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Determining upper limits on galactic ETI transmitting continuous beacon signals in the spectrum A Bayesian approach to the

Author: Mikael Flodin (810815-0171) [email protected]

Department of Physics Royal Institute of (KTH)

Supervisor: Lars Mattsson

June 14, 2019 Typeset in LATEX

TRITA-SCI-GRU 2019:305

© Mikael Flodin, 2019 Plurality must never be posited without necessity William of Ockham, 14th century Abstract

This thesis put constraints on the Fermi paradox by determining upper limits on the present time pop- ulation of galactic, radio-communicating, ETI civilizations, conditioned on the current null result of the joint SETI project. This is done via a approach, where five idealizing assumptions, (A1)-(A5), on ETI characteristics are adopted, in order to simulate the likelihood of detection in current and historical SETI surveys. The upper limits are then determined by conducting a Bayesian analysis on the simulated data. The combined result from 12 different submodels, regarding galactic geometry, Equivalent Isotropic Radiated (EIRP) and transmitting frequency, are analyzed and presented in the thesis. The main results, depending on the submodel, yields an upper limit (3σ) in the order of thousands KI civilizations (corresponding to an EIRP of 1016 W), currently transmitting in the . The result also suggests that the presence of radio-communicating KII and KIII civilizations are sparse or nonexistent. In the hypothetical event of detecting exactly one KI signal, it can be inferred that hundreds, or thousands, similar signals, that have eluded detection, exists at ’s position. A threshold EIRP is determined to approximately 1019 W, above which the upper limits are constant, and below which the upper limits are highly sensitive to the EIRP. The conclusion is that the Fermi para- dox is an actual paradox only if the real galactic ETI population obeys the idealized assumptions and simultaneously exceeds the determined upper limits.

Key words: Bayesian analysis, ETI-civilization, Fermi paradox, Radio communication, Monte Carlo simulation, Null result.

Sammanfattning

Denna avhandling unders¨oker Fermi-paradoxen genom att best¨amma ¨ovre gr¨anser f¨or nuv¨ardespopula- tionen av galaktiska, radiokommunicerande, ETI-civilisationer, betingat p˚adet nuvarande nollresultatet av det sammanlagda SETI-projektet. En Monte Carlo-metod anv¨ands, d¨ar fem idealiserade antagan- den, (A1)-(A5), g¨ors om ETI-civilisationers egenskaper, f¨or att simulera sannolikheten f¨or detektion i nu aktuella och historiska SETI-studier. De ¨ovre gr¨anserna best¨ams sedan genom att utf¨ora en Bayesiansk analys p˚asimulerad data. Det sammanlagda resultatet fr˚an12 olika delmodeller, som tar h¨ansyn till galak- tisk geometri, ekvivalent isotrop utstr˚aladeffekt (EIRP) och s¨andarfrekvens, analyseras och presenteras i avhandlingen. Huvudresultatet, beroende p˚amodel, ger en ¨ovre gr¨ans (3σ) motsvarande n˚agratusen KI-civilisationer (motsvarande en EIRP av 1016 W), som f¨or n¨arvarande s¨ander signaler i galaxen. Re- sultatet indikerar ocks˚aatt f¨orekomsten av radiokommunicerande KII- och KIII-civilisationer ¨ar sparsam eller obefintlig. I det hypotetiska scenariot att exakt en KI-signal detekteras, kan det konstateras att hundratals eller tusentals liknande signaler, som har undg˚att detektion, existerar vid jordens position. Ett tr¨oskelv¨arde f¨or utstr˚aladeffekt (EIRP) best¨ams till ungef¨ar 1019 W, ¨over vilket de ¨ovre gr¨anserna ¨ar konstanta, och under vilket de ¨ovre gr¨anserna ¨ar k¨ansligt beroende av EIRP-v¨ardet. Slutsatsen ¨ar att Fermi-paradoxen endast ¨ar en faktisk paradox om den verkliga galaktiska ETI-populationen f¨oljer de idealiserade antagandena och samtidigt ¨overskrider de best¨amda ¨ovre gr¨anserna.

Nyckelord: Bayesiansk analys, ETI-civilisation, Fermi-paradoxen, Radiokommunikation, Monte Carlo-simulering, Nollresultat. Glossary

EIRP – Equivalent Isotropic Radiated Power

ETI – Extraterrestrial

SETI – Search for Extraterrestrial Intelligence

GG – Galactic Geometry (model category)

EMW – Entire (submodel)

GHZ – (submodel)

RP – Radiated Power (model category)

SPO – Static Power Output (model category)

PPO – Progressive Power Output (model category)

TF – Transmitting Frequency (model category)

HI – line, 21 cm (submodel)

WHR – Water-hole Region, 1.1-1.9 GHz (submodel)

QR – Quiet Region, 1-10 GHz (submodel)

ATA –

GBT –

SKA –

VLA – Very Large

FFT – Fast Fourier Transform

RA – Right Ascension (unit hours)

DEC – Declination (unit degrees)

RFI – Radio Frequency Interference

Units

1 kpc – 1 kiloparsec = 3260 light

1 ly – 1 light = 9.46 × 1015 m

1 Jy – 1 = 10−26 Wm−2

1 GHz – 1 Gigahertz = 109 s−1 Foreword

Are we alone in the ? That is perhaps the oldest and most fundamental question in history. The answer, whatever it is, would have profound implications, not only to but for the human civilization at large. Questions about man and her position in cosmos have stimulated human philosophy for millennia. Anaximander (610-546 B.C.) discussed the concept of Cosmic Plurality, an idea of multiple, or even an infinite number, of harboring extraterrestrial . The atomist Metrodorus (331-277 B.C.) strongly argued that “To consider the Earth as the only populated world in infinite space is as absurd as to assert that in an entire field only one grain will grow.” During Medieval times and Renaissance, such thoughts were banned by the Catholic Church. Philosopher Giordano Bruno (1548-1600 A.D) challenged the dogmatic views of the church by claiming that “Innumerable exist; innumerable revolve around these suns in a manner similar to the way the seven planets revolve around our . Living beings inhabit these worlds.” Refusing to reject his ideas, Bruno was sentenced for blasphemy and burned at the stake in Campo dei Fiori, Rome, on February 17, 1600 (Angelo, 2007; Fitzgerald, 2017). Bruno was a philosopher and an advocate of free thought, however, it is only in our era these ideas have become a little bit more than mere speculation. Contents

1 Introduction 3 1.1 Introduction ...... 3 1.2 Purpose ...... 4 1.3 Outline of the Thesis ...... 4 1.4 Author’s contribution ...... 4

2 Background 5 2.1 DefiningETI ...... 5 2.2 The Fermi Paradox ...... 5 2.3 Solutions to the Fermi Paradox ...... 6 2.3.1 They exist and they are (were) here ...... 6 2.3.2 They exist but we have not yet established contact ...... 7 2.3.3 They do not exist ...... 9 2.4 The ...... 10 2.4.1 Two Examples ...... 10 2.4.2 Alternative Form ...... 11 2.5 Figure of Merits ...... 11 2.6 SETI Surveys ...... 13 2.7 The ...... 14

3 Theory 15 3.1 The Poisson Process ...... 15 3.1.1 The Nonhomogeneous Poisson Process ...... 15 3.1.2 A Birth and Death Process ...... 16 3.1.3 A Statistical Drake Equation ...... 17 3.2 Bayesian Analysis ...... 18 3.2.1 The Likelihood: Two Special Cases ...... 19 3.2.2 The Prior Distribution ...... 20 3.2.3 The Posterior Distributions ...... 20 3.3 Elements from ...... 21 3.3.1 Isotropic and Directional Antennas ...... 21 3.3.2 The Radiometer Equation ...... 21 3.3.3 Doppler Drift ...... 22 3.4 Galactic and Equatorial Coordinates ...... 22

4 Simulation overview 24 4.1 Description of the Simulation Algorithm ...... 24 4.2 Submodels...... 27 4.2.1 Galactic Geometry ...... 27 4.2.2 Radiated Power ...... 27 4.2.2.1 The PPO(I-III) Models ...... 27 4.2.2.2 The SPO(KI-KIII) Models ...... 28 4.2.3 Transmitting Frequency ...... 29 4.3 Filtering Conditions ...... 31 4.3.1 Causal Contact ...... 31 4.3.2 Sky Coverage ...... 31 4.3.2.1 All-Sky Surveys ...... 31

1 4.3.2.2 Targeted Searches ...... 32 4.3.3 Signal-to-Noise Ratio ...... 32 4.3.4 Frequency Coverage ...... 33 4.4 Boundaries ...... 33 4.4.1 The Birth Rate Parameter ...... 33 4.4.2 The Lifetime Parameter ...... 33 4.4.3 Sampling Domain ...... 34

5 Results 36 5.1 Initial Results ...... 36 5.1.1 Two Important Distributions ...... 36 5.1.2 The Significance of the Lower Cutoff ...... 36 5.2 Submodel Results ...... 37 5.2.1 Varying Galactic Geometry ...... 39 5.2.2 Varying Radiated Power ...... 40 5.2.2.1 SPO Models ...... 40 5.2.2.2 PPO Models ...... 41 5.2.3 Varying Transmitting Frequency ...... 42 5.2.4 Constraints on the Fermi Paradox ...... 42 5.3 Additional Results ...... 47 5.3.1 Mean Minimum Distance ...... 47 5.3.2 Relation between Population Parameters ...... 47 5.3.3 Future Limits ...... 47

6 Discussion and Conclusion 50 6.1 Discussion ...... 50 6.2 Conclusions ...... 52 6.2.1 Limitations of the Thesis ...... 53 6.2.2 Current and Future Research ...... 53 6.2.3 Is SETI Meaningful? ...... 54

7 Acknowledgements 55 Chapter 1

Introduction

1.1 Introduction

The and other observatories have discovered 4003 to date1. Approxi- mately 1-2 % of them are Earth-sized planets (0.5R⊕ ≤ R ≤ 2R⊕) orbiting in the habitable zone of their (Exoplanetarchive.ipac.caltech.edu, 2019). Extrapolations of the planetary data further suggest that there could be 40 billion habitable worlds in the Milky Way Galaxy (Petigura et al. 2013). In addition building blocks of life (such as aromatic hydrocarbons, isopropyl cyanide, and amino acids) have been detected in , meteorites, and in the (e.g. Belloche, 2014; Glavin et al. 2010). By the , this should imply the existence of life, and possibly, intelligent life elsewhere in the galaxy. Early probabilistic estimates suggested that the Milky Way harbors between a thousand and a million extraterrestrial intelligent civilizations (ETIs) (e.g. Shklovski & Sagan, 1966; Drake, 1993). Other estimates with only conservative assumptions show, however, that a single technological civiliza- tion, capable of , should be able to spread all over the Milky Way within just tens of millions of years, a minuscule fraction of the present age of the galaxy (Hart, 1975; Newman & Sagan, 1981). Nevertheless, not a single sign of such civilizations has ever been recorded. This discrepancy, between the high estimates of ETIs and the lack of evidence thereof, was addressed by physicist Enrico Fermi. In 1950, while doing research in Los Alamos National Laboratory, he spontaneously asked his colleagues at the lunch table: “[If there are so many of them] Where is Everybody?” (Jones, 1985). This remark has since become known as the “Fermi paradox.”2 The Search for ExtraTerrestrial Intelligence (SETI) began around 1960. In (1961), Drake observed two from the , and Epsilon Eridani, at frequencies around the H I line (21 cm). Since then, a number of surveys have been conducted in different parts of the radio spectrum, most notably: Project META (Horowitz & Sagan, 1993), Project BETA (Horowitz & Leigh, 1999), SERENDIP (Werthimer et al. 2010), Project (Cullers, 2000), ATA Project (Tarter, 2016) and (Enriquez, 2018; Breakthroughinitiatives.org, 2019). Although important progress has been made since the 1960s, SETI is often dependent on private funding. In addition to this, until recently, our technology and computational capabilities have been entirely eclipsed by the vast “cosmic haystack.” According to Wright et al. (2018), the total SETI search fraction to date is of the order 10−17, similar to the volume ratio of a large hot tub and all on Earth. Advocates of the SETI endeavor thus argues that since only an infinitesimal part of the actual search space has been processed, it would be overly optimistic to expect a result at this early stage (Wright et al. 2018; Tarter et al. 2010). However, others mean that the current null result of SETI clearly shows that this project is a waste of both time and money (Tipler, 1993). A study by Griffith et al. (2015) put weak constraints on the number of Type III civilizations from a sample of 100,000 . According to their study, no civilization harvest more than 85 % of the starlight in any of these galaxies, suggesting that advanced civilizations are rare. Physicist Stephen Webb (2014) argues that the discussion easily gets burdened by an anthropic bias, and, that it is impossible to make conclusions from just a single example. He stresses that Earth has undergone a series of very improbable events in its history, resulting in a unique evolutionary path towards intelligent, conscious, tool-making and communicative beings such as us. According to Webb, this observation, in combination with the empirical fact of no detection, tells us that our search for contact in the cosmos is nothing more than a search for ourselves.

106/14/2019 (Exoplanetarchive.ipac.caltech.edu, 2019). 2According to Webb (2014), the Tsiolkovsky–Fermi–Viewing–Hart paradox is a more proper name.

3 The aim of this thesis is to quantitatively determine the implications of the current SETI null result. Does the null result necessarily imply that galactic civilizations are extremely rare, or does it only have a modest influence on such predictions? Which constraints does the null result put on the number of galactic civilizations? How do the predictions change given the hypothetical event of exactly one detection? These are questions that this thesis will investigate.

1.2 Purpose

The main purpose of this thesis is to find upper limits on the galactic ETI population and to determine constraints on the Fermi paradox, given the current null result of SETI. This is done via a Monte Carlo method approach where idealized assumptions on ETI civilization characteristics are made in order to simulate the likelihood of detection in current and historical SETI surveys. Secondary purposes/motivations are: ˆ Try to determine whether SETI is a meaningful endeavor. ˆ Examine how Monte Carlo methods can be applied to gain insight into scientific problems where data is severely limited.

ˆ The “Where are they?” question has a particular appeal from a scientific and philosophical point of view. It interconnects a rich flora of disciplines, including , planetology, , radio astronomy, , cryptography, chemistry, , , , environmental science, archaeology, anthropology, sociology, psychology, history, and futurism. To contemplate this question is a healthy intellectual exercise.

1.3 Outline of the Thesis

The thesis begins by giving the reader some relevant background information associated with the Fermi paradox and the field of SETI. The following chapter contains some statistical theory and a brief intro- duction to radio astronomy. The simulation part of the thesis is described in detail in Chapter 4. Results for a variety of cases are presented in Chapter 5, followed by discussion and conclusions in Chapter 6.

1.4 Author’s contribution

This paper is, from idea to final thesis, a result of the author’s work. That includes all plots and tables, the development and implementation of the simulation model, all results presented, as well as all written text.

4 Chapter 2

Background

2.1 Defining ETI

By definition ETI, or ExtraTerrestrial Intelligence, is intelligence other than that of Earth. The concept of “intelligence” is in many contexts ambiguous, however, the SETI definition of intelligence is simply: “the ability to create technology and emit artificial radio waves.” The idea of ETI is currently a hypothetical concept, all SETI searches done to date has produced a null result (seticlassic.ssl.berkeley.edu, 2019). This thesis will adopt five idealized assumptions, (A1)-(A5), on the characteristics of galactic ETI civilizations. These assumptions are in some consistency with the historical SETI1 consensus (NASA, 1973; Drake, 1984), which in turn, should make them compatible with the SETI search parameters. The assumptions read as follows: A1. ETIs arise independently and randomly in space and time. A2. They transmit isotropic, or directed, beacon EM-signals as a means of galactic communication. A3. They communicate somewhere in the L-X-band (1-10 GHz) in a narrow-band 1 Hz-channel.

A4. They transmit on a 100 % duty cycle basis, either continuous or short-pulsed (T . 30 s) signals. A5. Their Equivalent Isotropic Radiated Power (EIRP) can be mapped onto the Kardashev scale. There are three main motives behind these assumptions:

ˆ By assuming a random distribution of ETIs simplifies the analysis considerably and limit the number of free model parameters to a minimum. ˆ By assuming ETI communication compatible with SETI yields a conservative estimate of the full population, according to: Circ ⊆ Crc ⊆ Cc ⊆ C, where C is a set of ETI civilizations of a certain characteristic, and where the indices run from the highly specific (isotropic radio communication) according to assumption (A1)-(A5), towards the general. The set C (without index) contains any ETI civilization, communicating or not.

ˆ Identifying the extremes makes it possible to interpolate conclusions based on more realistic assumptions.

2.2 The Fermi Paradox

A paradox is, by definition, a statement that rests on one or more seemingly reasonable – but false – assumptions that lead to a contradiction. When it comes to the Fermi paradox, it is currently unclear whether it is based on false assumptions, or given that the assumptions are correct, that these necessarily leads to a contradiction. It could therefore be argued that the Fermi paradox does not qualify as a paradox in the strict logical sense, but rather as an argument (Gray, 2015; Cirkovi´c,´ 2016). It should also

1The abbreviation SETI refers throughout this thesis to the general activity of Searching for ExtraTerrestrial Intelligence, not to any organization, such as the SETI Institute.

5 be noted that the Fermi paradox comes in different formulations and interpretations. This thesis will investigate what could be called the soft Fermi paradox (associated to the terms of the Drake equation and the Copernican principle) whereas Fermi’s original question rather represents a sharp Fermi paradox (e.g. Prantzos, 2000). The former interpretation, which hereafter will be referred to as just the Fermi paradox, can be summarized as follows: P1. There are billions of Sun-like stars in the galaxy harboring billions of Earth-size planets in the habitable zone (Petigura et al. 2013), on average 1-3 Gyr older than the Sun (Lineweaver, 2004). P2. Earth is not unique in the universe, thus, some of these planets have developed life.

P3. Life, in general, follow an evolutionary route towards increased complexity, thus, some of this life will eventually become intelligent. P4. Some of these civilizations will, in time, develop radio technology and communicate through space. There is already some evidence supporting assumption (P1). If also assumption (P2)-(P4) are true, then Fermi’s question is justified and requires an answer. On the other , if we learn that one or more of assumption (P1)-(P4) are false, then the Fermi paradox is a real paradox with an obvious and immediate solution. Section 2.3 presents some of the proposed solutions to the Fermi paradox, and as that section shows, this is not at all a trivial problem.

2.3 Solutions to the Fermi Paradox

There are many proposed solutions to the Fermi paradox. Webb (2014) addresses seventy-five of them, some more plausible than others. This section briefly summarizes some of the more interesting proposed solutions to the Fermi paradox. The purpose is to illustrate the richness and deeply non-trivial nature of the problem. Some of the points made here will be resumed in the closing section of the thesis. The solutions are naturally divided into three categories as follows: They exist and they are (were) here, They exist but we have not yet established contact, and, They do not exist.

2.3.1 They exist and they are (were) here : This is one of the more famous explanations to the Fermi paradox, proposed by John Ball (Ball, 1973). The idea is that the galaxy is controlled by a number of highly advanced civilizations. Just as we protect nature reserves, and primitive tribes in e.g. the Amazon, from the 21st-century civilization on Earth, so are the advanced civilizations protecting us from them. The purpose would be to study us on a distance and to not interfere with our development. Such interference would inevitably trace out our path in their direction and therefore impoverish the intellectual flora of the universe. On the other hand, by leaving us alone, other civilizations might learn something new by studying our development. One major shortcoming of this hypothesis is that it requires a “uniformity of motives.” All civilizations in the galaxy, separated by eons of time and space, have to submit to a “Prime Directive” and never break the rules. It is one thing not to make deliberate contact with new civilizations, a completely other thing to hide your presence altogether and avoid all sorts of leakage. Another problem, which is, unfortunately, the case of many proposed solutions, is that it is not falsifiable (within a reasonable time frame). Not even a thousand years of non-detection would neither confirm nor falsify the Zoo hypothesis. : A special case of the Zoo hypothesis might be the Panspermia theory. Studies of extremo- philes have shown that they are able to withstand the harsh environment of space for significant periods of time. is the deliberate spread of microbial life to suitable worlds by intelligent beings. This hypothesis would indeed explain the seemingly immediate appearance of life on Earth around 4 billion years ago. However, it does not explain how life arose in the first place. Nor does it offer any obvious means of testability, even if the detection of life, based on right-handed DNA, on several other planets/ in the would give it some credit. The panspermia hypothesis is intriguing since it suggests that we all are members of a cosmic family, but it is not a very strong candidate solution to the Fermi paradox.

6 2.3.2 They exist but we have not yet established contact The universe is large: The most immediate response to the Fermi paradox from the uninitiated reader is perhaps that the universe is large. The spacetime distance between us and them is simply far too great for communication, not to say, colonization. However, there have been many estimates of the time required for a single space-faring civilization to fully colonize the galaxy (e.g. Hart, 1975; Newman & Sagan, 1981). By Hart’s first estimate, a civilization would begin the colonization by sending out a space fleet to another system at speed 0.1c. When the fleet arrives it sends out, after some resting time, its fleet of its own to a third system, while the original civilization sends out yet another fleet to a fourth system. This colonization continues at an exponential rate and would result in full galactic colonization after just about 1 million years. It could be discussed whether it is realistic for a civilization to colonize the entire galaxy at a speed of 0.1c. It would without a doubt require an extraordinary determination for a very long time. However, more rigorous calculations with various assumptions, using e.g. a diffusion model (Newman & Sagan, 1981) or a population growth model (e.g. Jones, 1985), end up in the range 1 million to 100 million years. Even if this is a long time for human civilization, it is insignificant compared to the age of the galaxy. It takes only one single civilization to colonize the entire galaxy in a time-frame that is 0.01-1 % of the galaxy’s age. By this argument, the galaxy should have been colonized hundreds or thousand times over by now. Instead, not a single sign of their existence has been recorded. Perhaps we are one of the absolute first civilizations in the galaxy and that other civilizations have not had time to reach us yet? If this is true it would be a very peculiar coincidence since many Sun-like stars that harbor planets are several billion years older than our Sun.

