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\begin{document}

\title{Are we alone in the Universe? The Fermi Paradox}


\author[ ]{Matteo Cantiello}

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\textbf{Previous \href{https://www.authorea.com/users/2/articles/24715/_show_article}{"Astrobiology"}
 -- Next \href{https://www.authorea.com/users/2/articles/28104/_show_article
}{"Interactive Drake Equation"}}\\

With an estimated diameter of 93 billion light years and age of 13.7 billion years, our Universe is an astonishingly big place that's been around for a very long time. When you look up, you only get a short glimpse at a fraction of the hundreds of billions of stars that populate our Galaxy (which in turn is one of hundreds of billions in the cosmos), but it's enough to make you wonder: "Are we alone?" 
In the \href{https://www.authorea.com/users/2/articles/24715/_show_article}{previous post} we discussed the likelihood of the emergence of (intelligent) extraterrestrial life. Starting from the famous \href{https://www.authorea.com/11048}{Drake Equation} and using recent findings in astrophysics and some \href{https://www.authorea.com/24715}{astrobiology} arguments, we obtained a simple way to estimate $N$, the number of communicative civilizations in our Galaxy. This reduces to the product of the chance of emergence of intelligent life $f_i$ and the longevity $L$ (in years)  of a civilization's communicative phase: 

\begin{equation}\label{eq:Drake_simplified}
N \approx \,  \frac{1}{4}\, f_i \,  L \,. 
\end{equation}

  

Now we have some important observational constraints: we do not see alien spaceships landing on Earth and we have not detected E.T. signals coming from outer-space. \href{http://en.wikipedia.org/wiki/Search_for_extraterrestrial_intelligence}{SETI}? No signals. In \href{http://www.scientificamerican.com/article/alien-supercivilizations-absent-from-100-000-nearby-galaxies/}{recent news}, a space survey of 100,000 galaxies didn't find any clear sign of advanced alien civilizations. The observations could only rule out the presence of massive galactic colonization, however, with aliens using an amount of energy comparable to the total output of their galaxy \cite{Griffith_2015}. Advanced aliens might be more energy savvy, but 100,000 is also a lot of galaxies. 

In the words of Italian Nobel Laureate Enrico Fermi: "Where is everybody?" What Fermi meant is it's quite surprising we have seen no sign of  extraterrestrial intelligence, despite the fact the Universe is so vast and long-lived. This is the essence of the \textbf{Fermi Paradox}.

At this point I often hear saying: "Wait, but what about \href{http://en.wikipedia.org/wiki/Roswell_UFO_incident}{Roswell}, the \href{http://en.wikipedia.org/wiki/Wow!_signal}{WOW signal}, all those \href{http://en.wikipedia.org/wiki/List_of_reported_UFO_sightings}{UFO sightings}...?" I am not going into that. I will just state that the most economical explanations for the aforementioned stories have nothing to do with E.T. and that at this time there is no clear evidence  proving we had contact with alien life. Let's use \href{http://bit.ly/KDLFp6}{Occam's Razor} and throw away the conspiracy spoon.

Back to the main topic, the Fermi Paradox suggests that the number of communicative civilizations  $N$ in the Galaxy is small\footnote{Some peculiar solutions to the Fermi Paradox do not require $N$ to be small, see this \href{http://waitbutwhy.com/2014/05/fermi-paradox.html}{post} for a nice discussion}. Our revised version of the Drake Equation then implies two\footnote{Of course the third option is that both $f_i$ and $L$ are small} interesting alternatives: \begin{enumerate}
\item $f_i$ is a small number. \textbf{ Life is common in the Universe, but intelligent life is not}.
\item $L$ is a small number. \textbf{Intelligent life does not spend much time in the communicative phase}.
\end{enumerate}




\section*{Option 1: Intelligent life is rare}
On Earth, it took more than 3 billion years for life to evolve from single-celled bacteria to Homo Sapiens. This is a long time compared to the emergence of "simple" life-forms, an argument often used to conclude that $f_i \ll 1$. Scientists P. Ward and D. Brownlee in their book \href{http://bit.ly/1GGQt5B}{"Rare Earth"} claim that "Intelligent life on Earth relied on so many unlikely accidents that we are probably alone in the Universe". 

