MNRAS 000,1–17 (2018) Preprint 11th February 2020 Compiled using MNRAS LATEX style file v3.0

Physical modelling of galaxy cluster Sunyaev–Zel’dovich data using Einasto dark matter profiles

Kamran Javid1,2?, Yvette C. Perrott1,3, Clare Rumsey1, and Richard D. E. Saunders1,2 1Astrophysics Group, Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, UK 2Kavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, UK 3School of Chemical and Physical Sciences, Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand

Accepted XXX. Received YYY; in original form ZZZ

ABSTRACT We derive a model for Sunyaev–Zel’dovich data from a galaxy cluster which uses an Einasto profile to model the cluster’s dark matter component. This model is similar to the physical models for clusters previously used by the Arcminute Microkelvin Imager (AMI) consortium, which model the dark matter using a Navarro-Frenk-White (NFW) profile, but the Einasto profile provides an extra degree of freedom. We thus present a comparison between two physical models which differ only in the way they model dark matter: one which uses an NFW profile (PM I) and one that uses an Einasto profile (PM II). We illustrate the differences between the models by plotting physical properties of clusters as a function of cluster radius. We generate AMI simulations of clusters which are created and analysed with both models. From this we find that for 14 of the 16 simulations, the Bayesian evidence gives no preference to either of the models according to the Jeffreys scale, and for the other two simulations, weak preference in favour of the correct model. However, for the mass estimates obtained from the analyses, the values were within 1σ of the input values for 14 out of 16 of the clusters when using the correct model, but only in 6 out of 16 cases when the incorrect model was used to analyse the data. Finally we apply the models to real data from cluster A611 obtained with AMI, and find the mass estimates to be consistent with one another except in the case of when PM II is applied using an extreme value for the Einasto shape parameter. Key words: methods: data analysis – galaxies: clusters: general – cosmology: observations – cosmology: theory.

1 INTRODUCTION in Olamaie et al.(2012) (from here on referred to as MO12) and Olamaie et al.(2013) aims to predict; this physical modelling has Clusters of galaxies are the most massive gravitationally bound ob- been applied in for example Javid et al.(2018) (from here on KJ18) jects known in the Universe, and as such sample the Universe’s and Javid et al.(2019) (KJ19). The model presented in MO12 uses matter content. Some 85 to 90 percent (see e.g. Vikhlinin et al. a Navarro-Frenk-White (NFW) profile (Navarro et al. 1995) for arXiv:1809.03325v5 [astro-ph.CO] 9 Feb 2020 2006, Vikhlinin et al. 2009, Komatsu et al. 2011, Eckert et al. 2019 the dark matter component of the galaxy cluster, which is derived and references in these for data and issues) of cluster total mass from N-body simulations of galaxy clusters. Einasto(1965) derives is in (non-baryonic) dark matter. Stars, gas and dust in galaxies, an empirical profile for dark matter halos. Previous investigations as well as a hot ionised intra-cluster medium (ICM) make up the comparing the two dark matter profiles using simulated data (see e.g. remaining mass, with the latter being much the most massive ba- Dutton and Macciò 2014, Meneghetti et al. 2014, Klypin et al. 2016 ryonic component. The galaxies emit in the optical and infrared and Sereno et al. 2016) have shown that the Einasto model provides wavebands, the ICM emits in X-ray via thermal Bremsstrahlung, a better fit. In particular, Sereno et al.(2016) showed that, for weak and interacts with cosmic microwave background (CMB) photons lensing analysis of clusters, the NFW profile can overestimate virial via inverse Compton scattering. This last effect is known as the masses of very massive halos (≥ 1015 M /h where M is units Sunyaev–Zel’dovich (SZ) effect (Sunyaev and Zeldovich 1970). Sun Sun of solar mass and h ≡ h = H /(100 km s−1 Mpc−1) where H is It is this effect that the physical modelling of clusters described 100 0 0 the value of the Hubble constant now) by up to 10%. Errors of this magnitude are non-negligible and can have substantial effects on the estimates of parameters such as the normalisation of the matter ? E-mail: [email protected]

© 2018 The Authors 2 K. Javid et al. density fluctuations σ8, the matter density ΩM, and the density and sufficiently large to justify using a diffusion approximation for the equation of state of the dark energy field parameters ΩDE and w. scattering process. It is clear that at AMI observing frequencies 2 It is these previous analyses which have motivated us to derive hPν  mec where hP is the Planck constant, me is the mass of a physical galaxy cluster model for interferometric SZ data which an electron and c is the speed of light, the photons scatter in the uses the Einasto profile to model the dark matter component of the Thomson limit. In this limit the scattering rate is ∝ σTne where cluster. For a range of cluster model inputs we compare the physical σT is the Thomson scattering cross-section and ne is the electron parameter profiles of the two models, to see how the clusters they number density in the ICM. Thus the optical depth is given by represent deviate from one another. We then compare the parameter ¹ estimates and fits of the NFW and Einasto models for simulated τ = ne(r)σTdl, (3) cluster data created with both Einasto and NFW profiles (see e.g. Grainge et al. 2002), as well as real data for cluster A611 obtained where r is the radius from the galaxy cluster centre and the integral from the Arcminute Microkelvin Imager (AMI) radio interferometer is along the line of sight. The non-relativistic form for fν is given system (Zwart et al. 2008), (Hickish et al. 2018). This is to see how by flexible the models are in modelling clusters generated with different fν = X coth(X/2) − 4, (4) profiles, and test how well they fit real data. Section2 of this paper gives a brief overview of the theory where behind interferometry and the SZ effect. In Section3 we derive the h ν X = P . (5) physical model for interferometric SZ data which uses the Einasto kBTCMB profile, and describe the Bayesian methodology for fitting a model Here k is the Boltzmann constant. Referring back to equation2, to the data. Section4 present the results of our analysis, including B y is the Comptonisation parameter which is the number of colli- the radial profiles of physical parameters using both the model sions multiplied by the mean fractional change in energy of the presented here and the one derived in MO12 for a range of clusters. photons per collision, integrated along the line of sight. On average We also present the results of applying these models to simulated the electrons in the ICM transfer an energy k T (r)/m c2 to the and real cluster data using Bayesian analysis. B e e scattered CMB photons, where T (r). In the Thomson scattering A ‘concordance’ flat ΛCDM cosmology is assumed: Ω = e M regime described above this leads to 0.3, ΩΛ = 0.7, ΩR = 0, ΩK = 0, σ8 = 0.8, w0 = −1, and wa = 0. σ k ¹ The first four parameters correspond to the (dark + baryonic) matter, y = T B T (r)n (r) dl. (6) 2 e e the cosmological constant, the radiation, and the curvature densities mec respectively. σ8 is the power spectrum normalisation on the scale If an ideal gas equation of state is assumed for the electron gas of 8 h−1 Mpc now. w and w are the equation of state parameters 0 a then in terms of the electron pressure Pe(r), the Comptonisation of the Chevallier-Polarski-Linder parameterisation (Chevallier and parameter is given by Polarski 2001). σ ¹ y = T P (r) dl. (7) 2 e mec 2 MEASURING THE SZ EFFECT WITH AN Combining equations2,4 and7, we arrive at the following expres- INTERFEROMETER sion for δIν,cl in the non-relativistic limit 3 4 X ¹ For a small field size, an interferometer samples from the two- 2σT(kBTCMB) X e δIcl,ν = [X coth(X/2) − 4] Pe(r) dl. (8) dimensional complex visibility plane u, also known as the u-v h2c4(eX − 1)2 plane. At frequency ν, the samples correspond to the Fourier com- P ponents of the sky brightness distribution I˜ν(u). I˜ν(u) is given by Relativistic treatments of fν have been considered in e.g. Re- the weighted Fourier transform of the surface brightness Iν(x) phaeli(1995), Itoh et al.(1998), Challinor and Lasenby(1998), ¹ ∞ Nozawa et al.(1998), Pointecouteau et al.(1998), and more re- 2πiu·x 2 I˜ν(u) = Aν(x)Iν(x)e d x, (1) cently Chluba et al.(2012), by incorporating relativistic terms into −∞ the Kompaneets equation. Relativistic effects may be important in where x is the position in the sky relative to the phase centre and clusters where the ICM temperatures are high. Indeed Arnaud et al. Aν(x) is the primary beam of the antennas for a given frequency. The (1994) and Markevitch et al.(1996) have shown that electrons in the positions at which I˜ν(u) are sampled from is therefore determined ICM can reach energies above 10 keV. Challinor and Lasenby show by the physical orientation of the antennas. The change in CMB that these effects lead to a small decrease in the SZ effect. We have intensity δI, due to the thermal SZ effect in a galaxy cluster is given calculated the relativistic correction at 15.5 GHz using the SZPack by (see e.g. Birkinshaw 1999) code of Chluba et al.(2012) and find that to a good approximation ( − )/ − ∂Bν(T) δIcl,rel δIcl,non−rel δIcl,rel = 0.0034Te, where Te is the tem- δIcl,ν = TCMB y fν , (2) perature of ICM electrons, with the correction only reaching 5% at ∂T T=T CMB ≈ 15 keV. This is sub-dominant to other forms of error for AMI data where the last factor is the derivative of the blackbody spectrum and we neglect it for the current analysis, although it may become with respect to temperature evaluated at the absolute temperature of important for future instruments. the CMB, which at present is TCMB = 2.728 K (Fixsen et al. 1996). Bν(T) is the spectral radiance of blackbody radiation (given by f Planck’s law). The function ν expresses the spectral dependence 3 MODELLING INTERFEROMETRIC SZ DATA of the SZ signal and is derived from the Kompaneets equation (Kompaneets 1957). We first discuss how to model δIν,cl arising from the SZ effect, Rephaeli(1995) states that for the unmodified Kompaneets starting from input parameters (see Section 3.2 for more on what equation to be valid, the optical depth of the cluster τ, must be input parameters are) which describe some physical properties of

