The totally asymmetric simple exclusion process (TASEP) is a Markov process describing $N$ particles hopping forward on a one-dimensional lattice of $L$ sites. The periodic model evolving during a time $t$ has been studied recently (PRL 116 090601) on the relaxation time scale when $L$, $N$, $t$ go to infinity with finite density $\rho=N/L$, and finite rescaled time $\tau=t/L^{3/2}$ characteristic of KPZ universality. Exact expressions have been obtained for the average density profile, for the stationary two-point function, and for the probability density of current fluctuations for simple initial conditions. At small $\tau$, the distribution of current fluctuations converges to Tracy-Widom distributions. At large $\tau$, one recovers the stationary large deviation function of the current. The finite $\tau$ formulas have a nice interpretation in terms of a functional integral with the action of a scalar field in a linear potential.