If Loud Aliens Explain Human Earliness, Quiet Aliens Are Also Rare
Robin Hanson, Daniel Martin, Calvin McCarter, Jonathan Paulson
AA Simple Model of Grabby Aliens
Robin Hanson ∗ , Daniel Martin † , Calvin McCarter ‡ , Jonathan Paulson § February 3, 2021
Abstract
According to a hard-steps model of advanced life timing, humans seem puzzlinglyearly. We offer an explanation: an early deadline is set by “grabby” civilizations(GC), who expand rapidly, never die alone, change the appearance of the volumesthey control, and who are not born within other GC volumes. If we might soonbecome grabby, then today is near a sample origin date of such a GC. A selectioneffect explains why we don’t see them even though they probably control over athird of the universe now. Each parameter in our three parameter model can beestimated to within roughly a factor of four, allowing principled predictions of GCorigins, spacing, appearance, and durations till we see or meet them.
To a first approximation, there are two types of aliens: quiet and loud. Loud aliens lastlong, expand outward, and make visible changes. Quiet aliens fail on at least one of thesecriteria. Quiet aliens are harder to see, and so could be quite common, or not, forcingquite uncertain estimates of their density via methods like the Drake equation (1,2). Loudaliens, in contrast, could be quite noticeable if they exist in any substantial number perHubble volume.To study loud aliens, this paper focuses on a very simple model, one with only threefree parameters, each estimable to within roughly a factor of four. This allows for muchmore precise estimates than does the Drake equation. Yes, more complex models thanours are possible, but simple models have many advantages. ∗ Department of Economics, George Mason University, Fairfax, Va, USA, and Future of HumanityInstitute, Oxford University, Oxford, U.K. [email protected] † Centre for Particle Theory and Department of Mathematical Sciences, Durham University, Durham,DH1 3LE, U.K. [email protected] ‡ Machine Learning Department, Carnegie Mellon University, Pittsburgh, PA, U.S.A, and LightmatterInc., Boston, MA, U.S.A. [email protected] § Jump Trading LLC, U.S.A. [email protected] a r X i v : . [ q - b i o . O T ] F e b igure 1. These two diagrams each show a sample stochastic outcome from a grabby aliensmodel, in one (1D) and two (2D) spatial dimensions. Figure 1A shows space on the x-axis andtime moving up the y-axis. Figure 1B shows two spatial dimensions, with time moving downwardinto the box. A three-dimensional visualisation can be found at https://grabbyaliens.com .In both cases, randomly-colored grabby civilizations (GCs) are each born at a spacetime event,expand in forward-time cones, and then stop upon meeting another GC. The scenarios shownare for a “power” n of 6. If the expansion is at 50% lightspeed, then the 1D diagram shows 49GC across a spatial width of 18.8 Gly, while the 2D diagram shows 193 GC across a width of41.7 Gly, both widths applying at median GC birthdate, if that is our current date of 13.8 Gyr.We show coordinates co-moving in space and conformal in time, as explained in Section 8.
Ours is a model of “grabby” aliens, who by definition a) never die alone, b) expandthe volumes they control at the same speed, c) clearly change the look of their volumes(relative to uncontrolled volumes), and d) are not born within GC-controlled volumes.(See Figure 1 for examples of the space-time pattern produced by this model.)2ur first model parameter is the rate at which grabby civilizations are born. Weassume that we humans have non-zero chance of giving birth to a grabby civilization,and that, if this were to happen, it would happen within roughly ten million years. Wealso assume that our chance is space-time-representative, in that we have no good reasonto expect that our spacetime location is unusual, relative to other GCs. Given theseassumptions, and the fact that we do not now seem to be within a “clearly changed”alien volume, our current spacetime event becomes near a sample from the distributionof grabby civilization origins. That allows us to estimate the grabby birth rate to withinroughly a factor of two (for its inter-quartile range), at least for “powers” (explainedbelow) over 3. (Our assumptions reject the “zoo” hypothesis (3).)Yes, it is possible, and perhaps even desirable, that our descendants will not becomegrabby. Even so, our current date remains a data point. Surprised? Imagine that youare standing on a strange planet wondering how strong is its gravity. Your intuition tellsyou, from the way things seem to bounce and move around you, that you could probablyjump about 1.3 meters here, compared to the usual 0.5 meters on Earth. Which suggeststhat you are on a planet with a gravity like Mars. And it suggests this even if you donot actually jump . A counterfactual number can be just as valid a data point as a realnumber.Our second model parameter is the (assumed universal) speed at which grabby civi-lizations expand. Our model predicts that at typical grabby origin dates, a third to a halfof the universe is within grabby-controlled volumes. So if the grabby expansion speedwere low, then many such volumes should appear very noticeable in our sky. However,if their expansion speed were within ∼
25% of lightspeed, a selection effect implies thatwe would be more likely to not see than to see any such volumes (see Section 11). If wecould have seen them, then they would be here now instead of us. As we do not now seesuch volumes, we conclude that grabby aliens, if they exist, expand fast.Our third model parameter is the effective number of “hard steps” in the “great filter”process by which simple dead matter evolves to become a grabby civilization (4). It iswell-known that the chance of this entire process completing within a time duration goesas that duration raised to the power of the number of hard (i.e., take-very-long) steps (ortheir multi-step equivalents) in that evolutionary process. Using data on Earth historydurations, a literature estimates an Earth-duration-based power to lie roughly in the range3-9 (5, 6, 7, 8, 9, 10, 11, 12, 13).Such hard-steps power-law models are usually applied to planets. However, givensufficient variety in oasis origin times and durations, or a sufficient likelihood of pansper-mia (14), we will argue in Section 3 that such a power law can also apply to the chancesof advanced life arising within a larger volume like a galaxy. (At least over a modest timerange, and while such chances remain low.) This volume-based power is our third keyparameter.For each combination of