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

2. 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

3. Monopole : K
Fluctuation : ~ μK
Dipole : mK

4. WMAP 2003
COBE 1992
Planck 2013
Fluctuation : ~ μK

5. 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 : ~ μK

6. Theoretically we can calculate CMB power spectrum

7. 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

8. 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

9. What is SI violation

10. What is SI violation

11. Why do we consider SI violation
1.00 x
0.95 x x
0.95 x x

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

13. Why do we consider SI violation
1.00 x
0.95 x x
0.95 x x

14. 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

15. 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

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

19. 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

20. Analysis and Results
SI + 10 μ K
Consistent with 0

21. Analysis and Results
SI + 10 μ K
Consistent with 0

22. 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)

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

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

25. 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

26. Analysis and Results
We can recover all the input values properly

27. 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.

28. Thank you
for being awake
Any question?