellow. APPENDIX A: ANALYSIS OF THE MASSDEPENDENCE OF THE QUALITY OF THEFITS
Although we can see from Section 4.2.3 the skewness of thequality of fits as a whole, across all the data, it is instructiveto consider how the quality of the fit varies over the massrange of the samples. This can be seen graphically in FigureA1. Here we have plotted the contours for the percentilesassociated with what would be the 0.5, 1, 2, and 3 σ confi-dence regions were the data normally distributed against itsline-of-best-fit.The first thing to note is that there is a change over indata set at M (cid:39) M (cid:12) , on the left is the BAHAMASdata and on the right the MACSIS. This is of note simplybecause the data in the MACSIS set is less dense than that inthe BAHAMAS set, and this will contribute to the increased MNRAS , 1–21 (2019) E. Lee et al. errors we see to the right of the graph – the errors are drivenby lack of data as much as by the intrinsic scatter.Secondly we see, especially in T y , some anomalous re-sults at low masses, skewing the 2 and σ contours dramat-ically. In T m , we can see that the data is in fact roughly nor-mally distributed across the entire mass range, with roughlyconstant errors – this skew at low masses appears to be theonly changing factor. In fact, the σ region outside of thisskew is, if anything, underrepresented compared to a normaldistribution – that is, indicating smaller tails in the distribu-tion that would be expected. This may, however, be simplya limitation in the number of clusters in each mass bin tobe considered.In T y , however, we see this low-mass skew continuedstrongly in σ but still present to an extent across the en-tire range. This corroborates the long tail seen in the distri-bution of T sl in Figure 5 – however, it is worth noting thatthe skew appears to decrease to higher masses. A similar,but opposite, phenomena is seen in the T sl contours, werewe see a persistent and strong skew in the data to lowertemperatures. This indicates that although the fits modelwell the median and, even the σ variations, it would beinappropriate to consider this data as normally distributed. APPENDIX B: CYLINDRICAL PROFILES
The cylindrical profiles, or line-of-sight profiles, are perhapsof more value observationally than the radial profiles – tocreate radial profiles from observations, it is necessary to de-project the line-of-sight observations. As such it is of use toconsider these profiles alongside the radial profiles discussedin Section 5. To create these cylindrical profiles, for eachcluster the central sphere of radius R was first extractedfrom the simulation , and then cylindrical shells were slicedfrom this sphere along six maximally spaced lines of sightthrough the core of the cluster. These 6 lines-of-sight cylin-drical profiles were then averaged, to reduce the influenceof inhomogeneity between the viewing angles in each clus-ter. We would expect the cylindrical profiles to have similarqualities to the radial profiles, albeit smoothed.We can see this in Figure B1. Here we once again ob-serve that T y lies at systematically higher temperatures thanthe other temperature measures. Curiously however, we see(especially at lower masses) T m and T sl becoming somewhatindistinguishable. However, at larger radii, T m does alwaysrise above T sl which could lead to the observed volume av-erages. This is a result of the associated weightings in eachtemperature measure. T sl has a n dependence, which in line-of-sight averages will substantially upweight the central hot-ter regions of the cluster making the overall line of sightappear far greater relative to T m than one would naively as-sume from the radial profiles. We can also see here that core-excision removes a dramatic turn down observed in the T sl profile for higher mass clusters, which is not seen as clearlyin the other two temperature measures.Under redshift variation, these cylindrical profiles followalmost identical variation to that seen in the radial profiles, This causes some lack of precision very close to these edges asthe number of particles in each bin becomes small. . . . . T / T . < log m < . T y T m T sl . . . . . T / T . < log m < . r/r . . . . . T / T . < log m Figure B1.
The cylindrical profiles of the three different tem-peratures across the 5 different mass bins. This figure is arrangedas in Figure 7. so while tabulated in the Appendix C, these are not dis-cussed further here. By definition, in these cylindrical pro-files we do not have the outer regions of the clusters so wecannot compare their behaviours as we could in the previoussection for the radial profiles.
