In spite of recent rapid development of biology, chemistry, Earth science and astronomy, the origin of life (abiogenesis) is still a great mystery in science1,2,3,4,5. A prominent feature of life is the ordered information stored in DNA/RNA, and how such information appeared from abiotic processes is a crucial issue. The RNA world hypothesis6,7,8 postulates an early era when RNA played both the genetic and catalytic roles before the DNA-protein world came into being. This is widely accepted due to strong supporting evidence including catalytic activities of RNA, especially its central role in a ribosome. However, a more fundamental and unsolved problem is how an RNA polymer long enough to have a self-replicating RNA polymerase activity (i.e., RNA replicase ribozyme) emerged from prebiotic conditions and then triggered Darwinian evolution.

A key quantity is the minimum RNA length required to show a self-replicating ability. RNA molecules shorter than 25 nucleotides (nt) do not show a specified function, but there is a reasonable hope to find a functioning replicase ribozyme longer than 40–60 nt8,9. RNA polymerase ribozymes produced by in vitro experiments so far have a length longer than 100 nt10,11,12. Furthermore, formation of just a single long strand may not be sufficient to initiate an abiogenesis event. Instead a pair of identical strands may be necessary if one serves as a replicase ribozyme and the other as a template.

Polymerization of RNA in water is a thermodynamically uphill process, and hence reacting monomers need to be activated. Non-enzymatic reactions of adding activated monomers (e.g., imidazole-activated ribonucleotides) to an RNA oligomer have been experimentally studied1,4,13. Reactions at inorganic catalytic sites (e.g. surface of minerals such as montmorillonite clay) may be particularly efficient14,15. Some experiments yielded production of up to 40-mers of RNA16,17, which may be long enough to have some biological activities. However, these results have not been reproducible, and only short oligomers of up to 10-mers were produced conclusively in recent experiments, with the abundance rapidly decreasing with the oligomer length13,18,19,20. This trend is also consistent with the theoretical expectation for random adding of monomers (see below). An experimental difficulty is that aggregates may easily be mistaken for polymers, depending on detection methods13,21.

It is theoretically speculated that terminal ligation of oligomers may hierarchically produce further longer polymers22, but there is no experimental or quantitative demonstration of this starting from realistic prebiotic conditions. A report23 of experimental production of long polymers (>120 nt) by ligation has been subject to reproducibility and the aggregate/polymer discrimination problem13,21,24. A high concentration of oligomers is necessary for ligation to work efficiently, but this may be difficult because oligomer abundance rapidly decreases with oligomer length in polymerization by monomers, even if such a ligase activity exists.

If we consider only the conservative abiotic polymerization, i.e., statistically adding monomers, the probability of abiogenesis may be extremely low on a terrestrial planet. This case is not in contradiction with our existence on Earth, because we would find ourselves on a planet where abiogenesis happened. The life on Earth is believed to have descended from the single last universal common ancestor (LUCA) with no evidence for multiple abiogenesis events, and we do not know any life of a different origin in the universe. The emergence of life early in the history of Earth is often used to argue for a high abiogenesis rate, but an arbitrarily low rate cannot be robustly excluded25,26,27 because the chance probability t ab ∕t ⊕ ~ 0.1 is not negligible assuming a constant abiogenesis rate, where t ab is the time of abiogenesis elapsed from the birth of Earth28, and t ⊕ the present age of Earth. It may well be possible that early Earth was a more favorable environment for abiogenesis than present29,30. There may also be an anthropic selection effect favoring earlier abiogenesis on Earth, because an intelligent life must emerge before the increasing solar luminosity causes an end to Earth’s habitable state (estimated to be ~1 Gyr in future)31.

For the case of a low abiogenesis rate, the number of abiogenesis events is often considered in the Milky Way Galaxy containing about 1011 Sun-like stars32 or in the whole observable universe containing 1022 stars33 inside a spherical volume with a comoving radius of 46.3 Gly (or 13.8 Gly as a light travel time distance)26,34. However, the size of the observable universe is not related at all to its whole physical size. According to the widely accepted view of the inflationary cosmology35,36,37,38,39,40, the physical size of the universe created by an inflation event should be much larger, likely including more than 10100 stars (see below). In that case, even if the expected number of abiogenesis events is much less than unity in a volume size of the observable universe, it may still be consistent with our observations provided that abiogenesis is expected to occur somewhere in an inflationary universe.

The aim of this work is to examine this possibility quantitatively, assuming that the first biological RNA polymer was produced by randomly adding monomers. Koonin41 considered implications of the eternal inflation theory for the origin of life. In this scenario, most part of the universe inflates forever, self-reproducing many subregions that undergo a conventional inflation followed by a hot big-bang universe. Then an infinite number of stars and galaxies would be formed, and we expect emergence of life even if the abiogenesis probability is infinitely small. Though eternal inflation is a theoretically likely scenario42, it is difficult to confirm observationally, and a quantitative discussion is impossible. It is then interesting to ask if life can emerge within the homogeneous region in which we exist, assuming its minimal size necessary to explain observations. This work tries to give a quantitative answer to this question.