tokens: map in configuration space with a specific measurement error given. As possible noise sources, we take into account background fluctuations due to undetected clusters and the instrumental noise of ALMA itself. Furthermore, we include an estimate of the fluctuations introduced by primary CMB anisotropies. Since the different noise contributions are not correlated with each other, the total uncertainty is then given by Gaussian error propagation. 4.1 Background fluctuations due to undetected clusters Galaxy clusters unresolved by the telescope beam contribute a background noise level y bg that needs to be taken into account in all following calculations. Clusters are unresolved if they appear (much) smaller than the beam size. The background signal is thus dominated by low-mass clusters (Bartelmann 2001 ). Since an ideally homogeneous background could be removed from the data, we only have to consider the average background fluctuation level Δ y bg . The mean background level contributed by unresolved clusters with the differential mass function n ( M , z ) is where the mass- and redshift-integrations have to be carried out over that area in the mass–redshift plane where clusters are unresolved. The integrated Compton- y parameter Y ( M , z ) of a cluster with mass M at redshift z is where the angular diameter distance D d to the cluster appears and the total number of electrons N e . We choose the cluster mass function n ( M , z ) described by Sheth Tormen ( 1999 ), The number of background sources follows a continuous probability distribution. If one neglects any spatial cluster correlations, the rms due to background fluctuation is 4.2 Instrument noise Another relevant source of noise is the measurement noise of the telescope and the detector. We assume that the measurement error of the telescope has a Gaussian distribution with standard deviation σ Tel around zero, with σ Tel depending on the telescope and detector configuration and the setup of the specific observation. Since we need a resolution for our simulation that is comparable to other observations of massive galaxy clusters, we choose the characteristics of the Atacama Large Millimeter/submillimeter Array ( ALMA ) as an example telescope for achieving sufficiently precise thermal SZ observations. For all simulations presented in the remainder of this work, we use the following configuration of ALMA : All N ant = 32 antennas are assumed to be available with a baseline to be set. This configuration has then a given angular resolution in a frequency band chosen to be centred on ν = 116 GHz. Its bandwidth is assumed to be Δν = 7.5 GHz. According to the ALMA user manual (Lundgren 2012 ) the point-source sensitivity σ Tel of ALMA is where T sys is the temperature of the system, 1 η q the quantum efficiency (η q = 0.96), η c the correlator efficiency (η c = 0.88), A eff the effective area of an individual antenna, n p the number of polarization states ( n p = 2), and t int is the
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