CMB Bispectrum from Primordial Scalar, Vector and Tensor non-Gaussianities

TL;DR

This paper extends CMB bispectrum analysis to include primordial non-Gaussianities from vector and tensor modes, providing an all-sky formalism that preserves the angular and polarization structure of these perturbations. It develops general expressions for the CMB bispectrum sourced by scalar, vector, and tensor NG, and applies the framework to a Maldacena-type two-scalar–one-graviton coupling with coupling $g_{tss}$, yielding a rotationally invariant all-sky bispectrum built from spin-weighted harmonics and Wigner symbols. The calculation shows that the tensor-scalar-scalar bispectrum has a squeezed-like shape, with a reduced bispectrum scaling $|b^{(TSS)}_{III,\ell\ell\ell}+\dots| \sim \ell^{-4} \times 8\times 10^{-18} |g_{tss}|$, and forecasts Planck-level constraints $|g_{tss}|\lesssim 6$, illustrating the utility of including vector/tensor NG in CMB analyses. The approach is general and applicable to other sources of vector/tensor NG, offering a principled path to constrain early-universe physics beyond standard scalar NG models.

Abstract

We present an all-sky formalism for the Cosmic Microwave Background (CMB) bispectrum induced by the primordial non-Gaussianities not only in scalar but also in vector and tensor fluctuations. We find that the bispectrum can be formed in an explicitly rationally invariant way by taking into account the angular and polarization dependences of the vector and tensor modes. To demonstrate this and present how to use our formalism, we consider a specific example of the correlation between two scalars and a graviton as the source of non-Gaussianity. As a result, we show that the CMB reduced bispectrum of the intensity anisotropies is evaluated as a function of the multipole and the coupling constant between two scalars and a graviton denoted by $g_{tss}$; $|b_{\ell \ell \ell}| \sim \ell^{-4} \times 8 \times 10^{-18} |g_{tss}|$. By estimating the signal-to-noise ratio, we find that the constraint as $|g_{tss}| < 6$ will be expected from the PLANCK experiment.

Figures (4)

• Figure 1: (color online) Shape of $I / k_1$. For the symmetric property and the triangle condition, we limit the plot range as $k_1 \leq k_2 \leq k_3$ and $|k_1 - k_2| \leq k_3 \leq k_1 + k_2$.
• Figure 2: (color online) Absolute values of the CMB reduced bispectra of temperature fluctuation for $\ell_1 = \ell_2 = \ell_3$. The lines correspond to the spectra generated from tensor-scalar-scalar correlation given by Eq. (\ref{['eq:mal_cmb_red_bis']}) with $g_{tss} = 5$ (red solid line) and the primordial non-Gaussianity in the scalar curvature perturbations with $f^{\rm local}_{\rm NL} = 5$ (green dashed line). The other cosmological parameters are fixed to the mean values limited from WMAP-7yr data reported in Ref. Komatsu:2010fb.
• Figure 3: (color online) Absolute values of the CMB reduced bispectra of temperature fluctuation generated from tensor-scalar-scalar correlation given by Eq. (\ref{['eq:mal_cmb_red_bis']}) $(TSS)$ and the primordial non-Gaussianity in the scalar curvature perturbations $(SSS)$ as a function of $\ell_3$ with $\ell_1$ and $\ell_2$ fixed to some values as indicated. The parameters are fixed to the same values defined in Fig. \ref{['fig:SST_III_samel']}.
• Figure 4: (color online) Signal-to-noise ratio normalized by $g_{tss}$ as a function of the maximum value between $\ell_1, \ell_2$ and $\ell_3$, namely, $\ell$. Each parameter is fixed to the same values defined in Fig. \ref{['fig:SST_III_samel']}.