Draft version February 3, 2008
A Preprint typeset using L TEX style emulateapj v. 10/10/03
CROSS-CORRELATION ANALYSIS OF WMAP AND EGRET IN WAVELET SPACE
Xin Liu1 and Shuang Nan Zhang1,2,3,4
Draft version February 3, 2008
arXiv:astro-ph/0511550v1 18 Nov 2005
ABSTRACT We cross correlate the Wilkinson Microwave Anisotropy Probe (WMAP) ?rst year data and the di?use gamma-ray intensity maps from the Energetic Gamma Ray Experiment Telescope (EGRET) using spherical wavelet approaches. Correlations at 99.7% signi?cance level have been detected, at scales around 15? in the WMAP foreground cleaned W-band and Q-band maps, based on data from regions that are outside the most conservative WMAP foreground mask; no signi?cant correlation is found with the Tegmark cleaned map. The detected correlation is most likely of Galactic origin, and thus can help us probing the origins of possible Galactic foreground residuals and ultimately removing them from measured microwave sky maps. Subject headings: cosmic microwave background — cosmic rays — di?use radiation — methods: data analysis
The study of the anisotropies of the cosmic microwave background (CMB) is a powerful tool in cosmology. Results from the Wilkinson Microwave Anisotropy Probe (WMAP) provide us with the angular power spectrum and its cosmological implications (Bennett et al. 2003a; Spergel et al. 2003; Page et al. 2003). However unwanted signals due to foregrounds would contaminate any intrinsic signals, most importantly on large angular scales (Tegmark & Efstathiou 1996; de Oliveira-Costa et al. 2003). The WMAP team chose the CMB-dominated bands (two Q-band maps at 40.7 GHz, two V-band maps at 60.8 GHz and four W-band maps at 93.5 GHz) and combined them to give a signal-to-noise ratio enhanced map. Despite that the maps at selected frequencies are dominated by CMB, Galactic foregrounds as well as extragalactic point sources all contribute signi?cantly to the map, where the WMAP team performed a foreground template ?t (thermal dust from Finkbeiner et al. 1999; freefree from Finkbeiner 2003 and Schlegel et al. 1998; synchrotron from Haslam et al. 1982) to avoid the Galactic emissions and separated them from the underlying CMB signal according to the frequency-dependence property of foregrounds (Bennett et al. 2003b). Since the foreground ?uctuations depend on the multipole moment l as well (Bouchet et al. 1995), l-dependent statistical weights have been applied in Tegmark et al. (2003) where an independent foreground analysis was made. Several works of WMAP non-Gaussianity detection found residuals or the systematic e?ect correction as the sources (Chiang et al. 2003; Chiang & Naselsky 2004; Naselsky, Doroshkevich & Verkhodanov 2003, 2004; Eriksen et al. 2004;
1 Physics Department and Tsinghua Center for Astrophysics, Tsinghua University, Beijing 100084, China; email@example.com, firstname.lastname@example.org. 2 Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, P.O. Box 918-3, Beijing 100039, China 3 Physics Department, University of Alabama in Huntsville, Huntsville, AL 35899, USA 4 Space Science Laboratory, NASA Marshall Space Flight Center, SD50, Huntsville, AL 35812, USA
Hansen et al. 2004; Liu & Zhang 2005). Wibig and Wolfendale (2005) have shown evidence that the Tegmark cleaned map contains residual foregrounds possibly induced by cosmic rays, where the Energetic Gamma Ray Experiment Telescope (EGRET) diffuse gamma-ray intensity map (Hunter et al. 1997) was adopted as the Galactic tracer. Di?use Galactic gammaray emission is supposedly produced in interactions of Galactic cosmic rays with the interstellar gas and radiation ?eld, and thus provides us with an indirect measurement of cosmic rays in various locations in the Galaxy 5 In this paper we cross-correlate the WMAP ?rst-year data and the EGRET maps, which are based on more data and more complete point-source subtraction (Cillis & Hartman 2005), in wavelet space to probe the origins of potential residual foregrounds with characteristic scales. Both the WMAP combined foreground cleaned maps (Bennett et al. 2003b) and the Tegmark cleaned map (Tegmark et al. 2003) have been used.