our three model parameters, we can fully describe the stochas-tic spacetime patterns of GC activity across the universe, allowing us to estimate, forexample, where they are and when we would meet or see them. We will show in detail3ow these distributions change with our model parameters.Our grabby aliens model can explain a striking but neglected empirical puzzle: whydo we humans appear so early in the universe? Yes, a “galactic habitable zone” literatureoften finds our date to be not greatly atypical of habitable durations undisturbed bynearby sterilizing explosions, for both short and long durations (15, 16, 17, 18, 19, 20, 21,22, 23, 24).However, we will show that these calculations neglect the hard-steps power-law. Whenthat is included, and assuming powers above two and the habitability of stars less massivethan our sun, then humanity looks early. We will show this via another simple model, thisone of advanced life origin dates. This origin date model applies the hard-steps power lawto planets, and allows stars to form at different dates and to last for different durations.This model produces a time distribution over when advanced life should appear (if it everappears), and says that less than 10% of this distribution appears before today’s date,unless we make rather strong assumptions both about the hard-steps power and aboutthe habitability of stars that last longer than our sun. (See Figure 2.)Our grabby aliens model resolves this puzzle by denying a key assumption of this origindate model: that the birth of some advanced life has no effect on the chances that othersare born at later dates (25, 26). Our grabby alien model instead embodies a selectioneffect: if grabby aliens will soon grab all the universe volume, that sets a deadline bywhich others must be born, if they are not to be born within an alien volume. So we canexplain why we are early via the twin assumptions that (a) we could see but do not seealien controlled volumes, and (b) some of our descendants may soon become grabby.This paper will now review the robustness of the hard steps model, use a simple origindate model to show how the hard-steps process makes humanity’s birth date look early,describe the basic logic of our new model and how to simulate it, show how to changecoordinates to account for an expanding universe, and then describe our model’s specificpredictions for grabby alien civilization times, distances, angles, speeds, and more.
In 1983, Brandon Carter introduced a simple statistical model of how our civilizationmight rise from simple dead matter, via intermediate steps of life, complex life, etc., amodel that he and others have further developed (5,6,7,8,9,10,11,12,13). Carter posited asequence of required steps i , each of which has a rate 1 /a i per unit time of being achieved,given achievement of the previous step. The average of the duration t i to achieve step i is a i .Assume that this process starts at t = 0 when a planet first becomes habitable, andthat we are interested in the unlikely scenario where all of these steps are completed bytime t = T . (That is, assume (cid:80) i t i < T while (cid:80) i a i (cid:29) T .) Assume also for conveniencethat steps divide into two classes: easy steps with a i (cid:28) T , and hard steps with a i (cid:29) T .Conditional on this whole process completing within duration T , each easy step stillon average takes about a i , but each hard step (and also the time T − E − (cid:80) i t i left at4he end) on average takes about ( T − E ) / ( n + 1), regardless of its difficulty a i . (Where E = (cid:80) i a i for the easy steps.) And the chance of this unlikely completion is proportionalto T n , where n is the number of hard steps.This basic model can be generalized in many ways. For example, in addition to these“try-try” steps with a constant per-time chance of success, we can add constant timedelays (which in effect cut available time T ) or add “try-once” steps, which succeed orfail immediately but allow no recovery from failure. These additions still preserve the T n functional form.We can also add steps where the chance of completing step i within time t i goes as t n i , for a step specific n i . If all steps have this t n i form (try-once steps are n i = 0, whilehard try-try steps are nearly n i = 1), then the n in T n becomes n = (cid:80) i n i over all suchsteps i . (And this holds exactly; it is not an approximation.)For example, a “try-menu-combo” step might require the creation of a species with aparticular body design, such as the right sort of eye, hand, leg, stomach, etc. If the sizeof the available menu for each part (e.g., eye) increased randomly but linearly with time,and if species were created by randomly picking from the currently available menu foreach part, then the chance of completing this try-menu-combo step within time t i goes as t n i , where n i is the number of different body parts that all need to be of the right sort.We can also allow different planets (or parts of planets) to have different constantsmultiplying their T n , for example due to different volumes of biological activity, or dueto different metabolisms per unit volume. Such models can allow for any sorts of “oases”wherein life might appear and evolve, not just planets. And such models can accommodatea wide range of degrees of isolation versus mixing between the different parts of thesevolumes.In a slightly less simple model, if planet lifetimes are proportional to star lifetimes,and if only stars with lifetimes L < ¯ L are suitable for advanced life, then the probabilitydensity function α ( t ) of advanced life to appear at date t is given by α ( t ) = (cid:90) t x n − (cid:37) ( t − x ) (cid:104) H ( ¯ L ) − H ( x ) (cid:105) dx, (1)where (cid:37) ( t ) is the star formation rate (SFR) and H [ L ] is a cumulative distribution function(c.d.f.) over planet lifetimes. The c.d.f. of stellar lifetimes L goes as roughly H [ L ] = L / over an important range (up to a max star lifetime ¯ L ∼ × Gyr), because stellar mass m has c.d.f. that goes as m − . , while stellar lifetime goes as m − (down to ∼ . M (cid:12) ).Note that as Equation (1) has x < t , and as H [ ¯ L ] − H [ x ] is nearly constant for x (cid:29) ¯ L ,then t is nearly a power law when t (cid:28) ¯ L and (cid:37) ( t ) is a power law.Furthermore, for the purposes of this paper we actually only need the SFR to besufficiently well approximated by a power law over the actual range of times in whichgrabby civilizations are born. For example, we will see in Section 12 that for powers 3 orhigher, over 90% of GCs are born within the date range 0 .
