Cosmological simulations evolve a density field, specified at high redshift, forward in time. One of the tenets of the Inflationary Model is that the initial density fluctuations were created by quantum fluctuations within a fraction of a second during the Big Bang. Any density field can be described by the amplitudes of the fluctuations (the Power spectrum), which are determined by the cosmological model, and the phase information, which determines the specific peaks and dips in the density at different locations. A common way to parameterise this information is using Fourier modes.

In this paper, we use a new method for setting the initial conditions for cosmological simulations, building on earlier work by Adrian Jenkins, to specify the phase information. Instead of Fourier modes, we use orthogonal Octree basis functions, which have the benefit of being localised in space. By creating many variations of density fields, changing the phase information on different levels of the Octree (different spatial scales), we measure which scales in the very early universe effect the foŕmation of haloes we can observe today.

Dark matter density (left panel), an the difference in dark matter density in the same regions, when variations are introduced at different levels of the Octree. Ast the scale of variations in the initial conditions increases from left to right, the difference in the final dark matter density grows. Smaller haloes disappear (and new ones appear), while larger haloes change in mass and position.

We quantify, for example, on what scales the information for objects like the Milky Way or the Virgo Galaxy cluster is defined. We also quantify how perturbations of the density field on smaller scales affect observable properties of the system, such as the mass, concentration, or position.

The level of the Octree (top axis) and the equivalent length scale (bottom axis) at which the density field needs to be specified in order for haloes of different masses to be uniquely defined. For example, Milky Way mass haloes (green line) are defined by density fluctuations of approximately 1.6 Mpc.

Beyond the theoretical interest, this method of setting the initial conditions, and varying the density field has important practical applications. By iteratively randomising the phase information, we can explore the sensitivity of observables to the initial conditions, and create objects that closely match observations – all while preserving the Gaussian nature of the initial density field created by the quantum fluctuations of the Big Bang.