June 4, 2003
Afternoon discussion: What is the mass of a cluster?

What does a theorist mean by mass?

In particle based simulations, can use FOF or SO (spherical overdensity)

FOF:
 pro(?): unique assignment of particles/mass to clusters
 pro: varying b leads to nested hierarchy
 neg: as linking length b grows, merge dynamically distinct objects
 neg: must have discrete representation of mass field; although becomes isodensity threshold in continuum limit
 neg: produces objects of bizarre geometries which cannot be measured observationally

SO:
 neutral: non-unique mass assignment 
 pro: can have nested hierarchy
 pro: works in continuum limit (with filtering for centers)
 pro: easier to observe/measure quantities in spherical/circular region

==>SO currently favored

Mass M_D= 4pi/3 ro(z) r_d^3  d=delta
          where ro(z) is threshold density at some redshift

1. mean fixed 		ro(z)= d * ro_mean(z) where d~200
2. critical fixed 	ro(z) = d * ro_crit(z)
3. variable 		ro(z) = d_crit(Omega_m) * ro_crit(z)  where Omega_m is f(z)

third measure may not be useful; d_crit(Omega_m(z)) calculated from SO : area expands with universal expansion, then collapses. "Virialization" when collapse by factor of ~2 in size from maximum.

This behavior is not verified from simulations all that well.

"Press-Schecter + SO is dead"

Gus proposed we forget about definition (3) of mass

For mean fixed:
Nature seems to do accounting wrt to mean mass density
  Gobernato , Jenkins papers
mass fraction in collapsed objects (clusters, halos) has fixed form with three parameters
f(sig^-1(m)) ~ Aexp{|ln(sig^-1)+B|^eps}

Jenkins et al. looks at 4 very different cosmologies (LCDM,OCDM,SCDM,TCDM) and could fit functional form of mass fraction with SINGLE SET of parameters A,B,eps.


For critical fixed:
 dynamics are wrt to ro_crit(z)

sigma^2 ~ kT ~ (H(z) M_d) ^ 2/3

plot h(z)M_200 vs veldisp of DM get nice relation
sigma_DM = 1080 (h(z) M_200 / 10^15 Msun) ^1/3
  where M_200 is measured wrt mean. This gives nice cosmology-indep mass measure

How do sigma_gal , T_x map to sigma_DM ?


For variable:
Using ellipsoidal collapse (sheth-torman) reproduces mean_fixed ???



Pierpaoli: Is there a better mass definition that allows observable<->theoretical conversion?

M. White: What does M_vir refer to? Usually (3), but in NFW it is (2). So please define in papers.

If variance between observable and one mass definition is lower, then that is good reason to switch.