Photos of the black board after discussion on proper parameterization of scaling relations for cosmological analyses by Alexey Vikhlinin on 02/17/11. Discussion can be viewed here.

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Summary of the discussion (notes from Alexey Vikhlinin):

The purpose is to discuss the ways how to properly parameterize the evolution of the scaling relations when we observe departures from self-similarity. Let's focus on the soft-band X-ray luminosity vs. total mass relation. The self-similar expectation for this relation is very simple:

Lx ~ n^2 V ~ n*(nV) ~ rho_crit(z)*Mgas ~ Mtot*E(z)2

[A note to theorists: please use 0.5-2 keV luminosities in your models as this band most closely matches the response of most X-ray telescopes]

However, it is widely known that the slope of this relation is not 1, so it is not self-similar. E.g., if we write Lx ~ M
alpha, then for low-z clusters:
  • alpha=~1.6 measured in Vikhlinin et al. 2009 (V09; CCCP), Pratt et al. 2009 (P09; REXCESS); a similar slope found in Rykoff et al. 2008 (stacked maxBCG); the errorbar is approximately +- 0.1
  • alpha=1.3 in Mantz et al. 2009

The difference in the slope measurements can be attributed to several factors. 1) in M09, the clusters are of higher mass on average; 2) V09 and P09 estimate the cluster total mass from Yx, while M09 use Mgas (i.e., in their analysis Mtot = const*Mgas); if there is a trend of fgas with M, that would bias the slope low in M09 --- but it is unclear if there is a trend fgas(M) in this mass range

So, how should be parameterize the evolution given that the slope of the relation is not self-similar? M09 re-write the relation in the form
Lx/E(z) = [Mtot*E(z)]alpha, and refer to this evolution as self-similar even if alpha!=1. Since they measure alpha=1.3, their inferred evolution is E(z)2.3

V09 model the evolution as a free power of E(z), Lx ~ Malpha*E(z)gamma; they measure gamma=1.85 +-0.45. This is marginally consistent with the evolution factor used in M09, but a difference in the evolution factors of this type [E(z)2.3 vs E(z)1.85] is a big worry for the future big surveys, e.g. eRosita, when masses for most clusters will be estimated from Lx. For example, at z=0.65, E(z)=1.43, E(z)**0.45=1.175 and the mass inferred from Lx will be different by ~11%. Such a systematic uncertainty in mass would be bad.

So the questions are:
  • can we tie departures from self-similarity in the evolution of the relation to its slope, as in M09? --- the group was unable to come up with a physically motivated scenario for such a connection.
  • if we want to model non-self-similar evolution, should it be written as E(z)gamma, (1+z)gamma or what? --- several people felt that it's best to simply through in all possible departures and model them out. Others felt that this would degrade the cosmological constraints too much.

Eduardo Rozo suggested that for many relations, the E(z) factors in the evolution reflect only that the background density (with respected to which we measure M500) evolves, while physical conditions within the cluster remain constant. In this case, the E(z) factors should be sticked to the mass, so the baseline model for the relation should be something like Lx ~ [M*E(z)^2]**alpha.

We also discussed that it would be best to understand -- at least approximately -- the main physical mechanisms responsible for departures from self-similarity. This could suggest a better parameterization for non-self-similar evolution. For example, if departures from self-similarity reflect a systematic change in the relaxed vs. unrelaxed fraction as a function of mass, then a parametrization could use M*(z) rather than E(z) [M* here is a non-linear mass scale, not stellar mass]. If non-self-similarity is due to differences in the star formation efficiency (see Gonsalez et al.), the parameterization could use Mstar/Mgas as a function of z and Mtot.