Is there anybody out there? --\textit{ silence} --

\section*{Option 2: The Lifetime of Communicative Civilizations is short}
The other option is intelligent life is common, but the time an advanced civilization spends reaching out to potential galactic neighbors is short.  There could be all sorts of reasons for that, including transition to more efficient forms of communication beyond electromagnetic signals, singularity, and a loss of interest in exploration. However  self-annihilation through nuclear war or the exhaustion of natural resources definitely come to mind as viable options. Therefore, in this scenario the absence of contact tells us something important about the future of our own civilization, i.e. that there should be an important transition for our species happening in a short timescale $L$. We can't say for sure what will happen, beside that radio silence will follow. But we can estimate \textbf{when} that will happen. Below I show the date is not too far away: $L_{\rm Humankind} \approx L <$  \textbf{1800 years}\footnote{this number depends on some bold assumptions and should be considered  just a  ballpark estimate}. 

So in this scenario, the real message is the absence of a message. And it whispers "You're next". Scary.\selectlanguage{english}
\begin{figure}[h!]
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\includegraphics[width=1.00\columnwidth]{figures/Lcritical/Lcritical}
\caption{{\textit{Visualization of "spheres of communication" around the location of galactic civilizations. Communication is possible only if blue dots (representing host planets of communicative civilizations) are contained in at least two spheres. Absence of contact implies we could be living in the left scenario, where $L<L_c$.}%
}}
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\end{figure}

\subsection*{The real message is the absence of a message}
Let's define $n=N/V$, the number density of communicative civilizations in the Galaxy, where $N$ is the total number of such civilizations  existing at any given time in the Milky Way and $V$ is the total galactic volume. Assuming a homogeneous distribution across the Galaxy, the average separation $r$ between two neighbor communicative civilizations is $r\sim n^{-1/3}$. In order for two civilization to communicate, their separation  must necessarily  be smaller than their communicative lifetime multiplied by the speed of light $c$. This is assuming the latter is the maximum communication speed achievable. We can then ask what would be the minimum value for $L$ such that communication can occur (we call this value $L_c$). That is, we impose $r=L$, so that $L\approx (N/V)^{-1/3}$. Now we can use equation \ref{eq:Drake_simplified} 
to derive the minimum lifetime of the communicative phase in order for contact to occur:
\begin{align*}
L_c \sim \left[\frac{4\,V}{ f_i}\right]^{1/4}.
\end{align*}
%\begin{equation}
%L_c \sim \left[\frac{V}{2\,f_l\, f_i\, f_c}\right]^{1/4}.
%\end{equation}

Since we are discussing the second hypothesis, $f_i$ is not a small number. Let's assume, as Frank Drake did in his estimates, that $f_i=1$. Once life starts on a planet, one life-form always evolves to become intelligent. We obtain
%On the other hand our discussion of the Drake equation led to the conclusion that, roughly speaking, $N \sim L$ %(meaning the denominator in previous equation is $\sim$ 1) (but see the \href{http://en.wikipedia.org/wiki%%/Rare_Earth_hypothesis}{rare Earth hypothesis} for a different conclusion). Therefore we can further simplify and %obtain   
\begin{align*}
%L_c \sim \left(\frac{V}{L_c}\right)^{1/3} = V^{1/4}
L_c \sim (4V)^{1/4},
\end{align*}

which states that at any given time, the minimum value for the communicative phase lifetime required for contact to occur scales with the volume of the galaxy to the power 1/4. This is also the scaling for the maximum number of communicative civilizations in the Galaxy. The absence of a message implies $L<L_c \sim (4V)^{1/4}$. 

The galactic volume can be estimated as $\approx 10^{13}$ cubic light years: therefore in our Galaxy $L<L_c \approx 1800$ years.  \textbf{The lifetime in the communicative phase of galactic civilizations is smaller than a couple thousand years}.
It also means that in our Galaxy there should be currently less than $N\approx 450$ communicative civilizations.

Note that the calculations do not include "relic communications", messages received by a communicative civilization, but sent by a no longer communicative one. So the estimate above is an upper limit to $L$ and $N$. 
%Doomsday (or singularity) could happen even sooner.

\textbf{Previous \href{https://www.authorea.com/users/2/articles/24715/_show_article}{"Astrobiology"}
 -- Next \href{https://www.authorea.com/users/2/articles/28104/_show_article
}{"Interactive Drake Equation"}}


%and the lifetime of such civilizations is also usually shorter than 1000\...3000 years. This does not necessaril%y mean that the number of intelligent civilizations in the Galaxy is limited to this number, it just implies %that their communicative phase is limited to a short time. There could be different reasons for this, including %transition to more efficient forms of communication than electromagnetic signals, loss of interest, singularity %etc. (many of these possibilities have been extensively discussed in the context of the \href{http://en.wikipedi%a.org/wiki/Fermi_paradox}{Fermi paradox}). Nevertheless self-annhilation / destruction of natural resources is a %realistic possibility. Overall the absence of contact tells us that there should be an important transition in %the characteristic of human civilization occuring in a timescale shorter than about 1000\...2000 years.

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