  Planet formation in protoplanetary discs requires dust grains to coagulate
from the sub-micron sizes that are found in the interstellar medium into much
larger objects. For the first time, we study the growth of dust grains during
the earliest phases of star formation using three-dimensional hydrodynamical
simulations. We begin with a typical interstellar dust grain size distribution
and study dust growth during the collapse of a molecular cloud core and the
evolution of the first hydrostatic core, prior to the formation of the stellar
core. We examine how the dust size distribution evolves both spatially and
temporarily. We find that the envelope maintains its initial population of
small dust grains with little growth during these phases, except that in the
inner few hundreds of au the smallest grains are depleted. However, once the
first hydrostatic core forms rapid dust growth to sizes in excess of $100~\mu$m
occurs within the core (before stellar core formation). Progressively larger
grains are produced at smaller distances from the centre of the core. In
rapidly-rotating molecular cloud cores, the `first hydrostatic core' that forms
is better described as a pre-stellar disc that may be gravitationally unstable.
In such cases, grain growth is more rapid in the spiral density waves leading
to the larger grains being preferentially found in the spiral waves even though
there is no migration of grains relative to the gas. Thus, the grain size
distribution can vary substantially in the first core/pre-stellar disc even at
these very early times.
