You can do something similar to the Drake equation:
where Nlife is how many stars with life there are in the Milky Way and it is assumed that a) once self-replicating molecule is evolved it produces life with 100% probability; b) there is an infinite supply of RNA monomers, and c) lifetime of RNA does not depend on its length. In addition:
You can combine everything except Nbase and LRNA into one factor Pabio, which would give you an approximation of "sampling power" of the galaxy: how many base pairs could have been sampled. If you take assumption that parameters are distributed log-normally with lower estimated range corresponding to mean minus 2 standard deviations and upper range to mean plus 2 standard deviations (and converting all to the same units), you will get the approximate sampling power of Milky Way of
Using this approximation you can see how long an RNA molecule should be to be found if you take top 5% of Pabio distribution: 102 bases. Sequence of 122 bases could be found in at least one galaxy in the observable universe (with 5% probability).
In 2009 article https://www.science.org/doi/10.1126/science.1167856 the sequence of the RNA on the Fig. 1B contained 63 bases. Given the assumptions above, such an RNA molecule could have evolved 0.3 times - 300 trillion times per planet (for comparison, abiogenesis event on Earth' could have occurred 6-17 times in Earth's history, as calculated from the date of earliest evidence of life).
Small 16S ribosomal subunit of prokaryotes contains ~1500 nucleotides, there is no way such a complex machinery could have evolved in the observable universe by pure chance.
I read The Vital Question by Nick Lane a while ago and it was the most persuasive argument I've seen on abiogenesis. May be of interest to you. The argument made was that it could be fairly common based on proton gradients.
I'm not aware of an argument that there was only on abiogenesis event on Earth, just the observation that all known surviving lineages come from a universal common ancestor fairly early on. In principle that would be compatible with any number of initial events. It's just that once a given lineage evolved enough adaptions/improvements, it would spread and take over, and then no new lineage would be able to compete/get started.
Also, your scale for probability seems to be starting from assuming a single long self-replicating genome, but that isn't strictly necessary to bootstrap the evolution of a basic self-replicating metabolism. There are much shorter RNA strands (<200 base pairs) that have some catalytic activity including synthesizing additional RNA (though not copying themselves, AFAIK). Something like that could locally generate large numbers of shorter RNA strands, many with some form of catalytic activity of their own, collectively comprising some form of catalytic cycle that includes making more of all of them. Such a system would also be better able to cope with low copying fidelity b/c the individual strands that need to be copied correctly are shorter.
As far as going from bare RNA to a bacterium, I admit I don't know how this happen(ed? happens?). My naive initial thought is some RNA arising in this environment that could produce fatty acids, which could form a lipid layer around a cluster of RNA molecules spontaneously. Repeat and replicate enough times, and add in some endosymbiosis events and you're not too far off?
I'm not aware of an argument that there was only on abiogenesis event on Earth, just the observation that all known surviving lineages come from a universal common ancestor fairly early on. In principle that would be compatible with any number of initial events. It's just that once a given lineage evolved enough adaptions/improvements, it would spread and take over, and then no new lineage would be able to compete/get started.
Your observation is an argument for only one abiogenesis event, and your claim that one would spread and take over and no new lineag...
"Now, a pair of Scripps Research Institute scientists has taken a significant step toward answering that question. The scientists have synthesized for the first time RNA enzymes that can replicate themselves without the help of any proteins or other cellular components, and the process proceeds indefinitely." From https://www.sciencedaily.com/releases/2009/01/090109173205.htm
While cool, I didn't expect indefinite self-replication to be hard under these circumstances. The enzymes work by combining two halves of the other enzyme- i.e. they are not self-replicating using materials we would expect to ever naturally occur, they are self-replicating using bisected versions of themselves.
I've slightly downgraded my estimate for the minimum viable genome size for self-replicating RNA because I wasn't thinking about complicated groups of cross-catalyzing RNA.
OK, but how does this evolve into a bacterium? Won't it evolve into a local maximum of RNA enzyme replication efficiency and stay there?
The currently smallest self-replicating life form is a battle-hardened descendant of billions of years of attack by hostile lifeforms that have been actively trying to eat its ancestors. I would expect the task of surviving that (and the corresponding structure) to be much more complex than anything that developed during the process of abiogenesis.
As far as I'm aware, only one abiogenesis event has happened on earth- a place with copious amounts of the exact right molecules in cosmically unique quantity, density, and environment to form self-replicating RNA. If abiogenesis has happened anywhere else, it hasn't evolved into intelligent life whose work has intercepted our light cone. My current model for how abiogenesis went down in our world begins with the semi-random coalescence of more than a thousand nucleotides- for reference, the smallest known self-replicating bacterium has a roughly 580,000 base-pair genome. Assuming each additional required nucleotide introduces a 1/2 chance of failure, each random coalescence event has a less than (1/2)^1000 ~= 10^-300 chance of forming a viable self-replicator. This estimate is completely off the scale with respect to events that you can expect to occur in a 100 billion l.y. radius sphere.
This is my current idea, at least. Do you know of a more persuasive argument?
EDIT: For future readers, I now consider the Grabby Aliens paper to present a compelling alternate model.