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

Start with full sky, isotropic noise Assume alm Gaussian

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

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…

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

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

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

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

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

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

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

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.

Unbiased on average

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

Test with realistic mask kp2, use pseudo-Cl directly

Isotropic noise test ~ 143Ghz from science case red – same realisation analysed on full sky all 1 < l < 2001 Provisional CosmoMC module at http://cosmologist.info/cosmomc/CMBLike.html

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???

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

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

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

Also need all cross-terms…

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??

TT cov diagonal, 2 weights

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??

TE probably much more useful.. TE diagonal covariance

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

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

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

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

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