MNRAS 000,1–17 (2018) Physical modelling with Einasto profile 3 a cluster. Equation1 can then be used to replicate the quantity We can iteratively improve the solution for ρg by re-solving measured by an interferometer. the hydrostatic equilibrium equation using updated estimates of In Section 3.2 we discuss how to use Bayesian inference to perform M(r) = Mg(r) + Mdm(r) at each iteration until the solutions con- parameter estimation and model selection for comparison between verge. Figure1 shows the results of this improvement for a cluster at 15 different models using the NFW and Einasto dark matter profiles. z = 0.15 and with M(r200) = 1 × 10 MSun using an NFW profile for the dark matter; it is clear that the approximation works very well until small radii, and particularly near r200 and r500 which are 3.1 Cluster models where we use the model to solve for our normalisation factors, so we take the initial estimate of Mg(r) for our model. Note that a similar We now consider two cases for modelling physical properties of attempt at relaxing the mass approximation was attempted in Javid galaxy clusters, which we denote PM I and PM II. PM I is as (2019), that involved solving differential equations in M through described in MO12, but with the alteration in the mapping from r200 the hydrostatic equilibrium relation. The work in Javid(2019) gave to r500 (defined below) described in KJ19. PM II is qualitatively the same results found in this paper. similar to PM I, but with an Einasto profile replacing the NFW Below we describe the specific implementation of PM II, re- profile used for the dark matter component in PM I. ferring the reader to MO12 for more details of PM I. Three cluster input parameters are required for either PM: M(r200) which is the mass enclosed up to radius r200 from the cluster centre, fgas(r200) which is the fraction of the total mass 3.1.1 Dark matter profiles attributed to the gas mass enclosed up to radius r , and z, the 200 Assuming an Einasto profile (Einasto 1965), the dark matter density redshift of the cluster. A fourth input parameter is required for profile for a cluster ρdm,PM II is given by the PM II which we call the Einasto parameter αEin, which is also α described below. Note that in general the radius r∆ is the radius from  2  r  Ein  ∆ ρdm,PM II = ρ−2 exp − − 1 , (11) the centre at which the average total enclosed mass density is times αEin r−2 ρcrit(z), the critical density at the z of the cluster. ρcrit(z) is given 2 where αEin is a shape parameter, r−2 is the scale radius where by ρcrit(z) = 3H(z) /8πG where H(z) is the Hubble parameter and the logarithmic derivative of the density is −2 (analogue to rs in G is Newton’s constant. Note further that the total mass out to r∆ is r r ρ given by the NFW model, but note that in general −2 , s), and −2 is the density at this radius. The parameter αEin controls the degree 4π 3 of curvature of the profile. The larger its value, the more rapidly M(r∆) = ∆ρcrit(z)r∆. (9) 3 the slope varies with respect to r. In the limit that αEin → 0, the logarithmic derivative is −2 for all r. For comparison we state the Hence r200 can be calculated from M(r200). NFW dark matter profile used in PM I (Navarro et al. 1995) We assume spherical symmetry, hydrostatic equilibrium, and ρs ρdm,PM I(r) = , (12) that the cluster gas is an ideal gas. Both models follow the same    2 r 1 + r general computational steps, as follows. rs rs where ρs is a density normalisation constant, and rs is another scale (i) Given an input M(r ), z and fgas(r ), the normalisation 200 200 radius. It is tempting to assume that the Einasto profile is capable of of the dark matter mass profile is fixed by the requirement that providing a better fit due to the fact that the Einasto profile has an M (r ) = (1 − fgas(r ))M(r ) and the dark matter mass dm 200 200 200 extra degree of freedom (there are three for the Einasto profile, two profile M (r) is then fully specified. dm for the NFW), the shape parameter. However Klypin et al.(2016) (ii) We make the further assumption that we can calculate an claims that this is not strictly true, as the Einasto profile was seen initial estimate of the total mass profile of the cluster using only the to give a better fit to simulated dark matter haloes even with α dark matter model profile, i.e. making the approximation that all of Ein fixed. The asymptotic values of the logarithmic slope for the two the mass in the cluster is dark matter; or, equivalently, that the shape profiles are as follows: as r → 0 then d ln ρ (r)/d ln r → of the sum of the dark matter and gas mass profiles resembles the dm,PM I −1 and d ln ρ (r)/d ln r → 0. As r → ∞ then shape of the dark-matter-only profile dm,PM II d ln ρdm,PM I(r)/d ln r → −3 and d ln ρdm,PM II(r)/d ln r → −∞. ¹ r 02 0 0
0 The magnitude of αEin determines how quickly the slope changes M(r) = 4πr ρdm(r ) + ρg(r ) dr 0 between the two asymptotic values. Throughout this work when we ¹ r 0 refer to the NFW or Einasto model, we really mean the physical 02 ρdm(r ) 0 ≈ 4πr dr , (10) model which uses the NFW or Einasto model when considering the 0 1 − fgas(r200) dark matter density profile. where ρdm(r) and ρg(r) are the dark matter and gas density profiles Referring back to equation 11, the ratio r200/r−2 is defined as the respectively. concentration parameter c200 (and similarly c200 = r200/rs for the (iii) Now we can analytically solve the hydrostatic equilibrium NFW profile). Dutton and Macciò(2014) determine an analytical dPg(r) GM(r) form for c as a function of total mass and redshift (for the redshift equation = −ρg(r) 2 for the gas mass density profile 200 dr r [ ] ρg, given a template for the gas pressure Pg(r) and the initial dark- range z = 0, 5 ) for Einasto profiles based on simulations similar matter-only estimate for M(r). to those described in Macciò et al.(2007) and Macciò et al.(2008) (iv) We then numerically integrate the ρg solution to get the gas  M (r )  log (c ) = j(z) + k(z) log 200 , (13) mass profile Mg(r) and fix the gas mass normalisation at r200 using 10 200 10 12 −1 10 h MSun the input fgas(r200). (v) Finally, we derive the pressure profile normalisation from the where j(z) = 0.459 + 0.518 exp(−0.49z1.303) and k(z) = −0.13 + gas mass normalisation and calculate the SZ signal. 0.029z.

