The pollution would vastly outlive the polluters, so it would be detectable for some time after their demise, especially if anything they automated also polluted. That stuff would continue to pollute after its creators all died, at least a bit.
Also, it seems that the chance of detecting pollution would increase linearly as we continue to discover more planets.
...hopefully my point is made, now. I can't give details because they don't exist.
If a species produces sufficient pollution that it kills that species, it is a reasonable assumption that it was also produced at a rate far higher than the planets natural ecosystem could absorb it. It is also reasonable that the pollution also kills one or more species which were consuming or absorbing the pollution, slowing the rate at which it can be removed, or maybe even stopping that process completely. Pollution does not disappear at the same time pollutants stop being introduced into the atmosphere.
There will be a time after which no pollutants are produced and before no pollutants are detectable. That is a non-zero amount of time. That is why I say that pollution could one day be detected on a planet that has not had a living civilization for "a long time." "A long time" being "more than five minutes" if you really want me to nail it down.
Right. And if the argument depends on those details, then it's not a very convincing argument.
[1] https://en.wikipedia.org/wiki/Oxygen_cycle#Capacities_and_fl...
An alien society may produce pollutants which last a million years. Plastics can last millions of years.
Sufficient pollution could cause climate change severe enough to keep all kinds of things airborne for who knows how long. We are constantly surprised by what we learn about other worlds. It is silly to me to presume that anything we know about Earth pollution will be a given on a planet from another star system.
Pollution doesn't vanish when it isn't being made anymore.
So, if there is so much pollution that it both kills life on the planet, it will take a good amount of time (millions of years? How big of a planet are we talking about?) to be cleansed back to undetectable levels.
We were able to detect dinosaurs on our own planet 65 million years after they went extinct, and they didn't even pollute anything.
To your point getting strictly negative results would not necessarily mean anything one way or the other.