Exponential distribution for L

At its core, we are asking a statistical question – what is the likely age of a detected intelligence. The first requirement to make progress is to assign a probability distribution for L. The simplest lifetime model we can posit is an exponential distribution (Lawless Reference Lawless2011). We do not claim that this is necessarily the true distribution, and encourage the reader to treat this as an approximate-yet-instructive model for making analytic progress. Further discussion about the suitability of this model is offered in the 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 ourselves) and a much smaller number of long-lived counterparts. On this basis, one might naively posit that communication with another intelligence would surely be with one of the more abundant short-lived intelligences. We proceed by first writing down the probability density function of L given our exponential distribution assumption:

(1)$${\rm Pr}\lpar L\vert \tau\rpar = \tau^{-1}\, {\rm e}^{-L/\tau}\comma\; $$

where τ is the mean lifetime from the distribution. The exponential distribution assumes that the so-called hazard functionFootnote ¹ is constant over time – much like a decaying atomic nucleus. Certainly more sophisticated lifetime formulae have been suggested for species survival. For example, a Weibull distribution is a commonly used generalization of the exponential, that enables a time-dependent (specifically a power-law) hazard function (Lawless Reference Lawless2011), but comes at the expense of an extra unknown parameter.

We note that a power-law distribution has also been adopted in ecology studies (Pigolotti et al. Reference Pigolotti, Flammini, Marsili and Maritan2005), but we found it to exhibit several disadvantages over the exponential. First, it does not have semi-infinite support and thus requires truncation parameterized some additional bounding parameter, either a minimum lifetime or a maximum. Since no clear minimum exists, bounding at the maximum leads to a function which is only monotonically decreasing for indices between 0 and 1. This leads to an overly restrictive distribution compared to the exponential and for these reasons it is not used in what follows.

An exponential distribution, with its constant hazard function of 1/τ, could be criticized as being unrealistic since a longer lived species presumably has developed successful traits that improve its odds of future survival (Shimada et al. Reference Shimada, Yukawa and Ito2003). On the other hand, as technology advances, so too does an intelligence's capacity for self-destruction (Cooper Reference Cooper2013). If we consider the observed distribution for the life span of families obtained from Benton (Reference Benton1993) based on fossil evidence (see Fig. 1), the exponential distribution appears quite capable of describing the overall pattern out to 400 million years. Of course, intelligences producing technosignatures cannot be assumed to necessarily follow the same distribution as these fossils, although this gives confidence that the model is at least plausible. Although not a representative nor unbiased sample, we note that approximate lifetimes of past human civilizations are also well-described by an exponential distribution with τ = 336 years.Footnote ² In the absence of any other information, we invoke Ockham's razor in that the simplest viable model is the presently favoured one. Accordingly, we will adopt the exponential distribution in what follows.

Fig. 1. Probability distribution for the life span of families obtained from Benton (Reference Benton1993) using fossil evidence. On the left we fit an exponential distribution to the data shown, whereas on the right we show the more complicated Weibull distribution. Although we do not place any emphasis on the specific parameters recovered, the data show that an exponential distribution is a quite reasonable description of the overall distribution, which compounded with it's simplicity makes it attractive as a choice for modelling technosignature lifetimes.

Inferring the a-posteriori distribution of τ

Although we have a functional form for the probability distribution of L, it is governed 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 ₁, L ₂, …, L _N}^T (analogous to the data presented in Fig. 1), then we could write that the likelihood of measuring these values for a mean lifetime equal to τ would be

(2)$${\rm Pr}\lpar {\bf L}\vert \tau\rpar = \prod_{i = 1}^{N} \tau^{-1}\, {\rm e}^{-L_i/\tau}\comma\; $$

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 using Pr(τ|L) ∝ Pr(L|τ)Pr(τ).

