# A robust measurement of the first higher-derivative bias of dark matter halos

## TL;DR

This paper introduces a new simulation method to measure the first higher-derivative bias of dark matter halos, revealing a negative bias consistent with theoretical expectations and comparing well with other models.

## Contribution

The authors develop a novel amplification technique for specific density modes in simulations, enabling precise measurement of higher-derivative halo bias.

## Key findings

- The higher-derivative bias $b_{\nabla^2 \delta}$ is negative across all halo masses.
- Measured bias roughly follows the $-R_L^2(M)$ relation, with $R_L$ as the Lagrangian radius.
- Results agree with 1-loop power spectrum fits, previous studies, and peak theory predictions.

## Abstract

We present a new simulation technique in which any chosen mode $\mathbf{k}$ of the density contrast field can be amplified by an amplitude $\Delta$. These amplified-mode simulations allow us to study the response of the halo density field to a long-wavelength mode other than the DC mode. In this sense they are a generalization of the separate-universe simulations to finite-wavelength modes. In particular, we use these simulations to obtain robust measurements of the first higher-derivative bias of dark matter halos $b_{\nabla^2 \delta}$. We find a negative bias at all mass considered, roughly following the $-R_L^2(M)$ relation, the Lagrangian radius of halos squared, as naively expected. We compare our results with those obtained from a fit to the 1-loop halo-matter power spectrum, as well as with the recent results from Abidi and Baldauf (2018), and to the prediction from the peak theory.