Downloaded by guest on September 29, 2021 www.pnas.org/cgi/doi/10.1073/pnas.1808578115 a Grimaldi Claudio SETI to approach Bayesian SETI be to yet Earth, the cross power discovered. radiated comparable than more of typically signals that certainly almost requiring implies detectors, endeavor signal cur- Arecibo-like an one of kly, even within those such 40 finding than Conversely, no about higher telescopes. detect rent times of surveys thousands radius sensitivities all-sky a detector if to only up can out existence emitters signals ruled Arecibo-like The Way. detectable reasonably Milky potentially be of the Arecibo Galaxy in the the farther of in from that to coming analogous but power radar, radiated with Earth, over probability ing a typically with that from consistent 10% the kly still best exceeding 1 at nonetheless or about is emitters thereof, within galactic of scarcity signals lack the detecting pri- suggesting not while noninformative Earth, that fairly find signals Under Galaxy. we of the ors, detection of the crossing surveys signals or future radio in nondetection of either number assuming mean the Earth, of infer population sta- to galactic Bayesian model from a tistical the formulate Starting about we signals, ignorance thereof. electromagnetic nonnatural of detectable lack state of the current potentially abundance the is or future, the space foreseeable civilizations about search the extraterrestrial vast data in For the informative activity. sampled of bringing intense be fraction of significant to phase a expected new time, a first civi- through 2018) the galactic 20, going May hypothetical review is from for (received technosignatures lizations 2018 31, for August search approved and The NJ, Princeton, University, Princeton Bahcall, A. Neta by Edited 94720 CA Berkeley, California, of University opeesv erhee,adtepand“rdeo Life” of “Cradle planned most the and and largest the ever, 4), (2) search (1, initiatives. comprehensive Galaxy project SETI Listen” the of “Breakthrough revival in significant The a planets stimulated Earth-like recently inferred have the of and number (3) planets astronomical extrasolar of thousands of discovery to 0 from (2). ranging planets signals radio of being state number 10 possess- Earth total current stars the a of with our by consistent 0.01% illuminated of indeed to is 0.1% picture emitters of detectable vivid limit ing a upper result An gives this ignorance. 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C.G. contributions: Author fsgasppltn h Galaxy. entire number the total the populating larger from even signals an signals of with of Earth, the hundreds cross typically that Galaxy 100% almost (i.e., of neighborhood galactic ability the of from distance detec- signal a single within the than a contrast, powerful even In more of assumed. as emitters tion are such if radars telescopes, (SKA2), planetary SKA radio the the terrestrial future of on with 2 limit reached phase Galaxy. distance upper the be entire strong a the can a from limit to emissions places This out EM Earth detectable signals from of knowledge. detecting kly occurrence of 40 not level about that limited of show very results present the The the given entire of the Way, from probabilities emitted Milky Earth posterior crossing sensitivities the signals of the calculate number and average we account emitters detectors, into the the taking of of By distribution processes. luminosity emission the and the distributions, of of age longevity populations and the con- possible spatial We their considers survey. emitters, that SETI extraterrestrial model hypothetical signal statistical volume a a a the by of struct of sampled function Galaxy a detection as the range the of frequency radio or given nondetection a within either assuming by extraterrestrial Way. nonnatural, Milky of the in population extent, signals some galactic to wavelengths least possible at of sufficiently infer, a the to range providing data informative years, broad of following amount a the large in for expected Galaxy be to the has of fraction space. icant systematic search SETI a the technol- for of domain detector opportunities vast in unprecedented the in offer progress investigation 8), impressive (7, telescope with radio ogy together (SKA) Array 6), Kilometre (5, Square the of program owo orsodnemyb drse.Eal lui.rmlieflc rgeoff. or claudio.grimaldi@epfl.ch Email: addressed. [email protected] be may correspondence whom To 0 inl yial rs u lntfo h ik Way. Milky the from planet our cross typically of signals discovery detectable 100 The power. within of emission radiated Galaxy single of a entire range even wide the a in of absence emitters the show with We within patible emissions. signals detecting radio not nonnatural that outcome for null searches a of or all-sky number positive of a mean given the Earth, of of crossing probability signals formulation radio posterior Bayesian the a unprece- infer to construct civilizations an SETI we communicating on Here, of Earth. Galaxy evidence beyond find the to explore scale dented will (SETI) extraterrestrial for search intelligence the in initiatives future and Ongoing Significance ee erpr h eut faBysa nlssformulated analysis Bayesian a of results the report we Here, signif- a of exploration developments, rapid these of view In NSlicense.y PNAS ∼ 1 l rmErh it oapseirprob- posterior a to hints Earth) from kly PNAS . y ∼ l mle nta htover that instead implies kly 1 | ∼ o.115 vol. b 0kyfo at scom- is Earth from kly 40 eateto Astronomy, of Department www.pnas.org/lookup/suppl/doi:10. | o 42 no. | E9755–E9764

