Geometry of two colliding domains

Our results will be derived in the context of a completely homogeneous cosmology, with domains expanding according to a simple geometrical rule, which we now describe in the following list of assumptions and notation. This is an approximation by necessity, but as we illustrate in the appendix (by considering a map with over 40 000 galaxies), it accurately reflects a simple and plausible expansionistic behavior in the context of realistic cosmic structure.

1. 1. Spacetime is assumed to be a spatially flat and homogeneous Friedmann-Robertson-Walker (FRW) solutionFootnote ² . The scale factor is denoted by a(t), with a(t ₀) = 1. We use co-moving coordinates, so that the metric reads ds ² = −dt ² + a(t)² (dx ² + dy ² + dz ²), and we use units such that c = 1. All references to distances and volumes refer to co-moving distance and volume.

2. 2. In this homogeneous framework, the first civilization to reach a point in space is assumed to own its resources entirely, leading to a thin boundary between civilizations. This is motivated by considering spacecraft that reproduce exponentially upon reaching a new galaxy, quickly establishing a dominating presence throughout it. Because theoretical colonization times with self-reproducing spacecraft are shorter than intergalactic travel times Jones (Reference Jones1976), we do not expect an extended ‘thick’ boundary, and only a rare shared galaxy at the boundary.

3. 3. The expanding civilizations are taken to originate at coordinates {x ₁, y ₁, z ₁} = { − C, 0, 0} and {x ₂, y ₂, z ₂} = {C, 0, 0}, i.e. the co-moving distance between them is taken to be 2C. 2C is assumed to be a cosmological distance, i.e. larger than the homogeneity scale of the universe (≈0.25 Gly comoving).

4. 4. The first civilization begins expanding at cosmic time t = t ₁, while the second civilization begins expanding at the present cosmic time, t = t ₀. Be aware, this means t ₁ < t ₀.

5. 5. The civilizations expand outward in all directions with a constant velocity v in the co-moving frame of reference (i.e. stationary observers attached to galaxies would see the wave of expansion pass by at speed v in their own local frame of reference). This model is consistent with a number of aggressive expansion strategies (see the appendix as well as Olson (Reference Olson2015)), and it reflects the assumption that both civilizations will achieve the same practical limits to expansion speed. The co-moving distance from each origination point to its respective frontier at time t is given by $r_1 (t) = \int_{t_1} ^t (v/a(t^{\prime})){\rm d}t^{\prime}$ and $r_2 (t) = \int_{t_0} ^t (v/a(t^{\prime})){\rm d}t^{\prime}$ Footnote ³ . It is assumed that r ₁(t ₀) < 2C, i.e. that the first civilization has not overtaken the second one at the time it begins to expand.

At time t ₀, an expanding sphere of radius r ₁ (t ₀) has formed about the first civilization, while the second civilization is just beginning their own expansion. As the spheres expand, they meet at the domain boundary, which is equally distant from point {C, 0, 0} and the sphere of radius r ₁(t ₀) about point {–C, 0, 0}. This can be expressed as the points satisfying r ₁ − r ₂ = r ₁(t ₀), which defines a hyperboloid with foci at the origination points {–C, 0, 0} and {C, 0, 0}, and with semi-major axis A = (1/2)r ₁ (t ₀) – the domain boundary is the x > 0 sheet of this hyperboloid (see Fig. 1).

Fig. 1. Dimensions of a cosmic-scale collision of expanding civilizations: At time t ₀, civ 1 has expanded to radius 2 A, when civ 2 begins its own expansion a comoving distance of 2 C away. At t = ∞, the civilizations will have expanded to comoving radii of r ₁ (∞) and r ₂ (∞), with a hyperboloid forming the boundary between them.

Defining B ² ≡ C ² − A ², we express the hyperboloid in canonical form:

(1) $$\displaystyle{{x^2} \over {A^2}} - \displaystyle{{y^2} \over {B^2}} - \displaystyle{{z^2} \over {B^2}} = 1.$$

When bounded by the maximum expansion distance at r ₂(∞) from {C, 0, 0}, the region inside the x > 0 sheet of the hyperboloid represents the final co-moving volume of space occupied by the second civilization. This region may be integrated to give the final volume V ₂ occupied by the second civilization:

(2) $$V_2 = \displaystyle{{{\rm \pi} (C - A)(3r_2 (\infty )^2 (C + A) - (C - A)^2 (C + A) + 2 r_2 (\infty )^3 )} \over {3C}}.$$

The volume of the first and larger civilization, V ₁, is the volume of a sphere, minus that cut out by the hyperboloid:

(3) $$V_1 = \displaystyle{{{\rm \pi} (C + A)(3r_1 (\infty )^2 (C - A) - (C - A)(C + A)^2 + 2r_1 (\infty )^3 )} \over {3C}}.$$

These equations are valid for 2C ≤ r ₁ (∞) + r ₂ (∞). At greater separation distance, the civilizations never meet and the final volume is just that of a sphere.