Unfortunately, we do not have a sample of L _i values, and thus our likelihood function will certainly not be as constraining as this. Rather, we only know of N = 1 intelligence – ourselves. However, the problem is even worse than this because we do not even know L ₁ for this one datum. Human civilization has been producing a technosignature for an age $A_{\oplus }$ years, and the lifetime of this intelligence must at least exceed this value (i.e. $L_{\oplus } \geq A_{\oplus }$). We emphasize that it is somewhat unclear what numerical value to assign to $A_{\oplus }$ at this point. Although we have been transmitting radio signals for ~ 10² years, one might argue that an advanced civilization could remotely detect our settlements (Kuhn and Berdyugina Reference Kuhn and Berdyugina2015) and polluted atmosphere (Schneider et al. Reference Schneider, Léger, Fridlund, White, Eiroa, Henning, Herbst, Lammer, Liseau, Paresce, Penny, Quirrenbach, Roettgering, Selsis, Beichman, Danchi, Kaltenegger, Lunine, Stam and Tinetti2010; Lin et al. Reference Lin, Abad and Loeb2014) as unintentional technosignatures, which could increase $A_{\oplus }$. Regardless, we will proceed symbolically for the moment.

The likelihood of observing one civilization with $L_1\gt A_{\oplus }$, given that the mean lifetime is τ, is given by

(3)$$\eqalign{{\rm Pr}\lpar L_1{\gt }A_{\oplus}\vert \tau\rpar & = \int_{A_{\oplus}}^{\infty} {\rm Pr}\lpar L\vert \tau\rpar \, {\rm d}L\cr \qquad& = {\rm e}^{-A_{\oplus}/\tau}.}$$

In order to derive an a-posteriori distribution for τ, conditioned upon the constraint that $L_1\gt A_{\oplus }$, we first need to write down an a-priori distribution for τ. One is always free to choose any prior one wishes, but a strongly informative prior, such as a tight Gaussian, would naturally return a result which closely equals the prior. In other words, one has not really learned anything and no inference really occurred. Ideally, we wish to select a prior which is as uninformative as possible (Jaynes Reference Jaynes1968). This is not simply a flat prior, since such priors can place insufficient weight on small values, especially when the parameter has high dynamic range. Instead, we can define an objective Jeffrey's prior, which provides a means of expressing a scale-invariant distribution via the Fisher information matrix, ${\cal I}$ (Jeffreys Reference Jeffreys1946):

(4)$${\rm Pr}\lpar \tau\rpar \propto \sqrt{{\rm det}{\cal I}\lpar \tau\rpar }.$$

Evaluating the above, we obtain Pr(τ) ∝ τ^−1/2. Combining the likelihood and prior together, we obtain

(5)$$\eqalign{{\rm Pr}\lpar \tau\vert L_1{\gt }A_{\oplus}\rpar & \propto {\rm Pr}\lpar L_1{\gt }A_{\oplus}\vert \tau\rpar {\rm Pr}\lpar \tau\rpar \comma\; \cr \qquad& \propto \tau^{-1/2}\, {\rm e}^{-A_{\oplus}/\tau}.}$$

To normalize the above, one must define an upper limit on τ, for which we use the symbol τ_max. At this point, it is also convenient to work in temporal units of $A_{\oplus }$ in what follows, such that any timescales used will always be in that unit. Accordingly, the posterior is

(6)$${\rm Pr}\lpar \tau\vert L_1{\gt }1\rpar = { \tau^{-1/2}\, {\rm e}^{-1/\tau} \over 2 \sqrt{\tau_{{\rm max}}}\, {\rm e}^{-1/\tau_{{\rm max}}} - 2 \sqrt{\pi} {\rm erfc}\lsqb 1/\sqrt{\tau_{{\rm max}}}\rsqb }.$$

We plot the posterior, with comparison to the prior and likelihood, in Fig. 2.

Fig. 2. Comparison of the prior, likelihood and posterior distribution for τ (the mean lifetime of intelligences producing technosignatures) using τ_max = 10 Gyr, as an example. The mode of the posterior occurs at 2 as shown in the text.

