A Quantitative Assessment of Communicating Extra-Terrestrial Intelligent Civilizations in the Galaxy and the Case of FRB-like Signals
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A Quantitative Assessment of Communicating Extra-Terrestrial Intelligent Civilizations in theGalaxy and the Case of FRB-like Signals
Bing Zhang Department of Physics and Astronomy, University of Nevada Las Vegas, NV 89154, USA
ABSTRACTA formula is proposed to quantitatively estimate the signal emission rate of Communicating Extra-Terrestrial Intelligent civilizations (CETIs) in the Galaxy. I suggest that one possible type of CETIsignal would be brief radio bursts similar to fast radio bursts (FRBs). A dedicated search for FRB-likeartificial signals in the Galaxy for decades may pose a meaningful upper limit on the emission rate ofthese signals by CETIs. The Fermi-Hart paradox is answered in terms of not having enough observingtimes for this and other types of signals. Whether humans should send FRB-like signals in the farfuture is briefly discussed.
Keywords: fast radio bursts – astrobilogy DRAKE EQUATION AND ITS MANIPULATIONIn 1961, Frank Drake wrote the famous equation toestimate the number of actively Communicating Extra-Terrestrial Intelligent civilizations (CETIs) in the MilkyWay galaxy. The equation reads (see, e.g. Vakoch et al.2015) N = R ∗ · f p · n e · f l · f i · f c · L. (1)The meaning of each parameter, as defined in the book“The Drake Equation” (Vakoch et al. 2015), reads • R ∗ : Rate of formation of stars suitable for thedevelopment of intelligent life; • f p : Fraction of stars with planetary systems; • n e : Number of planets, per solar system, with anenvironment suitable for life; • f l : Fraction of suitable planets on which life actu-ally appears; • f i : Fraction of life-bearing planets on which intel-ligent life emerges; • f c : Fraction of civilizations that develop a technol-ogy that releases detectable signs of their existenceinto space; • L : Length of time such civilizations release de-tectable signals into space. [email protected] In this equation, the first three parameters, R ∗ , f p , and n e , can be constrained from astronomical observations,but the remaining four parameters are related to biology,sociology, and even philosophy, which cannot be quan-titatively assessed based on observations of the sole in-telligent life as we know it: our own (e.g. Burchell 2006;Vakoch et al. 2015). It is worth emphasizing that f l , f i ,and f c as appeared in the Drake equation should meanthe fractions that life, intelligent life and communicativeintelligent life that can eventually appear (even if in thefar future), not the fractions already appeared at thecurrent age of the universe (which will be discussed as f ′ l , f ′ i , and f ′ c later). To avoid ambiguity, I believe thatit would be better to define them as • f l : Probability of suitable planets that eventuallydevelop life; • f i : Probability of life-bearing planets that eventu-ally develop intelligent life; • f c : Probability that intelligent life that eventuallydevelop technology and become CETIs.In the following, I define n ceti e = n e f l f i f c , (2)which has the physical meaning • n ceti e : Number of planets, per solar system, withan environment suitable for life and eventually de-velop CETIs. Zhang
According to the so-called “astrobiological Copernicanprinciple” assumption (e.g. Westby & Conselice 2020),an Earth-like planet that are suitable for developing lifewill eventually develop life, intelligent life, and CETIs,so that f l f i f c should be close to unity, i.e. n ceti e . n e .The star formation rate R ∗ in Eq.