Secondary non-Gaussianity and cross-correlation analysis

Dipak Munshi 1 2 Alan Heavens 2 Asantha Cooray 0 Patrick Valageas 3 0 Department of Physics and Astronomy, University of California , Irvine, CA 92697 , USA 1 http://www.rssd.esa.int/index.php?project 2 http://map.gsfc.nasa.gov/ 3 Institut de Physique The orique, CEA Saclay 91191 , Gif-sur-Yvette , France A B S T R A C T We develop optimized estimators of two sorts of power spectra for fields defined on the sky, in the presence of partial sky coverage. The first is the cross-power spectrum of two fields on the sky; the second is the skew spectrum of three fields. The cross-power spectrum of the cosmic microwave background (CMB) sky with tracers of large-scale structure is useful as it provides valuable information on cosmological parameters. Numerous recent studies have proved the usefulness of cross-correlating the CMB sky with external data sets, which probes the integrated Sachs-Wolfe (ISW) effect at large angular scales and the SunyaevZel'dovich (SZ) effect from hot gas in clusters at small angular scales. The skew spectrum, recently introduced by Munshi & Heavens, is a powerful statistic, as it is optimized to study particular forms of non-Gaussianity, such as may arise in the early Universe, but in addition, it retains information on the nature of non-Gaussianity. As such, it allows a robust statistical analysis, where contributions from primordial and contaminating non-Gaussianity can be estimated. In this paper we develop the mathematical formalism for the skew spectrum of three different fields. When applied to the CMB, this allows us to explore the contamination of the skew spectrum by secondary sources of CMB fluctuations, in the case where the foreground contamination and the primary signal are not independent. After developing the analytical model we use them to study specific cases of cosmological interest which include cross-correlating the CMB with various large-scale tracers to probe the ISW and SZ effects for cross-spectral analysis. Next we use the formalism to study the signal-to-noise ratio for the detection of the weak lensing of the CMB by cross-correlating it with different tracers, as well as point sources for CMB experiments such as Planck. 1 I N T R O D U C T I O N 2 G E N E R A L I Z E D P S E U D O - C E S T I M AT O R F O R C R O S S - C O R R E L AT I O N A N A LY S I S : G E N E R A L I Z AT I O N T O A R B I T R A R Y D ATA S E T S 2.1 Estimator for ClXY alXm,Y = alXm,Y = X,Y ( i )wX,Y ( i ) pX,Y ( i )Ylm( i ). X,Y ( )Ylm() alXm,Y KlXm,lYm . Kl1m1l2m2 = w()Y l1m1 ()Y l2m2 ()d (1)m1 w l3m3 2.2 Covariances of pseudo-C values Cl Cl Cl M Here we have introduced the following notations: [wXwX]lm = [wXwY ]lm = d [wX()] 2Ylm(), d [wX()][w Y ()]Y lm(), Ylm(). Re [wY wY ]LM (w2 2)LXMX Re [wXwX]LM (w2 2)YLMY Re (wY wY )LM (w2 2)LXMX [MXY ]Ll1 C lXY C lXY [MXY ]L1l . S/N = fsky Cl Cl 3.1 Estimator for ClXY,Z 3.1.1 All-sky analysis alXmY = Y ()Y lm(); d Z( )Ylm(). Assuming homogeneity and isotropy, the correlation function CXY ,Z(, ) of XY ,Z: Cl CXY ,Z(, ) Z( ) = Z 1 alX1 mY1 al2m2 Yl1m1 ( )Yl2m2 ( ) = 4 (2l + 1)Pl [cos( )]C lXY ,Z. Y ()and the field Z(). In the absence alZm is a standard: Z() can be written in terms of alXmY alZm X Y dY l1m1 ()Y l2m2 ()Y lAmA ()Y lm(), dY l3m3 ()Y lB mB ()Y lm( ). 