  There are many astrophysical scenarios where extreme mass ratio inspiral
(EMRI) binaries can be surrounded by matter distribution. The distribution of
mass can affect the dynamical properties (e.g. orbital frequency, average
energy radiation rate, etc.) of the EMRI. In this matter distribution, instead
of Kepler-Newton potential, one may consider a more general form of potential
i.e. power law potential. Moreover, due to the power law potential, the
velocity profile of test particles does not fall as much as that predicted by
Kepler-Newton potential and this feature of the velocity profile may be
observationally important. In this study, we have obtained the first
post-Newtonian (1PN) expressions for dynamical quantities and the average
energy radiation rate from the circular orbit EMRI which is surrounded by a
matter distribution. We show that the energy radiation rate and orbital
frequency of EMRI can be significantly different in the presence of power law
potential as compared to that in the Kepler-Newton potential, signatures of
which may be observed in gravitational waves from EMRI.
