In a recent paper, which has been published in Nature, the LIGO Scientific Collaboration (LSC) obtained an upper limit on the stochastic gravitational-wave background of cosmological origin by using the data from a two-year science run of the Laser Interferometer Gravitational- wave Observatory (LIGO). Such an upper limit rules out some models of early Universe evolution, like the ones with relatively large equation-of-state parameter and the cosmic (super) string models with relatively small string tension arising from some String Theory's models. It results also an upper limit for the relic stochastic background of gravitational waves (RSBGWs) which is proposed by the Pre-Big-Bang Theory. On the other hand, the upper bound on the RSBGWs which is proposed by the Standard Inflationary Model is well known and often updated by using the Wilkinson Microwave Anisotropy Probe (WMAP). The potential detection of such a RSBGWs is the only way to learn about the evolution of the very early universe, up to the bounds of the Planck epoch and the initial singularity. This is a kind of information that is inaccessible to standard astrophysical observations. By using a conformal treatment, a formula that directly connects the average amplitude of the RSBGWs with the Inflaton field has been re- cently obtained in our paper Gen. Rel. Grav. 42, 5, 1323-1333 (2010). In this proceeding, by joining this formula with the equation for the characteristic amplitude hC for the RSBGWs, the upper bounds on the RSBGWs from the WMAP and LSC data will be translated in lower bounds on the Inflaton field. The results show that the value of the Inflaton field that arises from the WMAP bound on the RSBGWs is totally in agreement with the famous slow roll condition on Inflation, while the value of the Inflaton field that arises from the LSC bound on the RSBGWs could be not in agreement with such a condition.

To appear in the AIP Proceedings of the 8th International Conference of Numerical Analysis and Applied Mathematics, Hotel Rodos Palace, Rhodes, Greece, 19-25 September 2010