Non-Gaussian Halo Bias Beyond the Squeezed Limit

TL;DR

This work extends the non-Gaussian halo bias framework beyond the squeezed limit by developing a renormalized, tracer-agnostic bias expansion that remains valid at smaller scales. By representing the primordial bispectrum in a separable form and expanding in $k/k_1$, the authors derive leading and subleading contributions to the scale-dependent bias, expressed as $\Delta b(k)=\sum_\alpha A_\alpha [ b^{(\alpha,23)}_{010} S^{(1)}_\alpha(k) + 12 b^{(\alpha,23)}_{001} k^2 S^{(1)}_\alpha(k) + \text{cyclic}]$, with $S_\alpha^{(i)}(k)=F^{(i)}_\alpha(k)M_L(k)P_\phi(k)$. They show that capturing subleading effects requires a tri-variate bias parameter $\mu_*$ to absorb shape dependence and remove residual $R_L$-dependence; for local non-Gaussianity and universal mass functions, the subleading term becomes relevant at $k\gtrsim 0.02\,h\,\mathrm{Mpc}^{-1}$, while the combined leading+subleading terms remain accurate up to $k\lesssim 0.1\,h\,\mathrm{Mpc}^{-1}$. The results reconcile with conditional-PS mass-function results in the universal case and reveal additional scale-dependent contributions from nonlocal tracer formation, underscoring the need to include these effects when constraining $f_{NL}$ from intermediate-scale LSS data.

Abstract

Primordial non-Gaussianity, in particular the coupling of modes with widely different wavelengths, can have a strong impact on the large-scale clustering of tracers through a scale-dependent bias with respect to matter. We demonstrate that the standard derivation of this non-Gaussian scale-dependent bias is in general valid only in the extreme squeezed limit of the primordial bispectrum, i.e. for clustering over very large scales. We further show how the treatment can be generalized to describe the scale-dependent bias on smaller scales, without making any assumptions on the nature of tracers apart from a dependence on the small-scale fluctuations within a finite region. If the leading scale-dependent bias Δb \propto k^α, then the first subleading term will scale as k^{α+2}. This correction typically becomes relevant as one considers clustering over scales k >~ 0.01 h Mpc^{-1}.

Figures (3)

• Figure 1: Contributions to the scale-dependent bias from local non-Gaussianity ($f_{\rm NL}=1$) for halos with $M=2\cdot 10^{13}\,h^{-1}\,M_{\odot}$ at $z=0$ ($b_{100}\simeq 0.44$) and assuming a universal mass function from the Sheth-Tormen prescription sheth/tormen:1999, scaled by $(k/H_0)^2$ to yield a scale-independent value on large scales. The dashed line shows the leading term, the solid the leading plus subleading (order $(k/k_*)^2$) term, while the dash-dotted line includes the order $(k/k_*)^3$ term. The red long-dashed line shows the prediction from the conditional PS mass function (Sec. \ref{['sec:thr']}).
• Figure 2: Leading and sub-leading contributions to the scale-dependent bias from local non-Gaussianity, as in Fig. \ref{['fig:Db']}, but for different masses at $z=0$. The curves shown correspond to, from top to bottom, $M = 2\cdot 10^{14}$, $10^{14}$, $5\cdot 10^{13}$, $2\cdot 10^{13}$, $10^{13}$, $5 \cdot 10^{12}$, $2\cdot 10^{12}$, and $10^{12}\,h^{-1}\,M_{\odot}$, respectively.
• Figure 3: Fractional difference between the contributions in Eq. (\ref{['eq:Phtotloc']}), evaluated for a universal mass function, and the result for the conditional PS mass function Eq. (\ref{['eq:Db_thr']}) from long for local primordial non-Gaussianity. The black line solid line shows the residuals when including the leading and subleading (order $(k/k_*)^2$) contributions, while the green dash-dotted line also includes the order $(k/k_*)^3$ contribution from the last line of Eq. (\ref{['eq:Phtotloc']}). The dotted line shows a rough estimate of the order $(k/k_*)^4$ correction (see text). We have again assumed $M = 2\cdot 10^{13}\,h^{-1}\,M_{\odot}$ and $z=0$, although the results are essentially independent of the mass.