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arXiv:2006.08493v2 [physics.pop-ph] 6 Nov 2020 [email protected] a aay h qainras(e,e.g. (see, reads equation The Milky . the Way in (CETIs) Extra- civilizations Communicating Intelligent actively Terrestrial of number the estimate 2015 h enn fec aaee,a endi h book the in ( defined Equation” as Drake parameter, “The each of meaning The 1. n16,FakDaewoetefmu qainto equation famous the wrote Drake Frank 1961, In yee sn L using 2020 Typeset 10, November version Draft RK QAINADISMANIPULATION ITS AND EQUATION DRAKE • • • • • • • uniaieAssmn fCmuiaigExtra-Terrest Communicating of Assessment Quantitative A ) R n etbesgasit space. into signals tectable L existence space; their into of signs detectable releases that ogy f emerges; life ligent f appears; ally f life; for suitable environment f life; intelligent of development c i l p e egho iesc iiiain ees de- release civilizations such time of Length : ∗ rcino utbepaeso hc ieactu- life which on planets suitable of Fraction : rcino iebaigpaeso hc intel- which on planets life-bearing of Fraction : ie o hsadohrtpso inl.Wehrhmn hudse should humans Keywords: Whether signals. of discussed. types briefly term other is in future and upp answered meaningful is this paradox a for Fermi-Hart pose The times may CETIs. decades (FRBs) by for bursts signals radio Galaxy these fast the to in suggest similar signals bursts I artificial radio Galaxy. brief be the would in signal (CETIs) civilizations Intelligent Terrestrial rcino iiiain htdvlpatechnol- a develop that civilizations of Fraction : rcino tr ihpaeaysystems; planetary with of Fraction : ubro lnt,prslrsse,wt an with system, solar per planets, of Number : aeo omto fsassial o the for suitable stars of formation of Rate : oml spooe oqatttvl siaetesga msinr emission signal the estimate quantitatively to proposed is formula A N A = T E R X atrdobrt astrobilogy – bursts radio fast 1 ∗ eateto hsc n srnm,Uiest fNvd L Nevada of University Astronomy, and Physics of Department twocolumn · f p aohe l 2015 al. et Vakoch · n e · f l tl nAASTeX62 in style · f i aayadteCs fFBlk Signals FRB-like of Case the and Galaxy · f c · L. aohe al. et Vakoch ,reads ), igZhang Bing ABSTRACT (1) hc a h hsclmeaning physical the has which ntefloig define I following, the In twudb etrt en hmas them define to better be would it aohe l 2015 al. et Vakoch parameters, three first the equation, this In a uue,nttefatosarayappeared universe f the already of fractions age the current not future), far n elgn iea eko t u w (e.g. own our in- it: sole know we the as of life observations quan- telligent on be based cannot assessed which titatively philosophy, even biology, to and related sociology, are parameters four remaining the but and h rcin htlf,itlietlf n communicative that and life life intelligent intelligent life, that fractions the l ′ e 1 , a ecntandfo srnmclobservations, astronomical from constrained be can , f • • • • f i ′ and , c nevrnetsial o ieadeetal de- eventually CETIs. and velop life for suitable environment an n eeo ehooyadbcm CETIs. become and technology develop f life; intelligent develop ally f life; develop f c i l e ceti sapae nteDaeeuto hudmean should equation Drake the in appeared as rbblt fsial lnt hteventually that planets suitable of Probability : rbblt flf-ern lnt hteventu- that planets life-bearing of Probability : rbblt htitlietlf hteventually that life intelligent that Probability : ubro lnt,prslrsse,with system, solar per planets, of Number : f ilItlietCvlztosi the in Civilizations Intelligent rial c sVgs V814 USA 89154, NV Vegas, as ′ eiae erhfrFRB-like for search dedicated A . ae) oaodabgiy eiv that believe I ambiguity, avoid To later). fnthvn nuhobserving enough having not of s htoepsil yeo CETI of type possible one that rlmto h msinrt of rate emission the on limit er dFBlk inl ntefar the in signals FRB-like nd t fCmuiaigExtra- Communicating of ate .I swrhepaiigthat emphasizing worth is It ). a eventually can n e ceti = n wihwl edsusdas discussed be will (which e f l f i f per(vni nthe in if (even appear c , ucel2006 Burchell R ∗ , f p tthe at f and , l , (2) f i ; , 2 Zhang

