The "Inverse Problem" Of Dark Matter Is Insane

I saw the equations and suddenly, this is my gravitational lensing problem. He just said, "Oh, finally, you got it, 30 years after we found out." And it was quite astonishing how disruptive you think something could be. Dark matter. 85% of the universe's matter is supposedly missing. But what's the technical truth behind the popular science headlines? Dr. Jenny Wagner, a scientist at the Institute of Astronomy and Astrophysics and the Helsinki Institute of Physics, and winner of the Prize for Courageous Science, disagrees about how we got that number. This is because she proved mathematically that most of what we call evidence for dark matter is driven by the models we insert and not the data we collect.

On this channel, I, Curt Jaimungal, interview researchers regarding their theories of reality with rigor and technical depth. Today's episode is no different. Wagner found that when you strip out the model's assumptions, the only information that the data gives you is local. This means every grand dark matter map you've ever seen is extrapolation. Now, it's not wild speculation. That's not what she's saying at all. Specifically, she's saying a profusion of models can all fit the same data equally well. This is called lensing data. By the end of this podcast, Wagner makes the case that this is what Einstein predicted over a century ago. The homogeneous, isotropic, spherical cow model of the universe eventually needs to be replaced by the next level of detail. Today, she shows us how we may finally be there.

What do we actually know about dark matter? Dark matter. Dark matter is something that I wouldn't say it's something that we know about. But it's rather something, it's a term that came up because we missed matter when we looked at our observations. And so dark matter was the replacement or was the substitute for this missing mass in our explanations of certain observations. For instance, in observations of the cosmic microwave background in the very early universe, but also in the late universe where we have cosmic structures that seem to hold together in a gravitationally bound object. But then, if this object is actually gravitationally bound, then we miss a lot of matter because the luminous matter that we see, the baryonic matter, this is too little matter in order to keep this structure together. So if we do not have dark matter, then these structures would dissolve over cosmic time. Then we wouldn't see them as they are.

So what's the problem with dark matter? What's the matter with matter? What's the matter with dark matter? The matter with matter is that, I would say, in modern cosmology, people started out with observations, with observing stars, observing gas clouds, observing x-rays from the cosmos. And all of this seemed to be fine until we suddenly realized if we take a look at all of these things and we want to explain them in the way that we want to explain stuff on Earth, like to transfer the laws from Earth to the cosmos, we realize that matter is missing. And first people thought, okay, we can just improve our telescopes in order to get a better estimate of all of these missing masses. But on the other hand if we now estimate forward how much mass are we actually missing, it's 85% of the entire matter content of the universe that seems to be missing. And this is something that we cannot just make up by the stuff, for instance small planets or very faint gas that we haven't detected. So a lot of mass seems to be missing. And this is the puzzling thing, that we think we need dark matter in order to explain all of the structures, how they behave, and all of the cosmic structure evolution as well, how the galaxies formed.

So what does gravitational lensing have to do with any of this? I know you have a wine glass, so why don't you tell the audience, or explain, or visually show the audience, what is gravitational lensing, and then what the heck does it have to do with dark matter? So, if we take a look at all the cosmic structures that we have, then we usually need a lot of assumptions to find that missing matter. For instance, that some gas is in equilibrium or that a certain stage of evolution has been reached, that the structures are not merging, something like this. And strong gravitational lensing is a much more pure probe of the total matter content of such a structure, because it purely relies on general relativity, that heavy masses curve spacetime, and so the light will not go straight forward to us, but it will be bent along the so-called null geodesics. And in that sense, the only assumption we make to probe the entire mass of a structure is general relativity and how masses bend spacetime. And in order to visualize this, here you see a background source, like a pattern on the wallpaper. And here is my Earth-like gravitational lens. It's like an inhomogeneous, imperfect glass. And if I now slide as a foreground object, bending space-time, if I now slide this gravitational lens in front of the background source, you will see that the background source will not look the same to us as an observer, because now the light goes through the gravitational lens. And this means now that we do not see the source as it is, but we see a highly distorted, sometimes magnified and demagnified pattern that is a distorted image of the source. And strong gravitational lensing is even worse. It doesn't only create a single distorted image, but it creates multiple distorted images of the same background source. And so the good thing is if we have multiple images that are created by exactly the same background source, we can then correlate all the information from all of these objects together in order to infer properties of the source or properties of the lens. And I say or because if we know the lens perfectly, so if we can describe our lens, then we can reconstruct a source. On the other hand, if we know the source as it is naturally, without the lensing effect, then we can reconstruct the lens. But in cosmology, we neither know the source. It's an object in the early universe far, far away from us. That's one point. And the other point is our lens. If it really contains 85% of dark matter, we do not know where it is because it doesn't interact with light. So how do we know anything about the lens, about the mass distribution that curves spacetime, in order to reconstruct the source? So it's a chicken-egg problem. We neither know the source, nor do we know the lens.

Okay, so speaking of wine, many people, when they're younger, almost everyone, doesn't like wine. They say to their mom or to their dad, can I taste? And then they taste it, they're like, I don't like this. But then at some point that changes. And then you wonder, do I taste wine the same as I did when I was a kid and I just like it now? Or am I tasting something different? And had I tasted what I taste now as a kid, it would have tasted good. So a similar problem is there with pain perception. So if person A can feel the same amount of pain as person B and say I'm okay with it, but person B, that person A just thinks person B is a wimp. Well, you don't know. Is person A just having high pain tolerance, or do they actually experience the pain as less? And Daniel Dennett also talked about this. We don't actually know which one it is. We know the difference. So we know, say, A minus B equals 5, but we don't know what's the value of A, what's the value of B. Is that similar? Yes, yes. It's pretty similar as in, I would say, as in everything in this world. We can only measure or we can only experience changes, but we cannot have an absolute reference frame. I mean, if I see that my dad likes wine, I have the expectation because I like my dad, I may also like wine. So if you try the wine and then you have a certain expectation and it's completely different, you may not like it at first because it doesn't fit with your expectation because you have nothing else to refer to. And only if you have tasted a certain amount of different wines, maybe even different qualities, different ages, and all of this, then you will realize, oh, I like this wine more than I like that wine. Or you may find out, I've tried a lot of wines and I don't like any of them. But at first, you only have your expectation. You have to extrapolate from what you know or what you think other people know. That's your ground starting point. And then you gain experience. And then you realize, okay, I have a certain amount of experience and then I can relate one to the other. And I think this is the same in cosmology. The only thing we can measure is changes. And at first, this was Newton's great, I would say, great insight to transfer the expectation that laws in physics on Earth are exactly the same on the Moon or even in farther out space. This was Newton's idea. Let's transfer what we know from Earth, our expectations, into space. It could be different. I mean, nobody says that this is true. It could be that this wine is not good. But on the other hand, we found out in a lot of examples, with lots of experiences probing ourselves through the local universe, going to the larger scales, that actually a lot of these physical laws seem to be also plausible in farther out space. So it seems the same experiences that we make could be made at other positions in the universe.

Okay, and now the problem is what? Like, okay, sure, we don't know the mass distribution, but if we knew it, we could predict the lensing. And if we knew the lensing, then we could predict the mass distribution. The problem is what? That we don't know either and we have to infer both? Or what? Yes, the problem is when we know the source perfectly, we can reconstruct the lens. And when we know the lens perfectly, we can reconstruct the source. But the problem is in the universe, we are not sitting at the source position. So we do not know what the source looks like. And we do not know what the lens looks like either because the lens is also very far away. And in both of these cases, we have the problem that the source is only visible to us in a very distorted way. So we do not even know what's the morphology of it. And on the other side, the lens, this is something that we have even less knowledge of. Because we see, let's for instance take a galaxy cluster. We see a few galaxies in this cluster. But the rest, if we assume that dark matter exists, 85% of this cluster is most likely dark. And then, how do we make sense out of this lens? What is the distribution of the mass in this lens if I know less than 20% of this matter? I have a lot of room to wiggle to distribute the mass of the dark matter, and as well a lot of room to wiggle how the galaxies move that I actually see.

Okay so let's get to Jenny's large claim. What is your big claim? And then we spend the rest of this episode, the next hour or so, dissecting it and getting into the details. What's the punchline? The punchline that I hope to put forward is dare not to know. I would like to say if we take a look at the observables that we have, we should clearly separate our model assumptions from the data. And this implies, in the end, that we may not need as much dark matter as we currently think we need to describe the cosmic structures. So, I don't want to fill the knowledge gaps in our mass reconstruction with model assumptions. I would like to say, let's stick to the information that the data gives us, and that's it. And the rest is whatever. There might be something, there might be nothing, we don't know.

Okay, so the amount of dark matter is potentially drastically overestimated? Yes, it could be. I mean, the calculations that I do and my colleagues did, I mean, I'm not alone in this, we're a team. And so what we found out is that if we take a look at the multiple images in such a strong gravitational lensing event, then we can infer local properties of the lensing structure. And we can also infer how the source looks like up to an overall scaling constant. So we get the morphology of the source, like the relative view of it, and we get the local information of the light-bending object in space-time, but only the local properties of this lens, which means we know the local distortion directions, like how is the lens distorting each multiple. And we also know what is the relative size between these multiple images. So we know what's the relative power of the lens between the different positions. And all of this is the maximum information that is completely given by the data. So we do not make any additional assumptions how the dark matter could be distributed in the entire structure.

So I'm sure a critic would just say, look, we have so much converging evidence from simulations, rotation curves, the CMB, the bullet cluster, I'm sure it's always thrown out. How can you say that this is just model-driven? Because most of these, what you think is evidence, is actually model-driven in exactly the same way. Because we only have a limited amount of reasonable models, and we usually imply these because, on the one hand, if we start from the early days of modern cosmology, people tried to understand everything analytically. So the first models were spheres, ellipsoids, and maybe something a bit more complicated. And then in the 1970s, 1980s, computers started to take over. And this was the time when people said, hey, we can do this numerically. We can implement an entire universe in a computer. For instance, Volker Springel, he has one of the biggest cosmological simulations on Earth that he says, I can model an entire universe, including all galaxy clusters and everything, in a computer. And this is then numerical. But on the other hand, if you just take this numerical simulation, you have the question, what did you put in? You put in a certain amount of assumptions, like for instance, that your cluster is subject to Newtonian gravity, or that your cluster is taken together by Newtonian gravity and expanding on a certain cosmological model. You implement a lot of assumptions, and so based on these assumptions, you will see certain structures, or you will not see certain structures. This is why Volker has additional simulations to say, let's assume dark matter, for instance, has different properties. So let's play around. If dark matter is just a collisionless fluid, and these particles do not interact, what happens then? How do my structures look like? If I then let these particles collide, how would my structures look like then? And he finds out it's vastly different.

So I'm not a dark matterist, I'm not a relativist, I'm not a cosmologist or an astrophysicist. I'm just a fool. A fool with some good looks maybe, let's say. But they're not fools. So when you bring it up to other physicists, the model dependent argument, that must have occurred to them. What's the reception like? I would say it depends on whom you ask. I mean, I would say the community is divided, that half of the community says, yeah, of course we believe in dark matter because we can simulate it, we see that it's missing in all of these different probes together. But on the other hand, if you take a look, all of them have a very large overlap of the same assumptions, and everything is filling a lot of voids where we do not have data with models. This is the camp that I would say is in favor of whatever I model should actually be there in reality. And then there is the other camp, which I belong to, and these people, they say, if we now take a look at our observables, what have we actually measured? And if we now use a model to describe this, we know that this is not the reality. So we know that we make a simplification to reality. As George Ellis usually says, the cosmos is much more difficult and much more complex than just a spherical cow, because our cosmological standard model is a spherical cow. It's homogeneous, isotropic, so the same in every direction, and the matter density is the same everywhere. So this is something that is very, very simplistic. And if we use this model, we need to make sure that we know in how far does it fit our data. And if we get better data, is the model still appropriate to describe the degree of details that we have in the new data?

