  Sunyaev-Zel'dovich (SZ) cluster surveys are considered among the most
promising methods for probing dark energy up to large redshifts. However, their
premise is hinged upon an accurate mass-observable relationship, which could be
affected by the (rather poorly understood) physics of the intracluster gas. In
this letter, using a semi-analytic model of the intracluster gas that
accommodates various theoretical uncertainties, I develop a Fundamental Plane
relationship between the observed size, thermal energy, and mass of galaxy
clusters. In particular, I find that M ~ (Y_{SZ}/R_{SZ,2})^{3/4}, where M is
the mass, Y_{SZ} is the total SZ flux or thermal energy, and R_{SZ,2} is the SZ
half-light radius of the cluster. I first show that, within this model, using
the Fundamental Plane relationship reduces the (systematic+random) errors in
mass estimates to 14%, from 22% for a simple mass-flux relationship. Since
measurement of the cluster sizes is an inevitable part of observing the SZ
clusters, the Fundamental Plane relationship can be used to reduce the error of
the cluster mass estimates by ~ 34%, improving the accuracy of the resulting
cosmological constraints without any extra cost. I then argue why our
Fundamental Plane is distinctly different from the virial relationship that one
may naively expect between the cluster parameters. Finally, I argue that while
including more details of the observed SZ profile cannot significantly improve
the accuracy of mass estimates, a better understanding of the impact of
non-gravitational heating/cooling processes on the outskirts of the
intracluster medium (apart from external calibrations) might be the best way to
reduce these errors.
