Issue 
A&A
Volume 592, August 2016
The XXL Survey: First results



Article Number  A3  
Number of page(s)  16  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201526886  
Published online  15 June 2016 
The XXL Survey^{⋆,}^{⋆⋆}
III. Luminositytemperature relation of the bright cluster sample
^{1} School of Physics, HH Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 1TL, UK
email: P.Giles@bristol.ac.uk
^{2} ArgelanderInstitut fur Astronomie, University of Bonn, Auf dem Hügel 71, 53121 Bonn, Germany
^{3} School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK
^{4} MaxPlanck Institut fur Extraterrestrische Physik, Postfach 1312, 85741 Garching bei Munchen, Germany
^{5} Laboratoire AIM, CEA/DSM/IRFU/Sap, CEA Saclay, 91191 GifsurYvette, France
^{6} LAM (Laboratoire d’Astrophysique de Marseille) UMR 7326, AixMarseille Université, CNRS, 13388 Marseille, France
^{7} INAF, IASF Milano, via Bassini 15, 20133 Milano, Italy
^{8} INAF, Osservatorio Astronomico di Bologna, via Ranzani 1, 40127 Bologna, Italy
^{9} INFN, Sezione di Bologna, viale Berti Pichat 612, 40127 Bologna, Italy
^{10} Department of Physics and Astronomy, University of Victoria, 3800 Finnerty Road, Victoria, BC, Canada
Received: 3 July 2015
Accepted: 9 November 2015
Context. The XXL Survey is the largest homogeneous survey carried out with XMMNewton. Covering an area of 50 deg^{2}, the survey contains several hundred galaxy clusters out to a redshift of ~2 above an Xray flux limit of ~5 × 10^{15} erg cm^{2} s^{1}. This paper belongs to the first series of XXL papers focusing on the bright cluster sample.
Aims. We investigate the luminositytemperature (LT) relation for the brightest clusters detected in the XXL Survey, taking fully into account the selection biases. We investigate the form of the LT relation, placing constraints on its evolution.
Methods. We have classified the 100 brightest clusters in the XXL Survey based on their measured Xray flux. These 100 clusters have been analysed to determine their luminosity and temperature to evaluate the LT relation. We used three methods to fit the form of the LT relation, with two of these methods providing a prescription to fully take into account the selection effects of the survey. We measure the evolution of the LT relation internally using the broad redshift range of the sample.
Results. Taking fully into account selection effects, we find a slope of the bolometric LT relation of B_{LT} = 3.08 ± 0.15, steeper than the selfsimilar expectation (B_{LT} = 2). Our bestfit result for the evolution factor is E(z)^{1.64 ± 0.77}, fully consistent with “strong selfsimilar” evolution where clusters scale selfsimilarly with both mass and redshift. However, this result is marginally stronger than “weak selfsimilar” evolution, where clusters scale with redshift alone. We investigate the sensitivity of our results to the assumptions made in our fitting model, finding that using an external LT relation as a lowz baseline can have a profound effect on the measured evolution. However, more clusters are needed in order to break the degeneracy between the choice of likelihood model and masstemperature relation on the derived evolution.
Key words: Xrays: galaxies: clusters / galaxies: clusters: general / galaxies: clusters: intracluster medium
Based on observations obtained with XMMNewton, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA.
The Master Catalogue is available at the CDS via anonymous ftp to cdsarc.ustrasbg.fr (130.79.128.5) or via http://cdsarc.ustrasbg.fr/vizbin/qcat?J/A+A/592/A2
© ESO, 2016
1. Introduction
Under the assumption of selfsimilarity (Kaiser 1986), simple scaling laws can be derived between various properties of galaxy clusters. These scaling laws are advantageous as they provide a cheap way of measuring the masses of large samples of clusters, an important ingredient for cosmological studies using galaxy clusters (see Allen et al. 2011, for a review). One of the most explored scaling relations is that between the Xray luminosity (L) and temperature (T), expected to follow a relationship of L ∝ T^{2}. However, a multitude of studies have found that the slope of the LT relation is ~3 (e.g. Pratt et al. 2009; Eckmiller et al. 2011; Takey et al. 2013; Connor et al. 2014), steeper than the selfsimilar expectation. One of the main explanations for the deviation from this theoretical expectation is energy input by nongravitational processes during cluster formation at early times, including preheating, supernovae feedback, and heating from active galactic nuclei (AGN) at high redshift. These nongravitational processes should have the strongest effect in lower mass systems owing to their shallower potential wells that expel gas from the inner regions and suppress the luminosity. Observations have shown further steepening of the LT relation at the lowmass regime (e.g. Osmond & Ponman 2004; Sun et al. 2009). However, recent work, using methods to correct for sample selection effects, has suggested that the slope of the LT relation on group scales is consistent with massive clusters (e.g. Bharadwaj et al. 2015; Lovisari et al. 2015).
Xray flux limited samples suffer from two forms of selection bias, Malmquist bias, where higher luminosity clusters are preferentially selected out to higher redshifts, and Eddington bias, where in the presence of intrinsic or statistical scatter in luminosity for a given mass, objects above a flux limit will have above average luminosities for their mass. This effect is amplified by the steep slope of the cluster mass function, which results in a net movement of lower mass objects into a flux limited sample. The net effect on the LT relation is to bias the normalisation high and the slope low (see Allen et al. 2011, for a review). Therefore, taking these biases into account is paramount when modelling cluster scaling relations in order to uncover the true nature of any nongravitational heating driving departures from selfsimilar behaviour with mass or redshift. Although scaling relation studies have had a rich history, only a relatively small number of published relations attempt to account for selection biases (e.g. Stanek et al. 2006; Pacaud et al. 2007; Pratt et al. 2009; Vikhlinin et al. 2009; Andreon & Bergé 2012; Bharadwaj et al. 2015; Lovisari et al. 2015), while Mantz et al. (2010a) provides the most robust handling of selection effects to date.
In the selfsimilar model, with cluster properties measured within overdensity radii defined relative to the critical density, the evolution of the scaling relations can be parameterised by the factor E(z) (where ). In this framework, the evolution of the normalisation of the LT relation goes as E(z)^{γLT}, where γ_{LT} ≡ 1 if clusters scale selfsimilarly with both mass and redshift, or γ_{LT} ≈ 0.42 if the observed mass dependence of the cluster baryon fraction is also included (Maughan 2014). These two reference values for γ_{LT} assume the same underlying evolution driven by the changing critical density of the Universe; the difference arises purely from the algebraic combination of the scaling laws of gas mass, temperature, and gas structure with total mass that are used to construct the LT relation (Maughan 2014).
We refer to γ_{LT} = 1 as strong selfsimilar evolution, reflecting the fact that it is based on the assumption that clusters scale selfsimilarly with both mass and redshift, while γ_{LT} = 0.42 is referred to as weak selfsimilar evolution and assumes clusters only scale selfsimilarly with redshift. The latter is a more realistic prediction of the selfsimilar evolution against which to test for departures in the observed evolution of clusters in the LT plane.
Understanding how scaling relations evolve with redshift is important for two main reasons: (i) scaling relations must be wellcalibrated at high redshift to provide mass estimates for cosmology and (ii) the relations provide insight into the history of heating mechanisms in clusters. A consensus on the evolution from observations has yet to be achieved; various studies find an evolution consistent with selfsimilar (e.g. Vikhlinin et al. 2002; Maughan et al. 2006; Pacaud et al. 2007), while others find departures from the selfsimilar expectation (e.g. Ettori et al. 2004; Kotov & Vikhlinin 2005; Reichert et al. 2011; Hilton et al. 2012; Clerc et al. 2014). Again, the importance of taking into account selection effects is crucial for understanding evolution and recent work has shown that departures from selfsimilar evolution can be explained by selection biases (Pacaud et al. 2007; Mantz et al. 2010a; Maughan et al. 2012). Although these processes need to be understood to explain the differences between the observed and theoretical prediction, further considerations must be made to take into account the selection effects.
