 Thank you. So good morning. Nice to see at least a few still here so early in the morning of the last day now We made it yesterday until here So we still discussed a couple of things concerning inflation in particular. I tried to explain the horizon problem in a way which I hope was transparent and I think it's this business with the horizons is confusing and I think it's really important that you carefully Think about that yourself So therefore why I discussed it and I gave also these pictures from the review by Bauman now I will what I will do today as well as I will discuss a few simple things just a few basic formulae which we need Hopefully in a compact way so that you Have them for for those who see them maybe for the first time and for the others who have seen them Know them already a little bit as a reminder now What we'll discuss I mean the key reference here references here concern two classes of models for inflation one The so-called large field models of inflation And one is the star windsky model, which I will discuss later and one is this chaotic inflation by linda sort of one of Or maybe the standard model for inflationary cosmology also important is this one here Natural so-called natural inflation which has to do with a flat direction which you get from axiom field and Some potential generated not perturbatively. I will not discuss much about that, but I just mentioned it here Then what will be important for us is hybrid inflation and I will spend some time on that Then for reviews here much of that is in standard textbooks I listed I think two very good books here and I particularly recommend this Review by daniel bauman, which you can just download I think it's very good now The dynamics of inflation is a slowly rolling scalar field that maybe you have seen and everybody who had a course in General relativity knows how to deal with scalar fields in kerf space time so You know you have a certain action and then you have a Lagrangian which in the simplest case is just the real scalar with some potential Then what is important is to have the energy momentum tensor for that and then There's a standard way to derive that once you have the action and then you know we assume in the robot's vocal metric a certain geometry which is Well The following you have sort of the The space time we are looking at is say r the real axis corresponding to time times a maximally symmetric A three-dimensional space which is either open closed or flat and if you now have some So that's for the metric and that gives us robots a vocal metric and then you also have Of course a source for this metric and which via the einstein equations And determines the metric and that's energy momentum tensor And then of course has to have the same symmetries as what we assume for the metric And that gives you the sort of fluid form of the energy momentum tensor And that means the entire energy momentum tensor is determined in terms of two entries as for We had it before namely energy density Which is say the potential the kinetic term and the gradient energy And a pressure Which is essentially the same but with a very important minus sign here For the scalar field now we will always assume homogeneous scalar fields here So that means we drop the gradient term and then you have phi dot squared plus v and phi dot squared minus v And you see if the kinetic terms are small then Rho is essentially minus p And that's the equation of state which we need for exponential Expansion and therefore scalar field dynamics can realize what we want What we discussed yesterday for this shrinking That's the essence of that So we have now two sets of field equations We have the einstein equations which for this geometry boil down to the Friedmann equations And then we have an equation for the scalar field, which is this And you see it's a It's the usual equation Which we have for scalar field in Minkowski space plus this term, which is the friction term It is a proportional to the first derivative of the field and the Hubble Parameter as the friction and that is if you derive this equation of motion it comes from the volume term here Now What you can easily check yourself if you have done it yourself so far is the following You can have So-called slow roll solutions of these field equations which are characterized By the following the kinetic term is small compared to the potential and the modulus of the second Time derivative is small compared to the fourth term here and compared to the friction term So you can drop that And then you get in fact Two equations which i've written down here You get from the Scalar field equation just this remains and in the Friedmann equation You get this this is a Friedmann equation There's a potential because the energy density the kinetic energy does not contribute And what you can also check easily yourself is that these conditions Are equivalent that you can write them as conditions on the first and the second derivative of the potential So these are the so-called Slow roll conditions you have a quantity epsilon Which in units of the plant mass is the first derivative of the potential of the potential it's squared And the other is eta which is the second derivative of the potential the modulus of that And in units of the plant mass both have to be very small So these three this Is the Starting point for inflationary cosmologists So you see it's very easy if you make the one the model of the world yourself You just take two differential equations of first order and solve them put in some potential and play some games And once you have the code you can even download of course such codes from the internet. It's not very difficult But interesting So now one example and I think that was first really started with some of the physical implications by Linda many years ago is just a power here and Just a power and of course depending on what the power is The dimension of the coupling varies And then you get these large field inflation models and they have a very simple structure They always look like this You have a potential And you have the field And the potential looks something like this and the field rolls down here And from some value say phi Star it's usually called until