Where in the whole discovery is seen a connection with the scale at which inflation operated (10^16 Gev from what I see in the reviews). Is this energy value another output of the discovery? Why? Where?

I recommend a good textbook like Peacock "Cosmological principles" or the Tasi lectures on inflation for a derivation of this result:

arXiv:0907.5424

But yes, measuring a large tensor to scalar ratio corresponds to setting the energy scale at around the GUT scale.

Measuring the spectral tilt (which is related to the scalar ratio in simple single field models) will also typically constrain the form of the potential in various classes of models, which allows us to distinguish or rule out candidate theories.

It has to do primarily with the fact that the amplitude of the gravitational waves generated by inflation is directly proportional to the Hubble parameter squared, and the Hubble parameter is proportional to the energy density. So, by measuring the amplitude of B-modes, we are able to solve for the Hubble parameter at the time of inflation, giving us the energy scale.

I have been reading the Tasi lectures and I got stuck just in the same equation (in this text, 218). Here it says something like: "Since ∆2s is fixed and ∆2t ∝ H^2≈ V , the tensor-to-scalar ratio is a direct measure of the energy scale of inflation" and then he pulls the energy equation.

V^(1/4)=(r/0.01)^(1/4)*10^16gev

I have a couple of questions:

0 V is the inflaton potential, right?

1 where did the "^1/4" came from? I was expecting perhaps a "^1/2" due to some surely bad dimensional reasoning but I cant see that coming from the previous equations of the book.

2 where did the 10^16 gev came from? I was thinking that, as the book is in natural units, if r is a measure of energy then we have that energy = 1 means that in gev units it is equal to 10^18, and multiplying and dividing by 0.01 ( which appears also for first time in this expression) we get the mentioned 10^16. But seeing more closely, the 0.01 is inside the ""1/4" so my argument does not work. So: where did the 10^16 came from?

As always im looking forward for your help, thanks in advance and sorry for my english and, moreover, for my physics!

0) Yes
1) V is an energy density, hence V^{1/4} is the energy scale
2) The tensor/scalar ratio is the ratio of the tensor amplitude to the density perturbation amplitude on a given scale, [itex]r = A_T/A_S[/itex]. The scalar amplitude is measured to be [itex]\sim 10^{-9}[/itex] on CMB scales (say, at [itex]k = 0.002 h {\rm Mpc}^{-1}[/itex]). The tensor amplitude is known to be [itex]A_T = 16H^2/\pi[/itex] at lowest order in slow roll. Further, to lowest order in slow roll the Hubble parameter may be replaced by the height of the potential at the relevant scale, [itex]H^2 = 8\pi V/3[/itex]. Put that all together and shake vigorously.

Ok, thanks again. Its much more clear now. I will keep on reading the Tasi lectures. I am having different points of conflict doing that (the others are not so closely related to the topic of this thread) so I will probably be bothering all of you in the short term!