In order to address the points raised above, we have carried out the largest Xray survey undertaken by XMMNewton. The XXL Survey (Pierre et al. 2016, hereafter Paper I), is a 50 deg^{2} survey with an Xray sensitivity of ~5 × 10^{15} erg s^{1} cm^{2} (for pointlike sources), in the [0.5–2] keV band. With the aim of detecting several hundred clusters out to a redshift of ≈2, this survey provides a unique opportunity to constrain cluster scaling relations that fully accounts for selection effects, and in the future will place robust constraints on cosmological parameters (see Paper I).
In this paper, part of the first release of XXL results, we investigate the form of the luminositytemperature relation (LT) for a sample of the 100 brightest clusters detected in the XXL Survey. The LT relation will be derived taking the selection function of the cluster sample fully into account. The outline of this paper is as follows. In Sect. 2 we discuss the data preparation and sample selection. Section 3 outlines the cluster analysis. In Sect. 4 we present our results and derive the sample and biascorrected scaling relations. Our discussion and conclusions are presented in Sects. 5 and 6, respectively. Throughout this paper we assume a WMAP9 cosmology of H_{0} = 70 km/s/Mpc, Ω_{Λ} = 0.72, and Ω_{m} = 0.28 (Hinshaw et al. 2013).
2. Data processing and sample selection
The data processing and sample selection are fully detailed in Pacaud et al. (2016, hereafter Paper II), and is briefly summarised here. The XXL Survey contains 542 XMM pointings covering 50.8 deg^{2}. After lightcurve filtering (following Pratt & Arnaud 2002), rejection of bad pointings and exclusion of pointings with high background levels, the total XXL area spans 46.6 deg^{2} (from 454 pointings). Images, exposure maps, and detector masks were generated and processed using the Xamin pipeline (Pacaud et al. 2006; Clerc et al. 2012). SExtractor was then run to generate a conservative source catalogue and source masks, followed by a dedicated XMM maximum likelihood fitting procedure to determine likelihood ratios to assess the detection and source extent probabilities. Extended sources were defined with an extent larger than 5′′ and extension likelihood larger than 15. These extended sources were then separated into two classes, the C1 class with extension likelihood larger than 33 and detection likelihood larger than 32, and the C2 class with an extension likelihood 15 < EXT_LH < 32.
The sample used in this work is the 100 brightest clusters (hereafter XXL100GC^{1}) selected from the source list generated above. Count rates were estimated from a growth curve analysis (GCA, following Clerc et al. 2012) within a 60′′ aperture, and converted into fluxes using an energy conversion factor (assuming T = 3 keV, z = 0.3, and Z = 0.3) of 9.04 × 10^{13}, or 1.11×10^{13} for clusters falling on the damaged mos1 chip. The clusters were ranked in order of decreasing flux and the brightest 100 clusters were selected. We note that five clusters within the XXL100GC fell on observations with high periods of flaring. The five flared observations were not used and instead the next five brightest objects were included. The clusters XLSSC 113, 114, 115, 550, and 551 were replaced with XLSSC 091, 506, 516, 545, and 548. Taking this into account, our selection corresponds to a flux limit of F_{X,cut} = 3 × 10^{14} ergs s^{1} cm^{2}. The final XXL100GC sample consists of 96 C1 and 4 C2 clusters, respectively, with 51 falling within the northern field of the XXL footprint, and 49 within the southern field. Following a robust redshift validation (see Paper I), the XXL100GC span a redshift range of 0.04 ≤ z ≤ 1.05, with 98 spectroscopic and 2 photometric redshifts. The sample and the derived properties can be found in Table 1.
3. Analysis
In this section we describe the cluster analysis process used in this work.
The extent of the cluster emission was defined as the radius beyond which no significant cluster emission is detected using a threshold of 0.5σ above the background level. This is intended to provide a conservative estimate of the radius beyond which there is no significant cluster emission. A onedimensional (1D) surface brightness profile of each camera was produced. A background annulus was defined with an inner radius of 250′′ and modelled by a flat (particle) and vignetted (Xray) component. The two model components were then fit to the radial profile in the background region using the χ^{2} statistic and then summed to produce a total background model. The radial bins were constructed to be 15.4′′ in width, chosen to ensure greater than 20 counts per bin and with enough bins to perform the fitting procedure. The radial profile for each camera was then summed to produce an overall profile. An initial source extent was determined based on the 0.5σ radius, and the background fitting repeated using the extent as the inner radius for the background annulus. This process was iterated until the source extent radius changed by less than 1%.
To account for the background in the spectral analysis, local backgrounds were used. However, owing to the survey detection of the clusters, they are detected at a range of offaxis positions on the XMM cameras. Therefore, if possible, an annulus centred on the aimpoint of the observation (BG_{aim}) with a width equal to the diameter of the spectral extraction region was used. This ensures that the local background is taken at the same offaxis position as the cluster, which helps to reduce the systematic uncertainties due to the radial dependencies of the background components. To ensure that no cluster emission was included in the background subtraction, a region centred on the cluster with radius equal to the cluster extent (see above) was excluded. Figure 1 shows an example of this local background for one of our clusters. However, if this method was not possible (owing to close proximity to the aimpoint or a large cluster extent), an annulus centred on the cluster (BG_{clust}) with inner radius equal to the cluster extent and an outer radius of 400′′ was used for the local background. The clusters XLSSC 060 and XLSSC 091 had a source extent larger than 400; therefore, an outer radius of 500′′ and 800′′, respectively, was used for the cluster centred local background.
Fig. 1
Example of the background method used in the spectral analysis (see Sect. 3). The image of the cluster XLSSC 010 with the regions showing the spectral extraction region (black circle), cluster detection radius (dashed red circle), and the background annulus centred on the aimpoint of the observation (blue annulus). 
Cluster spectra were extracted for each of the XMM cameras and fits were performed in the 0.4–7.0 keV band with an absorbed APEC (Smith et al. 2001) model (v2.0.2) with the absorbing column fixed at the Galactic value (Kalberla et al. 2005). The spectra for each camera were fit simultaneously with the temperature of the APEC components tied together. The fits were performed using XSPEC (v12.8.1i) and the abundance table from Anders & Grevesse (1989). Because of the low number of counts for many of the clusters, the spectra were fitted using the cstat statistic. The background spectra were grouped such that they contained at least 5 counts per bin, and this grouping applied to the source spectra. A similar method was employed and justified in Willis et al. (2005) who analysed a sample of 12 galaxy groups and clusters in the XMMLSS. Throughout the spectral analysis we assumed a fixed metal abundance of Z = 0.3 Z_{⊙}.
The cluster temperatures are derived within 300 kpc for each cluster, denoted as T_{300 kpc}. This radius represents the largest radius for which a temperature could be derived for the entire cluster sample. The normalisations for each camera were free in the spectral fit, with the pn camera used to calculate the luminosity. We denote the luminosity within 300 kpc as , where the superscript XXL refers to the [0.5–2.0] keV band (cluster rest frame). Luminosities quoted within r_{500} are extrapolated from 300 kpc out to r_{500} by integrating under a βprofile assuming r_{c} = 0.15r_{500} and β = 0.667. The βprofile parameters were chosen to match those used in Paper II. We note that the uncertainties on the luminosity are scaled by this extrapolation, but do not include any uncertainty on the βprofile parameters. The impact of these assumptions is tested in Sect. 5.3.4. Values for cluster r_{500} are calculated using a masstemperature relation (r_{500,MT}, see below). We denote as the luminosity in the [0.5–2.0] keV band (cluster rest frame) within r_{500,MT}, and as the bolometric luminosity within r_{500,MT}. All the properties quoted include the cluster core because the exclusion of the cluster core is not possible for all 100 clusters. The results of our spectral analysis are given in Table 1. The cluster XLSSC 504 was dropped because of the unconstrained errors reported by XSPEC.
