Issue 
A&A
Volume 555, July 2013



Article Number  A82  
Number of page(s)  9  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201321433  
Published online  04 July 2013 
Optimal bispectrum estimator and simulations of the CMB lensingintegrated Sachs Wolfe nonGaussian signal
^{1}
Institut d’Astrophysique de Paris et Université Pierre et Marie Curie Paris
6,
98bis Bd. Arago,
75014
Paris,
France
email:
mangilli@iap.fr
^{2}
International Chair of Theoretical Cosmology, Lagrange Institute
(ILP), 98bis boulevard
Arago, 75014
Paris,
France
^{3}
INFN, Sezione di Padova and Dipartimento di Fisica e Astronomia G.
Galilei, Universitá degli Studi di Padova, via Marzolo 8, 35131
Padova,
Italy
Received:
7
March
2013
Accepted:
16
May
2013
We present the tools to optimally extract the lensingintegrated Sachs Wolfe (LISW) bispectrum signal from future cosmic microwave background (CMB) data. We implemented two different methods to simulate the nonGaussian CMB maps with the LISW signal: a nonperturbative method based on the FLINTS lensing code and the separable modeexpansion method. We implemented the Komatsu, Spergel, and Wandelt (KSW) optimal estimator analysis for the LISW bispectrum and tested it on the nonGaussian simulations for realistic CMB experimental settings with an inhomogeneous sky coverage. We show that the estimator approaches the CramerRao bound and that Wiener filtering the LISW simulations slightly improves the estimate of f_{NL}^{LISW} by ≤ 10%. For a realistic CMB experimental setting that accounts for anisotropic noise and masked sky, we show that the linear term of the estimator is highly correlated to the cubic term and it is necessary to recover the signal and the optimal error bars. We also show that the LISW bispectrum, if not correctly accounted for, yields an underestimation of the f_{NL}^{local} error bars of ≃ 4%. A joint analysis of the nonGaussian shapes and/or LISW template subtraction is needed to recover unbiased results of the primordial nonGaussian signal from ongoing and future CMB experiments.
Key words: cosmic background radiation
© ESO, 2013
1. Introduction
One of the most relevant mechanisms that can generate nonGaussianity from secondary cosmic microwave background (CMB) anisotropies is the coupling between weak lensing and the integrated Sachs Wolfe (ISW, Sachs & Wolfe 1967) and the Rees Sciama (RS) effects (Rees & Sciama 1968). This correlation gives in fact the leading contribution to the CMB secondary bispectrum with a blackbody frequency dependence (Goldberg & Spergel 1999; Verde & Spergel 2002; Giovi et al. 2005). Weak lensing of the CMB is caused by gradients in the matter gravitational potential that distorts the CMB photon geodesics. The ISW and the RS effects, on the other hand, are related to the time variation of the gravitational potential wells. The relevant mechanism is given by the late ISW, owing to the action of dark energy, which causes the decay of the gravitational potential wells as the Universe expands. Both the lensing and the ISW effect are therefore related to the matter gravitational potential and thus are correlated phenomena. This gives rise to a nonvanishing threepoint correlation function or, analogously, a nonvanishing bispectrum, its Fourier counterpart. The RS (also referred to as the nonlinear ISW) arises when the growth of structure in the evolving universe becomes nonlinear. Because it is a secondorder effect, the RS gives a smaller contribution to the signal than the ISW. The CMB bispectrum arising from the crosscorrelation between lensing and ISW/RS (from now on referred to as LISW) is expected to have a high signaltonoise ratio from ongoing and future CMB experiments so that it will be detectable in the near future with a high statistical significance (Verde & Spergel 2002; Giovi et al. 2005; Mangilli & Verde 2009; Lewis et al. 2011). A detection would open the possibility to exploit the cosmological information related to the latetime evolution encoded in the LISW signal. It is useful to stress that a significant detection of the LISW signal from ongoing CMB experiments such as Planck would be a powerful probe of dark energy from the CMB alone and would be a complementary probe of the latetime Universe with respect to the large scale structure and the CMB power spectrum analysis. Moreover, Mangilli & Verde (2009) and Hanson et al. (2009) showed that the LISW bispectrum can be a serious contaminant for estimating the primary local nonGaussianity from future data. Ongoing CMB experiment such as Planck (Planck Collaboration 2011) and future experiments such as COrE (The COrE Collaboration et al. 2011) will therefore require a detailed reconstruction of the LISW bispectrum either to be able to correctly remove the LISW contribution when estimating the local primary nonGaussian parameter f_{NL}, or to exploit the cosmological information encoded in the signal; therefore it becomes extremely important to know how to model and simulate it.
