The shape of the CMB lensing bispectrum

The shape of the CMB lensing bispectrum

Antony Lewis Department of Physics & Astronomy, University of Sussex, Brighton BN1 9QH, UK    Anthony Challinor Institute of Astronomy and Kavli Institute for Cosmology, Madingley Road, Cambridge, CB3 0HA, UK DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 OWA, UK    Duncan Hanson Institute of Astronomy and Kavli Institute for Cosmology, Madingley Road, Cambridge, CB3 0HA, UK Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena CA 91109, USA
August 3, 2019

Lensing of the CMB generates a significant bispectrum, which should be detected by the Planck satellite at the 5-sigma level and is potentially a non-negligible source of bias for estimators of local non-Gaussianity. We extend current understanding of the lensing bispectrum in several directions: (1) we perform a non-perturbative calculation of the lensing bispectrum which is more accurate than previous, first-order calculations; (2) we demonstrate how to incorporate the signal variance of the lensing bispectrum into estimates of its amplitude, providing a good analytical explanation for previous Monte-Carlo results; and (3) we discover the existence of a significant lensing bispectrum in polarization, due to a previously-unnoticed correlation between the lensing potential and -polarization as large as at low multipoles. We use this improved understanding of the lensing bispectra to re-evaluate Fisher-matrix predictions, both for Planck and cosmic variance limited data. We confirm that the non-negligible lensing-induced bias for estimation of local non-Gaussianity should be robustly treatable, and will only inflate error bars by a few percent over predictions where lensing effects are completely ignored (but note that lensing must still be accounted for to obtain unbiased constraints). We also show that the detection significance for the lensing bispectrum itself is ultimately limited to 9 sigma by cosmic variance. The tools that we develop for non-perturbative calculation of the lensing bispectrum are directly relevant to other calculations, and we give an explicit construction of a simple non-perturbative quadratic estimator for the lensing potential and relate its cross-correlation power spectrum to the bispectrum. Our numerical codes are publicly available as part of CAMB and LensPix.

I Introduction

The large-scale CMB temperature anisotropy has a contribution from the blue- and red-shifting of photons as they fall in and out of potential wells between the last-scattering surface and our observation. This integrated-Sachs-Wolfe (ISW) effect is not present during matter domination, but becomes important at redshift at which dark energy starts to affect the evolution of the matter perturbations. The CMB is also gravitationally lensed by structures along the line of sight, with most of the effect also coming from , so there is a correlation between the ISW signal and the CMB lenses. The effect of an overdensity is to magnify the last-scattering surface, effectively locally shifting the scale of the acoustic peaks. The variance over some range of scales is therefore changed by the magnification if the spectrum is not flat. This leads to a correlation between the small-scale CMB power and the large-scale lenses, and hence a correlation between the large-scale CMB temperature and the small-scale power. This corresponds to a ‘squeezed’ bispectrum shape — it is the correlation of one large scale with two much smaller scales. The lensing bispectrum falls off rapidly as the largest scale decreases, since the ISW contribution to the temperature falls rapidly on smaller scales. The existence of a significant temperature bispectrum is well known, and must be modeled when trying to detect small levels of local primordial non-Gaussianity Smith and Zaldarriaga (2006); Serra and Cooray (2008); Hanson et al. (2009); Mangilli and Verde (2009). It can also be used as a probe of the perturbation growth and expansion history of the universe at low redshift, and hence help to constrain the dark energy and curvature Seljak and Zaldarriaga (1999); Goldberg and Spergel (1999); Hu (2002); Giovi et al. (2003, 2005); Gold (2005).

Calculations of the temperature lensing bispectrum have until now been calculated at lowest order in the lensing effects, although some simulation work has also been done to verify that the effect of higher-order terms is small Hanson et al. (2009). In this work, we demonstrate how to extend these calculations non-perturbatively to higher-order by working in an ‘unlensed short-leg’ approximation, where we take one large-scale mode of the CMB temperature to be unlensed. This produces higher-order corrections to the usual lensing result which may be accurately reproduced simply by replacing the unlensed power spectra which appear in the lowest-order calculation of the lensing bispectrum with their lensed counterparts. This results in corrections to the lensing bispectrum which we verify using Monte-Carlo simulations.

