359 | Solo: Theories of Dark Energy

0:00:00.7 Sean Carroll: Hello everyone and welcome to the Mindscape podcast. I'm your host, Sean Carroll. I do hope everyone listening understands that they are witnessing an historic occasion. This is the first time ever that on Mindscape we've had a two-part episode. We always just make history in interesting ways here at Mindscape. I know that in the past I've been happy to just go on at great length in individual episodes, but I do think that this one, kind of naturally breaks into two different parts. And you don't have to listen to one part to get a lot out of the other one, so it made sense to do it this way. The overall topic, of course, is dark energy and the accelerating universe, theories of how dark energy works, what it might possibly be. So I planned out the episode and I realized I need to give background about vacuum energy and the cosmological constant, Einstein's old idea. And then, of course, that led to talking about the cosmological constant problem and the particle physics of it all, and so that made its own episode. That was the previous episode.

0:01:03.4 SC: And where we left off was, we have a discovery experimentally that the universe is accelerating. We can explain that by invoking a cosmological constant, much like Einstein did years ago for a slightly different reason. But it raises questions we don't know the answers to. The two big ones are, roughly speaking, the cosmological constant problem, which is why is the value so much smaller than you might expect it to be? If you think about effective field theory, which is an enormously successful paradigm to think about all of quantum field theory and particle physics, the cosmological constant appears as a number in the effective field theory, and we have expectations for how big that number should be. And the actual number is smaller than the expectation by something like 10 to the minus 122. So that's bad, and it might be a clue to some interesting physics underlying what's going on. And it certainly wasn't what a lot of particle physicists ever expected.

0:02:02.3 SC: The other problem is, okay, if you think that you've measured the vacuum energy, the energy density in empty space itself, and that's making the universe accelerate, et cetera, et cetera, then why are we so lucky that the value of the vacuum energy is approximately the same as the value of the matter energy density in the universe today? By approximately the same, we mean three or four times or five times as much, something like that, but not 10 to the 100 times as much. This is called the coincidence problem because the relative amounts of vacuum energy and matter density change as the universe expands. The vacuum energy stays constant as an energy density. The amount of matter in the universe as a density goes away as the universe expands. It dilutes away to zero. So if they're approximately the same order of magnitude today, in the past there was way more matter density than vacuum energy. In the future, there'll be way more vacuum energy than matter. Why is it that we got so lucky to be born at just the right time? It almost suggests that there is some kind of anthropic, human-centered explanation for this. And maybe there is, but maybe there isn't. Maybe we can do better in terms of thinking of new physical explanations.

0:03:22.4 SC: So when the idea of the accelerating universe became established in the late '90s, early 2000s, physicists immediately said we can't simply say it's the cosmological constant and stop there, that is causing the acceleration of the universe. Maybe it's something else. We should be open-minded. We were surprised once, we could be surprised again. And that led people to the idea of dynamical dark energy, something that's not quite the cosmological constant but looks that way, is a pretty good approximation. And that's something that we can look for experimentally as well as writing down theoretical models for what it might be. And so that is what we're getting to today. We're gonna talk about dynamical theories of what the dark energy could be. And as I said in the previous episode, I'm not doing 100% historically fair and balanced treatment of this subject by any stretch of the imagination. I was involved in writing papers and thinking about this for a long time, so I'm just telling you what I was thinking at the time and how that intersects with what other people were thinking at the time. And it's gonna cover a pretty decent amount of ground in terms of the different possibilities for what could be going on. We still don't know what is actually going on out there in the world, so in some sense I'm preparing you to think about new ideas as they get noticed and explained to people, or maybe even come up with some new ideas of your own. So with that, let's go.