(1) was introducedto describe a steady-state process. Logically it is notstraightforward to see how it enters the problem and how L is used to cancel out the “per unit time” dimensionintroduced in R ∗ . Since one is interested in the CETIswho are sending off signals to Earth “now” (correctedfor the light travel time from the source to Earth), it ismore reasonable to start with the number of stars in thegalaxy now, i.e. N ∗ , rather than R ∗ to estimate N , sothat N = N ∗ · f p · n ceti e · f ′ l · f ′ i · f ′ c , (3)where the three new f parameters are defined as • f ′ l : Fraction of planets that can in principle de-velop CETIs and have actually developed life now; • f ′ i : Fraction of the above-defined life-bearing plan-ets that have developed intelligent life now; • f ′ c : Fraction of the above-defined intelligent lifethat have become CETI now.Notice that they are different from f l , f i , and f c thatenter the Drake equation (1). Let us further define thefollowing four time scales: • L p : the average lifetime of Earth-like planets (fromits birth to death, likely associated with the deathof the host star). For Earth, it is at least ∼ . • L l : the average lifetime of life (from its birth todeath - probably associated with the death of theplanet). For Earth, it is at least ∼ . • L i : the average lifetime of the intelligent life (fromits birth to death - could be associated with thedeath of the planet, but could be sooner due to itsself-destroy). For Earth, it is at least ∼ years; • L c : the average lifetime of CETIs (from its birthto death - again could be associated with the deathof the planet or its self-destroy). For Earth, it isat least ∼ years, i.e. since humans have devel-oped the technology to send off artificial signals tospace.The term “average” in the above definitions refers tothe geometric mean rather than the arithmetic mean, or the average in the logarithmic space. This way, one canwrite the three fraction parameters defined above as f ′ l = L l L p , (4) f ′ i = L i L l , (5) f ′ c = L c L i , (6)so that the product f ′ l · f ′ i · f ′ c can be simply written asthe ratio L c /L p . The logic behind Eqs.(4)-(6) is that ina steady state and a random observing time, the prob-ability of seeing a short-duration event during a long-duration event should be the ratio between the durationsof the former and the latter.Plugging Eqs.(4)-(6) into Eq.(3), one can derive N = N ∗ · f p · n ceti e · L c L p = L MW L p · N ∗ L MW · f p · n ceti e · L c ≃ L MW L p · R ∗ · f p · n ceti e · L c ∼ R ∗ · f p · n ceti e · L c (7)where R ∗ ≃ N ∗ /L MW is the average star formation ratethroughout the MW history (with the assumption thatthe number of dead stars is much smaller than the num-ber of living stars, which is justified since low mass starsthat contributed to the vast majority of the total num-ber have not died yet, see Westby & Conselice 2020 fordetailed calculations), and L MW ∼ . L p is ∼ L MW (e.g.for Earth it is the lifetime of the Sun, i.e. ∼
10 Gyr), thelast step in Eq.(7) has a ∼ sign. One can see that laststep roughly reproduces the Drake equation (1) noticingEq.(2). A QUANTITATIVE ASSESSMENT OF THECETI SIGNAL EMISSION RATEThe number N of CETIs is not a direct measurablequantity. A human observer may be more interested inthe signal detection rate of CETI signals (e.g. in unitsof N s , o , wherethe subscripts ‘s’ and ‘o’ denote ‘signal’ and ‘observer’,respectively. Ultimately, one cares about the averagesignal emission rate per CETI, which I define as ˙ N s , e ,where the subscript ‘e’ denotes ‘emitter’. The emittedsignals of CETIs may not always be detected by humanson Earth. I therefore introduce a parameter ξ o to de-note the average fraction of the CETI signals that aredetectable by humans on Earth. One can then write˙ N s , o = N · ( ξ o ˙ N s , e )= N ∗ · f p · n ceti e · L c L p · ( ξ o ˙ N s , e ) . (8)where the first line of Eq.(7) has been used. This equa-tion is more helpful than Eq.(7) for a quantitative as-sessment of CETI signals. We now break down theterms introduced in Eq.(8) and discuss how they maybe constrained astronomically. The discussions on var-ious parameters of the Drake equation and other mod-ified forms can be also found in Vakoch et al. (2015)and many papers in the literature, e.g. Burchell (2006);Westby & Conselice (2020) and references therein. • N ∗ : the Sun’s distance from the Galactic center( d ⊙ ∼ . ± . v ⊙ ∼