We present the formalism and numerical implementation i) to generate simulated CMB maps containing the LISW signal and ii) to build and test the optimal estimator for the LISW bispectrum, accounting for both the cubic and the linear parts. The linear part for this specific kind of signal has been calculated and tested here for the first time. We implemented and tested the LISW signal with two methods for the CMB nonGaussian simulations: the separable modeexpansion method (Fergusson et al. 2010; Smith & Zaldarriaga 2011) and the nonperturbative approach described in Sect. 2.1.
It is important to have an optimal estimator for the LISW bispectrum to extract the signal optimally from future data and to separate it from other types of nonGaussianities, i.e. the local primary bispectrum, with which it is degenerated. Here, following Komatsu (2010) and Munshi et al. (2011), we implemented the KSW bispectrum estimator (Komatsu et al. 2005) for the LISW signal of a full sky, cosmicvariancelimited CMB experiment and a more realistic instrumental setting, similar to that of a spacebased experiment. Furthermore, for this realistic case, we investigated the statistical detection significance and the impact that the LISW bispectrum has on the estimation and on the variance of the primary local nonGaussian parameter f_{NL}.
The outline of this paper is as follows. In Sect. 2 we present the methods for simulating the nonGaussian CMB maps that contain the LISW bispectrum signal by using both the separable modeexpansion method and the nonperturbative covariance method. Section 3 provides the basics for building and implementing the optimal estimator for the LISW signal, including its linear part. It also includes a discussion of the implementation of the Wienerfiltered simulation algorithm. In Sect. 4 we present the relevant tests and results. In Sect. 5 we quantify the statistical detection significance of the LISW bispectrum and the impact on the error of primary nonGaussianity f_{NL} due to the presence of the ISW signal. Finally, in Sect. 6, we discuss the results and summarize the conclusions. Details on the simulations built with the covariance method and on the LISW crosscorrelation coefficients are given in the appendix.
2. Simulated nonGaussian CMB maps
In this section, we present the formalism for creating simulated CMB maps for the LISW bispectrum. We used two different methods: a nonperturbative approach, here named the “covariance method”, and the separable modeexpansion method (Fergusson et al. 2010 and Smith & Zaldarriaga 2011). The latter provides an efficient and easy to handle way of generating LISW maps, while the former method provides better insights into the physics related to the LISW bispectrum. In this case, in fact, the LISW signal is generated starting from the covariance matrix that represents the expected correlation between the lensing and the ISW/RS effects.
2.1. Covariance method
The LISW correlation is defined by the covariance matrix, (1)and the crosscorrelation coefficient, (2)Here, and are the CMB primary temperature power spectrum and the lensing power spectrum, where the lensing potential φ (the gravitational potential projection along the line of sight) is defined by (3)The term in the numerator, , is the power spectrum of the crosscorrelation between the lensing and the ISW/RS effect, see Appendix B for details.
After a Cholesky decomposition of the LISW correlation matrix C_{L  ISW}, the two new variables t_{ℓm} and z_{ℓm} are defined by where x_{ℓm} and y_{ℓm} are two independent random Gaussian fields. By definition, the new fields are such that: , and they have the nonzero crosscorrelation .
Fig. 1
LISW power spectrum from the covariance method simulation. The plot shows that the temperature power spectrum of the LISW simulations generated with the method described in Sect. 2.1 is compatible with the input theoretical power spectrum from CAMB and that the nonGaussian contribution is always subdominant. The temperature power spectrum from one simulated LISW realization is shown in black, the red line refers to the theoretical input from CAMB while the blue refers to the nonGaussian LISW contribution from the same realization. 

