Detection probability of non-natural signals in the Galaxy (this a simplified description of the model introduced here)
The discovery of thousands of extrasolar planets and the inferred astronomical number of Earth-like planets in the galaxy has recently fueled an intense activity in SETI projects. The “Breakthrough Listen” initiative, the largest and most comprehensive search ever, and the planned “Cradle of Life” program of the Square Kilometer Array radiotelescope offer the unprecedented opportunity of a systematic investigation in the vast domain of the SETI search space. These observational campaigns, and the empirical data they will deliver, can profoundly impact the current beliefs about the possible population of extraterrestrial signals in the galaxy.
But, what are those current beliefs? They basically range from those conceiving a universe teeming with extraterrestrial civilizations to others in which our planet is the only cradle of life. Precisely because of the lack of data (only a tiny fraction of the galaxy has been searched so far) at present we cannot exclude either of these two extreme possibilities. This implies that the number N of hypothetical E.T. signals that are travelling now at the speed of light in the Milky Way can be in principle any non-negative integer spanning several orders of magnitude.
The things look even worse by noting that N is not the only variable at play. Each communicating civilization (aka, the emitter) may have started to transmit a signal at a random time in the past and for a random duration, from an arbitrary location in the galaxy. A possible scenario could therefore appear as the one shown in the figure above, where the annular regions represent the space covered by signals (for the case of isotropic, non-directional emissions) and the red point represents the Earth. The outer radius of each spherical shell is proportional to the starting time of the emission, while the shell thickness is proportional to the duration L of the signal (the constant of proportionality is just the speed of light, c). The figure makes it clear that a necessary condition to detect E.T. transmissions is that the Earth falls within a region of the Galaxy covered by the signals. In other terms, the detection probability (or, the chances of success of any SETI program) is basically the fraction of the galactic volume occupied by the emitted signals.
To see why the number of signals does not specify completely the detection probability, consider the rather rural example of the figure above. There are two cows, with a different number of black spots on their body, and a fly that has a keen interest in cows but not in the particular pattern of their skin. The fractional area covered by the black spots gives the probability that the fly will randomly choose to lay on a black portion of the cow's skin. The fractional area (or area fraction) is the total area colored in black divided by the total area of the cow's skin. Therefore, as in the example above, if two cows have a different number of spots but equal fractional area of black skin, the probability that the fly will land on a black spot is the same for both cows.
The analogy with SETI and the probability of detecting E.T. signals becomes clear if we identify the cow with our galaxy, the black spots with the signals, and the fly with the Earth. The fractional black area of the cow's skin corresponds to the fractional volume, φ, of the galaxy covered by the signals. The probability of detecting a signal is therefore given simply by φ (the choice of this symbol is not casual. It often denotes the volume fraction occupied by a constituent in multi-phase hererogeneous materials). A large number of short-lived signals (large N, small L) may cover a similar galactic volume of just a few but long-lived signals (small N, large L). As in the example above, these two possibilities have comparable values of φ.
To calculate φ one would need to know the actual pattern of the signals covering the galaxy, which is of course information that nobody has. We can however use a statistical approach, in which different, possible patterns are considered. In this way, the detection probability is obtained by averaging over all these possibilities. Actually, there is no need to consider all imaginable signals, because those that are necessarily undetectable can safely be excluded from the calculation.
To understand this point, note that the Milky Way has a radius of about 60 kly (1 kly= 1'000 light years), while the distance of the Earth from the galactic center is about 27 kly (figure above, left panel). An hypothetical emitter in the Galaxy cannot be farther from us than RM ≈ 87 kly, and its signal would take tM=RM/c ≈ 87'000 years to reach the Earth. Any signal from this emitter that has been transmitted after a time tM before present cannot therefore interects the Earth's orbit, because the inner radius of its spherical shell signal has already surpassed us (figure above, right panel). Since this is true for any emitter, regardless of its distance from the Earth, the average is performed by considering only those signals that have been emitted within a time t<tM.
