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We investigate a cosmological scenario in which the dark matter particles can be created during the evolution of the Universe. By regarding the Universe as an open thermodynamic system and using non-equilibrium thermodynamics, we examine the mechanism of gravitational particle production. In this setup, we study the large-scale structure (LSS) formation of the Universe in the Newtonian regime of perturbations and derive the equations governing the evolution of the dark matter overdensities. Then, we implement the cosmological data from Planck 2018 CMB measurements, SNe Ia and BAO observations, as well as the SH0ES local measurement for $H_0$ to provide some cosmological constraints for the parameters of our model. We see that the best case of our scenario ($\chi_{{\rm tot}}^{2}=3834.40$) fits the observational data better than the baseline $\Lambda$CDM model ($\chi_{{\rm tot}}^{2} = 3838.00$) at the background level. We also see that this case results in the Hubble constant as $H_0 = 68.79\pm 0.59\,{\rm km\,s^{-1}\,Mpc^{-1}}$ which is greater than $H_0 = 68.20^{+0.42}_{-0.38}\,{\rm km\,s^{-1}\,Mpc^{-1}}$ given by the $\Lambda$CDM model, and hence we can alleviate the $H_0$ tension to some extent in our framework. Furthermore, the best case of our scenario gives a lower value for the best-fit of the $S_8$ parameter than the $\Lambda$CDM result, and therefore it also reduces the LSS tension slightly. We moreover estimate the growth factor of linear perturbations and show that the best case of our model ($\chi_{f\sigma_{8}}^{2}=40.84$) fits the LSS data significantly better than the $\Lambda$CDM model ($\chi_{f\sigma_{8}}^{2}=44.29$). Consequently, our model also makes a better performance at the level of the linear perturbations compared to the standard cosmological model.
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