APPENDIX C: BEST FIT VALUES
In the following tables we display the fits for all of the rela-tions mentioned above. For each scaling relationship, at eachredshift, we bootstrap our fits with 5000 iterations to gainfits for the binned medians of our data and the 16 th and 84 th percentiles in each of these bins. Hence, the errors on eachvalue are the bootstrapped errors in these median fits, 84 th and 16 th percentile bounding region edges. This allows theintercluster variance to be calculated – that is, for example,at some mass, M , the median y -weighted temperature atredshift z = is given by Equation (18), using A, B andC given by the first row of Table C1. However, the 68 percent confidence region of that value, given by the intrinsic MNRAS000
In the following tables we display the fits for all of the rela-tions mentioned above. For each scaling relationship, at eachredshift, we bootstrap our fits with 5000 iterations to gainfits for the binned medians of our data and the 16 th and 84 th percentiles in each of these bins. Hence, the errors on eachvalue are the bootstrapped errors in these median fits, 84 th and 16 th percentile bounding region edges. This allows theintercluster variance to be calculated – that is, for example,at some mass, M , the median y -weighted temperature atredshift z = is given by Equation (18), using A, B andC given by the first row of Table C1. However, the 68 percent confidence region of that value, given by the intrinsic MNRAS000 , 1–21 (2019)
Z temperature scalings intercluster variation can be found through using Equation(18) using parameters given by rows 4 and 7 of Table C1. C1 Volume Averages over R Tables C1 to C5 show the temperature–mass andtemperature–temperature volume averaged scalings for thesphere of radius R . C2 Volume Averages over R Tables C6 to C10 show the same as the previous section, butfor the sphere of radius R . C3 Volume Averaged Y Fits
We display the Y − M and Y − T y relations over spheres ofboth radii in tables C11 to C14. C4 Profile Fits
In tables C15 to C18, we display the fit quantities for theradial profiles of the median temperature measures, ( T / T )and variance, σ ( T y )/ T . The same quantities for the cylin-drical profiles are in tables C19 to C22.These mass bins are organised so that the highest massbin always corresponds to the MACSIS sample, hence thediscrepency in mass bin sizes. This paper has been typeset from a TEX/L A TEX file prepared bythe author.
Table C1.
The fit values for the medians, 84th and 16th per-centiles of each temperature measure at each redshift. The errorsare determined through bootstrap methods. The fit parameterscorrespond to those described in Equation (18). M = M A B C z = . , median T y . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . z = . , median T y . + . − . . + . − . . + . − . T m . + . − . . + . − . − . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . − . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . − . + . − . T sl . + . − . . + . − . − . + . − . z = . , median T y . + . − . . + . − . . + . − . T m . + . − . . + . − . − . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . − . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . − . + . − . T sl . + . − . . + . − . − . + . − . MNRAS , 1–21 (2019) E. Lee et al.
Table C2.
The fit values for the medians, 84th and 16th per-centiles of each temperature measure against T at each red-shift. The errors are determined through bootstrap methods. Thefit parameters correspond to those described in Equation (20). T rel = T A B C z = . , median T y . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . z = . , median T y . + . − . . + . − . . + . − . T m . + . − . . + . − . − . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . − . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . − . + . − . T sl . + . − . . + . − . − . + . − . z = . , median T y . + . − . . + . − . . + . − . T m . + . − . . + . − . − . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . − . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . − . + . − . T sl . + . − . . + . − . − . + . − . Table C3.
The fit values for the medians, 84th and 16th per-centiles of each temperature measure against T m at each red-shift. The errors are determined through bootstrap methods. Thefit parameters correspond to those described in Equation (20). T rel = T m A B C z = . , median T y . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . z = . , median T y . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . z = . , median T y . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . MNRAS000
The fit values for the medians, 84th and 16th per-centiles of each temperature measure against T m at each red-shift. The errors are determined through bootstrap methods. Thefit parameters correspond to those described in Equation (20). T rel = T m A B C z = . , median T y . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . z = . , median T y . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . z = . , median T y . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . MNRAS000 , 1–21 (2019)
Z temperature scalings Table C4.
The fit values for the medians, 84th and 16th per-centiles of each temperature measure for the Hot and Hot, Re-laxed Samples against M at each redshift. The errors are deter-mined through bootstrap methods. The fit parameters correspondto those described in Equation (18), taking C = .Hot Sample Hot, Relaxed Sample M A B A B z = . , median T y . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . T y . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . T y . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . z = . , median T y . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . T y . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . T y . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . z = . , median T y . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . T y . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . T y . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . Table C5.
The fit values for the medians, 84th and 16th per-centiles of σ ( T y ) at each redshift. The errors are determinedthrough bootstrap methods. The fit parameters correspond tothose described in Equation (18), with C = . ( σ ( T y )/ T y )( M ) A B z = . median . + . − . . + . − . . + . − . − . + . − . . + . − . . + . − . z = . median . + . − . . + . − . . + . − . − . + . − . . + . − . . + . − . z = . median . + . − . . + . − . . + . − . − . + . − . . + . − . . + . − . MNRAS , 1–21 (2019) E. Lee et al.
