Concentration-Mass Relation

For a decade the most massive lensing clusters appeared too tightly packed for their cosmology. Halo concentration, the ratio of virial radius to inner scale radius, is a fossil of formation epoch: earlier-forming halos are denser-centered, and ΛCDM predicts a shallow, gently declining concentration-mass relation for massive clusters. Strong-lensing measurements kept violating it: CLASH precursors and the strongest lens systems showed concentrations 50 to 100 percent above simulation predictions with a steep mass dependence, the overconcentration problem, suggesting cluster cores assembled earlier and denser than hierarchical growth allows, kin to the formation-epoch anomalies (alignment ages, BCG maturity) documented at the same scales.

The audit that followed is a methodological classic: concentration and mass estimates from lensing are anti-correlated, biasing fitted relations steep; strong-lens selection prefers intrinsically overconcentrated and elongated systems viewed along their long axes, inflating projected concentrations 20 to 30 percent at fixed mass; and when the CLASH program selected by X-ray morphology instead and the comparison simulations applied the survey's own selection function, measured and simulated relations agreed at the 90 percent confidence level (Meneghetti et al. 2014; Sereno et al. 2015). The headline tension substantially deflated, an instructive contrast to neighbors that survived their audits. What remains is residual and specific: the most extreme individual lenses still sit high after corrections, the relation's redshift evolution remains poorly tested, and the same coherence between mass and concentration errors complicates every new claim in both directions.

The standing is a largely domesticated anomaly with live edges: a cautionary success of selection modeling, persistent outlier systems, and Euclid's order-of-magnitude expansion of the lensing-cluster census about to retest the relation with selection functions designed in advance.

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