220 km / s as measured from the dipolemoment in the cosmic microwave background) al-lows one to estimate that the total mass withinthe solar orbit is M in = d ⊙ v ⊙ /G ∼ M ⊙ . De-ducting the contributions from dark matter, gasand dead remnants (e.g. black holes and neu-tron stars) that may not be helpful to harbor life,one can write the stellar mass within the solarorbit as M ∗ , in = f ∗ M in , where f ∗ < f in <
1. The totalstellar mass should be M ∗ = M ∗ , in /f in . Sincethe initial stellar mass function has a steep slope( N ( m ∗ ) dm ∗ ∝ m − . ∗ dm ∗ for m ∗ > . M ⊙ and N ( m ∗ ) dm ∗ ∝ m − . ∗ dm ∗ for 0 . M ⊙ < m ∗ < . M ⊙ (e.g. Kroupa 2001), the number of stars isdominated by those stars with the minimum mass m ∗ ,m = f m M ⊙ with f m <
1. The total number ofstars can be finally estimated as N ∗ ∼ M ∗ m ∗ ,m = f ∗ f m f in M in M ⊙ = f ∗ f m f in ∼ (10 − ) . (9)A commonly quoted number is N ∗ = 2 . × (e.g. Westby & Conselice 2020). • f p : Here the meaningful fraction should be thefraction of stars that can have habitable plan-ets (or habitable moons orbiting giant planets)in stable orbits long enough to develop life andCETIs. Modern exoplanet observations suggestthat planets may be ubiquitous in stellar systems(e.g. Howard et al. 2012; Burke et al. 2015). Since single stars are likely the ones to harbor stableplanet orbits, we assign f p as the fraction of singlestars, which ranges from 1/2 to 2/3 (Lada 2006). • n ceti e : This is the number with a large uncer-tainty. n e may be determined with precisionwhen survey observations of planets from nearbystellar systems are carried out, e.g. with theTransiting Exoplanet Survey Satellite (TESS) mis-sion (Ricker et al. 2015). Based on the Kepler(Borucki et al. 2010) observations, the fraction ofGK dwarfs with rocky planets in habitable zonesmay be around 0.1 (Burke et al. 2015). This maybe considered as the upper limit of n ceti e when con-sidering other factors (planet mass, metallicity, ex-istence of a magnetosphere, existence of a Jupiter-like large planet to deflect comets, etc., see, e.g.Lineweaver (2001)) that might be relevant for pro-ducing CETIs. Since it is far from clear what phys-ical conditions are essential for the emergence ofCETIs, this number should allow for a large uncer-tainty, from 0.1 all the way to very small numbers.We normalize this number to n ceti e ∼ − in thefollowing discussion, keeping in mind the large un-certainty involved. • L p : Humans, the only intelligent life we knowin the universe, emerge 4.54 billion years afterthe formation of the planet. One may conserva-tively assume that 4.5-billion-year is the minimumtimescale to develop CETIs. As a result, the hoststars of Earth-like planets to harbor SETIs shouldhave long lives, e.g. of the solar or later types(GK dwarfs or M dwarfs). The planets whereCETIs are harbored should also have survived fora comparable lifetime as their parent stars (seealso Westby & Conselice 2020). As a result, L p islikely at least several billion years, as is the caseof Earth. • L c : This is a parameter with the largest uncer-tainty, and cannot be estimated using availableastrophysical observations. Since humans have de-veloped communicating technology for more thana century on Earth, L c should be at least ∼ L c /L p is at least 10 − . However, ithas been widely speculated that CETIs can sur-vive much longer than this time scale (if theydo not destroy themselves) (e.g. Burchell 2006;Vakoch et al. 2015). Lacking any guidance, in thefollowing we normalize the ratio L c /L p (which isrelevant in the problem) to ∼ − , which corre-sponds to L c ∼ (10 − ) yr for L p ∼ (10 − ) Zhang yr. Notice that in principle, CETIs can “re-appear” after self-destroy. What matters in ourproblem is the total duration of the existence ofCETIs. The duration of each CETI and the num-ber of generations of CETIs in a planet is not rel-evant. As a result, L c defined here can be con-sidered as the average total duration of CETIs oneach planet in the Galaxy. • ˙ N s , e : This is the average signal emission rate perCETI (for detailed discussion of CETI signals fromthe emitter’s perspective, see Sect. 3). The CETIsmay