 Right, welcome everyone to our Latin American webinar on physics number 148. We are very excited because we are approaching the webinar number 150 and we will have a special talk on that day, so please stay put. We are very happy today to welcome our speaker, Kevin Crocker, who is an affiliate graduate faculty at the University of Hawaii, and he will be sharing his work on dark energy and black holes, astrophysical black holes and cosmology with us. So we are very excited about this talk. Please make sure that you share your questions or thoughts on the YouTube chat that you have on the side and we'll be happy to share your questions with the speaker at the end of his talk. Meanwhile, also please remember to subscribe to us in every single social network that you can, including Facebook, Twitter and YouTube, and be aware that there are many interesting talks coming up, so please be on the lookout for the announcements. With that, I'm happy to let Kevin take the baton, take it away, please. Thank you for being with us. Thank you one second. I actually, there is a sort of a troublesome typo with respect to something on one of the slides and I just fixed it. So if you just give me about 20 seconds, then I'll be able to pull up the correct time. This is when Roberto can entertain the audience, or do you have any announcement? Yeah, you can show your juggling skins. Don't you have any latrine thinking? I have this ball. They changed color. All right, I've got it fixed. So thank you for getting the entertainment for me. OK, so welcome, Kevin. Now, please take it away. All right, let's see if I can get this to work now. So any announcement? Yeah, you can show your juggling skin. Do you have any latrine thinking? I'm here. I'm doing an echo, actually. Do you have any? All right, I've got it fixed. Open also. No, I do not have anything. So welcome, Kevin. Now, please take it away. Walter, get this to work now. So can you guys see my screen? Yes. All right, so thank you, everyone, for having me as a guest on Latin American webinars on physics. It's a pleasure to be here and share with you today the work we've been doing in Hawaii on black holes and cosmological coupling, which I'll get into in a bit and how they could actually be an astrophysical source of dark energy. And so this is sort of a small milestone for a group. We've been working on these sort of topics and laying the theoretical foundation for this line of research and general relativity. Starting back in like 2017, even, our first papers came out around 2019. And so to sort of reach an observational milestone is we're pretty proud of it and we're really excited to share this. So there's a lot of people that were involved in this project. It was definitely a synthesis of many different many different areas of physics, observational, mathematical. And so I just wanted to highlight a couple of the of the names here. Duncan Farah and Sarah Petty definitely led the astronomy side of things and put the catalogs together and really had the idea for how to go about looking for these sorts of effects in astrophysical phenomenon. And Joel Weiner was was key in laying down the mathematical foundations with me. And then we had sort of bias contributions from Francisco Shankar and just a whole ton of people really, really working together to make this thing a reality. So we're going to be talking about black holes. And one of the most important things I think for just getting a getting a handle of what what this phenomenon is and what we're doing is to make a distinction in one's mind. And so this is the famous predator handshake. This is Carl Arnold Schwarzenegger on the right and Carl Withers on the left. And so what I'd like you to to to keep in mind is that we're going to make a very clear distinction in this talk between astrophysical black holes, which I will use that word or that phrase to mean the objects that actually exist in nature. So the things that are emerging that send gravitational wave waves to us that we observe in LIGO or Kagura that we get electromagnetic signatures for that we can observe when we're using event horizon telescope that we know we're getting, you know, the creation effects from, you know, A, G, N, those are the astrophysical objects. And those should be kept distinct in one's mind from the models that we have for these objects on paper, these black hole models, things like the Kerr model or things like other types of models that I'm going to get into later today. And, you know, our goal will always be to find a model that sort of best captures all of the relevant aspects of the astrophysical phenomenon. And with that said, I'd like to sort of give a little bit of historical introduction here, because I think it's pretty fun. And so we're going to start with a paper by Sakharov back from this tag of the submission date is March 2, 1965. And this is the initial stage of an expanding universe and the appearance of a nonuniform distribution of matter. And this was the paper when he laid out, like using quantum fluctuations to seed basically what would become structure in the universe that we observe today. So this is a pretty, pretty important paper. And in this paper, when he's discussing the initial condition that's that's suitable for this process, he has a very interesting footnote. And he says, E.B. Gleaner in a paper now in press called states with this initial condition that he wants in connection with the isotropic character of the four tensor proportional to the Dirac delta, a mu vacuum. And so he's talking here about the stress energy tensor. And he's saying if the stress energy tensor is proportional to the to the chronicle delta, sorry, that it's going to have a mu vacuum character. And what is this mu vacuum that he's talking about? And so we can find this cleaner paper and we can see that it was submitted actually to the same journal two months earlier. And it's called Algebraic properties of the energy moment of tensor and vacuum like states of matter. And I don't know about you guys, but I suspect it's the same thing. There's a few papers that I've experienced in my career that when I read them, I was just floored. I was like, wow, this this person really nailed it. Like really nailed the physics really was ahead of the curve here with something. This is one of those papers for me. And I just kind of want to read parts of this abstract. The physical interpretation of some algebraic structures of the energy moment of tensor allows us to suppose that there is a possible form of matter called the mu vacuum, which macroscopically possesses the properties of vacuum. And then he says down here, the space time of a mu vacuum is an Einstein space in the sense of Petrov's definition. And a uniform world of mu vacuum has the asymmetric. So in this cleaner paper, he's basically talking about the possibility of dark energy, theoretically. And he's not thinking of some uniform distributed stuff. He's thinking of it as sort of a possible form of a matter, right, which just has the same symmetries as vacuum. And this paper is just like a stunning to me. And this paragraph over here, just really here, the situation is not utterly unrealistic. An attempt to describe phenomenologically the structure of an elementary charge particle would lead to the conclusion that inside the particle, there must be a negative pressure which balances the electrostatic propulsion. This raises the thought that in an ultra dense state of matter with the baryons so compressed that the Mason fields which provide the interaction between them cannot be produced. A continuous medium is formed in which the conditions correspond to an attraction between material elements and are described phenomenologically by a negative pressure. For example, such a state might be reached in gravitational collapse. So you have that in this within this two month span, these two Russian physicists are interacting with each other. And one of them is using this material that is has the same symmetries as vacuum to seed structures that will eventually become the universe we observe. And the other one is saying this phenomenon is relevant at the depths of massive stars when you're, you know, forming what we would want to call a black hole these days. And then he says it would seem that a negative pressure should lead to an internal instability and that if there are no volume forces of the type of the electrostatic propulsion, it would lead to a contraction without limit. This is not true, however. And he goes on to make an argument with with conservation of stress energy of how this thing stabilizes itself. Now, this