Monte Carlo estimation of the probability of causal contacts between communicating civilisations
(cid:105)(cid:105) (cid:105) “manuscript” — 2020/7/14 — 1:29 — page 1 — (cid:105)(cid:105)(cid:105) (cid:105) (cid:105)(cid:105)
Submitted to InternationalJournal of Astrobiology
Key words:
SETI, Computer simulations, Statistics
Abbreviations:
SETI: Search for ExtraterrestrialIntelligence,CCN: causally connected node,SFC: surface of first contact,SLC: surface of last contact,DE: Discrete Event,GHZ: Galactic Habitable Zone
Author for correspondence:
Lares M, Email: [email protected]
Monte Carlo estimation of the probability ofcausal contacts between communicatingcivilisations
Lares M. , , Funes J. G. , & Gramajo L. , CONICET, Argentina Universidad Nacional de Córdoba, Observatorio Astronómico de Córdoba, Argentina Universidad Católica de Córdoba, Argentina
Abstract
In this work we address the problem of estimating the probabilities of causal contacts betweencivilisations in the Galaxy. We make no assumptions regarding the origin and evolution ofintelligent life. We simply assume a network of causally connected nodes. These nodes refersomehow to intelligent agents with the capacity of receiving and emitting electromagnetic sig-nals. Here we present a three-parametric statistical Monte Carlo model of the network in asimplified sketch of the Galaxy. Our goal, using Monte Carlo simulations, is to explore theparameter space and analyse the probabilities of causal contacts. We find that the odds to makea contact over decades of monitoring are low for most models, except for those of a galaxydensely populated with long-standing civilisations. We also find that the probability of causalcontacts increases with the lifetime of civilisations more significantly than with the number ofactive civilisations. We show that the maximum probability of making a contact occurs when acivilisation discovers the required communication technology.
1. INTRODUCTION
The Drake equation (Drake, 1962) provides a truly helpful educated guess, a rational setof lenses –the factors in the equation– through which to look at future contacts with tech-nologically advanced civilisations in the Milky Way. The equation quantifies the number ofcivilisations from whom we might receive an electromagnetic signal, using a collection of fac-tors that have been extensively discussed in the literature and whose estimated values are revisedcontinually. A comprehensive review and an analysis of each term of the equation are presentedin Vakoch & Dowd (2015). Optimistic estimates from the Drake equation contrast with theso-called Fermi paradox, which states the apparent contradiction between the expected abun-dance of life in the Galaxy and the lack of evidence for it (e.g. Hart, 1975; Brin, 1983; Barlow,2013a; Forgan, 2017a; Anchordoqui, Weber & Fernandez Soriano, 2017; Sotos, 2019; Carroll-Nellenback et al., 2019). There are many propositions aimed at solving this paradox, whichmake use of statistical (Solomonides et al., 2016; Horvat, 2006; Maccone, 2015) or stochas-tic approaches (Forgan, 2009; Bloetscher, 2019; Glade et al., 2012; Forgan & Rice, 2010).Regarding the Drake equation, analytical interpretations (Prantzos, 2013; Smith, 2009) or refor-mulations (Burchell, 2006, and references therein) have also been proposed. The absence ofdetections of extraterrestrial intelligent signals could be explained by astrophysical phenomenathat makes life difficult to develop (Annis, 1999). Besides the possible scarcity of life, alterna-tive scenarios have also been discussed (Barlow, 2013b; Lampton, 2013; Conway Morris, 2018;Forgan, 2017b). The large distances in the Galaxy and the likely limited lifetime of civilisationsmay play an important role in determining how difficult it would be to obtain evidence for otherinhabited worlds. The analysis of these scenarios is difficult due to the lack of data about thehypothetical extraterrestrial intelligences. Indeed, as Tarter (2001) pointed out, according to ourcurrent technical capabilities for the search of extraterrestrial intelligence (SETI), we have notreceived any signal yet. The absence of detections has also motivated alternative ideas for newSETI strategies (Forgan, 2019; Balbi, 2018; Loeb & Zaldarriaga, 2007; Maccone, 2010; Tarteret al., 2009; Enriquez et al., 2017; Loeb et al., 2016; MacCone, 2011; Lingam & Loeb, 2018;Wright et al., 2015; MacCone, 2013; Maccone, 2014b; Harp et al., 2018; Forgan, 2013, 2017b;Funes et al., 2019). a r X i v : . [ phy s i c s . pop - ph ] J u l (cid:105) “manuscript” — 2020/7/14 — 1:29 — page 2 — (cid:105)(cid:105)(cid:105) (cid:105) (cid:105)(cid:105) The discussion about the problem of the unknown abundanceof civilisations in the Galaxy has been organised around the factorsin the Drake equation (Hinkel et al., 2019). However, the uncer-tainties in these factors, specially the ones representing biologicalprocesses, make it less suitable to a formal study with the pur-pose of defining searching strategies or computing the estimatednumber of extraterrestrial intelligences. A number of studies pro-pose alternative formalisms for the estimation of the likelihoodof detecting intelligent signals from space. Prantzos (2013), forexample, proposes a unified framework for a joint analysis of theDrake equation and the Fermi paradox, concluding that for suffi-ciently long-lived civilisations, colonisation is the most promisingstrategy to find other life forms. Haqq-Misra & Kopparapu (2018)discuss the dependence of the Drake equation parameters on thespectral type of the host stars and the time since the Galaxyformed, and examine trajectories for the emergence of commu-nicative civilisations. Some modifications to the original idea ofthe Drake equation have been proposed, in order to integrate atemporal structure, to reformulate it as a stochastic process or topropose alternative probabilistic expressions. Temporal aspects ofthe distribution of communicating civilisations and their contactshave been explored by several authors (Fogg, 1987; Forgan, 2011;Balbi, 2018; Balb, 2018; Horvat et al., 2011), as well as effortson considering the stochastic nature of the Drake equation (Gladeet al., 2012).Several authors raise the distinction between causal contactsand actual contacts. In the first case, the determinations of metricsfor the likelihood of a contact are independent of considerationsabout the technological resources or the implicit coordination todecipher intelligent messages. In a recent work, Balbi (2018)uses a statistical model to analyse the occurrence of causal con-tacts between civilisations in the Galaxy. The author highlightsthe effect of evolutionary processes when attempting to esti-mate the number of communicating civilisations that might be incausal contact with an observer on the Earth. ´Cirkovi´c (2004) alsoemphasises the lack of temporal structure in the Drake equationand, in particular, the limitations of this expression to estimate therequired timescale of a SETI program to succeed in the detectionof intelligent signals. Balbi (2018) also investigates the chanceof communicating civilisations making causal contact within avolume surrounding the location of the Earth. The author arguesthat the causal contact requirement involves mainly the distancebetween civilisations, their lifespan and their times of appearance.This is important since the time the light takes to travel acrossthe Galaxy might be much lesser than the lifetime of the emitter.Balbi (2018) fixes the total number of civilisations and exploresthe parametric space that comprises three variables, namely, thedistance to the Earth, the time of appearance and the lifespanof the communicating civilisations. Each of these three variablesare drawn from a random distribution. The distances are drawnfrom a uniform model for the positions of civilisations withinthe plane of the Galaxy. For the distribution of the characteristictime of appearance, the author explores exponential and truncatedGaussian functions, while for the lifespans chooses an exponen-tial behaviour. It is important to point out that the estimation ofthe number of communicating civilisations vary with the choice ofthe statistical model for the time of appearance, as shown by Balbi(2018). For all analysed distributions, the author concludes thatthe fraction of emitters that are listened is