*** Full-sky, Cosmic-variance only chi-squared differences for the lensing-ISW bispectrum ***
April 9, 2012: E.Komatsu
Ref: Junk & Komatsu, arXiv:1204.3789

Here we provide a program for computing the full-sky, cosmic-variance 
only chi-squared differences between:

(1) the lensing-linear Integrated Sachs-Wolfe effect bispectrum and the null
(2) the lensing-non-linear Rees-Sciama effect bispectrum and the null
(3) the lensing-linear Integrated Sachs-Wolfe effect bispectrum and
    the lensing-non-linear Rees-Sciama effect bispectrum

Chi-squared is computed as

Chi^2 = sum_{2<=l1<=l2<=l3} [B1(l1,l2,l3)-B2(l1,l2,l3)]^2
                             /[C(l1)C(l2)C(l3)Delta(l1,l2,l3)].

where B2=0 for (1) and (2). 

<< CAUTION >>
Note that the covariance of the bispectrum appearing in the denominator of the above equation ignores contributions from the non-Gaussian signal (i.e., this covarianec is valid only when non-Gaussianity is absent). This leads to an over-estimation of chi^2. In particular, the values of chi^2 computed in this way exceed the fundamental limit imposed by the cosmic variance of the non-Gaussian signal at l>~2200. For details, see Lewis, Challinor & Hanson, JCAP, 03, 018 (2011).


To calculate the lensing-ISW and lensing-RS bispectra, we need the lensing-ISW and lensing-RS cross-power spectra, Q(l), as well as the power spectrum of the UNLENSED, primordial CMB, Cp(l). We provide the data for the ISW Q(l), "ell_ql_linear.txt", for the RS Q(l), "ell_ql_3pt.txt", computed from the 3rd-order perturbation theory, and for the unlensed C(l), "wmap5baosn_max_likelihood_scalCls.dat". 

Note that the leading-order expression for the lensing-ISW bispectrum using the unlensed Cl is accurate only up to 10% level beyond l~2000. One can improve the accuracy of the bispectrum in the squeezed limit, where one of the multipole is much smaller than the other two, by replacing the unlensed Cl by the lensed Cl. This was found by Lewis, Challinor & Hanson, JCAP, 03, 018 (2011).

To calculate the variance of the bispectrum (i.e., the denominator of the above equation), we need the lensed C(l), which is included here as "wmap5baosn_max_likelihood_lensedCls.dat".

These data were generatedfor 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). The codes for generating Q(l) are also available on CRL. 

- To compile and use the  program, edit Makefile and simply "make"
- It will generate an executable called "compute_chi2_lensingiswbispectrum"
- Run "./compute_chi2_lensingiswbispectrum", which will generate a data file 
"lmax_chi2_chi2lin_chi2nl_chi2diff_***.txt", which contains:
(1st colummn): maximum multipole, lmax
(2nd column): chi^2 difference between the lensing-ISW and the null signal
(3rd column): chi^2 difference between the lensing-RS and the null signal
(4th column): chi^2 difference between the lensing-ISW and the lensing-RS

For ***=leading, the unlensed Cl is used in the bispectrum.
For ***=accurate, the lensed Cl is used in the bispectrum.

For your convenience, the result of the calculation using the WMAP5+BAO+SN parameters is included as "results/lmax_chi2_chi2lin_chi2nl_chi2diff_leading.txt" and "results/lmax_chi2_chi2lin_chi2nl_chi2diff_accurate.txt"