MNRAS 000,1–17 (2018) 4 K. Javid et al.

Figure 1. Comparison of the first-order and iteratively improved solutions for Mg(r) (left) and fgas(r) (right), for a cluster at z = 0.15 and with M(r200) = 15 1 × 10 MSun. Vertical lines show r500, r200 and the radius at which we cut off our cluster model.

Following the method outlined above, we first calculate our first-order estimate for the total mass profile by integrating the Einasto profile

¹ r 02 0 0 M(r) ≈ 4πr ρdm,PM II(r ) dr 0 15 NFW M(r200) =1 ×10 MSun, z =0.15

3 α =0.05 3 18 Ein 4πρ− r  α 3/αEin 2 −2 Ein αEin =0.2 = exp (2/αEin) (14) α =2.0 16 Ein αEin 2 (Mpc) /   αEin  Sun 3 2 r 14

× γ , , M αEin αEin r−2

) in 12 r (

x dm ∫ a− −t ρ γ [a, x] = t 1e 10 where 0 dt is the incomplete lower gamma func- tion. The steps taken to get this result are given in AppendixA. 8 )) with

Equation9 can be evaluated at r200 and equated with equation 14 r ( evaluated at the same radius to obtain the following solution for ρ−2 dm 6 ρ ( 10 4

log −2.0 −1.5 −1.0 −0.5 0.0 0.5 log10(r/Mpc)  3 200 r200 1 ρ−2 = ρcrit(z) ×   3 r−  3/αEin 2 / ( / ) αEin 1 αEin exp 2 αEin 2 1 × . h  αEin i γ 3 , 2 r200 αEin αEin r−2

(15) Figure 2. Logarithmic dark matter density profiles as a function of log cluster radius using NFW and Einasto models. Three values of the Einasto profile are used: 0.05, 0.2, and 2.0. The additional input parameters used Equivalently, equation 14 can be evaluated at r200 and set equal to generate these profiles are: z = 0.15, M(r ) = 1 × 1015 M and to the known value of M(r200) to determine ρ−2. Note that ρ−2 is 200 Sun f (r ) = . the normalisation for our first-order approximation to the total mass gas 200 0 12. profile; the corresponding normalisation for the dark matter mass profile is (1 − fgas(r200))ρ−2. Figure2 shows the logarithmic dark matter density profiles as a 15 function of r for a cluster at z = 0.15 with M(r200) = 1 × 10 MSun f (r ) = . α and gas 200 0 12 for PM I and PM II for the Ein values: 3.1.2 Gas density and pressure profiles 0.05, 0.2, 2.0. It is clear that the Einasto profiles diverge the most from each other at low r and for the high αEin value at high r as As in PM I we follow Nagai et al.(2007) and assume a generalised- well. NFW (GNFW) profile to parameterise the electron pressure Pe as

MNRAS 000,1–17 (2018) Physical modelling with Einasto profile 5

a function of radius from the cluster centre, gives the following expression     P µg exp (2/α ) ei ( 3 ) Ein ( / )3/αEin ( ) Pe(r) = , (16) Pei = Gρ−2r−2 αEin 2 Mg r200  c    a(b−c)/a µe αEin r 1 + r rp rp 1 × , r h  0  a i ∫ 200 b r +c where P is the pressure normalisation constant and r is another 03 rp 0 ei p r a+b−c dr h  0  αEin i  0  c h  0  a i( a ) characteristic radius, defined by rp = r500/c500. The parameters 0 γ 3 , 2 r r 1+ r α α r rp rp a, b and c describe the slope of the pressure profile at r ≈ r , Ein Ein −2 p (20) r  rp and r  rp respectively. The slope parameters are taken to be a = 1.0510, b = 5.4905 and c = 0.3081. These ‘universal’ which must be evaluated numerically. Once Pei has been calculated, values were taken from Arnaud et al.(2010) and are the best fit the Comptonisation parameter as a function of projected radius on GNFW slope parameters derived from the REXCESS sub-sample the sky can be calculated using equation7 which in turn can be used (observed with XMM-Newton, Böhringer et al. 2007), as described to calculate the surface brightness using equation8. Finally this can in Section 5 of Arnaud et al. We also take the Arnaud et al. value be Fourier transformed to get the quantity comparable to what an of the gas concentration parameter c500 (note this is unrelated to interferometer measures, so that the physical model can be used to the concentration parameter associated with the dark matter profile) analyse data obtained with AMI. which is 1.177. Note that in MO12, KJ18 and KJ19 slightly different values derived for the standard self-similar case (Appendix B of Arnaud et al.) were used (a = 1.0620, b = 5.4807, c = 0.3292 and 3.1.3 Additional cluster parameters c500 = 1.156). It was shown in Olamaie et al.(2013) that PM I is not affected by which of these two sets of parameters is used. The cluster gas properties are fully determined by the model, and The analytical function used to convert from r200 to r500 in KJ19 is so other parameters not used for AMI data analysis can readily be specific to the NFW dark matter profile case and so is not applicable calculated. For example, as stated in Section 2 of MO12 the radial in PM II. We have not found an analytic fitting function for the profile of the electron number density is given by ne(r) = ρg(r)/µe. conversion in the case of an Einasto dark matter profile and so we Using the ideal gas assumption, the electron temperature Te(r) is obtain r500 iteratively as described in AppendixB. given by We can relate the gas pressure Pg(r), to the electron pressure through the relation 3 ! 4πµgGρ−2r h i −2 3/αEin Te(r) = (1/αEin) exp (2/αEin)(αEin/2) µgPg(r) = µePe(r), (17) kB h  αEin i γ 3 , 2 r where µ is the mean gas mass per electron and µ is the mean mass αEin αEin r−2 e g × per gas particle. Mason and Myers(2000) state that for a plasma r − with the cosmic helium mass fraction CHe = 0.24 and the solar   r  a   r  a  1 abundance values in Anders and Grevesse(1989), then µe = 1.146 × 1 + b + c , rp rp and µg = 0.592 in units of proton mass. Incorporating equations 14 and 17 into the hydrostatic equilibrium (21) equation gives the gas density which also equals Tg(r). This could be used to calculate relativistic corrections to the SZ signal. µ P 1 The gas mass can be determined numerically from equation 19, ρ (r) = e ei g 3 h i µg πGρ r 3/αEin 4 −2 −2 (1/αEin) exp(2/αEin) (αEin/2)   µe 1 Pei 1 r Mg(r) = h i × µg G ρ−2 ( /α ) ( /α )(α / )3/αEin r3 h  αEin i 1 Ein exp 2 Ein Ein 2 −2 γ 3 , 2 r200 α α r− a Ein Ein 2 ¹ r h  r0  i   b + c − − a+b−c 03 rp   c    a  a    a  × r r r r h  0 αEin i × 1 + b + c . γ 3 , 2 r rp rp rp 0 αEin αEin r−2   (18) a+b−c  r 0 c   r 0  a a × 1 + dr 0, rp rp Note that even though an analytical expression for ρg exists, within this model there is no such equivalent for the total gas mass as (22) which can also be used to determine f (r). ¹ r gas 0 02 0 Mg(r) = 4πρg(r )r dr (19) 0 3.2 Bayesian inference must be integrated numerically. Hence fgas(r) = Mg(r)/M(r) does not have a closed form solution. Nevertheless, we can use equa- The analysis of AMI data carried out in Section 4.2 is done using tions 18 and 19 to determine Pei since we know M(r200), fgas(r200) Bayesian inference. We now give a summary of this in the context and r200. Evaluating equations 18 and 19 at r200 and solving for Pei of both parameter estimation and model comparison.

MNRAS 000,1–17 (2018) 6 K. Javid et al.

