 Okay. Can you hear me? Yeah? Okay, so welcome this morning. We're going to start our first lecture on dark energy and modified gravity. So before I get there, so it's going to be a very theoretical course based mainly on field theory techniques applied for the development of dark energy and modified gravity models. We're not going to go too much on the observational constraint, but I'm going to tell you how we can shield these different type of models, how they could be possibly consistent with current observations without actually going through the data and proving that in detail. But first I'm going to give you a few motivations and this is going to be very quick before going into the core of it. So we know this is our cosmological picture of the evolution of the universe from a few seconds after the big bang. There was a period of inflation and I believe you have a whole lecture theory on inflation, the effective field theory of inflation. What we're going to be concerned about today is not the accelerated expansion that happened at the very beginning of the universe, but rather the second period of accelerated expansion that the universe is currently ongoing. So very recently we had someone for the past 20 years now we have very strong evidence that the evolution of the universe today is accelerating. So the data come from 20 years ago or more, but it was just a few years ago that the Nobel Prize of Physics 2011 was given to this gentleman for observations of supernovae that told us that the universe is currently expanding and accelerating. So I'm just going to show you a few quick slides on how this worked, what the physics behind this supernovae observations are and how from these supernovae observations we have some strong indication that the universe expansion is actually accelerating. So supernovae is an explosion of a star and one of the key thing with this type of supernovae is that they have a characteristic spectrum luminosity and so from having a luminosity and from the luminosity of its embedding galaxy we have some notion of the distance of this object from us. But also from its spectrum and in particular from the fact that its spectrum can be shifted by kind of Doppler shifting we have some strong indication of how fast this embedding galaxy, this object is moving away from us. So if we had a supernovae right now in this room, hopefully that wouldn't happen, this is the typical spectrum we would expect for this object. However, if the object is moving away from us or towards us it's going to be shifted and which way it gets shifted depends on whether it moves towards us or whether it moves away from us. But in most cases it's moving away from us and what we observe is a shift in its spectrum and so directly by analyzing the shift in its spectrum we can see, we can have a good indication on how fast the object is moving from away from us. So this is the level of the red shift we're just comparing we see something at a wavelength of 420 for instance when it should be at 400 so we can get what we call the red shift z and from the red shift z we get directly the speed of the object to be roughly 0.05 as compared to that of light. So it's moving at 5% of the speed of light it would look like and so that corresponds to a speed of let's say 1.5 10 to the 4 km per second. So it's huge, it's huge. Of course not every object is doing like that, it depends on the distance of the objects away from us and it's also an average effect but if we make a map of this supernovae and all the objects in the universe on what their red shift is or what the velocity away from us of that order of magnitude it was 1.5 10 to the 4 before as a function of the distance it is from us we see we see a trend like so and see so this really tells us that the universe is expanding it doesn't mean that we are in the center of the universe and everybody is away from going away from us it really means that that's not the picture we believe in but it really means that every at every point in the universe what we perceive is that all the objects are moving away from us wherever we are in the universe and so that means that the universe is expanding we're living in an expanding universe so this is the picture the hard big man picture where they may have been an explosion at the very beginning of the universe and and we're still within a period of expansion in here and that's just dominated by what the fluid was was doing but we're living in a world which is essentially gravitational there's a force of gravity and so even though the object started in an universe in expansion the universe is expanding like that the object started moving away from one another we would expect in the end of the day the force of gravity to at least to slow this expansion down a little bit so maybe there could be a phase of contraction at the end of this expansion or at the very least we would expect this expansion to slow down maybe not to get reverted but at the very least to slow down so if we go back to this Hubble diagram and we we try to understand what he means to have a universe which expansion will slow down it would mean that if we're looking at very far distances that corresponds to what happened before much earlier in the evolution of our universe in the history of our universe and so if the expansion is slowing down we would expect that what we see in the past would have been a higher expansion of the universe a higher expansion rate so if this is what we see at the moment it roughly looks like a straight line if you believe that but if you look at much larger distances you would expect if we had a deceleration if the expansion is slowing down that objects much further away from us would move faster because today the expansion we see this deceleration today the expansion would be would not be as fast as what he was has been in the past so that's what we would naively have expected but what we actually observe is quite the opposite the objects were slower actually in the past which means that the expansion of the universe is more rapid today which means that the expansion of the universe is actually accelerating and so that corresponds to the second phase of accelerated expansion in the history of our universe of course the scale the scales involved during this new phase of accelerating expansion in the universe are much smaller than what happened during the period of inflation and we'll go and we'll go through that and nowadays there's some strong evidence on the fact that the universe expansion is currently accelerating so supernova is one of them these are evidence from supernova that indicates that the universe is accelerating and sourcing this acceleration is some kind of fluid we call dark energy so in this plot we're not gonna this dark energy could be a cosmological constant or could be something else we just by this terminology here we're just indicating that there's some source that makes the expansion of the universe accelerating and then there's also some mass which is made out of the standard baryonic matter that we know that we made out of and then also maybe all of the dark matter present in universe so the key thing of this plot is to show that supernova indicates the existence of dark energy that makes the universe expansion accelerating but you we also have nowadays very independent type of observations that all converge towards that observation that universe