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0:05:11.6 SC: I really do wanna stress that, at least back in the late '90s, early 2000s, when people first started thinking about dynamical dark energy, there was essentially zero empirical reason to do so, by which I mean there was no data that said, oh, the cosmological constant doesn't really fit very well. From the start, the cosmological constant fit quite well. There's always a little bit of error bars and therefore tensions with your predictions, but nothing that was worrying people at all. Today, as we'll talk about very briefly because I'm actually not an expert on it and nobody is an expert because it's in flux, but there are tiny bits of empirical evidence that maybe something is changing and we do need dynamical dark energy, but I would say that they're too premature to get excited about right now. But back in the day, 25 years ago, people were definitely motivated by theoretical motivations. They were saying, look, we were wrong before about the naturalness of the cosmological constant. Maybe making things dynamical is even less natural, but nevertheless true, so we should be open-minded. And even, I think this was definitely a major motivation, if you have just a number, the vacuum energy, there's really nothing you can do with it, right? It's just sitting there. You can try to come up with some deep explanation for it, but you're not getting any extra data. It's not flexible. It's not surprising you in some dynamical, time-dependent way. Whereas if you have some new vibrant thing, something that is dynamical, something that can change with time, then maybe it can help you explain some of these puzzles that you had before, the cosmological constant problem, the coincidence problem, and so on.

0:06:57.2 SC: I think roughly speaking that ambition did not pan out. It's absolutely worth being ambitious, and people were. Try it out, see what happens, but I think roughly speaking it didn't really work. But maybe we're not clever enough. Maybe we just haven't come up with the right idea. So what do you need? So you know from the observations, let's recap the observational situation. There's something called the critical density of the universe. That's a theoretical number. Given Einstein's equation, given the way we relate the expansion rate of the universe to the stuff inside the universe, there is a certain amount of energy density that the universe could have as a function of its expansion rate, as a function of its Hubble constant, which would say, okay, you're exactly balanced between negative curvature and positive curvature. You have a flat... A geometrically flat universe that you live in, and if you have the right density for that, that's the critical density. If you have less than the critical density, then you're gonna be in a negatively curved universe. If you have more than the critical density, you'll be in a positively curved universe, a spatial section of the universe.

0:08:06.7 SC: And notice that those words and phrases have nothing to say about what kind of energy density it is. It could be cosmological constant, matter, radiation, something new, something different, whatever. But the observations... Let's group together all the observations we made between 1998 and 2005 or so, both from the supernova measurements of the Hubble diagram, velocity versus distance for faraway supernovae in different galaxies, and the microwave background, and there were other kinds of observations from large-scale structure and things like that. All of these, I'm not going into the details here because it's a whole story by itself how you go from measuring statistical properties of temperature anisotropies in the microwave background radiation leftover from the Big Bang, or statistical properties of galaxies and large-scale structure, and therefore saying, oh, I now know what the density of the universe is, or the Hubble parameter, or the cosmological constant, or something like that. But people did that. We're gonna group all those observations together in that time period, 25 or so years ago.

0:09:17.5 SC: And what they were telling us was we were at the critical density. The critical density of the universe that makes it spatially flat is the density of the universe. But that is not simply one component. As it turns out, it's divided up, about 70% cosmological constant in the simplest model, and about 30% matter. Now, matter to cosmologists, as we said, just means particles that are moving slowly compared to the speed of light. And why that's so important that they're moving slowly compared to the speed of light, if they're moving close to the speed of light or at the speed of light, like honest-to-goodness photons would be, then they lose energy as the universe expands because of the redshift as the universe gets bigger. But matter doesn't lose energy per particle. Its energy per particle is E equals mc squared, mass times the speed of light squared. There's a tiny bit of kinetic energy, but it's small compared to the rest energy because the particle's moving slowly compared to the speed of light. So, one-half mv squared, the kinetic energy, is negligible compared to mc squared, the rest energy. That's what matter is to cosmologists, and about 30% of it is... 30% of the total density of the universe today is in the form of matter. And it's about 25% dark matter, 5% ordinary matter. I haven't even mentioned dark matter yet, and I'm not gonna talk about dark matter, roughly speaking, because it's a completely different kind of thing than dark energy. It's an interesting story, an important story, but not a story that is relevant to the cosmological constant story. It's matter, that's the only thing that is relevant, and it's part of the 30% of the universe that is matter. But just because the word dark is there doesn't mean there's any necessary connection. I mean, maybe there is, again, that's something to speculate about as a theoretical physicist, but we have no reason to suspect that's true.