ASTRONOMY While our analysis focuses here on radio signals, the formalism can be extended to consider other wavelengths, like the optical and near infrared spectra searched by some SETI initiatives (9– 12). In the case of short wavelengths, however, absorption and scattering processes have to be considered, as briefly discussed in the concluding section. The Model In modeling possible galactic populations of nonnatural extrater- restrial signals, we start by considering a hypothetical techno- logical, communicating civilization (or emitter) located at some position vector ~r relative to the galactic center. We assume that at some time t in the past, the emitter started transmitting, either deliberately or not, an isotropic EM signal and that the emission process lasted a time interval denoted L. At the present time, the region of space occupied by the EM radiation is a spherical shell centered at ~r, with outer radius R = ct and thickness ∆ = cL, where c is the speed of light. A necessary condition for the detection of this signal is that, at the time of observation, the position vector of the Earth, ~ro , points to a location within the region occupied by the spherical Fig. 1. 2D schematic representation of a spherical shell signal of outer shell, which corresponds to requiring that (13, 14) radius R and thickness ∆. The spherical shell is centered at the emitter loca- tion identified by the position vector ~r relative to the galactic center, while the Earth (red circle) is located at ~ . In the figure, the Earth lies outside R − ∆ ≤ |~r −~ro | ≤ R, [1] ro the region covered by the shell, preventing the detection of the signal. The spherical shell signal intercepts the Earth only if the distance emitter–Earth, where |~r −~ro | is the distance of the emitter from the Earth |~r −~ro|, satisfies Eq. 1. (Fig. 1). The first inequality of Eq. 1 represents the condition that the last emitted signal of a spherical shell crosses Earth (15). Since the farthest possible position of a galactic emitter emission process started within a time tM + L before present) is is at the opposite edge of the galactic disk, its maximum con- therefore (13) ceivable distance from the Earth, RM ≈ 87 kly, is simply the ∗ Z Z tM +L Z sum of the galactic radius (≈ 60 kly ) and the distance of the dL ρL(L) dt ρt (t) d~rρs (~r)fR,∆(~r −~ro ) Earth from the galactic center (|~ro | ≈ 27 kly). Hence, any EM p = q 0 , [2] signal emitted before tM = RM /c ≈ 87, 000 y from present has Z Z tM+L already covered a distance larger than RM and is therefore Ns dL ρL(L) dt ρt (t) 0 absolutely undetectable at Earth. Since |~r −~ro | < RM , it follows also that the region filled by a spherical shell with outer radius where fR,∆(~r −~ro ) = 1 if Eq. 1 is satisfied and fR,∆(~r −~ro ) = 0 larger than RM + ∆ cannot contain our planet, regardless of the otherwise, and ρs (~r) is the star number density function. For the position of the emitter in the Milky Way. In temporal terms, moment, we do not need to specify its detailed form and require this means that any emission process lasting L years and that R only that d~rρs (~r) = Ns and that ρs (~r) has approximately a started at a time earlier than tM + L is unobservable and can be disk-like shape with a radius of about 60 kly. ignored. Assuming the steady-state condition that over a time span of order tM from present, the PDF of the starting time of emis- Probability of Shells at Earth. To calculate the probability of sig- sion, ρt (t), is essentially constant (13), the integrals over t in Eq. nals crossing Earth, we must consider all possible configurations 2 can be solved exactly and p reduces simply to (SI Appendix, of the spherical shells (their number, position, outer radius, and section I): 1 thickness) and identify those that satisfy Eq. . To this end, L¯ we first discard any unobservable signal by assigning to a ran- p = qλ ≡ q ¯ , [3] domly chosen star a probability q of harboring an emitter that L + tM has been actively transmitting some time within tM years from where the second equality defines the scaled longevity of the sig- ¯ R present. The mean number of such emitters is thus qNs , where nal, λ, and L = dLρL(L)L denotes the average duration of the Ns is the number of stars in the Galaxy. We make the additional signal. Finally, from Eq. 3 we obtain the mean number of signals assumption that the starting time and the duration of the emis- crossing Earth, ¯ sions (or, equivalently, the outer radius and the thickness of the k = qλNs , [4] spherical shells) are independent and identically distributed ran- which has to be understood as a statistical average over all dom variables, t and L, with probability density functions (PDFs) configurations of the emitted signals.† given by ρt (t) and ρL(L), respectively, and that the signal fre- k¯ is the quantity of main interest here for two reasons. First, quencies are distributed uniformly within a given range. The Eq. 4 expresses the two unknown quantities λ and q in terms resulting probability p that the Earth intersects a signal under of a single parameter, k¯, which, as shown in the following, can the condition that it is no older than tM (or, equivalently, that the be in principle inferred by observations. As emphasized in Fig. 2, knowledge of k¯, or at least plausible upper or lower bounds, would also enable via Eq. 4 an estimate of the mean number qNs *Here we adopt the presumption that stars that can potentially harbor emitters are rich of shell signals occupying the Galaxy as a function of the mean in heavy elements, so as to favor the formation of rocky planets. The galactic radius used here (60 kly) is approximately the radius of the thin disk (see Bayesian Analysis Applied to Existing and Upcoming SETI Detectors), which is a metal-rich component of the Milky Way. The contribution of farther stars (16, 17) and/or other galactic components (18) can †For example, values of ¯k smaller or much smaller than ∼ 1 imply that configurations nonetheless be incorporated by the present formalism. with signals crossing Earth are rare or very rare.

