Testing Primordial Non-Gaussianity in CMB Anisotropies

TL;DR

This work derives a general analytic CMB angular bispectrum for primordial non-Gaussianity with an arbitrary momentum-dependent kernel $f_{\rm NL}(\mathbf{k_1},\mathbf{k_2},\mathbf{k_3})$, rooted in second-order perturbation theory. It shows that the momentum dependence, encapsulated via a Legendre expansion, significantly affects the observed bispectrum beyond the standard constant-$f_{\rm NL}$ models. Numerical results indicate WMAP cannot detect the primordial signal, but Planck has the potential to reveal it, especially as high-$\ell$ computations and estimators are refined. The study also highlights the need for new estimators tailored to the derived kernel structure, and points to computational challenges that motivate future approximations such as flat-sky approaches for high-resolution data.$

Abstract

Recent second-order perturbation computations have provided an accurate prediction for the primordial gravitational potential $Φ(x)$ in scenarios in which cosmological perturbations are generated either during or after inflation. This enables us to make realistic predictions for a non-Gaussian part of $Φ(x)$, which is generically written in momentum space as a double convolution of its Gaussian part with a suitable kernel, f_NL(k1,k2). This kernel defines the amplitude and angular structure of the non-Gaussian signals and originates from the evolution of second-order perturbations after the generation of the curvature perturbation. We derive a generic formula for the CMB angular bispectrum with arbitrary f_NL(k1,k2) and examine the detectability of the primordial non-Gaussian signals from various scenarios such as single-field inflation, inhomogeneous reheating, and curvaton scenarios. Our results show that in the standard slow-roll inflation scenario the signal actually comes from the momentum-dependent part of f_NL(k1,k2), and thus it is important to include the momentum dependence in the data analysis. In the other scenarios the primordial non-Gaussianity is comparable to or larger than these post-inflationary effects. We find that WMAP cannot detect non-Gaussian signals generated by these models. Numerical calculations for l>500 are still computationally expensive, and we are not yet able to extend our calculations to Planck's angular resolution; however, there is an encouraging trend which shows that Planck may be able to detect these non-Gaussian signals.

Figures (3)

• Figure 1: Radial coefficients $\alpha^{(n)}_\ell(r)$ [Eq. (\ref{['eqn:losfactor']})] at the time of decoupling, $r_*$. From top to bottom we plot $(\ell-1)\ell(\ell+1)(\ell+2)\alpha^{(-4)}_\ell(r_*)$, $\ell(\ell+1)\alpha^{(-2)}_\ell(r_*)$, $\alpha^{(0)}_\ell(r_*)$, respectively.
• Figure 2: Radial coefficients $\beta^{(n)}_l(r)$ [Eq. (\ref{['eqn:beta']})] at the time of decoupling, $r_*$. On the left side, from top to bottom: $(\ell+ {1/2}) \beta^{(1)}_{\ell \ell}(r_*)$, $\ell(\ell+1)\beta^{(0)}_{\ell \ell}(r_*)$. On the right side, from top to bottom: $\beta^{(3)}_{\ell \ell}(r_*)$, $\beta_{\ell \ell}^{(2)}(r_*)$.
• Figure 3: Values of $f_{\rm NL}^{eff}(\ell_{\rm max})$ (Eq. [\ref{['eqn:fnleffdef']}]) in the standard single-field and inhomogeneous reheating scenarios. This parameter, $f^{eff}_{\rm NL}(\ell_{\rm max})$, represents $f_{\rm NL}$ in the usual parametrization of non-Gaussianity to reproduce the same level of non-Gaussianity predicted by our model for a given $\ell_{\rm max}$. For the standard single-field inflation the contribution to non-Gaussianity comes only from the post-inflation non-linear processing of perturbations, which is independent of the inflationary model. Thus the solid line in the plots also represents that part of the non-Gaussian signal which must be present in the CMB anisotropies, regardless of the considered inflationary model. The lower panel shows our results for an experiment with beam size and noise characteristics similar to WMAP. The upper panel shows the same analysis for Planck. We are considering $\ell_\textrm{max} = 500$, corresponding to the angular resolution of WMAP. A full analysis for Planck would require $\ell_\textrm{max} = 3000$, which is beyond the current computational power (see Sec \ref{['sec:numerical']}).