How Many Aliens Are There? The Drake Equation and the Fermi Paradox

0. Key points

• The Drake equation writes the number $N$ of civilizations in the Galaxy that are communicating right now as a product of seven factors. The equation itself is essentially correct and follows from the definitions; the hard part is the values of the factors.
• Even with plausible inputs, $N$ ranges from a few down to $10^{-8}$. Uncertainties in a product add up in the logarithm, so the answer is bound to swing by orders of magnitude.
• The heart of the paradox is not that “we hear no radio”. Even a ship travelling at about 1 % of the speed of light needs only some 20 million years to spread across the whole Galaxy, and that is 0.15 % of the age of the Galaxy.
• Conversely, hearing nothing means almost nothing. An Arecibo-class transmitter is detectable out to roughly 1000 light-years, and for even one civilization to lie inside that radius the Galaxy would have to hold about ten thousand civilizations simultaneously.
• Candidate answers fall into four families — “nobody is there”, “they died”, “they are silent”, “we have not noticed” — and each carries predictions that future observations can separate.

1. Motivation: a remark over lunch

Summer 1950, Los Alamos National Laboratory. On the way to lunch with his colleagues Emil Konopinski, Edward Teller and Herbert York, Enrico Fermi was discussing a cartoon that had appeared in The New Yorker. Public trash cans were disappearing from New York City at the time, and the cartoonist had pinned the blame on aliens carrying them off. The conversation drifted to flying saucers and faster-than-light travel, and then moved on to other things.

Some way into the meal, Fermi, who had been silent, is said to have suddenly asked:

Where is everybody?

According to the careful record assembled from the recollections of those present, Fermi then did one of his celebrated mental estimates on the spot and concluded that planets like the Earth cannot be rare, that on some of them life should become intelligent, that some of those should take up interstellar travel, and that they should therefore have reached the Earth long ago. Everyone thought “yes, quite” — and nobody had an answer.

The aim of this article is to translate that lunchtime aside into a problem that can be pinned down with numbers. Once the translation is done, something surprising happens. The question “are there aliens?” turns, for the most part, into questions of biology and social science, and what physics can answer stands out sharply. And the part physics can answer gives a strikingly definite conclusion.

This is not, incidentally, the first famous cosmic paradox. “Why is the night sky dark?” has exactly the same structure: knock down the naive assumptions one at a time and the shape of the universe emerges. For that one, read Why Is the Night Sky Dark?. The contradiction that follows from the naive assumptions is collected there as Olbers' paradox（Theorem 3.4）［Why Is the Night Sky Dark? Olbers' Paradox and the Finite Age of the Universe］.

2. Preliminaries: getting a feel for the size and the timescales of the Galaxy

Let us first line up the numbers we will use again and again.

Quantity	Approximate value
Number of stars in the Galaxy	$1 \times 10^{11}$ – $4 \times 10^{11}$
Diameter of the galactic disk	about $100{,}000$ light-years
Thickness of the disk (thin disk)	about $1000$ light-years
Sun to galactic centre	about $26{,}000$ light-years
Age of the oldest stars in the Galaxy	about $13$ billion years
Age of the Solar System	about $4.6$ billion years
Time since life appeared on Earth	about $4$ billion years
Time since multicellular life appeared	about $600$ million years
Time since Homo sapiens appeared	about $300{,}000$ years
Time since humans began emitting strong radio	about $90$ years

Numbers alone do not convey scale, so let us build a model. Shrink the $100{,}000$ light-year diameter of the Galaxy down to $1$ metre. Then $1$ light-year is $10\ \mu\mathrm{m}$.

• The $4.2$ light-years to the nearest star, Proxima Centauri, becomes $42\ \mu\mathrm{m}$ — half the thickness of a human hair (about $80\ \mu\mathrm{m}$).
• The Solar System (out to Neptune’s orbit) is about $0.001$ light-years across, so $10\ \mathrm{nm}$: one tenth the size of a virus.

So: on a disk one metre wide, a few hundred billion solar systems smaller than viruses are scattered at intervals about the thickness of a hair. That is the Galaxy. This model will pay off later, when we ask how far away we can be heard.

Note

How the distances and ages in the table were measured is covered in How Do We Know the Distances to the Stars? and The Edge and the Age of the Universe. Distances to remote objects are measured by stepping from one standard candle（Definition 2.3）［How We Know the Distance to a Star］ to the next, while the age of the universe comes out of the age formula for a flat universe（Theorem 3.6）［The Edge and the Age of the Universe］. Weighing the Galaxy itself from its rotation appears in the mass of the Milky Way（Example 3.2）［Dark Matter and Dark Energy］. Here we simply borrow the results.

3. The Drake equation

3.1. The idea of searching by radio

In 1959, Giuseppe Cocconi and Philip Morrison of Cornell University published a short paper in Nature. It asked which wavelength it would be rational to use for communication with a distant civilization. Their answer was the $21\ \mathrm{cm}$ line of atomic hydrogen, at a frequency of $1420\ \mathrm{MHz}$. The reasoning is transparent: every astronomer anywhere in the universe knows that frequency, and it sits in the band where the galactic background noise is lowest. Without knowing anything about the other party, you make “something they must also know” your meeting place.

The following year, in 1960, the young radio astronomer Frank Drake pointed a $26\ \mathrm{m}$ dish at Tau Ceti and Epsilon Eridani. This was Project Ozma. Nothing was found. Then in 1961 Drake convened a small meeting of eleven people at Green Bank. To organize the agenda, he wrote the following on the blackboard.

Definition 3.1（The Drake equation）

Let $N$ be the number of technological civilizations in the Galaxy that are, right now, emitting signals we could detect. Introduce the following seven quantities.

• $R_{*}$: the rate at which new stars form in the Galaxy (units: stars per year)
• $f_p$: the fraction of those stars that have planetary systems (dimensionless, $0 \le f_p \le 1$)
• $n_e$: the average number of bodies per planetary system with an environment where life could survive (dimensionless, $n_e \ge 0$)
• $f_{\ell}$: the fraction of such bodies on which life actually arises
• $f_i$: the fraction of life-bearing bodies on which life evolves all the way to intelligence
• $f_c$: the fraction of intelligent species that acquire a technology emitting signals detectable across space
• $L$: the length of time such a civilization keeps emitting signals (units: years)

Then

$$N = R_{*} \cdot f_p \cdot n_e \cdot f_{\ell} \cdot f_i \cdot f_c \cdot L$$

holds. This is called the Drake equation.