some end value So called end of inflation and in fact end of inflation Is defined by saying that the slow roll conditions no longer hold You can see that then the period of exponential expansion stops Now So you have something like this the so-called large field inflation models. They are large For a reason which we will see in a minute You can given such a potential you can easily work out For this example the relevant quantities which Characterize inflationary models. The first is the amount of expansion As we saw during inflation you have an exponential growth of the scale factor So if you start from some value a which corresponds to this field valued here Then you blow up And this growth is always characterized by the number of e-folds So you write that growth as a factor e to the n and then You can write this n Clearly as like this the number of e-folds As an integral over the hover parameter the hover parameter now the function of the field Because it depends on the potential which depends on the field and You can then rewrite it like this The potential of the derivative of the potential and for this particular example of the power Potential you just get phi squared over a Planck squared times 2 over 1 over 2 p and so You see if you for if you need n E-folds and we will see for reasons of that I have no time to discuss you need typically 50 or 60 it depends on the maximum temperature Which you have in the universe So for this Range you see that in fact this value of phi is pretty big. It is bigger than the Planck bars So inflation with this kind of potentials drives you to Transplankion field values. That's at the moment a hot topic in research whether that is consistent What one can think about it and how you can realize that in various models and but for these models, that's a fact now In fact qualitatively in order to reach these 60 50 or 60 e-folds You can do two things either you take say a potential as I had it this Phi to the p which is flat but not That flat and Then you are driven to this very large shield The other possibility is to have a potential which is even flatter a kind of plateau potential Then you can reach For smaller field values these number of e-folds. These are the so-called small field models to which I will come Then there are two indices related to this slow roll parameters Two linear combinations, which I've given here. We will see on the next slide why they are important That's a scalar and the tensor spectral index. He always given as a power Independent dependent on the power of p There's another quantity r Why this is written like that you cannot understand from here It comes with something which I will tell you on the next slide But it is just this quantity which is 16 epsilon the so-called tensor to scalar ratio, but this spectral indices And This value of r are the key quantities Which you always work out in models of inflation and which are needed to compare with data That's why I just repeat them here Now now we come back to this Microwave background and you know you have this enormous Information on the fluctuations. We saw that you saw that picture many times just a beautiful picture I can always look at it again And the latest and from that you extract a so-called power spectrum Now Just a second. I cannot Really explain how one gets from this picture to the spectrum that's Of course the key business in inflationary in inflationary model building and that's I think one of the most beautiful aspects Namely how you generate from quantum fluctuations These classical Anisotropy's in temperature and in other Things and that the first calculation of this kind was done by Shebizov and mukhanov many years ago, and I think this is really a very beautiful Thing, but I mean how you really get first of all how you calculate these quantum fluctuations And how you really get from there to this power spectrum, which you see here. That's takes A couple of lectures Say maybe one whole lecture five hour lecture course during such a school to really explain that so that you have to believe me Or look it up in one of these references. I gave you Now, but at least let me tell you what is really what the meaning is on this plot Of the various quantities. So this is here. This Quantity, which is plotted on this axis This is the following one looks at the sky and one looks now in different directions And one looks at say The temperature now the temperature is determined by the photon flux What one has checked is that this microwave background to very high accuracy as a plank spectrum and therefore it is It is Once, you know, it's a plank spectrum. You can do the following you can go to some frequency check for consistency and a few frequencies And just measure the number of photons which come from that direction. So then this photon flux Once, you know, it's a plank spectrum gives you the temperature So that's what is done. And then you see now if you vary the direction into which you look you see very tiny variations in this temperature And you can measure the temperature or you can measure another quantity. You can measure Say the polarization of the photons the flux of the photons of polarized photons, which come from some Direction that gives you other information for the so-called b-modes, which we briefly got to now Anyway, so what is plotted here then is you then do the two-point correlation function This is a temperature temperature correlation function. You look in say two directions and Then you do an analysis in terms of spherical harmonics And then the Legendre polynomials which enter their carries multiple Label L and you plot then These moments here as a function of L and you get this very characteristic spectrum And as you know that contain contains lots of important information on the moment of dark matter from the Elitified of these peaks amount of binary symmetry from the position of this you get that the universe is flat and so on Now So you have such power spectra And What goes into this is the following if you really compute that that's here an integral of a product of things One is the so-called primordial power spectrum This is what you calculate from inflation and the other the so-called transfer function That tells you and