A masstemperature (M_{WL}−T) relation is used in this work for two purposes: (i) to calculate r_{500,MT} for the XXL100GC and (ii) to convert the mass function to a temperature function (see Sect. 4.3) for the bias correction. The M_{WL}−T relation is presented in Lieu et al. (2016, hereafter Paper IV), based upon weak lensing cluster masses and the temperatures presented in this work. Briefly, the M_{WL}−T relation was determined using 37 XXLN clusters that fell within the CanadaFranceHawaii Telescope Lensing Survey (CFHTLenS) footprint, utilising the CFHTLenS shear catalogue (Heymans et al. 2012; Erben et al. 2013) for the mass measurements. In order to increase the statistics and the mass range covered, the XXL100GC is combined with 10 groups from the COSMOS survey (Kettula et al. 2013), and 50 massive clusters from the Canadian Cluster Comparison Project (CCCP, Hoekstra et al. 2015).
The combined M_{WL}−T relation is fitted with a powerlaw of the form (assuming selfsimilar evolution) (1)using the Bayesian code linmix_err (Kelly 2007). In Paper IV we find and . See Paper IV for further discussions on the results from the M_{WL}−T relation.
4. Results
Fig. 2
Left: LT relation for the XXL sample. The black line represents an unbiased fit to the data (see Sect. 4.1). The LT relations of Pratt et al. (2009) and Clerc et al. (2014), given by the red dotdashed line and the green dashed line, respectively, are overplotted; right: same as the left plot, but with the REXCESS clusters studied in Pratt et al. (2009) overplotted, split between coolcore (open red squares) and noncoolcore (filled red squares) clusters. In both plots, the errors on the temperatures for the XXL clusters have been transformed via the method described in Sect. 4. 
The results are presented in the form of a study of the LT relation, taking the selection effects fully into account. We present two implementations to account for the selection effects: (i) an updated method of the Pacaud et al. (2007) implementation, defined as the XXL likelihood; and (ii) a method based upon Mantz et al. (2010b), defined as the M10 likelihood. We first present the sample LT relation, not correcting for the selection effects. This allows for comparison of the LT relation when taking into account selection effects, what impact this has on the derived scaling relation, and also for the comparison with previously published relations that do not account for selection effects.
For the analysis of the scaling relations, we account for the fact that the likelihood curve for a measured temperature is approximately Gaussian in log space (consistent with the asymmetric errors usually found in temperature measurements). We use the method of Andreon (2012) to convert the generally asymmetric errors reported by XSPEC into a lognormal likelihood. We note that the errors on the temperature given in Table 1 are those reported by XSPEC.
Bestfitting parameters for the LT relations modelled in this work taking the form L/L_{0} = E(z)^{γLT}A_{LT}(T/T_{0})^{BLT}, where L_{0} = 3 × 10^{43} erg s^{1} and T_{0} = 3 keV.
4.1. Sample LT relation
Figure 2 (left panel) shows the L_{XXL}T relation for the XXL100GC. For simplicity, the L_{XXL}T notation explicity refers to the relation. A fit to the data using a power law of the form (2)was performed, where A_{LT}, B_{LT}, and γ_{LT} represent the normalisation, slope, and power of the evolution correction respectively. We note that the two clusters at low luminosity offset from the L_{XXL}T are the clusters XLSSC 011 and 527. These clusters are low S/N clusters at low redshift, 0.054 and 0.076, such that the 300 kpc region is large on the detector. The temperature measurement is likely affected by these factors. However, we still include them in the fit to the L_{XXL}T relation. The power law was fit to the data using the BCES orthogonal regression in base ten log space (Akritas & Bershady 1996) assuming selfsimilar evolution. The fit is given by the black solid line in Fig. 2, assuming L_{0} = 3 × 10^{43} erg s^{1}, T_{0} = 3 keV, and γ_{LT} = 1 (selfsimilar). We find a normalisation of A_{LT} = 0.90 ± 0.06, and slope B_{LT} = 3.03 ± 0.27. For comparison, we fit the LT relation to the 31 REXCESS clusters studied in Pratt et al. (2009, hereafter P09), and 52 clusters selected within the 11 deg^{2} XMMLSS survey (Clerc et al. 2014, hereafter C14). To fit for these LT relations we use the data from Table B1 (Cols. T_{1} and L[0.5−2]_{1}) in P09, and the data given in Table 1 (Cols. T_{X} and ) in C14. The P09 and C14 relations are hereafter denoted as L_{P09}−T and L_{C14}−T, respectively, and the fit parameters are given in Table 2. The fits to the P09 and C14 clusters are given by the red dotdashed line and green dashed line, respectively, assuming the same L_{0}, T_{0}, and γ_{LT} as for the XXL100GC fit. We find no significant difference between the XXL100GC L_{XXL}T relation and the L_{C14}−T relation. This is unsurprising as the C14 clusters are selected from the XMMLSS area, which has many clusters in common with the XXL100GC.
We find a 2.8σ difference in the normalisation as compared to the REXCESS clusters, with the XXL normalisation being lower. This is due to the presence of strong cool core clusters in the REXCESS sample. These clusters are apparent as highluminosity outliers in Fig. 2 (right), which shows the REXCESS clusters plotted on the XXL LT relation. Qualitatively, it is clear that in the absence of these strong cool core clusters, the remaining REXCESS clusters are consistent with the L_{XXL}T relation. As discussed in Sect. 5.2, the absence of strong cool core clusters from the XXL100GC sample is due to a combination of survey geometry and cool core evolution.
4.2. Selection function
Full details of the construction of the selection function are given in Paper II, but the key points are summarised here. The selection function takes into account three aspects of the XXL100GC selection, (i) the pipeline detection; (ii) the flux cut of the XXL100GC; and (iii) the survey sensitivity. The pipeline detection, i.e. the C1+2 classification in pointing p (denoted P_{C1 + 2,pi}(I  CR_{∞},r_{c},RA,Dec)), was studied in depth in Pacaud et al. (2006) and updated following the methodology of Clerc et al. (2012). We refer the reader to these works for full details. The modelling of the flux cut assumes that the errors on the GCA count rate can be modelled as a Gaussian distribution. The probability of a true aperture count rate (CR_{60}) being greater that the countrate cut (CR_{cut}) for a given pointing (p_{i}) is (3)where ϵ_{60}(r_{c}) = (1−[1 + (60′′/r_{c})^{2}]^{1.5−3β}) and CR_{∞} is the pipeline count rate extrapolated to infinity. The final ingredient is the modelling of pointing overlaps, where we assume the detection is independent over the different pointings. For a given position, pointings are sorted by increasing offaxis distance, reproducing the overlap crossmatching procedure of the survey. The combined selection function is then given by (4)
4.3. Likelihood
The observational data used in this study is the distribution of the XXL100GC clusters in L, T, and z. Our physical model assumes that the cluster population is described by a powerlaw correlation between L and T with lognormal scatter in L and evolution in L by E(z)^{γLT}. The XXL100GC data represent a subset of this population selected according to our selection function and with noisy measurements of L and T, denoted and . We neglect any measurement errors on z.
The number density of clusters in the survey volume is taken into account by using a mass function, assumed to be described by a Tinker mass function (Tinker et al. 2008). We then transform this to a temperature function, dn/ dT, using the M_{WL}−T relation given in Sect. 3. In the present analysis we neglect the intrinsic scatter in the M_{WL}−T relation, and the measurement errors on the parameters of the scaling relation, and assume that its evolution is selfsimilar (i.e. M ∝ E(z)T^{BMT}).
The total number of clusters N in the volume is then the integral over the temperature and redshift range considered, multiplied by the solid angle of the survey Ω: (5)The number of clusters predicted by our model to be observed in the subsample defined by our selection function is the integral of the mass function over the volume of the survey, weighted by the probability that a cluster of a given mass would be included in the subsample given the LT relation and the intrinsic and statistical scatter on the luminosity, (6)where θ stands for our full set of model parameters (A_{LT},B_{LT},γ_{LT},σ_{LT}) describing the LT relation. In this expression, the first probability, P(LT,z,θ), is the probability that cluster of temperature T has some intrinsically scattered luminosity. The second probability, P(IL,T,z), is the selection function, i.e. the probability that a cluster with a luminosity L and temperature T at a redshift z would be included in the subsample. This is summarised in Sect. 4.2. We note the change in notation as we are writing the likelihood in terms of the cluster properties L and T.
