 Okay, so thanks for the invitation. It's really good to be back in Trieste. And in this talk I'm going to try to answer this question that we asked in this paper with Elena Giuzarma from Rome, Olga Mena from Valencia, and Hector Ramirez was around in any case there's some nasty questions so you can ask him so he's there. So during this meeting, sorry, this is the outline. So I'm going to take you about some motivation while we ask this question. And then I represent the feed, the result, and there's no problem with this. And at the end, so I will comment on future constraints and inclusion. Okay, so during this meeting, so it's clear that we have some pretty good, pretty detailed picture of the universe, basically through three sets, three basic sets of experiments. So CMV, large-scale structure, and also NVIDIA simulation, which are also a kind of numerical experiment. And the whole picture that we call Lambda CDM holds together once we specify some initial conditions. And I am sure that you agree that among all the theories or the mechanism that we have, so basically inflation has a special place. These are a couple of slides just to remind you what inflation is. What kind of inflation I'm talking about. It's just single-field, slower inflation where we have a scalar field minimally coupled. This is the only sentence that you have to retain from this slide. Minimally coupled to gravity. And then once some conditions are satisfied, so we have really a set of prediction that are really impressive agreement with the data. So with respect to inflation, so the alternatives are less elegant. And there's some debate about that, but I'm sure most of you agree, except maybe Justin, who was here last week. But in any case, I'm going to focus on this simple picture and try to see whether this picture can be made more natural. So as I said, inflation has a special place among the early-inverse scenarios for initial conditions. However, it has some problems. For instance, as I showed you before, so we always assume that the inflow of scalar field is minimally coupled to gravity. And this is not the most general situation. Another problem that is related to this one that I will argue later on is the fact that we always assume that there is a normalizable potential and we don't include the normalizable interactions just because of simplicity. So basically, these terms are elected for convenience. So this is also a problem, a conceptual problem, but it should be addressed as a point if you want to understand inflation from an effective field theory upon the viewer or as a consistent theory. And this is hard to justify. In fact, if you consider shift symmetry, that is the only symmetry that you can think of when you're doing inflation, it's broken by the potential and also this nominal coupling that I will tell you about. So we ask this question whether this nominal coupling of the inflow is preferred or is it ruled out or what is the situation. So this is a list of partial list of papers of talented people that ask this question maybe in different context, but it's related to us. So this is the motivation. So the nominal couplings are expected in general and you can justify the presence in many ways. For instance, you can think of the inflaton as being coupled to a light degree to freedom that will hit at the end of the inflation, which is the usual picture. And these couplings will generate some non-trivial, non-zero, non-minimal coupling through the RGE. So once you include the one-loop corrections to this xi, the non-minimal coupling, so you will see that the running will generate a non-vanishing xi. And the property of this equation, which is very well known, except when you have a conformal coupling, when xi is one-sixth, so you have always a non-minimal coupling generated by the running and the magnitude is given by this basically. It's logarithmically sensitive to the scale. So you see that even if you start with a vanishing xi at some scale, so you generate it. The second argument is that, so one can say always that you can rephrase the dynamics. So suppose that you have this Lagrangian. I probably don't see that. So you start with this Lagrangian, which is a general one with a non-minimal coupling. But then, okay, so we consider, for instance, the working example that we will focus here in our talk. We will focus on the M squared, five-square scenario, which is the simplest one. You can always rephrase the dynamics in the Einstein frame. And Einstein frame is just making a way of scaling where we have this pre-factor for the metric. And then we are back to our familiar form with a new scalar field, calorific normalised, and with a new potential. So the new potential is related to the old one by the, this is you, by the way, is the old potential divided by the conformal factor, square of the conformal factor. And if we consider, for instance, this kind of, this kind of conformal factor where, I mean, this is just an expansion around with small side. So we see that if we expand it, we have new operators that are created and especially the non-minimal ones. So we can turn around this argument and say that if we have some theory, inflationary theory model with non-normalised operators, so we can always write it as a non-minimally coupled theory. So that's basically the motivation that led us to consider this non-minimally coupled. Okay, so