there's a lot of work in this transfer function That tells you if you start with a certain initial power, then what do you really see in the end? As a temperature fluctuation When the photons decouple and that involves all the physics of the The Plasma Before decoupling before the photons decoupled you have a plasma of say Charged protons of electrons of photons And which are in thermal contact and then their interactions they These If you now have density fluctuations then You get The interactions which have in these plasmas sort of the sound waves there they Are then important for what in the end you see as temperature fluctuations That's a big business in many codes analytically quasi analytically numerical which has been worked out now for many years And out of that in the end you get this Power spectrum Now the remarkable thing is That the outcome here Can be described now the say the physics between the primordial Fluctuations and what you see in the temperature fluctuations is something which In principle it's known one has to work it out, but in principle it's known What is farthest important is the input the primordial Spectrum that one characterized if you look at the fluctuations in Say the metric Then you can have a scalar component and the tensor component You can have different Lorentz contributions there and These are two kinds of fluctuations here characterized by these quantities and It turns out that these This input this primordial fluctuations are Almost flat that means they depend or they almost don't depend on this wavelength the moment the wave number here or the corresponding wavelength They are almost constant, but there is a tiny slope And so these tiny slopes is what you can calculate in inflation The remodels and what you can also calculate is a relative normalization of these two fluctuations This is this little art so called tensor tensor to scalar ratio Okay, so this is just to explain what it is And I cannot explain how you get there, but just again, that's why these numbers are important You have an s minus one which gives you the slope in one kind of Fluctuations and and t which is the slope in the tensor fluctuations And this is a relative normalization. These are the three quantities Which you have for these models and now You can go on and calculate And this is from this latest Paper of the plan collaboration, which shows you how for sheen how different models how well different models do So here you see now a plane This is the tenth set of scalar ratios is the quantity r Which I just explained what it is as a function not a function this is the One axis in this plane is This little r and the other axis is a scalar spectral index and then What you see here are one and two sigma Regions of what you get from the plank data The different colors here correspond to different data sets. So that's a complication We don't worry about Just always look at the dark blue here and the light blue here These are the two real. This is one sigma. This is two sigma And now the question is whether the predictions for little r and ns lie a say the one sigma or the two sigma region or not And this is here compared for different models For instance, the famous chaotic inflation m squared phi squared It gives depending on better for the number of e-folds you take 50 or 60 give you these two points So you see that lies outside the one sigma and the two sigma region. So it's now really disfavor. That doesn't work If you decrease the power Yes, so this was phi squared If you decrease the power say to one you get this So this looks better it is in the two sigma region, but also not in the one signal region Then You get a band here This is natural inflation That still overlaps a little bit with the two sigma region, but it doesn't look that good Anyway, the data may change a little bit. Don't worry here. So but this is the situation now And then you see something down here, which looks very good And this is a starovinsky model And I will come back to that. So this is this and then there is something Which connects this chaotic inflation or interpolates between chaotic inflation and the starovinsky model This is indicated by these yellow lines the so-called Alpha attractors. I will also talk about that Anyway, that's the situation now recent developments Of course, there are no key references now. There are also no reviews. I will just give a couple of examples So I will give you one example from a small field inflation In fact, something we have been working on because it has to be pushed To I think quite a detailed level I will say something about the impact of the bicep data And then I will show you what the potential problems are with large field inflation Or what the status is of that if you embed that in supergravity And then I will discuss the starovinsky model And okay, then I I forgot to say what I do here where you will see So that's the status Now, let me start with one Small field inflation example that's supersymmetric hybrid inflation That's uh The hybrid inflation also goes back. I gave you the earlier reference before This is some work done for a while with these people by these people And what is I think interesting about this is that this connects bariogenesis inflation and also dark matter So, uh You remember when we discussed leptogenesis yesterday, we had rather high temperatures So typical temperature say it was 10 to the 10 gv and uh now For many years, there is something which has been discussed a lot the so-called gravitino problem you say if you go to Super symmetric theories and you have such high temperatures in the early universe Then there is a danger that you produce too many gravitinos And that if the gravitinos are stable is the danger and there are the dark matter Is the danger to have too much dark matter If they are unstable and decay then because they decay late It gives you problems with uh nuclear synthesis. So that's a gravitino problem On the other hand, uh, I mean this has be calculated this rate and it is known that for numbers which you expect say supergravity Gravitino mass of 100 gv a gruino mass of a tv And now this reading temperature if you just multiply that together Then you get about a value of the observed