The lensing bispectrum should be detected soon at high-significance (e.g. in the data of the recently-launched Planck satellite Hanson et al. (2009)). In this regime, the cosmic variance of the lensing signal can have large effects on the expected error of the bispectrum amplitude. Calculation of the increase in error at first appears daunting as it involves a six-point function in the non-Gaussian, lensed CMB, however we will show how a heuristic interpretation of the lensing bispectrum estimator as a cross-correlation between the observed CMB temperature and a quadratic reconstruction of the lensing effects can be used to intuit an accurate approximation to the signal variance. This method also generalizes straightforwardly to a calculation of the increase in variance for other estimators of non-Gaussianity, where the bias due to lensing represents an additional effective source of noise. This increase has already been investigated numerically by Ref. Hanson et al. (2009) under the assumption that the amplitude of the lensing bispectrum is well constrained and may simply be subtracted from the data. We are able to reproduce this result analytically, as well as extend it to the case where the amplitude of the lensing bispectrum is treated as a free parameter and marginalized over directly from the data. Our discussion also leads to improvements to standard bispectrum estimators, which incorporate the signal variance appropriate to the lensing bispectrum.

Discussion of the CMB lensing bispectrum in the literature has focused on the temperature anisotropies since there is no direct analogue of the ISW effect in polarization. However, as we will show here, the large-scale polarization from reionization is also directly correlated with the matter distribution, giving a correlation between the -polarization and lensing potential at up to the level. This generates a significant polarized lensing bispectrum, detectable at with cosmic-variance limited data. We present the first calculations of these effects, and generalize our analytical non-perturbative bispectrum and variance calculations to the polarization case. Including this effect in a fit for the amplitude of the lensing bispectrum would increase the significance with which it is detected from to for Planck, or from to for an experiment which is cosmic-variance limited to .

The outline for this paper closely follows the description above. In Section II we review the quantitative description of lensing effects as a remapping by the gradient of a lensing potential , and derive the cross-correlation between the lensing potential and the CMB temperature and polarization. In Section III we then present calculations of the lensing bispectrum on the flat-sky, both at first order in the lensing potential as well as in the short-leg approximation which is effectively accurate at higher order as well. Use of the flat-sky expressions makes it straightforward to gain an intuition for the terms involved. In Section IV we proceed to give full-sky results for both temperature and polarization, which generalize straightforwardly from the flat-sky limit. In Section V we discuss the variance of estimators for the lensing bispectrum, and the increased variance for other non-Gaussian bispectra which occurs when marginalizing or subtracting the lensing contribution to avoid biases. Our conclusions are summarized in Section VI, and the details of several more involved calculations are contained in appendices. Throughout we assume a standard CDM cosmology, and for numerical examples use a constant spectral index spatially-flat model with , , , , , , and approximate the three neutrinos as massless.

Ii The lensing potential and its cross-correlation with temperature anisotropy and polarization

The effect of gravitational lensing is to alter the direction of propagation of photons such that when we look in direction we are actually seeing photons that originate from on the last-scattering surface, where is a deflection angle. Using the Born approximation, the deflection angle of a source at conformal distance is given in terms of the Weyl potential (i.e. the average of the Newtonian-gauge potentials) by the line-of-sight integral


where represents the angular derivative, equivalent to the covariant derivative on the sphere defined by . The quantity is the conformal time at which the photon was at position , and is the comoving angular-diameter distance. It is convenient to define the lensing potential,


so that the deflection angle is given by . From now on we write this simply as . For full derivations and review see Refs. Lewis and Challinor (2006); Hanson et al. (2010).

