This article is the script of the Rational Animations video linked above. This time, we go into the power law that constitutes the "core" of this model, how to generate simulations of the model, and how to generate predictions from simulations. The predictions covered are many, such as when we'll meet grabby aliens, how many galaxies they can grab, our chances of hearing alien messages, and becoming an interplanetary civilization ourselves. The model answers the Fermi paradox while estimating grabby civilizations' expansion speed: we don't see them in our skies because they travel very fast. High speeds constitute another selection effect: if we could see grabby aliens, they would be here now instead of us.

EA-Forum cross-post

Introduction/summary of the previous video

In the previous video, we talked about why humanity seems to have appeared very early in the universe: among the first civilizations that could ever appear. This estimate is the result of two considerations:

First: Before becoming a civilization, life has to go through a series of difficult phases called “hard steps” that are very unlikely to be completed as early as we did.
And second: Star systems with lifetimes of trillions of years might be habitable, making the overwhelming majority of future civilizations develop in those star systems in the trillions of years to come. 

Earliness constitutes a riddle. It makes humanity’s situation seem rather mysterious. Robin Hanson, who introduced the idea of the great filter, solves this riddle by postulating that civilizations that he calls “grabby” will soon fill the universe. They have three characteristics:

1. They expand from their origin planet at a fraction of the speed of light.
2. They make significant and visible changes wherever they go, and
3. They last a long time.

Once a grabby civilization has occupied a volume of space, it prevents other civilizations from being born in that volume. Therefore grabby civilizations set a deadline for other civilizations to appear. Every non-grabby civilization can only appear early because later, all the habitable planets will already be taken. This constitutes a selection effect.

Hypothesizing grabby aliens explains our apparent extreme earliness. It makes us look typical amongst the other non-grabby civilizations, which can only follow our same path of appearing early and then potentially becoming grabby and expanding into space.

As anticipated, grabby aliens are the main assumption of a detailed model describing how such aliens expand and distribute in the universe. The model makes many interesting predictions, such as when we’ll meet grabby aliens, our chances of hearing alien messages, and becoming an interplanetary civilization ourselves. It also answers why we don’t see aliens yet, considering the universe’s large number of stars and galaxies. 

A model simulation 

Here’s a simulation of the model in an expanding spherical region of space, starting with a diameter of half a billion light-years and containing 400 million galaxies. See these colored regions expanding? They represent the grabby civilizations. In this simulation, they are born at a particular rate, and then they expand at half of the speed of light. When they meet, they simply stop expanding. The model doesn’t tell us anything about what they’ll do after meeting each other. 

Civilizations that are born later take up less space, and if they are born very late, they occupy the crevices between bigger civilizations. Conversely, the first civilizations to appear end up controlling larger volumes.

At the end of the simulation, there are 87 grabby civilizations, 25 billion years have passed, and the diameter of the spherical region of space has expanded to 25 billion light-years. This is because the universe intrinsically expands. The distance between any two parts of the universe that are not gravitationally bound gets larger with time.

The power law and its variables: how we estimate the variables and their effects on the model.

The mathematical model underlying the simulation you just saw is relatively simple. It takes just three parameters: the rate at which grabby civilizations are born, the speed at which they expand, and the number of hard steps they have to go through. Hard steps are difficult phases that life has to go through before becoming a grabby civilization. Each of these steps has an extremely low probability of being completed per unit of time. For example, hard steps could be the creation of the first self-replicating molecules, the passage from prokaryotic cells to eukaryotic cells, and multicellularity. 

The appearance rate of grabby civilizations is denoted with “k” and the number of hard steps with “n”. They are parameters in a simple power law: (t/k)^n. This law takes a time t and returns the chance per co-moving volume that a grabby civilization has appeared by that time. A co-moving volume is a volume of space that expands at the same rate as the rest of the universe.

Note that as the number of galaxies per co-moving volume stays roughly constant in time, you can interpret this law as giving a chance per galaxy per unit time. Note also that this power law approximates the complicated function we showed in the previous video. Because this is just an approximation, the power “n” is close to but isn’t exactly the same as the number of hard steps. 

The power law and the expansion speed “s” are all the ingredients you need for the model. Once you choose a value for “k”, “n”, and “s”, you can run simulations such as the one you saw earlier, and you’re able to tell how grabby aliens expand over time and therefore make predictions.

We can estimate these parameters from data, which enables us to ground the model in real-world observations.

