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The existence of a bullet cluster (such as 1E 0657−56) poses a challenge to the concordance Λ cold dark matter (ΛCDM) model. Here we investigate the velocity distribution of dark matter (DM) halo pairs in large   N  ‐body simulations with differing box sizes ( 250  h  −1   Mpc  Gpc) and resolutions. We examine various basic statistics such as the halo masses, pairwise halo velocities (  v  12  ), collisional angles and pair separation distances. We then compare our results to the initial conditions required to reproduce the observational properties of 1E 0657−56 in non‐cosmological hydrodynamical simulations.  We find that the high‐velocity tail of the   v  12   distribution extends to greater velocities as we increase the simulation box size. We also find that the number of high   v  12   pairs increases as we increase the particle count and resolution with a fixed box size; however, this increase is mostly due to lower mass haloes which do not match the observed masses of 1E 0657−56. We find that the redshift evolution effect is not very strong for the   v  12   distribution function between   z = 0.0  and   z ∼ 0.5 .  We identify some pairs whose   v  12   resemble the required initial conditions, however, even the best candidates have either wrong halo mass ratios or too large separations. Our simulations suggest that it is very difficult to produce such initial conditions at   z = 0.0 , 0.296 and 0.489 in comoving volumes as large as (2   h  −1   Gpc)  3  . Based on the extrapolation of our cumulative   v  12   function, we find that one needs a simulation with a comoving box size of (4.48   h  −1   Gpc)  3   and  2240 3   DM particles in order to produce at least one pair of haloes that resembles the required   v  12   and observed masses of 1E 0657−56. From our simulated   v  12   probability distribution function, we find that the probability of finding a halo pair with   v  12 ≥ 3000  km s  −1   and masses  to be  2.76 × 10 −8   at   z  = 0.489. We conclude that either 1E 0657−56 is incompatible with the concordance ΛCDM universe or the initial conditions suggested by the non‐cosmological simulations must be revised to give a lower value of   v  12  .

Pairwise velocities of dark matter haloes: a test for the Λ cold dark matter model using the bullet cluster