Since the lensing potential is a weighted integral of the Weyl potential along the line of sight, it is correlated to the ISW contribution to the CMB temperature given by


where the dot denotes a conformal time derivative. In concordance CDM models, and are highly correlated (at above the 90% level) due to the similarity of their redshift kernels, which leads directly to a correlation between the total CMB anisotropy and the lensing potential. The full result for the angular power spectrum can easily be calculated numerically, and typical results are shown later in Fig. 3; in total the correlation is nearly at , but decreases rapidly with scale as the ISW contribution to the total diminishes, giving only a few percent correlation by .

Figure 1: The geometry of the polarization signal generated by scattering at reionization: the figure is roughly to scale in comoving distance, with reionization (shading) starting at , about 2/3 of the way to the last-scattering surface (dark red outer shell). After the universe has reionized, the probability of scattering falls off as the universe expands, so most scattering occurs between and of the distance to recombination. An electron at reionization sees its own last scattering surface as indicated by the red shell, and Thomson scattering of the quadrupolar component of the distribution of photons originating from this surface generates -polarization. For an electron at the start of reionization its last scattering shell extends from our last-scattering surface down to a redshift of about . Perturbations that generate a large-scale polarization signal will be correlated on large-scales, and therefore be correlated to perturbations at (see Fig. 2).
Figure 2: An illustration of the correlation between the quadrupole component of the polarization from reionization and the local density quadrupole, here generated by a superposition of two orthogonal plane waves.

The story with the polarization is rather different. The temperature quadrupole generated by the ISW can re-scatter leading to a correlated polarization signal; however as shown in Ref. Cooray and Melchiorri (2006) this signal is tiny because there is little scattering at the low redshifts where the ISW signal becomes significant. The dominant correlation is actually between the lensing potential and the large-scale polarization -modes generated by scattering at reionization. At redshift where reionization occurs, -mode polarization is generated by Thomson scattering of the local radiation quadrupole. This quadrupole has contributions from a wide range of redshifts (for the observer), overlapping with the region from which the CMB lensing potential is sourced, and is correlated over long distances. This is illustrated more concretely in Figs. 1 and 2, and plots of the cross-spectra and correlation coefficient are given in Fig. 3. The large-angle correlation is negative and so produces radial polarization around large-scale overdensities. Further discussion of the correlation is given in Appendix A where a simple analytic model which reproduces the main features of Fig. 3 is developed for the case of instantaneous reionization. With cosmic-variance limited full-sky and , could be detected as non-zero at approximately 2.5 sigma (compared to nearly 8 sigma for from and ).

Although the ISW effect does not directly generate the correlated -polarization signal (reionization occurs well before dark energy becomes dynamically important), there is nonetheless a significant indirect correlation between the ISW and because, as we have noted, the lensing potential is highly correlated to the ISW signal. Indeed the correlation at large scales is suppressed by about due to the (anti-)correlation between the ISW signal and the polarization. Note that the latter has the same sign as the correlation. Accurate numerical calculations of both and are now included in CAMB111 Lewis et al. (2000).

Figure 3: The power spectrum (left) and dimensionless correlation coefficient (right) for the correlation of the CMB lensing potential with the CMB temperature anisotropy (top) and -mode polarization (bottom) for a standard CDM cosmology with an optical depth to reionization (solid lines) and (dashed lines). Note that without reionization the -polarization power spectrum on large scales is very small, so although the correlation is still significant the actual size of the signal is very tiny. The correlation is due mainly to the ISW effect; the correlation to the reionization signal.

Iii Flat-sky CMB temperature lensing bispectrum

To understand the basic shape of the lensing bispectrum it is useful to start by considering the simple case of the CMB temperature in the flat-sky approximation. We follow the flat-sky notation and conventions of Ref. Lewis and Challinor (2006). Assuming statistical isotropy and that parity invariance holds in the mean, the reduced bispectrum can be defined as


where is the Fourier transform of the lensed temperature, and the delta-function ensures the triangle constraint. The reduced bispectrum is symmetric in its arguments and it is therefore convenient to restrict the values of , and such that , with other combinations obtainable by permutation.

iii.1 Leading perturbative result

Lensing remaps the temperature anisotropies so that the lensed temperature field is related to the unlensed field by . Fourier transforming and performing a series expansion to first order in gives222We are assuming the unlensed CMB is a single source plane at recombination governed by a single lensing potential. This is not quite correct on large scales because the ISW contributions are more local; however the bispectrum is only significant for small-scales of the lensed field, so we can neglect this complication to good accuracy.


