Just how likely is it that the galaxy is filled with technological civilizations? Kelvin F Long takes a look at the question using diffusion equations to probe the possible interactions among interstellar civilizations. Kelvin is an aerospace engineer, physicist and author of Deep Space Propulsion: A Roadmap to Interstellar Flight (Springer, 2011). He is the Director of the Interstellar Research Centre (UK), has been on the advisory committee of Breakthrough Starshot since its inception in 2016, and was the co-founder of Icarus Interstellar and the Initiative/Institute for Interstellar Studies, He has served as editor of the Journal of the British Interplanetary Society and continues to maintain the Interstellar Studies Bibliography, currently listing some 1400 papers on the subject.

by Kelvin F Long

Many excellent papers have been written about the Fermi paradox over the years, and until we find solid evidence for the existence of life or intelligent life elsewhere in the galaxy the best we can do is to estimate based on what we do know about the nature of the world we live in and the surrounding universe we observe across space and time.

Yet ultimately to increase the chances of finding life we need to send robotic probes external to our solar system to visit the planets around other stars. Whilst telescopes can do a lot of significant science, in principle a probe can conduct in-situ reconnaissance of the system to include orbiters, atmospheric penetrators and even landers.

Currently, the Voyager 1 and 2 probes are taking up the vanguard of this frontier and hopefully in the years ahead more will follow in their wake. Although these are only planetary flyby probes and would take tens of thousands of years to reach the nearest stars, our toes have been dipped into the cosmic ocean at least, and this is a start.

If we can send a probe out into the Cosmos, it stands to reason that other civilizations may do the same. As probes from different civilizations explore space, there is a possibility that they may encounter each other. Indeed, it could be argued that the probability of species-species first contact is more with their robotic ambassadors rather than the original biological organisms that launched them on their vast journeys.

However, the actual probability of two different probes from alternative points of origin (different species) interacting is low. This is for several reasons. The first relates to astrobiology in that we do not yet know how frequent life is in the galaxy. The second relates to the time of departure of the probes within the galaxy’s history. Two probes may appear in the same region of space, but if this happens millions of years apart then they will not meet. Third, and an issue not often discussed in the literature, is the fact that each probe will have a different propulsion system and so its velocity of motion will be different.

As a result, not only do probes have to contend with relativistic effects with respect to their world of origin (particularly if they are going close to the speed of light), but they will also have to deal with the fact that their clocks are not synchronised with each other. The implication is that for probes interacting from civilizations that are far apart, the relativistic effects become so large that it creates a complex scenario of temporal synchronization. This becomes more pronounced the larger the different species of probes, and the larger the difference in the respective average speeds. This is a state we might call ‘temporal spaghettification’, in reference to the complex space-time history of the spacecraft trajectories relative to each other.

An implication of this is that ideas like the Isaac Asimov Foundation series, where vast empires are constructed across hundreds or thousands of light years of space, do not seem plausible. This is particularly the case for ultra-fast speeds (where relativistic effects dominate) that do approach the speed of light. In general, the faster the probe speeds and the further apart the separate civilizations, the more pronounced the effect. In 2016 this author framed the idea as a postulate:

“Ultra-relativistic spaceflight leads to temporal spaghettification and is not compatible with galaxy wide civilizations interacting in stable equilibrium.”

Another consequence of ultra-fast speeds is that if civilizations do interact, it will not be possible to prevent the technology (i.e. power and propulsion) associated with the more advanced race from eventually emerging within the other species at some point in the future. Imagine, for example, if a species turned up with faster than light drives and simply chose to share that technology, even if for a price, as a part of a cultural information exchange.

Should such a culture refuse to share that technology with us, we would likely work towards its fruition anyway. This is because our knowledge of its existence will promote research within our own science to work towards its realisation. Alternatively, knowledge of that technology will eventually just leak out and be known by others.

There is also a statistical probability that if it can be invented by one species, it will be invented by another; as a law of large numbers. As a result when one species has this technology and starts interacting with others, eventually many other species will obtain it, even if it takes a long time to mature. We might think of this as a form of technological equilibration, in reference to an analogy to thermodynamics.