3.2.1 Parameter estimation or a wider domain results in a lower evidence value all other things being equal. Hence the evidence ‘punishes’ more complex models Given a model M and data D, one can obtain probability distribu- over basic (lower dimensionality / smaller input parameter space tions of the input parameters (also known as sampling parameters domains) ones which give an equally good fit to the data. Thus the or model parameters) Θ conditioned on M and D using Bayes’ evidence automatically implements Occam’s razor: when you have theorem: two competing theories that make exactly the same predictions, the Pr (D|Θ, M) Pr (Θ|M) Pr (Θ|D, M) = , (23) simpler one is the preferred. Jeffreys(1961) provides a scale for Pr (D|M) interpreting the ratio of evidences as a means of performing model where Pr (Θ|D, M) ≡ P (Θ) is the posterior distribution of the comparison (see Table1). A value of ln(Z1/Z2) above 5.0 (less model parameter set, Pr (D|Θ, M) ≡ L (Θ) is the likelihood func- than −5.0) presents strong evidence in favour of M1 (M2). Values tion for the data, Pr (Θ|M) ≡ π (Θ) is the prior probability dis- 2.5 ≤ ln(Z1/Z2) < 5.0 (−5.0 < ln(Z1/Z2) ≤ −2.5) present mod- tribution for the model parameter set, and Pr (D|M) ≡ Z is the erate evidence in favour of M1 (M2). Values 1 ≤ ln(Z1/Z2) < 2.5 Bayesian evidence of the data given a model M. The evidence can (−2.5 < ln(Z1/Z2) ≤ −1) present weak evidence in favour of M1 be interpreted as the factor required to normalise the posterior over (M2). Finally, values −1.0 < ln(Z1/Z2) < 1.0 require more in- the model parameter space: formation to come to a conclusion over preference between M1 and M ¹ 2. Z (D) = L (Θ) π (Θ) dΘ, (24) 3.2.3 Prior probability distributions where the integral is carried out over the N-dimensional parameter space. For the models using AMI data considered here, the input For both PM I and PM II we adopt the following approach (exclud- parameter set can be split into two subsets, (which are assumed to ing any mention of αEin in the former case). be independent of one another): cluster parameters, Θcl and radio- Following FF09, the cluster parameters are assumed to be independ- source or ‘nuisance’ parameters, Θrs. The set of cluster parameters ent of one another, so that is αEin, M(r200), fgas(r200), z, xc, and yc (where the former only π(Θ ) = π(α )π(M(r ))π( f (r ))π(z)π(x )π(y ). (27) appears for PM II). xc and yc are the cluster centre offsets from the cl Ein 200 gas 200 c c interferometer pointing centre, measured in arcseconds. The cluster Table2 lists the type of prior used for each cluster parameter and the prior probability distributions are given in Section 3.2.3. For more probability distribution parameters. The values used for z and αEin details on the radio-source modelling, please refer to Section 5.2 will be specified on a case by case basis in Section 4.2. The fgas(r200) of Feroz et al.(2009) (from here on FF09). For more information prior is based on Komatsu et al.(2011). We note that more recent on the likelihood function and covariance matrix used in the AMI observations support a higher value, e.g. Eckert et al.(2019) find a analysis, we refer the reader to Hobson and Maisinger(2002) and median fgas(r200) = 0.146 for a sample of high-mass, low-redshift Sections 5.3 of FF09 and 3.2.3 of KJ19. clusters; a more correct prior should take account mass-dependence (e.g. Dvorkin and Rephaeli 2015) but we leave this refinement for 3.2.2 Model comparison future work. The nested sampling algorithm, MultiNest (Feroz et al. 2009) Z (D) N calculates by making use of a transformation of the - 4 RESULTS dimensional evidence integral into a one-dimensional integral. The algorithm also generates samples from P (Θ) as a by-product, mean- 4.1 Cluster parameter profiles ing that it is suitable for both the parameter estimation and model We first present the results of using the Einasto model in the pro- comparison aspects of this work. Comparing models in a Bayesian filing of cluster dark matter for a range of different cluster input way can be done as follows. The probability of a model M, condi- parameters, along with the equivalent results from PM I. tioned on D can also be calculated using Bayes’ theorem 14 We consider two input masses, M(r200) = 1 × 10 MSun and Pr (D|M) Pr (M) M(r ) = 1 × 1015 M and use z-values of 0.15 and 0.9. We Pr (M|D) = . (25) 200 Sun Pr (D) take fgas(r200) = 0.12 following Komatsu et al.(2011), and con- α . , . , . Hence for two models M and M , the ratio of the probability of sider Ein values of 0 05 0 2 and 2 0 – see Figure2. Klypin et al. 1 2 α the models conditioned on the same dataset is given by (2016) find a positive correlation between Ein and the mass of a cluster, suggesting that the clusters considered here with relatively Pr (M |D) Pr (D|M ) Pr (M ) ( ) 1 = 1 1 , (26) large (small) values for αEin and small (large) values for M r200 Pr (M2|D) Pr (D|M2) Pr (M2) may be considered unphysical based on the findings of their paper. Nevertheless we proceed with our range of clusters as we would like where Pr(M2)/Pr(M1) is the a-priori probability ratio of the mod- els. We set this to one, i.e. place no bias towards a particular model to analyse the behaviour at these extreme values. We note that the − ≤ ( ) ≤ before performing the analysis. Hence the ratio of the probabilities same r range ( 2 log10 r 0.5 (where r is in units of Mpc)) of the models given the data is equal to the ratio of the evidence is considered for each cluster, and thus even though each parameter values obtained from the respective models (for brevity we define profile is self-similar in r with respect to mass and redshift, they are different for each cluster over the range of r considered here. Zi ≡ Pr (D|Mi)). The evidence is simply the average of the like- lihood function over the parameter space, weighted by the prior distribution. This means that the evidence is larger for a model if 4.1.1 Dark matter mass profiles more of its parameter space is likely and smaller for a model with large areas in its parameter space having low likelihood values. A Figure3 shows the dark matter mass profiles. The Einasto profiles larger parameter space, either in the form of higher dimensionality are calculated using equation 14 and the NFW profile from the

MNRAS 000,1–17 (2018) Physical modelling with Einasto profile 7

ln(Z1/Z2) Interpretation Probability of favoured model ≤ 1.0 better data are needed ≤ 0.75 ≤ 2.5 weak evidence in favour of M1 0.923 ≤ 5.0 moderate evidence in favour of M1 0.993 > 5.0 strong evidence in favour of M1 > 0.993

Table 1. Jeffreys scale for assessing model preferability based on the log of the evidence ratio of two models M1 and M2.

