> Can anyone explain

I'll try. The tl;dr is at the end before my own footnote. The author certainly doesn't make it easy, even for people familiar with the standard model of cosmology.

The very first sentence of the paper [ https://arxiv.org/abs/1905.01214 ] reads, "Dark matter (DM) may have its origin in a pre-big-bang epoch. It may have been produced, for example, by decays or annihilations of particles during the Big Bang, i.e. by the so-called ’freeze-in’ [1–3] mechanism, or by e.g. the misalignment mechanism which generated a non-zero DM abundance during cosmic inflation (see e.g. Ref.[4])." which while not strictly speaking self-contradictory is certainly far from clear.

The term "Big Bang" only appears in that first sentence, and the abstract.

If one does a case-independent search for "bang" in the paper's reference [1], there are no matches at all.

There is hope, however.

Reference [2] of the paper carefully uses only "big bang nucleosynthesis" and its abbreviation BBN.

BBN occurs after the universe has cooled via expansion ("adiabatic cooling") so that some of the hot, dense matter filling the universe earlier than BBN can "freeze" into atomic nuclei. That probably happened in steps: first quarks and gluons (and perhaps other particles feeling nuclear forces) could freeze into individual protons and neutrons, then those could join into atomic nuclei. It was still too hot for electrons to bind for long with these nuclei, so they were completely ionized.

Reference [3] uses "big bang nucleosynthesis" and "BBN" too, but also introduces "hot big bang cosmology". With respect to that new term, it defers to a further reference, which describes the typical picture of an arrangement of matter fields that undergo a phase transition wherein the result is BBN preceded by a plausible technical description of pre-BBN matter.

Reference [4] discusses a particularly speculative particle, the axion, and its role in the lead-up to BBN, compared with "the usual hot big bang"; it also leaves many of the details of "hot big bang cosmology" to other papers (e.g. at footnote 45).

I think it is fair to say that the widely circulated paraphrasings of the paper's first sentence are at best begging the question of whether the big bang is that of the standard model, or one of the variations or extensions in the first four of the paper's references. I also think it is fair to say that the author should have anticipated these paraphrasings, and that most readers would have even more trouble distinguishing exactly what is meant by "pre-big-bang" than working physical cosmologists.

For "professionals", the sentences immediately following equation (1) explain the picture: a field with very little mass gains mass during cosmic inflation, with the result that after inflation stops the field contents have the characteristics of a form of cold dark matter that interacts only gravitationally (it is a "free field", which is more amenable to modelling than an "interacting field" or a "self-interacting field" or a field that is both[a]). The paper considers constraints imposed by other observations, how generic a solution remains after considering those constraints, and that the entire idea would be obliterated by evidence favouring any sort of non-gravitational dark matter interaction (including non-gravitational interactions between DM and itself, or different types of DM).

Given this, one would tend to read "pre-big-bang" as used by the author as a region between the end of inflation and the beginning of big-bang nucleosynthesis. The epoch wherein one runs into conflicts between General Relativity and Quantum Field Theory is well before the end of the inflationary epoch, so one should feel free to completely ignore any sort of explanation which invokes things like the beginning of time, or even the differences in the nature of time in these two sufficiently-fundamental-for-these-purposes theories.

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[a] Strictly speaking the field is "minimally coupled to gravity"; it is non-interacting in the sense that there is no associated (non-gravitational) force-carrier, whereas interacting fields generally involve things like gauge bosons. Here because the end of inflation is so far from the part of the early universe that's hot and dense enough that quantum uncertainties and classical curvature cause problems, we can safely use textbook quantum field theory on curved spacetime -- the new physics is in the "decay" from a very light field to a massive field through the inflationary period, as well as the presence of a free field at all (no known fields are "free"). The mass-gaining mechanism is not described, but in the paragraph after the one containing eqn (21), the author claims that a wide range of possible mechanisms is allowed without conflict with other observations, and without conflicting with the central claim that dark matter experiences no non-gravitational interactions (including no self-interactions) after inflation.