75 to 1 .
15, if the median GCbirth date is set to 1. If the power law approximation fails at times outside this limitedrange, that just moves this range forward or back in time a bit, but preserves the validityof the model’s stochastic pattern of spacetime events within this range.5e thus see that instead of being a peculiar feature of a particular model of the originof advanced life, a t n power law time dependence may be a robust feature of the greatfilter not just for individual planets, but also for (limited early-enough periods of) muchlarger co-moving volumes that contain changing mixes of planets and other possible oases.In our grabby aliens model, we will thus assume that the chance for an advancedcivilization to arise in each “small” (perhaps galaxy size) volume by date t after the bigbang goes as t n . We assume that this t n form applies not just to the class of all advancedlife and civilizations, but also in particular to the subclass of “grabby” civilizations. A literature tries to estimate the number of (equivalent) hard steps passed so far in Earth’shistory from key durations. Here is an illustrative calculation.The two most diagnostic Earth durations seem to be the one from when Earth wasfirst habitable to when life first appeared ( ∼ . ∼ . e hard steps have happened on Earth so far (with no delays or easy steps), the expectedvalue for each of these durations should be ∼ . / ( e + 1). Solving for e using theobserved durations of 0 . . e values of 3 . .
5, suggesting amiddle estimate of near 6.The relevant power n that applies to our grabby aliens model differs from this e .It becomes smaller if evolution on Earth saw big delay steps, such as from many easysteps, in effect reducing Earth’s ∼ . e if there were hard steps before Earth (dueto panspermia), or if there will be future hard steps between us today and a futuregrabby stage. The enormous complexity and sophistication of even the simplest andearliest biology that we know also seems to suggest higher powers n , most likely viapanspermia (27, 28).In the following, we will take power n = 6 as our conservative middle estimate, andconsider n in 3 to 12 to be our plausible range, but at times also consider n as low as 1and as high as 50. Equation (1) above is a simple formula which can estimate the fraction of civilizations thatarrive before some date, if they ever arrive. While for early times the chance α ( t ) will goas a power law if the SFR function (cid:37) ( t ) is also a power law, a better SFR approximationcan draw from the large empirical SFR literature. While this literature embraces a widerange of functional forms, the most common seems to be (cid:37) ( t ) = t λ e − t/ϕ , a form that peaksat χ = λϕ . 6he most canonical parameter estimates in this literature seem to be power λ = 1 anddecay time ϕ = 4Gyr (29, 30, 31). However, as the SFR literature also finds a wide rangeof other decay times, we will consider three decay times: ϕ in 2 , , (cid:37) ( t ) = t λ e − t/ϕ to approximate the manyestimates produced in the “galactic habitable zone” (GHZ) literature. Such estimates areof the density at different times and places in our galaxy (and sometimes in other galaxies)of planets conducive to life or civilization. Such authors consider habitability not only dueto suitable planets and stars, but also due to sufficiently low rates of nearby sterilizingexplosions such as supernovae and gamma ray bursts (15, 16, 17, 18, 19, 20, 21, 22, 23, 24).While the GHZ literature has attended mostly to dates before today, it almost alwaysfinds peak dates χ much later than the canonical SFR peak of χ = 4Gyr, and often wellafter our current date of 13.8Gyr. We thus choose a habitable peak of χ = 12Gyr asroughly matching typical GHZ literature estimates, and combine this with three decaytimes: (cid:37) in 2 , , λ > λ in effect marks most earlystars as entirely unsuitable for life due to overly frequent nearby explosions. A moreprecise calculation would consider the specific duration lengths near each star betweensuch explosions.)Opinions in the literature vary regarding the relative habitability of lower mass stars.While the same planet should have the same metabolism at any star if its orbital radiusgives it a habitable temperature, smaller stars may or may not have smaller planets, orfewer planets within a habitable orbit range. Increased solar flares and tidal-locking mayalso be larger problems for planets around smaller stars.We use two model parameters to represent this variation in low mass star habitability.First, we allow variation in max planet lifetime ¯ L , which we let vary from half of our sun’slifetime of L (cid:12) ≈ ∼ × Gyr. Second, we multiply the usual star mass c.d.f. m − . by a factor m κ , to say that larger stars are this much higher of a habitability factor. This changesthe planet lifetime c.d.f. to go as roughly H [ L ] = L (3 − κ ) / . (When this is unbounded, weapply a low lifetime lower bound, which turns out not to matter.) We consider 0 and 3as values for the mass-favoring power (MFP) κ .In Figure 2, we show the percentile rank of today’s 13.8 Gyr date within the predicteddistribution of advanced life arrival dates, according to Equation 1. And we show how thisrank varies with four parameters: GHZ decay ϕ = 4, mass-favoring power κ , hard-stepspower n , and max planet lifetime ¯ L (Wolfram Research, Inc. ‘20). The appendix showshow results change as swe vary GHZ peak χ in 4 , ,
