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.