Presentation on theme: "CMB Power spectrum likelihood approximations Antony Lewis, IoA Work with Samira Hamimeche."— Presentation transcript:
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 << - 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 l min < l l min +1 ?
Binning: skewness ~ 1/ (number of modes) ~ 1 / (l Δ l ) - can use any Gaussian approximation for Δ l >> 1 Gaussian approximation with determinant: - Best-fit amplitude is - almost always a good approximation for l >> 1 - somewhat slow to calculate though Fiducial Gaussian: unbiased, - error bars depend on right fiducial model, but easy to choose accurate to 1/root(l)
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 - Exact on the full sky with isotropic noise - Use full covariance information - Quick to calculate Would like:
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 C f where Now generalizes to cut sky:
Other approximations also good just for temperature. But they dont 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-C l directly
Isotropic noise test ~ 143Ghz from science case red – same realisation analysed on full sky all 1 < l < 2001 Provisional CosmoMC module at
/data/maja1/ctp_ps/phase_2/maps/cmb_symm_noise_all_gal_map_1024.fits More realistic anisotropic Planck noise 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 w i define X=Y: n(n+1)/2 estimators; X<>Y, n 2 estimators in general
Covariance matrix approximations Small scales, large fsky etc… straightforward generalization for GPEs results.
Also need all cross-terms…
Combine to hybrid estimator? Find best single (Gaussian) fit spectrum using covariance matrix (GPE03). Keep simple: do C l 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? w 1 = uniform w 2 = inverse (smoothed with beam) noise Estimators like C TT,11 C TT,12 C TT,22 … For cross C TE,11 C TE,12 C TE,21 C TE,22 but Polarization much noisier than T, so C TE,11 C TE,12 C TE,22 OK? Low l TT force to uniform-only? Or maybe negative hybrid noise is fine, and doing better??
TT cov diagonal, 2 weights
TT hybrid diag cov, dashed binned, 2 weight (3est) vs 3 weights (6 est) vs 2 weights diag only (GPE) Noisex1 Does weight1-weight2 estimator add anything useful? Does it asymptote to the optimal value??
TE diagonal covariance TE probably much more useful..
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) chi-sq/2 not very good 3200 vs 2950 Somewhat cheating using exact fiducial model
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 N l, C-estimators, C l -fiducial, and Cov-fiducial - with good C l -estimators might even work at low l [MUCH faster than pixel-like] - seems to work but need to test for small biases