Tip

Let us check the units. $R_{*}$ is in “stars per year”, $L$ is in “years”, and everything else is dimensionless. The right-hand side therefore has units of (number per year) × (years) = number, matching the fact that $N$ on the left is a count. An equation whose units do not balance is wrong before you even consider its meaning.

flowchart TD
A["Stars born: R* per year"] --> B["Have planetary systems: fraction f_p"]
B --> C["Habitable bodies: n_e per system"]
C --> D["Life arises: fraction f_l"]
D --> E["Intelligence reached: fraction f_i"]
E --> F["Communication technology: fraction f_c"]
F --> G["Alive simultaneously: lifetime L years"]

The Drake equation reads as a funnel: it sieves the number of stars six times and leaves the number of civilizations alive right now

3.2. Why the lifetime $L$ appears as the last factor

The hardest part of the equation to swallow is that final $L$. Multiplying fractions together makes sense, but why does multiplying by a lifetime give “the number in existence right now”? This can be proved properly.

Proposition 3.2（A steady-state population equals birth rate times lifetime）

Suppose civilizations are born at a constant rate $R$ (units: per year), and that every civilization born emits signals for exactly $L$ years and then stops. If this situation has persisted for a long time $T \gg L$, then the average number $N$ of civilizations signalling at any given moment is

$$N = R \cdot L$$

Proof（Proposition 3.2）

Consider an observing window of length $T$. By definition, $RT$ civilizations are born during this window.

Each civilization signals for $L$ years, so a single civilization contributes $L$ years of “civilization alive” to the window. The total over the whole window is therefore

$$(RT) \times L = RTL \quad [\text{civilization} \cdot \text{years}]$$

(edge effects — the civilizations straddling the beginning and the end of the window — amount to a relative error of order $L/T$, which is negligible because $T \gg L$).

On the other hand, this same total is the integral over time of “the number alive at each instant”. Writing its time average as $N$, the total is $N \cdot T$. Equating the two,

$$N T = R T L \quad \Longrightarrow \quad N = R L$$

The part $R_{*} f_p n_e f_{\ell} f_i f_c$ of Definition 3.1 is precisely “how many communicating civilizations are newly born per year”. Identifying that with $R$, the Drake equation is exactly this proposition.

∎

This proposition is the same relation that queueing theory calls Little’s law. If two people per minute enter a noodle shop and each stays 15 minutes, there are on average 30 people inside. That is all there is to it. The last six factors of the Drake equation exist to estimate the “arrival rate”; the essential content is $N = RL$.

3.3. Putting in actual numbers

Example 3.3（Two calculations, optimistic and pessimistic）

The optimistic set. The present star formation rate of the Galaxy is roughly $1.5$ – $3$ solar masses per year, so take $R_{*} = 2$ per year. Exoplanet surveys show that stars with planets are the rule rather than the exception, so $f_p = 1$. Kepler statistics suggest that some $10$ – $20$ % of Sun-like stars have an Earth-sized planet in the habitable zone, so take $n_e = 0.2$. The rest is pure guesswork, but suppose life arises reasonably often once the conditions are met, $f_{\ell} = 0.1$; that the road to intelligence is narrower, $f_i = 0.1$; that a tenth of those take up radio, $f_c = 0.1$; and that the lifetime is $L = 10^{4}$ years.

$$\begin{aligned} N &= 2 \times 1 \times 0.2 \times 0.1 \times 0.1 \times 0.1 \times 10^{4} \\ &= 2 \times 0.2 \times 10^{-3} \times 10^{4} \\ &= 4 . \end{aligned}$$

Four in the Galaxy. This is the typical answer of the “aliens exist” camp.

The pessimistic set. Start from the same $R_{*} = 2$ and $f_p = 1$, but suppose that a “genuinely stable long-term environment” is rare, $n_e = 0.02$; that the origin of life is chemically very hard indeed, $f_{\ell} = 10^{-3}$; that the road from the eukaryotic cell to intelligence is narrower still, $f_i = 10^{-4}$; that $f_c = 0.1$; and that civilizations destroy themselves with their own technology after about $100$ years, $L = 100$ years.

$$\begin{aligned} N &= 2 \times 1 \times 0.02 \times 10^{-3} \times 10^{-4} \times 0.1 \times 100 \\ &= 4 \times 10^{-2} \times 10^{-7} \times 10^{1} \\ &= 4 \times 10^{-8} . \end{aligned}$$

One civilization per $25$ million galaxies. With about $2$ trillion galaxies in the observable universe, that gives some hundred thousand civilizations in the universe as a whole — and no chance whatsoever of our meeting one.

The same equation, both times with numbers that sound like “well, maybe”, gives $4$ and $4 \times 10^{-8}$: a difference of eight orders of magnitude. This is not an arithmetic error; it is the nature of the equation. The next section makes the reason explicit.

Remark 3.4

Drake himself said repeatedly that the equation is not a tool for producing an answer but “a way of organizing our ignorance”. Its value lies in having cut one enormous mystery, $N$, into smaller mysteries that separate disciplines can attack: astronomy ($R_{*}, f_p, n_e$), biology ($f_{\ell}, f_i$) and social science ($f_c, L$). And indeed, $f_p$ and $n_e$, completely unknown in 1961, are now pinned down observationally to within an order of magnitude. More than $5000$ exoplanets have been confirmed. From the left, the fog really is lifting.

4. The trap in a product: the mean is not the typical value

4.1. Uncertainties add in the logarithm

If seven quantities are multiplied together and each might be off by “oh, a factor of ten or so”, how far off is the product?

To measure the “order of magnitude” of a number we use the common logarithm. Taking $\log_{10}$ turns multiplication into addition, which simplifies matters.

Proposition 4.1（In logarithms, uncertainties propagate in quadrature）

Let $N = x_1 x_2 \cdots x_k$, where $x_1, \ldots, x_k$ are mutually independent positive random variables. Let $Y_i = \log_{10} x_i$ have mean $\mu_i$ and standard deviation $\sigma_i$. Then the mean and standard deviation of $Y = \log_{10} N$ are

$$E[Y] = \sum_{i=1}^{k} \mu_i, \qquad \mathrm{sd}[Y] = \sqrt{\sum_{i=1}^{k} \sigma_i^{2}}$$

In particular, if all the $\sigma_i$ equal a common value $\sigma$, then $\mathrm{sd}[Y] = \sigma\sqrt{k}$.