Assuming Gaussianity of the lensing potential and the CMB temperature anisotropies, to first order in we then obtain the three-point correlation Zaldarriaga (2000); Hu (2000)


This is the standard first-order result for the lensing bispectrum; as we shall see higher-order corrections result in corrections at the level.

iii.2 Unlensed short-leg approximation and non-perturbative result

As we have described, the physical origin of the bispectrum signal is the small-scale power changing due to (de)magnification and shearing by large-scale lenses. If we consider a lensed CMB sky, and add an additional large-scale lens, it will look substantially similar, but re-sized. We would therefore expect the power spectrum of the small-scale fluctuations over the extent of the large-scale lens to be determined by a shifted version of a lensed power spectrum. In the lowest-order result we calculated above, the expression for the bispectrum involved the unlensed small-scale temperature spectrum, but since the result is only lowest order we can expect higher-order corrections on small scales.

Figure 4: The fractional difference between the flat-sky lensed temperature power spectra and the unlensed power spectra for (top; dashed), (top; solid) and analogously for the polarization in the lower two plots (see Appendix C). The lensing bispectrum depends on , with to within about one percent.

Since the lensing potential correlations fall off rapidly at high , all of the bispectrum signal is at small (, where we restrict to ). Since the lensing effect on the temperature at is very small, to a good approximation we can calculate the bispectrum neglecting the lensing of the short leg , i.e. . As we will show, many of the results in this paper may be verified numerically with Monte-Carlo simulations, and in all squeezed-shape cases we have checked, short legs have proved more than adequate. A more general approximation may be required to accurately assess the lensing bias on non-squeezed bispectra.

Using the fact that the lensed temperature is linear in , and integrating the Gaussian expectation by parts, we have


where the ensemble average is taken over realizations of both the CMB and the lensing potential. The only approximation here is that is uncorrelated with the unlensed temperature modes that contribute to and . Using


where we introduced the lensed temperature gradient, and its Fourier transform, we then have


Here we have defined the power spectrum by


so that


The expression for then follows simply from Eq. (7). In the absence of lensing, reduces to the usual temperature power spectrum. With lensing, to the extent that gradients and lensing commute, is reasonably well approximated by the lensed power spectrum. Indeed, in Fig. 4 we show numerically that approximating is correct to about the percent level. For the temperature bispectrum this then gives


In Appendix B we show explicitly that this non-perturbative relation agrees with a direct perturbative calculation to third order in . Figure 5 shows the effect of the higher-order corrections, effectively smoothing out the lensing bispectrum at the 10% level; this may be important to estimate correctly the contribution of CMB lensing to estimators for other forms of non-Gaussianity, and also for using the lensing bispectrum to obtain cosmological constraints.

Note that Eq. (10) for the response of the lensed CMB covariance to a mode of the lensing potential differs from that which is usually derived at lowest order in the lensing potential, e.g. for quadratic estimators Okamoto and Hu (2003), in which the unlensed spectra appear rather than (effectively) the lensed spectra. The neglect of these higher-order contributions leads to a bias in standard quadratic lensing estimators, which is more rigorously calculated in Ref. Hanson et al. (2011). The non-perturbative response of the lensed covariance to a mode of the lensing potential which we present here provides a faster, more intuitive way to arrive at the same result.

