Contact Inequality -- First Contact Will Likely Be With An Older Civilization
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CONTACT INEQUALITY:FIRST CONTACT WILL LIKELY BE WITH AN OLDER CIVILIZATION
David Kipping , , Adam Frank and Caleb Scharf Department of Astronomy, Columbia University, 550 W 120th Street, New York, NY 10027 Center for Computational Astophysics, Flatiron Institute, 162 5th Av., New York, NY 10010 Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627
ABSTRACTFirst contact with another civilization, or simply another intelligence of some kind,will likely be quite different depending on whether that intelligence is more or lessadvanced than ourselves. If we assume that the lifetime distribution of intelligencesfollows an approximately exponential distribution, one might naively assume that thepile-up of short-lived entities dominates any detection or contact scenario. However,it is argued here that the probability of contact is proportional to the age of saidintelligence (or possibly stronger), which introduces a selection effect. We demon-strate that detected intelligences will have a mean age twice that of the underlying(detected + undetected) population, using the exponential model. We find that ourfirst contact will most likely be with an older intelligence, provided that the maximumallowed mean lifetime of the intelligence population, τ max , is ≥ e times larger thanour own. Older intelligences may be rare but they disproportionality contribute tofirst contacts, introducing what we call a “contact inequality”, analogous to wealthinequality. This reasoning formalizes intuitional arguments and highlights that firstcontact would likely be one-sided, with ramifications for how we approach SETI. Keywords:
SETI — technosignatures — interstellar communication INTRODUCTIONThe lifetime of a communicative civilization, L , plays a critical role in the DrakeEquation (Drake 1965; ´Cirkovi´c 2004; Maccone 2010; Glade et al. 2012). Little isknown about the possible range that this value can take (Burchell 2006). Our limitedtemporal existence provides a basis to estimate that L likely typically takes a valuegreater than or equal to modern civilization’s age thus far. Pessimists might suggest Corresponding author: David [email protected] a r X i v : . [ phy s i c s . pop - ph ] O c t Kipping et al. that the history of past human civilizations indicates that L will be brief, no greaterthan a few hundred years (Shermer 2002). Optimists could equally argue that we willsoon pass a critical juncture where comparable civilizations could ultimately enjoylong lifetimes, perhaps even billions of years (Grinspoon 2004).Although Drake cast L as the communicative lifetime, modern SETI has evolvedto include both deliberate and unintentional signatures of technology - “technosig-natures” (Wright 2017). We go further by relaxing the assumption that the tech-nosignature need originate from what we would recognize as a “civilization” - thesource is an intelligence of some kind (e.g. an artificial intelligence) which is capableof producing detectable technological signatures. In what follows, we consider L asrepresenting the lifetime over which techosignatures from this intelligence manifest.One basic question concerning this hypothetical intelligence is - what would firstcontact look like? This has been the playground of science fiction writers for gen-erations, and clearly this question has existential consequences for our way of life.Although we have no information about other intelligences yet, it is not unreasonableto assume that the nature of this contact will depend considerably upon the relativetechnological capabilities of this newfound entity. Humanity would surely treat com-munication with a comparably developed civilization in quite a different manner fromone with far greater technological capabilities. The longer lived an intelligence, L , thegreater the opportunity for technological development. Accordingly, the probabilitydistribution of the lifetime of detected intelligences will be of central importance toour decisions regarding contact.We note that it is of course possible for artifacts from an intelligence (or indeed acivilization) to persist far longer than the age of that entity. Sometimes referred toas artifact SETI, there is particular interest in applying to this Solar System objects(Freitas 1983; Lacki 2019; Wright 2018), for example. However, the detection of anartifact from a now extinct intelligence presents no opportunity for direct communi-cation or interaction (even if this is unclear from the initial detection).In this work, we therefore ask - what is the likely age of a detected and extantintelligence. Certainly speculation on this topic exists elsewhere. Carl Sagan famouslywrote that that civilizations were unlikely to be in technological lockstep with us(Sagan 1994) and thus would either be far less advanced or far more advanced. Sincethe less advanced ones would be undetected, this simple argument suggests contactwould be with an older intelligence. Similarly, Stephen Hawking warned that contactwould likely be with a more advanced and thus potentially dangerous entity. In whatfollows, we attempt to formalize the logic behind this problem and establish somestatistical results for L using a simple but plausible analytic model. A MODEL FOR TECHNOSIGNATURE LIFETIME2.1.