12 Gyr.Figure 2 suggests that GHZ decay ϕ and MFP κ make only modest differences relativeto the effect of the power law, which is often overwhelming. For κ = 0 our percentile rankis below 1% for max lifetimes ¯ L beyond a trillion years, no matter what the values ofother parameters. This also holds for κ = 3, when n > . n = 6, then even with a very restrictive lifetime ¯ L = 10Gyr,7 - - - - % % % % k k k k k k Max
Planet
Lifetime ( Gyr ) P o w e r n - - - - % % % % k k k k k k Max
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Lifetime ( Gyr ) P o w e r n - - - - % % % % k k k k k k Max
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Lifetime ( Gyr ) P o w e r n - - - - % % % % k k k k k k Max
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Lifetime ( Gyr ) P o w e r n - - - - % % % % k k k k k k Max
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Lifetime ( Gyr ) P o w e r n - - - - % % % % k k k k k k Max
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Lifetime ( Gyr ) P o w e r n MFP κ = MFP κ = Decay φ = Gyr
Decay φ = Gyr
Decay φ = Gyr
Figure 2.
Percentile rank of today’s 13.8Gyr date within the distribution of advanced lifearrival dates, as given by equation (1). Six diagrams show different combinations of MFP κ andGHZ decay ϕ , while each diagram varies power n and max habitable planet lifetime ¯ L . all ranks are < . n = 3, at the low end of our plausible range, all ranksare 10% for lifetime ¯ L = 15Gyr. And at power n = 2, all ranks are < .
4% for lifetime¯ L = 20 when κ = 0, and < .
4% when κ = 3.Thus we are roughly at least 10% “surprisingly early” for ( n, ¯ L ) combinations of { (6 , , (3 , , (2 , } . And modest increases in power n or max lifetime ¯ L beyondthese values quickly make our rank look very surprisingly early. So unless one is willingto assume rather low powers n , and also quite restrictive max planet lifetimes ¯ L , thereseems to be a real puzzle in need of explanation: why have we humans appeared so early?(As the appendix shows, assuming earlier GHZ peaks χ allows somewhat higher n or morerelaxed ¯ L , but not by a lot.) Our grabby aliens model offers such an explanation. While our grabby alien model should ultimately stand or fall on how well it directlyaccounts for observations, readers may want to hear plausibility arguments regarding itskey assumptions. 8e have already discussed (in Section 2) reasons to expect a power law time depen-dence in the chances to originate advanced life early in any one place, due to evolutionneeding to pass through many difficult steps. We’ve also discussed how this kind of de-pendence can apply to large volumes like galaxies, as well to smaller ones like planets.But why might there exist civilizations who expand steadily and indefinitely, changinghow their volumes look in the process?In Earth history, competing species, cultures, and organizations have shown consistenttendencies, when possible, to expand into new territories and niches previously unoccupiedby such units. When such new territories contain resources that can aid reproduction,then behaviors that encourage and enable such colonization have often been selectedfor over repeated episodes of expansion (32). (Note that individual motives are mostlyirrelevant when considering such selection effects.)In addition, expansions that harness resources tend to cause substantial changes tolocal processes, which induce changed appearances, at least to those who can sufficientlyobserve key resources and processes. While these two tendencies are hardly iron laws ofnature, they seem common enough to suggest that we consider stochastic models thatembody them.Furthermore, when uncoordinated local stochastic processes are aggregated to largeenough scales, they often result in relatively steady and consistent trends, trends whoseaverage rates are set by more fundamental constraints. Examples include the spreadof species and peoples into territories, diseases into populations, and innovations intocommunities of practice. Without wide coordination, local processes that induce local“death” seem unlikely to induce death correlated across very wide scales.Yes, expanding into the universe seems to us today a very difficult technical andsocial challenge, far beyond current abilities. Even so, many foresee a non-trivial chancethat some of our distant descendants may be up to the challenge. Furthermore, the largedistances and times involved suggest that large scale coordination will be difficult, makingit more plausible that uncoordinated local processes may aggregate into consistent overalltrends. Also, the spatial uniformity of the universe on large scales, and competitivepressures to expand faster, suggest that such trends could result in a steady and universalexpansion speed.Yes, perhaps there is only a tiny chance that any one civilization will fall into sucha scenario wherein internal selection successfully promotes sustained rapid overall expan-sion. Even so, the few exceptions could have a vastly disproportionate impact on theuniverse. If such expansions are at all possible, we should consider their consequences.