repeat their signals multiple times, and ˙ N s,e is defined as the total amount of signals emittedby a CETI divided by its entire lifetime L c , av-eraged over all CETIs. Maybe (and likely) dif-ferent CETIs attempt to communicate using dif-ferent types of signals. Maybe the same CETIattempts to communicate using several differenttypes of signals. So, ˙ N s,e is signal-type-dependent,and should be defined for each type of signal specif-ically. • ξ o : In order to connect the emission rate with thedetection rate, one should introduce this factorthat denotes the fraction of emitted signals de-tectable by astronomers on Earth. One may write ξ o ≡ ∆Ω e π (cid:18) d lim d MW (cid:19) , (10)which includes the average beaming factor ∆Ω e / π for the emitted signals and the flux limitation fac-tor ( d lim /d MW ) , where d lim is the maximumdistance from Earth the signal can be detectedand d MW ∼
10 kpc is the characteristic distancescale of the Milky Way galaxy. This second factordepends on the strength (luminosity) of the emit-ted signal and the sensitivity of the telescopes thatdetect these signals. Ideally, advanced CETIs mayemit signals detectable by all other civilizationsacross the Galaxy. For such signals, one takes( d lim /d MW ) ∼
1, so that ξ o ≃ ∆Ω e / π , whichonly depends on the average solid angle ∆Ω e ofthe emitted signals. • ˙ N s , o : This is the total detectable signal rate atEarth from all sky all time. Similar to ˙ N s , e , it A simple isotropic distribution is assumed here for order-of-magnitude estimation. A more careful study should account forthe MW structure and the anisotropic environment of the Earthneighborhood. should be defined specifically for each type of sig-nal. It is not the rate of the truly detected signals,which depends on the fraction of sky coverage andthe duty cycle of the telescopes. With dedicatedsurveys with certain sky and temporal coverage,the detected signal rate (or, very likely, its upperlimit) can be corrected to derive ˙ N s , o (or, verylikely, its upper limit). • N : Even though this is not a direct measur-able quantity, it is nonetheless interesting towrite down the estimated CETI number N in theGalaxy according to Eq.(7) with the normalizationvalues of each parameter as discussed above: N = 12500 (cid:18) N ∗ . × (cid:19) (cid:18) f p . (cid:19) (cid:18) n ceti e − (cid:19) × (cid:18) L c /L p − (cid:19) . (11)Notice that if one chooses L c ∼
100 yr, L p ∼ yr, and f p · n ceti e ∼ − (corresponding to the factor f L · f HZ · f M defined by Westby & Conselice 2020), oneobtains a minimum CETI number of 12.5, which is con-sistent with the minimum CETI number estimated byWestby & Conselice (2020). CETI SIGNALS FROM THE EMITTER’SPERSPECTIVEGreat efforts have been made in the Search-for-Extra-Terrestrial-Intelligence (SETI) community to speculatethe types of the CETI signals. Wright et al. (2018) de-fined an eight-dimensional model to describe CETI sig-nals and argued that the current SETI searches onlytouched a tiny phase space of this eight-dimention “cos-mic haystack”. Some of the dimensions defined byWright et al. (2018) (e.g. their dimension 1 [sensitivityto transmitted or received power] and their dimensions3-5 [distance and position]) are from the observer’s per-spective.In the following, I discuss a possible CETI signal from the emitter’s perspective by speculating what type of sig-nal a CETI may emit. Assuming that CETIs commu-nicate using an electromagnetic signal , I characterize aCETI signal by the following seven parameters: In principle, a multi-messenger channel may be also used.However, these messengers are technically challenging and noteconomical. For example, the generation of gravitational wavesis very expensive. Cosmic rays tend to be deflected by interstellarmagnetic fields. Even if neutrinos may be easier to generate thanthe other two messengers, they are very difficult to detect fromthe observer’s prospective due to the small cross section of weakinteraction. These channels may not be favored by CETIs andare, therefore, not discussed in this paper.
