We calculate how strongly one can put constraints on alternative theories of gravity such as Brans-Dicke and massive graviton theories with LISA .
We consider inspiral gravitational waves from a compact binary composed of a neutron star ( NS ) and an intermediate mass black hole ( IMBH ) in Brans-Dicke ( BD ) theory and that composed of 2 super massive black holes ( SMBHs ) in massive graviton theories .
We use the restricted 2PN waveforms including the effects of spins .
We also take both precession and eccentricity of the orbit into account .
For simplicity , we set the fiducial value for the spin of one of the binary constituents to zero so that we can apply the approximation called simple precession .
We perform the Monte Carlo simulations of 10 ^ { 4 } binaries , estimating the determination accuracy of binary parameters including the BD parameter \omega _ { \mathrm { BD } } and the Compton wavelength of graviton \lambda _ { g } for each binary using the Fisher matrix method .
We find that including both the spin-spin coupling \sigma and the eccentricity e into the binary parameters reduces the determination accuracy by an order of magnitude for the Brans-Dicke case , whilst it has less influence on massive graviton theories .
On the other hand , including precession enhances the constraint on \omega _ { \mathrm { BD } } only 20 \% but it increases the constraint on \lambda _ { g } by an order of magnitude .
Using a ( 1.4 + 1000 ) M _ { \odot } NS/BH binary of SNR= \sqrt { 200 } , one can put a constraint \omega _ { \mathrm { BD } } > 6944 , whilst using a ( 10 ^ { 7 } +10 ^ { 6 } ) M _ { \odot } BH/BH binary at 3Gpc , one can put \lambda _ { g } > 3.06 \times 10 ^ { 21 } cm , on average .
The latter is 4 orders of magnitude stronger than the one obtained from the solar system experiment .
These results are consistent with previous results within uncontrolled errors and indicate that the effects of precession and eccentricity must be taken carefully in the parameter estimation analysis .