Examining the Effect of the Map-Making Algorithm on Observed Power Asymmetry in WMAP Data

Transcription

1 Examining the Effect of the Map-Making Algorithm on Observed Power Asymmetry in WMAP Data P. E. Freeman 1, C. R. Genovese 1, C. J. Miller 2, R. C. Nichol 3, & L. Wasserman 1 1 Department of Statistics Carnegie Mellon University 2 National Optical Astronomy Observatory 3 Institute of Cosmology and Gravitation University of Portsmouth pfreeman@cmu.edu

2 INTRODUCTION Standard inflationary cosmology predicts the temperature field of the Cosmic Microwave Background (CMB) to be a Gaussian random field, i.e., when expressed using spherical harmonics T (θ, φ) = l=0 l m= l almylm(θ, φ), the modes are Gaussian random variables: alm N(0, < C 2 l >), where Cl = 1 2l + 1 l m= l alm 2. However, the first-year data of the Wilkinson Microwave Anisotropy Probe (WMAP) indicate possible irregularities: intrinsic CMB non-gaussianity? incomplete foreground modelling? instrument-related systematic error? map-making-related systematic error? We reproduce the WMAP map-making algorithm and explore how altering it affects observed power asymmetry between arbitrarily defined hemispheres. See Freeman et al. (astro-ph/ ) for more detail.

3 10 differential radiometers 5 frequency bands: K, Ka, Q, V, W 20 GHz (K) to 106 GHz (W) Radiometer: two horns, θsep 140 T between horns measured every 1.536/n sec (n = for bands K W) 366 days of TOD WMAP

4 CMB MAP-MAKING: THEORY Goal: make maximum likelihood estimate of the true difference field T sky given time-ordered differences T TOD : TTOD = M( Tsky) + ɛtod. M: sky pixel-to-time mapping matrix; ɛ: noise. Assumption: the noise is Gaussian. (During calibration, the noise is transformed from 1/f to white.) So: Tsky = (M T C 1 M) 1 M T C 1 TTOD. Assumption: C 1 I Assumption: M T M is diagonally dominant/the beam response is a δ-function. So: (M T M) 1 n 1 sky. where n sky,p is the number of observations of sky pixel p. See Hinshaw et al. (2003) for more details.

5 CMB MAP-MAKING: DETAILS Start with TOD T TOD and CMB field T CMB = 0. Iterate to solution. For each iteration i: Read T TOD sequentially. For each datum T TOD,t, determine the radiometer normal vectors (n A,n B )......Then subtract the Doppler-shifted monopole: TCMB,t = TTOD,t To [ β (na nb) + (β na) 2 (β nb) 2]. β: the velocity of the satellite in the CMB rest frame (expressed as a fraction of the speed of light c) To: the CMB monopole temperature (2.725 K)

6 Update map for next iteration: nsky,p A = nsky,p A + wt nsky,p B = nsky,p B + wt TCMB,t + (1 xim) TCMB,p TCMB,p A,i+1 = TCMB,p A,i+1 + B,i wt 1 + xim TCMB,t + (1 + xim) TCMB,p TCMB,p B,i+1 = TCMB,p B,i+1 + A,i wt 1 xim xim: tabulated loss-imbalance parameter ( 10 2 ) wt: statistical weight (we assume wt = 1). After the sequential read is complete, replace T CMB,i with T CMB,i+1 /n sky, and repeat. Convergence achieved in 20 iterations. Do not update the running sums for: either pixel if the data are flagged as bad; either pixel if a planet lies within θcut of n A or n B ; or pixel A (or B) if n B (or n A ) points to the Galactic plane..

7 OUR MAPS Q-BAND V-BAND W-BAND

8 WMAP MAPS (BENNETT ET AL. 2003a) Q-BAND V-BAND W-BAND

9 ASYMMETRY: DEFINITION We compute asymmetry in summed power between arbitrarily defined opposite hemispheres by: Subtracting foregrounds (see Bennett et al. 2003b). Combining the V- and W-band maps. Applying the Kp2 galactic mask. Computing the cut-sky pseudo-c l power spectra (e.g., MASTER algorithm of Hivon et al. 2002). Computing the ratio of summed power: Aobs(lgal, bgal) = l max l min l(l + 1)C l north l. max l min l(l + 1)C l south (l gal, b gal ): galactic longitude and latitude l: spherical harmonic multipole index C l : pseudo-c l power at multipole l for given hemisphere

10 ASYMMETRY: SIGNIFICANCE Directional asymmetry follows a lognormal distribution: Aobs(lgal, bgal) f(a, µ, σ) = [ ( 1 ln(a µ) exp 2π(A µ)σ 2σ ) 2 ]. We use fits of lognormal distributions to asymmetry observed in simulated data to estimate the directional significance: A obs αdir = min[ daf(a, µo, σo), 0 A obs daf(a, µo, σo)]. The global significance is not easily computed: values of α dir are correlated. (The Bonferroni correction multiplying the maximum observed significance α dir min by the number of directions examined is too conservative.) We find log(α dir min ) to follow an extreme-value distribution: log(α min dir ) dir ) 1 σ exp [ µ log(α min σ exp ( µ log(α min dir ) )] σ Similar to above, we use fits to estimate the global significance: [ ( µ o log(α dir min αglobal = 1 exp exp ) )]. σo.

