The case for simplifying the Drake equation

The Drake equation expresses the mean number of communicative civilizations in the Galaxy (Drake, Reference Drake, Mamikunian and Briggs1965), given by

(1)$$N_{\rm C} = R_{\star} \times f_{\rm P} \times n_{\rm E} \times f_{\rm L} \times f_{\rm I} \times f_{\rm C} \times L_{\rm C},\; $$

where $R_{\star }$ is mean rate of star formation, f _P is the fraction of stars that have planets, n _E is the mean number of planets that could support life per star with planets, f _L is the fraction of life-supporting planets that develop life, f _I is the fraction of planets with life where life develops intelligence, f _C is the fraction of intelligent civilizations that develop communication and L _C is the mean length of time that civilizations can communicate.

We note that many terms frequently used in the SETI literature are problematic (Almár, Reference Almár2011), including the term ‘civilization’ for its Eurocentric implications (Denning, Reference Denning2011; Smith, Reference Smith2012). In what follows, we favour the term extraterrestrial intelligences (ETIs), which although historically has been interpreted as ‘extra-terrestrial intelligences’, we here use to define ‘extraterrestrial technological instantiations’.

When discussing the Drake equation, it's important to emphasize that N _C is an expectation value. It does not represent the exact number of ETIs at any one time (Glade et al., Reference Glade, Ballet and Bastien2012), but rather the average over some time interval over which the dependent parameters are approximately stable (Kipping, Reference Kipping2021).

A common pedagogical exercise is to ask students to guess the various values and estimate N _C for themselves, and indeed these guesses are typically as reasonable as any other, since the latter four terms are wholly unknown (Sandberg et al., Reference Sandberg, Drexler and Ord2018). Indeed, this ignorance could be easily weaponized to dismiss the utility of the Drake equation (Gertz, Reference Gertz2021), to which proponents often counter that it was never intended to be used as a calculator like this – the original intent was merely to convene a meeting and ‘organize our ignorance’, to quote Jill Tarter (Achenbach, Reference Achenbach2020).

Although one must concede that a full set of accurate inputs cannot be supplied to the Drake equation (Sandberg et al., Reference Sandberg, Drexler and Ord2018), that does not mean it has no quantitative utility – the very framing of the problem implies certain statistical results. For example, Maccone (Reference Maccone2010) argued that the choice of which terms to include is somewhat debatable and one could reasonably conceive of longer lists of multiplicative parameters e.g. fraction of life forms that go on to develop multicellular life. If the list of parameters are treated as independently distributed then the central limit theorem dictates that N _C must follow a log-normal distribution, despite the fact we do not know what the probability distributions for each parameter even are.

The birth–death formalism

In our previous work (Kipping, Reference Kipping2021), it was suggested to imagine the reverse. Rather than trying to expand the Drake equation ad infinitum (Maccone, Reference Maccone2010), one may compress it to the most efficient form possible. It can be seen that the first six parameters in the Drake equation, including whatever additional intermediate terms one might wish to add (e.g. Molina Molina, Reference Molina Molina2019), all describe the net process of spawning communicative ETIs – the birth rateFootnote ¹. Indeed, these six terms together have units of inverse time. The final term is conceptually different in describing the death process – how long the communicative ETI lasts. We may thus re-write the Drake equation as simply

(2)$$N_{\rm C} = \underbrace{R_{\star} \times f_{\rm P} \times n_{\rm E} \times f_{\rm L} \times f_{\rm I} \times f_{\rm C}}_{\equiv\Gamma_{\rm C}} \times L_{\rm C}.$$

This birth–death formalism is flexible and easily interpretable. For example, it's trivial to exchange the terms to cover different variations of N. For example, if N _I is the number of ETIs (irrespective of their communicative intent), then N _I = Γ_IL _I. Further, it's also easy to change the volume from the entire Galaxy (á la Drake's original expression), to some relevant sample of interest (e.g. a specific cluster).

This formalism also dissolves many criticisms levied at the Drake equation. First, there is no longer a problem with temporal retardation effects as pointed out by Ćirković (Reference Ćirković2004). However in the original framing it's unclear as to whether $R_{\star }$ should be the current star formation rate (Gertz, Reference Gertz2021), or the value from 4.4 Gyr ago, or something in-between, that question is avoided completely by simply stating that there is some current birth rate at which ETIs emerge, and that's that. Second, it resolves the terracentric tunnel vision effect one might critique the original equation with (Ćirković, Reference Ćirković2007), which considers a singular pathway to intelligence that is perhaps ignorant of other roads (e.g. tidally heated moons of rogue planets; Abbot and Switzer, Reference Abbot and Switzer2011). Indeed, the apparent subjectiveness of which terms to include and which to not (e.g. Molina Molina, Reference Molina Molina2019) is moot here.