Implications of observing an expanding alien domain

We now specialize to the main case of interest. Suppose civ 2 is a young and ambitious technological species, perhaps comparable with humanity, who is close to embarking on rapid expansion into the universe. A short time prior to their launch of self-reproducing spacecraft, they perform a detailed galaxy survey and observe civ 1 at a very early stage of expansion, some cosmological distance away. What does this mean for civ 2's future ambitions?

This scenario imposes the relation $\int_{t1}^{t_0} (1/a(t^{\prime})) {\rm d}t^{\prime} = 2C$ – i.e. between their starting times, light has had just enough time to travel from civ 1 to civ 2. In this intervening time, civ 1 has also become much larger – the radius of civ 1 at time t ₀ is $\int_{t1}^{t_0} (v/a(t^{\prime})){\rm d}t^{\prime} = 2A$ , though civ 2 is unable to see this directly in their galaxy survey. Together, these relations mean that A = v C, meaning that we only need to specify the separation distance 2 C and the expansion velocity v, and we can immediately use the results from the previous section to find final volumes.

Figure 2 illustrates the final geometry of this situation for a separation distance 2C = 3 Gly with expansion velocities of v = 0.3, v = 0.6, v = 0.9 and v = 0.99. We can substitute A = v C into the equations of the previous section to get expressions for the final volumes:

(4) $$V_1 = \displaystyle{1 \over 3}{\rm \pi} (1 + v)\left( {3Cv^2 X_1^2 (1 - v) + 2v^3 X_1^3 - C^3 (1 - v)(1 + v)^2} \right)$$

(5) $$V_2 = \displaystyle{1 \over 3}{\rm \pi} (1 - v)\left( {3Cv^2 X_2^2 (1 + v) + 2v^3 X_2^3 - C^3 (1 - v)^2 (1 + v)} \right).$$

Fig. 2. Final geometry of four scenarios in which civ 2 (blue) observes civ 1 (yellow) at the earliest stages of expansion and simultaneously begins its own expansion. Separation distance is taken to be 3 Gly and the civilizations have a common expansion speed of: 0.3 in (a), 0.6 in (b), 0.9 in (c) and 0.99 in (d). Axes are in units of Gly.

In order to make the v-dependence explicit in the above equations, we have introduced $X_1 = \int_{t_1} ^\infty (1/a(t^{\prime})){\rm d}t^{\prime}$ and $X_2 = \int_{t_0} ^\infty (1/a(t^{\prime})){\rm d}t^{\prime}$ , so that $r_1 (\infty ) = v X_1 $ and $r_2 (\infty ) = v X_2 $ . Note that X ₁ = X ₂ + 2C in this scenario, so that the only place numerical calculations enter is in calculating X ₂, which is just the event horizon distance at the present time – for the cosmological parameters we have used, X ₂ ≈ 16.7 Gly. As before, these equations are valid so long as 2C ≤ r ₁ (∞) + r ₂ (∞), so that the civilizations actually meet.

Figure 2 shows that the consequences for civ 2 will depend heavily on the maximum practical speed of expansion. We make this more explicit in Fig. 3(a), which plots V ₂ as a function of expansion speed v. In absolute terms, note that scenarios with higher practical limits to technology (higher v) give a greater final volume, up to some maximum, whereupon higher limits to technology actually reduce the second civ's final volume, due to the increasingly dominant presence of the first civ.

Fig. 3. Dependence of final volume V ₂ on expansion speed. (a) Higher practical limits to technology (higher v) are only beneficial to the observing civilization up to a certain point. (b) As a fraction of the volume the observing civ could occupy without competition, higher practical speed limits always correspond to greater diminishment by the observed civilization.

We can also examine V ₂ relative to the final volume civ 2 would have occupied in the absence of civ 1 – we call this quantity V ₂ (Competitor)/V ₂ (No Competitor). This fraction is plotted in Fig. 3(b). Notice that for sufficiently small v, the fraction is exactly 1 – this corresponds to an expansion speed slow enough that the two civs are never able to meet, though they eventually see one other.

