Title: On the Apparent Power Law in CDM Halo Pseudo Phase Space Density Profiles,
Abstract: We investigate the apparent power-law scaling of the pseudo phase space
density (PPSD) in CDM halos. We study fluid collapse, using the close analogy
between the gas entropy and the PPSD in the fluid approximation. Our
hydrodynamic calculations allow for a precise evaluation of logarithmic
derivatives. For scale-free initial conditions, entropy is a power law in
Lagrangian (mass) coordinates, but not in Eulerian (radial) coordinates. The
deviation from a radial power law arises from incomplete hydrostatic
equilibrium (HSE), linked to bulk inflow and mass accretion, and the
convergence to the asymptotic central power-law slope is very slow. For more
realistic collapse, entropy is not a power law with either radius or mass due
to deviations from HSE and scale-dependent initial conditions. Instead, it is a
slowly rolling power law that appears approximately linear on a log-log plot.
Our fluid calculations recover PPSD power-law slopes and residual amplitudes
similar to N-body simulations, indicating that deviations from a power law are
not numerical artefacts. In addition, we find that realistic collapse is not
self-similar: scale lengths such as the shock radius and the turnaround radius
are not power-law functions of time. We therefore argue that the apparent
power-law PPSD cannot be used to make detailed dynamical inferences or
extrapolate halo profiles inward, and that it does not indicate any hidden
integrals of motion. We also suggest that the apparent agreement between the
PPSD and the asymptotic Bertschinger slope is purely coincidental.