1. Duration: Shorter durations may be preferredfrom economical considerations, but the signalshould be long enough for other civilizations todetect.2. Peak luminosity: Since any signal has a rising andfading phase, the luminosity at the peak time isa relevant parameter to consider. This parameterwill define how far the signal can be detected byother civilizations given a certain detector sensi-tivity.3. Emission spectrum: This concerns the central fre-quency and the bandwidth of the transmissionsignal, which has been discussed by Wright et al.(2018) as their dimensions 2 & 6.4. Polarization: The polarization properties of aCETI signal may carry additional information.This is the dimension ξ o dis-cussed in Eq.(10).7. Repetition rate: This defines the ˙ N s,e in Eq.(8),which is the average value of all CETIs in theGalaxy. The dimension N s,o in Eq.(8).Since no CETI signal has been detected, one can onlytake humans’ own communicating signals for compari-son. The most famous signal was the “Arecibo message”broadcast in 1974 (Staff at the National Astronomy & Ionosphere Center1975). The properties of the signal in connection withthe seven dimensions discussed above are: • Duration: . • Luminosity: 450 kilo-Watts or 4 . × erg s − ; • Spectrum: At frequency 2,380 MHz with an ef-fective bandwidth of 10 Hz, with modulations byshifting the frequency by 10 Hz; • Polarization: No information is available. • Lightcurve: Consisting of 1,679 binary digits (ap-proximately 210 bytes) that encode rich informa-tion such as numbers, atomic numbers, DNAs, ahuman image, the solar system, and the Areciboradio telescope. • Solid angle: A narrow beam of about 1 square arc-minute pointing toward the globular cluster M13 ∼ ,
000 light years away. • Repetition rate: Not repeated with the same con-figuration according to the public record.One may estimate the detectability of this signal byother CETIs. First, aliens in all other directions outsidethe narrow beam would never know that such a signalwas emitted. Second, at the distance of M13, the fluxdensity of this signal is ∼ . × − Jy, which is 21orders magnitude fainter than the sensitivity (10 Jy) ofthe current humans’ own narrow-band SETI search in-strument, the Allen Telescope Array (Welch et al. 2009).Suppose that there are indeed aliens ∼ . FRB-LIKE ARTIFICIAL SIGNALS BY CETISFast radio bursts (FRBs) are frequently detected,millisecond-duration, radio bursts that originate fromcosmological distances (Lorimer et al. 2007; Petroff et al.2019; Cordes & Chatterjee 2019). Since their physicalorigin is unknown, an alien connection was speculatedby some authors. For example, when placing phys-ical constraints on FRBs, Luan & Goldreich (2014)discussed the possibility that the observed FRBs areartificial signals sent by aliens and concluded that amodest power requirement is needed. Lingam & Loeb(2017) interpreted FRBs as radio beam signals producedby extra-galactic advanced civilizations to launch lightsails for interstellar travels. These suggestions intended
Zhang to interpret at least a fraction of the observed FRBs asof an artificial origin.I believe that the observed cosmological FRBs are dueto astrophysical origins. Indeed, many theoretical mod-els have been proposed in the literature to interpret var-ious observational properties of FRBs (e.g. Platts et al.2019). Rather, I speculate that if CETIs do exist andindeed have the intention to communicate, they maysend off FRB-like artificial signals to the Galaxy, likelyalong the Galactic plane. There are two main reasonsfor this speculation. First, intelligent civilizations knowthat FRBs happen very frequently (of the order ∼ per day all sky for bright ones) in the universe and thatother civilizations must be monitoring these events allthe time. The same observational facilities designed todetect FRBs (e.g. wide-field and sensitive radio tele-scope arrays) could easily spot artificial signals theywould send. Second, probably more importantly, FRBshave radio frequencies and extremely short durations(but still long enough for detection). They are rela-tively easy to mimic from the economical point of view.In contrast, it is for example more