dark matter That means if uh the um Temperature sort of the maximum temperature in the early universe would be about that what you have in leptogenesis Then maybe that would just fit together with uh What you need for dark matter But then why should that be? I mean, how can it be let's say the maximum temperature which you have in the early universe is about the temperature of leptogenesis Now, uh, there is a funny thing you remember or maybe a curiosity, but maybe also interesting You remember yesterday that's a typical number of the heavy neutrino Which made leptogenesis was about 10 to the 10 gv and this effective neutrino mass was about 0.01 ev If you calculate the then the decay widths of this heavy neutrino you find it's 10 to the 3 gv Now imagine The evolution of the universe is such That you go through a phase where say after inflation That you are dominated for a while the energy density is dominated for a while by the By by the by the mass density of such heavy neutrinos then Their decays would reheat the universe and give you Will be crucial for the heating process and then you can compute what's the heating temperature is that means up to which temperature You would reheat The universe and you find out that that temperature is about 10 to the 10 gv So that means If you have such a scenario in cosmology, then Maybe you can really understand why The maximal temperature of the universe is such that you can also explain the dark matter in terms of gravitinos Now, how does this work? In fact, you can make a model like this not a model But I would say in hybrid inflation that picture can be naturally realized So you start now from Superpotential or that you know now from the lectures of marcos luti This is what I had yesterday written without superfields just for fermions and hicks You have in unification here these couplings of these 10 plates To n5 star plates to 2 hicks fields and this is a term for the right hanging neutrinos Now, what happens in In supersymmetry is the following if you want to break To get also As you get the masses for these fields the mass Of these heavy right-handed neutrinos if you want to get them from spontaneous symmetry breaking You need a field whose vacuum expectation value breaks b minus l that's denoted here is this and then To realize that in supersymmetry you need a partner field a second one and you need a singlet which Tells you in the end that's the vacuum expectation values Are given by this graph which you need for b minus l breaking So this superpotential was first the sort of the basic Superpotential for discussing symmetry breaking first used by faillet many years ago for sq2 breaking. This is well known Now it happens that What you then have This superpotential for quarks and leptons the usual neutrino couplings and so on and this one. This is just The superpotential of hybrid inflation Which was considered independent of this before So that means if you start from neutrino physics and you want to Spontaneously break b minus l then The superpotential for b minus l breaking gives you automatically an inflat on candidate Maybe the singlet and The scenario of hybrid inflation Now what is interesting about this is that you don't have any parameters In fact, all the parameters here are essentially fixed from the masses and from leptogenesis So you have one more here for this strengths and Just this sort of minimally supersymmetric framework Including a spontaneous breaking of barion minus lepton number includes What you need in cosmology now all together namely inflation led to the genesis and dark matter now You get now from this the typical potential of hybrid inflation, which is well known This is the potential now as a function on the one hand of this field s this Which breaks the symmetry and the inflat on field here Which moves in this direction and the characteristic feature of hybrid inflation is that the inflat on direct in the inflat on direction the potential is very flat and in the other The curvature changes. So at early times The symmetry breaking field the field which breaks b minus l symmetry has a steep potential And then it becomes flatter and flatter and eventually the curvature becomes negative It's like a second order phase transition And you get a spontaneous symmetry breaking And in this spontaneous symmetry breaking that has been where it started You have a process which is called tachyonic preheating So you generate rather quickly You go from a homogeneous inflationary phase To the phase of b minus air breaking In fact, the situation has been even simulated on the lattice by these people already for a simplified model number of years ago And this shows you how the expectation value of the field breaking b minus l approaches its equilibrium value and this shows you During this process during this process of tachyonic preheating what kind of particles you produce and These particles as a function of their momentum And you see sort of this is all This is big only a small momentum So what you essentially produce during this process is many particles, which are rather soft now starting from here You can then work out After this stage of tachyonic preheating you remember in the if you think back yesterday we had this Plot where the co-moving Hubble radius was shown as a function of the scale factor which is essentially time and first it went down And then up again So the point at the bottom is the point where inflation ends and where the hot big bank starts And that is here Okay, so with this method of tachyonic preheating you can then calculate the abundances of your particles and What comes out of this phase transition at the end of inflation and then after that You can study the time evolution of the system during the hot big bank by means of standard bolzmann equations But you have a couple of particles and you have to look at the interactions and but these are known techniques and that was done you can look at This chain of bolzmann equations and write a code for that and solve these equations and so on And then you find what the time evolution of the system is in the beginning The energy density is dominated by these heavy scalars corresponding to be minus air breaking which you would use And then