With the assumption that civilization continued for million of generations / while the sun provided a suitable environment,
what would the average population growth per year or per generation be?
Going to extremes with the assumption that in a billion years (40 million current generations) the entire mass of the planet consists only of intelligent beings weighing 1 kg each (i.e., on the order of 6 times 10 to the 24 individuals), the average annual rate of growth from a starting base of 7 billion would be less than 0.000001%, (about 3 millionths of a percent or about 200 per year more born than dying at the current level). Assuming there would only be 1 individual left, the average annual rate of population reduction would likewise be less than 0.000001%.
- It only enforces the argument - there are lower limits, too: sentient individuals, with conscious experience, would be expected to require useful energy. We don't understand how consciousness arises, but most expect that it requires the processing of information. If, as seems likely, the second law of thermodynamics describes reality adequately, processing of information requires the consumption of useful energy. Human brains require roughly 10-20 Watts. Arbitrarily assuming that a fraction of that, say 1 Watt, is required for (an adequate level of) consciousness, on the order of 5 times 10 to the 16 conscious individuals at any one time could be supported by solar energy if no energy would be used for anything else. This is about 9 million times current human population, but more than 100 million times less than in the above example.
- This assumes suface temperature and albedo are roughly maintained. While a Dyson Sphere construct would allow much more power and thus individuals, these could not be on this planet if temperature be maintained, so with the assumed power requirement of 1 W per individual this is an upper limit to the conscious population on the planet, and even if different sources of energy (such as fusion power or unknown sources) were added, it could not be surpassed unless temperature would be allowed to rise.
- The argument supports the notion that the consumption of useful energy comes at an absolute cost: less sentient life is possible if energy is used any other way.
Fluctuations in between, either way, need to cancel each other out in the long run. On comparison, with an average growth of 1% (roughly the average world population growth between 1800 and 2010), it would need less than 3,500 years (the pyramids are much older) to reach the mass of the earth with individuals weighing only 1 kg from the current number of humans, and only about 3000 years to reach the limit provided by conscious individuals requiring only 1 W of power consumption each.
Continued exponential growth over long periods is impossible. Life will become very tough by external factors containing growth unless we find ways to contain it, and humanity will quickly be extinct unless there are mechanism to keep its population from falling too low.