2. CROSS-CORRELATION IN WAVELET SPACE
A measure of the correspondence of two data sets on the sphere is the angular cross-correlation function CCF (θ), which represents how the two measures of the sky separated by an angle θ are correlated. Previous works have performed the cross correlation of the CMB data and the nearby galaxy density tracers in search for the Integrated Sachs-Wolfe e?ect (ISW) (Boughn & Crittenden 2002, 2004; Nolta et al. 2004; Fosalba & Gazta? aga 2004; Fosalba et al. 2004; Afshordi et al. n 2004; Vielva et al. 2004b). As being performed in Vielva et al. (2004b), cross-correlation studies can also be made in wavelet space. Wavelet approach is very useful for detecting signals with a characteristic scale that a most optimal detection can be made by ?ltering the data at a given scale, thus structures at that scale are ampli?ed. It has been adopted in the CMB-related analysis for non-Gaussianity studies (Hobson et al. 1999; Barreiro et al. 2000; Aghanim et al. 2003; Cay?n et al. 2001, 2003; Mart? o ?nez5 See Bertsch et al. 1993 and Hunter et al. 1997 for threedimensional modeling.
X. Liu and S. N. Zhang
Gonz?lez et al. 2002; Mukherjee & Wang, 2004; Vielva a et al. 2004a; McEwen et al. 2004; Liu & Zhang 2005). In wavelet space, the cross-correlation covariance at a given scale a is de?ned as (Vielva et al. 2004b): CovW ?E (a) = 1 Na ωCM B (a, p)ωEGRET (a, p),
where ωCM B (a, p) and ωEGRET (a, p) are the wavelet coef?cients of the WMAP and EGRET data at position p on the sky map, and the sum p is extended over all the pixels (Na ) that are not masked by the Galactic mask “Kp0”. Equation (1) gives the auto-correlation covariance (ACC) when the two data sets are the same. To make the analysis more robust and less sensitive to any discrepancies between CMB data and simulations, we use the dimensionless and normalized cross-correlation coef?cient CW ?E (θ) as the test statistics, which is given by 2 CW ?E (θ) = CovW ?E (θ)/(σW σE ), where σW = ACCW 2 and σE = ACCE are the WMAP and EGRET autocorrelation covariances respectively. The wavelet coe?cients are obtained by convolving the map with a certain spherical wavelet basis at a given scale: ωD (a, p) = d?′ D(p + p′ )ΨS (θ′ , a), (2)
where ΨS (θ′ , a) is the spherical wavelet basis and D(p + p′ ) is the data set to be analyzed. The spherical wavelet can be obtained from the Euclidean Wavelet counterpart following the stereographic projection suggested by Antoine & Vanderheynst (1998). Mart? ?nez-Gonz?lez et a al. (2002) has described the projection for the Spherical Mexican Hat Wavelet (SMHW) as well as its properties, whereas the Spherical Morlet Wavelet (SMW) has been applied for non-Gaussianity detection in the WMAP data (McEwen et al. 2004, Liu & Zhang 2005). In this correlation study we have adopted both wavelets, of which the SMW seems to be more sensitive than the SMHW. We only present results from the SMW analysis here, even though similar correlations are also found when applying the SMHW. The preprocessing pipelines for the WMAP CMB data and Monte Carlo simulations are the same as in Liu & Zhang (2005), whereas the gamma-ray intensity maps are degraded to the same resolution with the CMB data to be cross-correlated, all in equi-angular spherical grid pixalisation. The preprocessed maps are shown in Fig. 1. 10,000 simulations of the WMAP data have been applied, following the cosmological model given by the Table 1 of Spergel et al. (2003), to obtain the signi?cance levels of any correlation detection. We obtain these levels in a robust way by taking into account the probability distribution of the wavelet cross-correlation coe?cient at each scale. Although the map-making algorithm presented by Cillis and Hartman (2005) possibly produced some systematic e?ects in the EGRET data, these can be calibrated out by simulations.
Fig. 1.— Preprocessed EGRET di?use gamma-ray intensity map (> 1 GeV) (Cillis and Hartman 2005) and CMB anisotropy maps from the WMAP map (Bennett et al. 2003a) and the Tegmark cleaned map (Tegmark et al. 2003) to be analyzed in the crosscorrelation study. Note that we have analyzed the EGRET di?use gamma-ray intensity maps of di?erent energies separately, i.e., 30? 100 MeV, 100 ? 300 MeV, 300 ? 1000 MeV and > 1 GeV, where only the > 1 GeV map is shown here. All maps are plotted in Galactic coordinates with the Galactic center (l, b) = (0, 0) in the middle and Galactic longitude l increasing to the left.