Ultimately, this implies that it is not possible to contain the information associated with the technology forever once species-species interaction begins. Indeed, it has been discovered that even the gravitational prisons of light (black holes) are leaky through Hawking evaporation. The idea that there is no such thing as a permanently closed system was also previously framed as a second postulate by this author:

“No information can be contained in any system indefinitely.”

Adopting analogues from plasma physics and the concept of distribution functions, we can imagine a scenario in which within a galaxy there are multiple populations, each sending out waves of probes at some average velocity of expansion rate. If most of the populations adopted fusion propulsion technology, for example, as their choice of interstellar transport, then the average velocity might be around 0.1c (i.e. plausible speeds for fusion propulsion are 0.05-0.15c) and this would then define the peak of a velocity distribution function.

The case of human-carrying ships may be represented by world ships traveling at the slow speeds of 0.01-0.03c. In the scenario of the majority of the populations employing a more energetic propulsion method, such as using antimatter fuel, the peak would shift to the right. In general, the faster the average expansion speed, the further to the right the peak would shift, since the peak represents the average velocity.

The more the populations interacted, the greater the technological equilibration over time, and this could see a gradual shift into the relativistic and then ultra-relativistic (>0.9c) speed regimes. Yet, due to the limiting factor of the speed of light limit (~300,000 km/s or 1c), the peak would start to move asymptotically towards some infinite value.

There is also the special case of faster-than-light travel (ftl), but by the second postulate if any one civilization develops it then eventually many of the others will also develop it. Then as the mean velocity of many of the galactic populations tends towards some ftl value, you get a situation where many civilizations can now leave the galaxy, creating a massive population expansion outwards, as starships are essentially capable of reaching other galaxies. That population would also be expanding inwards to the other stars within our galaxy since trip times are so short. Indeed, ships would also be arriving from other galaxies due to the ease of travel. But if this were the case, starships would be arriving in Earth orbit by now.

In effect, the more those civilizations interact, the more the average speed of spacecraft in the galaxy would shift to higher speeds, and eventually this average would begin to move asymptotically towards ftl (assuming it is physically possible), which is an effect we might refer to as ‘spatial runaway’ since there is no longer any tendency towards some equilibrium speed limit. In addition, the ubiquity of ftl transport comes with all sorts of implications for communications and causality and in general creates a chaotic scenario that does not lean towards a stable state.

This then leads to the third postulate:

“Faster than light spaceflight leads to spatial runaway, and is not compatible with galaxy wide civilizations interacting in stable equilibrium.”

Each species that is closely interacting may start out with different propulsion systems so that they have an average speed of population expansion, but if technology is swapped there will be some sort of equilibration that will occur such that all species tend towards some mean velocity of population diffusion.

The modeling of a population density of a substance is borrowed from stochastic potential theory, with discrete implementation for the quantization of space and time intervals by the use of average collision parameters. This is analogous to problems such as Brownian motion, where particles undergo a random walk. This can be adopted as an analogy to explain the motion of a population of interstellar probes dispersing through the galaxy from a point of origin.

Modeling population interaction is best done using the diffusion equation of physics, which is derived from Fick’s first and second law for the dispersion of a material flux, and also the continuity equation. This is a second order partial differential equation and its solution for a population that starts with some initial high density and drops to some low density. It is given by a flux equation which is a function of both distance and time. This equation is proportional to the exponential of the negative distance squared.

Using this physics as a model, it is possible to show that the galaxy can be populated within only a couple of million years, but even faster if the population is growing rapidly, as for instance via von Neumann self-replication. A key part of the use of the diffusion equation is the definition of a diffusion coefficient which is equal to ½(distance squared/time), where the distance is the average collision distance between stars (assumed to be around 5 light years) and time is the average collision time between stars (assumed to be between 50-100 years for 0.05-0.1c average speed). These relatively low cruise speeds were chosen because the calculations were conducted in relation to fusion propulsion designs only.