Percolation theory: A perhaps more plausible version of the former explanation was proposed by Landis (2007). The percolation model assumes that colonization through large distances is difficult. Fur- thermore, it assumes that a civilization has probability p of being interested in colonization. Thus, a mother civilization will expand within a given bubble in space containing a number of stars. Daughter civilizations will continue this expansion outwards as long as they are colonizing civiliza- tions. However, sooner or later this process will stop when a daughter civilization is uninterested in further colonization (which happens with probability 1 − p). Landis’ simulations show that there are three distinct scenarios, either p < pc, p = pc or p > pc, where pc is a critical probability. In the first scenario, civilizations will spread to a certain limited size and then stop spreading. In the second scenario, civilizations will occupy a fractal-like region across the galaxy with many empty voids in it. In the third scenario, civilizations will occupy all space with very few exceptions. If scenario one is true, then it could possibly solve the Fermi paradox. Scenario two would imply that Earth is situated in one of the unexplored regions, surrounded by other civilizations. However, if this is true one have to explain why we have not seen a single evidence of their existence. It is easy to find shortcomings with the percolation theory, just as for any other solution, however, it is nevertheless one of the better candidate solutions for the Fermi paradox, partly since it does not assume the usual “uniformity of motives”. Another indication that percolation theory might have some bearing is that the human colonization of Earth, as seen from space, also follows a percolation pattern, i.e. large cities surrounded by uninhabited forests, deserts and mountain regions. Predator civilizations: According to this scenario, in the early days of the Milky Way, a rather nasty intelligent species occurred. It developed space technology and spread like a virus, exploiting every in its path. All young and less advanced civilizations were annihilated by the predator civilization in order to take over their resources and to avoid competition. The only hope for a new civilization to survive was to keep silent and not reveal its presence. If this is the situation today, our galaxy is ruled by a regime of silence and fear. The predator civilization keeps silent in order to surprise their prey. All other civilizations keep silent in order to survive. One often mentioned objection to this scenario is that it seems unlikely that an aggressive and violent species could avoid self-destruction and develop into an old advanced civilization. However, if the virus analogy is true, it is possible that the aggression is only directed towards the “host organism”, i.e. other civilizations, and not towards the predators themselves. It could also be noted that racism and xenophobia are quite common also in the human community. Physicist , among others, has pointed out that caution should be preferred and that it is unwise to send messages (METI) into space before we know what is out there. On the other hand, if they really existed, should they not already be here? They have had thousands of years to discover the human progress and to wipe us out in our infancy. Why wait until we have developed radio technology?

7 We are unable to detect their signals: Technological signals of extraterrestrial origin might pass Earth un- noticed. This could be due to several reasons. For instance, ETIs might transmit signals in fre- quencies that are not covered by our instruments. Gamma-rays, instead of radio, could increase the information content of the signal by many orders of magnitude. There is also a possibility that ETIs, instead of electromagnetic waves, exploit other means of communication, such as neutrinos, gravitational waves, or some kind of exotic particles, unknown to us. But even if ETIs use radio as their primary method of communication, it is not certain that we would detect or recognize the signal. To date, only frequencies up to 10 GHz have been surveyed to a satisfactory degree, while the radio spectrum comprises frequencies up to 300 GHz. However, large parts of this spectrum are exposed to heavy galactic noise and could thus be regarded as less likely transmitting frequencies. Historically, the waterhole region, 1-2 GHz, has been considered as a prime candidate frequency band for galactic communication, due to its absence of cosmic interference and to its symbolic value. There have, however, been many other proposals of “magic” frequencies, e.g. kT0/h = 56.8 GHz (T0 = 2.73 K), proposed by Drake and Sagan in 1973. Assuming that we are listening to the right frequency, there are still obstacles. The signal must be narrow-band to stand out against the background noise, and if the signal is subject to some modulation, it must be of a kind that our FFT-algorithms can handle. It would be even more difficult, not to say impossible, if the signals were encrypted, to avoid detection by an outside party, or directed via or a high gain antenna. There are plenty of ways to hide a signal in the cosmic background noise. However, if there are some ETIs that have the slightest interest in galactic communication, it might seem natural to assume that they want to make their presence heard and thus utilize all relevant communication channels.

They do not communicate: There could be argued that communication is just a transient phase in the of a technological civilization. The purpose of communication is to exchange information. But a civilization on a sufficiently high scientific, sociological and technological level may have surpassed the need for new information. Maybe because they are not curious anymore. Maybe because they already know everything of interest. Of course, they could still transfer information internally, but such signals, not aimed at our direction, and possibly of very complex composition, would be difficult for us to ever detect. In our current era of developments in artificial intelligence (A.I.) and converging of technology and biology, the idea of a postbiological evolution becomes more and more likely. If this is a typical evolutionary process of all intelligent civilizations, the singularity (the moment in time when A.I surpasses the intellect of its creator) could mark the endpoint of their biological civilization, and mark the beginning of something completely different. It is very likely that postbiological, possibly nonphysical, beings have very different emotions and driving forces than us and that their existential ponderings just might fade away. In our case, it has been estimated that the singularity could come within 50 years. However, although the singularity certainly will have a fundamental impact on society it seems very unlikely that would transform into emotionless robots overnight. We will still be humans for the foreseeable future, even though incremental changes with each generation could have a substantial effect over time. The Doomesday argument: Technological progress goes hand in hand with increasing existential threats. The development of A.I. is one such potential risk to our long future. On one hand, A.I. could be of great benefit to mankind by creating great wealth, releasing us from work we find boring and solving difficult problems. However, it is also an obvious risk that A.I will lead to severe unemployment and increased class divisions. It is also a risk that an uncontrolled A.I. regards humanity as a threat, or that the A.I. invents its own agenda that it unscrupulously tries to pursue. For the last 75 years, humanity has lived on the brink of nuclear war. Even though this risk had its last peak during the Cold War, very little speaks for a safe future. A global nuclear war would most certainly annihilate humanity (if not from the immediate blast, so from radiation-induced cancers) and have devastating effects on Earth’s for a long time. Biotechnology, biological weapons, and could pose other threats. Examples are artificially created viruses for which there is no cure, or self-replicating nanorobots that run amok and converts our world to a “grey goo”2. Global warming is another example. Even though global warming itself does not pose an immediate threat to our existence, uncontrolled global warming, of say 5-10 degrees, could have dramatic consequences to billions of people, through hurricanes, flooding, drought, starvation, migration, and possibly war. If ETIs have the self-destructing tendencies we have, it could be expected that a large partition of them destroy themselves within a century after they have started radio communication. If so, their lifetime is short, and the probability for us to detect them limited. 2See Eric Drexler, Engines of Creation (1986).

8 2.3.3 They do not exist We live inside a simulation: The computer era has enabled us to construct simulations of the real world. As computing power has increased, the simulations have been more detailed and advanced. Consid- ering the millions of simulations we have constructed to date, and the rate at which they improve, is there a possibility that our complete world indeed is a simulation? Advocates of this idea argue that it is far more likely that our world is just another simulation, since there undeniably already exists millions of them, than something else that could be considered a real world. Some long-lasting questions would certainly get their explanation by this hypothesis. For instance, why are there nat- ural laws that can be described by mathematics? Where did the initial conditions of the universe come from? Why do we exist when our presence seems so unlikely? In the simulation scenario, the answer is that we exist (or think we exist) because the constructor of the simulation defined the initial conditions in such a way for intelligence to emerge. If that is the case, it is not completely surprising that we are alone inside the simulation (universe). However, a large, high-resolution simulation demands a lot of energy. We know that the resolution of our world is roughly the Planck p 3 −35 length, `P = ~G/c ≈ 1.6 × 10 m. So, what energy would be needed to create the world we see? The Bekenstein Bound3 implies that the energy required for creating the full extent of our solar system in Planck length resolution equals the energy output of a whole galaxy. However, the Voyager 1 probe left the solar system in 2012 seemingly without bouncing into a wall. Thus, either do we live in the real world, or the simulation is created by a super civilization harnessing the power of at least 100 billion suns devoted to power one single simulation. If the latter is true one must ask the questions: Who are they? Them or us? And, most importantly, how large is their electric bill?

Rare Earth Hypothesis: This hypothesis was originally proposed by Peter Ward (geologist) and Donald E. Brownlee () in the book : Why Complex Life is Uncommon in the Universe (2000). The idea comprises the fields of astronomy, chemistry, biology, and geology. The can be seen as the opposite of the Copernican principle, the former claiming that Earth is rare or unique, the latter claiming that Earth-like planets must be common due to the sheer size of cosmos. The general assumption of the Rare Earth hypothesis is that our history has followed a route through at least five crucial and improbable steps (or filters). The first crucial step in the Rare Earth scenario is the event of biogenesis. It is still an open question, given the right conditions, how likely this event is. Studies of the oldest fossils show that life appeared on Earth around 4 billion years ago, only shortly after the planet was formed. This would suggest that biogenesis is common. However, only by detecting microbial life on , or elsewhere in the solar system, that is unrelated to life on Earth would settle this question. Other crucial and improbable steps in the evolution of life, according to the Rare Earth hypothesis, would be the transition from prokaryotic to eukaryotic cells, as well as the transition from single-cellular to multi-cellular organisms, the latter occurring around 1.2 billion years ago, according to fossil records. The most recent crucial evolutionary step on Earth could have been the transition from non-intelligence to intelligence4. There is no consensus regarding the likelihood of any of these events, however, the proponents of the Rare Earth hypothesis argue that they are all unlikely. Rare Earth proponents bases their arguments on that our specific position in the galaxy, our position in the solar system, the orbit of the , and many random historical events, such as the impact 66 million years ago, all have had a crucial influence on Earth’s evolutionary trajectory. The final step on life’s evolutionary journey could be the transition from planet-bound lifeforms to an interstellar species, capable of traveling between the stars. If this last step is unlikely, possibly due to self-destruction, advanced cosmic civilizations are unlikely. From a human perspective it would be of interest to ask which steps are most unlikely, the ones in our past or the one in our future. Although the Rare Earth hypothesis could explain the Fermi paradox it is in some senses a problematic idea. If the Rare Earth hypothesis indeed is the explanation to the Fermi paradox then the conclusion is that the evolution towards intelligence is extremely sensitive to a myriad of improbable and random events. Proponents often argue that humans had never evolved being it not for an asteroid impact 66 million years ago. Yes, that is most probably true, but that does not imply that an asteroid impact at the exact right moment is necessary for the evolution of any kind of intelligence at some point in time.

3See Webb (2014) p. 69 for further details. 4As defined by SETI (see Chapter 2.1).

9 2.4 The Drake Equation

The Drake equation was proposed by radio astronomer at the first SETI meeting in 1961 (Drake & Sobel, 2010). It is a probabilistic argument that summarizes what we know, and most impor- tantly do not know, in order to estimate the number of extraterrestrial civilizations in the Milky Way with whom we could communicate. The purpose of the equation was, according to Drake, to stimulate scientific dialogue on the subject, rather than to predict an exact number. The equation in its most common form5 (Shklovskii & Sagan, 1966) reads as follows:

N = [R~fpne][flfifc] L = R⊕fbtL, (2.4.1) where N is the present number of communicating civilizations in our galaxy, R~ is the average number of stars formed per year, fp is the fraction of those stars that have planets, ne is the number of those planets hospitable to life, fl is the fraction of those planets that develop life, fi is the fraction of those planets that develop intelligent life (civilizations), fc is the fraction of civilizations that develop radio technology, L is the length of time of their communicating phase. The Drake equation links seven parameters associated with a wide range of disciplines such as astron- omy, biology, and sociology. The first three, denoted R⊕, can be regarded as astronomical parameters, today relatively well-known thanks to Kepler and other telescopes. The following three parameters, de- noted fbt, often referred to as bio-technical parameters, are less known, although at least fl could be within grasp in the near decades (if e.g. microbial life is found in the Martian soil or in the deepwater oceans of or ). The last parameter, L, the lifetime of a communicating civilization, is however completely unknown. One of the main critiques related to the Drake equation is that this lack of knowledge creates a large bias, i.e. anyone can adjust the input in order to get the desired result, making it impossible to draw any reliable conclusions. The collected criticism of the Drake equation can be summarized as: ˆ The equation does not take into account that N is a random variable (e.g. Maccone, 2012).

ˆ The equation does not take into account the temporal aspects of N (e.g. Cirkovi´c,´ 2004). ˆ The equation does not provide any margin of error of N (e.g. Glade, 2012). ˆ The equation assumes that it is possible to make point estimates of parameters that can vary many orders of magnitude (Sandberg et al. 2018). ˆ The equation assumes that other galactic civilizations, like us, use electromagnetic telecom- munication as a final stage of technological evolution (Wilson, 1984). ˆ The equation assumes that galactic civilizations appear randomly in space as a consequence of planet specific biological evolution, i.e. it does not take colonization into account (Shostak, 2019). ˆ The equation does not weigh in all available knowledge, such as the current null result of SETI. This thesis will address a number of these issues. In e.g. Chapter 3 a statistical Drake equation is derived resulting in a formula for error estimation. Furthermore, and more importantly, a Bayesian analysis will be pursued in order to put constraints on the present time population, N, imposed by six decades of SETI.

2.4.1 Two Examples One problematic feature of the Drake equation is that a seemingly small change in input can result in a large change in output. This property will here be demonstrated by two not implausible examples. The first slightly “optimistic” example follow the original Drake estimate from the first SETI conference in 1961. The second example, slightly more “pessimistic”, follows Engler et al. (2018) and is based on three assumptions/principles: the principle of insufficient reason, ergodicity of biological evolution (evolution on Earth is typical), and, the Copernican principle. The astronomical parameters, R⊕, the rate at which habitable planets form in the galaxy, varies somewhat in the literature but is often taken be in the interval 10−1, 101. In modern estimates values in the lower half of the interval is preferred (Shklovskii & Sagan, 1966; Fukugita et al. 2004; Petigura et al. 2013; Licquia et al. 2014; Tuomi et al. 2014; Engler et al. 2018).

5 Drake’s original equation consists of three factors, R⊕, fbt, L (Shklovskii & Sagan, 1966).

10 Example 1 : Drake and his colleagues assumed that 10 stars form in the galaxy each year; that 50 % of these stars form planets and that, on average, two planets in each system is suitable for life. They furthermore assumed that life appear in 100 % of the cases, given the right conditions; that this life becomes intelligent with 50 % chance, given enough time, and that intelligence always develops radio communication technology. Finally, they assumed that these civilizations communicate through space for, on average, 10,000 years. These numbers result in: N = 10 × 0.5 × 2 × 1 × 0.5 × 1 × 104 = 5×104 communicating civilizations present in the galaxy. This figure was probably an incitement when starting the SETI program in the early 1960s. Example 2 : The principle of insufficient reason states that if there are n possibilities then, without any further knowledge, the probability of each possibility is 1/n. Thus, by the principle of insufficient reason, there is a 50 % chance that life arises on a planet, given the right conditions (either life appear or it does not). By the Copernican principle a technological communicating civilization, with Earth as an example, appears in one case of roughly one billion (about one billion species have lived on Earth, of which one has developed radio technology). Assuming (author’s assumption) that these civilizations, on average, are transmitting for 1000 years yields (with R⊕ = 1): N = 1 × 0.5 × 10−9 × 103 = 5 × 10−7, which should mean that there is, on average, less than one communicating civilization per one million galaxies. This could be regarded as a small number, however, it would nonetheless imply that there are 5 × 104 civilizations in the visible universe. It is obvious from the examples that N can vary many orders of magnitude with equally reasonable input values. In the examples the result span over 11 orders of magnitude. However, these values should by no means be interpreted as extremes. As Sandberg et al. (2018) have pointed out, there are good reasons to assume that some of the parameters in the Drake equation have far greater uncertainty than has been expected in the SETI literature. For instance, the validity of the Copernican principle, used in the second example, is highly uncertain and would be in contradiction to e.g. the Rare Earth scenario. This work addresses this uncertainty by exploiting stochastic input parameters instead of making point estimates.

2.4.2 Alternative Form

By introducing two free model parameters, λb ≡ R⊕fbt, and, λ` ≡ L, (in line with e.g. Glade et al. 2012; Engler et al. 2018) it is possible to rewrite the Drake equation in a more compact form:

Ne = λbλ`, (2.4.2) where λb now represents the (time independent) birth rate of ETI civilizations in the galaxy, and where λ` are their average lifetime in the communicating phase. Eq. (2.4.2) is the usual time-independent Drake equation with index “e” emphasizing that N represents a statistical equilibrium population. The Monte Carlo simulation model developed in this thesis will, however, take this a couple of steps further. Firstly, the parameters, λb, λ`, are considered stochastic, thus randomly sampled from a predefined parameter space, and (in the former case) time-dependent. Secondly, they are used as input parameters in the Monte Carlo simulation.

2.5 Figure of Merits

A Figure of Merit is a measure used to compare the performance of a certain method. In the context of SETI surveys, the Figure of Merit is measured in terms of the search volume of the “cosmic haystack”. This is of great importance since it makes comparisons between SETI studies of different approaches and search strategies much easier. The Drake Figure of Merit (DFM), proposed by Drake et al. (1984), is a measure composed of three relevant search parameters, according to: ∆ν · Ω DFM = , (2.5.1) 3/2 Smin where ∆ν is the frequency coverage of the study, Ω is the sky coverage in steradians or square degrees, Smin is the minimum detectable flux from the source, usually measured in Janskys. The denominator is motivated as follows. The minimum detectable flux is proportional to the inverse −2 square distance to the most remote detectable source, i.e. Smin ∝ rmax. Furthermore is the searched 3 spherical space volume proportional to the distance cubed, Vmax ∝ rmax. Therefore, the Drake Figure of Merit can be rewritten as the product, DFM = ∆ν · Ω · Vmax. Next section uses the DFM to compare some major SETI searches in order to filter out those of greatest importance.

11 Table 2.1 – A number of searches and corresponding technical parameters. ◦ 1 05 +35 − 0 . 0 ≤ 10 10 - 3 3 8 3 ± δ 25 10 0.6 1.7 692 300 × × 10.6 GBT all sky 1.1-1.9 ≤ 2 4 42 . . ◦ 1 1 AO (305 m) Enriquez (2018) SERENDIP V.v +2 Breakthrough Listen Werthimer et al. (2010) ◦ ◦ 4 0 +20 +89 − 008 . 10 ◦ ≤ ≤ 0 10 1 - - δ δ × 12 90 0.6 1.7 0.7 ± 131 × 1080 ATA ∼ 4 ≤ ≤ 20,000 all sky 0 . 42 ◦ ◦ > . 7 1 Tarter (2016) Parkes (64 m) 90 32 1-1.5 (max 1-10) − − > Southern SERENDIP Stootman et al. (2000) Survey ATA ◦ ◦ 3 3 +60 +89 − − 6 ≤ ≤ 10 10 δ δ × 15 93 0.5 0.5 6.5 0.7 102 884 664 × × 9293 ATA 3 ≤ ≤ all sky 1.4-1.7 5 4 . . ◦ ◦ 5 1 Harp (2016) 1-9 (1-1.606) 30 32 Project BETA − − Horowitz & Leigh (1999) SETI Observations ... ATA Harvard-Smithsonian (26 m) ◦ ◦ 20 2 +90 5 − 0002 . 4 ≤ − ≤ 10 0 − ------10 800 δ δ 20 10 4 × 0.05 ± 10 ∼ ≤ ≤ all sky all sky 1.3-1.7 7 10 . ◦ ◦ 42 1 . 1 (Shuch, 2019) 80 90 Project META II − − SETI Legue (Argus) Colomb et al. (1995) 5000 amateur telescopes Radioastronom´ıaantenna (30 m) ◦ 4 1 +60 0 − − . 0002 . 3 ≤ 10 10 0 - δ − 30 30 20 18 1.0 884 800 × × 0.05 ± 2 ≤ . all sky 3 5 . . 1 ◦ 42 2 1 . 2/3/8/13/14 1 30 2.2/42.9/99.65 Project META Project Phoenix 138/195/276/552 − AO/Parkes/NRAO 14 Horowitz & Sagan (1993) Harvard-Smithsonian (26 m) Cullers (2000), Enriquez (2018) ] ] 2 2 / / ] ] 3 3 2 2 − − Jy Jy ) [Hz] ) [Hz] ) [s] ) [s] · · τ τ δν δν 2 2 deg deg · · S/N S/N Telescope Telescope SEFD [Jy] SEFD [Jy] Number of Stars Number of Stars Beam Width [arcmin] Beam Width [arcmin] Integration Time ( Integration Time ( Frequency Coverage [GHz] Frequency Coverage [GHz] Sky Coverage [Dec or deg Sky Coverage [Dec or deg Spectral Resolution ( Spectral Resolution ( 12 Max DFM [GHz Max DFM [GHz 2.6 SETI Surveys