Properties of the posterior

There are several useful properties of the posterior above that we highlight. First, equation (6) has a maximum at $\hat {\tau } = 2$ (the mode), irrespective of τ_max, which can be demonstrated through differentiation of the expression and setting to zero. If we set $\tau = \hat {\tau }$, then the mean lifetime of an intelligence would be twice of that of ourselves. But it is important to remember that this is the entire lifetime of this intelligence, not its age at the time of their detection, A. Assuming that the technosignature is no more or less likely to be detected at any point during its manifested lifetime, then $A \sim {\cal U}\lsqb 0\comma\; L\rsqb$ (where ${\cal U}$ denotes a uniform distribution). Accordingly, if $\tau = \hat {\tau }$, then the mean age at the time of detection would be = 1, i.e. our current age. Of course, fixing $\tau = \hat {\tau }$ does not correctly account for the broad posterior distribution of τ, but this 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 as an interpretable summary statistic. The expectation value of a distribution provides better intuition as to the ‘typical’ value of the distribution. This can be seen by simple consideration of the exponential distribution. Its mode is zero but the average draw will be around the mean of the distribution, not zero. We may calculate the a-posteriori expectation value for τ using

(7)$$\eqalign{E\lsqb \tau\vert L_1{\gt }1\rsqb & = \int_{\tau = 0}^{\tau_{{\rm max}}} \tau \, {\rm Pr}\lpar \tau\vert L{\gt }1\rpar \, {\rm d}\tau\cr \qquad& = \mu\comma\; }$$

where we define the symbol

(8)$$\mu \equiv {1\over 3} \left({\tau_{{\rm max}}\over 1 - \sqrt{\pi} \tau_{{\rm max}}^{-1/2}\, {\rm e}^{1/\tau_{{\rm max}}} {\rm erfc}\lsqb 1/\sqrt{\tau_{{\rm max}}}\rsqb } - 2 \right)\comma\; $$

where erfc[x] is the complementary error function. One may show that μ ≃ τ_max/3 for τ_max ≫ 1. The dependency upon τ_max can be understood by the fact that although the mode of the distribution does not depend on τ_max, pushing the upper limit ever higher naturally drags the tail out and thus pulls the expectation value over.

Marginalized distribution for L

Given that we have now obtained an a-posteriori distribution for τ, we need to propagate that into a posterior distribution for L. As discussed earlier, simply fixing τ to an a-posteriori summary statistic, like $\hat {\tau }$ or E[τ|L ₁ > 1], is inadequate, as it does not propagate the (considerable) uncertainty on τ into the resulting distribution. This propagation can be conducted through marginalizing out τ (i.e. integrating over τ):

(9)$$\eqalign{{\rm Pr}\lpar L\vert L_1{\gt }1\rpar & = \int_{\tau = 0}^{\tau_{{\rm max}}} {\rm Pr}\lpar L\vert \tau\rpar {\rm Pr}\lpar \tau\vert L_1{\gt }1\rpar \, {\rm d}\tau\comma\; \cr \qquad& = { \sqrt{\pi} {\rm erfc}\lsqb \sqrt{L + 1}/\sqrt{\tau_{{\rm max}}}\rsqb \over 2 \sqrt{L + 1} \big(\sqrt{\tau_{{\rm max}}}\, {\rm e}^{-1/\tau_{{\rm max}}} - \sqrt{\pi} {\rm erfc}\lsqb 1/\sqrt{\tau_{{\rm max}}}\rsqb \big)}.}$$

The above represents the probability distribution of the lifetime of technosignature-producing intelligences, given the singular constraint imposed by humanity's existence. It has a maximum at L → 0, which is a property shared by the original exponential distribution used for Pr(L). We also note that the expectation value satisfies E[L|L ₁ > 1] = μ.