How did Mark Gorenstein react? Oh, he was... When I told him that I think that it's the local information of gravitational lenses that is actually the information in the data, and anything else is just model dependent, he just said, "Oh, finally, you got it. 30 years after we found out that lensing is degenerate and lensing models can fit the data equally well." So this was for us, this was a puzzle. And funny enough, he came from astrophysics and then he moved on to biophysics and he looked also a lot into the optical analogy between strong gravitational lensing and optical stuff. And I came from biophysics and I moved into astrophysics. So we had some kind of overlap, which was quite funny. So in the end, I think he could have found it as well, but he didn't see that to get rid of the model. He thought, okay, let's just probe a lot of models and then we will find out what is the underlying principle. So forward modeling to insert different models and finding out are the observables still the same, or does the model predict different observables? He thought, if we do this long enough, we will find it out. But actually I came the other way around. I said we need to get rid of the model. We need to find what is the common thing that the formalism itself tells us without inserting any model. And I found out the formalism is local. Every equation that is there has an X there, meaning position of the multiple image. There's always an X. It's the potential of X. It's the deflection angle depending on the position. Everything is depending. So I thought, okay, it can only be something local. It cannot be something global. And then I found out if we leave out the model, the only thing that remains is the local information.

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Okay, so walk me through, what does that mean physically? What does local information physically mean? Local information physically means that usually people want to determine the entire mass of a gravitational lens, like of a big object. And so they are after making an assumption based on the multiple images that they see, what is the total mass of this object? So they are inferring a huge global mass for one object based on a few data points. But if you now say the local information is the maximum information, not the information you have, it's obvious, you cannot get the entire mass. All you can get is some directions where the lens is distorting. You only know the properties of the lens at these positions, and this doesn't give you any information about the mass directly, not even a mass at these image positions.

Wouldn't the critic just say, hey, we always have incomplete data for everything, not just astrophysically, down on Earth we have incomplete data, we have incomplete data abound. And so what you're articulating is just fallibilism, we could be wrong, or some form of epistemic humility. But at some point, one has to commit. Like every measurement, yeah, has model dependencies, but at some point, you just say, okay, well, this is the model and this model works, or something like that. So tell me what you think. Yes, yes, yes. I mean, in principle, when I realized that it's only local information that we have, then I had the problem, okay, this model that we have, we insert it. So on the one hand, we cannot say much without a model, but then how good of a quality are our models that we have? And there, if you go back to what's their origin, it started as some idea that we have. It's called the singular isothermal sphere, meaning it's a gas in equilibrium that's spherically distributed, because this comes out as a solution of an analytic problem. How would a gas cloud look like that is very idealized? And that comes from like the first calculations that we did for astrophysics when we said we want to know how stars form, so we consider a gas cloud that is collapsing under its own gravity and all of this. So we had these power laws which is like one over r to the power of something. And the singular isothermal sphere happened to be one over r squared. This was one of the first, I would say, like structure descriptions that people had. And then they said, hey, if we now have some structure that is not an ideal gas cloud, let's just stick to these power laws, and let's just fit for whatever goes one to the power of alpha. And then they found this still doesn't fit. And then they did a more elaborate model based on simulations to fit for something that is in the end a heuristic mass density profile that seems to come up in simulations. So you can now ask, we have a model, and we could say, okay, we validated it in some sense by simulations. But do I believe that these simulations are actually mimicking reality? That's one point. And the second point is, a theoretical, fundamental, physical explanation, unfortunately, is still missing. This is why I also started to develop a derivation of such mass density profiles in order to reason why this power law is actually missing, that is not necessarily reasonable, and why we should use it, why we should trust it. And this is very complicated, of course, because a lot of people suffer here that normally we have statistical mechanics. That's what we've learned since 1900-something. But the problem is that this statistical mechanics is not suitable for gravity. And I think Roger Penrose, he would definitely agree to this. He also propagates this quite a lot of times, that he says gravity is exactly the opposite of what we do. We think statistical mechanics happens. The entropy should always increase, and then you have like a uniform distribution of everything. But what does gravity do? Gravity takes exactly the opposite direction. Everything that is distributed is just collapsed into a single point. This is, I would say, maximum order instead of maximum entropy in that sense. So how do you use a statistical mechanics approach that is doing the opposite of what gravity does in order to describe gravity? And this was my problem when I started to set out, okay, how can we describe a mass density profile? How can we derive this? And I'm not allowed to use statistical mechanics. And so I called it DEMON for Dark Emergent Meta-halo Ontological Nexus... And that was my approach that I put forward. And I found out that it is possible to reason why this power law is good, but not using statistical mechanics, to just say, okay, we start with Newtonian gravity, and then we say Newtonian gravity is scale-free. And this scale-freeness is obviously something that leads to power laws. This is what we also see in nature on Earth. And then you can reason why a power law should be a good approach. And then depending on how you coarse-grain your galaxy distribution into some density, you can then argue why you're going to go to the next level, a certain experiment is reasonable or not, and in which limit you can get this. So I'm more and more convinced that these power laws are actually reasonable. But still, I mean, we need to further understand from a fundamental physical perspective which model is actually good. And this is what I would like to see in the near future for astrophysicists. Don't use black boxes. I mean, simulations are important, but don't use them as black boxes and just heuristically fit. Try to understand, to bridge the gap, so we can bridge the gap between theory and simulation in order to get a better understanding of what we're actually doing.

Why don't you give us an intuition as to what a power law is and then what scale-free means, and why it's reasonable to think of Newtonian gravity as scale-free, and how that may imply power laws? What does this all have to do with one another? I mean, power laws is an inverse proportionality in that sense, but to a higher exponent. So you have 1 over r to the power of alpha. And this means that if you have the curve, I guess you can show the graph of the curve, how this looks like for one over r squared or something like this. And then you see that the farther you go away from your origin, the less important this becomes. But the point is, it is still there. It is not going exactly to zero. It's not really being damped by quite a lot. So this is the power law that you can, for instance, describe what is the richness of people, so what is the wealth of people, and then this usually follows a power law that you have a lot of people who earn hardly anything, and then you have like your middle class, and then you have the bunch of millionaires, but obviously it's not zero. And you cannot say if you have 10 millionaires who have a certain amount of money that you now know exactly what's the total amount of wealth that a person can have. Because there's always the surprising moment that somebody, like a billionaire, could step into the room and say, here's my paycheck, I earn much more. And this is, I would say, the perk of a power law, that sometimes there could be, depending on the exponent, there is no mean, there is no variance, and this means that the universe could keep on surprising us that it's not something that we can fully grab by, okay, here's the mean and here's the variance, and now anything else would be completely challenging everything. And the perks of these power laws is there is always a chance of surprisal, that if we find a structure that is too big to be true, in a power law statistics, this can come up, this can happen. And it's not unusual in the sense that it's completely challenging the world view immediately. This is why I like this idea of power law statistics. And Newtonian gravity is one example, because it has a gravitational law. The force is one over R squared. So this means that the force only depends on the distance between two objects. It doesn't depend on the time at which these two objects come together. It doesn't depend on any other property than just the masses, which are in the numerator, and this R squared in the denominator. And this is the nice thing of scale-free gravity, like this Newtonian law.

Now, I have the pleasure of speaking with you before this call, and so I have a quote from you I want to read out. Yes, yes, sure. I hope I wrote it down correctly. It's roughly correct. The entire community suffers from the problem that people like to model. Forward models can be generated by every bachelor student in the first week. It's an entire machinery. But if you want to solve the inverse problem, then it's much harder. It requires much more math. It's ugly. It may take months or years. I hope I've roughly captured you correctly. Yes. I hope I didn't lens, I hope I didn't distort what you said. So anyhow, please explain the difference between forward modeling and then inverse modeling, because it's extremely important. Yes, I think that forward modeling is something that everybody intuitively does. It's also in everyday life. When you see something, you immediately try to find a cause and a reason, and how everything happens, what happens from the start to the point at which you observe something. And so forward modeling is something that everybody does, and it's something that we build up a certain expectation, and the moment that we get a confirmation from our forward model, that it was predicting something correctly, we are happy. So this is something that gives us some endorphins, if you want to see it like this. But on the other hand, the moment that we forward model, which is quite easy to do, obviously, to just think from start and then think, how do I get to the point that I observe myself in my scenario. This is the point that when you say, okay, I get the prediction, but now if the prediction is wrong, nature tells you, "Sorry, no, this is not the way that I work." Then you try a different model. And then if you get another rejection, you get another rejection. And then another, another, another. And this is where we stand in cosmology right now. We ask, okay, we have tensions in cosmology. We have a lot of things, a lot of stuff we don't understand. So is it this direction? And nature tells us no. Then we ask, okay, can we wiggle around with this other parameter? No. Or can we wiggle around here? No. And if this is not solving the tension, we don't get a direction where to go. And I don't know why the community likes to do this. I guess only it's because you think you're close to the solution and you want to guess it right. Because it seems that from a philosophical point of view, having a forward model and then predicting something and getting the approval of nature, yes, this is right, this is supported by the experiment, this is something that everybody values as it's a good scientific theory, because it made a prediction and it turned out to be true. But on the other hand, if you do the inverse problem, you see something, and then you say, based on what I see, how can I reason? How could I have gotten there in all possible ways that is there? Or what is the necessary model that gets me there? If a mug lies on the floor and is in a thousand pieces, it doesn't matter whether it was the dog who put it over the table, or whether it was the kid, or whether it was myself. It doesn't matter. It just matters. The necessary assumption is the mug is in a thousand pieces on the floor. It must have fallen from the table. This is the necessary model that I have there if I see the table and if I see the mug. Okay. But this is all. This is all. And this is the point that I say, if we have this necessary model, we know that this is something that is required by the data. And if I now say, okay, I know the mug is broken, I know it must have fallen from the table, or maybe it was pushed from the table. Then I look, could it have been the dog? Could it have been my daughter? Could it have been my husband? Whoever. Then I start to look, who could it be? Like looking for the culprit after I fixed the point, I can go to the next part of the model in order to make my causal relation working. And I think that this is a more positive way of doing science, to start with the necessary thing, because if I see it must have fallen off the table, I have a positive idea. Okay, the mug is broken, but I have a positive idea of I have a model that must be true. The next step is now to find who pushed it, if it was pushed at all. Maybe the window was open and it fell by itself. But all of these additional assumptions are for the first point not necessary. But for the next point of my causal relation. But if I'm wrong that it wasn't my dog, then it doesn't, I don't fall back to square zero. I just fall back to it must have fallen from the table, and then I just look for the next possible reason.

