How Much SETI Has Been Done? Finding Needles in the *n*-Dimensional Cosmic Haystack

Many articulations of the Fermi Paradox have as a premise, implicitly or explicitly, that humanity has searched for signs of extraterrestrial radio transmissions and concluded that there are few or no obvious ones to be found. Tarter et al. (2010) and others have argued strongly to the contrary: bright and obvious radio beacons might be quite common in the sky, but we would not know it yet because our search completeness to date is so low, akin to having searched a drinking glass's worth of seawater for evidence of fish in all of Earth's oceans. Here, we develop the metaphor of the multidimensional "Cosmic Haystack" through which SETI hunts for alien "needles" into a quantitative, eight-dimensional model, and perform an analytic integral to compute the fraction of this haystack that several large radio SETI programs have collectively examined. Although this model haystack has many qualitative differences from the Tarter et al. (2010) haystack, we conclude that the fraction of it searched to date is also very small: similar to the ratio of the volume of a large hot tub or small swimming pool to that of the Earth's oceans. With this article, we provide a Python script to calculate haystack volumes for future searches and for similar haystacks with different boundaries. We hope this formalism will aid in the development of a common parameter space for the computation of upper limits and completeness fractions of search programs for radio and other technosignatures.