is accelerating so these are observations from the cosmic microwave background and this is from baryonic acoustic oscillations we see that this type of observations are quite independent the physics involved is quite different the scales involved is very different and yet they all seem to converge towards that point in the concordance model of cosmology where the universe is dominated by some type of fluid around 70 something like that which makes the universe expansion accelerating and so you probably seen that pie chart before where we see that in terms of energy budget today it is the universe is dominated by around 70 percent 71 percent of dark energy 24 percent of dark matter and the standard baryonic matter that we know of only makes out of roughly five percent of the energy budget of the universe and typically this is sold to us as being one of the big puzzles today of why so much of the universe 95 percent of the energy budget of the universe is made of dark stuff we need to understand this dark stuff but what we'll see today though is that while this is surprising that there's some dark energy what is even more surprising is that it doesn't dominate more the energy budget of the universe very naively from a particle physics perspective we would have expected to have much more vacuum energy which leads to something called dark energy which leads to an accelerated expansion of the universe so we'll have a look at that today but in parallel what we'll see today and you probably know that this 70 percent of the universe energy project that leads to an accelerated expansion could very well be corresponding to nothing other than a cosmological constant and a cosmological constant was something that was introduced by Einstein from the very well not from the very beginning but that was introduced in Einstein's equations so these are just stuck in a in a wall a graffiti in a wall would say the Einstein's equation here was so the curvature the Einstein tensor in there and on the right hand side the matter component present universe and yeah so very early on at the time so there was a hundred years ago it was believed that the universe was roughly static there wasn't this notion of the evolution of the universe from the herbal diagram yet and so that was bothering Einstein the fact that if you have some matter present in the universe it will automatically leads to an evolution of the universe and so his original idea was to introduce a cosmological constant so as to make the universe static so as to compensate the effect from the matter present in the universe so while this in principle could happen as a particular solution it's actually a very unstable solution because if you add a tiny little bit more matter then it would make the whole universe collapse and if you add a little bit more of a cosmological constant it will make the whole universe actually expand and accelerate so adding a cosmological constant is not a very good solution to making the universe static you know in a stable way but that's actually a good thing because the universe is not static the universe is not only expanding it's also its expansion is also accelerating today and what we know today is that this cosmological constant could be one of the most natural source for the accelerated expansion of the universe and so dark energy could in principle be nothing other than a cosmological constant there is some level of theoretical wonder of why this cosmological constant would have the order of magnitude that we observe today to explain the level of accelerated expansion that we are observed today and this is that the origin of the cosmological constant problem that we'll discuss about today so there is between 10 to the 56 and 10 to the 120 orders of magnitude discrepancy between the naively anticipated order of magnitude of the cosmological constant we would have expected and the observed value for this constant and that's one of the biggest discrepancy was the biggest discrepancy of physics that we know of so if we like the cosmological constant to be the source for dark energy then we'll need to understand this order of magnitude discrepancy between observations and theoretical anticipations let's say so nowadays there's many surveys many missions that try to capitalize a lot of the physics of what can be making dark energy of today and in particular there's some observations as I said that lead to the belief that the universe is accelerating not only coming from supernovae but also coming from all the sources and so there are data from supernovae as I said from baryony cacosic oscillation from the cosmic microwave background from galaxy power spectrum just looking at the the spectrum of galaxies how many galaxies are mapping the galaxies from weak lensing so seeing how the structure in the universe leads to weak lensing and so all of these data all of this information come in at different scales for the universe and so give us a lot of information which is much more than just having one number telling us this cosmological constant is of that value and in particular we could mimic the equation of state parameter so if w is exactly equal to minus one that would correspond to a cosmological constant we can look at how that parameter would vary as a function of scale since we have various different probes and in particular as a function of scale we can say as a function of redshift we use redshift as our notion of time if you want our notion of distance and so the if we just had a cosmological constant we would have expected this equation of state parameter to be exactly equal to minus one throughout the redshift so throughout the recent history of the universe but if you combine all of these data that rely on quite different physics we see that there's actually some starting to be some some chances of evidence for something which is actually much more dynamical so not necessarily having w being exactly minus one for this region in z but something which varies and if you take into account the error bars of these observations this looks at least from this observation that there's something dynamical we should call dynamical dark energy is preferred over a cosmological constant at 3.5 sigma level so 3.5 sigma level that tells us our our confidence on how likely these measurements is really something physical as opposed to just coming from the statistics or just coming from one chance in 2049 that something like that could have happened so to put that in other percentage number we would think that there's 0.5 percent chances that this is just a statistical flick that there's nothing physical in there so if we put it in at that order many two within one there's something really some strong physics in there and I'm not an observer but we do know that through the history of observation particularly cosmological observations it's not the first time that we have the impression to see something at the two sigma level or the three sigma level so some strong evidence for something in the universe and at the end of the day it just comes in from the fact that all of these measurements are extremely complicated to make and maybe some of the error balls should be a little bit bigger than than what we thought they would be so there seems to be some evidence that the dark energy could be dynamical rather than a cosmological constant but this level of confidence is not strong enough yet to