0:11:09.5 SC: Okay, so we have the critical density of matter 70%... Critical density of energy in the universe. 70% of it is in the form of the cosmological constant or something like it. So, to be like it, what you need is something that is almost constantly spread throughout space, right? Because if this new energy, this dark energy, were clumping into galaxies and clusters of galaxies, then you would see it in the microwave background, in large-scale structure, in gravitational lensing, in the orbits of stars around galaxies, in a million different local ways. That's all exactly what we missed before the supernovae came along. The real reason why supernovae were such a good way to measure the cosmological constant, and indeed to measure the total density of the universe, was that they were really responsive to the whole amount of energy in the universe, not just the amount of energy in an individual galaxy or cluster, which you would then try to extrapolate to larger scales.

0:12:11.8 SC: That was exactly the hope of the Supernova Cosmology Project and the High-Z Supernova Team, and that hope was borne out. So you need something to be dark energy that is almost smooth throughout space and also almost constant throughout time. Remember, if you go back to the Friedmann equation and we said that there is this sort of complicated explanation, if you have an energy density that is constant, it leads to a constant Hubble parameter, which gets seen, visibly, as an accelerating universe. So all of those sentences still go through if you replace constant with almost constant. If you have an almost constant energy density, then you get an almost constant Hubble parameter and an almost... Well, you still get an accelerating universe. It's not quite exponential, okay? It's sub-exponential, but still accelerating. The function, which is called the scale factor, telling you how big the relative distances are between galaxies as a function of time, is still curved upwards like a smile, not curved downward like a frown, as a function of time. That's what it means for the universe to be accelerating.

0:13:20.3 SC: So that's not that hard. You want something that is smooth over space and constant over time. Of course, the constant, the cosmological constant, the vacuum energy is perfectly good at this. And again, there's no experimental reason to go beyond that, but let's just broaden our horizons by thinking a little bit more carefully. So let me let you in on a little jargon that cosmologists use to discuss this issue. It's called the equation of state parameter. Remember that the thing about the cosmological constant, interpreted as a form of vacuum energy, is that it has a pressure that is negative, that is in fact minus the energy density of the vacuum. Of course, I shouldn't say the pressure is negative just because it's the cosmological constant. We can imagine cosmological constants that are themselves negative, and then the pressure would be positive. That's what you get in anti-de Sitter space, okay? If you're interested in the AdS/CFT correspondence, et cetera, that's a universe with a negative cosmological constant, and therefore there's a positive pressure. That doesn't seem to be relevant to explaining our world as we observe it right now, so let's put that aside, and I'm gonna talk about a positive energy density, a negative pressure.

0:14:34.1 SC: So for the strict cosmological constant, P equals minus rho. Pressure is minus the energy density. So for something that is not quite the cosmological constant, what we can do is we can say, let P equal w times rho, where w is some number, a number that would be minus one if it were exactly the cosmological constant, and it would be not quite minus one, but something close to it, if it were something that were close to the cosmological constant, but not quite. So P equals minus rho... Sorry, P equals w times rho, and w is this equation of state parameter, and we're trying to think of that as a new measurable quantity that we can go out and look at the universe and ask, what is w? We know that there's dark energy, but it fits the data to have w be minus one, but is it exactly minus one? Any data always has error bars, and maybe it's minus 0.9 or something like that, okay? And that sounds a little, not immediately tangible, like, how do we know what the pressure of the dark energy is? How are you gonna measure that separately from measuring its energy density? But there's a very nice and immediate connection between the equation of state parameter w and the rate of change of the energy density in the dark energy as the universe expands. If w is exactly minus one, then the energy density stays precisely constant, that's the cosmological constant. If w is greater than minus one, so what that means is minus 0.9, minus 0.8, something like that, right? Those numbers are greater than minus one because minus one is a negative number.

0:16:15.0 SC: Then the dark energy density gradually fades away, decreases over time. You can remember this because w equals zero is saying that there's no pressure at all and that's just matter, right? And matter fades away quite quickly. For radiation, for a gas of photons or something like that, w would be one-third. P equals one-third rho. So for vacuum energy, w is minus one. For something that is not quite vacuum energy, maybe it's minus 0.9, minus 0.8, minus two-thirds, who knows? Depends on the details. You could also imagine w being less than minus one. So it could be minus 1.2 or something like that. That would mean that the dark energy density is growing with time. Okay? So not just the amount of dark energy growing with time because the universe is expanding. That happens with any dark energy model. Energy is not conserved in general relativity if what you mean by energy is the energy on the right-hand side of Einstein's equation. It's sort of, by coincidence, conserved if there's matter in the universe and nothing else. Because it's just E equals mc squared, the number of particles per cubic centimeter goes down as the universe expands, but the energy per particle stays the same and the number of cubic centimeters goes up and exactly cancels. But if the universe is full of radiation, the energy is not conserved. It goes down because the individual radiation particles redshift away some of their energy. If it's dark energy, the total energy goes up. So for w being less than minus one, we're saying something more dramatic than that. We're saying that the energy density goes up. That might bother you. Maybe it should bother you a little bit. We'll get back to whether or not it really should bother you, but it's certainly allowed to plug into the equations and ask what happens.