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As described in Appendix A, the map that contains the desired LISW bispectrum is given by the coefficients (6)where and are the unlensed primary and the lensing angular coefficients, and corresponds to the lensing expansion terms only. Note that by construction y_{ℓm} has the same phases as and the same as x_{ℓm}, which is necessary for building a map with the desired bispectrum signal.
Figure 1 shows in black the temperature CMB power spectrum of one simulated LISW map, , built from Eq. (6). The nonGaussian contribution, in blue in the figure, is always subdominant and the are consistent with the theoretical input (red line) obtained with CAMB (Lewis et al. 2000)^{1}. As throughout the paper, the reference cosmological model used is the ΛCDM model with parameter values defined in Komatsu et al. (2011).
2.2. Separable modeexpansion method
Following Fergusson et al. (2010) and Smith & Zaldarriaga (2011), the nonGaussian part of the CMB angular coefficients can be defined starting from a given reduced bispectrum. For the LISW signal, the method can be used because this kind of signal is separable, so that (7)From the expression of the LISW reduced bispectrum in Eq. (13) and by factorizing the ℓ dependence, the explicit form of the nonGaussian contribution to the a_{ℓm} from the LISW cross correlation is given by (8)Here, (9)The maps with a δ^{2} prefix are given by, e.g., ; they correspond to the maps of Eq. (9) multiplied by the − ℓ(ℓ + 1) factor. The final solution containing the LISW signal is (10)where is the Gaussian part.
In Fig. 2, we show the CMB temperature power spectra from the Gaussian and the nonGaussian map, as defined in Eq. (8). The nonGaussian contribution is always subdominant, as expected.
Fig. 2
LISW power spectrum from the separable modeexpansion method simulation. The temperature power spectrum of the LISW simulation generated with the method described in Sect. 2.2 is compatible with the input theoretical power spectrum from CAMB and the nonGaussian contribution is always subdominant. The temperature power spectrum from one simulated LISW realization is shown in black, the red line refers to the theoretical input from CAMB while the blue refers to the nonGaussian LISW contribution from the same realization. 

