On the evolution of cluster scaling relations
  Understanding the evolution of scaling relations between the observable
properties of clusters and their total mass is key to realizing their potential
as cosmological probes. In this study, we investigate whether the evolution of
cluster scaling relations is affected by the spurious evolution of mass caused
by the evolving reference density with respect to which halo masses are defined
(pseudo-evolution). We use the relation between mass, M, and velocity
dispersion, sigma, as a test case, and show that the deviation from the M-sigma
relation of cluster-sized halos caused by pseudo-evolution is smaller than 10%
for a wide range of mass definitions. The reason for this small impact is a
tight relation between the velocity dispersion and mass profiles, sigma(<r) =
const * (GM(<r) / r)^1/2, which holds across a wide range of radii. We show
that such a relation is generically expected for a variety of density profiles,
as long as halos are in approximate Jeans equilibrium. Thus, as the outer
"virial" radius used to define the halo mass, R, increases due to
pseudo-evolution, halos approximately preserve their M-sigma relation. This
result highlights the fact that tight scaling relations are the result of tight
equilibrium relations between radial profiles of physical quantities. We find
exceptions at very small and very large radii, where the profiles deviate from
the relations they exhibit at intermediate radii. We discuss the implications
of these results for other cluster scaling relations, and argue that
pseudo-evolution should have a small effect on most scaling relations, except
for those that involve the stellar masses of galaxies. In particular, we show
that the relation between stellar-mass fraction and total mass is affected by
pseudo-evolution and is largely shaped by it for halo masses of less than 10^14
solar masses.