Finally, the formalism is indisputable. Worlds inhabited by species using technology must sometimes emerge (since we are one) and thus one can always define the number of newly birthed examples per unit time. The mechanism of this emergence is irrelevant in this framing, it could be via natural evolution (Kipping, Reference Kipping2020), directed panspermia (Crick and Orgel, Reference Crick and Orgel1973) or something else entirely – all pathways simply sum into this one birth rate. Similarly, one can always define the number of ETIs that terminate per unit time. Of course, these advantages come at the expense of a disadvantage – the birth–death formalism is not as useful for organizing a scientific conference, the original purpose of the Drake equation (Gertz, Reference Gertz2021).

Lessons from a previous attempt at the stationary Drake equation

In Kipping (Reference Kipping2021), it was argued that the birth–death formalism of the Drake equation allows one to say something about the distribution of the number of ETIs over timeFootnote ². Although N _C represents the mean number of communicative ETIs, let's define n _C as the time varying number – the stochastic value which should fluctuate around N _C. Accordingly, one may write that E[n _C] = N _C. The birth process is characterized by a mean number of communicative ETIs birthed per unit time, Γ_C, which rigorously defines a Poisson process (Glade et al., Reference Glade, Ballet and Bastien2012). For the death process, an exponential distribution was assumed in Kipping (Reference Kipping2021), following the arguments made in Kipping et al. (Reference Kipping, Frank and Scharf2020). It was then demonstrated that the product of a Poisson and an exponential process yields another Poisson process. Hence, n _C ~ Po[N _C].

However, one feature of this approach seems in error and reveals an apparent weakness in the birth–death simplification – indeed a weakness that extends to the original Drake equation too. The two terms, Γ_C and L _C, have extreme permissible ranges (Lacki, Reference Lacki2016) and thus could be varied such that N _C exceeds the number of stars in the Galaxy (or whatever volume one is considering). This also rings true for the original Drake equation, since L _C is naively unbounded and could likewise be engineered to yield more ETIs than stars. This might seem like an edge effect; after all, few would reasonably argue for a value of L _C > 10¹¹ years that would create this issue since this exceeds the age of the Universe (Planck et al., Reference Planck, Aghanim, Akrami, Ashdown, Aumont, Baccigalupi, Ballardini, Banday, Barreiro, Bartolo, Basak, Battye, Benabed, Bernard, Bersanelli, Bielewicz, Bock, Bond, Borrill, Bouchet, Boulanger, Bucher, Burigana, Butler, Calabrese, Cardoso, Carron, Challinor, Chiang, Chluba, Colombo, Combet, Contreras, Crill, Cuttaia, de Bernardis, de Zotti, Delabrouille, Delouis, Di Valentino, Diego, Doré, Douspis, Ducout, Dupac, Dusini, Efstathiou, Elsner, Enßlin, Eriksen, Fantaye, Farhang, Fergusson, Fernandez-Cobos, Finelli, Forastieri, Frailis, Fraisse, Franceschi, Frolov, Galeotta, Galli, Ganga, Génova-Santos, Gerbino, Ghosh, González-Nuevo, Górski, Gratton, Gruppuso, Gudmundsson, Hamann, Handley, Hansen, Herranz, Hildebrandt, Hivon, Huang, Jaffe, Jones, Karakci, Keihänen, Keskitalo, Kiiveri, Kim, Kisner, Knox, Krachmalnicoff, Kunz, Kurki-Suonio, Lagache, Lamarre, Lasenby, Lattanzi, Lawrence, Le Jeune, Lemos, Lesgourgues, Levrier, Lewis, Liguori, Lilje, Lilley, Lindholm, López-Caniego, Lubin, Ma, Macías-Pérez, Maggio, Maino, Mandolesi, Mangilli, Marcos-Caballero, Maris, Martin, Martinelli, Marténez-González, Matarrese, Mauri, McEwen, Meinhold, Melchiorri, Mennella, Migliaccio, Millea, Mitra, Miville-Deschênes, Molinari, Montier, Morgante, Moss, Natoli, Nørgaard-Nielsen, Pagano, Paoletti, Partridge, Patanchon, Peiris, Perrotta, Pettorino, Piacentini, Polastri, Polenta, Puget, Rachen, Reinecke, Remazeilles, Renzi, Rocha, Rosset, Roudier, Rubiño-Martín, Ruiz-Granados, Salvati, Sandri, Savelainen, Scott, Shellard, Sirignano, Sirri, Spencer, Sunyaev, Suur-Uski, Tauber, Tavagnacco, Tenti, Toffolatti, Tomasi, Trombetti, Valenziano, Valiviita, Van Tent, Vibert, Vielva, Villa, Vittorio, Wandelt, Wehus, White, White, Zacchei and Zonca2020). Nevertheless, it rigorously demonstrates that the Drake equation is incomplete, missing some detail that leads to the correct asymptotic behaviour.

