Effects of interactions on the critical temperature of a trapped Bose gas.

We perform high-precision measurements of the condensation temperature of a harmonically trapped atomic Bose gas with widely tunable interactions. For weak interactions we observe a negative shift of the critical temperature in excellent agreement with mean-field theory. However for sufficiently strong interactions we clearly observe an additional positive shift, characteristic of beyond-mean-field critical correlations. We also discuss nonequilibrium effects on the apparent critical temperature for both very weak and very strong interactions.