at condensation threshold is $P_{{\mathrm{th}}} = 79~{\mathrm{W}}{\mathrm{cm}}^{-2}$. Note that the determination of the excitation power density at threshold is quite complex, e.g. due to the coexistence of intense emission from uncondensed polaritons for $P \gtrsim {P_{{\mathrm{th}}}}$, but significant for the comparison with theoretical calculations discussed in Sec. \[sec:fit\_of\_exp\_data\]. Details for the experimental determination of ${P_{{\mathrm{th}}}}$ can be found in the Supplemental Material, Sec. \[sec:SM\_Pth\].

In all cases investigated here, the condensate emission is distributed dispersion-less at horizontal lines in $k$-space with maximum intensity between the LPB dispersion, which is visible in the far-field PL images (cf. Fig. \[fig:Disorder\]) for low excitation power $P \gtrsim {P_{{\mathrm{th}}}}$. This indicates a weak expansion of the condensed polaritons due to the background potential induced by the excitation spot, whose size is similar or even larger than the polariton propagation length. [@Franke.2012; @Wouters.2008] For the lowest excitation power shown here, $P = P_{{\mathrm{th}}}$, the emission intensity from the uncondensed polaritons and the condensate are of same order which prevents a clear distinction. With increasing excitation power the BEC states undergo a blueshift due to the increasing interaction potential, and we observe several states with different energy. Previous studies in the literature on this multimode behavior 