Table C6.
The fit values for the medians, 84th and 16th per-centiles of each temperature measure at each redshift. The errorsare determined through bootstrap methods. The fit parameterscorrespond to those described in Equation (18). M = M A B C z = . , median T y . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . z = . , median T y . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . − . + . − . T sl . + . − . . + . − . − . + . − . z = . , median T y . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . − . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . Table C7.
The fit values for the medians, 84th and 16th per-centiles of each temperature measure against T at each red-shift. The errors are determined through bootstrap methods. Thefit parameters correspond to those described in Equation (20). T rel = T A B C z = . , median T y . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . z = . , median T y . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . z = . , median T y . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . − . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . MNRAS000
The fit values for the medians, 84th and 16th per-centiles of each temperature measure against T at each red-shift. The errors are determined through bootstrap methods. Thefit parameters correspond to those described in Equation (20). T rel = T A B C z = . , median T y . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . z = . , median T y . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . z = . , median T y . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . − . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . MNRAS000 , 1–21 (2019)
Z temperature scalings Table C8.
The fit values for the medians, 84th and 16th per-centiles of each temperature measure against T m at each red-shift. The errors are determined through bootstrap methods. Thefit parameters correspond to those described in Equation (20). T rel = T m A B C z = . , median T y . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . z = . , median T y . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . z = . , median T y . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . T y . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . Table C9.
The fit values for the medians, 84th and 16th per-centiles of each temperature measure for the Hot and Hot, Re-laxed Samples against M at each redshift. The errors are deter-mined through bootstrap methods. The fit parameters correspondto those described in Equation (18), taking C = .Hot Sample Hot, Relaxed Sample M A B A B z = . , median T y . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . T y . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . T y . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . z = . , median T y . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . T y . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . T y . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . z = . , median T y . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . T y . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . T y . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . MNRAS , 1–21 (2019) E. Lee et al.
Table C10.
The fit values for the medians, 84th and 16th per-centiles of σ ( T y ) at each redshift. The errors are determinedthrough bootstrap methods. The fit parameters correspond tothose described in Equation (18), with C = . ( σ ( T y )/ T y )( M ) A B z = . median . + . − . . + . − . . + . − . − . + . − . . + . − . . + . − . z = . median . + . − . . + . − . . + . − . − . + . − . . + . − . . + . − . z = . median . + . − . . + . − . . + . − . − . + . − . . + . − . . + . − . Table C11.
The fit values for the medians, 84th and 16th per-centiles of T y to Y at each redshift. The errors are determinedthrough bootstrap methods. The fit parameters correspond tothose described in Equation (21). This is a replica of Table 6found in Section 4.2.4. T Y − Y A B C z = . median . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . z = . median . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . z = . median . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . Table C12.
The fit values for the medians, 84th and 16th per-centiles of T y to Y at each redshift. The errors are determinedthrough bootstrap methods. The fit parameters correspond tothose described in Equation (21). T Y − Y A B C z = . median . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . z = . median . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . z = . median . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . Table C13.
The fit values for the medians, 84th and 16th per-centiles of Y to M at each redshift. The errors are determinedthrough bootstrap methods. The fit parameters correspond tothose described in Equation (18), with Y in the place of T . Y − M A [× − ] B C z = . median . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . z = . median . + . − . . + . − . − . + . − . . + . − . . + . − . − . + . − . . + . − . . + . − . − . + . − . z = . median . + . − . . + . − . − . + . − . . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . MNRAS000
The fit values for the medians, 84th and 16th per-centiles of Y to M at each redshift. The errors are determinedthrough bootstrap methods. The fit parameters correspond tothose described in Equation (18), with Y in the place of T . Y − M A [× − ] B C z = . median . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . z = . median . + . − . . + . − . − . + . − . . + . − . . + . − . − . + . − . . + . − . . + . − . − . + . − . z = . median . + . − . . + . − . − . + . − . . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . MNRAS000 , 1–21 (2019)
Z temperature scalings Table C14.
The fit values for the medians, 84th and 16th per-centiles of Y to M at each redshift. The errors are determinedthrough bootstrap methods. The fit parameters correspond tothose described in Equation (18), with Y in the place of T . Y − M A [× − ] B C z = . median . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . z = . median . + . − . . + . − . . + . − . . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . z = . median . + . − . . + . − . . + . − . . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . MNRAS , 1–21 (2019) E. Lee et al.
Table C15.