Our basic model sits in a cosmology that is static relative to its coordinates. That is,galaxies sit at constant spatial position vectors v in a D -dimensional space, time movesforward after t = 0, and constant speed movement in the x coordinate direction satisfies s = ∆ x/ ∆ t for speed s . (We are mainly interested in D = 3, but will sometimes consider9 in 1 , v, t ). Bydefinition, GC volumes look clearly different, and expand in every direction at a constantlocal speed s until meeting volumes controlled by other GC. Once a volume is controlledby any GC, it is forever controlled by some GC.Note that we allow the possibility of non-grabby civilizations (NGC), perhaps even agreat many of them, whom GCs may or may not leave in peace. But our assumptions dorequire that NGCs only rarely block GCs from the activities that define them: expandingand changing volume appearances. And the larger a ratio we postulate between NGCsand GCs, the smaller a transition rate we must postulate for NGC to give birth to GC.For example, if in the history of the universe there have so far been a million times moreNGC than GC, then, on the timescales seen so far, on average no more than one in amillion NGC can give birth to a GC descendant.Each “small” (perhaps larger than galaxy-sized) volume has the same uniform per-volume chance of a GC being born there, a chance that is independent of the chances inother volumes. Over time the chance of birth by t at some position v rises as ( t/k ) n , apower n of time since t = 0 divided by a timescale constant. Except that this chance goesto zero as soon as the expanding volume of another GC includes this position v . As wediscussed above, this power law dependence may be a robust feature of many models ofthe origin of life and civilization.And that is our whole model. It has three free parameters: the speed s of expansionand the constant k and power n of the birthing power law. It turns out that we canestimate each of these parameters reasonably well.Specific examples of the spacetime distribution resulting from this process are shownin Figure 1. Notice how smaller GC tend to be found at later origin times in the “crevices”between larger earlier GC. (For simplicity, these examples show each GC retaining controlof its initial volume after meetings. Our analysis only depends on this assumption whenwe calculate distributions over final GC volumes.) A simple deterministic model gives a rough approximation to this stochastic model in onedimension.Assume a regular array of “constraining” GC origins that all have the same origin time t = x , and which are equally spaced so that neighboring expansion cones all intersect at t = 1. (See Figure 3.) If these cones set the deadline for the origins of other “arriving” GC,we can then find a distribution over arriving GC origin times that results from integrating t n − over the regions allowed by the constraining GC.The key modeling assumption of this simplified heuristic model is to equate the con-straining GC origin time x with the percentile rank r of the resulting distribution of10 igure 3. Illustration of the heuristic math model of Section 7 .arriving GC origins. This assumption implies1 − rr (cid:90) x t n − dt = (cid:90) x t n − − t − x dt (2)and is independent of speed s or constant k . This math model captures two key symme-tries of our stochastic model, which are described in Section 9.For each power n , there is some rank r where this heuristic model’s prediction (1 − x ) /x equals a 1D simulation result for the ratio of median time till meet aliens to the medianorigin time. This rank is ∼ .
88 at n = 1, falls to a minimum of ∼ .
61 at n = 4, andthen rises up to ∼ .
88 again at n = 24. This simple heuristic math model thus roughlycaptures some key features of our full stochastic model. Our model seems to have a big problem: its cosmology has things staying put, yet ouruniverse is expanding. Our rather-standard solution: a change of coordinates.Usually, using ordinary local “ruler” spatial distances dv and “clock” times dt , themetric distance d between events is given by d = dt − dv . We instead use “model”coordinates, which are co-moving spatial positions du = dv/a ( t ) and conformal times dτ = dt/a ( t ). Here a ( t ) is a “scale factor” saying how much the universe has expandedat time t relative to time t = 1. Metric distance then becomes d = a ( t ) · ( dτ − du ).In terms of our model spatial coordinates u , galaxies tend to stay near the samespatial positions. However, in an expanding universe a freely-falling object that starts atan initial speed ∆ u/ ∆ τ , and has no forces acting on it, does not maintain that ∆ u/ ∆ τ coordinate speed as the universe expands. It instead slows down (33). Does this show11hat a GC which might in a static universe expand at a constant clock speed ∆ u/ ∆ τ , notexpand at constant model speed ∆ u/ ∆ τ in an expanding universe?No, because the frontier of an expanding civilization is less like an object thrown andmore like the speed of a plane; a given engine intensity will set a plane speed relative tothe air, not relative to the ground or its initial launch. Similarly, a civilization expandsby stopping at local resources, developing those resources for a time, and then using themto travel another spatial distance (34). As this process is relative to local co-movingmaterials, it does maintain a constant model speed ∆ u/ ∆ τ .We thus run our simulations in a static model space u , and in model time τ . Toconvert results from model time τ to clock time t , it suffices to know the scale factorfunction a ( t ). This scale factor a ( t ) went as t / during the “radiation-dominated” erafrom the first second until about 50,000 years after the big bang, after which it went as t / during the “matter-dominated” era. In the last few billion years, it has been slowlyapproaching e Ω t as dark energy comes to dominate.Assume that the scale factor is a power law a ( t ) = t m , and that we today are atpercentile rank r in the distribution over GC origin dates. If so, we can convert frommodel time τ to clock time t via t = t ∗ ( τ /τ ) / (1 − m ) and r = F ( τ ), where t = 13 . F ( τ ) is the c.d.f. over GCmodel origin times. We can also convert between clock time power n and model timepower n via n (cid:48) = n/ (1 − m ).Now, a least-squared-error fit of a power law to the actual a ( t ) within 0-20 Gyr afterthe Big Bang gives a best fit of m ≈ .
