Quantifying the Fermi paradox via passive SETI: a general framework

In this paper we consider the extent to which a lack of observations from SETI may be used to quantify the Fermi paradox. Building on previous research, we construct a geometrical model to compute the probability of at least one detection of an extraterrestrial electromagnetic (EM) signal of galactic origin, as a function of the number $N$ of communicative civilizations. We show how this is derivable from the probability of detecting a single signal; the latter is $\approx 0.6 \delta/R$, where $\delta$ is the distance between the initial and final EM signals and $R$ is the radius of the Milky Way, for $\delta/R \ll 1$. We show how to combine this analysis with the Drake equation $N = \mathscr{N} \delta /c$, where $c$ is the speed of light; this implies, applying a simplified toy model as an example, that the probability of detecting at least one signal is $>99 \%$ for $\delta / c \gtrsim 10^{2.8}$ years, given that $\mathscr{N} = 1$. Lastly, we list this toy model's significant limitations, and suggest ways to ameliorate them in more realistic future models.