The (unnormalised) likelihood of a cluster i in our sample having the observed properties (,) is given by (7)The inclusion of ⟨N⟩ normalises the temperature function to a probability distribution for an arbitrary cluster to have a temperature T at redshift z. The probability of P(LT,z) is as defined above, and the remaining terms are the probability of each of the observables using the measured uncertainty for that observable. The joint probability of the full set of observed cluster properties is the product of over all N_{det} observed clusters in the sample.
Equation (7) is an improper probability because P(IL,T,z) does not integrate to unity, and it does not penalise the model for predicting the existence of clusters in parts of the L,T,z space where they are within the selection function but are not observed. In other words, the model is not penalised for excess probability density in regions of the space where the lack of detections disfavours the existence of clusters. This is resolved by normalising Eq. (7) by the integral over the observed space to give the final likelihood (8)This normalisation penalises the model for excess probability density in regions of the parameter space where the data disfavour the existence of clusters. For example, we consider a set of model parameters that give a good fit to the properties of the observed clusters, but also give a high probability of cool, highly luminous clusters, when none are observed. This would lead to a larger denominator in Eq. (7), and hence a lower overall likelihood compared with an alternate set of parameters that describes the properties of the observed clusters equally well, but does not predict the unobserved clusters. This likelihood is referred to as the XXL likelihood.
We also consider an alternative construction of the likelihood set out by M10. In this approach, the final likelihood for the sample of clusters and their observed properties is the product of a Poisson likelihood of N total clusters (detected plus undetected) given the model prediction ⟨N⟩, a binomial coefficient accounting for the number of ways of drawing N_{det} detected clusters from the total N, the joint probability of the set of observed cluster properties (the product of Eq. (7) over the N_{det} clusters), and the probability of not detecting the remaining N−N_{det} clusters. Neglecting terms not dependent on the model parameters, the likelihood simplifies to (9)where for the ith cluster.
The principal difference between Eqs. (8) and (9) is that the latter has a stronger requirement that the model must accurately predict the number of observed clusters in addition to their distribution in (L,T,z). This has the advantage that it uses additional information to constrain the model, and is a requirement when the analysis is being used to constrain cosmology in addition to the form of the scaling relations, as in M10.
However, when the aim of the analysis is to infer the form of the scaling relations under an assumed cosmology, Eq. (8) has certain advantages. This implementation is insensitive to systematics affecting the numbers of clusters (such as the normalisation of the mass function and hence σ_{8}) or the normalisation of the MT relation, and so gives more robust measurements of the scaling relation parameters. Furthermore, since the number of observed clusters is not used to constrain the model parameters in this approach, the number of detected clusters can be used as a posterior predictive check.
4.4. Inference of model parameters
The likelihood of Eq. (7) was combined with priors on each of the model parameters to compute the posterior probability. The priors used were uniform in the range (− ∞,∞) for A_{LT},B_{LT}, and γ_{LT} and uniform in the range (0.01,2.0) for σ_{LT} (expressed in natural log, so representing fractional scatter).
The posterior distribution was analysed using the Bayesian inference package Laplace’s Demon^{2} within the R statistical computing environment (R Core Team 2014). The posterior distribution was first explored using a Laplace approximation for computational efficiency before refining the fit using an MCMC algorithm. We used the Adaptive MetropoliswithinGibbs algorithm in Laplace’s Demon for this purpose, and used four parallel chains of 50 000 iterations each, initialised to randomised starting values (near the mode of the posterior identified by the Laplace approximation). The stationary parts of the chains were compared using the Gelman and Rubin (1992) convergence diagnostic, and the largest value of the 95% upper bound on the potential scale reduction factor was 1.01, giving a strong indication that the chains had converged. The stationary parts of the chains were then concatenated giving an effective sample size of at least 500 for each parameter.
Fig. 3
L_{XXL}T relation with the bestfitting model. The light blue circles show the XXL100GC clusters; the bestfitting model is shown as the solid black line the 1σ uncertainty represented by the grey shaded region. 
4.5. The L_{XXL}T relation
The L_{XXL}T relation is plotted in Fig. 3 along with the bestfitting model (black line). The bestfitting parameter values and their uncertainties are then summarised by the mean and standard deviation of the posterior chains for each parameter. The values are given in Table 2, and illustrated with the scatterplot matrix in Fig. 5. We focus our attention on the XXL likelihood method, noting that the results do not change significantly between the XXL and M10 methods. Using the XXL likelihood method, we find a normalisation and slope of A_{LT} = 0.71 ± 0.11 and B_{LT} = 2.63 ± 0.15, respectively. We find a shallower slope than that found when using the BCES regression fit (which did not account for biases, see Sect. 4.1), although the difference in slope is only significant at the 1σ level. The L_{XXL}T relation is consistent with recent results which also accounts for selection biases. Bharadwaj et al. (2015) studied the LT relation of 26 groups (spanning the 0.6 <T< 3.6 keV range) and found a slope of B_{LT,B14} ≈ 2.7, converting their LT relation from bolometric to the soft band (see Sect. 4.7). The comparison is complicated, however, owing to the differing implementations of the biascorrection and fitting methods used in Bharadwaj et al. (2015) and in our work. Interestingly, they find that the slope of the sample LT relation is shallower than the biascorrected fit, opposite to what we find in this work. However, this could be driven by the large number of strong cool core (SCC) systems in their sample (≈50%); the SCC LT relation shows a change in slope from 2.56 ± 0.22 to 3.60 ± 0.22 when correcting for biases.
Next we focus on the evolution of the L_{XXL}T relation, γ_{LT}, where the evolution is expressed as E(z)^{γLT}. For the first time we are able to measure the evolution of the LT relation using clusters drawn from a single homogeneous survey, fully accounting for selection biases. As introduced previously, we expect γ_{LT} = 1 for strong selfsimilar evolution, and γ_{LT} = 0.42 for weak selfsimilar evolution. We find γ_{LT} = 1.64 ± 0.77, fully consistent with both the the strong and weak selfsimilar evolution models. Figure 4 plots the evolution of the LT relation as inferred from our bestfitting model. The bestfit evolution is given by the black solid line along with the 1σ uncertainty, the strong and weak selfsimilar expectations are given by the red and blue dashed lines, respectively. Our bestfit model appears to lie below all the highredshift clusters, due to the fit being driven by the larger number of lower redshift clusters. However, the evolution favoured by our bestfit model is in tension with other recent results, and is discussed further in Sect. 5.1.
Fig. 4
Evolution of the L_{XXL}T relation for the XXL100GC. The XXL100GC are represented by the light blue circles and the bestfitting model is given by the black solid line; the grey shaded region highlights the 1σ uncertainty. The “strong” and “weak” selfsimilar expectations are given by the red dashed and blue dashed lines, respectively. 
Fig. 5
Scatterplot matrix for the fit of the L_{XXL}T relation of the XXL100GC sample. The posterior densities are shown along the diagonal; the 1σ, 2σ, and 3σ confidence contours for the pairs of parameters are shown in the upper right panels. The lower left panels show the Pearson’s correlation coefficient for the corresponding pair of parameters (text size is proportional to the correlation strength). 
Fig. 6
