Deep learning insights into cosmological structure formation

TL;DR

A deep learning framework is built to learn this non-linear relationship between the mass of dark matter and dark matter halos, and techniques to physically interpret the learnt mapping are developed.

Abstract

The evolution of linear initial conditions present in the early universe into extended halos of dark matter at late times can be computed using cosmological simulations. However, a theoretical understanding of this complex process remains elusive; in particular, the role of anisotropic information in the initial conditions in establishing the final mass of dark matter halos remains a long-standing puzzle. Here, we build a deep learning framework to investigate this question. We train a three-dimensional convolutional neural network (CNN) to predict the mass of dark matter halos from the initial conditions, and quantify in full generality the amounts of information in the isotropic and anisotropic aspects of the initial density field about final halo masses. We find that anisotropies add a small, albeit statistically significant amount of information over that contained within spherical averages of the density field about final halo mass. However, the overall scatter in the final mass predictions does not change qualitatively with this additional information, only decreasing from 0.9 dex to 0.7 dex. Given such a small improvement, our results demonstrate that isotropic aspects of the initial density field essentially saturate the relevant information about final halo mass. Therefore, instead of searching for information directly encoded in initial conditions anisotropies, a more promising route to accurate, fast halo mass predictions is to add approximate dynamical information based e.g. on perturbation theory. More broadly, our results indicate that deep learning frameworks can provide a powerful tool for extracting physical insight into cosmological structure formation.

Figures (10)

• Figure 1: $N$-body simulations of cosmological structure formation can accurately compute the gravitational evolution of dark matter over cosmic time, but do not provide a physical understanding of how cosmic structures arise from the initial conditions. We train a CNN model to learn the relationship between the initial density field and the final dark matter halos, given examples from $N$-body simulations. The inputs to the CNN are given by the initial density field surrounding each dark matter particle and the outputs are the mass of the dark matter halos to which each particle belongs at $z=0$. The aim is to interpret the mapping learnt by the CNN in order to gain physical insights into dark matter halo formation.
• Figure 2: (A) The CNN makes predictions for simulation particles that occupy different regions of the initial conditions of the simulation. These particles end up in halos which differ not only in their mass, but also in their formation history, large-scale environment, and amount of sub-structure within the halos. The CNN must identify from the initial density field the features that impact the final mass of the resulting halos. (B) Halo mass predictions returned by a CNN trained on the initial density field surrounding each dark matter particle's initial position. The predictions are shown against the ground truth halo mass values as a two-dimensional histogram in the top panel, while the bottom panel shows the residuals $\log(M_{\rm pred}/M_{\rm true})$. The errorbars in the top (bottom) panel show the median and 68% confidence interval of the predictions (residuals) in bins of ground-truth mass values.
• Figure 3: (A) We re-train the model on inputs where the density in the initial conditions is averaged over shells so that any anisotropic information is removed. The two models each return a set of predictions for the test set particles. (B) We compare the predictions returned by the raw-density training set model and the averaged-density training set model. The histograms show the difference between the predicted and the true log halo mass for particles split into three mass bins of halos. The bands of the histograms capture the scatter in the predictions of each model trained with four different random seeds. The two models show similar residual distributions, except for a slightly smaller variance in the residual distribution of particles in the mid-mass range of halos.
• Figure 4: Mutual information between predicted and ground truth halo mass values for the raw-density and averaged-density models. The horizontal grey lines show the value of the MI for mock predicted halo mass values constructed by adding Gaussian noise to the ground truth values with standard deviation of 1.5, 1., 0.75, 0.5 dex, respectively.
• Figure 5: Halo mass predictions for particles that reside in different locations inside the halos: those located in the inner region of the halo ($r \leq 0.3 \, r_{\rm 200m}$; left panel), those in a intermediate region ($0.3 < r/r_{\rm 200m} \leq 0.6$); middle panel), and those in the outskirts of halos ($r > 0.6 \, r_{\rm 200m}$; right panel). The MI between predicted and ground truth halo mass values is indicated in the legend box of each panel. While the change in MI between raw-density and averaged-density models is statistically significant, the decrease in the scatter of the predictions is not sufficient to provide a qualitative improvement in the initial conditions-to-halo mass mapping.