Angular Trispectrum of CMB Temperature Anisotropy from Primordial Non-Gaussianity with the Full Radiation Transfer Function

Abstract

We calculate the cosmic microwave background (CMB) angular trispectrum, spherical harmonic transform of the four-point correlation function, from primordial non-Gaussianity in primordial curvature perturbations characterized by a constant non-linear coupling parameter, $f_{\rm NL}$. We fully take into account the effect of the radiation transfer function, and thus provide the most accurate estimate of the signal-to-noise ratio of the angular trispectrum of CMB temperature anisotropy. We find that the predicted signal-to-noise ratio of the trispectrum summed up to a given $l$ is approximately a power-law, $(S/N)(<l)\sim 2.2\times 10^{-9}f^2_{\rm NL}l^2$, up to the maximum multipole that we have reached in our numerical calculation, $l=1200$, assuming that the error is dominated by cosmic variance. Our results indicate that the signal-to-noise ratio of the temperature trispectrum exceeds that of the bispectrum at the critical multipole, $l_c \sim 1500~(50/|f_{\rm NL}|)$. Therefore, the trispectrum of the Planck data is more sensitive to primordial non-Gaussianity than the bispectrum for $|f_{\rm NL}|\gtrsim 50$. We also report the predicted constraints on the amplitude of trispectrum, which may be useful for other non-Gaussian models such as curvaton models.

Figures (2)

• Figure 1: Signal-to-noise ratio squared, $(S/N)^2$, of the angular trispectrum summed over all $L$, $L \le 10$, and only $L=1$ modes, respectively, as a function of the maximum multipole, $l_{\rm max}$. Here, $L$ denotes the multipole of the diagonal of a trispectrum configuration. We assume $f_{\rm NL}=1$ and $f_2=1$ and use the full radiation transfer function. This figure shows that the summation over $L$ needs to be done only up to $L\simeq 10$.
• Figure 2: Predicted signal-to-noise ratio squared, $(S/N)^2$, of the angular trispectrum and bispectrum with the full radiation transfer function for $f_{\rm NL}=50$ and $f_2=1$. Power-law fits are also shown. Note that the power-law fits break down at $l_{\rm max}\gtrsim 3000$, where the gravitational lensing effects become important: $(S/N)^2$ ceases to grow at $l_{\rm max}\sim 3000$KS01BZ04.