CMB Power spectrum likelihood approximations

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CMB Power spectrum likelihood approximations Ant
ony Lewis, IoA Work with Samira Hamimeche
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(No Transcript)
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• Start with full sky, isotropic noise

Assume alm Gaussian
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Integrate alm that give same Chat
- Wishart distribution
For temperature
Non-Gaussian skew 1/l
For unbiased parameters need bias ltlt
- might need to be careful at all ell
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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
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Do you get the answer right for amplitude over
range lmin lt l lmin1 ?
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• Binning skewness 1/ (number of modes)

1 / (l ?l)
- can use any Gaussian approximation for ?l gtgt 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 gtgt 1
- somewhat slow to calculate though
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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

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Matrices or vectors?

• Vector of n(n1)/2 distinct elements of C

Covariance
For symmetric A and B, key result is
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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
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Likelihood approximation
where
Then write as
where
Re-write in terms of vector of matrix elements
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For some fiducial model Cf
where
Now generalizes to cut sky
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Other approximations also good just for
temperature. But they dont generalize.
Can calculate likelihood exactly for azimuthal
cuts and uniform noise - to compare.
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Unbiased on average
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T and E Consistency with binned likelihoods
(all Gaussian accurate to 1/(l Delta_l) by
central limit theorem)
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Test with realistic maskkp2, use pseudo-Cl
directly
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Isotropic noise test 143Ghz from science
casered same realisation analysed on full
skyall 1 lt l lt 2001
Provisional CosmoMC module at http//cosmologist.i
nfo/cosmomc/CMBLike.html
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More realistic anisotropic Planck noise
/data/maja1/ctp_ps/phase_2/maps/cmb_symm_noise_all
_gal_map_1024.fits
For test upgrade to Nside2048, smooth with
7/3arcmin beam.
What is the noise level???
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Science case vs phase2 sim (TT only, noise as-is)
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Hybrid Pseudo-Cl estimatorsFollowing GPE 2003,
2006 ( numerous PCL papers)slight
generalization to cross-weights
For n weight functions wi define
XY n(n1)/2 estimators XltgtY, n2 estimators in
general
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Covariance matrix approximationsSmall scales,
large fsky
etc straightforward generalization for GPEs
results.
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Also need all cross-terms
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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,22but
  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??
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TT cov diagonal, 2 weights
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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 asymptoteto the optimal value??
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TE probably much more useful..
TE diagonal covariance
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Hybrid estimator cmb_symm_noise_all_gal_map_1024.f
its sim with TT Noise/16 N_QQN_UU4N_TT fwhm7ar
cmin2 weights, kp2 cut
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l gt30, tau fixedfull sky uniform noise exact
science case 153GHz avgvs TT,TE,EE polarized
hybrid (2 weights, 3 cross) estimator on sim
(Noise/16)
Somewhat cheatingusing exactfiducial model
chi-sq/2 not very good3200 vs 2950
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Very similar result with Gaussian approx and
(true) fiducial covariance
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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 -----gt same likelihood approx probably
OK when information loss is small
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Conclusions

• Gaussian can be good at l gtgt 1-gt 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