Okay, I’m about a month late and the thread has moved on, but I said I would comment about these papers, so here I am.
This is an interesting paper from a theoretical perspective, but the practical implications are really only addressed in later papers. The authors conclude that the allele frequency cannot uniquely identify the actual demographic history, but as noted in the third cited paper, the example they give is not biologically plausible.
Note: I know this paper fairly well, since I shared an office with two of the authors (Simon and Nick) at different times, including while they were writing the paper. The third author is a math heavyweight they had to bring in to get past a sticky bit. Nick had some trouble getting the paper published, not because there was anything wrong with it, but because a reviewer sat on the paper on the paper for well over a year. I was at a mathematical genetics meeting in Durham (where I really did not belong), where Nick gave a talk. He ended by pointing out that this paper had been out for review forever, that the reviewer was probably in the audience, and could he please do his job? He got the reviews back a few weeks later.
This paper does indeed consider a population bottleneck followed by exponential increase in size, and concludes that there are fundamental limits on how accurately such a demography can be reconstructed solely from the site frequency spectrum. It is important to note, however, that the difficulty they demonstrate is in reconstructing demographic events prior to the bottleneck, not the existence of the bottleneck itself. This is clear from their discussion section: “Intuitively, as the severity of the bottleneck increases, the population is increasingly likely to find its most recent common ancestor (MRCA) during that time; farther back in time than the MRCA, no information is conveyed concerning the demographic events experienced by the population.” Similarly: “Additionally, an interesting aspect of our work is that our minimax lower bounds do not depend on the number n of sampled individuals; increasing n is not enough to overcome the information barrier imposed by the presence of a bottleneck.”
This is the most interesting of the papers. It shows that the results of the first paper apply approximately for much more realistic demographies, and that tight bottlenecks can be invisible when just looking at the frequency spectrum. Again, though, there’s an important point to note: the bottleneck they simulate is still, compared to what we’ve been talking about, quite old: 2.5 times the usual 2N generations. Certainly well over a million years ago for humans. I have no trouble believing that demographic signals from that era can be erased. What I have always found implausible is that a signal from less than 0.5 x 2N could be erased, since it leaves insufficient time to accumulate new mutations and get them to high frequency.