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Currently, the discussions and investigations for the vacuum energy isdrawing great both theoretical and experimental attention. The vacuum states ofvariety of fields, subject to special boundary conditions, may contribute tonon-trivial macroscopic vacuum energy, i.e. the Casimir effect, which become aninterdisciplinary subject and plays an important role in a variety of fields ofphysics. We adopt Schwinger's source theory and study the Casimir effect due tothe quantization of gravitation, i.e. the gravitational Casimir effect (GCE),in inspiraling neutron star (NS) binaries with wide separation of $10^9 \rm m$.By considering gravitoelectromagnetism (GEM) arising from the spiral-in orbitalmotion and evaluating the contributions of GEM to the vacuum energy ofgravitons radiated during the orbital decay, we demonstrate that, when theorbital separation of the binary decay a distance of $L$ in radial direction,the GEM results in a small Casimir correction to the gravitational vacuumenergy, which contributes to an attractive gravitational Casimir force to thebinary, in addition the gravitational force. The gravitational waves (GWs),emitted from wide inspiraling NS binaries, locate in the low-frequency band of$10^{-4}-1$ Hz. For a characteristic GW frequency of $10^{-3}$ Hz, thegravitational Casimir correction to the signals is estimated as of the order of$\sim10^{-24}$, which corresponds to a force of $10^{-20}$ N. By consideringthat the sensitivity of space-based gravitational wave observatory, LISA/eLISAand Taiji, can be reduced to $10^{-24}$, we would expect that LISA/eLISA andTaiji with sensitivity improvements give the powerful tool to detect GCE in thenear future.

Gravitational Casimir Effect in Inspiralling Neutron Star Binary