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3. The optimal KSW estimator for the lensingISW/RS bispectrum
In this section we present the formalism for the KSW estimator (Komatsu et al. 2005) for the LISW bispectrum signal.
3.1. Definition
The a_{ℓm} probability distribution function (PDF) in the limit of weak nonGaussianity (i.e., truncated at the bispectrum level) is given by (Babich 2005; Taylor & Watts 2001; Komatsu 2010) (11)where ⟨ a_{ℓ1m1}a_{ℓ2m2}a_{ℓ3m3} ⟩ is the angular bispectrum. Here, we are interested in the LISW case, for which the angular bispectrum, parametrized by the amplitude parameter , is (12)where (13)is the reduced bispectrum and are the LISW crosscorrelation coefficients. According to Komatsu et al. (2005), for a small departure from Gaussianity, the optimal estimator for the LISW amplitude parameter is given by (14)where (F^{1}) is the inverse of the LISW Fisher matrix (15)For a realistic CMB experimental setting, the noise, N_{ℓ}, and the beam window function, w_{ℓ}, are accounted for so that . In this case, the bispectrum is also convolved with the beam transfer function w_{ℓ}, . For a mask M(p), the observed sky fraction f_{sky} is defined as (16)where is the number of pixels in the map, N_{s} is the map resolution, and the sum ∑ _{p} is done over the pixels.
Assuming that the only relevant nonGaussian contribution is coming from the LISW term, which is the case if the local primordial nonGaussianity is small and foregrounds and point sources have been correctly removed and masked, S_{L  ISW} is given by the data as (17)By factorizing the ℓ_{i} dependence, this becomes (Komatsu 2010) (18)where the maps , , etc. are the same as defined in Eqs. (9) and, in the case of a realistic experiment, they are convolved with the experimental window function w_{ℓ} so that, for example, .
In Eq. (18), the first two lines refer to the cubic part of the estimator, while is the linear part, which corrects for anisotropies and must be included if rotational invariance is not preserved. Details on the analytic expression of the LISW linear term and on its numerical implementation are given in the next Sect. 3.2 and in Sect. 4.
3.2. Linear term
The linear term of the estimator is given by (19)By using the explicit form of and by factorizing the ℓdependence, one obtains (20)where ⟨ ⟩ _{MC} indicates the Monte Carlo (MC) averages and the different maps are defined in Eq. 9. They are convolved with the experimental window function w_{ℓ}, so that , etc.
3.3. Wienerfiltered maps
The optimal bispectrum estimator as described in Eqs. (17) and (18) involves products of inversevariancefiltered maps, C^{1}a = (S + N)^{1}a, where S and N are the signal and the noise covariance matrix, respectively. A bruteforce calculation of such an expression is impractical for modern highresolution experiments because it involves the inversion of two matrices that are too large to be stored and processed as dense systems. If the noise covariance can be described in terms of a simple power spectrum in spherical harmonic space, the calculation simplifies significantly. However, this approach is no longer exact for experiments with anisotropic noise distribution or reduced sky coverage, leading to an increase in the error bars of the estimates.
Here, we used Wiener filtering as a basis for the exact evaluation of terms involving C^{1}a. We applied the iterative scheme of Elsner & Wandelt (2013) to calculate the Wiener filter a^{WF} ≡ S(S + N)^{1}a, the maximum a posteriori solution if signal and noise are Gaussian random fields. After a^{WF} has been successfully computed, we finally obtain the inversevariancefiltered map by normalizing the spherical harmonic coefficients of the Wienerfilter solution by the CMB power spectrum multiplied with the beam window function, .
4. Results
In this section we present the results for the numerical implementation of the optimal estimator and the methods presented in Sects. 2.1 and 2.2 to build the CMB maps that contain the LISW bispectrum. In particular, we processed the simulated LISW maps through the estimator pipeline to obtain the amplitude parameter of Eq. (14). We considered two main settings:

a fullsky cosmicvariancelimited CMB experiment up to a maximum multipole ℓ_{max} ≃ 1000, and

a more realistic experimental setting that consisted of a onechannel CMB experiment with a Gaussian beam with a FWHM θ_{b} = 7′, a galactic mask leaving ≃80% of the sky, and anisotropic uncorrelated noise. These settings are visualized in Figs. C.3, C.2 and details are given in Sect. C.
Fig. 3
Largescale contribution to the nonGaussian LISW signal. Upper panel: the map contains the LISW coefficients and enters the LISW estimator Eq. (18). The ℓfilter acts as a filter that suppresses the small scales (lower panel). 

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Fig. 4
Intermediatescale contribution to the nonGaussian LISW signal. Same as Fig. 3 but, for the map (upper panel) and its corresponding filter . The factor ℓ(ℓ + 1) dominates at high ℓ, defining more smallscale features than the previous Q map. 

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Fig. 5
values for 100 simulated nonGaussian maps obtained with the covariance method of Sect. 2.1. The lensing part has been computed with the FLINTS code (Lavaux & Wandelt 2010). The straight line refers to the averaged from these simulations, the dashed line to the averaged 1σ error. Here ℓ_{max} = 1000. 

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Fig. 6
Same as Fig. 5, but for 100 simulations built with the separable modeexpansion method (Eq. (8)). 