Since Drake initiated SETI in 1960, a total of about 140 different SETI surveys have been conducted. Approximately 100 of which have been radio searches. Of these, about 50 were made over at least 100 observation hours, while only 10 were made over at least 10,000 observation hours (Technosearch.seti.org, 2019). Typically, Radio SETI has listened for narrow-band signals around the spectrum lines of H and OH in the L-band (1-2 GHz), the so-called water-hole region. Narrow-band signals stand out against the background of known natural processes (≥ 300 Hz), making them potential signs of artificial origin. The water-hole region provides a quiet window in the electromagnetic spectrum, relatively free from galactic background emissions and quantum noise. The water-hole region is also protected by international agreements which reduce atmospheric interference. Radio SETI is performed via two main search strategies: targeted searches and all-sky searches (Nasa, 1973; Drake, 1984). A targeted search makes it possible to monitor single stars for an arbitrary time span. This limits the number of targets but increases sensitivity. An all-sky search scans the entire field of view for bright beacon signals. This makes the potential search volume immense, but with an imminent risk of missing weak closeby signals due to limited sensitivity. In this thesis, a total of 13 SETI projects (including several more substudies), made over roughly 250,000-300,000 observation hours, are taken into consideration. The surveys were selected mainly on basis of their DFM value, and to what extent the studies mutually complement each other, but also other aspects, like the survey’s historical importance, has been taken into account. This section briefly describes some of these surveys. Project META: The Megachannel ExtraTerrestrial Assay (META) ran between 1986 and 1991 (Horowitz & Sagan, 1993). The project was an all-sky search of the northern sky (−30◦ ≤ δ ≤ +60◦) with the Harvard/Smithsonian 26 m radio telescope instantaneously monitoring 8.4 million very narrow 0.05-Hz channels near the H I line (1420 MHz) and its second harmonic. The observing frequency was corrected for Doppler shift effects due to Earth’s rotation and with respect to three different astronomical inertial frames. During the 5 year mission, a total of 6 × 1013 spectral channels were searched, of which 37 candidate events exceeding the detection threshold were found. The strongest of them emerged from the . None of the candidate signals were detected upon re- observation. In 1995 META was succeeded by META II, a follow-up survey with the same search parameters in the southern hemisphere, Argentina (Colomb et al. 1995). The survey found 10 candidate events, but none of them gained later confirmation. Project BETA: The Billion-channel ExtraTerrestrial Assay (BETA) project was the first all-sky and all- waterhole search for narrow-band artificial microwave signals (Horowitz & Leigh, 1999). It was an upgraded continuation of the META project with much wider frequency coverage (1.400-1.720 GHz), but with lower sensitivity, which automatically scanned the northern sky (−30◦ ≤ δ ≤ +60◦) four times. One important innovation implemented in the BETA system was the adaptive radio frequency interference (RFI) filtering scheme, which helped to reduce the problem of interference. The project was canceled in 1999 when a storm made great damage on the dish of the 26-meter radio telescope. The result of the 3.5-year survey was negative, but it contributed to putting strong upper limits on the number of galactic ETIs transmitting in the waterhole region. SERENDIP: Search for Extraterrestrial Radio Emissions from Nearby Developed Intelligent Populations (SERENDIP) was a SETI program that started in 1979 by the Berkeley SETI Research Center. The idea behind SERENDIP is (with help of an extra receiver) to “piggy-back” on other obser- vation programs at NRAO telescope (91 m), Green Bank, or at the (305 m), Puerto Rico. This enables SERENDIP to work almost 24 hours a day, although without any au- thority to decide the observation target. The SERENDIP program at Arecibo scans the sky in a band between +2◦ ≤ δ ≤ +35◦ (±16◦ from Arecibo’s zenith). The data collected is partly analyzed by SETI@home which is a citizen science project that borrows computer time from (cur- rently) 100 thousand home computers (SETI@home, 2019). The latest generation of instruments (SERENDIP V.v) currently in operation at Arecibo is a 128 million-channel digital spectrometer covering 200 MHz of instantaneous bandwidth (1.56 Hz/channel) near the . An ex- tension to 7 beams×300 MHz, covering 2.1 GHz of instantaneous bandwidth, has been planned for the future. To date, no evidence of artificial signals has been detected, although several interesting signals have been reported. (Werthimer et al. 2010; MacRobert, 2019; Technosearch.seti.org, 2019)

13 Project Phoenix: Project Phoenix was a SETI Institute program (originally initiated by NASA) conducted between 1995 and 2004 at the sites of Arecibo (Puerto Rico), (Australia) and NRAO (Virginia, USA). It was the most sensitive targeted search of the time systematically observing 800 nearby (< 150 ly) Sun-like stars in the 1.2-3.0 GHz band at a resolution of 0.7 Hz. The major breakthrough of Project Phoenix was the technological achievements, the high sensitivity of the search and the high resistance to RFI, where only 1/3819 false positives occurred at the Parkes antenna. The conclusion was that none of the observed stars showed any continuously present signal stronger than 0.1 TW EIRP. (Cullers, 2000; Harp et al. 2016).

Breakthrough Listen: The Breakthrough Listen is a $100 million program funded by science philanthropist and investor . The aim of the program is to perform the most sensitive and extensive search to date for signals of extraterrestrial intelligence. The program, stationed at GBT, Parkes and Berkeley SETI Research Center, will comprise the 1000,000 nearest stars, the galactic plane including the , as well as the 100 nearest galaxies. In a first study part of the program, conducted by Enriquez et al. (2018), 692 nearby stars were observed in 1.1-1.9 GHz at the GBT site. The study concluded that none of the observed systems host high-duty-cycle transmitters of 13 EIRP 10 W, and above, in the given frequency range, and that . 0.1 % of all systems within 50 pc host such transmitters (Enriquez, 2018; Breakthroughinitiatives.org, 2019).

In Table 2.1 the technical specifications of ten previous SETI surveys (conducted between 1993 and 2018) are summarized. A majority of them are all-sky surveys while the rest are targeted searches. It should be noted that the performance of a search, in terms of its DFM, on average has increased by a factor of ten per decade, indicating an exponential advancement over time (this trend holds back to the 1960s).

2.7 The Kardashev Scale

Russian astrophysicist (1964) proposed a classification system for hypothetical ETI civilizations. This system is based on the amount of energy that is controlled and exploited by the civilization. According to Kardashev there are three significant levels: Type I, Type II and Type III, defined as follows: I. A civilization of Type I can utilize the total energy of a planet, corresponding to approximately 1016 W. It is likely that a Type I civilization has developed nuclear and space technology and has begun harvesting natural resources on nearby planets, moons or in their solar system. II. A civilization of Type II possess the energy equal to that of a star, corresponding to approximately 1026 W. A Type II-civilization may have interest in large scale astroengineering projects, such as construction of space habitats (e.g. O’Neill cylinders), Dyson spheres (Dyson, 1966), Matrioshka (Bradbury, 2004) and interstellar travel (e.g. probes, panspermia or stellar arks). III. A civilization of Type III control the energy of an entire galaxy, corresponding to approximately 1036 W. Characteristics of a Type III-civilization may include astroengineering on galactic scales, utilizing super-massive black holes for energy production, interstellar travel, and communication. Sagan expanded Kardashev’s discrete concept to include intermediate levels of civilization progress. This is described by the formula: log P − 6 K = 10 (2.7.1) 10 This formula indeed recovers the values for K = 1, 2, 3 when P = 1016 W, 1026 W, 1036 W respectively6. As a comparison, where does the human civilization end up on the Kardashev scale? According to the International Energy Agency (Iea.org, 2019) the total energy output of the human civilization in the year 2016 corresponded to P = 1.8 × 1013 W, which imply K = 0.73. Thus, humanity has to expand its total energy output by a factor of about 500 to obtain Type I-status (which will happen in about 300 years at the current yearly growth rate). The Kardashev scale concept is used in the simulation model in order to distinguish civilizations of different power output.

6However it also claims that K = 0 for P = 106 W, which may seem arbitrary. This can be justified as follows. At the beginning of a civilization’s lifespan, there are neither cities nor technology, the only energy consumption comes from collecting and eating food ( and vegetables) from the natural environment. The power output of a man in rest is approximately 100 W. For a civilization of, let say, 10,000 individuals, this would correspond to a figure of 106 W.

14 Chapter 3

Theory

This chapter is dedicated to some theoretical concepts used in the thesis. The first section defines the Poisson process which is part of assumption (A1) of the thesis. The following section provides the theory of Bayesian inference which is a central idea throughout the thesis. The final sections of the chapter concern some relevant concepts of radio astronomy, the radiometer equation, and methods of coordinate transformation.

3.1 The Poisson Process

A Poisson process is a type of stochastic counting process where “events” occur continuously and inde- pendently defined by, N (t), a monotone increasing discrete function, counting the total number of events at time t ≥ 0. Examples of Poisson processes are: the total number of people that has entered a store in the time interval [0, t] (an event occurs when a person enters the store); the number of children born in a city between 00:00 am and t hours later, January 1st; the number of goals made by a soccer player t minutes into the game (t ≤ 90 min).

3.1.1 The Nonhomogeneous Poisson Process The Nonhomogeneous Poisson process is a special case of a Poisson process in which the event rate is a function of time. Examples may include total number of people that have entered a store in the time interval [0, t], at different times of the day, the number of children born between January 1st and day d later in the year (the birth rate varies during the year). Compared to a counting process in general, the nonhomogeneous Poisson process has the conditions: 1. N (0) = 0. 2. The process has independent time increments, i.e. N (s + t) − N (s) is independent of N (s). 3. The number of events at time t ≥ 0, is Poisson distributed with intensity function λ (t) = λf (t), t t ≥ 0, and mean value function Λ (t) = 0 λ (s) ds, according to: ´ [Λ (t)]n P {N (t) = n} = e−Λ(t) , n = 0, 1,... (3.1.1) n! where also E [N (t)] = V [N (t)] = Λ (t). One of the most important properties of the Poisson process regards the waiting time between two events. If Tn denotes the waiting time until the nth event, then P {Tn ≤ t} = P {N (t) ≥ n}, since in both cases at lest n events has occurred in time interval (0, t]. With aid of Eq. (3.1.1) this gives the cumulative probability function for Tn:

∞ X Λk (t) P {T ≤ t} = P {N (t) ≥ n} = e−Λ(t) . n k! k=n

15 By differentiating with respect to k gives the probability density function for Tn:

∞ X Λk−1 (t) Λk (t) f (t) = λ (t) e−Λ(t) − n (k − 1)! k! k=n Λn−1 (t) = λ (t) e−Λ(t) , (n − 1)! where the last step follows from the telescoping property of the sum. The distribution of T1, i.e. the waiting time between two events, is thus given by:

−Λ(t) f1 (t) = λ (t) e .

In the case of a homogeneous Poisson process the intensity function is constant and f1 (t) reduces to λe−λt. This means that for a homogeneous Poisson process the waiting time between two events follows an exponential distribution with mean 1/λ. During a sufficiently short time interval [t, t + h], where h ≤ 1/λ (t), the nonhomogeneous Poisson process can be treated as a homogeneous Poisson process with mean 1/λ(t). It hereby follows that the nonhomogeneous Poisson process can be regarded as a sequence of exponentially distributed interarrival times with mean {1/λ (t1) , 1/λ (t2) ,..., 1/λ (tn)}.

3.1.2 A Birth and Death Process This section implements the Poisson process model to establish a stochastic birth and death process of galactic ETI civilizations. The outcome of this birth and death model is the present time population of galactic civilizations, N0, its expectation value, N¯0, as well as the proportion of them detectable to us. The present time mean population, N¯0, is calculated as the arithmetic mean of parallel histories. First, by letting C (t) represent the cumulative number of civilizations at time t, then the birth of a new civilization, C → C + 1, can be seen as an event in a nonhomogeneous Poisson process characterized −1 by the intensity function λb (t) [year ]. By Condition 1 of the Poisson process there exists a starting time, t0, at which there are no civilizations, i.e. C (t0) = 0. Furthermore, by Condition 3, it follows that the probability of C birth events in time interval [0, t] is given by Eq. (3.1.1). Most importantly, the events of the process (births of new civilizations) are separated by independent time increments, ∆Tj ∼ Exp (1/λb (t)), where no two events occur simultaneously. The instant at which civilization j is Pj born can thus be written as Tj = t0 + i=1 ∆Ti. The death process of a civilization is linked to the lifespan of the civilization, Lj. If the lifespan is a random variable of some survival function, L, such that E [L] = λ`, then the civilization death rate is given by the inverse, 1/λ`. Furthermore, the instant at which civilization j dies is given by Tj + Lj. The model states that civilizations are born at a time-dependent rate, λb (t), independent of each other (and independent of the number of civilizations present at time t). They also die independent of each other at rate, 1/λ`, but the number of civilizations that dies a given time is also a function of the instantaneous number of civilizations, N (t), present at that time. In conclusion, the number of civilizations will initially increase, and then gradually adapt to the cumulative birth rate and the death rate. Depending on the interrelationship between these two variables the instantaneous population either increases, decreases, or reaches a plateau where an equal number of civilizations are born and die per unit time. The average behavior of this stochastic birth and death process is governed by the differential equation: dN 1 = λb (t) − N, dt λ` with general solution (following Condition 1 for a Poisson process, N (t0) = 0):

t 0 N (t) = e−t/λ` λ (t0) et /λ` dt0. (3.1.2) ˆ b t0 The Milky Way galaxy has undergone significant physico-chemical evolution since its formation around 13 Gyr ago. This suggests several factors that may have influenced the birth rate function, λb (t), in the galactic history. The current rate (STR) in the galaxy has been estimated to 1-3 M~ year−1 (Licquia et al. 2014). However, studies suggest that the STR could have been up to three times greater in the past (Twarog, 1980). Supernovae produces vast amounts of cosmic rays, gamma rays, and x-rays that could potentially end life on nearby planets. As the SFR, and the fuel content, in the

16 galaxy has declined over time, so has the rate. According to Lineweaver (2004), the threat of supernova events could have been 20 times greater 10 Gyr ago compared to today, thus increasing the prospects of life over time. Another potentially important factor for the development of life is the aggregated metal concentration, e.g. [Fe/H], in the galaxy. Metals are formed at the end of the life-cycle of heavy stars and spread through the galaxy via supernova explosions. Simulations (Lineweaver, 2004) show that this process of buildup has increased the [Fe/H] ratio by at least two orders of magnitude, within a radial distance of 8 kpc from the galactic center, during the last 10 Gyr. The combined effect of these factors, including the assumption of 4±1 Gyr as a characteristic time required for the evolution of complex life, is analyzed by Lineweaver (2004). The result of this analysis is encapsulated by a birth rate function of the form:

 1 h i λ (t) = λ0 exp 1 − (t¯− t)2 , (3.1.3) b b 2σ2 where the variance of the appearance (in time) of a civilization is given by σ2 = 7/2 Gy2 and the mean t¯ = −1 Gy (i.e. the point in time for which the birth rate is maximum). The present birth rate, 0 0 λb (0) = λb , is a free model parameter chosen somewhere in the range 0 < λb < R⊕. The birth rate function, λb (t), as well as the instantaneous population, N (t) (Eq. 3.1.2), are visualized in the graphs 0 −6 −1 below, with a present birth rate arbitrary chosen to λb = 10 year .

0 3 −1 Figure 3.1.1 – Left: The birth rate function plotted for λb = 10 Gyr . Right: The instantaneous population, N (t), for three diffenent choices of λ`.

In the case when the civilization lifespan, λ`, is short compared to the time scale of the evolutionary processes of the galaxy, i.e. for λ` ≤ 1 Gyr, the present time mean population converges to that of ¯ a time-independent model, i.e. limλ`→0 N0 → Ne = λbλ` (the statistical equilibrium population). For instance, in the case when λ` = 1 Gyr there is only a 3.9 % deviation between N¯0 and Ne. This means that the present time mean population is approximately equal to the statistical equilibrium population, i.e. N¯0 ≈ Ne, for all λ` ≤ 1 Gyr. This is expected since t = 0 happens to coincide with a period in time when ˙ the time derivative of the birth rate function, λb, is comparably small. This, in turn, is a consequence of the evolution of Sbc-type spiral galaxies, such as the Milky Way, where the galaxy typically enters a phase of “quasi-constant” development over billions of years (Kennicutt & Evans, 2012; Prantzos, 2013). On the other hand, for the extreme value λ` = 10 Gyr, the approximate equality does no longer hold, with a deviation increased to a factor of three, i.e. Ne/N¯0 ≈ 3. Therefore, in order to avoid unnecessary restrictions on λ`, as well as for purely academic reasons, the time-dependent treatment of the ETI population will continue throughout the thesis (with an exception for the next section).

3.1.3 A Statistical Drake Equation This section will provide an alternative, statistical, form of the Drake equation. As in the previous section the birth and death of an ETI civilization is considered to be a stochastic process. However, unlike the

17 0 previous section the parameters, λb = λb and λ`, will, for the sake of simplicity, be regarded as time- independent. It should be noted, however, that the result from a time-independent analysis is generally applicable as long as λ` ≤ 1 Gyr (which can be regarded as a quite weak constraint). From these premises it can be shown, via the moment generating function (for proof see e.g. Koroliouk, 1978; Foata & Fuchs, 2002), that given a Poissonian birth process, ∼ Po (λb), with arbitrary stochastic survival function, L (E [L] = λ`), the population variable, N, itself is given by a Poisson distribution with parameter λbλ`. That is: N ∼ Po (λbλ`) .

By replacing λrλ` with the statistical equilibrium population, Ne, leads to an explicit statistical Drake equation1: N N P (N; N ) = e exp {−N } , (3.1.4) e N! e where Ne and N¯0 are interchangeable for λ` ≤ 1 Gyr. What is the probability for us being alone in the Milky Way given a certain Ne? By letting N = 0, this probability can be calculated as P (alone) = exp {−Ne}. For small integer values of Ne ≥ 0, this corresponds to a substantial probability, e.g. Ne = 2 gives P (alone) = 0.14. This implies that we could be alone even if Ne > 0. The special case when Ne & 15 can easily be shown to produce an alternative form of Eq. (3.1.4) (see e.g. Chapter 5.1). According to the Central Limit Theorem, a Poisson distribution approaches a normal distribution such that Po (Ne) → N (Ne,Ne) , where the mean and the variance is the statistical equilibrium population, Ne, and where the standard deviation is the square root of the√ variance. An approximate 95 % probability interval for N, centered at Ne, is given by IN = Ne ± 1.96 Ne.

3.2 Bayesian Analysis

This section describes the Bayesian analysis framework of the thesis. The section loosely follows the work of Grimaldi et al. (2018), however, while his work uses a simplified theoretical approach, this thesis adopts a more rigorous Monte Carlo simulation approach that accounts for real data. In Bayesian analysis, all unknown parameters are treated as random variables, rightfully described by their probability distribution. This is opposed to the frequentist interpretation where a parameter is considered unknown but fixed (e.g. there is only one true mean). Via Bayes’ theorem, the method of Bayesian analysis provides a technique to update an initial hypothesis on the basis of new evidence. Any such analysis consists of three main parts:  1. All knowledge available from earlier experience is captured in the prior distribution, fπ N¯0 .  2. Information provided by new evidence is captured in the likelihood function, L Dk| N¯0 . The 2 likelihood function expresses the likelihood of the observed evidence, Dk , given the population parameter, N¯0. 3. Prior knowledge and new evidence (observation, data) are linked via Bayes’ theorem and sum- ¯  marized in the posterior distribution, f N0 Dk . The posterior distribution thus represents the updated knowledge with respect to both prior knowledge and new data.

According to Bayes’ theorem, the posterior distribution for the population parameter, N¯0, given obser- vation Dk, can be written as:

L D | N¯  f N¯ D  = k 0 · f N¯  , (3.2.1) 0 k ¯ 0  ¯ 0  ¯ 0 π 0 L Dk| N0 fπ N0 dN0 ´ where the denominator can be regarded as a normalization constant since it is integrated over, and thus independent, of the population parameter, N¯0. The stochastic data variable, Dk, the number of detections for observation k (1 ≤ k ≤ n), is defined PC as the sum of C independent Bernoulli trials, i.e. Dk ≡ j=1 Djk, where Djk ∼ B (p). The sum, Dk,

1A log-normal statistical Drake equation is derived by Maccone (2012). In this derivation, each factor in the original Drake equation is assumed to be uniformly distributed, after which CLT is applied. 2Index k refers to real or simulated observations/data. All real data, corresponding to the SETI null result, can be grouped into e.g. k = 0, while each simulation output can be related to a specific k > 0.

18 in turn, can be empirically shown to converge to a Poisson distribution (Chapter 5.1), which gives an expression for the probability of making Dk number of detections: ∆Dk P (Dk; ∆) = exp {−∆} , Dk! where ∆ is an unknown distribution parameter. In order to find an expression for ∆, in terms of Dk, one can adopt the method of maximum likelihood, which gives the most likely value of ∆, given observation Dk. By definition, in this case, the likelihood is written as:

n P n Dk Y ∆k=1 L (∆) ≡ P (Dk| ∆) = n exp {−n∆} . Q k=1 Dk! k=1 By furthermore putting dL/d∆ = 0, yields a ML-estimate of ∆: n 1 X ∆ = D , n k k=1 thus, ∆ is (not surprisingly) the arithmetic mean of the number of detections of n different observation campaigns (e.g. n different simulation outputs). The mean number of detections is in turn a function of the population parameter, N¯0. From these observations it is now possible to define a new important parameter, δS , according to: ∆ δS ≡ . (3.2.2) N¯0 This parameter represents the detection probability given a mean number of detections, ∆, and a present time mean population, N¯0. The detection probability, δS = δ (S) where 0 ≤ δS ≤ 1, is a function of a given set of model settings, S, that is calculated via the Monte Carlo simulation.

3.2.1 The Likelihood: Two Special Cases  The likelihood function, L Dk| N¯0 , represents the weight of observational data in the posterior probabil- ity. In this thesis, data refers to the simulated number of detections, Dk, made by the joint SETI project (year ≥1960), combined with the “actual” number of detections (which can be assumed or arbitrary chosen). Two cases will be considered as follows: 0: The event of non-detection, which is the current SETI null result. 1: The event of exactly one detection.

In the case when the detection probability, δ (S), is a constant function of N¯0 (i.e. independent of N¯0) the likelihood functions have simple analytical expressions. The likelihood for the case of non-detection, 0, properly normalized over N¯0 space, can be written as:   L 0| N¯0 = δS exp −δS N¯0 (3.2.3) whereas in the case of exactly one detection, 1: ¯  2 ¯  ¯ L 1| N0 = δS N0 exp −δS N0 . (3.2.4) It should be noted that the likelihood representing the event of non-detection, Eq. (3.2.3), is a simple  declining exponential function with maximum at N¯0 = 0 (for which L 0| N¯0 = 0 = δS ). This is of course expected since data (no detection) suggests that the most probable value for N¯0 is zero. It is also obvious that the length of the tail is inversely proportional to the detection probability, δS . This makes larger values of N¯0 more probable as the detection probability decreases. However, unlike the case of zero detections, the likelihood representing the event of one detection has a maximum for N¯0 > 0. This maximum can be found by differentiating Eq. (3.2.4) with respect to N¯0, which yields: 1 Nˆ0 = . δS

This means that in the event of exactly one detection the most probable value, Nˆ0, for N¯0 is the inverse of the detection probability. A small detection probability (which would be the case of a weak transmitter) in combination with exactly one detection should therefore imply a large ML-estimate of N¯0.

