What is given
What is given? Humans have a practical problem, they need living space.
But for how many? How fast population grow?
As for the first question, some believe that for Earth 1 billion ($10^{9}$) people are ok, and what is above is excess, there are a bunch of other believers. But solving the problem is to solve the problem, so export 0.1% of the population isn't solving it, 50% - yes maybe it's a significant change. As for today, 50% is 3 billion people. This way - how many? - is probably billion(s) of people.
How fast the population grows, as for 2012 average grow was 1.1% per year. There where higher numbers in the past, 1.8%, 2.2% , World population. There are different beliefs about the growth rate, how it will change in the future, and why it is happening. But let us go with 1.2%. I will skip immortality as insignificant for that problem, as the problem will exist without immortality, and as a factor which consequences I can not predict. I think with immortality grow rate may be at 0% easy (because mortality is one of the driving factors for having children).
As note: grow rate isn't something fixed, as we may see, boiling frog works and people are happy to accept even negative growth under some circumstances, with immortals it will be 0% grow so using fixed value is more for illustrating purposes.
- population, 11 billion, growth rate 1.2% per year, desired(optimal) Earth population 10 billion, and let say 10 billion per each terraformed planet.
Interstellar
There are some factors that are important and are pro- early interstellar colonization.
One star system is limited in resources, like energy, heavy elements, and because of the exponential nature of population grow at some point grow will outcome ability to send people (sustain one human needs power roughly 10kW (24.7.365), sending it needs more even with almost free Bussard jet engine.) And at that point civilization is almost stuck, until it loses some weight or finds some other solutions.
Let's say in each star system we will find, on average, 10 bodies, which we can terraform, this way average population per star system will be 100 billion people.
So if we send at early stage 1 billion people in each star system, and left with 1 billion people in the solar system. For the next 400 years problem is solved, as with 1.2% growth per year, it needs 200 to 10x multiply in population, and 400 years to 100x multiply. $\small 1.012^{200}=10.87$
But is that really a solution, or better to say longer to lasts solution, because expansion isn't the solution at exponential growth, because the exponent is damn fast, it beats everything we might suggest as a solution, at least for now.
Starting at 11 billion population, and have sent 1 billion to each of 10 stars nearby, after 400 years we will end with the need to send people to 10000 stars, 1 billion people for each star as we did it before - to buy another 400 years. (return to the state with 1 billion per system)
When does this strategy stop working for the solar system? When it has to send people farther than 400 light-year - they send 1 billion and they will arrive as 100 billion. So a star system has to send let say 100kk, just to give them some time after arrival to do something before they will have the need to send another pack of people themselves.
- they multiply at travel, because otherwise, it makes no sense, they could stay at home, and not multiply there. They travel because they wish to multiply. (and for simplicity of the picture)
Good thing is, with that strict policy of 100 billion per star system they will have plenty of energy to use it to solve that population problem, they are not stuck in the system, they have plenty of resources to just fly in another galaxy if they wish, or time and resources to find another solution. One of the problems is that each of them will begin to solve that problem at a different time and they will have a different starting point and different optimal strategies.
But how fast solar system will have the need to send at 400+ ly?
Stellar density
- The true stellar density near the Sun is estimated as 0.004 stars per cubic light-year
So 400 ly distance is 1'000'000 stars in that volume, given 100 billion per each star, starting population 10 billion - this volume with radius 400 ly will be inhabited in 1400 years.
There can be better strategies in sending which propagates this time a bit. For example, sending at the beginning right to the maximum distance (let's say 400 ly), and they will send also to the maximum distance.
Sending at a greater speed to have time dilation effects as an example, this way we might send 1 billion people and they will arrive as 1 billion people. Hibernation through relativistic effects.
Local system, not interstellar, not terraforming
There are some difficulties in sending people interstellar, and most important is: what will they find there? Green worlds or mostly rocks which they have to convert to habitable worlds. Most likely a bunch of planets not suitable for terraforming.
But let see what is our system's maximum capability by supporting human life.
Sun Power is 3.828×1026 W
And if we assume 100kW of it per person, and that is enough for food and lifestyle we(some of us) enjoy now it's enough for 3.828×1021 population.
For how long it might last with 1.2% growth per year, for 4166 years.
Problems overview, Critique planetary, and terraforming approaches.
Many of those, who are not lazy to think about non-FTL future (and those who consider interstellar travel in generation ships - all of them) forget about a simple fact - to make a generation ship or just a ship which traves a long time, the ship needs to be habitable in the full sense of that meaning. Every detail we enjoy or consider as the reason to terraform a planet - have to be on that ship. Period.
It has to be this way, not only for the duration of travel but in the duration of terraforming a planet.
Terraforming is Art and Exam which may take 1000's years to accomplish.
Obviously, I'm not sure about 1000's years, and I may imagine something which may terraform planets in let's say 50 years or maybe less - with perfect knowledge about what it doing, perfect knowledge about properties internal structure of the planet, perfect prediction of important moments of how the biosphere will affect that planet, and how that will affect the biosphere. Not sure if that perfect knowledge is possible at all, because of the nature of processes involved - waggly, unstable, changeable - on a fundamental level.
The nice thing about a biological system is that it will adapt and find its own equilibrium in most cases - but will we be happy about that equilibrium or will we complain like here Jellyfish Invading Japan. just a jellyfish, which can be eaten, not something we can't eat, and taste is fine, but no no, we do not like that.