90. However, as this m value tends to give moreextreme results, we conservatively use m = 2 / m = 0 . r (i.e., that we could beany percentile rank in the GC origin time distribution), we can convert distributions overmodel times τ (e.g., an F ( τ ) over GC model origin times) into distributions over clocktimes t . This in effect uses our current date of 13.8Gyr to estimate a distribution overthe model timescale constant k . If we use instead of F ( τ ), the distribution F ( τ ), whichconsiders only those GCs who do not see any aliens at their origin date, we can also applythe information that we humans do not now see aliens.In the rest of this paper we will show spacetime diagrams in terms of model coordinates( r, τ ), but when possible we will talk and show statistics and distributions in terms of clocktimes, including clock time powers n . Our stochastic model scales in two ways. That is, two kinds of transformations preserve itsstochastic pattern of GC space-time origins. First, halving the speed s of expansion halvesthe average spatial distance between GC, but otherwise preserves their pattern. Second,changing timescale k changes the median GC origin time, but preserves the pattern once12imes and distances are rescaled by the same factor to give the same median origin timeas before.So simulations need only vary dimension D and power n , and repeatedly sample, to seethe full range of stochastic GC origin patterns that can be produced by this model. Thissimplifies our task of simulating this model. We can fix expansion speed at s = 1, focuson a unit time range [0 ,
1] and unit volume [0 , D , and use a “wrap around” (toroidal)metric which identifies x = 0 with x = 1, etc. in all spatial dimensions. (The appendixchecks for robustness to these assumptions.) We generate N candidate GC origins ( u, τ )as (uniformly) random positions v within this unit volume, paired with random times τ drawn from a c.d.f that goes as τ n on [0 , s = 1, spacetime event A “precludes” event B if A ’s time is earlierand if the spatial distance between them is less than their time difference. Given a set of N candidate origin events, we filter out any members precluded by other members, andthe remaining set C of origins then defines a stochastic sample from our model. (It helpsto test earlier candidates first, each against the test-passing origins collected so far.)Except that we rescale all times and distances in C by the same factor to make themedian origin time be one. We can then transform such a sample into a sample with adiffering speed s by rescaling all distances. And we can transform it into samples withdiffering timescales k by rescaling the median origin time. (Such “samples” may describethe same basic stochastic pattern over larger or smaller spatial volumes, in essence holdingmore or less “copies” of the basic stochastic pattern.)We know that a spacetime event is controlled by some GC if it is precluded by anyGC origin. While a larger sample N of candidate GC origins tends to induce a largernon-precluded set C , eventually C stops increasing, giving a “full” sample. Lightspeed c can be varied relative to s to calculate who can see what in such a sample.
10 Simulation Statistics
Imagine one has a sample of simulation runs, each of which produces a set of C grabby civ-ilization at origins ( u, τ ) within the model box [0 , . Here are some interesting statisticsthat one can calculate within each run (and average over multiple runs).The following statistics depend only on the power n :A) For each GC origin we can pick a (uniformly) random position u in the [0 , D volume, combine that with this GC origin time τ , and see if that event ( u, τ ) isprecluded by any GC. Repeating this estimates the volume fraction of the universewhich is controlled by GCs at that GC origin time τ . Repeating for all GCs canshow how this volume fraction varies with GC origin rank.B) We can collect a simulated distribution F ( τ ) over model GC origin times τ .C) For each GC origin position, we can find the time τ at which a speed s = 1 travelerwould arrive at that position from each other GC origin event. The minimum of13hese is the arrival time when the first other GC expansion wave would, if allowed,arrive at this GC position.D) The average of that min arrival time and this GC origin time is the meet time , whenthe two GC expansion waves will collide. The first GC it meets is also the first oneto arrive.The following statistics depend on both power n and speed ratio s/c :E) Take the subset of GC who don’t see any other GC-controlled-volume at their origin.For each such GC, if we assume Earth today has its rank r , that gives a constant forconverting all model times into clock times. If we then assume a uniform distributionover Earth rank r (expressing the assumption that we have a representative chanceof birthing grabby descendants), we can convert any distribution over model timesinto a distribution over clock times. We can also take any distribution over pairsof model times into a distribution over clock durations between those times. Foreach GC origin, we so far have three model times: origin, meet, and arrival. Wecan convert the model origin times into clock origin dates. And we can obtain clockdurations between origin times and when they will meet aliens, or when aliens wouldarrive there.F) Model box [0 , corresponds to ordinary physical volume (13 . s/cτ ) Gly if today’sdate of 13 . τ . Also, there are today ∼ .
