  The one-dimensional totally asymmetric simple exclusion process (TASEP) with
$N$ particles on a periodic lattice of $L$ sites is an interacting particle
system with hopping rates breaking detailed balance. The total time-integrated
current of particles $Q$ between time $0$ and time $T$ is studied for this
model in the thermodynamic limit $L,N\to\infty$ with finite density of
particles $\overline{\rho}=N/L$. The current $Q$ takes at leading order a
deterministic value which follows from the hydrodynamic evolution of the
macroscopic density profile by the inviscid Burgers' equation. Using
asymptotics of Bethe ansatz formulas for eigenvalues and eigenvectors, an exact
expression for the probability distribution of the fluctuations of $Q$ is
derived on the relaxation time scale $T\sim L^{3/2}$ for an evolution
conditioned on simple initial and final states. For flat initial and final
states, a large deviation function expressed simply in terms of the Airy
function is obtained at small rescaled time $T/L^{3/2}$.