E9756 | www.pnas.org/cgi/doi/10.1073/pnas.1808578115 Grimaldi and Marcy Downloaded by guest on September 29, 2021 Downloaded by guest on September 29, 2021 rmliadMarcy and Grimaldi Galaxy, the in present 2. Fig. etlydtcal sln sisdsac rmteErhis Earth the from instru- is distance emitter its the the as to source, long that such proportional the as from inversely detectable distance is mentally the detector tele- of the radio square denoted the by flux (i.e., received detectable detector flux minimum the a range has frequency scope) same luminos- the intrinsic at has in frequencies emitter radio an of sensitiv- that ity range detector assume given the we a by this, within and by see emitter specified To the the sur- radius case, ity. of of all-sky this Earth power at In an radiated centered sky. selected, the sphere the a is of is directions stars space all search target in principle in searches, of covers targeted set vey to within discrete signals Contrary a radio frequencies. for which of sky range entire given the scan a to SETI a designed which is in case search the here consider we search signals, extraterrestrial the and detectors, the sig- of the of sensitivity strategy. wavelength detec- minimum the their the power, distance signals, radiated the nals, their as the such emitters, by factors the of covered number of a region on depends a actually in tion that lies hypothesis the planet in tele- Even our dedicated surveys. observational terrestrial, in of to used means scopes referring by without detectability signals actual isotropic their of populations galactic ble Detectability. in Signal emitters active presently informed of An Galaxy. abundance the 20). potential (19, the equation about Drake of fac- the estimate probability in different appearing the tors incorporates effectively particular, emitters, In with I). coincide section to Appendix, (SI are signals that emitters of number Earth. average the cross not do signals the of majority vast as large qN inllneiy o example, For longevity. signal planet. our cross possibly could that through Earth, oilsrt o hs atr nunetedtcaiiyof detectability the influence factors these how illustrate To hypothesis, steady-state the under Second, s L o given for ntt ecnue with confused be to (not eainbtenteaeaenme fshrclselsignals shell spherical of number average the between Relation ∼ 1000 ¯ L qN k ¯ sdtrie by determined is s if ol rn hrfr aubeknowledge valuable a therefore bring would = L/τ ¯ L ¯ k ¯ qN ∼ ndrvn Eq. deriving In ( ¯ L L + 1) where (14), s 100 n h ensga longevity signal mean the and , ≥ t M 4π )/ sasmd lhuhi hscs the case this in although assumed, is y ¯ where L, k ¯ | ~ r h vrg ubro inl crossing signals of number average the , − k ¯ h inllneiy n that and longevity) signal the L, ∼ ~ r 0 τ 1 | t ehv osdrdpossi- considered have we 4, −1 2 M currently S mle that implies min steaeo h letsignal oldest the of age the is h vrg itrt of birthrate average the , . k ¯ aitn isotropic radiating onie ihthe with coincides ~ k ¯ r S qN min httransmits that a eshown be can ¯ h au of value The L. s ic the Since . a eas be can [5] rmwihw eoe Eq. recover we which from ucin orsodn opsil ucmso EIsearch. SETI a following of Eq. outcomes the possible use In to corresponding will model. functions we our of analysis, definition Bayesian the completes which an Eq. π of from instead significantly differ plane, to galactic expected the not a is around survey, sky from all-sky the resulting of probability survey take detection SETI we luminosity since the that disk-like, Note Earth. intersects shell emitted the the if even undetectable, instrumentally is Earth on centered sphere radius a of outside and value, is that maximum emitter an strictly that a implies not than larger Although Eq. nosities probability. in assumed have detection we necessary, luminosity the is the therefore, replace, We Eq. process. in given emission probability detection the (com- of luminosity function), duration the luminosity of denoted PDF monly a introducing lumi- by intrinsic probabilistically emitter, the the detector, of the nosity of characteristics instrumental Earth. from emitter an make to enough to reduces signal single a of probability detection luminosity t intrinsic with emitter an L which beyond distance the is inl eue oabnma distribution: binomial the a to in reduces involved signals exactly telescope detects a survey that all-sky probability the function, nosity h vrg ubro inl htcnb eetdb h survey the by detected therefore be can is that signals of number average The ubro inl tErhgvni Eq. in given of Earth value at the signals of number therefore Eq. (2), as billions long of as tens of order the 10 in is telescope space Kepler x 2 iin n utb iutnosyflle.Teresulting The multiply fulfilled. to simultaneously amounts be probability must con- 5 that detection therefore and requires 1 signal isotropic ditions an of detection The o < nEq. in sisrmnal neetbe fe h nerl over integrals the After undetectable. instrumentally is by Given Although (R a ecneinl prxmtdb oso distribution Poisson a by approximated conveniently be can π and 0, o L θ ∗ (R (R n suigta h mteshv h aelumi- same the have emitters the that assuming and ), L 2 L ∗ p k | − r efre ne h taysaecniin the condition, steady-state the under performed are = ) 0 and = P N S p ~ r min q s 0 N k N p − 1 λπ = =0 s nerdfo h nlsso h aafo the from data the of analysis the from inferred π (k s R ~ r o Z N sakonprmtrta eed nthe on depends that parameter known a is o o |π kp k L ¯ q p 0 where , | s where ), L o igrthan bigger r uhsalrthan smaller much are (k (k λ ∗ sa nnw uniy hc etreat we which quantity, unknown an is L, = ) d | Z R | π R π Lg L o

o d L ∗ = ) = ) (L) ~ r = 3 N = k ρ s ycosn ausof values choosing by π r s k r θ ! (π p Z o ( (x ~ ,1 2 1, 0, = r 0 p 4π N o 4π k )θ d R sasothn oainfor notation short-hand a is PNAS 0 7 1 = ) k ¯ ! k s ~ r M 8 S (R L ) L (1 S by = ρ k 11 min ∗ that min h aiu itneof distance maximum the , s ρ e L − ( π g p s ~ −π r | | − o (r) odrv h likelihood the derive to needn fthe of independent (L), if 0 )θ p k ¯ = L o.115 vol. g . . . o 0 ial,ntn that noting Finally, 4. where , ~ x r ∗ (R ) (L) k ¯ q N ob approximately be to nti a,Eq. way, this In . , N − ≥ f , λπ s R L −k s N ~ r 0 aihsfrlumi- for vanishes ,∆ | − ≈ o o s . |), (R and ( | ~ 10 ~ r r peia shell spherical k ¯ o 42 no. − L − L/S 10 ∗ stemean the is θ ~ where ), ~ r r ewrite We . o (x o min | ) ) 0 = ) | nEq. in L large E9757 [11] [10] and 8. [7] [6] [9] [8] if 8

