  We introduce "Simulations Beyond The Local Universe" (SIBELIUS) that connect
the Local Group to its cosmic environment. We show that introducing
hierarchical small-scale perturbations to a density field constrained on large
scales by observations provides an efficient way to explore the sample space of
Local Group analogues. From more than 60 000 simulations, we identify a
hierarchy of Local Group characteristics emanating from different scales: the
total mass, orientation, orbital energy and the angular momentum are largely
determined by modes above $\lambda$ = 1.6 comoving Mpc (cMpc) in the primordial
density field.
  Smaller scale variations are mostly manifest as perturbations to the MW-M31
orbit, and we find that the observables commonly used to describe the Local
Group -- the MW-M31 separation and radial velocity -- are transient and depend
on specifying scales down to 0.2 cMpc in the primordial density field. We
further find that the presence of M33/LMC analogues significantly affects the
MW-M31 orbit and its sensitivity to small-scale perturbations. We construct
initial conditions that lead to the formation of a Local Group whose primary
observables precisely match the current observations.