Okay so if I'm understanding this correctly, you're explaining some of my psychology. So I very much like cop shows or true crime shows. And I think that's because I'm so theoretical that I constantly do forward modeling. So I care about what is the cause and then I try to produce what the effect would be. So I'm thinking, okay, here's an equation, here's a cannonball. As a first student you think of balls that are thrown into air and then you think what would happen if you did this. But you're thinking about it experimentally from a data analysis point of view. No, the universe doesn't work from our model and then runs it forward. The universe gives us data, and we're trying to get the model that accurately matches the data. So this is what cops do, this is what detectives do. They have the effect, they have the then, and they're trying to infer the if. Whereas theorists start with the if, and then they do the then. Yes, CSI cosmology, if you want to see it like that. Right. Great, great. Okay, so explain how that forward versus inverse, which every student does forward. The inverse is the reason why it's messy, and it's not a one-to-one map, it's a one-to-many map. Right. So, as you gave the example with the shattered glass on the floor, you look at that, it could have come from multiple sources. There are many coarse grainings that are consistent with that shattering. So it could have been your dog, could have been your daughter, could have been your husband, could have been you, and when you're sleepwalking, blah, blah, blah. So how does this now, this abstract concept of forward versus inverse modeling, relate to dark matter? And some of the problems that you think we have with modeling dark matter, maybe we've overestimated the amount of dark matter. I mean, the biggest problem that I see is we are not short of models. I mean, since Zwicky had the idea that there could be some missing mass in the Coma cluster, which is one of our neighboring clusters, and he found out there seems to be missing mass. He was actually more convinced that it's just a missing mass that we have from, I mean, that hasn't been discovered. It wasn't a big thing. But then when people saw that there is something, some potential for something new, of course, the particle physicists immediately jumped at this. They said, okay, if this is a new particle, if this is some new substance, then we can go and look for it at CERN, we can go and look for it at Fermilab and everywhere. So theoreticians in particle physics got excited about this. And then there are now, by now, myriads of ideas which particles it could be. And of course, CERN, Fermilab, SLAC, they all went and tried to find a particle. And then the string theoreticians, I mean, they got their hearts up that it could be something like a minimally coupled thing from supersymmetry. They said, okay, maybe this is finally our minimum stable particle in supersymmetry. They were happy about this. But then, after all of the particle physicists brought their idea forward and went to measure, there hasn't been any detection. So there are lots of models. People tried a lot of models. So far, no clear evidence this is a dark matter particle. And then people said, okay, maybe it's not a particle. People like Mordehai Milgrom, or there are several other people who said, actually, maybe we're just misguided that we're not missing mass, that we are actually having something wrong in our gravity. Worst statement, Einstein was wrong. The best clickbait. Einstein was wrong. It's not missing mass, it's actually missing gravitational understanding. And so this is why Mordehai Milgrom, he started with the modified Newtonian dynamics, because you think that most of these cosmic structures in their own vicinity can still be described more or less quasi, that you say I can say to a good approximation Newtonian physics still works for them. And so Milgrom thought, how about we modify Newtonian dynamics because our best probe that we have of how gravity works is the Pioneer satellites. And then he said the scale on which they have probed the universe, we can be sure that the gravitation works as Newton said. But if we now go out, we don't know. It could be different. So let's modify Newton's law based on a scale that is larger than what we have measured with our satellites. It's a totally valid assumption. And so he went on and he found there is actually, there seems to be a law that seems to work. And MOND has been there. There was a conference, MOND at 40, meaning the 40th anniversary of MOND. And of course it suffers from problems. It's Newtonian, it's not general relativistic. Then people said, yeah, it suffers from certain problems. You have one scale parameter in there. You cannot argue where it comes from. You fix it by fitting galaxies, for instance, or you're fitting some galaxies and then you get the scale parameter based on the data you measure. There are lots of issues, and people say, this is why we don't want to have this modified Newtonian dynamics as an alternative to dark matter. But on the other hand, MOND survived 40 years, and it was able to solve some of the problems to extend the theory. And I would say if we compare the ideas, missing mass to use dark matter, whatever dark matter model it may be, or modifying gravity, we're still missing mass, and it seems both of them have their issues. And we could say both of them are in some sense not sufficient. There still is some question here. But on the other hand, as long as we do not know, it can't be a particle, but is it really some modification of gravity? Maybe not really. We cannot know what is actually out there. And I think as long as we don't know for certainty, both of these ideas, both of these explanations are still somehow compatible with the data within their realms. I mean, both of them have problems, but on the other hand, I do not see that I could now, I cannot definitely refute MOND on something that I see, and I cannot definitely refute dark matter on something that I see. So I'm happy. The data tell me both of them have problems, but in some sense, both of them are compatible with most of the data that I have. So for me, I don't see any problem. Why should I now go for dark matter or go for MOND or go for anything else in favor and leave the other stuff aside? I cannot say this works better than that one, purely based on the data that I see.

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Okay, so to stick and get this point fully across about forward versus inverse modeling, as it's super, super important. And one of the reasons why I'm so excited to speak with you is that you've developed these data analysis techniques that not only will be useful for cosmology, but potentially, we spoke about this before, but off-air, potentially for particle collider physics, particle physics, and for interdisciplinary work. You have some... Biophysics. Yes. Biophysics, biomedical, whatever. Yeah, your background, you have a background in machine learning and cancer of various sorts. And so it's super interesting, because it's a template that, I don't know if you can also help solve crimes. But it's a template. Yes, in principle, I would say it's also valid for CSI.

Okay, so I'm in Toronto, and there are condos all about. And so let's just imagine, let me give some analogy and see if it's roughly on track. Suppose I'm trying to model that building's foot traffic. And the way I can't, it's too difficult for me to do so, it's intrusive, I don't want to, but I can at least measure doors opening and closing. I'm going to use that as some proxy. Then I have some model of what the residents do, their work schedules, the concierge's work schedules, visitors, and so forth. Then imagine that the doors open, I collect that data, more than my model predicts. I could postulate an invisible occupant called dark tenants, which... Mr. X. Mr. and Mrs. X. Yes, yes. You would get a lawsuit if you refer to your tenants as dark tenants, especially if they look like me. But let's just say... So it could be that there's some unknown residents, it also could just be delivery drivers, it could be that there's automatic doors opening and closing. You could call them phantom people. Yes, great, great. So another one is, I think I did this a while ago, I was trying to measure my own electricity bill a decade ago, and then I said, okay, my fridge takes up this much electricity, and so on, so on. And then the bill was a bit higher than I thought. And so I could have posited some dark electricity, or it could just be that my model of the fridge, that it's nonlinear in the compressor, or that there was some other source, or maybe the electric bill is wrong. So is that what you're saying? It's like, we need to be careful about what entities we're positing to exist. Yes, that's exactly the point. I mean, if you try to model who goes in and out the next door condos, then first and foremost, the way you described your model. Have a look. How many parameters did you put in there? I mean, you had to look, you need to know the working schedule of the people, you need to know how many people live there, you need to know, do these people have family who visit quite often? I mean, all of this, this kind of like, it makes your parameter space explode. You need to keep track of a lot of things. And now imagine you just are wrong in one of these things by a factor of two or so. Then your model would be completely off because you have to look at the same time, you have to look at all of the families in this building, and then if you just with one estimate you're a bit off, this adds up. And this is, as you said, your nonlinear effect in the fridge electricity. And then you can see that all of this forward modeling is really fragile. And this is why I also think that having the inverse model is easier. It's actually, okay, it's mathematically more challenging, but it's easier in the sense of you only have fewer amounts of parameters, and reducing stuff to the necessary ingredient, it gives you a clearer understanding of what is actually there. You do not need to know all of the family members of each condo. You do not need to know what's the working schedule of them. You just need to know, is there a bug in the door openers that when some bee flies by the door is open? Or is there something that the moment that the automatic light goes on the door goes open? You just need to understand when does the door open? Because this is what you're interested in. So your inverse model would be, instead of trying to assemble all information you can about all the inhabitants in the condo, you just need to understand how does this door work? And this is a much, I would say, a much easier problem to solve if you have figured out that this is what you actually want. Now going on MOND versus dark matter, wouldn't the dark matterist, the proponent of dark

Matter, say, well, look at the CMB. That has nothing to do with lensing, nothing to do with galaxy models, or rotation curves. And there's plasma oscillations from 400,000 years or so after the Big Bang, and that's independent physics. There's one of those acoustic peak... Sorry, sorry, acoustic peak... Baryon acoustic oscillations. Yes, yes, BAO, BAO, yes. BAO, yes. All this is, again, it's independent evidence. It's not just we're looking at the galaxy and saying that it's spinning too fast, or not in the right places or what have you. So what do you say to that?

Yes, I agree. I mean, the CMB is one of the, I would say, the most complicated things to analyze. And it's definitely a complicated and complementary probe. So you have the very early universe where we think it was more or less a rather homogeneous soup. And then we see these tiny wiggles on top of it, these fluctuations on the CMB. So in that sense, we have an easy game to do with the CMB, to describe this cosmic microwave background with our cosmological standard model. So you would say it's a really clear and clean probe that you have something that is more or less homogeneous and isotropic. If you now fit something homogeneous and isotropic plus epsilon, you would get really good constraining power with the data. And we have an all-sky survey with the highest resolution we could get from the Planck satellite after years of expertise from COBE, WMAP, and all of this. So CMB is really something that we would say, yes, that is really a high precision probe.

But to see the CMB, we need to get through all of the foregrounds. I mean, the CMB was the baby picture of the universe. Until this light reaches us, it has to go through a lot of stuff. And this means, have we corrected the foregrounds correctly? I mean, there is the band of the Milky Way. And usually we see the CMB picture, like the fluctuation picture. We usually see this as an all-sky map, but this is actually not what we measure. What we measure is something that is rather at the upper parts and the lower parts, and in the middle there is this band from the Milky Way that is usually with a less high significance. The signal-to-noise ratio there is lower. So what people do is they improve on this data by simulations and also by extrapolations and by additional data. And all of this goes through the data. Then everybody says that's a very clean probe. But I would say the thing itself, yes. But the data processing that you need to do in order to actually get to this is much more intricate. So in that sense you could say we are fitting something clear, we get a high precision, admittedly, but still, have we gotten all of the foregrounds right?

And concerning dark matter here, for instance, we could say okay we have a parametric model, we fit this parametric model to our observables. So we find that these peaks, that relative height between the second and the third peak, which is supposed to give us the amount of dark matter that we know based on our models again. So this means we fit a model to data and then we find if we fit this model we are missing so much mass. So now the question is, and this is definitely a very, very hard task, can we explain the cosmic microwave background without dark matter, or what would then be the replacement for dark matter? This is something that maybe we missed an understanding there. This is possible, but on the other hand, I say the CMB is the hardest thing to explain in terms of getting rid of dark matter. This is definitely true. But on the other hand, I would say if we start from us on Earth, we haven't seen dark matter. And if we then go further out in the local universe, it doesn't seem that we need a lot of dark matter there either. So how come that suddenly, the farther away we go from our own position, where we have lots of data, how is it possible that suddenly, when we go far away, that we need a lot of dark matter, or that we miss mass? Is it rather, I would say it could also be, well, possible, that we are actually missing something in our models. That we are not missing mass, but just that nature is more complicated than we think, and the models that we have are not complex enough to capture that.

I mean, you mentioned the bullet cluster. The bullet cluster is one of the famous examples where people said, this is the evidence for dark matter. The smoking gun. The smoking gun, in the literal sense, this is why it's called the bullet cluster. Because you see that you have two big clumps of matter that have been merging, have been colliding, and then they fly away. So they flew through each other, and now they fly away at a certain distance. And then you see that you have in the x-ray signal, you have the baryonic hot plasma that also interacted in the collision. And then because there is a lot of physics going on there, like the plasma interactions and all of them, this merger is then delayed for this x-ray signal. And then you see two bullets, one of them is really clearly like a bullet, flying out again after this collision. And then people say, if we now take a look at the position of this x-ray cloud, or the two x-ray clouds, and the position of the galaxies involved in this merger, we see that there is an offset between the optical image and the image that we have. And of course also the dark matter part that has been reconstructed from lensing again around these optically detected galaxies. This is one part, and this is offset to the x-ray signal that is the bullets, the bullets in the gas cloud. And this offset then was taken as an evidence that here dark matter does not follow the luminous matter, the baryons, because we have this offset. So the total mass is not where the mass of the visible things are, mostly. This was propagated from 2004 onwards. And a recent study just last year in 2025, there was a big NASA press release from Sangjun Cha et al. Pardon my bad Korean. And these persons, they had a look at the intracluster light, meaning all of the stars that are not bound into the individual galaxies of the cluster. So because if you have a violent merger then of course not only hot gas is emitted but there are also like galaxies that lose their stars. And these stars they go into a collective cloud of stars which is called the intracluster light in the intracluster medium. And with James Webb it became possible to track this intracluster medium because the infrared wavelength at the center of the cloud which James Webb observes, makes it now possible to look at this. And so they took a map of this gas cloud of the stellar cloud, and they found that this merger is much more complicated than just two clumps colliding and moving apart again. They said it could have been that the biggest clump actually underwent an earlier merger, and then they saw that there was an elongated arm going in the direction of the merger. And so it wasn't just two spherical clumps colliding. There was no offset in that sense between the luminous mass and the dark matter mass, because all of the stars in this intracluster light were actually nicely following this merging structure. And so they said this cannot be a clear hint that dark matter is offset to luminous matter. And again, refuting MOND. And so they said, if we now take a closer look with new data that's actually possible now for the first time, we see we can reconcile everything we know with the new data here. So there is nothing spectacular that the dark matter would be completely detached here.

