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

Introduction  CMB is a 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 ( yrs)  Radiation is highly isotropic  Temperature fluctuations of the CMB are at level Time Inflation Present CMB observer yrs 3x10 5 yrs LSS W. Hu 2002

Introduction  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) Colder radiation Hotter radiation W. Hu 2001

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

Objectives  Find a semi-analytic formulation of the cross power spectra C l TT, C l TE, C l EE, C l BB  Compute the cross power spectra using computationally fast pseudo-C l estimator  Correct systematic effects due to  Non-circularity of instrument beam response  Foreground emissions  Instrumental noise Multipole C l TT C l TE C l EE C l BB (r=0.1) C l BB (r=10 -4 ) C l BB (lensing) CMB angular power spectra Rosset C. 2005

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

Foreground emissions Mask function Instrumental noise Beam function Measured T True T Planck first image Bennett et al

Methodology  Decompose Stokes parameters into spin-two harmonics  True power spectra Pseudo-C l estimators

Methodology  The expectation values of pseudo-C l is given by (Mitra et al. 2008) Bias matrix

CPU time for caculating C l TT bias matrix 1000 dual core CPUs l max =3000 m beam =2 Preliminary results Expectation values of pseudo-C l estimator for full sky and non-circular beam Mitra et al. (2008) 8 weeks CPU time

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

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

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

Conclusion  Pseudo-C l method provides computationally fast cross power spectra estimation at small angular scale (l max =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

References

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