We study gravitational wave memory effect in the FRW cosmological model with
matter and cosmological constant. Since the background is curved, gravitational
radiation develops a tail part arriving after the main signal that travels
along the past light cone of the observer. First we discuss first order
gravitational wave sourced by a binary system, and find that the tail only
gives a negligible memory, in accord with previous results. Then we study the
nonlinear memory effect coming from induced gravitational radiation sourced by
first order gravitational radiation propagating over cosmological distances. In
the light cone part of the induced gravitational wave we find a novel term
missed in previous studies of the cosmological memory effect. Furthermore, we
show that the induced gravitational wave has a tail part that slowly
accumulates after the light cone part has passed and grows to a sizeable
magnitude over a cosmological timescale. This tail part of the memory effect
will be a new component in the stochastic gravitational wave background.
