On the noise effect of test mass surface roughness in spaceborne gravitational wave detectors

Spaceborne gravitational wave detection mission has a demanding requirement for the precision of displacement sensing, which is conducted by the interaction between the laser field and test mass. However, due to the roughness of the reflecting surface of the test mass, the displacement measurement along the sensitive axis suffers a coupling error caused by the residue motion of other degrees of freedom. In this article, we model the coupling of the test mass residue random motion to the displacement sensing along the sensitive axis and derived an analytical formula of the required precision of the surface error for the spaceborne gravitational wave detectors. Our result shows that this coupling error will not contaminate the picometer displacement sensing for the test masses in the LISA pathfinder.