0:17:54.3 SC: Okay, so one of the very first things that people started doing after the acceleration of the universe was discovered was asking themselves, could we constrain this parameter, w, the equation of state parameter for the dark energy? So this is a sort of what we call a phenomenological approach to doing physics, because we're not saying I have a theory, right? I'm not saying here is my model of dark energy and here's what it predicts. We're saying, very specifically, I don't have a theory. I'm parameterizing the possibilities of my ignorance, right? I'm saying there's some theory out there, it's not quite the cosmological constant. How close is it or how far away is it? So you have a new parameter that you add to your model fitting and that gives you more room to play. And this is exactly what scientists know something about and are pretty good at dealing with this kind of situation. So I actually got to be a co-author on one of the first papers to do this. Not the very first paper. I think there was a paper by maybe Seljak and White, perhaps, that did it, but one of the first papers from the supernova groups to do this. And by do this, I mean not just find the best fit value of the cosmological constant, but find the simultaneous best fit value for the energy density of the dark energy and its equation of state parameter. For just the cosmological constant, you choose the equation of state as exactly minus one, and now you're letting it vary. So you have more room to play, like we said. And like I said in the previous episode, I was friends with some of the people on the supernova teams, in particular the High-Z Supernova Team led by Brian Schmidt at the time. And Peter Garnavich was yet another friend of ours who at the time had become a professor at Notre Dame, and he was a member of the team and he led the effort to write a paper, to do the analysis and write a paper saying, okay, we've discovered the universe is accelerating, what can we say about the equation of state parameter? And so they had a question, the group, I was friends with all of them and so they had this question, could the equation of state parameter be less than minus one? Could w be less than minus one? That, it seemed like there's no problem to them putting it in their equations and their plots and things like that.

0:20:27.7 SC: But maybe there was something physically not allowed about it, and they didn't know the answer. So they asked me, in part because I knew something about general relativity and stuff like that. And so I had never thought about that question before, and I did think about it, and I came up with a sort of weasel-worded answer, and I wrote a couple of paragraphs and added... We added them to the paper. And what I said, if I'm recalling correctly, I haven't read the paper in a long time, but roughly speaking at that time, so it was near, I don't know, 1999, maybe 2000, so everything was still new and shiny and exciting at the time. And I said, look, w less than minus one from a general relativity point of view violates energy conditions. There's a very traditional thing you do in general relativity because general relativists care about the left-hand side of Einstein's equation, right? That's the part where you have the curvature of spacetime and it's all fun and it's interesting. The right-hand side, where you have energy and pressure and heat and all those things, that's harder to understand because it depends on your model of what the matter is or whatever. So what general relativists do is they invent energy conditions. They basically say, well, something like energy density should be positive, or pressure should not be bigger than the energy density in absolute magnitude, things like that. And there's not any law of physics that say these energy conditions must be true, but they sort of guarantee that gravity isn't repulsive and things are stable and stuff like that. So I explained that w being less than minus one would violate energy conditions. That's not a guarantee that it can't be done, but it provides you a license to say, okay, we're not gonna consider that possibility. So in fact, in the paper, Garnavich et al., 1999, I think, we didn't consider the possibility. We cut off the values of w we were looking at at minus one. I might have said, but if you wanna look for w less than minus one, you should be allowed to do that because looking at the data is looking at the data. You shouldn't be too blinded by theoretical prejudice. I know that I had that thought. I'm not sure if it actually ever made it into the paper. And then the funny thing about that paper, astronomers at the time were still learning their way around large collaborations, right?

