The eventful life journey of galaxy clusters - I. Impact of dark matter halo and intracluster medium properties on their full assembly histories

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Volume 699 (July 2025)

A&A, 699 (2025) A1

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Table 1.

Fitted relations for determining the formation redshift, z_form, from each of the assembly state indicators (in each row) introduced in Sect. 2.3.

Parameter	Fit	Scatter	Validity region	R²
Centre offset, Δ_r	z_form(Δ_r) = 0.75 − 1.33Δ_r	σ_{z_form}(Δ_r) = 0.18 + 1.15Δ_r	Δ_r∈ [0, 0.25]	0.55
Virial ratio, η	z_form(η) = 1.07 − 1.17(η − 1)	σ_{z_form}(η) = 0.23 + 0.10(η − 1)	η∈ [1.1, 1.6]	0.84
Mean radial velocity, $⟨ {\tilde{v}}_{r} ⟩$ $\langle \tilde v_r \rangle$	$z_{form} (⟨ {\tilde{v}}_{r} ⟩) = 0.73 - 0.89 ⟨ {\tilde{v}}_{r} ⟩$ $z_{\mathrm{form}}(\langle \tilde v_r \rangle) = 0.73{-}0.89\langle \tilde v_r \rangle$	$σ_{z_{form}} (⟨ {\tilde{v}}_{r} ⟩) = 0.24 + 0.53 ⟨ {\tilde{v}}_{r} ⟩$ $\sigma_{z_{\mathrm{form}}}(\langle \tilde v_r \rangle) = 0.24+0.53\langle \tilde v_r \rangle$	$⟨ {\tilde{v}}_{r} ⟩ \in$ $\langle \tilde v_r \rangle \in$ [0, 0.25]	0.04
Sparsity, $s_{200 c}^{500 c}$ $s_{200c}^{500c}$	$z_{form} (s_{200 c}^{500 c}) = 1.06 - 0.82 (s_{200 c}^{500 c} - 1)$ $z_{\mathrm{form}}\left(s_{200c}^{500c}\right) = 1.06{-}0.82\left(s_{200c}^{500c}-1\right)$	$σ_{z_{form}} (s_{200 c}^{500 c}) = 0.27 - 0.07 (s_{200 c}^{500 c} - 1)$ $\sigma_{z_{\mathrm{form}}}\left(s_{200c}^{500c}\right) = 0.27{-}0.07 \left(s_{200c}^{500c}-1\right)$	$s_{200 c}^{500 c} \in$ $s_{200c}^{500c} \in$ [1, 2]	0.91
Ellipticity, ϵ	z_form(ϵ) = 1.10 − 1.71ϵ	σ_{z_form}(ϵ) = 0.38 − 0.47ϵ	ϵ∈ [0, 0.4]	0.87
Substructure fraction, f_sub	z_form(f_sub) = 0.83 − 8.35f_sub	σ_{z_form}(f_sub) = 0.34 − 4.60f_sub	f_sub∈ [0, 0.08]	0.89
Spin parameter, λ_Bullock	z_form(λ_Bullock) = 0.96 − 4.63λ_Bullock	σ_{z_form}(λ_Bullock) = 0.33 − 1.51λ_Bullock	λ_Bullock∈ [0, 0.1]	0.86
Combined indicator, χ	z_form(χ) = 0.30 + 0.37χ	σ_{z_form}(χ) = 0.42 − 0.17χ	χ∈ [0, 1.5]	0.95

Notes. The third column gives the 1σ uncertainty in the determination of z_form of an individual halo. The fourth column informs about the validity regions for each fit, while the fifth column gives the coefficient of the determination R² as a measure of the goodness of the fit z_form(X).

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