Cross correlation with the EGRET maps in wavelet space has been performed to both of the Q-V-W combined WMAP map and the Tegmark cleaned map to
detect foreground residuals (here and throughout, all WMAP CMB maps used are foreground removed maps). Also we have performed the analysis to: (a) the WMAP map in each band separately, i.e. Q, V, and W; (b) systematic beam and noise maps from subtracting maps of the receivers at the same frequency; and (c) a foreground map almost free of CMB, which is made by subtracting the two receivers of Q band and the two receivers of V band from the four receivers of W band (Vielva et al. 2004a). For every CMB related map, we have performed the analysis using EGRET maps in di?erent energy ranges separately, i.e. 30 ? 100 MeV, 100 ? 300 MeV, 300 ? 1000 MeV and > 1 GeV, and only the results from the > 1
WMAP and EGRET Cross-Correlation GeV map are shown here because: (a) it has been stated by Cillis & Hartman (2005) that the > 1 GeV intensity map has the best statistical accuracy compared with others from lower energy ranges, where possible correlations can be smeared out by statistical uncertainty ?uctuations; and (b) the analysis with di?erent gamma-ray energy ranges stated above generally give similar correlations and the signi?cance level seems to increase a little with the energy of the adopted gamma-ray data. Note this variance of correlation signi?cance does not con?ict with the power-law energy spectrum, since the cross-correlation coe?cient is a normalized quantity. The cross-correlation coe?cients CW ?E (θ) are illustrated in Figs. 2(a) and 2(b), where the corresponding size θ on the sky can be obtained from equation (3) in Liu & Zhang (2005); we have analyzed scales from 1? to 90? , where we are only concerned about results at scales > 4? due to the angular resolution limit in the EGRET data. We have shown the ACCs (normalized by the map dispersion σ0 in the real space) of the WMAP combined map, the Tegmark cleaned map, the foreground-component map and the di?use gamma-ray intensity map in Fig. 2(c), which indicate that the general patterns of CW ?E (θ) in Figs. 2(a) and 2(b) are caused by the convolution of the gamma-ray data and the wavelets, not by systematical artifacts. Note at all the concerned scales the foreground map correlates more signi?cantly with the EGRET map, consistent with the assumption that most of the foregrounds have been removed from the CMB maps. Results from analyzing the systematic beam and noise maps are not shown since their correlation coe?cients are all consistent with zero values within statistical ?uctuations. For the WMAP CMB maps, correlations at 99.7% signi?cant level are detected at scales from about 14? to 16? , in the W-band and Q-band maps; less signi?cant correlations (96.8%) are found at larger scales, from about 43? to 48? , only in the W-band map. In sum, the W-band and Q-band maps exhibit more signi?cant correlations compared with those from the V-band map, at all the concerned scales. This frequency dependence feature can be easily understood according to Fig. 10(a) in Bennett et al. (2003b), showing evidence that the detected correlations are caused by residual foreground signals. In order to test whether the detected signal has a Galactic origin, we have shown in Fig. 2(d) the cross-correlation coe?cients as a function of the galactic latitude around θ = 15? . Note that the sum of the cross-correlation coe?cients at all the latitudes in Fig. 2(d) corresponds to the CW ?E (15? ) in Figs. 2(a). Here the oscillating pattern is given by the SMW, and we have tested that the outline pro?le is caused by the EGRET intensity maps. Tests at other scales also present similar pro?les. The results show some consistency with the co-sec law for the Galactic components, and also present a north-south asymmetry where the correlation is stronger in the southern hemisphere. Analysis of the Tegmark cleaned map does not show any signi?cant correlation at all the concerned scales, as an evidence that it is “cleaner” than the WMAP combined map.
