Observation of even a single massive cluster, especially at high redshift, can falsify the standard cosmological framework consisting of a cosmological constant and cold dark matter ($\Lambda \mathrm{CDM}$) with Gaussian initial conditions by exposing an inconsistency between the well-measured expansion history and the growth of structure it predicts. Through a likelihood analysis of current cosmological data that constrain the expansion history, we show that the $\Lambda \mathrm{CDM}$ upper limits on the expected number of massive, distant clusters are nearly identical to limits predicted by all quintessence models where dark energy is a minimally coupled scalar field with a canonical kinetic term. We provide convenient fitting formulas for the confidence level at which the observation of a cluster of mass $M$ at redshift $z$ can falsify $\Lambda \mathrm{CDM}$ and quintessence given cosmological parameter uncertainties and sample variance, as well as for the expected number of such clusters in the light cone and the Eddington bias factor that must be applied to observed masses. By our conservative confidence criteria, which equivalently require masses 3 times larger than typically expected in surveys of a few hundred square degrees, none of the presently known clusters falsify these models. Various systematic errors, including uncertainties in the form of the mass function and differences between supernova light curve fitters, typically shift the exclusion curves by less than 10% in mass, making current statistical and systematic uncertainties in cluster mass determination the most critical factor in assessing falsification of $\Lambda \mathrm{CDM}$ and quintessence.

• Received 3 November 2010

DOI:https://doi.org/10.1103/PhysRevD.83.023015