Finally we can easily construct an accurate approximation for the lensing bispectrum which is non-perturbatively correct if the short-leg approximation holds, and also agrees with the perturbative result to leading order even if it is violated:


iii.3 Squeezed limit

Since rapidly becomes small on small scales, the bispectrum is nearly zero unless is small. However the lensing deflection angles are small, a few arcminutes, so the lensing only has a significant effect on on small scales (). Hence almost all of the bispectrum signal is in squeezed triangles with . If we consider the ultra-squeezed limit we can define and expand in the small quantity giving the leading terms for the reduced bispectrum,

Figure 5: The CMB temperature lensing reduced bispectrum using the squeezed-limit approximation of Eq. (15) with the unlensed (thin blue dot-dashed) and lensed (thin blue solid) small-scale power spectrum, compared to the full spherical result of Eq. (23) using the lensed small-scale spectrum (thick red). For comparison the thick black line shows the result for a local-model primordial bispectrum with (which has not itself been lensed here; see Ref. Hanson et al. (2009)); note the difference in phase and scale-dependence.

The partly quadrupolar dependence on the angle between the large-scale and small-scale modes is very different from the isotropic squeezed limit expected from primordial modulations (e.g. the local model), making the quadrupole part of the lensing signal orthogonal. For and parallel, the signal is proportional to , reflecting the change in small-scale power due to shifting of scales by lensing (de)magnification and shearing. Since the spectrum has acoustic oscillations, the derivative term oscillates in , with a phase shift compared to the power spectrum. For (i.e. ) the derivative term is small and the bispectrum is generally of much smaller amplitude and has the same phase of acoustic oscillations as the power spectrum. The phase shift of the dominant lensing bispectrum signal compared to the phase of the acoustic oscillations is rather distinctive, and different from that expected for any primordial bispectrum of adiabatic perturbations. The strong scale-dependence (very little signal for ) is also different from standard local non-Gaussianity models; see Figs. 5 and 6. However as we shall see the isotropic part of the lensing bispectrum signal does have significant overlap with the local model, so although it is easily distinguished it is also important to model when studying local primordial non-Gaussianity.

Figure 6: Contour plot of (with non-linear intervals) for ISW-lensing (left) and local-model primordial non-Gaussianity (right; different overall scale). Although both are peaked for squeezed configurations, there are large phase and shape differences.

We can also derive the squeezed limit following an argument similar to Refs. Maldacena (2003); Creminelli and Zaldarriaga (2004a, b) by considering one fixed very large-scale lensing mode of the magnification matrix , where


Here, is the convergence and is the symmetric, trace-free shear. Since for a local displacement we have , it follows that taking the average with fixed we have


Expanding to first order in the convergence and shear matrix gives Bucher et al. (2010)


If there are also small-scale lensing modes then approximately the same result is obtained, with the power spectra replaced by the lensed CMB power spectra, in agreement with Eq. (15) when correlated with . Equation(18) makes clear the different shape dependence of the convergence and shear effects: a scale-invariant spectrum looks the same under uniform magnification, but shear introduces observable distortion to the hot and cold spots (only the term contributes if ); a white-noise spectrum looks the same after shearing, but the noise amplitude is changed under magnification (only the term contributes if )333The squeezed-limit form of the bispectrum here disagrees with Ref. Creminelli and Zaldarriaga (2004b) which has incorrect factors in the anisotropic term. The result given in Ref. Boubekeur et al. (2009) is in agreement in the matter-dominated Sachs-Wolfe limit..

Iv General full-sky CMB lensing bispectra

We now present the generalization of the lensing bispectrum calculation of the previous section to the full-sky and polarization. Further details are contained in Appendix C. Following Ref. Okamoto and Hu (2003) the lensed field is given by where the leading-order lensing correction for is




Note that is only non-zero for even and is then symmetric in and ; is only non-zero for odd and is then antisymmetric in and . We shall assume there are no unlensed modes, so that is due entirely to lensing. The non-perturbative flat-sky derivation generalizes directly to the curved-sky case; we implement the result here by simply using the lowest-order series-expansion result and then replacing the unlensed power spectra with their lensed counterparts. The leading-order lensing-induced 3-point function, using the lensed power spectra for the small scales to reproduce accurately the non-perturbative calculation, is then given by


where and , and , and . Equation (23) includes a sum over all six permutations of . The bispectrum then follows from


Note that under interchange of a pair of arguments, e.g. , the bispectrum changes by a factor . If the bispectrum involves a parity-odd combination of fields, e.g. , parity-invariance in the mean requires non-zero bispectra to have odd and hence to change sign under interchange of a pair of arguments. Furthermore, non-zero bispectra with odd are necessarily imaginary.