According to the so-called “astrobiological Copernican the average in the logarithmic space. This way, one can principle” assumption (e.g. Westby & Conselice 2020), write the three fraction parameters defined above as an -like planet that are suitable for developing life ′ Ll will eventually develop life, intelligent life, and CETIs, fl = , (4) ceti Lp so that flfifc should be close to unity, i.e. ne . ne. The formation rate R∗ in Eq.(1) was introduced ′ Li fi = , (5) to describe a steady-state process. Logically it is not Ll straightforward to see how it enters the problem and how ′ Lc fc = , (6) L is used to cancel out the “per unit time” dimension Li introduced in R∗. Since one is interested in the CETIs so that the product f ′ · f ′ · f ′ can be simply written as who are sending off signals to Earth “now” (corrected l i c the ratio Lc/Lp. The logic behind Eqs.(4)-(6) is that in for the light travel time from the source to Earth), it is a steady state and a random observing time, the prob- more reasonable to start with the number of stars in the ability of seeing a short-duration event during a long- galaxy now, i.e. N∗, rather than R∗ to estimate N, so duration event should be the ratio between the durations that of the former and the latter. · · ceti · ′ · ′ · ′ N = N∗ fp ne fl fi fc, (3) Plugging Eqs.(4)-(6) into Eq.(3), one can derive where the three new f parameters are defined as · · ceti · Lc N = N∗ fp ne • ′ Lp fl : Fraction of planets that can in principle de- velop CETIs and have actually developed life now; LMW · N∗ · · ceti · = fp ne Lc Lp LMW • f ′: Fraction of the above-defined life-bearing plan- i ≃ LMW · · · ceti · ets that have developed intelligent life now; R∗ fp ne Lc Lp ceti • ′ ∼ R∗ · fp · n · Lc (7) fc: Fraction of the above-defined intelligent life e that have become CETI now. where R∗ ≃ N∗/LMW is the average star formation rate

Notice that they are different from fl, fi, and fc that throughout the MW history (with the assumption that enter the (1). Let us further define the the number of dead stars is much smaller than the num- following four time scales: ber of living stars, which is justified since low mass stars that contributed to the vast majority of the total num- • Lp: the average lifetime of Earth-like planets (from ber have not died yet, see Westby & Conselice 2020 for its birth to death, likely associated with the death detailed calculations), and LMW ∼ 13.5 billion is of the host star). For Earth, it is at least ∼ 4.54 the current lifetime of the Milky Way galaxy. Since the billion years; average lifetime of earth-like planets Lp is ∼ LMW (e.g. for Earth it is the lifetime of the , i.e. ∼ 10 Gyr), the • Ll: the average lifetime of life (from its birth to last step in Eq.(7) has a ∼ sign. One can see that last death - probably associated with the death of the step roughly reproduces the Drake equation (1) noticing ∼ planet). For Earth, it is at least 3.8 billion years; Eq.(2). • Li: the average lifetime of the intelligent life (from 2. A QUANTITATIVE ASSESSMENT OF THE its birth to death - could be associated with the CETI SIGNAL EMISSION RATE death of the planet, but could be sooner due to its The number N of CETIs is not a direct measurable self-destroy). For Earth, it is at least ∼ 104 years; quantity. A human observer may be more interested in • L : the average lifetime of CETIs (from its birth the signal detection rate of CETI signals (e.g. in units c ˙ to death - again could be associated with the death of # per all sky), which I define as Ns,o, where of the planet or its self-destroy). For Earth, it is the subscripts ‘s’ and ‘o’ denote ‘signal’ and ‘observer’, at least ∼ 102 years, i.e. since humans have devel- respectively. Ultimately, one cares about the average ˙ oped the technology to send off artificial signals to signal emission rate per CETI, which I define as Ns,e, space. where the subscript ‘e’ denotes ‘emitter’. The emitted signals of CETIs may not always be detected by humans The term “average” in the above definitions refers to on Earth. I therefore introduce a parameter ξo to de- the geometric mean rather than the arithmetic mean, or note the average fraction of the CETI signals that are 3 detectable by humans on Earth. One can then write single stars are likely the ones to harbor stable planet orbits, we assign fp as the fraction of single ˙ · ˙ Ns,o = N (ξoNs,e) stars, which ranges from 1/2 to 2/3 (Lada 2006). · · ceti · Lc · ˙ = N∗ fp ne (ξoNs,e). (8) • ceti Lp ne : This is the number with a large uncer- tainty. ne may be determined with precision where the first line of Eq.