Lifetime weighting

The distribution Pr(L|L ₁ > 1) describes the probability distribution of the lifetime of intelligences producing detectable technosignatures. This is the underlying true population – but it does not represent the intelligences that we are most likely to detect. It is worth pausing to clearly distinguish between detection and contact. If and when an intelligence is detected, that detection may either be in the form of a directed attempt at communication on their behalf, or it may simply be passive detection of their technology on our behalf. Regardless, humanity's decision as to whether to send a message back – to initiate contact – will be likely somewhat dependent on the technological development and, by proxy, age (A) of said intelligence.Footnote ³ If the technosignature itself provides little information regarding the age, we would be left with the a-priori distribution – which is the focus of this paper. Yet this distribution will not simply equal P(A|L ₁), since a critical selection effect sculpts our observations that we will account for here.

The start time of these other intelligences is presumably arbitrary (except when one pushes into timescales of $\gg$Gyr, over which time variability is expected for the rates of 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, a longer 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 sensitive to how long the technosignature persists. An equivalent statement is that at any single snapshot in time (representing our current epoch, e.g.), the fraction of worlds that go on to produce long-lived intelligences may be relatively rare, but their persistence through time means that one must account for their overrepresentation amongst the extant intelligences. This is simply a product of their longevity and is independent of their activities or behaviour. This situation is analogous to the ages of trees in an old growth forest – if we assigned a unique identity to each tree that will ever live, 1000+ year old trees are rare amongst the ensemble, perhaps representing just 1%, yet a visit to the forest will show them to be seemingly more common due to their longevity, for example, comprising 10% of the extant trees.

Accordingly, we will assume that the probability of detecting an intelligence's technosignature is proportional to its lifetime, L. The validity of this assumption is discussed later in the Discussion section, as well as an explanation as to why distance does not affect the results presented hereafter.

This simple weighting will substantially change the picture, meaning that the long tail of rare long-lived intelligences will have a considerable increase on their relative probability of detection. We write that the probability distribution of L, conditioned upon both a mean lifetime, τ, and the assumption of detection, ${\cal D}$, is

(10)$${\rm Pr}\lpar L\vert \tau\comma\; {\cal D}\rpar \propto L \, {\rm Pr}\lpar L\vert \tau\rpar \comma\; $$

or after normalization

(11)$${\rm Pr}\lpar L\vert \tau\comma\; {\cal D}\rpar = {L\over \tau^{2}}\, {\rm e}^{-L/\tau}.$$

Since we have already learnt τ from before, we can use this acquired information to express a marginalized posterior for L conditioned upon both ${\cal D}$ and the fact L ₁ > 1, using:

(12)$$\eqalign{{\rm Pr}\lpar L\vert L_1{\gt }1\comma\; {\cal D}\rpar & = \int_{\tau = 0}^{\tau_{{\rm max}}} {\rm Pr}\lpar L\vert \tau\comma\; {\cal D}\rpar {\rm Pr}\lpar \tau\vert L_1\gt 1\rpar \, {\rm d}\tau\comma\; \cr \qquad& = W \Bigg({L\, {\rm e}^{-L/\tau_{{\rm max}}}\over 4 \lpar L + 1\rpar ^{3/2}}\Bigg).}$$

where

(13)$$\eqalign{W = \Bigg({2 \sqrt{\lpar {L + 1}\rpar /{\tau_{{\rm max}}}} + \sqrt{\pi}\, {\rm e}^{\lpar L + 1\rpar /\tau_{{\rm max}}} {\rm erfc}\lsqb \sqrt{L + 1}/\sqrt{\tau_{{\rm max}}}\rsqb \over \sqrt{\tau_{{\rm max}}} - \sqrt{\pi}\, {\rm e}^{1/\tau_{{\rm max}}} {\rm erfc}\lsqb 1/\sqrt{\tau_{{\rm max}}}\rsqb }\Bigg)}$$

