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

CMB Power spectrum likelihood approximations Antony Lewis, IoA Work with Samira Hamimeche. Start with full sky, isotropic noise. Assume a lm Gaussian. Integrate alm that give same Chat. - Wishart distribution. For temperature. Non-Gaussian skew ~ 1/l. For unbiased parameters need bias <<.

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

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

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

  3. 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

  4. 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…

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

  6. ~ 1 / (l Δl) Binning: skewness ~ 1/ (number of modes) - 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

  7. 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

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

  9. 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

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

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

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

  13. Unbiased on average

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

  15. Test with realistic maskkp2, use pseudo-Cl directly

  16. 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???

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

  18. Hybrid Pseudo-Cl estimatorsFollowing 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

  19. Covariance matrix approximationsSmall scales, large fsky etc… straightforward generalization for GPE’s results.

  20. Also need all cross-terms…

  21. 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,22but 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??

  22. TT cov diagonal, 2 weights

  23. 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 asymptoteto the optimal value??

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

  25. Hybrid estimator cmb_symm_noise_all_gal_map_1024.fits sim with TT Noise/16 N_QQ=N_UU=4N_TT fwhm=7arcmin2 weights, kp2 cut

  26. l >30, tau fixedfull sky uniform noise exact science case 153GHz avgvs TT,TE,EE polarized hybrid (2 weights, 3 cross) estimator on sim (Noise/16) Somewhat cheatingusing exactfiducial model chi-sq/2 not very good3200 vs 2950

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

  28. 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

  29. 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

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