Proof（Proposition 4.1）

Repeated use of $\log_{10}(ab) = \log_{10}a + \log_{10}b$ gives

$$Y = \log_{10}(x_1 x_2 \cdots x_k) = \sum_{i=1}^{k} \log_{10} x_i = \sum_{i=1}^{k} Y_i$$

The mean (expectation) is linear on sums, that is, $E[A+B] = E[A]+E[B]$ holds without assuming independence, so $E[Y] = \sum_i E[Y_i] = \sum_i \mu_i$ follows.

For the variance, in the case of two variables

$$V[Y_1 + Y_2] = V[Y_1] + V[Y_2] + 2\,\mathrm{Cov}(Y_1, Y_2)$$

holds in general. If $Y_1$ and $Y_2$ are independent, the covariance $\mathrm{Cov}(Y_1,Y_2)$ vanishes, so $V[Y_1+Y_2] = V[Y_1]+V[Y_2]$. Applying this repeatedly to $k$ variables gives $V[Y] = \sum_i \sigma_i^{2}$. The standard deviation is the positive square root of the variance, so $\mathrm{sd}[Y] = \sqrt{\sum_i \sigma_i^2}$.

If all $\sigma_i = \sigma$, then $\sqrt{k\sigma^2} = \sigma\sqrt{k}$.

∎

Apply this to Definition 3.1. Here $k = 7$, and suppose each factor is uncertain by “a factor of ten”, that is, $\sigma_i = 1$ order of magnitude. Then

$$\mathrm{sd}[\log_{10} N] = \sqrt{7} \approx 2.65 \ \text{orders of magnitude}$$

The band of $\pm 1$ standard deviation — the perfectly ordinary range — already spans $2 \times 2.65 = 5.3$ orders of magnitude, a factor of $200{,}000$. The eight orders of magnitude in Example 3.3 arose because $f_{\ell}$ and $f_i$ are uncertain by far more than a single order; if anything, that example was restrained.

4.2. “Twenty on average” and “zero half the time” are compatible

There is something worse. When uncertain quantities are multiplied, the expectation is not the typical value.

Proposition 4.2（The arithmetic mean is at least the geometric mean）

Let $N$ be a random variable taking each of the positive values $N_1, N_2, \ldots, N_m$ with probability $1/m$. Then

$$E[N] \;=\; \frac{1}{m}\sum_{j=1}^{m} N_j \;\ge\; \left(\prod_{j=1}^{m} N_j\right)^{1/m} \;=\; 10^{\,E[\log_{10} N]}$$

holds, with equality if and only if $N_1 = N_2 = \cdots = N_m$.

Proof（Proposition 4.2）

First check the rightmost equality. Since $E[\log_{10}N] = \frac{1}{m}\sum_j \log_{10} N_j = \frac{1}{m}\log_{10}\left(\prod_j N_j\right)$, raising $10$ to both sides gives

$$10^{\,E[\log_{10}N]} = 10^{\frac{1}{m}\log_{10}(\prod_j N_j)} = \left(\prod_{j} N_j\right)^{1/m}$$

which is the geometric mean.

The remaining inequality is the AM–GM inequality itself. The case $m = 2$ can be verified directly: for $a, b > 0$,

$$\frac{a+b}{2} - \sqrt{ab} = \frac{a - 2\sqrt{ab} + b}{2} = \frac{(\sqrt{a}-\sqrt{b})^{2}}{2} \ \ge\ 0$$

with equality only when $\sqrt a = \sqrt b$, that is, $a = b$. The same inequality (AM–GM) is known to hold for general $m$, again with equality only when all the values coincide.

∎

Example 4.3（When the probability of life arising is a coin flip）

Take the optimistic set of Example 3.3 and treat $f_{\ell}$ alone as follows. The origin of life is a problem of chemistry, and we have no idea how hard it is. So suppose it is either “essentially certain once conditions allow ($f_{\ell} = 1$)” or “so hard that there is hardly another instance in the universe ($f_{\ell} = 10^{-10}$)”, each with probability one half.

The optimistic set used $f_{\ell} = 0.1$ and gave $N = 4$, and $N$ is proportional to $f_{\ell}$, so

$$N = \begin{cases} 4 \times \dfrac{1}{0.1} = 40 & (\text{probability } 1/2) \\[2mm] 4 \times \dfrac{10^{-10}}{0.1} = 4\times 10^{-9} & (\text{probability } 1/2) \end{cases}$$

The expectation is

$$E[N] = \frac{40 + 4\times10^{-9}}{2} \approx 20$$

But by Proposition 4.2 the geometric mean is

$$\sqrt{40 \times 4\times10^{-9}} = \sqrt{1.6\times10^{-7}} = 4\times10^{-4}$$

which is only one fifty-thousandth of the expectation. And looking inside, half the time the Galaxy contains nobody but us. The statement “the Galaxy contains on average 20 civilizations” and the statement “if the coin comes up tails we are utterly alone” came out of exactly the same model.

Remark 4.4

Sandberg, Drexler and Ord carried this argument through carefully for all seven factors in their 2018 paper “Dissolving the Fermi Paradox”. Instead of point estimates, they assigned each factor a probability distribution spanning the range the current scientific literature allows, ran millions of Monte Carlo samples, and obtained a probability of 53 – 99.6 % that we are alone in the Galaxy, and 39 – 85 % that we are alone in the observable universe. Their claim is not “there are no aliens” but “once uncertainty is handled properly, there was never much reason to be surprised by silence”. Half the paradox, on this view, was an illusion manufactured by the bad habit of using point estimates.

5. The paradox stated precisely: why “nobody comes” is the problem

We have now seen that a small $N$ is nothing to wonder at. So has Fermi’s question been answered? It has not. The genuinely sharp part of the paradox has nothing to do with radio at all.

5.1. The time it takes to cross the Galaxy

Picture a ship hopping from star to star. Each civilization sends ships to nearby systems; on arrival they gather resources, get established, and from there launch ships to the next systems. Let us compute how fast this wave spreads.

Proposition 5.1（The spreading time of a colonization wave）

Suppose a civilization spreads from star system to star system by the following rule.

• Each hop covers a distance $d$ (the mean distance to the neighbouring system).
• Ships travel at a constant speed $v$.
• After arriving at a new system, a preparation time $\tau$ elapses before the next ship departs.

