  Active matter exhibits various forms of non-equilibrium states in the absence
of external forcing, including macroscopic steady-state currents. Such states
are often too complex to be modelled from first principles and our
understanding of their physics relies heavily on minimal models. These have
mostly been studied in the case of "dry" active matter, where particle dynamics
are dominated by friction with their surroundings. Significantly less is known
about systems with long-range hydrodynamic interactions that belong to "wet"
active matter. Dilute suspensions of motile bacteria, modelled as
self-propelled dipolar particles interacting solely through long-ranged
hydrodynamic fields, are arguably the most studied example from this class of
active systems. Their phenomenology is well-established: at sufficiently high
density of bacteria, there appear large-scale vortices and jets comprising many
individual organisms, forming a chaotic state commonly known as bacterial
turbulence. As revealed by computer simulations, below the onset of collective
motion, the suspension exhibits very strong correlations between individual
microswimmers stemming from the long-ranged nature of dipolar fields. Here we
demonstrate that this phenomenology is captured by the minimal model of
microswimmers. We develop a kinetic theory that goes beyond the commonly used
mean-field assumption, and explicitly takes into account such correlations.
Notably, these can be computed exactly within our theory. We calculate the
fluid velocity variance, spatial and temporal correlation functions, the fluid
velocity spectrum, and the enhanced diffusivity of tracer particles. We find
that correlations are suppressed by particle self-propulsion, although the
mean-field behaviour is not restored even in the limit of very fast swimming.
