Let us now find out how much subtracting the correlated temperature map, Tcorrrec, from the original CMB temperature map would improve ISW measurements with currently available CMB and LSS data. In chapter 4, we will compute Tcorrrec and Tuncorrrec together with the covariance of the correlated temperature map, Ccorr, for the WMAP data, and we will already use these results here. We transform the covariance matrix Ccorr to multipole space, which gives us the non-diagonal matrix Ccorrl,m,l′,m′ since our problem is not isotropic. This is due to the inhomogeneous noise in the polarization map and the fact that we we have to mask out the Galactic plane in the polarization data (cf. chapter 4). However, we only consider the diagonal, Clcorr,m,l,m, and take the average over all multipole components m for every fixed l:
Ccorrll ≡ 1
2l+1
X
m
3.6 Improvement for currently available data 77 1 1.05 1.1 1.15 1.2 1.25 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 rel S / N zmax (S / N)polav / (S / N)st (S / N)polav / (S / N)tav 0 1 2 3 4 5 6 7 8 9 S / N (S / N)pol av (S / N)tav (S / N)st
Figure 3.9: Comparison of the signal-to-noise ratios versus the maximal redshift zmax of the galaxy
survey. Top panel: Average signal-to-noise ratio of the optimal polarization method (S/N)avpol (solid), of the optimal temperature method (S/N)avτ (dashed) and signal-to-noise ratio of the standard method (S/N)cc (dotted). Bottom panel: Ratio of the signal-to-noise of the optimal polarization method with
the one of the standard method (dotted) and with the one of the optimal temperature method (dashed). We see that with polarization data included, the signal-to-noise is significantly enhanced even for low redshifts.
In the upper panel of Fig. 3.10, we plot this ‘averaged power’ of the observed Trec
corr together with the theoretical power spectrum of the correlated temperature map, CTcorrrec
l ≈(C T,Ecmb
l )
2/CEcmb
l , and the power in the CMB temperature fluctuations, CTl . The theoretical power spectrum, (CTl,Ecmb)2/ClEcmb, is the upper limit of the reduction of the variance that we can obtain per multipole (cf. Fig. 3.4). The ratio Cllcorr/[(ClT,Ecmb)2/ClEcmb] is shown in the bottom panel of Fig. 3.10. For WMAP, this ratio is around 50 per cent for the lowest multipoles and falls offto a value below 1 per cent above l≈10.
We now compute the signal-to-noise ratio we would obtain with the WMAP data for our optimal polarization method and compare it to the one of the standard method. We do not create an ISW template from LSS surveys to compute the correct signal-to-noise ratio, but we rather perform a crude estimate in multipole space. To this end, we estimate the signal-to-noise ratio of the optimal polarization method for WMAP data, (S/N)avpol,W MAP, by simply substituting the term (Clprim,Ecmb)2/ClEcmb in eq. (3.85) by Cllcorr. We plot the ratio of (S/N)avpol,W MAP to the signal-to-noise ratio of the standard method, (S/N)cc, for a given depth of the LSS survey in Fig. 3.11. This is analogous to what is plotted in the bottom panel of Fig. 3.9. Note that we have again neglected the shot-noise in the galaxy distribution. We see that for galaxy surveys with zmax . 0.6, we do not gain more than 5 per cent in detection significance for WMAP. With currently available data, our optimal polarization method and the standard method for ISW detection thus still yield approximately the same detection significance. Given that the standard method has been applied to all currently available LSS data sets (see Ho et al. 2008; Giannantonio et al. 2008, and references therein), we decide not to proceed and apply our method to WMAP data.