low if they are spreaded in time and with limited lifetimes. An analytical explanation ofthese concepts are presented in Grimaldi (2017), who considers astatistical model for the probability of the Earth contacting otherintelligent civilisations, taking into account the finite lifetime ofsignal emitters, and based on the fractional volume occupied byall signals reaching our planet.The quest for a formal statistical theory has also lead to impor-tant progress in the mathematical foundations of SETI. Recently,Bloetscher (2019) considers a Bayesian approach, still motivatedby the Drake equation, to estimate the number of civilisations inthe Galaxy. To that end, the authors employ Monte Carlo MarkovChains over each factor of the Drake equation, and combine themean values to reach a probabilistic result. It is worth mentioningthat the author proposes a log-normal target distribution to com-pute the posterior probabilities. This study concludes that there is asmall probability that the Galaxy is populated with a large numberof communicating civilisations. Smith (2009) uses an analyticalmodel to gauge the probabilities of contact between two randomlylocated civilisations and the waiting time for the first contact. Theauthor stresses that the maximum broadcasting distance and thelifetime of civilisations come into play to produce the possiblenetwork of connections.On this topic a number of works consider numerical simula-tions (Forgan et al., 2016; Vukoti´c et al., 2016; Murante et al.,2015; Forgan, 2009, 2017b; Ramirez et al., 2017). Although thisapproach does not rely on values that can be measured from obser-vations (like the fraction of stars with planets), it depends on thedefinition of unknown or uncertain parameters required to carryout the simulations. In another numerical approach, Vukoti´c &´Cirkovi´c (2012) propose a probabilistic cellular automata mod-elling. In this framework, a complex system is modelled by alattice of cells which evolve at discrete time steps, according totransition rules that take into account the neighbour cells. Theauthors implement this model to a network of cells which rep-resent life complexity on a two–dimensional region resemblingthe Galactic Habitable Zone (GHZ), an annular ring set betweena minimal radius of 6 kpc and a peak radius of 10 kpc. Thesesimulations represent the spread of intelligence as an implemen-tation of panspermia theories. Within this framework, Vukoti´c &´Cirkovi´c (2012) also make Monte Carlo simulations and analyseensemble-averaged results. Their work aims at analysing the evo-lution of life, and although it does not account for the network ofcausal contacts among technological civilisations, it offers a toolto think SETI from a novel point of view.This study is conceived as an introduction to a simple proba-bilistic model and a numerical exploration of its parameter space.With these tools, we address the problem of the temporal and spa-tial structure of the distribution of communicating civilisations.This approach does not require asumptions about, e.g., the ori-gin of life, the development of intelligence or the formation ofhabitable planets according to stellar type. Instead, we assumemonoparametric function families to model the appearance ofpoints in the disk of the Galaxy and over the time, which wecall nodes . These nodes represent the locations of ideal intelligentagents that are able to receive and emit signals with perfect effi-ciency in all directions. Then we analyse the network of causallyconnected nodes limited to have a maximum separation represent-ing the maximum distance a signal can travel with an intensityabove a fixed threshold. A node is causally connected to other (cid:105) “manuscript” — 2020/7/14 — 1:29 — page 3 — (cid:105)(cid:105)(cid:105) (cid:105) (cid:105)(cid:105)
Probability of causal contacts nodes if it is within their light cone. We adopt as a definition thata light cone is the region in spacetime within the surface gen-erated by the light emanating from a given point in space in agiven period of time. For example, two nodes separated by 100light years which appear with a difference in time of 80 yearsand last 50 years each, will be causally connected for 30 years.This connection is not, though, bidirectional. In fact, the secondnode will see the first one for 30 years, but the first node willnever acknowledge the existence of the second. An scheme of thisexample is shown if Fig. 1. This simple tought experiment exem-plify the importance of the time variable to analyse the structureof connections. The model can be described with three param-eters, namely, the mean separation between the appearance ofnodes, the mean lifetime of the nodes, and the maximum sepa-ration to allow an effective causal contact. The model presentedhere comprises the stochastic networks of constrained causallyconnected nodes, together with the three parameters and addi-tional hypotheses. Hereafter, we refer to this model as the SC3Netmodel, standing for ”Stochastic Constrained Causally ConnectedNetwork“. Fig. 1. : Example of a configuration of an emitter and a receiver toproduce a causal contact. The observers are separated by 100 lightyears and appear with a difference in time of 80 years. Oneobserver is active between t = t =
50 yr. The otherobserver is also active for 50 yr, between t =
80 yr and t =
130 yr.The causal connection occurs between t =
100 yr and t =
130 yr.Meaningful times are marked with vertical dashed lines.We provide a framework to explore, through a suite of numer-ical simulations, the parameter space of three unknown observ-ables. This allows to discuss possible scenarios and their con-sequences in terms of the probability of making contacts. Themethod we use for simulating a stochastic process is an approxi-mation that allows to study the behaviour of complex systems, byconsidering a sequence of well defined discrete events. The sim-ulation is carried out by following all the variables that describeand constitute the state of the system. The evolution of the processis then described as the set of changes in those variables. In thiscontext, an event produces a specific change in the state that canbe triggered by random variables that encode the stochastic natureof the physical phenomenon. For example, when a new contact isproduced between two entities in the simulated galaxy, the num-ber of active communication lines is increased by one. Also, if it is the first contact for that nodes, then the number of communi-cated nodes increases by two. When a new node becomes active,the sistem has an increase of one in the number of nodes, althoughthe number of communcations does not necessarily change. Theprocess then involves following the changes on the state of the sys-tem, defining the initial and final states. This is done by defining amethod that allows to keep track of the time progress in steps andmaintaining a list of relevant events, i.e., the events that producea change in the vriables of interest. With this method we aban-don the frequentist approach of the Drake equation to computethe number of civilisations, providing instead its statistical distri-bution. More importantly, we are interested on the probability ofcontacts, which depend critically on the time variable. This is anexploratory analysis that aims at developing a numerical tool todiscuss the different scenarios based on statistical heuristics. Theapproach proposed here should be considered as a compromisebetween the uncertainties of the frequentist estimations and thedetailed recipes required on the numerical simulations. It is worthnoticing that the SC3Net model is not intended for a formal fit atthis stage due to the lack of data, but it can help to understand howunusual it would be to actively search for intelligent signals for 50years without possitive detections.This paper is organized as follows. In Sec. 2. we introduce themethods and discuss the candidate distributions for the statisti-cal aspects of the times involved in the communication process.We present our results in Sec 3., with special emphasis on thestatistical distributions of the duration of causal contacts in andthe distribution of time intervals of waiting for the first contact.This quantities are considered as a function of the three simulationparameters. In Sec. 4. we discuss our results and future researchdirections.