Results for the polarization bispectra have been derived before (e.g. Ref. Hu (2000)), however previous calculations have invariably set the large-scale , missing a signal detectable at several sigma with cosmic-variance limited data, and the power spectra have usually been the unlensed ones, giving a systematic error of . We show several slices through the temperature and polarization bispectra in Figs. 7, 8, 9, using both analytical calculations as well as simulations using the Monte-Carlo procedure outlined in Appendix D, testing the accuracy of the unlensed short-leg approximation and the use of lensed power spectra rather than e.g. . We demonstrate in Appendix C that for the polarization case, the non-perturbative calculation involves a new spectrum as well as and . Terms involving this spectrum are missed in the approximation of replacing e.g.  by . However, this is harmless since is of similar magnitude to and the error from neglecting such terms is small compared to the change in due to lensing (which is the dominant correction to the leading-order bispectra).

For the temperature, the reduced bispectrum is defined so that


where is taken to be zero for odd. This generalizes straightforwardly to bispectra involving only and/or since parity-invariance forces the bispectra to vanish for odd. However, there does not appear to be a standard equivalent definition for the bispectra involving a product of fields with net odd parity. For sufficiently sensitive data, these “odd-parity” bispectra are well measured because of the expected absence of small-scale primordial -modes; this is equivalent to the lensing reconstruction from correlations having lowest statistical noise, and hence correlating well with the large-scale temperature (and polarization). Since gravity waves decay on sub-horizon scales, the squeezed “odd-parity” CMB lensing bispectra are yet another way in which the CMB bispectra are very different from any primordial source. With sufficiently low noise, the large-scale lensing potential can be reconstructed very well using the small-scale and polarization Okamoto and Hu (2003); Hirata and Seljak (2003), so it would be straightforward to project the correlated component out of the temperature and polarization data and thereby remove CMB lensing as a source of contamination for other signals.

Figure 7: The reduced bispectra for various temperature and -polarization combinations. Left: Thin solid (red) lines show the lowest-order result for the CMB lensing bispectrum; thick solid (blue) show the approximate non-perturbative result using the lensed CMB power spectrum. Dash-dotted (black) lines for comparison show the result for a local-model primordial bispectrum with . Right: The difference between the non-perturbative bispectrum and the lowest-order result, in units of the maximum of the absolute value of each bispectrum. Smooth (red) lines show the theoretical approximation of Eq. (23); noisy (blue) lines are results from 1000 simulations, smoothed over .
Figure 8: Same as Fig. 7 but now showing , where results on the right are from 700 simulations and are unsmoothed.
Figure 9: The “odd-parity” bispectra (left) and (right) comparing the theoretical result of Eq. (23) (smooth line; red) to the result (noisy line; blue) of 700 (left) and 150 (right) simulations.

V Estimators, variance and bias

We now turn to a discussion of optimal estimators for the lensing bispectrum and their variance. As reviewed in Appendix E, in the case of an isotropic survey (full sky and uniform noise) the optimal estimator for the amplitude of a bispectrum template is, for small signals, given by


where is the vector of distinct elements of with covariance matrix , is the total cross-power spectrum including noise, and is the inverse of the Fisher error in the limit of no non-Gaussianity, given by


Here (with no implicit sums over -labels): is 6 if , 2 if two of the indices are equal, and 1 otherwise.

