Space races: Settling the universe Fast

Space races: settling the universe fast
Anders Sandberg
Future of Humanity Institute,
Oxford Martin School,
University of Oxford
January 24, 2018
Abstract
This report examines the issue of the resource demands and constraints
for very fast large-scale settlement by technologically mature civilizations.
I derive various bounds due to the energy and matter requirements for
relativistic probes, and compare with bounds due to the need to avoid
collisions with interstellar dust. When two groups of the same species
race for the universe, the group with the biggest resources completely
preempts the other group under conditions of transparency. Expanding
nearby at a lower speed in order to gain resources to expand far at a high
speed is eective. If alien competitors are expected this sets a distance
scale aecting the desired probe distribution.
This report examines the issue of the resource demands and constraints for
very fast large-scale settlement. If a technologically mature species wishes to
settle as much of the universe as possible as fast as possible, what should they
do? If they seek to preempt other species in colonizing the universe, what is
the optimal strategy? What if the competitor is another fraction of their own
civilization?
1 Motivation and earlier work
In an earlier paper [2] we outlined how a civilization could use a small fraction of
a solar system to send replicating probes to all reachable galaxies, and then re-
peat 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, even beyond the discussions of galactic colonization
found in the SETI literature.
A key question is whether it is rational to use up much of the available
resources for this expansion, or if even very aggressively expanding species would
leave signicant resources behind for consumption. Hanson has argued that a
species expanding outwards would have economic/evolutionary pressures to use
more and more of resources for settlement, "burning the cosmic commons"
[1]. In this model the settlement distances are small, so there are numerous
generations and hence evolutionary pressures could exert much in
uence. In
contrast, our model of long-range colonization [2] uses limited resources and has
1
few generations but is not optimized for speed: internal competitive pressures
might hence lead to dierent strategies.
The related model in [5] describes civilizations converting matter under its
control to waste heat, examining the cosmological eects of this. Here the energy
costs of the probe expansion is not explicitely modelled: the conversion is just
assumed to be due to consumption.
1.1 Assumptions
We will assume civilizations at technological maturity can perform and auto-
mate all activities we observe in nature. In particular, they are able to freely
convert mass into energy1in order to launch relativistic objects able to self-
replicate given local resources at target systems. The report also assumes a
lightspeed limit on velocities: were FTL possible the very concept of "as early
as possible" becomes ill-dened because of the time-travel eects. This is similar
to assumptions in [2, 5].
The discussion of dust and radiation (3.1) issues assumes some physical limits
due to the fragility of molecular matter; if more resilient structures are possible
those constraints disappear.
The universe is assumed to be dark-matter and dark-energy dominated as the
CDM model, implying accelerating expansion and eventual causal separation
of gravitationally bound clusters in 1011years or so.
The civilization has enough astronomical data and prediction power to reli-
ably aim for distant targets with only minor need for dynamical course correc-
tions.
We will assume 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.
2 Energy constraints
Assume the species has available resource mass Rand can launch probes of mass
m. If they use a fraction fofRto make probes and the rest to power them
(through perfect energy conversion) they will get N=fR=m probes.
The energy it takes to accelerate a probe to a given
-factor isEprobe =
(
1)mc2. The energy to accelerate all probes is E= (
1)mc2fR=m =
(
1)Rfc2. The available energy to launch is E=Rc2(1f), so equaling the
two energies produces

= 1 +1f
f=1
f: (1)
The speed as a function of gamma is v(
) =cp
11=
2. The speed as a
function of fis hence:
v(f) =cp
11=(1 + (1f)=f)2=cp
1f2: (2)
1Whether dark matter or only baryonic matter is usable as mass-energy to the civilization
does not play a role in the argument, but may be important for post-settlement activity.
2
Neatly, the speed (as a fraction of cand function of f) is a quarter circle.
2.1 Simple constraints
There is a minimum probe mass, required in order to slow down the payload2at
the destination. Using the relativistic rocket equation we get the requirement
mmpayloadetanh1(v=c): (3)
The multiplicative factor is 3.7 for
= 2, 5.8 for
= 3 and 19.9 for
= 10.
Note that this assumes a perfect mass-energy photon rocket ( Isp=c= 1).
For less eective rockets such as fusion rockets ( Isp=c= 0:119) or ssion rockets
(Isp=c= 0:04) the mass requirements are far higher, mmpayloade(c=Isp) tanh1(v=c):
There is a trivial bound NR=mpayload , corresponding to the smallest,
slowest and most numerous possible probes (which would require external slow-
ing, for example through a magsail or Hubble expansion).