difficult to make ar-tificial gamma-ray bursts or fast optical bursts.In terms of the seven dimensions discussed in Section3, the properties of an FRB-like CETI signal may bespeculated as follows: • Duration: An FRB-like signal should have a du-ration of the order of milliseconds, i.e. ∆ t =(1 ms) ∆ t ms . • Luminosity and energetics: How bright the signalsare emitted depends on the technological level ofthe aliens, but an advanced CETI who is eagerto broadcast its existence may try to emit at aluminosity such that the detected flux level bya typical Milky Way observer (like humans onEarth) is comparable to that of a cosmologicalFRB. For a 1 ms-Jy signal with a characteristicdistance in the Milky Way, the isotropic luminos-ity should be ∼ (10 erg s − ) d . Considerthe beaming factor (which is also the probabilityfactor for the observer to see the signal), ξ o =∆Ω / π . The true emission power should be ∼ (10 erg s − ) ξ o, − d or (10 W) ξ o, − d ,where ξ o is normalized to 10 − . The trueenergy is ∼ (10 erg) ξ o, − ∆ t ms d or ∼ (10 J) ξ o, − ∆ t ms d . This energy is ∼ times of the emitted energy in the Arecibo mes-sage, which is comparable to the rest mass energy This is much greater than the one estimated byLuan & Goldreich (2014). of ∼
100 kg matter. Emitting such a signal isbeyond the current technological capability of hu-mans. On the other hand, there is no fundamentalphysical barrier to prevent this from happening. • Spectrum and dispersion measure (DM): FRBshave been detected from ∼
400 MHz to 8 GHz(Petroff et al. 2019). The observed FRB flux typ-ically show a steep drop at high frequencies. Thelow frequency range also suffers from a few sup-pression effects such as plasma scattering andabsorption, which are particularly severe in theGalactic plane. One natural frequency CETIs mayconsider to broadcast artificial FRB signals wouldbe around the hydrogen 21cm line frequency (1.42GHz). Its “resonances” (e.g. 1.5 times or 2 times)may be also possible. In order to mimic an FRBsignal, the CETIs may consider to emit the sig-nal in a wide enough bandwidth so that a DMcould be measured. In order to draw attention toobservers, they may also add an extra DM (e.g. ∼
500 pc cm − , a typical value for cosmologicalFRBs) in excess of the Galactic value by placing acold plasma along the path of signal propagation.Such a signal can be easily picked up by observersusing the standard FRB-searching algorithms. • Polarization: So far, the FRB polarization prop-erties do not show well-defined common charac-teristics. The polarization properties are also notessential for FRB detections. There is no need todesign certain polarization properties in the artifi-cial signals unless some extra intelligent informa-tion can be carried. For example, a CETI may de-cide to emit an FRB with 100% linear polarizationwith a Faraday rotation measure (RM) greatly ex-ceeding the local value in the Galaxy through gen-erating a strong magnetic field in the emission site.They may also vary RM significantly to show thatthe signals are indeed artificial. • Lightcurve: This is how CETIs deliver informa-tion to show the intelligent nature of their sig-nals. There are many possibilities, which we re-frain from speculating. In any case, one mayexpect a series of FRB-like signals that encodeprofound information understandable by other ad-vanced civilizations. • Solid angle: Advanced CETIs may like to broad-cast their signals as wide as possible, so that anall-sky (4 π ) signal would be most ideal. In prac-tice, it would be easier to send collimated signalsfor economical and technical reasons. Since theGalactic plane has the highest probability for othercivilizations to detect the signals, a CETI in theGalactic plane may send a fan beam with the ver-tical angle defined by the height to radius ratio ofthe Galactic plane, which is of the order of 10 − .If the azimuthal angle is 2 π , the beaming factor is∆Ω / π ∼ − . • Repetition rate ˙ N s,e : This is something not easyto estimate, but can be in principle constrainedfrom the data (see Section 5 below for detaileddiscussion). However, since sending these signalsconsumes a lot of energy, the emission rate maynot