Exponential Distribution for L ontact Inequality likely age of a detectedintelligence. The first requirement to make progress is to assign a probability distri-bution for L . The simplest lifetime model we can posit is an exponential distribution(Lawless 2011). We do not claim that this is necessarily the true distribution, andencourage the reader to treat this as an approximate-yet-instructive model for makinganalytic progress. Further discussion about the suitability of this model is offered inthe Discussion.With such a model, amongst the ensemble of all intelligences that will ever arise,there would be a large number of short-lived intelligences (potentially such as our-selves) and a much smaller number of long-lived counterparts. On this basis, onemight naively posit that communication with another intelligence would surely bewith one of the more abundant short-lived intelligences. We proceed by first writ-ing down the probability density function of L given our exponential distributionassumption: Pr( L | τ ) = τ − e − L/τ , (1)where τ is the mean lifetime from the distribution. The exponential distributionassumes that the so-called hazard function is constant over time - much like a de-caying atomic nucleus. Certainly more sophisticated lifetime formulae have beensuggested for species survival. For example, a Weibull distribution is a commonlyused generalization of the exponential, that enables a time-dependent (specifically apower-law) hazard function (Lawless 2011), but comes at the expense of an extraunknown parameter.We note that a power-law distribution has also been adopted in ecology studies(Pigolotti et al. 2005), but we found it to exhibit several disadvantages over theexponential. First, it does not have semi-infinite support and thus requires truncationparameterized some additional bounding parameter, either a minimum lifetime or amaximum. Since no clear minimum exists, bounding at the maximum leads to afunction which is only monotonically decreasing for indices between 0 and 1. Thisleads to an overly restrictive distribution compared to the exponential and for thesereasons it is not used in what follows.An exponential distribution, with its constant hazard function of 1 /τ , could becriticized as being unrealistic since a longer lived species presumably has developedsuccessful traits that improve its odds of future survival (Shimada et al. 2003). Onthe other hand, as technology advances, so too does an intelligence’s capacity forself-destruction (Cooper 2013). If we consider the observed distribution for the lifespan of families obtained from Benton (1993) based on fossil evidence (see Figure 1), The hazard function is defined as the probability that an observed values lies between t and t + d t , given that it is larger than t for infinitesimal d t . Kipping et al. ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ × - × - × - [ Myr ] p r o b . d e n s i t y ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ × - × - × - [ Myr ] p r o b . d e n s i t y Exponential distribution Weibull distribution
Figure 1.