Interesting. I didn't know about that paper. So I'll place that paper on screen and the link will be in the description. And for those who are driving and just listening to this, maybe running or what have you, but you didn't see it and you were thinking, oh, I wish I could see it, you can. That will just rewind. We'll place the visuals of that on screen. Okay, so what's the reception to that paper been like then? I'm sure as with anything, philosophy, math, physics, everywhere, there's usually counters to, not usually in math, sorry, in physics or philosophy. But there's usually counters to when someone puts up a position. So what's the counter there?

I haven't heard much about objections or so. I would rather think that it's, I would say it's a clear evidence that this cluster is much more complicated. And this comes actually in a row of several other findings that have now become possible with better technologies and better data. That people say, okay, first, back in the 90s, and I mean the first gravitational lens was found in 1979. And so we do not have that much experience in lensing when we go to 1990, so everything was pretty coarse-grained and preliminary. But now, today, after all of these years, we have a lot of data and we have a lot of experience with different galaxy clusters. So if we now go back to the findings of the earlier times, and we re-look at all of these galaxies or galaxy clusters, we find much more detail. And this detail is then the thing that resolves the puzzle. Like, for instance, here in the bullet cluster, we have this intracluster stars, which now we solve the discrepancy between dark matter and luminous matter. And the same happened for a galaxy cluster that my team and I, we were investigating. It was Abell 3827. This is another very beautiful galaxy cluster. And in the center, you have four luminous galaxies. They're almost equally bright. And from this very unusual configuration, it's pretty obvious that this is a dynamically very active cluster. Normally people like to look at galaxy clusters that are relaxed, as we say, meaning we have one brightest cluster galaxy and the rest is then just nicely, hopefully isotropically distributed. So the ideal case would be a spherical galaxy cluster with one very bright galaxy in the center. But this one here is exactly the opposite. Four luminous galaxies in the center and then we have strong gravitational lensing around these four galaxies. And this lensing almost forms a complete ring. This was why it was such a really beautiful, huge gravitational lensing event around these four galaxies. So people had a look at this, and then it was pretty obvious at the start when you do forward modeling, you need to impose a lot of dark matter. And when you reconstruct your total mass distribution with such a model, you find again that the dark matter seems to be at different places than these four luminous galaxies. And there, it was exactly the same process. People first thought this is an evidence that dark matter and luminous matter can be decoupled from each other. So what physicists call light does not trace mass, meaning you can have the luminous stuff at a different position. And then people first thought this was yet another bullet cluster. But with more and more observations, you could see how this discrepancy was shrinking, even in modeling. And then my colleagues and I, we analyzed this cluster with our model-independent approach. And there we could definitely prove that this offset between the luminous part and the dark matter part was completely driven by the models. And this is the perk of our approach that we can say we have a look at the local lens properties, and we can track how fast these properties change over our multiple images. So we had huge, huge multiple images in this case. It's a very beautiful example. And in each multiple image we found star-forming regions. And so we could use these star-forming regions to chop up the multiple image, each multiple image, into several parts. And then we mapped only these parts onto each other in all of these multiple images to get the local lens properties. And so we saw that over the area of the entire multiple images, the properties were changing. And this for us meant, okay, our approach relies on the fact that over a certain area that we look at, that the properties do not change. So we assume that we have constant properties in the area where we look at. And if we now have patches across a multiple image and we see each patch has a different property for this local lens property, sorry, that each patch gives us a different value for a certain local lens property, then we know, okay, for instance, the mass density needs to change, or the distortion strength needs to change over the multiple image. And this means the lens is quite turbulent. It's changing. So if we now assume we have a huge dark matter halo that is smooth and hardly changing, then we can see that this is not fitting the actual reality that is given by the data. And so we could say, if the mass density already changes over a single multiple image, how can we extrapolate these lensing properties into a region that is much more than just a single image? We have more data, and it's much farther away from these multiple images than just an epsilon, than just the individual patches that we aligned together.

Can you explain to me, in an elementary fashion, how is it that your approach can be called a model-independent approach? How can you do science if you're not using models? What does that look like?

The only truly model-independent thing that I can think of is just you say, here's the data. And you just list the data with conjunctions. You just say this and this and this. Right, right. You always have to specify which model it is independent of. And in this sense, I would say it's independent of a lens model. So our formalism is independent of an assumption how the global mass distribution of such an object, such a gravitational lens, looks like. So we just want to have a look at the pure lensing formalism, which still makes assumptions. I mean, which still describes how this light is curved around an object. But what we do is, in this lensing formalism, we track the light bundles that are emitted from the source. And then these light bundles are more or less shot through the gravitational lens, and then the observables that we get, this is what the lensing formalism tells us, it's these observables depending on the individual positions. And people who want to use a lens model, they now use the formalism. And in this formalism, we have, for instance, that we need to specify a gravitational potential. And then they say, my gravitational potential is of this and that form. So at position X, it has this form, at position Y it has this form and that value. And so this is how this lens model is inserted into the formalism. And I just work on the basis of the formalism. So for me, I have a gravitational potential at a position X. And what I do then is to say, how does it change to a position Y? Or what are the changes that can be directly tracked in the observables? While other people say, I want to make it more complicated, I use a model. And then this model gives me what is the potential at position X, and what is the potential at position Y? And then, of course, if you have a model, you know how it changes from X to Y, but then this is a prediction of your model.

Why don't you sum this up into a core thesis that holds this all together? So what's occurring to me is something like data constrains local information. Everything else, every dark matter map that people have seen, it's model-driven extrapolation. It doesn't mean it's speculation or just unconstrained surmising and conjecturing. It is exactly that. It is exactly that. You got it 100% right. You got it 100% right. That's exactly how you describe it. So the summary is that the local properties that we extract from these gravitational lensing data sets, that they are the maximum information we can gain about this gravitational lensing object, about, for instance, a galaxy cluster or a galaxy. And what we know is that these local information only encode the local shearing strength. So how much does a lens distort objects? And the relative power, how much can a lens enlarge the image at a certain position with respect to another position? This is the information we can get from the data. And now the point is, the farther away we go from the data, of course, the more any model assumption will take over. Like a real-time model assumption will take over. In mathematics, when you say I have a certain regularization function that I say if I go far away from the data I want to have a flat, for instance if you say what the gravitational potential in the vicinity of these images, if you say if I go just far away from these multiple images I don't want to have any potential, let's assume I just set it to zero or I set it to some constant. Then this is a regularization that you make. And of course the farther away you go from these images the more all of these assumptions will take over to describe your global lens or your global mass density profile. Where you have data you have information. Where you do not have data you have extrapolation, speculation, prediction, whatever you want to call it. But in principle, this is the point. Local evidence and I would say global extrapolation. Because the moment you use a model, you want to have this based on fundamental principles. So you thought about what makes sense to combine all of these data points into some overall model. So in that sense, I would say you make an extrapolation. Physically informed extrapolation, if you want. Is it like this? Yes, sure. Is that the same as curve fitting? So you have discrete data points and then you have a curve that you can place through it? Yes, yes, that's exactly it. So would the analogy be, the data is the local part, and then the curve fitting, the model that you suppose produces that data, of course in the discrete amounts, but that is the global? Yes, yes. I mean, you're a mathematician. You can understand if I have a sample of data points, and I want to know what is the curve going through all of these data points? Then you can assume, if I have sampled at a certain frequency, then I can reconstruct my signal, my smooth signal, or my curve, if you want. But if you do not have enough data points, then you have ambiguities. So what could be the actual signal? It's the Nyquist-Shannon theorem. That you say, if I have sampled at twice the frequency, and I have a signal of finite bandwidth, then I can exactly reconstruct my signal. If I lack data points, if I don't have the equidistant sampling, or if I do not have finite bandwidth, I immediately get ambiguities, but I can resolve them if I have an idea, if I have prior assumptions how my signal looked like. So then, using additional assumptions, I may still uniquely fix my signal. I still can uniquely reconstruct it. If I do not want to use these assumptions, that's the inverse problem method, I don't want to use any assumption, then I have to live with the ambiguities. As good as it gets, I have different curves that fit the same data equally well. If I have more idea what my curve actually was, then I can use the assumptions to uniquely fix the curve. So the universe doesn't give you these equidistant samples? Yes, unfortunately not. And you don't know the bandwidth? You don't know what spatial frequencies the mass distribution has? No. I mean, can we observe the entire universe? We can't. I mean, usually you need to see the entire signal to know that it has a finite bandwidth. And the only thing we can do, and this is why I'm an observational cosmologist, I say I have limited bandwidth physically by my particle horizon. Everything that I have ever observed, or that we can theoretically ever observe, because the speed of light is finite. So we have a light cone that we can observe, and all of this in our past that we can have observed. That's our finite bandwidth in that sense. But we do not know what's beyond that. So in that sense, we could do it, but we don't know the rest. There's always the philosophical boundary of what comes behind the horizon.

Now your breakthrough, if I recall correctly, came from reading someone else's thesis in a completely different field. I believe it was quantum chemistry or electrodynamics or something like that. Yes. Yes. Yes. Yeah, okay. And it had to do with the Laplace operator and Green's functions. Tell me about that please.

Yes. I mean, in principle I did my master's at CERN in particle physics. Then I moved on to cancer research and machine learning because astrophysicists didn't want to take on machine learning at that time. And afterwards I moved to astrophysics. But in principle all I've ever done was analyzing data under mathematical principles. And then a friend of mine, he graduated in quantum chemistry, and so he asked me to have a look at his thesis, which was more or less the mathematics of how to form molecules. And all of them, all of the prescriptions, how we can understand which molecules can form based on all of the principles that we know. And if I then had a look at it, I realized, gravity and electrodynamics have a lot of things in common. Most importantly, the mathematics. And I saw the equations and suddenly, I mean, he also described everything as local positions of electrons, and then you had the, how do you call this, the ions in the molecules where you have the positive ions, and then you have the electron cloud around this. And when I saw all of this, I thought, this is my gravitational lensing problem. I saw the Laplace operator, I saw all of the functions and how to solve it, and suddenly it was pretty obvious how to solve my problem to describe all of the degeneracies in this lensing formalism. So how can I wiggle around the potential to keep all my observables invariant? And then I saw in this thesis that the mathematics is exactly the same. So I went back to my math book that I still have on the bookshelf here. And I found exactly the theorems in functional analysis to describe my lensing degeneracies. So all I had to do was copy the theorems from mathematics and translate them into the language of this gravitational lensing problem.

So why didn't others see what you saw? I think it was 30 years between when you made an application to an adjacent field. It's not even that far, it's adjacent. I mean, it's not directly, but it was 30 years or so. So what did you see that others missed? And what allowed you to see it?