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We built a set of 100 CMB simulations for each of the two methods described in sections 2.2 and 2.1 for a cosmicvariancelimited CMB experiment with full sky coverage. For the covariance method, we used the FLINTS code (Lavaux & Wandelt 2010) to generate the lensing coefficients and the lensing potential coefficients φ_{ℓm} needed to build the nonGaussian as described in Sect. 2.1. In both cases, we analyzed the LISW simulated CMB maps with the LISW estimator up to ℓ_{max} = 1000. According to the definition of , the expected value is 1 with 1σ error predicted from theory for ℓ_{max} = 1000 of ≃0.64. For the separable expansions mode method, the simulations give a mean with averaged 1σ error ≃0.69. With the simulations built with the covariance method, we obtain a mean with averaged 1σ error of 0.67. The results are summarized in Figs. 6 and 5. These estimates are compatible with the theoretical predictions. The error bars are slightly suboptimal because of numerical noise and the fact that we assumed a diagonal covariance matrix so that (C^{1}a)_{ℓm} = a_{ℓm}/C_{ℓ}.
Fig. 7
Same as Fig.6 but for a more realistic CMB experiment with a 7′ FWHM Gaussian beam, anisotropic noise, and a 20% galactic mask. Here ℓ_{max} = 1500. The dashed lines are the 1σ averaged error bars from simulations, while the dotted lines are the expected Fisher errors. 

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To test the estimator on a more realistic case, we built a set of 100 simulations with the separable modeexpansion method considering a realistic experimental setting. This consists of a CMB onechannel experiment with a Gaussian beam FWHM θ_{b} = 7′, a galactic mask with f_{sky} = 0.78, and anisotropic noise, as previously described. In this case, we run the estimator up to ℓ_{max} = 1500. The expected theoretical 1σ error on for this experimental setting and up to ℓ_{max} = 1500 is ≃0.49. This estimate accounts for a ≃10% percent increase in the error bar because the lensing is intrinsically nonGaussian and gives an extra contribution to the variance, as shown in Lewis et al. (2011). We obtain a mean with averaged 1σ error ≃0.55. In this case we computed both the cubic and linear part of the estimator. In particular, the linear term has been tested with a set of 100 Monte Carlo (MC) averages generated for each map product in equation (Eq. (20)). In the presence of anisotropic noise and a sky cut, the linear part of the estimator is necessary to recover the expected estimation of and error bars. The linear contribution to is strongly anticorrelated with the cubic part. This behavior is summarized in Fig. 8. In the plot are shown the linear and the cubic contributions to the total amplitude . We also checked that with 100 MC averages the linear term converges and is stable: for this specific experimental setting we find that the results do not improve when increasing the MC averages to 200.
Finally, to test optimality, we Wienerfiltered the 100 LISW simulations and processed them through the LISW estimator pipeline. The maps were produced following Elsner & Wandelt (2013), as described in section Sect. 3.3. We used the same experimental settings as inputs as was described previously. The linear term was computed with 100 Wienerfiltered MC simulations. We found that the improvement over the non Wiener filtered simulations is small (<10%) for our particular settings. However, this does not exclude that the Wiener filtering may have a more noticeable impact for a more realistic experimental setting and noise covariance.
Fig. 8
The linear term of the estimator reduces the error bars for anisotropic data. The plot shows the linear (solid black line) and the cubic (dotdashed black line) contributions to the total (red line) for a CMB experiment with anisotropic noise and a 22% galactic mask. 