We find that in the limit of τ_max ≫ 1, this distribution peaks at 2. The expectation value is given by

(14)$$\eqalign{E\lsqb L\vert L_1{\gt }1\comma\; {\cal D}\rsqb & = \int_{\tau = 0}^{\infty} L {\rm Pr}\lpar \tau\vert L{\gt }1\comma\; {\cal D}\rpar \, {\rm d}L\comma\; \cr \qquad& = 2\mu.}$$

For comparison, without the conditional ${\cal D}$, the a-posteriori expectation value was μ but including it doubles it. We plot the posterior ${\rm Pr}\lpar L\vert L_1{\gt }1\comma\; {\cal D}\rpar$, and compare it to Pr(L|L ₁ > 1), in Fig. 3.

Fig. 3. Comparison of the marginalized posterior probability for the age of intelligences, L, for the ensemble population (Pr(L|L ₁ > 1)) and the detected population (${\rm Pr}\lpar L\vert L_1{\gt }1\comma\; {\cal D}$)). 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 an intelligence at the end of its lifetime, L, but rather one drawn randomly from across its lifespan. In other words, an intelligence's age (at the time of detection) does not equal its lifetime. If we assume that the age at the time of detection is uniformly distributed from 0 to L, then

(15)$$\eqalign{{\rm Pr}\lpar A\vert L_1{\gt }1\comma\; {\cal D}\rpar & = \int_{\tau = 0}^{\tau_{{\rm max}}} {\cal U}\lsqb 0\comma\; L\rsqb {\rm Pr}\lpar L\vert L_1{\gt }1\comma\; {\cal D}\rpar \, {\rm d}L\comma\; \cr \qquad& = { \sqrt{\pi} {\rm erfc}\lsqb \sqrt{A + 1}/\sqrt{\tau_{{\rm max}}}\rsqb \over 2 \sqrt{A + 1} \big(\sqrt{\tau_{{\rm max}}}\, {\rm e}^{-1/\tau_{{\rm max}}} - \sqrt{\pi}{\rm erfc}\lsqb 1/\sqrt{\tau_{{\rm max}}}\rsqb \big). }}$$

Equipped with our final form for the a-posteriori probability distribution of the age of detected intelligences, we can deduce several basic properties. First, it is interesting to ask whether the civilization is likely to be older or younger than our own. The probability that the civilization is older is given by

(16)$$\eqalign{{\rm Pr}\lpar A{\gt }1\vert L_1{\gt }1\comma\; {\cal D}\rpar = & \int_{A = 1}^{\infty} {\rm Pr}\lpar A\vert L_1{\gt }1\comma\; {\cal D}\rpar \, {\rm d}A\comma\; \cr \qquad = & \big(\sqrt{\tau_{{\rm max}}}\, {\rm e}^{-1/\tau_{{\rm max}}}\cr \qquad& - \sqrt{2\pi}\, {\rm e}^{1/\tau_{{\rm max}}}{\rm erfc}\lsqb \sqrt{2}/\sqrt{\tau_{{\rm max}}}\rsqb \big)\cr \qquad& / \big(\sqrt{\tau_{{\rm max}}} - \sqrt{\pi}\, {\rm e}^{1/\tau_{{\rm max}}}{\rm erfc}\lsqb 1/\sqrt{\tau_{{\rm max}}}\rsqb \big).}$$

A useful summary statistic to interpret the above is the median – above which half the cases will lie. This may be solved for by setting the above to 0.5 and numerically solve for τ_max, which gives the result that if τ_max > 2.6776 · · · ≃e, then the age of a detected intelligence will most likely exceed that of our own, i.e. ${\rm Pr}\lpar A{\gt }1\vert L_1{\gt }1\comma\; {\cal D}\rpar \gt 0.5$. In other words, if this condition is true then we are most likely to detect an older intelligence than ourselves.