be very frequent unless CETIs are in greatdesire to communicate (e.g. sending S.O.S signalsfor help). A QUANTITATIVE ASSESSMENT OF CETI’SFRB-LIKE ARTIFICIAL SIGNALSWith the operations (or planned operations) of agrowing number of radio antenna arrays (e.g. TheCanadian Hydrogen Intensity Mapping Experiment[CHIME] (Bandura et al. 2014), the Deep Synoptic Ar-ray 2000-antenna [DSA-2000] (Hallinan et al. 2019),and the Square Kilometre Array [SKA] (Johnston et al.2008)) to detect FRBs, one can start to place a con-straint on ˙ N s , o of FRB-like artificial signals from theMilky Way. Non-detection of any FRB-like artificialsignal from the Galactic plane, when corrected for thesky coverage and duty cycle for progressively longerobservational times, can place progressively tighter con-straints on ˙ N s , o , which would place constraints on theaverage signal emission rate of CETIs, ˙ N s , e , for FRB-like signals based on Eq.(8). For example, an upperlimit of ˙ N s , o < . − (e.g. all sky no detection in adecade) can lead to a constraint˙ N s , e < (0 .
008 yr − ) ˙ N s , o < . − ! (cid:18) N ∗ . × (cid:19) − (cid:18) f p / (cid:19) − (cid:18) n ceti e − (cid:19) − (cid:18) L p /L c (cid:19) (cid:18) ξ o − (cid:19) − , (12)where the parameters are normalized to the character-istic values as discussed in Section 2. With the non-detection upper limit in decades, one may then make aquantitative statement that with the fiducial values ofthe parameters, CETIs on average emit less than oneper century FRB-like artificial signals that can cross theentire Milky Way. FERMI-HART PARADOX One commonly discussed question in the SETI com-munity is the “Fermi-Hart paradox” regarding “where iseverybody?” (as elaborated in detail by Hart (1975), seealso Brin (1983)). Even though the question was aboutwhy there is no evidence for extraterrestrial intelligencevisiting Earth, another version was to address why wehave not received signals from CETIs. One general typeof answer to this question is that humans have not ob-served the universe long enough to allow any detection.Indeed, Wright et al. (2018) showed that humans onlysearched a tiny parameter space in a multi-dimensional“cosmic haystack” through blind searches.If one focuses on one specific type of signal, e.g. theFRB-like artificial signal discussed in this paper, the“haystack” search volume greatly shrinks. I argue thateven for such specific signals, the answer to the “Fermi-Hart paradox” is still “we simply have not observedlong enough yet” . One can quantitatively show this us-ing Equation (12). For FRB-like signals, even if onecan achieve the ˙ N s , o < . − upper limit (whichrequires dedicated efforts from wide-field radio arraysworking for decades), for fiducial parameters, one canonly set a moderate upper limit on the signal emis-sion rate of highly advanced CETIs who can broad-cast their existence across the entire Milky Way, i.e.˙ N s , e < .
008 yr − . It is impossible that all CETIs havesuch a capability to broadcase across the Milky Way(e.g. humans do not). If CETIs are less advanced, ξ o would be greatly reduced since the ( d lim /d MW ) factorwould have to be included. For example, for ξ o ∼ − (corresponding to the case that detectable CETI signalscan only reach a distance of d lim ∼
100 pc), one can onlyset a limit of ˙ N s , e <
80 yr − for ˙ N s , o < . − and typ-ical parameters. It is hard to imagine that an averageCETI so desperately communicates with the universe bysending signals at a rate >
80 yr − . As a result, thereis essentially no “paradox”.This argument applies not only to the FRB-like sig-nal but also other specific signals, which are probablyeven more difficult to generate by CETIs. If one doesnot specify a signal type, the search parameter volumewould be increased exponentially, so that the “paradox”is further diminished (Wright et al. 2018, and referencestherein). The lack of detection of any CETI signals nowis naturally expected. SUMMARY AND DISCUSSIONThe points made in this paper can be summarized asfollows: • I presented a derivation of Equation (Eq.(7)) basedon a probability argument, which is consistentwith the original Drake equation (Eq.(1)).
