The Hubble Tension

[Auto-generated transcript. Edits may have been applied for clarity.] So I, I would say. Uh. Thank you. Thank you, everyone, for, uh, coming to the event. And, uh, I thought I'd tell you about what is called a crisis in cosmology. And it goes by the name of the Hubble tension. And, uh, maybe introduce you to what?

What people are talking about when when you see these words reported in the media and also, uh, show you what, what it would take to solve the Hubble tension, what we might learn from it, uh, and really do spoil the punch line. Uh, we don't actually have any satisfactory solutions, so we don't really know what's going on. I give you a few solutions and show you why they're not really satisfactory in any sense. So this is a version of, uh, of the of the timeline, Edward, showing us.

Uh, this is, uh, beginning of the universe on the left. And maybe I have a pointer here somewhere. Um. Uh. And as time goes on, the universe is expanding. And the epoch we are going to focus on, uh, is, uh, relatively late in cosmology compared to what Ed was talking about. And so we're going to, uh, focus on the epochs after recombination.

So this is depicted in this picture as, uh, the surface here, uh, and all the way to today, uh, in a, in the cosmology, the epoch of recombination is the epoch where, uh, as Joe was describing, protons and electrons came together to form neutral hydrogen. And so the universe became transparent to photons. And so that's, uh, that's sometimes called the surface of last scattering. Most photons encountered the last scattering at that surface, and the universe became transparent.

So we can actually see, uh, down, uh, all the way back to that surface. And really, we'd actually be focusing on most of the, um, focusing mostly on the physics of the surface of last scattering, uh, the CMB, uh, itself and physics around, uh, current times and not so much about the intervening, uh, dark edges. Um, so, uh, uh, before we get into the details, let's just, uh, say what the hub of tension is. Uh, uh, there's this parameter in cosmology called the Hubble parameter.

Uh, and, uh, the thing that we are focussed on is the Hubble parameter today. So this is called h. Not usually not denotes, uh, things that we're talking that, uh uh uh, not denote the time today. So denote as time today H nought is the Hubble constant Hubble parameter today. And uh, this blue uh, uh, this blue point here represents, uh, the Hubble constant today. It's not as deduced from measurements of the cosmic microwave background by the Planck satellite.

And it's a really nice measurement. You can see it's, uh, uh, it's about, uh, a percent level measurement of the Hubble constant. And this red, uh, here is a measurement what is often called a local measurement of the Hubble constant, which is we look out to, uh, nearby or, uh, relatively nearby galaxies and, uh, and supernovae and, and, uh, calculate the Hubble uh, parameter, uh, uh, using that method. Uh, and that has been, uh, that has been an industry for a very long time.

That was indeed the first way we discovered that the universe, uh, that's expansion of the universe was accelerating. So this is also a very mature science, and that gives us a value that is insignificant discrepancy with the value that we deduce from the CMB. Uh, these yellow, uh, points are relatively recent analysis, uh, using James Webb data. And then you can see sort of they're not quite there yet in terms of precision to really, uh, say what's going on.

Uh, and they themselves have a little bit of a spread between them. So, so the situation is a bit murky. Certainly these two measurements, which, uh, which have been really thoroughly vetted and investigated over, over many, many years by many, many different groups are incomplete, uh, statistical, uh, tension with each other with very high significance. And we don't really know, uh, who's right, so to speak.

Okay. And, uh, but I introduce more elements of what we mean by the Hubble constant and so on as we go with them. So, uh, this so at one glance, it might seem that there's sure there's some there's some disagreement, uh, in some experimental data. So why is that really so significant? So one point that is significant is that this is one of the six parameters in the standard model of cosmology,

and we don't have that many opportunities to measure it. And so people have really spent a long time measuring the this quantity. And the discrepancy in this really, uh, might be pointing out that something is going on in our understanding. So my interest in the subject, uh, relates to this aspect, uh, from the era of recombination until today, uh, our universe is, uh, has been dominated by, uh, the so-called dark components, uh, of the, of the universe.

Uh, most of the matter density that we know of, uh, we know that it's in the form of a dark matter. And today, if you look at the energy budget of the universe, most of the energy is actually in form of dark energy that causes the accelerated expansion of the universe. So, um, we have very we have relatively firmly established the existence of these objects, the dark some some kind of dark energy is required to drive the accelerated expansion.

Some kind of dark matter is required to, to ensure that, uh, matter clumps at the right, uh, right time and right edge, uh, to form galaxies. But we don't know very much about them. Uh, as may be, uh, hinted at by the word dark in front of them. And so we don't we really don't know any properties about dark matter and dark energy.