Parameter Prior distribution 4.2 Bayesian analysis of AMI data 00 00 xc N(0 , 60 ) 00 00 yc N(0 , 60 ) We now focus our attention on applying the PM II to simulated and z δ(z) real AMI data, to compare the parameter estimates and Bayesian 14 14 M(r200)U[log(0.5 × 10 MSun), log(50 × 10 MSun)] evidences with those obtained from the PM I. fgas(r200)N(0.12, 0.02) αEin δ(αEin) 4.2.1 Simulated AMI data Table 2. Cluster parameter prior distributions, where the normal distribu- tions are parameterised by their mean and standard deviations. Sereno et al.(2016) study the errors associated with fitting NFW profiles to Einasto dark matter halos and vice versa for weak lensing studies. We conduct similar work in the context of simulated SZ observations. The simulations were carried out using the in-house equivalent relation given in MO12 (equation 5). Note that these AMI simulation package Profile, which has been used in various are proportional to the first-order total mass solutions, i.e. M(r) ≈ forms in e.g. Grainge et al.(2002), Olamaie et al.(2013) and KJ19. (1 − fgas(r200))Mdm(r). As before we consider Einasto profiles with the αEin values 0.05, 14 The αEin = 2 case always converges quickly as the density 0.2, and 2.0 plus an NFW profile. each with M(r200) = 1×10 MSun 15 rapidly falls to zero, while the other three profiles including the or M(r200) = 1 × 10 MSun, z = 0.15 or z = 0.90 and fgas(r200) = NFW show divergent behaviour at the largest radii considered here. 0.12. Note for all of these simulations no radio-sources, primordial The high mass inputs result in similar profiles for the αEin = 0.05, CMB or confusion noise were included, and instrumental noise was αEin = 0.2 and NFW cases, whereas the low mass inputs result in set to a negligible level. the αEin = 0.05 case diverging somewhat more rapidly than the We first compare the posterior distributions for the input para- others. meters (except those with δ-function priors). The posterior distri- butions are plotted using GetDist1, and the contours on the two- dimensional plots represent the 95% and 68% mean confidence intervals. Table C1 in AppendixC summarises the input and output 4.1.2 Gas density profiles values of the 16 simulations. The output values are the marginal- Figure4 shows the gas density profiles. The Einasto profiles are ised posterior mass mean estimates and standard deviations. The calculated using equation 18 and the NFW profile from the equi- first column gives the model used to simulate the cluster, with the valent relation given in MO12 (equation 6). Note that these are the following two columns giving the mass and z input values. For each analysed first-order solutions for ρg(r), i.e. assuming M(r) ∝ Mdm(r) when simulation, we the data using two models, one using the solving the hydrostatic equilibrium equation. NFW profile and one using an Einasto profile. For data simulated The plots show that the profiles are similar for all inputs of using an NFW profile, when analysing the data with an Einasto pro- α = . mass and redshift, with the αEin = 0.2 Einasto profile again most file we used Ein 0 2. For data simulated using an Einasto profile, α resembling the NFW profile. However, the αEin = 2.0 profile has when analysing the data with an Einasto profile we set Ein equal the highest gas density at high r for both masses and both z values. to the value used as the input for the simulation. We also repeated each simulation 10 times with different noise realisations to check the statistical significance of our results. We firstly note that the Bayesian evidence values for the Einasto 4.1.3 Gas mass profiles and NFW analyses are the same within the errors in almost all cases, with only the α = 2.0 and M(r ) = 10 × 1014 M Figure5 shows M (r) as a function of cluster radius. As in Figure3 Ein 200 Sun g cases showing a weak preference for the correct model. This can with the dark matter mass profiles, the high mass inputs correspond be explained as follows. Both models implement a GNFW profile to divergent behaviour at large r. But for α = 2.0 the profile of Ein for the pressure distribution, with the physical model providing the M (r) also shows a more noticeable such divergence. Furthermore, g characteristic scale and normalisation parameters r and P (see in all four input parameter cases, α = 2.0 shows more divergent p ei Ein equation 16). Therefore the SZ signal is entirely described by these behaviour than other values of α and the NFW profile in gas Ein two parameters, and provided that the correct pair of values can be mass, which is in contrast to the dark matter mass profiles. reached using the physical model used for the analysis, the SZ signal can be described equally well by either model. This is illustrated in Figure7, where we show the prior on rp and Pei induced by PM I 4.1.4 Gas temperature profiles for the two redshifts of our simulations, over-plotted with the true values for each of the simulations. It can be seen that the αEin = 2.0 Gas temperature profiles are shown in Figure6. The αEin = 2.0 is very distinctive, always peaking at much higher r than the other three and also always much more sharply. 1 http://getdist.readthedocs.io/en/latest/.

MNRAS 000,1–17 (2018) 8 K. Javid et al.

14 NFW 15 NFW 1e14 M(r200) =1 ×10 MSun, z =0.15 1e15 M(r200) =1 ×10 MSun, z =0.15 2.5 αEin =0.05 1.4 αEin =0.05

αEin =0.2 αEin =0.2 α =2.0 α =2.0 Ein 1.2 Ein 2.0

1.0 ) )

Sun 1.5 Sun 0.8 M M )( )( r r

( ( 0.6 1.0 dm dm M M 0.4

0.5 0.2

0.0 0.0 −2.0 −1.5 −1.0 −0.5 0.0 0.5 −2.0 −1.5 −1.0 −0.5 0.0 0.5 log10(r/Mpc) log10(r/Mpc)

14 NFW 15 NFW 1e14 M(r200) =1 ×10 MSun, z =0.9 1e15 M(r200) =1 ×10 MSun, z =0.9 3.5 αEin =0.05 1.8 αEin =0.05

αEin =0.2 αEin =0.2 α =2.0 1.6 α =2.0 3.0 Ein Ein

1.4 2.5

) ) 1.2 Sun Sun 2.0 1.0 M M )( )(

r r 0.8

( 1.5 ( dm dm 0.6 M M 1.0 0.4

0.5 0.2

0.0 0.0 −2.0 −1.5 −1.0 −0.5 0.0 0.5 −2.0 −1.5 −1.0 −0.5 0.0 0.5 log10(r/Mpc) log10(r/Mpc)

Figure 3. Dark matter mass profiles as a function of log cluster radius using NFW and Einasto models. Values of αEin = 0.05, 0.2, and 2.0 are used as inputs. 14 15 Top row has z = 0.15, bottom row has z = 0.9. Left column has M(r200) = 1 × 10 MSun, right column has M(r200) = 1 × 10 MSun.

MNRAS 000,1–17 (2018) Physical modelling with Einasto profile 9

14 NFW 15 NFW M(r200) =1 ×10 MSun, z =0.15 M(r200) =1 ×10 MSun, z =0.15 16 αEin =0.05 17 αEin =0.05 3 3 αEin =0.2 αEin =0.2

αEin =2.0 αEin =2.0 15 16 (Mpc) (Mpc) / / Sun Sun 14 15 M M ) in ) in r r

( 13 ( 14 g g ρ ρ

12 13 )) with )) with r r ( (

g 11 g 12 ρ ρ ( ( 10 10

log 10 log 11 −2.0 −1.5 −1.0 −0.5 0.0 0.5 −2.0 −1.5 −1.0 −0.5 0.0 0.5 log10(r/Mpc) log10(r/Mpc)

14 NFW 15 NFW M(r200) =1 ×10 MSun, z =0.9 M(r200) =1 ×10 MSun, z =0.9 17 αEin =0.05 18 αEin =0.05 3 3 αEin =0.2 αEin =0.2 α =2.0 α =2.0 16 Ein 17 Ein (Mpc) (Mpc) / /

Sun 15 Sun 16 M M

14 15 ) in ) in r r ( ( g g

ρ 13 ρ 14

12 13 )) with )) with r r ( ( g g

ρ 11 ρ 12 ( ( 10 10

log 10 log 11 −2.0 −1.5 −1.0 −0.5 0.0 0.5 −2.0 −1.5 −1.0 −0.5 0.0 0.5 log10(r/Mpc) log10(r/Mpc)

Figure 4. Logarithmic gas density profiles as a function of log cluster radius using NFW and Einasto models. Values of αEin = 0.05, 0.2, and 2.0 are used as 14 15 inputs. Top row has z = 0.15, bottom row has z = 0.9. Left column has M(r200) = 1 × 10 MSun, right column has M(r200) = 1 × 10 MSun.

MNRAS 000,1–17 (2018) 10 K. Javid et al.

14 NFW 15 NFW 1e13 M(r200) =1 ×10 MSun, z =0.15 1e14 M(r200) =1 ×10 MSun, z =0.15 3.5 αEin =0.05 1.6 αEin =0.05

αEin =0.2 αEin =0.2 α =2.0 1.4 α =2.0 3.0 Ein Ein

1.2 2.5

) ) 1.0

Sun 2.0 Sun

M M 0.8 )( )(

r 1.5 r ( ( g g 0.6 M M 1.0 0.4

0.5 0.2

0.0 0.0 −2.0 −1.5 −1.0 −0.5 0.0 0.5 −2.0 −1.5 −1.0 −0.5 0.0 0.5 log10(r/Mpc) log10(r/Mpc)

14 NFW 15 NFW 1e13 M(r200) =1 ×10 MSun, z =0.9 1e14 M(r200) =1 ×10 MSun, z =0.9 3.5 αEin =0.05 2.0 αEin =0.05

αEin =0.2 αEin =0.2 α =2.0 α =2.0 3.0 Ein Ein

1.5 2.5 ) )

Sun 2.0 Sun

M M 1.0 )( )(

r 1.5 r ( ( g g M M 1.0 0.5

0.5

0.0 0.0 −2.0 −1.5 −1.0 −0.5 0.0 0.5 −2.0 −1.5 −1.0 −0.5 0.0 0.5 log10(r/Mpc) log10(r/Mpc)

Figure 5. Gas mass profiles as a function of log cluster radius using NFW and Einasto models. Values of αEin = 0.05, 0.2, and 2.0 are used as inputs. Top row 14 15 has z = 0.15, bottom row has z = 0.9. Left column has M(r200) = 1 × 10 MSun, right column has M(r200) = 1 × 10 MSun.