Contours of the simulated clusters generated from the bestfitting model are plotted in the L,z (left panel); T,z (middle panel); and L,T (right panel) planes, along with points indicating the observed clusters. 
4.6. Posterior predictive checks
While the modelling process described above determines the bestfitting values of the model parameters for the chosen model, it does not guarantee that the model is a good description of the data. For this we use three types of posterior predictive check to assess how well the final model describes the data.
First, we make visual comparisons of the population of clusters predicted by the bestfitting model and those observed. To do this, a large number of model clusters were generated using the bestfitting model parameters, and including the selection function. Figure 6 shows contours of the simulated population along with the observed clusters in different projections of the observed properties. Figure 7, instead, shows histograms of the simulated and observed properties. In all cases the visual agreement is good.
The second test was to compare the number of clusters predicted to be observed by the bestfitting model with the number observed. The model predicted 116.7 clusters, so the probability of observing a sample as discrepant as or more discrepant than the 99 clusters observed is the sum of the Poisson probabilities P(N< 99  116.7) + P(N> 134  116.7) = 0.11. The number of observed clusters is thus reasonably consistent with the number predicted. We emphasise that this is not trivial, as the model is not required to reproduce the number of observed clusters in our likelihood.
The final test we make is to measure the posterior predictive pvalue (Meng 1994). This is done by measuring the discrepancy of the data with the model for a set of the model parameters taken from the posterior distribution, and doing the same for a simulated data set generated from the same model parameters. This is repeated for a large number of sets of parameters sampled from the posterior, and the fraction of instances in which the simulated points are more discrepant than the observed data is the posterior predictive pvalue. A small fraction indicates that the model is a poor description of the data.
To implement this, we used the χ^{2} statistic to measure the discrepancy between the observed or simulated data and the model (as in e.g. Andreon 2012). We scale the luminosity of each point by E(z)^{γLT} for the value of γ_{LT} currently considered and use the modified χ^{2} described in Press et al. (1986), which includes the uncertainties in both L and T. For each sample of parameter values, a simulated population of 99 clusters was generated and measurement errors were assigned by taking the logspace errors from a cluster randomly selected from the observed data; the corresponding statistical scatter was then added. This sampling of errors is justified, since we find no dependency of the size of the errors on the measured parameters.
This procedure was repeated for 1000 samples from the posterior; in 19% of those iterations the simulated data were more discrepant than the observed data. We thus conclude that there is no strong evidence for the data to reject our model.
4.7. The L_{bol}T relation
So far we have been considering the scaling of softband luminosity with temperature, but it is often useful to refer to the bolometric luminosity, L_{bol}. We can convert the L_{XXL}T relation to an L_{bol}−T relation by using a kcorrection.
Using XSPEC simulations, we find this kcorrection can be approximated by (10)with A_{k} = 0.587 and B_{k} = −0.450. This power law approximates the kcorrection to within ≲3% over the temperature range of our sample. Substituting into the LT relation we get the bolometric relation: The L_{bol}T relation is shown in Fig. 8; the bestfitting model coefficients are given in Table 2.
5. Discussion
In this work we have shown that the evolution of the LT relation is consistent with the strong and weak selfsimilar expectation (see Sect. 4.5). However, our bestfit evolution is not in agreement with previous studies, and therefore warrants further discussion. We will compare our results with other observational and theoretical work and examine systematics in our analysis that could influence the measured evolution.
Fig. 7
Histograms of the redshift (left panel), luminosity (middle panel), and temperature (right panel) of the observed clusters are plotted along with the density distribution of the same properties from the simulated clusters generated from the bestfitting model. The histograms and curves are normalised to a total area of 1. 
Fig. 8
L_{bol}T relation with bestfitting model. Our bestfitting model is shown as the solid black line with the grey shading indicating the 1σ uncertainty. The bestfitting model is transformed from the soft band to bolometric via the steps described in Sect. 4.7. 
5.1. Comparison with other observational studies
We found that the normalisation of the L_{XXL}T relation evolves as E(z)^{1.27 ± 0.82} (Sect. 4.5). This would indicate that clusters are more luminous as a function of increasing redshift for a given temperature. Our result, however, does not agree with recent studies that find a negative evolution (Reichert et al. 2011; Hilton et al. 2012; C14).
The discrepancy with C14 is surprising, given the overlap in the clusters used from the XMMLSS and that they account for selection effects. Clerc et al. (2014) find a negative evolution when using various local baseline LT relations. Assuming the normalisation of the LT relation evolves as E(z)(1 + z)^{αLT}, they find α_{LT} = −2.5 ± 0.4 and α_{LT} = −1.6 ± 0.4 when using the P09 and Maughan et al. (2012) LT relations as local baselines. The differences in the assumptions made in C14 and this work are the likely cause of the discrepancy. These assumptions are (i) the local relation used as a reference for comparison to the highredshift clusters; (ii) the choice of MT relation to determine r_{500} and hence L_{500}; C14 use the MT relation of Sun et al. (2009), whereas we use the internally calibrated relation in Sect. 3; (iii) the value of r_{c}, where Clerc et al. (2014) use r_{c} = 180 kpc and we use r_{c} = 0.15r_{500}; and (iv) C14 use the likelihood from Pacaud et al. (2007) for their bias correction, which does not account for the mass function.
To test the dependence of the fit to these assumptions, we repeated the fit using assumptions close to those in (i)–(iii). For this comparison we use the LT relation of P09 for a local baseline, which gave the strongest change in evolution found in C14. We found that we recover a consistent, strong negative evolution on LT as found in C14 when using the same assumptions on the local LT relation, r_{c}, and MT relation. However, given the small dependence of the results on the assumed x_{500} (defined as the ratio of r_{c} and r_{500}) and MT relation (see Sect. 5.3 below), the difference in evolution is likely driven by the assumed local LT relation. This shows that differences in the local LT relation, of less than 3σ (see Sect. 4.1), can change the form of the LT relation evolution from positive to negative. As we discuss in Sect. 5.2, this effect appears to be driven by the differing cool core populations in the samples used for the local LT relation.
Many studies investigating the evolution of the LT relation, use an external local baseline with which to compare the normalisation of the LT relation. This complicates the interpretation of departures from selfsimilar evolution, and can change the form of the evolution (see above). Hilton et al. (2012) investigated the evolution of the LT relation using a complete sample of 211 clusters drawn from the XMM Cluster Survey (XCS Romer et al. 2001). Using an internally calibrated LT relation, they find a negatively evolving LT relation of the form (1 + z)^{− 1.5 ± 0.5}. This is again in disagreement with the positive evolution found in this work; however, we note that the Hilton et al. (2012) results do not account for selection biases as we have done here.
Bestfitting parameters for the LT relation while varying some assumptions in the model.
5.2. Impact of cool cores
The evolving cool core population in clusters complicates the interpretation of the evolution in the LT relation. Cooling is not present in the selfsimilar model, and so a reduction in the number or strength of cool cores at high redshift (McDonald et al. 2014) would manifest as weaker than selfsimilar evolution. As we saw in Sect. 5.1, using an external sample which contains a larger number of strong cool core clusters (such as P09; see our Fig. 2, right plot) strongly affects the magnitude and sign of the measured evolution. It is thus possible that previous measurements of negative evolution in the coreincluded LT relation mainly reflect the decreasing contribution of cool cores to cluster luminosities at high redshift. An evolving massdependence of the prevalence and strength of cool cores would further complicate the interpretation.
