Bayesian Estimation of the isotropy violation in the
Bayesian Estimation of the isotropy violation in the CMB sky Santanu Das University of Wisconsin, Madison & Fermi National Accelerator Laboratory
SITool. Box Estimate the isotropy violation in the CMB sky Applicable to the polarization maps
1965 1990 2003 2013 COBEand satellite measured temperature fluctuations in CMB data release WMAP measured CMB fluctuation Penzias Wilson. Planck discovered cosmic microwave background
Fluctuation Dipole : 3. 538 : 60 -70 m. K Monopole 2. 726 K ~ μK
WMAP COBE Planck 1992 2013 2003 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
2010 Take SI map Scan using WMAP beam and scan pattern 2011 2013 2014 Reconstract the map WMAP-7 detected Statistica is. Isotropy in the CMB sky It is proved that SI violation comingviolation from WMAP scan strategy Das et. al. 2014, Joshiet. al. 2010 2012 Bennet
Observed CMB sky is actually not SI 1. Weak lensing 2. Doppler anisotropy 3. Non circular Satellite Beam and scan pattern 4. Anisotropic Noise 5. Masking 6. Nonstandard topology of the universe The data analysis has to be proper
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
Why do we consider SI violation Bipo. SH Hajian & Souradeep, 2003
Why do we consider SI violation 1. 00 x 0. 95 x + 0. 05 x
Is it sufficient to calculate only Bipo. SH-es? Expand in spherical harmonics : Jointly Calculate : Bayesian measurement of the Bipo. SH is important i. e. we need the average and the posterior of the Bipo. SH
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 x 2 Take pi and evaluate on time Evaluate x 1
All these matrices are really big 107 x 107 inverting the matrices are difficult
White Noise matrix is diagonal in pixel space Assume diagonally dominated Expand it in the Taylor series Das 2018, Shaikh et. al. 2018
Analysis and Results SI + 10 μ K SI map + 10 μK Calculate Bipo. SH by minimizing all Bipo. SH 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 SI + 10 μ K n. SI + 10 μ K (Anisotropic) Take SI map Scan using WMAP beam and scan pattern Reconstract the map
Analysis and Results SI + 10 μ K n. SI + Scan + Anisotropic noise Noise : 10μK Noise : 30μK
Analysis and Results SI + 10 μ K n. SI + 10 μ K (Anisotropic) n. SI + Scan + Anisotropic noise + Mask Take SI map Scan using WMAP beam and scan pattern Reconstract the map
Analysis and Results SI + 10 μ K n. SI + Scan + Anisotropic noise + Mask Noise : 30μK
Analysis and Results SI + 10 μ K Dipole modulated map Mask Noise variance Noise Sample SI + Scan + Anisotropic noise ( 1 + x ) Dipole modulation + 10 μ K Input map to SITool. Box A recovered sample map Das 2018
Analysis and Results Doppler Boost Mukherjee et. al. , 2015 Generate NSI map using Co. NIGS Mukherjee et. al. , 2013 10 μK noise
Analysis and Results We can recover all the input values properly
Summary 1. Cl is not sufficient to provide full information about the cosmological model in case of Statistical isotropy violation. We need the Bipo. SH coefficients also. 2. We develop a formalism to calculate Cl and Bipo. SH coefficients together from a noisy map using complete Bayesian technique. 3. We use Hamiltonian Monte Carlo for sampling the posterior. 4. We apply our algorithm on different simulated maps and recover the Bipo. SH coefficients appropriately. 5. The codes are made public through github. They are readily applicable to the polarization skymap and other random data over a sphere.
Thank you for being awake Any question?
- Slides: 34