Concentration Mass Relation

The concentration-mass relation c(M) describes how NFW halo concentration varies with total virial mass across the galaxy, group, and cluster mass scales. ΛCDM simulations predict a specific slope and normalization for c(M) — lower concentration at higher mass, with a weak power-law slope — calibrated from large cosmological N-body simulations. Observational measurements of the c(M) relation from weak lensing stacks, X-ray hydrostatic equilibrium, and strong lensing arc statistics show a normalization that is systematically higher than simulations by 20–40 percent across the full mass range, and a slope that may differ from the simulated prediction at the low-mass and high-mass ends. This excess concentration is one manifestation of the broader tension between the amplitude of gravitational clustering observed in the real universe versus what ΛCDM predicts.

Successive Collision Theory resolves the concentration-mass relation normalization excess through the gravitational superposition mechanism. The overlapping gravitational fields of nested comoving frames contribute an effective gravitational influence beyond what counting discrete mass particles provides, adding a smooth, distributed mass-equivalent component to the total gravitational potential of every structure at every mass scale. This superposition contribution preferentially compresses the inner regions of halos — where the depth of frame overlap is greatest — while having a smaller effect on the outer profiles. The result is a systematic inward shift of the characteristic NFW scale radius relative to what pure mass-particle gravity would produce, increasing the concentration parameter by a factor consistent with the observed 20–40 percent excess across all mass scales simultaneously. This is a single, mass-scale-independent normalization shift rather than a slope change, matching the character of the observed discrepancy.

The angular momentum inherited from the collision debris provides a complementary contribution to the concentration-mass relation through the centrifugal barrier mechanism. At each mass scale, the inherited specific angular momentum j sets a minimum collapse radius below which infalling material is centrifugally supported, effectively adding a floor to how small the scale radius can become. Since the angular momentum scaling law J ∝ M^(5/3) means that lower-mass structures have relatively higher specific angular momentum, the centrifugal barrier is most effective at small masses — truncating the scale radius at a larger fraction of the virial radius and producing concentrations that are higher than pure CDM accretion would generate. The combined effect of superposition boosting concentration at all masses and angular momentum selectively elevating concentrations at low mass produces a c(M) relation with both the correct normalization excess and the observed flattening of the slope at low halo masses.