The P09 sample appears to contain a population of strong coolcore clusters, which could drive the regression fit to a higher normalisation not seen in the XXL100GC sample. This is likely due to a combination of the evolving cool core population and the geometries of the two surveys. The REXCESS clusters were selected from REFLEX (Böhringer et al. 2001), a fluxlimited (F × [ 0.1−2.4 keV ] ≥ 3 × 10^{12} ergs s^{1} cm^{12}) wide area (≈14 000 deg^{2}) cluster survey sensitive primarily to highluminosity clusters at redshifts z ≲ 0.25, while XXL has a much smaller local volume sensitive to lower luminosity clusters than REFLEX. This result was found in Paper II (see Fig. 16), where our data was compared to REFLEXII (Böhringer et al. 2013).
To explain the apparent lack of strong cool core clusters in the XXL100GC sample, we make the simplifying assumptions that the cool core clusters in the P09 sample are uniformly distributed in volume out to z = 0.2 (the limit of the REXCESS sample). The total volume of the REXCESS sample selection was calculated from the conditions given in Table 1 of Böhringer et al. (2007). From the ten cool core clusters classified in P09, we would expect an average of one cool core cluster per 1.1 × 10^{8} Mpc^{3}. When applying this to the 50 deg^{2} area of XXL out to z = 0.2, we would expect ≈0.03 strong cool core clusters.
Thus, the evolution measured in the L_{XXL}T relation is not confounded by the changing cool core population, and arguably probes the evolution of the baryon content of the clusters more cleanly than studies which compare highz clusters from small area surveys with lowredshift samples from wide area surveys.
5.3. Systematic effects
Here we investigate the dependence on the model fit to assumptions made in this work.
5.3.1. The choice of x_{500}
One assumption that could have an effect on the L_{XXL}T relation and its evolution is the relation between r_{c} and r_{500,MT} (where we adopt x_{500} = 0.15). To test the dependence of the fit on the assumed x_{500}, we repeated the fit for x_{500} = 0.1 and x_{500} = 0.2. The results are given in Table 3 and show no significant difference in the fit parameters when x_{500} is varied in this range. However, x_{500} is assumed to be independent of mass and redshift, but this definition introduces a dependence of the physical size of r_{c} on mass and redshift, in line with selfsimilar expectations. Given the decreasing fraction of sharply peaked coolcores with increasing redshift (Vikhlinin et al. 2007; Santos et al. 2011; Mann & Ebeling 2012; McDonald et al. 2014), one could expect that the average r_{c} value would increase with redshift. Hudson et al. (2010) found that there was a trend, albeit with large scatter, between r_{c} and the central cooling time (CCT) for their sample of 64 HIFLUGS clusters. The CCT is a robust proxy for the presence of a coolcore, and therefore this trend indicates that r_{c} increases from CC to NCC clusters (see Fig. 6b in Hudson et al. 2010). Furthermore, Hudson et al. (2010) note a temperature dependence on r_{c} such that cooler clusters appear to have smaller values of x_{500} compared to hotter clusters. Figure 6 shows the temperaturez distribution of our clusters, highlighting how the median cluster temperature in the sample increases with redshift. Depending on the strength of the x_{500}temperature relation, we could be artificially introducing an evolution on x_{500}. Coupled with the r_{c}CC/NCC dependance, we are most likely underestimating the value of x_{500} for the highredshift clusters. This would lead us to underestimate the luminosities of the highredshift clusters, steepening the slope of the LT relation. This would lead to the evolution of the LT relation being underestimated.
5.3.2. The choice of MT relation
The choice of MT relation will also have an impact on our results. The MT relation is used (i) to calculate r_{500,MT} for the cluster sample and (ii) to convert the mass function to a temperature function (using the M_{WL}−T relation, see Sect. 4.3). Throughout, we have used the M_{WL}−T relation presented in Paper IV based on XXL+COSMOS+CCCP clusters. To test the dependence of the fitted LT relation to the choice of MT relation, we repeat the fit using the MT relation of Arnaud et al. (2005), which has a similar slope to our M_{WL}−T relation but is ≈20% lower in normalisation at 3 keV. The results are given in Table 3. We find that the fit using the XXL likelihood does not change significantly when using the Arnaud et al. (2005)MT relation. However, the fit performed using the M10 likelihood with the Arnaud et al. (2005)MT relation, would imply weaker than selfsimilar evolution of the LT relation. Because of the large errors on γ_{LT}, the difference is only significant at the ~1σ level. Furthermore, as a result of the overprediction of the number of clusters in the M10 fit (a requirement of the M10 likelihood), this fit is not an accurate description of the data. The predicted number of clusters from the M10 fit using the Arnaud et al. (2005)MT relation, 114 clusters, does not agree with the observed number of 99 found using the Poisson calculation as above (see Sect. 4.6). We note that the XXL likelihood strongly disfavours the use of the Arnaud et al. (2005)MT relation in the context of the overall model because of the 178 predicted clusters. This shows that the XXL method is less sensitive to changes in the MT relation than the M10 fitting method. However, because of the large errors, drawing conclusions on the effects of the choice of MT relation and likelihood model on γ_{LT} will require many more clusters than contained in the XXL100GC sample.
Fig. 9
Left: comparison of the luminosity extrapolated out to r_{500} () to that determined from a spectral analysis within r_{500} (). Right: comparison of the temperature derived within 300 kpc (T_{300 kpc}) to the temperature derived within r_{500} (T_{500}). The black line represents a 1:1 relation. 
5.3.3. The role of scatter
Even with our efforts to model the impact of selection bias on the LT relation and its evolution, we have made the simplifying assumption that the scatter in the LT relation is independent of both mass and redshift. There is evidence to suggest that the scatter in luminosity for a given temperature or mass decreases for higher redshift systems (Maughan 2007, although this analysis did not account for selection effects). This can be understood from the decline of strong cool core systems towards higher redshift (e.g. McDonald et al. 2014).
Because the higher redshift clusters in our sample have higher temperatures on average, the decreasing cool core fraction leads to a decreasing scatter in L at a given T for the higher redshift part of our sample. Therefore, our model, which assumes a constant scatter, may overestimate the scatter in the highz population. This would lead to the evolution of the L_{XXL}T relation being underestimated since a lower scatter would require stronger positive evolution to reproduce the observed luminous highredshift clusters.
5.3.4. The extrapolation out to r_{500}
Throughout this work we have used a spectral extraction region of 300 kpc for the cluster analysis. This was chosen because a spectral analysis within r_{500} could not be achieved for the majority of the cluster sample. The luminosities are determined within 300 kpc, and extrapolated out to an estimate of r_{500} by integrating under a βprofile (see Sect. 3). We investigate here what systematic effect this could have on the derived luminosities by comparing the extrapolated luminosities () with those determined via a spectral analysis within r_{500} () for the brightest clusters in the sample. We also compare the temperature derived within 300 kpc (T_{300 kpc}) with that derived within r_{500} (T_{500}) to check for any systematic effects.
We estimate r_{500} via an iterative process based on the MT relation in Paper IV. An initial temperature of 2 keV is used to calculate an initial r_{500}, a spectrum is extracted, and a temperature determined. A new r_{500} is calculated from this temperature, and the process is iterated until r_{500} changes by less than 1%. The left plot in Fig. 9 compares to for the ten highest flux clusters in the XXL100GC for which the iteration process could be performed. We find good agreement between the luminosities for lowluminosity clusters (≲7×10^{43} erg s^{1}). However, for higher luminosity clusters (≳7×10^{43} erg s^{1}) we find evidence that could be underestimated.
