Selection function of clusters in Dark Energy Survey year 3 data from cross-matching with South Pole Telescope detections

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Table 1.

Summary of the models we consider, listed by their abbreviations.

Name	Intrinsic scatter	Projection effects	Contamination	Free parameters	Results
plain	log-normal, constant
	+ Poisson term,	no	no	A_λ, B_λ, C_λ, σ_λ
	Equation (6)

cont	same as plain	no	Equation (20)	plain,	cont. consistent with zero,
				b₀^cont, b_z^cont, a₀^cont	absent at $\hat{λ} ≳ 100$ $\hat\lambda \gtrsim 100$

mass dep	log-normal with mass
	trend + Poisson term,	no	no	plain,	constant scatter for
	Equation (6,18)			log₁₀M_break, B_σ	log₁₀M_200, c ≳ 14.2

prj	same as plain	from simulation			preferred over
		see Fig. 2	no	plain	plain, cont and mass dep

				plain,
prj+cont	same as plain	same as prj	same as cont	b₀^cont, b_z^cont, a₀^cont	same as cont

prj+	same as plain	prj with free amplitude	no	plain α₀	consistent with prj
		Equation (19)

		prj with free amplitude,
prj++	same as plain	richness & redshift trend	no	plain,	possible redshift trend in
		Equation (19)		α₀, α_λ, α_z	strength of projection effects

Notes. We report the intrinsic richness model assumed, the treatment of projection effects and contamination, their free parameters of the richness–mass relation, and the main result we found in each model. All models share the same treatment of photometric scatter, Equation (9).

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