The Fermi paradox juxtaposes the fact that we’ve heard no signal from extraterrestrial life against claims that life must be common in the universe. The paradox, named after physicist Enrico Fermi, is not so inscrutable given 1) the immense distances in interstellar space; 2) the likelihood that few civilizations have survived long enough to attain a high level of advancement; and 3) the slim chance (until recently, perhaps) of detecting messages from a distant civilization.

These points are addressed in recent paper by Evan Solomonides and Yervant Terzian of Cornell University entitled “A Probabilistic Analysis of the Fermi Paradox” (the full PDF is available for download on the upper right at the link). The authors note that the first broadcast from Earth that could be detected beyond the planet took place roughly 80 years ago. Unfortunately, it was Hitler’s commentary on the superiority of arian athletes at the 1936 Olympics. That seems a shame, but then it’s unlikely that the recipients can interpret the signal at all. Still, the authors estimate that by now, that broadcast will have reached over 8,000 stars and over 3,500 earth-like planets within an 80 light-year sphere around the Earth. That might sound like a lot, but it is an infinitesimal fraction of the Milky Way galaxy. There is roughly a zero chance that an advanced civilizations exists within a population of potentially habitable planets that small.

How Common Is Life?

Solomonides and Terzian extend their analysis using a version of the so-called Drake equation, which relies on a series of assumed probabilities to calculate the number of of active, intelligent civilizations in the galaxy, call it N. However, the authors assert that their version of the Drake equation yields an estimate of N for the entire history of the galaxy, but it looks very much as if they’ve simply redefined the equation’s terms. They use the equation to specify a relation between 1) the average length of communication history for all intelligent civilizations; and 2) the total area of all spheres of coverage, which is dependent on N. They assert that as of today, this combined area must encompass less than one-half of the galaxy because that implies that we are less likely to have heard a signal from extraterrestrials than not to have heard one. And we haven’t. However, this seems like a thin foundation from which to draw implications, and it is based on an expectation. Even if those spheres covered 90% of the galaxy or more, it would not rule out the silence we’ve “heard” to date. Nevertheless, the authors use this inequality to derive a lower and upper bound on the number of intelligent, communicating civilizations in Milky Way history.

Expanding Broadcast Spheres

Again, Solomonides and Terzian define “spheres of communication” extending into space as far in light years as the length of a civilization’s broadcasting history in earth-years. Earth’s sphere of communication now extends outward by 80 light years. But imagine that a long-extinct civilization on the other side of the galaxy sent messages about 30,000 years ago. The signal might be reaching us just now. But if that civilization’s broadcasting history lasted 1,000 years before an untimely extinction, its broadcast sphere would be like a hollow, expanding dumpling, now almost as wide as the galaxy itself but with dough walls a constant 1,000 light-years thick. Only the walls contain broadcast information, so the areas of spheres like these are not simply additive. The hollow “inner spheres” must be subtracted to get the total broadcast area.

Revisiting the Duration of Broadcasts

The communication spheres defined by Solomonides and Terzian are “disks” rather than spheres, because collapsing the galaxy into two dimensions simplifies the analysis. They do not appear to allow for the sort of hollowness implied by an older, advanced civilization with a limited survival time. That matters in terms the history relevant to our failure to detect signals thus far. If a civilization’s broadcasting history is of short duration relative to its distance from Earth, then its communication sphere has thin broadcast walls. If that now-extinct civilization originated signals on the other side of the galaxy more than about 35,000 years ago, the signals would be irrelevant to “our” Fermi paradox because by now, Earth is almost certainly inside the wall of its expanding broadcast “dumpling”. The signal passed us by before we were advanced enough to have thought about it. So the relevant history of the galaxy, for purposes of identifying the broadcasting histories of civilizations we might have heard from by now, goes back a bit farther than the total width of the galaxy, which is roughly 32,600 light years. The relevant history might be 35,000 years, give or take, not the 13 billion years since the galaxy’s birth. At least that limits one dimension of the problem.

I find the following sentence somewhat troublesome because Solomonides and Terzian seem to focus on the length of broadcasting histories only with reference to the present time:

“The planar area of the galactic disk reached by communication from any intelligent civilization (assuming such civilizations are uniformly dispersed throughout the galaxy) can be modeled as the area of N disks with radius Lh (average length of broadcasting history in years)…“

Some of those N disks are probably hollow. The idea that civilizations are commonplace may not mean that they all exist contemporaneously, and while the authors must understand that point, the description above implies that all disks are saturated with broadcasted information from center to outer rim rather than hollow.

Solomonides and Terzian then attempt to place the radius of earth’s own minuscule broadcast disk within the hypothetical distribution of all such discs within the galaxy:

“… we know that humanity was almost certainly not the (or even one of the) first species in the galaxy to develop broadcasting technology, nor one of the last. Put statistically, it can be said with a high degree of confidence that humanity is somewhere in the median 90% of the population of galactic species as far as broadcasting history is concerned. That is to say, we are not among the first nor last 5% of civilizations to develop this technology. … Taking a very conservative estimate, we posit that we have been broadcasting for 5% as long as the average communicative species has been, and as such this upper limit on the average is approximately 1600 years. This can be substituted back into the inequality derived previously to give an idea of the frequency of life that our apparent loneliness suggests.“

A couple of notes on this statement: First, it assumes that the lengths of broadcasting histories are distributed uniformly from 0 to 1,600 years. Second, it seems to preclude any history of broadcasts prior to 1,600 years ago. If that is the case, then the number of civilizations they consider are what I’d call “near contemporaneous”. The authors do not seem to be accounting for all history after all.