I think, I mean, most astrophysicists that I have met, they are more phenomenologists. They look at something and they have an intuition. If I have twice the mass, I have like that many lens power. Or if I have twice the mass, these things should run double fast or something like this. And for me, I cannot say much until I have written down the equations. So for me, it's the mathematical framework that, in the end, gives me the reasons to interpret physical things. So I think that this is why I found this degeneracy, because I saw the mathematical formalism and I saw I can one-to-one translate this here. And only afterwards I found out that this actually makes sense in physics. So I first solved the equations and I knew this is the solution. This must be right. But then I said, okay, I have the equations, but now they need to get some physical meaning. Because most people, they either live in the world of mathematics and then they have like variables, parameters, but then these parameters are usually called in a certain name. And then they say, but this is lambda. Like, for instance, the cosmological constant, it has this lambda name. And then people say, yeah, but this is lambda. This is a mathematical term. I want to know what the physical realism behind all of this is. What's the physical interpretation in this model? And after I had the equations I then went on to understand what do all of my parameters and variables mean in my equation. Carlo Rovelli said you should not write down a single thing that you cannot attribute a clear physical meaning. And this is something that I really took to my heart. And I found out that the mathematical formalism completely shows you the degeneracies because it's obvious. If I have a parameter that is called the reduced shear, I mean, it's a small g. This is what's in the equation. And then I thought, what does it mean? And it means that I can only constrain the local shearing power. So what is the local shearing power? But this shearing power is independent of the mass that it takes to shear this. It's just, okay, this is the amplitude of the shear and this is the direction. But how much mass physically it takes to create that shear is irrelevant. And so I saw that this is something that makes total sense physically because I do not know the total mass. That was obvious. So I cannot constrain anything that is related to the mass. But I can constrain properties that are more or less something with respect to a certain mass. You always see ratios in these equations. And you can ask why. Well, it's always something divided by the mass. So this is the getting rid of the degeneracies that we cannot constrain. So this was the, I would say, the nice and beautiful part when I realized mathematics works out, and then I can learn something about the physics from these equations.

Can you tell me who else inspires you, or has inspired you? So for instance, you mentioned Carlo Rovelli, with the gist of it being that don't write down anything that doesn't have a clear physical motivation, something like that. Who else? What else?

Well, my mathematics professor in my first few semesters, he was really great, Professor Yeager, Professor Willi Yeager. And he had several honorary doctorates. And in lesson number three, you immediately knew why. He was a mathematician doing calculus, functional analysis, and in this direction, so more the, I would say the numerical part of and the practical part of applied mathematics. And he taught me calculus, functional analysis, and also a little bit of finite element theory. And whenever he did something, like proof, theorem, whatever he did, he first explained what is it good for. And we had a lecture with a lot of people from biology, chemistry, physics, everybody was sitting there, also mathematicians. But he always made sure that we knew the practical applications and what is it good for. What is the problem in the proof where we have to really take care that reality and the proof still matches? And this was really inspiring for me. And this is, I think, what went through all of my data analysis that I always remembered that you need to make sense out of this and your mathematics should not, I mean, it should not be somewhere in an abstract space. You need to be sure that all the requirements of your proof are actually fulfilled in your physical problem. This is from the mathematical side. And another very inspiring person is George Ellis from Cape Town. He is the one who brought forward, based on, he did his PhD in Cambridge with Dennis Sciama. And I think Dennis Sciama was one of the first persons in modern cosmology who tried to ask this inverse problem question. And George Ellis and all of the collaborators from the inhomogeneous cosmology community that I very much like and appreciate, they have brought forward that this inverse problem approach should be pursued further. And I find this very inspiring, and this is half of the camp of the cosmologists who say, let's have a look what's in the data and not model too much.

You said something that stuck with me. You said that you can place infinitely many black holes, you can stack them in a null set. What are you talking about?

Yes. Yes, this is something that completely struck me. When I tried to derive the lensing degeneracies mathematically, this all lives in the very abstract notion of a Sobolev space in mathematics. And when I was studying Sobolev spaces and all of these, what does a function require to be integrated or what does a function require to be differentiated? As a student I thought, who the hell needs this? Why on earth should I care? And as a physicist in third, fourth term, you do not encounter these issues because you're not as deep in the research that you would actually care. But then, when I had this formalism to solve, and I suddenly realized it makes a difference if my function is smooth. It's just differentiable twice. Is it even continuous? I mean, what do I know about this lensing potential? I have no clue. So I wanted to have a function for my lensing potential that is the most agnostic in terms of what do I have to put in the lensing potential. And suddenly I realized, oh, it makes suddenly sense to say my function, my potential, should just be integrable or maybe I want it to be smooth. Let's assume it's smooth. What can I say then? Or then let's just be completely agnostic. What happens if this function is not even continuous or so? And then I discovered suddenly, if I say the function should just have the minimum requirements so that it's integrable in my formalism, I suddenly end up in these Sobolev spaces. And in this Sobolev space, if a function is integrable, you can still say that if you change the contents under the integral, meaning your function, by a so-called null set, then you wouldn't change the integral. And since all you care for in this formalism is the final result of the integral, you are allowed to change your potential by this null set. And what is a null set? A null set could be in physics. Now you go from mathematics to physics, what does it mean in physics? It could be a countable set of black holes, a countable set of point masses. And if you now think in a physical sense, a potential that is really very nice and smooth, that's something completely different than a potential that can have a lot of point masses everywhere and is actually humpy bumpy full of black holes. And this is something that quite struck me back then when I thought, suddenly this makes sense because this was this one sentence in one of the books on strong gravitational lensing. "Oh, and we can also put some black holes in this potential. It doesn't matter." And I thought, where's this coming from? I don't know. And for me, this sounded disruptive. I couldn't imagine this. But the moment that I worked through the mathematics, it was obvious why I could do this. But this came from a much more, I would say, fundamental sound mathematical theory that I suddenly understood where it's coming from. But physically, I would say, does this make sense? Could it really be that I have an infinite amount, as long as it's countable, an infinite amount of black holes in my potential? So I would say mathematically, it's clear you can have it, you have this freedom. But physically, you could ask, is this reasonable? The formalism gives it to you, so let's zoom out.

You and Subir Sarkar, who I spoke to on this channel, I'll place the link on screen, you both go against the standard model of cosmology, but in different respects. So, where do you agree with him and where do you differ?

Well, I mean, I've also talked to Subir quite a lot, and he's a great discussion partner, and he's very knowledgeable. And this is why I think that the stuff that he has put forward makes a lot of sense. And I would definitely trust the results. So whatever he found is there. We cannot discuss it away. And he comes from particle physics. So he says anything below this five sigma that he also mentioned in your interview is nothing because it could go away as just a side effect or just some fluke. But everything above this five sigma, whatever it means, however you redefine your statistics, as he also said, is something that has to be taken seriously. And in several works he has shown that he has an issue to reconcile the early universe with the late universe in terms of a fundamental reference frame. That he says, if I'm now on Earth and I look into the cosmic microwave background, of course we do not see the nice pictures we've already spoke about. We do not see this nice homogeneous isotropic picture. But we have to boost ourselves into this reference frame, meaning we are moving with respect to this reference frame. And now, if Subir comes and says, if this reference frame is fundamental, I should also see it in a later part of the universe. So if I take a look at far away quasars, they live in a bit later universe, then I could say I should do exactly the same boost and I should still see a homogeneous and isotropic distribution of these quasars in this reference frame. And so far I'm with him, but now the question is, how do you now make sure that you are actually boosting yourself in the right reference frame? I mean, how do you make sure that you found all the quasars at the right redshift? That it's really exactly that, in this shell of our cosmic time? Because normally, like 10 years ago or 15 years ago, it was quite expensive to make these estimates of how far these galaxies are away, like at which redshift are they. And he also had a study to show that there was some contamination of low redshift quasars in his high redshift samples, so he had to get rid of them. And on the other hand, it's also a question, how do you make sure that you have actually sampled homogeneously or that you really got all the quasars in order to say, I boost myself into this reference frame and I'm sure that I have enough quasars at the right places. I mean, he also argues that he needed more than a million quasars in order to get his probe to be significant. But the question is still, have we found all of them? Have we really found all of the quasars and not something else? Because there could also be contamination from other similarly looking sources, for instance. And here I would be a bit more cautious to say he found an effect that is definitely something there. But the question is the statistics that we now apply to say this was really a five sigma detection, or is it really something that is just as he analyzed it? Because there are also people who say it could be that these quasars change over time. And if these quasars change over time, that could also create this effect. I mean, for me I would say it is something that is definitely to be taken seriously and to be investigated. But I also see the other camp saying, yeah, but what about this? What about the change in the quasars? What about this? What about that? You will always have the, I would say, the underdetermination that you can say I see this as a hint for an inhomogeneous cosmology or for some breakdown of Lambda CDM. I would say the breakdown of Lambda CDM is not as bad as breaking down the cosmological principle. This is even the better point and the stronger probe, I would say. That most people just try to say something is rotten in Lambda CDM. Meaning we have our standard cosmological model and this has a certain set of parameters for one specific class of models, which is homogeneous and isotropic. And now Subir even says, it's not that something is rotten in Lambda CDM. It's that something is rotten in the entire class of these models. So it's not just that we have a problem that, for instance, a certain parameter has this value and somebody else says no, it has a lower value or a higher value. It's the entire class of these models, this very simple assumption that the universe is homogeneous and isotropic, that this is actually at stake. So, if his effect is actually true, we may not only have to abandon Lambda CDM, we will also have to abandon the entire class of these spherical cow models. So this is, I would say, the more groundbreaking thing. But on the other hand, the inhomogeneous cosmology community, like George Ellis always says, the universe is more complex than that, actually is waiting for this because, I mean, I would say Einstein, 1917, already knew that this assumption that he made in his first cosmology paper, to say, let's assume that the universe is homogeneous and isotropic, or statistically even homogeneous and isotropic, that's what he wrote. He wrote clearly, "I assume this. But I do not think that this is true. Because even with my limited knowledge of stars and a little bit of a galaxy, I see that there are stars and there is nothing. So I know that I make a simplified assumption, and I hope that with more data we will overcome this simplified assumption and we will get to the next level of detail." So this is the interesting part. So an opponent can always say to Subir, hey, your quasars are more variable than you think. You haven't completed your sky coverage. There may be other prior assumptions. Yes. So do you think that his model has more wiggle room, then Lambda CDM? And is that the core here? That if a theory is flexible and it's difficult to kill, then that's not such a great theory.

I think Lakatos, I don't know if you know who the philosopher of science Lakatos was, but he said a degenerating research program is one that can absorb away these issues because you don't want your science, your theories to bend so much and accommodate everything or potentially accommodate. Yes, yes. I mean, Subir is not making any, well, he's just saying if I interpret everything in Lambda CDM within this framework, I run into an inconsistency. So in principle he is, or not only Lambda CDM, he's even saying I use the cosmological principle that the universe is the same around every point in the universe, homogeneous and isotropic. And then he says, okay, if this is true, then I can do the boost in the same way in the early and in the late universe. I end up at the same homogeneous and isotropic distribution. And now he says I don't find this. So in that sense he doesn't have a more complex model or so. He just shows that our current model has an inconsistency. So he's just proving our model wrong if all of his assumptions are true. That's his statement.

Didn't you give a talk about Lambda CDM where you played both sides? Like, forward and then played devil's advocate? That's a rare quality to be able to do it, and then also to do it publicly. So tell me about that.