was, you know, broad, very broad strokes looking at the algebraic properties of the of the stress tensor linear operator. And it took about 27 years, but 27 years later, one of these solutions was written down by Irina Demnikova, who was a student of Bliener, if I get the history correct. And I'd really like to highlight Irina's contributions here because she's really been a pioneer in this field. There's so many solutions that she's worked on in this sort of vein, these non-singular types of objects where there's pressure, negative pressure in the inside of the object, it's strong. And here's what the paper says. Sphereally symmetric vacuum stress energy tensor with one assumption concerning its specific form generates the exact analytic solution of Einstein's equations, which for large are coincides with the Schwarzschild solution. And for small art, behaves like this solution and describes a spherically symmetric black hole singularity free everywhere. And this was just the tip of the iceberg. So these sorts of solutions with with strongly negative pressure are not, you know, there's not just like three of them characterized by mass spin in charge. There's a whole zoo of these things. And they've been studied for more than half a century now. So here's some examples. There's this junctional solution of Demnikova, which is noteworthy because it's smooth. And then there's junction solutions for people who work on sort of stitching an interior solution to an exterior solution. Matt Visser down in New Zealand has worked on these things in Francisco Global. There's condensed matter approaches for how such an object could form and be interpreted by this interesting paper by Chaplin. Bob Laughlin is from the fractional quantum Hall effect Nobel Prize is actually on this paper. Probably the most familiar or widely known model for this type of object is called the gravistar. And it's from sort of motivated by quantum field theoretic considerations by Mauser and Matoa. They have two versions. I would want to call an old gravistar and a new gravistar. The new gravistar is definitely the one to be paying attention to. This solution is pretty interesting because it's horizon free as well as being singularity free. So the entire solution is regular and it's in causal contact with the with the universe exterior to it. And that horizon free and causal contact nature will become very important for what we're about to discuss. People have considered assembly. So Paolo Gondolo and his grad student Philip Ultracchi at University of Utah have been working on sort of understanding dynamically how this thing these sorts of objects could form. And, you know, the great thing about these things is they're all Lego consistent. So they mimic or they mimic or they're indistinguishable from the Schwartz child Kerr simple black hole models on the outside. They mimic or they're indistinguishable from Schwartz child Kerr. Is that a good thing? Maybe that's a bad thing. We're trying to do physics here. And so we need ways to distinguish these sorts of solutions from the the simpler black hole solutions. If they're, you know, if they're observationally indistinguishable and the solutions are much more complicated, then why don't we just continue working with Schwartz child Kerr? And what I'm going to do now is ask you to take your glasses off. If you've not seen this, I can't read the audience's response right now because I can't see your faces. But this is from a John Carpenter movie called They Live. If you've not seen it, I do suggest seeing it. It's pretty entertaining. So I would like you to get ready and buckle up for this. What? How do we get some signatures for these objects? So as we've all taken cosmology, either sort of in our, you know, late in our undergraduates or in graduate school, we definitely encountered the discussion that as the universe is sort of forming structures, there's initial over densities. These over densities begin to contract gravitationally. And at some point, we are told these these condensations, these gravitational condensations, their self gravity somehow beats the cosmological influence and these regions detach in some sense from the rest of the universe. And they just sort of evolve independently of whatever the rest of the cosmology is doing. It turns out this intuition is just false. It comes from studies of dust only. So P defined to be zero models from about the 50s. I think it goes back to Bondi. But again, I don't know if I have the history exactly right there. And it's shown false in 2007 by this interesting paper by Valerio Faroni and Abhijak. And I would just want to say here that new exact solutions are presented, which describe black holes perfectly co-moving with a generic Friedman universe. And that sounds strange given this intuition that we've developed, but they just write it down. So here's this Nolan interior solution, which is you take a cosmological solution, an arbitrary cosmology on the outside and you stitch something on the inside that is regular and makes sense. And this was written down in 1933 for the cosmological solution on the outside by McVitie. And Brian Nolan wrote down the interior solution, I believe in 1995. And here you go. The star service is co-moving with the cosmic substratum and the proper curvature radius of the star is linearly scales with the scale factor of the arbitrary Roberts and Walker cosmology. And therefore we have a local relativistic object with strong field, which is perfectly co-moving at all times. And in this case, the cosmic expansion wins over the local dynamics. And so the lesson to take from this is that when you have realistic cosmological boundary conditions, your solutions in general relativity can have all sorts of new phenomenon, interesting phenomenon and useful phenomenon. So there's this co-moving radius solution by Faroni and Abhijak, which to my knowledge is the first paper to have this effect. Subsequent studies have constructed solutions with dynamical mass. All of this Guadalcancha solution in 2012 and there's another paper by Masiel in 2016. I should have put up here. But in this dynamical, so you'd think that if the co-moving radius for like a black hole object is expanding with the universe since R is 2M, you'd expect that the mass of the object would also be like increasing with the scale factor of the universe. And so this dynamical mass solution heads in that direction. But for this solution, the relationship between the expansion of the object and the rate of mass growth is not constrained. And so the relationship between these was not so clear. Our work in Hawaii started laying down how you could work out that relationship. But I just wanted to point out that very recently there's a paper by Kedani et al. with some researchers at Gran Sasso and INFN in Italy that constructed a definitely coupled solution with the mass that does grow proportionate the scale factor. And that paper is very recent. So this came out I think just a few weeks ago. So now that we have objects which can interact with a cosmology, we might be able to see this effect over a long time scales. So we have a way to distinguish these sorts of solutions perhaps from current child exteriors if we are able to hook into the cosmological behavior. And so let's see if we can find evidence or we can examine the hypothesis of Gleaner that these objects should be collapsing to form strong P equals negative row interiors with things that we would call dark energy today. And so to get a handle on how to do this like to get the quantitative prediction out of this this is what we did in Hawaii back in 2019. So the usual approach in cosmology when you're building a cosmological model starts with the matter. So you observe that the matter seems isotropic and homogenous on large scales. And then you assert from that observation that the stress energy tensor only depends on time and also has the symmetries of it perfectly. Then you conclude then that the Robertson-Walker metric model for gravity is required for consistency. You plug these things in, you get Friedman's equations and you fit for what this time dependent row and time dependent PR. Now this, excuse me, this procedure has some problems that have been known for quite a few decades. The largest of these is called the averaging problem. And it's how do you compute this time, this position independent stress tensor from the actual position dependent stress tensor that we have in our universe? If you just look around you, right? Like your room is not homogenous and isotropic and neither are you. And so we need some way to go from the