(7) has been used. This equa- when survey observations of planets from nearby tion is more helpful than Eq.(7) for a quantitative as- stellar systems are carried out, e.g. with the sessment of CETI signals. We now break down the Transiting Survey Satellite (TESS) mis- terms introduced in Eq.(8) and discuss how they may sion (Ricker et al. 2015). Based on the Kepler be constrained astronomically. The discussions on var- (Borucki et al. 2010) observations, the fraction of ious parameters of the Drake equation and other mod- GK dwarfs with rocky planets in habitable zones ified forms can be also found in Vakoch et al. (2015) may be around 0.1 (Burke et al. 2015). This may ceti and many papers in the literature, e.g. Burchell (2006); be considered as the upper limit of ne when con- Westby & Conselice (2020) and references therein. sidering other factors (planet mass, , ex- istence of a magnetosphere, existence of a - • N∗: the Sun’s distance from the Galactic center like large planet to deflect comets, etc., see, e.g. (d⊙ ∼ 8.0 ± 0.5 kpc) and its veloc- Lineweaver (2001)) that might be relevant for pro- ity (v⊙ ∼ 220 km/s as measured from the dipole ducing CETIs. Since it is far from clear what phys- moment in the cosmic microwave background) al- ical conditions are essential for the emergence of lows one to estimate that the total mass within 2 11 CETIs, this number should allow for a large uncer- the solar orbit is M = d⊙v /G ∼ 10 M⊙. De- in ⊙ tainty, from 0.1 all the way to very small numbers. ducting the contributions from dark matter, gas We normalize this number to nceti ∼ 10−3 in the and dead remnants (e.g. black holes and neu- e following discussion, keeping in mind the large un- tron stars) that may not be helpful to harbor life, certainty involved. one can write the within the solar orbit as M∗,in = f∗Min, where f∗ < 1 is the • Lp: Humans, the only intelligent life we know fraction of mass attributed to stars. Consider in the universe, emerge 4.54 billion years after that the fraction of stars in the Milky Way that the formation of the planet. One may conserva- is within the solar orbit is fin < 1. The total tively assume that 4.5-billion-year is the minimum stellar mass should be M∗ = M∗,in/fin. Since timescale to develop CETIs. As a result, the host the initial stellar mass function has a steep slope stars of Earth-like planets to harbor SETIs should ∝ −2.3 N(m∗)dm∗ m∗ dm∗ for m∗ > 0.5M⊙ and have long lives, e.g. of the solar or later types ∝ −1.3 N(m∗)dm∗ m∗ dm∗ for 0.08M⊙ < m∗ < (GK dwarfs or M dwarfs). The planets where 0.5M⊙ (e.g. Kroupa 2001), the number of stars is CETIs are harbored should also have survived for dominated by those stars with the minimum mass a comparable lifetime as their parent stars (see m∗,m = fmM⊙ with fm < 1. The total number of also Westby & Conselice 2020). As a result, Lp is stars can be finally estimated as likely at least several billion years, as is the case

M∗ f∗ Min of Earth. N∗ ∼ = m∗,m fmfin M⊙ • Lc: This is a parameter with the largest uncer- f∗ = 1011 ∼ (1011 − 1012). (9) tainty, and cannot be estimated using available fmfin astrophysical observations. Since humans have de- 11 veloped communicating technology for more than A commonly quoted number is N∗ = 2.5 × 10 a century on Earth, Lc should be at least ∼ 100 (e.g. Westby & Conselice 2020). −8 yr, so that Lc/Lp is at least 10 . However, it • fp: Here the meaningful fraction should be the has been widely speculated that CETIs can sur- fraction of stars that can have habitable plan- vive much longer than this time scale (if they ets (or habitable moons orbiting giant planets) do not destroy themselves) (e.g. Burchell 2006; in stable orbits long enough to develop life and Vakoch et al. 2015). Lacking any guidance, in the CETIs. Modern exoplanet observations suggest following we normalize the ratio Lc/Lp (which is that planets may be ubiquitous in stellar systems relevant in the problem) to ∼ 10−4, which corre- 5 6 9 10 (e.g. Howard et al. 2012; Burke et al. 2015). Since sponds to Lc ∼ (10 −10 ) yr for Lp ∼ (10 −10 ) 4 Zhang

