In response to Lawrence,
---
It is my understanding the inflaton field is similar, though not identical, to the Higgs field. A lot of work these days focuses on the slow rolling of the field, which then enters into an m phi^2/2 phase. So the inflaton field is most likely not perfectly quadratic in the phi field, though it might be argued that near the minimum it is approximately so.
---
The inflaton is likely a scalar degree of freedom (like the Higgs) but it is very different from the Higgs, in that it must couple weakly to all other kinds of matter in order to preserve the flatness of the potential (which will otherwise be corrected by loop terms).
The m^2 phi^2/2 is a toy model, and it is not clear how good it is -- you are right that any minimum will look quadratic, but in models which try to work out the inflaton potential "honestly" this quadratic region is often fairly small -- but the basic dynamics is still representative, making it good enough for my purposes.
---
I will have to read your arXiv:0805.2154[astro-ph] to understand more completely your argument leading to figure 1. I will say that based on your paper it does appear that you have clearly argued for the quadratic inflaton potential.
---
We are not arguing FOR the quadratic potential, but were really exploring how the potential can be constrained with data for general inflationary models. We plotted the predictions of a couple of well known models primarily as a service for the reader, not because we think any of them have been "proven" (although it is well know that the quartic model is not a good fit).
---
Yet I can't help but think we might be able to peer beyond the opaque boundary of the CMB. If we get very good at neutrino astronomy or in detecting gravitational radiation from the earliest moments of the universe we could in principle observe the universe up to the inflationary period.
---
There are several things going on here. The CMB decouples from the rest of the universe relatively late, when compared to neutrinos (which do not interact with the rest of the universe from a minute or so after the big bang) and gravitational waves (which can be sourced at any time, but do not interact significantly when the density is significantly below the Planck scale). However, the initial conditions for the CMB (in particular the amplitude and scale dependence of the primordial fluctuations) are set during inflation, maybe 10-30 seconds after the big bang (if you just naively extrapolate back in time). The "peaks" in the CMB power spectrum arise much later, but it is relatively straightforward to back those out and recover the primordial spectrum. So in that sense, the CMB is directly sensitive to inflationary physics, despite the late decoupling time relative to neutrinos or gravitons.
However, if you consider a moment during inflation at which the universe will grow (say) 100100 times large larger before inflation came to an end, a graviton which existed then with a wavelength larger than the Planck length would now have a wavelength far larger than the visible universe (thanks to the redshift factor) - and the total inflationary growth can be far larger than 100100. So even if the graviton is "there" in some sense, it is not possible to detect it. We ARE sensitive to very long wavelength gravitational waves (ie modes whose wavelength is roughly the same as the current size of the visible universe), since these directly source the polarization of the CMB, and this signal is being actively searched for. But significantly beyond that limit, it does not seem reasonable to think of a detection.
Hope this helps,
Richard