actual distribution of stress energy to this idealized thing for the model we're trying to build. And that prescription is not well defined. You can see sort of it's a chicken and the egg problem. I need a say I wanna compare a tensor quantity at different places on the manifold because I wanna take an average and I wanna integrate over some region. Well, how do I make that comparison? How do I push my tensor to like some point P on the manifold so I can compare them in a meaningful way? If I had a metric around maybe I could try geodeusics or maybe I could try something else in parallel propagation. But this is the object I need to solve for the metric for Einstein's equations. And so I do have a chicken and the egg problem here. How do we make progress? This isn't the only issue with this approach though. The other issue is that it's perturbatively inconsistent in that intuition from small scales has been traditionally used to argue that late-time pressure is zero. Usually when cosmological models have been built there was this idea that there was some sort of kinetic theory notion of pressure that you have a bunch of galaxies and they're very rarefied, they're very dilute. And they're not sort of interacting with each other too often, like they're not directly scattering. And so then the pressure for this sort of aggregate fluid should be zero. And then you'll have see arguments where people will use like Birkoff's theorem and they'll try to say, well, outside of this object, I only perceive a point mass. And the problem with all of these things is that you're using intuition from small scales and you're using intuition from flat space time to make this argument. And at zero order, the universe has no notion of inside or outside of anything. That's only a notion that comes into play at first order and then to get down to the scales like compact objects, like who knows what order you have to take the calculation to. To do a consistent perturbative calculation, you have to get the background first before any position dependent equations even exist. And then you take that solution you found for the scale factor and you stick it into your first order equations. And then this intuition from small scales has the wrong boundary conditions. We've got Minkowski boundary conditions and those are only gonna be valid for some amount of time when whatever time scale I'm looking at for my phenomenon is short compared to the expansion rate at that epoch. And so what we did at Hawaii to get around these problems is we inverted the way that you usually build a cosmology. And so we started with the metric model. And so we said our observation is that we have no control over what's actually in the universe. And our job here is to reconstruct what T mu nu is based on observations. And so we're going to, instead of assert the standard model, we're going to assert the perturbed Robertson Walker model as I've written it here. And this is now an explicit choice that we're consciously making as model builders. And I would like to point out that other choices are okay too. We just picked this choice because this is the standard starting point for the usual analysis. And so it was the easiest way to build a bridge from the previous understanding to this other approach we're going to take now within GR. And then when you do this, you notice something very interesting happens. And you find that the action, the Einstein-Hilbert action itself, the integral globally constrains what the stress energy tensor needs to be doing in order to match T mu nu. And what I mean by that is that the averaging problem is automatically solved. You get unambiguous averages over the space-time manifold that come directly from the principle of stationary action. They come directly from your integration over the Lagrange density. And so there's no argument anymore about how does one average contributions from the actual stress tensor, it comes directly from the action. These equations are perturbatively consistent as well because there's no assumptions about T mu nu are required to build consistent equations of motion. The stress tensor is whatever it is. And then the averages that you need come directly from the action. So the result of all this is that the Friedman equations and the Bardin first order equations are unchanged. So the equations you've been working with for 34 years, they haven't changed at all. But what does change is the relation between the micro physical stress tensor and the parameters that enter your Friedman's equations that enter your Bardin equations. These parameters become constrained by the action principle. And by parameters, I mean things like the equation of state. I mean things like the sound speed at first order, those parameters become altered and they depend on the micro physics. So we get a GR prediction. And so that was a lot of material. I apologize. So I've given you these QR codes if you'd like to grab them. This is sort of a very succinct description, three pages in publishing PRD of how to go about making these sorts of computations within a classical field theories. And this is the nitty-gritty details within general relativity specifically at background order. So this one applies to arbitrary orders. This one applies to background order. The upshot of all of this is we get a prediction out of general relativity that was not recognized before. If a species of relativistic bodies contributes non-zero principal pressures to Friedman's equations, then the individual masses will shift analogously to the photon redshift. So here MI is the mass when the relativistic body becomes cosmologically coupled. In our context, this will be when stellar gravitational collapse happens and baryons are pushed into this dark energy configuration that was first hypothesized by Gleaner. AI is the scale factor at which this gravitational collapse takes place and K is the strength of the coupling. I've written this as K to just make it clean the way things work out as this is negative three times the equation of state at background order for the species under consideration. So this effect is not restricted to black holes because the equation of state for neutron stars is also significant, but it's much more, much more the effect is much larger for these non-singular horizon free black holes. So if we wanna search for this effect, we need the population of relativistic bodies the ones we're going to use your black holes but you can ask me about neutron stars if you'd like, we can talk about that. But we need the ability to observe them for a long time because this is sort of a cosmological effect. If we're looking on short time scales things are probably going to look like Kerr or Schwarzschild and we need something that's going to take place over giga years and so that's challenging. So where did we go about looking for this? We looked in old galaxies. So there's galaxies are very complicated objects. There's a lot going on here and they are not my expertise but I will do my best to try to summarize that how we constructed the study. So you have early type galaxies, things that formed very early and got done with the revolution and then you have late type galaxies. We're gonna work at these early type galaxies called massive ellipticals. We're not going to look at these guys, no spirals. So these early type galaxies they have a lot of old stars because all the young stars, the big stars burned themselves out already. So you have the very red. We looked at elliptical galaxies that had very low interstellar medium. So there's no fuel to build new stars. We looked at a population of ellipticals that had star formation rates that were like suppressed five times relative to sort of their pure group at that epoch to sort of support that notion that yes, the gas is depleted. We're not making new stars. And these things are really massive. So we looked at objects that had maybe 10 to the 10.5, 10 to the 11 solar masses in stellar mass at least. And so the idea here was that we were going to try to find galaxies that were toward the end of their evolutionary or at the end of their evolutionary progression. And what do galaxies usually do? Like how do they build up? And so this simulation, this sort of graph from simulations captures the essential notions of galaxy assembly. So on the y-axis, you're looking at mass of the central black hole. And on the x-axis, you're looking at mass of the stellar masses. And the idea here is that processes which tend to build up stellar mass tend to also build up black hole mass. So you could consider if you pull in a bunch of gas into your