Abstract：  
I will give a very brief overview of the rigid mathematical proof of one basic question in cosmological simulation: on what space and time scales Newtonian cosmological simulations can be trusted to approximate relativistic cosmologies?  
We resolve this question by investigating Einstein-Euler systems with positive cosmological constant and Poisson-Euler systems under a small initial data condition. Informally, we establish the initial data set in the meaning of cosmological scale which solves constraint equations and construct the existence of 1-parameter families of $\epsilon$-dependent solutions to Einstein-Euler systems with a positive cosmological constant that:  
(1) are defined for $\epsilon \in (0,\epsilon_0)$ for some fixed constant $\epsilon_0>0$,  
(2) exist globally on $(t,x^i)\in[0,+\infty)\times \mathbb{R}^3$, % and are geodesically complete to the future,  
(3) converge, in a suitable sense, as $\epsilon \searrow 0$ to solutions of the cosmological Poison-Euler equations of Newtonian gravity, and  
(4) are small, non-linear perturbations of the FLRW fluid solutions (via conformal singular formulation of Einstein-Euler system).    
This talk originates from a joint work with Todd Oliynyk, see arXiv:1701.03975 and arXiv:1711.10896.