And so it's very plausible that as we increase the precision of cosmology, uh, observations, that we start learning about these quantities and they don't behave exactly like, uh, what we our first naive estimate is. And so, uh, what we know about dark matter is that it should have small interactions with us. It shouldn't interact with light very strongly. Uh, and it should be non-relativistic.

It should be what is called, uh, cold, uh, in, in cosmological parlance, uh, for dark energy, we what the only thing we know about it is it should be relatively constant. Should behave like, uh, almost like a cosmological constant. But we don't know anything about it beyond that. We don't know whether these components interact with each other with themselves. Uh, whether the cosmological, uh, whether the dark energy is a cosmological constant or itself an evolving field.

Energy density is slowly changing with time. Um, and there may be other contributions, uh, beyond just these kinds of contributions that, that we might not have, uh, yet seen. But we are starting to see, starting to see now as we crank up the precision in our cosmological measurements. So, uh, some of these questions I've explored in these papers, I won't have, uh, time to go through much of the details of these particular papers. Maybe I'll come back to one of the ideas in these papers.

But if you're interested in following this thread, uh, I'd encourage you to look at these, uh, these papers. Okay, so I wanted to just give you, uh, at least a cartoon version of what are these two kinds of measurements, uh, that are made and how how do they come about. And so to see what are the moving parts of this discrepancy? Um, good. So, uh, to set the stage, uh, the background cosmology, we we have very good.

We have very good evidence, uh, that it's dictated by, uh, that, uh, the universe will live as in homogeneous and isotropic. Um, we also have pretty strong evidence that it's spatially flat. So if you take any given time, slice, uh, the universe, uh, looks like ordinary Euclidean three, uh, three dimensional space.

Uh, and such a universe is described by the FLW metric, where the line element, the distance between any two points in space time, uh, given by something that looks very similar to the Minkowski metric, uh, except for this, uh, scale factor of T, and the Hubble parameter that we have been talking about is nothing but just, uh, the logarithmic derivative of the scale factor of T over E.

And so in some sense, uh, uh, the expanding universe that that cartoon I was showing you at the beginning of the talk, uh, is, uh, encapsulated by just a function f of t, which is growing with time. Uh, and all of cosmology in some sense can be encoded all of at least the background cosmology can be encoded, uh, in this function, one function of time, uh, and Hubble then becomes the rate of expansion of the universe.

Uh, an intuitive way to think about. Uh, think about the scale factor is the following. Imagine two galaxies that are very far separated from each other so that they have negligible pull on each other. So they would just follow geodesics in spacetime in these coordinates T and x. They will just they will not move. So if you put the put first galaxy at some coordinate x1 and the other galaxy at some coordinate x2, uh, they're just they just retain their coordinates. They're going on geodesics.

That's what these geodesics look like. Uh, and the fact that the universe is expanding tells you that, uh, the physical distance between them is changing, and that's changing due to this presence of a scale factor. So to go from coordinate distance, which in this case is fixed to physical distance, which in this case is given by eight times delta uh given by eight times delta x. We see that these galaxies would appear to be receding from each other in terms of physical distance.

Okay. Well, how do we how do we find out what this function have? It looks like. Um, we we have a metric. If we have some energy density in the universe that we know about. We can solve Einstein's equation. Einstein's equation famously tells us that, uh, geometry, the metric follows from energy density. And so, uh, in this context, these ancient equation look like, uh, look like this.

These are called Friedmann equations in the context of cosmology. Uh, and this is one of the Friedmann equations. Um, on the left hand side, you essentially have this graph of the Hubble parameter, and that is proportional to the total energy density of the universe. Rho. Here is the energy density of the universe, uh, and the energy density of the universe, as we said, can be modelled as ideal fluids. Um, um, and these fluids are essentially characterised by the equation of state.

So what kind of fluids they are, what is their pressure relative to their energy density? And equivalently, uh, they're characterised by their, their scaling of energy with respect to air. Uh, so let's take an example. Let's think about uh, matter. So, uh, again, as I was showing this, the energy density matter, uh, the total energy density rho is uh, the mass. Uh, I have it, uh. Look, I have it here as it would be, satisfy this equation that energy and matter was the mass of the dark matter.

A mass of matter times the number density. And again, if this if these matter particles were on geodesics, they would just stand there on their coordinates wherever you put them. They're non-relativistic, remember. So they're essentially at rest. So in a comoving volume, uh, the number density remains constant, which means that in the physical volume the number density decreases as a cubed. So these matter particles are valued as a cubed. And that means that energy density dilutes as a cubed.

