  Gravitational Wave (GW) observations of coalescing compact binaries will be
unique probes of strong-field, dynamical aspects of relativistic gravity. We
present a short review of various schemes proposed in the literature to test
General Relativity (GR) and alternative theories of gravity using inspiral
waveforms. Broadly these schemes may be classified into two types: model
dependent and model independent. In the model dependent category, GW
observations are compared against a specific waveform model representative of a
particular theory or a class of theories like Scalar-Tensor theories, Dynamical
Chern-Simons theory and Massive graviton theories. Model independent tests are
attempts to write down a parametrised gravitational waveform where the free
parameters take different values for different theories and (at least some of)
which can be constrained by GW observations. We revisit some of the proposed
bounds in the case of downscaled LISA configuration (eLISA) and compare them
with the original LISA configuration. We also compare the expected bounds on
alternative theories of gravity from ground-based and space-based detectors and
find that space-based GW detectors can test GR and other theories of gravity
with unprecedented accuracies. We then focus on a recent proposal to use
singular value decomposition (SVD) of the Fisher information matrix to improve
the accuracies with which Post-Newtonian (PN) theory can be tested. We extend
those results to the case of space based detector eLISA and discuss its
implications.