ASTRONOMY Bayesian Analysis Galaxy in which a SETI survey is successful, the larger is the likely Bayes’ theorem provides a recipe for updating an initial hypoth- number of broadcasting emitters in the Milky Way. esis about the probability of occurrence of an event in response ¯ to new evidence (21). Here, we take the initial hypothesis that Prior Distribution. To obtain the posterior probability p(k|E), ¯ the Earth intersects with a prior probability distribution p(k¯) an E = E0, E1, and E 0, we need to specify p(k), the prior prob- average number k¯ ≥ 0 of signals emitted from communicating ability distribution of k¯. Presently, we lack generally accepted civilizations in the Galaxy, regardless of whether we detect them arguments to estimating either the fraction q of stars in the or not. Galaxy that may harbor communicating civilizations or the mean Let us suppose that new evidence on the number of detected signal longevity L¯. Possible values of k¯ may therefore range signals (evidence E) emerges from the acquisition of new data in from k¯ = 0, as argued by proponents of the rare Earth hypoth- a SETI survey. Bayes’ theorem states that the posterior proba- esis (22), to a significant fraction of Ns , in the most optimistic bility that there are in average k¯ signals intercepting our planet scenarios. taking into account the evidence E is A natural choice of p(k¯), befitting our ignorance about even the scale of k¯, would be taking a prior PDF that is uniform in p(E|k¯)p(k¯) log(k¯), which corresponds to p(k¯) ∝ k¯−1, to give equal weight p(k¯|E) = , [12] ‡ p(E) to all orders of magnitude (21, 23). Although the log-uniform prior appropriately expresses our state of ignorance, it fails where p(E) = R dkp¯ (E|k¯)p(k¯), the marginal likelihood of E, is to take into account that, after all, various past SETI sur- a normalization constant and p(E|k¯) is the likelihood function veys have been conducted for several decades (24), with null defined as the conditional probability that the event E occurs results. Likewise, there have been no serendipitous detection of given the initial hypothesis about k¯. nonnatural extraterrestrial signals since the invention of radio telescopes. Likelihood Terms. For the sake of simplicity, we shall not dis- To allow the prior PDF to reflect the so-far lack of detection, cuss here the occurrence of false-positive or false-negative results we introduce a prior luminosity detection probability defined as prior prior prior from an all-sky SETI survey, and we consider the only two possi- πo = πo (RL∗ ), where RL∗ , the prior observational radius, ble outcomes—that is, a negative result for signal detection or a is representative of the distance accessible by past SETI surveys positive evidence for the existence of communicating civilizations for given values of L∗ and frequency range. The likelihood of represented by the detection of one signal. nondetection, Eq. 13, immediately suggests that a natural way In the first case, we assume that an observational campaign as to inform the prior about past SETI negative results is to update ¯ ¯ ¯−1 prior ¯ the one described in The Model has detected no signals within p(k) using Bayes’ theorem, leading to p(k) ∝ k exp(−πo k). the entire sky (and within a depth set by RL∗ ). Let E0 denote In so doing, we are simply adopting as prior the posterior this evidence. The corresponding likelihood function, p(E0|k¯), is PDF resulting from the nondetection of signals of past SETI obtained by setting k = 0 in Eq. 11: initiatives. Finally, to make the prior distribution proper (i.e., normalizable), we introduce a lower cutoff in k¯ so that p(k¯) = ¯
−πo k ¯ ¯ p E0|k¯ = e . [13] 0 when k < kmin. The normalized prior distribution becomes therefore Quite intuitively, Eq. 13 shows that for πo 6= 0 the probability of prior ¯ k¯−1e−πo k E0 occurring decays exponentially with k¯, implying that values of ¯ ¯ ¯ p(k) = prior θ(k − kmin), [16] k¯ much larger than 1/πo can be ruled out by the nondetection E1(πo k¯min) of signals. R ∞ −t In considering the case that an all-sky SETI survey detects where E1(x) = x dt e /t is the exponential integral. The value ¯ a nonnatural, extraterrestrial signal, we need to distinguish of kmin can be chosen so as to satisfy some appropriate criterion. ¯ between two possibilities depending on whether the gathered evi- Here we adopt the requirement that kmin gives the least infor- dence can exclude or not the existence of detectable signals from mative prior probability that at least one signal from the entire ¯ prior other emitters within RL∗ (besides the one already detected). Galaxy intercepts Earth’s orbit, leading to kmin ' 0.14πo (SI This distinction has to be made because the detection may occur Appendix, section II). before the sky has been entirely swept out, not excluding there- fore the possibility that there may be other detectable signals Posterior Probabilities. The normalized product of the prior prob- from emitters within RL∗ . In this case, the evidence, denoted ability distribution 16 with each of the three likelihood func- E 0, is that there is at least one detectable signal emitted within a tions 13, 14, and 15 gives the respective PDFs resulting from distance RL∗ . The associated likelihood is p(E 0|k¯) = 1 − p(E0|k¯) the events E0 (nondetection), E 0 (at least one detection), and because E 0 is the negation of E0, the event of nondetection E1 (exactly one detection). To keep the analysis as general considered above. Hence, as possible, for the moment we treat the luminosity detection probability, πo , as an independent variable ranging between 0 ¯
−πo k p E 0|k¯ = 1 − e . [14] and 1. We shall restore its full dependence upon the observa- tional radius RL∗ in the section dedicated to the discussion of The second possibility is that the evidence, denoted E1, amounts present and future SETI surveys. For illustrative purposes, we prior −5 to detect exactly one emitter in the entire sky within a depth RL∗ , also take πo constant and equal to 10 , a value not far as it would be the case if no other signals have been detected from our subsequent estimates of the prior luminosity detection upon the completion of the survey. The likelihood term in this probability. case is therefore given by Eq. 11 with k = 1: Fig. 3 compares the PDFs of k¯ (solid lines), calculated for dif- ferent values of πo , with the prior distribution (dashed lines). ¯
−πo k p E1|k¯ = πo ke¯ . [15] There are three relevant features worth being stressed. First, all

Since the likelihoods 14 and 15 are significant when, respec- ¯ ¯ ¯ tively, πo k & 1 and πo k ∼ 1, large values of k have to be expected ‡At first sight, a prior PDF uniform in ¯k appears to reflect our state of ignorance. It is, when πo is small. In other terms, the smaller the fraction of the however, a highly informative prior because it strongly favors large values of ¯k (23).