Another trivial constraint is that fis bounded by N= 1, that is, fm=R .
Hence the highest possible speed is
vN=1=cp
1(m=R )2; (4)
when all resources are spent on accelerating a single probe.
2.2 Expanding across the universe
Given an energy budget, how should probes be allocated to distance to reach
everything as fast as possible? Here "as fast as possible" means minimizing the
maximum arrival time.
Most points in a sphere are closer to the surface than the center: the average
distance is a= (3=4)r. In co-moving coordinates the amount of matter within
distanceLgrows asL3. Hence most resources are remote. This is limited by
access: only material within dh5 Gpc can eventually be reached due to the
expansion of the universe, and only gravitationally bound matter such as some
superclusters (typically smaller than 20 Mpc [4]) will remain cohesive in future
expansion.
Given the above, reaching the surface of the sphere needs to be as fast as
possible. If interior points are reached faster, that energy could be reallocated
to launching for the more distant points, so hence the optimal (in this sense)
expansion reaches every destination at the same time.
In the simple non-expanding universe case the necessary velocity of an in-
dividual probe aimed at distance risv(r) =cr=dh, whereis the maxi-
mal fraction of lightspeed possible due to other constraints. This corresponds
to
(r) = 1=p
1k2r2wherek==dh. The total kinetic energy for send-
ing probes to the entire sphere is Etotal =Rdh
04r2d(
(r)1)mc2drwhere
d= 3fR=4md3
hdenotes the destination density of the N probes (assumed to
be constant throughout the sphere). Expanding,
Etotal= 4dmc2Zdh
0r21p
1k2r21
dr
2Potentialmpayload mentioned have been 500 tons, 30 grams, or 8 107grams [2].
3
Figure 1: Plot offas a function of under the requirement that Etotal=E. Points
under the curve represents feasible choices.
=fRc2
23"
3 sin1() +3(3)p
1223#
(5)
The bracket represents the geometric part of the distribution. The dh's cancel,
since the expression only distributes the N probes; a settlement program trying
to reach the entire volume would aim for N=fR=m/d3
h.
The highest value of compatible with having E=R(1f)c2Joules of
energy to accelerate can be found by equating the EtotalwithEand expressing
it as a function of f. The result is an arc descending from = 0;f= 1
to= 1;f= 2=(32)0:2694, independent of m(gure 1). Assuming
f= 0:2694 gives
= 1=f= 3:7. Building more or heavier probes (higher f)
than this curve leaves less energy to power them than is needed for the intended
.
If remote destinations are more important to reach early than nearby ones
(or Hubble expansion is a major factor), then a nonlinear velocity distribution
v(r) =c(r=dh)with > 1 would ensure faster arrival at the frontier. The
overall eect is to reduce the overall energy demand (roughly as 1 =), since the
average velocity is lower. The extra energy can of course be allocated to increase
to 1, but beyond that it is likely rational to add
to the interior probes.
2.3 Expansion of the universe
In an expanding universe an initial high velocity will be counteracted by the
expansion: in co-moving coordinates the initial velocity v0declines as v0=a(t)
wherea(t) is the cosmological scale factor. For a de Sitter universe (an approx-
imation to late cosmological eras in universes dominated by dark energy) this
goes asv(t) =v0eHt. The distance traversed behaves as s(t) = (v0=H)(1
eHt), and the arrival time at distance Listarrive =(1=H) ln(1HL=v 0).
4
Figure 2: (Top) Maximum reachable co-moving distance when traveling at constant
co-moving speed or with a given initial velocity. Note that arrival occurs at t=1to
the right. (Bottom) Dierence in conformal volume that can be settled between the
methods.
While this means a probe going suciently far does not have to use resources to
brake on arrival, it slows arrival times and causes a maximal reachable distance
v0=H.
If a probe were to re-accelerate halfway to its original speed it would arrive
at timet0
arrive =(2=H) ln(1HL= 2v0)< tarrive . With more and more
waypoints the probe approaches a velocity v0=v0+ (t=d) wheretis the stop
time anddthe average distance between stops. This term can be very small for
intergalactic d. The reachable distance scales as ( v0=c)dh.