Probability distribution for the life span of families obtained from Benton (1993)using fossil evidence. On the left we fit an exponential distribution to the data shown, whereason the right we show the more complicated Weibull distribution. Although we don’t place anyemphasis on the specific parameters recovered, the data show that an exponential distribution isa quite reasonable description of the overall distribution, which compounded with it’s simplicitymakes it attractive as a choice for modeling technosignature lifetimes. the exponential distribution appears quite capable of describing the overall patternout to 400 million years. Of course, intelligences producing technosignatures cannotbe assumed to necessarily follow the same distribution as these fossils, although thisgive confidence that the model is at least plausible. Although not a representativenor unbiased sample, we note that approximate lifetimes of past human civilizationsis also well-described by an exponential distribution with τ = 336 years .In the absence of any other information, we invoke Ockham’s razor in that thesimplest viable model is the presently favored one. Accordingly, we will adopt theexponential distribution in what follows.2.2. Inferring the a-posteriori distribution of τ Although we have a functional form for the probability distribution of L , it isgoverned by a shape parameter, τ (mean lifetime), which one needs to also assign.This would typically be handled through statistical inference. For example, if we had N known examples of intelligences with lifetimes L = { L , L , ..., L N } T (analogous tothe data presented in Figure 1), then we could write that the likelihood of measuringthese values for a mean lifetime equal to τ would be Using the lifetimes reported at EnergySkeptic.com ontact Inequality L | τ ) = N (cid:89) i =1 τ − e − L i /τ , (2)where one can see that the above is a straight-forward extension of Equation (1).Conventionally, one would then apply Bayes’ theorem to constrain/measure τ usingPr( τ | L ) ∝ Pr( L | τ )Pr( τ ).Unfortunately, we do not have a sample of L i values, and thus our likelihood func-tion will certainly not be as constraining as this. Rather, we only know of N = 1intelligence - ourselves. However, the problem is even worse than this because wedo not even know L for this one datum. Human civilization has been producinga technosignature for an age A ⊕ years, and the lifetime of this intelligence must atleast exceed this value (i.e. L ⊕ ≥ A ⊕ ). We emphasize that it’s somewhat unclearwhat numerical value to assign to A ⊕ at this point. Although we’ve been transmit-ting radio signals for ∼ years, one might argue that an advanced civilization couldremotely detect our settlements (Kuhn & Berdyugina 2015) and polluted atmosphere(Schneider et al. 2010; Lin et al. 2014) as unintentional technosignatures, which couldincrease A ⊕ . Regardless, we will proceed symbolically for the moment.The likelihood of observing one civilization with L >A ⊕ , given that the mean life-time is τ , is given by Pr( L >A ⊕ | τ ) = (cid:90) ∞ A ⊕ Pr( L | τ )d L = e − A ⊕ /τ . (3)In order to derive an a-posteriori distribution for τ , conditioned upon the constraintthat L > A ⊕ , we first need to write down an a-priori distribution for τ . One isalways free to choose any prior one wishes, but a strongly informative prior, suchas a tight Gaussian, would naturally return a result which closely equals the prior.In other words, one hasn’t really learned anything and no inference really occurred.Ideally, we wish to select a prior which is as uninformative as possible (Jaynes 1968).This is not simply a flat prior, since such priors can place insufficient weight on smallvalues, especially when the parameter has high dynamic range. Instead, we can definean objective Jeffrey’s prior, which provides a means of expressing a scale-invariantdistribution via the Fisher information matrix, I (Jeffreys 1946):Pr( τ ) ∝ (cid:112) det I ( τ ) . (4)Evaluating the above, we obtain Pr( τ ) ∝ τ − / . Combining the likelihood and priortogether, we obtain Kipping et al. prior, Pr ( τ ) prior, Pr ( τ ) likelihood, Pr ( L > | τ ) likelihood, Pr ( L > | τ ) posterior, Pr ( τ | L > ) posterior, Pr ( τ | L > ) τ [ A ⊕ ] p r o b . d e n s i t y Figure 2.