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| 0 , 1/π o 1/π E n,det h uofat cutoff the to due and, E 0 o ABC ABC 1 opeetr uuaiedsrbto ucin CDs of (CCDFs) functions distribution cumulative Complementary rbblt itiuinfntos(Ds ftema ubro hl inl rsigEarth, crossing signals shell of number mean the of (PDFs) functions distribution Probability and , o .In ). and satcptdb h ieiodtr 3 substan- A 13. term likelihood the by anticipated as , π o h otro CFbcmsporsieysalrta h ro as prior the than smaller progressively becomes CCDF posterior the A, k ¯ osga eeto (event detection signal no (A) from resulting PDFs posterior and prior the respectively, represent, lines solid and dashed The . < E E 1 k 1/π 1 ¯ n o h ro,cnrigta h latter the that confirming prior, the not and ) fcnetaigtewih of weight the concentrating of , ∼ k ¯ o 1/π hra in whereas , −1 p R ( k ¯ π L ftepir osqety for Consequently, prior. the of o | o ∗ E E .Teeoe ncs fdetec- of case in Therefore, 3C). (Fig. ,telre stepoaiiyta the that probability the is larger the ), → 0 0 ,ad(C and ), ) 0 and π π o C n eitsms rmi when it from most deviates and o π k o,euvlnl,tesalrthe smaller the equivalently, (or, h egto h otro D scnetae otyaround mostly concentrated is PDF posterior the of weight the ¯ p infiatylre than larger significantly o π π xcl n eetbesga (event signal detectable one exactly ) > ( → k ¯ 1/π |E π nti ii,tecorre- the limit, this In 0. p o 1 ( ) o iiihs hshsthe has This diminishes. k ¯ r prxmtl con- approximately are | ntelklho term likelihood the in E 0 π ) o | ssalr For smaller. is to 3C) (Fig. p B ( k ¯ and |E k ¯ 1 k ¯ π ) . which , E π do C) nthe in 0 π 1/π tlatoedtcal inl(event signal detectable one least at (B) ), o prior o k ¯ k ¯ = o ifrn auso h uioiydtcinprobability detection luminosity the of values different for > o 10 , . −3 E 1 .In ). h otro rbblt htteeaemr hn10sgasintercepting signals 100 than more are there that probability posterior the , nieGlx,i agrthan the larger the from is that transmitted Galaxy, probabilities Earth, entire intersecting updated shells the of give number which mean CCDFs, the late at nescsmn hl inl te hnteoealready one the than other signals shell detected. many intersects Earth h CF r ie,rsetvl,by respectively, given, are CCDFs the fFg .TepirCD dse ie)i bandb setting by obtained is lines) (dashed CCDF prior The 3. Fig. of Eqs. shows 4 Fig. P P P yitgaigtepseirPF from PDFs posterior the integrating By A ( π ( ( k ¯ k k ¯ ¯ and o | |E |E nrae,adi aihsepnnilyfor exponentially vanishes it and increases, E 0 0 1 h otro Dsaesalrta h ro hn respectively, when, prior the than smaller are PDFs posterior the B, = ) = ) = ) π k ¯ = h ahdadsldlnsrpeet epciey h prior the respectively, represent, lines solid and dashed The . Z Z Z E k k k ¯ ¯ ¯ 1 ∞ ∞ ∞ (π E d d d 1 o prior k ¯ k k k ¯ ¯ ¯ 17–19 (π ∼ 0 0 0 k ¯ p p p o prior 1/π o ifrn auso h uioiydetection luminosity the of values different for , ( ( ( k | ¯ k k k ¯ ¯ ¯ min 0 0 0 o |E | |E k sldlns o h aevle of values same the for lines) (solid ¯ E . ) ) 0 1 0 − − = ) = ) ) E 0 k ¯ E E ,ad( and ), o ahevent each For . e 1 1 E π −(π [(π [(π PNAS E 1 [(π 1 o o [(π o C + + o xcl n eetbesignal detectable one exactly ) + | o + π π π + o o o.115 vol. prior prior π o prior o prior π k ¯ o prior > k ¯ ) ) )( k k 1/π ¯ ¯ π ) to min ] o k k ¯ ¯ ) ahcregives curve Each . E | min k ¯ o − 0 ecalcu- we ∞, In . ] o 42 no. ] , E , k ¯ ] 0 E min tleast at (B) ), , π B 0 and , and ) | . the C, E9759 [17] [18] [19] E π 1 o ,

ASTRONOMY AB

Fig. 5. (A) Probability πo that an emitter is within an observable sphere of radius RL∗ for the cases in which the emitter luminosity function is either a single ∗ ∗ Dirac-delta peak centered at L or a uniform distribution extending up to L . Inset shows that within the galactic neighborhood (RL∗ . 1 kly) πo scales as 3 13 RL∗ .(B) Radius of the observable sphere for several telescopes as a function of the intrinsic luminosity of an emitter in units of LArecibo = 2 × 10 W, the√ EIRP ∗ of the Arecibo radar. The values of the minimum detectable flux (Smin) are reported in Table 1 for each telescope. Note that from Eq. 9 RL∗ scales as L .