galaxy, some of that gas will end up as stars and some of that gas will end up on the center of black hole. And so you'll tend to grow stellar mass and black hole mass together. And so you're going to sort of scoot diagonally. Similarly for mergers, if I have mergers between two galaxies that are approximately peers, then I'm going to double the amount of stellar mass and if the black holes find each other what they probably do and if they merge, which is a different question, but it's probable, then the central black hole mass will effectively double as well. So again, you're going to be moving along a diagonal in this sort of diagram. The thing to note here in these sort of simulations is this is for all galaxies in their sim, it's not just for these massive elliptical early type things. And if you look at these very difficult to see colors, I apologize, in Z2, Z1 and Z0, you'll see that there's not so much evolution really in this distribution. These are sort of stacked on top of each other at late redshift. There's a little bit here. These are pretty stacked and there's a little bit here. And so most of the time you don't really expect much to happen after the size of the galaxy gets really large, you, okay. So we looked and we looked at this sample of elliptical galaxies, samples of elliptical galaxies selected in a way that I previously described. So we looked for these very old systems that are very large. There's no recent evidence of mergers. Like they're very nice, smooth footballs and they're red and dead. And so here's a very old sample from the cosmos, cosmos population taken in median redshift of about 1.61 to roughly 9 billion years ago. And even though the sample is small, you can see this sort of, this diagonal band structure already in the sample. Here's another sample from the WISE survey with a median redshift of about 0.85. And again, you can see the band already emerging. And it looks like you maybe shifted a little to the right and shifted a little up. But if we look at a sample of local elliptical, massive elliptical galaxies, so this is at the present day. Again, you can see this band structure, but it looks like it shifted again to the left and dramatically up. And just to get a sort of sense of how dramatic the shift is, here's the population medians. This is like a shift of 20 decks. That's a lot. So sorry, 20 decks, yeah, yeah, yeah. This is, yeah, 1.2 decks. So that's pretty big, a factor of 20. That's what I wanted to say. 20 decks would be crazy. So 1.2 decks is a really big shift. Now, was this our analysis? Can we say, yo, we're done. You found some evidence of this weird GR effect. And of course not. That was not the hard part of the paper. The hard part of the paper was being able to confidently and consistently compare all of these different galaxy samples taken at widely different times in the universe, measured with completely different techniques. Like how do you make a sensible comparison between all these different surveys? And so there's tons of biases. There's selection biases, there's measurement biases. And that was really the hard part. Was figuring out the uncertainties on each of these data points that are very large that I've just not shown for clarity in this diagram. How do you fold into all of these individual measurement uncertainties and these biases and be able to sort of search for a measurement of how we did these things evolve in time? And so we worked really hard. We teamed up with Francesco Schankar to cross-check our biases and put together a bias model to sort of account for all the measurements in stellar masses and measurements in black hole masses. And this is what we found. So I'd first like to draw your attention to the bottom of this graph. What you're looking at here are shifts. These towels, these are translations in this plane. How much do you have to translate the sample up and down and left and right? And the goal for the translation is to line these samples up with the local sample. The idea being that these are old objects that haven't done anything for the past nine billion years. There's no evidence of mergers. They're at the top of the line for stellar masses. They're really nicely, nice uniformly shaped. They don't have any ability to grow stars or black hole mass because they're not merging and they don't have any gas anymore. And so what we tried to do was perform the selection so that we were sampling the same population of objects but just at different times. So the expectation would be that there wouldn't be any shifting around in this diagram at all. What we found is that the stellar masses are consistent with that expectation. When you correct for all the biases, then they're consistent with zero shift. The colors in this diagram are the same as the previous colors. So this is that high redshift Cosmos sample. This is the medium redshift wide sample. This green one is a cross check sample where we used a different AGN measurement technique, one that's used at the high redshift. We used it at the low redshift to compare against our other low redshift elliptical masses to make sure that we were measuring black hole mass consistently and de-biasing that correctly. And what you find up here is that the offsets persist. In fact, the offsets grow larger when you de-biased the populations. And so you grow by about 0.1 dex even relative to the naive expectation here. So is this a signature of cosmologically coupled growth? Is this the signature of this mass gain that you expect when the equation of state becomes relativistic? And to do this, to ask about this, we split our Y sample into two different redshift bins. This one was measured with a hydrogen beta line. This one was measured with a magnesium two spectral line. We pulled in an additional sample as a cross check from the Sloan survey and did the same separation into measurement bins with hydrogen beta and magnesium two to make sure we were getting a handle of systematic effects. Here's the Cosmos sample. And for this study, each object has its mass being shifted by this relation. Whereas in this study, we were just shifting the populations monolithically around the plane. For this study, each galaxy's position is gonna scoot up based on its spectroscopically measured redshift within the survey. And so this is a different study, but using the same populations. And you'll notice that the coupling strength we extract is about three each time. So we scoot these things around based on their redshift and some candidate value of coupling strength. And we find that those coupling strengths are clustering around three. If we aggregate the measurements, this is what we find. And so the thing to first pull from this result is that zero coupling is excluded in these elliptical galaxies, supermassive black holes at pretty high confidence. So this is greater than 3.9 sigma. And the value that we find for preferred value for K, this coupling strength is about three. So 3.11 plus or minus about 1.2. These error bars are 90% confidences, they're not sigmas. So this is why this is tighter than what you might first think. So what does that mean? If K is three, then each black hole's mass is growing about proportional to the scale factor cubed. So each black hole's mass is growing with the volume of the universe, but black holes are objects inside of cosmology. And so their number density is diluting with the size of the universe. So the total energy density of black holes remains constant, presuming you're not making new ones. But what does a constant energy density cosmologically mean? And we can just see that very quickly just putting this into the conservation of stress energy relation at free bin order. And so if row is constant, this term vanishes. And in order for this equation to be satisfied, the pressure has to equal to the negative of the energy density, which means black holes in aggregate contribute as a dark energy species, which is what was predicted back in 1965 and written down as exact solutions starting in 1992. So we thought that was pretty interesting. So the question now is, all right, here's some observational evidence for this sort of coupled behavior in astrophysically realistic black holes that are aware of the cosmology in which they reside. What could we do with this? So we know that the vacuum energy now interiors with non-singular and horizon-free, and that's really important for the mathematical formulation we sort of argue these effects within, whether these black hole models can cosmologically couple. So we've established that. They don't need to, but they can. And so it's something