really determine with yes yes to be honest I I don't know I this I don't know I don't know at all yeah this is combined data for I'm not at all an observer I'm not at all dealing with this data yeah yeah so I would I would recommend so this is a strong motivation to look at that paper okay so this is just to give you some motivation for starting to look at dynamical dark energy as opposed to a cosmological constant and that's what we're going to do throughout this four next lectures so the the plan of the lectures would be to give you some theoretical motivations for looking at dynamical dark energy as opposed to just considering that the evolution of the the accelerated evolution of the universe is just due to a cosmological constant so we're going to look at some theoretical motivations and then we're going to go through different types of models of dark energy the first few ones that came up some 20 15 years ago and then we're going to go through a more formal classification of the different type of dark energy models that we can think of particularly based on their level of of how they how they behave within within observations how they can how they can still be consistent with observations and there's of course a thin line between what we mean by dark energy and what we mean by modified gravity so it's not like a model is it's very it's always very clear whether we should put it in one category or another category but I'll still go into something slightly disconnected to the vanilla dark energy type of models which are more within modified gravity where we really start modifying the way the gravitational force behaves maybe at large distances or maybe even the way gravitational waves would evolve and how that may modify the way dark energy could lead to acceleration of the universe or maybe even remove the need of dark energy altogether so we'll see how that goes 10 yeah 10 years ago in 2008 with Andrew Tolly we gave a summer school on dark energy and at the time we gave a different classification of the different models that people were thinking at the time so that was 10 years ago on whether we still believed in gr has been the correct description of gravity on all cosmological distance scales and then of course you could have a cosmological constant you could have back reaction where it's just coming from the structure of the universe leading to an acceleration of the universe a landscape of different vacuums so that there could be some anthropic arguments on selecting the different one of the vacuums in this in this landscape we could have dynamical dark energy and this dynamical dark energy how it decouples to matter or high couples to the curvature would lead to different type of scenarios either quintessence which we're going to talk about later today or brand sticky or type of f of our modifications of gravity but it's really gravity in a scalar field that we'll also discuss either later today or or tomorrow or we could have more dynamical dark energy and not really from an additional degree of freedom separated to the gravitational sector but really within the gravitational sector itself and so for instance the gravity term would not be a spin to massless particle it could effectively have a mass like a resonance or in different ways to breaking locality or through breaking Lorentz invariance leading to a whole set of different type of models so that was 10 years ago and just recently I thought well that's that's really now quite all just to give you a sense of how dynamical this whole field is and how many ideas have come up since then I just looked at the different models that we have had more recently and so I looking forward to see what's going to happen in in the next five years what's interesting and I'm not going to talk too much about is in parallel of developing this accelerated rate rate of model the observations are also doing a pretty good job at discriminating between the different type of models and so there are some current observations that allow us to tell us that some of these models could not any longer play the role for dark energy but still we have more and more models that could potentially help us understanding what dark energy is let alone what dark matter could be so I'm not going to go through all of them in detail but this is really just to give you a sense of how bubbling this this field is and I'll stop here for the the slides part and now we're going to go into the the core of the matter any questions about this yes yes yeah yeah I think this is the contribution just from this is just from dark energy so this yeah there's some prior assumption coming in there and exactly this is one of the key issue in a in a in putting many of this data together is that some of them to be to being able to extract some information about dark energy there's some prior which is there already and a lot of the time what is taking is lambda cdm so cosmological constant and called dark matter to explain the background evolution of the universe and we're trying to extract some of the behavior of dark energy itself so so it's it's it's not at all a very clean thing to do where there's a unique answer that there's a lot of prior there's some biases coming in into the game yeah anything else yes oh yes that's a good question that's a very good question it looks it looks like a at ten sigma no if you put all of that together that's a that's a very good question and yeah does anyone know actually do you know each one is three separately yeah yeah yeah but all together i don't know but it's uh it must be it must be quite a lot yeah i mean if you just if you just look at it from the arabas rise yes this is this is one two sorry this is uh yeah one two three for each one of them so one two three here but compared to that this is this is huge yeah yeah anything else should i have a light here so the plan is in should be available online let me know if it's not the case and and different dark energy models as well so we're gonna start today with the motivations and some of you may already be quite familiar with a lot of what we're gonna be doing today particularly the coincidence problem the cosmological constant problem but i think it's good to put everybody at the same level and also to put down the the the language we're gonna be using for the next three lectures so we're gonna go today through the coincidence problem and then through the old and new cosmological constant problem and so these two things are motivations for looking at alternatives of just having a cosmological constant as the driver of the accelerated expansion today so let's throw this for most of today and tomorrow we'll be working in four dimensions this may seem obvious to you that we live in three plus one dimension but we're gonna relax this assumption later on this week and we're working in a minus plus plus plus signature so we have a e-time unit is the diagonal of that and standard rectangular coordinates and for most of what we'll be doing we'll assume laurence invariance assume laurence invariance and a lot of the time we'll assume locality okay so let's let's start by fr w or fr w the friemann-röbert-röbertsson-lemet Walter metric and i'll just write it including the laps because it will help us deriving the friemann equation so i'm not gonna derive that for you i'm gonna work under the assumption that you have all derived that earlier in your life and