Yes, yes. I mean, the point is, I mean, I'm also, I mean, I started at CERN as a particle physicist, so I also think anything below 5 sigma is maybe worth looking at, but it's not worth something that we say, okay, now we take this seriously. So my colleagues and I, we wrote a paper, we decided to write a paper to collect all of the issues that we have, not only for the Lambda CDM tensions, but also for the entire tensions of the cosmological principle, like one of the probes is Subir's matter dipole. So to say, let's have a look, what do the data tell us about the cosmological principle? Like about this homogeneous and isotropic universe that Einstein assumed in 1917? I mean, we wrote this paper in 2023, so more than 100 years later. And Einstein back then said, "We have to overcome this. There must be the next level of detail." And we wanted to ask, have we reached that level yet? And so we teamed up with 22 or 23 scientists and we collected in a review paper all of the hints that seemed to violate the cosmological principle. And Subir's probe was one of them. And this was one of the few that actually reached the five sigma for the statistics. Another one, which is also at the same time in the other way that I think is the bulk flow of Kostas Migkas. So Kostas Migkas, he analyzed galaxy clusters and he wanted to know in how far these galaxy clusters on the sky, how they are moving as an entire entity. Do they have some so-called bulk flow? Do they move together in a certain direction? Or does every cluster move in a certain direction and statistically the motion is averaging out? This started as a PhD thesis and he thought, okay, I now prove that nothing happens here and then I can move on to my actual project. No. He didn't set out to find this, but he found a certain bulk flow on the sky that he said there is something that is moving in a certain direction. It doesn't average out. And he turned out, after hundreds of pages of papers that he wrote and his PhD thesis and a lot of different data sets that he analyzed, this effect was also at a five sigma significance. And so in our paper we found, we had Kostas as a co-author, of course. So we found that there are actually many effects that hint at a violation of this cosmological principle. So Subir's is one of the most prominent with five sigma. Kostas Migkas' bulk flows is also one at five sigma. And then we had several other effects as well that were not at five sigma, but maybe at 3 or 4 sigma. For instance, Alexia Lopez and Roger Clowes, they found what was called back then the giant arc on the sky. A really very big structure on the sky that seemed to be like clustering, but in a larger scale. And by now, they even found a giant ring. So the whole structure is growing and growing. It's getting larger and larger. And this means now, if we have a finite time from the Big Bang till now, how can we assemble such a huge structure? The counter argument is, this is a chance alignment. This can happen. And you just have one of these structures. Or maybe like, let's say we have three of these structures. On a statistical basis, this doesn't challenge our universe. That's one of the arguments. But on the other hand, if you take this in context, that we have the cosmic matter dipole, that we have these galaxy clusters that are flowing in a certain direction, that we have two large structures that seem to be challenging our standard model, that we have additional bulk flows of other things, not only galaxy clusters, that seem to challenge our model. And we have a lot of these probes. The CMB, I mean, you think it's a clear probe. We talked to CMB scientists and we found out there are also axes of evil in the CMB. And there are also anisotropies or even like asymmetries in the CMB. And all of these things together, I mean, this was for me one of the key moments to say, either our universe is more complex than a spherical cow. Or all of these probes are missing something. And it's not even the same thing. Because they all seem to miss something different. Like, do we get all of these calibrations wrong? In each probe, do we have different calibrations? And all of them do not seem to be right? This was something for me that I found most convincing to say we are at the stage that Einstein said, now we see the next level of detail. Now we should look at what is the next level of detail for our cosmology. And I would see it positively. I mean, I wouldn't say it's cosmology in crisis. It's, we have, after a hundred years of research, we have reached a stage where we can say we are now ready to move on. After a hundred years of this first standard model that we have, we are now ready to see the next level of detail. I mean, homogeneous and isotropic, that's the second most boring thing you can do. And as long as your data is bad enough, you will not see the next level of detail. But if you increase your data, if you increase your quality, you will see more details than homogeneous and isotropic things. You will start to see dipoles. You will start to see directions that are preferred. It seems that we are at this point. I would say it's rather a very positive thing to think of than a negative thing.

Now, those rings that you mentioned. Roger Penrose claims that those CMB rings are echoes from a previous universe. One that was not a part of the Big Bang. A prior Big Bang. And then prior Big Bang ad infinitum, his conformal cyclic cosmology. You work with that kind of data. You just referenced it. Is he onto something?

You sound like a typical forward modeling speculator. Do we have something here? And that's, I think, that's the fun with forward modeling that you can kind of like speculate and gamble. Is he onto something? And here, this is a brilliant example again of forward modeling. What do you want to believe? I mean, Roger's idea goes far beyond anything that goes currently to our observational points. The first picture that we have is the cosmic microwave background. And he now claims we have space and time before the Big Bang. This is revolutionary. And then you could say, but how do you test this with our data? Here, which I find is really great, is that

He makes predictions that can actually be tested in our own data that we have at the moment. And he says that in this cosmic microwave background there should be these remnants of the black hole mergers, the so-called circles of low variance. And he and another person, they also teamed up in order to measure these, to find these rings. And in the WMAP data after seven years of data taking, they claimed that they found four of these concentric rings.

But now again the question is, the other camp says, and there is another paper says, yeah, he claims that he has significant evidence. They do not state how many sigma, if I'm not mistaken. But the other camp says, yeah, but what about the foregrounds to the CMB? You need to take this into account as well. And then the significance changes. And on the other hand, they claim, if you now try to say it's a significant detection, with respect to what? And then Roger Penrose and his colleague, they used a certain kind of statistics that they said, this is significant. But the other team says, we used a different ground truth, a different simulation in order to see, is this actually significant? And if you add everything up, they say, at the moment in this data that we have, we don't find significant evidence for this. So it doesn't mean that Roger is wrong. It doesn't mean that Roger's theory is supported. It means the jury is still out on this, and we need more data, better data from the cosmic microwave background in order to now see, is this theory supported or not? But the problem here is, this theory just stands now at the moment as is. There is neither evidence against it nor evidence supporting it. So it's another theory that we can gamble on.

Does this mean that you found an alternate way to test Roger's theory? No, I mean, I didn't work on this at all. I'm more of a late universe person. But in principle, the question is always, if you try to find the statistical significance, you need to compare to simulated universes that you think are in accordance or consistent, as Neil Turok would say, with the data that we have and all the knowledge and our theory that we currently have. And here, this is the difference between these two camps. That they say, we do not have different data. We just have a different way of interpreting the data.

Speaking of Neil Turok, he's also been on this channel and you referenced him before as well. He pushes back, much like yourself, against exotic dark matter. He thinks that it may just be neutrinos. You're both minimalists in this sense. Do you find your work converging with his? Or do you see yourself as coming from completely different directions? Oh yeah. Neil is a great guy. We even met at a conference and we talked to each other. I very much support his view that if we do not see more particles coming from detectors, we should stick to the contents that we have. And then we should make a self-consistent picture of the universe with the stuff we have without going to look for mysterious things. And in that sense, I think the idea he's proposing with the right-handed neutrinos is pretty cool. Because we have already seen neutrinos, just left-handed. But it looks reasonable to make a small extension in our sector to say we allow for the right-handed ones as well. So in that sense, we're definitely converging. And I also agree that we should try to find a theory that does not use any exotic stuff. But where I see myself a little bit diverging from him is his idea how to solve the cosmological constant problem. So at the moment, he says, we have explanations for two of his five parameters. But for three, he still needs to have phenomenological fitting functions. And for dark matter, this is still one fitting function, but he puts forward half of an explanation that he says, we still need to make more sense out of this, but the right-handed neutrinos might do the job. That's good. But for the cosmological constant, he says he might have an explanation of why this constant is so low. And on the other hand, why it comes from the vacuum energy of quantum field theory. And here Subir Sarkar brought forward this explanation has a lot of problems. And if Neil finds a solution, that's great. But in my opinion, lambda is coming from a classical theory. It came into the world based on some classical invariants. So I think we should find an explanation of lambda which is first based on classical mechanics or classical theories. And only then say, okay, there might also be an attempt to unify, and we can identify gravity and quantum afterwards to say there is also an explanation for lambda in quantum theory.

Ah, okay. So you think that the vacuum catastrophe is somehow misplaced or misconceived or ill-conceived because you're trying to come up with an explanation for this lambda which stems from general relativity, a classical theory. Yes. Yes. Yes. Yes. Yes. And you're trying to find its roots in a quantum theory. Yes. Yes. Now, of course, the counter would just be, well, anything that's classical emerges from the quantum. So that's a natural predilection. Yes, yes. I mean, that's a thing that would be nice to have, to say we can argue from quantum fields to the classical regime. But on the other hand, it seems that there is a problem there. I'm not sure how it can be resolved. I mean, it's not my field of expertise. But Subir says that it seems pretty hard to solve it from the quantum point of view. So I would say maybe to go a step backward and to say it came in classically. Let's try to solve it classically, that might be the more reasonable first approach, in my opinion. I mean, people like Thomas Buchert or Subir Sarkar, we all try to make sense out of this also from the classical point of view. To see, is the data maybe splittable or partitionable in a different way so that lambda is a phenomenological parameter that is actually standing for some inhomogeneous spacetime curvature or something else that can come from some classical explanation. And not immediately, at least some quantum explanation.

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Yes, see, many people, when challenging something as widely held as Lambda CDM, which is not just held because scientists are obstinate or recalcitrant and they're fools and they're just unwilling to move on. That's foolish. That's not why that's held. There's great reasons for it. There's great evidence for it and so forth. But when someone comes along like a Subir or a you and puts cracks in it, sometimes it's thought that that is a dismal view of science. But actually, you're saying, no, no, no. It's a hopeful view. We're on the precipice of something. Yes, yes. I mean, I always see, I don't know why this is the case. I mean, Subir always says particle physicists try to weird down their models. They try to find something new. They want to be proven wrong. And in astrophysics, it seems to be the other way around. I wouldn't say that. But I think that people get used to things. I mean, humans are like, they want to have like traditions. They want to have stuff that is predictable. And if we now say, wait a moment, our universe is surprising. It may give us something that is unexpected all the time. And we cannot control this. Maybe it's a reason to panic. On the other hand, after 40 years of getting used to dark matter, I mean, there are people alive who fought for dark matter to be in the picture. And who were the first ones to do these simulations to show dark matter actually makes sense? They fought for it to be there. And then do they want to see it die again after like 40 years of research and all of this? And it's always the question, did we do all of this in vain?

What's super interesting is you're more radical for being conservative. So for instance, you're more radical for being conservative in a way to do it. For instance, when we were speaking before, you were talking to me about a colleague of yours who invented or proposed some crazy form of dark matter. And that the more that the data was collected, the less there was speculation and then the less of that sort of dark matter was permitted. And you're actually sobering them. But this sobering process, if people have been used to being intoxicated, can feel like, well, I want to stay where I was before. But you're the more conservative one, at least in this respect. Yes, yes. I mean, I have, because I have a problem of trying to chase something. I mean, I'm willing for crazy ideas. I mean, I also investigated naked singularities and how we could detect them. I mean, I also tried to write a paper on that. It's not that I don't like crazy ideas. I have a video about that, actually. Oh, really? Yeah, it's a crazy thing. So a whole podcast with JB Manchak about naked singularities and then pathological spacetime. Because many people will say the word that a closed timelike curve spacetime or one that doesn't have a, one that's not globally hyperbolic is pathological. Well, we have rotating black holes and so forth. And they're not globally hyperbolic. But it's also somewhat of a sociological word to say pathological. Einstein thought black holes were pathological in and of themselves and potentially the Big Bang. I think people even doubted their existence until we got this first picture of a black hole from the Event Horizon Telescope, that people said, okay, finally, we have a proof that there is something experimental that we can attribute to these spacetime singularities, whatever you want to call this. And I mean, when I saw, it was just a conference during the pandemic. And then there was a guy who said, okay, I think I can explain dark energy with primordial black holes. And then he went on and he thought, to repel each other with a certain charge, then I could explain the spacetime expansion. He tried to understand dark energy in terms of repelling black holes. So, and repelling black holes, then he tried to calculate the charges, and it turned out they were naked singularities. And so I thought, because everybody was skeptical, and so I thought, hey, it's winter break, I have nothing to do. How about I just prove this wrong in like 14 days? So I sat on my desk and I took his work. I took the charges and the masses that he predicted. And I went on to conservatively think, if I had such black holes in my universe, what would be the consequences for this? Like, for what modeling? I do this. And I found out I cannot kill this theory. Because you think of naked singularities, I'm thinking, yeah, naked singularities are disruptive things in spacetime. It turns out, if you have a naked singularity and you send a light ray through a plasma cloud, and you think, okay, what happens to this plasma, sorry, if you have a naked singularity and this is a charged singularity and you have a plasma cloud in this vicinity, what happens to the charges in the plasma cloud? I thought this would just be disruptive. You would even have like proton decay. Or I mean, I was thinking about the worst things. It turns out, even naked singularities, you need to take this plasma cloud very close to it in order to actually dissolve it. And for normal settings, or for when you say this is likely to be observed, you wouldn't find this thing. Because it's a singularity. It's really a point-like thing. You need a very perfect alignment between us, a plasma cloud, and something to observe an effect. And the other thing is, I mean, I'm a gravitational lensing person. So, normally you have a point mass. And a point mass is a very easy model to describe the lensing signals. And then you would say, I have a mass and the Einstein ring is proportional to the square root of the mass. So if I have a star that sends light over this point mass, I would see an Einstein ring around this point mass. And it's square root of mass. So if I have an exorbitantly large mass, I should have a huge Einstein ring. And then I looked at the equations and I found out, no, actually, because it's a naked singularity and it also carries a charge, this huge Einstein ring doesn't occur because the equations tell me that the Einstein rings are actually much smaller and I couldn't observe them. And it was quite astonishing how disruptive you think something could be, and it's so hard to disprove that theory. And in the end, I talked to an observer from the Westerbork radio telescope, and he said, "Yeah, I've seen many of these really exorbitantly crazy ideas and models that couldn't be proven wrong. You cannot prove something wrong. That's really difficult." And he also said, "Yeah, I definitely believe you that you tried your best, but it's really hard to challenge something like this." And with dark matter, it's even more intricate, as you said. I mean, you have a theory. Dark matter is just a word. So what do you call dark matter now? Is it just the missing mass in terms of clouds that you can't observe? Or do you want to have a new particle? And then, depending on what kind of dark matter, you can always wiggle yourself around. If there is a new observation, you could say, "Ah, sorry, dark matter wasn't collisionless. I see an offset. So it has to be collisional." So then we need to constrain the offsets. So you can always amend your theory.

