*** Cross-power spectrum of CMB lensing potential and the non-linear Rees-Sciama effect, Q(l), from the 3rd-order perturbation theory  ***
April 8, 2012: E.Komatsu

This program reads in a non-linear matter power spectrum calculated from the 3rd-order perturbation theory (3PT), and calculates the cross-correlation power spectrum of the CMB lensing potential and the non-linear Rees-Sciama effect:

Q(l) = 2 \int_0^z* dz [r(z*)-r(z)]/[r(z*)r^3(z)] [dPpsi(k,z)/dz]_{k=l/r(z)}

where r(z) is the comoving distance, z*=1090 is the redshift of the photon decoupling epoch, and Ppsi(k,z) is the power spectrum of Newtonian gravitational potential.

This reads in the pieces of the 3PT non-linear matter power spectrum, P11(k), P22(k), and P13(k), as a function of wavenumbers k [in units of h/Mpc], and computes the power spectrum as well as its redshift derivative as

P(k,z) = D^2(z)*P_11(k) + D^4(z)*[P_22(k)+2P_13(k)]
dP(k,z)/dz = 2*D(z)*dD/dz*{ P_11(k) + 2*D^2(z)*[P_22(k)+2P_13(k)] }

where P_11(k), P_22(k), and P_13(k) are pre-computed at z=30 and D(z) is the linear growth factor between z and z=30. We provide the data for these spectra in "wavenumber_pkd11_pkd22_pkd13_at_z=30.txt", which were generated by "compute_pk11pk22pk13", available also on CRL.

*** P(k) is in units of h^-3 Mpc^3, and k is in units of h Mpc^-1 ***
(This is the same as CAMB definition.)

The 3PT spectra were computed from the input linear power spectrum, "wmap5baosn_max_likelihood_matterpower.dat," which was generated using CAMB code for the maximum likelihood parameters given in Table I of Komatsu et al.(2008) [WMAP 5-year interpretation paper] with "WMAP5+BAO+SN". The input file for CAMB is also provided (wmap5baosn_max_likelihood_params.ini). NOTE THAT THIS POWER SPECTRUM IS COMPUTED AT Z=0.

Another power spectrum, evolved back to z=30, is provided as "wmap5baosn_max_likelihood_matterpower_at_z=30.dat".

- To compile and use the program, edit Makefile and simply "make"
- It will generate an executable called "compute_ql_linear"
- Running "compute_ql" generates a file called "ell_ql.txt", which contains:

(1st column) multipole, l
(2nd column) Q(l)