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

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 approximationCan 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 Cf

where

Now generalizes to cut sky:

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 maskkp2, use pseudo-Cl directly

/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 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

Covariance matrix approximationsSmall 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

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 asymptoteto the optimal value??

TE diagonal covarianceTE probably much more useful..

Hybrid estimatorcmb_symm_noise_all_gal_map_1024.fits

sim with TT Noise/16N_QQ=N_UU=4N_TTfwhm=7arcmin2 weights, kp2 cut

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)

chi-sq/2 not very good3200 vs 2950

Somewhat cheatingusing exactfiducial 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 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