πo = 0 in Eq. 17. As anticipated by the analysis of the PDFs, the 99% of finding a star at a distance of 60 kly from the galactic posterior CCDFs resulting from the nondetection or a detec- center.§ tion of a signal differ significantly from each other. While the response to E0 (Fig. 4A) becomes progressively smaller than the Luminosity Function. Since we ignore what a plausible PDF of L ¯ prior as πo increases and getting negligibly small for k > 1/πo , looks like and whether it even exists, modeling the luminosity the posterior CCDFs resulting from the detection of a signal function g(L) unavoidably requires making some assumptions. −α (either event E 0 or E1) depart abruptly from the prior CCDF Previously, a power-law distribution of the form g(L) ∝ L has as soon as πo =6 0, as shown in Fig. 4 B and C. In particular, been proposed as a vehicle to assess, through the choice of the −3 for a relatively small value of πo (say, for example, ∼ 10 ), the exponent α, the interplay between the proximity of a detectable detection of at least one signal (event E0) implies that the pos- emitter and its spectral density (33, 34). Here, we limit our anal- terior probability that the Earth intersects typically more than ysis to the effect on the detection probability of the spread of 3 k¯ ∼ 1/πo ∼ 10 signals exceeds ∼ 80%. This probability drops to the luminosity distribution by considering some limiting forms about 35% if exactly one signal is detected in the entire sky for of g(L). To this end, we take g(L) to be given either by a sin- gle Dirac-delta peak centered at some characteristic luminosity the same value of πo (event E1, Fig. 3C). Since P(k¯|E0) and ¯ L∗, g(L)= δ(L − L∗), or by a uniform distribution ranging from P(k|E1) represent, respectively, an upper and lower limit for ∗ ∗ ∗ the posterior probability in the case of detection, the probability L = 0 up to L : g(L)= θ(L − L)/L . In either case, the depen- that there are on average more than 103 galactic signals crossing dence of πo on the width of the luminosity function can be the Earth is therefore comprised between ∼ 80% and ∼ 35% in conveniently expressed in terms of the maximum detectable dis- this example. tance, RL∗ (Eq. 9), as illustrated in Fig. 5A. While at distances of about RM ∼ 90 kly or larger, πo saturates to 1 due to the finite Bayesian Analysis Applied to Existing and Upcoming SETI size of the Galaxy, in the galactic neighborhood (RL∗ . 1 kly), −3 3 Detectors the function πo is smaller than about 10 , and it scales as RL∗ We now apply our Bayesian formalism by considering existing (Fig. 5 A, Inset). and planned radio telescope to infer the posterior probabilities Observational Radius. To determine the observational radius R ∗ E E E L following the potential occurrence of events 0, 0, and 1 in (Eq. 9) of an all-sky SETI survey, we must specify the character- a SETI survey. The quantity governing the response to these istic luminosity of the emitters, L∗, and the minimum detectable events is πo (RL∗ ), the probability that an emitter is within a flux, Smin. The latter quantity is determined by the characteris- R ∗ 8 distance L from the Earth. According to Eq. , this quantity tics of the telescope used in the SETI search and the intrinsic ρ (~r) depends on the number distribution of stars, s , and the lumi- bandwidth of the transmitted signal. Here, we consider the case g(L) nosity function of the emitters, . The latter identifies also the of a signal bandwidth narrower than the spectral resolution of observational radius, RL∗ , once a specific telescope sensitivity is the telescope, which reduces Smin to (1, 26) assigned. r ∆ν Number Density of Stars. We take the number density function Smin = mSsys , [21] ρs (~r) to have a cylindrical symmetry of the form: t where m is the desired signal-to-noise ratio, t is the integration −r/rs −|z|/zs time in seconds, ∆ν is the receiver channel bandwidth (expressed ρs (~r) = ρ0e e [20] in Hz), and Ssys is the system equivalent flux density (SEFD), which depends on the system temperature of the receiver and where r is the radial distance from the galactic center, z is the on the effective collecting area of the telescope. Table 1 lists the height from the galactic plane, and ρ0 is a normalization factor R ensuring that d~rρs (~r) = Ns . We assume that the emitters are potentially located in the thin disk of the Milky Way, whose star §We have considered also the possibility that the distribution of stars that can potentially distribution follows approximately Eq. 20 with rs = 8.15 kly and harbor life has an annular shape, as in the galactic habitable zone proposed in ref. 32 zs = 0.52 kly (31). The resulting ρs (~r)/Ns gives a probability over (SI Appendix, section 3).

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Table phase study the in still operational fully or only yet FAST) not and are either (MeerKAT SKA2 are and telescopes these FAST, as MeerKAT, indicative, to attributed those while ro Parameters. Prior kly, 1 make about to to assumed be up only probability distances the probe where can telescope) SKA the radar, where GBT, VLA, between Parkes, comprised ATA, radio frequencies nonnatural of for of searches values targeted signals past SEFD to the refer Arecibo 1, and Table In s. n n lne aiiis(,6 52)adtecorresponding the and 25–29) 6, (1, facilities S planned and ing of values SKA. of precursors and 2, phase See 1, (30). phase telescope SKA the of 2 astronomers.skatelescope.org/documents phase the of completion upon geted b a rmliadMarcy and Grimaldi The (30). SKA2 and and (29), hun- (FAST) Five Telescope of the (28), Spherical values telescope Array Aperture Karoo Arecibo Large meter Meer the Very the dred (1), (27), Jansky (GBT) (MeerKAT) Telescope the Telescope Bank (1), Green Array the telescope 26), Parkes (1, (VLA) the telescope 25), (1, (ATA) Array Parkes ATA Telescope radio forthcoming and current of sensitivities telescope Threshold 1. Table rqec ag – H iha IPeuvln ota of that to equivalent EIRP an radar. with Arecibo GHz the 1–2 range frequency a within emitters from nals 10 VLA FAST Arecibo MeerKAT GBT SKA2 ¶ EDfr2 antennas. 27 for SEFD siaeo h olvleteSF feunyrange (frequency SEFD the value goal the of Estimate oprsn steAeiordri h otpwru ai rnmte nEarth. on transmitter radio powerful most the here L is considered radar adopt Arecibo coverage the nonetheless frequency as we observational comparison, the GHz), outside 1–2 is (about which GHz, 2.38 of lhuhti au fteER eest h rcb aa rnmtiga frequency a at transmitting radar Arecibo the to refers EIRP the of value this Although min H.Fo i.5 Fig. From GHz. i.5B Fig. SEFD, −23 t = nWm in ¶ 10 L Wm p 0min. 10 S vntems estv eevr(h lne hs of 2 phase planned (the receiver sensitive most the even Arecibo S −24 sys L min S hw h ausof values the shows ∗ n orsodn sensitivity, corresponding and , −2 sys /L −2 r acltdfo Eq. from calculated are W/m eaerln u h xsec fdtcal sig- detectable of existence the out ruling are we , Arecibo nJ 1Jy (1 Jy in 2 = acltdfor calculated oasg h probability the assign To 2 × ntefeunyrne12Gz(4 35–40). (24, GHz 1–2 range frequency the in A π 10 l.I te em,b adopting by terms, other In kly. π o and o 13 significant. ssal while small, is S ∼ 664 steER mte yteArecibo the by emitted EIRP the is W 0.3 sys =10 1.2 8.6 43 18 10 3 eseta for that see we B, 40 Jy , b a R m −26 yfo at httasi within transmit that Earth from ly L 15, = ∗ Arecibo 21 httetlsoe nitdin enlisted telescopes the that Wm o ute ouetto on documentation further for assuming S = ∆ν S min −2 min L 2 ∗ × S ·Hz o h le Telescope Allen the for , denoted , /L 0 = min prior 10 π ∼ Arecibo 13 o prior −1 m .5 L 1 saueu emof term useful a as W S = 10 = ∗ fafwexist- few a of ) z and Hz, min 10 /L ∼ and 15, hthst be to has that 10 , −22 1 Arecibo & −23 − 287 ∆ν S 18.6 100 ∼ min −26 prior H)tar- GHz) 2 0.13 7.8 0.5 1.3 3.7 4.3 W/m R = S 2 Wm t min L prior prior W/m Hz, 0.5 a to has . 600 = https:// ∼ ∗ that , GHz, 2 0.1, 1 −2 2 to = − = , eutn rmevents from resulting Posteriors. above. of discussed value responding uioiyi ml n olw oe a.FraDirac- a For law. estimate power we emitter a an 2.6 function, follows detecting luminosity and of delta probability small the is which luminosity at distance the r optdb dpigadlaDrclmnst ucinfrteemit- the for radar. results function Arecibo The L luminosity text. at delta-Dirac the dashed centered a in The ters adopting center. described galactic by as the calculated computed to CCDF are distance prior 27 the the within is denotes mil- or which line project, one (B), Listen Earth about Breakthrough from the containing kly by radius targeted stars observational nearby an lion to corresponding (A), GHz, same 2 the and to referring GHz as 1 understood between be range. frequency must frequencies (dashed probability posteriors to prior the the refers calculate to lines) used take radius can events vational probability limiting posterior the the between that values (L the encompasses radar Arecibo the delta eutn rmteeet fnneeto (E ( nondetection nal of events the than from larger resulting is Earth crossing signals 6. Fig. vnfor Even × E 0 10 ,adeatyoedtcal inl(E signal detectable one exactly and ), g otro CF iigtepoaiiista h ennme of number mean the that probabilities the giving CCDFs Posterior B A (L) −8 (L i.6sostepseirCDsof CCDFs posterior the shows 6 Fig. n nER fteeitr qa ota of that to equal emitters the of EIRP an and L ∗ ∗ /L ∗ 100L = = Arecibo L Arecibo ∗ k Arecibo E ¯ ) min 0 = 3/2 hr L where , , L E Arecibo 0 .Tecor- The III). section Appendix, (SI olw from follows and , h chosen the , E 0 k ¯ Arecibo h oi uvsrfrt h CCDFs the to refer curves solid The . and PNAS .Teae ooe ngray in colored area The ). E 1 = E 0 optduigaDirac- a using computed 1 | ,a es n eetbesig- detectable one least at ), 2 ic h ro obser- prior the Since . 1 × ihn50l rmEarth from ly 500 within ) o.115 vol. 10 π k R ¯ min o prior 13 L prior ∗ steER fthe of EIRP the is W ' | = k ¯ 0.14π swl within well is o 42 no. π sldlines) (solid o (R o prior L prior | ∗ E9761 as , ) ∼

