Power spectrum 1-halo term integral disagreement with CHOMP?

[tomshanks20]

The 1 halo term for P(k) in I_0^2 appears to be missing a factor of (M /rho)^2. In both source and docs, CCL gives
I_0^2=int dM n(M,A)<u(k,a|M)v(k,a|M)>

whereas it should be
I_0^2=int dM (M/rho)^2n(M,A)<u(k,a|M)v(k,a|M)>

see eg Eq 10 of White, M (2001 MNRAS 321 1)

P_mm(k)= ∫ dnuf(nu)/rho u(k|m)^2 = ∫ n(M)dM (M/rho)^2 u(k|m)^2 since f(nu) * dnu = nu(M)dM*(M/rho)

or eq 7 of Jain, Scranton and Sheth (2003) which gives:-

P_mm(k)=∫ dM n(M) (M/rho)^2 u(k|M)^2

But maybe it's something to do with me misunderstanding CCL u(k,a|M) normalisation?

Anyway it leads to results that disagree with CHOMP eg for projected halo-halo 2-point function w_mm - see attached plot.

[rreischke]

Dear @tomshanks20 , the mass dimension is carried by the profile itself (we do not use normalised profiles which approach unity as $k\to 0$), the mean density is taken care of by the normalisation in the profile_base (see HaloProfileMatter).

I am not familiar with CHOMP, but are you using the exact same settings, e.g. the concentration mass relation, mass function, profile, halo bias, etc.?

[tomshanks20]

Both CCL and CHOMP are assumiing Tinker 2010 halo mass function and halo bias, the Duffy08 concentration- mass relation, NFW profile. Both integrate over 10^8-10^16 Msol mass range. Both start with a linear P(k) (ie no use of HALOFIT).

CHOMP is at https://github.com/morriscb/chomp

It was used by SDSS team of Scranton et al (2005 ApJ 633 589) based on Jain, B., Scranton, R., & Sheth, R. K. (2003, MNRAS, 345, 62) and refs therein. It hasn't had much support since 2011 but I have been checking it out for the past year or so. Did find some problems in trying to apply it to QSOs as lenses at z~1.5 (now fixed in our version) but nothing that would upset its operation for lower z lenses. The n(z) we use is from equ 8a of Scranton et al for both CHOMP and CCL.

We suppressed CHOMP 1-halo term in the usual way to avoid any issue as k->0 but it didn't make much difference.

The difference appears at the earliest P(k)_mm stage so nothing to do with Limber formula etc. We go via P(k)_mm to C_l and then to w_gg(theta) using pyccl.correlation in CCL but no problem here either, from previous experience.

Is it easy for you to check the above w_mm result using CCL?

[tomshanks20]

In second last line it should be w_mm(theta) rather than w_gg(theta) - Typo!

[tomshanks20]

Reducing the CHOMP delta_V, i.e. the virial over density, by a factor of 3 to closer to the CCL value from Bryan & Norman (2008) helps a little as does making the CHOMP virial radius comoving by multiplying r_vir by 1+z=1.2. Doing both gets CHOMP a little closer to CCL(see attached) but there still seems something else needed to change either in CHOMP or CCL to improve the agreement in 1-halo term.

[tomshanks20]

After further debugging of CHOMP and noting that MassDefVir is with respect to rho_crit in CCL and is with respect to rho_bar in CHOMP etc I got the result below. Delta_V=103 for CCL ,applying to rho_crit, is from Bryan and Norman (1998) while Delta_v=340, applying to rho_bar, is from Jain et al 2003 and is close to the CCL value when rho_crit_rho_bar difference is accounted for.. The Delta_V=178 is what was in CHOMP to begin with but it was being applied to rho_crit rather than rho_bar etc and not converted to comoving radii etc. The graph shows what I get for wgg and wmm in these various cases and they are close enough that I think there is no CCL problem to resolve here. Let me know if you agree and you/we can then close down this issue. Thanks for the help and apologies for any inconvenience caused1