Some things are clear: we don’t seem to be in a part of the universe that has been visibly changed by aliens. Less clearly, but still plausibly, we seem to be on a path to perhaps becoming grabby ourselves. Let’s say that if this happens, it will happen in less than 10 million years. We don’t have any reason to believe that our location in space and time might be unusual compared to other grabby civilizations’ origin locations. From here, we can tentatively say that we can treat our current date as a random sample from all the dates at which a grabby civilization could be born. Therefore, in expectation, our origin date falls in the middle between the other possible origin dates. The fact that we expect to be in the middle of the distribution of grabby aliens’ origin dates constrains the rate at which they must be born, giving us an estimate for k within roughly a factor of two. 

Now let’s talk briefly about the number of hard steps again. As we saw in the previous video, their number can be easily estimated considering two hard step candidates. First: the time between now and when Earth will first become uninhabitable, which is 1.1 billion years. This leads to an estimate of 3.9 hard steps. And second: The time between when Earth first became habitable to when life first appeared, which is 0.4 billion years. This would lead to an estimate of 12.5 hard steps. These two durations suggest a middle number of at least n = 6.

So, now we have k and n. How do we estimate “s”, the speed at which grabby civilizations expand? Surprisingly, we can estimate it by simply observing that we don’t see any trace of other civilizations. The model predicts that, on average, when a grabby civilization is born, a third to a half of the universe is already filled with other grabby civilizations. Since we expect our date to be among grabby aliens’ origin dates, we should see a significant portion of the universe already occupied by other civilizations. And yet, we don’t see that. As we are about to explain, from this observation alone, assuming that grabby aliens exist, we infer that they expand very fast. In short, we don’t see them because if we could have seen them, they would be here now instead of us. 

This is a selection effect in the same vein as the reasoning for why we’re early. Let’s dive into it in more detail.

This is our backward light cone:

Each point falling on the red lines is a coordinate in space-time: simply a date and a place in the universe. From Earth, we can only see the coordinates falling on the red lines. For example, we can now see events that happened three years ago, three light-years away from us. But we have already seen events that happened four years ago and two light-years away, and we still have to see events that happened two years ago, but four light-years away. Inside the cone, there are events that we have already seen. Outside of the cone, there are events that we haven’t seen yet. 

Now, let’s add a grabby civilization that expands fast, almost at the speed of light. 

 

See the yellow region? All of the coordinates in that region can’t be origins for other grabby civilizations. If they were, those grabby civilizations would be here instead of us. They would have prevented our existence. 

For us to be able to see other grabby civilizations, they would have had to be born at the coordinates in these smaller green regions.

The faster the aliens are, the bigger the yellow region and the smaller the green regions. Therefore the faster the aliens are, the smaller is the chance that we could see them.

Since we don’t see them, we conclude that they are probably expanding at higher speeds.

Look at this graph:

The colors and the numbers here represent likelihood ratios. They tell us how well each combination of speed and number of hard steps predict our evidence of “seeing no aliens” compared to the other combinations. The combinations of “n” and “s” falling into bluer areas poorly predict the evidence that we see no aliens relative to the other combinations. That means that those combinations of s and n make seeing no aliens more unlikely. Colors that are more yellow mean the opposite: combinations that correspond to them better predict the evidence that we see no aliens. That means that they make seeing no aliens more likely.

For example, the probability of seeing no aliens assuming 8 hard steps and that they travel at 10% of the speed of light is more or less a hundred times smaller than the probability that we see no aliens assuming 8 hard steps and speeds of 70% of the speed of light.

Therefore the hypothesis that grabby civilizations travel at 70% of the speed of light gains a lot of credence relative to the hypothesis that aliens travel at 10% of the speed of light.

Notice how speed influences the colors way more than the number of hard steps here. Colors are going from blue to yellow horizontally much more than vertically. This is also clear by looking at the numbers. A takeaway from this graph is that speeds of less than one-third of the speed of light predict our evidence of “seeing no aliens” very poorly. That means that they place our chances of seeing no aliens very low. Therefore, they lose a lot of credence compared to higher speeds.

This whole reasoning about speed is similar to the reasoning at the basis of the model. The grabby aliens model explains why we are early because if we weren’t, grabby aliens would have already occupied Earth instead of us. In the same way, we don’t see aliens because otherwise, they would have already occupied our planet. Both hypotheses, grabby aliens and high speeds, make us look more typical: not seeing aliens is more common if grabby civilizations expand at high speeds.