then something else that i should have mentioned is that we're gonna be working with a flat spatial metric the main reason for that is because we want to really focus on the component of the arc energy as opposed to the spatial curvature so this will just be um or three invariant just the x1 squared plus dx2 squared plus dx3 squared so if i have this frw metric and i put it into my einstein action so this is my antonic action squared over two ribbon curvature and then i'll have my matter like range so this is the matter like range and when i say matter i don't really mean matter dark matter as opposed as opposed to radiation as opposed to massless particle i mean all of the stuff that lives on our spacetime so this is all the all the species that live on our spacetime they will be living into this matter like range and that's a metric g in frw is given by that and these are all my species so this is my action for gr and now let me work on frw so if i have the einstein scalar curvature on frw after integration by parts this is um on frw this is n a cube and then if i integrate a few things by parts to remove the time derivatives that would act on my lapse n what we get from here and there is minus 6 a dot squared over n so there's an n squared that one of n squared that comes from here but then it gets cancelled with one of the n that come from there and then from the matter like range and part plus this thing will have a cube n and then minus rho of a so this may not be the way you see it usually but i'll i'll derive it like that so these are all my matter fields so i there and then on a frw i will just assume wow this just happens to be the case that the contribution from all the matter fields led to uh are captured by the energy density of every one of this matter field which is a function of how the universe evolves which is captured by the del factor a so rho is the energy density of these fields another moment it really corresponds to the sum of all uh the matter fields but we'll specialize on a particular case in a in a second so the reason i write it like that is because now we can vary easily just on top of our head derive the friemann equation which you all um sure you have seen in that language the friemann equation is nothing other than a constraint it's a Hamiltonian constraint constraint is derived from a Lagrange multiplier now Lagrange multiplier here is something which has no is not dynamical it has no dot acting on it it's a little bit more concrete than that but that's what we can take at this level and it's nothing other than the lapse so if we vary this action with respect to the lapse we get the constraint which is the friemann equation yeah at the moment this is everything at the moment this is everything you can think of everything that you think should live in our universe and then we'll focus on special cases so at the moment this is the abstract we have when there could be everything that you believe is going to live in your universe yeah yeah because this is this is at the level of the action so the pressure will come in into how this energy density will will evolve but the at the action sorry at the level of the Lagrangian you've seen that the Lagrangian in principle corresponds to the kinetic energy minus the potential this is just giving you the potential the the effective energy density that lives in into this Lagrangian where the pressure comes in into that language is how this evolves it's in the equation of state of of these different fluids and then to understand how that works how we will need to categorize between the different ones yeah including i can't hear sorry in the metric yes yes i can't hear yes this is the adm formalism yes yeah so n is a lapse so n is a lapse this is in adm you want this is uh the lapse it's in uh it's an frw adm if you want and this is a scale factor and the fact there's an assumption that comes in here which is actually um the right thing to do but the fact that we're just gonna put this assumption into the answer equation that corresponds to the modular space approximation yes yes the integration by parts yes absolutely so when i told you this was the action for gr i lied to you this is actually not the action for gr the action for gr is that plus the boundary terms that would be there to remove precisely this so in here what we have is things that involve second derivatives acting on the metric what we should have been doing really is just write this down into first and to um do the integration by parts first this is what the action for gr would be and then in that formalism when we have the second derivative acting on a metric in here we have the um on the boundary we have the hooking given boundary terms these are the hooking given boundary terms this is the action for gr and so the when you integrate by parts the boundary terms precisely cancels those ones so that we don't need to worry about the boundary terms this is what it should be with these boundary terms the action for gr is not this thing without the boundary terms yeah yes uh some notion of the of the fact you probably seen things where you can set the the the laps to one you probably seen anything like that and on purpose i kept the laps here uh not be equal to one because i can do it but because also there's a lot of physics involved into these laps in particular we can derive the freeman equation directly by varying respect to the laps it corresponds to a Lagrange multiplier you can always perform a feedback definition to a new time coordinates so as to set the laps to one and this is quite trivial in in this case you can always say well i don't like to be working with n squared dt squared i want to be working with a new notion of time where this is equal to minus dt tilde squared for instance and so you can easily integrate that out whatever function you had in there you had decided for yourself you can say now you'll have your d tilde over udt is equal to n of t and so you can integrate that out and in particular you can work in conformal coordinates uh so that this is exactly equal to a squared of t so this may be a conformal coordinates this is not what we want to do today we want to keep for ourselves for a second just a level of different variance or time reparameterization in the variance of action because it would allow us to directly derive the constraint and because i'm allowed to do it sometimes you set this to one sometimes you set this to a squared just the fact that some people may want to do one thing some people may want to do the other thing just gives me a strong motivation for being quite agnostic about it and leave it arbitrary for now okay now we see that there's no dynamics in this lapse and so if i vary with respect to this lapse i'll get my friend my equation here comes in as an inverse power and here comes in as a positive power so it's quite simple to derive them here i'll have a minus sign coming from the fact it's one of a n we have this um m plank squared and then on the right hand side we'll have a cube yeah sorry this should have been i put two minus signs this is just one minus sign so this is uh correct like that okay so let me rewrite this equation as a dot squared over n squared a squared is equal to one over three m plank squared does that remind you of anything possibly yeah this is the freedman equation where this is the Hubble parameter the Hubble parameter square so