cmb power spectrum likelihood approximations antony lewis, ioa work with samira hamimeche
DESCRIPTION
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 biasTRANSCRIPT
![Page 1: CMB Power spectrum likelihood approximations Antony Lewis, IoA Work with Samira Hamimeche](https://reader035.vdocuments.us/reader035/viewer/2022062316/56814d60550346895dbaa635/html5/thumbnails/1.jpg)
CMB Power spectrum likelihood approximations
Antony Lewis, IoA
Work with Samira Hamimeche
![Page 2: CMB Power spectrum likelihood approximations Antony Lewis, IoA Work with Samira Hamimeche](https://reader035.vdocuments.us/reader035/viewer/2022062316/56814d60550346895dbaa635/html5/thumbnails/2.jpg)
![Page 3: CMB Power spectrum likelihood approximations Antony Lewis, IoA Work with Samira Hamimeche](https://reader035.vdocuments.us/reader035/viewer/2022062316/56814d60550346895dbaa635/html5/thumbnails/3.jpg)
• Start with full sky, isotropic noise
Assume alm Gaussian
![Page 4: CMB Power spectrum likelihood approximations Antony Lewis, IoA Work with Samira Hamimeche](https://reader035.vdocuments.us/reader035/viewer/2022062316/56814d60550346895dbaa635/html5/thumbnails/4.jpg)
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
![Page 5: CMB Power spectrum likelihood approximations Antony Lewis, IoA Work with Samira Hamimeche](https://reader035.vdocuments.us/reader035/viewer/2022062316/56814d60550346895dbaa635/html5/thumbnails/5.jpg)
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…
![Page 6: CMB Power spectrum likelihood approximations Antony Lewis, IoA Work with Samira Hamimeche](https://reader035.vdocuments.us/reader035/viewer/2022062316/56814d60550346895dbaa635/html5/thumbnails/6.jpg)
Do you get the answer right for amplitude over range lmin < l lmin+1 ?
![Page 7: CMB Power spectrum likelihood approximations Antony Lewis, IoA Work with Samira Hamimeche](https://reader035.vdocuments.us/reader035/viewer/2022062316/56814d60550346895dbaa635/html5/thumbnails/7.jpg)
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)
![Page 8: CMB Power spectrum likelihood approximations Antony Lewis, IoA Work with Samira Hamimeche](https://reader035.vdocuments.us/reader035/viewer/2022062316/56814d60550346895dbaa635/html5/thumbnails/8.jpg)
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:
![Page 9: CMB Power spectrum likelihood approximations Antony Lewis, IoA Work with Samira Hamimeche](https://reader035.vdocuments.us/reader035/viewer/2022062316/56814d60550346895dbaa635/html5/thumbnails/9.jpg)
Matrices or vectors?
• Vector of n(n+1)/2 distinct elements of C
Covariance:
For symmetric A and B, key result is:
![Page 10: CMB Power spectrum likelihood approximations Antony Lewis, IoA Work with Samira Hamimeche](https://reader035.vdocuments.us/reader035/viewer/2022062316/56814d60550346895dbaa635/html5/thumbnails/10.jpg)
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
![Page 11: CMB Power spectrum likelihood approximations Antony Lewis, IoA Work with Samira Hamimeche](https://reader035.vdocuments.us/reader035/viewer/2022062316/56814d60550346895dbaa635/html5/thumbnails/11.jpg)
Likelihood approximation
where
Then write as
where
Re-write in terms of vector of matrix elements…
![Page 12: CMB Power spectrum likelihood approximations Antony Lewis, IoA Work with Samira Hamimeche](https://reader035.vdocuments.us/reader035/viewer/2022062316/56814d60550346895dbaa635/html5/thumbnails/12.jpg)
For some fiducial model Cf
where
Now generalizes to cut sky:
![Page 13: CMB Power spectrum likelihood approximations Antony Lewis, IoA Work with Samira Hamimeche](https://reader035.vdocuments.us/reader035/viewer/2022062316/56814d60550346895dbaa635/html5/thumbnails/13.jpg)
Other approximations also good just for temperature. But they don’t generalize.
Can calculate likelihood exactly for azimuthal cuts and uniform noise - to compare.