That is kinda sarcasm, but a tiny difference will make significant a significant impact and create a system that we might not enjoy. All discussion I have seen about terraforming, in recent times in connection to Mars stuff - they forget a simple Fact - In a place where we can live, in the place can live everything that lives on Earth at the moment.
On a planet-scale, to undo some errors -it may be impossible without the destruction of the planet. I mean microbiological live mostly, especially one which lives deep in the soil, and which is important for the cycle of life and death. But with other nonmicrobial life, it is also not so simple, at least at our current level of technologies, which sure is subject of change in the future, when we research and will gain more knowledge, but still, it might be not an easy task to control them.
Terraforming task is way much bigger than just blasting a few nuclear bombs on poles to release ...
I consider disassemble of planets an ordinary task, it needs just force, tool, energy - and ok, done.
Terraforming, in a way we might enjoy the result, and accept as equivalent or better than Earth variant is opposite of that, it is almost no force, but pure Knowledge.
So talks about difficulties getting the atmosphere, water - it is just a minuscule part of terraforming, the easiest part of it. As an example, growing a forest - it requires 100's of years, just for the first generation and as result, it will be not a forest but a park - it needs more than that until it will begin to be a forest, especially in case if we do not prepare the soil for it to be like in a forest, thus it might take thousands of years just for the thing to grow and create look and feel of a forest. I even do not talk about 1000 years old trees and forests out of them. Problems like that may be solved, but we are even not close atm for being able to do that.
Why not go to other star systems as places to live
The problem with planets in other star systems is one problem, I would say the least important one. There is another more important problem, in that non-FTL future - the speed of information transmission.
FTL is possible or not possible - not known at the moment. And to know it and probably build it, if it is possible - we need to do a lot of researches, lots of information have to be classified, data exchanges, a lot of work have to be done.
Placing parts of our civilization in places with 400-year ping - does not help in that. Yes, there are partial solutions for that information exchange problem - as an example nodes can be more independent in terms of information, each works at its slice of problems, or in some direction, or randomly picking problems from possible problems, but it does not make everything better it makes it just less worse or same in computing power sense, but anyway, important changes will distribute themselves slower and overall solving different problems like as an example FTL problem it will be slower. There are also other social identification problems, with groups - less likely to agree to work on common problems, and sense of what is the common problem will drift - work can be done, but that long ping and consequence do not make solving faster and easier.
And a planet-oriented civilization(which expands and uses only planets) has 1400 years of its expansion, or less if there are not enough plants for them to use. If civilization relies on a just solar system and space habitat's and energy/matter delivery they have 4000+ years for their expansion with the same population growth rate. After that, both have to change their habits and try to maintain the 0-growth rate. Space habitat systems might be established in other star systems too, but the speed of information exchange is a problem here as well.
How it might be done better.
- do not terraform planets just for a living - it's a very inefficient use of available materials, losing scientific information about the universe, which may be important for our future survival.
- Learn how to live in space, build space habitats - way much easier than terraforming planets, and needed anyway for long trips.
- learn how to use resources in the solar system on a large scale
- send tech seeds to other nearby systems, to research these star systems
- grow seed to be able to supply our system with energy and heavy materials we might need or are used here - the process of energy/material transfer can be combined, and be a very efficient one.
- test that supply system on nearby systems, and begin to extend the system with 0.9c define collection points - make energy material transfer grid-system.
- establish an interstellar transportation system for those who would like to take nice pictures on vacation trips, send those people across the galaxy at 0.999999c.
This way it is possible to grow influence sphere almost with the speed of light, store energy for future use from many stars, do large scale projects, make efficient travel system(with almost 0 energy loss and at desired time dilation)
Do different researches with high energy demands, or if building FTL needs a lot of energy, then this way we might have needed energy. We all will have common goals and fast information exchange in a more compact group - and it might help to solve problems.
Storing the energy of stars is very simple, just by disassembling those stars as an example. Such a process needs a lot of energy and there are lots of useful stuff besides H and He. Assembly them back to get energy back (not very efficient, but better than just losing lots of energy as it is now).
The most efficient and profitable way is to store energy in traveling people and ships(as kinetic energy) by building a global travel network, getting that energy back at the arrival point, or extracting it as needed in any system. More energy we will conserve at early stages more freedom-mobility we will have in the future.
Each second of sun energy, conserved, will allow 3 more people to travel in the future with 10000-times dilation. Each year - 100kk.
At the moment galaxy is just wasting the energy which we might use.
Exponential grow
Exponential growth will catch up with any strategy, even the FTL strategy. Not a big deal trough - as may be seen on any creature living on Earth - humans included.
For how long it will last with 1.2% growth and expanding the sphere of influence at 0.9c speed, which collects all resources - energy, heavy materials - and send's them back to Earth(solar system). Without energy conservation for about 4000 years with the resources of the one-star system and additional 1600-1700 years with resources collected by the influence sphere.
It might take a longer time, with gradually slowing down the growth rate. But overall it's limited by the speed of light, and after 6000-7000 years might be something like a 0.02% growth rate per year.
Expanding with non-FTL more than at 1000-2000 years distance makes almost no sense except travel and information gathering and establishing a galactic transport system. I mean, further expansion - it makes sense for all reasons except solving the overpopulation problem. That's your great filter if you wish so.