4. CONCLUSIONS AND DISCUSSIONS
0.8 0.6 CW?E(θ) 0.4 0.2 0 ?0.2 (a) 10
WMAP combined WMAP Q WMAP V WMAP W Tegmark foreground
15 θ (degrees)
(b), same legends as in (a) ?1 40 50 60 θ (degrees) 70 80
0.8 ACC(θ)/ σ0 × 103
WMAP combined Tegmark foreground EGRET × 102
0.2 (c) 0 20 40 60 θ (degrees) 80
x 10 4.5 4 3.5 3
WMAP combined WMAP Q WMAP V WMAP W Tegmark
2.5 2 1.5 1 0.5 0 ?0.5 90
(d), θ=15 degrees
60 30 0 ?30 ?60 ?90
Galactic Latitude b (degrees)
Fig. 2.— (a) and (b): cross-correlation coe?cients CW ?E (θ) of the WMAP and EGRET data in wavelet space. The acceptance intervals at the 68% (solid), 95% (dashed) and 99% (dashdotted) signi?cance levels given by 10,000 Monte Carlo simulations are shown, respectively. The error bars are too small compared to the coe?cient values and can be ignored in the ?gures; (c): normalized auto-correlation covariances, representing that the general patterns in (a) and (b) are caused by the convolution of the gammaray data and the wavelet basis, not by systematical artifacts; (d): cross-correlation coe?cient as a function of the galactic latitude around θ = 15? , testing the Galactic origin of the detected correlations.
X. Liu and S. N. Zhang the Galaxy. The detected foreground residuals can be located in the coe?cient map at a certain scale. It is therefore worthwhile to make correlation study with cosmic ray’s spatial and spectral distributions in detail. Note that the di?use gamma-ray emission has not been well understood that at energies > 1 GeV the observed intensity in inner Galaxy displays a GeV excess at a level of 60% compared with predictions (Hunter et al. 1997), which has been interpreted in several models (Strong et al. 2000; de Boer 2005). Further cross-correlation work must be done to fully understand the nature of foreground residuals and ?nally remove them from the CMB maps completely, in order to minimize the impacts of foreground residuals to the cosmological studies of CMB. Acknowledgement: We are very grateful to the anonymous referees whose extremely detailed and insightful comments and suggestions allowed us to clarify several issues and improve the readability of the paper. This research has made use of the Astrophysical Integrated Research Environment (AIRE) which is operated by the Center for Astrophysics, Tsinghua University. We acknowledge the use of LAMBDA, support for which is provided by the NASA O?ce of Space Science, the YAWtb (http://www.fyma.ucl.ac.be/projects/yawtb) toolbox developed by A. Coron, L. Jacques, A. Rivoldini and P. Vandergheynst, the software packages HEALPix (http://www.eso.org/science/healpix), developed by K.M. Gorski, E. F. Hivon, B. D. Wandelt, J. Banday, F. K. Hansen and M. Barthelmann., and CMBFAST (http://www.cmbfast.org), developed by U. Seljak and M. Zaldarriaga. This study is supported in part by the Special Funds for Major State Basic Research Projects and by the National Natural Science Foundation of China. SNZ also acknowledges partial funding support from the US NASA Long Term Space Astrophysics program.