You know, I was going to say that I've seen a pattern that the more restrained the theory, unexpectedly, you get more critiqued. Because you would think that it would be more, the more speculative you are, the more crazy you are, as you mentioned, the crazy dark matter, that that would be less accepted. But someone like yourself, someone like Subir, someone like Jacob Barandes and Tim Maudlin on the quantum mechanics side, who are more restrained, actually, or they have realist interpretations of quantum mechanics, they get critiqued. But then I also realized, everyone gets critiqued. There was this comedian who was saying that people think everyone hates America. If when you're in America you feel like every other country hates you. And then he said, yeah, but every country hates every other country. And you always hate your neighbor more than you hate America. Just everyone, everyone critiques everyone. And everyone always feels like they're the ones that's on the defense and they're the underdogs. So I think it's just par for the course. But I think that's exactly the point. I mean, I grew up, I thought, okay, math and physics, that's something that has something absolute. There is some truth in it. Some absolute truth. You can say one plus one is two. There is math that you cannot be wrong in that. No, one plus one is not two. It depends on the system you are in. One plus one can be something else. I mean, if it's in the binary system, you would add up differently. So it depends on your reference frame. And in physics, you have more room to wiggle. It's not math even. It's even less, I would say, with rigor. But on the other hand, I usually feel like a lawyer. I have a certain case, then I need to take some sides. What do I think is the most reasonable thing? Then I defend this. I try to find arguments in favor of this way to see the world. I mean, this is how I see physics. It's not absolute anymore for me. It's more, I'm a lawyer in that sense, to say I have now chosen a certain camp, the camp of wanting the least amount of magic in my universe, and the most amount of things that I can actually experience and I can see, like empiricism. Yes. And then to say, how can I keep this worldview, or how can I live in this worldview, given the input from outside?

Let's get into applications. Yes. So, particle physics has a similar issue where there's a huge amount of data, huge amounts of model dependence, and triggering and reconstructing events and background and subtracting away the background. So can your methods transfer to collider physics? In principle, yes. I mean, there's lots of data. So in principle, you would actually have the luxury to say, for the inverse problem, we just try to find what are all the possibilities that we have tracked here. So in that sense, that's even better. Because this is what I think that cosmology lacks. Cosmology is merely observational science. So we cannot make experiments. So we cannot interact with the things that we think are there. We set up models and then we can only watch. But we cannot say, we believe that this mechanism is happening, and now I go into the lab and I show, if I apply all of the requirements, I can create something. And I think maybe this is the reason, now that I think of it, this is why forward modeling is so popular. Because usually we have a chance of being the agent of producing it, given these prerequisites. So on Earth, being a particle physicist, it makes sense to forward model, to say, if I do this and that, if I collide this particle with that particle, I will see this collision, then there will be this particle and that particle going out. I will have some hadron jets. And in the end, I will see here, there's my Higgs. And this is why forward modeling makes sense, because I can prove myself wrong or right by exactly performing my experiment. And then I know this formalism leads to this. But in cosmology, we have a different setting. We cannot just go into the lab and say, here I have my forward model, and let me rerun the universe. Let me, yes, exactly. So, now I want to have modified Newtonian dynamics for sure. So let me rerun the universe. And the simulations, they try to replace this. But obviously, simulations are different in the sense that we face, it's on a computer. We have numerical instabilities. It's not the real thing. It's just a surrogate of a real thing. So in that sense, you can always say simulations are as good as it gets, but it's still not 100% like an experiment. And so in that sense, I would say maybe particle physics does not need that much of inverse problem solving, because they have lots of data and they have this agency that they can just create particles as they want, or do the collisions and not find a particle, which has also happened. In that sense, maybe particle physics is not the best target. But I would say I'm coming from cancer research and biophysics, and there I saw exactly the same problem. For instance, for cancer, there are more than 100 types of cancer, and you think, again, like in a forward modeling approach, you have certain processes that you say, if I have for instance a certain DNA that is replicated, or so, if I then translate this in the wrong way, proteins, then cancer can grow. And this is the signaling pathway approach. You start from different models, cancer can evolve through different pathways, on the different enzymes, on the different stages of all of your cells, on the different stages which your cells are in, and then it can happen that you have a certain signaling pathway where you say, okay, this type of cancer is growing in this way. But suddenly you have a patient where you say, nothing of this fits, and the medication we apply doesn't work. Why? Oh, there is a second signaling pathway in which this cancer can also grow. And then you suddenly start to realize that you have different options to end up in what you think is the same state. And of course, then you could also apply the inverse modeling to ask, what is the necessary ingredient in order to get to this stage, or hopefully in order to prevent this cancer? In that sense, I would say there is a lot of potential there as well.

So what happens next? Where is this research leading? I mean, for me, I would say I've realized that after almost 10 years in lensing, I've realized lensing isn't good for anything unless you couple it to other data. And this is why in the last two years I've started to look into kinematics, because if I have a structure that is a lens, then I would like to know, can I constrain the structure from the inside a bit better? Because now I know how to shoot light rays around it. But inside, for instance, in a galaxy cluster, I have galaxies and these galaxies move, and I can measure part of the movement. So now I'm trying to get a more holistic approach to say I have local information from lenses, and funny enough, math is your friend. I found a way that actually I can transfer my local lensing approach to the kinematics, to the description of the kinematics. And so I hope in the future I can also get local kinematics information about such a structure. And then to patchwork all of this, as we said, I want the positive way of science, meaning I know from the lensing the local stuff. I hopefully then know from the kinematics, and I have a lot of stuff. So I will more or less reveal, shed more light into the structure by all of these individual information pieces that I will puzzle together. But I know that I can only fall back to the local information as my ground truth, as the stuff that I have already validated by my approaches. So that's the next step for this one. But on the other hand, it also needs funding. And I think that our community, I would really like to see some light at the end of the tunnel, that some more funding flows in our direction. Because I get rejections with words like, nobody has done progress in 30 years on this project, or in this research direction. Why should you? And then I think, yeah, give me a chance to do it. Or at least give me a chance to move a little bit forward. Maybe I'm not reaching the final goal, but it's a direction that needs more attention. Because if I take a look, people say, for instance, I don't like artificial intelligence, or I'm against artificial intelligence. I'm not. But I see that in cosmology we have a hard time to actually succeed with artificial intelligence. Demis Hassabis, he won the Nobel Prize for AlphaFold, for this protein folding. And when he gave his Nobel lecture, he had three criteria, what makes a successful AI application. Number one is your feature space, and it should be really large. Like a human cannot understand it, you really need a huge computing form to resolve all of the parameters you have in your problem. And the parameter space is huge. You cannot do everything. And the second is your goal function. What is my function I want to optimize for? With AlphaFold it was pretty obvious. You have the free energy should be minimized. And then you know how the protein is folding. And the third part is you need lots of data in order to train your artificial intelligence. And if you now look at cosmology, what do we have? None of the three. Do we know our feature space? No. Because we don't have like 23 amino acids that we can combine to proteins if we just assemble them long enough. We do not know what dark matter is. So what are we looking for? What's our feature space? Is it a particle? Is it a fluid? What is it?

Wait, what is a feature space? What does a feature space mean? A feature space is the space of the ingredients that you have. Like, for instance, I would say, if I have a galaxy cluster, I would say, okay, what do I need? I need galaxies. I need the positions of the galaxies. I need the velocity of the galaxies in order to describe such a gravitationally bound structure. But I also need the dark matter in this galaxy and the positions and the velocities. So, if I want to understand how a galaxy cluster is taken together by artificial intelligence, I would need the position and the velocity of dark matter. If I do not know what it is, I cannot reserve part of this feature space for it. Then the second part is the goal function. What is the goal? I mean, do we really understand gravity to a degree that we say we can write down this optimization function? I mean, Demis Hassabis used the free energy for the proteins. What is energy in general relativity? Masses have to be defined, and the definition may not be unique. So, there we struggle from the other side to say, from the GR point of view, from Einstein's general relativity, how can we write down such a goal function to describe, for instance, a galaxy cluster? Like I said, also statistical mechanics to describe it is not going to cut it. And then third, data. We can use simulations. We have lots of simulations, and in simulations we have, I would say, enough data to train. But this is very costly to store on the one hand, very costly to produce from the starting point. And the big question is, is it actually realistic? Are we training on something that makes sense? For instance, for strong gravitational lensing, there was an AI approach. And this AI approach trained on simulations because we do not have enough data from strong lensing from the observational point of view to train such a machine. So they used simulated data, and they found out that when they tried to recover the simulations, great rates, 100%, I mean, not 100%, but 95% recovery rate of the data and correct descriptions. But then they went to the observations. And then the rates went pretty much down. Because there was something missing in the simulations that was in the data. And so the training process was not realistic enough, and you can't see it enough so that the machine could actually find all of the lensing events that were there, or that we actually know they were there. So, there is something missing here. And last but not least, if you take the observations, like, for instance, DESI has now found 11 million galaxies in our local universe, like 11 million galaxies. That's not half of the population of Ghana. So, just to put in a context, and you have to take a certain redshift in a huge cosmological volume. And now compare this to what we know on Earth. I mean, the population of Ghana, if I'm not mistaken, is 37 million people. So, in a very small country in Africa. So, if we now compare this to the 11 million galaxies in this cosmic volume, you know how the sampling is. I mean, we have a lot of data. We have a high degree of detail. But I don't think that it's enough to train artificial intelligence. Leave aside the, we don't know what's actually in your observations due to dark matter, dark energy, and the unknown ingredients of the universe. And this is why I think we first need to understand ourselves. What are we talking about? What is dark matter? What is dark energy? And when we have these questions, I think we can use artificial intelligence. But so far for the exploration process, we may get some hints where to go or not. But we cannot expect, for instance, an AI to understand the global mass distribution in a strong lensing event when I've already shown that only local information is the one we get from the data. So, in that sense, the AI would just speculate better, or give us a more sophisticated model, but we still don't know why this model, okay, it fits the data, but as do all other models, as I've shown. So, what is the gain using artificial intelligence, for instance, in this problem, if all we get back is a more complex model, and even maybe a less transparent model that we cannot understand. I mean, if we get a more complex model and not just a power law, how do we now argue that this is reasonable physically? The machine may give us something that fits the data, but it's just one option of many. This is what we know. But then, how do we argue that this is the true model? This is the truth that's actually out there. When we already know from the data, the inverse problem, we cannot know this.

