The rise and fall of a challenger: the Bullet Cluster in $\Lambda$ Cold Dark Matter simulations

The Bullet Cluster has provided some of the best evidence for the $\Lambda$ cold dark matter ($\Lambda\rm{CDM}$) model via direct empirical proof of the existence of collisionless dark matter, while posing a serious challenge owing to the unusually high inferred pairwise velocities of its progenitor clusters. Here we investigate the probability of finding such a high-velocity pair in large-volume N-body simulations, particularly focusing on differences between halo finding algorithms. We find that algorithms that do not account for the kinematics of infalling groups yield vastly different statistics and probabilities. When employing the ROCKSTAR halo finder that considers particle velocities, we find numerous Bullet-like pair candidates that closely match not only the high pairwise velocity, but also the mass, mass ratio, separation distance, and collision angle of the initial conditions that have been shown to produce the Bullet Cluster in non-cosmological hydrodynamic simulations. The probability of finding a high pairwise velocity pair among haloes with $M_{\rm halo}\geq10^{14} M_{\odot}$ is $4.6\times 10^{-4}$ using ROCKSTAR, while it is $\approx 34\times$ lower using a friends-of-friends (FOF) based approach as in previous studies. This is because the typical spatial extent of Bullet progenitors is such that FOF tends to group them into a single halo despite clearly distinct kinematics. Further requiring an appropriately high average mass among the two progenitors, we find the comoving number density of potential Bullet-like candidates to be on the order of $\approx10^{-10} \,{\rm Mpc}^{-3}$. Our findings suggest that $\Lambda\rm{CDM}$ straightforwardly produces massive, high relative velocity halo pairs analogous to Bullet Cluster progenitors, and hence the Bullet Cluster does not present a challenge to the $\Lambda\rm{CDM}$ model.