EXTRACTING CLUSTER PHYSICAL PROPERTIES FROM RECONSTRUCTED SZ MAPS
4.1 Bulk Cluster Properties
4.1.1 Least-Squares Fitting
C h a p t e r 4
EXTRACTING CLUSTER PHYSICAL PROPERTIES FROM
50 normalization 𝑃0, an isothermal temperature component𝑇𝑒, a single bulk velocity component𝑣𝑧, the scale radius𝑟𝑠, an𝑥 , 𝑦 position on the sky, and, in certain cases, the GFNW slope parameters𝛼, 𝛽, and𝛾. We assume the clusters are spherical for simplicity. In addition, we assume that the beam and transfer function have been fully characterized in advance, so we do not parameterize them in the fitting process.
The reported value of𝜏is the average optical depth within a 10radius.
We chose this model because it is straightforward to implement and fit. More- over, fitting a model in this way enables us to robustly characterize non-Gaussian uncertainties, in contrast to Fisher matrix methods; see Section 4.1.2 below for details. In addition, the form of the ICM model is a good match to the average shape of hydrodynamically simulated clusters. While this approach does ignore deviations from smooth radial profiles, we will show using simulated clusters that this approach does not create significant biases.
We consider two different types of mock observations, each of which uses a different description of the cluster: (a) a smooth, analytical, radial profile, referred to as the “analytical case,” and (b) a map of a hydrodynamically simulated cluster lacking azimuthal symmetry. The analytical case serves as a useful starting point for which the computational machinery has already been implemented. The second case, the “simulation-based case,” which can be viewed as a generalization of the first, will be necessary for process validation and for application to real observations.
It is natural to discuss the analytical case first, as the machinery for fitting in the simulation-based case builds upon that for the analytical case.
In the analytical case, it suffices to perform a least squares fit using the analytical model described above. The goal of the fitter is to minimize the value of the 𝜒2:
𝜒2=Õ
𝑖
(𝐷𝑖− 𝑀𝑖)2 𝜖2
𝑖
, (4.1)
where𝐷𝑖is the data vector,𝑀𝑖is the model vector, and𝜖𝑖is the noise RMS per pixel;
the index𝑖runs over all map pixels in all frequency channels. The data vector𝐷𝑖is the output of the mock observation pipeline, including noise and all contaminants, while the model vector 𝑀𝑖 consists of the projected SZ signal of the cluster alone.
While it is necessary in real observations to allow the noise rms 𝜖𝑖 to vary from pixel to pixel, we assume for simplicity in this work that𝜖𝑖is a constant function of position on the sky and varies only with observing frequency.
We explored two choices for the fitting algorithm, namely, the Levenberg- Marquardt (Levenberg, 1944; Marquardt, 1963) and Nelder-Mead (Nelder et al.,
1965) algorithms. The Levenberg-Marquardt (L-M) algorithm works by estimating the gradient of the 𝜒2, while the Nelder-Mead algorithm uses the values of the 𝜒2 directly.1 The L-M algorithm is a popular choice because it converges quickly and fitters generally provide uncertainty estimates. The Nelder-Mead algorithm is more robust, but it tends to converge more slowly and, since it does not calculate derivatives, does not automatically give uncertainty estimates.
In the end, we chose to use the Nelder-Mead algorithm for its added robust- ness. The L-M algorithm failed to converge in enough cases that it impacted our constraints derived from bootstrapping over many noise realizations as described in Section 4.1.2. The downside of slower convergence was mitigated by using high- performance computing (HPC) resources, and, in any case, the least-squares fit was not the computational bottleneck in the analysis pipeline.2 In addition, we relied on noise resampling to determine the uncertainties (see Section 4.1.2), so we did not need uncertainty estimates based on derivatives.
We have made some effort to adapt the least-squares fitter to work well in the simulation-based case, that is, when the mock data map is more complicated than a simple radial profile. Our tests of this regime have used SZ effect maps derived from the IllustrisTNG simulations (D. Nelson et al., 2019). Although the GNFW model was designed to agree well with hydrodynamical cluster simulations (Nagai et al., 2007), achieving convergence has proven to be somewhat nontrival without manually guiding the fitter, which would be intractable when analyzing a large number of clusters, particularly when using bootstrap resampling as described in the following section. This difficulty appears to be due to the behavior of the fitting algorithm and not a deficiency in the freedom of the GNFW parameterization.
To help automate convergence, we considered (but have not tested) the approach of fitting a one-dimensional profile to determine the shape parameters of the cluster’s ICM before attempting to fit the full multiband map, in the hope that the reduced degrees of freedom will improve the convergence of the fit. Here, we describe the proposed steps in detail. First, locate the cluster center in a map by fitting a cluster model using GNFW slope parameters𝛼, 𝛽, and𝛾 fixed to the values given in Arnaud et al. (2010), which should be a good match to the general case on average. For simplicity, it should suffice to pick a single frequency band for the fit.
Next, calculate an approximate profile by averaging the map in single-pixel-wide
1Nelder-Mead is also known as the amoeba method. IDL, for example, uses this nomenclature.
2The bottleneck was generally the source cleaning algorithm (Section 3.1).
52 annular bins. Fit this profile with a projected cluster model to obtain estimates of the slope parameters along with the scale radius 𝑟𝑠 and the amplitude. There is some degeneracy among these parameters, so it may be necessary to fix one or more of them for the fit to converge. Allowing ellipticity in the GNFW model may also be necessary for certain clusters in practice; one could calculate a profile by averaging the map on ellipses instead of circles, and the ellipses’ axis ratios could be determined by alternating with a 2D map fit. Finally, with this estimate of the cluster shape, one may reattempt fitting the cluster model, with its shape parameters fixed, to the full multiband map as described above. The full process—fitting the profile, then fitting the full map—may be iterated to improve the fit.
4.1.2 Characterizing Parameter Uncertainties from Least-Squares Fits