A universal model for the concentration-mass relation
Concentration is defined as the ratio of an outer radius to the scale radius, e.g., c200c = R200c / rs. An NFW profile can be parameterized by only a mass (or radius) and concentration, making the concentration-mass relation an important ingredient in modeling the density profiles of halos when only their mass is known.
Traditionally, the c-M relation has been described in two ways. First, concentration is tightly related to halo age, meaning that a prediction for the age of halos as a function of mass can be turned into a prediction for concentration (e.g., Wechsler et al. 2002). Second, the c-M relation is more or less a power law at fixed redshift and cosmology, leading to numerous numerical calibrations (e.g., Dutton & Maccio 2014).
There is, however, a third way to model concentrations: by considering the properties of the density peaks they originated from. Concentration is almost universal (i.e., independent of redshift) when expressed in units of peak height rather than mass (e.g., Prada et al. 2012). This is demonstrated in the figure below: the redshift evolution of the concentration-peak height relation (left) is much less dramatic than that of the concentration-mass relation (right).
We show that the residual non-universality is due to the shape of the power spectrum, and build a model that is a function of only peak height and the power spectrum slope (dashed lines in the figure above). This model matches our ΛCDM results (for different cosmologies) to 5% accuracy up to z = 6, and matches scale-free Ωm=1 models to 15% accuracy. The model also reproduces the low concentrations of Earth-mass halos at z = 30, meaning it extrapolates correctly over 16 orders of magnitude in halo mass.
We note that, according to our model, there is no well-defined floor in the concentration values. Instead, the minimum concentration depends on redshift: at fixed peak height, halos at higher redshift experience steeper power spectrum slopes, and thus have lower minimum concentrations. Our model can be evaluated with the python code Colossus.