That something is carrying capacity. In any environment, be is a Petri dish or a galaxy, there is a finite carrying capacity of that volume to host the life forms under consideration – a fact understood in ecology since at least the 1840s (Sayre, Reference Sayre2008). Microbial colonies can't grow ad infinitum in a test tube. In what follows, we build upon our previous work (Kipping, Reference Kipping2021) to account for this effect, within the birth–death paradigm.

A steady-state Drake (SSD) equation

In this work, we concern ourselves with the steady-state condition, where the mean number of births per unit time equals the mean number of deaths per unit time. This equilibrium point will of course drift over aeons due to the evolution of the cosmos (e.g. the star formation rate), but we assume that the extant population is in such a steady state. The conditions for this statement to hold are explicated later.

Let us write that the mean number of births per location over a time interval, δt, is given by λ _Bδt. Note that already we have deviated from the parameterization of Kipping (Reference Kipping2021), who consider the birth rate summed over all worlds (Γ), not per world as done so hereFootnote ³. Also note our deliberately vaguely defined ‘location’ phrasing above. To some degree, how one defines a location is up to the user and is inextricable with how one defines an occupied location. It could be planets, rocky objects, star systems or even cubic parsecs. We'll use the term ‘seats’ in what follows, as in potential seats for occupation.

Consider that there exists N _A available seats, and thus we have an expectation value of N _Aλ _Bδt births over the interval δt, where the actual birth number in any given time interval will vary stochastically about this expectation value following a Poisson distribution (Kipping, Reference Kipping2021). The number of available seats will not be equal to the total number of seats, N _T, since we assume here that a birth cannot occur within a seat that is already occupied. Hence, we write that N _A = (N _T − N _O), where N _O is the mean number of occupied seats.

The mean number of deaths over a time interval, δt, will equal the mean number of occupied seats multiplied by the fraction that are expected to perish. Let us write that over a time interval δt, the fraction of occupied seats that will extinguish is λ _Dδt. Thus λ _D represents the death rate (of the current epoch)Footnote ⁴ . We may now balance the births and deaths as

(3)$$( N_{\rm T}-N_{\rm O}) \lambda_{\rm B} = N_{\rm O} \lambda_{\rm D},\; $$

where the δt time intervals have cancelled out. This expressions defines our condition for a steady state. Re-arranging, we obtain

(4)$$F = { \lambda_{\rm BD} \over 1 + \lambda_{\rm BD} },\; $$

where we have used the substitutions F ≡ (N _O/N _T) (i.e. the occupation fraction) and λ _BD ≡ (λ _B/λ _D). We refer to equation (4) as the SSD equation in what follows. Figure 2 depicts 50 Monte Carlo simulations where we set δt = 0.01, λ _B = 3 and λ _D = 0.2 for N _T = 1000 total seats. Since we have a discrete number of seats and we know the mean number of births/deaths in each interval, then the stochastic realizations are simply drawn from binomial distributions. Using equation (4), we expect the simulation to stabilize at a steady state of N _O = 937.5, marked by the horizontal red line.

Figure 2.

Using the parameters δt = 0.01, λ _B = 3, λ _D = 0.2 for N _T = 1000, we perform 50 Monte Carlo simulations of birth/death actions (wiggly black lines). The horizontal red line represents the steady-state value of N _O, as predicted by equation (4), which matches the simulations. The characteristic time folding time to reach steady state is given by τ. Finally, the blue line shows the predictive growth towards steady state using equation (13).

Before continuing, we highlight some nuances in the above. First, we have only stated that there exists some rate at which ETIs are born, and correspondingly die. The mechanism of these processes is not relevant to the mere act of defining that a pair of such rates must exist. Notably, it could happen via interaction or not; either way one can always still define such quantities. Second, our primary assumption is that a steady-state condition exists. Again, this could occur via interaction models or not, our only assumption is that the population is in a steady state regardless. We will explore what it takes to violate this assumption later in the paper (such as certain interaction behaviours), as well as the consequences. Third, our model also assumes a finite number of seats for potential occupation, but this can readily be interpreted as the number of cubic parsecs, and thus can be defined in a way robust against manipulation. Put together, our reference to a ‘carrying capacity’ refers to the finite number of seats, but we acknowledge that this can elicit confusion as limits to growth exist along other axes too, such as nutrition, resources and energy (which would all feed into the birth rate). Our only claim with respect to ‘carrying capacity’ is that the number of occupied seats cannot exceed the number of available seats, a third assumption of this work.