now this is the fit that we did. So basically we took M squared, five-square model that we see here in red dots, and we add a non-minimal coupling. And we see what are the bounds on this non-minimally coupled by using Monte Carlo chains. So we see as we know that, for instance, the M squared, five-square scenario is already under tension with respect to the new data. It depends on the number of e-folds, but for instance, for 50 e-fold, it's outside the two sigma confidence levels. And we see that the positive non-minimally coupling is favored, because as you see here, this represents the magnitude of this non-minimally coupling. And for negative xi, so you follow this trajectory, while for positive one, so you go back to the one sigma confidence level of black for it to snow. And you see that the magnitude of this xi is of other 10 to minus three. Okay, more details. So you see now that we have a slight preference for non-managing xi. So this is the pdf of xi, and this is of other 10 to minus three, while also you have also a non-vanishing also for r, for tensor to scalar ratio. Okay, so this is to be compared with the situation without non-minimally coupling, for instance with Planck's 2515, we see that the pdf of r, I mean visually it doesn't show any preference for non-managing r. So in the context of this model, it's easy to understand because this model, there is a non-trivial correlation between ns and r, so you cannot push r low without having a non-managing xi. Okay, so there is a slight preference for non-managing non-minimally coupling. Let's see how much. So we see that, okay, so these are the results for some set of parameters, for a number of e4, 60, and 50, and we see that basically the central value is around 10 to minus 3. So this is 0.03, or 3 by 10 to minus 3. While the tensor to scalar ratio is around 0.04, 0.46. Okay, so in order to answer the question that we are asking in the title of the talk, so we must be more quantitative, and in order to be so, we should compare the distributions for both vanishing and non-managing non-minimally coupling. The simplest thing to do is to compare the chi-square, and we compare the chi-square for a minimally coupled and non-minimally coupled theory. By taking into account that there is an additional degree of freedom in xi, and we see that obviously the chi-square decrease, however it decreases in such a way that there is a statistical significance to that. So the xi is favored to be non-zero, and as the p-value with this set of data indicates it's very significant. So that's 99% confidence level. Okay, so it's interesting also to look at the excursion. So one can imagine that since we are considering some more complete theory of m-square phi-square, so the excursion might be smaller. So it turned out that for the values that we found, the effect of xi, non-minimally coupled, is not so relevant because it's so small. So basically phi and bar phi are related by this usual expression. And for the values of xi that we find, the excursion is still super-planked. So it doesn't fix the problem of super-planked excursion. I mean, in this sense it's still not satisfactory. Okay, what are the future constraints on this scenario? So in the context of Planck 2015, so there is this slight preference, what can we say in the future constraints? So we can construct a combination of first-order observables, and we see that, okay, in the case of m-square phi-square, this combination vanishes at order 1 over n on a cube. And if we are able to measure the left-hand side of this equation at this precision, which is the one that these observations are aiming to, we can probe xi at the level of 10 to the minus 4. Also, we can combine this with accurate measurement of ns because, as I said, there is a non-trivial correlation between ns and r in this kind of scenarios. So suppose that the bounds of r go down by one order of magnitude, so 0.1, so this will push ns outside the one sigma, the measure, for instance, the central value of Planck, and this will rule out completely the model. So this is one way of falsifying it. The second way is to look at the running. So this is the running with respect to the value of the non-minimal coupling. And we see that, okay, so this is the m-square phi-square prediction. So for negative xi, there is a significant running. However, as I said that for negative xi, this higher r, so it's already ruled out. However, so we can hope that with some more accurate measurement of the running, so if you measure running bigger than that, so because we are here, so the preferred values are here, we can say something about this model. And as an example, so there's a lot of experiment as an example, so there is a proposal in Caltech called Cypher X, and it promises to reach this impressive number, so it's like 6 by 10 to minus 4 in alpha s. And this is in this way, so one can hope to probe, okay, among the other models also this kind of scenario and to say something about it. Okay, so these are the conclusions. So the answer to the question is yes, the current data have a preference for a nominal by a couple phi-square, and the next round of observation will falsify or ruled out the scenario, especially combined with the measurement of NS and R because of the non-trivial correlation I explained, and more futuristic ones like 21 cm will certainly answer this question more accurately.