that should be looked for. Let's assume the decretion is subdominant to cosmological coupling. In that case, then the energy density will remain constant and the black hole population mimics a cosmological constant. Can stellar collapse black holes explain all of omega lambda? So let's assume for the moment that there is no actual cosmological constant. There's not like an Einstein's equations or what they are. There's no little term proportional to G that's put in by hand. Let's assume the black holes are the only source of this stuff. So no quintessence, no caissons. And let's assume the cellular gravitational collapse is the only source of black holes. So there's no direct collapse objects at high redshift. There's no primordial objects. We can get into details of why those sort of scenarios are disfavored later on. What do we find in this case? And is the sort of star formation history of the universe is that enough to, will that give you enough dark energy density? And the take home message is yes. So there's a lot going on in this figure. So please bear with me and I'm gonna try to break it down in little stages. So on the left-hand side, you're looking at a star formation of a density. So it's suns per cubic mega-parsec co-moving per year. On the bottom, you're looking at redshift. So to the right, we're going back in time to an assumed stellar first light at Z25. This is just pulled from James Webb, early science release, sort of inferences of when this time could be from a Harakane paper. Up top is the look-back time. If you prefer to use that measurement as it corresponds to the same locations. What we're showing here is what are candidates star formation rates that are capable of giving a cosmological constant that's the Planck measured value, say 0.68. The star formation rate is not enough to predict that. You also need to know, what's your distribution of stellar masses at birth? And you need to know how do you go from stellar masses to black hole masses? And you need to know, once I've formed black holes, how efficient are they at creating matter? So there's a lot of astrophysical uncertainties that are going to go into producing this figure here. What I'd like to point out though, is that everything in this figure assumes this coupling of three. So this dark energy, exact dark energy coupling. So there's no, the width of this viable region is not coming from some parameter that we can play with in this coupling scenario. This is coming from actual astrophysical uncertainties in what is the distribution of stellar masses at birth? Or what is the rate of accretion onto newly formed black holes at very high red shift? And how long does that last? And so what we've done here, is we've considered a couple of like, observationally plausible things that have been proposed in literature. So there's a Krupa initial mass function. So that's this sort of mass function that basically looks like mass to the negative 2.33, but sort of tails off at the low end, toward, you know, brown dwarf type masses. And then we considered what I call it here, a Harikane IMF, which is a similar power law, mass of the negative 2.35, but the domain on which this negative 2.35 is defined is much larger. So the masses here go from 50 to 500 solar masses. So this is sort of motivated by this population three stars that are very pure, very element poor. And so they don't burn very efficiently. They get very large. They don't capture radiation well. And so they burn large, burn hard, die fast. And most of that is expected to matter, is expected to collapse directly to black holes. So we considered the sort of bounding regions of the stellar birth distribution. This is sort of a low red shift motivated thing. This is a high red shift motivated thing. And then we considered just an Eddington accretion model onto the formed black hole. So here's a line that has no accretion. So this assumes this top heavy initial mass function, no accretion. It uses the star formation rate that's been measured by made out on Freigos by aggregating many other people's measurements up to C4 and then assumes this slope power law for the star formation rate. That is a viable route to get a 0.68 omega lambda. Here are some ones that assume some, this Krupa IMF, but then you have quite a bit of accretion. And then here's this top heavy IMF, this Harakane one with a more reasonable amount of accretion. This is the Eddington factor. So this is actually how much over Eddington accretion are you acquiring? So this is like a super Eddington thing. So this is 10x Eddington. This is no accretion at all. And these values like I said are literature standard values for obscurity creation and AGN for instance. So this is a duration 20 meg a year that's standard in the literature or within standard values I should say. And again, these Eddington factors are also within literature expectations. So the point here is that this is very early science release James web data that I've shown here from last summer. Here's an Alma point that the star formation rates that are coming in from the data are definitely compatible with the types of things that would give rise to all the cosmological constant today. And the parameters, the sort of the parameter, the viable space is very large from data at present but will become constrained within the next five years by James web. And also 21 centimeter I should mention up here which can provide some very interesting constraints by extra galactic background light as well. But we did not factor those constraints into the production of this figure. So the next question you'd wanna ask is, all right if you're gonna make a bunch of black holes in population three, how massive are they gonna be? They're probably gonna be pretty massive. If there's no accretion, then you expect something that's gonna be like a high mass intermediate mass black hole 10 to the five, 10 to the six solar masses. If there is accretion, you're gonna make a larger black hole on. So that's a lot of big black holes. In fact, if you wanna get dark energy today it's 0.7 of the critical density in black holes is that observationally viable? And the answer is surprisingly yes. Now, I put the asterisk up here to point out something that's happening that we don't have time to get into. And remember how I said that cosmological coupling doesn't change the evolution equations at zero order or first order in your perturbation theory. But what it does change is the relationship between the micro physical stress that's actually present in your universe and the parameters that enter the theory, the equations of state, the sound speeds, whether there's like an anisotropic stress that shows up in your late-time stress tensor. All of those now become questions that have to be answered using a local model for your object. In order for the scenario to be viable, you have to have a sound speed contribution or an anisotropic stress contribution at first order in order for the objects to not undergo accelerated clumping. And you're going to undergo accelerated clumping because the masses are going up. And so the orbits, if you have bound orbits, they're going to adiabatically shrink. And so you need some sort of effect at first order that will act against this desire to clump acceleratedly. And the way to do that in the simple models that we considered in 2020, it's spin. There's a spin effect that takes place when you're going from the framing that makes sense for describing a stress tensor to the framings that make sense for extracting cosmological predictions. If the object is spinning very rapidly, you can get that the sound speed goes from around negative one up to one third. And then the fluid of black holes in aggregate at first order will disperse like a radiation fluid and tend toward uniformity. That is the scenario that I'm presenting to you in these graphs. And so you can see in that scenario where the objects are all spinning very rapidly and they've dispersed, you're totally okay with constraints on massive compact halo objects. Here I'm displaying constraints from ultrafine dwarf existence, these sort of heating arguments. They've got a bunch of massive objects. They would impart kinetic energy to the stars and then the stars would proof out and disperse. That's actually the most stringent one in this scenario because we're dealing with the high mass end. Large man-alienic cloud microlensing events and like halo wide binaries are extremely not constraining in this scenario. Incidentally, this is fraction of halo mass. So this is not like a DNDM. It's not a mass function for the present day distribution