Here is the scenario from wikipedia
According to inflation theory, the inflaton is a scalar field that is responsible for cosmic inflation in the very early universe. A quantized particle for this field is expected, similar to other quantum fields, called an inflaton. The field provides a mechanism by which a period of rapid expansion from 10^−35 to 10^−34 seconds after the initial expansion can be generated, forming the universe.
Note that before 10^-35 seconds the inflation is the steady one from the original Big Bang.
The basic process of inflation consists of three steps:
Prior to the expansion period, the inflaton field was at a higher-energy state.
Random quantum fluctuations triggered a phase transition whereby the inflaton field released its potential energy as matter and radiation as it settled to its lowest-energy state.
the rapid inflation results in the increase of kinetic energy as the inflaton field settled to its lower energy level
This action generated a repulsive force that drove the portion of the universe that is observable to us today to expand from approximately 10^−50 metres in radius at 10^−35 seconds to almost 1 metre in radius at 10^−34 seconds.
So it is the time necessary for the inflaton field to fall from the higher to the lower energy level.
The model continues here
At the end of the early universe's inflationary period, all the matter and energy in the universe was set on an inertial trajectory consistent with the equivalence principle and Einstein's general theory of relativity and this is when the precise and regular form of the universe's expansion had its origin (that is, matter in the universe is separating because it was separating in the past due to the inflaton field).
According to measurements, the universe's expansion rate was decelerating until about 5 billion years ago due to the gravitational attraction of the matter content of the universe,
So it is the gravitational attraction that takes over once the inflaton field is at its lowest energy state.A lot of kinetic energy ended up as potential gravitational balancing the expansion
after which time the expansion began accelerating.
and is currently doing so.
In order to explain the acceleration physicists have postulated the existence of dark energy which appears in the simplest theoretical models as a cosmological constant. According to the simplest extrapolation of the currently-favored cosmological model (known as "ΛCDM"), this acceleration becomes more dominant into the future.
As I understand it the subject is still under research, theoretically and experimentally