you've probably seen the Hubble parameter without the n in it now that we have derived that from an equation we have no more use for n at this level if you were i'm allowed to make change of time if i wanted to set n is equal to one if i wanted to but also you can keep it here you know that if you choose conformal time or if you choose the physical time the Hubble parameter will have a different level of n in here so sometimes some of you may have seen that the Hubble parameter is equal to a dot over a but some of you may have seen that the Hubble parameter is equal to a prime of a squared it's not like we're doing different different things we're dealing with different universes this is exactly the same thing in one case n is one and in the other case n it has been set to equal to a we can do either things and sometimes one time parameterization is better than the other anyways i didn't want to to spend too much time on that so this is the freedman equation and if we derive now vary with respect to the scale factor a we get what is called the rachadori equation which kind of tells us how h dot behaves i'm not going to write it down exactly because we don't is not going to be of any relevance for us today but if you combine this equation with the variation of the freedman equation you would end up with the conservation of energy equation so which is not independent which tells us how rho dot behaves so this is going to be equal to minus three h rho plus p and the pressure comes in here into the variation so the pressure is related the variation of the the energy density and at this level so this is we can put the energy density for all the fluids if we wanted to and the associated pressure for all the fluids if we wanted to we're not we're not going to specify right now what we want but what we could do is for a particular fluid we can define the equation of state parameter the pressure is equal to w times the energy density and so if we put out here we'll get that rho dot is equal to minus three h rho one plus w how much can we trust in that last equation ah at the moment this is an assumption so so these i can always write if it's f r w i can always write there's an assumption if the if you assume and this is constant that's why the assumption lies in if this dependent on time then there's no assumption there right then it's just saying that something that depends on time is related to something else that depends on time so this you will see different scenarios okay yeah yeah so for stating that um dust or radiation behave behaves as a perfect fluid with a constant equation of state is a very good assumption yeah uh deciding that dark energy is a fluid which behaves with a constant um equation of state parameter that's not necessarily a very good assumption yeah and as we see in the w for dark energy may not necessarily be a constant equal to one okay so now if we can look at that cosmological constant so what we'll do right now is just assume that the universe is dominated by a cosmological constant it doesn't mean that that matter and radiation and standard matter is not present but we're just going to look at the contribution from a cosmological constant to see what that would do on the evolution of the universe and in that case the energy density would be a constant let's just call it uh lambda and then we see directly here that corresponds to an equation of state parameter equals to minus one for a cosmological constant and if we put this cosmological constant into the Friedman equation we get a Friedman equation that tells us that the Hubble parameter is equal to uh one over three ampline squared oh sorry that cosmological constant lambda and so if this is constant it leads to a constant Hubble parameter lambda is constant h is a constant but recall that h is equal to uh let me now set the lapse to be one if I work in a gauge where the lapse is one I can always choose my time to do that now it'll just be easier it doesn't change anything if you don't do that I will have the time variation of the lapse sorry of the Hubble parameter is error is equal to a double dot over a minus the squared minus h squared and so we see that the acceleration of the scale factor is equal to the constant Hubble the constant Hubble parameter which is positive so we have an acceleration an accelerated expansion of the universe can you see if I write here no okay so having a cosmological constant leads to the acceleration of the scale factor being positive and so this is an accelerated expansion so really all is good we observe that universe expansion is accelerating and the first thing we can think of putting a cosmological constant there leads to an accelerated expansion of the universe so everything is great right five more hours to feel so shall we just stop here or no okay so so up here this is fine but there's a few issues that that that we'll discuss about so this is if the universe is dominated by a cosmological constant clearly that was and what happened throughout the history of the universe there was a whole big bang history the universe underwent a period of radiation dominated era and then a period of matter domination dominated era and so let's see how the evolution of the universe had the Hubble or the scale factor would evolve during that period or the energy the local energy density so if we were dominated by by radiation do you know how the energy density would behave during the period of radiation era energy density constant yes one of a to the four so you all seem to be aware of that I'm not going to go through the derivation of this and then if we are during the matter dominated era the energy density with that scale with a scale factor yeah okay very good so this one is probably the easiest to understand if it's dust if you if you just think of particles then the volume of your box scales like that distance cube and so the density get reduced by the distance cube and then for radiation we have an additional contribution coming from the redshift so the the wavelength get larger as well and so let me call that so if we look throughout the age of the universe let me put it a here too much the total contribution the function of a to the energy density so let's imagine we have a total contribution throughout the age of the universe which is that of radiation that started dominating the universe but then we see that it gets deleted very quickly matter then then starts to dominate because it gets deleted less quickly and then the cosmological constant which remains constant throughout the age and then maybe there's some special curvature but we have ignored that so so let's not go into that so we'll have to start with some this is how the energy density for radiation with scale let's say as one of a to the four and so since it deletes more rapidly there will be a time where it will be sub-dominant as compared to matter and so at some point matter starts taking over it still decreases this is not to scale don't take me or quote me on that and then there's a final time where the matter has diluted enough that what starts taking over is the cosmological constant which has remained constant throughout the age of the universe so we see that today we're living in a universe where we have roughly 30 percent of matter and roughly 70 percent of dark energy so the amount of matter and the amount of dark energy even though we're dominated by by a cosmological