The optimist's fine-tuning problem

The term λ _BD is unknown to us, and could in fact span many orders of magnitude a priori (Lacki, Reference Lacki2016). Plotting F as a function of λ _BD on a linear-log scale (see Fig. 3), one sees a familiar S-shaped curve characteristic of population growth models with finite carrying capacity (Pearl and Reed, Reference Pearl and Reed1920). The saturation at high λ _BD reflects the finite carrying capacity – increasing λ _BD any further has little effect on F. In contrast, at low λ _BD, we asymptotically approach a lonely universe.

Figure 3.

Left: Occupation fraction of potential ‘seats’ as a function of the birth-to-death rate ratio (λ _B/λ _D), accounting for finite carrying capacity. In the context of communicative ETIs, an occupation fraction of F ~ 1 is apparently incompatible with both Earth's history and our (limited) observations to date. Values of λ _B/λ _D ≪ 1 imply a lonely cosmos, and thus SETI optimists must reside somewhere along the middle of the S-shaped curve. Right: As we expand the bounds on λ _B/λ _D, the case for SETI optimism appears increasingly contrived and becomes a case of fine-tuning.

It is this curve that reveals the basic dilemma facing SETI optimists. We first establish that it would seem most improbable that F ≈ 1. Although our historical SETI surveys are woefully incomplete (Wright et al., Reference Wright, Kanodia and Lubar2018), there presently exists no reproducible, compelling evidence for other technological entities. Everything we observe about the cosmos appears consistent with this unsettling prospect. A counter argument is that we may be the subject of some grand conspiracy, namely the ‘zoo hypothesis’ (Ball, Reference Ball1973), which is potentially compatible with F ≈ 1. However, maintaining a monolithic culture at galactic scales given the finite speed of causality makes such a scenario highly contrived (Forgan, Reference Forgan2017).

Moreover, F ≈ 1 is simply incompatible with Earth's history. Most of Earth's history lacks even multicellular life, let alone a technological civilizationFootnote ⁵ (also see Kipping, Reference Kipping2020). We thus argue that F ≈ 1 can be reasonably dismissed as a viable hypothesis, which correspondingly excludes λ _BD ≫ 1 by equation (4) – although we revisit this point later. We highlight that excluding F ≈ 1 is compatible with placing a ‘Great Filter’ at any position, such as the ‘Rare Earth’ hypothesis (Ward and Brownlee, Reference Ward and Brownlee2000) or some evolutionary ‘Hard Step’ (Carter, Reference Carter2008).

If one concedes that $F\not \approx 1$, which we will take as granted in what follows, then what are the remaining scenarios? One lies somewhere along the steeply ascending S-curve, corresponding to λ _BD ≈ 1. This is what we consider to be the SETI optimist's scenario (given that F ≈ 1 is not allowed). Here, F takes on modest but respectable values, sufficiently large that one might expect success with a SETI survey. For example, modern SETI surveys scan N _T ~ 10³–10⁴ targets (e.g. Enriquez et al., Reference Enriquez, Siemion, Foster, Gajjar, Hellbourg, Hickish, Isaacson, Price, Croft, DeBoer, Lebofsky, MacMahon and Werthimer2017; Maire et al., Reference Maire, Wright, Barrett, Dexter, Dorval, Duenas, Drake, Hultgren, Isaacson, Marcy, Meyer, Ramos, Shirman, Siemion, Stone, Tallis, Tellis, Treffers and Werthimer2019; Price et al., Reference Price, Enriquez, Brzycki, Croft, Czech, DeBoer, DeMarines, Foster, Gajjar, Gizani, Hellbourg, Isaacson, Lacki, Lebofsky, MacMahon, Pater, Siemion, Werthimer, Green, Kaczmarek, Maddalena, Mader, Drew and Worden2020; Ma et al., Reference Ma, Ng, Rizk, Croft, Siemion, Brzycki, Czech, Drew, Gajjar, Hoang, Isaacson, Lebofsky, MacMahon, de Pater, Price, Sheikh and Worden2023; Margot et al., Reference Margot, Li, Pinchuk, Myhrvold, Lesyna, Alcantara, Andrakin, Arunseangroj, Baclet, Belk, Bhadha, Brandis, Carey, Cassar, Chava, Chen, Chen, Cheng, Cimbri, Cloutier, Combitsis, Couvrette, Coy, Davis, Delcayre, Du, Feil, Fu, Gilmore, Grahill-Bland, Iglesias, Juneau, Karapetian, Karfakis, Lambert, Lazbin, Li, Li, Liskij, Lopilato, Lu, Ma, Mathur, Minasyan, Muller, Nasielski, Nguyen, Nicholson, Niemoeller, Ohri, Padhye, Penmetcha, Prakash, Qi, Rindt, Sahu, Scally, Scott, Seddon, Shohet, Sinha, Sinigiani, Song, Stice, Tabucol, Uplisashvili, Vanga, Vazquez, Vetushko, Villa, Vincent, Waasdorp, Wagaman, Wang, Wight, Wong, Yamaguchi, Zhang, Zhao and Lynch2023), so for such a survey to be successful one requires F to exceed the reciprocal of this (i.e. F⩾10⁻⁴), but realistically greatly so (i.e. F ≫ 10⁻⁴) since not every occupied seat will produce the exact technosignature we are searching for, in the precise moments we look, and at the power level we are sensitive to. This arguably places the SETI optimist is a rather narrow corridor of requiring $N_{\rm T}^{-1} \ll \lambda _{\rm BD} \lesssim 1$.

