  Kinetic and hydrodynamic theories are widely employed for describing the
collective behaviour of active matter systems. At the fluctuating level, these
have been obtained from explicit coarse-graining procedures in the limit where
each particle interacts weakly with many others, so that the total forces and
torques exerted on each of them is of order unity at all times. Such limit is
however not relevant for dilute systems that mostly interact via alignment;
there, collisions are rare and make the self-propulsion direction to change
abruptly. We derive a fluctuating kinetic theory, and the corresponding
fluctuating hydrodynamics, for aligning self-propelled particles in the limit
of dilute systems. We discover that fluctuations at kinetic level are not
Gaussian and depend on the interactions among particles, but that only their
Gaussian part survives in the hydrodynamic limit. At variance with fluctuating
hydrodynamics for weakly interacting particles, we find that the noise variance
at hydrodynamic level depends on the interaction rules among particles and is
proportional to the square of the density, reflecting the binary nature of the
aligning process. The results of this paper, which are derived for polar
self-propelled particles with polar alignment, could be straightforwardly
extended to polar particles with nematic alignment or to fully nematic systems.
