Abstract
The non-Gaussianity of the cosmic microwave background is a powerful probe of the physics of the early Universe. However, we do not expect all of the observed non-Gaussianity to be of primordial origin. The late-time non-linear effects described in the previous chapters will generate some degree of non-Gaussianity even in the absence of a primordial signal. This results in the emergence of an intrinsic CMB bispectrum, the topic of this chapter, which can bias the primordial signal and is interesting in its own. We first derive the formula needed to compute the intrinsic bispectrum starting from the second-order transfer functions, which is now fully implemented in our numerical code, SONG, and discuss the most popular templates for the primordial bispectrum: local, equilateral and orthogonal (Sect. 6.2). We then set to compute the signal-to-noise ratio of the intrinsic bispectrum and the bias that its presence induces in the measurements of the primordial non-Gaussianity. To do so, we adopt a Fisher matrix approach (Sect. 6.3), and find that the amplitude of the temperature intrinsic bispectrum is beyond the sensitivity of the Planck CMB survey, with a signal-to-noise ratio of \(\sim \)1/3 and biases smaller than the error bars (Sect. 6.4). We conclude the chapter with a number of numerical and analytical checks on SONG ’s results ( Sect. 6.5). These include extensive convergence tests on the most important numerical parameters in SONG and a successful comparison which the well-known analytical limit for the squeezed configurations of the bispectrum.
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- 1.
Note that from now on we shall omit writing the time dependence. This does not create ambiguity as the transfer functions \(\mathcal {T} \) are always evaluated today, \(\tau _{\small {\text {0}}} \), and the potentials \(\varPhi \) at the initial time \(\tau _\text {in} \).
- 2.
Note that we have also assumed that the Dirac delta function does not depend on \(\varvec{\hat{k}_3}\,\); we shall prove this point later in the comment to Eq. 6.30
- 3.
The adjective “angle-averaged” comes from the fact that, using Eq. A.36, \(\,B_{{\ell _1} {\ell _2} {\ell _3}} \,\) can be written as
- 4.
More precisely, our uncertainties are about \(15\!-\!20\,\%\) smaller than Planck’s. The reason is that the error budget in Planck’s analysis includes uncertainties from more subtle effects such as incomplete foreground removal. By setting \(f_\text {sky}=0.74\,\) in our Fisher matrix estimator, we obtain a percent-level match.
- 5.
Cyril Pitrou, private communication (2013).
- 6.
Note that the inclusion of the non-scalar modes should not affect the \(S/N\) of the primordial templates, because we assume that the vector and tensor modes vanish at first order. However, we can see from the Fisher matrix in Eq. 6.84 that there are differences of the order \(5\,\%\) for the local template. The reason for this discrepancy is purely numerical: in order to compute the intrinsic bispectrum for the \(m\ne 0\) modes we have adopted a different \(\ell \)-grid that contains only configurations where \({\ell _1} +{\ell _2} +{\ell _3} \) is even, as the bispectrum formula (Eq. 6.36) vanishes otherwise. The local template is the most affected one by this slightly worse grid because it is very peaked for squeezed configurations.
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Pettinari, G.W. (2016). The Intrinsic Bispectrum of the CMB. In: The Intrinsic Bispectrum of the Cosmic Microwave Background. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-21882-3_6
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