Comparison of the prior, likelihood and posterior distribution for τ (the meanlifetime of intelligences producing technosignatures) using τ max = 10 Gyr, as an example. Themode of the posterior occurs at 2 as shown in the text. Pr( τ | L >A ⊕ ) ∝ Pr( L >A ⊕ | τ )Pr( τ ) , ∝ τ − / e − A ⊕ /τ . (5)To normalize the above, one must define an upper limit on τ , for which we use thesymbol τ max . At this point, it is also convenient to work in temporal units of A ⊕ inwhat follows, such that any timescales used will always be in that unit. Accordingly,the posterior is Pr( τ | L >
1) = τ − / e − /τ √ τ max e − /τ max − √ π erfc[1 / √ τ max ] . (6)We plot the posterior, with comparison to the prior and likelihood, in Figure 2.2.3. Properties of the posterior
There are several useful properties of the posterior above that we highlight. First,Equation (6) has a maximum at ˆ τ = 2 (the mode), irrespective of τ max , which can bedemonstrated through differentiation of the expression and setting to zero. If we set ontact Inequality τ = ˆ τ , then the mean lifetime of an intelligence would be twice of that of ourselves.But it’s important to remember that this is the entire lifetime of this intelligence, notit’s age at the time of of their detection, A . Assuming that the technosignature is nomore or less likely to be detected at any point during its manifested lifetime, then A ∼ U [0 , L ] (where U denotes a uniform distribution). Accordingly, if τ = ˆ τ , thenthe mean age at the time of detection would = 1 i.e. our current age. Of course,fixing τ = ˆ τ does not correctly account for the broad posterior distribution of τ , butthis exercise provides some intuition as to why the modal value of τ occurs at 2.Although the mode can be solved for independent of τ max , it is somewhat limited asan interpretable summary statistic. The expectation value of a distribution providesbetter intuition as to the “typical” value of the distribution. This can be seen bysimple consideration of the exponential distribution. Its mode is zero but the averagedraw will be around the mean of the distribution, not zero. We may calculate the a-posteriori expectation value for τ using E [ τ | L >
1] = (cid:90) τ max τ =0 τ Pr( τ | L>
1) d τ = µ, (7)where we define the symbol µ ≡ (cid:32) τ max − √ πτ − / e /τ max erfc[1 / √ τ max ] − (cid:33) , (8)where erfc[ x ] is the complementary error function. One may show that µ (cid:39) τ max / τ max (cid:29)
1. The dependency upon τ max can be understood by the fact that althoughthe mode of the distribution does not depend on τ max , pushing the upper limit everhigher naturally drags the tail out and thus pulls the expectation value over.2.4. Marginalized distribution for L Given that we have now obtained an a-posteriori distribution for τ , we need topropagate that into a posterior distribution for L . As discussed earlier, simply fixing τ to an a-posteriori summary statistic, like ˆ τ or E[ τ | L > τ into the resulting distribution. Thispropagation can be conducted through marginalizing out τ (i.e. integrating over τ ):Pr( L | L >
1) = (cid:90) τ max τ =0 Pr( L | τ )Pr( τ | L > τ, = √ π erfc[ √ L + 1 / √ τ max ]2 √ L + 1 (cid:0) √ τ max e − /τ max − √ π erfc[1 / √ τ max ] (cid:1) . (9) Kipping et al.
The above represents the probability distribution of the lifetime of technosignature-producing intelligences, given the singular constraint imposed by humanity’s exis-tence. It has a maximum at L →
0, which is a property shared by the originalexponential distribution used for Pr( L ). We also note that the expectation valuesatisfies E[ L | L >
1] = µ . OBSERVATIONALLY WEIGHTING THE MODEL3.1.
Lifetime weighting
The distribution Pr( L | L >
1) describes the probability distribution of the lifetimeof intelligences producing detectable technosignatures. This is the underlying truepopulation - but it does not represent the intelligences that we are most likely todetect. It’s worth pausing to clearly distinguish between detection and contact. If andwhen an intelligence is detected, that detection may either be in the form of a directedattempt at communication on their behalf, or it may simply be passive detection oftheir technology on our behalf. Regardless, humanity’s decision as to whether tosend a message back - to initiate contact - will be likely somewhat dependent onthe technological development and, by proxy, age ( A ) of said intelligence . If thetechnosignature itself provides little information regarding the age, we would be leftwith the a-priori distribution - which is the focus of this paper. Yet this distributionwill not simply equal P ( A | L ), since a critical selection effect sculpts