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Gravitational-wave observations by the Laser InterferometerGravitational-Wave Observatory (LIGO) and Virgo have provided us a new tool toexplore the universe on all scales from nuclear physics to the cosmos and havethe massive potential to further impact fundamental physics, astrophysics, andcosmology for decades to come. In this paper we have studied the sciencecapabilities of a network of LIGO detectors when they reach their best possiblesensitivity, called A#, and a new generation of observatories that are factorof 10 to 100 times more sensitive (depending on the frequency), in particular apair of L-shaped Cosmic Explorer observatories (one 40 km and one 20 km armlength) in the US and the triangular Einstein Telescope with 10 km arms inEurope. We use a set of science metrics derived from the top priorities ofseveral funding agencies to characterize the science capabilities of differentnetworks. The presence of one or two A# observatories in a network containingtwo or one next generation observatories, respectively, will provide goodlocalization capabilities for facilitating multimessenger astronomy andprecision measurement of the Hubble parameter. A network of two Cosmic Explorerobservatories and the Einstein Telescope is critical for accomplishing all theidentified science metrics including the nuclear equation of state,cosmological parameters, growth of black holes through cosmic history, and makenew discoveries such as the presence of dark matter within or around neutronstars and black holes, continuous gravitational waves from rotating neutronstars, transient signals from supernovae, and the production of stellar-massblack holes in the early universe. For most metrics the triple network of nextgeneration terrestrial observatories are a factor 100 better than what can beaccomplished by a network of three A# observatories.

Characterizing Gravitational Wave Detector Networks: From A$^\sharp$ to Cosmic Explorer