1. CMB Power spectrum likelihood approximations
Antony Lewis, IoA
Work with Samira Hamimeche

3. Start with full sky, isotropic noise
Assume alm Gaussian

4. Integrate alm that give same Chat
- Wishart distribution
For temperature
Non-Gaussian skew ~ 1/l
For unbiased parameters need bias <<
- might need to be careful at all ell

5. Gaussian/quadratic approximation
Gaussian in what? What is the variance?
Not Gaussian of Chat – no Det
fixed fiducial variance
exactly unbiased, best-fit on
average is correct
Actual Gaussian in Chat
or change variable, Gaussian in log(C), C-1/3 etc…

6. Do you get the answer right for amplitude over range lmin < l lmin+1 ?

7. Binning: skewness ~ 1/ (number of modes)
~ 1 / (l Δl)
- can use any Gaussian approximation for Δl >> 1
Fiducial Gaussian: unbiased,
- error bars depend on right fiducial model, but easy to choose accurate to 1/root(l)
Gaussian approximation with determinant: - Best-fit amplitude is
- almost always a good approximation for l >> 1
- somewhat slow to calculate though

8. New approximation
Can we write exact likelihood in a form that generalizes for cut-sky estimators?
- correlations between TT, TE, EE.
- correlations between l, l’
Would like:
Exact on the full sky with isotropic noise
Use full covariance information
Quick to calculate

9. Matrices or vectors? Vector of n(n+1)/2 distinct elements of C
Covariance:
For symmetric A and B, key result is:

10. For example exact likelihood function in terms of X and M is
using result:
Try to write as quadratic from that can be generalized to the cut sky

11. Likelihood approximation
where
Then write as
where
Re-write in terms of vector of matrix elements…

12. For some fiducial model Cf
where
Now generalizes to cut sky:

13. Other approximations also good just for temperature
Other approximations also good just for temperature. But they don’t generalize.
Can calculate likelihood exactly for azimuthal cuts and uniform noise - to compare.

14. Unbiased on average

15. T and E: Consistency with binned likelihoods (all Gaussian accurate to 1/(l Delta_l) by central limit theorem)

16. Test with realistic mask kp2, use pseudo-Cl directly

17. Isotropic noise test ~ 143Ghz from science case red – same realisation analysed on full sky all 1 < l < 2001
Provisional CosmoMC module at

18. More realistic anisotropic Planck noise
/data/maja1/ctp_ps/phase_2/maps/cmb_symm_noise_all_gal_map_1024.fits
For test upgrade to Nside=2048, smooth with 7/3arcmin beam.
What is the noise level???

19. Science case vs phase2 sim (TT only, noise as-is)

20. Hybrid Pseudo-Cl estimators Following GPE 2003, 2006 (+ numerous PCL papers) slight generalization to cross-weights
For n weight functions wi define
X=Y: n(n+1)/2 estimators; X<>Y, n2 estimators in general

21. Covariance matrix approximations Small scales, large fsky
etc… straightforward generalization for GPE’s results.

22. Also need all cross-terms…

23. Combine to hybrid estimator?
Find best single (Gaussian) fit spectrum using covariance matrix (GPE03). Keep simple: do Cl separately
Low noise: want uniform weight - minimize cosmic variance
High noise: inverse-noise weight - minimize noise (but increases cosmic variance, lower eff fsky)
Most natural choice of window function set? w1 = uniform w2 = inverse (smoothed with beam) noise
Estimators like CTT,11 CTT,12 CTT,22 …
For cross CTE,11 CTE,12 CTE,21 CTE,22 but Polarization much noisier than T, so CTE,11 CTE,12 CTE,22 OK?
Low l TT force to uniform-only? Or maybe negative hybrid noise is fine, and doing better??

24. TT cov diagonal, 2 weights

25. Does weight1-weight2 estimator add anything useful?
TT hybrid diag cov, dashed binned, 2 weight (3est) vs 3 weights (6 est) vs 2 weights diag only (GPE) Noisex1
Does it asymptote to the optimal value??

26. TE probably much more useful..
TE diagonal covariance

27. fwhm=7arcmin 2 weights, kp2 cut
Hybrid estimator
cmb_symm_noise_all_gal_map_1024.fits sim with TT Noise/16
N_QQ=N_UU=4N_TT
fwhm=7arcmin 2 weights, kp2 cut

28. l >30, tau fixed full sky uniform noise exact science case 153GHz avg vs TT,TE,EE polarized hybrid (2 weights, 3 cross) estimator on sim (Noise/16)
Somewhat cheating using exact fiducial model
chi-sq/2 not very good 3200 vs 2950

29. Very similar result with Gaussian approx and (true) fiducial covariance

30. What about cross-spectra from maps with independent noise? (Xfaster?)
- on full sky estimators no longer have Wishart distribution. Eg for temp
- asymptotically, for large numbers of maps it does > same likelihood approx probably OK when information loss is small

31. Conclusions
Gaussian can be good at l >> 1 -> MUST include determinant - either function of theory, or constant fixed fiducial model
New likelihood approximation - exact on full sky - fast to calculate - uses Nl, C-estimators, Cl-fiducial, and Cov-fiducial - with good Cl-estimators might even work at low l [MUCH faster than pixel-like] - seems to work but need to test for small biases