of population three black holes or black holes altogether actually because we studied this all the way through the present day. But you'll see that about 35% are sitting in of dark energy is sitting in 10 to the five, 10 to the six solar mass objects. So future directions and tests. So we should be wrapping up. Cosmological coupling impacts diverse astrophysical systems and these can all be used to further test the hypothesis. Some examples which we can go into later detail if you're curious is the CNB. So like I mentioned before these first order parameters that you would expect to be zero or not present in your models if you're dealing with coupled species can become present. So things like that the sound speed can change and you can also pick up anisotropic stresses. And if you get anisotropic stresses at late time those will enter as an ISW effect in your CNB and they will enter at a low multiple ISW effect in your CNB. And so that's a pretty interesting signature. You gotta make sure you don't ruin agreement with the CNB but maybe you also can predict some things in the CNB as well. That's something we're working on right now. There are intentions with the stellar mass black hole population and this is one of the coolest things that has happened in the past four months is there's been a flurry of research interest in this and people with expertise in stellar mass black holes have observational and simulation expertise have really weighed in and it seems like the coupling strength from stellar populations really can't be much larger than like K one and really K one half seems to be coming into play a lot. And so the reasons for this are very interesting and sort of I have some sort of suspicions as to why I think this is taking place but I would just like to point out that the objects if they're cosmologically coupled within the GR framework we've put together they can't have horizons. And so if you don't have a horizon anymore when you're doing gravitational collapse you don't necessarily have a tov limit anymore. You don't have that Tolman Oppenheimer Volkov maximum neutron star mass minimum black hole mass that doesn't happen anymore necessarily because as you're collapsing you're converting baryons into this new vacuum into this dark energy material and that's a new physics process there's dynamics there and it's in causal contact during conversion. And if you read the Mazur Matola 2015 paper or more recent work you have to globally conserve entropy. And if you're forming a blob of dark energy that's a very low entropy configuration it's a condensate. And so all of that entropy has got to go somewhere. And so they propose that it would actually have a really loud explosion which could help carry away a lot of the mass that you would subsequently expect to fall onto a newly forming remnant. So these are ideas that are completely fresh they're super new, there's people working on them we don't know where it's gonna go yet but we're trying to sort of reconstruct what could a plausible distribution of these cosmological coupled remnants be that's consistent with all of the data data say from Gaia data from globular clusters and data from the elliptical supermassive black holes. The stochastic gravitational wave background should be impacted. I guess the papers came out now so it doesn't have as much impact me saying this but we expect to the amplitude to be higher but with respect to particular because there's more merging systems because the number one the black holes are heavier but also they merge more often because they can find each other because their masses are going up. So we expected the stochastic gravitational wave background to be a little louder with respect to quantitative predictions we weren't able to make them it's very hard because you don't know the underlying population as well. So it's also difficult to say sort of what's the relative the relative change in amplitude for like say frequency at reciprocal year and frequency at reciprocal 10 year. So but that's something that people are trying to look into right now. It's really good for early quasars. You can make a lot of make a lot of high mass black holes at very early redshift but that's a two-sided that's a double-sided sword. There we go because you can overproduce your black hole mass too excuse me and you can make black holes that are too large and so checking consistency between the, you know the objects that we observe at redshift six redshift seven and probably further back as we start using James Webb to dig deeper and the current population of massive black holes is an important check. I should note that it's kind of a difficult check because there could be significant selection biases. It could be that very massive black holes don't accrete luminously anymore. There's a paper by King I believe in 2015 that sort of discusses this possibility. The smoking gun signature here for this scenario really is enhanced orbital period decay in a black hole pulsar binary because the pulsar you'd be able to get exquisite timing measurements from this of course from pulsing and they don't, if there's coupled growth in neutron stars it's extremely small. You can show that. And so the growth would really be dominant in the black hole and you can compute that rate of orbital decay. And if you watched a system like this I think for around five years you'd be able to see it. Finally, you'll expect a correlation of the look back time with star formation instead of things like that. So you can look for precision signatures in the Hubble rate. And for instance some numbers the dark energy spectroscopic instrument has a precision on the Hubble rate down to about 0.3 or 0.4% given the background equation of state contribution for the cosmological black hole fluid. This effect could be as large as a 1% effect it could be as small as a 0.3% effect. We're working on a paper right now that sort of goes into the details of this. So it could be observable within the next five years. So in summary, thank you for your attention today and I'm going a little long but I still think I'm under an hour. A known black hole model suggests that Friedman's equations allow black holes to cosmologically couple and gain mass proportion scale factor. This anomalous supermassive black hole mass growth in mass politicals is consistent with this GR effect and we can exclude zero coupling at high confidence. The measured value of the coupling strength is consistent with K3 which implies that black holes contribute as a cosmological constant or as a dark energy species. A stellar collapse remnant black holes with population three stars can easily produce the observed value today given the measured star formation rate and current literature best estimates of initial mass function and decretion properties. And there's tons of ways to confirm or refute the measurement. So I would ask next time you look at a mass physical black hole, don't judge a book by its cover. Thank you. Thank you very much Kevin for this interesting talk and now we're going to open the floor for questions. So while I check on YouTube to see if there are any questions, is there any question from the audience, from the guys connected on Zoom? Thank you Kevin for this very nice webinar. I was not aware of any of this and I actually never thought about this type of coupling with cosmological. So thank you. And just thinking about the stellar black holes, I would imagine, I mean, I don't know if this is a strong assumption that all black holes in the universe belong to the same species in some sense. So stellar mass black holes and then the current black holes that I can observe with LIG or EHC are not that far away in some sense but then the ones in the past should belong to the same theory. And in that sense, everything that we see is consistent and then you claim, okay, there could be these mimicers that even though they might be the boundaries might be coupled cosmologically, I still have a good. So my question is more or less like, how bad is this approximation just to continue carry on assuming? Yes, I know our universe is not asymptotically flat but it's super consistent with everything else. And I'm also curious because you said you have some ideas speculative about that. Why when you do this for stellar mass black holes, K could be 1.5, which in some sense would be a tension with that sort of. Yeah, definitely. Okay, so I think there's a couple of questions there. And the first question is, I think was it an assumption that we assume that all black holes are the same species? And yes, so that's sort of the parsimony or the