The requirement for such fine-tuning forms the basis of our concern. It might be argued that as we increase N _T, the contrivance decreases and thus the fine-tuning problem dissolves. In a relative sense, we agree that the contrivance certainly diminishes as N _T grows, but in an absolute sense the contrivance may still be extreme nonetheless. The basic problem is that λ _BD has no clear lower limit, and could be plausibly be outrageously small. For example, the probability of spontaneously forming proteins from amino acids has been estimated be ~10⁻⁷⁷ (Axe, Reference Axe2004), to say nothing of the many subsequent steps needed to produce living creatures – let alone technology development.

One might try to argue that our very existence demands $F \geq 1/N_{\star }$, where $N_{\star }$ is the number of stars in the observable Universe. But truly this limit is not justifiable, since the Universe appears to be much larger than the Hubble volume (Planck et al., Reference Planck, Aghanim, Akrami, Ashdown, Aumont, Baccigalupi, Ballardini, Banday, Barreiro, Bartolo, Basak, Battye, Benabed, Bernard, Bersanelli, Bielewicz, Bock, Bond, Borrill, Bouchet, Boulanger, Bucher, Burigana, Butler, Calabrese, Cardoso, Carron, Challinor, Chiang, Chluba, Colombo, Combet, Contreras, Crill, Cuttaia, de Bernardis, de Zotti, Delabrouille, Delouis, Di Valentino, Diego, Doré, Douspis, Ducout, Dupac, Dusini, Efstathiou, Elsner, Enßlin, Eriksen, Fantaye, Farhang, Fergusson, Fernandez-Cobos, Finelli, Forastieri, Frailis, Fraisse, Franceschi, Frolov, Galeotta, Galli, Ganga, Génova-Santos, Gerbino, Ghosh, González-Nuevo, Górski, Gratton, Gruppuso, Gudmundsson, Hamann, Handley, Hansen, Herranz, Hildebrandt, Hivon, Huang, Jaffe, Jones, Karakci, Keihänen, Keskitalo, Kiiveri, Kim, Kisner, Knox, Krachmalnicoff, Kunz, Kurki-Suonio, Lagache, Lamarre, Lasenby, Lattanzi, Lawrence, Le Jeune, Lemos, Lesgourgues, Levrier, Lewis, Liguori, Lilje, Lilley, Lindholm, López-Caniego, Lubin, Ma, Macías-Pérez, Maggio, Maino, Mandolesi, Mangilli, Marcos-Caballero, Maris, Martin, Martinelli, Marténez-González, Matarrese, Mauri, McEwen, Meinhold, Melchiorri, Mennella, Migliaccio, Millea, Mitra, Miville-Deschênes, Molinari, Montier, Morgante, Moss, Natoli, Nørgaard-Nielsen, Pagano, Paoletti, Partridge, Patanchon, Peiris, Perrotta, Pettorino, Piacentini, Polastri, Polenta, Puget, Rachen, Reinecke, Remazeilles, Renzi, Rocha, Rosset, Roudier, Rubiño-Martín, Ruiz-Granados, Salvati, Sandri, Savelainen, Scott, Shellard, Sirignano, Sirri, Spencer, Sunyaev, Suur-Uski, Tauber, Tavagnacco, Tenti, Toffolatti, Tomasi, Trombetti, Valenziano, Valiviita, Van Tent, Vibert, Vielva, Villa, Vittorio, Wandelt, Wehus, White, White, Zacchei and Zonca2020), and thus most Hubble volumes could be devoid of sentience – we necessarily live in the one where it occurred via the weak anthropic principle (Carter, Reference Carter and Longair1974).

Probabilistic steady state

The SSD equation relates the occupation fraction, F, as a function of a single input parameter, λ _BD. However, it does not speak to the probability distribution of F. For that, we need to adopt some a-priori distribution for λ _BD and then evaluate the implied distribution in F-space.

