Quantifying the Fermi paradox via passive SETI: a general framework

TL;DR

This work introduces a geometric, two-dimensional framework to quantify the Fermi paradox by linking the probability of detecting at least one extraterrestrial signal to the Drake equation. By modeling the Milky Way as a disk with civilizations at various distances and signal fronts expanding at light speed, the paper derives a per-source detection probability $P_i$, which in the small-$\delta/R$ limit is approximately $P_i \approx 0.6\,\delta/R$. Aggregating across $N$ civilizations via $\mathcal{P} = 1 - \prod_{i=1}^N (1 - P_i)$, and adopting a homogeneous $P_i = P$, the authors show how null SETI results constrain the Drake parameter space through $N = \mathscr{N} l$, yielding actionable bounds on $l$ for given $\mathcal{P}$ and $\mathscr{N}$. A toy example indicates that with $\mathscr{N} \approx 1$, achieving $\mathcal{P} \gtrsim 0.99$ requires $l \gtrsim 10^{2.8}$ years, illustrating how null results can inform the prevalence and longevity of EM-emitting civilizations. The paper also candidly discusses significant limitations (e.g., 2D geometry, attenuation effects, and emission intermittency) and outlines concrete paths for making the framework more realistic and constraining the $(\mathscr{N},l)$ space more tightly in future work.

Abstract

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 $\approx 0.6 δ/R$, where $δ$ is the distance between the initial and final EM signals and $R$ is the radius of the Milky Way, for $δ/R \ll 1$. We show how to combine this analysis with the Drake equation $N = \mathscr{N} δ/c$, 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 $δ/ c \gtrsim 10^{2.8}$ 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.

Figures (6)

• Figure 1: This figure conceptually describes the geometrical model used in this paper. The essential idea is that the greater the number of years during which a civilization has emitted signals, the greater the likelihood of observing those signals. This likelihood is proportional to the signal area, which is denoted with yellow diamonds. Here, the signal is being sent from a source "$S$", which is a distance "$r_0$" from the center of the MW. The first and last signals have radii $r_B$ and $r_A$. Thus, $l$ in Equation \ref{['Drake']} can be connected to the probability $\mathcal{P}$ of observing at least one signal.
• Figure 2: Here, we depict $\log_{10} N$ vs. $\log_{10} P$ (Equation \ref{['PNrel']}) in the homogeneous probability model, for $\mathcal{P} = 0.1$ (in red), $0.5$ (in blue), and $0.99$ (in black). These curves are overlayed with the small-$P$ approximation Equation \ref{['PNrelSmallP']}, shown in green dashed lines, for each $\mathcal{P}$.
• Figure 3: Here, we plot $\log_{10} \mathcal{N}$ vs. $\log_{10} l$ (Equation \ref{['curlyN-l']}) for $\mathcal{P} = 0.99$, $0.5$, and $0.01$ (blue, orange, and green, respectively). These curves assume that $\ll 1$. We also plot $\mathcal{N} \approx 0.1$ and $\mathcal{N} \approx 1$ (magenta and red horizontal lines, respectively). The rightmost part of the figure corresponds to $\log_{10} l = 4$, implying that $\log_{10} \left( \frac{1}{2} \cdot 10^5 \right) = 4 \Rightarrow < 1/5$ in this figure.
• Figure 4: Here, we plot $\log_{10} l$ vs. $\mathcal{P}$, as per Equation \ref{['curlyNequals1Curve']}. Thus, here we presume that $\mathcal{N} \approx 1$. This shows that, given the limitations of our toy model, it is unlikely that there are about $10^{2.8}$ civilizations or more unless $\mathcal{N}$ is considerably smaller than $1$.
• Figure 5: Here, we depict the basic geometry of our model. The MW galaxy has a radius "R". The two light-fronts, emanating from an extraterrestrial source "S", are labelled "A" and "B". The source is a distance $r_0$ from the center of the MW galaxy, and $\tilde{r}_A$ is the distance from the origin to a point on light-front A. The distance of the Earth from the center of the Milky Way is labeled "$r_E$", and this figure depicts the condition defined by Equation \ref{['Ptestdef']}.