Numerical integration due to Stuart Armstrong using a realistic scale factor
a(t) shows that for probes moving at 0 :5ca single acceleration launch can reach
1:24109pc, while an extra stop every billion years increases it to 2 :31109
(86% more) and the continuous limit is 2 :36109(90% more). For 0 :8cthe
gains are 60% and 61% respectively; while an improvement, the upper limit
is set by the reachability horizon dh4:71109pc corresponding to travel
atc. Plotting the reachability limits as a function of v0(gure 2) shows the
lower bound of a single acceleration (blue) versus the upper bound of constant
reaccelerations (red). The reacceleration method gives the largest payo in
terms of extra reachable volume for velocities near 0 :8c; for extremely relativistic
probes stopping provides less benet than for moderately relativistic probes.
This supplements the "near foraging" approach in section 4.4, suggesting
that constructing way-points where local resources are used to boost long-range
travel is a rational strategy for long-range intergalactic settlement (within galaxy
clusters the expansion has negligible eects), assuming that other constraints
prevent ultrarelativistic velocities. Actual waypoint distances are going to be
aected by issues of where convenient mass-sources are available3and how much
3The cosmic web suggests that distances on the order of void sizes (100 Mpc) may be
5
sideways deviation acceptable.
It is worth noting that the resources needed are fairly modest, since only
one4probe needs to be accelerated.
2.4 Summary of constraints
Together these bounds (plus the dust bound discussed in section 3.1 and heating
bound in section 3.2) produce a region of feasibility in the ( v;m) plane, as shown
in gure 3.
For large values of fthere is not enough energy to accelerate probes strongly.
Asfdecreases the main bounds for numerous and small probes are primarily the
deceleration bound (especially if photon rockets are not feasible) and possibly
dust/radiation (depending on astrophysical environment). Heavier probes are
strongly aected by the deceleration constraint, typically making the dust issue
irrelevant.
3 Dust and radiation
Traversing long distances makes the probes vulnerable to impacts with dust
grains and interstellar gas; the rst produces an explosion, the second acts as a
high energy proton beam.
For a detailed analysis of these issues for a mildly relativistic probe (0.2c)
traveling to Alpha Centauri, see [11].
3.1 Dust
3.1.1 Individual dust grains
A probe of cross section area traversing a distance Lwill sweep out a volume
L. If dangerous dust has density , the probability of no impact during transit
iseL, and the required redundancy to ensure at least one probe reaches the
destination iseL. This increases exponentially with a length scale set by
1=(), see below.
The upper limit of survivable dust impacts depends on the maximum energy
impact that can be withstood, Elim. As an approximation, Elimmust be lesser
than the total binding energy5of the probe: Elimkm. Hence, if a probe of
massmneeds to survive encountering a grain of size mdustit has to travel at a
speed below

lim= 1 +km=mdustc2: (6)
In reality the impact would generate a cone with initial opening angle h2i

2, widening as spallation products interacted with the probe; the volume
aected is hence smaller than the entire probe but potentially more strongly
aected. Various shielding congurations can be considered.
reasonable, although there are signicant numbers of galaxies and stars in the voids.
4Or enough redundant probes to replenish expected losses on the next leg so that at least
one probe will arrive.
5For diamond kis about 800 kJ/mol, or 66 MJ/kg. This is likely near the upper limit for
molecular matter.
6
Figure 3: Illustration of the basic constraints for interstellar or intergalactic probes.
Herempayload = 30 g,R= 71022kg (lunar mass), f= 0:65 (selected arbitrarily)
andmdust = 2:5109kg (plausible common navigational hazard dust). The red
line indicates the maximum speed the probes can be accelerated to given the existing
energy budget. As fincreases 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
dierent numbers of destinations.
7
3.1.2 Dust size distributions
The total interstellar dust density is about total6:21024kg/m3. While
the typical interstellar dust particle diameter is below 0.1 and has mass on
the order of 1015kg, there is a potential tail of heavier gravel. The Purcell
theory of dust extinction suggests that very large >10m grains (mdust>1011
kg) cannot contribute much mass, and the MRN distribution also assumed an
upper cut-o at 0 :25m. Using these cut-os make
limessentially irrelevant
compared to the deceleration constraint. However, recent results suggest that
the tail may be heavier than expected [7].
Considering interstellar objects (asteroids, comets etc) larger than 1 km,
various estimates of number densities range from 1010to 103per cubic AU
(with the lower estimates dominating recently) [8]. Assuming a number density
104per cubic AU and a cross section of 3 :1106square meters gives a impact
risk of 1:41019per AU, making the mean free path far longer than the width
of the galaxy. Even the most dense estimate in [8], 5 :4102per cubic AU,
gives a negligible risk.
