Physical origin of the inflaton field? I have two particular questions regarding the inflationary scenario. They are:

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*What is the physical origin of the inflaton field?


*Why has the potential of the inflation field its particular form?
 A: The source of inflation is the inflaton, but no-one knows what the inflaton is!
The inflaton potential is calculated by looking at the universe and fiddling with the potential to get something that fits observations.
In other words, there are no fundamental theories that predict the inflaton properties. At the moment the theory is purely phenomenological. However this does not mean it lacks predictive power, as we get more information out of the theory than we have to put in as parameters. Hopefully the Planck satellite will give us more information about the theory.
A: As @Rennie states, no-one knows what the inflaton is. The current state of affairs is that it is generally accepted that a period of exponential expansion took place during the early universe. This explains many of the features of the observable universe that are otherwise extremely hard to explain. One of the big industries in cosmology is to try to build a sensible and well motivated model of inflation that agrees with the experimental data. To date there are hundreds of such models, but many of them share a common feature which is that the inflation is produced by the potential of a scalar field. When this is the case, the particular scalar field in the model is known as the inflaton. Examples of models include:


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*Higgs inflation where the standard model higgs boson plays the role of the inflaton. Since the Higgs is the only fundamental scalar to have been observed so far, it is an important question whther it could be the inflaton.

*GUT inflatons. In grand unified theories, there are a lot of extra scalar fields which must be present to break the GUT symmetries, these could play the role of the inflaton.

*SUSY inflation. In Supersymmetric models, scalar fields abound and many extensions of the MSSM require additional scalar fields, any of these could play the role of the inflaton.


There are many many more models, all have various pros and cons. The important point is whether the scalar field in the model has a potential that could produce inflation and how the predictions from the specific model agree with experiment. As CMB data gets more refined, some models will be ruled out but unambiguously identifying which scalar field in nature is the inflaton is a long long way off.
The potential of the inflaton field has to have a particular form to produce inflation. Inflation requires a negative-pressure vacuum energy density. This is generically produced by a scalar field as long as $\dot\phi^2 < V(\phi)$. So the requirement boils down to having the scalar field sit at a point where the potential takes a large value and is not too steep. This can either be a local minimum (false vacuum) and the field eventually tunnels out, thus ending inflation or a flat potential where the field slowly rolls down the potential. Slow roll inflation is now generally preferred since false vacuum inflation has problems with reheating.
Remember, at the moment, producing models of inflation is very much in the realm of model building and there are still models that do not even invoke an inflaton field.
A: There no need of 'inflaton/inflationary scenario' at all.
The inflation era is needed in the BBT framework where space expands and is not needed in the opposite viewpoint: particles shrink thru time.
Assume that particles were created long time ago evenly all over universe at some point in time in the past due to a sudden and general change of state of the vacuum.
In the beginning the atoms were much larger than the ones we see around (the ones that we use to measure distance/mass/time and anything else but simple counting).
If they are larger then the redshift of light is a natural effect, provided c is constant, and the distant galaxies are not moving away at all (except local/peculiar motions).
As time goes by the particles give back their energy to the vacuum (it is established that electrostatic/gravitic energy spread away from the particles) and, as any other physical process where the effect is proportional to the source, such as radioactivity, it obeys an exponential decreasing law.
Thus we have a physically motivated exponential decrease of something, as opposed to 'exponential increase of space amount'.  
The theory is fully derived in this paper at vixra: A self-similar model of the Universe unveils the nature of dark energy , and it matches all the fundamental measures of the universe. The dark energy, inflation, cosmological constant, etc.. are necessary artifacts of a bad model (BBT). 
Nothing in physics says that the atom is an invariant. We work in loop: define units of measure with an atom (and c) and then we calculate the size of anything. 
In conclusion : the size/mass of any object is some fraction/multiple of an atom, and we are blinded to any variation of the unit of measure.
(see Poincaré sphere-world )
An example:
Suppose we want to measure the change in length of a copper bar in function of temperature change. If we put a graduated copper bar inside the oven we are deceived and we will conclude: here's proof that the heat has no effect on the length of the bodies.
edit add:
2nd Thermodynamic law, gravity and an homogeneous universe:
start with  temperature 0ºK , move 1 atom and temperature will rise. Video see at 8min20 Stephen_Hawking__The_Story_of_Everything (start at 7min), and the equations are posted here.  
