Abstract
One of the fundamental hypotheses in observational cosmology is the validity of the so-called cosmic distance-duality relation. In this paper, we perform Monte Carlo simulations based on the method developed in Holanda, Gon\ifmmode \mbox{\c{c}}\else \c{c}\fi{}alves, and Alcaniz [J. Cosmol. Astropart. Phys. 06 (2012) 022] to answer the following question: what is the number of galaxy clusters observations ${N}_{\text{crit}}$ needed to check the validity of this relation at a given confidence level? At $2\ensuremath{\sigma}$, we find that ${N}_{\text{crit}}$ should be increased at least by a factor of 5 relative to the current sample size if we assume the current observational uncertainty ${\ensuremath{\sigma}}_{\text{obs}}$. Reducing this latter quantity by a factor of 2, we show that the present amount of data would be already close to the required number to check the validity of the cosmic distance-duality relation at $2\ensuremath{\sigma}$.
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