MNRAS 000,1–17 (2018) Physical modelling with Einasto profile 11

14 NFW 15 NFW M(r200) =1 ×10 MSun, z =0.15 M(r200) =1 ×10 MSun, z =0.15 4.5 αEin =0.05 18 αEin =0.05

αEin =0.2 αEin =0.2 4.0 16 αEin =2.0 αEin =2.0

3.5 14

3.0 12

2.5 10

) (keV) 2.0 ) (keV) 8 r r ( ( g g

T 1.5 T 6

1.0 4

0.5 2

0.0 0 −2.0 −1.5 −1.0 −0.5 0.0 0.5 −2.0 −1.5 −1.0 −0.5 0.0 0.5 log10(r/Mpc) log10(r/Mpc)

14 NFW 15 NFW M(r200) =1 ×10 MSun, z =0.9 M(r200) =1 ×10 MSun, z =0.9 6 αEin =0.05 25 αEin =0.05

αEin =0.2 αEin =0.2

αEin =2.0 αEin =2.0 5 20

4 15

3 ) (keV) ) (keV) r r

( ( 10 g g

T 2 T

5 1

0 0 −2.0 −1.5 −1.0 −0.5 0.0 0.5 −2.0 −1.5 −1.0 −0.5 0.0 0.5 log10(r/Mpc) log10(r/Mpc)

Figure 6. Gas temperature profiles as a function of log cluster radius using NFW and Einasto models. Values of αEin = 0.05, 0.2, and 2.0 are used as inputs. 14 15 Top row has z = 0.15, bottom row has z = 0.9. Left column has M(r200) = 1 × 10 MSun, right column has M(r200) = 1 × 10 MSun.

MNRAS 000,1–17 (2018) 12 K. Javid et al.

Figure 7. Prior induced on the rp and Pei parameters describing the gas pressure profile using PM I (black contours; 68 and 95% levels), for the two simulation redshifts, overplotted with the true values for our simulation set. Black dots are PM I; red empty triangles, circles and squares are PM II with αEin = 0.05, 0.2, 2.0 respectively. The set of points with lower rp/Pei values 14 correspond to the M(r200) = 1 × 10 MSun simulations and the set with 14 higher values to the M(r200) = 10 × 10 MSun simulations.

14 and M(r200) = 10 × 10 MSun cases are the only ones for which the true value is significantly outside the prior, explaining the reduced evidence values. Figure 8. Posterior distributions for cluster simulated with αEin = 2.0, 15 However, despite the identical evidence values, the mass con- M(r200) = 1 × 10 MSun and z = 0.9, modelled with: Einasto dark matter straints are clearly biased when using the wrong model as the map- profile (black filled contours and solid lines), and NFW dark matter profile ping from M(r200) and fgas(r200) to rp and Pei differs depending on (magenta empty contours and dashed lines). True values are shown with the model. When simulating and analysing with the same model, we black stars and vertical lines. find the mean of the mass posterior is within 1σ of the input value in 14 out of 16 cases (82% over all the noise realisations); this is higher than the expected 68% due to the additional information provided by the physical model with a given value of αEin on rp and Pei, by the prior. When analysing the simulation with the wrong model, as shown in Figure 10. The constraint on αEin depends entirely on we find this in only 6 out of 16 cases (42% over all the noise realisa- whether the (rp, Pei) parameter pair can be reached using PM II with tions). In Figure8 we show the posteriors produced when analysing a given α value; the case of α = 0.2, M(r ) = 10×1014 M ( ) × 14 Ein Ein 200 Sun the PM II simulation with αEin = 2.0, M r200 = 10 10 MSun, and z = 0.9 is the only one where the priors induced by the different z = 0.9 with PM II and PM I. The correct rp and Pei values are physical models are quite separate and therefore the only one with recovered in both cases, however the mass and gas fraction posteri- a strong constraint. The difference between the priors on rp and Pei ors are strongly biased from their true values when analysing with at z = 0.15 and z = 0.9 is simply produced by the differences in PM I. The very low fgas(r200) value, far outside the prior, can be c200 and ρcrit at the different redshifts. understood by considering Figure7; the rp/Pei prior is ‘thickened’ We therefore see that although α can be constrained using ( ) Ein by allowing a greater range in fgas r200 , so to reach the rp/Pei para- SZ data in some cases, in general it is unconstrained and varying ( ) meter pair outside the prior, fgas r200 must be dragged outside its it can introduce biases in the one-dimensional marginalised mass prior range. estimates. Referring back to the finding by Klypin et al.(2016) that Finally, we tried running the Bayesian analysis on eight of the αEin and cluster mass are positively correlated, it makes sense in Einasto simulated clusters with uniform analysis priors on αEin. future work to either incorporate a joint prior on M and αEin, or to These clusters corresponded to the simulations with input values of include a functional form between the two variables in the cluster either αEin = 0.2 or αEin = 2.0. models. In both cases we analysed the simulations with a uniform prior on αEin, U[0.05, 3.5]. The results fell into three general categor- ies, examples of which are shown in Figure9. In the first category, 4.2.2 Analysis of real AMI observations of A611 αEin was completely unconstrained and the mass posterior was just widened by the marginalization over αEin. This was generally the Following MO12 we conduct Bayesian analysis on data from ob- case for the lower-mass clusters with αEin = 2.0, which falls more to- servations with AMI of the cluster A611 at z = 0.288, which has ward the centre of the allowed prior range (Figure9a). In most other been studied through its X-ray emission, strong lensing, weak lens- cases, large, curving degeneracies were seen between the paramet- ing and SZ effect (see Schmidt and Allen 2007, Donnarumma et ers which induced biases in the one-dimensional marginalised mass al. 2011, Romano et al. 2010 and Rumsey et al. 2016 respectively). constraints, although the true value was correctly located within the These studies suggest that there is no significant contamination from two-dimensional constraints; a particularly severe example of this radio-sources and that the cluster has similar weak lensing and X- is shown in Figure9b. In only one case, the cluster with αEin = 0.2, ray masses and is close to the TX–TSZ relation for clusters close to 14 M(r200) = 10 × 10 MSun and z = 0.9, a strong constraint with hydrostatic equilibrium. little degeneracy was produced on all parameters (Figure9c). These Referring back to Section 3.2.3, we take z = 0.288 and for the ana- results can again be understood by considering the priors induced lysis with the PM II we consider three different cases separately –

MNRAS 000,1–17 (2018) Physical modelling with Einasto profile 13

(a) (b) (c)

14 Figure 9. Posterior distributions of Einasto model input parameters for: (a) αEin = 2.0, M(r200) = 1 × 10 MSun and z = 0.9, (b) αEin = 0.2, M(r200) = 14 14 10 × 10 MSun and z = 0.15 simulated cluster, and (c) αEin = 0.2, M(r200) = 10 × 10 MSun and z = 0.9 simulated clusters.