3.1.2 Partial sky coverage On further simplification using spherical decomposition of individual fields, we can express the composite harmonic coefficient alXmY in terms of the individual ones: X Y Z Our aim is to compute the cross-correlation power spectra of the product field we can relate the multipoles alXmY with multipoles alX1m1 and alY2m2 : alXmY = dY lm() Y () = X Y dY lm()Y l1m1 ()Y l2m2 ( ) Z (). Using a harmonic decomposition Y ()w A()Y lm()d ; Z( )wB ()Y lm()d . Mll ClXY ,Z. simulations in an accompanying paper. This analysis is complementary to the work by Chen & Szapudi (2006) where a similar suboptimal estimator was used to study non-Gaussianity. 4 C M B S E C O N D A R Y B I S P E C T R U M : R E V I E W O F E X I S T I N G M O D E L S Expanding the respective terms in spherical harmonics, we can write alm () Ylm(); BlP1l,2Ll3,S (T P)l1m1 (TL)l2m2 (TS)l3m3 = BlP1l,2Ll3,S = bl1l2l3 Il1l2l3 , Il1l2l3 5 E S T I M AT I O N O F S K E W S P E C T R A : G E N E R A L I Z AT I O N T O T H E C A S E O F M I X E D S E C O N D A R Y S K E W S P E C T R A 1 S Bl(m2) = alm , Cl Cl 1 AB,C (2l + 1)Cl2,1 = bl1l2l3 = dr[fl1 (r)gl2 (r) + cyc.perm.]. Real (AB)lmClm + cyc.perm. = m l1l2 and from this compute the skew-spectrum by carrying out the line-of-sight integration: 5.3 General expression 1 Nfact ali1 bli2 cli3 + cyc.perm. Cl 5.4 Cross-contamination from point sources and primary non-Gaussianity S = S = 6.1 One-point estimator: mixed skewness Q[X , Y , Z ] Q [X , Y , Z ] alXmalYm alZm , Q[xi ] E[xi ] = [C XX]l1m1,l4m4 [C Y Y ]l2m2,l5m5 [C ZZ]l3m3,l6m6 + cyc.perm. . 1 1 ! ClX1 X ClY2Y ClZ3Z l1l4 l2l5 l3l6 + cyc.perm. . XX XXl1l2Xl3X , 6.2 Two-point estimators: mixed skew spectrum 1 QL[X , Y , Z ] 6 BLXlYlZ 1 lXmQ L[Y , Z ] Ll 6 1 lYmQ L[X , Z ] 6 BLXlYl Z BLXlYl Z E LX,Y Z[xi ] = [F 1]LXL,Y Z QL [xi ] [xj ]lm ljmQL [xi ] [F ]LL = l1m1 QiL[x ] Cl1 m11,l2m2 li1m1 QL[x ] Cl1 m11,l2m2 [C XX]LM,L M [C Y Y ]l1m1,l1m1 [C ZZ]l1 m1 ,l2 m2 + [C XY ]LM,l2m2 [C Y Z]l1m1,l2 m2 [C ZX]l1 m1 ,L M + [C XZ]LM,l2m2 [C ZY ]l1m1,l2m2 [C Y X]l1m1,L M + [C XX]LM,L M [C ZY ]l1m1,l2m2 [C Y Z]l1m1,l2m2 + [C Y Y ]l1m1,l2m2 [C XZ]LM,l2m2 [C ZX]l1m1,L M + [C ZZ]l1m1,l2m2 [C XY ]LM,l2m2 [C Y X]l1m1,L M CLXY BLXlYl ZBLXlYl Z CLX1X Cl1YY Cl1ZZ + BLXlYl ZBLXlYlZ CLX1X + BLXLY ZlBLX YlLZ [xi ]lm lmQL [xi ] MC) . i=1,2,3 6.2.1 Special case (A): Z() = The estimator in this case corresponds to ElX,Y 2 . It represents the cross-correlation of a squared field Y () 2 against another field 1 1 1 ! BLXlYl Y BLXlYl Y CLXX ClY Y ClY Y Cross-terms involving CLXY contribute to the off-diagonal elements of the Fisher matrix. In case [ClY X]2 ClXXClY Y these terms can be ignored. The expression simplifies a lot if we simply consider only the first term, which depends on the mixed bispectra and the power spectra of individual data sets. 6.2.2 Special case (B): Z() = Y () = BLXlXlXBLXlXlX BLXLXXlBLXLXXl 6.2.3 Special case (C): Z() = 1 lYmQ L[X , Y ] 6 1 & lXmQ L[X , Y ] 6 [F 1]LL QL [xi ] [xi ]lm limQL [xi ] MC) . 7.1 Lensing reconstruction 7.1.1 One-point estimator CL 7 S P E C I F I C E X A M P L E S The discussion so far has been completely general. We specialize now for a few practical cases of cosmological importance. These correspond to the study of mixed bispectra associated with lensing-induced correlation of secondaries and CMB as well as frequency-cleaned SZ catalogues against the CMB sky. Slens l1m1 l2m2 l3m3 [C]1 l1m1,l2m2 l3m3 . fl1l2l3 Cl2 + fl2l1l3 Cl1 [C]l11m1,l4m4 [C]l21m2,l5m5 [C ]l31m3,l6m6 . 7.1.2 Estimators for the skew spectrum Instea (...truncated)