Few Neighbors Or Many?

The authors go on to calculate lower and upper bounds on the number of communicating civilizations in the Milky Way, but again, in light of the quotes above, the implied existence of civilizations seems to be near-contemporaneous. Perhaps that’s okay for arriving at a lower bound. Again, Solomonides and Terzian assume that our 80 years of communication history puts us below 95% of all other communicating civilizations. Therefore, the longest history among such civilizations would be just 1,600 years. Because we have heard nothing, there must be great distances between relatively few civilizations: only about 210, according to Solomonides and Terzian.

In light of the relevant history of the Milky Way, 1,600 years seems outrageously short for the longest communication history. To my way of thinking, broadcast histories, whatever their number, must be distributed over the entire galaxy and over a time span of about 33,000 years. If extinctions shorten the duration of those histories, then it is possible that we’ve simply missed the outer walls of a number of broadcast disks that have already reached us. In that case, civilizations must be sparse both spatially and over time. Unfortunately, the authors’ lower bound for the number of communicating civilizations must be taken as an estimate for civilizations whose existence is near-contemporaneous with our own. However, that does not fit as neatly into an explanation of the Fermi paradox as the authors would like.

For an upper bound on the number of communicating civilizations, the authors assume that we are on the verge of hearing from another civilization in response to our initial communication. If so,  then we have a very close, neighboring civilization about 40 light years away, which implies an outrageously high frequency of civilizations in the galaxy: about 78 million, according to the authors. The upper bound relies on an assumption that it’s necessary for Earth to receive a response to our communication, as opposed to receiving an independent communication from afar. Perhaps the signal/response requirement is imposed for reasons of estimating a more densely populated galaxy for the upper bound.

The lower and upper bounds imply that life is either rare or ubiquitous; the authors claim that either is an unreasonable violation of the so-called “mediocrity principle”, which posits that our civilization is “run-of-the-mill”: the first is a violation because we are rare; the second is a violation because we’ve somehow managed to avoid hearing anything despite the denseness of communicating civilizations in the Milky Way.

Great Filters

It’s reasonable to question the assumption that an advanced civilization’s broadcast history would be of relatively short duration. The galaxy is a hazardous place, however, presenting extreme natural threats to any planet finding itself in a “Goldilocks zone” near its host star and capable of harboring life over an extended period. Threats range from interloping space rocks to variations in a planet’s exposure to radiation. Then, there are hazards to life arising from natural conditions on the planet itself, such as extreme volcanic activity and perhaps natural toxins. Finally, the development of technology brings hazards as well, including the possibility of chemical, biological and nuclear calamities. All of these constitute “Great Filters” that may prevent civilizations from reaching a stage of advancement sufficient for interstellar travel and colonization of other worlds.

Can such hazards be expected to put a halt to a representative civilization’s broadcasting, and within how many earth centuries? In some cases, it’s likely to be as few as 10 or 20 centuries, but even if extinction is common, there are also likely to be a few civilizations making it to the far right tail of the survival distribution. Those few civilizations, or even one, could have begun broadcasting so long ago that their communication spheres are much larger than the galaxy itself. We might just hear them if they exist, but perhaps that argues that they do not.

Detection and and Understanding

Beyond the limits of communication spheres, another compelling reason for our failure to detect signals from other civilizations is signal degradation over great distances. According to Solomonides and Terzian, signal strength weakens with the inverse square of distance. Even today, messages of extremely distant origin might be impossible for us to discern, let alone understand.

“Though a handful of these signals have been designed to be picked up by extraterrestrial intelligence (…. i.e. Fibonacci, the prime numbers, the squares, all broadcast in binary), the vast majority would be indecipherable. This is because an alien civilization would need to first decode binary into sound (and figure out our tone encryption method) or video (with very specific, inconsistent formats), and if they could somehow do that, they would then need to decode the resulting 3,000 human languages … into something they could parse successfully.“

Given sufficiently well-equipped listening centers here on Earth, detection becomes something of a mathematical exercise. Over the past 10-15 years, there have been advances in developing algorithms to extract signals from an otherwise noisy background.

Conclusion

With plausible assumptions, the Drake equation yields the conclusion that the galaxy may be populated with a large number of intelligent civilizations, larger still if we count those existing at any time over the past 35,000 years. The Solomonides and Terzian paper shows that the lack of detection on Earth is not very surprising, but in a limited context. The silence might be even less surprising if many of the historical civilizations had a broadcasting age of limited duration, generating hollow broadcasting spheres, because the walls of many of those spheres would have passed us by long before our own radio age. Therefore, the Fermi paradox does not seem to be such a paradox after all.