And so that tells us that there's a contribution to rho which is proportional to uh e to the minus three times some proportionality constant. That depends on your initial conditions that you set up. Uh, a similar kind of argument can be made for radiation, where the number density of radiation, number density of photons would dilute as a cubed as well. Uh, but the wavelength also redshifts. And so the total energy density of photons actually redshifts says uh at minus four.

And then there's components like this dark energy component which is essentially constant. So various components of the universe, the particles that we know and love, um, either fall into the category of radiation, like photons and neutrinos when they're relativistic matter, like baryons, protons, uh, or the cold dark matter or this unknown dark energy component.

So this, um. This sets up some, uh, equations that we can figure out what the Hubble rate of the universe is, given what the relative fraction of the universe is in radiation matter dark. And we can see that if we, uh, if we choose some epoch. Uh, so, so if you choose some epoch where a is really small, remember the, um, the convention is that we set a equals one today. And so in the past, in the past it was really, really small.

And so you can see that in the past the radiation becomes a much more important component, uh, and eventually, uh, as it increases, that becomes less important. And that gives away to matter becoming a more important component because it it scales slower than, uh, radiation with respect to a and eventually, uh, uh, for today, uh, dark energy becomes important as well.

So this is this this equation also gives, uh, gives us this idea that in the early universe, the universe was radiation dominated. Uh, eventually it became matter dominated. And today it's dark energy dominated. If this was a question at possible lectures. Uh, possible question, but, uh, maybe.

Okay. Another thing that will be. So one other ingredient that would be useful for us to, uh, set up the measurement of the Hubble constant is, uh, different, uh, uh, different notions of time as the related question that was asked earlier. Uh, uh, the time coordinate that we chose could, uh, could, uh, is not unique. We could have chosen other parameterisations of the FLW metric.

And one thing that one particular, uh parameterisation that would be really useful to us is the so-called conformal time, which is defined, uh, just as such, uh, that DT is just eight times the ETA. But the advantage of that is that, uh, the metric, uh, the line element in this coordinate looks something like this. So it looks basically like a Minkowski metric, uh, multiplied by some overall factor squared. And, uh, but we're going to be studying photons travelling in the universe for photons.

Uh, DS squared is zero. Geodesics of photons have, uh, photons travel along null rays. And so the photons don't see in these coordinates don't see this factor squared error at all. As you can see that if I said d squared equals zero, the pre factor uh factors out. And so for photons in these coordinates uh the universe looks like Minkowski space. And the photons literally travel along straight lines in these coordinates. And so that hence the hence the uh utility of these coordinate system.

Um, another, another uh, kind of uh, clock that appears uh, often in cosmology is called redshift. Uh, and it's, uh, it's related to the fact that the photon wavelength, uh, um, is proportional to, uh, is proportional to the scale factor. And so, uh, this is a definition that's often used in cosmology where, uh, one plus z is defined as one over a. For part, partially for historical reasons, really.

But um, so this is again related to a question about, uh, time that was asked earlier, which time really it is. Uh, and there's a time in the early universe is the time in the early universe the same as time today. And you see from from at least from, uh, this aspect, you can see that, um, the frequency of a given photon in the early universe, uh, with redshift.

And so if you, uh, if you emit some photon at a particular known frequency in the universe today, we measure its frequency to be, uh, redshifted by a certain amount. So certainly, if you're using atomic clocks to keep time, the notion of time in the early universe, the notion of time to day would be redshifted to each other, similar to gravitational redshift or relativistic redshift.

Um, so in the end, there's many, many, uh, equivalent descriptions of the cosmological clock and could use t the time coordinate we started with. We could uh, use the conformal time, or equivalently, the scale factor itself, or the redshift or the temperature of the of the photons, or as, uh, cosmological clocks, uh, basically because each of them have, uh, monotonic relation to the other. And so we could equivalently convert between any one of them as our clock.

Okay, so, uh, the way we calculate the Hubble constant is using different distance measures in cosmology. Um, and the two, the two set of distance measures that come to play in these two ways of calculating the Hubble constant. I relate it to the standard ruler way of doing it in the standard, uh, candle way of doing it. Uh, and so here's how that, uh, here's why these things are, uh, these things. Let us, uh, calculate the Hubble constant. Uh, so the first thing is the standard ruler method.

Uh, and it goes by the name of angular diameter distance. The idea is that if you know, if, you know, uh, um, at least in Euclidean geometry, we are familiar with the fact that if you have something that, you know, the length of, like there's a metre stick and I know the length of that with a stick, I look at it and I see a certain angle that is certain to me.

I can figure out how far it is to from me, just from geometry. And so, uh, we want to use the same idea but apply to this evolving cosmology. Luckily, we have, uh, at our disposal this conformal time coordinate in which light rays really do travel that straight line. So we can apply this Euclidean geometry logic. Right. Uh, directly to this, uh, to this problem. So imagine, uh, there's some, uh, imagine a picture in which there's an object for which, you know, that physical distance.

