Cosmic velocity, density and halo mass function: Insights from deep learning

We discuss an implementation of a deep learning framework to gain insight into dark matter (DM) structure formation. We investigate the contribution of velocity and density field information to the construction of the halo mass function (HMF) in cosmological N-body simulations. We train a Convolutional Neural Network (CNN) on the initial snapshot of a DM-only simulation to predict the halo mass that individual particles fall into at $z = 0$, in the halo mass range of $10.5 < \log (M/M_{\odot}) < 14$. We show that for the standard $\Lambda$CDM cosmology with amplitude of initial perturbations $A_s = 2 \times 10^{-9}$, the initial velocity and density fields have equivalent information, as expected in the linear regime, and manifest the power of our CNN to diagnose the redundant information. To investigate the non-linear effects, we increase the initial power spectrum. In the linear regime, this is equivalent to decreasing the initial redshift. The CNN model trained on the simulation snapshots with large $A_s$ shows a considerable improvement in the HMF prediction when adding the velocity field information. Using our CNN map without further physical assumptions, we precisely evaluate when these non-linear effects become vital. Eventually, for the simulation with $A_s = 8 \times10^{-8}$, the model trained with only density information shows at least an $80\%$ increase in the mean squared error relative to the model with both velocity and density information. Our work shows the interpretability and ability of CNNs to read higher-order information from simple images, making them an excellent tool for cosmological studies.