  The basic constituent of interferometric gravitational wave detectors -- the
test mass to test mass interferometric link -- behaves as a differential
dynamometer measuring effective differential forces, comprising an integrated
measure of gravity curvature, inertial effects, as well as non-gravitational
spurious forces. This last contribution is going to be characterised by the
LISA Pathfinder mission, a technology precursor of future space-borne detectors
like eLISA. Changing the perspective from displacement to acceleration can
benefit the data analysis of LISA Pathfinder and future detectors. The response
in differential acceleration to gravitational waves is derived for a
space-based detector's interferometric link. The acceleration formalism can
also be integrated into time delay interferometry by building up the
unequal-arm Michelson differential acceleration combination. The differential
acceleration is nominally insensitive to the system free evolution dominating
the slow displacement dynamics of low-frequency detectors. Working with
acceleration also provides an effective way to subtract measured signals acting
as systematics, including the actuation forces. Because of the strong
similarity with the equations of motion, the optimal subtraction of systematic
signals, known within some amplitude and time shift, with the focus on
measuring the noise provides an effective way to solve the problem and
marginalise over nuisance parameters. The $\mathcal{F}$-statistic, in
widespread use throughout the gravitation waves community, is included in the
method and suitably generalised to marginalise over linear parameters and noise
at the same time. The method is applied to LPF simulator data and, thanks to
its generality, can also be applied to the data reduction and analysis of
future gravitational wave detectors.