2. METHODS AND WORKING HYPOTHESES
Simulations are suitable tools to analyse systems that evolvewith time and involve randomness. An advantage of a numer-ical approach is that it usually requires fewer assumptions andsimplifications, and can be applied to systems where analyticalmodels are hard or impossible to develop. In particular, a suitabletool to model complex stochastic processes through the changesin the state of a system is the discrete–event (hereafter, DE)simulation approach. A system described with the DE paradigmis characterised by a set of actors and events. Actors interactcausally through a series of events on a timeline and process themin chronological order (Ptolemaeus, 2014; Chung, 2003; Ross,2012). Each event modify the variables that define the process,producing the corresponding change in the state of the simulatedsystem. This method is well suited for the particular case of thediffusion of intelligent signals in the galaxy and allows to exploreseveral models easily. We simulate the statistical properties of a setof points in space and time that have a causal connection at lightspeed and are separated by a maximum distance. We refer to thisnodes as ”Constrained Causal Contact Nodes“ (hereafter, node).We choose this generic name in order to stress the fact that inthis analysis only the causal contact is considered, independentlyof any broadcasting or lookout activity. The system we proposeis ideal, in the sense that it considers the special case of a fully (cid:105) “manuscript” — 2020/7/14 — 1:29 — page 4 — (cid:105)(cid:105)(cid:105) (cid:105) (cid:105)(cid:105)
Fig. 2. : Space–time diagrams showing a schematic representationof the different stages in the development of a constrained causalcontact node. Panel (a) represents the region R in space andtime which is causally connected to the emitter (E, left vertex),following an “A” type event (“Awakening”) in which the nodeacquires the communication capability. The sphere of the first con-tact (SFC) is the sphere centred on the emitter that grows until itsradius reaches the D max distance, in which the power of the signalwould equal the detectability threshold. This sphere is representedby the left triangle of the region R . The surface of the last contact(SLC) is another sphere that grows from a “D” event (“Dooms-day”). The region which is causally connected to the emitter isthen limited by these two spheres, and has the shape of a sphereor of a spherical shell, depending on the time. The temporal inter-vals for the communication between two nodes are representedin the panel (b). The node E can listen to signals from the nodeE , since the “Contact” event (t=C ) up to the “Blackout” event(t=B ). Similarly, the node E is in causal contact from the nodeE , in the time interval (C , B ).efficient node that emits and receives isotropically. A causal con-tact node, considered as a broadcasting station that has the abilityto detect signals through an active search program, could have alesser than one efficiency factor, which is not included in the cur-rent analysis. Also, it is worth mentioning that this is a generalapproach, and not necessarily a node is the host of an intelligentcivilisation. It can be associated with a planet where life has devel-oped, became intelligent, reached the skills required to find theright communication channel, sustained a search and establisheda contact. Alternative message processing entities could be con-sidered, for example interstellar beacons where intelligent life hasceased to exist but continue with its emission, or communicationstations established by probes or left by intelligent beings (see,e.g., Peters, 2018; Barlow, 2013b). In principle, these strategies could affect our results since it would be easier to configure a clus-ter of nodes that spread in time. However, we do not consider thesespeculative alternatives at this point. For the purpose of the presentanalysis, only the communication capability is relevant, since westudy the causal contacts between the locations. The system isdefined by a number of actors that represent nodes and appearat different instants in time, generating events that produce mean-ingful changes in the variables that describe the system, i.e., in thearrangement of nodes and their network of causal contacts. Forexample, the appearance of a new node in a region filled with asignal emitted by another node, will increase the number of activenodes and the number of pairs of nodes in causal contact. Assum-ing some simple hypotheses, the discrete events method can beperformed taking into account a small number of variables, whichallow to analyze the variation of the results in the SC3Net modelparameter space.In what follows, we outline the experimental setup adopted toestimate the probabilities of causal contacts and several derivedquantities. This is done in terms of three independent parameters,namely, the mean time span between the appearance of consecu-tive nodes, τ a , their mean lifetime τ s , and the maximum distancea signal can be detected by another node ( D max ). Intuitively, theprobability of the existence of causal contacts between pairs ofnodes would be larger for smaller τ a parameters, higher τ s orhigher D max parameter values. We also propose theoretical dis-tribution functions for both the lifespans ( τ s ) and the number ofnodes per unit time (Maccone, 2014a; Sotos, 2019). The later isrelated to the time span between the appearance of consecutivenodes (since when τ a is shorter, it produces a greater density ofnodes). The analytical expresions for these distributions are set toa fixed law, as discussed in Sec. 2.1..Space–time diagrams, where time and space are representedon the horizontal and vertical axes, respectively, are suitable toolsto describe the causal connections among different nodes. Weillustrate in the Fig. 2 the schematic representation of the regioncausally connected to a given node. For the sake of simplicity,we show in the plot the light travel distance, i.e., the distancetraversed by any signal spreading from the emitter at light speed(for example, any electromagnetic signal). In this scheme, a lightpulse would follow a trajectory represented by a line at 45 degreesfrom the axes, given the units of time and space axes are years andlight years, respectively. Panel (a) represents the region R (shadedpolygon) which is causally connected to the emitter (left ver-tex). This region develops after an event (hereafter dubbed A-typeevent) in which the node acquires the communication capability,or becomes ”active”. The sphere of first contact (SFC) is centredon the emitter, represented by the left angle of the region R . Thissphere grows until its radius reaches the D max distance, in whichit would be no longer detectable due to the decrease in the energyper unit area, which falls under the assumed detectability thresh-old. Similarly, the surface of last contact (SLC) is another spherethat grows from an event in which the nodes ceases to possessthe communication capability (D type event), and carries the lastsignal produced by the emitter. The region which is causally con-nected to the emitter is then limited by these two spheres andtherefore has the shape of a filled sphere (before the D event)or of a spherical shell (after the D event). The temporal intervalsfor the communication between two nodes are represented in thepanel (b) of the Fig. 2. In this scheme, the entire lifetime of a (cid:105) “manuscript” — 2020/7/14 — 1:29 — page 5 — (cid:105)(cid:105)(cid:105) (cid:105) (cid:105)(cid:105) Probability of causal contacts node