you have also this is an algorithmic scale you have also contributions immediately from radiation You have some right-handed neutrinos You then you have gravitinos from the beginning but then More and more during decays more and more energy is pumped into the plasma from slowly decaying heavy particles which break b minus a and Then the temperature increases and then So you get a shift from an equation of state which is metadominated to one which is radiation dominated and in the end The hot big bank starts here. It's dominated just by radiation but as a cause of The various processes which took place the plasma you have also generated an asymmetry b minus a And you have generated a gravity non-amber density So this is what gives you dark matter and this is what gives you the various In the way discussed yesterday by a leptogenesis Actually just a curiosity What is interesting is that during this period this transition period from So the first time Scales After the end of inflation the temperature of the system essentially does not change I mean you shift the distribution of the energy density from matter to radiation, but the temperature Remains more or less constant which makes the calculations reliable. So you have Essentially a plateau now You can work all this out and That's a lot of stuff and I just give you some Results for that. There are weak, but there are implications also for LHC physics And that depends on what the dark matter is The dark matter here comes from gravitinos and there are still two possibilities depending on the masses of the gravitinos Either the gravitinos themselves are the dark matter In this case here, you get a lower bound on the gravitino mass of about 10 gv As a function of this effective neutrino mass and that gives you an hc phenomenology some constraints Alternatively, if you have heavier gravitinos say a hundred tv They can decay into hexinos or winos And then you have neutrino Dark matter non-termally produced With and what this picture then gives you is upper bounds On the masses of these particles. So this is something to be searched for at LHC Now these models were believed for a long time to have some problem namely That these hybrid inflation models could not really get the right spectral index of 0.96 It was believed that the spectral index is always higher and So as inflationary models, they were not so Good However, it turns out that if you look at that more closely And if you take into account the effect of supersymmetry breaking then The inflationary model is significantly modified and you can also get The right spectral index So Let me just tell you in supergravity if you work it out, you know now from marcos lute's lectures What the scalar potential is in such a theory you have basically a constant that what drives inflation You have a logarithmic corrections to that that what gives you a little slow and makes in platon field move You have a supergravity correction and then you have here Something which is proportional to m3 halves the gravitino mass which generates in fact a linear term in the scalar potential And that makes the Hybrid influence the hybrid inflation model into a two-field model of inflation You can work that out. You have now trajectories You have a complex inflat on field with a real and Imaginary component and in this plane inflationary trajectories on which inflation Proceeds are now given by certain lines And that means now that an observable like the scalar spectral index Depends on the trajectory on which inflation proceeds that means on the initial condition And there are a couple of trajectories here if you move to the right for which in fact you get do get the right spectral index 0.96 And if you further pursue that There was also another problem which was often mentioned that in hybrid inflation models There's a danger that you produce too many cosmic strings And cosmic strings are in principle interesting because this is something you can also look for an observation But if you have too many of them, that's also excluded Now it turns out if you are in a parameter space, this is coupling lambda of this b minus l breaking part And this is the width of b minus l breaking then And The region here in parameter space where the spectral index comes out, right is this green bed And you see that then the constraint for which comes from cosmic string production is automatically Satisfied So this is just an example of hybrid inflation which has been worked out to some detail in fact with some effort over a while And about a year ago we were about to finish to to publish say the Latest paper in a series of all that And the spectral index Sorry the tensor to scalar ratio which you get here is about For that you find that it has to be smaller than about 10 to the minus 6 So at the time when we were essentially finished The bicep paper came out Okay, so we had this prediction of 10 to the minus 6 And then there were these bicep data That you may have seen I'm sure you have seen that I heard a colloquium on that where the indication was that this Tensor to scalar ratio was 0.2 Which would mean that all the stuff which I showed you so far was irrelevant The data looked very convincing So it seemed that the small field inflationary models were sort of ruled out And one needed large field models like in chaotic inflation or alternatives and Okay For reasons of time Let me not really spend on that you have seen that or this is a cosmology part to explain What here e-modes and b-modes are you get a world pattern of polarization lines down here and here you get tangential or radial polarization pattern I will not Explain that here you can find a nice discussion of that also this review by bauman How this pattern emerges Let me just tell you what the present situation is with that This bicep paper caused a lot of activity Well, I think among The experimenters I think to find out what is true among the other collaborations and also among the theorists Now to find out whether one could construct large field models Which were nice and theoretically acceptable So I discussed some of that now, but first let me mention what the present experiment situation is Here, uh, in fact, uh, as you may know, there was Earlier