The Fisher error in Eq. (27) was calculated for Gaussian , but in the presence of lensing the variance is necessarily larger since there is a guaranteed signal and this itself has some variance. We will motivate an expression for this increase in variance by recasting the estimator for the lensing bispectrum as a cross-correlation between a quadratic estimate of the lensing potential and the CMB itself. We begin for simplicity in Section V.1 by considering the temperature-only case. Polarization is a straightforward generalization and is presented in Section V.2. Then in Section V.3 we combine these results to determine the significance with which the lensing bispectrum may be detected. In Section V.4 we generalize our results further, to the case where the lensing bispectrum is used in a joint analysis with other bispectra, as a source of bias to be subtracted or marginalized over.

v.1 Temperature

Equation (26) for the amplitude of the lensing bispectrum from temperature data can be rewritten as


The term in square brackets is (proportional to) a quadratic estimator for the lensing potential  Okamoto and Hu (2003). To see this, consider taking the expectation value of this term over noise, small-scale modes of the unlensed temperature, and the modes of the lensing potential with . We have that


which is correct to first order in modes of at and non-perturbatively correct in its other modes. Here,


which is related to the squeezed limit of the lensing bispectrum by . It follows that there is a quadratic estimator for of the form




This estimator satisfies to first-order in but is non-perturbatively correct in the modes of . It is a non-perturbative version of the usual quadratic estimator Okamoto and Hu (2003), avoiding the low- (‘’) bias in the standard estimator that was identified by Ref. Hanson et al. (2011) and generalizing the perturbative corrections of Ref. Hanson et al. (2011) to a non-perturbative form by simply using the lensed small-scale power spectra in the filter functions. In the Gaussian limit, the variance of the estimator is simply and the weighting in and in Eq. (31) can be shown to minimise this Gaussian variance subject to the estimator being unbiased.

We can now rewrite the estimator of the bispectrum amplitude in terms of the reconstruction as


and the normalization


Recasting the bispectrum estimator in this form leads directly to an understanding of the contribution to the error from signal variance. The estimator depends on the empirical cross-power between the reconstruction and the large-scale observed temperature:


Since , each is an unbiased estimate of the bispectrum amplitude. As with any other power spectrum estimator, has uncertainty both from reconstruction noise and from signal/cosmic variance:




where and the usual zero-signal variance term is which comes from the term involving in Eq. (36). In the standard estimator, Eq. (33), the are weighted with the and the normalisation is accordingly . In the presence of a non-zero signal, we can reduce the variance by weighting the with the full inverse variance. This defines a lower-variance estimator for the bispectrum amplitude,


which has variance given by where


When is large, so that the lensing modes are reconstructed with a small error, the contribution of the signal variance terms become important, ensuring that the total signal-to-noise never exceeds that expected from the cosmic-variance limit on the cross-correlation. Neglect of the signal variance term would lead to an overestimation of the significance for a detection of the lensing bispectrum (a similar effect happens with primordial non-Gaussianities444To account for the signal variance we have used an -dependent weighting in Eqs. (38) and (39); Creminelli et al. Creminelli et al. (2007) use a single realization-dependent change to the overall estimator normalization, which should be less optimal. The argument for lensing here can straightforwardly be generalized for estimation of local non-Gaussianity, using a quadratic estimator for the small-scale primordial power modulation rather than the lensing potential Hanson and Lewis (2009); the corresponding estimator may be a fast nearly-optimal alternative to a fully Bayesian method Elsner and Wandelt (2010) if the non-Gaussianity were large.  Creminelli et al. (2007)).

The optimal Fisher variance in Eq. (39) can easily be related to that for an optimal measurement of the cross-correlation, giving


This is exactly the same result as obtained from an optimal estimator of the amplitude of the cross-correlation using the estimator for the lensing potential.

Note that here we have only discussed the optimal estimator from the measured cross-correlation (bispectrum). If auto-spectra (power spectrum and lensing trispectrum) are also included the variance can be reduced further555We thank the referee for raising this point.; however since the correlation is always , and where the signal-to-noise peaks even with no noise, the gain from a more optimal joint estimator is rather modest, being . We do not discuss joint estimators further here, but a likelihood analysis of actual data should of course properly account for the full covariance structure of the estimators used.

v.2 Polarization

The arguments above carry over quite directly to polarization. The original estimator for the bispectrum amplitude, Eq. (26), can be written as


which involves the quadratic estimator