can be represented by a polygon. This polygon is limitedin the spatial axis by the maximum distance of the signal D max and in the time axis by the time span between the A-type event(concave vertex) and the D-type event (convex vertex). This repre-sentation allows to visualize the important events that result fromthe presence and communications of two nodes. The points in timewhere the nodes acquire their communicating capacity are dubbed“Awakening” events ( A and A ). Similarly, the instants in timewhere the nodes lose their communicating capacity are dubbedâ ˘AŸâ ˘A ´ZDoomsdayâ ˘A ´Zâ ˘A ´Z events ( D and D ). The points inspace–time where the first contact is produced for each one ofthe nodes, are defined as “Contact” events, shown as C and C .Finally, the two points where the contact is lost for each one thenodes, are denominated “Blackout” events, presented as B and B . The receiver node E can listen to signals from the emitternode E , from the first contact event (t=C ) until the last one(t=B ). Similarly, the receiver node E can listen to signals fromthe emitter node E , from t=C up to t=B . The time intervalsfor the open communication channel are determined by the “C”and “B” type events. It is noteworthy the fact that there is a timedelay between the contact events of the two nodes involved in thisanalysis, and also a time delay between the two blackout events.Therefore, the time range when a bidirectional contact is possibleoccurs between the maximum time of the contact points and theminimum time between the blackout points.The temporal structure that emerges from the experimentalsetup implies that, as a fundamental property of the simulations,a causal connection can be produced without requiring that thetwo nodes are active at the same time. This property of the systemarises as a consequence of the large spatial and temporal scales,where a message is transmitted at the (relatively small) light speed.Although a node could be active for a large enough period to trans-mit a message at large distances, the limited power of the messageand the dilution that depends on the squared distance from thesource imposes a detectability limit. As a consequence of this limi-tation and of their finite lifetime, considered as the period betweenthe acquisition and loss of communicating capacity, each node willfill a spherical shell region of the galaxy, limited by two concen-tric spherical surfaces. The leading front, or surface of first contact(SFC) grows from the central node until it reaches the maximumdistance D max . Following the end of the civilisation, there is stilla region which is filled with the emitted signals. This approach hasbeen also considered in other statistical models (e.g., Smith, 2009;Grimaldi, 2017; Grimaldi et al., 2018). The trailing front, or sur-face of last contact (SLC) also grows from the central node, witha delay with respect to the SFC equivalent to the lifetime of thenode, and produces a spherical shell region. Any other node withinthis region will be in causal contact with the originating node, evenif it has disappeared before the time of contact. This region willgrow if the surface of first contact has not yet reached the maxi-mum distance D max , and will shrink after a D-type event until thesurface of last contact reaches D max , producing as a result the lossof all signal from the central node. In our approach, we consider amodel galaxy where the width of the disk is negligible with respectto the radius of the disk. In the 2D simulation only the intersectionof the communicating spherical shells with the plane of the galaxyis relevant, and produce the corresponding circles or rings for thefilled spheres or annular regions of the spherical shells, respec-tively. The initially growing communicating sphere is shown over space–time diagrams, where time is represented on the horizontaldirection, and space is represented in the vertical direction. In theFig. 2 the two emitters in panel (b), E and E , reach each otherat different times. The time span for E i is (A i , D i ) , for i = 1 , .Emitter i can listen to emitter j between C ij and B ij . The typeand length of causal contact in both directions depend on the dis-tance and time lag between the awakening events, the maximumdistance that a signal can reach and the time period in which eachemitter is active.In our experimental configuration the simulation starts assum-ing that the stochastic process is already stable, and finishes beforeany galactic evolution effect could modify the fixed values ofthe variables. Likewise, we assume that the probability for theappearance of a node is homogeneous over the GHZ. The adoptedgeometry of the GHZ in all the simulations is given by a two-dimensional annular region, with an inner radius of 7 kpc andan outer radius of 9 kpc (Lineweaver et al., 2004). Although theGalaxy has a well-known spiral structure, the nodes are assumedto be sparse (otherwise the Galaxy would be full of life) and thespiral structure would not, in that case, produce significant dif-ferences. If the distribution of nodes is not sparse, as it could bethe case if the spiral arms host most of the nodes, then contactswould be more frequent between closely located nodes. In sucha case our results underestimate the number of contacts betweenclose pairs of nodes within the same spiral arm, and conversely,overestimate the number of contacts between separated nodes. Wealso limit the possibilities of life or other types of civilisations tothe usually stated hypotheses for the definition of the GHZ (Dayalet al., 2016; Gonzalez et al., 2001; Lineweaver et al., 2004; Gon-zalez, 2005; Morrison & Gowanlock, 2015; Haqq-Misra, 2019;Rahvar, 2017; Gobat & Hong, 2016; Rahvar, 2017) and considerthat habitability remains constant over time (see, however, Gon-zalez, 2005; Dayal et al., 2016; Gobat & Hong, 2016). This meansthat we set aside civilisations that could survive in severe condi-tions or unstable systems, which would prevent the appearance oflife as we know it.We stress the fact that we are considering causal contactsinstead of actual contacts. In more realistic scenarios, there areseveral sources of “signal loss” with respect to the ideal case.Among them, we can mention temporal and signal power aspectsand direction dependent communication capabilities. The resultsmust then be interpreted as the case of ideal nodes, with a perfectefficiency in the emision and reception of signals. The probabili-ties of contacts presented here are then upper limits to the numberof communications between civilisations in the Galaxy, given theimplemented hipotheses in the model. The use of light conesas causal contact regions is inspired by the fact that light-speedtraveling messengers like electromagnetic radiation, gravitationalradiation or neutrinos are often considered as possible mes-sage carriers (Hippke, 2017; Wright et al., 2018). This excludesmessages sent with mechanical means or physical objects (e.g.,Armstrong & Sandberg, 2013; Barlow, 2013b), or through someunknown technology that violates the known laws of physics.As part of this benchmark, we assume that the capacity to emitand receive signals occur at the same time. Although there areseveral reasons to think that this could not be the case, at largetime scales it can be considered that both abilities occur roughlyat coincident epochs. In the ideal setup, this would be equivalent tonodes that send messages isotropically and scan the local skies on (cid:105) “manuscript” — 2020/7/14 — 1:29 — page 6 — (cid:105)(cid:105)(cid:105) (cid:105) (cid:105)(cid:105) independent variables (parameter space) min. value max. value Nbins τ a Mean temporal separation between consecutive awakenings 5 · yr 40500 yr 21(linear grids) 5 · yr 10 yr 51 τ s Mean lifetime of a node 5 · yr 40500 yr 21(linear grids) 5 · yr 10 yr 51D max Maximum reach of a message 100 pc, 500 pc, 1 kpc, 5 kpc, 10 kpcfixed variables assumptions valuestatistical properties of all nodes equally distributedPoint process for the distribution in time homogeneousf s The scan of the sky fully efficient 1f p panspermia or colonization absent 0shape of the Galactic Habitable Zone two-dimensional ringR min GHZ
