 The first speaker and the second part is going to be Ipucratis Salthas and he is speaking about the non-running Hicks inflation from an ERG perspective. Thank you Daniel. Let me thank a lot the organizers for giving the opportunity to speak here today and I would also like to apologize to the organizers for the slight change in the title of my talk while the content is still the same. But I will be speaking about Hicks inflation. You might probably, most of you, not all of you heard about it, but I'm going to assume that not all of you are familiar with the main idea of the model so I'm going to motivate it a little bit and explain very briefly how it works. Okay, good, so we think we know, we think we understand physics reasonably well up to the electro-weak scale while there are still some big mysteries at the energies well below that but up to the electro-weak scale we think we have a good understanding of physics and one of the most important questions is how the standard model of particle physics looks like for energies beyond the electro-weak scale. So cosmology gives us a really powerful opportunity to test physics beyond these energies because as we look very early back in time in the cosmological evolution and when I'm talking about very early on, I'm really essentially talking about the big bank or sufficiently close to it, we are probing very high scale, very high energy scales of very small distances if you like and we currently believe that the universe started with a rapid acceleration which happened essentially after a fraction of time after the big bank and during this rapid acceleration, this rapid acceleration solves quite some problems of the standard big bank cosmology but one of the most important ones which I will be talking about today as well is the generation of the seeds of structures as we observe them around us through the tiny primordial fluctuations. Now these primordial fluctuations have two forms, scalar and tensor fluctuations, the scalar fluctuations we observe on the cosmic microwave background, they manifest themselves as variations of temperature in the in the CMB, tensile cosmic gravitational waves we haven't observed yet, however this thing itself puts a bound on their amplitude. Now primordial inflation happens, is expected to happen around the gut scale and it's a very interesting observational window for particle physics. Now Higgs inflation is the simply the idea that the Higgs, the standard model Higgs plays the role of the Inflaton. Now the Higgs has some very nice properties which could make it a good candidate, first of all it is a scalar, we need the Inflaton to be a scalar so we don't spoil the isotropy of the universe and the model where Higgs, the standard model Higgs is the Inflaton is extremely economical because we don't, we really don't need to introduce any new field content into our theory as you might know in cosmology people usually introduce many different fields in order to explain the different observations but here we have a very good motivation to consider the Higgs as the Inflaton. Now the implementation of the model is pretty straightforward, one couples minimally standard model Higgs sector with the Einstein-Hilbert action and this is the model introduced, this is a model of chaotic type introduced by Andrei Lindey in 1985. Now in this view the Inflaton, the Higgs starts at very trans-flagged and filled values, some sufficiently high energy scale, rolls down its potential and when I'm talking about rolling down this potential I'm talking about rolling down the potential sufficiently slow, this is what's usually called the slow roll approximation, so the energy density of the universe is dominated by the energy density of the Inflaton and in order to describe, to understand, to do physics in this context if the Inflaton is really the Higgs we need to have a handle of how the potential looks like at this high energy scale. Now what one usually does in this case is one starts from the initial conditions of the electro-higgs scale, uses the standard model but the RG equations evolves up to the inflationary scale and looks like how this implies about observations. Now doing that and assuming that the Inflaton is the Higgs, one finds that there is a huge disagreement with observations. Now this is because there is a disagreement with the amplitude of gravitational waves, the amplitude of gravitational waves for which we have a bound of about less than 10 to the minus 10 probes directly the amplitude of the potential and if this is the the quartic coupling of the Higgs that is about 0.1 or 0.0 on the initial conditions of the electro-higgs scale and you can see that the fluctuations are huge to agree with observations but there has been a twist some years ago where the model has been modified in a way that can achieve agreement with observations and this is the introduction of a new coupling not present in the standard model. This is a non-minimal coupling between the scalar and the curvature. In this view, the scalar and the metric are kinetically mixed because five couples to derivatives of the metric through the richest color and one can perform a diagonalization of the kinetic term of the two fields which is done through a conformal redefinition of the metric and one arrives at the so-called Einstein frame action which is classically equivalent to the previous one. Now in this view, there is an essential redefinition of the scalar as well to canonically normalize this kinetic term and then we end up with this potential which makes the healing effect of this coupling evident. In this view, we get a potential that looks like this. The field starts from this flat region again at a field value of the order more or less of the order of the Planck scale and it has to cover a particular number of refoldings so that it at least the observable universe is reproduced and the energy scale of this part of the potential or the amplitude of the potential, here you can see that it's suppressed by a factor of Xi squared. Now Xi is a free parameter which we are free to tune using observations and in fact as you can see here Xi is going to depend on cosmological parameters like the number of refoldings, the amplitude of scalar or tensor fluctuations if you like but also there is an implicit dependence on it from on electro weak physics like the top cork mass and this is if one evaluates the cosmological parameters in this relation arrives to something that looks like this. So Xi, the coupling Xi, the scale, the inflationary scale has to be 10 to the quarter times square root of lambda at the scale of inflation. So this lambda is evaluated by evolving the standard model better functions up to the inflationary scale. So actually from the potential you can see that for sufficiently large values we get this constant regime here, this plateau but as the filter rolls down then we get the filter is going to roll at its true minimum and hopefully there's going to be a graceful exit. Now what I would like to, if I would like to pass