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Table 4: Expected contamination on $f_{NL}$ measurements from secondary bispectra (see Section 3.9.3) and from the effects described in Section 3.9.5. The expected bias of the primordial $f_{NL}$ estimate is given for local and equilateral shapes (if not reported, that means that the corresponding shape has not been studied). All of the estimates above are for an experiment with the angular resolution of the Planck satellite (i.e., ${\ell }_{max}~2000$). All of the effects above have been proved to be negligible, when compared to primordial $f_{NL}$-error bars, in the WMAP case. The expected biases have to be compared to the primordial error bars estimated from Fisher matrix forecasts for a Planck-like experiment. These are, as reported in the table, ${\mathrm{\Delta }}_{f_{NL}}^{local}\simeq 5$ and ${\mathrm{\Delta }}_{f_{NL}}^{equil}\simeq 60$. Some effects, namely, ISW-lensing bispectra and the three-point function from inhomogenous recombination, have been studied by different authors. In these cases all of the results obtained in different works are reported. While a good agreement is found for ISW-lensing estimates, some discrepancy between different studies is present for inhomogenous recombination calculations. Note how asymmetric beams and residuals of destriping change the correlation properties of the CMB temperature field, but leave it Gaussian. For this reason they can in principle affect the final error bars, but they cannot produce any bias. In addition to the effects summarized in this table, another potential source of contamination comes from foreground residuals (see Section 3.9.1). This has been shown to be negligible for WMAP, while its impact for Planck has not been discussed yet in the literature and will be assessed in the forthcoming Planck data release. The effect of point sources on WMAP $f_{NL}$ estimates has been studied in [1], where it was found that bias from unresolved point sources generates an additional contribution to the error bars of order $5$ and $22$ for local and equilateral shapes, respectively (to be compared to WMAP primordial $f_{NL}$-error bars ${\mathrm{\Delta }}_{f_{NL}}^{local}\simeq 20$ and ${\mathrm{\Delta }}_{f_{NL}}^{equilateral}\simeq 100$). Note finally how the contribution from the propagation of cosmological parameter errors on the $f_{NL}$ estimate is dependent on the measured value of $f_{NL}$ (like for point sources, the effect of cosmological parameters error propagation is to bias the estimator; however the sign and exact magnitude of this bias cannot be computed; that produces an additional uncertainty and correspondingly an additional contribution to the error bars.).

Local $\mathrm{\hspace{0.17em}\hspace{0.17em}}\mathrm{\hspace{0.17em}\hspace{0.17em}}\mathrm{\hspace{0.17em}\hspace{0.17em}}\mathrm{\hspace{0.17em}\hspace{0.17em}}({\mathrm{\Delta }}_{f_{\text{NL}}}\simeq 5)$ Equil. $({\mathrm{\Delta }}_{f_{\text{NL}}}\simeq 60)$ ISW-lensing Serra and Cooray [106] $f_{\text{bias}}\simeq 10$ $f_{\text{bias}}\simeq -3$ Hanson et al. [108] $f_{\text{bias}}\simeq 10$ ISW$+$RS-Lensing Mangilli and Verde [107] $f_{\text{bias}}\simeq 10$ Unres. Point Sources (PS) Serra and Cooray [106] $f_{\text{bias}}\simeq 1$ SZ number density modulation Babich and Pierpaoli [111] $f_{\text{bias}}\simeq -1.0$ $f_{\text{bias}}\simeq 0$ PS density modulation Babich and Pierpaoli [111] $f_{\text{bias}}\simeq -0.4$ $f_{\text{bias}}\simeq 0$ PS lensing magnification Babich and Pierpaoli [111] $f_{\text{bias}}\simeq 0.3$ $f_{\text{bias}}\simeq 0$ SZ lensing magnification Babich and Pierpaoli [111] $f_{\text{bias}}\simeq 0.02$ $f_{\text{bias}}\simeq 0$ Inhomogenous recombination Khatri and Wandelt [109] $f_{\text{bias}}\simeq -$0.1 $f_{\text{bias}}\simeq 0$ Senatore et al. [110] $f_{\text{bias}}\simeq -3.5$ $f_{\text{bias}}\simeq 0$ Boltmann-Einstein (BE) “1st $\times$ 1st” Nitta et al. [104] $f_{\text{bias}}\simeq 0.5$ BE “Early times” Pitrou et al. [105] $f_{\text{bias}}\simeq$5 $f_{\text{bias}}\simeq$ 5 Cosm. parameters uncert. Liguori and Riotto [112] $|f_{\text{bias}}|\simeq 0.05\mathrm{\hspace{0.17em}\hspace{0.17em}}{f}_{\text{NL}}^{\text{local}}$ $|f_{\text{bias}}|\simeq 0.05\mathrm{\hspace{0.17em}\hspace{0.17em}}{f}_{\text{NL}}^{\text{equil}\mathrm{\hspace{0.17em}\hspace{0.17em}}}$ Asymmetric beams Donzelli et al. [113] ${\mathrm{\Delta }}_{f_{\text{NL}}}\simeq 0$ ${\mathrm{\Delta }}_{f_{\text{NL}}}\simeq 0$ Residuals of Destriping Donzelli et al. [113] ${\mathrm{\Delta }}_{f_{\text{NL}}}\simeq 0$ ${\mathrm{\Delta }}_{f_{\text{NL}}}\simeq 0$