Inner radius of the GHZ Lineweaver et al. (2004) 7 kpcR max
GHZ
Outer radius of the GHZ Lineweaver et al. (2004) 9 kpct max
Time span of the simulation 10 yr - 10 yrnumber of random realizations for each point in the parameter space 1 - 50discrete events affected variablesA event Awakening: a node starts its communication capabilities Number of active nodesB event Blackout: the end of the communication channel stops Number causal channelsC event Contact: a new causal contact is produced Number of causal channelsD event Doomsday: a node ends its communication capabilities Number of active nodesTable 1.: Definition of independent variables and adopted values for fixed parameters that are part of the simulation. Variable parametersdefine the spatial and temporal structure of the process and the maximum reach of the messages.all directions with a perfect efficiency. Another essential assump-tion is that all nodes use the same signal power, so that there isa maximum distance out to which it can be detected. In such asystem we compute probabilities of a random node making con-tact with another node, i.e., they are not specific for the case ofcontacts with the Earth. Regarding the extent of the signals, weknow that the distance from which a signal from Earth could bedetected using the current technology is about few parsecs, giventhat the signal was sent to a specific direction. It is straightforwardto propose and implement a distribution of maximum distances,although this would increase the model complexity at the cost ofa larger uncertainty.In our simulations, we assume the simplest configuration forthe growth of the sphere of first contact. In particular, we do notconsider the possibility of stellar colonisation (e.g. Newman &Sagan, 1981; Walters et al., 1980; Starling & Forgan, 2014; Bar-low, 2013a; Jeong et al., 2000; Maccone, 2011). We also assumethat communication is equally likely in all directions, i.e., weassume isotropic communication in all cases. Different commu-nication efficiencies or detection methods are straightforward tocarry out in the simulations for more detailed and complex sce-narios. This approach, however, is beyond this work because itobscures the experimental setup and make the results less clear.The assumptions we accept imply that the results are independentof whether intelligent agents are organic or artificial. Moreover,the causes of the limited lifetime of a civilisation can be natu-ral (astrophysical phenomena), caused by auto destruction or byexternal factors, to name a few. However, we assume that theseevents are sufficiently numerous in the Galaxy so that a statisti-cal model is plausible. The failure of this hypothesis would implyextreme values for the parameters that represent the density of thenodes (i.e., τ a ). The temporal structure of the process is defined by two distribu-tion parameters. One of them represent the mean time interval thata node can emit and receive signals (its lifetime), and the other themean time interval between the emergence of consecutive nodes.The spatial structure of the simulation is given by the size andshape of the Galactic Habitable Zone and the maximum distancea signal can travel to be detected (D max ). The parameters for thetemporal distributions also determine the spatial properties, sincethe density of active nodes in the galaxy depend on these twoparameters. For example, small τ a and a large τ s will producea densely populated galaxy (in this context, a galaxy is an ele-ment in a statistical ensemble, not the Milky Way). Also, somehypotheses regarding the shapes of the distributions of the tempo-ral parameters must be made in order to complete the simulation.Forgan (2011) argues that the times at which different civilisa-tions become intelligent follow a Gaussian distribution, and thenthe distribution of inter-arrival times is an inverse exponential. Weassume that the distribution of the times of A events is a sta-tionary Poisson process, and then the distribution of the timesbetween the appearance of new consecutive nodes is exponential.Regarding the duration of a node, we propose that its distributionis a stationary exponential distribution. There are no clear argu-ments to conclude a statistical law for the later distribution, sothat we propose it as a working hipothesis. This heuristic does notmake any consideration about the origin of life, although differentapproaches are possible. For example, Maccone (2014b) arguesthat this distribution should be a log-normal. Preferentially, a theo-retical statistical distribution of the lifespan of civilisations wouldrely on the basis of the underlying astrophysical and biologicalprocesses (Balbi, 2018).The power law and exponential statistical distributions areamong the most common patterns found in natural phenomena. (cid:105) “manuscript” — 2020/7/14 — 1:29 — page 7 — (cid:105)(cid:105)(cid:105) (cid:105) (cid:105)(cid:105) Probability of causal contacts For example, the distribution of the frequency of words in manylanguages is known to follow the law of Zipf (which is a powerlaw). These distributions arise any time a phenomenon is charac-terised by commonly occurring small events and rarely producedlarge events (e.g. Adamic, 2000). Zipf law also describes pop-ulation ranks of cities in various countries, corporation sizes,income rankings, ranks of the number of people watching thesame TV channel, etc. The magnitudes of earthquakes, hurri-canes, volcanic eruptions and floods; the sizes of meteorites orthe losses caused by business interruptions from accidents, arealso well described by power laws (Sornette, 2006). Additionalexamples include stock market fluctuations, sizes of computerfiles or word frequency in languages (Mitzenmacher, 2004; New-man, 2005; Simkin & Roychowdhury, 2006). Power laws havealso been widely used in biological sciences. Some examples arethe analysis of connectivity patterns in metabolic networks (Jeonget al., 2000) or the number of species observed per unit area inecology (Martín & Goldenfeld, 2006; Frank, 2009). More exam-ples can be found in the literature (Martín & Goldenfeld, 2006;Maccone, 2010; Barabási, 2009; Maccone, 2014a,b). This distri-bution family is suitable for the statistical description of the D max parameter, although in this work we assume a uniform distribu-tion for simplicity. The power law family of functions is also agood candidate for the description of the temporal variables inthe model. However, we prefer the exponential model. The expo-nential distribution of lifespan and waiting times is justified byconsidering the hypothesis that the process of appearance of lifein the galaxy is homogeneous and stationary. That is, there is nota preferred region within the GHZ for the spontaneous appear-ance of life, and the emergence of a node is independent of theexistence of previous nodes in the galaxy. These seem to be sim-ple conditions, and allow to propose a distribution family withoutknowing the details of the underlying process. The