Zhang • Based on Eq.(7), I proposed a new equation(Eq.(8)) to connect the observed CETI signalrate ˙ N s,o with the average signal emission rate˙ N s,e by CETIs. Subject to uncertainties of severalparameters, this equation allows one to use obser-vations to directly infer the value (or, very likely,the upper limit) of the average CETI signal emis-sion rate. The equation applies to specific signalsrather than unspecified blind-search signals. • After characterizing CETI signal properties inseven dimensions from the emitters’ perspective,I suggested that FRB-like artificial signals couldbe one type of CETI signals for good reasons. Us-ing Eq.(8), I derive a constraint one may pose withthe detection/non-detection of FRB-like artificialsignals from the Milky Galaxy. • The ˙ N s,e constraint derived from the detection/non-detection of FRB-like artificial signals (Eq.(12))is taken as an example to quantitatively showwhy the “Fermi-Hart paradox” is not a concern.Even for one particular type of signal and underthe most optimistic assumption (i.e. an averageCETI is able to broadcast FRB-like artificial sig-nals across the entire Milky Way), one would notexpect to detect any signal now and probablystill not even after decades or centuries of dedi-cated monitoring. This strengthens the argumentagainst the Fermi-Hart paradox by Wright et al.(2018) for blind searches.Finally, one natural question is whether humansshould send FRB-like artificial signals in the future when technology is advanced enough. This is definitelya topic subject to debate (see, e.g. Gertz (2016) for ageneral discussion on the pros and cons on Messaging toExtra-Terrestrial Intelligence (METI)). Optimists maythink that CETIs are eager to find out whether they arealone in the universe and would be happy to remotelycommunicate with other civilizations. Pessimists, on theother hand, would believe that it is very dangerous toexpose ourselves to more advanced civilizations as theywould invade Earth to snatch resources . In any case, Ibelieve that this should be a decision to be made by theentire humanity, not by a small group of “elites”. Mypersonal recommendation is: Do not do anything untilone can develop the technology to emit FRB-like signals(this may take some time, e.g. hundreds, thousands oreven millions of years); keep watching whether “others”have emitted any such signal along the way; and make adecision then! If non-detection of FRB-like signals per-sists for a long time (e.g. after thousands of years), thenthe Fermi-Hart paradox may become more a concern.The CETIs may be also taking a pessimistic approachlike us, so that the paradox may find an answer alongthis reasoning in the far future.I thank Qiang Yuan for an important remark on anearlier version of the paper, Jason Steffen for discussingthe current status of exoplanet searches, and MauraMcLaughlin for discussing the current status of GalacticFRB-like signal searches.REFERENCES Bandura, K., Addison, G. E., Amiri, M., et al. 2014,Society of Photo-Optical Instrumentation Engineers(SPIE) Conference Series, Vol. 9145, Canadian HydrogenIntensity Mapping Experiment (CHIME) pathfinder,914522Borucki, W. J., Koch, D., Basri, G., et al. 2010, Science,327, 977Brin, G. D. 1983, QJRAS, 24, 283Burchell, M. J. 2006, International Journal of Astrobiology,5, 243 This point was raised by Stephen Hawking in his 2010 doc-umentary series and delineated in Liu Cixin’s famous science fic-tion trilogy “Remembrance of Earth’s Past” (or “The Three-BodyProblem” series). See also Gertz (2016) for a critical review.