19 3.2.2 The Prior Distribution  The prior, fπ N¯0 , contains all previous knowledge about the population parameter N¯0. This should be interpreted as the available knowledge at a time before SETI started in 1960 (since the likelihood function corresponds to data from all significant SETI studies dating back to 1960). A proper prior that would capture the lack of knowledge at this time could be what is known as the noninformative prior distribution. This is a prior that is uniform over the physical region of the parameter space, i.e. when N¯0 ≥ 0. However, since N¯0 can vary many orders of magnitude it could be argued that the  n n+1 integral fπ N¯0 dN¯0 should be constant when N¯0 is integrated from e.g. 10 to 10 (Cowan, 1998). That is to´ say that the prior is uniform in log N¯0, which has the property of giving equal weight to all orders of magnitude between some chosen limits. By performing the integration, it turns out that the non-normalized noninformative prior distribution can be written as:  ¯ ¯ 0, 0 < N0 < Nmin ¯   fπ N0 = 1/N¯0, N¯min ≤ N¯0 < N¯max  0, N¯0 ≥ N¯max

11 Here the upper cutoff value is set to N¯max ≡ 10 , approximately corresponding to the number of planetary systems in the Milky Way (which could be considered a maximally optimistic a priori estimate). The lower cutoff value is chosen to be some N¯min > 0 in order to make the posterior distributions normalizable. What would be a reasonable value for N¯min? To choose N = 1 would be to assume that there is a high probability (37 %) that we are alone in the galaxy. To choose N = 10−11 would be to assume that there is a high probability that we are alone in the entire visible universe (among 1011 galaxies). And to choose e.g. N = 10−23 would likewise be to assume that there is a high probability that we are alone in one trillion hypothetical . Because of the obvious uncertainty of N¯min a set of values will be tested, ranging from the more pessimistic to the more optimistic. That is: ¯ −n Nmin ≡ 10 , where {n ∈ N : 5 ≤ n ≤ 23} , −5 where the (upper) minimum value, N¯min = 10 , approximately corresponds to that of Grimaldi (2018).

3.2.3 The Posterior Distributions The aim of the Bayesian analysis is the posterior distributions and their cumulative distribution functions (CDF). The posterior distribution is, according to Bayes’ theorem Eq. (3.2.1) the normalized product of the likelihood and the prior distribution. There are two main cases in this study, the case for non- ¯  detection for which the posterior can be written as f N0 0 , and the case of exactly one detection for ¯  which the posterior can be written as f N0 1 . The corresponding CDF in the former case (0) is given by: N¯0 ¯  ¯ 0  ¯ 0 P N ≤ N0 0 = f N 0 dN , (3.2.5) ˆ 0 0 Nmin and in the latter case (1):

N¯0 ¯  ¯ 0  ¯ 0 P N ≤ N0 1 = f N 1 dN . (3.2.6) ˆ 0 0 0 The CDF in the case of non-detection does not give rise to an analytical expression and must be evaluated numerically. However, the CDF in the case of one detection can be expressed analytically, whenever  δS 6= δS N¯0 , thanks to the vanishing denominator. The normalized CDF in this case is thus given by: ¯   ¯ P N ≤ N0 1 = 1 − exp −δS N0 .

By e.g. putting Nˆ0 = 1/δS gives 1 − 1/e ≈ 0.63. That is, there is a 63 % probability that the present time mean population is smaller than Nˆe = 1/δS , given that exactly one ETI has been detected. In the ¯ Q case of non-detection, 0, the quantiles of the population parameter, N0 (0), are calculated numerically by putting the corresponding CDF equal to Q according to: ¯  P N ≤ N0 0 = Q1,2, where e.g. Q1 = 0.95 (2σ) and Q2 = 0.9987 (3σ). This procedure is also applicable if the detection  probability is a function of the population parameter, i.e. if δS = δS N¯0 .

20 3.3 Elements from Radio Astronomy

This section addresses some fundamental and necessary concepts from radio astronomy related to inter- stellar communication. This regards the physical laws of transmitting and receiving antennas, the radio- meter equation and the influence of Doppler drift.

3.3.1 Isotropic and Directional Antennas

All electromagnetic radiation attenuates as it propagates over distance. Specifically, the power PO of an isotropically transmitted radio signal is uniformly distributed over a sphere of radius R. Thus, it will obey the inverse square law given by: P A = 2 , (3.3.1) PO 4πR where P is the received power at an antenna of effective area A, and where R is the interstellar distance between the transmitter and the receiver. For instance, according to Eq. (3.3.1), an isotropic signal that e.g. traverses from the center of the galaxy to Earth is subject to a ∼ 1042-fold intensity drop. A directional antenna is an antenna that radiates all the electromagnetic energy in a specified direction. If a directional antenna, of effective area AD, is made to radiate through a solid angle Ω, a fraction of 4π sr, then the power output PD = POΩ/4π is equivalent to the isotropic (omnidirectional) power PO in the specified direction. The quotient, g ≡ 4π/Ω = PO/PD = AD/AO, turns out to be an important measure, called the antenna gain. By the Antenna Theorem (Nasa, 1973 p. 37), the effective area of an 2 isotropic antenna is related to the square of the transmitted or received wavelength, AO = λ /4π. The gain of a directional antenna can thus be related to its effective area, AD, through: 4πA r g g = D or l ∼ λ, λ2 D 4π where the latter expression relates the length scale of the antenna to its gain and the wavelength. Typically the gain has a quite large value for directional antennas, e.g. the gain of the Arecibo telescope is roughly 109 (90 dBi) at 10 GHz.

3.3.2 The Radiometer Equation It is often a challenge to distinguish a signal originating from an astronomical source from the thermal noise from the galactic background and the receiver (radiometer).√ The superposition of the noise and signal temperature fluctuate as a function of time, with amplitude, 2Tsys, and with a typical timescale, 3 −1 tN , inversely related to the bandwidth of the receiver , tN = (2∆ν) . Sampling system temperature measurements during some integration time, τ, thus yields a number of independent measurements,√ n = τ/tN = 2τ∆ν. According to a well-known relation the uncertainty of a parameter√ scales as 1/ n, which applied to this case, gives that the noise temperature, N, is proportional to Tsys/ τ∆ν. Thus an increased integration time gives a decreased system noise in the receiver, and thus an enhanced source signal. By comparing the source signal temperature, S = Tsrc, and the noise temperature gives the radiometer equation: S T √ = src τ∆ν. (3.3.2) N Tsys ∗ −1 The spectral flux density, Sν , of a source is the spectral power, Pν [W·Hz ], divided by the effective area, Ae, of the receiving antenna. The spectral flux density is often measured in Janskys, where 1 Jy = 10−26·W·m−2·Hz−1. By the Antenna Theorem it can be deduced (Berkeley.edu, 2019) that the spectral power in turn be written in terms av the source temperature and the Boltzmann constant, Pν = 2kBTsrc, which gives an expression for the spectral flux in terms of source temperature:   ∗ 2kB Sν = Tsrc, Ae where the coefficient in the parentheses is the antenna gain. Furthermore, it is often convenient to define the System Equivalent Flux Density (SEFD) of a telescope system as:   2kB Tsys ∗ SEFD ≡ Tsys = Sν . Ae Tsrc

3 See Nyquist rate; two samples are statistically independent if they are separated by a time increment ≥ tN .

21 This gives an alternative form of the radiometer equation, expressed in terms of the spectral flux density of the source and the SEFD of the observing system: S S∗ √ = ν τ∆ν. N SEFD By rearranging terms one have a useful relation for the minimum detectable flux density from a source given a certain integration time and minimum signal-to-noise ratio: √   ( ∗ S SEFD 1/ ∆ν, ∆ν ≥ ∆νt Sν,min = √ · √ (3.3.3) N min npτ δν/δνt, δν < δνt

Here np is the number of photon polarizations that the observing system can distinguish, either np = 1 or np = 2. In the case when the spectral resolution of the observation, δν, is less than the transmitted bandwidth of the source, δνt, the observable flux is reduced by the factor δν/δνt, as expressed by the second relation. Assuming a SEFD of the Arecibo telescope, 2.2 Jy, a S/N of 25, an integration time of 300 s, and a transmitted and observed bandwidth of 1.0 Hz, gives a minimum detectable flux density of 2.2 Jy. This would e.g. correspond to an isotropic monochromatic signal with luminosity, L = 1013 W/Hz4, transmitted from a distance of 0.2 kpc from Earth (about 2 % of the Earth-Milky Way Center distance). By contrast, a directed signal of equal luminosity (1013 W/Hz) and gain, g = 100 dBi, would be detectable by Arecibo from a distance of 6.2 Mpc. A distance that comprises roughly one thousand galaxies.

3.3.3 Doppler Drift Stars and planets are subject to relative motion within the galaxy. Due to the Doppler effect, this causes periodic frequency shifts in any astronomical signal which in turn has effects on the capability to detect narrow-band signals (Drake, 1984). The rate of this drift,ν ˙, is proportional to the frequency, ν, through: a ν˙ = − · ν, (3.3.4) c where a is the acceleration between the emitter and observer. If b is the spectrometer channel width and t is the time it takes for the drifting signal to pass through the channel, then t = b/ν˙. But t must not be smaller than the response time, τ, inversely proportional√ to the channel width, i.e. t > τ and b/ν˙ > 1/b. √ ∗ This yields a minimum usable bandwidth, bmin < ν˙ ∝ ν. This has implications for Sν,min, Eq. (3.3.2), which will be proportional to the square root of the narrow-band signal frequency. That is:

r ν S ∝ S∗ · . (3.3.5) ν,min ν,min 1 GHz

This Doppler drift correction is within one order of magnitude (. 3), and can thus be regarded small as compared to other uncertainties. However, although a small effect, it is being taken into consideration in order to reduce overall uncertainty as much as possible.

3.4 Galactic and Equatorial Coordinates

In an Euclidean 3-space, defined by Cartesian coordinates, an arbitrary point can be defined as R = (x, y, z). Its vertical projection on the xy−plane is then given by Rxy = (x, y, 0). If another point is placed at xe along the x-axis, then it is located at the point Re = (xe, 0, 0). Question: What is the longitudinal, l, and latitudinal angle, b, between origo/xy-plane and R as seen from Re? Given the definition of dot product of two Euclidean vectors it is possible to calculate these angles in terms of R, Re and Rxy. In a right handed coordinate system the equations for (l, b) become: ( (R − R ) · (−R ) l → l if y ≤ 0 cos l = xy e e , where |Rxy − Re| · |Re| l → 2π − l if y > 0 ( (R − R ) · (R − R ) b → −b if z ≤ 0 cos b = xy e e , where |Rxy − Re| · |R − Re| b → b if z > 0

4Total power output of human civilization.

22 These equations gives the galactic coordinates of a point R as seen from Earth (point Re). The Euclidean 3-distance between R, Re is given by D = |R − Re|. In the case when an observation (survey) is defined in equatorial coordinates (which is often the case), rather than galactic coordinates, a coordinate transformation is required. In order to transform galactic longitude and latitude, (l, b), to RA and Dec, (α, δ), the following set of equations is used:

cos δ cos (α − α0) = cos b cos (l − l0)

cos δ sin (α − α0) = cos b sin (l − l0) cos β0 − sin b sin β0

sin δ = cos b sin (l − l0) sin β0 + sin b cos β0,

◦ ◦ ◦ where the constants, α0, β0, l0 = 282.3 , 62.6 , 33.0 , corresponds to the J2000 epoch reference frame. It should be noted that the equations, for (α, δ), are bivariate and thus depend on both galactic longitude and latitude. By e.g. putting (l, b) = (0, 0) in the above equations gives the equatorial coordinates of the galactic center. Calculation gives a declination δ = −28.9◦, and a right ascension α = 17h 40m.

23 Chapter 4

Simulation overview

This chapter describes the general components of the Monte Carlo simulation model as well as the process of its implementation. The model has been developed exclusively by the author in MATLAB®. It consists, in essence, of a 3+1-dimensional Earth-centered Milky Way model in which ETI civilizations pop into existence in a random position in space and time. Model input is two free parameters described in Chapter 2.4.2. These parameters are sampled randomly from a predefined parameter space (Chapter 4.4). The simulation program allows the user to choose between a variety of submodels to analyze and compare the outcome.

4.1 Description of the Simulation Algorithm

The following Monte Carlo simulation models the birth and survival process of galactic ETI civilizations. It includes various submodels for different galactic geometries as well as different ETI signal characteris- tics. By running the simulation repeatedly, with randomly chosen input parameters (1/λb, λ`), predictions on the likelihood of detection in existing Earth-based radio telescopes (e.g. ATA, GBT and VLA) are obtained. The outcome is used to determine quantiles of the N¯0 parameter, conditioned on two different hypotheses: the current SETI null result, and, on the event of one detection. Here follows a compact description of the simulation algorithm. More details on the submodels and the filtering conditions are provided in Chapter 4.2 and 4.3.

Time of Birth Vector: Specifies the points in time when a new ETI civilization is born. A while-loop 5 9 is fed by a seed time defined by: Tstart ≡ − min 10 + 15λ`, 5 × 10 < 0 (unit years). It then iterates the instants, T1 = Tstart + ∆T1,T2 = T1 + ∆T2,... , Tn+1 = Tn + ∆Tn+1, where ∆Ti ∼ Exp(1/λb (Ti−1 + 1/λb (Ti−1))), until Tn+1 ≥ 0 at which time the process is terminated. The final result is a random vector, T , of length, n ∼ Po {(Tn − T1) λb}, corresponding to the total number of civilizations born in [T1,Tn].

Position Matrix: The position, Ri, of civilization i is specified by the GAL (n, BOOL) function that models the ETI spatial probability density, ρ (ri, θi, zi), via the EMW(0) and GHZ Models (1). The result is a position matrix, R, of dimensions n × 3. The position matrix is furthermore transformed into a corresponding matrix in Cartesian coordinates in order to calculate distances, |Ri − E|, between civilization i and Earth. In case of the GHZ Model, points outside the interval 5 ≤ ri ≤ 11 [kpc] 0 0 0 are subtracted resulting in T and Rj of length n ≤ n. Lifetime Vector: The lifetime of a civilization, i.e. the time spent in the radio communicating phase, is 0 specified by Li ∼ Exp (λ`) . The lifetime vector, L, is thus a random vector of length n or n .

Galactic and Equatorial Coordinates: Position Ri of civilization i is transformed to galactic coordinates via a scheme described in detail in Chapter 3.4. The galactic longitude, li, and latitude, bi, is in turn transformed to equatorial coordinates, (li, bi) → (αi, δi), also described in Chapter 3.4. Submodel: The simulation consists of a set of model categories, S = GG ⊗ RP ⊗ TF , in turn linked to an array of submodels preselected by the user. The Galactic Geometry Model category (GG) consists of two submodels, generated by the GAL (n, BOOL) function: the Entire Milky Way Model (EMW) and the Galactic Habitable Zone Model (GHZ). The Radiated Power Model category (RP) consists of the following submodels: the Static Power Output KI/K1.15/KII/KIII Models (SPO) (4

24 different) and the Progressive Power Output I/II/III Models (PPO) (3 different). The third model category defines the Transmitting Frequency (TF) which consists of three submodels: the H I Line Model (1.4200±0.0002 GHz), the water-hole Region Model (WHR: 1.1-1.9 GHz) and the Quiet Region Model (QR: 1-10 GHz). Overall, the submodels yield a total of 42 different combinations. All submodels are described in Section 4.2. Experiments, Filtering Conditions and Detections: A total of 13 unique SETI surveys are taken into con- sideration in this simulation (of which three are part of the Phoenix project). All surveys having an impact > 10−1 on the final result are included. Each survey is associated with a number of unique technical search parameters that results in a number of filtering conditions, giving logical 1 (true) or 0 (false) as a result. For a single survey in the simulation to make a detection, i.e. Djk = 1, the logical conjugation of all filtering conditions must be equal to one (i.e. 1 ∧ 1∧,..., ∧1 = 1). The first condition concerns the Causal Contact criterion (further discussed in Section 4.3.1) where it is checked whether or not the ETI spacetime position is part of the past light cone of Earth. The second condition regards sky coverage and checks if criterion (αi, δi) ∈ Ωs is met (where Ωs is the sky coverage of a certain SETI survey; see Section 4.4.2). In case of targeted searches an additional S condition checks whether criterion (αi, δi) ∈ j dΩj is met (where dΩj represents the field of view around a specific target j; see Section 4.3.2 for details). The third condition checks whether the flux density of the source (i.e. the spectral power per unit area) is greater or equal to the minimal detectable flux density of a given survey, i.e. Sν ≥ Sν,min (see Chapter 3.3 and 4.3.3). The fourth condition checks the transmitting frequency in relation to the frequency coverage of a SETI survey, i.e. checks whether criterion νt ∈ [νmin, νmax] is met (see Section 4.3.4 for further details).

Input and Output: All values are generated by the function call: >> N (1/λb, λ`), where the input param- eters are randomly sampled from a parameter space (Section 4.4). The output values from a single run are: Ne, NCC , NIRS, N0 and Dk, where Ne is the statistical equilibrium population, NCC is the population of civilizations with whom we have causal contact, NIRC is the number of spherical radio shells intersecting any point of the Milky Way galaxy, N0 is the present time population, and Dk is the number of unique civilizations detected. The output from n runs (function call:  >> runN) is the detection probability, δ (S), from which e.g. the likelihood function, L 0| N¯0 , ¯  and the posterior cumulative distribution function (CDF), P N0 ≤ N 0 , can be determined.

Table 4.1 – Sketch of the Civilization Data Matrix

# T L TOD R E3D RP ν Sν CC-test Dik-test

C1 T1 L1 T1 + L1 R1 D1 P1 ν1 Sν1 0 0

C2 T2 L2 T2 + L2 R2 D2 P2 ν2 Sν2 1 0

C3 ...... 0 0

C4 ...... 1 1 ......

Cn Tn Ln Tn + Ln Rn Dn Pn νn Sνn CCn Dn P P P P n T0 + ∆Ti ≤ 0 N0 ≡ {Ti + Li :≥ 0} NCC ≡ CCi Dk ≡ Dik

In Table 4.1 a schematic example of the Civilization Data Matrix is shown. The T - and L-columns are the Time of Birth and Lifetime vectors respectively. The elements of the Time of Death column (TOD) is the sum of the corresponding elements in T and L. Counting the number of elements in the TOD-column that is equal or larger than zero gives the present time population, N0. Column R represents the position matrix (n × 3) in Cartesian coordinates. The elements of the Euclidean 3-distance column (E3D) are the distances between point Ri and Earth. The Radiated Power column (RP) is the Equivalent Isotropic Radiated Power (EIRP) of civilization i given by one of the RP-models described in Section 4.2.2. The elements of the Frequency column, ν, is the transmitting frequency of civilization i restricted by the frequency range, [ν1, ν2], defined by one of the models in Section 4.2.3. The Flux density column (Sν ) calculates the flux density of the signal on the distance given by the elements in the E3D-column. The second last column, CC-test, checks whether the Causal Contact condition is met, and the sum of the column corresponds to the number of civilizations with whom we have causal contact at present. The last column, Dik-test, is calculated by, firstly, take the logical conjunction of all filtering conditions set by each survey (including the CC-test), and secondly, take the logical disjunction of the results from different surveys (e.g. 0 ∨ 1∨,..., ∨0 = 1). The sum of this column corresponds to the simulated number of unique civilizations detected, Dk, by the sum of all SETI surveys.

25 Figure 4.1.1 – Flowchart of the N and runN algorithm.

26 4.2 Submodels

This chapter provides a description of the 2 + 3 + 7 = 12 submodels, and 2 × 3 × 7 = 42 submodel combinations, used in the simulation. Section 4.2.1 presents and motivates two Galactic Geometry Models (GG): EMW and GHZ. Section 4.2.2 presents the Radiated Power Models (RP) that defines the EIRP of an ETI: SPO and PPO. The last section treats three Transmitting Frequency Models (TF): H I, WHR and QR.

Figure 4.2.1 – Visualization of S-space.

4.2.1 Galactic Geometry The Galactic Geometry Models (GG) models the ETI spatial probability density in the Milky Way galaxy. This is done in two ways: by using an Entire Milky Way Model (EMW), and, by using a galactic model subject to a galactic habitable zone (GHZ). Both are time-dependent1, stochastic models characterized by a spherical/cylindrical symmetry with exponential radial scale, following McMillan (2011), Reid et al. (2014), and Lineweaver (2004) respectively. Both models are displayed in Figure 4.2.2. EMW: The Entire Milky Way Model assumes that the entire galaxy is habitable for intelligent life (following e.g. Forgan, 2009; Prantzos, 2000). The disk is modeled as a highly flattened structure composed of three principal components. The spiral arms: Scutum-Centaurus, Carina-Sagittarius, Orion-Cygnus (containing Earth), Perseus and Outer arm, specified by the parameters in Reid et al. (2014); a spherically symmetric central bulge with exponential radial scale length of Rb = 0.5 kpc; a galactic diffuse with exponential radial scale length of Rd = 2 kpc and scale height of zd = 0.3 kpc. The density profile of each component is adjusted in order to match the overall density profile of the Milky Way. GHZ: The Galactic Habitable Zone Model, following Lineweaver (2004) and Grimaldi (2017), as- sumes that some parts of the galaxy are more likely to be habitable due to high metallicity, high star formation rate and a low frequency of catastrophic events, such as supernovae. According to the GHZ hypothesis there exists an annulus region, 5 kpc ≤ R ≤ 11 kpc (the time-averaged 68 % contour), concentric around the galactic center, with the highest poten- tial of harboring life. Consequently, the GHZ Model is defined by an annulus with sharp boundaries at 5 kpc and 11 kpc. Within the ring, it follows the density profile of the EMW Model. This geometry, combined with the overall exponential radial density profile, reduces the GHZ Model mean population to 9.5 % in comparison to the EMW Model. This discrep- ancy is accounted for by normalizing the GHZ Model such that equal local population density is obtained in the two models.

