A deeper look into the structure of {\Lambda}CDM haloes: correlations between halo parameters from Einasto fits

We used high resolution dark matter only cosmological simulations to investigate the structural properties of Lambda Cold Dark Matter ($\Lambda$CDM) haloes over cosmic time. The haloes in our study range in mass from $\sim 10^{10}$ to $\sim 10^{12} \mathrm{M}_\odot$, and are resolved with $10^5$ to $10^7$ particles. We fit the spherically averaged density profiles of DM haloes with the three parameter Einasto function. For our sample of haloes, the Einasto shape parameter, $\alpha$, is uncorrelated with the concentration, $c$, at fixed halo mass, and at all redshifts. Previous reports of an anti-correlation are traced to fitting degeneracies, which our fits are less sensitive to due to our higher spatial resolution. However, for individual haloes the evolution in $\alpha$ and $c$ is anti-correlated: at redshift $z=7$, $\alpha \simeq 0.4$ and decreases with time, while $c\simeq 3$ and increases with time. The evolution in structure is primarily due to accretion of mass at larger radii. We suggest that $\alpha$ traces the evolutionary state of the halo, with dynamically young haloes having high $\alpha$ (closer to a top-hat: $\alpha^{-1}=0$), and dynamically relaxed haloes having low $\alpha$ (closer to isothermal: $\alpha=0$). Such an evolutionary dependence reconciles the increase of $\alpha$ vs peak height, $\nu$, with the dependence on the slope of the power spectrum of initial density fluctuations found by previous studies.