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
Best Answer
The expansion of the Universe is driven by the dominant forms of energy and matter density. This is essentially an expression of a famous statement by John Wheeler (from memory, apologies if I get the exact wording wrong): "Spacetime tells matter how to move, matter tells spacetime how to curve". In this case, the matter is telling the Universe how to expand. Mathematically, Einstein's equations describing gravity, when restricted to homogeneous and isotropic Universes (relevant for cosmology), reduce to the Friedmann equations which relate the expansion rate to the matter content.
There are lots of different kinds of energy density, but for cosmology there (to zeroth order) three that really matter:
Note that ordinary matter (relativistic and non-relativistic) do not have a constant energy density. As the Universe expands, the energy density (energy per unit volume) goes down, because there are the same number of particles but a larger volume. (Well, it can be that the number of particles changes, but at the level of this discussion that's a detail that is not so important, the key thing is that the energy per unit volume decreases for ordinary matter).
So something weird is needed in order to provide constant energy density as the Universe expands. Physically we can imagine that there is a certain amount of energy associated with every volume of space -- then as space expands, in a sense creating more space, there is also more energy associated with that space. This is essentially the cosmological constant. Another possibility is that there is a dynamical field (a "scalar field") which permeates space, and an associated potential energy that is approximately the same anywhere.
With all that background, the current generally accepted model of cosmology is that:
While this is sort of a standard picture, not all points are on equally solid footing. The first two bullet points are not universally accepted. The first bullet point is called "inflation", and (if true) explains why the Universe we see is spatially flat, and why we observe the CMB to be at the same temperature, even though the different patches that make up the CMB were not in causal contact with each other before the Big Bang in models without inflation. The inflationary period also explains the origin of the spectrum of the CMB. The second bullet point is a mysterious process that is thought to involve some combination of the decay of scalar particles into other matter ("reheating") or some dynamical process where the scalar field coherently pumps energy into other fields ("preheating").
Points 3-5, however, are on quite solid observational footing. We have observations throughout the Universe's history that probe the Universe during these different eras. While we do not fully understand what is driving the Universe today, a perfectly satisfactory model that explains all the observations is that there is a very small constant energy density that has always been pervading the Universe throughout its entire history, but we are only seeing it now because we had to wait until the Universe became dilute enough to see its effects.