Then the time $T$ for the wave to reach a point at distance $D$ from the starting point is

$$T = \frac{D}{d}\left(\frac{d}{v} + \tau\right) = \frac{D}{v} + \frac{D}{d}\,\tau$$

and the effective speed of the wavefront $v_{\mathrm{eff}} = D/T$ is

$$v_{\mathrm{eff}} = \frac{d}{\dfrac{d}{v} + \tau} = \frac{v}{1 + \dfrac{v\tau}{d}}$$

In particular, $v_{\mathrm{eff}}$ does not depend on $D$.

Proof（Proposition 5.1）

Dividing the distance $D$ by the hop length $d$, the number of hops required is $n = D/d$.

One hop takes the flight time $d/v$ plus the preparation time $\tau$ on arrival, so $d/v + \tau$. The hops occur in series one after another, so the total time for $n$ hops is

$$T = n\left(\frac{d}{v}+\tau\right) = \frac{D}{d}\left(\frac{d}{v}+\tau\right)$$

Expanding the bracket gives $T = D/v + (D/d)\tau$.

The effective speed is by definition $v_{\mathrm{eff}} = D/T$, so substituting the above $T$,

$$v_{\mathrm{eff}} = \frac{D}{\dfrac{D}{d}\left(\dfrac{d}{v}+\tau\right)} = \frac{d}{\dfrac{d}{v}+\tau}$$

Dividing numerator and denominator by $d$ gives $v_{\mathrm{eff}} = v/(1 + v\tau/d)$. Since $D$ does not appear in this expression, $v_{\mathrm{eff}}$ is a constant independent of distance.

The real spreading is spherical rather than along a line, but the wavefront advances by the same rule in every direction, so the arrival time measured radially from the centre is exactly the calculation above.

∎

Here is where it becomes interesting.

Corollary 5.2（However fast the ships, the waiting time sets the limit）

In the setting of Proposition 5.1, if $v\tau \gg d$ (equivalently $d/v \ll \tau$), then

$$v_{\mathrm{eff}} \approx \frac{d}{\tau}, \qquad T \approx \frac{D}{d}\,\tau$$

In particular, even in the limit $v \to \infty$ (instantaneous travel), $T$ never falls below $\dfrac{D}{d}\tau$.

Proof（Corollary 5.2）

In the expression $v_{\mathrm{eff}} = d/(d/v + \tau)$ from Proposition 5.1, the assumption $d/v \ll \tau$ lets us approximate the denominator by $\tau$. Hence $v_{\mathrm{eff}} \approx d/\tau$ and $T = D/v_{\mathrm{eff}} \approx (D/d)\tau$.

More precisely, in $T = D/v + (D/d)\tau$ the first term $D/v$ can be made as small as we like by increasing $v$, but the second term $(D/d)\tau$ contains no $v$ at all. So in the limit $v \to \infty$ we get $T \to (D/d)\tau$, and this is a lower bound.

∎

The significance of this corollary is considerable. No warp drive and no antimatter engine is required. The waiting time $\tau$ and the hop length $d$ alone fix a lower bound on the time to cross the Galaxy.

Example 5.3（What modest numbers give）

From the mean spacing of stars in the solar neighbourhood take $d = 10$ light-years; let the ships travel at 1 % of light speed, $v = 0.01c$; and allow $\tau = 1000$ years of preparation between arriving in a new system and launching the next ship. A thousand years means resting, every single time, for as long as it took humanity to get from the Middle Ages to the present. Taking the distance across the Galaxy as $D = 10^{5}$ light-years, Proposition 5.1 gives

$$\begin{aligned} \frac{D}{v} &= \frac{10^{5}\ \text{ly}}{0.01\ \text{ly/yr}} = 10^{7}\ \text{years}, \\ \frac{D}{d}\tau &= \frac{10^{5}}{10}\times 10^{3} = 10^{4}\times10^{3} = 10^{7}\ \text{years}, \end{aligned}$$

and therefore

$$T = 10^{7} + 10^{7} = 2\times10^{7}\ \text{years} = 20\ \text{million years}$$

The effective speed is $v_{\mathrm{eff}} = 10^{5}/(2\times10^{7}) = 5\times10^{-3}$ light-years per year, that is, $0.5$ % of the speed of light.

Moreover, by Corollary 5.2, making the ships infinitely fast cannot bring $T$ below $10^{7}$ years, or 10 million years. Conversely, stretching the preparation time to $\tau = 10^{4}$ years (twice the whole recorded history of humanity) still only gives $T = 10^{7} + 10^{8} \approx 1.1\times10^{8}$ years.

Compared with the age of the Galaxy, $1.3\times10^{10}$ years, $2\times10^{7}$ years is $0.15$ %. Even $1.1\times10^{8}$ years is only $0.85$ %.

The time to cross the Galaxy against the age of the Galaxy. The horizontal axis is a common linear scale.

5.2. Stating the paradox precisely

Definition 5.4（The Fermi paradox）

Consider the following four claims.

1. The Galaxy contains many stars billions of years older than the Sun, and most of them have planetary systems.
2. Even if the probability of life and intelligence arising is very small, there are $10^{11}$ stars, so we should expect at least one case in which a technological civilization appeared hundreds of millions to billions of years before us.
3. If interstellar travel is possible in principle, such a civilization can reach the whole Galaxy in $10^{7}$ to $10^{8}$ years, a tiny fraction of the age of the Galaxy (Proposition 5.1, Corollary 5.2).
4. Yet neither in the Solar System nor on Earth has any trace of a visit by an extraterrestrial civilization, nor any of its machines or structures, ever been found.

The situation that claims 1 through 4 cannot all hold at once — that one of the premises must be false — is called the Fermi paradox.

Put this way, it becomes clear where the substance of the paradox lies. The problem is not that “we hear no radio” but that nobody has come. Michael Hart (1975) and Frank Tipler (1980) pushed this argument hard. Tipler further pointed out that if a civilization launches machines capable of building copies of themselves (von Neumann probes), then even after the civilization itself has perished the probes keep multiplying and fill the Galaxy. One probe builds two, each of those builds two more; repeat this $40$ times and you have $10^{12}$ of them.

In short: if a single expansion-minded civilization arose even once, anywhere, during the 13 billion years of galactic history, its traces should be in the Solar System. That is the blade of the paradox.