Uh, physical size. Sorry. Um, and, uh, so the physical size that, you know, is some quantity called D, and you look at that object and it turns on your detector an angle theta. You measure that the it's some angle. In cosmology we're always just measuring angles. The angle that you measure it sometimes is theta. Uh and so the angular diameter distance is defined as the quantity such that this equation is correct.

In other words the physical distance, physical size of that object, uh, physical size of the object divided by the angle is substance d r detector. So here's a picture where the picture is now drawn in conformal time coordinates. So this is a coordinate space picture that's remember that's the that's the space in which light travels in a straight line. So if you want to draw this picture and take it literally we should we should work in coordinate space and convert it to physical space.

Uh, so let's say in coordinate space the size of the object is x, uh, and that is the distance to the object from us is r. And so here we are allowed to use Euclidean geometry. So r is just uh so theta is just x over r. And now we want to convert it into this form to actually figure out what the angular diameter distance would be related to the physical distances. So the physical size of the object, uh, would be the scale factor at the time of emission.

So a at ETA being the conformal time uh times uh the quarter distance x. Um, since uh, I've set C to one here. So the time light takes to travel from a coordinate distance R to us is the same as the conformal time, uh, at the conformal time today zero. So the total time taken is the same as the total distance when c is one. And so this uh the coordinate distance r is the same as that.

So so we can we can use the simple geometric picture and actually get to a, uh, an equation for what the angular diameter distance is. Uh, x over r is nothing but d over eight times eta e. And so that tells us that the angular diameter distance is, uh, the scale factor at the time of admission times the total conformal time to the time, uh, between us and the time of admission. This can be converted using that using the definition of the conformal time into an integral over time.

Remember data was uh ada eta was dt. So I've just written eta as integral d over a and you can again convert use any other clock that you'd like. If you want to convert this to uh, convert this integral over time to integral over a, you can use uh, multiply and divide by a dot over a, uh, and convert into integral over a, or you can uh, convert into a integral over redshift.

So this is just coming again from definitions, uh, of what we, what we said a wasn't relative to uh, ETA was whether it, relative to time or H uh or Z was relative to uh, an instance. Uh, so the upshot is, uh, that this angular diameter distance, uh, is given by an integral over, say, here redshift, uh, from the red uh, from the redshift information uh, to today, redshift uh zero is today, uh, or equivalently the scale factor dimension to uh the scale factor today.

So this angular diameter distance is something that's sensitive to the, to the entire cosmology, cosmological evolution of the Hubble constant from the time of uh, emission, uh, to the time of observation. Usually we would find something, uh, in cosmology for which we know the physical size or some, some standard ruler, and we would measure theta very precisely.

So that would give us, uh, the angular diameter distance to that object, which in turn gives us this integral over, uh, the Hubble parameter, at least from the time of emission till today. And so in this way, we would be sensitive, uh, to the Hubble parameter. Today particularly this integral is actually dominated by redshifts close to zero. So this integral is actually quite sensitive to the, uh, Hubble parameter today.

But it's also sensitive to what's happening between the emission and, uh, emission. And today okay. Uh, so this is one way. Uh, and we'll come back to this. This is the way CMB actually measures, uh, the, uh, Hubble parameter. The other distance measure is luminosity distance, which is two with which again in Euclidean space. You would say, if I knew there was 100 watt bulb, uh, if I if I see it, it's brighter and I reduce that, I'm closer to it.

If I think it's dimmer and I'm far further from it. So if I know the intrinsic luminosity of an object, and I know the apparent luminosity of an object, I can figure out how far that is. And again, we want to do this in, uh, in the context of uh, uh, uh, expanding cosmology, but the same, same idea and applying the same idea. The definition of luminosity distance is the following. There's some intrinsic luminosity of an object, and there's a measured flux.

But luminosity distance is defined to be the, uh, to be such that this equation is true. Um, we can again run the same kind of argument I, uh, I was telling you about for, uh, the angular diameter distance, um, in, in conformal time coordinates, again, light travels and on straight lines. So the area in in these coordinates, the area of a sphere at distance r is just four pi r squared. And so the dilution of flux as you go to a distance r is just one over four pi r squared.

R is the coordinate distance. And we can convert it to a physical distance. And the upshot is that we can get another expression for uh this so-called luminosity distance, which uh uh, looks similar, but it's not quite exactly the same as the angular diameter distance. But again, it's given by an integral over the cosmology from the time of emission until the time of observation.