yr. Notice that in principle, CETIs can “re- should be defined specifically for each type of sig- appear” after self-destroy. What matters in our nal. It is not the rate of the truly detected signals, problem is the total duration of the existence of which depends on the fraction of sky coverage and CETIs. The duration of each CETI and the num- the duty cycle of the telescopes. With dedicated ber of generations of CETIs in a planet is not rel- surveys with certain sky and temporal coverage, evant. As a result, Lc defined here can be con- the detected signal rate (or, very likely, its upper sidered as the average total duration of CETIs on limit) can be corrected to derive N˙ s,o (or, very each planet in the Galaxy. likely, its upper limit). • N: Even though this is not a direct measur- • N˙ s,e: This is the average signal emission rate per CETI (for detailed discussion of CETI signals from able quantity, it is nonetheless interesting to the emitter’s perspective, see Sect. 3). The CETIs write down the estimated CETI number N in the Galaxy according to Eq.(7) with the normalization may repeat their signals multiple times, and N˙ s,e is defined as the total amount of signals emitted values of each parameter as discussed above: ceti by a CETI divided by its entire lifetime Lc, av- N∗ f n N =12500 p e eraged over all CETIs. Maybe (and likely) dif- 2.5 × 1011 0.5 10−3       ferent CETIs attempt to communicate using dif- L /L × c p . (11) ferent types of signals. Maybe the same CETI 10−4 attempts to communicate using several different   10 types of signals. So, N˙ s,e is signal-type-dependent, Notice that if one chooses Lc ∼ 100 yr, Lp ∼ 10 · ceti ∼ −2 and should be defined for each type of signal specif- yr, and fp ne 10 (corresponding to the factor ically. fL · fHZ · fM defined by Westby & Conselice 2020), one obtains a minimum CETI number of 12.5, which is con- • ξo: In order to connect the emission rate with the sistent with the minimum CETI number estimated by detection rate, one should introduce this factor Westby & Conselice (2020). that denotes the fraction of emitted signals de- tectable by astronomers on Earth. One may write 3. CETI SIGNALS FROM THE EMITTER’S PERSPECTIVE 2 ∆Ωe dlim Great efforts have been made in the Search-for-Extra- ξo ≡ , (10) 4π d Terrestrial-Intelligence (SETI) community to speculate  MW  the types of the CETI signals. Wright et al. (2018) de- which includes the average beaming factor ∆Ωe/4π fined an eight-dimensional model to describe CETI sig- for the emitted signals and the flux limitation fac- nals and argued that the current SETI searches only 1 2 tor (dlim/dMW) , where dlim is the maximum touched a tiny phase space of this eight-dimention “cos- distance from Earth the signal can be detected mic haystack”. Some of the dimensions defined by and dMW ∼ 10 kpc is the characteristic distance Wright et al. (2018) (e.g. their dimension 1 [sensitivity scale of the Milky Way galaxy. This second factor to transmitted or received power] and their dimensions depends on the strength (luminosity) of the emit- 3-5 [distance and position]) are from the observer’s per- ted signal and the sensitivity of the telescopes that spective. detect these signals. Ideally, advanced CETIs may In the following, I discuss a possible CETI signal from emit signals detectable by all other civilizations the emitter’s perspective by speculating what type of sig- across the Galaxy. For such signals, one takes nal a CETI may emit. Assuming that CETIs commu- 2 2 (dlim/dMW) ∼ 1, so that ξo ≃ ∆Ωe/4π, which nicate using an electromagnetic signal , I characterize a only depends on the average solid angle ∆Ωe of CETI signal by the following seven parameters: the emitted signals.