simplest assumption that it's a, this gravitational collapse is a universal phenomenon and produces some universal type of astrophysical phenomenon. Does that have to be the case? And the answer is no, right? So general relativity is a nonlinear partial differential equation system and the boundary value problems do not have unique solutions in general. So every time you write down a solution in general relativity to like a compact object, that's telling you something that general relativity permits in your universe but it's never telling you something else is excluded. So that's a really important distinction that needs to be made between say electrodynamics where you have existence and uniqueness versus a nonlinear theory like general relativity. So could there be different types of gravitational collapse objects? The answer is yes. But we did not consider that yet. And we hope we don't have to go because there's a lot of possibilities. So that was I believe one of your questions. Then the next question was, could you ask the next question again because I lost track of it? Yeah, I guess my point is more, you mentioned this tension that could be K 1.5 for a stellar math course and when they were in some sense, you have to reconcile. But I'm also curious about your response to the previous question because GR in some sense is a scale-free theory as opposed to any other theories. Like so the nonlinearity that you mentioned in your answer is actually in some sense ill-defined. If I just go to Newtonian limit, it's a scale-free also in that sense as a linear theory and then I can have strong gravity but then the behavior is qualitatively the same as weak and strong in Newtonian gravity. So you can have something that has more gravity but it's qualitatively the same. In GR, that limit for nonlinearity is actually ill-defined in some sense because you don't know when nonlinearity will turn on and in cosmological settings you can assume also the universe is in strong gravity all the time and then black holes around them. So that's Kelly and that thing is like... So this reminds me, so the question was like how good is the fidelity of something like a courage for child solution in sort of this more general context? And I think what can help to develop some intuition there is just to look at the time scales that we have data on these objects. So if we're looking at like mergers for the stellar masses we've seen so far the merger event itself that we perceive is taking place over fractions of a second like maybe like 200 milliseconds or something like that. If we're watching stars taking stellar dynamical mass measures of a black hole maybe we're watching these stars near the black hole for months or years, time scales. If we're watching sort of active galactic nucleus variability and we're trying to get reverberation mass-mapped measures of AGN from distant objects we're getting these things at this variability scales on the order of days like seven or 10 days. And if we're doing sort of event horizon telescope the variability is similar to that scale as well. So the data that we have on black holes at present really is sort of constrained or confined to be terrestrial scales and for the actual measurements of what's going on and those scales are extremely rapid compared to cosmological time scales. And so in order to be able to have any hope of seeing these effects we need to be looking over cosmological time scales or have something of exquisite precision that can get down to like 13, 14 decimal places like that binary black hole pulsar system that I discussed at the end there. So I guess so putting it back into theoretical context is I guess what you wanted to what you were going for. So I can take a Roberts and Walker cosmology and so maybe I will share my screen just a moment again if I am I allowed to, yeah. That's of course. Okay, and let's rewind. Sorry, there we go. Let's rewind back to here. There we go. So we can take this Roberts and Walker cosmology that I've written here and I can pick some epoch of interest to myself, right? So fix some eta at eta naught and then do a Taylor expansion here and just take the first term. Then I have a metric that is proportional to Minkowski this, you know, a squared at eta naught. And so that's like a unit redefinition so I can just absorb that factor and then I have some correction factors that are gonna go is order sort of the Hubble times the duration of conformal time that I want to consider. And that's an error term that I have to be willing to work with when I'm performing these calculations. And so I'm allowed to use asymptotically flat models just fine within the framework that we've adopted in this construction of cosmology within GR that we've performed, but the clock is ticking. And what happens is at some point that error term becomes so large that it swamps out the effects that I care about and actually this becomes sort of like coordinated singular. And so the best you can do using asymptotically flat models is actually have this consider a sequence of asymptotically flat models. But when I have that sequence of models, then I'm stuck. I have a mass parameter say I'm doing a sequence of Schwartz child models. I have a mass parameter for the first one in the sequence. I have a mass parameter for the second one and the third one and so on. But without knowledge of the global solution, I don't know how to stitch those solutions together in time in order to get something that's gonna allow me to make a good prediction. And that's really the issue here. Did that help? Yes, yes. Thank you. This is fascinating. And I think in the context of future life of Virgo where we will be able to serve all the black holes in the universe or- Absolutely. Yeah. Which is really fascinating. Thank you, Karen. All right. Let me read a couple of questions from YouTube so that we have a chance to answer. And so Shantanu asks, if black holes couple to the scale factor, would the helens downscurve singing PTA datasets be affected? Ooh, I do not know. What is, I just, could you, can I see the comment or can you type it somewhere? Like I don't, I just don't know the name. All right. So let me write this down. Let's see, let me repeat it again or maybe I can just put it in the chat so that- Yeah, and then I can just jump in the chat real fast and see it. Okay, so it is on the zoom. So if black holes couple to the scale factor, would the helens downscurve singing PTA datasets be affected? Okay. So I do- Mention a little bit about the, yeah. So Kastik- So the PTA I'm guessing is Pulsar timing arrays, right? So this is related, I guess to the- Last Thursday announcement, I guess. Yeah, yeah, yeah, yeah. So I will need to look into what the helens down curve is. I was just looking at the one that I looked at was appendix A1 figure in one of the papers that the collaboration put up, which sort of had the expectations over the past 25 papers or expectations for 25 papers of what the amplitude of the spectrum would be, let's say at reciprocal one year and reciprocal 10 year. And so I sort of spot checked that figure to see if I thought that the amplitudes were larger and it looked like they were larger than was expected. But again, so the good and quantitative predictions here is very challenging. And so if I can't speak to what helens down is yet, unless the person on YouTube wants to tell me somehow, oh, I can read about it though. But the idea I guess to getting a prediction for like the stochastic background and why it's so challenging is that the distribution in delay times becomes really complicated or not really complicated, but it becomes more difficult to compute independent of population parameters in a cosmologically coupled scenario. And the issue there is that the delay time doesn't just depend on the initial semi-major axis separation and the initial masses and mass ratio and eccentricity. What you get is that the, or say let's do circularized orbits and throw away eccentricity. What you get is that when the object formed is really important for how long the system will stay before it collapses. Because if you form a binary at some separation, let's fix the separation at 10 AU. This system almost certainly if the masses are stellar mass would not merge within a Hubble time. But in a cosmologically coupled scenario, not only are the masses going up but that orbit is shrinking because we've argued that angular momentum has to be preserved in this setting because it's a cosmological effect and there's no preferred directions. So to preserve angular momentum, the orbit collapses quite rapidly. And so systems