this year There was a joint a publication from bicep this bicep collaboration and plug On the status of these b-mode signal And the problem in these b-mode signal is that these polarization patterns which you see there you can get On the one hand from primordial Uh Tens of fluctuations, but you can also get it from dust So the question is on the wave between The last scattering surface of the cnb and today How much dust is there and the bicep had collaboration and assumed that there was essentially no dust And plunk now had a dust map And so they tried to combine Their information and to extract from that Reduced data set so here's the points which you see here. They are That is The bicep data without the dust subtraction what you see here The points below is what you get if the dust is subtracted and the red line Is a theoretical prediction from for this B-mode contribution if you just include so-called lensing contributions Now so you see That is rather well Consistent so there is no At the moment looking at this there is no you don't really have to worry About these b-modes and large field inflation anymore On the other hand they also made a statistical analysis these collaborations and came to the conclusion That there is almost a two sigma effect which Uh remains this is a likelihood plot and the preferred value For this tensor to scalar ratio is in fact about 0.05 So maybe there is still hope and maybe There still is something in the data So I think this remains a very important and hot topic and we can hope maybe still During this year for some more information for sure within the next few years And it would be wonderful if Something would be found But at the moment there is no Evidence really and therefore the small field inflationary models are still okay So this brings me now to an example one example, which I want to discuss concerning large field inflation And a large field inflation is say the typical example is chaotic inflation in square five squared You want if you get these large field values to want to embed it in supergravity And in fact A lot of work on that has recently be done in the context of string theory Because people say if I get such large field values, I have to know about the ultraviolet completion of the theory and So I should do some string theory and the number of string models have been constructed for that Now what that means is that at the end of the day, which I mean First already a number of years ago. There was axion monotony by silverstein invest file. There were aligned axions there is There are now various models in f theory by a number of groups. So there's a lot of interesting activity But what but strings Theory as you know is complicated And you always can study only certain aspects of that and what that boils down to in the end Is that you make certain assumptions on your Compactification manifold on the stabilization of moduli and so on And in the end you are left with an effective supergravity description of your inflaton field That's what you do in the end. So in the end you do some supergravity model Now what we were interested in was the following If you this large Field inflation models are realized then the energy density during inflation is large It's of the order of the gut Scale to the force power this energy density And then you really have to worry about the destabilization of moduli the supersymmetry braking and so on And now the simplest model you can Or the simplest scenario, which you can study is in fact where you have say at least the volume modulus of your Moneyfold or which you compactify and supersymmetry braking fields field Which brings you from an entire center vacuum, which we typically have To start from back to minkowski space and then you have your inflaton Now So we studied that for a couple of for some examples of moduli stabilization And i'll just illustrate that by one example Maybe the most well known example for moduli stabilization, which is a kkt mechanism. There, you know You have or you may not know or i don't know whether Vakus Lutti had time enough to Discuss it but it's sort of a well known modeling supergravity how to stabilize such a modulus field You have a certain keller potential there, which is this And then you have a super potential, which is this and then You Well, you study the stabilization First what you get is a so-called anti if you just take these things you get an anti They sit the vacuum down here Then you need something to an uplift mechanism which moves you up back here to minkowski space And one thing which you can do is you can use a poloni field Which means that you add essentially such a term to the potential and then you are You get something like this So That means if you have a potential like this you need You have a modulus field and you have something for supersymmetry braking then You add the You add the inflat on and for the inflat on you also need a keller potential and One which you essentially need for these large heat models is something which was proposed by By yarangida and friends a number of years ago where you have for the inflat on field A shift to which So putting this together you can then works this out And see what happens and for which parameters and field they use the thing is consistent And what you find in this case is something a little bit unusual You can first of all, you need large supersymmetry braking So no hope To see then anything at the hc the gravitino mass here Which you obtain Should be larger than about 10 to the 14 gv. So you get supersymmetry braking somewhat below the scale of grand unification And the potential here in the plane of modulus and Inflat on looks like this So you have here a valley for the inflat on but this valley Disappears For some field where you can't really Read that I think here. This is Should be a field you have about 20 minutes of the plant mass for the inflat on field If you have such large field values Then as the local minimum for the modulus disappears, you cannot stabilize anymore your modules And on the way up here This chaotic inflation potential Is flat you get a Correction term which is negative and has a higher power in fire So if you then work this out how inflation would look like