6. What the silence means: how much have we actually listened to?

Before turning to the blade, let us fix with numbers how weak the “we hear nothing” evidence is. This is the part physics answers cleanly.

Definition 6.1（Detection horizon）

Let $P$ be the equivalent isotropically radiated power (EIRP) of a transmitter. This is the power of a hypothetical transmitter that radiates the same signal strength uniformly in all directions; it is larger than the actual transmitted power by whatever factor the antenna concentrates the beam.

If $S_{\min}$ (units: $\mathrm{W/m^2}$) is the smallest power flux density the receiver can detect, the maximum distance $r_{\max}$ at which this transmitter can be detected is called the detection horizon.

Proposition 6.2（The detection horizon scales as the square root of the power）

When absorption of radio waves in interstellar space is negligible, the detection horizon of Definition 6.1 is

$$r_{\max} = \sqrt{\frac{P}{4\pi S_{\min}}}$$

In particular, $r_{\max}$ is proportional to $\sqrt{P}$.

Proof（Proposition 6.2）

Consider a sphere of radius $r$ centred on the transmitter. By conservation of energy, in the absence of absorption the electromagnetic energy crossing this sphere per unit time equals $P$ regardless of distance. Since we have reduced everything to the isotropic equivalent, we may treat the flux as uniform over the sphere, whose area is $4\pi r^{2}$; the power crossing unit area (the power flux density) is therefore

$$S(r) = \frac{P}{4\pi r^{2}}$$

This is the inverse-square law（Proposition 2.2）［How We Know the Distance to a Star］. Detection requires $S(r) \ge S_{\min}$, that is,

$$\frac{P}{4\pi r^{2}} \ge S_{\min} \quad\Longleftrightarrow\quad r^{2} \le \frac{P}{4\pi S_{\min}} \quad\Longleftrightarrow\quad r \le \sqrt{\frac{P}{4\pi S_{\min}}} .$$

The last equivalence uses $r > 0$. The right-hand side is $r_{\max}$.

Note that in the $1$ – $10\ \mathrm{GHz}$ band both absorption by the interstellar medium and the galactic background are small, which is why this band is called the “cosmic window”. It is also why Cocconi and Morrison chose $1420\ \mathrm{MHz}$, and it makes neglecting absorption legitimate in this band.

∎

Example 6.3（How far does the Arecibo planetary radar reach?）

The $305\ \mathrm{m}$ dish of the Arecibo Observatory in Puerto Rico carried a powerful radar transmitter for observing asteroids. Its EIRP was about $P = 10^{13}\ \mathrm{W}$.

Assume a comparable telescope on the receiving end. Restricting to a narrow band (bandwidth of order $1\ \mathrm{Hz}$) and integrating for a long time gives a detection limit of roughly $S_{\min} = 10^{-26}\ \mathrm{W/m^2}$. From Proposition 6.2,

$$r_{\max} = \sqrt{\frac{10^{13}}{4\pi \times 10^{-26}}} = \sqrt{\frac{10^{13}}{1.26\times10^{-25}}} = \sqrt{7.96\times10^{37}} = 8.9\times10^{18}\ \mathrm{m} .$$

Dividing by $1$ light-year $= 9.46\times10^{15}\ \mathrm{m}$,

$$r_{\max} \approx 940\ \text{light-years} \approx 1000\ \text{light-years}$$

That is only 1 % of the $100{,}000$ light-year diameter of the Galaxy. In the scale model of section 2, the region within earshot of us is a circle of radius $1\ \mathrm{cm}$ on a disk one metre across.

Incidentally, the Arecibo telescope collapsed under its own weight in December 2020. Humanity does not currently possess even the transmitting capability used in this calculation.

Example 6.4（Is the Earth shouting into space?）

One often hears that “Earth’s television broadcasts have spread out to a radius of 90 light-years”. They have spread, but whether anyone can hear them is another matter.

The effective radiated power of a strong UHF television transmitter is at most about $P = 10^{7}\ \mathrm{W}$. Using Proposition 6.2 with the same $S_{\min} = 10^{-26}\ \mathrm{W/m^2}$,

$$r_{\max} = \sqrt{\frac{10^{7}}{1.26\times10^{-25}}} = \sqrt{7.96\times10^{31}} = 8.9\times10^{15}\ \mathrm{m} \approx 0.94\ \text{light-years}$$

It does not even reach one light-year. Put a telescope like ours at the nearest star, Proxima Centauri ($4.2$ light-years; how that distance is measured appears in the distance to the nearest star（Example 3.3）［How We Know the Distance to a Star］), and Earth’s television is inaudible. Worse, real television signals are broadband, so the sensitivity of narrowband searches does not apply, making the situation worse still.

So the intuition that “if aliens existed they would have noticed Earth’s radio long ago” is simply wrong. And human radio leakage is currently decreasing: terrestrial broadcasting is being replaced by optical fibre, and satellites by highly directional beams. If other civilizations follow the same path, the “time spent emitting radio” behind $f_c$ and $L$ may be startlingly short.

6.1. How close would they have to be for us to hear them?

Proposition 6.5（Typical spacing of civilizations scattered through the galactic disk）

Suppose $N$ civilizations are distributed roughly uniformly over a disk-shaped region of radius $R_g$. If the thickness of the disk is much smaller than the spacing between civilizations, the typical distance $\ell$ between neighbouring civilizations is

$$\ell \approx R_g\sqrt{\frac{\pi}{N}}$$

Proof（Proposition 6.5）

Since the thickness of the disk is much smaller than the spacing, we may treat the distribution as a two-dimensional problem. The area of the disk is $\pi R_g^{2}$. Shared among $N$ civilizations, the area allotted to each is

$$A_1 = \frac{\pi R_g^{2}}{N}$$

Regarding this area as a square of side $\ell$ gives $\ell^{2} = A_1$, that is,

$$\ell = \sqrt{\frac{\pi R_g^{2}}{N}} = R_g\sqrt{\frac{\pi}{N}}$$

For a completely random (two-dimensional Poisson) arrangement, the mean nearest-neighbour distance works out to $1/(2\sqrt{n}) = \ell/2$ in terms of the number density $n = N/(\pi R_g^{2})$. That is half the estimate above, but the order of magnitude is unchanged. In what follows we use $\ell$, conscious of this factor-of-two ambiguity.