Of course, if you believe speeds higher than one-third of the speed of light to be highly unrealistic in the first place, then you have a reason not to believe at least some parts of the grabby aliens model. For example, you might disbelieve that grabby civilizations ought to make very visible changes wherever they go and say that this is why we don’t currently see them in our skies. Maybe they produce only subtle changes that we will notice in the future. The Grabby Aliens model would still apply, but we wouldn’t be able to say much about grabby civilizations’ expansion speed. 

If Grabby Aliens produce changes that we will only notice with better tools, then the model predicts that we will see many civilizations.

And they would be big! Most grabby-aliens-occupied volumes we could see would appear to us to be much larger than the full moon.

Predictions and how to generate them from the simulations

Now that we have this simple law and know how to roughly estimate the parameters, we can describe and simulate how grabby civilizations expand and distribute in the universe. This allows us to make many specific predictions. 

For example, there’s something interesting to say about how fast grabby aliens appear in the universe. The power law implies that they’ll all appear in a relatively short time window. 

This graph means that they all appear within a time window between five and fifty billion years long, depending on the power “n” we choose to plug in the model. 

The horizontal axis represents grabby civilizations origin times, the vertical axis represents the percentile.

Here’s an example that will help interpret it: let’s say we assume 3 hard steps. Let’s take 20 billion years as an origin date on the horizontal axis. It corresponds to a percentile of almost 0.8. That means that 80% of civilizations will have appeared by the time the universe is 20 billion years old. The other 20% will appear later: till the universe is at least 50 billion years old. A percentile of 0.8 also means that any random grabby civilization has an 80% chance of having appeared by the time the universe is 20 billion years old.

See how sharply these functions rise? The reason is that the power-law implies that civilizations tend to appear just before the deadline. In general, they are very unlikely to appear; therefore, the few lucky ones that do, appear as late as possible. And soon after, they preclude everyone else from appearing, and the functions become flat again. 

It is interesting to understand how such graphs are generated from simulations. 

In general, a simulation of the model, such as the one you saw at the beginning of the video, is generated in this way: potential grabby civilization origins are placed uniformly within a volume at random positions. The positions are paired with random times drawn from a power law (t/k)^n with its own k and n. So, basically, you generate random origins in space and time, but there are more origin points at the times where the power law rises faster. Then, all the origins that would be precluded by the expansion of the other grabby civilizations are eliminated. And then you run the simulation.

Now, let’s try to understand how the graph you just saw is generated from simulation runs. You pick one value for “n”, for example 6, and then you run many simulations with n set to 6. Then you average out all of these simulation runs to get an average simulation. The resulting curve you see in the graph is generated from this average simulation.

Consider this one-dimensional version of a simulation:
 

Space has just one dimension, the horizontal axis. Time is the vertical axis. The “cones” you see are the grabby civilizations. They are born at a certain point, and then they expand till they meet another civilization. Now consider the curve we just saw. You can link each origin point in this one-dimensional simulation to a point in the function. Each of the points corresponds to a different time and percentile. But this is a simplified example. We are doing this with a one-dimensional simulation, but the graphs are generated using 3D simulations.

Having understood all this, we can also better characterize how the parameter “k” of the model is estimated. Maybe you’ve noticed that we haven’t ever mentioned a specific value for “k” during the video. This is because we don’t need to calculate “k” explicitly to generate predictions from simulations. Given a particular simulation, we can iterate through each of its civilization origins, and set that date to be our current date of 13.8 billion years, adjusting all the other dates in that simulation proportionally. This iteration gives us a distribution of answers to whatever question we might be asking. This process in effect assumes that humanity’s rank among the grabby aliens is equally likely to be any particular rank. That is, we are just as likely to be at rank 18% as at rank 72% Therefore “k” is given implicitly using this method.

Now that we have understood more or less how the model predictions are generated, let’s see a few more of them:
 

This graph shows how much of the universe grabby civilizations will collectively control at each particular origin date rank. For example, suppose our origin date is in the middle among other origin dates. In that case, that means other grabby civilizations have already taken about 30% to a half of the universe, depending on the number of hard steps we have plugged into the model. 

We can also say how many galaxies each grabby civilization controls when it meets another grabby civilization. In this case, “galaxy” means a clump of more than 1 million stars. The average galaxy has a hundred million stars, and our milky way galaxy has 100 billion stars.
 

As you can see, grabby civilizations will control between 10 and 10 billion galaxies by the time they meet another one, depending on how lucky they are and the number of hard steps. Without including the more unlikely extremes, the number falls between 1000 and 1 billion galaxies. So we can expect to become quite a big civilization ourselves by the time we meet aliens - provided that we don’t suffer an extinction event or something that permanently curtails our potential. 
 