As a scale parameter, let's take the common approach of adopting a log-uniform prior on λ _BD, equivalent to a uniform prior on $\gamma _{\rm BD} = \log \lambda _{\rm BD}$. We bound γ _BD between γ _BD,min and γ _BD,max, such that

(5)$$\hbox{Pr}( \gamma_{\rm BD}) \, {\rm d}\gamma_{\rm BD} = {1\over \gamma_{{\rm BD}, {\rm max}} - \gamma_{{\rm BD}, {\rm min}}}\, {\rm d}\gamma_{\rm BD}.$$

Since F = λ _BD/(λ _BD + 1) (equation (4)), then by re-arrangement we have

(6)$$\lambda_{\rm BD} = {F\over 1-F},\; $$

and thus

(7)$$\gamma_{\rm BD} = \log\left({F\over 1-F}\right).$$

We can calculate the probability distribution of F by noting that

(8)$$\eqalign{\hbox{Pr}( F) \, {\rm d}F & = \hbox{Pr}( \gamma_{\rm BD}) \, {\rm d}\gamma_{\rm BD},\; \cr & = {1\over \gamma_{{\rm BD}, {\rm max}} - \gamma_{{\rm BD}, {\rm min}}}\, {{\rm d}\gamma_{\rm BD}\over {\rm d}F}\, {\rm d}F,\; }$$

where we have used equation (5) on line two. The derivative on the right-hand side is easily calculated from equation (7), giving

(9)$$\eqalign{{{\rm d}\gamma_{\rm BD}\over {\rm d}F} & = {{\rm d}\over {\rm d}F} \log\left({F\over 1-F}\right),\; \cr & = {1\over F( 1-F) }.}$$

Substituting this result into equation (8), we finally obtain

(10)$$\hbox{Pr}( F) \, {\rm d}F = {1\over \log\lambda_{{\rm BD}, {\rm max}} - \log\lambda_{{\rm BD}, {\rm min}}} {1\over F ( 1-F) }\, {\rm d}F,\; $$

which is essentially the Haldane prior. A subtle difference between Haldane's prior is that the original version lacks the normalization terms present here, and is defined over the interval F of [0,  1], whereas here the bounds are F _min and F _max, which can be found by equation (4) to yield

(11)$$\hbox{Pr}( F) \, {\rm d}F = {1\over \log\big({F_{{\rm max}}}/{( 1-F_{{\rm max}}) }\big)- \log\big({F_{{\rm min}}}/{( 1-F_{{\rm min}}) }\big)} {1\over F ( 1-F) }\, {\rm d}F.$$

Note that integrating the above from F _min to F _max equals unity. In other words our version of the Haldane prior is proper, resolving the common criticism of the original Haldane prior.

Haldane = SSD

This derivation resolves the underlying connection between the Drake equation and the Haldane perspective. Starting from the Haldane perspective, which seems intuitionally appropriate for the case of astrobiology as argued earlier, we would conclude that the cosmos is either very lonely or very crowded, whereas as intermediate values appear contrived. Since a crowded cosmos appears incompatible with observations, SETI optimists find themselves in the fine-tuning valley (see Fig. 1). This matches the conclusion found using the SSD equation – given the enormous potential range of the birth-to-death rate ratio, λ _BD, the cosmos is likely nearly empty or fully occupied, with intermediate states requiring fine-tuning. We have shown why these two appear to show the same thing, they are in fact equivalent under the assumption that λ _B/λ _D is a-priori log-uniformly distributed – the generic uninformative prior of a scale parameter. In the Appendix, we show that one can also engineer a Jaynes prior to manifest, instead of the Haldane, by adopting a uniform prior in $-2\sin ^{-1}( 1 + \lambda _{\rm BD}) ^{-1/2}$ (instead of $\log \lambda _{\rm BD}$). However, we argue there is no clear justification for favouring this over a log-uniform prior.

Conditions for a steady state

To arrive at this result, we have assumed that the current population of extant ETIs exists in a steady state. We here investigate under what conditions this assumption is appropriate. If we initialize at some time t = 0, the number of occupied seats will clearly not be in such a state and will require some time to grow and reach equilibrium. The relationship between the evolution of the occupied states is given by

(12)$$\eqalign{{{\rm d} N_{\rm O}\over {\rm d}t} & = -{d{\rm }N_{\rm A}\over {\rm d}t},\; \cr & = \lambda_{\rm B} N_{\rm A} - \lambda_{\rm D} N_{\rm O},\; \cr & = \lambda_{\rm B} N_{\rm T} - ( \lambda_{\rm B} + \lambda_{\rm D} ) N_{\rm O},\; }$$

subject to the initial condition N _O(0) = 0. Solving this, it can be shown that this growth follows

(13)$$N_{\rm O}( t) = \left({N_{\rm T} \lambda_{\rm BD}\over 1 + \lambda_{\rm BD}}\right)\Bigg(1 - \exp\big(-( \lambda_{\rm B} + \lambda_{\rm D}) t\big)\Bigg).$$

Accordingly, the timescale for settling into the steady state is several τ ≡ (λ _B + λ _D)⁻¹. Recall that $\lambda _{\rm D}^{-1}$ corresponds to the mean lifetime of occupation, and one might reasonably argue that this must be much smaller than the age of the Universe, $H_0^{-1}$, hence $\lambda _{\rm D}^{-1} \ll H_0^{-1}$. If $\lambda _{\rm D}^{-1} \ll \lambda _{\rm B}^{-1}$, then this argument guarantees that the steady-state condition has been satisfied already.

