We performed positron annihilation spectroscopy measurements on uranium dioxide irradiated with 45 MeV $\alpha$particles. The positron lifetime was measured as a function of the temperature in the 15-300 K range. The experimental results were combined with electronic structure calculations of positron lifetimes of vacancies and vacancy clusters in UO$_2$. Neutral and charged defects consisting of from one to six vacancies were studied computationally using the DFT + $U$ method to take into account strong correlations between the 5$f$ electrons of uranium. The two-component density functional theory with two different fully self-consistent schemes was used to calculate the positron lifetimes. All defects were relaxed taking into account the forces due to the creation of defects and the positron localized in the vacancy. The interpretation of the experimental observations in the light of the DFT + $U$ results and the positron trapping model indicates that neutral V$_U$ + 2V$_O$ trivacancies (bound Schottky defects) are the predominant defects detected in the 45 MeV $\alpha$ irradiated UO$_2$ samples. Our results show that the coupling of a precise experimental work and calculations using carefully chosen assumptions is an effective method to bring further insight into the subject of irradiation induced defects in UO$_2$ .