Any proof that inflatons exist? According to the theory of cosmic inflation, the particles that cause inflation are called inflatons. My question is how strongly a cosmologist working on inflation proves in the existence of such particles.  
 A: There is no proof, and no specific evidence of the Inflaton field, nor its particle excitations, the Inflaton. 
The Inflaton field is a hypothetical scalar quantum field that has been postulated as a possible way by which inflation came about, as it decayed from a high energy state to a lower one. There have been and are various models of the Inflaton field, with important parameters such as the decay time and others. The current one that best matches the data observed is slow roll inflation. It also imparted energy to the universe as it came down its high potential energy state, kicked off that unstable equilibrium  state due to quantum fluctuations.
A simple article on it is at Wikipedia, maybe a way to start with other references.
See also a more detailed treatment of the theory of the Inflaton field. It also has a section on observational data relevant to the inflation theory. See it here.
A: There's no deductive proof that any particle exists. Theories with inflatons are much more plausible than theories without inflatons, since the former predict a flat universe with once causally connected patches (i.e. generically agree with experimental data without fine-tuning, solving the flatness and horizon problems). The latter, on the other hand, require miraculous tuning.
A: Proofs in physics are different than proofs in mathematics.
In mathematics, there are axioms and mathematical statements based on the axioms and mathematical statements  can be proven with absolute certainty as true or false.
Physics starts with data, and these data force a "shape" on a mathematical model, picking a subset of the possible solutions that fit the data.
So the inflation period and the need for an inflaton does not come out of the blue, it comes because it is an attempt to explain mathematically the observations, in a consistent mathematically physics model.
Particularly the Cosmic Microwave Backround (CMB) radiation, measured   with large accuracy, displays a uniformity over the known universe that was  inconsistent with the first cosmological Big Bang model. Any uniformity in matter and energy distributions in classical physics can be very accurately described by thermodynamics. In the case of the CMB, which is a picture of the early universe, a classical thermodynamic model cannot be applied because of general and special relativity.
The regions in the early universe could not interact over the whole spacetime because of the limits of the velocity of light, and thermodynamics depends on the electromagnetic interactions of atoms and radiation, i.e. on the velocity of light. There were regions in the early universe that could not communicate between them to reach a thermodynamic equilibrium, which is what the CMB uniformity implies.
In addition, the first Big Bang version had a singularity at the beginning of the universe, which everybody expected to be taken into account by the quantization of gravity (an ongoing research program). Thus phenomenologists decided to use an effective quantization of gravity for the early universe, which gave the inflation period, where the universe was just a "ball" of energy , described quantum mechanically , expanding rapidly as seen here . The quantum fluctuations homogenized the early universe and explain the observed CMB distribution, as well as other features of the cosmos.

Inflation theory was developed in the early 1980s. It explains the origin of the large-scale structure of the cosmos. Quantum fluctuations in the microscopic inflationary region, magnified to cosmic size, become the seeds for the growth of structure in the Universe (see galaxy formation and evolution and structure formation). Many physicists also believe that inflation explains why the Universe appears to be the same in all directions (isotropic), why the cosmic microwave background radiation is distributed evenly, why the Universe is flat, and why no magnetic monopoles have been observed.  

Quantum physics models have interactions and exchange particles as a basis, and the name given to the gravitational scalar  particle ( still an object of research for the exact model) is inflaton.
Thus the "proof " that the inflaton exists is that it is needed in order to model the observed universe, its imprint is in the astrophysical data. It is more complicated than claiming proof of the existence of the  electron,  but both "proofs" depend on mathematically modelling data.
