Cubic Halo Bias in Eulerian and Lagrangian Space

TL;DR

This work advances halo bias modeling by extending a quadratic-field method to cubic fields to measure bias parameters up to cubic order in both Eulerian and Lagrangian spaces. By orthogonalizing cubic-field cross-correlations to remove UV-sensitive components and introducing a Taylor counterterm around the halo smoothing scale, the authors obtain robust detections of local and non-local cubic bias, and directly constrain protohalo Lagrangian biases. The results reveal a non-local Lagrangian tidal bias and show that Eulerian cubic non-local terms are in tension with predictions from local Lagrangian bias evolution, implying a significant role for gravity-driven co-evolution. The methodology enables precise one-loop halo statistics and offers a path to probing cubic primordial non-Gaussianity, though it requires careful handling of smoothing-scale dependencies and higher-derivative terms for massive halos.

Abstract

Predictions of the next-to-leading order, i.e. one-loop, halo power spectra depend on local and non-local bias parameters up to cubic order. The linear bias parameter can be estimated from the large scale limit of the halo-matter power spectrum, and the second order bias parameters from the large scale, tree-level, bispectrum. Cubic operators would naturally be quantified using the tree-level trispectrum. As the latter is computationally expensive, we extent the quadratic field method proposed in Schmittfull et al. 2014 to cubic fields in order to estimate cubic bias parameters. We cross-correlate a basis set of cubic bias operators with the halo field and express the result in terms of the cross-spectra of these operators in order to cancel cosmic variance. We obtain significant detections of local and non-local cubic bias parameters, which are partially in tension with predictions based on local Lagrangian bias schemes. We directly measure the Lagrangian bias parameters of the protohaloes associated with our halo sample and clearly detect a non-local quadratic term in Lagrangian space. We do not find a clear detection of non-local cubic Lagrangian terms for low mass bins, but there is some mild evidence for their presence for the highest mass bin. While the method presented here focuses on cubic bias parameters, the approach could also be applied to quantifications of cubic primordial non-Gaussianity.

Figures (15)

• Figure 1: Theoretical predictions of bias parameters are obtained from the co-evolution of the local Lagrangian bias model (LLB) and the co-evolution of the Lagrangian bias model with the non-zero tidal term $b^{\text{L}}_{s2}$ at second order. The mass dependence of the initial Lagrangian bias is defined in Eq. \ref{['Ltidal']}.
• Figure 2: Diagrammatic representation of $\langle \mathcal{D}_{2}(\boldsymbol{k})|\delta^{(2)}(\boldsymbol{k}')\rangle'$.
• Figure 3: Perturbative expressions for one-loop, and two-loop irreducible and two-loop reducible terms of $\langle \mathcal{D}_{3}(\boldsymbol{k})|\delta_{\text{NL}}(\boldsymbol{k}') \rangle$ are shown in diagrammatic form. The propagators are represented by linear power spectra $P_{\text{lin}}$, the cubic field kernel $\mathcal{D}_{3}$ is represented by the hatched square. Finally, empty squares correspond to the gravitational kernel $F_{3}$. Loops correspond to integrals over all wavenumbers $\boldsymbol{q}$ or $\boldsymbol{p}$ and arrows represent the flow of momentum.
• Figure 4: Diagrams contributing to the correlation of cubic fields with the halo field in Eq. \ref{['D3dh']}. The triangles represent the linear, quadratic and cubic bias kernels. The straight lines are used to describe the density field, whereas halo fields are described by wiggly lines. The Feynman rules are discussed in detail in Baldauf:2010vn.
• Figure 5: Cross-correlations of cubic fields with the orthogonalized fields contain only the two-loop irreducible diagram. The figure shows PT diagrams for $\langle D_3|\widetilde{F}_3 \rangle$ (left) and $\langle D_3|\widetilde{\mathcal{O}}_{3}^{j} \rangle$(right).