So both these distance measures are, um, uh, sensitive to h nought the Hubble parameter today through an integral over, uh, uh, over the Hubble parameter between emission and uh, um, and today. So maybe I should say, uh, for the CMB, the relevant, uh, redshift of emission would be about 1000. So that's when the at the surface of last scattering, when the CMB was generated, um, the universe was about a thousand times smaller than it is today.

So this integral would go from about a thousand to, to zero. Uh, and um, for, for the luminosity distance, which would be the relevant, uh, quantity that supernovae, uh, the local measurement measurements use. Uh, these measurements go out to redshift of order one, 1 or 2. So this integral would go um from say 2 to 0 or something like that, whatever that, uh, redshift of uh, the supernovae happen to be. Um, and so, so it's not exactly entirely the same quantity they're measuring.

One is measuring the universe at really late times between redshift of two and one, and the other one is measuring the integral of the Hubble constant from redshift to to today. So we might wonder whether that that's a hint towards what the solution could be.

Okay. So this let's first look at how the, uh, how the supernovae measurement measurements are done, uh, using the so-called distance ladder method, uh, to calculate, uh, essentially calculate the, this luminosity distance and the Hubble constant. And so the idea is to build up, um, the idea is of use, known distance measurement, measurement techniques, uh, to calibrate and then use uh, the calibration to then move that distance measurement technique further.

So how does that work? Uh, the the best way we know in cosmology and astrophysics to measure distances is using parallax. Uh, we well, uh, we we can we can wait for it for half a year. Uh, as the earth goes around the sun and we can look at a certain star. And in this particular case, the star that is looked, looked at is a separate variable star. And we can look at how that star moves in the field of some very background, distant stars.

And uh, assuming the distances are right, the actual that, that motion of that, uh, separate star in the background of the distant stars is visible and measurable. And so that, that, that this geometry gives you a measurement of, uh, of the distance to the star. So close for close by things, parallax is the best way to measure distances. And that's what we use the in some region around. So this is, this is us and some region around us about, say, uh, 10,000 light years or so.

Anything around, uh, to that distance we measure, use parallax to measure distances so we can measure distances of field stars. And what what we find is actually that, um, the reason we use stars is because variable stars have a certain periodicity, and, and it's empirically seen that their luminosity, their intrinsic luminosity has a higher degree of correlation with their periodicity.

We can also measure the distance using this parallax method. So if you see a spade star we can measure its periodicity. We can measure its luminosity. But we know how far it is because of this, uh, just this parallax method. Uh, and so we can figure out what its intrinsic luminosity is. I should say that. And on distances of such a small, uh, the distances that are this small compared to cosmology, the fact that we are living in an expanding universe are irrelevant.

Just like when we when we calculate anything about the solar system, we don't worry about the fact that we live in an expanding universe. So this is still, uh, small enough distances that we can work in the approximation that we live in a flat, uh, flat space time. And we don't worry about that. We can include it, but it would be a small correction to the to to to anything we calculate using just a flat surface limit.

So using this parallax we can figure out the distances field which tells us what the intrinsic luminosity of the surface is. And then we find galaxies which host both, uh, separate variable stars as well as, uh, supernovae. We don't have very many supernovae around us, but we do have galaxies further away which whole host would and supernovae, the surface. We know the intrinsic luminosity for we know their apparent luminosity.

So we can figure out their the luminosity distance just using the thing that we just went through. Uh, so that gives us a calibration of the distance to these galaxies. Uh, and then we look at the supernovae and, uh, look at their apparent, uh, luminosity, uh, and using the distance inferred from Cepheids, we can figure out their intrinsic luminosity as well. And so these, uh, supernovae are called standard candles.

They have, uh, they have a very strong relationship between their so-called light curve and their intrinsic luminosity. So this gives us a way to calculate the, uh, intrinsic luminosity of the supernovae. And then very far away, we don't see speeds anymore, but we do see it supernovae, uh, for which we have deduced the intrinsic luminosity. So now using these supernovae, we can figure out, uh, distances, uh, to these very, very far, uh, far off, uh uh, um, um, cosmological objects.

Um, so this is this this is to give you a sense of how this calculation is actually done, but also to see this is a rather intricate thing which relies on a bunch of astrophysics, some, some empirical, uh, estimation of these standardised lines that we don't have first principle and the first principle, understanding of the variability of surface stars. We don't have first principle, uh, truly first principle, uh, analysis of why, uh, supernovae are standard candles.

Uh, but we can build this run of, uh, rungs up the distance ladder. And again, this is a this is a very mature, uh, piece of experimental physics that has been going on for a while. And this is the measurement by which we find the Hubble constant today is 72km per second per megaparsec.

