Measurement of the cosmic microwave background bispectrum on the COBE DMR sky maps

TL;DR

This study measures all independent configurations of the CMB bispectrum on COBE DMR four-year sky maps to test Gaussianity of primordial fluctuations and constrain non-Gaussianity. By analyzing 466 bispectrum modes up to $l_3\le 20$ and using Monte Carlo Gaussian realizations, the authors show the normalized bispectrum $B_{l_1l_2l_3}/\sqrt{C_{l_1}C_{l_2}C_{l_3}}$ is largely Gaussian, with no significant mode-by-mode detections and the previously claimed $l_1=l_2=l_3=16$ signal attributable to statistical fluctuation. They fit a primary inflationary bispectrum parameterized by $f_{\rm NL}$ and account for foregrounds from Galactic dust and synchrotron, finding no significant foreground contamination and only a weak constraint on $|f_{\rm NL}|$ (of order $10^3$). The work highlights the impact of incomplete sky coverage on bispectrum sensitivity and anticipates stronger constraints from upcoming missions like MAP and Planck. Overall, the DMR maps remain consistent with Gaussian primordial fluctuations, reinforcing the standard slow-roll inflation paradigm while setting the stage for tighter future tests of non-Gaussianity.

Abstract

We measure the angular bispectrum of the cosmic microwave background (CMB) radiation anisotropy from the COBE Differential Microwave Radiometer (DMR) four-year sky maps. The angular bispectrum is the harmonic transform of the three-point correlation function, analogous to the angular power spectrum, the harmonic transform of the two-point correlation function. First, we study statistical properties of the bispectrum and the normalized bispectrum. We find the latter more useful for statistical analysis; the distribution of the normalized bispectrum is very much Gaussian, while the bare bispectrum distribution is highly non-Gaussian. Then, we measure 466 modes of the normalized bispectrum, all independent combinations of three-point configurations up to a maximum multipole of 20, the mode corresponding to the DMR beam size. By measuring 10 times as many modes as the sum of previous work, we test Gaussianity of the DMR maps. We compare the data with the simulated Gaussian realizations, finding no significant detection of the normalized bispectrum on the mode-by-mode basis. We also find that the previously reported detection of the normalized bispectrum is consistent with a statistical fluctuation. By fitting a theoretical prediction to the data for the primary CMB bispectrum, which is motivated by slow-roll inflation, we put a weak constraint on a parameter characterizing non-linearity in inflation. Simultaneously fitting the foreground bispectra estimated from interstellar dust and synchrotron template maps shows that neither dust nor synchrotron emissions significantly contribute to the bispectrum at high Galactic latitude. We conclude that the DMR map is consistent with Gaussianity.

Figures (8)

• Figure 1: Comparison of the variance of the normalized bispectrum, $\left<B_{l_1l_2l_3}^2/\left(C_{l_1}C_{l_2}C_{l_3}\right)\right>$, with that of the bare bispectrum, $\left<B_{l_1l_2l_3}^2\right>/ \left(\left<C_{l_1}\right>\left<C_{l_2}\right>\left<C_{l_3}\right>\right)$. The top-left panel shows the case of full sky coverage, while the rest of panels show the cases of incomplete sky coverage. The top-right, bottom-left, and bottom-right panels use the $20^\circ$ cut, the extended cut, and the $25^\circ$ cut, respectively.
• Figure 2: Distribution of the bispectrum drawn from the Monte--Carlo simulations for the $20^\circ$ Galactic cut (solid lines). $B_{l_1l_2l_3}/\left(\left<C_{l_1}\right> \left<C_{l_2}\right>\left<C_{l_3}\right>\right)^{1/2}$ is plotted, where the brackets denote the ensemble average over realizations from the Monte--Carlo simulations. The dashed lines plot Gaussian distributions calculated from r.m.s. values. Each panel represents a certain mode of $(l_1,l_2,l_3)$ as quoted in the panels.
• Figure 3: Distribution of the normalized bispectrum, $B_{l_1l_2l_3}/\left(C_{l_1}C_{l_2}C_{l_3}\right)^{1/2}$, drawn from the Monte--Carlo simulations for the $20^\circ$ Galactic cut. The meaning of the lines is the same as in figure \ref{['fig:dist_bare']}.
• Figure 4: $P$ distribution (Eq.(\ref{['eq:significance']})). $P$ is the probability of the CMB normalized bispectrum, $B_{l_1l_2l_3}/\left(C_{l_1}C_{l_2}C_{l_3}\right)^{1/2}$, measured on the COBE DMR $53+90~{\rm GHz}$ sky map, being larger than those drawn from the Monte--Carlo simulations. There are 466 modes in total. The thick dashed, solid, and dotted lines represent the three different Galactic cuts as quoted in the figure. The thin solid line shows the expectation value for a Gaussian field. The top panel shows the $P$ distribution, while the bottom panel shows the cumulative $P$ distribution, for which we calculate the KS statistic. The KS statistic gives the probability of the distribution being consistent with the expectation for Gaussianity as 6.7%, 73%, and 77% for the three Galactic cuts, respectively.
• Figure 5: Constraint on the non-linear coupling parameter, $f_{\rm NL}$, which characterizes non-linearity in inflation (Eq.(\ref{['eq:modelreal']})). The dashed, solid, and dotted lines represent the three different Galactic cuts as quoted in the figure. The thick vertical lines plot the measured values of $f_{\rm NL}$ from the COBE DMR maps, while the histograms plot those drawn from the Monte--Carlo simulations for each cut. 68% confidence limits on $f_{\rm NL}$ are $\left|f_{\rm NL}\right|<1.6\times 10^3$, $1.5\times 10^3$, and $1.7\times 10^3$ for the three cuts, respectively.