2 In principle, a multi-messenger channel may be also used. • N˙ s,o: This is the total detectable signal rate at ˙ However, these messengers are technically challenging and not Earth from all sky all time. Similar to Ns,e, it economical. For example, the generation of gravitational waves is very expensive. Cosmic rays tend to be deflected by interstellar magnetic fields. Even if neutrinos may be easier to generate than 1 A simple isotropic distribution is assumed here for order-of- the other two messengers, they are very difficult to detect from magnitude estimation. A more careful study should account for the observer’s prospective due to the small cross section of weak the MW structure and the anisotropic environment of the Earth interaction. These channels may not be favored by CETIs and neighborhood. are, therefore, not discussed in this paper. 5

1. Duration: Shorter durations may be preferred • Lightcurve: Consisting of 1,679 binary digits (ap- from economical considerations, but the signal proximately 210 bytes) that encode rich informa- should be long enough for other civilizations to tion such as numbers, atomic numbers, , a detect. human image, the , and the Arecibo radio telescope. 2. Peak luminosity: Since any signal has a rising and • fading phase, the luminosity at the peak time is Solid angle: A narrow beam of about 1 square arc- a relevant parameter to consider. This parameter minute pointing toward the globular cluster M13 ∼ will define how far the signal can be detected by 25, 000 light years away. other civilizations given a certain detector sensi- • Repetition rate: Not repeated with the same con- tivity. figuration according to the public record.

3. Emission spectrum: This concerns the central fre- One may estimate the detectability of this signal by quency and the bandwidth of the transmission other CETIs. First, aliens in all other directions outside signal, which has been discussed by Wright et al. the narrow beam would never know that such a signal (2018) as their dimensions 2 & 6. was emitted. Second, at the distance of M13, the flux density of this signal is ∼ 2.7 × 10−20 Jy, which is 21 4. Polarization: The polarization properties of a orders magnitude fainter than the sensitivity (10 Jy) of CETI signal may carry additional information. the current humans’ own narrow-band SETI search in- This is the dimension #8 of Wright et al. (2018). strument, the (Welch et al. 2009). Suppose that there are indeed aliens ∼ 25,000 years later 5. Lightcurve: This includes how the luminosity of an on a planet orbiting one of the stars in M13, they need individual signal rises and falls as well as how mul- to use a telescope with a collecting area much greater tiple signals group together to display intelligent than the size of the planet itself in order to detect the information. This is somewhat discussed as the signal. The telescope should be also pointing toward dimension #9 by Wright et al. (2018) as “modu- the direction of Earth during the . 3 minutes of time lation”. when the passes by the planet. The chance is negligibly small. Finally, since no other star is along 6. Solid angle: This describes how wide the CETI the path towards M13, the only civilizations other than signal beam is, which is directly related to ξo dis- those in M13 who could in principle receive the signal cussed in Eq.(10). would be the even more distant ones behind M13. The flux on their planets are even much lower and even more ˙ 7. Repetition rate: This defines the Ns,e in Eq.(8), advanced technology is required to catch the signal. which is the average value of all CETIs in the So, if CETIs do exist and indeed have the intention Galaxy. The dimension #7 of Wright et al. (2018) to broadcast their existence in the Galaxy, they need to concerns the detected repetition rate, which is re- emit signals that are many orders of magnitude more ˙ lated to Ns,o in Eq.(8). powerful and in a much larger solid angle than the Arecibo message signal humans sent. Since no CETI signal has been detected, one can only take humans’ own communicating signals for compari- 4. FRB-LIKE ARTIFICIAL SIGNALS BY CETIS son. The most famous signal was the “Arecibo message” Fast radio bursts (FRBs) are frequently detected, broadcast in 1974 (Staff at the National Astronomy & Ionosphere Center millisecond-duration, radio bursts that originate from 1975). The properties of the signal in connection with cosmological distances (Lorimer et al. 2007; Petroff et al. the seven dimensions discussed above are: 2019; Cordes & Chatterjee 2019). Since their physical • Duration: . 