Download presentation

1
**CMB Power spectrum likelihood approximations**

Antony Lewis, IoA Work with Samira Hamimeche

3
**Start with full sky, isotropic noise**

Assume alm Gaussian

4
**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

5
**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…

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 >> 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

8
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

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**

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

12
**For some fiducial model Cf**

where Now 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.

14
Unbiased on average

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 < 2001 Provisional CosmoMC module at

18
**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???

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-weights For n weight functions wi define X=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.

22
**Also need all cross-terms…**

23
**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??

24
**TT cov diagonal, 2 weights**

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) Noisex1 Does it asymptote to the optimal value??

26
**TE probably much more useful..**

TE diagonal covariance

27
**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

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 model chi-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
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

Similar presentations

OK

Geology 5670/6670 Inverse Theory 21 Jan 2015 © A.R. Lowry 2015 Read for Fri 23 Jan: Menke Ch 3 (39-68) Last time: Ordinary Least Squares Inversion Ordinary.

Geology 5670/6670 Inverse Theory 21 Jan 2015 © A.R. Lowry 2015 Read for Fri 23 Jan: Menke Ch 3 (39-68) Last time: Ordinary Least Squares Inversion Ordinary.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google

Ppt on metro cash and carry Ppt on standing order edinburgh Ppt on machine translation software Download ppt on coal and petroleum for class 8 Ppt on cross-sectional study advantages and disadvantages Ppt on need for conservation of natural resources Ppt on pivot table in excel 2007 Doc convert to ppt online Ppt on college website Ppt on db2 architecture and process