4.3. Reconstruction without slow-roll
Eventually, in order to get the best possible constraints on inflation one will want to circumvent the slow-roll approximation completely, and this can be done by computing the power spectra (first the scalar and tensor spectra, and from them the induced microwave anisotropies) entirely numerically. Such an approach was recently described by Grivell and Liddle , and represents the optimal way to obtain constraints on inflation from the data (though at present it has only been implemented for single-field models).
Figure 1. The 4 potential as seen by the PLANCK satellite. In the upper panel, the dashed line shows the true potential, and the full lines show a series of Monte Carlo reconstructions, which differ in the realization of the observational errors. In reality we get only one of these curves. The lower panel shows the combination V'/V3/2, which is much better determined. Scalar field values when scales equalled the Hubble radius during inflation are shown, roughly corresponding to the range of PLANCK.
In this approach, rather than estimating quantities such as the spectral index n from the observations, one directly estimates the potential, in some parametrization such as the coefficients of a Taylor series. An example is shown in Figure 1, which shows a test case of a 4 potential as it might be reconstructed by the PLANCK satellite - see Ref.  for full details. Twenty different reconstructions are shown (corresponding to different realizations of the random observational errors), whereas in the real world we would get only one of these. We see considerable variation, which arises because the overall amplitude can only be fixed by detection of the tensor component, which is quite marginal in this model. However, there are functions of the potential which are quite well determined. The lower panel shows V'/V3/2 (where the prime is a derivative with respect to the field), which is given to an accuracy of a few percent on the scales where the observations are most efficient. This particular combination is favoured because it is the combination which (at least in the slow-roll approximation) gives the density perturbation spectrum.
No doubt, when first confronted with quality data people will aim to determine n, r, and so on along with the cosmological parameters such as the density and Hubble parameter. However, if we become convinced that the inflationary explanation is a good one, this direct reconstruction approach takes maximum advantage of the data in constraining the inflaton potential.