We have performed cross-correlation analysis of the WMAP ?rst-year data with the EGRET di?use gammaray intensity maps in wavelet space, ?nding correlations in the WMAP foreground cleaned maps, based on regions that are outside the most conservative WMAP foreground mask. Analysis of the WMAP W-band and Qband maps exhibits correlation at 99.7% signi?cant level at scales around 15? in the sky, and at scales around 45? less signi?cant correlation (96.8%) is found only in the WMAP W-band map. The Tegmark cleaned map seems to be compatible with pure CMB simulations at all the concerned scales. These cross-correlation signals are not caused by systematic beams nor noise because: (a) the analysis of the systematic beam and noise map show almost zero correlation results; and (b) the correlations are detected at the scales where noise and beam e?ects can be ignored (Tegmark et al. 2003). We thus conclude that these cross-correlation signals are most probably caused by foreground residuals because: (a) correlations from the Q and W bands are more signi?cant than those from the V band, consistent with the frequency dependence nature of foreground; (b) the correlation coe?cients from the CMB maps present similar patterns with those from the foreground map, whereas random simulations do not show correlations at the the detected level; (c) the Tegmark cleaned map, which has been commonly believed to be cleaner than the WMAP combined map in terms of the foreground removal, shows no signi?cant correlation with the EGRET map in the analysis; and (d) the detected cross-correlation coe?cient as a function of the galactic latitude appears to be consistent with the co-sec law as an evidence for the Galactic origin. It is possible that these foreground residuals may be induced by cosmic rays, since the Galactic di?use gammaray emission is supposedly produced in interactions of cosmic rays with gas and ambient photon ?elds and thus can be an indicator of cosmic rays in various locations in
REFERENCES Afshordi, N., Loh, Y.-S., & Strauss, M. A. 2004, Phys. Rev. D, 69, 3524 Aghanim, N., Kunz, M., Castro, P. G., & Forni, O. 2003, A&A, 406, 797 Antoine, J.-P., & Vandergheynst, P. 1998, J. Math. Phys., 39, 3987 Barreiro, R. B., et al. 2000, MNRAS, 318, 475 Bennett, C. L., et al. 2003a, ApJS, 148, 1 Bennett, C. L., et al. 2003b, ApJS, 148, 97 Bertsch, D. L., et al. 1993, ApJ, 416, 587 Boughn, S. P., & Crittenden, R. G. 2002, Phys. Rev. Lett., 88, 21302 Boughn, S. P., & Crittenden, R. G. 2004, Nature, 427, 45 Bouchet, F., et al. 1995, Space Science Review, astro-ph/9410004 Cay?n, L., et al. 2001, MNRAS, 326, 1243 o Cay?n, L., et al. 2003, MNRAS, 339, 1189 o Chiang, L.-Y., Naselsky, P. D., Verkhodanov, O. V., & Way, M. J. 2003, ApJ, 590, L65 Chiang, L.-Y., & Naselsky, P. D. 2004, astro-ph/0407395 Cillis, A. N., & Hartman, R. C. 2005, ApJ, 621, 291 de Boer, W. 2005, New Astron.Rev., 49, 213 de Oliveira-Costa, A., et al. 2003, Phys. Rev. D, 68, 083003 Eriksen, H. K., Novikov, D. I., Lilie, P. B., Banday, A. J., Gorski, K. M. 2004, ApJ, 612, 64 Finkbeiner, D. P. 2003, ApJS, 146, 407 Finkbeiner, D. P., Davis, M., & Schlegel, D. J. 1999, ApJ, 524, 867 Fosalba, P., & Gazta?aga, E. 2004, MNRAS, 350, 37 n Fosalba, P., Gazta?aga, E., & Castander, F. 2004, ApJ, 597, L89 n Hansen, F. K., Cabella, P., Marinucci, D., & Vittorio, N. 2004, ApJ, 607, L67 Hartman, R. C., et al. 1999, ApJS, 123, 79 Haslam, C. G. T., Salter, C. J., Sto?el, H., & Wilson, W. E. 1982, A&AS, 47, 1 Hobson, M. P., et al. 1999, MNRAS, 309, 125 Hunter, S. D., et al. 1997, ApJ, 481, 205 Liu, X., & Zhang, S. N. 2005, ApJ, 633, 542 Mart? ?nez–Gonz?lez, E., et al. 2002, MNRAS, 336, 22 a McEwen, J. D., et al. 2005, MNRAS, 359, 1583 Mukherjee, P., & Wang, Yun 2004, ApJ, 613, 51 Naselsky, P. D., Doroshkevich, A., & Verkhodanov, O. V. 2004, MNRAS, 349, 695 Naselsky, P. D., Doroshkevich, A., & Verkhodanov, O. V. 2003, ApJ, 599, L53 Nolta, M. R., et al. 2004, ApJ, 608, 10 Page, L., et al. 2003, ApJS, 148, 233 Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, ApJ, 500, 525 Spergel, D. N., et al. 2003, ApJS, 148, 175 Tegmark, M., & Efstathiou, G. 1996, MNRAS, 281, 1297 Tegmark, M., de Oliveira-Costa, A., & Hamilton, A. J. S. 2003, Phys. Rev. D, 68, 123523 Vielva, P., et al. 2004a, ApJ, 609, 22 Vielva, P., Mart? ?nez–Gonz?lez, E., & Tucci, M. 2004b, a astro-ph/0408252 Wibig, T., & Wolfendale, A. W. 2005, MNRAS, 360, 236