Discreteness effects in $N$-body simulations of warm dark matter

In cosmological $N$-body simulations of warm dark matter, thermal velocities of dark-matter particles are sometimes taken into account by adding random initial velocities to the particles of simulation. However, a particle in the $N$-body system represents a huge collection of dark-matter particles, whose average thermal velocity is very close to zero. We consider justification of the procedure of adding thermal velocities in $N$-body simulations and build a simple model of their influence on the power spectrum. Our model captures the physical effect of suppression of the power spectrum at small wave numbers and also explains its artificial enhancement at large wave numbers, observed in numerical simulations with added thermal velocities. The cause of this enhancement is the disturbance of the growth rate of the density profile introduced when adding random initial thermal velocities. Specifically, the model predicts a turnover in the behavior of the simulated power spectrum at a certain wave number $k_{*}$, beyond which it grows as $P\left(k\right)\propto k^2$. Our treatment is generalized to a system consisting of several matter components with different thermal velocity dispersion. We also estimate the effects of discreteness related to the bulk velocity field and establish the conditions under which these effects dominate over those of thermal velocities.