We report on high-field electron spin resonance (ESR) studies of magnetic excitations in the spin-1/2 triangular-lattice antiferromagnet Cs$_2$CuBr$_4$. Frequency-field diagrams of ESR excitations are measured for different orientations of magnetic fields up to 25 T. We show that the substantial zero-field energy gap, $Deltaapprox9.5$ K, observed in the low-temperature excitation spectrum of Cs$_2$CuBr$_4$ [Zvyagin $et~al.$, Phys. Rev. Lett. 112, 077206 (2014)], is present well above $T_N$. Noticeably, the transition into the long-range magnetically ordered phase does not significantly affect the size of the gap, suggesting that even below $T_N$ the high-energy spin dynamics in Cs$_2$CuBr$_4$ is determined by short-range-order spin correlations. The experimental data are compared with results of model spin-wave-theory calculations for spin-1/2 triangle-lattice antiferromagnet.
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