Crowded or Lonely? The Statistics of Alien Life

Source: Crowded or Lonely? The Statistics of Alien Life, Cool Worlds, 20:21, uploaded 2024-06-19, category Mathematics, playlist index 951.

Cool Worlds opens by making fun of the usual alien headline. A video about an ambiguous signal can always suggest that contact is close, often with Michio Kaku’s face somewhere in the thumbnail. David Kipping takes the opposite route. A paper written with Geraint Lewis has led him towards a result that makes the search for intelligent life look bleak: a galaxy is likely to be almost empty or almost full, whilst an intermediate population occupies a narrow and finely tuned range.

Edwin Jaynes and the flasks of water

The starting point comes from a thought experiment that Edwin Jaynes proposed in 1968. Imagine a table covered with flasks of water. Each flask has roughly the same volume, and the room keeps roughly the same temperature and pressure. Small differences remain. Someone hands you a random chemical, compound X, and asks in what percentage of the flasks it will dissolve.

With no prior information, Jaynes argues that a guess near 0% or 100% makes sense. A small variation between flasks can decide the outcome when the chemical almost always dissolves or almost never does. A guess around 50% carries a stranger implication. It says that the small differences between the flasks happen to sit near the threshold between dissolving and failing to dissolve, as if the whole room had been set up at exactly the point where those minor variations matter. That fine tuning deserves a lower probability.

The resulting distribution has high probability near both ends and a valley in the middle. Jaynes later gave this intuition a rigorous form, and the video places the result among the foundations of objective Bayesian statistics. Kipping’s connection to astrobiology follows from changing the objects in the experiment. Star systems take the place of the flasks, and the question becomes the fraction that eventually develops intelligent life. Jaynes’s argument then points towards a fraction close to 0% or close to 100%. Kipping quotes Arthur C. Clarke’s view that both possibilities are equally terrifying.

The Drake equation and its missing ceiling

The immediate objection concerns the Drake equation. Frank Drake’s equation describes the number of communicative civilisations in the galaxy as a product of seven input variables. Kipping thinks the equation has earned much of its bad reputation through misuse. Someone can guess values for the terms, multiply them together and announce a precise number of civilisations, although the equation itself cannot supply those guesses.

There is a structural problem as well. In its usual form, the equation has no upper bound. Set every fraction to one, give a civilisation a ten-billion-year lifetime and use a star-formation rate of 100 stars per year. The multiplication produces a trillion civilisations. The Milky Way contains fewer than a trillion stars, so the result has already escaped the population it is meant to describe. The number of terms also looks arbitrary. A model could add fractions for multicellular life, language, or any other step that someone regards as necessary.

Kipping has therefore spent years trying to reduce the equation to a simpler relation between births and deaths. The birth-death formalism applies to any volume, from a galaxy to a supercluster. Some civilisations appear during a given period and some disappear. After the population settles, the birth rate balances the death rate.

Let the birth rate per possible location be $b$ and the death rate per occupied location be $d$. If a volume contains $N$ possible locations and $O$ occupied ones, the birth rate is $b(N-O)$ and the death rate is $dO$. Equilibrium requires those quantities to match. The occupation fraction $F=O/N$ then depends on the ratio of birth to death rates, which Kipping calls $\Lambda_{\mathrm{BD}}$:

The formalism assumes a steady population, although it stays agnostic about the mechanism. Civilisations might arise spontaneously or through empire building. Their deaths might follow internal conflict or an outside event. The locations could be stars, planets, or cubic light-years. The balance relation remains the same within the model.

The S-curve and the narrow middle

When $\Lambda_{\mathrm{BD}}$ is very small, civilisations emerge rarely and disappear quickly. The galaxy stays almost empty. As the ratio grows, the occupation fraction rises. Eventually most locations are occupied, so a further increase in the birth rate has little effect. The graph has the familiar S shape of a process with a finite substrate.

Kipping gives mould on bread as one example. The mould starts with a small population, spreads quickly, and slows once the bread runs out. Technology adoption has the same shape in the video’s telling. Early adopters buy the first iPhone, the mainstream follows and growth slows once Apple has approached the available market.

The S-curve becomes sharper when the horizontal axis extends across the full range of possible birth-death ratios. A wide range of values leaves the galaxy nearly empty. Another wide range leaves it nearly full. The transition between them occupies a thin neck. Intermediate occupation therefore requires a narrow choice of $\Lambda_{\mathrm{BD}}$.

This is where Jaynes’s result returns. The paper adopts a log-uniform prior for $\Lambda_{\mathrm{BD}}$, which the video describes as the usual uninformative prior for a scale parameter. Under that prior, the paper formally derives a distribution for $F$ that follows Jaynes’s earlier result. The Drake-style birth-death picture and the flask experiment reach the same shape: intelligent life is likely to fill the galaxy or remain close to absent, whilst the middle looks contrived.