Uh, the other, uh, the the other way, the, the other half of the measurement, which was the CMB measurement comes from, uh, comes from observations from, say, Planck, uh, where the Planck satellite observes the, the sky and sees the cosmic microwave background that looks like this. Uh, if you look at if you Google Planck CMB, then this is the picture that they throw up on their website. But this picture is false for many reasons.

First of all, if you actually look at observation, there should be a huge mosque in the middle for the galaxy that they don't actually have data. There's also mosques elsewhere because of that, uh, instrumental reasons. So this is and, uh, yeah.

So this is an image that's actually generated, uh, generated and filled in these regions of, uh, space, which we actually don't observe, but we observe enough of the, of the whole sky to actually be able to reliably, um, reliably fill in this, uh, picture. Uh, sorry, I should say, uh, with this, uh, I assume everybody's seen this picture before, but what this is showing is, uh, temperature on the sky. Uh, and, uh, more precisely, what we have taken out is, uh, uniform temperature in the sky.

So if you look out in the sky, you actually. What would you see is, uh, uniform three Kelvin, black body radiation coming from everywhere, which is equal from every direction. You subtract the average out, what you would see is a huge ten to the minus three, huge ten to the minus three dipole, uh, that is associated with us moving through the rest frame of the CMB. So you take that out as well.

And then what you're left with, uh, is a temperature contrast of order ten to the minus five, uh, on top of the, the background levels. So this is that subtracted picture and then filled in where we don't actually have observations. And this is what this picture looks like. And the idea is that the cosmic microwave background actually provides us with, uh, kind of a standard ruler. And the standard ruler is one thing we can use to measure distances, uh, as well.

So, um. So what? Where do these cosmological fluctuations come from? There was, uh. Um. As Joe was, as Joe was telling us, if there were these these cosmological fluctuations are the reasons. Eventually, uh, galaxies form and stars form and planets form.

And so the existence of interesting stretches of the universe to this initial perturbations, even though they start off as ten to the minus five, eventually gravitational wells make them grow and uh, and, and do interesting things like have star formation and then nuclear fusion inside the star, inside those stars. Um, the initial conditions, uh, for these things are set by inflation, as Joe was mentioning.

Uh, and there's some initial condition that inflation gives us that there was some fluctuation in the universe at some epoch, depending on their wavelength, they start, uh, they start oscillating. And further, for the for our purposes, what is relevant is that in the CMB epoch, when the, uh, when the photons and uh, uh, the photons and protons and electrons are tightly bound to each other.

So this is right before recombination, right before decoupling. Uh. And then eventually the crossed the temperature wet foot where dominantly all the, uh, protons electrons bound together in neutral hydrogen. Uh, and then this, uh, the photons can scatter. So that is the epoch we're interested in, where we go from a tightly coupled baryon photon fluid, plasma, uh, to a transparent universe with photons streaming to us from the surface of last scattering.

And so, uh, for, uh, for our purposes, what is relevant is these fluctuations lead to fluctuations in these plasma and sound waves in this plasma. So what we see in the CMB is really these sound waves. We have some you take a plasma and you set up some initial fluctuations and you let it oscillate. And these waves are just oscillate. Uh, these are just sound waves. And the, uh, peaks and troughs of these sound waves.

Is that what you see as, uh, peaks and troughs of temperature in the in the CMB? Um. And so if you, uh, if you look at that CMB map that I showed you, but instead of looking at in real space, we look at in Fourier space are more, more, more precisely spherical harmonics space.

Uh, you see this feature rather rather rather clearly that, uh, you see these peaks and troughs which are, which are associated with, uh, uh, which are associated with, uh, fluctuations, uh, that, uh, for example, a peak would be associated with a fluctuation that has just reaches extrema as the surface of last scattering happens.

So so again, let's imagine this plasma where there's a bunch of sound waves and various fluctuations at various wavelengths are, uh, are, uh, are oscillating, uh, and at some point, therefore this, uh, uh, plasma decouples and the photons can stream through. So it's sort of like the universe taking a photo of that oscillating plasma.

And at that instance of taking of the photograph, whichever fluctuations were at the, uh, whether at the extrema, they would appear as peaks in the spectrum, they would have most, uh, density contrast, uh, and, uh, the fluctuations that were sort of in the middle of the oscillation would appear as troughs. So, so, so this picture tells us properties of, uh, sound waves, particularly, uh, which wavelengths had reached their maximum, uh, by the time that decoupling happened.

You could spend an hour talking about the fluctuations CMB. But but the rough picture is the rough picture is that this understanding of the sound waves in in the CMB plasma, as the photons are decoupling, give us a standard ruler. We can we can figure out what is the distance scale at which, uh, this peak should occur. And we look again, in cosmology, we measure angles and we measure this angle, the position of this peak rather precisely.

