# The Effect of Thermal Velocities on Structure Formation in N-body Simulations of Warm Dark Matter

## TL;DR

This paper examines how adding thermal velocities in warm dark matter N-body simulations introduces numerical noise, affecting early-time structure formation and halo properties, with effects diminishing at later times.

## Contribution

It provides a detailed analysis of the impact of thermal velocities on simulation accuracy, highlighting the numerical noise introduced and its influence on structure formation.

## Key findings

- Thermal velocities cause significant early-time deviations in power spectra.
- Increased resolution shifts discrepancies to smaller scales.
- Spurious haloes appear at higher masses when thermal velocities are included.

## Abstract

We investigate the impact of thermal velocities in N-body simulations of structure formation in warm dark matter models. Adopting the commonly used approach of adding thermal velocities, randomly selected from a Fermi-Dirac distribution, to the gravitationally-induced velocities of the simulation particles, we compare the matter and velocity power spectra measured from CDM and WDM simulations, in the latter case with and without thermal velocities. This prescription for adding thermal velocities introduces numerical noise into the initial conditions, which influences structure formation. At early times, the noise affects dramatically the power spectra measured from simulations with thermal velocities, with deviations of the order of $\sim \mathcal{O}(10)$ (in the matter power spectra) and of the order of $\sim \mathcal{O}(10^2)$ (in the velocity power spectra) compared to those extracted from simulations without thermal velocities. At late times, these effects are less pronounced with deviations of less than a few percent. Increasing the resolution of the N-body simulation shifts these discrepancies to higher wavenumbers. We also find that spurious haloes start to appear in simulations which include thermal velocities at a mass that is $\sim$3 times larger than in simulations without thermal velocities.