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. We compute the probability of at least one detection of an extraterrestrial electromagnetic (EM) signal of Galactic origin, as a function of the number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>N</mml:mi> </mml:math> of communicative civilizations. We show how this is derivable from the probability of detecting a single signal; the latter is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo>≈</mml:mo> <mml:mn>0.6</mml:mn> <mml:mi>δ</mml:mi> <mml:mi>/</mml:mi> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>δ</mml:mi> </mml:math> is the distance between the initial and final EM signals and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>R</mml:mi> </mml:math> is the radius of the Milky Way, for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>δ</mml:mi> <mml:mi>/</mml:mi> <mml:mi>R</mml:mi> <mml:mo>≪</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . We show how to combine this analysis with the Drake equation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mi>𝒩</mml:mi> <mml:mi>δ</mml:mi> <mml:mi>/</mml:mi> <mml:mi>c</mml:mi> </mml:mrow> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>c</mml:mi> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo>&gt;</mml:mo> <mml:mn>99</mml:mn> <mml:mi>%</mml:mi> </mml:mrow> </mml:math> for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>δ</mml:mi> <mml:mi>/</mml:mi> <mml:mi>c</mml:mi> <mml:mo>≳</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>2.8</mml:mn> </mml:msup> </mml:mrow> </mml:math> years, given that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>𝒩</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . Lastly, we list this toy model’s significant limitations, and suggest ways to ameliorate them in more realistic future models.