exponentialdistribution for the separations in time is equivalent to proposinga Poisson process for the emergence of nodes, given the relationbetween the number of events in time or space and the wait-ing time or separation, respectively (e.g., Ross, 2012). That is,these are two alternative approaches to describing the same pro-cess, a Poisson distribution for the number of events implies anexponential distribution for their separations, and vice versa. Itshould be emphasised that the exponential laws used in this workare assumed as part of the working hypothesis, and instead ofanalysing results from a particular parameter chosen ad hoc, weexplore the parameter space and analyse the impact of the valuesof these parameters on the results. In this Section we discuss the degree of complexity in the model,considering a compromise between the accuracy of the model andthe number of parameters that are free or with a high uncertainty.Firstly, we emphasise that the odds of a causal contact betweentwo nodes should not be considered as the odds of a contactbetween two intelligent civilisations, and in fact the latter could bemuch lesser than the former. Indeed, in order to establish a con-tact between any two entities, a minimum degree of compatibilitymust be accomplished without any previous agreement, makingthe possibility of a contact with a message that could be deci-phered highly rare (see e.g. Forgan, 2014). Besides the trade–off
Fig. 3. : Empirical cumulative distributions of the number con-tacts for nodes in different samples and with D max =10 kpc. Theupper panel shows the variation of the distribution as a func-tion of the τ a parameter, including the values 0.1, 0.5, 1, 2, and5 kyr. All the curves correspond to models with τ s =
40 kyr andD max = 10 kpc. The bottom panel corresponds to τ a = 1 kyr andD max = 10 kpc, and the following values of τ s : 5, 10, 20 and50 kyr. Cumulative distributions can be used to visualise the prob-ability of a random node of having more than M contacts in agiven model. For example, in the upper panel the probability of arandom node having more than 10 contacts is nearly one for themodel with the smallest τ a value, and nearly zero for the modelwith the largest τ a value.between the simplicity and the complexity of the experiments,further analysis could be performed following this frameworkin order to explore possible implications of the results for moredetailed configurations. For example, the communication method(isotropic, collimated, serendipitous) can affect the observables,making it necessary to implement a correction factor. Taking this (cid:105) “manuscript” — 2020/7/14 — 1:29 — page 8 — (cid:105)(cid:105)(cid:105) (cid:105) (cid:105)(cid:105) into account, our results regarding the probability of causal con-tact should be considered as upper limits for effective contacts,since they depend on the efficiency of both the emitter and thereceiver to broadcast and scan the sky for intelligent signals,respectively (Grimaldi, 2017). A correction by a coverage ratio inthe detection and by a targeting ratio in the emission could be eas-ily implemented in the simulation, although the effect of reducingthe probability of contact is basically the product of the efficiencyratios and thus such implementation is not necessary. Therefore,the values of the probabilities could be modified by a constantcorrection factor equal to the combined emission/reception effi-ciency (Smith, 2009; Anchordoqui & Weber, 2019; Forgan, 2014)or for beam-like transmissions (Grimaldi, 2017). Other consider-ations include the effects of alignments, the use of stars as sourcesor amplifiers (Edmondson & Stevens, 2003; Borra, 2012), or thenature of the message carrier. Another improvement could be theuse of a spatial distribution that resembles the spiral shape andthe width of the disc of the Galaxy. Regarding the distributions ofthe parameters, different distribution functions for the mean life-time of nodes can be implemented as alternatives. Regarding thedegree of idealisation, the model admits different efficiencies forthe sphere of the causal region of each node, featuring differentsearching strategies (Hippke, 2017). It is also possible to con-sider that D max is different for different nodes. For example, apower law where a powerful emission is rare and a low emissionis common could be an improvement to the model. In this workwe choose not to implement this for the sake of simplicity. Finally,the role of the message contents could influence on the lifespan ofa node that receives a message, although the implementation ofsuch behaviour would increase the number of free parameters andwould be more speculative. We limit the scope of this work to asimple version of the model. Once the model has been defined, itcan be implemented as a discrete event simulation, as described indetail in the next Section. A discrete event simulation is performed for a given model, inthis work M ( τ a , τ s , D max ) , by keeping track of a set of vari-ables that change each time an event happens. In addition, themodel comprises elements that are fixed for all simulations, forexample, the functional forms of the statistical distributions andthe adopted values of particular variables (see 1). The main vari-ables that follow the evolution of the simulation are: the positionsof stars, which are sampled randomly within the GHZ; the timeof the awakening of each node (A event); and its time of disap-pearance (D event). The variables that can be deduced from theprevious ones include: the number of nodes in casual contact withat least another node at a given time; the number of nodes as afunction of time; the number of nodes that receive at least onemessage; the number of nodes that receive a message at least onetime and successfully deliver an answer; and the number distribu-tion of waiting times to receive a message. All these quantities areupdated each time one of the four events (A, B, C, D) occurs. In order to make a reproducible project, we developed the toolsthat allow to run the simulations and obtain the results shown in this work. The simulations were implemented on a Python-3 code, dubbed
HEARSAY (Lares et al., in preparation), whichis publicly accessible through the GitHub platform a under theMIT-license. The project is in the process of registration withthe â ˘AŸâ ˘A ´ZAstrophysics Source Code Libraryâ ˘A ´Zâ ˘A ´Z (ASCL,Allen & Schmidt, 2015; Allen et al., 2020) From the viewpointof a user, HEARSAY is an object-oriented package that exposesthe main functionalities as classes and methods. The code fulfillsstandar quality assurance metrics, that account for testing, style,documentation and coverage. In the configuration step, the userprepares the set of simulation parameters through an initialisationfile. Since configuration files and simulation results are persistent,it is straightforward to keep track of different experiments and theexperiments can be revisited easily. An in-depth description of themethods can be found in the documentation, which is automati-cally generated from
HEARSAY docstrings and made public in theread-the-docs service b . Since the simulation setup is configurable,the time required for a simulation to complete depends on the sim-ulation parameters. It also depends on the hardware that is used tomake the run, and on whether the parallel option is set. However,it is simple to make a local run of limited versions of the experi-ments to have a sense of the time required by the code to completethe simulations. More information on this can be found on thedocumentation of the software.
