  One of the primary sources for the future space-based gravitational wave
detector, the Laser Interferometer Space Antenna, are the inspirals of small
compact objects into massive black holes in the centres of galaxies. The
gravitational waveforms from such Extreme Mass Ratio Inspiral (EMRI) systems
will provide measurements of their parameters with unprecedented precision, but
only if the waveforms are accurately modeled. Here we explore the impact of
transient orbital resonances which occur when the ratio of radial and polar
frequencies is a rational number. We introduce a new Effective Resonance Model,
which is an extension of the numerical kludge EMRI waveform approximation to
include the effect of resonances, and use it to explore the impact of
resonances on EMRI parameter estimation. For one-year inspirals, we find that
the few cycle dephasings induced by 3:2 resonances can lead to systematic
errors in parameter estimates, that are up to several times the typical
measurement precision of the parameters. The bias is greatest for 3:2
resonances that occur closer to the central black hole. By regarding them as
unknown model parameters, we further show that observations will be able to
constrain the size of the changes in the orbital parameters across the
resonance to a relative precision of 10% for a typical one-year EMRI
observation with signal-to-noise ratio of 20. Such measurements can be regarded
as tests of fundamental physics, by comparing the measured changes to those
predicted in general relativity.
