After an alien civilization gets to the point of asymptotic technology, that is, science is over and done with, they have a number of choices to make. One of them is how much population they wish to maintain. The choice is directly related to how long their resources will last, as for a given level of recycling, twice as many aliens use twice as much resources per time period.
One aspect of this question relates to the process for reducing population. The direct and immediate solution is to simply gestate fewer aliens and allow the population to shrink at whatever level they could choose. It could be as drastic as going from a billion to a million over a few centuries or generations of alien life. The choice of what target to use is related to their view of themselves and their role in the universe. Is it to simply go extinct, or do they plan to go to some other solar system, in one of the many ways possible? If they choose to travel, there is a minimum population necessary to build the ship or ships. If they just are content to go extinct, an unlikely alternative, they could do it quickly with lots of aliens, or slowly, with only a few. Neither is very pleasant, as resource shortages do not make for high living standards.
This question is interesting, but also interesting is the process for getting the population down. There is one question that stands out: what about all the infrastructure? They don't need all the infrastructure that a larger population needs, and they probably don't want to spend the additional energy and resources to maintain it if it is only there for a ghost population.
Consider first fungible architecture. By this time, resource pressure, or at least the knowledge that it will be happening in the future, has mandated that the aliens will live in large arcologies, where recycling is pretty much total. One arcology might be almost identical with another, so there is no reason to keep the second one going if the first one can handle the population post reduction. In an advanced alien civilization, recycling will be part of everyone's life, and everything will be recyclable, even the entire arcology. So over a period of time, the second arcology might be taken apart, and fed into the recycling system of the first one, supposing they are not too far apart so transportation costs are not a significant factor. This adds to the longevity of the resource base.
In order to make an arcology recyclable, it would have to be divisable into parts, so there would not need to be any crowding of a double population into an arcology. Time for this could be stretched out, as by this time the civilization will have figured out it might have a million years on their home planet, so there is no rush to do anything in a short time. If gestation cycles are a hundred years or so, spending a few of them combining arcologies will not affect much over the long term. Aliens in the superfluous arcology could be given the choice of moving to the remaining one, or staying in the part of it which was not yet taken down. By this time, the arcologies would be self sufficient, with their own power plants, industrial sources of nutrition, air filtration, internal transportation, and everything else necessary to have a comfortable life for the alien population. There could well be some residual agriculture, for specialty products or for the amusement of aliens who wanted to be involved with farming for a period of time, and these would simply be reduced according to the population level drop.
What about the non-fungible parts of the infrastructure? Would there be any monuments, historic places, unique but antique buildings? There might have been, but the lifetime of any of these might only be a few thousand years at the highest, and after that, deterioration unless it was periodically rebuilt. Suppose they had preserved something from their earliest eras, before technology was greatly developed, and this was a part of their culture and something they used to maintain their heritage. With a drastic population reduction, as in the example above of a billion to a million, or something proportionate to this, a few heritage sites might be maintained, but not a large number.
It might not be appreciated that heritage could be a very important part of the alien civilization. Heritage is the reason they decide, over and over again in each generation, that they think their civilization is worth preserving and should not be allowed to become extinct. Each generation would have the possibility to reverse the decisions of previous generations, and, for example, stop work on a starship and instead use the capital to change their activities, to, perhaps, have more jet aircraft and spend much more time flying over the landscape and visiting unique sites on their planet, in person. Without heritage sites, the pressure to keep on track with previous generations might grow less, and reach a tipping point.
So, perhaps there is another factor which comes into play when an alien civilization is thinking about its target population for the long, long term. Having sufficient population to build starships may be one, but another might be having enough to maintain a critical number of very important heritage sites.
Almost everything else would have been conquered by technology. There might not need to be a minimum number to maintain the automated operations which provide energy and the standard of living to the population. This particular factor is not clear, but it could very well be that star travel and heritage sites are the only things which feature in the choice of population numbers.
What this means to us is that if these two problems of minimum population result in numbers fairly small, there might only be one arcology on an alien planet. When we have our huge telescopes, able to focus in on planets in different solar systems, looking for one with a thousand dots of light, probably infrared only, this may not be the signature of an alien civilization that we can find. There might be one dot of light, where they all live, and the rest of the planet has been returned to nature. This would be the situation for almost all of the million years of existence of the alien civilization, whereas the huge populations might be only a thousand or less, meaning a tenth of a percent chance of seeing them during this phase. This is one more signature of an alien civilization that needs to be carefully thought through.
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