∎

Corollary 6.6（The silence is exactly what we should expect）

Let $r_{\max}$ be the detection horizon and $R_g$ the radius of the galactic disk. For the nearest civilization to lie inside the detection horizon, that is, for $\ell \le r_{\max}$, it is necessary that

$$N \ \ge\ \frac{\pi R_g^{2}}{r_{\max}^{2}}$$

For $R_g = 5\times10^{4}$ light-years and $r_{\max} = 10^{3}$ light-years, this condition reads $N \ge 7.9\times10^{3}$: it requires about ten thousand civilizations to exist in the Galaxy simultaneously.

Proof（Corollary 6.6）

By Proposition 6.5, $\ell = R_g\sqrt{\pi/N}$. The condition $\ell \le r_{\max}$ is equivalent to

$$R_g\sqrt{\frac{\pi}{N}} \le r_{\max} \quad\Longleftrightarrow\quad \frac{\pi R_g^{2}}{N} \le r_{\max}^{2} \quad\Longleftrightarrow\quad N \ge \frac{\pi R_g^{2}}{r_{\max}^{2}}$$

(both sides are positive, so squaring does not reverse the inequality). Substituting the numbers,

$$N \ge \frac{\pi \times (5\times10^{4})^{2}}{(10^{3})^{2}} = \frac{\pi \times 2.5\times10^{9}}{10^{6}} = 7.9\times10^{3}$$

Including the Poisson correction mentioned in the proof of Proposition 6.5 (which halves $\ell$) gives $N \ge 2\times10^{3}$ or so; the order of magnitude is unchanged.

∎

Even the optimistic set of Example 3.3 gave only $N = 4$. In that case the distance to the nearest civilization, from Proposition 6.5, is

$$\ell = 5\times10^{4}\times\sqrt{\frac{\pi}{4}} = 5\times10^{4}\times0.886 \approx 4.4\times10^{4}\ \text{light-years}$$

which is more than $40$ times the detection horizon. A round-trip message would take $90{,}000$ years. Even if the optimists are right, hearing nothing is the normal outcome for us.

Remark 6.7

There is also research quantifying the coverage of the search itself. In a 2018 paper, Wright, Kanodia and Lubar treated the SETI search space as a multidimensional haystack — direction on the sky, frequency, sensitivity, polarization, modulation and time — and evaluated what fraction of its volume humanity has actually examined. Their conclusion was that the fraction searched is a tiny sliver of the whole, comparable to a bathtub of water set against all the oceans of Earth. Even in the data the Breakthrough Listen programme released in 2020, the number of nearby stars examined carefully was 1327. That is 1327 out of $10^{11}$ stars in the Galaxy.

The strong narrowband signal that the Big Ear telescope at Ohio State University caught for just $72$ seconds in 1977 (called the “Wow!” signal because its discoverer wrote that word in the margin of the printout) has never been seen again, however hard anyone has looked. What it was remains unknown.

7. Candidate “answers”

So one of the four claims in Definition 5.4 is false. Which? Dozens of solutions have been proposed, but they sort into four broad families.

7.1. Nobody is there in the first place

This position denies claim 2. In their 2000 book Rare Earth, Peter Ward and Donald Brownlee argued that while single-celled organisms may be commonplace, complex animals require an extraordinary number of conditions: a moon of the right size to stabilize the spin axis, Jupiter-class planets to sweep up comets, plate tectonics to run the carbon cycle, and a position in the Galaxy where radiation is not too intense and heavy elements not too scarce. If missing even one of these is fatal, then $n_e$ and $f_i$ become dramatically small.

Another version is the “we are first” hypothesis. The universe is still young. Sun-like stars live for $10$ billion years, but red dwarfs — 75 % of all stars — live for more than $10^{12}$ years. Most of the “time suitable for life” that the universe will ever experience has not yet arrived. In a 2016 paper, Loeb and collaborators evaluated the probability of life as a function of cosmic time and argued that it peaks around low-mass stars far in the future. On this view we are guests who arrived at the cosmic party far too early, while the others are still getting ready.

7.2. They were there, and they are gone

Definition 7.1（The Great Filter）

Consider the sequence of steps leading from inanimate matter to a technological civilization spread across the Galaxy: planet formation, the origin of life, complex cells, multicellularity, intelligence, technology, expansion into space. Given that the Fermi paradox is a fact, somewhere along this sequence there must be a barrier with an extremely small probability of being passed. This barrier is called the Great Filter.

When the filter lies at a step earlier than humanity’s present position, we say it is behind us; when it lies at a later step, that it is ahead of us.

The concept was introduced by Robin Hanson in 1998. The logic is simple. The fact the paradox reveals — that the Galaxy is empty — implies that some point along the path is extremely narrow. We do not know where the narrow point is, but there must be one.

Here we reach the famous and unsettling consequence Nick Bostrom stated in 2008: if independently originated life were found on Mars or Europa, it would be bad news for humanity. The reasoning goes as follows. If the origin of life ($f_{\ell}$) turns out to be easy, then the filter must lie after the origin of life, which raises the probability that it lies ahead of us. Conversely, if no second origin of life is found anywhere in the Solar System, the filter is more likely to be a barrier behind us, one we have already passed, and humanity’s future is bright. Astrobiology thereby becomes the strange discipline in which finding nothing is the reassuring outcome.

Caution

This is an argument about redistributing probabilities, not a claim that “Martian microbes will destroy humanity”. It is Bayesian reasoning about how the probability of a hypothesis $H$ is updated given an observation $E$. It is easily misread, so read it carefully.

Candidates commonly proposed for a filter ahead of us include nuclear war, irreversible destruction of the environment, artificial intelligence that escapes control, and engineered pathogens. In short, the claim is that $L$ is short. As Example 3.3 shows, $N$ is proportional to $L$, so whether the mean lifetime of a civilization is $100$ years or a million changes the answer by a factor of $10{,}000$. The rightmost factor of the Drake equation is a factor about ourselves.

7.3. They are there, and they are silent

This position rereads “no traces” in claim 4 as “traces are being kept from us”. The zoo hypothesis, proposed by John A. Ball in 1973, holds that we are treated like organisms in a nature reserve: observed, but with contact forbidden. David Brin organized and criticized hypotheses of this kind in a 1983 review.

A darker version is the dark forest hypothesis, made widely known by the novels of Liu Cixin. When you cannot know another party’s intentions, whoever reveals their position first is at a disadvantage; therefore every rational civilization stays silent. As a story it is superb; as a scientific hypothesis it has a weakness.

