Acoustic Signatures in the Primary Microwave Background Bispectrum

TL;DR

This work computes the primary CMB bispectrum using the complete radiation transfer function and a local $f_{NL}$-type primordial non-Gaussianity model. The reduced bispectrum $b_{l_1l_2l_3}$ reveals acoustic peaks with sign changes, and its amplitude scales as ${b^{primary}_{l_1l_2l_3}\propto f_{NL}}$, with typical size ${\sim 10^{-17}}$ at high multipoles. The authors show that the bispectrum provides a far more powerful probe of non-Gaussianity than skewness, and they quantify detectability for COBE, MAP, Planck, and an ideal experiment via a Fisher-matrix analysis, including secondary foregrounds such as SZ-lensing and point sources. They demonstrate that Planck can separate the primary bispectrum from secondary bispectra by shape, while the simplest inflationary models typically predict very small ${f_{NL}}$, so a detection would challenge standard slow-roll scenarios. Overall, the primary bispectrum serves as a stringent test of inflation and a window into early-universe non-linear physics, with practical implications for foreground separation and data analysis in upcoming CMB experiments.

Abstract

If the primordial fluctuations are non-Gaussian, then this non-Gaussianity will be apparent in the cosmic microwave background (CMB) sky. With their sensitive all-sky observation, MAP and Planck satellites should be able to detect weak non-Gaussianity in the CMB sky. On large angular scale, there is a simple relationship between the CMB temperature and the primordial curvature perturbation. On smaller scales; however, the radiation transfer function becomes more complex. In this paper, we present the angular bispectrum of the primary CMB anisotropy that uses the full transfer function. We find that the bispectrum has a series of acoustic peaks that change a sign, and a period of acoustic oscillations is twice as long as that of the angular power spectrum. Using a single non-linear coupling parameter to characterize the amplitude of the bispectrum, we estimate the expected signal-to-noise ratio for COBE, MAP, and Planck experiments. We find that the detection of the primary bispectrum by any kind of experiments should be problematic for the simple slow-roll inflationary scenarios. We compare the sensitivity of the primary bispectrum to the primary skewness and conclude that when we can compute the predicted form of the bispectrum, it becomes a ``matched filter'' for detecting the non-Gaussianity in the data, and much more powerful tool than the skewness. We also show that MAP and Planck can separate the primary bispectrum from various secondary bispectra on the basis of the shape difference. The primary CMB bispectrum is a test of the inflationary scenario, and also a probe of the non-linear physics in the very early universe.

Figures (5)

• Figure 1: This figure shows $b_l^L(r)$ (Eq.(\ref{['eq:bLr']})) and $b_l^{NL}(r)$ (Eq.(\ref{['eq:bNLr']})), the two terms in our calculation of the primary CMB angular bispectrum, as a function of $r$. Various lines in the upper panel show $\left[l(l+1)b_l^L(r)/2\pi\right]\times 10^{10}$, where $r=c\left(\tau_0-\tau\right)$, at $\tau=0.4,0.6,0.8,1.0,1.2,1.4$, and $1.6\times \tau_*$ (decoupling time), while $\left[b_l^{NL}(r)f^{-1}_{NL}\right]\times 10^{10}$ are shown in the lower panel. $\tau_0$ is the present-day conformal time. Note that $c\tau_0=11.8\ {\rm Gpc}$, and $c\tau_*=235\ {\rm Mpc}$ in our cosmological model chosen here. The thickest solid line in the upper panel is the CMB angular power spectrum $\left[l(l+1)C_l/2\pi\right]\times 10^{10}$. $C_l$ is shown for comparison.
• Figure 2: The primary CMB angular bispectrum (Eq.(\ref{['eq:almspec']})) divided by the Gaunt integral ${\cal G}_{l_1l_2l_3}^{m_1m_2m_3}$ (Eq.(\ref{['eq:gaunt']})). The upper panel shows $\left[l_2(l_2+1)l_3(l_3+1)\left< a_{l_1m_1}^{NL} a_{l_2m_2}^L a_{l_3m_3}^L\right>f_{NL}^{-1} \left({\cal G}_{l_1l_2l_3}^{m_1m_2m_3}\right)^{-1}/(2\pi)^2\right] \times 10^{19}$, while the lower panel shows $\left[l_1(l_1+1)l_2(l_2+1)\left< a_{l_1m_1}^{L} a_{l_2m_2}^L a_{l_3m_3}^{NL}\right>f_{NL}^{-1} \left({\cal G}_{l_1l_2l_3}^{m_1m_2m_3}\right)^{-1}/(2\pi)^2\right] \times 10^{19}$. Those are shown as functions of $l_3$ for $(l_1,l_2)=(9,11),(99,101),(199,201)$, and $(499,501)$.
• Figure 3: The upper panel shows the primary CMB skewness (Eq.(\ref{['eq:skewness']})) summed up to a certain $l_3$, $-S_3(<l_3)f_{NL}^{-1}\times 10^{15}$. The lower panel shows the noise (Eq.(\ref{['eq:skewvar']})) summed up to $l_3$, $\sigma_{S_3}(<l_3)\times 10^{15}$. Solid line represents the zero-noise ideal experiment, while dotted lines show COBE, MAP, and Planck experiments.
• Figure 4: The predictions of the signal-to-noise ratio, $S/N$, for COBE, MAP, and Planck experiments (see Eq.(\ref{['eq:sn']})). The differential $S/N$ at $\ln l_3$ interval is shown in the upper panel, while the cumulative $S/N$ up to a certain $l_3$ is shown in the lower panel. Both are in units of $f_{NL}$. Solid line represents the zero-noise ideal experiment, while dotted lines show the realistic experiments mentioned above. The total $(S/N)f^{-1}_{NL}$ are $1.7\times 10^{-3}$, $5.8\times 10^{-2}$, and $0.19$ for COBE, MAP, and Planck experiments, respectively.
• Figure 5: The comparison of the signal-to-noise ratio summed up to a certain $l_3$, $S/N(<l_3)$, for the detection of the bispectrum (upper panel; Eq.(\ref{['eq:sn']})) and the skewness (lower panel; Eq.(\ref{['eq:skew_sn']})) in units of $f_{NL}$ for COBE, MAP, and Planck experiments (dotted lines), and the ideal experiment (solid line). See table \ref{['tab:fnl']} for values of $f_{NL}$ in order to obtain $S/N>1$.