Bayesian Estimation of the isotropy violation in the CMB sky Santanu Das UW Madison

1965 1990 2003 2013 Penzias and Wilson discovered cosmic microwave background  WMAP measured CMB fluctuation COBE satellite measured temperature fluctuations in CMB Planck data release

Monopole : 2.726 K Fluctuation : 60-70 ~ μK Dipole : 3.538 mK

WMAP 2003 COBE 1992 Planck 2013 Fluctuation : 60-70 ~ μK

Statistical Isotropy Fluctuation : 60-70 ~ μK Any fluctuation on a sphere can be expanded in terms of spherical harmonics are Gausisan So we need some statistical measure The angular power spectrum of the CMB sky can be calculated as Statistical Isotropy Fluctuation : 60-70 ~ μK

Theoretically we can calculate CMB power spectrum

Observed CMB sky is actually not SI Weak lensing Doppler anisotropy Non circular Satellite Beam and scan pattern Anisotropic Noise Masking Nonstandard topology of the universe The data analysis has to be proper

If there is any pattern in the sky map then that is SI violated What is SI violation If there is any pattern in the sky map then that is SI violated

What is SI violation

What is SI violation

Why do we consider SI violation 1.00 x 0.95 x + 0.05 x 0.95 x + 0.05 x

Why do we consider SI violation BipoSH Hajian & Souradeep, 2003

Why do we consider SI violation 1.00 x 0.95 x + 0.05 x 0.95 x + 0.05 x

Is it sufficient to calculate only BipoSH-es? Expand in spherical harmonics : Jointly Calculate : Bayesian measurement of the BipoSH is important i.e. we need the average and the posterior of the BipoSH

We use Hamiltonian Monte Carlo method for sampling the Posterior Observed Sky Temp Original Sky Temp Noise Observed sky : Spherical Harmonics basis : We use Hamiltonian Monte Carlo method for sampling the Posterior

What is HMC sampling x2 x1 Evaluate Take pi and evaluate on time

Calculate BipoSH by minimizing all BipoSH coefficients upto L=2 Analysis and Results SI + 10 μ K SI map + 10 μK Calculate BipoSH by minimizing all BipoSH coefficients upto L=2

Analysis and Results SI + 10 μ K Consistent with 0

Analysis and Results SI + 10 μ K Consistent with 0

Analysis and Results Take SI map Scan using WMAP beam and scan pattern Reconstract the map SI + 10 μ K SI + 10 μ K (Anisotropic)

SI + Scan + Anisotropic noise Analysis and Results SI + 10 μ K SI + Scan + Anisotropic noise SI + 5 μ K SI + 10 μ K

SI + Scan + Anisotropic noise Analysis and Results BipoSH Signal recovery is very well SI + 10 μ K SI + Scan + Anisotropic noise ( 0.95 + 0.05 ) Dipole modulation + 10 μ K

Generate NSI map using CoNIGS Analysis and Results Doppler Boost Mukherjee et. al., 2015 Generate NSI map using CoNIGS 10 μK noise Mukherjee et. al., 2013

Analysis and Results We can recover all the input values properly

Summary Cl is not sufficient to provide full information about the cosmological model in case of Statistical isotropy violation. We need the BipoSH coefficients also. We develop a formalism to calculate Cl and BipoSH coefficients a together from a noisy map using complete Bayesian technique. We use Hamiltonion Monte Carlo for sampling the posterior. We apply our algorithm on different simulated maps and recover the BipoSH coefficients appropriately.

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