  Spectral properties of the turbulent cascade from fluid to kinetic scales in
collisionless plasmas are investigated by means of large-size three-dimensional
(3D) hybrid (fluid electrons, kinetic protons) particle-in-cell simulations.
Initially isotropic Alfv\`enic fluctuations rapidly develop a strongly
anisotropic turbulent cascade, mainly in the direction perpendicular to the
ambient magnetic field. The omnidirectional magnetic field spectrum shows a
double power-law behavior over almost two decades in wavenumber, with a
Kolmogorov-like index at large scales, a spectral break around ion scales, and
a steepening at sub-ion scales. Power laws are also observed in the spectra of
the ion bulk velocity, density, and electric field, both at magnetohydrodynamic
(MHD) and at kinetic scales. Despite the complex structure, the omnidirectional
spectra of all fields at ion and sub-ion scales are in remarkable quantitative
agreement with those of a two-dimensional (2D) simulation with similar physical
parameters. This provides a partial, a-posteriori validation of the 2D
approximation at kinetic scales. Conversely, at MHD scales, the spectra of the
density and of the velocity (and, consequently, of the electric field) exhibit
differences between the 2D and 3D cases. Although they can be partly ascribed
to the lower spatial resolution, the main reason is likely the larger
importance of compressible effects in a full geometry. Our findings are also in
remarkable quantitative agreement with solar wind observations.
