Part of my upcoming book explores the long term trajectory of a future human civilisation. This inevitably runs into the issue of the Fermi paradox; if we have a future as an expanding interstellar civilisation, then presumably any intelligent species that evolved before us could also have had that future, and we should see evidence of them. The questions of our own future and the present lack of clear technosignatures from astronomy are inseparable.

Once humanity has escaped the gravity well, I expect mass value to go exponential, because having more mass under ones command allows more energy to be collected, and having more energy allows one to liberate more mass from celestial bodies and do things with it. Energy consumption would then also go exponential. An unchecked exponential civilisation would fill the galaxy pretty quickly; in a few thousand years at single digit percent growth. However, it will in fact be checked over time.

Ergovores

Consider the energy consumption of an interplanetary civilisation made of up lots of disparate parts; planets, habitats of all possible types with different cultures and economies. The civilisations power (its rate of energy consumption) at any time can be expressed as

Where each g is the growth factor of a part of the civilisation and each A is a constant representing its energy consumption rate at time t=0. Over time, the total energy consumption will tend towards being equal to the fastest growing component only. This means that, as distant observers over long enough timescales, we can treat each civilisation as a monolithic energy consumer growing at a single rate.

We do not need to speculate about what happens to the other parts of the civilisation – if they are wiped out, assimilated, or if they coexist. Its simply that they don’t matter from the perspective of total civilisational energy consumption. Taking this very zoomed out view removes the need for excess speculation about what extra terrestrial civilisations might actually be like in terms of politics, economics etc. It might be the case that a particular civilisation has no part of it that grows exponentially; that doesn’t matter either – as we simply won’t see such a civilisation. From the perspective of the Fermi paradox all that really matters is an ‘ergovore’ civilisation that is driven to expand its energy base exponentially, and which would make itself known by through large technosignatures e.g. the dimming of stars due to the construction of Dyson swarms. That is what we need to explain that lack of.

An energy hungry civilisation that is capable of interstellar travel would expand outwards looking for new stars to power itself. It would have some maximum speed it could expand – physically this is the speed of light but practically would likely be slower – and so the energy it could gather in this manner would increase over time in proportion to the cube of this speed, as the spherical volume of stars it could potentially access expands. And here is the crux of this discussion – an exponential function always outpaces a cubic function.

Below is a figure to illustrate when this crossover happens; a civilisation begins with 1 solar luminosity of power (i.e. a Kardashev level 2 civilisation) and then expands into a region with a constant power density of 1/200th of a solar luminosity per cubic lightyear; roughly what we see in the region of our Sun. Once this crossover is reached, then as a matter of physics the exponential expansion ends. I’ll discuss more about what happens at that stage later; for now let’s just say the civilisation cannot continue growing outwards.

The solid line is represents expansion at the speed of light. The dotted lines represent expansion at 0.9, 0.75, 0.5 and 0.1c.

What this shows is that even with very high speeds, and very modest growth rates (compared to what is likely possible for spacefaring civilisations) the crossover is at a fairly short distance on galactic scales – Earth is about 30,000 light years from the centre of the Milky Way to give an impression of the scale. The above plot is also quite generous in allowing cubic growth to continue for thousands of light years. In reality, the density of stars drops off as you get further from the disk, so even on the scales shown here the increase in power available would drop from a cubic to being closer to quadratic.

This model essentially simplified the Fermi paradox by limiting the volume over which it is required to operate. We only have to explain the lack of aliens within a relatively small bubble instead of the entire galaxy or the entire universe. Of course, we can’t know for sure the maximum growth rate of a civilisation to input into the model – but we can perhaps do the reverse, define an volume in which we are confident there are no signatures of Kardashev 2+ civilisations, and then figure out what combinations of growth rates and speeds to exclude.

The Centre Cannot Hold

What exactly happens at the crossover point is hard to say exactly. It’s tempting to think there is a dramatic collapse, thanks both to Asimov and his Foundation series, and the fact Western civilisation hasn’t really got over the fall of Rome.

It’s been 1,500 years. Time to move on

That narrative is certainly possible – it’s made no more or less so by how much it appeals to us intellectually – but there are other courses it could follow. Maybe there is a stagnation, or a secular decline in population. Maybe new technology eases the needed to acquire more energy. We can’t generalise about all possible alien civilisations.

What I think can be said is that it is not possible to pass the crossover by reducing growth rate such that it does not overtake the cubic energy supply. The reason is that the issue of overshoot is a global property of the entire light years wide civilisation; whereas exponential growth is a phenomenon local to each solar system, because solar systems will not be fully exploited as the sphere of the civilisation expands. The original formulation will apply locally to each solar system, producing an ergovore. To prevent this is a coordination problem between parts of a civilisation that are causally separated by years or perhaps centuries. It’s reasonable to assert such a problem hasn’t been solved in our galaxy.

It’s Not The End Of The World

This might seem like a gloomy prognosis for the otherwise hopeful project of space colonisation. But I don’t see it that way. There is plenty of room to grow until then, and the end of such growth is not necessarily the end of everything. Even if it were so – does it matter that civilisation is mortal on a long timescale? We already know that as a result of cosmology. Eventually the universe will just be too cold to support anything. I don’t find any despair in this either though.

I am planning to work in this model; perhaps apply it to an actual model of our galaxy to get a feel for how stellar density drop off as you expand outside the disk changes things. I also welcome any feedback or discussion on this idea.

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