our observationsthat we will account for here.The start time of these other intelligences is presumably arbitrary (except when onepushes into timescales of (cid:29) Gyr, over which time variability is expected for the ratesof star formation and high-energy astrophysical phenomena e.g. SNe, AGNs, GRBs).A start time 10 million years ago is just as a-priori likely as 100 years ago. Thus, alonger lived intelligence is more likely to be detected than one which is very short lived,since the requirement for contemporaneity (modulo the light cone) is clearly sensitiveto how long the technosignature persists. An equivalent statement is that at anysingle snapshot in time (representing our current epoch for example), the fraction ofworlds that go on to produce long-lived intelligences may be relatively rare, but theirpersistence through time means that one must account for their overrepresentationamongst the extant intelligences. This is simply a product of their longevity and isindependent of their activities or behavior. This situation is analogous to the ages oftrees in an old growth forest - if we assigned a unique identity to each tree that willever live, 1000+ year old trees are rare amongst the ensemble, perhaps representingjust 1%, yet a visit to the forest will show them to be seemingly more common dueto their longevity, for example comprising 10% of the extant trees. Age is subtlety distinct from lifetime and the difference between the two is expounded uponmore rigorously in the next subsection. ontact Inequality L . The validity of this assumption isdiscussed later in the Discussion section, as well as an explanation as to why distancedoes not affect the results presented hereafter.This simple weighting will substantially change the picture, meaning that the longtail of rare long-lived intelligences will have a considerable increase on their relativeprobability of detection. We write that the probability distribution of L , conditionedupon both a mean lifetime, τ , and the assumption of detection, D , isPr( L | τ, D ) ∝ L Pr( L | τ ) , (10)or after normalization Pr( L | τ, D ) = Lτ e − L/τ . (11)Since we have already learnt τ from before, we can use this acquired information toexpress a marginalized posterior for L conditioned upon both D and the fact L > L | L > , D ) = (cid:90) τ max τ =0 Pr( L | τ, D )Pr( τ | L > τ, = W (cid:32) Le − L/τ max L + 1) / (cid:33) . (12)where W = (cid:32) (cid:113) L +1 τ max + √ πe ( L +1) /τ max erfc[ √ L + 1 / √ τ max ] √ τ max − √ πe /τ max erfc[1 / √ τ max ] (cid:33) (13)We find that in the limit of τ max (cid:29)
1, this distribution peaks at 2. The expectationvalue is given by E [ L | L > , D ] = (cid:90) ∞ τ =0 L Pr( τ | L> , D ) d L, = 2 µ. (14)For comparison, without the conditional D , the a-posteriori expectation value was µ but including it doubles it. We plot the posterior Pr( L | L > , D ), and compare itto Pr( L | L > Kipping et al. Pr ( L | L > ) Pr ( L | L > ) Pr ( L | L > ) Pr ( L | L > ) [ A ⊕ ] p r o b . d e n s i t y Figure 3.
Comparison of the marginalized posterior probability for the age of intelligences, L , for the ensemble population (Pr( L | L > L | L > , D )).Here we adopt τ max = 10. The age distribution of detected intelligences
The final step is to account for the fact that detection would not occur with anintelligence at the end of its lifetime, L , but rather one drawn randomly from acrossits lifespan. In other words, an intelligence’s age (at the time of detection) does notequal its lifetime. If we assume that the age at the time of detection is uniformlydistributed from 0 to L , thenPr( A | L > , D ) = (cid:90) τ max τ =0 U [0 , L ]Pr( L | L > , D )d L, = √ π erfc[ √ A + 1 / √ τ max ]2 √ A + 1 (cid:0) √ τ max e − /τ max − √ π erfc[1 / √ τ max ] (cid:1) . (15)Equipped with our final form for the a-posteriori probability distribution of the ageof detected intelligences, we can deduce several basic properties. First, it is interestingto ask whether the civilization is likely to be older or younger than our own. Theprobability than the civilization is older is given by ontact Inequality A> | L > , D ) = (cid:90) ∞ A =1 Pr( A | L > , D )d A, = (cid:16) √ τ max e − /τ max − √ πe /τ max erfc[ √ / √ τ max ] (cid:17) / (cid:16) √ τ max − √ πe /τ max erfc[1 / √ τ max ] (cid:17) . (16)A useful summary statistic to interpret the above is the median - above which halfthe cases will lie. This may be solved for by setting the above to 0.5 and numericallysolve for τ max , which gives the result that if τ max > . ... (cid:39) e , then the age of adetected intelligence will most likely exceed that of our own i.e. Pr( A> | L > , D ) > .