However, assuming a diameter power law distribution aimplies a total
cross section of (a) =(1km)(a=1km)+2. For asteroids, estimates of
range from 2.3 to 3.5. The uncertainties are partially due to measurements on
dierent subpopulations, where the upper value is the equilibrium distribution
for fragmentation processes. Accepting this somewhat doubtful model gives a
risk of 1018to 1015/AU for 1 meter objects, 2 1018to 1013for 0.1 meter/1
kg objects, and starts to become a navigation hazard over interstellar distances
only if there is a sizable fraction of mass in the 0.1 mm range or below. This dust
mass is, however, also likely within the range that can be shielded (as above),
assuming a modest
.
The largest dust grain expected to hit during a long
ight when Ngrains
distributed as aare encountered scales as amax/N1=(1). Forin the
2.3-3.5 range this implies a scaling as N0:4toN0:76. Hence a probe designed for
a trip through the entire plane of the Milky Way should expect grain diameters
between 7,000 and 2 107times the median grain encountered over 1 AU.
3.1.3 Redundancy
In order to reach the destination with high probability enough probes must be
sent so that at least one arrives: N >eL. This produces a constraint
f=m>eL=R: (7)
This constraint is relatively binary: either there is enough resources or a su-
ciently safe trip to do it with few probes and probes can be made very large,
or the risk is so high that the probes must be made as small and redundant as
possible (f1;m0). Intermediate cases require ne tuning of the parame-
ters.
Another simple model of speed-related risk would take the rate of failure to
be proportional to
, where1. The probability of surviving distance d
isekd
wherekis a vulnerability constant. The characteristic distance that
could be traveled without extreme redundancy is dsafe= 1=k
, or conversely,
the safe speed for going the distance would be
safe = (kd)1=. The arrival
timed=vwould scale nearly linearly with distance until close to dsafe, where
8
it has a singularity and diverges. Hence the behavior is roughly binary in the
same way as the redundancy issue: there exists a (environment, construction
dependent) characteristic distance or speed beyond which requirements increase
without bound, but below it the requirements are fairly modest. Unfortunately
estimatingk;requires fairly specic models of the probe and the environment.
3.2 Radiation
The radiation constraint is that the probe needs to survive incident high energy
radiation throughout the
ight. Each cubic meter traversed will host about
600,000 protons. The probe will experience a proton kinetic energy irradiance
of
I(
) =
(
1)c2v (8)
where1021kg/m3is the interstellar gas density (which can be a few orders
of magnitude higher in gas clouds). The rst
factor is due to the time dilation
and the second the proton kinetic energy.
This is 4:6104W/m2for
= 2 (0.86c) and 1 :5105W/m2for
= 3 (0.94c).
The exact damage depends in a nontrivial manner on the interaction with the
probe structure since the proton generates a particle shower that is slowed down
by the material and deposits its energy in the interior. Above
= 1:3 pion
production from proton-proton collisions begin to occur, and above
= 9:1
anti-protons are produced [6]. For modest
a thick shield or de
ection system
could in principle handle this6, but clearly there exists an upper limit where the
induced thermal vibrations7, secondary particles or broken bonds produce an
error cascade. This constraint is generally dominated by the dust constraint for
small probes.
The more exotic concerns due to interactions with the cosmic microwave
background discussed in [6] occur at signicantly higher velocities than the
likely dust and radiation limits. Above
= 108CMB photons induce pair
production, and the viscous drag force due to scattering the photons becomes
signicant.
4 Racing for the universe
What if there are two groups starting at t= 0;r= 0, attempting to claim as
much as possible of the surrounding universe?
4.1 Trivial case
In the simplest case of two groups with dierent resources R1;R2the trivial
answer is that the group with the most resources wins. They simply send
probes of the same mass to the same destinations as the other, but use the
extra resources to give them extra
.
6Personal communication with Eric Drexler.
7The equilibrium temperature due to direct heating of a probe with front cross section S
and total radiating area A, assuming full absorption of proton energy, is T= [SI(
)=A]1=4
and scales asp
. Geometry matters signicantly: a cubical 1 meter probe reaches the melting
temperature of iron 1800 K at
= 12:2, while a 1 meter cylinder probe with 1 cm2cross section
can go up to
= 96.
9
4.2 Strategy
A strategy shift would be to send fewer but faster probes: the less resource-
rich group could aim for a few desirable targets, giving up enough probes in
favor of speed to ensure rst arrival. Clearly, if the other group knew this
they could perform the same reallocation and still win (although at the price
of not getting all intended targets, at least at rst). This demonstrates that
information asymmetries and randomized strategies are going to be important.