11 ASYMMETRY: RESULTS Direction of maximum asymmetry in galactic coordinates (top two panels), the significance α dir (third panel), and the global significance α global (bottom panel), as a function of l. The horizontal dotted line in the top two panels indicates the position of the ecliptic NP, while the dot-dashed line in the bottom panel marks α global = The additional dotted lines in the bottom panel indicate the values of α global if we combine (nearly) contiguous significant ranges.

12 ASYMMETRY: RESULTS Over the multipole range l = [2,64]: α dir,min obs = towards (l gal, b gal ) = (72,9 ) α global = Clear evidence of power asymmetry! Similar results hold for other combinations of data (adding the Q-band data, looking at single bands, etc.): asymmetry is frequency-independent! Intriguing results at large scales: Within the ranges l = [2,3] and [6,7], α global 0. We find the contours of least directional significance to be oriented along great circles inclined relative to the ecliptic (Sun-Earth) plane!

13 Contour plots showing directional significance of asymmetry as a function of NP location in (l gal, b gal ), for four significant multipole ranges. The direction of maximum directional significance is labeled with a triangle, the cross and solid line indicate the ecliptic NP and plane, respectively, and the asterisk and dashed line indicate the supergalactic NP and plane, respectively. All panels exhibit contours of least significance that appear to follow great circles. The dotted and dashdotted lines represent great circles inclined relative to the ecliptic and galactic planes, respectively. The lines are placed by eye lgal lgal b gal l=[8,10] l=[11,14] b gal l=[2,3] l=[6,7] ASYMMETRY: RESULTS

14 ASYMMETRY: EFFECT OF MAP-MAKING Algorithmic aspects that are too complex to address here: Assuming Gaussian noise and least-squares methodology Assuming diagonal dominance of (M T M) 1 Algorithmic aspects that are not currently addressable: The WMAP data calibration algorithm The statistical weights for each datum Algorithmic aspects that we examine: The assumption that To = K The assumption that β = ( 0.88, 8.13,9.16) 10 4 The effect of loss-imbalance parameters The effect of galactic masking The interpolation scheme used to estimate (n A, n B ) The effect of changing the planetary cut (θcut = 1.5 ) The effect of changing foreground map normalization

15 ASYMMETRY & MAP-MAKING: RESULTS Reasonable changes to the map-making parameters have negligible effect, except that... Increasing the magnitude of the dipole velocity by 1-3 σ ( 2-6 km/s) markedly reduces asymmetry at large scales. Note that the test dipole velocities differ from that used to calibrate the data, which affects quantitative interpretation. Plots of α global as a function of shifted dipole velocity (1 σ 1.87 km/s), for the multipole ranges displayed in each panel. The dot-dashed line in each panel represents α global = 0.05.

16 CONCLUSIONS We produce our own CMB sky maps using the first-year WMAP data. We analyze these maps and find: Significant (α global 10 4 ), frequency-independent power asymmetry between arbitrarily defined hemispheres. At large scales (l 16), we can fit contours of least significance with great circles inclined to the ecliptic plane. Altering the map-making algorithm has negligible effect, except......that increasing the dipole velocity magnitude (a frequency-independent change) markedly reduces α global at the largest scales (l 8). We conclude that the use of an incorrect dipole vector, in combination with a systematic or foreground process associated with the ecliptic, may help explain the observed power asymmetry.

17 ACKNOWLEDGMENTS Our parallelized map-making software was run on the NCSA/Teragrid Linux Cluster (grant number MCA04N009, PI Roy Williams), supported by NSF under the following programs: Partnerships for Advanced Computational Infrastructure, Distributed Terascale Facility (DTF) and Terascale Extensions: Enhancements to the Extensible Terascale Facility. This work was also supported by NSF grants ACI , Statistical Data Mining in Cosmology, and AST , Nonparametrical Statistical Methods for Astrophysical and Cosmological Data. REFERENCES Bennett, C. L., et al. 2003a, ApJS, 148, 1 Bennett, C. L., et al. 2003b, ApJS, 148, 97 Freeman, P. E., Genovese, C. R.,Miller, C. J., Nichol, R. C., & Wasserman, L. 2005, ApJ, in press (astro-ph/ ) Hinshaw, G., et al. 2003a, ApJS, 148, 63 Hivon, E., Górski, K. M., Netterfield, C. B., Crill, B. P., Prunet, S., & Hansen, F. 2002, ApJ, 567, 2