that would never have merged before cosmological coupling can now contribute to your signal. The complication there is that how quickly that gap is now closed. It depends on when your double compact object was formed. If it was formed at high redshift, the Hubble rate is much larger than it is at sort of Z.7 at its nadir, right? When you switch to accelerated expansion. So if you form something deep in matter domination at the same fixed separation, it's gonna collapse much faster than that same sort of system formed later. And so to figure out the, how the delay time distribution changes, you need more information from your underlying population. You need to know a detailed distribution of semi-major axes. You need to know to a lesser extent but only slightly lesser the distribution of masses. And you need to know the distribution of eccentricities to like a more moderately lesser extent. But it's really that semi-major axis and mass that becomes folded into the final predictions for what you're going to get for like an aggregate spectrum. And just without better understandings of what the underlying populations are, it's hard to make that prediction. All right, excellent. Adila Afzal asks, well, she says, they say thanks for the nice talk. Thank you. I want to ask what, when there were no black holes in the universe, the universe still underwent accelerated expansions for instance, during inflation. So all black holes can be the only source or the source for dark energy. Okay, that's a really good point. And so I should have exempted the inflationary initial condition from the existence of this material. Sort of from the viewpoint of inflation, right? Or even this process that we're sort of invoking that should be happening when you're undergoing gravitational collapse. The way that I like to think of it, these microphysically is the time reversal of whatever inflationary reheating is taking place. So you have some process at inflation that's going from a dark energy or a condensate state into kinetic degrees of freedom into particles during reheating. And so if you're going to like just run that microphysics backwards during gravitational collapse, that's like the simplest model that you could start from for how one of these things could form at late times. Now locally are sort of like simple time reversal, I think is legitimate, right? With our theories, but macroscopic, whether the entropy is correct is a much larger question. And that's sort of that discussion that I made, right? If you're going to form one of these objects, you should probably globally conserve entropy. And so you're going to have to be giving off a whole lot of entropy if you're going to form a very low entropy configuration. And so that sort of folds into, you know, why should the remit spectrum be looking different for the stellar mass objects and like why you don't expect a top limit to be coming into play. But definitely, yeah, the inflationary initial condition should have been a dark energy condition that I should have said. So thank you. Thank you. And then the Sardinian cosmologist asks, do you have a comment on the recent paper with constraints on K from the AWS team? And in general about the possibility that K can differ from zero, but not necessarily be equal to five, two, three. Okay, so specifically which, so there's been a few preprints that have showed up lately that have been talking about James Webb, James Webb stuff. So which one did you have in mind? Because there's different aspects of different papers. Let's see if they answer. Oh, and they might have asked the question a while ago. Okay, so I guess I can answer another question. Okay, sure. Alexander Bonilla says, great talk, Kevin. I'm very interested. Thank you. The proposal you make and solve the tension, the tensions on the Hubble tension basically and the normalism CMB, could you make some comments about it? So he's asking if this could be applied to the Hubble tension. So the Hubble tension is actually, in my opinion, more challenging. So the trouble there is that we have measurements not on, we have measurements on omega M H squared, right? There's constraints on what, you know, at least little joint probability bananas of omega M little H squared and whatever other cosmological parameter you wanna be looking at. So it's really hard to change little H squared without changing omega M. Now, the tricky thing there is, is that the omegas are measured relative to critical density today, but you're making measurements, like those that omega M is being inferred from say physics that happened long ago, right? And flatness that happened that was present long ago. So whether or not you can get like an 11% shift in the Hubble rate with this is, I think it's difficult. So if you try to say form a bunch of, convert a bunch of baryons into dark energy at very late times, so like near the peak of star formation, say, which would sort of not be consistent with our understanding of star formation and cellular gravitational collapse, but you could just try to do it anyway. You find that to get the magnitude of the discrepancy, at least not to close the airbar gap, but to actually shift the medians that you need to eat too many baryons or maybe just a little bit too many baryons. The uncertainty on the late-time baryon density is actually quite large. It's a paper by McCourt in 2020 and there was a nature paper in 2022. I believe, I don't remember the author is, please forgive me. But so it seems unlikely to me that it could give rise to the magnitude of the discrepancy that's currently observed in H naught, but I don't have a definitive answer on that yet. With respect to the, but I should say that they're getting sort of 0.3% or 1% effects, you can definitely get those. And so those are within the sort of, within reach of current instrumentation in the next five years, all right. So he asked about two parameters about H naught and he asked about something else. What else? Sigma A and the other anomalies in CMB. Oh, Sigma A. Okay, so I'm not an expert in Sigma A and so I don't feel qualified to discuss that, but I can find some people that are and we can make that happen. With respect to sort of low multi-pole anomalies in the CMB that's actually currently what I'm looking into. You know, there's definitely some wiggles and some lack of correlation, but it seems mostly within cosmic variants, right? And because it's mostly within cosmic variants, it's the question like how much, how much can we really do with this? So the interesting target might be the really low quadrupole power and the jury's out, we're working on it right now. So we hope to have some results for you soon. All right, just coming back to one last question before we close this webinar. The question regarding the James W. James Webb, yes. The person responded the one showing K more or less zero, like close to zero. Yeah, so there's one with negative K. Okay, so there's a, there was a people, that paper was a Chinese team with quite a few authors. So I think it's still a preprint. I'm not the astronomy experts on the team, but my sense of that paper is that the comparison is unreasonable. To take an object that looks like an elliptical at extraordinarily high redshift and then to assert that there's nothing further has taken place over the intervening history of the universe is a far bigger assertion than even the large assertion we were making that there hasn't been any subsequent evolution over the past nine billion years. So we leaned on the sort of knowledge of the red sequence that currently exists today. I guess as James Webb continues and surveys more objects that sort of knowledge of galaxy evolution will change. But the sample was like three objects and I think two of them had accretion rates that were too high to be considered even quiescent based on the selections that we used. We're getting into specifics, I'm not quite sure, but I'm not putting too much attention on that preprint yet. Okay. All right, there are a couple more questions. I invite everyone to reach out to Kevin and I'm sure that he'd be happy to engage and answer your questions. Thank you all for connecting today. Thank you, Kevin, for sharing your work with us. It was a very exciting and entertaining talk. Thank you for having me. To all who are connected, we hope to see you again in our next webinar will be in August the second. And so be in the lookout for the announcements and subscribe to the channel on YouTube to our main list and the other social network and we'll see you all very soon. Have a great rest of the week. Goodbye. All right, I think that.