4.2.2 Radiated Power This section provides a description of the Radiated Power Models (RP), associated with the equivalent isotropic radiated power (EIRP) of a radio communicating civilization. This model consists of two subcategories, the PPO Models and the SPO Models, which in turn consists of a total of 7 submodels.

4.2.2.1 The PPO(I-III) Models The Progressive Power Output Models (PPO) is based on the assumption that the civilization power output is positively correlated to the lifetime of the civilization, i.e. that an older civilization possesses 1Constrained by the galactic star formation rate, metallicity, supernovae rate and timescale of complex life evolution, discussed in Lineweaver (2004).

27 more energy than a younger civilization. This assumption is at least true for the only technological civilization we know: the human civilization. However, it could be argued that the same laws of progress should apply to any technological civilization, at least for some time of their history, since all technology arises as a product of incremental improvement over time. In case of the human civilization, the energy consumption has been intimately linked to the human population growth which, in turn, has been roughly exponential ever since the agricultural revolution (10,000 years ago), when the world population was approximately 1-10 million (Bureau, 2019). Three different submodels are considered corresponding to three different exponential mean power ¯ L output progression rates, Pκ,L = P0κ1,2,3, calibrated to the 1950’s power output of the human civilization 13 (P0 = 10 W). In consistency with work done by others (e.g. Drake, 1984; Shostak, 2000; Enriquez, 2018), the power output vector, P , where element i correspond to the power output of civilization i, follows an exponential distribution (for a given mean, L), which is generated by a random vector such  36 that Pκ,L,i ∼ Exp P¯κ,L . In addition to this, a ceiling of 10 W has been set in each model, corresponding to the power output of a whole galaxy. The following expression gives the Kardashev K-value in the case of the three submodels: κ − 1 K (κ ,L) = 0.7 + 1,2,3 L, κ − 1 ≡ 10{−4,−3,−2}, (4.2.1) 1,2,3 10 ln 10 1,2,3 where L is the time (in years) that the civilization has spent in the radio communicating phase, and where κ1,2,3 is the annual growth rate defined by one of the three submodels. −4 PPO(I): The submodel corresponds to a conservative power output progression rate, with κ1−1 = 10 (i.e. 0.01 % yearly growth rate). This is half the estimated average annual growth rate since the Palaeolithic period2 (Iea.org, 2019; Wou.edu, 2003; Gibbons, 1993; Cook, 1971). At this rate, advancing one step on the Kardashev scale takes 105 ln 10 years ≈ 230,000 years.

−3 PPO(II): The submodel corresponds to a medium power output progression rate, with κ2 − 1 = 10 (i.e. 0.1 % yearly growth rate). This roughly equals the average annual growth rate since the agricultural revolution, 10,000 B.C (Iea.org, 2019; Bureau, 2019; Wou.edu, 2003; Cook, 1971). At this rate, advancing one step on the Kardashev scale takes 23.000 years.

−2 PPO(III): The submodel corresponds to an optimistic power output progression rate, with κ3 −1 = 10 (i.e. 1 % yearly growth rate). This equals the present annual growth rate of global energy consumption (Iea.org, 2019). At this rate, advancing one step on the Kardashev scale takes 2300 years.

4.2.2.2 The SPO(KI-KIII) Models The Static Power Output Models (SPO) are defined such that the civilization power output is static over time, lingering at one of the Kardashev levels, assumption (A5). As in the case of the PPO Models, the power output is generated by exponentially distributed random numbers, however, in this case with mean given by four predefined discrete values corresponding to a level on the Kardashev scale. For short, the submodels considered are denoted: KI, K1.15, KII, and KIII. Each model represents the spectral power of an isotropic or directed signal. In the case of a long-lasting, continuous, isotropic signal it is reasonable to assume that only a civilization of type K(N + 1) or higher could generate a signal of a power corresponding to KN. However, in the case of a directed signal, with the corresponding EIRP, this restriction could be significantly weakened. It should e.g be feasible for a KI civilization to construct a space-based rotating directional transmitter, covering the galactic plane, with a gain g = 100 dBi, and thus reducing the transmitting power by the same factor3.

SPO(KI): The KI submodel corresponds to an EIRP corresponding to the power output of a Type I civilization on the Kardashev scale, i.e. P = 1016 W (equal to that of a planet). Such a signal could be maintained by a Type I civilization using a directional transmitter or a Type II civilization using an isotropic transmitter.

2These calculations inevitably become somewhat speculative, however, the energy consumption per capita in terms of food intake has been roughly constant during the history. Thus, assume that the Palaeolithic man (with no other energy sources available) consumed 2000 kcal per day, averaging to 100 W. Further, according to some evidence there occurred a genetic bottleneck event around 70,000 B.C. when the population dropped to around 1000-10,000 (Gibbons, 1993). This gives an upper estimate of the world energy consumption at this time of ∼1 MW (in agreement with the “−6” in Eq. 2.7.1). Given a world power consumption today of 1.8 × 1013 W (Iea.org, 2019) gives a lower estimate of the average annual power consumption growth rate (since the Palaeolithic period) of just over 0.02 %. 3The Arecibo antenna (D = 305 m) has g ≈ 109 at 10 GHz.

28 SPO(K1.15): The K1.15 submodel corresponds to a signal power equal to the power output of an inter- mediate Type 1.15 civilization on the Kardashev scale, i.e. P = 1017.5 W, 32 times that of a Type I civilization (equal to that of e.g. a partial ). Such signal could be maintained by a Type I civilization using a directional transmitter or a Type II civilization using an isotropic transmitter.

SPO(KII): The KII submodel corresponds to a signal power equal to the power output of a Type II civilization on the Kardashev scale, i.e. P = 1026 W (equal to that of a star). Such signal could be maintained by a Type II civilization using a directional transmitter or a Type III civilization using an isotropic transmitter.

SPO(KIII): The KIII submodel corresponds to a signal power equal to the power output of a Type III civilization on the Kardashev scale, i.e. P = 1036 W (equal to that of a galaxy). Such signal could possibly be maintained by a Type II4 or a Type III civilization using a directional transmitter or a very hypothetical Type IV civilization using an isotropic transmitter.

4.2.3 Transmitting Frequency This section provides a background to the Transmitting Frequency Models (TF). The SETI search strategy was outlined in (Nasa, 1973) and then in e.g. the SETI Working Group Report (Drake, 1984). The main idea was that a technological civilization interested in galactic communication probably makes rational choices in order to maximize the benefit-cost ratio of their communication strategy. It is, of course, impossible to know what kind of technology a highly advanced civilization has developed, but the argument is that all civilizations at least should be limited by the physical laws of nature. The Project Cyclops report (Nasa, 1973) establishes radio waves as a natural choice for interstellar communication. Radio waves are cost-effective, easily produced, easily detected, they traverse the universe at the speed c, and they can propagate through the interstellar medium. In order to minimize dispersion or scattering in the interstellar medium a narrow-band strategy could be the preferred choice. A nearly monochromatic signal have the benefits of maximizing the signal-to-noise ratio and reducing the risk of it being mistaken for naturally occurring phenomena, such as astronomical masers (∆ν ∼ 300 Hz). However, a too narrow bandwidth would limit the information content of the signal and increase the problems imposed by Doppler drift, as seen in Chapter 3.3.3. The water-hole region provides a relatively quiet, noise-free, window in the radio spectrum (Drake, 1984). According to the SETI consensus, any technological civilization interested in interstellar commu- nication should be aware of the “”, and possibly gather there, like animals on the savanna. The water-hole region is characterized by two distinct emission spikes: neutral hydrogen atoms undergoing hyperfine transition (H I line) at 1420 MHz, and radiation lines from interstellar hydroxyl ions (OH line) at 1660 MHz (thus, H + OH = H2O). Three principal emission components surrounds the region: galactic synchrotron emissions at low frequencies (22 MHz-10 GHz), T ∝ ν−2.6 (Kogut et al. 2009), the cosmic microwave background, TCMB = 2.73 K, and quantum-effect emissions, T ∼ hν/kB, at high frequencies (& 100 GHz). By taking into account the influence of Doppler drift (Eq. 3.3.4) the region between 1-2 GHz, which forms a minimum in the noise spectrum, stands out (with no previous knowledge) as particularly promising. The TF Models are based on assumption (A3), i.e. that civilizations send narrow-band signals within a frequency band, νi ∼ U (νmin, νmax), with boundaries defined by three different submodels. The signals are further assumed to be of monochromatic nature with bandwidth δνi = 1 Hz. The TF Models, each corresponding to one ETI transmitting strategy, are defined as:

HI: The H I Line Model (1.4200 ± 0.0002 GHz) is confined by the spectral bounds νmin = 1.4198 GHz and νmax = 1.4202 GHz, corresponding to the 21 cm hydrogen emission line. WHR: The Waterhole Region Model (1.1-1.9 GHz) is confined by the spectral bounds νmin = 1.1 GHz and νmax = 1.9 GHz, corresponding to the local noise minima around the H I and OH lines.

QR: The Quiet Region Model (1-10 GHz) is confined by the spectral bounds νmin = 1 GHz and νmax = 10 GHz.

4This would, however, require a transmitting antenna of astronomical scale, ∼ 105 km, for a 10 GHz signal.

29 Figure 4.2.2 – 3D-plot of the models EMW (left) and GHZ (right). Earth’s position marked with red plus sign.

Figure 4.2.3 – Sky coverage of some SETI surveys specified in Table 2.1.

30 4.3 Filtering Conditions

This section discusses four so-called filtering conditions, i.e. criteria that must be fulfilled in order for an Earth-based radio telescope to detect a simulated ETI-signal following a characteristic defined in the last section. The first criterion regards the Causal Contact condition that ensures that Earth is currently in the proper spacetime position with respect to the signal. The following three criteria (Sky Coverage, Signal-to-Noise Ratio, and Frequency Coverage) corresponds to the instrumental bias imposed by all SETI surveys considered in this thesis. If a signal fails to meet one or more conditions it will pass the detector system unnoticed as white noise.

4.3.1 Causal Contact A minimal condition in order to achieve interstellar contact is that Earth, at present, is in causal contact with a hypothetical transmitter. This means that the worldline of a past omnidirectional transmitter (sending radio waves) must intersect the surface of our past light cone. In any other case, the signal passes Earth either in our past or in our future. Assume that a civilization, located at distance ∆r from Earth, initiate isotropic radio transmissions at time t before present, according to assumption (A2) and (A4). As time goes by their transmissions will spread isotropically, at constant speed c, in a spherical space bubble. Further, assume that this civilization (for some reason) end their transmissions after time L. This would result in a further expanding spherical shell with thickness cL. For us, at present, to detect such signal requires that Earth is in a region of space occupied by the shell, i.e. that Earth is between the inner and outer radius of the shell. This implies the following inequalities: ( ct − cL ≤ ∆r

ct + cLE ≥ ∆r, where LE is the time during which we have been able to detect radio signals from . By assuming that LE is insignificant compared to t (i.e. LE << t) one arrive at the following condition for causal contact: ct − cL ≤ ∆r ≤ ct, (4.3.1) where the lower and upper bound corresponds to the inner and outer radii of the spherical shell. This condition implies, for instance, that any transmission older than ∼ 80 thousand years will never be detected on Earth (since 80 kly is the maximum possible distance between Earth and a galactic ETI).

4.3.2 Sky Coverage All SETI surveys are limited by their sky coverage. No survey covers the full sky, but all surveys cover some portion of the sky, which often is the hemisphere in reach to a telescope at a certain latitude. Typically, an all-sky survey covers 30-70 % of the sky, while targeted searches cover significantly less, although with much greater “depth” due to higher sensitivity. This section describes the filtering conditions that arise from these two cases.

4.3.2.1 All-Sky Surveys A telescope performing an all-sky survey scans the entire sky in thin segments as at some constant drive rate, ω ≥ 0.004◦/s (Earth’s rotation). All-sky surveys have great potential to detect signals, in a wide frequency band, anywhere in the sky, but on the expense of limited sensitivity, typically in the order of ∼ 10−23 W/m2. Examples of all-sky SETI surveys are the META- and BETA projects at the Agassiz station as well as SERENDIP performed at Arecibo, Puerto Rico. Equatorial coordinates is an often used coordinate system for astronomical observations. In the case for all-sky surveys, they are usually specified by their minimum and maximum declination, δmin and δmax (determined partly by the latitude location of the telescope). The sky coverage condition in the case of an all-sky survey checks whether point, i, meets the criterion:

δmin ≤ δi ≤ δmax, (4.3.2) however, in order to make this test it is first necessary to make the proper coordinate transformation (in the case of this thesis from galactic to equatorial coordinates, a procedure described in Chapter 3). When the coordinate transformation is done, point i (corresponding to civilization i) can be subject for the test, Eq. (4.3.2).

31 It is of interest to ask how large part of the sky a certain survey covers. This can be calculated by integrating the surface of a sphere with radius R = 1 between altitude angle (declination) δmin and δmax, which yields the following useful formula:

Ωs = 2π (sin δmax − sin δmin) , where 0 ≤ Ωs ≤ 4π is measured in steradians. As an example, one can insert the values for Arecibo ◦ ◦ (+2 ≤ δ ≤ +35 ) and get Ωs = 3.4 sr, which corresponds to 27 % of the whole sky (however, as can be seen in Figure 4.2.3 it misses a large portion of the galactic plane, including the galactic center).

4.3.2.2 Targeted Searches A targeted search is conducted by observing each target for a time τ ∼ 102 s (i.e. a factor 100 longer than for sky surveys). This increases the sensitivity to about 10−25-10−27 W/m2, which allows detection of sources 1000 times fainter compared to all-sky searches. The weakness of targeted searches is, on the other hand, the limited number of targets, historically in the order of 100-1000. This demands a careful selection of promising targets, which in the case of SETI, is a difficult task. In the case of the Project Phoenix (Cullers et al. 2000), 800 nearby Sun-like stars were selected for a targeted six-year search. In the recent Red Dwarf Survey (Tarter, 2016) at ATA, 20,000 of the nearest (4-724 ly) red dwarfs were selected for a full 1-10 GHz search. The sky coverage of a targeted search is determined by the instantaneous beamwidth of the telescope, θB, (i.e. the telescope resolution ∼ λ/D) and the number of targets, n. If the frequency range is narrow, as in the case for most targeted searches, the beamwidth will be approximately constant. The beamwidth, θB, can furthermore be regarded as the diameter of a circle of area dΩj (the field of view). In summary, this gives an expression for the field of view around target j: π dΩ = θ2 , j 4 ln 2 B where ln 2 in the denominator holds for Gaussian beams (cv.nrao.edu, 2019). From this follows, given that the minimum and maximum declination of the search is known, that the probability for a point i to be within the covered portion of the sky, ∆Ω, can be approximated5 as:  n ∆Ω dΩj P∆Ω ≡ P {i ∈ ∆Ω} = = 1 − 1 − . (4.3.3) Ωs Ωs The assumption is now that not only the targeted objects in the search are subject to investigation, but all background stars within the covered sky portion (which along the galactic plane can be of the order of thousands). The sky coverage condition in the case of a targeted search thus checks whether the condition defined by Eq. (4.3.2) is met and then uses Eq. (4.3.3) to calculate the probability for (αi, δi) being in ∆Ω. The probability is then used to check the condition P∆Ω ≥ X, where X ∼ U (0, 1) is a random number. Let Project Phoenix serve as the first example. The specific beamwidth in this case (Table 2.1) is ¯ −6 roughly θB = 8 arcmin, which gives dΩj = 6 × 10 sr, and (with n = 800) a probability P∆Ω ≈ −4 n · dΩj/Ωs = 4 × 10 . Now, let the ATA Red Dwarf Survey serve as the second example. By using that ¯ ◦ −4 the ATA beamwidth (in the 1-10 GHZ range) is roughly θB = 1 , one have dΩj = 3 × 10 sr, and (with n = 20, 000) a probability P∆Ω = 0.5. This makes the Red Dwarf Survey comparable to an all-sky survey in terms of sky coverage.

4.3.3 Signal-to-Noise Ratio

The Radiometer Equation (3.3.4), states that the minimum detectable flux density, Sν,min, from a source is a function of the minimum signal-to-noise ratio, the SEFD of the instrument, the integration time and the bandwidth of the search. These are search specific parameters limited by time, technology and economics. These limitations have in turn influenced our expectations about the characteristics of ETI signals. However, little is known about hypothetical motives and capabilities of an ETI. Thus, the RP-Models (Chapter 4.2.2) cover a wide range of such motives in order to reduce any possible bias. The signal-to-noise criterion calculates the flux density, Sν , for an ETI signal at Earth, given any choice of RP-Model and random distance to the source, and compare it to the minimum detectable flux

5Point i is not uniformly distributed over the sky.

32 density, Sν,min, determined by the radiometer equation. It is here assumed that Sν,min is constant over the field of view, dΩ. Thus: Sν ≥ Sν,min. (4.3.4) Any radio signal that passes the test is in principle detectable in an Earth-based telescope.

4.3.4 Frequency Coverage Radio SETI has primarily focused on the frequency band 1-10 GHz, with particular emphasis on the waterhole region at the lower end of the interval. The arguments for this are mentioned in e.g. Section 4.2.3 and can be summarized as a combination of restrictions posed by physical laws, economy, and soci- ological speculations. The fourth and last filtering condition of the simulation compare the transmitting frequency, νt, of an ETI according to the TF Models (Section 4.2.3), to the frequency coverage of a SETI survey, [νmin, νmax]. That is: νmin ≤ νt ≤ νmax (4.3.5) According to the definition in Chapter 2, a true ETI signal is stable in nature and would therefore be detected repeatedly upon re-observation. This eliminates the risk that a signal is rejected as RFI. This also assumes that the problem of false-positives and false-negatives are properly handled.

4.4 Boundaries

The purpose of this section is to determine some boundaries on the two free model parameters, the birth rate parameter and the lifetime parameter, in addition to the interval on N¯0 discussed in Chapter 3. The boundaries will, in turn, be used in order to randomly sample the free parameters in the Monte Carlo simulation. The problem of potential bias is handled by adopting generous margins to the boundaries.

4.4.1 The Birth Rate Parameter

An upper bound estimate of the birth rate parameter, λb, follows from the example in Chapter 2.4.1. −1 The star formation rate (SFR) in the Milky Way is of the order 1-3 M~ year (e.g. Licquia et al. 2014), while the proportion of stars with planets lies in the span 0.5-1, with some studies suggesting a value in the upper end of this interval (Tuomi et al. 2014). With a proportion of solar systems with habitable planets estimated to be approximately 0.4 (Petigura et al. 2013), gives an order of magnitude estimate of −1 R⊕ ∼ 0.1-1 year . Furthermore, the observation that fbt ≤ 1, yields an approximate upper bound for the birth rate parameter, i.e. λb ≤ 1. It is more difficult to argue for a lower bound of λb. Engler et al. (2018) suggest that the birth rate −10 −8 parameter should lie in the interval 6.7×10 ≤ λb ≤ 2.7×10 , however, their argument relies on three crucial assumptions, of which the Copernican principle is one. Without making any assumptions on λb, + clearly, the only thing certain is that there exists a limit where fbt → 0 (the lower end of the physical region). This, in turn, implies that there is no explicit lower bound for the birth rate parameter, λb. For instance, any of the values 1, 10−10 or 10−100 could, in principle, be equally correct. In summary, these observations gives the following restrictions on the birth rate parameter:

R 0  −1 lim 10 < λb ≤ 10 years (4.4.1) R→−∞

This huge uncertainty of λb is the main reason for choosing a noninformative prior distribution in the Bayesian analysis, i.e. a distribution that is uniform in log N¯0. However, the need for a lower cutoff value of N¯0 will inevitably impose a lower bound of λb. Therefore, it is of no concern whether the lower bound of λb is undefined at this point. As it turns out, this will not affect the simulation result.

4.4.2 The Lifetime Parameter

Very little is known about the mean lifetime, λ`, of radio-communicating civilizations. The only existing example – us – provides no more than qualified guesses. As of 2019, our civilization has used radio technology for about a century. At the current rate of technological advancement, how long will this continue? Based on the knowledge of a phenomenon’s current age and the assumption that we observe the phenomenon at a random point in time, Gott (1993) argues that it is possible to predict its remaining lifetime. Thus, if we randomly observe a phenomenon of age T , then its remaining lifetime is, according

33 to Gott, given by the confidence interval [T/39, 39T ] at a 95 % C.L. Knowing that the human species is around 200,000 years old, Gott estimates the total lifetime of the human civilization to somewhere between 0.2 and 8 million years at a 95 % confidence level6. It is interesting to note that the upper end of this interval is of the same order as the typical lifespan of a species on Earth before extinction, 5-10 million years for invertebrates and 1-2 million years for (Mace, 1998). Based on a similar statistical argument Gott also provides an upper limit on the mean lifetime of radio-communicating civilizations, corresponding to λ` < 12.000 years at a 95 % C.L. (Gott, 1993). However, although Gott successfully predicted the remaining lifetime of the Berlin Wall in 19697, his method is not without critique (e.g. Caves, 2008). It is questionable whether Gott’s argument applies to the lifetime of species/civilizations since: 1) we are not necessarily random observers of ourselves, 2) the lifetime of living things are governed by the biophysical laws of nature (e.g. a human being has a highly predictable lifespan of . 100 years). A common assumption among the optimistic minded proponents of SETI in the 1960s (e.g. Sagan and Drake) was that the mean lifetime, λ`, lacks a strict upper bound (Cirkovi´c,´ 2004). It could be argued that a civilization that becomes sufficiently advanced would be immune to all childhood diseases and become immortal. This should e.g. be the case for a Type III civilization capable of colonizing a large portion of the galaxy. Even if some colonies disappear, other colonies will soon recolonize the depopulated areas. However, it is important to distinguish between a civilization’s lifetime per se and its lifetime, λ`, in the radio-communicating phase. Even if a civilization survives for billions of years, it is not at all certain that it will pursue radio-communication, or any communication, the full extent of its lifetime. Cirkovi´c(2004)´ argues that it seems “preposterous” that it would be possible to communicate to a civilization, say, 1 Gyr older than us. He, therefore, suggests adding an extra factor, ξ, to the Drake equation, corresponding to the portion of the civilization’s lifetime that is devoted to radio-communication, i.e. a “communication window”. The consequence is that a long lifetime not necessarily means a large N. To the contrary, N could be near zero if ξ is sufficiently small. Although this is an interesting and fruitful discussion, it should not be seen as more than pure speculation. The bottom line is: there exist no data that has bearing on λ`. The purpose of this thesis is to address this limited data problem by sampling the full parameter space instead of choosing some preferred value. So based on logical grounds, what are the minimum and maximum mean lifetime of a radio-communicating civilization, given that it has arisen? The minimum value would apply if all civilizations invent radio-communication and immediately thereafter blow themselves up, say, after 1 = 100 year. The maximum value would apply if all civilizations survive indefinitely. However, even such civilizations would be restricted by the age of the universe, 13.8 Gyr, and they would surely need at least a couple of billion years to evolve, which implies a maximum (current) lifetime of 1010 years. In accordance with this argumentation, the following boundaries are applied for the lifetime parameter:

0 10 10 ≤ λ` ≤ 10 [years] .