a statement, today is the fact that one has to worry about radiative corrections to these pictures so I have been, I have not been told, all the discussions so far is classical but one has to worry about radiative corrections and this is what I will be explaining is about corrections coming from Higgs and Graviton loops in particular whether these can affect the standard picture, the standard dynamics of inflation in this context. So we are using the exact RG, I'm not going to explain anything about this equation, I'm sorry, all of you are experts on. So the action, the effective action, the answer for the effective action is one where Riggs' color is coupled with the general function of F, this is just for generality, the general scalar potential plus a gauge fixing term and a Gauss term which comes after exponentiating the residual determinant, the path integral. Now expanding the field, we use a background field method, we expand the field into background and fluctuating piece and the inverse propagator entries take, acquire this minimal form after restricting to a Euclidean sphere background where R and Phi are constant. Now during inflation this is a reasonable choice because due to the slow rolling of the field in the slow roll approximation, field gradients are negligible. Now we get three types of entries for the inverse propagator, Graviton, Graviton, Graviton scalar and scalar scalar. Good. Now one has to work out a method for cutting off, for regularizing the trace in the Vetterich equation and for that I'm using a type one cut-off where the cut-off acts to modify the eigenvalues of the Laplacian and for the regulator function an optimized or lithium regulator is used. Now fixing the gauge simplifies things a lot. I have fixed the gauge to be another gauge and one arrives at the flow equation that looks like this. Notice that these derivatives here come after the, manifest the non-perturbative character of this flow equation. I will be talking about just the one loop approximation so this will go, this will flow to zero and now expanding this equation to project into operators, I'm getting a set of two equations for the non-minimal coupling function and the potential. Now what is particular in this case is that although for the Ritz scalar I'm using the usual asymptotic expansion of the Ritz scalar where the Ritz scalar in units of the cut-off goes, is expanding around zero, this is not true for the Higgs field which I assume it acquires a non-trivial web. So I'm expanding around a non-trivial web for the Higgs field and this is going to appear as an external parameter if you like in my equations. Now the equations, the better functions look like this. I'm interested in the set of these three better functions, the quadratic, the non-minimal and unit of z. So what is particular about these equations is that the non-trivial vacuum expectation value of the Higgs during inflation acts, brings in some particular threshold effects which manifest themselves into these non-trivial denominators here and in fact for sufficiently large psi and sufficiently large web for the scalar they are to suppress the running of the couplings. In particular what it turns out to be, right. So however in order to, if I want to, if I want to evaluate, if I want to understand properly how this, if I want to estimate in order of magnitude for these quantities I really need to know the value of the cut of during inflation because this is going to determine how what phi tilde is going to do. So now remember that phi, that during inflation the amplitude of the potential is approximately constant at this plateau. So I'm using the fact that my expansion, my, the asymptotic expansion for the Ritz scalar which is used to calculate the trace integrals has to be less or order one maximum. So using this inequality together with the slow roll approximation which tells me that the potential probes, the amplitude of the potential is effectively approximately equal to the Ritz scalar or the Hubble square parameter if you like. It gives me an equality of the cut of scale during inflation. And now that I can translate to the, to evaluate for the dimensionless field during inflation and doing that what one finds is that G is the higher order corrections in G are sufficiently small. So G is in the classical regime and Newton's it doesn't run but most importantly the lambda and psi running receive the, receive corrections of the order lambda over psi to some positive power. Now what this means is that psi effectively the sufficiently large value of psi has another healing effect. Then actually in combination with the non-trivial field vacuum expectation value of the scalar the effect is to introduce threshold effects which effectively suppress both the standard, the quantum gravitational corrections but also the standard, the standard perturbative standard standard, the use of standard model terms here. So you can see that the use standard the lambda square term in the quality couplings beta function receives a correction of the order psi cubed. Now thinking that psi is then to the fourth this is really a huge separation. Now of course one has to worry about observables, one has to worry whether any observable is going to acquire any running in this context and the amplitude of the potential which is an observable or the fractional change of the amplitude of the potential at leading order is of the order lambda over psi cubed. Now for lambda order 0.1 or 0.01 psi order then to the fourth this is again extremely small number and the amplitude of the potential is effectively runs but it's unobservably, unobservably small. Now in the post evolutionary era as the field flows down to its true minimum the threshold effects is to exceed we one recovers the standard equations particularly the standard the usual form for the for the beta functions for psi and lambda. Now I don't have time to talk about the asymptotic safety but I'm going to just stand in some important as I would think open questions so the stability of the electro weak vacuum is crucial in this case I have not talked about it however any successful Higgs inflation this context has to take that into account. Now the initial conditions for the model are very important because the value for psi for the non-minimal coupling required is particularly high and one should come with an explanation about how that comes about and of course the inclusion of the Yuccava sector this analysis after including fermionics and Yuccava sector is important together as well as the study of the frame dependence of the model. So of time calculation the Jordan frame but although the two frames are equivalent classical it's that's not always true at the quantum level. Now if I would stand to just a final remark the running from the Higgs and gravitons during Higgs inflation appears to be sufficiently suppressed and the flatness of the potential is is preserved the ERG provides pretty an elegant quite an elegant description of the quantum dynamics of the model thank you.