What evidence is there that universe will not shrink again? There are theory that universe eternally expanding and shrinking, so that every event and our lives will repeated again. At the other hand some scientists thinks that it will not shrink again because they noticed that expansion is accelerating for now. But how they can know that expansion will not slow down in the future ? May be universe is like bubble - it expanding with acceleration for now, but will shrink in the future ?
 A: Until dark energy (and dark matter) are properly understood, it is impossible to be certain of the future fate of the universe.
The concordance $\Lambda$CDM model, deduced from observations of distant supernovae, from the cosmic microwave background and from baryon acoustic oscillations suggest that the expansion of the universe is accelerating and that there is no prospect of a future big crunch.
The big unknown here is the equation of state for dark energy - that is, its pressure as a function of its energy density. If $p/\rho = w$ and $w < -1/3$ then the expansion accelerates. If $w<-1$ then that acceleration will be so severe that the universe will end in a "big rip". At present, the data are consistent with a cosmological constant with $w=-1$. Here the dark energy density remains constant as the universe expands, whilst the matter density falls. Ultimately, dark energy completely dominates and the asymptotic behaviour of the universe is a perpetual exponential expansion.
But without knowing for sure what dark energy is, one couldn't rule out its disappearance in the future or some drastic change in character. 
If dark energy just disappeared tomorrow, the amount of gravitating matter in the universe is insufficient to pull it back to a big crunch. But other ideas for how dark energy evolves (e.g. Kaloper & Padilla 2015) suggest that it may change its nature in the future, causing the universe to collapse again.
A: I am not an expert to give the most proper answer but  according to Hubble's law , the most distant galaxies are moving away from us faster than the speed of light, due to space expansion. This fact makes me think that also the gravitational effect of these galaxies is lost forever, and this looks to be the future of our universe. Even if a 'big crunch' would suddenly  start now for some reason , it wouldn't lead to an exactly inverse sequence of the 'big bang'.
A: Well, the observations of the acceleration of the universe's expansion is perhaps the strongest argument against the Big Crunch argument. 
Nonetheless though, it is by no means a disproven theory- we cannot very accurately predict the future until we understand the 95% of the universe that isn't baryonic matter, the 95% that current theories have as the explanation for the acceleration in the first place. 