4.4.3 Sampling Domain By considering the intersection between the intervals determined in the previous two sections and the interval for N determined in Chapter 3.2.2, a sampling domain, D, for λ` and λb can be defined. This domain can be formally written as:

 + −23 10  + −33 0  + 0 10 D ≡ λ`, λb ∈ R : 10 ≤ N ≤ 10 ∩ λb ∈ R : 10 ≤ λb ≤ 10 ∩ λ` ∈ R : 10 ≤ λ` ≤ 10 ,

10 −33 where the regions N > 10 and λb < 10 have been excluded by the intersection. For convenience, the number N used in this definition, could be chosen to e.g. N ≡ Ne = λ`λb. It should however be noted that careful handling of Ne and N¯0 is required when it comes to the simulation output. In case of  the PPO Models, where δS = δS N¯0 , it turns out that the full sampling domain D does not perform optimally. This can be solved by adding an additional condition to D such that the population parameter, Ni, is varied over its entire regime (like before), but held stepwise fixed in each run. A new domain with S i i this property is defined by DN ≡ D , where each subdomain, D ⊂ DN , contain such λ`, λb-pairs i N N  for which N = Ni is fixed. This gives a more efficient tool for generating the function values of δS N¯0 .

6According to the current best estimate H. sapiens is 315.000 years old based on the evidence from the Jebel Irhoud site, Marocco (Hublin, 2017). This should increase Gott’s estimate to ∼ [0.2, 12] million years. 7Based on the present age of the Berlin Wall in 1969, 8 years, Gott estimated that the wall should fall before 1993 at a 50 % C.L. (it fell in 1989).

34 The sampling procedure proceeds as follows. Firstly, in order to get λ`, λb uniformly distributed in log-space it is necessary to randomly sample their corresponding exponents from a continuous uniform distribution. Secondly, the exponents of λ` are used as input in the sampling process of the exponents of λb, according to the definition of the sampling domain. The following code defines this approach in case of D:

s = unifrnd(0,10,n,1); t = -unifrnd(max(0,s-10),min(s+23,33),n,1); λl = 10.^s; λb = 10.^t;

i And in case of DN : s = unifrnd(max(0,log10(N)),10,n,1); t = s-log10(N); λl = 10.^s; λb = 10.^t; Here n is an arbitrary number that determines the number of runs of the simulation. It is typically chosen somewhere between 103 and 108 depending on the size of N. A small N requires a large n in order to acquire a sufficient amount of data. An important observation is that whenever the detection probability, δS , is independent of N¯0, any of the sampling domains D or DN are applicable. And further, if it can be  experimentally confirmed that the condition δS 6= δS N¯0 holds for a model (e.g. the SPO Models), then i it follows that a single subdomain, DN , is a sufficient sampling domain. Figure 4.1.1 below illustrates i the two sampling domains, D in black and DN in red, where each line represents a subdomain, DN .

Figure 4.4.1 – Sampling domain D (black) and DN (red).

35 Chapter 5

Results

This chapter presents the results of the simulation project. These are presented in diagrams and tables, followed by appropriate interpretations. The chapter is divided into three sections. The first section presents the results important for understanding the following sections. The next section is dedicated to the various simulation models of the study. A final section provides some relevant additional results.

5.1 Initial Results

The purpose of this section is to present some initial results that give an important background to the following sections. It analyses the two distributions, associated with the present time population and the number of detections, and the significance of the lower cutoff value which partly defines the prior distribution in the Bayesian analysis.

5.1.1 Two Important Distributions The theoretical framework described in the Bayesian analysis (Chapter 3.2) is partly based on a Poissonian ansatz, which gave rise to analytical expressions for the likelihood and posterior distributions. This is not a crucial premise for the result but it simplifies the computations significantly. It also helps emphasize the differences between the simulation submodels by attributing a specific value, δS , (or function) to each submodel. This ansatz is, however, not taken out of the blue but based on a stringent analysis of the stochastic behavior of the present time population, N0, and the distribution of the number of detections, Dk. This analysis is presented in Figure 5.2.1.  The upper panels in Figure 5.2.1 show the distributions P N0; N¯0 in the cases when N¯0 = 1, 2, 10, 100. The red curve in each graph represents the probabilities of a Poisson distribution of mean N¯0. As the result  show there is a clear agreement between the Poisson probabilities and the simulated data, P N0; N¯0 . As expected by the theoretical predictions made in Chapter 3.1.3, when N¯0 → ∞ the distribution converges ¯ 2 ¯ into a normal distribution of mean µN0 = N0 and variance σN = µN0 = N0. The lower panels in Figure 0  5.2.1 show the distributions of Dk, P (Dk; ∆), corresponding to the N¯0 of P N0; N¯0 above in the same column. The red curve in each graph represents, once again, the probabilities of a Poisson distribution of mean ∆ = D¯ k. Based on their similarities it is clear that the number of detections distribution, P (Dk; ∆), to good approximation follow a Poisson distribution of mean ∆. In conclusion, this verifies that both the present time population, N0, and the number of detections, Dk, are indeed Poisson distributed (or close to). This is an important observation that will be exploited in the further analysis.

5.1.2 The Significance of the Lower Cutoff

The choice of prior distribution poses a potential problem linked to the lower cutoff value, N¯min. This cutoff value is, of course, necessary in order to make the posterior distribution (in the case of non- detection) normalizable, but it is also a manifestation of uncertainty. In Chapter 3.2.2 this problem was  −23 −5 addressed by choosing N¯min from a set of values spanning over 18 orders of magnitude, 10 , 10 . This section provides an analysis of how the choice of N¯min affects the quantiles of the posterior distributions. ¯ Q Figure 5.2.2 shows the quantiles of the present time mean population, N0 , with Q = 0.9987 and Q = 0.95, as a function of N¯min in the case of two representative submodels. Black and red squares

36 correspond to calculated data points, while the dashed and dotted lines are regression lines fitted to the −23 −11 −5 data. The vertical gray lines correspond to the three principal values N¯min = 10 , 10 , 10 , where the lower end value correspond to the numerical limitations imposed by Matlab. ¯ Q ¯ 2 N0 exhibit, to good approximation, a log-linear dependence of Nmin (R ∼ 0.985-0.999). Although not visible in the figure, a weak non-linear behavior in log-space is present, with slopes decreasing as + N¯min → 0 . Most notable is that a higher quantile imply a flatter regression line (lower k-value), ¯ ¯ Q ¯ and therefore weaker dependence of Nmin; and vice versa, N0 is increasingly sensitive to Nmin as Q decreases. Additionally, regression lines corresponding to the same quantile but different submodel are almost parallel, showing that the N¯min dependence is insignificantly influenced by the model choice. For −2 the upper quantile, Q = 0.9987, the slope, k0.9987, is roughly 10 in both cases. Quantitatively this ¯ 0.9987 ¯ means that N0 decreases by a factor of 1.25 for every ten orders of magnitude decrease of Nmin. −2 In the case of Q = 0.95, the slope has grown by a factor of about 5-6, i.e. k0.95 ∼ 6 × 10 . This is ¯ 0.95 ¯  −23 −5 equivalent to a one order of magnitude variation of N0 for Nmin ∈ 10 , 10 . For lower quantiles, −1 e.g. Q = 0.50, the slope increases to k0.50 = 5 × 10 . This corresponds to a one order of magnitude ¯ 0.50 ¯ decrease of N0 whenever Nmin decreases by two orders magnitude, which is a substantial effect. As for the opposite extreme, however, kQ → 0 as Q → 1 , which implies that the N¯min dependence vanish altogether. There is, of course, disadvantages of choosing Q too high since it could significantly increase the margin of error. It is therefore important to choose Q such that the following opposite criteria hold: ¯ Q ¯ 1) N0 is kept within tolerable limits for varying Nmin, 2) to not inflate the margin of error. In conclusion, by choosing e.g. Q = 0.95, 0.9987 the influence of the lower cutoff, N¯min, can be kept within tolerable limits, without having a catastrophic effect on the margin of error (which can be seen in ¯ Q the next section). The present time mean population, N0 , corresponding to this choice of quantiles can thus be regarded as relatively robust with respect to the lower cutoff value.

5.2 Submodel Results

This section provides results for a major part of S-space defined in Chapter 4. In each case, upper limits on the present time mean population, N¯0, given the current null result (0), are determined. Additionally, maximum likelihood estimates of N¯0, under the assumption of exactly one detection (1), are calculated (Nˆ0). Each section varies the submodel within one of the three model categories (GG, RP, TF) while keeping the other constant. The figures display a limited subset of S-space, selected with respect to clarity −11 and/or generality, while the tables contain a larger portion of S. The lower cutoff value N¯min = 10 has been chosen as the principal value, corresponding to the assumption that we are alone in the visible −23 −5 universe, although the results corresponding to N¯min = 10 and N¯min = 10 has also been included in all tables. All submodel results are presented in 2×3 graphs. Columns contain the normalized likelihood, posterior and cumulative density functions. Rows contain graphs corresponding to the two cases: null result (0) and exactly one detection (1), respectively. Each graph, in turn, contains at least two submodels ¯ 0.95 for comparison. The vertical red dashed lines show the N0 value when 0 is assumed, while vertical blue dashed lines show the maximum likelihood estimate of N¯0 when 1 is assumed. The prior function is plotted as a red dash-dotted line. The robustness of the results has been investigated by testing four different classes of lifetime distribu-√ tions (for sampled λ`): an exponential distribution, L ∼ Exp(λ`), a normal distribution, L ∼ N (λ`, λ`), a uniform distribution, L ∼ U(0, 2λ`), and a delta peak function, δ(L − λ`). This analysis shows that the choice of distribution indeed affects the expectation value of the current age distribution, E [L0], i.e. the mean age of all civilizations alive at time T0. For an exponentially distributed lifetime: E [L0] = λ`; for a normally distributed lifetime: E [L0] = λ`/2; for a uniformly distributed lifetime: E [L0] = 2λ`/3; and for a delta peak lifetime: E [L0] = λ`/2. However, the choice of lifetime distribution does not affect the ¯ Q detection probability, δS , or the estimations on the upper limits, N0 . Figure 5.2.3 shows the detection probability, with corresponding uncertainties, as a function of N¯0 for some arbitrary chosen SPO Models (left) and PPO Models (right). The data clearly suggests that the detection probability, δS ≡ D¯ k/N¯0, indeed is a constant function of N¯0 in the case of the SPO Models and a monotone ramp-like function in the case of the PPO Models. The left graph also shows that the margin of error increases exponentially as N¯0 decreases (running time increases by a factor of five per  data point, read from right to left). These observations lead to the conclusion that, when δS 6= δS N¯0 , i a sampling domain of type DN , combined with a large N, is preferable over the full sampling domain, D, in terms of time complexity and margin of error. In the tables, a 95 % C.L, that accounts for the uncertainty of δS , is provided for each value.

37 Figure 5.2.1 – Distributions of N0, for varying mean, with the corresponding Dk distributions (K1.2 civilization).

Figure 5.2.2 – Significance of N¯min, larger k means stronger dependence. Right: kQ as a function of Q.

Figure 5.2.3 – Detection probability, δS , as a function of N¯0. Left: SPO Models. Right: PPO Models.

38 5.2.1 Varying Galactic Geometry The Galactic Geometry Models, defined in Chapter 4.2.1, can be considered as two extremes in terms of galactic habitability, none of which are completely realistic. However, being extremes, it should be easy for the reader to interpolate any intermediate value of choice. Varying only the Galactic Geometry Model implies that the Radiated Power Models (RP) and the Transmitting Frequency Models (TF) are kept unchanged, where the RP Model and TF Model of choice are SPO(KI) and WHR, respectively. The latter model represents the center value between the two extremes. The EMW Model is composed of three principal components (spiral arms, diffuse disc, central bulge) where the central bulge is the densest region, accounting for 1/3 of all stars/civilizations. This implies an ETI-Earth distance distribution, D, with the shape of a spike (at 8.1 kpc) with mean 8.6 kpc (the Earth-galactic center distance) and standard deviation 2.4 kpc. The Galactic Habitable Zone Model is a truncated version of the EMW Model with equal population density in the interval 5 ≤ R ≤ 11 [kpc], but zero otherwise. This property gives a much broader ETI-Earth distance distribution with two significant peaks at approximately 3 kpc and 13 kpc, with an overall mean of 10.1 kpc and a standard deviation of 4.7 kpc. Both distributions are displayed as PDFs in the left graph in Figure 5.3.1. Figure 5.2.4 shows the likelihood, posterior probability and the cumulative density function in the cases of the EMW and GHZ Models. The likelihood function encapsulates the data provided by all significant SETI searches done to date, represented by the detection probability (a function of the com- bined conditions: causal contact, sky coverage, sensitivity, and frequency coverage) and the on of the two cases 0 or 1. In the case of non-detection (upper panels), the likelihood function has a maximum at the endpoint N¯0 = 0, reflecting that N¯0 ∼ 0 is the most probable region given the null result. As N¯0 increases the likelihood exhibit a constant behavior until N¯0 ∼ 1/δS when data forces the likelihood towards zero. As seen from Figure 5.2.4 this sharp turn occurs for ∼ 10 times smaller N¯0 in the GHZ Model, compared to the EMW Model, as a result of its roughly 10 times higher detection probability, δS . The posterior distribution reflects the combined effect of the likelihood and the prior distribution. The prior, in turn, encapsulates the pre-SETI knowledge of the galactic ETI population, where it is assumed that N¯0 is uniformly distributed in log-space for N¯0 ∈ [Nmin,Nmax]. The upper-middle graph in Figure 5.2.4 shows that the posterior converges to the prior, f(N¯0) → fπ(N¯0), as N¯0 → 0. This behavior is seen for small N¯0 as long as the likelihood (data) does not object. However, the data has increasing ¯ ¯ difficulties in supporting larger N0, and, as in the case for the likelihood, when N0 & 1/δS the posterior makes a sharp turn towards zero. The cumulative density function (CDF) for the two models, in the case of non-detection, can be seen in the upper-right graph Figure 5.2.4. The prior, the red dash-dotted diagonal line, has endpoints in (Nmin; 0), (Nmax; 1). As for the posterior, the prior is the limiting case for the CDF:s when δS → 0, that is, when the influence of SETI approaches zero. It is important to stress that the behavior of the CDF:s are governed purely by the data and Nmin, and not by Nmax, as long as δS  1/Nmax (which is the case for all submodels). The red dashed vertical lines corresponds to the ¯ 0.95 N0 values for the GHZ and EMW Model. As expected, the calculated upper limit for the GHZ Model is significantly lower than the corresponding upper limit for the EMW Model, due to the lower δS of the ¯ Q latter. It is a general feature that the model difference in N0 roughly equals the inverse model difference in δS . In the (hypothetical) case of exactly one detection the result turns out rather differently (lower panels). 0 The likelihood has a global maximum at N¯0 = 1/δS , at which L (N¯0) = 0, and approaches zero for small and large values of N¯0. Therefore, the data support distinctly nonzero values, N¯0  0. In the case of 3 2 the EMW and GHZ Models (KI/WHR) the expectation values are Nˆ0 = 1.7 × 10 and Nˆ0 = 1.6 × 10 respectively, the former a factor ∼ 10 larger than the latter. As expected, given that one detection has been made, a small value of δS results in a large expectation value, Nˆ0. The posterior distributions are shown in the lower-middle graph in Figure 5.2.4. As a result of the choice of prior, the posterior distribution in the case of one detection happens to be identical to the likelihood in the case of non- detection, albeit with a different interpretation. As opposed to the case of non-detection, there is no longer any obvious relationship between the prior and posterior distributions, due to the one detection that gives increased weight to the data. As N¯0 → 0 the posterior approaches the constant, δS, due to  canceling factors in the likelihood and the prior, while the prior approaches infinity, fπ N¯0 → ∞. Data dictates that the posterior distributions, like in the former cases, makes a sharp turn towards zero for ¯ ¯ N0 & 1/δS . The GHZ Model posterior meets this threshold for smaller N0, due to a larger δS , followed by the EMW Model posterior for larger N¯0. Unlike the CDF corresponding to non-detection, 0, that ¯ 0 ¯ gives significant probabilities, P (N0 ≤ N0), in the full region [Nmin,Nmax], the CDF corresponding to 1, performs an almost stepwise transition from 0 to 1 around the expectation value Nˆ0 = 1/δS (marked with a blue dashed vertical line).

39 The qualitative description above is to a great extent applicable to all submodels in the study. In the specific case of the EMW and GHZ Model (KI), a 10 times larger detection probability has been noted for the GHZ Model compared to the EMW Model, affecting the upper limit estimates by the same order. This number is no coincidence, but follows from the fraction of the two population numbers in the EMW and GHZ Model, described in Chapter 4.2.1. This is somewhat expected since both models are normalized to the same population density in the annulus region, giving the same number of detections, D¯ k, if the EIRP is sufficiently small, whereas the total population number, Ntot ∝ N¯0, differ by a factor of 10.5. By definition, δS ≡ D¯ k/N¯0, which transfer the 10.5-factor to the detection probability in cases where the maximum detection distance is less than the half width of the GHZ region. More generally, 16 whenever the EIRP . 10 W, it holds that δS (GHZ)/δS (EMW) ≈ 1/0.095 = 10.5. However, as the EIRP increases, the situation turns out different. Table 5.1 gives a more complete picture of the overall differences between the EMW and GHZ Model. For higher EIRP the difference between the two galactic geometries decreases, and in the case of KII- and KIII-signals the EMW and GHZ Model exhibits the opposite relation: the EMW Model is associated with a higher detection probability than the GHZ Model. This might seem somewhat counterintuitive. However, it could probably be attributed to three combined factors. Firstly, due to the large EIRP, detections of type KII- and KIII-signals are less sensitive, or insensitive, to the Earth-ETI distance. Secondly, as seen in Figure 5.3.5, the ETI-Earth distance distribution in the GHZ Model is widely dispersed compared to that of the EMW Model, where a large portion of the signals originates from the galactic center. Thirdly, it is more likely that the filtering conditions favor a geometry where most signals originate from the central region in the galactic plane, given that this region is sufficiently covered, rather than a geometry where signals originate from random locations in the periphery.

5.2.2 Varying Radiated Power The Radiated Power Models are defined in Chapter 4.2.2. These models are in turn divided into two subcategories: the SPO Models and the PPO Models, adding up to a total of 7 submodels. Varying the Radiated Power Models implies that the Galactic Geometry Models (GG) and the Transmitting Frequency Models (TF) are held constant. The GG and TF Models of choice are the EMW and WHR Model, respectively. The corresponding likelihood, posterior and CDF are seen in Figure 5.2.5 (SPO) and Figure 5.2.6 (PPO).

5.2.2.1 SPO Models A general description of the qualitative behavior of the likelihood, posterior and CDF was given in the previous section. These considerations also apply to the RP Models. This section, therefore, provides a more concise description of the concrete differences between the KI, K1.15, KII and KIII Models. As expected, different EIRP results in very different values of δS. Signals of high radiated power are associated with a high detection probability and signals of low radiated power are associated with a low detection probability. For example, as the EIRP increases from 1016 W (KI) to 1036 W (KIII), the value −4 −1 of δS, increases from 6.0 × 10 to 6.9 × 10 , i.e. a range of more than three orders of magnitude. Less expected is the insignificant (or nonexistent) difference between the detection probability associated to 26 KII- and KIII-signals, indicating that δS reaches an upper limit for some EIRP ≤ 10 W. This could be attributed to a decreasing sensitivity to distance as the EIRP becomes large. So, for which threshold EIRP does δS start to deviate significantly from its maximum value? It turns out that this occurs in a narrow region between K1.15 and K1.3 corresponding to an EIRP between 3×1017 W and 1019 W. Thus, a rather distinct threshold value exists slightly above K1.15 (somewhat dependent on the model), and moreover, this threshold value is more than 10 orders of magnitude lower than the expected maximum EIRP of a KII or KIII civilization. As seen in Table 5.1 the wide range of δS in the SPO Models, spanning over three orders of magnitude, has dramatic effects on the estimated upper limits on N¯0 (in the case of non-detection), as well as on ˆ ¯ Q the expectation values, N0 (in the case of exactly one detection). The upper limits, N0 , on KII- and KIII-signals are confined to less than unity in all cases for Q = 0.95, increasing to 1.0 × 101 for Q = 0.999 in the most generous model (EMW/KIII/QR). In the event of exactly one detection of a KII- or KIII- signal, this signal is with almost a 100 % probability one of, at most, 5.0 similar signals in the galaxy, and with a relatively high probability, it is the only such signal in the entire galaxy. As regards K1.15- and KI-signals the prospects are somewhat better. As the EIRP decreases towards ¯ Q the threshold value the detection probability declines abruptly, resulting in larger estimates of N0 . ¯ Q −1 1 Regarding K1.15 and Q = 0.95 the value of N0 range between 8.9 × 10 and 7.2 × 10 , while in the

40 ¯ Q 3 case of KI, N0 range between 8.3 and 1.6 × 10 . The higher quantile, Q = 0.999, leads to a maximum upper limit value of 1.1 × 103 for K1.15 and 2.9 × 104 for KI. The ratio of the maximum number of “low” ¯ 0.999 ¯ 0.999 3 EIRP signals and high EIRP signals, N0 (KI)/N0 (K1.3 ≤), is in the order of ∼ 10 in the EMW 2 Models and ∼ 10 in the GHZ Models. The expectation value, Nˆ0, corresponding to the most probable value of N¯0 given 1, also vary greatly depending on the submodel of choice. Consequently, detection of one single KI- or K1.15-signal implies that there, on average, exists between 5.0 (H I) and 1.4 × 104 (QR) undetected signals of equal characteristics in the entire galaxy (where the lower figure only applies to signals where specifically ν = 1420 MHz). Thus, a reasonable estimate of Nˆ0 in the case of K1.15 and KI is in order of hundreds, or thousands, if the signal characteristics are minimally restricted by the model. On the other hand, in the case of non-detection of a KI signal, the probability of us being alone in the galaxy, N¯0 < 1, range between 74 % and 88 % (or between 95 % and 99 % for K1.3-KIII signals).