For probes that eventually manufacture another probe on average (i.e., not fully self-reproducing), this might be seen as analogous to a critical nuclear state. Where the probe reproduction rate drops to less than unity on average, this is like a sub-critical state and eventually the probe population will fall-off until some stagnation horizon is reached. For example, calculations by this author using the diffusion equation show that with an initial population as large as 1 million probes, each traveling at an average velocity of 0.1c, after about ~1,000 years the population would have stagnated at a distance of approximately ~100 light years.

If however, the number of probes being produced is greater than unity, such as through self-replicating von Neumann probes, then the population will grow from a low density state to a high density state as a type of geometrical progression. This is analogous to a supercritical state. For example, if each probe produced a further two probes on average from a starting population of 10 probes, then by the 10th generation there would be a total of 10,000 probes in the population.

Assume that there are at least 100 billion stars in the Milky Way galaxy. For the number of von Neumann probes in the population to equal that number of stars would only require a starting population of less than 100 probe factories, with each producing 10 replication probes, and after only 10 generations of replication. This underscores the argument made by some such as Boyce (Extraterrestrial Encounter, A Personal Perspective, 1979) that von Neumann-like replication probes should be here already. The suggestion of self-replicating probes was advanced by Bracewell (The Galactic Club: Intelligent Life in Outer Space, 1975) but has its origins in automata replication and the research of John von Neumann (Theory of Self-Reproducing Automata, 1966).

Any discussion about robotic probes interacting is also a discussion about the number of intelligent civilizations – such probes had to be originally designed by someone. It is possible that these probes are no longer in contact with their originator civilization, which may be many hundreds of light years away. This is why such probes would have to be fully autonomous in their decision making capability. Indeed, it could be argued that the probability of the human species first meeting an artificial intelligence-based robotic probe is more likely than meeting an alien biological organism. It may also be the case that in reality there is no difference, if biological entities have figured out how to go fully artificial and avoid their mortal fate.

Indeed, when considering the future of Homo Sapiens and our continued convergence with technology the science and science fiction writer Arthur C Clarke referred to a new species that would eventually emerge, which he called Homo Electronicus. He depicted it thus:

“One day we may be able to enter into temporary unions with any sufficiently sophisticated machines, thus being able not merely to control but to become a spaceship or a submarine or a TV network….the thrill that can be obtained from driving a racing car or flying an aeroplane may be only a pale ghost of the excitement our great grandchildren may know, when the individual human consciousness is free to roam at will from machines to machine, through all reaches of sea and sky and space.” (Profiles of the Future, 1962).

So even the idea of separating a biological organism from a machine intelligence may be an incorrect description of the likely encounter scenarios of the future. A von Neumann robotic spacecraft could turn up in our orbit tomorrow and from a cultural information exchange perspective there may be no distinction. It is certainly the case that robotic probes are more suited for the environment of space than biological organisms that require a survival environment.

Consider a thought experiment. Assume the galaxy’s disc diameter is 100,000 light years and consider only one dimension of space. A population of probes starts out at one end with an average diffusion wave speed of around 10 percent of the speed of light (0.1c). We assume no stopping and instantaneous time between populations of diffusion waves (in reality, there would be a superposition of diffusion waves propagating as a function of distance and time). This diffusion wave would take on the order of 1 million years to cross from one side of the galaxy to the other. We can continue this thought experiment and imagine that the same population starts at the centre and expands out as a spherical diffusion wave. Assuming that the wave did not dissipate and continued to grow, then the time to cover the galactic disc would be approximately half than if it had started on one side.

Now imagine there are two originating civilizations, each sending out populations of probes that continue to grow and do not dissipate. These two civilizations are located at opposite ends of the galaxy. The time for the galaxy to be covered by the two populations will now be half of a single population starting out on the edge of the disc. We can continue to add more numbers of populations n=1,2,3,4,5,6….and we get t, t/2, t/4, t/6, t/8, t/10…and we eventually find that for n>1 it follows a geometrical series of the form tn=t0/2(n-1), where t0 is the galactic crossing timescale (i.e. 1 million years) assumed for an initiating population of probes derived from a single civilization which is a function of the diffusion wave speed.