78 Optimal methods for detecting the integrated Sachs-Wolfe effect 0.1 1 10 100 1000 10000 0 5 10 15 20 25 l ( l +1)/2 π C l [ µ K 2 ] l ClT (ClTE)2 / ClE Cllcorr 0.001 0.01 0.1 1 0 5 10 15 20 25
ratio to theoretical power
l
Cllcorr / ((ClTE)2 / ClE)
Figure 3.10: Reduction of the variance of ISW detection from including polarization data for WMAP and for 2 limiting estimates for Planck, which are explained in the text. Top panel: ClT (dotted), the theoretical reduction of variance per multipole, (CT,Ecmb
l )
2/CEcmb
l (dashed), and the variance reduction
achieved by WMAP/Planck Cllcorr(solid). The three cases shown for Cllcorrare: The estimate for Planck with the WMAP noise covariance scaled down by a factor of 1 per cent (upper thin line), the same for a scale-factor of 10 per cent (middle line), and the estimate for WMAP (bottom thick line). Bottom
panel: The ratio Ccorrll /((CT,Ecmb
l )
2/CEcmb
l ), which is roughly the ratio of the variance reduction achieved
by WMAP/Planck over the theoretically achievable reduction of variance per multipole. The three cases
3.6 Improvement for currently available data 79 1 1.05 1.1 1.15 1.2 1.25 1.3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 rel S / N zmax
Planck, noise scale factor = 1%
Planck, noise scale factor = 10%
WMAP
Figure 3.11: Ratio of the signal-to-noise ratio of the optimal polarization method to the one of the standard method, for WMAP (dotted line) and for the two limiting estimates for Planck as explained in the text: WMAP noise covariance scaled down by a factor of 10 per cent (dashed line,α=0.7) and by a factor of 1 per cent (solid line,α= 0.9). For Planck, we will obtain an enhancement of the detection significance of at least 10 per cent, even for the more conservative estimate.
However, the Planck satellite (Tauber 2000), which was launched in May 2009, will provide us with much more accurate polarization measurements than WMAP. Let us therefore do a similar rough estimate of the potential improvement of ISW detection with Planck data. We first need to find a way to estimate Cllcorr for Planck, which we attempt to do based on the polarization noise covariance matrix of WMAP. For Planck, the Galactic foregrounds will certainly be the limiting factor for the quality of the polarization data. We thus assume the detector noise to be negligible in comparison to the residual foregrounds. We can furthermore expect the foreground removal for
Planck data to be more accurate than the one for WMAP, due to the broader range of frequencies
covered by Planck. In the following, we therefore assume the covariance due to residual foregrounds for Planck to be between 5 and 50 per cent of the one for WMAP. For the WMAP polarization data, the foregrounds contribute about 20 per cent to the diagonal of the noise covariance matrix NP in pixel space. In order to obtain a noise estimate for the two limiting cases of the Planck foreground removal, we do not change the shape of the total noise covariance, NP, but simply scale it down with a factor of 1 per cent or 10 per cent, respectively. This will only give us a very crude estimate, of course, since the covariance matrix of the residual foregrounds should differ strongly from the detector-noise dominated covariance of WMAP. In particular, the contribution of the foregrounds to the total noise covariance matrix of WMAP should depend on the multipole, which we ignore by just downscaling the total noise covariance. Nevertheless, for the very rough estimate we are trying to obtain, this assumption should be good enough. We show the resulting power spectra and their ratio in Fig. 3.10. The ratio Ccorrll /[(CTl,Ecmb)2/ClEcmb] for the lowest multipoles is around 0.7 for the WMAP noise covariance scaled down by a factor of 10 per cent, and around 0.9 for the noise covariance scaled down by a factor of 1 per cent. For the higher multipoles, this ratio falls offquite rapidly, which is simply due to the fact that the WMAP polarization data contain so little information at the higher multipoles. For Planck, we assume that the quality of the polarization data
80 Optimal methods for detecting the integrated Sachs-Wolfe effect
does not notably drop until l≈100. That is, we assume a constant ratio
α≡Ccorrll /[(ClT,Ecmb)2/ClEcmb], (3.88) and we take α to be 0.7 and 0.9, for the two limiting cases described above. For these two cases, we again compute the signal-to-noise ratio of the optimal polarization method by substituting (Clprim,Ecmb)2/ClEcmb in eq. (3.85) byα(Cpriml ,Ecmb)2/CEcmbl , and plot its ratio to the signal-to-noise ratio of the standard method in Fig. 3.11. The improvement we obtain is already around 10 per cent for low redshift surveys of zmax ≈ 0.3 for the more conservative estimate. We can thus expect that the improvement of the ISW detection significance for the optimal polarization method will be at least 10 per cent for Planck, even with currently available LSS surveys.