So again, we're in the situation where we know a physical distance scale, the standard ruler, which is some wavelength that we that is supposed to have reached its maximum. And we also, um, um, and we also know that the angle that, uh, that this uh, corresponds to. So we can figure out the distance, this angular diameter, distance to the CMB using these two parameters.

So this is, this is this is that statement again in uh, uh, in equation, uh, there's an angle associated with the associated with the first peak in the CMB that we measure to exquisite precision. Uh, remember when we're talking about the Hubble tangent, we're talking about quantities with a discrepant to order, uh, 7 to 10%. So that's that's sort of the, uh, going rate of things we are tracking. And I think that's better. Much better. Much more precisely measured is, is is just almost exact.

Uh, and uh, this angle again is given by some quantity, which is related to the wavelength of the, these, uh, oscillations in the plasma, uh, that we can predict. Divided by some, uh, divided by the distance, uh, the angular diameter distance between us and the CMB. And that distance is again an integration over the redshift, uh, at the CMB, which is about 1000 till today. Off this parameter, uh, the of this, uh, one over the Hubble parameter.

Okay. Uh, so we measure this, uh, we have a good prediction of this depending on, uh, our model model of early cosmology. And so we can deduce this. And this is how CMB reduces the value of the Hubble constant that it measures. So, um. Right. So. So imagine that we said we took the I think this is the sort of two paths one could take.

One could take say can we can we investigate what is wrong with maybe something wrong, or maybe there's something, something weird going on with local measurements. So can we take the supernovae measurements, which are 72km per second per megaparsec, and investigate what is, uh, what can be made to make them move from their value? And this is a whole industry. This industry involves figuring out how the distance ladder was actually built.

Is there a systematic error that were underestimated if there was some biases introduced, uh, in any of those things? Another, as another half of the community is focussed around thinking about, well, let's take the supernova measurement, the 72km per second per megaparsec at face value and see can can there be some new physics in that in the dark sector? Can there be some new physics in the CMB that can change the CMB prediction?

Uh, from 67 to 72. So that's the, uh, that's the path I'm going to present very quickly. Uh, just to show you what kind of solutions uh, I presented. And so this is something that we measured very precisely that cannot be changed. And so this ratio, we better keep fix whatever we're doing to whatever we're doing to cosmology. So there are two classes of solutions. People have proposed the so-called late solutions or early solutions. In the late solution you don't change the value of Rs.

So this, uh, um, but uh, this physical distance scale and the CMB, uh, the standard ruler is left unchanged, which must mean that you should also leave, uh, DM unchanged because we've measured the ratio very precisely. And so the only way to change it's not the value of Hubble today without changing the M is to make sure somehow that this integral stays the same, but its shape changes dramatically so that it's endpoint changes, uh, from what it was before.

And so those are the so-called solutions where you have something happening in cosmology such that the integral of this over this, uh, whole epoch is left unchanged. Uh, but the actual final value, h nought today is different. And then in better agreement, uh, I'll show you what better agreement means.

Better and better agreement with with with uh uh with uh, uh, 72km per second per megaparsec are the early, early solutions are uh, for say that we can keep that as fixed but change both R's and M by the same ratio about 10%. Uh and don't change the shape of Z. Just scales everything up and you get uh, you would get H not to be larger. So, um. So the so called solutions, which are not really solutions.

Uh, look, look, look like the following. I'll, I'll spare the details of the solution itself, but I just wanted to give you an example of the flavour of what the solutions proposed of AI look like. Uh, so the so-called solution, uh, I remind you, was that we keep this integral fixed. Uh, but we want to change the shape of h of Z such that the integral these are over h of Z is the same, but it's not as it is different.

So there's there is some, uh, there's some model where you can imagine there's some new the dark energy and dark matter and not just some cold, some cold dark matter particle and some cosmological constant, but they actually interact with each other. And that has some, some motivation from, uh, from theoretical constructions, uh, elsewhere. Uh, but for our purposes, I just want to show you what that kind of looks like. Uh, what I've plotted here is a function of redshift.

Uh, as the Hubble in this model that we get and the Hubble that you get in the standard cosmology, Lambda CDM. So one would be just the lambda CDM value, uh, and uh, and you can see sort of in this value because dark energy and dark matter interact, the Hubble that we have a Lambda CDM has a different shape. And indeed it has some dip and then it has a rise.

