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 can result in tight cosmological constraints by using small and intermediate clustering scales of the large scale structure (LSS) of the galaxy distribution. By focusing on the redshift dependence, the AP distortion can be distinguished from the distortions produced by the redshift space distortions (RSD). In this work, we combine the tomographic AP method with other recent observational datasets of SNIa+BAO+CMB+H 0 to reconstruct the dark energy equation-of-state w in a non-parametric form. The result favors a dynamical DE at z 1, and shows a mild deviation ( 2σ) from w = −1 at z = 0.5 − 0.7. We find the addition of the AP method improves the low redshift (z 0.7) constraint by ∼ 50%.

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