I think that there is 3 ways to present these ideas in more rigorius form.
Use Gott formula to estimate probability distribution P(N) that total number of civilizations on Earth will be N based on the fact that our rank number in all known civilization is n. (And in our case n=1, so N=2 has 50 per cent probability, N=4 has 25 per cent probability etc.) See the same calculation for original Doomsday argument. https://en.wikipedia.org/wiki/Doomsday_argument#Gott.27s_formulation:_.27vague_prior.27_total_population
Use the fact that we don’t know anything about past civilizations to put constrains on the informational traces T. T is function of civilisational technological level L and time distance to it t. So T(L,t) must be below some level of noticeability. T function is unknown to us but could be estimated as L/t which means that high tech and recent civilization will be more notable. Any risks from previous civilizations will also decay with time. So we could start to create math model form here.
We could look on existing scientific literature. A lot of literature use observational data trying to explain original Fermi paradox, but it is surprising not true for past civilizations. There is no analog for “SETI search” for rare isotopes changes which could sign of civilization 100 million years from now here on Earth—or I don’t know about this literature. I also don’t know what is the rate of publishing of theoretically inappropriate results if someone randomly finds something which seems to be strange. There is attempt by late Russian author Kalandadze to collect evidences that some other hominids used fire here: http://www.evolbiol.ru/document/915 The work is controversial. I don’t have special knowledge to assess it.
I think that there is 3 ways to present these ideas in more rigorius form.
Use Gott formula to estimate probability distribution P(N) that total number of civilizations on Earth will be N based on the fact that our rank number in all known civilization is n. (And in our case n=1, so N=2 has 50 per cent probability, N=4 has 25 per cent probability etc.) See the same calculation for original Doomsday argument. https://en.wikipedia.org/wiki/Doomsday_argument#Gott.27s_formulation:_.27vague_prior.27_total_population
Use the fact that we don’t know anything about past civilizations to put constrains on the informational traces T. T is function of civilisational technological level L and time distance to it t. So T(L,t) must be below some level of noticeability. T function is unknown to us but could be estimated as L/t which means that high tech and recent civilization will be more notable. Any risks from previous civilizations will also decay with time. So we could start to create math model form here.
We could look on existing scientific literature. A lot of literature use observational data trying to explain original Fermi paradox, but it is surprising not true for past civilizations. There is no analog for “SETI search” for rare isotopes changes which could sign of civilization 100 million years from now here on Earth—or I don’t know about this literature. I also don’t know what is the rate of publishing of theoretically inappropriate results if someone randomly finds something which seems to be strange. There is attempt by late Russian author Kalandadze to collect evidences that some other hominids used fire here: http://www.evolbiol.ru/document/915 The work is controversial. I don’t have special knowledge to assess it.