Systematic effects in cosmic microwave background polarization and power spectrum estimation

1. Systematic effects in cosmic microwave background polarization and power spectrum estimation Fidy A. RAMAMONJISOA University of KwaZulu-Natal School of Mathematical Science PhD project supervised by Prof Subharthi Ray SKA 2010 Postgraduate Bursary Conference, Stellenbosch Institute for Advanced Study 30/11/10

2. Introduction Time • CMB is a 2.725 K blackbody radiation composing the majority of the radiation of the universe in mm-cm wavelength • CMB photons are emitted from the last scattering surface (LSS) at z=1100 (379 000 yrs) • Radiation is highly isotropic • Temperature fluctuations of the CMB are at 10-5 level 1010 yrs 3x105 yrs CMB observer Inflation Present LSS W. Hu 2002

3. Introduction Colder radiation • Polarization first detected by the Degree Angular Scale Interferometer (DASI) in 2002 • Due to Thomson scattering the fluctuations are polarized at 10% level • Polarization is decomposed into • E-mode (scalar/tensor perturbations due to density fluctuations) • B-mode (tensor perturbations due to gravity waves) Hotter radiation W. Hu 2001

4. Stokes parameters • CMB polarization are defined by Stokes parameters • For CMB photons: V=0, Q and Ucharacterize linear polarization Incident waves Electron Electric field Linearly polarized radiation Mukhanov V. 2005

5. ClTT CMB angular power spectra ClTE ClEE ClBB (lensing) ClBB (r=0.1) ClBB (r=10-4) Multipole Rosset C. 2005 Objectives • Find a semi-analytic formulation of the cross power spectra ClTT, ClTE, ClEE, ClBB • Compute the cross power spectra using computationally fast pseudo-Cl estimator • Correct systematic effects due to • Non-circularity of instrument beam response • Foreground emissions • Instrumental noise

6. Beam asymmetry Planck 100 GHz • Non-circularity of beam assumption is essential at small angular scales (higher l) • Assume Gaussian window function ClTT ClTE ClEE Multipole l Errors in power spectrum estimation as a function of beam ellipticity (beam ellipticity parameter: deviation of the beam from circularity) Folsaba et al. 2002

7. Foreground emissions Bennett et al. 2003 • Foreground emissions Instrumental noise Mask function Beam function Measured T True T Planck first image http://www.scientificamerican.com/media

8. Methodology • Decompose Stokes parameters into spin-two harmonics • True power spectra Pseudo-Cl estimators

9. Bias matrix Methodology • The expectation values of pseudo-Cl is given by (Mitra et al. 2008)

10. 8 weeksCPU time Preliminary results • Expectation values of pseudo-Cl estimator for full sky and non-circular beam Mitra et al. (2008) CPU time for caculating ClTT bias matrix 1000 dual core CPUs lmax=3000 mbeam=2

11. Preliminary results Limiting case of full sky and non-circular beam Bias matrix for TE power spectra Beam distortion parameter Beam function Clebsch-Gordon coefficients Wigner-d function 3j symbol

12. Preliminary results Limiting case of full sky and non-circular beam • Bias matrix for EE and BB power spectra

13. Future works • Introduce mask function to account for cut-sky • Write codes to compute bias matrix and power spectra • Run our codes using CHPC facilities • Estimate the covariance matrix errors due to beam asymmetry and incomplete sky coverage • Match theory with upcoming Planck data

14. Conclusion • Pseudo-Cl method provides computationally fast cross power spectra estimation at small angular scale (lmax=3000) • Systematic effect corrections are crucial for the Planck-like high resolution CMB experiment • Detection of B-mode polarization is a direct probe of gravitational waves predicted by inflationary models • B-mode polarization detection is challenging

15. References

16. Acknowledgements • I acknowledge the South African Square Kilometre Array Project for financial support of this project.