07 galaxies (each with mass > M (cid:12) ) per Mpc , which is 2 × per Gly (Conselice et al. 2019). Thus if galaxies are conserved, then for any time rank ofhumans in the model, the model box holds G = 2 × (13 . s/cτ ) galaxies. So ifsim run finds C civilizations, it has G/C galaxies per GC at sim end. At percentile r GC origin date, this is (
G/rC ) ∗ (volume fraction at r ). (There are ∼ > M (cid:12) .)G) If at some date, the model volume wherein a single GC is first to arrive is V < V ∗ G galaxies at that date, at least if we assume that GCs who meetsimply stop and retain control of their existing volumes. Iterating through the GC,a distribution over galaxies per GC can be found for different times.H) A visible GC-controlled-volume would (unless it had already collided with anotherGC) appear as a disk in the sky with angle θ given by tan( θ/
2) = x/ ( c ( b − x )) where x solves d = x + ( c ( b − x )) . We can thus find a distribution over the max angle that each GC can see. (If it sees none, its max angle is zero.)I) Consider two GC origins. The later origin τ can see the earlier origin if their spatialdistance d is less than cb , where b is their time difference. We can find a distributionover GCs of the number of other GC origins that each one can see at its origin. (Thisignores the possibility that opaque GC volumes might block the view of others.)14 igure 4. Illustration of selection effect. When expansion speeds s are near lightspeed c , alienorigin events in most of our backward lightcone would have created a GC that controls the eventfrom which we are viewing, preventing here-now from becoming a candidate GC origin. J) If a GC has not yet seen any other GC, it will first see the GC that it will first meet,and see it before their meeting. That when see time is τ = ( d + τ + cτ ) / (1 + c ),where τ is the viewer’s origin time, and τ is the viewed’s origin time.K) We can convert origin and view times from model times into clock times, take thedifference and get a distribution over clock time until see aliens. (That duration iszero if aliens can already be seen at GC origin.)Code to simulate the grabby alien model and to compute the above statistics can befound at https://github.com/jonathanpaulson/grabby_aliens .
11 Estimating Expansion Speed
Our grabby aliens model has three free parameters, and we have so far discussed empiricalestimates of two of them: the power n (in Section 3) and the power law constant k (in Section 8). The remaining parameter, expansion speed s , can also be estimatedempirically, via the datum that we humans today do not see alien volumes in our sky.We will see in Figure 10 that visible alien volumes are typically huge in the sky, muchlarger than the full moon. So there is only a miniscule chance of a visible volume beingtoo small to be seen by the naked eye, much less by our powerful telescopes. So if alienvolumes looked at all different, as we assume in our GC definition, and if their volumesintersected with our backwards light cone, we would clearly see them.We will also see in Figure 7 that, averaging over GC origin dates, over a third of thevolume of the universe is controlled by GCs. So from a random location at such dates,one is likely to see large alien-controlled volumes. However, if the GC expansion speedis a high enough fraction of the speed of light, a selection effect, illustrated in Figure 4,makes it unlikely for a random GC to see such an alien volume at its origin date.15 igure 5. Likelihood ratios, for ( n, s/c ) parameter pairs, regarding the observation that wesee no large alien-controlled volumes in our sky. To compute a posterior distribution over thesepairs, multiply this ratio by a prior for each pair, then renormalize.
Figure 5 shows how our evidence that we do not now see alien volumes updates ourbeliefs about the chances of various power and speed-ratio pairs ( n, s/c ). It shows theratio of the number of GC origin events that see no alien volumes in their sky, divided bythe number that do see alien volumes. Speed ratios of s/c < ∼ /
12 Simulation Results
This section contains many graphs showing how distributions over key statistics vary withpower n , and sometimes also with speed ratio s/c . Unless stated otherwise, all numbersand curves shown average over 5 simulation repetitions, each with L = 1 , s = 1 , c = 1 , and 10 sample GC origin events. Correctness of code has been checked by comparingindependent implementations.Figure 6 shows clock GC origin dates, and regarding those origin events Figure 7 showsvolume-controlled fractions, Figure 8 shows clock durations til a descendant meets aliens,Figure 9 shows how many alien volumes are seen, Figure 10 shows the largest alien volumeangle seen, and Figure 11 shows clock durations until a descendant sees aliens. Figure12 shows how the number of galaxies per GC at the simulation end (when GCs fill allvolumes) varies with power n, and Figure 13 shows many statistics collected together ina single diagram. 16 igure 6. C.d.f.s over GC origin clock-times, assuming a uniform distribution overhumanity’s rank in this distribution. Asthis is for speed ratio s/c = 1, it ignoresinfo that we see no aliens.
Figure 7.
Fraction of universe volume con-trolled by GCs, as a function of rank of GCorigin time.
Figure 8.
C.d.f.s over clock-time un-til some of our descendants directlymeet aliens, assuming we give birthsoon to a GC who has a uniform dis-tribution over rank among GC origintimes.
Figure 9.
C.d.f.s over how manyother GCs each one sees at its origin.At speed ratio s/c = 1, no GCs see anyothers at their origin. igure 10. C.d.f.s over largest an-gle in sky of GC seen from GC ori-gins. Red line is our Moon’s diameter(29 (cid:48) (cid:48)(cid:48) ). Figure 11.
C.d.f.s over clock time tillsome GC descendant sees aliens. Fors/c = 0.1, almost all GC have alreadyseen aliens.
