Transcript CMB Power Spectrum Likelihoods

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
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
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
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 mask
kp2, use pseudo-Cl directly
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???
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 wi define
X=Y: n(n+1)/2 estimators; X<>Y, n2 estimators in general
Covariance matrix approximations
Small 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
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??
TE probably much more useful..
TE diagonal covariance
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
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
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