  Compressive plasma turbulence is investigated at sub-ion scales in the solar
wind using both the Fast Plasma Investigation (FPI) instrument on the
Magnetospheric MultiScale mission (MMS), as well as using calibrated spacecraft
potential data from the Spin Plane Double Probe (SDP) instrument. The data from
FPI allow a measurement down to the sub-ion scale region ($f_{sc}\gtrsim 1$ Hz)
to be investigated before the instrumental noise becomes significant at a
spacecraft frame frequency of $f_{sc}\approx 3$Hz, whereas calibrated
spacecraft potential allows a measurement up to $f_{sc}\approx 40$Hz. In this
work, we give a detailed description of density estimation in the solar wind
using the spacecraft potential measurement from the SDP instrument on MMS.
Several intervals of solar wind plasma have been processed using the
methodology described which are made available. One of the intervals is
investigated in more detail and the power spectral density of the compressive
fluctuations is measured from the inertial range to the sub-ion range. The
morphology of the density spectra can be explained by either a cascade of
Alfv\'en waves and slow waves at large scales and kinetic Alfv\'en waves at
sub-ion scales, or more generally by the Hall effect. Using electric field
measurements the two hypotheses are discussed.