5. In other words, if this condition is true than we are most likely to detect an olderintelligence than ourselves.It’s important to remember that τ max does not represent the maximum allowedlifetime of a civilization, L - rather it’s simply the maximum a-priori mean lifetime.Fundamentally, there is no obvious reason why τ max could not be many billions ofyears (Grinspoon 2004), and thus detection would almost always occur with an oldercivilization, however one defines A ⊕ .The expectation value for the intelligence’s age is given by simply µ , whereaswe found the expectation value for their lifetime to be 2 µ . Since E[ A | L >
1] =E[ L | L > /
2, then we can see that the effect of including this observational biasis that the mean age of detected, and thus contacted, intelligences is twice that of theoverall population - as expected.3.3.
Contact inequality
Using our results, it is instructive to compare the underlying age population,Pr( A | L > A | L > , D ). Re-call that A is age of the intelligence at the time of detection/contact, whereas L isthe total lifetime of said intelligence. The fact that older intelligences are assumed inthis work to be more likely to be detected, and thus contacted (by virtue of havingsimply more opportunities to do so), introduces an inequality. The rare long-livedintelligences make a disproportionate number of contacts.This “contact inequality” can be thought of as being analogous to wealth inequalityin economics. One way to quantify the degree of inequality comes from the Gini co-efficient (Gini 1909), which takes the value of 1 for a maximally unequal distribution,and 0 for a fully equal one. It may be calculated for a probability density functionPr( x ) using G = 12 µ (cid:90) ∞ (cid:90) ∞ Pr( x )Pr( y ) | x − y | d x d y. (17)2 Kipping et al.
Although we were not able find a closed-form solution to the above using Pr( x ) =Pr( A | L > , D ), one may numerically integrate the expression for a specific choiceof τ max .We argue here that a conservative choice of τ max is one which causes our currentage to be the median age of the entire population of technosignature producing in-telligences. This is a form of the mediocrity principle, since we posit humanity livesclose to the center of the age-ordered list of intelligences in the cosmos (Gott 1993;Simpson 2016). It requires us to solve τ max such that (cid:90) A =0 Pr( A | L >
1) = 12 . (18)We solved the above numerically and obtain τ max = 9 .
43. This also somewhat passesthe astronomer’s logic of going up by an order-of-magnitude as one’s upper limit on avariable. However, we suggest here that this limit is somewhat conservative though,since it makes million-/billion-year intelligences essentially non-existent, which is itselfa strong assumption.Nevertheless, using τ max = 9 .
43, we compute a Gini coefficient of 0.57. The valuedoesn’t grossly change by varying τ max . For example, setting τ max = 10 increases G to 0.63, and decreasing it to τ max = 1 yields G = 0 .
52. Interestingly, we findthat in the limit of τ max → , G → .
5. Thus, under the assumptions of our simple model, we findthat G ≥ .
5, which is similar to the wealth inequality of many developed nations.To visualize the inequality, we show a stacked histogram of the a-posteriori agedistribution of intelligences in Figure 4 using τ max = 9 .
43. Specifically, one can seethe effect of the bias weighting longer lived intelligences. We find that the top 1%of the oldest intelligences are over-represented in the fraction of first contacts by afactor of 4. DISCUSSIONIn this work, we have suggested a simple model for the lifetime distribution ofcivilizations (or more generally intelligences) producing technosignatures - specificallyan exponential distribution. This is motivated by its monotonic, single-parameterform and is a simple but effective description of the lifetime of biological familieson Earth. Amongst these hypothetical intelligences, we may plausibly detect theirtechnosignatures in the coming years, which may either take the form of direct contactor open the door for us to contact them. We have argued that the fact that longerlived intelligences simply have had more time available to them makes them morelikely to be detected - and thus the contacted population is weighted towards olderintelligences. We also note that lim τ max →∞ G = 1. ontact Inequality underlying population detected population → → → → → → → → → → → → → → → → → → → →
20 50.0%16.9%9.5%6.1%4.2%3.0%2.2%1.7%1.3%1.0%0.78%0.63%0.50%0.41%0.33%0.27%0.22%0.18%0.15%0.13% 26.3%16.3%11.5%8.5%6.6%5.2%4.2%3.4%2.8%2.3%1.9%1.6%1.4%1.1%0.97%0.83%0.71%0.61%0.52%0.45% ratio a g e / a g e o f hu m a n i t y li n e o f e q u a li t y L o r e n z c u r v e underlying population (cumulative) d e t e c t e d p o pu l a t i o n ( c u m u l a t i v e ) graphical comparison youngest oldestyoungestoldest Figure 4.
