Primordial non-Gaussianity, scale-dependent bias, and the bispectrum of galaxies

TL;DR

This work provides a comprehensive treatment of how local-type primordial non-Gaussianity imprints on the galaxy bispectrum. By applying the MLB peak formalism and a local bias expansion, the authors derive a complete halo bispectrum that includes five physically distinct contributions, notably introducing the new f_NL^2 (τ_NL) and g_NL terms from the primordial trispectrum. They show that these non-Gaussian terms produce their largest signals in squeezed configurations and can dominate over Gaussian and nonlinear terms at high redshift, challenging the practice of substituting scale-dependent bias into the bispectrum. The results offer a powerful avenue to constrain inflationary models, particularly multi-field scenarios, by measuring the squeezed-bispectrum and testing τ_NL versus f_NL relations, with high-redshift surveys providing the most leverage. Overall, the paper demonstrates that the galaxy bispectrum is a more sensitive probe of primordial non-Gaussianity than previously recognized and cannot be captured by a simple scale-dependent bias alone.

Abstract

We calculate the bispectrum, B_g(k_1,k_2,k_3), Fourier transform of the three-point function of density peaks (e.g., galaxies), using two different methods: the Matarrese-Lucchin-Bonometto formula and the locality of galaxy bias. The bispectrum of peaks is not only sensitive to that of the underlying matter density fluctuations, but also to the four-point function. For a physically-motivated, local form of primordial non-Gaussianity in the curvature perturbation, we show that the galaxy bispectrum contains five physically distinct pieces: (i) non-linear gravitational evolution, (ii) non-linear galaxy bias, (iii) f_nl, (iv) f_nl^2, and (v) \gnl. While (i), (ii), and a part of (iii) have been derived in the literature, (iv) and (v) are derived in this paper for the first time. Our finding suggests that the galaxy bispectrum is more sensitive to f_nl than previously recognized, and is also sensitive to a new term, g_nl. For a more general form of local-type non-Gaussianity, the coefficient \fnl^2 can be interpreted as τ_nl, which allows us to test multi-field inflation models. The usual terms from Gaussian initial conditions, have the smallest signals in the squeezed configurations, while the others have the largest signals; thus, we can distinguish them easily. We cannot interpret the effects of f_nl on B_g(k_1,k_2,k_3) as a scale-dependent bias, and thus replacing the linear bias in the galaxy bispectrum with the scale-dependent bias known for the power spectrum results in an incorrect prediction. As the importance of primordial non-Gaussianity relative to the non-linear gravity evolution and galaxy bias increases toward higher redshifts, galaxy surveys probing a high-redshift universe are particularly useful for probing the primordial non-Gaussianity.

Figures (14)

• Figure 1: Visual representations of triangles forming the bispectrum, $B(k_1,k_2,k_3)$, with various combinations of wavenumbers satisfying $k_3\le k_2\le k_1$.
• Figure 2: Shape of the bispectrum, $B(k_1,k_2,k_3)$. Each panel shows the amplitude of the bispectrum as a function of $k_2/k_1$ and $k_3/k_1$ for a given $k_1$, with a condition that $k_3\le k_2\le k_1$ is satisfied. The amplitude is normalized such that it is unity at the point where the bispectrum takes on the maximum value. For the visual representations of the triangle names such as the squeezed, elongated, folded, isosceles, and equilateral, see Fig. \ref{['fig:triangles']}. (Top Left) The bispectrum from the non-linear gravitational evolution, $B_m^G$ (Eq. (\ref{['eq:bmg']})), for $k_1=0.01~h~{\rm Mpc}^{-1}$. (Top Right) $B_m^G$ for $k_1=0.05~h~{\rm Mpc}^{-1}$. (Bottom Left) The bispectrum from the non-linear galaxy biasing, $P_R(k_1)P_R(k_2)+(2~\hbox{cyclic})$ (the second term in Eq. (\ref{['eq:bisp_nG2']})), for $k_1=0.01~h~{\rm Mpc}^{-1}$. (Bottom Right) $P_R(k_1)P_R(k_2)+(2~\hbox{cyclic})$ for $k_1=0.05~h~{\rm Mpc}^{-1}$.
• Figure 3: Same as the top panels of Fig. \ref{['fig:bkG']}, but for $B_m^G/[P_R(k_1)P_R(k_2)+(2~\hbox{cyclic})]$ (Eq. (\ref{['eq:qh']})).
• Figure 4: Same as Fig. \ref{['fig:bkG']}, but for the terms proportional to $f_\mathrm{NL}$. (Top) the $B_{m}^{nG}$ term (Eq. (\ref{['eq:BmnG']})), (Middle) the $B_{f_\mathrm{NL}}^{nG0}$ term (Eq. (\ref{['eq:bfnl']})), and (Bottom) the $B_{f_\mathrm{NL}}^{nG1}$ term (Eq. (\ref{['eq:BfnlnG1']})). Note that the non-Gaussian terms diverge in the exact squeezed limit, $k_3\to 0$; thus, we show these terms normalized to be unity at $k_3/k_1=10^{-2}$. In order to facilitate the comparison better, we draw the dotted contour for all six panels.
• Figure 5: Same as Fig. \ref{['fig:bkG']}, but for (Top) the $g_\mathrm{NL}$ term (Eq. (\ref{['eq:bgnl']})), and (Bottom) the $f_\mathrm{NL}^2$ term (Eq. (\ref{['eq:bfnl2']})). Note that the non-Gaussian terms diverge in the exact squeezed limit, $k_3\to 0$; thus, we show these terms normalized to be unity at $k_3/k_1=10^{-2}$. In order to facilitate the comparison better, we draw the dotted contour for top panels.