The various speed limits discussed earlier imply that the fewer but faster
probe strategy has a limit speed: once probes are at the limit, neither side
has any advantage. At this point launching might just be a gamble that one's
probe arrives faster or that the competitor suers a failure in transit. This may
actually motivate groups to launch probes with fairly high probability of mission
failure. The high-resource group can force the low-resource group to use risky
probes/targets, while themselves using excess resources to settle guaranteed
targets. This issue also makes launching rst more tempting (but see below).
As noted in [2, 3] 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.
4.3 Acceptable delays in competition
If the travel occurs over distance Lfrom the origin at speed v1, a ship launched
with delaydand speedv2>v1will arrive faster if L=v 1>d+L=v 2. It is hence
rational to delay in order to get faster speed if
L1
v11
v2
>d: (9)
OverL109ly hence dierences in von the order of 109are worth waiting
one year for, if traveling at c-like speeds. This remains approximately true for
much smaller v, with longer delays acceptable.
Hence it can be rational for a group belonging to a technologically immature
species to delay launching if they expect their technology to advance signicantly
in the future [10]. Conversely, a competing group with more resources would
be irrational in launching early unless they calculate that the speed gains from
further technology will be smaller than their resource advantage. This might
imply that rational competing groups will wait to launch until their species
reaches technological maturity, in which case the trivial case above applies. Since
that disfavors the weaker group, stealthy early launching may be desirable.
4.4 Near-far strategy
What if group 1 launches far, while group 2 launches for nearby resources, takes
control over them and then launches far? The amount of new resources that
can be reached in time tisR2(t) = (4=3)Rv3
2t3(hereRdenotes resource
density). Assuming the launch of an equal number Nof far probes as group
1, they get
(t) = 1 +R2(t)=Nm , allowing a faster speed v0
2(t). This ignores
the probe mass cost, assuming it to be negligible compared to the large energy
gains. We will also assume the size of v2tis small compared to dso we can treat
it as being essentially at distance dfrom the ultimate destinations.
10
Figure 4: Eect of the near-far strategy. Group 1 (red) has 80% of the initial resources
of a solar system and use it to reach a destination 200 ly away in 240 years by going
at 0.83c. Group 2 can only move at 0.55c and would arrive after 361 years, but if
it gathers resources (0.004 solar systems worth per cubic lightyear) for a time before
pooling the resources for a dash towards the destination it can reach it in 214 years.
Note that gathering resources for less than 7 years will not be enough to beat group
1, while spending more than 40 years gives it too much of a head start.
In order to overtake group 1, group 2 needs v0
2so thatt+d=v0
2tdelay:
Among competing technologically mature groups starting from the same
location and time the group with the most resources wins, unless it is sur-
prised. It is rational to do an initial resource-harvesting near settlement step
to gather energy for faster travel to suciently remote destinations. Intermit-
tent resource-harvesting during very long trips is rational for relativistic but not
ultra-relativistic travel.
13
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= 1010(stars in the galaxy), 1015(in a supercluster), or 1022
(in reachable universe). For
= 3:7 probes of mass mthe total mass-energy
resources needed to launch Rprobes to each target is
NRm ; a solar system
worth of resources is enough for Rm5:41019probe-kilograms towards the
galaxy, 5:41014towards a supercluster, and 5 :4106towards the reachable
universe.
The amount of resources required for the maximum feasible speed may be
signicant, but the scaling of the harvesting volume is r/
1=3
lim. This means
that the total amount of resources used for powering the speed of the expansion
is very small compared to the size of space that can be reached. Using very
large numbers of probes can in principle eat up any amount of resources, but
this is only rational (1) if one is trying to breach the redundancy constraint, (2)
are in a competitive situation at the speed limit and trying to win by gambling
(probes as "lottery tickets" with a win probability R1=R2).
While this report does not support the view that large-scale space settlement
will lead to the waste of signicant resources, it does not follow that settlement is
so innocuous that it can be allowed to happen in any form. When occurring over
large distances the ability to enforce coordination disappears, and this means
that if no or bad ground rules were established before causal contact was lost
there can be major waste. As argued in [9], most value of the resources in the
universe resides in using them in the far future: using up resources too early
can lose a factor of 1030. Early con
icts or incentives to use resources before
the late eras can lead to massive losses.
Acknowledgments
Nick Bostrom "nerd-sniped"11me with the original question and some of the
game-theory issues. I want to thank Stuart Armstrong for doing extra calcula-
tions on long-range colonization for this report. Stuart, Toby Ord, Eric Drexler
and Milan Circovi c have provided invaluable comments.
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