tokens: Via the introduction of a physical model of gravitational structure growth, the inference of the non-linear dark matter distribution today can be posed as a statistical initial condition problem. To do so, the borg algorithm seeks to infer the cosmic initial conditions from which present 3D structures in the distribution of galaxies have formed via non-linear gravitational mass aggregation. The borg algorithm explores the posterior distribution of large-scale structures, consisting of a Gaussian prior for the initial density field at an initial scale factor of a = 10 −3 and a choice of bias model and likelihood metric at scale factor a = 1. The evolution of the initial density fields can be related to the present galaxy (or halo) distribution via a first- or second-order Lagrangian perturbation theory (LPT or 2LPT) or a full particle mesh model of gravitational structure formation (for details, see Jasche Wandelt 2013 ; Jasche Lavaux 2019 ). By exploiting non-linear structure growth models, the borg algorithm naturally accounts for the filamentary structure of the cosmic web typically associated with higher order statistics induced by non-linear gravitational processes. Furthermore, the posterior distribution accounts for systematic and stochastic uncertainties, such as survey geometries, selection effects, unknown noise, and galaxy biases as well as foreground contamination (see e.g. Jasche Wandelt 2013 ; Jasche et al. 2015 ; Lavaux Jasche 2016 ; Jasche Lavaux 2017 ). In this work, we use the borg algorithm to sample the initial conditions of the dark matter density field using LPT to evolve the field to the dark matter conditions today and then rely on the neural bias model to sample the parameters of the NPE to provide a field that is most informative about the halo mass distribution function inferred from a halo catalogue. A detailed schematic of the interconnection between parts of the model is shown in Fig. 1 . The stochastic uncertainties are assumed to be Poissonian using the likelihood in equation ( 16 ), and further study into direct learning of the deviations from Poissonity will be conducted in an upcoming work. Figure 1. Schematic of the borg algorithm with the neural bias model. Initial conditions for the density field in Fourier space, , are drawn from a prior given a cosmology , . These are then evolved forward using a deterministic prescription, in this example using LPT. The evolved field is then transformed further using the NPE that requires parameters that are drawn from a prior . This provides a field from which the halo mass distribution function can be described using the MDN with parameters drawn from a prior . This halo mass distribution function is then compared to the masses of haloes M obs from the observed halo catalogue. 5 RESULTS To examine the techniques developed in this paper, we will consider a relatively simple mock run using a simulated dark matter distribution and minimal working neural bias model.
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