Danger

Both the zoo hypothesis and the dark forest hypothesis predict “not being found” as such, and are therefore consistent with any observational result whatsoever. An unfalsifiable hypothesis may happen to be true, but it does not move the scientific discussion forward. When examining these proposals, always ask: “if this were false, what observation should we obtain?”

7.4. They are there, and we have not noticed

This position blames claim 4 on the limits of our searching. As Example 6.3 and Corollary 6.6 show, this position is overwhelmingly strong where radio is concerned. We can listen only out to about 1 % of the diameter of the Galaxy, and within that we have examined only a few thousand stars carefully. If they use lasers rather than radio, or some method we have not imagined, our net never had a chance of catching them.

Against claim 3 of Definition 5.4, however, this position is powerless. The structures of a civilization that has filled the Galaxy would be far harder to hide.

7.5. The overall picture

Hypothesis	Core idea	Which part of the paradox it explains	Observations that could bear on it
Rare Earth	The conditions for complex life are extraordinarily strict	$n_e, f_{\ell}, f_i$ are extremely small	Searches for atmospheric biosignatures on exoplanets
We are first	The universe is young; the prime time for civilizations lies ahead	Explains small $N$ by timing	Theory of stellar lifetime distributions and cosmic chemical evolution
Great Filter (behind)	The origin of life or of eukaryotes is exceedingly rare	Why nobody comes	Searches for a second origin of life on Mars, Europa, Enceladus
Great Filter (ahead)	Technological civilizations are short-lived	Why $L$ is small	Our own history (nuclear weapons, climate, AI)
Zoo hypothesis	We are observed but contact is forbidden	Both the silence and the absence	Hard to falsify; little predictive power
Dark forest	Exposure is suicide, so nobody transmits	The silence	Searching for artefacts rather than transmissions
Poor searching	The search space is vast and mostly unexplored	The silence only	Systematic expansion of the searched volume

Laid out this way, the structure of the question becomes visible. There is no shortage of hypotheses explaining the silence, and most of them become unnecessary the moment one accepts Corollary 6.6. What genuinely demands explanation is the absence: the fact that a Galaxy which should fill up in $10^{7}$ years is empty. And the explanations for absence reduce, in practice, to three: “nobody is there in the first place”, “the Great Filter”, and “declining to expand is a universal choice”.

Fermi’s question is still open $76$ years later. But its shape has certainly grown sharper.

8. Exercises

Exercise 8.1Easy

In Definition 3.1, hold every factor except $L$ fixed.

(1) If $L$ changes from $100$ years to $10^{6}$ years, by what factor does $N$ change?

(2) For the optimistic set of Example 3.3 ($L = 10^{4}$ years, $N = 4$), find $N$ when $L = 100$ years.

Solution

(1) On the right-hand side of Definition 3.1, $L$ appears to the first power, so $N$ is proportional to $L$. Hence the factor is

$$\frac{10^{6}}{100} = 10^{4}$$

that is, $10{,}000$.

(2) By proportionality,

$$N = 4 \times \frac{100}{10^{4}} = 4\times10^{-2} = 0.04$$

which means one civilization per $25$ galaxies.

However optimistic our estimates of the astronomical and biological factors, if a civilization emits radio for only $100$ years then the Galaxy effectively contains nobody but us. That is why the rightmost factor of the Drake equation is said to be “a question about ourselves”.

Exercise 8.2Standard

In the setting of Proposition 5.1, take $D = 10^{5}$ light-years, hop length $d = 20$ light-years, preparation time $\tau = 5000$ years, and ship speed $v = 0.1c$.

(1) Find the crossing time $T$ and the effective speed $v_{\mathrm{eff}}$.

(2) Find $T$ in the limit of infinitely fast ships ($v \to \infty$) and compare with (1).

(3) What percentage of the age of the Galaxy, $1.3\times10^{10}$ years, is the $T$ from (1)?

Solution

(1) Substitute into $T = D/v + (D/d)\tau$ from Proposition 5.1. Since $v = 0.1c = 0.1$ light-years per year,

$$\frac{D}{v} = \frac{10^{5}}{0.1} = 10^{6}\ \text{years},$$$$\frac{D}{d}\tau = \frac{10^{5}}{20}\times 5000 = 5000 \times 5000 = 2.5\times10^{7}\ \text{years}.$$

Hence

$$T = 1.0\times10^{6} + 2.5\times10^{7} = 2.6\times10^{7}\ \text{years}$$

The effective speed is

$$v_{\mathrm{eff}} = \frac{D}{T} = \frac{10^{5}}{2.6\times10^{7}} = 3.8\times10^{-3}\ \text{light-years/year} = 0.0038\,c$$

The ships fly at 10 % of the speed of light, yet the wavefront advances at only 0.38 % of it.

(2) By Corollary 5.2, as $v \to \infty$ we get $T \to (D/d)\tau = 2.5\times10^{7}$ years. Compared with the $2.6\times10^{7}$ years of (1), that is a reduction of only $4$ %. In this setting the waiting time already dominates, and improving the ships’ performance is pointless.

(3)

$$\frac{2.6\times10^{7}}{1.3\times10^{10}} = 2.0\times10^{-3} = 0.2\ \%$$

The whole Galaxy fills up in $0.2$ % of its history. Put the other way round: if an expansion-minded civilization had appeared at any moment in the past 13 billion years, its traces should have persisted through $99.8$ % of that time.

Exercise 8.3Standard

Use Proposition 6.2, with the receiver’s detection limit fixed at $S_{\min} = 10^{-26}\ \mathrm{W/m^2}$.

(1) If the transmitter’s EIRP is increased by a factor of $100$, by what factor does the detection horizon $r_{\max}$ increase?

(2) Find the EIRP needed to reach the far side of the Galaxy ($r = 10^{5}$ light-years $= 9.46\times10^{20}\ \mathrm{m}$).

(3) Compare the value from (2) with humanity’s total primary energy consumption, $2\times10^{13}\ \mathrm{W}$. If the antenna gain is the same as Arecibo’s (EIRP is $10^{7}$ times the actual transmitted power), what actual transmitted power is required?

Solution

(1) By Proposition 6.2, $r_{\max} \propto \sqrt{P}$, so the factor is $\sqrt{100} = 10$. Distance is expensive: reaching $10$ times farther takes $100$ times the power.