3 minutes; origin is unknown, an alien connection was speculated by some authors. For example, when placing phys- • Luminosity: 450 kilo-Watts or 4.5 × 1012 erg s−1; ical constraints on FRBs, Luan & Goldreich (2014) discussed the possibility that the observed FRBs are • Spectrum: At frequency 2,380 MHz with an ef- artificial signals sent by aliens and concluded that a fective bandwidth of 10 Hz, with modulations by modest power requirement is needed. Lingam & Loeb shifting the frequency by 10 Hz; (2017) interpreted FRBs as radio beam signals produced by extra-galactic advanced civilizations to launch light • Polarization: No information is available. sails for interstellar travels. These suggestions intended 6 Zhang to interpret at least a fraction of the observed FRBs as of ∼ 100 kg matter. Emitting such a signal is of an artificial origin. beyond the current technological capability of hu- I believe that the observed cosmological FRBs are due mans. On the other hand, there is no fundamental to astrophysical origins. Indeed, many theoretical mod- physical barrier to prevent this from happening. els have been proposed in the literature to interpret var- • ious observational properties of FRBs (e.g. Platts et al. Spectrum and dispersion measure (DM): FRBs ∼ 2019). Rather, I speculate that if CETIs do exist and have been detected from 400 MHz to 8 GHz indeed have the intention to communicate, they may (Petroff et al. 2019). The observed FRB flux typ- send off FRB-like artificial signals to the Galaxy, likely ically show a steep drop at high frequencies. The along the Galactic plane. There are two main reasons low frequency range also suffers from a few sup- for this speculation. First, intelligent civilizations know pression effects such as plasma scattering and that FRBs happen very frequently (of the order ∼ 104 absorption, which are particularly severe in the per day all sky for bright ones) in the universe and that Galactic plane. One natural frequency CETIs may other civilizations must be monitoring these events all consider to broadcast artificial FRB signals would the time. The same observational facilities designed to be around the 21cm line frequency (1.42 detect FRBs (e.g. wide-field and sensitive radio tele- GHz). Its “resonances” (e.g. 1.5 times or 2 times) scope arrays) could easily spot artificial signals they may be also possible. In order to mimic an FRB would send. Second, probably more importantly, FRBs signal, the CETIs may consider to emit the sig- have radio frequencies and extremely short durations nal in a wide enough bandwidth so that a DM (but still long enough for detection). They are rela- could be measured. In order to draw attention to tively easy to mimic from the economical point of view. observers, they may also add an extra DM (e.g. ∼ −3 In contrast, it is for example more difficult to make ar- 500 pc cm , a typical value for cosmological tificial gamma-ray bursts or fast optical bursts. FRBs) in excess of the Galactic value by placing a In terms of the seven dimensions discussed in Section cold plasma along the path of signal propagation. 3, the properties of an FRB-like CETI signal may be Such a signal can be easily picked up by observers speculated as follows: using the standard FRB-searching algorithms. • Duration: An FRB-like signal should have a du- • Polarization: So far, the FRB polarization prop- ration of the order of milliseconds, i.e. ∆t = erties do not show well-defined common charac- (1 ms) ∆tms. teristics. The polarization properties are also not essential for FRB detections. There is no need to • Luminosity and energetics: How bright the signals design certain polarization properties in the artifi- are emitted depends on the technological level of cial signals unless some extra intelligent informa- the aliens, but an advanced CETI who is eager tion can be carried. For example, a CETI may de- to broadcast its existence may try to emit at a cide to emit an FRB with 100% linear polarization luminosity such that the detected flux level by with a Faraday rotation measure (RM) greatly ex- a typical Milky Way observer (like humans on ceeding the local value in the Galaxy through gen- Earth) is comparable to that of a cosmological erating a strong magnetic field in the emission site. FRB. For a 1 ms-Jy signal with a characteristic They may also vary RM significantly to show that distance