Because of the extrapolation out to r_{500}, the values of would be underestimated by either i) underestimating the value of x_{500} and hence r_{c} or ii) overestimating the value of β. Section 5.3.1 explored recent evidence for a trend of r_{c} with increasing temperature, hence increasing luminosity. The increase in r_{c} with increasing luminosity would explain the underestimate of compared to , hence leading to an underestimate of the evolution of the LT relation (as stated in Sect. 5.3.1).
The right plot in Fig. 9 compares T_{300 kpc} to T_{500}. We find no systematic differences between the two temperatures; therefore, we can assume that T_{300 kpc} represents the global temperature.
5.4. Do the data require γ_{LT} as a free parameter?
Fig. 10
Contours of the simulated clusters generated from our freeevolution (shaded contours), strong selfsimilar (black solid contours), and weak selfsimilar (red dashed contours) evolution models (see Sect. 5.4). Solid black points indicate the observed clusters in our sample. 
Our bestfitting evolution is consistent with strong selfsimilar evolution, but only stronger than the weak self similar prediction at the ~1σ level. One may question whether we are justified in fitting for the evolution at all. In other words, compared with a model with fixed selfsimilar evolution, is the improvement in the fit when we add the additional flexibility to the model to fit the evolution sufficiently large to justify the extra complexity in the model?
There are several ways to address this in a Bayesian framework, and we adopted the deviance information criteria (DIC Spiegelhalter et al. 2002; Liddle 2007). The DIC can be computed for different models and the model with the lowest DIC is preferred by the data with a degree of support determined by the difference in DIC between competing models. The deviance is defined as D = −2lnℒ and the DIC is then given by (13)where is the mean of the deviance computed over the MCMC chain, and p_{D} is a measure of the effective number of parameters of the model. The latter term penalises more complex models, and is given by (14)The term is the deviance computed at the mean parameter values from the MCMC chain.
The DIC was computed for models with γ_{LT} as a free parameter fixed at the strong and weak selfsimilar values of 1 and 0.42, respectively. In both cases, the difference between the freeevolution model and the fixed evolution models was greater than − 1. Conventionally, differences in DIC more negative than − 5 are taken as strong evidence of the more complex model. Our analysis suggests therefore that while the data are better described by the freeevolution model (i.e. is lower), the improvement is not sufficient to give strong support for including evolution as a free parameter.
While the data as a whole do not strongly support fitting for the evolution, we do observe that the freeevolution model appears to provide the best description of the more distant clusters. This is most clearly seen in the T,z plane plotted in Fig. 10, which compares the distribution of the observed clusters in this plane with distributions of large populations simulated from the bestfitting models for each evolutionary scenario. The filled contours show the distribution of simulated clusters from the freeevolution model; these are the same as those shown with the contours in Fig. 6 (middle plot), but with fewer contour levels for clarity. The black solid and red dashed contours show the distributions of clusters simulated from the strong and weak selfsimilar evolution models, respectively. In all cases the contour levels enclose 90%, 50%, and 10% of the density of simulated clusters.
Figure 10 suggests that while the overall modelling of the XXL100GC data do not strongly support fitting for the evolution of the LT relation, the fixed evolution models are not successful at reproducing the observed population of z> 0.6 clusters. These clusters appear to prefer stronger than (weak) selfsimilar evolution, in order that they are sufficiently luminous to be included in the XXL100GC sample. With this in mind, it is worth considering the interpretation and implications of a strongerthanselfsimilar evolution in the LT relation.
5.5. Interpreting γ_{LT}
The evolution of the LT relation is best expressed as a combination of the evolution and slopes of the LM and MT relations (e.g. Maughan 2014), (15)where B_{LM} and γ_{LM} are the slope and power of the evolution term of the LM relation, respectively. We note that B_{TM} is the reciprocal of the slope of the MT relation, B_{MT}.
Thus, strongerthanselfsimilar evolution in the LT relation implies strongerthanselfsimilar evolution in the LM relation, or weakerthanselfsimilar evolution in the TM relation. Strongerthanselfsimilar evolution in the LM relation requires either that the baryon fraction within a fixed overdensity be higher at highz (disfavoured by simulations, e.g. Planelles et al. 2013; Battaglia et al. 2013) or that it be driven by higher density regions of ICM at highz. In the absence of cool cores, such regions could perhaps be associated with the larger degree of substructure typically seen in higher redshift clusters as we approach the epoch of cluster assembly (e.g. Jeltema et al. 2007; Maughan et al. 2008).
Alternatively, weakerthanselfsimilar evolution of the TM relation would require clusters to be cooler for a given mass at high redshift. This could be caused by incomplete virialisation of the ICM at higher redshift. Given that our analysis assumes selfsimilar evolution of the TM relation, a weaker evolution would mean that we underestimate the mass and hence r_{500} for higher redshift clusters in our Bayesian analysis. This in turn would mean that the luminosity extrapolated to r_{500,MT} would be underestimated and that the core radius (defined via x_{500}) would be underestimated. An underestimation of the core radius for highz clusters would generally mean that we overestimate the detection probability for those clusters. This would lead us to underestimate the evolution in LT, since fainter clusters would have a higher detection probability than if the core radius were larger. These arguments together imply that any inference of weakerthanselfsimilar (i.e. more negative) evolution of the TM relation could be regarded as an upper limit, in the sense that the true evolution could be weaker still.
5.6. Comparisons with simulations
Since we find a positive evolution of the L_{XXL}T relation, we compare our data to simulations in order to determine what physical processes could be at play that give rise to our observed evolution. Short et al. (2010) studied the evolution of scaling relations using simulations from the Millennium Gas Project (Springel et al. 2005). They employ three different models in their simulations, a gravityonly (GO) control model, a simple preheating model (PC), and a model using feedback from the energy input due to SNe and AGN (FO). Firstly, they show that the local LT relation for the PC and FO runs compare well to the observed LT relation, finding B_{LT,S10} ≈ 3. Secondly, they find that the evolution of the LT relation behaves differently for the PC and FO simulations. The LT relation evolves negatively for the PC simulation, whereas the FO simulation leads to a positively evolving LT relation. The evolution found using the FO simulation follows the positive evolution of the L_{XXL}T relation found in this work.
As shown in Sect. 5.5, the interpretation of the evolution of the LT relation is complicated owing to the dependence of the evolution on the LM and TM relations. Short et al. (2010) found that the evolution of the TM relation evolves negatively, a result implied by the observed positive evolution of the L_{XXL}T relation. Negative evolution of the TM relation was also found in Pike et al. (2014), using simulations taking into account radiative cooling, star formation, supernovae feedback, and AGN feedback. This provides evidence that the positive evolution observed in the L_{XXL}T relation is underestimated (see Sect. 5.5). Furthermore, Short et al. (2010) and Pike et al. (2014) find a positively evolving LM relation with a slope steeper than the selfsimilar expectation. A positively evolving LM relation with a steeperthanselfsimilar slope and negatively evolving TM relation (the slope of the TM relation from the simulations agree with the selfsimilar expectation) would lead to a positively evolving LT relation. Therefore, the evidence of a positively evolving L_{XXL}T relation (see Sect. 4.5) is in line with the expectation from simulations.
6. Conclusions
We have presented a detailed analysis of the brightest 100 clusters detected in the 50 deg^{2} XMMXXL Survey; the sample spans a wide range of redshift (0.05 <z< 1.05), temperature (0.6 <T< 7.0 keV), and luminosity (9 × 10^{41}<L< 5 × 10^{43} erg s^{1}). The LT relation has been studied in detail and we present the first measurement of the evolution using a single sample with an internal local baseline LT relation and fully accounting for selection biases. Our main conclusions are as follows.