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5. f_{NL} error estimation
This section summarizes the results for the impact of the LISW signal on the error estimation of f_{NL} from the localtype nonGaussianity. If the only contribution to f_{NL} were coming from the primary localtype nonGaussianity, the error on this parameter would be simply given by (21)i.e., the inverse of the Fisher matrix of the localtype nonGaussian contribution (22)where f_{sky} refers to the observed sky fraction. The noise, N_{ℓ}, and the beam, b_{ℓ}, can be accounted for, so that . In this case the bispectrum is also convolved with the beam transfer function b_{ℓ}: .
However, the LISW can be a serious contaminant of the local primary signal (Mangilli & Verde 2009; Hanson et al. 2009), so that it is important to quantify the effect on the expected f_{NL} error as well. If the LISW signal is present, the error matrix will be given by a nondiagonal Fisher matrix of the form (23)where (24)is the crosscorrelation term and F^{L  ISW} is the the Fisher term of the LISW signal of Eq. (15). The expected error on the local f_{NL} is (25)i.e., the inverse of the full Fisher matrix containing the crosscorrelation between the primary local nonGaussianity and the LISW signal. The difference between the error estimation on f_{NL}primary with and without the LISW contribution is (26)To quantify the level of correlation between the two signals, one can define the correlation coefficient as (27)We found that the effect of the local nonGaussianity on the LISW is negligible. Therefore, the 1 − σ error of the LISW amplitude parameter is given by .
For a realistic CMB experiment as described in Sect. 4 and in Appendix C, we found that the correlation between the two signals is r = 0.20 at ℓ_{max} = 1500, and r = 0.27 at ℓ_{max} = 2000, and that the expected detection significance of the LISW signal (1/σ^{L  ISW}) is at ≃2 and ≃3σ. The effect on the f_{NL} local error due to the contamination is in the range between ≃3% to ≃5% for ℓ_{max} from 1000 to 2000, depending on how strongly the two signals are correlated. For ℓ_{max} = 2000, if the LISW signal is not accounted for correctly, the f_{NL} error bars are overestimated by ≃4%.
6. Discussion and conclusions
We have presented the formalism and numerical implementation to build the optimal KSW estimator for the lensingintegrated Sachs Wolfe (LISW) bispectrum. Moreover, we have tested the estimator on simulated CMB maps containing the LISW nonGaussian signal and on the Wienerfiltered simulations to test optimality. We implemented and tested two methods for the simulations: a nonperturbative approach to simulate CMB sky maps with the LISW signal, which is based on the FLINTS lensing code (Lavaux & Wandelt 2010), and the perturbative separable modeexpansion method calculated for this specific signal. We provided the analytical expression and numerical implementation of the linear term of the estimator for this specific type of bispectrum. For a realistic CMB experimental setting that accounts for anisotropic noise and masked sky, the linear term gives a relevant contribution that is highly anticorrelated with the cubic part, and it is necessary to recover the signal and optimal error bars. To achieve optimality, we also tested the estimator on the Wienerfiltered LISWsimulated CMB maps. In this case we recovered the signal with error bars that saturate the theoretical CramerRao bound, with a small improvement of <10% with respect to the nonWienerfiltered simulations. Finally, we estimated that, if not correctly accounted for, the LISW effect also has an impact on error bars, leading to a bias and an overestimation of ≃4%, in agreement with Lewis et al. (2011). Thus a joint analysis of nonGaussian shapes and/or LISW template subtraction will be needed to recover unbiased minimum variance results of