  Detections of gravitational wave (GW) stimulate the discussion of how GWs
propagate in the expanding Universe. General relativity predicts that GWs are
massless and propagate at the speed of light with no extra friction term, which
relates to the attenuation of GWs, while some modified gravities may predict a
different behavior. The mass and speed terms can be tightly constrained by the
GW150914-like and GW170817/GRB 170817A events, respectively. However, the
friction term remaining unconstrained. In this paper, we quantize the
nonstandard propagating gravitational waves with nonzero friction term in the
cosmological background, and study the influence of the friction term on the GW
luminosity distance in quantum level, and the initial conditions of
perturbations given by inflation. We find the quantum nature of the difference
between GW and electromagnetic luminosity distance is graviton particle number
non-conservation. For the initial conditions, we obtain an analytical
expression of the power spectrum with nonzero friction term for the de Sitter
background. In observations, both the GW luminosity distance and primordial GWs
can be used to constrain the friction term.
