tokens: 3 DM particles (Rasera et al. 2010 ; Alimi et al. 2012 ; Bouillot et al. 2015 ). For the flat ΛCDM model, the six key cosmological parameters were set at the central values determined by the Wilkinson Microwave Anisotropy Probe Seven Year data (WMAP7) (Spergel et al. 2007 ; Komatsu et al. 2011 ). For the other two DE models, the values of the seven cosmological parameters (the six key parameters + w ) were chosen to satisfy the same WMAP7 constraints. Table 1 lists the values of the matter density parameter, Ω m , linear power spectrum amplitude on the scale of , σ 8 , dimensionless Hubble parameter, h , dark energy equation of state, w ≡ w 0 + w a (1 − a ) with scale factor a ( t ) (Linder 2003 ), and mass of individual DM particles, m p , for the three DE models. For a more detailed description of the DEUS, we refer the readers to Alimi et al. ( 2012 ) as well as the DEUS webpage. 4 4 All data from the DEUS are publicly available at http://www.deus-consortium.org/ . Table 1 Initial Conditions and the Particle Mass Resolution A phantom DE ( w CDM) is characterized by its negative kinetic energy that exerts stronger repulsive force than Λ (Caldwell et al. 2003 ). Whereas, a quintessence model is characterized by the potential shape of a scalar field DE. For the DEUS, the Ratra–Peebles potential, whose shape is expressed as powers of linear combinations of the exponentials of the scalar field (Ratra Peebles 1988 ), was specifically chosen. Although the distant-field diagnostics based on the linear observables like the CMB temperature spectra and SN Ia luminosity–distance relations cannot distinguish among the three models (Bouillot et al. 2015 ), an efficient near-field probe based on the nonlinear observables might be able to pull it off since the formation and evolution of the nonlinear structures proceed in a different way among them (e.g., see Alimi et al. 2010 ). Table 2 Halo Spin Transition Zones The halo catalog from the DEUS comprises the gravitationally bound objects identified at various redshifts by the Friends-of-Friends (FoF) halo finder with a linkage parameter of 0.2 (Roy et al. 2014 ) and provides information not only on the center of mass position, , and virial mass, M h , of each object but also on the positions and velocities of its constituent DM particles, which allow us to compute its angular momentum vector, . The well-resolved FoF objects in the mass range of at z = 0 are selected for a sample of the galactic halos to be used for the current analysis. The low-mass FoF objects with are excluded from our analysis on the ground that the angular momentum vectors of the halos composed of less than 300 DM particles are likely to suffer from inaccuracy caused by the shot noise (Bett et al. 2007 ). The routine we go through for the investigation of the DE dependence of the halo spin transitions is similar to that described in Lee et al. ( 2020 ), the summary of which is provided in the following. 1. Divide the simulation volume
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