3. RESULTS: EXPLORING THE PARAMETER SPACE
We implemented the simulation of a regular grids of models vary-ing over the parameter space, which covers 5204 models. For eachmodel, we simulated several realizations with different randomseeds, adding up a total of 158546 simulation runs. The numberof random realzations varies from one (for the densest populatedmodels) up to 50 (for the sparsely populated models). The param-eters for the temporal aspects of the simulation (the mean waitingtime for the next awakening, τ a , and the mean lifetime, τ s ) coverthe ranges 10 -10 yr, with two linear partitions of 21 values (10 -40500 yr) and 51 values ( · -10 yr) for each parameter. Thispartition was chosen after the requirement of the software to takelinear bins, aiming at a better sampling of the low τ a and low τ s region of the parameter space. For the D max parameter, we takethe values 100 pc, 500 pc, 1 kpc, 5 kpc and 10 kpc. In the Table 1we show the three variable parameters, the ranges of their valuesand the number of bins that have been explored in the numericalexperiments. We also show the set of fixed parameters that takepart in the simulation, their values and the hypotheses that theyrepresent.As a product of the simulations, several quantities can beobtained. Some of them are directly derived from the discreteevents, namely, the ID of emitting and receiving nodes and theposition in the galaxy. We also save the times of each of the eventsthat are relevant to keep track of the number of nodes for eachsimulation, i.e., the times of the four types of events. The times ofC-type and B-type events are used to derive the number of con-tacts. We also obtain quantities that represent the properties of thenodes, for example the total time elapsed between the A-type and a https://github.com/mlares/hearsay b https://hearsay.readthedocs.io/en/latest/ (cid:105) “manuscript” — 2020/7/14 — 1:29 — page 9 — (cid:105)(cid:105)(cid:105) (cid:105) (cid:105)(cid:105) Probability of causal contacts Fig. 4. : The fraction of constrained causal contact nodes that never make contact (listening) as a function of τ a and τ s , for D max =1 kpc(left panel), D max =5 kpc (middle panel), and D max =10 kpc (right panel). The values of τ a and τ s are in the range 5 · to 10 yr. Thematrix is shown as obtained from the simulations, and the level curves are shown for the smoothed matrix. Fig. 5. : The fraction of constrained causal contact nodes that make contact at the moment of the awakening (i.e., t A = t C ), as a functionof τ a and τ s , for D max =1 kpc (left panel), D max =5 kpc (middle panel) and D max =10 kpc (right panel). The values of τ a and τ s arein the range 5 · to 10 yr. The matrix is shown as obtained from the simulations, and the level curves are shown for the smoothedmatrix.the D-type of each node, which represent their corresponding life-times. The time span of a node listening another or being listenedby another node can also be derived by keepig track of the timesof the events in a simulation. This way we can also compute thedistribution in the simulated galaxy of nodes that reach contact,the distributions of the waiting times until the first contact or the distributions of waiting times until the next contact. The followingproperties of the population of nodes can also be derived: the frac-tion of the lifetime a node is listening to at least another node (i.e.,within their light cone), the age of contacted nodes at first contact,the fraction of nodes where the first contact is given at the awak-ening, the distribution of the number of contacts for each node, (cid:105) “manuscript” — 2020/7/14 — 1:29 — page 10 — (cid:105)(cid:105)(cid:105) (cid:105) (cid:105)(cid:105)
10 Lares, Funes & Gramajo
Fig. 6. : Histograms of the mean waiting times for the first contact,for several models. Upper panel shows the histograms for sev-eral values of τ a , and τ s =10 kyr and D max =10 kpc. Bottom panelshows the histograms for several values of τ s , with τ a =10 kyr andD max =10 kpc. The arrows indicate the sample and highlight thevalues of the first waiting time bin, which has a clear excess forthe models with larger τ s values. The choice of these models wasmade in order to show the trends in the results as a function of thetwo temporal parameters.the distribution of the number of contacts as a function of the ageof the node, the number of contacts as a function of time in thegalaxy, the fractions of nodes that succeed in making contact, andthe distribution of distances between contacted nodes. Anotheruseful derived quantity is the duration of two-way communicationchannels or the fraction of contacts that admit a response. It is alsopossible to analyze the relations between the distance to node vs.the time of two-way communications, the distance to node vs. theage of contacted node, the age of a node and the maximum num-ber of contacted nodes before the D–type event, or the lifespan of a node vs. the maximum number of contacts. All these quantitiescan be analyzed as a function of the simulation parameters. In Fig. 3 we show the empirical cumulative distributions of thenumber contacts for nodes in six different samples, includingshort and long lifetimes, dense and sparse spatial distributionand D max =10 kpc. As it can be seen, the mean lifetime is moredeterminant than the mean awakening rate (dense and sparse, rep-resented by a different shade) for the number of contacts. Themodel with a dense awakening in the timeline (low τ a ) maxi-mizes the number of contacts, reaching a maximum of more than300 contacts for a single node. This case, however, requires thata new node appears in the Galaxy every 100 years on average.Similarly, the model with long lifetimes has the maximum num-ber of contacts, reaching nearly 100 contacts for each node in itsentire lifetime. This is considerably larger than any model with ashorter lifetime, which produce a number of contacts of at mostthe order of ten contacts per node. It is expected then that a modelwhere nodes appear with a high frecuency and have very long life-times can reach tens of contacts on the full time period between t A and t D . On the other side, a model where the activation ofnew nodes requires a large waiting time and the survival time isshort, contacts are extremely rare. We should point out that thisplot has a logarithmic scale on the x-axis, and it is the cumulative,not differential, empirical distribution. Therefore, the differencesin the number of contacts for different models are large. This isa consequence of the wide range in both τ a and τ s covered in thesimulations. With this ranges, the fraction of nodes with no contactranges from nearly zero up to one. This analysis is made in order toexplore the behavior of the SC3Net model. On the Fig. 4 we show2D color maps with the fraction of nodes in the simulations thatnever make contact (i.e., never listen to another node), as a func-tion of the mean lifetime ( τ s , in the range 5 · to 10 yr) and themean awakening time ( τ a , in the range 5 · to 10 yr), for threedifferent values of the maximum signal range, D max =1 kpc (leftpanel), D max = 5 kpc (middle panel), and D max = 10 kpc (rightpanel). A clear pattern emerges, showing that the probability fora node of making causal contact with at least another node duringtheir entire lifetime, increases with increasing D max , increasing τ s and decreasing τ a , following a roughly linear dependence withthe three parameters. The results of the simulations that comprisethe full range of values for τ a and τ s from 5 · yr to yr (notshown in the Figures) maintain a similar trend. The number ofnodes that do not succeed in reaching the causal contact regionsof other nodes is a useful indicator of the degree of isolation. Onthe other hand, there is also the chance that a number of nodesare already in the causal contact region of other nodes, at the timeof their A-type events. The fraction of nodes that make the firstcontact at the awakening event is shown in the Fig. 5, as a func-tion of the mean lifetime ( τ s , in the range 5 · to 10 yr) andthe mean awakening time ( τ a , in the range 5 · to 10 yr). Thethree panels correspond to different values of the maximum signalrange: D max = 1 kpc (left panel), D max = 5 kpc (middle panel),and D max = 10 kpc (right panel). The dependence of this metricwith the three parameters is roughly linear in all cases. (cid:105) “manuscript” — 2020/7/14 — 1:29 — page 11 — (cid:105)(cid:105)(cid:105) (cid:105) (cid:105)(cid:105) Probability of causal contacts In this subsection we analyze the distribution of the waiting timesfor a first contact. Such distribution can be considered to computethe probability for a random node to carry out the searching ofother nodes and spend a given time until the first contact is made,under the hypotheses of the experiment. In the Fig. 6 we showthe histograms of the mean waiting times for the first contact, forseveral models. Upper panel (a) panel shows the histograms forseveral values of the mean awakening time τ a , with the mean sur-vival time τ s of 10 kyr. Lower panel (b) shows the histograms forseveral values of τ s , with τ a =10 kyr. In both panels the value ofD max is set to 10 kpc. As it can be seen, there is a clear trendwhere the number of nodes that require a time t to make the firstcontact decreases exponentially with the time t . The fraction ofthe nodes that have made at least one contact, as a function ofthe elapsed time since the awakening, is computed with respectto the total number of nodes that make causal contact at least onetime in the time range from the A–type up to the D–type events.From a frequentist approach, the cumulative fraction of contactsfrom the awakening ( t = t =0 yr. That is, fora short period of time the initial moment is the most promis-ing for making a (causal) contact, for a given technology. Thisis given by the fact that at the awakening event, many nodes arealready on the light cones of other nodes, so that the awakeningtime offers the best chance of making contacts. This offers a newapproach to SETI programs, where the search for new commu-nication technologies or the exploration of new communicationchannels has a fundamental role and could be more efficient thanlong observation programs.