So that for a high number of initiating populations where n ? infinity, the interaction time between populations will be low so that tn ? 0, and the probability of interaction is therefore high. However, for a low number of initiating populations where n ? 0, the interaction time between populations will be high, so that tn ? infinity; thus the timescales between potential interactions are a lot larger and the probability of interaction is therefore low.

It is important to clarify the definition of interaction time used here. The shorter the interaction time, the higher the probability of interaction, since the time between effective overlapping diffusion waves is short. Conversely, where the interaction time is long, the time between overlapping diffusion waves is long and so the probability of interaction is low. The illustrated graphic below demonstrates these limits and the boxes are the results of diffusion calculations and the implications for population interaction.

As discussed by Bond & Martin (‘Is Mankind Unique?’, JBIS 36, 1983), the graphic illustrates two extreme viewpoints about intelligence within the galaxy. The first is known as Drake-Sagan chauvinism and advocates for a crowded galaxy. This has been argued by Shklovskii & Sagan (‘Intelligent Life in the Universe’, 1966), Sagan & Drake (The Search for Extraterrestrial Intelligence, 1975). In the graphic this occurs when n ? ? , tn ? 0, so that the probability of interaction is extremely high.

Especially since there are likely to be a large superposition of diffusion waves overlapping each other. This effect would become more pronounced for multiple populations of vN probes diffusing simultaneously. We note also that an implication of this model for the galaxy is that if there are large populations of probes, then there must have been large populations of civilizations to launch them, which implies that the many steps to complexity in astrobiology are easier than we might believe. In terms of diffusion waves this scenario is characterised by very high population densities such that ?(S,t) ? ? which also implies that the probability of probe-probe interaction is high p(S,t) ? ?. This is box (d) in the graphic.

The second viewpoint is known as Hart-Viewing chauvinism and advocates for a quiet galaxy. This has been argued by Tipler (‘Extraterrestrial Intelligent Beings do not Exist’, 1980), Hart (‘An Explanation for the Absence of Extraterrestrials on Earth’, 1975) and Viewing (‘Directly Interacting Extraterrestrial Technological Communities’, 1975). This occurs when n ? 0, tn ? ?, so that the probability of interaction is extremely low. In contrast with the first argument, this might imply that the many steps to complexity in astrobiology are hard. This scenario is characterised by very low population densities such that ?(S,t) ? 0 so that few diffusion waves can be expected and also that the probability of interaction is low p(S,t) ? 0. This is box (a) in the graphic.

In discussing biological complexity, we are referring to the difficulty in going from single celled to multi-celled organisms, but then also to large animals, and then to intelligent life which proceeds towards a state of advanced technological attainment. A state where biology is considered ‘easy’ is when all this happens regularly provided the environmental conditions for life are met within a habitat. A state where biology is considered ‘hard’ may be, for example, where it may be possible for life to emerge purely as a function of chemistry but building that up to more complex life such as to an intelligent life-form that may one day build robotic probes is a lot more difficult and less probable. This is a reference to the science of astrobiology which will not be discussed further here. However, since the existence of robotic probes would require a starting population of organisms it has to be mentioned at least.

Given that these two extremes are the limits of our argument, it stands to reason that there must be transition regimes in between which either work towards or against the existence of intelligence and therefore the probability of interaction. The right set of parameters would be optimum to explain our own thinking around the Fermi paradox in terms of our theoretical predictions being in contradiction to our observations.

As shown in the graphic it comes down to the variance ?2 of the statistical distribution for the distance S of a number of probe populations ni within a region of space in a galaxy (not necessarily a whole galaxy), where the variance is also the square root of the standard distribution ? relative to a mean distance between population sources ?S. In other words whether the originating civilizations that initiated the probe populations are closely compacted or widely spread out.