it would mean that the potential goes like this and then this correction becomes important You never reach the top here Of this potential where this becomes flat because saddle point because Already before the minimum for as a modulus disappears so if you are want to have together modular stabilization and large field inflation You have to be very careful and it becomes difficult the spectral index comes out nice and the tensor to scalar ratio is A little big but still okay, so phenomenologically these things Could work You can summarize these you can summarize these Results also for the other more other molds in the following This effect of flattening of these potentials due to modular effects is generic That you always find in all these molds and then you can compare the The situation with the The plant data the predictions with the plant data This is the game this plane of r the tensor to scalar ratio and the scalar spectral index And these are the plant data one in two sigma regions and You for comparison what you have here this purple band is what you get a natural inflation These models which I discussed now chaotic inflation with modular effect are this green band So similar to this natural inflation and then for comparison power potentials for different powers Give you this band So up here you have chaotic inflation and then smaller powers you have down here and so that gives you this band Now there is one important piece of Information which I should Give you to be honest I mean when This plot was made That was in january this year all that was available were the published plant data at that time And then this was a one sigma region and this was two sigma region And so Things looked reasonably compatible In the meantime Now the uh, plant data had been the latest plant data are available And essentially what happened is that the old One sigma region now became the two sigma region So that means There's a predictions of these models which I discussed here and also all the power potential large field models Look a little bit disfavored. I mean they look they are outside of the two sigma region So we will see what happens either with the further development of these models Or with the change in data But this is status Okay, so I've discussed two aspects now Say a typical small field model with hybrid inflation type model with implications connections to dark matter and Diogenesis and now some aspects of a large field inflationary model now, let me explain To you what the star windski model is Which is somewhat intriguing. I think although it's not so clear what it means Somehow the this model is it's not explicitly written But it's essentially contained in a paper by star windski in 1980 Maybe one of the first papers on inflation at all On which then which was then also the basis for shibis of and mukanov to calculate the density fluctuations Now that paper Just looks at a modified gravity Lagrangian. So it takes this And then instead of just the einstein-hiewald term here. It takes r minus an r square term And for dimensional reasons the scale is needed here Now there is a standard thing in gravity used in many Circumstances namely you can do a vial escalation Rescaling a vial rescaling that means you switch from the metric g to a metric g to a metric g twitter times five Then you can just work out what happens with this Lagrangian And it is just an identity That you get this Lagrangian Now in fact what you know is If you have an r square term if if the Lagrangian has just Zeinstein the gravity Lagrangian has just r then you have a Lagrangian which is quadratic in time derivatives If you add r squared then you get something which has four times and you know and it is known that Then there are more propagating degrees of freedom and in fact you get one more you get a scalar degrees of freedom And to make that manifest is that's a well known technique to rewrite such a Lagrangian in terms of The metric and this additional field in fact you can do it not just for this But you can do it for any function f of r And then what you get is you get the einstein term for The modified metric and you get in an in addition here Scalar which has a non kinetic non canonical kinetic term and some potential Now you can Make a field transformation such that scalar field has a canonical kinetic term That is such a field redefinition here And if you now rewrite the Lagrangian in terms of this should be The small phi in terms of the field which has canonical kinetic term Then the potential is this so it is It is a say m squared the scale times m Planck squared and Then you have this And then you have an exponential squared so Now you see what this potential does is this potential For large field values goes to a constant This here Really has this slope It goes up to What game I think and the other Goes like this So you go to a constant and you reach a certain plateau In fact, that's similar to Hybrid inflation in hybrid inflation for very large field values. You also Reach a plateau. Well, it depends on how you do hybrid inflation if you do I should say If you do this hybrid inflation model, which I described to you then the potential will increase logarithmically at very large values But whether that happens or not depends really on the cannot kinetic term What has been discussed is say d term hybrid inflation in connection I have no time to discuss that here, but in connection with super conformal symmetry Then you have a different Keller potential for this field you get a different kinetic term and in such models in fact it happens that the potential asymptotically also goes to a constant to a plateau and then you get some modification of that and Then this is something which I had no time to discuss but then you also effectively get this potential and one model which has been discussed a lot where You also get such a potential is Higgs inflation In Higgs inflation if you add to the Lagrangian A term which is say the Higgs potential squared times Some factor which is very big Then you can also by means of field redefinition Transform this into such a potential. So it's quite a curious potential And what is interesting is here it comes out just from This second term in gravity, which