When binary black holes are embedded in a gaseous environment , a rotating disk surrounding them , the so-called circumbinary disk , will be formed .
The binary exerts a gravitational torque on the circumbinary disk and thereby the orbital angular momentum is transferred to it , while the angular momentum of the circumbinary disk is transferred to the binary through the mass accretion .
The binary undergoes an orbital decay due to both the gravitational wave emission and the binary-disk interaction .
This causes the phase evolution of the gravitational wave signal .
The precise measurement of the gravitational wave phase thus may provide information regarding the circumbinary disk .
In this paper , we assess the detectability of the signature of the binary-disk interaction using the future space-borne gravitational wave detectors such as DECIGO and BBO by the standard matched filtering analysis .
We find that the effect of the circumbinary disk around binary black holes in the mass range 6 M _ { \odot } \leq { M } \lesssim 3 \times 10 ^ { 3 } M _ { \odot } is detectable at a statistically significant level in five year observation , provided that gas accretes onto the binary at a rate greater than \dot { \it M } \sim 1.4 \times 10 ^ { 17 } [ { g s ^ { -1 } } ] { \it j } ^ { -1 } ( { \it M } / 10 { % \it M } _ { \odot } ) ^ { 33 / 23 } with 10 \% mass-to-energy conversion efficiency , where j represents the efficiency of the angular momentum transfer from the binary to the circumbinary disk .
We show that O ( 0.1 ) coalescence events are expected to occur in sufficiently dense molecular clouds in five year observation .
We also point out that the circumbinary disk is detectable , even if its mass at around the inner edge is by over 10 orders of magnitude less than the binary mass .