the localtype primordial nonGaussian signal. The unbiased nonGaussian results for the local shape from Planck data can be found in Planck Collaboration (2013a). A detailed description of different methods for an unbiased analysis of CMB nonGaussianity can be found in Kim et al. (2013). After subtracting the LISW bispectrum signal, there is no evidence of primordial local nonGaussianity in Planck data, since (Planck Collaboration 2013a). It is important to note that the KSW bispectrum approach to the estimation of the LISW is complementary to the lensing reconstruction estimator of Lewis et al. (2011). In principle, the KSW estimator can offer advantages over other methods because the bispectrum has a unique shape and has been shown to be robust to foreground contamination (Yadav & Wandelt 2010), therefore it can be measured by using a larger sky fraction. In addition, including the LISW in the framework of bispectrum analysis gives an unified approach to testing for primordial nonGaussianity. The tools presented here enable the optimal analysis of this important signal from CMB data and can be used for the Planck data analysis as described in Planck Collaboration (2013a,b). The LISW signal is detected at 2.7σ in the Planck temperature data, and we expect that the statistical significance will be increased by ≃15% when adding the polarization data. As pointed out in Lewis et al. (2011), the LISW bispectrum signal is expected to be detected at ≃9σ by combining the temperature and polarization data from future CMB fullsky and cosmicvariancelimited experiments such as COrE.
We found that the amplitude on the nonlinear effect estimated in Mangilli & Verde (2009) (as well as probably in Giovi et al. 2005) compared to Lewis (2012); Junk & Komatsu (2012) was due to an interpolation problem. Even a small numerical effect at the interpolation scale can propagate through the lineofsight integration to a relevant effect on the nonlinear transition scale of the . Note, however, that the coefficients are very sensitive to the cosmology parameters related to the latetime evolution, Ω_{Λ}, w, σ_{8}, and to the modeling of the nonlinearities (e.g. Verde & Spergel (2002)), so extra care must be taken when comparing results from different authors.
Acknowledgments
This work was supported in part by NSF grants AST 0708849 and AST 0908902, and by NASA/JPL subcontract 1413479; and through Ben Wandelt’s ANR Chaire d’excellence ANR10CEXC00401. A.M. acknowledges Guilhem Lavaux for the FLINTS lensing simulations, Licia Verde for useful comments and discussion, and the University of Illinois for the use of the curvaton computers.
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Appendix A: simulated LISW CMB bispectrum from the covariance method
This appendix describes the covariance method used to build the LISW simulated maps and described in Sect. 2.1. It is straightforward to check that the coefficients give the desired bispectrum by calculating . The lensing coefficients can be expressed analytically, at first order in the lensing expansion, as (A.1)where the primary and the harmonic coefficients of the lensing potential φ^{L}. Since, according to the new variables definition (z_{ℓm},t_{ℓm}) of Eq. (6), , the can be written as (A.2)Here, to simplify the notation, , so that at first order. The explicit expression for takes the form (A.3)From this the only nonzero term is (A.4)From this only survives (A.5)By using the definition of in Eq. (2) and the approximation for which , since by construction , , ⟨ x^{2} ⟩ = 1, ⟨ y^{2} ⟩ = 1 and ⟨ xy ⟩ = 0, we recover the expected signal (A.6)
Fig. C.1
Noise simulations. Example of one simulated anisotropic noise map realization (bottom panel) and its correspondent noise power spectrum in red on the upper panel. The black line corresponds to the power spectrum from the same LISW simulation. 

