Even a car constantly accelerating for a long time was still at t0.
The expansion is an expected behaviour of space time, and is predicted by general relativity given a certain parametrization of the density of matter an energy of the universe. To explain the acceleration, though, a large energy parameter is required, which has not been yet identified with any known process, although many theories exhist; as placeholder the parameter is known as dark energy. It is of course possible, but generally considered unlikely, that general relativity is incomplete and another theory could explain the acceleration without dark energy.