Ω 0 and substructure in galaxy clusters

We examine the theoretical relationship between Ω 0 and substructure in galaxy clusters that are formed by the collapse of high‐density peaks in a Gaussian random field. The radial mass distributions of the clusters are computed from the spherical accretion model using the adiabatic approximation following Ryden & Gunn. For a cluster of mass M ( r , t ), we compute the quantity Δ M   M — at a cosmic time t and within a radius r , where Δ M is the accreted mass and M — is the average mass of the cluster during the previous relaxation time, which is computed individually for each cluster. For a real cluster in three dimensions we argue that Δ M   M — should be strongly correlated with the low‐order multipole ratios, Φ int l /Φ int 0 , of the potential due to matter interior to r . Because our analysis is restricted to considering only the low‐order moments in the gravitational potential, the uncertainty associated with the survival time of substructure is substantially reduced in relation to previous theoretical studies of the ‘frequency of substructure’ in clusters.  We study the dependence of Δ M  /  M — on radius, mass, Ω 0 , λ 0  = 1 − Ω 0 , redshift and relaxation time‐scale in universes with cold dark matter (CDM) and power‐law power spectra. The strongest dependence on Ω 0 (λ 0  = 0) occurs at z  = 0, where Δ M  /  M —  ∝ Ω 1/2 0 for relaxation times ∼ 1–2 crossing times and only very weakly depends on mass and radius. The fractional accreted mass in CDM models with Ω 0  + λ 0  = 1 depends very weakly on Ω 0 and has a magnitude similar to the Ω 0  = 1 value. Δ M  /  M — evolves more rapidly with redshift in low‐density universes and decreases significantly with radius for Ω 0  = 1 models for z ≳0.5. We discuss how to optimize constraints on Ω 0 and λ 0 using cluster morphologies.  It is shown that the expected correlation between Δ M  /  M — and Φ int l  / Φ int 0 extends to the two‐dimensional multipole ratios Ψ int m  / Ψ int 0 , which are well‐defined observables of the cluster density distribution. We describe how N ‐body simulations can quantify this correlation and thus allow Δ M  /  M — to be measured directly from observations of cluster morphologies.