We develop a deep-learning technique to infer the nonlinear velocity field from the dark matter density field. The deep-learning architecture we use is a “U-net” style convolutional neural network, which consists of 15 convolution layers and 2 deconvolution layers. This setup maps the three-dimensional density field of 323 voxels to the three-dimensional velocity or momentum fields of 203 voxels. Through the analysis of the dark matter simulation with a resolution of 2h −1 Mpc, we find that the network can predict the the nonlinearity, complexity, and vorticity of the velocity and momentum fields, as well as the power spectra of their value, divergence, and vorticity and its prediction accuracy reaches the range of k ≃ 1.4 h Mpc−1 with a relative error ranging from 1% to ≲10%. A simple comparison shows that neural networks may have an overwhelming advantage over perturbation theory in the reconstruction of velocity or momentum fields.

The tomographic Alcock-Paczynski (AP) method utilizes the redshift evolution of the AP distortion to place constraints on cosmological parameters. It has proved to be a robust method that can separate the AP signature from the redshift space distortion (RSD) effect, and deliver powerful cosmological constraints using the 40h −1 Mpc clustering region. In previous works, the tomographic AP method was performed via the anisotropic 2-point correlation function statistic. In this work we consider the feasibility of conducting the analysis in the Fourier domain and examine the pros and cons of this approach. We use the integrated galaxy power spectrum (PS) as a function of direction,P ∆k (µ), to quantify the magnitude of anisotropy in the large-scale structure clustering, and use its redshift variation to do the AP test. The method is tested on the large, high resolution Big-MultiDark Planck (BigMD) simulation at redshifts z = 0−1, using the underlying true cosmology Ω m = 0.3071, w = −1. Testing the redshift evolution ofP ∆k (µ) in the true cosmology and cosmologies deviating from the truth with δΩ m = 0.1, δw = 0.3, we find that the redshift evolution of the AP distortion overwhelms the effects created by the RSD by a factor of ∼ 1.7 − 3.6. We test the method in the range of k ∈ (0.2, 1.8) h Mpc −1 , and find that it works well throughout the entire regime. We tune the halo mass within the range 2 × 10 13 to 10 14 M , and find that the change of halo bias results in 5% change inP ∆k (µ), which is less significant compared with the cosmological effect. Our work shows that it is feasible to conduct the tomographic AP analysis in the Fourier space. Subject headings: large-scale structure of Universe -dark energy -cosmological parameters