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\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 $L$ 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 $r=Lc$, $r=L\,c$,  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}.