Now, I know you said you will be talking about this issue and your colleague may push back and you say, just give me a chance. Maybe I can't take the ball all the way to the end of the football field, but I could take it a few feet further. So, allow me to ask, what changes if you get to the end of the football field? Paint that picture for me. What changes in cosmology? What changes in cancer research? What does the future look like if your method of prioritizing inverse problems over forward problems, and your solutions via these inverse problems, takes off? Well, I mean, inverse problem solving is, I would say, a typical thing that you use when you solve criminal cases. So now let's assume that we use it also more in biomedical, in astro, in cosmology, wherever we have, like, complicated problems with maybe non-unique answers. So, if we do this, I would say we have to rethink our scientific method completely. If we replace this forward modeling with this inverse problem-solving approach, it would mean that we change the way we think about science. And in my opinion, this would be a better world because we would have a more positive way of knowledge gaining. And I would like to show you how I mean this. I mean, imagine you now have this inverse problem-solving method and you build a tree of necessary models from the trunk to the branches to the leaves. And every higher level is based on the assumptions the lower levels. So in that sense you now think, I want to solve a problem, I have already built the trunk, now from one branch I want to go to a smaller twig. So to extend my model a little bit further. If I guess a little bit wrong, I can only fall back to the level beneath this one. But I will not fall down the entire tree to the ground. This is what forward modeling sometimes does, you end up at square zero, you do not know how to climb up again. And this way here we avoid this problem. And so I hope this would be more positive, encourage more people to care about science, if they have already built their trunk of knowledge, a certain part, they only need to extend their knowledge a little bit. So that's more positive. And on the other hand, this is even more important, if we now take a look at how science is done, it's usually investing a lot of money into a high-risk, high-gain science research. If we now say we already have a solid trunk and we just need to extend it a little bit, it takes out the high risk of all of this. And another point that comes with it, if you now climb up the tree, the further you climb, the less choices you should get. It's like in a criminal case, the more evidence is presented, the more people you should be able to kick off your suspect list. So, in the end, you narrow down the choices. And then this means you need fewer and fewer resources the higher you climb up. So, this idea is highly efficient. It's resource saving. And in that sense, I think you could even fund more projects than before, at a lower risk, of course. And so, well, I think if you build up all of this, it sounds more positive for me than gambling on certain forward models. And in the end, you may not even be able to find counter evidence or evidence in favor of something.

This is super interesting because theoretical physics is, in large part, model generation. So, on the arXiv, almost every day is a new model. Here's how the universe works. Here's how, even on this channel, theories of everything. New model. Well, here's how quantum theory works. Here's how it doesn't work. Here's how blah, blah, blah, blah, blah. Here's how GR works. Here's how you combine them. Here's what spacetime is. Here's what spacetime isn't. And then the question is, well, look, in absence of data, it's always caveated like that. In absence of new data, we don't have anything that's beyond the standard model. In absence of new data, what else are we to do as theoretical physicists? All we know how to do is just generate models, and we don't exactly have data to go by. So, in your analogy, it sounds like what you're saying is that would be the equivalent of just adding more suspects to your criminal list. Yes. Yes. Yes. And that's the opposite of what should be done. You should be narrowing suspects down. Yes. Yes. But, okay, I'm sure you've spoken to theoreticians about this, and this goes against the whole ethos. And again, like I mentioned, it's caveated with, in absence of evidence. So, what do you say to this, throwing models to the wall and seeing what sticks? It seems like that's all that can be done. Yeah, but on the other hand, I mean, there are people like Neil Turok who say, we have a lot of data, so let's use it in a different way. Let's try to be minimalistic. And in that sense, to clean up. I mean, Bjørn Hein also said, for particle physics, that when they found all the new particles, and they always inserted yet another field, yet another particle and so on, so their model just grew. And he said that the discovery was so quick. They just added and added and it somehow worked out. But he said we were always hoping that a few years in the future somebody would come and clean up that mess. So, they were obviously putting together a model with a really hot needle because they had so many discoveries. But they thought, in the end, hopefully somebody will look from top down in order to make sense out of this from a higher viewpoint and then to clean up the mess. And I think it's the same with all of these models that we have in cosmology or astrophysics. We're not short of models. But to pair them with each other, find what do they have in common, where do they differ, why do they differ, all these things are the important things to do. It's not coming up with another model. It's trying to embed the model that we now have, we want to put forward the landscape of all the other models. And in addition to relate them to the other models, say here is where I differ, here is where I agree with the others. And this is what I did with my approach, to say the maximum information is where all models agree. And then it's pretty obvious where the models will differ from each other and will differ from the information in the data.

What question keeps you up at night? What question has been keeping me up at night since I was 16? I mean, I would really like to understand what is gravity. That's the thing. I do not know why, but this is something that completely struck me the first time I heard it. When I first heard about Einstein's general relativity, I thought, what is this? This is really bizarre. And then I realized, gravity, this is a force, but on the other hand, it seems to be spacetime. So, what is it now? What is real? What is not real? And how can I understand this really? This is something that's, it's not just like electromagnetism that lives on a background. It seems to be the background. And this is something that is still keeping me going. I mean, academia is not the easiest way to earn money. Actually, it's the hardest, and it's the most stressful one. But this is the only reason why I'm doing this. I want an answer.

What's some concept? So, maybe it's this one. But what I was going to ask is, there must be some concept, whether it's mathematical or physical, that you couldn't understand. Maybe you tried for months, even years to understand it. Then all of a sudden it made sense. And so what was that concept or idea, or what have you, and what made it click? I think the best, there were several, of course. I mean, when you study, you learn a lot of things and you always ask yourself, why? Why do I need to know that? And then there is a certain point when you're suddenly confronted in your research project with something, you think, wait a second, I've heard about this during my lectures, and back then I wondered what is it good for. Now I know it. And one of the key moments in that direction for me was the Sobolev space. Because when I studied functional analysis, I mean, I had the best professor in the world to do this. He was really trying to make it applicable and to show all of the things that he knew from his applications. But it was always something that I never connected much to because he was more of engineering problems and all of that. And suddenly when I saw in my lensing formalism, Sobolev spaces physically mean you can have an infinite amount of black holes in your spacetime. And math doesn't care about this. This was something that I found completely intriguing, but on the other hand also bothersome. And I thought, okay, finally I understand why this concept was invented, or discovered or needed, in mathematics in order to tackle cases like this. That was super interesting.

Now, where can people find out more about you? Potentially some funders who are listening, who are watching, who want to help this sort of research. And then also maybe some other researchers who would like to collaborate with you. Where should people go to find out more about you? Oh, first and foremost, I'm really happy to collaborate with anybody on the entire planet. So the sun never sets over my projects. I really love this internationality. And during the pandemic, I really liked this, the review that we wrote on the cosmological principles. For example, it was written from researchers all over the world. And everybody has met everybody on this project. It was just collaboration. So if you want to find out, at my webpage. And I also have a Wikipedia page. But all of them contain my email address. So drop me an email. I'm always happy to answer. I'm always happy to go on podcasts, talk about all of these things. Because I think that the more people know about this and the more people know a little bit more of the details, people might be more convinced.

And lastly, what's a life lesson that you wish you could impart to your younger self? Well, there are many. But I would say the most important one is don't listen to others. Like, make your own judgment. Because I would say, based on the experiences that I've made, and I draw a lot on the philosophers of cosmology that I have been collaborating with, they taught me that in principle, there are many possible ways of living a self-consistent and fully reasonable life. To say, I have a theory, I have a way of living, I have a way of interpreting things that I see, to make sense out of everything. But not everybody would agree on the interpretations and how to do this. And I think I grew up a little bit in the wrong bubble. And it took me a hard time to get out of this bubble and to meet with the people. For instance, last year I met Thomas Buchert from Lyon. He is the guy who says that if we take GR seriously, then we always need to intertwine the cosmological background with the matter on top of it. And as Einstein said, both of them are one. There is an equal sign in the equation. So this means I cannot decouple my cosmology from the evolution of the masses. And he says that matter has a back reaction on the spacetime. And we need to take this into account. And when I met him, he said, "Yeah, I'm also skeptical of dark matter and of dark energy, because in the end, it's all the underdetermination problem, that you chop up your entire signal into parts. Part of it is, let's say, like the baryonic physics. Part of it is dark matter. Part of it is dark energy. But the question is, if you have several unknowns that are hard to capture, you can redistribute your entire signal, because it's hard to say, under which assumptions you assume this is a dark matter property, this is a dark energy property." And then he also told me, "Yeah, it's really hard to kill dark matter or to kill dark energy, because there are always the two camps in this underdetermination to say, this is definitely a sign against dark energy, and we don't have to take a certain amount of dark matter against the cosmological principle. Others say, no, it's an evolution in time. Or, no, it's actually a special property of dark matter that we have missed." And I would say, you have to make up your own way. You have to find your own way that you say, this is something that I find reasonable. And not just take on what other people say just because you think, yeah, I accept them as authorities or think that they know more than I do. I would say being critical, also in what happens on Earth, it's the same. Being critical and making your own judgment, I think, is the most important thing.

Thank you. Thank you for spending so long with me and the audience. Yeah. Could I ask a question? I saw that you have, like, always mostly 30,000 to 100,000 viewers or so. This is, I mean, it's highly impressive. And you have like half a million subscribers. So, who's your target audience? I mean, can you see, like, from which countries or so, what people are mostly watching your blog? Yeah, well, those are two different questions. So, the target audience, we always aim this podcast to skew toward the research direction. So, we aim it toward researchers. We aim it to be technical, as if this was behind the closed doors of academia. And it's just professors speaking. I'm not a professor, but you get the idea. So that's the target. And the audience skews technical. But the majority of people are actually, well, they're artists. They're people who are computer scientists, philosophers. It's a mix. And these people are interested in deep questions. In fundamental questions. And what is this place? What is time? What is reality? What is consciousness? So, it's people who are probing and they want something more than the mysticism that they've heard before from many popularizers of science, like the particles in two places at once. The cat's dead and alive at the same time. Isn't it cool? Isn't it cool? Isn't it cool? That's the same hundred times that I tell you over and over. And they're like, okay, but what's, what else is there? What's the actual math behind that? What does the math say? Does the math validate that? Does the physics validate it? People want to know. Can the cat really be dead and alive at the same time? Because this doesn't, physically, this doesn't make sense. So what does it mean? Yeah. And is that exactly what a superposition says? Yeah, yeah, yeah. So, in principle, no, that's interesting. So, you really capture the entire scope of people. It's not just the engineers, which are usually the ones who go into that direction. So it's really more great. Yeah. I aim toward researchers and academics. I aim that. And I think we have a large proportion toward that, more so than the average, because we're technical. If it limits the audience, and it does limit the audience, but the bulk of the audience, yeah, I imagine, are scientists and logicians and sorts. And just lay people. Truck drivers, not like their developers, nurses. That's cool. That's cool. That's cool. I mean, if you could get a chance, I mean, it might be interesting to get George Ellis for the podcast, because he does cosmology, he does philosophy of science, and recently he abandoned all of this, because I think he understood cosmology really to the fullest, and he said, "Now it's boring, gravity is the same everywhere, I don't care." And then he said, "Now I'm doing neuroscience, because now I want actually to see where is consciousness coming from?" And he's also, like, driven by mathematics. And he says the mathematics is the same everywhere. And he won the Templeton Prize. This very famous prize. It gets more than a Nobel Prize. For applying mathematics of cosmology to the real estate market in South Africa in order to improve the conditions for the underprivileged people there. And he won the Templeton Prize in principle for doing good in his own country, for the people. But in principle, I mean, he says the math is the same. And this is why he could actually apply the same equations to one and that one in order to help people on Earth with what he learned from the cosmos. So he's also interdisciplinary in that sense. Yeah, I would like to speak to him. Anyhow, I want to thank you once more, and the audience, you as well. Thank you for having me. It was a really lovely chat, highly inspiring.

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