4. DISCUSSION
We have presented a stochastic model (SC3Net) to analyse thenetwork of constrained causally connected nodes in a simplifiedGalaxy. It represents an idealised scenario of perfectly efficientemitters and receivers with the restriction of a maximum distanceseparation. These emitters and receivers correspond to nodes thatform a communication network whose properties depend on theirdensity and mean survival time. The statistical analysis of themodel allows to estimate the probabilities for a random node andfor a given model instance of making contacts with other nodes,along with the waiting times and durations of such contacts. Usingnumerical simulations, we implemented this model to explore thethree-dimensional parameter space, considering a grid of meantime separations between the activation of new nodes in the range500–10 yr. Additionally, we explore a grid of the mean survivaltimes between 500 yr and 10 yr, and values for the maximumdistance range of signals of 100 pc, 500 pc, 1 kpc, 5 kpc and10 kpc. The simulation of each parameter point was performed several times with different random seeds in order to improve theconfidence of the derived quantities.Although the simulations use several hypotheses, we argue thatthe model is not worthy of further complexity, given the lim-ited knowledge about the origin and persistence of life in theGalaxy. The implementation of more detailed or sophisticatedmodels would increase the number of free parameters withoutany improvement in their predictive power. Thus, we take advan-tage of the simplicity of the model to explore the parameter spacein order to gain insight on the consequences of different scenar-ios for the search of intelligent life. Our analysis is not centredin obtaining the odds for the Earth to make contact with anothercivilisation. Instead, we focus on obtaining a statistical, parameterdependent description of the possible properties of the communi-cation networks that comprise sets of nodes with broadcasting andreception capabilities. This causally connected nodes are sparselydistributed in both space and time, making analytical treatmentsdifficult and justifying the simulation approach.Under the hypotheses of our experiments, we conclude that acausal contact is extremely unlikely unless the galaxy is denselypopulated by intelligent civilisations with large average lifetimes.This result is qualitatively similar to the results presented by sev-eral authors, which state that a contact between the Earth andanother intelligent civilisation in the Galaxy is quite unlikely, pro-vided the maximum distance of the signal and the lifetime of theemitter are not large enough. This analysis supports the idea that,in order to increase the possibilities of a contact, more activestrategies of the emitter would be required. Some proposals inthis direction include intertellar exploration, colonization and set-tlement (Brin, 1983; Došovi´c et al., 2019; Galera et al., 2019),although it would require large temporal scales. Došovi´c et al.(2019) use probabilistic cellular automaton simulations to explorethe parameter space of a model with colonization and catastrophicevents. According to the timescales involved, their results couldexplain the Fermi paradox. Although our work does not take intoaccount the colonization hypothesis, it does consider catastrophicevents implicitly in the mean value and distribution of the lifetime, τ s . Other strategies could also increase the probability of con-tacts, for example panspermia (e.g., Starling & Forgan, 2014) orself–replicating probes (e.g., Barlow, 2013b), although they wouldbe too slow to make a significant impact on the communicationnetwork among intelligent civilisations. Our results are also con-sistent with those presented by Grimaldi (2017), who estimatesan upper limit for the mean number of extraterrestrial civilisationsthat could contact Earth using Monte Carlo simulations, from astatistical model where the width of the Galactic disk is not neg-ligible. Unlike most of the studies that make use of statisticalmodels or simulations ( ´Cirkovi´c, 2004; Smith, 2009; Bloetscher,2019), our approach does not relay on the Drake equation. Thus,it does not need a detailed description or modeling of the physicalprocesses that give rise to intelligent life. However, we argue thatit is a valid empirical formulation to discuss the probabilities ofcontact and the time scales involed in the problem. Ours is an alter-native to the method proposed by Balbi (2018), who performs ananalysis based on the Earth with a different model. Despite thosedifferences, their results and ours are in general agreement.We have also fond that there is a balance between the densityof nodes and the mean lifetime since, as expected, a lower densitycan be compensated by a longer active time period. However, a (cid:105) “manuscript” — 2020/7/14 — 1:29 — page 12 — (cid:105)(cid:105)(cid:105) (cid:105) (cid:105)(cid:105)
12 Lares, Funes & Gramajo large number of nodes does not easily compensate their short livesto reach the same probability of causal contact than in the caseof a less populated galaxy but with very ancient civilisations. Inall cases, for a short period of time (for instance, the time SETIprograms have been active on Earth), the maximum probabilityof making a contact occurs at the moment of the awakening. Thissuggests the possibility that an alternative SETI strategy could bethe search for alternative message carriers, for the case in whichthe search has not been performed on the adequate channels. Then,if a contact is produced for the first time, the origin of the signal ismore likely to be very old. Also, the chances of entering the causalconnected zone of a node does not grow linearly, but favours thefirst period of 10 years.In the approach of this work, we have used computer simula-tions to address the problem of the probabilities of causal contactsbetween locations in the Galaxy with the possibility of sendingand receiving messages. Instead of making a number of assump-tions, we have explored the parameter space, reducing the problemto only three parameters and a few simple hypotheses to perform acomplete model for the population and communication network inthe galaxy. This allows to consider the Fermi paradox from a newperspective, and to propose an alternative treatment for the num-ber of intelligent emitter/receivers. If the time intervals betweenthe rise and fall of civilisations are short compared to the timerequired for an electromagnetic signal to travel the large distancesin the Galaxy, then the number of contacts would be limited to alow number.The short time interval between the rise and fall of civilisa-tions, compared to the age and extension of our Galaxy, is afundamental limitation for the number of contacts. The temporaldimension, which is missing in the Drake equation, is a key factorto understand the network of contacts on different scenarios. Acknowledgement.
This work was partially supported by the ConsejoNacional de Investigaciones Científicas y Técnicas (CONICET, Argentina),the Secretaría de Ciencia y Tecnología, Universidad Nacional de Córdoba,Argentina, and the Universidad Católica de Córdoba, Argentina. This researchhas made use of NASA’s Astrophysics Data System. Plots and simulationswere made with software developed by the authors the python language. Plotswere postprocessed with inkscape. Simulations were run on the Clementecluster at the Instituto de Astronomía Teórica y Experimental (IATE).
Disclosure statement.
No competing financial interests exist.
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Probability of causal contacts13