Primordial non-Gaussianity in the Bispectrum of the Halo Density Field

TL;DR

The paper develops a diagrammatic, tree-level framework to predict the halo bispectrum including non-linear clustering, non-linear bias, and local primordial non-Gaussianity. It shows that non-Gaussian corrections to bias parameters amplify the tree-level halo bispectrum, yielding larger signals in squeezed configurations and enabling stronger f_NL constraints than the power spectrum. The authors validate the approach against simulations and demonstrate that the halo bispectrum can achieve sigma_fNL around 5 for optimistic surveys, significantly improving over power-spectrum analyses. The method combines a multivariate bias expansion with peak-background split and a consistent perturbative treatment, making the halo bispectrum a promising probe for inflationary physics.

Abstract

The bispectrum vanishes for linear Gaussian fields and is thus a sensitive probe of non-linearities and non-Gaussianities in the cosmic density field. Hence, a detection of the bispectrum in the halo density field would enable tight constraints on non-Gaussian processes in the early Universe and allow inference of the dynamics driving inflation. We present a tree level derivation of the halo bispectrum arising from non-linear clustering, non-linear biasing and primordial non-Gaussianity. A diagrammatic description is developed to provide an intuitive understanding of the contributing terms and their dependence on scale, shape and the non-Gaussianity parameter fNL. We compute the terms based on a multivariate bias expansion and the peak-background split method and show that non-Gaussian modifications to the bias parameters lead to amplifications of the tree level bispectrum that were ignored in previous studies. Our results are in a good agreement with published simulation measurements of the halo bispectrum. Finally, we estimate the expected signal to noise on fNL and show that the constraint obtainable from the bispectrum analysis significantly exceeds the one obtainable from the power spectrum analysis.

Figures (15)

• Figure 1: Left panel: Poisson factor $\alpha(k)$ relating density and potential via $\delta(k)=\alpha(k) \Phi(k)$. On scales of $k \approx 0.1 \ h\text{Mpc}^{-1}$ the potential is smaller than the density by a factor of $10^5$, whereas they are equal on very large scales of $k\approx 2\times 10^{-4} \ h\text{Mpc}^{-1}$. The non-Gaussian corrections generally scale as $f_\text{NL}/\alpha(k)$ and are thus suppressed on high $k$'s. Note that the Poisson equation in Newtonian gauge receives general relativistic corrections as $\alpha(k)$ approaches unity. Right panel: Skewness $C_{3}=\sigma S_3$ and its first and second mass derivatives evaluated for $f_\text{NL}=1$ and $\varphi_\text{l}=0$.
• Figure 2: Eulerian multivariate bias parameters for $f_\text{NL}=100$, $g_\text{NL}=0$. Left panel: Bias parameters for $\delta$. Right panel: Bias parameters for $\varphi$. For $g_\text{NL}=0$ one has $b_{02}\propto f_\text{NL}^2$, $b_{01}\propto f_\text{NL}$, $b_{11}\propto f_\text{NL}$ and thus the rescaled bias parameters are independent of $f_\text{NL}$. $b_{02}$ shown by the dashed line shows a pronounced minimum for $M\approx 1\times 10^{14}\ h^{-1} M_\odot$ that is multiplied by $f_\text{NL}^2$ and can thus lead to a large contribution. The red lines are derived from the ST mass function, whereas the blue thin lines are derived from the LV mass function including the explicit $\varphi_\text{l}$ correction of Eqs. \ref{['eq:db01']}, \ref{['eq:db11']} and \ref{['eq:db02']}.
• Figure 3: Symbols use to represent the fields and power spectra. From left to right, the primordial Gaussian potential $\varphi$, the matter density field $\delta_\text{m}$, the galaxy/halo density field $\delta_\text{h}$, and the power spectrum $P(k)$ arising from two linked fields.
• Figure 4: Non-linear gravitational clustering leads to a convolution integral weighted by a $F_i$ kernel, which we symbolize by a square shaped vertex. From left to right we show the first, second and third order mode coupling contributions to the matter density field. Note that $F_1\equiv 1$.
• Figure 5: The multivariate bias expansion can be expressed by the triangular vertices shown above. The interaction of two fields on a vertex corresponds to a convolution integral over the ingoing $k$-vectors without any weighting. The vertex is connected to the outer point by the wiggly halo propagator. Ingoing potentials are always primordial because the coupling of long and short modes, and thus the enhancement of the short wavelength variance, happens in the early Universe.