3 Start with full sky, isotropic noise Assume alm Gaussian
4 Integrate alm that give same Chat - Wishart distributionFor temperatureNon-Gaussian skew ~ 1/lFor 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 Detfixed fiducial varianceexactly unbiased, best-fit onaverage is correctActual Gaussian in Chator 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 >> 1Fiducial 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 approximationCan 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 noiseUse full covariance informationQuick 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 whereThen write aswhereRe-write in terms of vector of matrix elements…
12 For some fiducial model Cf whereNow 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.
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 < 2001Provisional CosmoMC module at
18 More realistic anisotropic Planck noise /data/maja1/ctp_ps/phase_2/maps/cmb_symm_noise_all_gal_map_1024.fitsFor 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-weightsFor n weight functions wi defineX=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.
23 Combine to hybrid estimator? Find best single (Gaussian) fit spectrum using covariance matrix (GPE03). Keep simple: do Cl separatelyLow noise: want uniform weight - minimize cosmic varianceHigh 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) noiseEstimators 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??
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) Noisex1Does it asymptote to the optimal value??
26 TE probably much more useful.. TE diagonal covariance
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 modelchi-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 ConclusionsGaussian can be good at l >> 1 -> MUST include determinant - either function of theory, or constant fixed fiducial modelNew 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