5.2.2.2 PPO Models The underlying assumption in the PPO Models is that the EIRP of a civilization, as in the case of the human civilization1, increases exponentially as a function of the civilization’s lifetime. This, in turn, results in a detection probability, δS, that is a function of the present time mean population, N¯0, i.e. δS = δS(N¯0), where the choice has been made to calculate δS for a series of stepwise fixed N¯0. As seen in Figure 5.2.3 it turns out that δS(N¯0) to a first order approximation is a linear ramp function in log N¯0. ¯ ∗ ¯ It approaches δS → 0 as N0 → −∞, in the lower end, and δS → δS as N0 → ∞, in the upper end. The ∗ upper limit, δ , turns out to be equal to the corresponding limit, δS , in the SPO Model for large EIRP S  and equal GG- and TF-submodel settings. In order to quantitatively identify the properties of δS N¯0 , it has been approximated by a ramp function of the form:

¯    ¯ ∗ δS N0 = min max α + β log N0, 0 , δS , where the model specific parameters α and β have been determined by a least square fit to the simulation 2  data (with R ∼ 0.984-0.999). It should, however, be noted that δS N¯0 exhibits some weak, but obvious,  non-linear behavior that is determined by the galactic geometry of choice. In the GHZ Model δS N¯0 exhibit an almost perfect linearity (R2 ∼ 0.999), whereas it in the EMW Model exhibit some tendency of a zigzag behavior (R2 ∼ 0.984). The error estimations of α and β are given in Table 5.2. As seen in Figure 5.2.6 the likelihood and posterior distributions of the PPO Models considered show a similar qualitative and quantitative behavior as in the case of the corresponding limiting SPO Model. A striking observation regarding the different PPO Models is that, despite corresponding to progression rates varying two orders of magnitude, and despite altering the galactic geometry, all results, as seen in Table 5.2, only vary within a factor of two (for illustrative purposes the two opposite extremes of the studied cases is shown i Figure 5.2.6). It seems that the, by far, most important factor for the outcome ∗ ∗ is the limiting value, δS , of the model, which in turn affects α but not β. Consequently, since δS is close ¯ Q to one for all PPO Models in Table 5.2, the corresponding upper limits, N0 , become close to unity. In ¯ 0.999 the case of the PPOI Model (EMW/WHR) the upper limit N0 is 5.9, which is the largest value of the investigated submodels. In the case of PPOIII (EMW/WHR) the corresponding upper limit is 4.2. By altering from EMW to GHZ Models results in a very small change in the upper limit estimate (since ∗ both models have almost identical δS ). The smallest upper limit value of the considered cases is given ¯ 0.999 by the GHZ/PPOIII/WHR Model for which N0 = 3.7. In the PPO Models, the detection probability, δS = δS (N¯0), is a function of the present mean pop- ulation, N¯0. However, since the PPO and SPO Models show a very similar qualitative behavior in the likelihood and posterior distribution it could be suspected that there should exists a constant, an equivalent detection probability, δE, that generates the same result as the function δS (N¯0). Investiga- tion of the likelihood function indeed reveals that there is such a constant. By taking the lower limit, ¯ limN¯→0 L(N0), one arrives at a general formula for the equivalent detection probability:

∞ 1/δ = exp −δ N¯ 0  N¯ 0 dN¯ 0 , E ˆ S 0 0 0 0 where the right hand side is the likelihood normalization factor. One can also note that this formula recovers the relation, δE = δS , in cases when the detection probability is a constant. The values of δE can be seen in Table 5.2 for each submodel. 1It could be argued that this applies primarily on the western civilizations rather than on the human civilization at large.

41 5.2.3 Varying Transmitting Frequency The Transmitting Frequency Models are defined in Chapter 4.2.3. Varying the Radiated Power Models implies that the Galactic Geometry Models (GG) and the Radiated Power Models (TF) are kept con- stant. The GG Model and TF Model of choice are EMW and SPO(KI), respectively. The corresponding likelihood, posterior and CDF are seen in Figure 5.2.7. As opposed to the PPO Models the TF Models comes with no major surprises. As expected, the ten- dency is a gradual decreasing detection probability, δS , for a progressively increasing allowed transmitting frequency bandwidth, going from H I to WHR to QR. It is however not possible to infer the resulting numbers a priori. The H I Model comprises frequencies between 1.4198 GHz and 1.4202 GHz, that is, a bandwidth of 0.0004 GHz. The bandwidth of the WHR Model, where frequencies between 1.1 GHz and 1.9 GHz are covered, is 0.8 GHz, that is 2000 times the bandwidth of the H I Model. This implies that if all SETI had been conducted exclusively around the H I line, as was the case in the 1960s, then the detection probability, δS , would be 2000 times higher in H I Model compared to the WHR Model. The actual figures are seen in Table 5.1, which gives δS (H I)/δS (WHR) ≈ 1.5-3  2000. The conclusion of this is that the joint SETI effort has covered both regions to almost the same degree, although with some particular focus on the H I region as expected. As regards the QR Model, which comprises transmitting frequencies between 1 GHz and 10 GHz, the situation is a little different. The QR Model bandwidth is thus 9 GHz or 11.25 times broader than the WHR Model bandwidth. A comparison analog to the former case gives δS (WHR I)/δS (QR) ≈ 3-8, i.e. not very far from 11.25 which would be the case if no SETI were conducted between 1.9 GHz and 10 GHz. The conclusion is that SETI has covered the water-hole region well, but more work remains to be done in the 2-10 GHz region. ¯ Q These observations imply the highest estimates of the upper limits, N0 , in the QR Model, and 1- 2 orders of magnitude lower estimates in the WHR and H I Models. The maximum upper limit is ¯ 0.999 4 associated to the EMW/KI/QR Model with N0 = 2.2 × 10 while the minimum upper limit for KI ¯ 0.999 1 civilizations is associated to the GHZ/KI/H I Model with N0 = 8.6 × 10 , i.e. a factor of 250 less. The minimum upper limit for any type of civilization is associated with e.g. the EMW/KII/H I Model, ¯ 0.999 0 where N0 = 1.64 × 10 .

5.2.4 Constraints on the Fermi Paradox Section 5.2.2.1 provided results for the SPO Models showing a relationship between the EIRP and the corresponding upper limits. Specifically, it was concluded that there exists a threshold EIRP, in the region 3 × 1017 W to 1019 W, above which the estimated upper limits are constant, and below which ¯ Q the upper limits are highly sensitive to the EIRP. Figure 5.2.8 shows the quantiles N0 , for Q = 0.95, and Q = 0.999, as a function of EIRP, where vertical red dashed lines indicate the principal Kardashev ¯ Q levels corresponding to KI, KII and KIII civilizations. The two curves of N0 divide the full N-EIRP- space into three subdomains where the Fermi paradox is more or less pronounced. In the domain where Q ≥ 0.999 the present time mean population, N¯0, is beyond what would be expected in 99.9 % of the cases, which means that the domain could be excluded at a 3σ level, given assumption (A1)-(A5) and the constraints of SETI. Consequently, if the real value of N¯0 belongs to the Q ≥ 0.999 domain one arrive at a Strong Fermi paradox. In such a scenario assumption (P1)-(P4) are all valid and the number of radio- communicating civilizations is very large compared to the expected (dictated by data). Any explanation that could account for this discrepancy should therefore be in the realm of the Zoo hypothesis. If N¯0 belongs to the domain where 0.95 ≤ Q ≤ 0.999, denoted Weak Fermi paradox, then it is somewhat less peculiar that no detections have been made, it could be a reflection of bad luck. However, in such a scenario it should be just a matter of a few years before we achieve our first contact. If, on the other hand, N¯0 is in the domain where Q ≤ 0.95, denoted No Fermi paradox, it should be expected that SETI to date has produced a null result since the number of communicating civilizations in the galaxy is too low for allowing any detections at our current technological level. I such a scenario assumption (P4), and possibly (P3), become more unlikely and the Fermi paradox is dissolved. It should, however, be noted that a very low number of N¯0 would conflict the Copernican principle, particularly if it turned out that we are alone in the entire galaxy. There is an interesting corollary to Figure 5.2.8 with implications on the civilization mean lifespan parameter, λ`. By assuming a PPO Model, with exponentially distributed lifetime, and by exploiting the most optimistic scenario, PPO(III), yields a lower bound for the lifespan parameter of λ` ≥ 34 years. This bound will increase as the upper limits are lowered in the future. A similar argument for the upper 4 bound gives λ` . 10 years, however, this value is associated with much higher uncertainty due to its sensitive dependence on the PPO(I) Model parameter.

42 ¯ ±11 Figure 5.2.4 – Varying the Galactic Geometry Model while keeping RP/TF Models constant. Nmax/min = 10 .

¯ ±11 Figure 5.2.5 – Varying the Radiated Power Model while keeping GG/TF Models constant. Nmax/min = 10 .

43 −11 Figure 5.2.6 – Varying the PPO Model (showing two opposite extremes). N¯min = 10 .

¯ ±11 Figure 5.2.7 – Varying the TF Model while keeping GG/RP Models constant. Nmax/min = 10 .

44 Table 5.1 – SPO Models

−11 −5 −23 Model 95 % C.L. N¯min = 10 N¯min = 10 N¯min = 10 ¯ 0.95 ¯ 0.999 ¯ 0.999 ¯ 0.999 ˆ EMW/H I δS N0 N0 N0 N0 N|1 KI 1.90(2) × 10−3 7.15(7) × 101 1.13(1) × 103 1.36(1) × 103 8.2(2) × 102 5.26(6) × 102 K1.15 3.04(1) × 10−2 5.24(2) × 100 7.27(2) × 101 8.94(3) × 101 5.3(1) × 102 3.29(1) × 101 KII 1.000(.1) × 100 1.964(0) × 10−1 2.308(0) × 100 2.93(0) × 100 1.64(1) × 100 1.00(0) × 100 KIII 1.000(.1) × 100 1.964(0) × 10−1 2.308(0) × 100 2.93(0) × 100 1.64(1) × 100 1.00(0) × 100

¯ 0.95 ¯ 0.999 ¯ 0.999 ¯ 0.999 ˆ GHZ/H I δS N0 N0 N0 N0 N|1 KI 1.86(1) × 10−2 8.33(4) × 100 1.18(1) × 102 1.45(1) × 102 8.6(2) × 101 5.38(3) × 101 K1.15 2.01(2) × 10−1 8.87(8) × 10−1 1.12(2) × 101 1.41(3) × 101 8.1(3) × 100 4.98(5) × 100 KII 1.000(.1) × 100 1.964(0) × 10−1 2.308(0) × 100 2.93(0) × 100 1.64(1) × 100 1.00(0) × 100 KIII 1.000(.1) × 100 1.964(0) × 10−1 2.308(0) × 100 2.93(0) × 100 1.64(1) × 100 1.00(0) × 100

¯ 0.95 ¯ 0.999 ¯ 0.999 ¯ 0.999 ˆ EMW/WHR δS N0 N0 N0 N0 N|1 KI 6.00(5) × 10−4 2.12(2) × 102 3.52(3) × 103 4.22(4) × 103 2.63(4) × 103 1.67(1) × 103 K1.15 1.47(1) × 10−2 1.04(1) × 101 1.49(1) × 102 1.82(1) × 102 1.08(2) × 102 6.80(5) × 101 KII 6.88(4) × 10−1 2.79(2) × 10−1 3.34(2) × 100 4.23(3) × 100 2.37(2) × 100 1.45(1) × 100 KIII 6.88(.1) × 10−1 2.791(0) × 10−1 3.338(1) × 100 4.226(1) × 100 2.37(2) × 100 1.45(0) × 100

¯ 0.95 ¯ 0.999 ¯ 0.999 ¯ 0.999 ˆ GHZ/WHR δS N0 N0 N0 N0 N|1 KI 6.10(8) × 10−3 2.38(3) × 101 3.56(4) × 102 4.32(5) × 102 2.59(5) × 102 1.64(2) × 102 K1.15 7.29(3) × 10−2 2.30(1) × 100 3.06(2) × 101 3.79(3) × 101 2.22(3) × 101 1.37(4) × 101 KII 6.80(3) × 10−1 2.82(1) × 10−1 3.38(1) × 100 4.27(1) × 100 2.40(2) × 100 1.47(1) × 100 KIII 6.80(1) × 10−1 2.82(1) × 10−1 3.38(0) × 100 4.27(1) × 100 2.40(2) × 100 1.47(0) × 100

¯ 0.95 ¯ 0.999 ¯ 0.999 ¯ 0.999 ˆ EMW/QR δS N0 N0 N0 N0 N|1 KI 7.1(3) × 10−5 1.59(6) × 103 2.9(1) × 104 3.5(1) × 104 2.2(2) × 104 1.41(6) × 104 K1.15 1.9(1) × 10−3 7.1(4) × 101 1.1(1) × 103 1.4(1) × 103 8.0(2) × 102 5.3(3) × 102 KII 2.2(1) × 10−1 8.1(4) × 10−1 1.0(1) × 101 1.3(1) × 100 7.1(2) × 100 4.5(2) × 100 KIII 2.2(1) × 10−1 8.1(4) × 10−1 1.0(1) × 101 1.3(1) × 101 7.1(2) × 100 4.5(2) × 100

¯ 0.95 ¯ 0.999 ¯ 0.999 ¯ 0.999 ˆ GHZ/QR δS N0 N0 N0 N0 N|1 KI 7.3(3) × 10−4 1.76(7) × 102 2.9(1) × 103 3.5(1) × 103 2.1(2) × 103 1.37(6) × 103 K1.15 1.49(9) × 10−2 1.03(6) × 101 1.47(9) × 102 1.8(1) × 102 1.1(2) × 102 6.7(4) × 101 KII 1.96(8) × 10−1 9.1(3) × 10−1 1.15(6) × 101 1.4(1) × 101 8.2(2) × 100 5.1(2) × 100 KIII 1.96(9) × 10−1 9.1(3) × 10−1 1.15(6) × 101 1.4(1) × 101 8.2(2) × 100 5.1(2) × 100

Table 5.2 – PPO/WHR Models

−11 Model 95 % C.L. 95 % C.L. 95 % C.L. 95 % C.L. N¯min = 10 ∗ ¯ 0.95 ¯ 0.999 EMW/WHR α β δE δS N0 N0 PPOI 3.26(6) × 10−1 6.3(1) × 10−2 3.68(6) × 10−1 6.88(.1) × 10−1 5.3(1) × 10−1 5.9(1) × 100 PPOII 4.14(9) × 10−1 6.2(1) × 10−2 4.42(9) × 10−1 6.88(.1) × 10−1 4.4(1) × 10−1 5.0(1) × 100 PPOIII 5.0(1) × 10−1 5.9(1) × 10−2 5.3(1) × 10−1 6.88(.1) × 10−1 3.7(1) × 10−1 4.2(1) × 100

∗ ¯ 0.95 ¯ 0.999 GHZ/WHR α β δE δS N0 N0 PPOI 3.92(5) × 10−1 8.4(1) × 10−2 4.45(2) × 10−1 6.80(1) × 10−1 4.4(1) × 10−1 4.9(1) × 101 PPOII 4.90(4) × 10−1 8.1(1) × 10−2 5.26(4) × 10−1 6.80(1) × 10−1 3.7(1) × 10−1 4.2(1) × 100 PPOIII 5.76(7) × 10−1 7.1(1) × 10−2 6.04(4) × 10−1 6.80(1) × 10−1 3.2(1) × 10−1 3.7(1) × 100

45 ¯ Q Figure 5.2.8 – N0 as a function of the Equivalent Isotropic Radiated Power (EIRP) of a civilization. The quantiles divide the N¯0-EIRP-space into three distinct subdomains, corresponding to: a Strong Fermi paradox (excluded region), a Weak Fermi paradox (restricted region), and, No Fermi paradox (possible region). The blue dashed line shows the threshold EIRP of 1019 W. The red dashed lines show the EIRP corresponding to the principal Kardashev levels of KI, KII and KIII civilizations.

46 5.3 Additional Results

This final section provides some additional results that have relevance for the topic of this work. The results of this section, obtained via small modifications of the Monte Carlo simulation model, comprises estimates of the distance to the nearest ETI, given one detection, establishes the relation between pop- ulation parameters, as well as extrapolations of future limits. Each section is associated with one of the figures on page 48.

5.3.1 Mean Minimum Distance This study has provided a straightforward technique for calculating the most probable ETI population size in the hypothetical event of one signal detection. By identifying the signal characteristics of this one signal the detection probability, δS, can be estimated. The expectation value of the present time mean population of ETIs of similar characteristic, Nˆ0, is then easily calculated through the inverse detection −2 probability, corresponding to the maximum of the likelihood. If e.g. δS = 10 for a given signal 2 characteristic, then Nˆ0 = 10 . In such scenario one could ask, how far away is the nearest communicating civilization2? The right graph in Figure 5.3.1 gives the mean minimum distance to an ETI, for a given 2 population size, Nˆ0, in the case of the EMW and GHZ Model. Pursuing the example, for which Nˆ0 = 10 , the EMW Model gives 2.6 kpc while the GHZ Model gives 0.9 kpc as the minimum distance. Therefore, since these galactic geometries represent two opposite extremes (in terms of the ETI density distribution) −2 it is very likely that, given one detection for which δS = 10 , the distance to our nearest (communicating or not communicating) neighbor is less than 1.75 ± 0.85 kpc, regardless of the distance to the one ETI −3 3 detected (which, by the way, could be difficult to establish). If instead δS = 10 , implying Nˆ0 = 10 , the corresponding interval decreases to 0.65 ± 0.25 kpc.

5.3.2 Relation between Population Parameters Figure 5.3.2 shows the relation between the four population parameters briefly discussed in Chapter 4.1. These are the statistical equilibrium population, Ne; the mean population of civilizations with whom we currently are in causal contact, N¯CC ; the mean number of isotropic radio shells intersecting any point of the Milky Way galaxy, N¯IRS, and, the present time mean population, N¯0. There are at least three things noteworthy. Firstly, somewhat surprisingly, the present time mean population, N¯0, and the population currently in causal contact, N¯CC , are equal in magnitude. This should however not be interpreted as the two populations being identical, they constitute different subsets where a member of one not necessarily is a member of the other. Secondly, the total number of isotropic radio shells intersecting the galactic plane, N¯IRS, produced by now living, and long gone, civilizations deviates strongly from the other population 4 parameters for Ne ≤ 10 . For e.g. Ne = 1 the ratio N¯IRS/N¯0 = 2700 in the EMW Model. As Ne grows larger, N¯IRS converges to N¯0 and N¯CC . Thirdly, the two galactic models, EMW and GHZ, produce two curves for N¯IRS, but one curve for N¯0 and N¯CC .

5.3.3 Future Limits The technological advancement since the early days of SETI has been remarkable. In project Ozma (1961) Drake monitored two stars, one at a time, in one channel. In the Red Dwarf Survey at ATA (Tarter, 2018), hundreds of stars were monitored simultaneously in billions of channels. One way of quantifying this development is to study the Drake Figure of Merit (DFM) defined by Eq. (2.5.1). In the 1960s a typical DFM was in the order of 10−6 while a typical value today is around 1, that is, a roughly one order of magnitude improvement per decade. As for the future, it is reasonable to assume that this development will continue. The Five hundred meter Aperture Spherical Telescope (FAST), operational in 2016 in Guizhou, Kina, will, among other things, search for ETI signals. It has a minimum detectable flux that is about three times that of Arecibo. The Allen Telescope Array (ATA) has been in operation since 2007 with its current 42 dishes. The future plan, although dependent on funding, is to increase the number of dishes to 350 which would add substantially to the sensitivity and the instantaneous sky coverage. The Square Kilometer Array (SKA) in the southern hemisphere, with stations in Australia and South Africa, is an international collaboration planned for full operation in 2030 (www.skatelescope.org, 2019). When completed, it will be 50 times more sensitive than any current system, and due to its location, it will be the first highly sensitive telescope to cover a substantial part of the galactic plane including the central parts of the Milky Way. 2Observe that “communicating” does not implies detectable.