ASTRONOMY The results shown in Fig. 6A are computed for an obser- ing on whether we assign the signal detection to event E1 or 5 8 vational radius containing about one million stars (RL∗ = 500 event E 0, k¯ is bounded from above by ∼ 3 × 10 or ∼ 10 , ly), which is the number of nearby targeted stars that the respectively. Breakthrough Listen program will search for radio emissions. Extending the observational radius up to the galactic cen- Since the fractional volume of the Galaxy encompassed by ter (RL∗ = 27 kly; Fig. 6B) changes drastically the responses to −5 this value of RL∗ is very small [πo (RL∗ = 500 ly) ∼ 10 ; Fig. events E0, E 0, and E1. Not detecting signals out to 27 kly implies ¯ 5A], the posterior CCDF resulting from the nondetection of that there are practically zero chances that k & 3, and no more signals (event E0) is not significantly smaller than the prior. than typically 10 detectable signals are expected to populate the ¯ 5 While the inferred upper limit of k (∼ 1/πo ∼ 10 ) is about Galaxy if instead exactly one signal is discovered within that two orders of magnitude smaller than that derived from our radius (event E1). prior, the posterior probability that k¯ ≥ 1 (∼ 33%) is reduced The impact of the observational radius highlighted in Fig. 6 is by a factor of only 1.4. On the contrary, the posterior CCDFs best illustrated in Fig. 7, where the posterior probabilities that resulting from the discovery of a signal within 500 ly differ con- k¯ ≥ 1 (A and B) and k¯ ≥ 100 (C and D) are plotted as a func- siderably from the prior. We find a probability exceeding 97% tion of RL∗ for Dirac-delta luminosities peaked at LArecibo (A 3 that more than 10 signals typically cross our planet. Depend- and C) and 100LArecibo (B and D). The red vertical lines are the

AB

CD

¯ ¯ ∗ ∗ Fig. 7. Posterior probability that k ≥ 1 (A and B) and k ≥ 100 (C and D) for emitters with characteristic luminosity L /LArecibo = 1 (A and C) and L /LArecibo = 13 100 (B and D), where LArecibo = 2 × 10 W is the EIRP of the Arecibo radar. Dashed lines denote the prior probabilities, while the solid curves are posterior probabilities as a function of the observational radius RL∗ resulting from the events of nondetection (E0), at least one detectable signal (E0), and exactly ∗ one detectable signal (E1). The results have been obtained by assuming a Dirac-delta luminosity function centered at L . The red vertical lines indicate the values of RL∗ that are accessible to the telescopes listed in Table 1.

