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Register a new userFree-form modelling of galaxy clusters: a Bayesian and data-driven approach
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A new method is presented for modelling the physical properties of galaxy clusters. Our technique moves away from the traditional approach of assuming specific parameterised functional forms for the variation of physical quantities within the cluster, and instead allows for a free-form reconstruction, but one for which the level of complexity is determined automatically by the observational data and may depend on position within the cluster. This is achieved by representing each independent cluster property as some interpolating or approximating function that is specified by a set of control points, or nodes, for which the number of nodes, together with their positions and amplitudes, are allowed to vary and are inferred in a Bayesian manner from the data. We illustrate our nodal approach in the case of a spherical cluster by modelling the electron pressure profile Pe(r) in analyses both of simulated Sunyaev-Zeldovich (SZ) data from the Arcminute MicroKelvin Imager (AMI) and of real AMI observations of the cluster MACS J0744+3927 in the CLASH sample. We demonstrate that one may indeed determine the complexity supported by the data in the reconstructed Pe(r), and that one may constrain two very important quantities in such an analysis: the cluster total volume integrated Comptonisation parameter (Ytot) and the extent of the gas distribution in the cluster (rmax). The approach is also well-suited to detecting clusters in blind SZ surveys.
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I compare the mass values obtained with data taken from the Arcminute Microkelvin Imager (AMI) radio interferometer system and from the Planck satellite. The former of these uses a Bayesian analysis pipeline that parameterises a cluster in terms of its physical quantities, and models the dark matter & baryonic components of a cluster using Navarro-Frenk-White (NFW) and generalised-NFW profiles respectively. I also analyse simulated AMI data with input values based on PwS mass estimates. I then compare three cluster models using AMI data for the 54 cluster sample. The two observational models considered only model the gas content of the cluster. To compare the physical and observational models I consider their posterior parameter estimates, including the calculation of a metric defined between two probability distributions. The models fit to the cluster data is evaluated by looking at the Bayesian evidence values. Improvements to the physical modelling of galaxy clusters are then considered, either by relaxing some of the assumptions underlying the physical model, or by introducing a new profile for the dark matter component of clusters. The final part of the cluster analysis work focuses on Bayesian analysis using a joint likelihood function of data from both AMI and the Planck satellite simultaneously. Finally, a new Bayesian inference algorithm based on nested sampling is presented. The algorithm, named the geometric nested sampler, is an adaption of the Metropolis-Hastings nested sampler and makes use of the geometrical interpretation of sets of parameters to sample from their domains efficiently. The geometric nested sampler is tested on several toy models as well as a model representing the emission of gravitational waves from binary black hole mergers.
We present a Bayesian hierarchical inference formalism to study the relation between the properties of dark matter halos and those of their central galaxies using weak gravitational lensing. Unlike traditional methods, this technique does not resort to stacking the weak lensing signal in bins, and thus allows for a more efficient use of the information content in the data. Our method is particularly useful for constraining scaling relations between two or more galaxy properties and dark matter halo mass, and can also be used to constrain the intrinsic scatter in these scaling relations. We show that, if observational scatter is not properly accounted for, the traditional stacking method can produce biased results when exploring correlations between multiple galaxy properties and halo mass. For example, this bias can affect studies of the joint correlation between galaxy mass, halo mass, and galaxy size, or galaxy color. In contrast, our method easily and efficiently handles the intrinsic and observational scatter in multiple galaxy properties and halo mass. We test our method on mocks with varying degrees of complexity. We find that we can recover the mean halo mass and concentration, each with a $0.1$ dex accuracy, and the intrinsic scatter in halo mass with a $0.05$ dex accuracy. In its current version, our method will be most useful for studying the weak lensing signal around central galaxies in groups and clusters, as well as massive galaxies samples with $log{M_*} > 11$, which have low satellite fractions.
A multiply-lensed galaxy, MACS0647-JD, with a probable photometric redshift of $zsimeq 10.7^{+0.6}_{-0.4}$ is claimed to constitute one of the very earliest known galaxies, formed well before reionization was completed. However, spectral evidence that MACS0647-JD lies at high redshift has proven infeasible and so here we seek an independent lensing based geometric redshift derived from the angles between the three lensed images of MACS0647-JD, using our free-form mass model (WSLAP+) for the lensing cluster MACSJ0647.7+7015 (at $z=0.591$). Our lens model uses the 9 sets of multiple images, including those of MACS0647-JD, identified by the CLASH survey towards this cluster. We convincingly exclude the low redshift regime of $z<3$, for which convoluted critical curves are generated by our method, as the solution bends to accommodate the wide angles of MACS0647-JD for this low redshift. Instead, a best fit to all sets of lensed galaxy positions and redshifts provides a geometric redshift of $zsimeq 10.8^{+0.3}_{-0.4}$ for MACS0647-JD, strongly supporting the higher photometric redshift solution. Importantly, we find a tight linear relation between the relative brightnesses of all 9 sets of multiply lensed images and their relative magnifications as predicted by our model. This agreement provides a benchmark for the quality of the lens model, and establishes the robustness of our free-form lensing method for measuring model-independent geometric source distances and for deriving objective central cluster mass distributions. After correcting for its magnification the luminosity of MACS0647-JD remains relatively high at $M_{UV}=-19.4$, which is within a factor of a few in flux of some surprisingly luminous $zsimeq 10$--$11$ candidates discovered recently in Hubble black field surveys.
In the near future, ultra deep observations of galaxy clusters with HST or JWST will uncover $300-1000$ lensed multiple images, increasing the current count per cluster by up to an order of magnitude. This will further refine our view of clusters, leading to a more accurate and precise mapping of the total and dark matter distribution in clusters, and enabling a better understanding of background galaxy population and their luminosity functions. However, to effectively use that many images as input to lens inversion will require a re-evaluation of, and possibly upgrades to the existing methods. In this paper we scrutinize the performance of the free-form lens inversion method Grale in the regime of $150-1000$ input images, using synthetic massive galaxy clusters. Our results show that with an increasing number of input images, Grale produces improved reconstructed mass distributions, with the fraction of the lens plane recovered at better than $10%$ accuracy increasing from $40-50%$ for $sim!!150$ images to $65%$ for $sim!1000$ images. The reconstructed time delays imply a more precise measurement of $H_0$, with $lesssim 1%$ bias. While the fidelity of the reconstruction improves with the increasing number of multiple images used as model constraints, $sim 150$ to $sim 1000$, the lens plane rms deteriorates from $sim 0.11$ to $sim 0.28$. Since lens plane rms is not necessarily the best indicator of the quality of the mass reconstructions, looking for an alternative indicator is warranted.
We present an analysis of observations made with the Arcminute Microkelvin Imager (AMI) and the Canada-France-Hawaii Telescope (CFHT) of six galaxy clusters in a redshift range of 0.16--0.41. The cluster gas is modelled using the Sunyaev--Zeldovich (SZ) data provided by AMI, while the total mass is modelled using the lensing data from the CFHT. In this paper, we: i) find very good agreement between SZ measurements (assuming large-scale virialisation and a gas-fraction prior) and lensing measurements of the total cluster masses out to r_200; ii) perform the first multiple-component weak-lensing analysis of A115; iii) confirm the unusual separation between the gas and mass components in A1914; iv) jointly analyse the SZ and lensing data for the relaxed cluster A611, confirming our use of a simulation-derived mass-temperature relation for parameterizing measurements of the SZ effect.
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