Also of interest is the dependence on the separation distance, 2C. As shown in Fig. 4, dependence on the separation distance is surprisingly weak over realistic distancesFootnote ⁴ , particularly for high-v scenarios. The difference between observing an expanding civilization just outside our own supercluster and observing one at the maximum plausible distance (8 Gly) amounts to less than a factor of 2 in final volume over a wide range of expansion speeds.

Fig. 4. Dependence of final volume V ₂ on the initial separation distance, 2C. From ‘next door’ to 8 Gly, the distance to the observed civilizations amounts to only a factor of ≈2 in final volume to the observing civilization.

Two visible expanding domains

In the case that two expanding domains are observed, the observing civ is left with a region that is the intersection of the interior of two hyperboloids, and is less convenient for obtaining simple volume equations (we use numerical integration to find the final volume). However, a relevant phenomenon – trapping – is easy to analyze. We describe a civilization as ‘trapped’ if they reach a domain boundary with another civilization in every direction of their expansion, i.e. they cannot reach their maximum expansion radius of r(∞) in any direction. It is very possible that observing two early-stage expanding domains at a cosmological distance implies being trapped by them. Figure 5(a) and (b) illustrate two non-trapped scenarios, while Fig. 5(c) represents a critically trapped scenario and 5d is a trapped scenario – the only difference in these cases is the expansion velocity.

Fig. 5. Final geometry of four scenarios in which civ 2 (blue) observes civs 1a and 1b just as it begins expansion. Angular separation of civs 1a and 1b is 90° and the initial distance to 1a and 1b is 3 Gly. Scenarios (a) and (b) are non-trapped, while (c) is critically trapped (at v = 0.75765) and (d) (at v = 0.9) is trapped. Civ 2 reaches its greatest expansion distance in the plane of the three civilizations, which is shown here.

We will assume that two early civilizations (civs 1a and 1b) are visible to civilization 2 at the earliest stages of their expansion and that they appear at the same distance from civ 2. We will work in the plane formed by the civs, as civ 2 will reach its maximum expansion distance in this plane. For a given separation of 2 C (from civ 2) and angular separation θ between 1a and 1b (as viewed by civ 2), we want to know the expansion speed above which civ 2 will be trapped. The critical speed v _trap occurs when the three spheres of radii r _1a (∞), r _1b (∞) and r ₂(∞) all intersect one another at a single point – i.e. civ 2 has just barely managed to reach r ₂ (∞) when further progress would be cut off anyway by civs 1a and 1b (see Fig. 5(c)).

We express the maximum expansion radii again as r _1a (∞) = r _1b (∞) = vX ₂ + 2vC and r ₂ (∞) = vX ₂, where X ₂ is again taken to be $X_2 = \int_{t_0} ^\infty (1/a(t^{\prime})){\rm d}t^{\prime}$ . The equations for three spheres intersecting at a point then allows us to solve for v _trap, giving:

(6) $$ v_{{\rm trap}} = \displaystyle{{\sqrt {\sqrt 2 X_2 \cos ({\rm \theta} /2)\sqrt {8C^2 + 8CX_2 + X_2^2 \cos ({\rm \theta} ) + X_2^2} + (2C + X_2 )^2 + X_2^2 \cos ({\rm \theta} )}} \over {2(C + X_2 )}}.$$

As expected, v _trap decreases from 1 (the speed of light) at θ = 0 to a minimum value of $\sqrt {C/C + X_2} $ at θ = 180° – the curve is illustrated in Fig. 6. As before, if civ 2 corresponds to humanity or some other species on the cusp of cosmic expansion at t ₀, then X ₂ ≈ 16.7  Gly for our cosmological parameters.

Fig. 6. Dependence of the critical trapping speed on the angular separation of two visible expanding civilizations.

Though we do not have a simple volume equation, we can numerically calculate the final volume of civ 2, which appears in Fig. 7, describing scenarios with a constant angular separation of θ = 90°. The final volume is much more limited than the single-observation scenario, the optimal value for v is lower, and one can see that the final volume becomes rapidly cut off as v exceeds the critical trapping speed.

Fig. 7. Final co-moving volume of civ 2 in the case of two visible domains, with an angular separation of θ = 90°. Both observed domains appear at the same distance 2C from civ 2.