The C-index is this: how close would we have to be to an identical twin Earth to detect them with our SETI searches, given our current detection technology and our (their) emissions? Many argue that we would identify zero twin Earths this way because to detect them, they'd have to be within 3 LY or so - closer than the closest star.
But this post is asking a different question: if modern archaebacteria were seeded onto Hadean prebiotic Earth (say, an iron-sulfur species like the ones that our best guesses show were the last universal common ancestor of all life on Earth) - would they run rampant and colonize the whole world, or would they collapse, relying on some pre-existing network of metabolites produced by other cells? This is relevant to the question of passive colonization of simple organisms from Earth to nearby stars over arbitrary time scales. Archaebacteria in particular are a concern for NASA in terms of contaminating other planets.
This idea has been floated multiple times, including by astronomer and writer Fred Hoyle. Here is how the process of passive colonization could work. Asteroid impacts are sometimes powerful enough to eject surface material at greater than escape velocity. This is how we have over 100 fragments of Mars on Earth right now. If this happens, some bacteria may survive the initial shock, heat, then freezing and dehydration (some bacteria can survive these conditions; and in any event it doesn't have to be many.) Some of these meteorites will escape Earth orbit. If in vacuum and cold they're stable for arbitrarily long periods, they'll just accumulate in the solar system as septic Earth-meteorites over time. Some percentage of these fragments will interact with a solar system body (e.g. Jupiter) and be accelerated to solar escape velocity (like Oamuamua was in its native system.) Some percentage of those will enter another solar system (this is likely to happen once every 15 billion years, based on calculations inspired by Oamuamua.) Some percentage of these fragments will pass through a solar system with "primitive" planets with a liquid water-CO2 atmosphere like the early Earth. Some percentage of fragments will actually strike those planets, and some percentage of bacteria will make it to the surface intact.
Of course there are many unknowns and I only cited one number. We need to know the bacterial "burden" blasted out of orbit per unit time - none of those in our lifetimes, probably not even Tunguska; the percent chance of survival; the stability over time once frozen (being in solid phase is certainly not absolute protection against radiation). The frequency of planets is roughly known, but not the frequency of terrestrial CO2-water worlds, although reasonable bounds could be placed. As to surviving re-entry, this tends to raise the most eyebrows - but it's worth repeating that roundworms on the Columbia did in fact survive uncontrolled re-entry and were found alive on the ground weeks later. Inside a large iron-silicate rock they may be even better protected.
The most uncertain part of this list of attrition factors is the last one we've now come to, the chance of the bacteria fluorishing on the new planet (lack of metal ions for enzymes, or presence of cyanide, low volcanic activity if we're relying on the iron-sulfur archaebacteria; etc.) But we can start actually filling in the values for passive ("dumb") colonization - the equation to show how fast Fred Hoyle's "lifecloud" and Arrhenius's panspermia would actually occur. Note that this is a different concept from the organic von Neumann probes that could also unintentionally seed life as a side effect, although mechanics would be the same, and we would still be looking on watery low-gravity bodies for evidence of them, like comets and asteroids.
Initially I was tempted to make a stab at setting bounds on 50% chance of colonizing another star, but many of the probabilities would be just guesses. It's clear that this is quite a list of attrition factors, reducing the chance nearly to zero for any one cell or asteroid strike on Earth to seed a future alien ecosystem. But over geologic time there have been quite a few of these strikes, and assuming vacuum-frozen surviving bacteria are stable for a long time (a relative straight-forward thing to test), then the solar system has been slowly filling up with septic asteroids - some of which no doubt have been ejected. So for the near-term, this is unlikely to produce lots of passively seeded worlds, but over arbitrary time, the universe would be accumulating archaebacteria from every place that life evolved. If we think of the Sun as a second-generation main sequence star, then planets of third generation stars are more likely to have been seeded by second-generation ecosystems - and may have more metals available in the ashes of the second generation stars from which they're built.
Other quantitative predictions: a one-third chance of life on Europa.
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