The lonely galaxy and the problem for SETI

The nearly full case sits badly with what astronomers observe. Decades of radio searches have produced silence. Astronomers see no engineered stars or obvious laser beams, and the ordinary observations of the galaxy remain explainable through natural processes. Earth’s own history creates another problem for the high-occupation scenario. Technological civilisation has existed for a tiny part of Earth’s lifetime, so a galaxy in which almost every star hosts a technological civilisation would require an unusual alignment of timing.

Once the filled galaxy is set aside, the model leaves the S-curve’s narrow neck and the nearly empty plateau. A SETI search that hopes to find a signal implicitly places its bet on the narrow region where intermediate occupation occurs. Kipping says that this conclusion troubled him because the search for extraterrestrial intelligence is one of the things he most wants to work on. He grew up dreaming about alien life, and he does not want the result to become an argument for abandoning the field.

The discomfort leads him to attack his own paper. Scientific work should survive that treatment, and he tries two ways to loosen the conclusion.

A larger survey and an unknown lower bound

The first possibility accepts a mostly empty galaxy and relies on scale. If very few civilisations exist, a larger survey should eventually find one. Modern SETI surveys cover around 10,000 stars, while the Milky Way contains roughly 100 billion. Perhaps the answer is to examine more of them.

The mathematics sets a difficult condition. To find another civilisation in a sample, $\Lambda_{\mathrm{BD}}$ must be greater than roughly the reciprocal of the sample size. It must also remain below about one, since values near one lead towards the filled-galaxy case. Even a survey of the whole galaxy would leave the lower end open. There is no useful theory in the video for how small the birth-death ratio can become.

Kipping illustrates the problem with $\Lambda_{\mathrm{BD}}=10^{-30}$. If that were the true value, surveying all $10^{11}$ stars would still give a probability of about $10^{-19}$ of finding another civilisation. A preference for a larger value remains possible as a personal judgement. The video treats it as an evidential claim only when observation supports it, and says that the current evidence supplies no such lower limit.

Breaking the steady-state assumption

The second possibility rejects the main assumption of the formalism: the galactic population has reached a steady balance between births and deaths. A population could be declining, although Kipping finds that hard to motivate because the universe has become more hospitable over time. The early universe had more gamma-ray bursts, supernovae and rapid phases of star formation. An increasing population therefore seems easier to imagine.

An expanding galactic empire could raise $\Lambda_{\mathrm{BD}}$ through colonisation. The idea has persuasive advocates, yet it creates another timing problem. Even probes travelling at speeds available to us should cross the galaxy in much less time than the galaxy’s current age. On cosmic timescales, a colonisation wave would look like a rapid phase transition.

That leaves three temporal positions. We might live before the colonising civilisation emerges, in which case there is nobody to contact. We might live during its expansion, which occupies a brief interval in cosmic history. Or we might live after colonisation, which conflicts with the apparent emptiness of the galaxy. Kipping cannot make this route fit the observations without another finely chosen circumstance, so the pessimistic reading of the paper remains.

Why search anyway

The result concerns technological civilisations. Simple life could still exist around almost every star. The video says that current observations place no useful limit on that possibility, and Earth’s history remains consistent with life becoming common. Life might often develop technology in forms that our searches cannot recognise.

That possibility changes the meaning of an empty sky. The instruments may be looking for a narrow class of signals produced by a narrow class of civilisation. A better response could involve methods that search for different forms of activity, rather than increasing the number of stars in the same survey without limit.

Kipping gives a practical reason to continue SETI even under the pessimistic model. The probability of success might be small, while the discovery would carry an enormous scientific return. Stopping guarantees that the search produces no detection. He also keeps open the filled-galaxy scenario for simple life and for technologies whose traces remain outside current methods.

The final position stays provisional. Kipping says his view has changed over several years in public, and he expects it to change again. Jaynes’s old result gives him a useful way to approach the question, while the search itself keeps the question alive.

Limits

The note follows David Kipping’s presentation and the English captions for the 20-minute video. The description names a paper by Kipping and Geraint Lewis, “Do SETI Optimists Have a Fine-Tuning Problem?”, as submitted to the International Journal of Astrobiology. The video presents the paper’s derivation, its priors and its numerical examples. The paper’s peer-review status, working calculations and underlying observational data are outside this note.

The birth-death formalism depends on a steady-state population, which the video identifies as its main assumption. The claims about SETI survey sizes, the number of stars in the galaxy, the speed of colonisation and the absence of technological signatures appear as reported figures or arguments in the video. They are useful for reconstructing the reasoning, while the captions and description do not provide enough source detail to treat each one as independently verified here.

Further reading / references

• Kipping and Lewis, “Do SETI Optimists Have a Fine-Tuning Problem?”, submitted to the International Journal of Astrobiology, as linked in the video description.
• Edwin Jaynes, “Prior Probabilities”, IEEE Transactions on Systems Science and Cybernetics, 4, 227, 1968.