(2) Solving $r_{\max} = \sqrt{P/(4\pi S_{\min})}$ for $P$ gives $P = 4\pi r_{\max}^{2} S_{\min}$. With $r_{\max} = 9.46\times10^{20}\ \mathrm{m}$,

$$r_{\max}^{2} = 8.95\times10^{41}\ \mathrm{m^2}, \qquad 4\pi r_{\max}^{2} = 1.12\times10^{43}\ \mathrm{m^2},$$$$P = 1.12\times10^{43} \times 10^{-26} = 1.1\times10^{17}\ \mathrm{W}.$$

(3) The ratio to humanity’s primary energy consumption is

$$\frac{1.1\times10^{17}}{2\times10^{13}} \approx 5.6\times10^{3}$$

about $5600$ times all the energy humanity uses. But this is the isotropic equivalent; concentrating the beam with an antenna reduces the actual transmitted power a great deal. With a gain of $10^{7}$,

$$P_{\text{actual}} = \frac{1.1\times10^{17}}{10^{7}} = 1.1\times10^{10}\ \mathrm{W} = 11\ \text{billion}\ \mathrm{W}$$

roughly the output of ten large nuclear power stations. Technically that is not an impossible figure.

One caveat, though. Because the beam is narrow, it illuminates only one very small direction of the sky at a time. Running a permanent beacon aimed at the whole Galaxy would require as many transmitters as there are directions. This is the source of SETI’s basic asymmetry: listening is far cheaper than transmitting. And if everyone does that calculation and everyone chooses to listen, the universe stays quiet.

Exercise 8.4Hard

Suppose fossil microbes are found beneath the surface of Mars, and it is certain that they arose independently of life on Earth.

(1) Which factor of Definition 3.1 does this inform us about?

(2) Is this discovery good news or bad news for humanity’s future? Argue using the terms “behind” and “ahead” from Definition 7.1.

Solution

(1) It informs us about $f_{\ell}$, the fraction of habitable bodies on which life actually arises. If life arose twice independently within a single planetary system, the natural conclusion is that $f_{\ell}$ is very large (close to $1$).

(2) It is bad news. Let us take it step by step.

First, what Definition 5.4 establishes is the observational fact that the Galaxy is not filled with expanding civilizations. By Definition 7.1, this fact means that somewhere along the road from inanimate matter to a galactic civilization there is an extremely narrow barrier. That the barrier exists somewhere is not negotiable.

Next, suppose we learn that $f_{\ell}$ is large. Then the step “the origin of life” was not the narrow barrier. Since the barrier is somewhere, by elimination the probability rises that it lies in one of the remaining steps. Those are eukaryotic cells, multicellularity, intelligence, technology, and the long-term survival of a technological civilization. Of these, the only one humanity has not yet passed is the last. Hence the probability that the filter lies ahead of us rises.

Conversely, if no second origin of life is found anywhere in the Solar System, the possibility remains that $f_{\ell}$ is extremely small. In that case the filter is behind us, a barrier humanity has already passed, and the probability that one waits ahead goes down. This is the substance of Bostrom’s claim that we should hope to find nothing.

Note that this argument is Bayesian reasoning about how observations update probabilities; it does not mean that Martian microbes are themselves dangerous. Note also that it presupposes Definition 5.4 as a fact. As Corollary 6.6 shows, the “silence” half of the paradox is adequately explained by insufficient searching, so the argument really bites only on the “absence” half.

References

1. G. Cocconi and P. Morrison, “Searching for Interstellar Communications”, Nature 184 (1959), 844–846. — The two-page original paper proposing $1420\ \mathrm{MHz}$.
2. E. M. Jones, “Where Is Everybody?” An Account of Fermi’s Question, Los Alamos National Laboratory report LA-10311-MS (1985). — A primary source collecting interviews with those present at the 1950 lunch.
3. M. H. Hart, “An Explanation for the Absence of Extraterrestrials on Earth”, Quarterly Journal of the Royal Astronomical Society 16 (1975), 128–135. — The paper that made the paradox rigorous via colonization timescales.
4. S. Webb, If the Universe Is Teeming with Aliens … WHERE IS EVERYBODY? Seventy-Five Solutions to the Fermi Paradox and the Problem of Extraterrestrial Life, 2nd ed., Springer, 2015. — A comprehensive catalogue of the proposed solutions.
5. A. Sandberg, E. Drexler and T. Ord, “Dissolving the Fermi Paradox” (2018), arXiv:1806.02404. — Treats the uncertainties of the Drake equation as distributions.
6. J. T. Wright, S. Kanodia and E. Lubar, “How Much SETI Has Been Done? Finding Needles in the n-Dimensional Cosmic Haystack”, The Astronomical Journal 156 (2018), 260. arXiv:1809.07252 — Quantifies the coverage of the search.

Appendix: Searching galaxy by galaxy

The idea of looking for traces. Radio is emitted only when a civilization decides to transmit. But there is something a civilization cannot help emitting once it grows large: heat. By the second law of thermodynamics, energy that has been used is finally discarded as infrared radiation. This was Freeman Dyson’s point in 1960: a civilization exploiting stellar energy on a large scale should appear anomalously bright in the mid-infrared.

The Kardashev scale. In 1964, Nikolai Kardashev proposed classifying civilizations by the scale of the energy they use. Type I uses all the starlight reaching its planet (about $10^{16}\ \mathrm{W}$), Type II the entire output of its star (about $10^{26}\ \mathrm{W}$), and Type III the output of a whole galaxy (about $10^{37}\ \mathrm{W}$). Humanity has not yet reached Type I; at about $10^{13}\ \mathrm{W}$ we sit around Type $0.7$.

And they were not there. If a Type III civilization existed, its galaxy should stand out in catalogues as faint in visible light and anomalously bright in the mid-infrared. In 2015, Griffith, Wright and collaborators examined about 100,000 galaxies from the catalogue of the WISE infrared satellite and reported finding not one galaxy that reprocesses the bulk of its starlight into the mid-infrared. Their limit was that no galaxy re-radiates more than $85$ % of its starlight.

The Fermi paradox is therefore not confined to our own Galaxy. In none of 100,000 galaxies is there a civilization that has remade its galaxy. The calculation in Proposition 5.1 concerned the interior of the Galaxy, but 13 billion years is also enough to cross the intergalactic distances of a few million light-years. The scale of the question has grown far beyond what Fermi had in mind over lunch.