constant today they are roughly comparable so we're living in a region here where roughly the amount of sorry of matter is maybe point I don't know what it would be a few percent from three it's not point three but let me say roughly like that of dark energy so it's roughly of that amount of the cosmological constant but throughout the whole history of the universe this is a very special point before that just before that we were very much dominated by matter and before that we were very much dominated by radiation and throughout the history of the universe the cosmological constant has been has been sub-dominant for almost all of his history all the way up to today so how come we happen to be living I precisely at the time of our universe where dark matter is roughly the same order of magnitude as dark energy okay it's not quite the same but it's not suppressed by 50 orders of magnitude let's say when throughout the age of the history this typically wasn't what was happening dark that kind of energy was very sub-dominant throughout the age of the of the universe so this the fact that we precisely living in that sweet spot corresponds to the coincidence problem so not you think this is an issue may depend on your taste I'm gonna remain agnostic about that but some people had developed some models to try to explain the coincidence problem and so I'm just gonna review them it may or may not be something that you consider an issue you may think well maybe that's just a question of scale as well if I change my if I change my axis there I can make that a little bit bigger I can make that a bit smaller who is to judge but I'm gonna remain agnostic okay this is one of the motivation that people have been considering to to to to study models of dark energy where instead of having something which has been constant and very small and very sub-dominant throughout the age of the universe maybe would have tracked the the energy density of radiation and maybe later the energy density of matter so that there's no coincidence in the fact that we almost at the same level of dark matter and dark energy today so that's the essence of the coincidence problem and the motivation for looking at tracker solution which I'll briefly mention maybe so I'm going much slower than I thought I would but I like to do now is look at the cosmological constant problem which to my mind is a much deeper issue that it may be a question of of our taste so the cosmological constant problem is coming up when we're looking at the numbers we've seen that if our cosmological constant it leads to the acceleration of the universe the Hubble parameter today h today is what it has error here to say it's today as opposed to inflation or yeah 70 70 kilometers per second and then per megaparsec so if our supernova was one mega one megaparsec away from us it will move on average at a with a with a velocity of 70 kilometers per second which is pretty fast if we put that back into energy scale so I'll be working in units where h bar is equal to the speed of light is equal to one so that means that a scale of energy is equal to a scale of mass and that's the inverse as a scale of distance a speed is dimensionless this is roughly 10 to the minus 33 electron oh yes thank you thank you thank you thank you yeah thank you yeah so if you compare with the energy scales that we're used to dealing with when we at the LHC for instance what the typical energy scale at the LHC tv yes yeah the Higgs mass the mass of the Higgs is roughly 125 giga electron volt so that's roughly 10 to the 9 giga is 9 and then 10 11 electron volt this is pretty small as an energy scale compared to to that even for the neutrinos if they have a mass it's of the order of a milli electron volt so 10 to the minus 3 electron volt there's still 30 orders of magnitude difference between the scale we're talking about here and the standard particle physics scale okay so that in principle would be okay if we hadn't had the the the prior that from a field theory perspective any field that we leave that leaves on a spacetime if he has a mass would naturally contribute to a vacuum energy which is constant which acts like as a cosmological constant but with an order of magnitude which scales at the very least like the mass of this particle to the power four so we would expect contribution to lambda that go like 10 uh like the mass of the particle to a power four for every particle of mass so for instance the Higgs has a mass of around 125 giga electron volt we would expect it to give a contribution to the cosmological constant and therefore to give a contribution to the rate of the acceleration of the universe today which would scale like it's a mass to the power four and then we'll have to divide by the plan square but still we have something which is much larger than what we are observing today to put that more concretely we can do two things i'll just give you some incentive to do our calculation and then i'm not going to carry it through because it would take us an hour but what you can do is have a look at gravity and a scalar field it doesn't need to be a scalar field necessarily you can take fermions you can take a vector field it doesn't matter too much but it matches you if you do it with a scalar field and if you do it with anything else so let's imagine you have your action for g r which is m plan squared over two and the scalar curvature and then you put a scalar field minus a half let me put it this is the inverse metric d mu phi d mu phi for your scalar field minus v of phi so this scalar field couples to gravity it sees gravity through this factor here and through this factor there so there's an interactions between gravity and your scalar field yes yes i'm gonna i'm gonna show you yes let me show you yeah so you can have a look at that let's see uh you could take just a massive scalar field so that's where the mass comes in what you can do is do perturbations on flat spacetime it would be easier this is just a trick that you can use it will be easier if you're usually you write a perturbation like so another h menu of m plank right typically you write things like that i can if i'm just going to do perturbation if this is small it's actually actually easier if i do it if i wrote it like that for this calculation where this h square menu is h mu new h sorry h mu alpha h mu beta eta alpha beta so this is just a change of variables let me write it like that it doesn't matter too much then what you could do is expand this scalar curvature or this anstein term in here in fluctuations in there what you would get at leading order is h mu new let me just do it exactly uh square root minus g m plank squared other two r this is actually equal to leading order to one of the four h mu new epsilon alpha beta mu new h alpha beta where this is the lichnador rates operator and that has been we have done integration by parts here so this is a symmetrization factor i'm working with a convention that if i have two symmetrized indices that corresponds to a half d sorry a menu plus a new mu there's actually a two here plus d mu d new h where h is a trace of h menu so you take h menu and you contract it with a eta menu and then we have minus eta menu box h minus d alpha d beta h alpha beta so that's not very important for us right now but we'll use it later on this lecture these lectures okay so this is the can you read in here that's okay yeah okay