Figure 12.
Average number of galaxies within volume controlled by each GC, after all volumeis controlled by GCs. We apparently live on a one-in-ten-million-plus-galaxies “rare Earth” (15). igure 13. C.d.f.s of many GC statistics for four powers n and four speed ratios s/c. Themappings used between model and clock times all update on the fact that we don’t now seealiens, but the c.d.f.s show all GCs. Each c.d.f. here is based on a single simulation run.
13 Conclusion
A literature has modeled the evolution of life on Earth as a sequence of “hard steps”,and compared specific predictions of this model to Earth’s historical record. This modelseems to fit, and supports inferences about the number of hard steps so far experiencedon Earth. As far as we can tell, this model seems widely accepted, without publishedcriticisms.We argue that this standard hard-steps model has not been taken sufficiently seriously.For example, the literature on the “galactic habitable zone”, which often estimates thetiming of the appearance of advanced life in a galaxy, has never included the key hard-steps effect that the chance of success by a deadline goes as a power law. In Section 4we have shown a simple model which includes this effect, wherein humanity today seemsto be early, unless one assumes both a rather low power and a very restrictive limit onhabitable planet lifetimes.Related literatures have also apparently not considered applying this hard-steps-basedpower law to larger volumes like galaxies, instead of to just planets. Some authors havesuggested that big sterilizing explosions may allow a scenario wherein galaxies long stayedempty, and have only recently been filling up with civilizations as such explosions havewaned. But these authors do not seem to have realized that, with a sufficiently high power,a volume-based hard steps power law produces a similar scenario (if on a larger scale)without invoking sterilizing explosions. Nor have they noticed that a scenario whereinadvanced civilizations grab most of the available volumes soon seems required to explainour early arrival.To formalize this argument, we have presented in this paper a simple model of whatwe call “grabby” civilizations (GC), who are born according to a volume-based power lawand who once born simply expand at a constant speed relative to local materials. Thisspeed and the two parameters of this power law are the only three parameters of ourmodel, each of which can be estimated to within roughly a factor of four.The hard-steps literature helps estimate the power, and our current date helps estimatethe power law timescale. Furthermore, the fact that we do not now see large alien-controlled volumes in our sky, even though they should control much of the universevolume now, gives us our last estimate: that aliens expand at over half of lightspeed. Givenestimates of all three parameters, we have in this paper shown many model predictionsregarding alien timing, spacing, appearance, and the durations until we see or meet them.Being especially simple, our model is unlikely to be an exact representation of reality.So future research might explore more realistic variations. For example, one might betteraccount for the recent exponential expansion of the universe. Instead of being uniformacross space, the GCs birth rate might be higher within galaxies, more within largergalaxies, and follow their typical spatial correlations. The expansion might take a duration20o bring its full effect to any one location, and the expansion speed might vary and dependon local geographies of resources and obstacles. Finally, GC subvolumes might sometimesstop expanding or die, either spontaneously or in response to local disturbances.
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Figure 14 varies box length L , holding constant s = 1, to see if border effects causeproblems due to the wrap-around metric. L = 1 seems sufficiently large. Figure 14.
Testing if L=1 is sufficiently large, when s=1.
Scenario: Scenario: s/c = 1 / , n = 6 s/c = 3 / , n = 12 Percentile 1% 25% 75% 1% 25% 75%
Origin 0.810 0.955 1.036 0.893 0.977 1.019MinArrival 0.932 1.022 1.076 0.962 1.011 1.038MinSee 0.855 0.937 0.985 0.936 0.984 1.014Origin (Gyr) 8.99 15.25 21.24 10.20 13.46 16.02MinTillMeet (Gyr) 0.019 0.488 2.226 0.006 0.188 0.882MinTillSee (Gyr) 0 0 0 0 0 0.425MaxAngle 0 0.132 0.908 0 0 0.313% Empty 0.010 0.320 0.830 0.010 0.290 0.810
Table 1.
Specific numbers for two scenarios, ( n, s/c ) = (6 , / , (12 , / For those frustrated by difficulties in reading numbers off our many graphs, Table 1gives specific numbers for two scenarios.As discussed in Section 8, most of our simulations have assumed a power law cosmo-logical scale factor of a ( t ) = t m , with m = 2 /
3. Figure 15 shows how some of our resultschange when m = 0 . n . Figure 16 shows how both of these power law approximations compare to theactual scale factor a ( t ). 24 igure 15. Comparing results for cosmological scale factor powers of m = 2 / m = 0 . t a ( t ) Universe scale factor
Our universe a ~ t / , same a [ ] a ~ t , same a [ ] Figure 16.
Cosmological scale factor over time, in reality and as assumed in this paper.
Figure 17 shows robustness of our earliness estimates to varying the GHZ peak, bycomparing the three values of peak χ in 4 , ,
12 Gyr, all given MFP κ = 0.26 igure 17. Percentile rank of today’s 13.8Gyr date within the distribution of advanced lifearrival dates, as given by equation (1), assuming MFP κ = 0. Nine diagrams show differentcombinations of GHZ peak χ and decay ϕ , while each diagram varies power n and max habitableplanet lifetime ¯ L ..