In reality, λ _B and λ _D will also vary over cosmic history due to the changing environment of the Universe itself, such metallicity enrichment and evolving star formation rates. However, if the characteristic timescale over which this cosmic evolution occurs is much greater than the equilibrium timescale (τ), then the number of occupied seats will remain in a steady state, albeit one for which the equilibrium point slowly drifts.

Paths to optimism within the SSD framework?

First, we currently have no lower bound on λ _BD and thus the fact it can be arbitrarily (even outrageously) small underpins the entire fine-tuning argument made here. As shown earlier, the SETI optimist requires that $N_{\rm T}^{-1}\ll \lambda _{\rm BD}\lesssim 1$, but current knowledge is fully compatible with $\lambda _{{\rm BD}, {\rm min}} \lll N_{\rm T}^{-1}$ (Lacki, Reference Lacki2016) – hence the conclusion that SETI optimists live in a narrow corridor. Accordingly, one path to salvage hope would be to place a firm lower limit on λ _BD that is comparable to, or greater than $N_{\rm T}^{-1}$ i.e. we require $\lambda _{{\rm BD}, {\rm min}} \gtrsim N_{\rm T}^{-1}$. In practice, it's difficult to imagine how we could ever place a constraint on this parameter besides a successful SETI detection – even null detections have little power here since there are innumerable ways in which a ETI could be missed (Kipping and Wright, Reference Kipping and Wright2024).

Of course, we are always free to increase N _T i.e. perform ever larger SETI surveys. This will increase the relative odds of success in direct proportion to N _T. However, without a lower bound on λ _BD,min this doesn't, in general, lead to optimism in an absolute sense. For example, consider that λ _BD = 10⁻³⁰. In this case, a case perfectly compatible with everything we know about our cosmos, increasing N _T from say a value of 10⁴ typical of modern SETI surveys (e.g. Enriquez et al., Reference Enriquez, Siemion, Foster, Gajjar, Hellbourg, Hickish, Isaacson, Price, Croft, DeBoer, Lebofsky, MacMahon and Werthimer2017; Maire et al., Reference Maire, Wright, Barrett, Dexter, Dorval, Duenas, Drake, Hultgren, Isaacson, Marcy, Meyer, Ramos, Shirman, Siemion, Stone, Tallis, Tellis, Treffers and Werthimer2019; Price et al., Reference Price, Enriquez, Brzycki, Croft, Czech, DeBoer, DeMarines, Foster, Gajjar, Gizani, Hellbourg, Isaacson, Lacki, Lebofsky, MacMahon, Pater, Siemion, Werthimer, Green, Kaczmarek, Maddalena, Mader, Drew and Worden2020; Ma et al., Reference Ma, Ng, Rizk, Croft, Siemion, Brzycki, Czech, Drew, Gajjar, Hoang, Isaacson, Lebofsky, MacMahon, de Pater, Price, Sheikh and Worden2023; Margot et al., Reference Margot, Li, Pinchuk, Myhrvold, Lesyna, Alcantara, Andrakin, Arunseangroj, Baclet, Belk, Bhadha, Brandis, Carey, Cassar, Chava, Chen, Chen, Cheng, Cimbri, Cloutier, Combitsis, Couvrette, Coy, Davis, Delcayre, Du, Feil, Fu, Gilmore, Grahill-Bland, Iglesias, Juneau, Karapetian, Karfakis, Lambert, Lazbin, Li, Li, Liskij, Lopilato, Lu, Ma, Mathur, Minasyan, Muller, Nasielski, Nguyen, Nicholson, Niemoeller, Ohri, Padhye, Penmetcha, Prakash, Qi, Rindt, Sahu, Scally, Scott, Seddon, Shohet, Sinha, Sinigiani, Song, Stice, Tabucol, Uplisashvili, Vanga, Vazquez, Vetushko, Villa, Vincent, Waasdorp, Wagaman, Wang, Wight, Wong, Yamaguchi, Zhang, Zhao and Lynch2023) to every star in the galaxy (10¹¹) increases the relative probability of success by 10 million. However, despite this increase, the absolute probability of success is still minuscule, just 10⁻¹⁹. In contrast, the suppose λ _BD = 10⁻¹⁰ instead, which again is equally plausibly, then the expectation value becomes 10 technospheres amongst the ensemble. Of course, these are just examples and a rigorous quantification would require knowledge of shapes (e.g. Jeffreys (Reference Jeffreys1946), Lacki (Reference Lacki2016), etc.) and bounding limits (e.g. see Spiegel and Turner, Reference Spiegel and Turner2012) of a prior on λ _BD, over which one could then marginalize to compute expectation values. A core argument of this work is that there is no sensible lower limit to λ _BD with present information, and thus it extends down into the abyss of extremely and arbitrarily small values.