so this way you'll be using later all you have to remember is that this has second derivatives on each menu it will give you what the propagator but we're not going to look too much at what happens from that what we want to look at is the communication is the coupling between your scalar field phi your massive scalar field five and your metric so if you look at this thing here and you have square root minus g minus a half d phi squared minus a half m squared phi squared if you're living in a metric like that that will correspond to a coupling between little h menu and your scalar field that goes like h menu of a m plank t menu this is the at leading order this is the stress energy terms associated with your scalar field if you were to live on flat spacetime and then you have an infinite number of corrections that come in here so this is the first order term then you have something that goes like h menu h alphabeta of a m plank squared t menu alphabeta etc so what i'm not putting in here is things which would scale like m h cube so you have contributions like that at leading order this t menu is the stress energy tensor for your scalar field if you were to live on flat spacetime you can't read here right can everybody here read no okay so what is that it's a d mu phi d mu phi minus this living on flat spacetime is c time menu a half d phi squared plus the potential in our case the potential is a half m squared phi squared so at this level you don't even need to think too much about quantizing your back your tensor field h menu but in principle this is a quantum field h menu but your scalar field is also a quantum field we're living quantum field theory h menu t menu and h menu h alphabeta t menu alphabeta etc they will lead to corrections to uh or you can look at Feynman diagrams associated with that and so from that you have an h menu coupled to two phi's i can construct from that a diagram where i have an h menu as an external leg and then a loop of matter a loop of phi field you have a phi living in here from these two phi in here or in here or in there and but you're not only have this type of interaction you also have interactions with two h's and interactions with three h's and four h's and everything so you have in total contribution to from looking at one loop contribution of the scalar field on what is your graviton or your tensor mode which menu that goes like that then you have something that goes like this where you have a loop of phi so this is coming from an interaction like that in here or that vertex and an interaction like that at that vertex but you also have something that goes uh from here so two h's connecting to two phi's in here so that's going to give a contribution in the one loop effective action for h to integrate out the one loop of phi the contribution on h effectively so on the effective action for h that will have to scale like h what do we mean by that is the trace of h plus corrections so if I just look at the low energy limit of this contribution and what I mean by that is if this external leg for each menu had zero momentum or in the limit where they would have zero momentum the contribution to this one loop effective action will go like that and it would scale if you look at this one loop of the scalar field it will scale like roughly the mass of the particle to the full at the very least if you put a cutoff for your particle you will have something that would scale even larger amount but at the very least it would scale like that from this contribution you would have something that would go like h squared minus h squared what we mean by that is h mu nu h mu nu this is this is the trace of the square of that tensor and then this is the trace h squared the trace of h is what we wrote here so it's h mu nu minus h alpha alpha h beta beta you see that this would contribute something running so there would be a log running if you put if you work in beam rag you would have a scale associated with this running and I don't want to go too much into that right now but I like you to see that you'll have contribution that would scale like this and contribution that would scale like that and this is a calculation I recommend you do it so that you you see what you really get by working in that language if you were to do all of this sum of loops so you'll have something with just one h something with two h's something with three h's would come in here what you would have had there is something that would go like m to the four and then h cube minus three h h squared plus two h cube the square brackets are the traces of the tensors so if I have a tensor m mu nu the square bracket of a tensor m mu nu is the trace of that tensor so for instance m to the 26th is you multiply you have this thing this matrix to the 26th and then you take etc 26 times you'll have that you can go to quartic order so you have something that goes like that plus something with three and one and then something with that you could go and compute those exactly and you would get something that would go like h to the four and then a particular combination so why do I go through all of that is that if you could do this calculation you would actually see that it would truncate exactly at that order if you go to any higher order the result vanishes at low energy it vanishes exactly and what I mean by low energy is if the momentum associated with the external leg for your h mu nu have zero momentum and then the contribution for any higher contribution to the one loop effective action of your of your graviton would be exactly zero so it truncates at that order and then if you were to look at the contribution from this from that from that and then from here what you would get is exactly what a cosmological constant is in that language so if you are working in that language to all orders in h what is square root minus g so you could check it out but it's one plus the trace of h of m plank plus the trace of h squared minus h squared of m plank squared then you have exactly that cubic interaction then you have a quartic contribution and then that's it it stops at that order at cubic order and then something a quartic order and then it stops yes yes yes the scale factor yes so we're working about flat spacetime indeed yes so you can do that but here here you're working on flat spacetime so there's no scale factor yes exactly you just get one you just get one exactly so the square root of minus g here yes so this contribution gives you one and then from this perturbations it gives you this this four terms yeah it stops it stops here this is the way I write my metric and it in this I write it like that so it's an exact square so that I can write it as eta mu nu plus h mu nu of m plank squared and so that's when I take the square root it only gives me a finite number of terms yes yes so this is this is my whole metric whatever the metric is no matter how complicated I can always write it down like that always this is just at that level just we have a very efficient very definition but of course we're working under the assumption that it's small fluctuation living on flat spacetime and so this would be small and so at leading order it doesn't matter whether you want to work with h mu nu or with square mu nu but I would say it's a good choice to be working with that so that the whole expansion of the determinant of the metric and square root truncates at the finite order