3. Scientific instruments
These would be nearly as good as an actual message if
we found them, as they would not only prove the
existence of alien intelligence, they would tell us something
about what the beings were interested in, and
possibly allow us to communicate with them if they are
somehow monitoring the instruments. However, finding
instruments would be much harder than finding a message,
as the aliens would have no incentive to make an
instrument easy to spot. Even a large radio telescope on
the far side of the moon (to shield it from potential radio
traffic from any future terrestrial technological society)might be just a few tens of meters across. Furthermore, an
instrument would likely be coated in lunar dust after a
few hundred years, making it had to distinguish from a
mundane geological feature. The best way to identify an
active instrument would probably be to look for signs of
its power source. Solar panels, if kept clean enough to
keep the instrument powered, would probably appear as
large black geometric shapes. Radiogenic sources would
likely produce detectable gamma radiation, as there is
little incentive for the aliens to go to the effort of shielding
the reactor. More exotic power sources might also produce
detectable radiation, although we cannot currently
predict what these might be, so there is no good way to
actively search for them.