Ein =20

0

arcsec

c 0 Figure 10. Prior induced on the rp and Pei parameters describing the gas pressure profile using PM II, for a representative set of αEin values, 0.2

(solid magenta), 1.0 (transparent blue) and 2.0 (empty black), where the contours shown are the 68% levels, for the two simulation redshifts. The Sun 0 ) true values for the simulations analysed varying αEin are overplotted with 200 ( 0 circles (αEin = 0.2) and squares (αEin = 2.0). The set of points with lower 14 rp/Pei values correspond to the M(r200) = 1 × 10 MSun simulations and 14 the set with higher values to the M(r200) = 10 × 10 MSun simulations. 0

) 0 200 ( 0 gas f

00

0 0 0 0 0 0 0 00 0 0 0 carcsec carcsec (200)Sun fgas(200)

αEin ∈ {0.05, 0.2, 2.0} – so that in total there are four sets of results to compare for A611 (including the NFW model). The Bayesian analysis was conducted in the same way as it was for the simula- tions in the previous Section. Figure 11. Marginalised posterior distributions of physical model input α = . The means and standard deviations of the four analyses are given in parameters for the NFW and Ein 2 0 models applied to real A611 data. The contour plots are the two-dimensional marginalised plots of the Table3. As in Section 4.1, α = 0.05 and α = 0.2 show similar Ein Ein parameters named in the corresponding row / column. The line plots are the ( ) results to PM I. αEin = 2 gives a different estimate for M r200 , fully marginalised posterior distributions. and its posterior distribution is shown in Figure 11 along with that obtained with the NFW profile. The mean mass estimates are within one combined standard deviation away from each other. However, (Z /Z ) as seen in Table3 the value of ln Ein NFW imply that ‘no model 5 FORECASTING is favoured by the data’ according to the Jeffreys scale. This is en- tirely consistent with the results obtained from the simulations in Our simulations have shown that SZ data are generally unable to the previous Section. distinguish between these physical models based on the thermal SZ

MNRAS 000,1–17 (2018) 14 K. Javid et al.

14 Model xc (arcsec) yc (arcsec) M(r200) (×10 MSun) fgas(r200) ln (Z) NFW 24.7 ± 12.4 13.9 ± 11.5 7.84 ± 1.24 0.129 ± 0.020 38629.4 ± 0.3 αEin = 0.05 22.7 ± 12.5 13.1 ± 12.6 7.45 ± 1.24 0.130 ± 0.019 38629.2 ± 0.3 αEin = 0.2 25.5 ± 12.8 14.9 ± 13.0 7.67 ± 1.27 0.127 ± 0.017 38629.7 ± 0.2 αEin = 2.0 24.3 ± 12.4 14.3 ± 13.2 6.17 ± 1.12 0.130 ± 0.017 38629.2 ± 0.2 Table 3. Marginalised posterior distribution mean values and standard deviations of physical model input parameters and Bayesian evidences associated with each model, applied to real A611 data.

effect alone; our simulations have a thermal noise level ≈ 100× smal- clusters (compared to 14 out of 16 for the correct model). In certain ler than a typical AMI observation and include no primordial CMB cases, the gas mass fraction at r200 is also inferred incorrectly, or radio source confusion noise (or other instrumental systematics). which in turn is due to the true values of the GNFW scale and However, recently it has been shown that Planck cluster constraints normalisation parameters (rp and Pei) laying outside of the model’s may be biased by the relativistic SZ corrections which depend on priors for these parameters. the temperature of the cluster (e.g. Erler et al. 2018), and including • Looking at the Bayesian evidence values for the simulations, these corrections will become crucial for forthcoming instruments for 14 of the 16 clusters no model is preferred over the other one with higher sensitivity and angular resolution such as CCAT-prime according to the Jeffreys scale. (e.g. Parshley et al. 2018). Our models provide a consistent physical • The two simulations which did show preference towards a mechanism for modelling and including the relativistic corrections. model picked the correct one, with ‘weak preference’ according It has been shown in Mittal, de Bernardis & Niemack(2018) that as- to the Jeffreys scale. Both of these simulations were generated with suming isothermality biases cluster constraints based on simulated PM II (and showed preference to PM II over PM I). From inspection CCAT-prime data, so given the very different temperature profiles of the priors on rp and Pei for PM I for these two clusters, it was produced by our models (see Figure6), forthcoming instruments apparent that the true value was far away from the prior peaks, thus may be able to distinguish between these models based on spatially- explaining the low evidence values for these runs. resolved relativistic SZ constraints, although the effects of cooling • When αEin was allowed to vary in the analysis, it was found flow and merger activity must also be considered. to be mostly unconstrained (except in one exceptional case), and the large, curving degeneracies between αEin and the other cluster parameters could produce biases in the one-dimensional mass con- straints. To safely allow variation in α , a physically-motivated 6 CONCLUSIONS Ein relationship between M and αEin could be introduced such as that Based on the physical model introduced in Olamaie et al.(2012) found by Klypin et al.(2016). (PM I) which uses an NFW profile (Navarro et al. 1995) to model Using real data from cluster A611 we found, consistent with the the dark matter content of galaxy clusters, we derive a new physical simulations: model (PM II) which models the dark matter with an Einasto profile (Einasto 1965). The Einasto profile has an additional degree of • The αEin = 0.05 and αEin = 0.2 models gave very similar res- freedom compared to the NFW profile, which dictates the shape ults to the NFW model; the αEin = 2 model however underestimates of the dark matter density as a function of radius. For different M(r200) relative to the other three models. values of αEin we have investigated the profiles of several physical • The Bayesian evidence values calculated from these four ana- properties of a cluster, namely the dark matter density, dark matter lyses were roughly equal, suggesting no model provided a statistic- mass, gas density, gas mass and gas temperature. We have also ally more significant fit relative to the others. provided the equivalent profiles in the NFW case. From this we found the following. In a forthcoming paper Perrott et al.(2019), Bayesian analysis will be performed on joint AMI-Planck datasets. • Of the three values of αEin considered, αEin = 0.2 gave the most similar profile to that given by the NFW model (as discussed in Dutton and Macciò 2014), with the main discrepancy between the two arising in the peak amplitude of the gas temperature. ACKNOWLEDGEMENTS • αEin = 2.0 showed the most convergent behaviour in Mdm(r) at high r, but the most divergent in Mg(r) in the same limit. We are grateful to the anonymous referee for a wealth of well-judged, • The gas temperature profiles were somewhat different for the constructive and thorough comments that have much improved this αEin values considered here. This suggests that if one can carefully paper. This work was performed using the Darwin Supercomputer of measure the temperature profile of a cluster, then one could infer the University of Cambridge High Performance Computing (HPC) αEin and use this in the model presented here (though one has to be Service (http://www.hpc.cam.ac.uk/), provided by Dell Inc. aware of cooling flow and merger activity). using Strategic Research Infrastructure Funding from the Higher Education Funding Council for England and funding from the Sci- Next we applied Bayesian analysis to simulated and AMI datasets ence and Technology Facilities Council. The authors would like to using PM I and PM II, to compare the models’ parameter estimates thank Stuart Rankin from HPC and Greg Willatt and David Tit- and fits to the data. Simulating clusters with either NFW or Einasto terington from Cavendish Astrophysics for computing assistance. dark matter profiles, which were then ‘observed’ by AMI, we found They would also like to thank Dave Green for his invaluable help the following. using LATEX. Kamran Javid acknowledges an STFC studentship. • When the wrong cluster model is used in the analysis, the Yvette Perrott acknowledges support from a Trinity College Junior correct mass value is inferred to within 1σ for only 6 of the 16 Research Fellowship and Rutherford Discovery Fellowship.