1.
The sample L_{XXL}T relation has a slope of B_{LT} = 3.01 ± 0.27, not considering the effects of selection biases. This is consistent with previous studies of the observed steep slope of the LT relation when compared to the selfsimilar expectation.

2.
When taking into account the selection effects utilising a Bayesian approach using two forms of the likelihood we find a slope of the softband L_{XXL}T relation of B_{LT} = 2.63 ± 0.15 and a bolometric slope of B_{LT} = 3.08 ± 0.15.

3.
After taking into account the selection effects, our data show that the L_{XXL}T relation prefers an evolution of the form E(z)^{1.64 ± 0.77}. This is consistent with the expected “strong” selfsimilar evolution; however, is marginally stronger than the “weak” selfsimilar expectation.

4.
Comparisons to clusters detected in the XMMLSS (a precursor to XXL), which favour a negatively evolving LT relation, can be explained by the different assumptions made for the local baseline LT relation, the assumed masstemperature relation, and the value of the core radius.

5.
Investigating the impact of the assumed masstemperature relation and the x_{500} parameter, we find that they do not have a significant impact on the L_{XXL}T relation when using the XXL likelihood model. However, because of the large errors on the evolution, there is a degeneracy between the choice of MT relation, likelihood model, and evolution.

6.
We find that small changes in the comparative local baseline LT relation can change the inferred evolution of the L_{XXL}T relation. This appears to be due to the differing cool core populations in the samples used for the local LT baseline, which are affected by the evolving cool core population and the geometries of the surveys used to draw the cluster samples used for the LT baselines.

7.
By comparing our results with those determined from simulations, we find that the positive evolution favours models of cluster formation that include feedback from energy injection from SNe and AGNs.
Our results show that the evolution of the LT relation is strongly affected by the choice of comparative local baseline scaling relations. The XMMXXL Survey has allowed us for the first time to study in detail the evolution of the LT relation, fully accounting for selection biases, using a single homogeneous sample of clusters. Furthermore, using the data from the ≈450 clusters detected in the XXL survey, we will be able to place robust constraints on cosmological parameters.
A master XXL100GC cluster catalogue will also be available in electronic form at http://cosmosdb.isafmilano.inaf.it/XXL/ and via the XMM XXL DataBase at http://xmmlss.in2p3.fr
Acknowledgments
XXL is an international project based around an XMM Very Large Programme surveying two 25 deg^{2} extragalactic fields at a depth of ~5 × 10^{15} erg s^{1} cm^{2} in the [0.5–2.0] keV band for pointlike sources. The XXL website is http://irfu.cea.fr/xxl. Multiwavelength information and spectroscopic followup of the Xray sources are obtained through a number of survey programmes, summarised at http://xxlmultiwave.pbworks.com/. P.A.G., B.J.M., M.L., G.P.S., T.P. and J.D. acknowledge support from the UK Science and Technology Facilities Council. G.P.S. acknowledges support from the Royal Society. F.P. acknowledges support from the BMBF/DLR grant 50 OR 1117, the DFG grant RE 14626 and the DFG Transregio Programme TR33.
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All Tables
Bestfitting parameters for the LT relations modelled in this work taking the form L/L_{0} = E(z)^{γLT}A_{LT}(T/T_{0})^{BLT}, where L_{0} = 3 × 10^{43} erg s^{1} and T_{0} = 3 keV.
Bestfitting parameters for the LT relation while varying some assumptions in the model.
All Figures
Fig. 1
Example of the background method used in the spectral analysis (see Sect. 3). The image of the cluster XLSSC 010 with the regions showing the spectral extraction region (black circle), cluster detection radius (dashed red circle), and the background annulus centred on the aimpoint of the observation (blue annulus). 

In the text 
Fig. 2
Left: LT relation for the XXL sample. The black line represents an unbiased fit to the data (see Sect. 4.1). The LT relations of Pratt et al. (2009) and Clerc et al. (2014), given by the red dotdashed line and the green dashed line, respectively, are overplotted; right: same as the left plot, but with the REXCESS clusters studied in Pratt et al. (2009) overplotted, split between coolcore (open red squares) and noncoolcore (filled red squares) clusters. In both plots, the errors on the temperatures for the XXL clusters have been transformed via the method described in Sect. 4. 

In the text 
Fig. 3
L_{XXL}T relation with the bestfitting model. The light blue circles show the XXL100GC clusters; the bestfitting model is shown as the solid black line the 1σ uncertainty represented by the grey shaded region. 

In the text 
Fig. 4
Evolution of the L_{XXL}T relation for the XXL100GC. The XXL100GC are represented by the light blue circles and the bestfitting model is given by the black solid line; the grey shaded region highlights the 1σ uncertainty. The “strong” and “weak” selfsimilar expectations are given by the red dashed and blue dashed lines, respectively. 

In the text 
Fig. 5
Scatterplot matrix for the fit of the L_{XXL}T relation of the XXL100GC sample. The posterior densities are shown along the diagonal; the 1σ, 2σ, and 3σ confidence contours for the pairs of parameters are shown in the upper right panels. The lower left panels show the Pearson’s correlation coefficient for the corresponding pair of parameters (text size is proportional to the correlation strength). 

In the text 
Fig. 6
Contours of the simulated clusters generated from the bestfitting model are plotted in the L,z (left panel); T,z (middle panel); and L,T (right panel) planes, along with points indicating the observed clusters. 

In the text 
Fig. 7
Histograms of the redshift (left panel), luminosity (middle panel), and temperature (right panel) of the observed clusters are plotted along with the density distribution of the same properties from the simulated clusters generated from the bestfitting model. The histograms and curves are normalised to a total area of 1. 

In the text 
Fig. 8
L_{bol}T relation with bestfitting model. Our bestfitting model is shown as the solid black line with the grey shading indicating the 1σ uncertainty. The bestfitting model is transformed from the soft band to bolometric via the steps described in Sect. 4.7. 

In the text 
Fig. 9
Left: comparison of the luminosity extrapolated out to r_{500} () to that determined from a spectral analysis within r_{500} (). Right: comparison of the temperature derived within 300 kpc (T_{300 kpc}) to the temperature derived within r_{500} (T_{500}). The black line represents a 1:1 relation. 

In the text 
Fig. 10
Contours of the simulated clusters generated from our freeevolution (shaded contours), strong selfsimilar (black solid contours), and weak selfsimilar (red dashed contours) evolution models (see Sect. 5.4). Solid black points indicate the observed clusters in our sample. 

In the text 
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