in the Milky Way, the isotropic luminos- the signals are indeed artificial. ∼ 32 −1 2 ity should be (10 erg s ) d10kpc. Consider the beaming factor (which is also the probability • Lightcurve: This is how CETIs deliver informa- factor for the observer to see the signal), ξo = tion to show the intelligent nature of their sig- ∆Ω/4π. The true emission power3 should be ∼ nals. There are many possibilities, which we re- 29 −1 2 22 2 (10 erg s )ξo,−3d10kpc or (10 W)ξo,−3d10kpc, frain from speculating. In any case, one may −3 where ξo is normalized to 10 . The true expect a series of FRB-like signals that encode ∼ 26 2 ∼ energy is (10 erg)ξo,−3∆tmsd10kpc or profound information understandable by other ad- 19 2 ∼ 11 (10 J)ξo,−3∆tmsd10kpc. This energy is 10 vanced civilizations. times of the emitted energy in the Arecibo mes- • Solid angle: Advanced CETIs may like to broad- sage, which is comparable to the rest mass energy cast their signals as wide as possible, so that an all-sky (4π) signal would be most ideal. In prac- 3 This is much greater than the one estimated by tice, it would be easier to send collimated signals Luan & Goldreich (2014). for economical and technical reasons. Since the 7

Galactic plane has the highest probability for other One commonly discussed question in the SETI com- civilizations to detect the signals, a CETI in the munity is the “Fermi-Hart paradox” regarding “where is Galactic plane may send a fan beam with the ver- everybody?” (as elaborated in detail by Hart (1975), see tical angle defined by the height to radius ratio of also Brin (1983)). Even though the question was about the Galactic plane, which is of the order of 10−3. why there is no evidence for extraterrestrial intelligence If the azimuthal angle is 2π, the beaming factor is visiting Earth, another version was to address why we ∆Ω/4π ∼ 10−3. have not received signals from CETIs. One general type of answer to this question is that humans have not ob- • ˙ Repetition rate Ns,e: This is something not easy served the universe long enough to allow any detection. to estimate, but can be in principle constrained Indeed, Wright et al. (2018) showed that humans only from the data (see Section 5 below for detailed searched a tiny parameter space in a multi-dimensional discussion). However, since sending these signals “cosmic haystack” through blind searches. consumes a lot of energy, the emission rate may If one focuses on one specific type of signal, e.g. the not be very frequent unless CETIs are in great FRB-like artificial signal discussed in this paper, the desire to communicate (e.g. sending S.O.S signals “haystack” search volume greatly shrinks. I argue that for help). even for such specific signals, the answer to the “Fermi- Hart paradox” is still “we simply have not observed 5. A QUANTITATIVE ASSESSMENT OF CETI’S long enough yet”. One can quantitatively show this us- FRB-LIKE ARTIFICIAL SIGNALS ing Equation (12). For FRB-like signals, even if one −1 With the operations (or planned operations) of a can achieve the N˙ s,o < 0.1 yr upper limit (which growing number of radio antenna arrays (e.g. The requires dedicated efforts from wide-field radio arrays Canadian Hydrogen Intensity Mapping Experiment working for decades), for fiducial parameters, one can [CHIME] (Bandura et al. 2014), the Deep Synoptic Ar- only set a moderate upper limit on the signal emis- ray 2000-antenna [DSA-2000] (Hallinan et al. 2019), sion rate of highly advanced CETIs who can broad- and the Square Kilometre Array [SKA] (Johnston et al. cast their existence across the entire Milky Way, i.e. −1 2008)) to detect FRBs, one can start to place a con- N˙ s,e < 0.008 yr . It is impossible that all CETIs have ˙ straint on Ns,o of FRB-like artificial signals from the such a capability to broadcase across the Milky Way Milky Way. Non-detection of any FRB-like artificial (e.g. humans do not). If CETIs are less advanced, ξo 2 signal from the Galactic plane, when corrected for the would be greatly reduced since the (dlim/dMW) factor −7 sky coverage and duty cycle for progressively longer would have to be included. For example, for ξo ∼ 10 observational times, can place progressively tighter con- (corresponding to the case that detectable CETI signals ˙ straints on Ns,o, which would place constraints on the can only reach a distance of dlim ∼ 100 pc), one can only ˙ −1 −1 average signal emission rate of CETIs, Ns,e, for FRB- set a limit of N˙ s,e < 80 yr for N˙ s,o < 0.1 yr and typ- like signals based on Eq.