The fit values for the medians of the radial temperature profiles, T / T , at z = . The errors are determined throughbootstrap methods – errors written as 0.00, correspond to very small values, < − . The fit parameters correspond to those described inEquation (22). m = M / M (cid:12) . z = T r t a b c r cool a cool T min log m < . T y . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < . T y . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < . T y . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < . T y . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m T y . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + − . T m . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . + − T sl . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . MNRAS000
The fit values for the medians of the radial temperature profiles, T / T , at z = . The errors are determined throughbootstrap methods – errors written as 0.00, correspond to very small values, < − . The fit parameters correspond to those described inEquation (22). m = M / M (cid:12) . z = T r t a b c r cool a cool T min log m < . T y . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < . T y . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < . T y . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < . T y . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m T y . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + − . T m . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . + − T sl . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . MNRAS000 , 1–21 (2019)
Z temperature scalings Table C16.
The fit values for the medians of the radial temperature profiles, T / T , at z = . . The errors are determined throughbootstrap methods – errors written as 0.00, correspond to very small values, < − . The fit parameters correspond to those described inEquation (22). z = . T r t a b c r cool a cool T min log m < . T y . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < . T y . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < . T y . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m T y . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . Table C17.
The fit values for the medians of the radial temperature profiles, T / T , at z = . The errors are determined throughbootstrap methods – errors written as 0.00, correspond to very small values, < − . The fit parameters correspond to those described inEquation (22). z = T r t a b c r cool a cool T min log m < . T y . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < . T y . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < . T y . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m T y . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . MNRAS , 1–21 (2019) E. Lee et al.
Table C18.
The fit values for the medians of the radial profiles of σ ( T y )/ T across all redshifts. The errors are determined throughbootstrap methods – errors written as 0.00, correspond to very small values, < − . The fit parameters correspond to those described inEquation (22). σ ( T y ) T r t a b c r cool a cool T min z = m < . . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < . . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < . . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < .
78 0 . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . + − z = . m < . . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . . + . − . . + . − . . < log m < . . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < .
55 0 . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . z = m < . . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < . . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < . . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . Table C19.
The fit values for the medians of the cylindrical temperature profiles, T / T , at z = . The errors are determined throughbootstrap methods – errors written as 0.00, correspond to very small values, < − . The fit parameters correspond to those described inEquation (22). z = T r t a b c r cool a cool T min log m < . T y . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < . T y . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < . T y . + . − . . + . − . . + . − . . + − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < . T y . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m T y . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . MNRAS000
The fit values for the medians of the cylindrical temperature profiles, T / T , at z = . The errors are determined throughbootstrap methods – errors written as 0.00, correspond to very small values, < − . The fit parameters correspond to those described inEquation (22). z = T r t a b c r cool a cool T min log m < . T y . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < . T y . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < . T y . + . − . . + . − . . + . − . . + − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < . T y . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m T y . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . MNRAS000 , 1–21 (2019)
Z temperature scalings Table C20.
The fit values for the medians of the cylindrical temperature profiles, T / T , at z = . . The errors are determined throughbootstrap methods – errors written as 0.00, correspond to very small values, < − . The fit parameters correspond to those described inEquation (22). z = . T r t a b c r cool a cool T min log m < . T y . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < . T y . + . − . . + . − . . + . − . . + − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < . T y . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m T y . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . Table C21.
The fit values for the medians of the cylindrical temperature profiles, T / T , at z = . The errors are determined throughbootstrap methods – errors written as 0.00, correspond to very small values, < − . The fit parameters correspond to those described inEquation (22). z = T r t a b c r cool a cool T min log m < . T y . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < . T y . + . − . . + . − . . + . − . . + − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < . T y . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m T y . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T m . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T sl . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . MNRAS , 1–21 (2019) E. Lee et al.
Table C22.
The fit values for the medians of the cylindrical profiles of σ ( T y )/ T across all redshifts. The errors are determined throughbootstrap methods – errors written as 0.00, correspond to very small values, < − . The fit parameters correspond to those described inEquation (22). σ ( T y ) T r t a b c r cool a cool T min z = m < . . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < . . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < . . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < .
78 0 . + . − . . + . − . − . + . − . + − . + . − . . + . − . . + . − . . + . − . . < log m . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . z = . m < . . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < . . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < .
55 0 . + . − . . + . − . − . + . − . . + − . . + . − . . + . − . . + . − . . + . − . . < log m . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . z = m < . . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < . . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . < log m < . . + . − . . + . − . − . + . − . . + − . . + . − . . + . − . . + . − . . + . − . . < log m . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . MNRAS000