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  1.   1.  Surveys
    1.   1.1  Surveys results :
    2.   1.2  Correlations matrices :
    3.   1.3  Survey eigenvalues :
  2.   2.  Small nside tests
    1.   2.1  Monte_carlo VS analitical signal matrix construction
    2.   2.2  spectra + var ns=4 :
    3.   2.3  Condition number Signal Matrix :
    4.   2.4  range tests
    5.   2.5  Symetric matrix ?

1.  Surveys

1.1  Surveys results :

  • /!\ fsky=0.5 ! => more leakage, and more cosmic variance compared to the fsky=0.7 case.
  • first panel : estimator vs model
  • second panel : residues (1000 simulations)
  • third panel : spectrum error
LiBBIC
 

1.2  Correlations matrices :

  • FULL (LiB) , Ns4,
  • 46 vs 62 bins,
  • SlmaxA = 3*ns-1 VS SlmaxB = 4*ns-1
  • fwhmdegA=0.5 VS fwhmdegB=4.0 deg
fsky0.5

1.3  Survey eigenvalues :

  • Left : LiteBird, nside=16, fsky=0.7, svd(S) blue, svd(S+N) green at 0.1 muK
  • Right : BICEP, nside=128, svd(S) blue, svd(S+N) green at 0.1 muK
  • v3 or v4 are the eigenvalues of S (lmax = 3nside-1 or 4nside-1)
  • vC3 or vC4 are the eigenvalues of C (lmax = 3nside-1 or 4nside-1)
 
fwhmdeg0.5, fsky0.7

Smoothing effect :

  • axis label are inverted (x axis = fwhm [deg])
LiteBird ns16BICEP ns128
 

Comparison smoothing vs lmax :

  • Smoothing allow to neglect the change on diag S. As a consequence, it also impact the inversibility that procure the addition of new modes in the construction of the signal matrix.
  • nex step : compute the impact on the spectrum variance !

2.  Small nside tests

2.1  Monte_carlo VS analitical signal matrix construction

nside= 4, flat spectrum fiducial model

  • 100000 simulations
  • flat cl = 1.0
 Slmax = 3*ns-1Slmax = 4*ns-1Slmax = 5*ns-1Slmax = 10*ns-1
fwhm = 0.5 + pixwin
fwhm = 0.0 + no pixwin
  • observations :
    • for a flat spectrum, the analytical pixel covariance matrix correspond exactly to the monte-carlo generated one.
    • this is valid even for lmax > 4*nside !?

nside= 4, planck model spectrum fiducial model

  • 100000 simulations
  • planck model cl
 Slmax = 3*ns-1Slmax = 4*ns-1Slmax = 5*ns-1Slmax = 10*ns-1
spectra
fwhm = 0.5 + pixwin
  • observations :
    • for a flat spectrum, the analytical pixel covariance matrix correspond exactly to the monte-carlo generated one.
    • this is valid even for lmax > 4*nside !?
    • Mainly the level of the diagonal is slightly impacted by the S lmax. The other features from the matrix remain constant.
 anayticamc
S eigen values
  • observations :
    • condition number improve from lmax = 3ns to 4ns. Less improvement for higher lmax. As already shown (other plots downstair) .

nside= 8 eigen values

S ana eigen values
  • observations :
    • condition number decrease with resolution (nside).
    • but noise level increases, thus improving the condition number of C = S + N

2.2  spectra + var ns=4 :

  • markers are the spectra estimation
  • solid line = analytical (fisher) spectrum error
  • solid line low opacity = mc spectrum error
 fwhm=15.0 degfwhm=0.5 deg
Nbins=10, Slmax=3*ns-1, lmax simu = 3ns-1
Nbins=10, Slmax=4*ns-1, lmax simu = 4ns-1
Nbins=10, Slmax=4*ns-1, lmax simu = 3ns-1
Nbins=14, Slmax=4*ns-1, lmax simu = 4ns-1
Nbins=14, Slmax=4*ns-1, lmax simu = 3ns-1
  • Pushing lmax simulation to 4ns seems to bias the spectrum
  • Pushing lmax in the signal covariance amtrix S seems to boost the variance at high multipole

New Correlations matrices :

  • FULL (LiB) , Ns4,
  • 10 vs 14 bins,
  • SlmaxA = 3*ns-1 VS SlmaxB = 4*ns-1
  • fwhmdegA=0.5 VS fwhmdegB=15.0 deg
fsky0.8
fsky1.0

Old Correlations matrices :

  • FULL (LiB) , Ns4, Nbins14, Slmx=4ns-1, fwhmdeg=15.0, r=0.001, fsky0.8

2.3  Condition number Signal Matrix :

2.4  range tests

Condition number versus lmax.

  • Colors are map beams change (fhwm = [0.5, 1.0, 2.0, 3.0] deg ). We see that there is little variation of the condition number when changing the beam.
  • Condition number is mainly impacted by the fksy and lmax value.

2.5  Symetric matrix ?

Nside = 8, fsky=0.7, 0.1 muK

  • note : WAB = FAB (fisher matrix)
sym ElAB at l=0 ?False
sym ElAB at l=1 ?False
sym ElAB at l=2 ?False
sym ElAB at l=3 ?False
sym ElAB at l=4 ?True
sym ElAB at l=5 ?True
sym ElAB at l=6 ?True
sym ElAB at l=7 ?True
sym ElAB at l=8 ?True
sym ElAB at l=9 ?True
sym ElAB at l=10 ?True
sym ElAB at l=11 ?True
sym ElAB at l=12 ?True
sym ElAB at l=13 ?True
sym ElAB at l=14 ?True
sym ElAB at l=15 ?True
sym ElAB at l=16 ?True
sym ElAB at l=17 ?True
sym ElAB at l=18 ?True
sym ElAB at l=19 ?True
sym ElAB at l=20 ?True
sym ElAB at l=21 ?True
sym ElAB at l=22 ?True
sym ElAB at l=23 ?True
sym ElAB at l=24 ?True
sym ElAB at l=25 ?True
sym ElAB at l=26 ?True
sym ElAB at l=27 ?True
sym ElAB at l=28 ?True
sym ElAB at l=29 ?True
sym ElAB at l=30 ?False
sym ElAB at l=31 ?True
sym ElAB at l=32 ?True
sym ElAB at l=33 ?True
sym ElAB at l=34 ?True
sym ElAB at l=35 ?True
sym ElAB at l=36 ?True
sym ElAB at l=37 ?True
sym ElAB at l=38 ?True
sym ElAB at l=39 ?True
sym ElAB at l=40 ?True
sym ElAB at l=41 ?True
sym ElAB at l=42 ?True
sym ElAB at l=43 ?True
sym ElAB at l=44 ?True
sym ElAB at l=45 ?True
sym ElAB at l=46 ?True
sym ElAB at l=47 ?True
sym ElAB at l=48 ?True
sym ElAB at l=49 ?True
sym ElAB at l=50 ?True
sym ElAB at l=51 ?True
sym ElAB at l=52 ?True
sym ElAB at l=53 ?True
sym ElAB at l=54 ?True
sym ElAB at l=55 ?True
sym ElAB at l=56 ?True
sym ElAB at l=57 ?True
sym ElAB at l=58 ?True
sym ElAB at l=59 ?True
sym CAA ?True
sym invCAA ?True
sym WABb ?True
sym invWAB ?True
sym GABb ?True