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(dashed that lack probability probability the prior case, posterior the former the of event In sion kly. detection, 1 signal about of than larger or smaller tr a oei e.2.W aeipoe nti approach the this of on function improved luminosity have the We sensitivity, 25. detector including ref. by in done was stars smaller times 10 threshold galactic detection observa- than prior effective the the An relatively of beyond radius. choices are different tional results to well respect Our with region S3). also to robust Fig. the responses Appendix, in (SI the neighborhood affected Specifically, probabilities 7. slightly posterior and only the 6 qualitatively Figs. change of entire not the does to delta patch small a from grows sky sphere. celestial the of between portion ined ranges discovered (∼ be powerful signal, 100 a 7D than discovers Fig. more telescope from this follows that 7 it assume Fig. and in example shown tive as Galaxy, entire C the encompasses sphere to response able in significant remain they that chances there (L Galaxy. Arecibo- that from 10% emitters Earth crossing than of signals like larger 100 distance than probability more a a typically are to with of consistent out state still signals current is detecting the reflects not correctly knowledge, prior our that prob- posterior assumption the consider that we if ability significantly, not but 7 reduced, (Fig. probability prior the 1. 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S ). E min fw aeaanteSA eecp sa illustra- an as telescope SKA2 the again take we If D. prior 0 sdtcal within detectable is ) L 10 = k ¯ ∗ k ¯ 100L = ≥ ≥ R ssoni i.7 Fig. in shown as 100, −23 ∗ R L 100 ∗ = L R ∗ ausacsil otedtco facilities detector the to accessible values ). L L Arecibo & Wm ∗ rpepnnilyt for 0 to exponentially drop Arecibo 30 xed elbyn h aatcneigh- galactic the beyond well extends httepoaiiyta hr r still are there that probability the that L −2 − Arecibo k ¯ E 0 ≥ 100L oae nteentire the in located 7C) Fig. ; 40 is S2 Figs. Appendix, (SI assumed is A osntipyadaai revi- dramatic a imply not does , fet nytepseirresult- posterior the only affects E k R ¯ 1 0 and ≥ L l Fg 7 (Fig. kly ∼ ∼ sntsalrta bu half about than smaller not is a tl nesc h Earth the intersect still may ) ∗ or Arecibo hl ntefre aethe case former the in While . 100 )that 20%) n %a h exam- the as 0% and 52% 146(kly .Ti atri somewhat is factor This B). E E 1 1 E ra es n signal one least at or ) rp ool bu 3% about only to drops 0 olpe o1ee for even 1 to collapses eetbeeitr to emitters detectable ) k ¯ vnwe h observ- the when even C /R ≈ ≥ and 1 1 A L ∼ ∗ k ¯ l,wihdrops which kly, eue oless to reduces ) and 1 ≥ 3 E ∼ ne the Under D. l mle a implies kly 0 R inl inter- signals 1 1 100L L and .I fol- It B). ∗ Fg 7A), (Fig. l differs kly & Arecibo ∼ 10 E R 1 1 L ∗ kly, are kly is ) rk,Eii niuz rcJ opl,Ade imo,Jl Tarter, Jill Siemion, Andrew Korpela, discussions. J. fruitful for Eric Werthimer Dan and Enriquez, Tellis, Nathaniel Emilio Drake, ACKNOWLEDGMENTS. data. SETI empirical from possible directly number a civ- Drake’s galactic enabling of of estimate (14), number emitting mean of the number currently also mean ilizations gives the Earth that crossing at emphasize signals signals we of shell remark, amount last typical a the As data appropri- Earth. from nonnatural most infer the of to be of tool to population ate appears terms approach galactic in Bayesian the A deliver emitters. about potentially data can informative surveys SETI forthcoming coefficient, emissions. spa- opacity the the of the anisotropy consider and aforementioned dust should the with galactic model taken together the the are of case, distribution medium this tial interstellar In the account. of into processes optical scattering the renewed and aborption at signals, a that signals requires of gained wavelengths radio probability micrometer–submicrometer recently detection for the have Modeling search (10–12). (9) interest the spectra near-infrared in and put directed efforts are SETI they unless marginal (13). is number us toward total Earth deliberately the crossing to beam- signals contribution of of their fraction the difficult, a although is not includes emissions, these that is like model of It a One processes. formulate made. to emission however, have the we of isotropy that presumed assumptions few a mean ing the on 2. depending Fig. planet in shown our as cross longevity, to signal fraction expected a is only signals because them of larger of even number be total may Galaxy The the Earth. populating the intersect than 100% power a more radiated almost average, implies on radar, that, Arecibo probability the of that detected to of comparable signal radius a be a priors, to likely within noninformed is reasonable Earth the Under crossing systems, signals large. star of number nearby average least from if inferred emission at only extraterrestrial of out natural radius ruled a reasonably within be observed Earth. can for are Arecibo that planet signals the possibility as our no The powerful cross Galaxy. as con- emissions the radar EM region of nonnatural sampled fraction galactic, the significant however, unless a has, value tains nondetection exist informative of frequencies moderate datum comparable a The probabil- at distances. the further datum transmitting framework, at input emitters statistical an that Bayesian as a ity used a within be within infer, can and Earth, to frequencies, from of distance window certain certain a signals within no detected that Evidence are out- surveys. an potential SETI extensive assuming However, future, allow by of orbit. possible comes still Earth’s to is intersects approach insufficient theoretical Way a Milky signals is EM the nonnatural from knowledge that emitted probability the of of estimate state informed present The Conclusions than smaller than larger lifetimes is lower planet signal our a For 2. represents 000 Fig. 7 87, in Fig. illustrated in as number, shown total the Earth to bound crossing signals of posterior accurate more 23). a (21, gives detection log- likely of the probabilities here, as such used prior, PDF Galaxy. uninformative the uniform an in of stars use of the function Furthermore, number density the and emitters, ncnlso,w hn hti stm oatcpt what anticipate to time is it that think we conclusion, In planned and current that importance the Notwithstanding relax- by formulation present the improve to possible is It non- genuinely a detects survey SETI a that hypothesis the In number mean inferred the that stressing worth is it Finally, ,tetpclaon fglci inl htd o cross not do that signals galactic of amount typical the y, ≈ etakAee ab,Toa abl,Frank Basbøll, Thomas Balbi, Amedeo thank We 1 l rmErh mte iha EIRP an with emitted Earth, from kly k ¯ . qN s fsgasppltn h Galaxy, the populating signals of , PNAS | ∼ o.115 vol. 100 inl fsimilar of signals | o 42 no. ∼ | 40 t E9763 M kly '

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