![Page 14: CMB Power spectrum likelihood approximations Antony Lewis, IoA Work with Samira Hamimeche](https://reader035.vdocuments.us/reader035/viewer/2022062316/56814d60550346895dbaa635/html5/thumbnails/14.jpg)
Unbiased on average
![Page 15: CMB Power spectrum likelihood approximations Antony Lewis, IoA Work with Samira Hamimeche](https://reader035.vdocuments.us/reader035/viewer/2022062316/56814d60550346895dbaa635/html5/thumbnails/15.jpg)
T and E: Consistency with binned likelihoods (all Gaussian accurate to 1/(l Delta_l) by central limit theorem)
![Page 16: CMB Power spectrum likelihood approximations Antony Lewis, IoA Work with Samira Hamimeche](https://reader035.vdocuments.us/reader035/viewer/2022062316/56814d60550346895dbaa635/html5/thumbnails/16.jpg)
Test with realistic maskkp2, use pseudo-Cl directly
![Page 17: CMB Power spectrum likelihood approximations Antony Lewis, IoA Work with Samira Hamimeche](https://reader035.vdocuments.us/reader035/viewer/2022062316/56814d60550346895dbaa635/html5/thumbnails/17.jpg)
/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???
![Page 18: CMB Power spectrum likelihood approximations Antony Lewis, IoA Work with Samira Hamimeche](https://reader035.vdocuments.us/reader035/viewer/2022062316/56814d60550346895dbaa635/html5/thumbnails/18.jpg)
Science case vs phase2 sim (TT only, noise as-is)
![Page 19: CMB Power spectrum likelihood approximations Antony Lewis, IoA Work with Samira Hamimeche](https://reader035.vdocuments.us/reader035/viewer/2022062316/56814d60550346895dbaa635/html5/thumbnails/19.jpg)
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
![Page 20: CMB Power spectrum likelihood approximations Antony Lewis, IoA Work with Samira Hamimeche](https://reader035.vdocuments.us/reader035/viewer/2022062316/56814d60550346895dbaa635/html5/thumbnails/20.jpg)
Covariance matrix approximationsSmall scales, large fsky
etc… straightforward generalization for GPE’s results.
![Page 21: CMB Power spectrum likelihood approximations Antony Lewis, IoA Work with Samira Hamimeche](https://reader035.vdocuments.us/reader035/viewer/2022062316/56814d60550346895dbaa635/html5/thumbnails/21.jpg)
Also need all cross-terms…
![Page 22: CMB Power spectrum likelihood approximations Antony Lewis, IoA Work with Samira Hamimeche](https://reader035.vdocuments.us/reader035/viewer/2022062316/56814d60550346895dbaa635/html5/thumbnails/22.jpg)
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??
![Page 23: CMB Power spectrum likelihood approximations Antony Lewis, IoA Work with Samira Hamimeche](https://reader035.vdocuments.us/reader035/viewer/2022062316/56814d60550346895dbaa635/html5/thumbnails/23.jpg)
TT cov diagonal, 2 weights
![Page 24: CMB Power spectrum likelihood approximations Antony Lewis, IoA Work with Samira Hamimeche](https://reader035.vdocuments.us/reader035/viewer/2022062316/56814d60550346895dbaa635/html5/thumbnails/24.jpg)
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??
![Page 25: CMB Power spectrum likelihood approximations Antony Lewis, IoA Work with Samira Hamimeche](https://reader035.vdocuments.us/reader035/viewer/2022062316/56814d60550346895dbaa635/html5/thumbnails/25.jpg)
TE diagonal covarianceTE probably much more useful..
![Page 26: CMB Power spectrum likelihood approximations Antony Lewis, IoA Work with Samira Hamimeche](https://reader035.vdocuments.us/reader035/viewer/2022062316/56814d60550346895dbaa635/html5/thumbnails/26.jpg)
Hybrid estimatorcmb_symm_noise_all_gal_map_1024.fits
sim with TT Noise/16N_QQ=N_UU=4N_TTfwhm=7arcmin2 weights, kp2 cut
![Page 27: CMB Power spectrum likelihood approximations Antony Lewis, IoA Work with Samira Hamimeche](https://reader035.vdocuments.us/reader035/viewer/2022062316/56814d60550346895dbaa635/html5/thumbnails/27.jpg)
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
![Page 28: CMB Power spectrum likelihood approximations Antony Lewis, IoA Work with Samira Hamimeche](https://reader035.vdocuments.us/reader035/viewer/2022062316/56814d60550346895dbaa635/html5/thumbnails/28.jpg)
Very similar result with Gaussian approx and (true) fiducial covariance
![Page 29: CMB Power spectrum likelihood approximations Antony Lewis, IoA Work with Samira Hamimeche](https://reader035.vdocuments.us/reader035/viewer/2022062316/56814d60550346895dbaa635/html5/thumbnails/29.jpg)
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
![Page 30: CMB Power spectrum likelihood approximations Antony Lewis, IoA Work with Samira Hamimeche](https://reader035.vdocuments.us/reader035/viewer/2022062316/56814d60550346895dbaa635/html5/thumbnails/30.jpg)
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