The earlier a grabby civilization appears, the more galaxies it can grab. The earliest civilizations will be able to take as much as 10 billion galaxies. Civilizations appearing very late will have to settle for the crevices between other grabby civilizations, which could mean way less than a single galaxy to 1 million galaxies, depending on the number of hard steps.

Given our appearance date of 13.8 billion years, and if the number of hard steps is our middle estimate of six, we can expect to take between ten thousand and 100 million galaxies. Cosmic-humanity might be quite vast! 

When will this happen? The model predicts that we’ll meet another grabby civilization in about 50 million to 50 billion years. One hundred million to 10 billion years if we exclude the extreme cases.

SETI and probability of becoming grabby 

Another two very important things that we can try to understand are: 

1. How likely is it that we will become grabby? And 2. How likely is it that we will observe signs of other non-grabby civilizations? Such as alien messages or signatures of alien technology? It turns out these questions are intimately related.

Let’s say that in the universe, the number of non-grabby civilizations is R times the number of grabby civilizations. Then, R is the ratio between non-grabby and grabby civilizations.

The number of non-grabby civilizations includes all the non-grabby civilizations ever to exist, even the ones that at some point transition to grabby. For the purpose of calculating this ratio R, the set of non-grabby civilizations contains the set of grabby civilizations.

Therefore, we can say that 1 out of R is the chance any non-grabby civilization becomes grabby in the absence of other information.

For example, if non-grabby civilizations are 100 times more common than grabby ones, then each non-grabby civilization has only a one in a hundred chance of becoming grabby.

The more non-grabby civilizations there are, the higher is R, because the number of grabby civilizations is fixed by our choice of model parameters. But the more non-grabby civilizations there are, the higher is our chance to succeed in the search for extraterrestrial intelligence.

This means that the higher our chances of finding signs of aliens are, the lower is our chance of becoming grabby ourselves. Thus It would be very bad news to hear alien messages or find other non-grabby civilizations, for example, by spotting signatures of alien technology. That would mean R is high and that our chance of becoming grabby is low.

Why should we care about our chances of becoming grabby being low? Because many scenarios in which we don’t become grabby happen because we become extinct or suffer a catastrophe that permanently limits our potential. 

This reasoning reminds us of the original great filter paper and of arguments already made by Hanson. He argues that finding any sort of extraterrestrial life that is not yet big and visible on an astronomical scale would increase our probability estimate that the great filter is ahead of us. Therefore it would increase our probability estimate that we’re headed for extinction. 

Consider this: Let’s pretend for a moment that grabby aliens expand at the speed of light. Then, if we estimate six hard steps, to expect any non-grabby civilization to have ever been active in our galaxy, a ratio of over 10-thousand is required. This ratio gets smaller with lower speeds. For example, assuming our lower-bound speed of ⅓ of the speed of light, the ratio would be 27 times smaller: roughly 400. Still pretty high. But if we allow even lower speeds, then R becomes even smaller, and the conflict between the probability of finding signs of aliens and of surviving becomes less and less pronounced.

But there’s more: to expect just one other non-grabby civilization to be active now in our galaxy, a ratio of over one million is required. This is valid assuming that non-grabby civilizations last 1 million years. If they last 10 times less, then a ten times larger ratio is required, and if they last more, the ratio required goes down. But it doesn’t get lower than the ratio required for another non-grabby civilization to have ever been active in our galaxy.

Therefore, if higher speeds are possible and if you believe that we are not alone in our galaxy, you also need to accept that the chance to become grabby ourselves is vanishingly small. And if you believe that our chances of becoming grabby are good, then we are almost surely alone in our galaxy. Pick one.

Conclusion

And this is it. Now, you should have a pretty clear picture. Notice how this is not just speculation. The strength of this model is that its basic assumption, grabby civilizations, stems from a very precise observation: that humanity looks early. And as we’ve seen, the mathematics of the model can be grounded in observations too. To strongly reject the model, you need to reject that we’re early or find a much better explanation than grabby aliens. Otherwise, the model looks reasonably likely. My gut tells me that grabby aliens have a decent chance of actually being a correct assumption. 

If grabby aliens as described in this video are true, then the universe is about to enter a new phase. Up until now, cosmology has been about dead stuff. But soon, everything might be filled with life that restructures everything it touches according to its goals. What will become of the universe will be decided by life itself rather than by dead processes. 

If this is all true, I hope that humanity will survive to join the feast. But if we don’t manage to survive, grabby aliens might come across our ruins, so remember to write respectful comments, just in case they discover this video.