(8). For example, an upper ical parameters. It is hard to imagine that an average ˙ −1 limit of Ns,o < 0.1 yr (e.g. all sky no detection in a CETI so desperately communicates with the universe by decade) can lead to a constraint sending signals at a rate > 80 yr−1. As a result, there is essentially no “paradox”. ˙ −1 −1 Ns,o N∗ This argument applies not only to the FRB-like sig- N˙ s,e < (0.008 yr ) < 0.1 yr−1 2.5 × 1011 !   nal but also other specific signals, which are probably − −1 −1 even more difficult to generate by CETIs. If one does fp n Lp/Lc ξo e , not specify a signal type, the search parameter volume 1/2 10−3 104 10−3         would be increased exponentially, so that the “paradox” (12) is further diminished (Wright et al. 2018, and references therein). The lack of detection of any CETI signals now where the parameters are normalized to the character- is naturally expected. istic values as discussed in Section 2. With the non- detection upper limit in decades, one may then make a 7. SUMMARY AND DISCUSSION quantitative statement that with the fiducial values of The points made in this paper can be summarized as the parameters, CETIs on average emit less than one follows: per century FRB-like artificial signals that can cross the entire Milky Way. • I presented a derivation of Equation (Eq.(7)) based on a probability argument, which is consistent 6. FERMI-HART PARADOX with the original Drake equation (Eq.(1)). 8 Zhang

• Based on Eq.(7), I proposed a new equation when technology is advanced enough. This is definitely (Eq.(8)) to connect the observed CETI signal a topic subject to debate (see, e.g. Gertz (2016) for a rate N˙ s,o with the average signal emission rate general discussion on the pros and cons on Messaging to N˙ s,e by CETIs. Subject to uncertainties of several Extra-Terrestrial Intelligence (METI)). Optimists may parameters, this equation allows one to use obser- think that CETIs are eager to find out whether they are vations to directly infer the value (or, very likely, alone in the universe and would be happy to remotely the upper limit) of the average CETI signal emis- communicate with other civilizations. Pessimists, on the sion rate. The equation applies to specific signals other hand, would believe that it is very dangerous to rather than unspecified blind-search signals. expose ourselves to more advanced civilizations as they would invade Earth to snatch resources4. In any case, I • After characterizing CETI signal properties in believe that this should be a decision to be made by the seven dimensions from the emitters’ perspective, entire humanity, not by a small group of “elites”. My I suggested that FRB-like artificial signals could personal recommendation is: Do not do anything until be one type of CETI signals for good reasons. Us- one can develop the technology to emit FRB-like signals ing Eq.(8), I derive a constraint one may pose with (this may take some time, e.g. hundreds, thousands or the detection/non-detection of FRB-like artificial even millions of years); keep watching whether “others” signals from the Milky Galaxy. have emitted any such signal along the way; and make a decision then! If non-detection of FRB-like signals per- • The N˙ s,e constraint derived from the detection/non- detection of FRB-like artificial signals (Eq.(12)) sists for a long time (e.g. after thousands of years), then is taken as an example to quantitatively show the Fermi-Hart paradox may become more a concern. why the “Fermi-Hart paradox” is not a concern. The CETIs may be also taking a pessimistic approach Even for one particular type of signal and under like us, so that the paradox may find an answer along the most optimistic assumption (i.e. an average this reasoning in the far future. CETI is able to broadcast FRB-like artificial sig- nals across the entire Milky Way), one would not expect to detect any signal now and probably still not even after decades or centuries of dedi- cated monitoring. This strengthens the argument against the Fermi-Hart paradox by Wright et al. I thank Qiang Yuan for an important remark on an (2018) for blind searches. earlier version of the paper, Jason Steffen for discussing the current status of exoplanet searches, and Maura Finally, one natural question is whether humans McLaughlin for discussing the current status of Galactic should send FRB-like artificial signals in the future FRB-like signal searches.

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