So that actually turns out that this integral one over H is left rather is left reasonably constant, except you end up with a much somewhat larger value of Hubble today, which is Z of zero. The larger value, uh, here that you actually find is, uh, not much larger than the CMB value. So again, CMB says 67, uh, supernovas at 72, this sort of drops in somewhere in the middle of the, of the two. So by no means you would call this a resolution of the paradox of the Hubble tension.

Uh, and indeed, uh, people have done, um, people have included more data sets, such as the Baryon Acoustic Oscillation data set, which is looking at clustering of galaxies. Uh, and these kinds of LED solutions are very, very unlikely to work because changing the shape very dramatically is very, very constrained. So even this sort of gets you a little bit of the way. But to actually get this, I mean, to do is virtually impossible with all the data sets that you might have.

Um, in terms of early recombination. You would want to change this quantity, which was the standard ruler, and then you also change this denominator by the same ratio, about 7%. Uh, and then you, you would uh, if that, if you did that, then you would indeed get h nought uh, 7% higher. But the standard ruler is standard for a reason. It's very hard to change this. Uh, this quantity is set by the makeup of the universe with the CMB, uh, constraints rather, uh, severely.

Uh, and one, sort of one compelling possibility would have been that if there were some new degrees of freedom which are relativistic of the CMB beyond the Standard Model, uh, and these could be something like an extra generation of neutrinos, uh, some new kinds of, uh, particles, like axion, but relativistic, like I was talking about. There could even be gravitons, but they were produced relativistic in some phase transition.

Uh, these would actually serve to increase, uh, the deduced value of h nought from the CMB. And so one could try to see how much extra, how much, how many extra degrees of freedom would you need for this to be the case? Um, so that usually it's measured in this, uh, parameter called an effective for whatever reasons, again, historical reasons, where it's sort of counting the number of degrees of freedom of neutrinos.

And there are three nutritional standard models. So the standard model is close to three. Again, close and not exactly three for historical definitional reasons. So what you would need is about, uh, any factor to be about 4.2.

So about an one extra degree of freedom on top of the neutrinos, say for, for you to have, uh, for you to have something like this work, which by itself is not something which is, uh, which, which is uh, uh, such a hard thing to believe, except that once you actually add this much amount of an effective the the fit to the CMB spectrum, uh, becomes much, much worse.

And so you you do you can satisfy all these other constraints, but you actually find that, uh, there's this, uh, large, uh, large l damping tail of the CMB, which I can show you here. Uh, here. Sorry. So for at large l values, the CMB starts damping down. These oscillations start damping down. And that damping has to do with how many relativistic degrees of freedom you have. And we have measured this damping tail be rather high precision as well as you can see.

And if you had that many degrees of freedom, this damping there doesn't like it at all. So you can't actually add those degrees of freedom. Uh, in the early measurement, which is to say, in general, in general, it's actually very hard to change this value, which is a standard ruler, without messing up this very nice fit to the CMB as against people pace, place then to try and inject some energy, these relativistic degrees of freedom.

But they do it in a way that. Doesn't dump the the Daylesford sort of discriminatory injection right at the right time of the epoch that you needed to change ours but not change the dumping tale. And so this goes by the name of early dark energy, which, um, if it's not clear from my tone, is not a very compelling, compelling solution at all.

And so, uh, even this is this works to some extent, but you really need to sort of by hand, put in some energy and make in your model such that you actually get exactly what you want, and then you get some better fit with the, with the, with a suspension. So by no means something that, that seems like the universe would do for, for no reason at all.

So, so these are the, there's, there's various versions of these latent early solutions that I've just sort of drawn cartoons of, um, but really there's no real compelling solution that you can point to. Uh, the experiments go back and, uh, the observational observations go back and check. And, uh, various groups have done cross checks and nobody can find a bug. And so this is just a problem that's in our hand. We don't know if there's something simple going on.

Somebody forgot a factor. There's some population of stars that looks different from this other population of stars. And there's some systematic bias that we're introducing or that we're learning something deep about cosmology. Just because this is the first time that we've sort of gone to percent level precision, uh, in cosmology in the future.

One one great hope was that neutron star mergers could completely break this degeneracy, because that would be a completely third new way to find, uh, to do to find a it's not, uh, we haven't been seeing as many neutron star mergers as we really hoped.

So the actual statistics we would need to build up seem to say that this is not a short term proposition, and people are obviously pursuing constantly new ideas to look for different markers in astrophysics to, uh, like the, the, the yellow, uh, intermediate yellow, uh, error bars that I showed you that were, uh, interpolating between the, uh, CMB measurement and the and the and the supernova measurement.

And those will get refined hopefully over time. And so maybe they will shed light on what's going on. But at the moment, Hubble tension remains a mystery. Uh, waiting for resolution. Thank you.