Let us now move to illustrate the results of this approach. We start by assuming that there exists only one emitter in the Galaxy. The probability of detecting its signal is obtained by considering all possible instances in which its spherical shell intercepts the Earth's orbit conditioned to the requirement that t<tM. The average over the possible shells is simply an average over the emitter's position, the duration L of the signal, and t (under the condition that t<tM). After performing these averages, It turns out that the detection probability of a single signal has the following expression:
which is just the ratio between ⟨L⟩, the mean duration time of the signal, and the oldest possible time that a potentially detectable signal has been first transmitted (see the right panel of the figure above). This ratio is also defined as the scaled longevity of the signal λ, which is a convenient quantity to characterize the signal duration that is relevant for SETI. For example, a signal with mean longevity much smaller than tM ≈ 90'000 years (small λ) covers only a small fraction of the Galaxy and it has a small detection probability, while a signal with mean longevity larger than tM (λ ≈1) encompasses most of the Milky Way and it has a large probability to intersect the Earth's orbit.
Let us assume now that there are N isotropic signals in the Galaxy that are no older than tM. The N signals are independent of each other, and their spherical shells have outer radii and thicknesses identically distributed. Since each signal has detection probability λ, the probability that the Earth does not intersect any of them is simply (1-λ)N, and so the probability to detect at least one signal is:
Despite its simplicity, Eq. (2) is a non-trivial result showing that the dependence of φ on any complex pattern of (independent) signals covering the Galaxy reduces to just two parameters: the scaled signal longevity λ, Eq. (1), and the number N of signals. Equation (2) explicitly shows that it is the combination of these two parameters that ultimately determines the chances of detection. As for the spotted cows discussed above, very different values of λ and N can give similar values of φ. For example, the detection probability of a single signal with a mean longevity of 10'000 years (λ ≈ 0.103) equals the probability that the Earth itersects at least one among 1'000 signals with ⟨L⟩ ≈ 9.5 years (λ ≈ 0.000109).
The detection probability of Eq. (2) gives the likelihood that, given λ and N, the Earth intersects a region of the Galaxy populated by signals, but it does not tell us how many of those signals we could possibly detect. We can however easily calculated the mean number, ⟨k⟩, of signals that are impinging upon our planet by summing the detection probabilities of each signal:
In general, an increase of the fractional volume occupied by the signals (i.e., the detection probability) does not imply a proportional increase of ⟨k⟩, which instead grows more slowly. This can be easily seen by combining Eq. (2) and (3). Indeed, if we take the natural logarithm of 1-φ and use ln(1-λ) ≤ -λ for any 0 ≤ λ ≤ 1, we obtain that the mean number of detectable signals is bounded logarithmically from above:
This inequality is specially important because it allows us to set an upper limit to the typical number of detectable signals directly from our assumptions about φ. This can have interesting implications on our perception of the likelihood of detecting E.T. signals. For example, Eq. (4) implies that even for detection probabilities as large as 60%, the typical number of detectable signals is below one. The region covered by hypothetical extraterrestrial signals can thus comprehend a significant fraction of the galaxy without us even noticing it. This is perhaps the most compelling argument that the so-called Fermi paradox is, actually, not a paradox.
In conclusion, we don't know if our Galaxy is populated by extraterrestrial signals, waiting for us to be detected. Using simple geometric-probabilistic arguments, it is possible however to draw some general conclusions : (i) any detectable signal cannot be older than about 87'000 years and (ii) more than 60% of the Galaxy must be filled by E.T. signals in order that at least one of them in average could be detected.
Search for Extraterrestrial Intelligence
This is a quantity useful to test the plausibility of various hypotheses about the galactic population of E.T. signals. For example, if there were 1'000 signals in the Galaxy lasting well over 10'000 years (λ ∼ 1) and strong enough to be clearly discernible from the background noise, then Eq. (3) implies that, in average, about 1'000 signals are expected to reach us at this moment. Since there is no evidence that we are under such intense bombardment, this hypothesis is likely to be wrong. However, if we assume that the mean longevity of 1'000 signals is only about 100 years, in average we could potentially detect only one signal in the entire galaxy (⟨k⟩≈ 1.15). In this case, the lack of detection would appear less surprising.
An interesting aspect of this latter example is that, althought the values of N and λ are such that ⟨k⟩ ∼ 1, the detection probability is actually quite large (φ ≈ 0.68). In other terms, almost 70% of the galaxy is covered by the signals, but only one intersects our planet in average. Mathematically, this is due to the different ways in which φ and ⟨k⟩ depend upon N (φ is a power function of N while ⟨k⟩ is linear in N). A more intuitive understanding is given by the figure below which shows that even for a rather dense pattern of shell signals the area spanned by the regions where two or more signals overlap is small.