MNRAS 000,1–17 (2018) Physical modelling with Einasto profile 15

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Romano, A., and 27 colleagues 2010. Abell 611. I. Weak lensing analysis where the last equality follows from the definition of the incom- ∫ x with LBC. Astronomy and Astrophysics 514, A88. γ [a, x] = ta−1e−t plete lower Gamma function 0 dt. Including the Rumsey, C., and 9 colleagues 2016. AMI observations of 10 CLASH galaxy constant factor in equation A3 leads to the result clusters: SZ and X-ray data used together to determine cluster dynamical /α   αEin  states. Monthly Notices of the Royal Astronomical Society 460, 569. 4πρ−2  αEin 3 Ein 3 2 r Schmidt, R. W., Allen, S. W. 2007. The dark matter haloes of massive, M(r) = exp(2/αEin) γ , . αEin 2 αEin αEin r−2 relaxed galaxy clusters observed with Chandra. Monthly Notices of the Royal Astronomical Society 379, 209. (A5) Sereno, M., Fedeli, C., Moscardini, L. 2016. Comparison of weak lensing by NFW and Einasto halos and systematic errors. Journal of Cosmology and Astro-Particle Physics 2016, 042. APPENDIX B: DETERMINING r500 ITERATIVELY AMI Consortium, T., and 18 colleagues 2011. Sunyaev-Zel’dovich observa- tions of LoCuSS clusters with the Arcminute Microkelvin Imager: high Evaluating equations9 and 14 at r500 and equating we get X-ray luminosity sample. arXiv e-prints arXiv:1101.5590. 3/α Sunyaev, R. A., Zeldovich, Y. B. 1970. The Spectrum of Primordial Radi- 4π 3  αEin  Ein 3 500ρcrit(z)r = 4πρ−21/αEin exp(2/αEin) r− ation, its Distortions and their Significance. Comments on Astrophysics 3 500 2 2   αEin  and Space Physics 2, 66. 3 2 r500 Vikhlinin, A., and 6 colleagues 2006. Chandra Sample of Nearby Relaxed × γ , . αEin αEin r−2 Galaxy Clusters: Mass, Gas Fraction, and Mass-Temperature Relation. (B1) The Astrophysical Journal 640, 691. Vikhlinin, A., and 10 colleagues 2009. Chandra Cluster Cosmology Project If we let R = r500/r−2, then we can determine r500 by solving the III: Cosmological Parameter Constraints. The Astrophysical Journal following for R 692, 1060. Zeldovich, Y. B., Sunyaev, R. A. 1969. The Interaction of Matter and Ra- R3 diation in a Hot-Model Universe. Astrophysics and Space Science 4, h i γ 3 , 2 RαEin 301. αEin αEin (B2) Zwart, J. T. L., and 60 colleagues 2008. The Arcminute Microkelvin Imager. /α 1 3ρ−2  αEin 3 Ein exp (2/αEin) Monthly Notices of the Royal Astronomical Society 391, 1545. − = 0 ρ (z) 500 2 α Zwicky, F. 1933. Die Rotverschiebung von extragalaktischen Nebeln. Hel- crit Ein vetica Physica Acta 6, 110. by some iterative root-finding method e.g. Newton-Raphson. We Zwicky, F. 1937. On the Masses of Nebulae and of Clusters of Nebulae. The use the starting point R = 2r200 which usually results in the 0 3r−2 Astrophysical Journal 86, 217. algorithm converging in O(10) iterations.

We now show that equation B2 only has one solution for a APPENDIX A: EINASTO MASS INTEGRAL given r−2. We start by considering both sides of equation B1 as two different functions, and ignore constant terms for simplicity (this From equations 11 and 14 we have that does not affect the truth of the final result), i.e. we consider the two ¹ r functions M(r) = 4πr 02 ρ (r 0) dr0 dm,PMII   αEin  0 3 2 r f (r ) = r3 , g(r ) = γ , 500 . ¹ r   0 αEin  500 500 500 (B3) 02 −2 r 0 αEin αEin r−2 = 4πρ−2 exp (2/αEin) r exp dr . 0 αEin r−2 We first note that f (0) = g(0) = 0, and differentiate both functions (A1) with respect to r500 Using the substitution d f dg  2  r αEin  ∝ 2 ∝ 2 − 500 − r500, r500 exp 1 . (B4) 23/αEin r 03 3 × 23/αEin r02 dr500 dr500 αEin r−2 u = ⇒ = 0 / du / dr (A2) α3 αEin r3 α3 αEin 3 Setting these two derivatives equal to each other yields one solution Ein −2 Ein r−2 at r500 = r−2 for all αEin , 0, meaning the derivatives only intersect g f then equation A1 becomes once. Furthermore d tends to zero for large r whilst d is dr500 500 dr500 3/αEin 3 a monotonically increasing function, meaning the former must be 4πρ−2 exp (2/αEin) α r− M(r) = Ein 2 larger before the two intersect. This coupled with the fact that f (0) = 3/α 3 × 2 Ein g( ) = g(r ) > f (r ) / (A3) 0 0 means that 500 500 until some point (which has 23 αEin r3 ¹ u= 3/α to be after the derivatives intersect) when the two intersect, after α Ein r3  α /3 × Ein −2 −u Ein . exp du which f (r500) > g(r500) as g(r500) flattens off. This proves that u=0 equation B2 only has one root and that equation B1 only has one αEin/3 Finally, using the substitution t = u so that dt = solution in r500 for fixed r−2. αEin αEin/3−1 3 u du, then the integral in equation A3 (ignoring the con- stant factor) becomes

α α /3 2r Ein ¹ u Ein ¹ α 3 −α / −t 3 α r Ein /α − −t u1 Ein 3e dt = Ein −2 t3 Ein 1e dt αEin 0 αEin 0  3 2  r αEin  = γ , , αEin αEin r−2 (A4)

MNRAS 000,1–17 (2018) Physical modelling with Einasto profile 17 ) NFW 3 3 2 2 . . . . 3 3 4 3 3 3 2 2 2 2 2 2 /Z 0 0 0 0 ...... 0 0 0 0 0 0 0 0 0 0 0 0 ± ± ± ± Ein ± ± ± ± ± ± ± ± ± ± ± ± 1 1 1 1 . . . . 5 0 5 4 2 0 1 1 1 2 0 5 (Z 0 0 0 0 ...... − − − − ln 2 0 2 0 2 2 3 3 2 0 2 1 1 2 0 2 0 2 0 1 0 2 0 2 0 2 2 0 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (Z) ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ln 3 4 1 1 1 1 3 2 8 7 8 8 1 6 8 4 ...... NFW 2 29982 2 30208 22 29995 2 30009 30022 2 30060 2 30003 12 30033 30088 1 29967 2 30098 1 30059 2 29991 2 30034 22 30061 30114 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± (Z) 8 4 0 0 6 5 5 1 8 9 9 3 6 7 8 8 ...... ln Ein . model ) simulate Sun modelling M 14 10 ) (× 200 r ( 34 30208 3542 30009 30025 26 30004 19 30088 M . . . . . 35 29982 19 29995 19 30060 12 30033 15 29967 13 30098 13 30059 09 29991 11 30034 1015 30061 30114 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± is the mean and standard deviation ) 93 03 30 75 06 . . . . . 71 52 48 01 22 27 08 03 94 96 46 ...... 200 NFW out r ( are the input values used to create the M z ) Sun M 14 10 and Input ) ) (× is the natural log Bayesian evidence corresponding to 200 r 200 ( r 30 10 31 11 20 9 17 14 ( (Z) M . . . . 31 9 27 16 19 9 1120 1 10 12 1 12 1 12 1 12 1 10 0 1012 0 1 ...... 0 0 0 0 is the natural log ratio of the two evidences obtained. ln M 0 0 0 0 0 0 0 0 0 0 0 0 ) ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 24 00 06 01 . . . . NFW 82 85 44 94 96 13 95 99 02 95 95 07 ...... 8 10 10 9 9 10 10 0 9 1 0 0 1 0 0 1 Ein out /Z Ein z (Z Table C1: Input and outputmatter profiles. values The of first simulations column is usingthe what NFW cluster. dark and matter Input Einasto profile was dark used to simulation for the given model. Ein out of the posterior distribution obtained inferred using an Einasto profile to the cluster. Ein the inference. NFW... isln as before but using an NFW profile in the Input ) Sun M 14 10 ) (× 200 r ( M X file prepared by the author. E T A 10 0.90 1010 0.90 0.90 10 0.15 10 0.15 10 0.15 1 0.90 1 0.90 1 0.90 1 0.15 11 0.15 0.15 X/L E 05 2 0 05 0 2 05 0 2 05 2 0 ...... 0 0 2 0 2 0 0 2 0 0 0 2 ======Model Input Ein Ein Ein Ein Ein Ein Ein Ein Ein Ein Ein Ein α α α α α α α α α α α α NFW 10 0.90 NFW 10 0.15 NFW 1 0.90 NFW 1 0.15 APPENDIX C: SIMULATION RESULTS TABLE This paper has been typeset from a T

MNRAS 000,1–17 (2018)