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Fig. C.2
Experimental setting: beam window function and anisotropic noise map. The Gaussian beam with a 7′ FWHM is shown in the upper panel, while the bottom panel shows the dipolelike anisotropic noise covariance matrix map. 

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Fig. C.3
Mask. Galactic mask cut with f_{sky} = 0.78 (upper panel) obtained from thresholding the smoothed 100 μm IRAS map (bottom panel). 

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Appendix B: LISW crosscorrelation coefficients
The definition of the CMB lensingISW/RS crosscorrelation coefficients is (Spergel & Goldberg 1999; Verde & Spergel 2002; Giovi et al. 2005): (B.1)where, r(z) is the comoving conformal distance and P_{φ}(k,z) is the gravitational potential power spectrum that accounts for both the linear and nonlinear contributions. The nonlinear regime RS contribution to the signal is tiny, in agreement with Lewis (2012) and Junk & Komatsu (2012)^{2}. Considering both the linear ISW and the Rees Sciama effect improves the variance and signaltonoise ratio by a few percent (≃2%) with respect to the linearonly case calculation. In this work we considered both contributions for completeness. As a template for both the simulations and the estimator, we used the late ISWlensing crosscorrelation coefficients of Eq. (B.1). This is a good approximation since this effect provides the main contribution. However, for a detailed description see Lewis (2012).
Appendix C: experimental setting
Figures C.2 and C.3 summarize the experimental settings used in the simulations. These settings are inspired by a spacebased experiment such as WMAP or Planck, with a variation of the noise with the ecliptic latitude. We considered a onechannel CMB experiment with a Gaussian beam with a FWHM θ_{b} = 7′, a galactic mask leaving ≃80% of the sky and anisotropic uncorrelated noise. In particular, we obtained a galactic type mask from the IRAS^{3} 100 μm map smoothed at 5 angular degrees resolution and with a threshold of 12 MJy/sr. We considered a dipoletype anisotropic noise covariance matrix, which accounts for the anisotropies owing to, e.g., the scanning strategy. An example of an anisotropic noise realization and the corresponding power spectrum is given in the bottom and upper panels of Fig. C.1. The noise starts to dominate from ℓ ≃ 1300.
All Figures
Fig. 1
LISW power spectrum from the covariance method simulation. The plot shows that the temperature power spectrum of the LISW simulations generated with the method described in Sect. 2.1 is compatible with the input theoretical power spectrum from CAMB and that the nonGaussian contribution is always subdominant. The temperature power spectrum from one simulated LISW realization is shown in black, the red line refers to the theoretical input from CAMB while the blue refers to the nonGaussian LISW contribution from the same realization. 

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In the text 
Fig. 2
LISW power spectrum from the separable modeexpansion method simulation. The temperature power spectrum of the LISW simulation generated with the method described in Sect. 2.2 is compatible with the input theoretical power spectrum from CAMB and the nonGaussian contribution is always subdominant. The temperature power spectrum from one simulated LISW realization is shown in black, the red line refers to the theoretical input from CAMB while the blue refers to the nonGaussian LISW contribution from the same realization. 

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In the text 
Fig. 3
Largescale contribution to the nonGaussian LISW signal. Upper panel: the map contains the LISW coefficients and enters the LISW estimator Eq. (18). The ℓfilter acts as a filter that suppresses the small scales (lower panel). 

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In the text 
Fig. 4
Intermediatescale contribution to the nonGaussian LISW signal. Same as Fig. 3 but, for the map (upper panel) and its corresponding filter . The factor ℓ(ℓ + 1) dominates at high ℓ, defining more smallscale features than the previous Q map. 

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In the text 
Fig. 5
values for 100 simulated nonGaussian maps obtained with the covariance method of Sect. 2.1. The lensing part has been computed with the FLINTS code (Lavaux & Wandelt 2010). The straight line refers to the averaged from these simulations, the dashed line to the averaged 1σ error. Here ℓ_{max} = 1000. 

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In the text 
Fig. 6
Same as Fig. 5, but for 100 simulations built with the separable modeexpansion method (Eq. (8)). 

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In the text 
Fig. 7
Same as Fig.6 but for a more realistic CMB experiment with a 7′ FWHM Gaussian beam, anisotropic noise, and a 20% galactic mask. Here ℓ_{max} = 1500. The dashed lines are the 1σ averaged error bars from simulations, while the dotted lines are the expected Fisher errors. 

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In the text 
Fig. 8
The linear term of the estimator reduces the error bars for anisotropic data. The plot shows the linear (solid black line) and the cubic (dotdashed black line) contributions to the total (red line) for a CMB experiment with anisotropic noise and a 22% galactic mask. 

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In the text 
Fig. C.1
Noise simulations. Example of one simulated anisotropic noise map realization (bottom panel) and its correspondent noise power spectrum in red on the upper panel. The black line corresponds to the power spectrum from the same LISW simulation. 

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In the text 
Fig. C.2
Experimental setting: beam window function and anisotropic noise map. The Gaussian beam with a 7′ FWHM is shown in the upper panel, while the bottom panel shows the dipolelike anisotropic noise covariance matrix map. 

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In the text 
Fig. C.3
Mask. Galactic mask cut with f_{sky} = 0.78 (upper panel) obtained from thresholding the smoothed 100 μm IRAS map (bottom panel). 

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In the text 
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