  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.
