Weak lensing of the primary CMB bispectrum

The bispectrum of cosmic microwave background (CMB) anisotropies is a well-known probe of the non-Gaussianity of primordial perturbations. Just as the intervening large-scale structure modifies the CMB angular power spectrum through weak gravitational lensing, the CMB primary bispectrum generated at the last scattering surface is also modified by lensing. We discuss the lensing modification to the CMB bispectrum and show that lensing leads to an overall decrease in the amplitude of the primary bispectrum at multipoles of interest between 100 and 2000 through additional smoothing introduced by lensing. Since weak lensing is not accounted for in current estimators of the primordial non-Gaussianity parameter, the existing measurements of $f_{\mathrm{NL}}$ of the local model with WMAP out to $l_{\max }\sim 750$ is biased low by about 6%. For a high resolution experiment such as Planck, the lensing modification to the bispectrum must be properly included when attempting to estimate the primordial non-Gaussianity or the bias will be at the level of 30%. For Planck, weak lensing increases the minimum detectable value for the non-Gaussianity parameter of the local type $f_{\mathrm{NL}}$ to 7 from the previous estimate of about 5 without lensing. The minimum detectable value of $f_{\mathrm{NL}}$ for a cosmic variance limited experiment is also increased from less than 3 to $\sim 5$.

Figure 1

The CMB bispectrum for the equilateral case ($l_1=l_2=l_3=l$) with (solid line) and without (dashed line) lensing. Here we plot $l^4{B}_{lll}^{\theta }/(2\pi {)}^2$ as a function of the multipole $l$ for one of the sides. We assume $f_{\mathrm{NL}}=1$. The lensing effect can be described as a decrease in the amplitude of the bispectrum when $l\lesssim 1700$ with an increase at higher multipoles.

Figure 2

The squeezed configurations ($l_1\sim l_3\gg l_2$) of the CMB bispectrum with (solid line) and without (dashed line) lensing. The left panel is for $l_2=10$ and right panel is $l_2=100$. We vary $l=l_1$ with $l_3=l+l_2$ in both cases and plot $l^4{B}_{l,l_2,l+l_2}/(2\pi {)}^2$ as a function of $l$. Again, we take $f_{\mathrm{NL}}=1$. In these configurations, the lensing effect can be described as an overall decrease in the amplitude of the bispectrum when $l\lesssim 1200$. This suggests that lensing by the intervening large-scale structure leads to a less non-Gaussianity in the CMB map.

Figure 3

The signal-to-noise ratio for a detection of the CMB bispectrum with (thick lines) and without (thin lines) lensing. The long-dashed lines show the case for WMAP and dot-dashed lines for Planck. The left panel shows the signal-to-noise ratio as a function of $l_3$, while the right panel shows the cumulative signal-to-noise ratio below $l_3$ in the x-axis. Note the overall reduction in the signal-to-noise ratio (when $l_3\sim 1500$) in the case of lensing relative to the case where lensing is ignored.

Figure 4

Contour plots of $d(S/N{)}^2/d\log l_{\max }d\log l_{\min }$ [Eq. (40)] as a function of $l_{\max }$ and $l_{\min }$ for the primary bispectrum (top panel) and the lensed primary bispectrum (bottom panel). We take $f_{\mathrm{NL}}=1$.

Figure 5

Contour plot of the difference $d[(S/N{)}_{\mathrm{lensed}}^2-(S/N{)}_{\mathrm{unlensed}}^2)/d\log l_{\max }d\log l_{\min }$ as a function of $l_{\max }$ and $l_{\min }$ (same as the difference between bottom and top panels of Fig. 4).

Figure 6

The bias in the primordial non-Gaussianity parameter $f_{\mathrm{NL}}$ when lensing is ignored in the estimation. $(+)/(-)$ shows the sign of $\Delta f_{\mathrm{NL}}/f_{\mathrm{NL}}$ between the true and measured values (see text for details).