A region of space which had a high probe population density (not spread out or sharp distribution function) would be characterised by a low variance. A region with a low probe population density (widely distributed or flattened distribution function) would be characterised by a high variance. The starting interaction time to of two separate diffusion waves from independent civilizations would then be proportional to the variance and the diffusion wave velocity vdw of each population such that to is proportional to ?2/vdw.

Going back to the graphic there comes a point where the number of populations of probes becomes less than some critical number n<nc, the value of which we do not know, but as this threshold is crossed the interaction time will also increase past that critical value tn>tc. In box (c) of the graphic, biology is ‘hard’ and so despite the low variance the population density will be less than some critical value ?(S,t)<?c(S,t) which means that the probability of probe-probe interaction will be low p(S,t) ? 0. This is referred to as a low spatio-temporal distributed galaxy. Whereas for box (b) of the graphic although biology may be ‘easy’, the large variance of the populations makes for a low population density of the total combined and so also a low probability of probe-probe interaction. This is referred to as a high spatio-temporal distributed galaxy.

Taking all this into account and assessing the Milky Way, we don’t see evidence of a crowded galaxy, which would rule out box (d) in the graphic. In this author’s opinion the existence of life on Earth and its diversity does not imply (at least) consistency with a quiet galaxy (unless one is invoking something special about planet Earth). This is indicated in (a). On the basis of all this, we might consider a fourth postulate along the following lines:

“The probability of interaction for advanced technological intelligent civilizations within a galaxy strongly depends on the number of such civilizations, and their spatial-temporal variance.”

Due to the exponential fall-off in the solution of the diffusion wave equation, the various calculations by this author suggest that intelligent life may occur at distances of less than ~200 ly, which for a 100-200 kly diameter galaxy might suggest somewhere in the range of ~500-1,000 intelligent civilizations along a galactic disc. Given the vast numbers of stars in the galaxy this would lean towards a sparsely populated galaxy, but one where civilizations do occur. Then considering the calculated time scales for interaction, the high probability of von Neumann probes or other types of probes interacting therefore remains.

We note that the actual diffusion calculations performed by this author showed that even with a seed population of 1 billion probes, the distance where the population falls off was at around ~164 ly. This is not too dissimilar to the independent conclusion of Betinis (“On ETI Alien Probe Flux Density”, JBIS, 1978) who calculated that the sources of probes would likely be somewhere within 70-140 ly. Bond and Martin (‘A Conservative Estimate of the Number of Habitable Planets in the Galaxy’ 1978) also calculated that the average distance between habitable planets was likely ~110 ly and ~140 ly between intelligent life relevant planets. Sagan (‘Direct Contact Among Galactic Civilizations by Relativistic Interstellar Spaceflight’, 1963) also calculated that the most probable distance to the nearest extant advanced technical civilization in our galaxy would be several hundred light years. This all implies that an extraterrestrial civilization would be at less than several hundred light years distance, and this therefore is where we should focus search efforts.

When it comes down to the Fermi paradox, this analysis implies that we live in a moderately populated galaxy, and so the probability of interaction is low when considering both the spatial and temporal scales. However, when it comes to von Neumann probes it is clear that the galaxy could potentially be populated in a timescale of less than a million years. This implies they should be here already. As we perhaps ponder recent news stories that are gaining popular attention, we might once again consider the words of Arthur C Clarke in this regard:

“I can never look now at the Milky Way without wondering from which of those banked clouds of stars the emissaries are coming…I do not think we will have to wait for long.” (‘The Sentinel’, 1951).

The content of this article is by this author and appears in a recently accepted 2022 paper for the Journal of the British Interplanetary Society titled ‘Galactic Crossing Times for Robotic Probes Driven by Inertial Confinement Fusion Propulsion’, as well as in an earlier paper published in the same journal titled ‘Unstable Equilibrium Hypothesis: A Consideration of Ultra-Relativistic and Faster than Light Interstellar Spaceflight’, JBIS, 69, 2016.