Using τ max = 9 .
43, one can set the a-posteriori distribution of intelligence agessuch that humanity lives at the median (far left). The exponential distribution assumed heavilyweights the population towards younger civilizations, most of which will not progress into olderones. However, older intelligences have more opportunities to contact others, simply by theirgreater age, which skews the distribution of the contacted population (mid-left). Taking theratio of the two (mid-right), the “contact inequality” is apparent - which can also be visualizedas a Lorenz curve (far-right).
Another framing of the above is that at any given time, the number of extant long-lived intelligences is disproportionality represented simply by the fact they persistlonger than their short-lived counterparts.We are able to establish that the expectation age of a contacted intelligence is twicethat of the ensemble, without any assumption about the maximum mean lifespan ofthis population. Further, we show that if the maximum mean lifespan of intelligencesis any greater than ∼ e times our current age, then we will most likely detect an olderintelligence than ourselves.Finally, we use this simple model to show that a “contact inequality” should ex-ist, where the older intelligences represent a disproportionate fraction of galactic firstcontacts. Using this analogy, we can define a Gini coefficient to quantify the inequal-ity, which we show must be greater than 0 . Validity of the employed model
First, we fully acknowledge here that the exponential distribution model is indeedextremely simplistic and may not fully describe the true distribution. The hazardfunction is a constant with respect to age and it’s deeply unclear whether a moreadvanced intelligence poses a greater risk to itself through emerging technologies4
Kipping et al. (e.g. Cooper 2013), or, on the other hand, is more likely to persist due to their trackrecord of survival thus far. The lifespan of biological families from fossil evidenceshows that an exponential distribution may not always be the best fit, but it doesbroadly capture the overall behavior (see Shimada et al. 2003 and Figure 1). Italso satisfies the basic expectation of a monotonically decreasing smooth function.Without any other evidence in hand, we argue that at present there is no justificationfor invoking a more complex model.The assumption of lifetime-weighted contact also deserves scrutiny. In this work, wehave very simply assumed that the longer an intelligence lasts, the more opportunitiesit has to be spotted. For example, if a civilization builds a beacon which lasts for aninterval L at some random point in the Universe’s history, the probability that wewill detect that beacon must be directly proportional to L . But of course one couldchallenge this picture from both the direction of increased or decreased detectability .For example, as an intelligence becomes more advanced, it could construct morepowerful beacons, with greater range, at lower cost, and in greater number (Benfordet al. 2008), even sending them out between the stars to add coverage. Those areintentional contact scenarios, but even unintended technosignatures might be arguedto become more detectable as intelligences advance, such as the production of Dyso-nian artifacts (Dyson 1960). On this basis, one might conclude that our assumptionhere that the probability of contact is proportional to L greatly underestimates thetrue value. If so, then older intelligences would dominate the number of first contactsby an even more extreme degree, raising the Gini index yet higher . This fundamen-tally does not change our hypothesis that a contact inequality likely exists, in fact itexacerbates the inequality.On the other hand, one might argue that as intelligences develop, their detectabil-ity decreases. Science fiction writer Karl Schroeder captures this hypothesis in histwist on Arthur C. Clarke’s famous line “Any sufficiently advanced civilization is in-distinguishable from nature” (Schroeder 2003). They might also simply lose interestin communicating with far less advanced intelligences and elect to hide themselves(Smart 2012; Kipping & Teachey 2016). If their detectable presence is suddenly elimi-nated altogether, then they are technically no longer a member of the assumed under-lying population - which is specifically one which produces (potentially detectable)technosignatures. Thus, they are effectively extinct and thus don’t actually affectthe arguments laid out here. However, if the detectability of intelligences diminisheswith age, in particular in a way such that the time-integrated probability of detectionculminates in a scaling of L α where α <