It is important to remember that τ_max does not represent the maximum allowed lifetime of a civilization, L – rather it is simply the maximum a-priori mean lifetime. Fundamentally, there is no obvious reason why τ_max could not be many billions of years (Grinspoon Reference Grinspoon2004), and thus detection would almost always occur with an older civilization, however one defines $A_{\oplus }$.

The expectation value for the intelligence's age is given by simply μ, whereas we found the expectation value for their lifetime to be 2μ. Since E[A|L ₁ > 1] = E[L|L ₁ > 1]/2, then we can see that the effect of including this observational bias is that the mean age of detected, and thus contacted, intelligences is twice that of the overall population – as expected.

Contact inequality

Using our results, it is instructive to compare the underlying age population, Pr(A|L ₁ > 1), with the population which goes on to be detected, ${\rm Pr}\lpar A\vert L_1{\gt }1\comma\; {\cal D}\rpar$. Recall that A is age of the intelligence at the time of detection/contact, whereas L is the total lifetime of said intelligence. The fact that older intelligences are assumed in this work to be more likely to be detected, and thus contacted (by virtue of having simply more opportunities to do so), introduces an inequality. The rare long-lived intelligences make a disproportionate number of contacts.

This ‘contact inequality’ can be thought of as being analogous to wealth inequality in economics. One way to quantify the degree of inequality comes from the Gini coefficient (Gini Reference Gini1909), 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 function Pr(x) using

(17)$$G = {1\over 2\mu} \int_{0}^{\infty} \int_{0}^{\infty} {\rm Pr}\lpar x\rpar {\rm Pr}\lpar y\rpar \vert x-y\vert \, {\rm d}x\, {\rm d}y.$$

Although we were not able find a closed-form solution to the above using ${\rm Pr}\lpar x\rpar = {\rm Pr}\lpar A\vert L_1\gt 1\comma\; {\cal D}\rpar$, one may numerically integrate the expression for a specific choice of τ_max.

We argue here that a conservative choice of τ_max is one which causes our current age to be the median age of the entire population of technosignature producing intelligences. This is a form of the mediocrity principle, since we posit humanity lives close to the centre of the age-ordered list of intelligences in the cosmos (Gott Reference Gott1993; Simpson Reference Simpson2016). It requires us to solve τ_max such that

(18)$$\int_{A = 0}^{1} {\rm Pr}\lpar A\vert L_1\gt 1\rpar = {1\over 2}.$$

We solved the above numerically and obtain τ_max = 9.43. This also somewhat passes the astronomer's logic of going up by an order-of-magnitude as one's upper limit on a variable. However, we suggest here that this limit is somewhat conservative though, since it makes million-/billion-year intelligences essentially non-existent, which is itself a strong assumption.

Nevertheless, using τ_max = 9.43, we compute a Gini coefficient of 0.57. The value does not grossly change by varying τ_max. For example, setting τ_max = 10³ increases G to 0.63, and decreasing it to τ_max = 1 yields G = 0.52. Interestingly, we find that in the limit of τ_max → 0 (which would make humanity an incredibly long lived civilization),Footnote ⁴ G → 0.5. Thus, under the assumptions of our simple model, we find that G ≥ 0.5, which is similar to the wealth inequality of many developed nations.

To visualize the inequality, we show a stacked histogram of the a-posteriori age distribution of intelligences in Fig. 4 using τ_max = 9.43. Specifically, one can see the 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 a factor of 4.

Fig. 4. Using τ_max = 9.43, one can set the a-posteriori distribution of intelligence ages such that humanity lives at the median (far left). The exponential distribution assumed heavily weights the population towards younger civilizations, most of which will not progress into older ones. However, older intelligences have more opportunities to contact others, simply by their greater age, which skews the distribution of the contacted population (mid-left). Taking the ratio of the two (mid-right), the ‘contact inequality’ is apparent – which can also be visualized as a Lorenz curve (far-right).