yes yeah so that you can plug in here and you will see that it's exactly that but what you see is that the contribution from the one loop of your scalar field of mass n if you sum all of that it's all of these terms but there's a finite number of terms that you need to consider they would resum to give you precisely these contributions precisely that and then with a scale in front of it which is associated with the mass of that scalar field we're just dealing with a scalar field here for simplicity so the one loop effective action for that scalar field contribution to how the what the what the the graviton sees really is something that looks like a cosmological constant it's exactly a cosmological constant it has all of these terms so I haven't written them those are on here and those one there but you could check and correspond exactly to this one and exactly to that one not up to a factor of two or anything exactly really exactly and so the contribution that you have from at that level a scalar field of mass m in the low energy limit so in the limit where the momentum of the external leg goes to zero is exactly a cosmological constant so the one loop effective action integrating out matter fields gives you precisely a cosmological constant and the scale associated with our cosmological constant is related to the mass of the scalar field so by integrating out I could do it more probably if I if I have um if I wouldn't let me do it in euclidean the one loop for my graviton if I consider that to be the integration of a mass scalar field I put a minus here because I want to be in euclidean just for a second it'll be easier to make the the argument is the integration of that dr plus the scalar field if I were to integrate out at one loop I would get a contribution which goes like a cosmological constant which is described in minus g scaled by the mass of the field to the power four and then you have a log running the silver m and then corrections that come in from momentum of the external legs and these corresponds to derivatives acting on the metric and that they would have to be packaged into something which is covariant which is for instance a curvature term with some scale in front of it maybe an m squared in front of it and then an army new army new squared and this type of corrections so this they're going to be not important as compared to the contribution that you already have in gr with scales like like the power and plank to and plank squared which is going to be bigger than that so that's not very important but what we see is that this correspond to an effective cosmological constant that scales like the mass of the particle to the power four of course it's not an accident that we get something that repackages exactly as a cosmological constant because of covariance whatever the result should have been it would have ended up being covariant so the only thing we could have had is something which is a cosmological constant but it's good to do it once just for to know that we're not trying to sell you something which doesn't quite work like that and to see that the scaling when you complete the loop would actually go like at the very least the mass of the particle to the four if not higher if you had put another type of regularization like cutoff regularization or something like that so you get at least a power and m to the four contribution like the scales like m to the four if you're doing dimensional regularization so i think i'm out of time is all right i'm running yeah i'm sorry about that so yeah yeah so so that tells you how the graviton is going to contribute to the to the effective action for your scalar field but actually this is very suppressed because it's in blank corrections as compared to the scalar field so you could consider that as well yeah yeah yeah so you want to have external legs of the scalar field in here for the graviton loop yeah you want to have an external leg of the scalar field yes quarter gravity on x yes so you can consider as well you can consider a diagram like that with just a scalar leg or something like that with a scalar leg on top of it that's going to correspond to modifications on things like that but it's just going to be in blank corrections to that to what you already have yeah so so at the level of i mean here we're only integrating out that we only consider a loop of the scalar field and only graviton external legs because that's the question we're interested in but you can look at what the corrections are for from that and so you you integrate out only the internal graviton and scalar field you could do that that that is going to lead to corrections to that but which are not very relevant for the question we want to answer yeah so definitely you're going to start generating things which are corrections to this but you're always going to have at least one graviton external legs so it's going to be always some plank suppressed gr here in a what is not renormalizable now so this is perfectly renormalizable this this is this is we're just doing effective field theory we're just working very at energy scale very low compared to the plank scale and at that level i can this other contribution i i get and depending on here i'm working in the one loop effective picture so i'm integrating out this is the contribution i get if you want it instead you could think of this gives you some divergences you want to add counter terms to remove these divergences and perturbatively order by order i can do that there's absolutely no issue yeah that's right it's only if you want to consider the effective field theory of gravity at a scale comparable to the cutoff to the plank scale that then will start having issues but perturbatively at low energies there's absolutely no issues yeah yes in principle you could do that so in principle you could do that but then that means that what you end up having well that you cancel that thing you would need to have an extremely tuned you you need to have two huge numbers put them together with an extremely tuned accuracy to cancel you could in principle yes you could in principle just remove it away yes but in in field theory usually when you have something which is out of that order of magnitude you have something which is of that order of magnitude you you expect it to be there when you when you at this level you just computing the one loop effective action so you haven't decided that your action for gr would be exactly that suppress for all the particles and then you're going to integrate over all the particles and get the contribution so as to remove it usually you have a leftover which is of the same order of magnitude because you will need to do that for the one loop and then you will have a similar contribution coming at two loops and that's three loops etc so you will need to know precisely what the contribution from all the particles from all the loops would be to decide precisely what contribution you should have had put them into your bear like ranjan that would have removed all of these contributions are priority so in principle yes you could do it but but usually from a from an effective field theory point of view that's not that's not quite what happens you you you have something which is of the same order of magnitude you have a running of that contribution can I suggest to continue the discussion during the break because it's already late uh let's thank the