The SSD equation only involves these two parameters, λ _BD and N _T, yet neither provides the leverage to dismiss the fine-tuning argument in a practical sense. Under the assumptions of this work, efforts to salvage optimism must thus turn to violating either (1) the SSD itself (2) the assumption that $F\not \approx 1$. We believe these are only the only two pathways to retaining plausible optimism for SETI.

Violating the SSD

Consider (1) first – violating the SSD. As explored earlier, the steady-state condition is not guaranteed and strictly assumes that the elapsed time within the window of cosmic habitability exceeds several τ ≡ (λ _B + λ _D)⁻¹. If this is false, equilibrium has not yet been achieved and the number of occupied seats is presumably in ascent. A decline is also possible, but this assumes the conditions of the cosmos are more hostile to life now than previously, which is difficult to justify. An ascent would be compatible with some roaming ETI colonizing the galaxy (Bracewell, Reference Bracewell1960; Freitas, Reference Freitas1980; Tipler, Reference Tipler1980), thus themselves engineering λ _B exponentially upwards over time. However, we note that the settlement front is expected to fill the Galaxy $\lesssim$ Gyr even for ‘slow’ probes (30 km s⁻¹), aided by the motion of the stars themselves (Carroll-Nellenback et al., Reference Carroll-Nellenback, Frank, Wright and Scharf2019). Accordingly, there is arguably another fine-tuning problem that we find ourselves living during the mid-point of this relatively rapid transition – too early and there's no-one to talk to, too late and we shouldn't be here to think about it. A similar argument can be made for any process in which λ _B undergoes rapid growth (e.g. directed panspermia; Crick and Orgel, Reference Crick and Orgel1973). If, instead, slow growth is favoured, then we fall back into the steady-state condition.

Allowing for F ≈ 1

An alternative solution is (2) – violating the assumption that $F\not \approx 1$. Such a position is challenged by our observations of the local universe and why it was treated as forbidden in much of this work. However, a possible way around this is to invoke the ‘Grabby Aliens’ hypothesis (Hanson et al., Reference Hanson, Martin, McCarter and Paulson2021) and the weak anthropic principle (Carter, Reference Carter and Longair1974). Here, one might imagine that ETIs emerge rarely, but when they do, they often proceed to colonize their surrounding region in short order. In such a universe, most regions are filled and thus F ≈ 1. The fact that we don't see F ≈ 1 locally is because humanity must necessarily have emerged in a volume of space where this wave has not yet reached, via the weak anthropic principle. We note that this solution falls into tension with the cosmological principle (Keel, Reference Keel2002). Such a scenario lends itself to inverting the normal view of SETI – rather than looking locally, we should be looking at regions greatly separated from us. Such a hypothesis has the advantage that it is, in principle, verifiable via extragalactic SETI (Griffith et al., Reference Griffith, Wright, Maldonado, Povich, Sigurdsson and Mullan2015; Zackrisson et al., Reference Zackrisson, Calissendorff, Asadi and Nyholm2015).

There are other ways to allow for F ≈ 1, such as invoking a universal technological ceiling. This is not a ‘Great Filter’, since that speaks to the death rate. Instead, we have here a universal limit to technological development for all ETIs, and that limit is coincidentally not far off humanity's current level of advancement. In this picture, the fact that we see no evidence of technosignatures littering the sky is because ETIs never develop to a point where their footprint would be noticeable. In our view, this is a strong, contrived and unfounded assumption to assert. Similarly, co-ordinated ETI behaviour to avoid contact (the ‘zoo hypothesis’; Ball, Reference Ball1973) is challenged by the finite speed of light, the sheer scale of the Galaxy and likely heterogeneity of emergent behaviours (Forgan, Reference Forgan2017). Similar challenges exist for ‘Dark Forest’ arguments (Yu, Reference Yu2015), with the added issue as to why Earth was sterilized at some point in its long 4.5 Gyr history already.

In summary, the case of F ≈ 1 may be in tension with our observations of the local universe when considering SETI, but it is perfectly consistent with our observations when considering life more generally, especially if we treat the seats as star systems or Earth-like planets. By the SSD equation, we would here argue for precisely the Haldane perspective – a cosmos either teeming with life or almost devoid of it, both of which are compatible with current observations. We therefore emphasize that our argument for pessimism does not extend to life more broadly.