Over the next hundreds years or so humanity should be able to develop molecular nanotechnology, advanced nuclear fusion for power generation and space propulsion and the harnessing of anti-matter. This might be a technological plateau which might not be exceeded if faster than light travel or wormhole travel is not physically possible.
In previous work the Future of Humanity Institute outlined how a civilization could use a small fraction of a solar system to send replicating probes to all reachable galaxies, and then repeat the process to colonize all stars inside them. The model was not optimized for speed but rather intended as an existence proof that it is physically feasible to colonize on vast scales.
Anders simplifies the analysis and assumes that among competing technologically mature groups starting from the same location and time the group with the most resources wins, unless it is surprised.
It is therefore rational to do an initial resource-harvesting near settlement step to gather energy for faster travel to sufficiently remote destinations. Intermittent resource-harvesting during very long trips is rational for relativistic but not ultra-relativistic travel.
They assumed claimed resources are inviolable. One reason is that species able to convert matter to energy can perform a credible scorched earth tactic: rather than let an invader have the resources they can be dissipated, leaving the invader with a net loss due to the resources expended to claim them. Unless the invader has goals other than resources this makes it irrational to attempt to invade.
Only material within 5 billion parsecs can eventually be reached due to the expansion of the universe, and only gravitationally bound matter such as some superclusters (typically smaller than 20 million parsecs) will remain cohesive in future expansion.
Using a realistic scale factor shows that for probes moving at 0.5c a single acceleration launch can reach 1.24 billion parsecs, while an extra stop every billion years increases it to 2.31 · billion parsecs (86% more) and the continuous limit is 2.36 · billion parsecs (90% more). For 0.8c the gains are 60% and 61% respectively; while an improvement, the upper limit is set by the reachability horizon dh ≈ 4.71 · billion parsecs corresponding to travel at c. The reacceleration method gives the largest payoff in terms of extra reachable volume for velocities near 0.8c; for extremely relativistic probes stopping provides less benefit than for moderately relativistic probes.
Illustration of the basic constraints for interstellar or intergalactic probes. Here mpayload = 30 g, R = 7 · 1022 kg (lunar mass), f = 0.65 (selected arbitrarily) and mdust = 2.5 · 10−9 kg (plausible common navigational hazard dust). The red line indicates the maximum speed the probes can be accelerated to given the existing energy budget. As f increases the energy constraint moves left; it can be moved right by launching fewer probes. The blue curve indicates the slowdown constraint for an ideal rocket; dashed blue curves represent weaker, more realistic engines. The purple curve indicates where dust collisions have energy enough to disrupt all bonds in the probe; the dashed purple curve is 10% of this energy. The yellow curve indicates the constraint of keeping the probe under 1800K, assuming a cylindrical shape with radius 1 meter. The green curve represents the constraint of sending at least one probe. The black dashed curves indicate the number of probes needed for reaching different numbers of destinations.
It is possible to permanently outrun others if one can settle beyond the reachability horizon for the other group. This can occur even for lower expansion velocities if there is enough of a spatial or temporal head-start.
The distance to reach for depends on expected civilization density. If it is assumed to be zero, the extreme limit is the reachability horizon. If other civilizations are expected their density sets an optimal distance. If the density is uncertain a probabilistic strategy favors allocating resources in a power-law fashion out to the reachability horizon.
The number of targets an advanced civilization may want to reach are on the order of N = 10 billion (stars in the galaxy), 1000 trillion (in a supercluster), or ten sextillion 10^22 (in reachable universe).
For γ = 3.7 probes of mass m the total mass-energy resources needed to launch R probes to each target is γNRm; a solar system worth of resources is enough for Rm ≈ 5.4 · 10^19 probe-kilograms towards the galaxy, 5.4 · 10^14 towards a supercluster, and 5.4 · 10^6 towards the reachableuniverse.