0, then this would reverse our conclusion -contact would likely occur with less advanced members of the population. Moreover, it would be interesting to consider non-monontonic models with functional dependen-cies of detectability versus lifetime (but not to be confused with age). We did attempt to repeat our study assuming a proportionality of L n , where n is a free-index,but were not able to make analytic progress. ontact Inequality τ max is an order-of-magnitude greaterthan our current age.Together, whilst we accept that our model is surely an oversimplification, the qual-itative result that older intelligences should be overrepresented in the ensemble ofdetections may actually be quite robust.4.2. A note on distance
Our detection bias model assumes that the probability of detection is proportionalto an intelligence’s lifetime, but the distance to that intelligence does not feature.Why not? Certainly, closer intelligences will be more likely to be detected than moredistant ones, since signals generally decrease as 1 /d . But this work only concernsitself with the lifetime distribution of detected intelligences, not their distance (whichcan be thought of as being marginalized over). The real question for this work is -do we expect there to be some off-diagonal covariance between lifetime and distanceof the detected population? More simply, is there any reason to suspect that theintrinsic lifetimes of detected intelligences is dependent upon their displaced locationfrom the Earth?As discussed in detail in the last section, one could invoke an argument that longerlived intelligences are more detectable, which would exacerbate the contact inequalityresult of this work. Only if detectability rapidly diminished with time would our basicconclusion change.A separate aspect to the distance issue, is not with detectability per say, but ratherwith intrinsic lifetimes varying with distance. Do we expect an intelligence’s lifetimeto depend upon how far away from us they are? At distances of hundreds, eventhousands, of light years - the answer is no. There is nothing inherently special aboutwhere we live and thus a civilization emerging a few hundred light years should not6 Kipping et al. have any particular reason to live longer or shorter than ourselves. Extending furtherafield, where effects such as galactic chemical gradients (Gonzalez 2001), supernovaerates (Lineweaver et al. 2004), active galactic nuclei (Balbi & Tombesi 2018; Lingamet al. 2019), stellar encounter rates (McTier et al. 2020) may vary, would indeedrequire formally building a model which described this covariance. Accordingly, theresults of our work should be understood to be formally only applicable to wherecases where L is not expected to be intrinsically linked to location, such as our localstellar neighborhood. 4.3. Implications
Let’s proceed under the assumption that the hypothesis is correct: probabilistically,we are more likely to make first contact with an intelligence that is considerablyolder than ourselves. It should be noted that this age difference could be quiteextreme, perhaps millions or even billions of years, in principle. Although age does notnecessarily ensure greater technological advancement, that is the obvious expectationfrom such a scenario. Of course, we may never detect any technosignatures and thusnever have the opportunity for first contact, but under the premise that we will oneday succeed, it is interesting to ask what the implications of our suggested contactedinequality are.Some have voiced concerns that humanity’s historical record of encounters betweensocieties of different technological capabilities generally ends poorly for the less ad-vanced entity. Of course, it’s unclear that human behavior can be extrapolated toanother intelligence that is far older than ourselves. Accordingly, we prefer to avoidspeculating about the impact of such a contact directly.However, the contact inequality hypothesis does have significant bearing on our ownactive searches for technosignatures. Focusing on searching for technology similarto that of our own may be unlikely to lead to success. If an intelligence is muchmore advanced than us, then planet-integrated transient signatures associated withdisequilibrium (such as climate change and pollution) are less likely to be the meansof detection, since they are simply unsustainable for a long-lived entity.Further, to ensure their own survival, such intelligences may have relocated or ex-panded their presence off-world, thus favoring technosignatures associated with suchactivities. Ultimately, this work concerns itself with a formalism for establishing thehypothesis rather the consequences of it. But from our work, we encourage the for-malism and prediction established here to be considered in future efforts to seek outtechnosignatures, including more detailed exploration of the assumptions and analyticforms of civilization longevity and technological age. ontact Inequality