47 The left graph in Figure 5.3.3 shows the minimum detectable flux, Smin, in all Radio SETI surveys 0.54 from 1960 until today. On average, Smin decreases by a factor of 10 = 3.5 per decade. Combining this result with the definition of DFM and its factor of 10 increase per decade implies an improvement of the frequency coverage, ∆ν, and sky coverage, Ω, by a factor of, on average, 1.24 per decade. This will continue until the frequency coverage, ∆ν, comprises the full 1-10 GHz region, and Ω = 4π sr. On the basis of these assumptions, the right graph in Figure 5.3.3 has been constructed. It shows a prognosis of the constraints on N¯0 in future SETI (the year 2040, 2060 and 2080), given the current technological trend. There are several things noteworthy in the figure. As of 2040, given that ATA and SKA are operating, the full 1-10 GHz region (in 1 Hz bins) will most certainly be covered in all directions, Ω = 4π sr. This, ¯ 0.999 in turn, yields an upper limit N0 converging to a minimum value of 2.3 that will never be surpassed. Another notable√ result shown in Figure 5.3.3 is the progression of the threshold EIRP, decreasing by a factor of ∼ 10 per decade. It shows that by 2080, given that we still have not made a detection, the threshold EIRP has moved to 1016 W, corresponding to the EIRP of a KI civilization (the right vertical ¯ 0.999 red dashed line in the figure). This implies that the upper limit N0 for KI civilizations has decreased to roughly 2.3 by the year 2080. Extrapolating even further into the future, given the same premises, as of 2140, we will be able to put an upper limit on Arecibo-like transmitters (EIRP = 1013 W) of 2.3 in the entire galaxy. Black squares in Figure 5.3.3 correspond to limits of KI civilizations for intermediate decades.

48 Figure 5.3.1 – Left: Distance distribution, f (D). Right: Mean distance to nearest civilization, given N¯0.

Figure 5.3.2 – Relation between different population parameters Ne, N¯0, N¯CC and N¯IRS in the EMW/GHZ Models.

¯ 0.999 Figure 5.3.3 – Left: Minimum detectable flux per decade. Right: N0 as a function of EIRP, year 2020-2080.

49 Chapter 6

Discussion and Conclusion

6.1 Discussion

This thesis has provided new analytical and numerical results concerning some important aspects of SETI. It has also provided a useful template for further research. Most notably, it is the first study known to the author to put constraints on the galactic population of ETI civilizations, conditioned on real SETI data. However, this analysis relies on five crucial assumptions, (A1)-(A5), regarding the characteristics of ETIs. These assumptions are briefly discussed below. The first assumption, (A1), states that ETIs arise independently and randomly in space and time. This is an assumption that lies at the heart of the original Drake equation, as well as in the stochastic birth and death process model of this thesis. Indeed, in a Copernican, non-panspermian, scenario, it follows that the building blocks of life are present throughout the cosmos and, through chemical and biological processes, develops into (intelligent) life occasionally. One central objection to assumption (A1) could be that it seems to rely heavily on the Copernican principle. This is partly true, but this problem is effectively addressed by sampling the input parameters from a very large sampling space reducing the bias to a minimum. It could also be questioned whether intelligent life, once it has arisen, will remain on its home planet indefinitely. The human species has already visited another celestial body (the Moon), and in the coming centuries, humanity will probably colonize the entire solar system. It is, therefore, reasonable to assume that ETI civilizations many millennia ahead of us have had time to colonize many solar systems. Such colonization would not follow a random Poisson process as stated by assumption (A1), but if the civilization as a whole, including all of its colonies, is regarded as one communicating ETI civilization, this assumption turns out more plausible. The second assumption, (A2), states that the ETI transmissions are isotropic, or directed, beacon EM-signals intended for galactic communication. An advanced civilization consisting of many colonies is likely to conduct some form of communication in order to maintain contact within the civilization. Such communication would with all likelihood be directed and utilize technology far beyond our capabilities. It is therefore highly unlikely that we would be able to eavesdrop on the internal communication of another civilization. However, if this multi-stellar civilization is curious in nature and interested in contact with other intelligent civilizations, then they might reveal their presence by using loud omnidirectional or directional transmitters. is a hallmark of intelligence and seen in several species on Earth (e.g. , dolphins, crows). It is difficult to imagine any scientific or technological progress without curiosity. However, it is not certain that other intelligence exhibits the same type of curiosity as we do, and in the case of a post-biological civilization it is very probable that the driving forces are very different from ours. Nonetheless, if galactic technological civilizations are common, it can be suspected that at least some of them exhibit curiosity similar to ours. The third and fourth assumption, (A3)-(A4), states that the ETI communication is conducted some- where in the L-X-band (1-10 GHz) in a narrow-band 1 Hz-channel, and that the signals are either continuous or short-pulsed (T . 30 s), transmitted on a 100 % duty cycle basis. If there is a subset of civilizations that are interested in galactic communication, which technology would be most suitable? To maximize the chances of contact it would be reasonable to use a simple transmitting technology that is cheap and produces easily detected signals. From our perspective radio waves falls into this category. By conducting communication in a frequency band where there is a minimum of noise, such as the water-hole region, and by using narrow-band signals, which are resistant to dispersion and scattering in the interstel- lar medium, radio waves becomes an even more promising candidate. A continuous or short-pulsed signal

50 would, of course, be preferred if the goal is to maximize the chances of contact, but it could also be costly. These assumptions, however, rely on the premises that ETIs are interested in making contact at all, that radio waves is a reasonably good choice compared to any alternative, and that other civilizations have undergone roughly the same technological development as we have. It is quite possible that ETIs indeed communicate, but put more focus on the information content of the signal rather than its detectabil- ity. By the Shannon-Hartley theorem, an optimal information bearing signal has properties similar to Gaussian white noise (Kardashev, 1964; Harp, 2010) and therefore resembles many natural phenomena. Further, if the signal is subject to some complicated modulation it would be almost impossible to detect. That ETIs should communicate in the 1-10 GHz region is also less than certain. The transparent window as seen through Earth’s is approximately 1-30 GHz, and the galactic transparent window is even broader. Furthermore, if civilizations typically use gamma rays, gravitational waves, neutrinos, or some other hypothetical particles, this could be the principal communication channel among galactic civilizations. It could, however, be argued that it is unlikely that we are unique in thinking of radio waves as a means of galactic communication, even if other means of communication are more common. The fifth assumption, (A5), states that the ETI signal EIRP can be mapped onto the Kardashev scale. Among contact optimists, it is usually assumed that our searches to date have been too limited and too insensitive for any intelligent signal of extraterrestrial origin to be detected. However, any advanced civilization interested in galactic communication is, of course, aware of the power needed to traverse the galactic distances. Assuming that a KII civilization employs a directional rotating transmitter covering the galactic plane, it turns out that an EIRP of 1026 W is more than enough to be detectable all over the galaxy with much more primitive instruments than ours. If the directional transmitter has a gain of 100 dBi (ten times that of Arecibo in 10 GHz) the actual power of that transmitter would be 1016 W, i.e. the power received by an entire planet from its star. Is this too much for a KII civilization? A quick calculation shows that a power of 1016 W to a KII civilization would translate to a power of 1 kW to us. This could be compared to one single air traffic control radar that typically has a power of 100 kW. And there are many such devices continuously transmitting on Earth. Thus, if there are KII civilizations in our galaxy, they should by a good margin be able to maintain a continuous, or short-pulsed, signal EIRP of 1026 W. It is, therefore, reasonable to assume, given assumption (A1)-(A4), that the signal EIRP of a civilization could be of the same order as their level on the Kardashev scale. The results of this thesis show that continuous, or short-pulsed, monochromatic radio signals in the L-X-band with an EIRP of 1019 W (K1.3), and above, are sparse or nonexistent in the galaxy. This is in line with Griffith et al. (2015) that concluded that among 100,000 investigated galaxies, no one exhibits obvious signs of being inhabited by a KIII civilization. However, the results also give an upper limit −7 −1 (somewhat dependent on the submodel) in the order of thousands on KI civilizations (n . 10 star ) currently transmitting in the galaxy This is a significantly tighter bound than that found in e.g. Horowitz & Leigh (1999) where n < 5 × 10−6 star−1. The large discrepancy between the upper limits on strong and weak EIRP signals is first and foremost an artifact of instrumental bias and should not necessarily be interpreted as KI civilizations being more common in the galaxy. It should also be clearly stressed that the upper limits, in the case of non-detection, are strongly dependent on the choice of confidence level (3σ). As long as the null result prevails, the likelihood function favors zero as the most probable population number in our galaxy. For instance, the CDF median, in the case of the most generous model (EMW/KI/QR), is only 2.8 × 10−4. Further, the same submodel settings yield a probability of 74 % that we are alone in the galaxy (88 % in the most conservative KI submodel). This result is compatible with Sandberg et al. (2018), whose main conclusion was that there is a substantial probability (53 % to 99.6 %) that we are alone in the galaxy. However, the most pessimistic result of Sandberg et al. (2018) is hard to reproduce without making rather extreme assumptions on Nmin. In the hypothetical event of detecting one single KI signal, we can infer that there are hundreds, or possibly thousands, of others, unique, undetected signals in the galaxy (consistent with Grimaldi, 2018). This could, of course, be seen as a strong argument for the contact optimists. Thus, if we detect one KI signal, it is just a matter of time before we detect another. Then we can conclude that our search strategy is correct. In such a scenario we can also conclude that the civilization birth rate parameter, λb, must be comparably large. However, this thesis strongly suggests that K1.3 signals, or higher, are very sparse in the galaxy. Thus, what implications would the detection of a single KI signal have? One possibility is that more advanced civilizations (K1.3≤) abruptly stop sending radio signals. Another possibility is that they stop communicating altogether. Yet another possibility is that civilizations typically have a short lifespan and that more advanced civilizations simply do not exist. Given that there are hundreds of radio-communicating KI civilizations, which is the conservative conclusion if we detect only one, this last, most pessimistic, possibility may seem the most plausible.

51 If we, on the other hand, do not detect any signal in the coming decades or centuries, then we can conclude by the year 2080 that there probably are no KI civilizations, satisfying assumption (A1)-(A5), in the entire galaxy. And by the year 2140, we can conclude, with high probability, that there are no continuous Arecibo-like transmitters in the entire galaxy. Would this favor a Rare Earth scenario? Possibly, but it can not be ruled out that ETIs still exist but that they are uninterested in galactic communication, that they use elusive gaussian, information bearing signals, that their signals are directed in a different direction than ours, that their signals are subject to encryption, impossible to decrypt without the right key, that they conduct communication in a completely different frequency band than radio, or exploit something else than photons for transmitting information. Paradoxically, without any detection, there is less support for a small λ`, and hence, less support for the Doomesday argument. The future prospects of humanity is therefore still an open question.

6.2 Conclusions

The purpose of this thesis has been to determine some upper limits on the present time mean population of galactic ETI civilizations and to find the constraints on the Fermi paradox, conditioned on data generated from the joint SETI project. This has been realized by exploiting five idealized assumptions, (A1)-(A5), on ETI characteristics, consistent with the historical SETI consensus, in order to construct a Monte Carlo simulation model comprising the ETI birth and death process in 3+1 dimensions, two galactic models, various signal characteristics, as well as detection biases imposed by the most significant SETI experiments conducted between 1993 and 2018. ¯ 0.999 As expected, the estimated upper limits, N0 , depends greatly on the submodel of choice, and range between 2.3, for KII and KIII civilizations transmitting on the H I line (1.42±0.0002 GHz), and 2.9 × 104 for KI civilizations transmitting somewhere in the QR region (1-10 GHz). The upper limit ratio of KI civilizations to K1.3-KIII civilizations is in general ∼ 102-103, allowing for KI civilizations to be quite −7 −1 frequent in the galaxy (. 10 star ), while KII and KIII civilizations, satisfying assumption (A1)-(A5), are sparse or non-existent. The results have been shown to be robust with respect to the lower cutoff value, Nmin, in the prior. As for an adequate error estimation of N0, given that the upper limits regards the present time mean population, N¯0, the confidence interval derived in Chapter 3.1.3 could be applied. Although the result allows for a rather large population of KI civilizations, the probability for us being alone in the galaxy, given the current null result, has been estimated to between 74 % and 88 %. In addition to the upper limits, the analysis has investigated the implications of the hypothetical event of exactly one detection. The result shows that, given one detection of a ETI signal, it is probable that several more exist. The most likely number of communicating galactic civilizations in such scenario is dependent on the characteristics of the detected signal. If the signal is of type KI, transmitting in the QR region, it can be inferred that between 103 and 104 other similar signals exist at our position in the galaxy. On the other hand, if the signal is of type KII or KIII, transmitting on the H I line, then this is to a high probability a unique signal. However, a detection of a KII or KIII civilization should give us hope of finding numerous other less advanced civilizations. In case of detecting a KI/QR signal, the analysis suggests that our nearest neighbor is located at a distance of 0.65 ± 0.25 kpc from Earth, and in case of detecting a KII or KIII signal, its origin is most likely to be 9.1 ± 1.0 kpc from Earth. Another significant result of this thesis is the finding of a threshold Equivalent Isotropic Radiated Power (EIRP), above which the upper limits are constant, and below which the upper limits are highly sensitive to the EIRP. At the current stage of SETI, this threshold value is located at an EIRP of 3×1017- 1019 W, corresponding to that of a K1.15-K1.3 civilization. This implies that all calculated upper limits for KII and KIII civilizations also apply to K1.3 civilizations and above. On basis of this analysis, a diagram has been constructed defining the boundaries for which the Fermi paradox can be considered “strong”, “weak” or nonexistent. The conclusion is that the (soft) Fermi paradox is an actual paradox only if the real galactic ETI population obeys the idealized assumptions, (A1)-(A5), and simultaneously exceeds the determined upper limits. In future SETI, given the present rate of technological advancement, we will lower the current limits significantly. By year 2080 the analysis suggests that we have a minimum upper limit on galactic KI civilizations of 2.3. As a final remark, the results produced in this thesis is highly dependent on the five assumptions, (A1)-(A5), stated in Chapter 2.1. If these assumptions are representative of the real universe, then the results could be considered accurate. If one or more of the assumptions are too rigid, then these results should be regarded as conservative estimates of the upper limits.

52 6.2.1 Limitations of the Thesis The principal assumptions, (A1)-(A5), of this work, were specified in Chapter 2.1. It is very important to stress that these assumptions follow from the limitations of SETI, and should therefore not be considered as limitations of this work. A meaningful analysis has been possible only by adopting SETI-compatible assumptions. As mentioned earlier, some of the more severe limitations of SETI to date is: the limited frequency band covered (1-10 GHz), which constitutes only 1/30 of the full radio spectrum (1-300 GHz), the inability to detect long pulsed signals (> 30 s.); the inability to filter out anything else than monochro- matic, or simply modulated broadband, signals from the background noise; and the inability to detect directed signals not deliberately pointed toward Earth. Another aspect worth mentioning is that SETI, as in all science, has a reobservation criteria that must be fulfilled in order to announce a detection. It is quite possible that hundreds of recorded “suspicious” signals, in fact, are real ETI signals but not con- firmed as such since they have only been observed once. One such example could be what is known as the “Wow!” signal detected on August 15, 1977 (Kraus, 1994). Nowadays similar signals have been recorded in excess, formally known as Fast Radio Bursts (FRBs). Although the most plausible explanation of FRBs is that they are absolutely natural in origin, the physics behind them is still poorly understood. One of the most obvious limitations specific to this work is that it only considers surveys done in the field of Radio SETI. In doing so the thesis only addresses a special case of the Fermi paradox, the “soft” Fermi paradox, that considers the absence of detectable radio signals from other civilizations. Radio SETI is certainly the oldest branch of SETI, but during the last decades, other branches have emerged. Examples are Optical SETI (OSETI), SETI performed in the visible light spectrum, searches for anomalies in databases (e.g. The VASCO network, 2019), and searches for signs of galactic and extragalactic , such as Dyson spheres. Nor does the thesis take into account results from other non-SETI surveys covering large portions of the sky, e.g. SDSS or Gaia. Combining the null result of all branches of SETI, and non-SETI, would, of course, lower the upper limits determined in this thesis to some extent. However, Radio-SETI is still the predominant, most specialized, search strategy, so any contribution from other branches of SETI should be limited. Another limitation of the thesis is the assumption that narrow-band signals are negligibly dispersed and scattered in the interstellar medium. Although this is a standard assumption within SETI, it can not be ruled out that the interstellar medium has some non-negligible effect on narrow-band signals traversing the full extent of the galaxy. A follow-up study could investigate this effect in more depth.

6.2.2 Current and Future Research Kent Cullers (2000) concluded in the Project Phoenix report: “The future of SETI as a part of a compu- tationally driven radio astronomy revolution is bright indeed.” This thesis has shown that a significant amount of work has been done in the search for narrow-band signals in the L-X-band. However, it also stresses the need for further surveys on broadband signals, the spectrum above 10 GHz, and along the galactic plane with focus on the central regions. Continuous narrow-band signals carry only one bit of information which is not optimal for interstellar communication. Harp (2010) describes a new promising search strategy based on an autocorrelation algorithm for detecting new classes of broadband signals that could contain arbitrary large amounts of information (Harp, 2010). This search will benefit greatly from the improved sensitivity offered by e.g. ATA and SKA in the coming decade. Apart from new developments in Radio-SETI (covered in this work), the future looks promising in many other branches of SETI. That includes Optical SETI, Laser SETI (https://www.seti.org/laser-seti), search in the gamma spectrum, search for outlier phenomena, such as disappearing stars, stars glowing in the , or abnormal galaxies (The VASCO network, 2019), that could be signs of extraterrestrial activity. SETI will also benefit indirectly by new space-based telescopes searching for and characterizing exoplanets, such as TESS, CHEOPS, JWT, PLATO and ARIEL (Sci.esa.int, 2019). The James Webb Telescope (NASA), scheduled for launch 2021, will be able to determine the properties of . This could e.g. reveal atmospheric that could be signs of life or artificial pollutants that could be signs of industrial activity of a civilization. In addition to the observational part of SETI, this work opens up for several follow-up studies. Chapter 2.2 described the percolation simulation model by Landis (2007). This simulation could be extended using the more realistic EMW Model as a galactic model. This work put constraints on ETI civilizations in the Milky Way galaxy. A follow-up study could apply the same recipe on nearby galaxies in order to put constraints on KII and KIII civilization. It would also be valuable to investigate the influence of the interstellar medium on remote ETI signals.

53 6.2.3 Is SETI Meaningful? Are we alone in the universe? For millennia, this was a question for religion and philosophy. In the era of science, this is no longer the case. It is a testable hypothesis where there exist only two distinct possi- bilities: we are alone or we are not alone. It would furthermore only take one single positive observation to confirm whether the latter is true. SETI is this joint scientific effort to search for extraterrestrial intelligence. The only way to eventually find the answer is to continue on the present path. We could get the answer tomorrow. Or it could take a million years. But either way, only by asking the question will take us closer to the answer. If we detect only one KI civilization, then we know for certain that there are hundreds, or possibly thousands, similar civilizations in the galaxy, and many more in our neighboring galaxies. This would give us a new perspective on the universe, and not the least, on ourselves. The discovery of other intelligence in the universe would be the single most important human discovery of all time and a triumph of science. However, a discovery of a KI civilization, but no more advanced civilizations, would also cast a disturbing shadow on our own future. If SETI continues to produce null results for the decades or centuries to come, despite ever-increasing sky coverage and sensitivity, this would be an equally profound result as the opposite. Although it would be hard to exclude the existence of ETIs, it would indeed strengthen the Rare Earth hypothesis, possibly implying that we are a unique species. In any case, the future of the human species appears as even more important, increasing our respon- sibility towards ourselves and the planet. This implies that we must take care of our Earth, we must stop polluting the atmosphere, the forests and the seas, and we must learn to combine economic growth with thriving and biological diversity. SETI is an endeavor with low stakes and high yield. Whatever answer it may produce, it will be of greatest importance.

54 Chapter 7

Acknowledgements

I want to thank myself for coming up with an interesting idea and materializing it into a thesis. I also want to thank Lars Mattsson, Josefin Larsson, Bj¨orn Ahlgren and Felix Ryde for reviewing the thesis.

55 Sometimes I believe we are alone. Sometimes I do not. But either way, the implications are staggering. Arthur C. Clarke, 20th century

56 List of Figures

0 3 −1 3.1.1 Left: The birth rate function plotted for λb = 10 Gyr . Right: The instantaneous population, N (t), for three diffenent choices of λ`...... 17

4.1.1 Flowchart of the N and runN algorithm...... 26 4.2.1 Visualization of S-space...... 27 4.2.2 3D-plot of the models EMW (left) and GHZ (right). Earth’s position marked with red plussign...... 30 4.2.3 Sky coverage of some SETI surveys specified in Table 2.1...... 30 4.4.1 Sampling domain D (black) and DN (red)...... 35

5.2.1 Distributions of N0, for varying mean, with the corresponding Dk distributions (K1.2 civilization)...... 38 5.2.2 Significance of N¯min, larger k means stronger dependence. Right: kQ as a function of Q.. 38 5.2.3 Detection probability, δS , as a function of N¯0. Left: SPO Models. Right: PPO Models. . 38 ¯ 5.2.4 Varying the Galactic Geometry Model while keeping RP/TF Models constant. Nmax/min = 10±11...... 43 ¯ 5.2.5 Varying the Radiated Power Model while keeping GG/TF Models constant. Nmax/min = 10±11...... 43 −11 5.2.6 Varying the PPO Model (showing two opposite extremes). N¯min = 10 ...... 44 ¯ ±11 5.2.7 Varying the TF Model while keeping GG/RP Models constant. Nmax/min = 10 ..... 44 ¯ Q 5.2.8 N0 as a function of the Equivalent Isotropic Radiated Power (EIRP) of a civilization. The quantiles divide the N¯0-EIRP-space into three distinct subdomains, corresponding to: a Strong Fermi paradox (excluded region), a Weak Fermi paradox (restricted region), and, No Fermi paradox (possible region). The blue dashed line shows the threshold EIRP of 1019 W. The red dashed lines show the EIRP corresponding to the principal Kardashev levels of KI, KII and KIII civilizations...... 46 5.3.1 Left: Distance distribution, f (D). Right: Mean distance to nearest civilization, given N¯0. 49 5.3.2 Relation between different population parameters Ne, N¯0, N¯CC and N¯IRS in the EMW/GHZ Models...... 49 ¯ 0.999 5.3.3 Left: Minimum detectable flux per decade. Right: N0 as a function of EIRP, year 2020-2080...... 49

57 List of Tables

2.1 A number of searches and corresponding technical parameters...... 12

4.1 Sketch of the Civilization Data Matrix ...... 25

5.1 SPOModels...... 45 5.2 PPO/WHR Models ...... 45

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62