you add and so that may make you think whether this has some universal property I mean you can derive this kind of potential in from a couple of other explicit models You can get this and then you can rewrite it this way Now you can work out And because of this plateau This model is the perfect model of inflation inflation You know Start somewhere here and then it usually ends somewhere here Where the slow roll Conditions are no longer satisfied and you can work out What the predictions are for this model You get in fact what is interesting you get the scalar spectral index is the same As for a chaotic inflation it is this which works nicely the tensor to scalar ratio is smaller It's this value and so Significantly smaller than chaotic inflation, but it may be not impossible to reach it Say within the next 10 years depending on what people are doing But if something like if this would be true, then maybe it could also Be realized now it also has just one parameter here the model. So it is somewhat intriguing It is somehow I mean if you are very naive it's sort of the first model Ever written down and it's now the one which works best But you of course you still have to ask the question. Why is this and what is the significance of that? Is this just an accident or not now That brings me finally to a little Curiosity or something by the paper ashore in the sky written recently by Kaloshen Linde Which in fact interpolates between chaotic inflation and the star bien scheme on So You see this is again This plane which you now see in all these inflationary models where you have the tensor to scalar ratio here and the Scalar spectral index here and these blue line Contours here the 1 and 2 sigma region of the plant data Now As we saw already on a in a previous plot chaotic inflation You have here m squared phi squared And the star winsky model Down here Actually, I should say one has to be careful with this because if I say the star winsky model is down here It has to do with the scale. Of course, this is a linear scale And of course you Everything which is smaller than say 0.05 looks like zero here, but of course are many many The different possibilities of what is the value of our little r really is Now so there what you can do is you can write down now You can look at say chaotic inflation. However with a non-minimal kinetic term So the minimal kinetic term would just be d phi squared But you can just multiply it by this in fact a structure like this you get in all these modes with super conformal symmetry you get Denominator like this and it has a feature that The field range in these modes is limited And you see if you Of course you you want to have Phi equal zero as a possibility but you can Make phi not too big. Otherwise you hit the singularity here And so you feel where your phi is As a boundary so this You can now again do the trick which we did by the star winsky model. You can do essentially as here You can do a field redefinition which looks almost exactly like this one And then Is this Lagrangian in terms of the field with the kinetic canonical kinetic term looks like this We have this and this Then you have a tangent Hyperbolic tangent Of the xi it's the same as in the star winsky model And you have here m squared multiplied by alpha The phenomenology is fine In fact independent of alpha the scalar spectral index is always 1 minus 2 over n which is this 0.96 also it works very well Therefore you get here a straight line And this value of r. However the density scalar ratio now depends on alpha You see if alpha is 1 You get what I showed you previously You get 12 over n squared Which we had If you you can pick also value for alpha where you get the prediction of chaotic inflation and everything in between so you get such a line I mean One should say however that the model interpolates not just between these two points It could go up also here and could go down further here to smaller values But it's the class of potentials which Have as two points chaotic inflation and the star winsky model Now what is but what does that mean? And now what is now what is interesting in About this paper I've never seen before on the archive a paper with so many nice colorful pictures as you can find them here and What they now argue that's a title asher in the sky Is that in fact that is essentially the non canonical kinetic term? Where does it come from? So the question is what is the meaning of this parameter alpha? And what you can now do is this is just a certain metric here, of course This factor which we had for this kinetic term and it corresponds to a metric of a hyperbolic just Two-dimensional hyperbolic space as you usually learn what it is when you study When you do general relativity And you want to work out Maximill asymmetric spaces Then for you get either the sphere and the compact space of the hyperboloid And the hyperboloid is characterized by the distance here from the bottom to here And that's but now this So This hyperbolic geometry has now one interesting thing Which is you can map the whole hyperboloid to a disc In a certain way by a certain projection And this disc is called Poincaré's disc Okay, this is Just a well-known result in mathematics And in fact the radius of this Is a free parameter Is the radius of this disc and it's related to the distance from here to here And now there is this Dutch artist Escher Who apparently made for all kinds of mathematical Constructions He made some interesting pictures and they are very colorful and You can find the references in this paper by Kaloche and Linda where to find These pictures and how they are made and what the story is behind them and so on so in fact He made a picture For the Poincaré disc This which should reflect some of the Poincaré disc this artist, which is this picture This I don't know what to say about the picture. Anyway, this is just a picture and If you're interested look up the paper you can see all the other pictures and Well, I hope that looking at this and thinking a little bit about the geometry maybe of the metric the kinetic term What could possibly mean from for inflation and so on that this may Inspire future work About the early universe, so that brings me to the end of my lecture