0:22:47.5 SC: There weren't a lot of large collaborations in astronomy like there are in particle physics. So in particle physics, everyone is very happy just putting the author list alphabetical. Because you can't... If you have a 1000 people or 5000 people on your author list, you're not gonna keep track of exactly who has done what amount of work. So just alphabetical order makes things simple. And even in theoretical physics, where we have only a tiny amount of authors, we just do alphabetical order. Most of the papers I've ever written have been in alphabetical order. Usually benefits me, sometimes I lose out. That's okay, that's life. So what the High-Z Supernova Team had as their strategy for the author list was they would say that, okay, for any given paper, someone is basically in charge of making sure everything works, right? Someone is the lead author on the paper. They're carrying out the analysis, they're checking everything very carefully. There's contributions from everyone in the collaboration, but there's one person who's the boss of that particular paper. And different papers will have different bosses. So Peter Garnavich was the boss of this paper. And so in their strategy, the boss, their name comes first, and then the whole rest of the team comes in alphabetical order. Okay? As it would have turned out, though, if we had done that for this paper that I became a co-author on, Peter Garnavich would be first, and then I would have been second, just because of alphabetical order, right? And then the whole rest of the High-Z Supernova Team. But I was clearly not a real member of the High-Z Supernova Team. I didn't do... It's work to be a member of one of those teams. You have to sort of earn your bones or whatever, putting in a lot of effort to earn your right to be an author on those papers and then go to the Nobel Prize ceremony, et cetera. And I hadn't done any of that.

0:24:33.8 SC: So the strategy for our paper was Peter Garnavich came first, the whole rest of the High-Z Supernova Team came next in alphabetical order, and then I came last, just to let everyone know I did a little bit of work but not too much work. And I thought that was completely, entirely fair. I have no problems with it. Anyway, in our paper, what we found, and in subsequent papers have found very similar things, w equals minus one, the equation of state parameter corresponding to a real cosmological constant, fits great. No problems with it at all, perfect fit. But there is some wiggle room there, and indeed back in 1999, there was a good amount of wiggle room. So there's room to play, and the data tell you exactly how much room you have to play. You need to increase the energy density in the vacuum energy, in the dark energy, a little bit and then compensate for it by having it fade away a little bit, and you can still fit the data pretty well. And now one of the games to be played, if you're in this, and it's a vibrant, active subset of cosmology, is not just playing that game, but deciding, okay, what data counts, right? So you have the supernova data, but that's not all you have. You have data from large-scale structure, from the cosmic microwave background. These days, you have data from baryon acoustic oscillations and maybe some lensing statistics or stuff like that. There's a bunch of different things that you have. So different analyses will put different sets of data into their analysis, and then it becomes very complicated to figure out what the paper is actually saying.

0:26:06.8 SC: So all of these, by the way, I should say that the paper we wrote, Garnavich et al., and the earlier paper, Seljak and White, I think it was... I hope it's those guys, I really should look it up, sorry. But this idea started the experimental program of constraining the properties of the dark energy. And that has become a huge deal in cosmology. Many experiments, satellites, and ground-based experiments were motivated by the idea, we are going to probe the dark energy. We're gonna learn about the dark energy. And I kind of have mixed feelings about that, just to be super duper honest. It's like a, on this hand, on the other hand, on the third hand situation. On the one hand, of course, you should probe the dark energy. You should try to learn something about it. That's a good, useful scientific thing. On the other hand, if it is the cosmological constant, then we're done probing it. We're not gonna learn any more by these probes. We're just gonna measure this one number to increasing precision, but the precision doesn't really tell us that much about what the underlying physics is. It's still just a number that we have no explanation for. So in some sense, this program of probing and experimentally constraining the dark energy is a bit overblown because there's not that much to constrain if it is the cosmological constant. But if it's not, then you got to do it. Okay? And so I think it's a good thing that people are doing it, especially because, and I think this is the killer argument here, maybe we learn something completely different.

0:27:46.6 SC: Maybe even though we're motivating doing these experiments by testing theories of dark energy, we're still doing cosmology. We're still collecting data. We're learning about supernovae and structure and early galaxies and whatever it is. And I think that's good. So that's wonderful. So it's not like we're simply targeting one thing and only learning about that one thing and it's kind of unlikely to come true, that's just not how observational cosmology generally works. So these days, there's a lot of effort. Satellites, the Nancy Grace Roman satellite is recently up there. The Rubin telescope, which used to be the Large Scale Synoptic Telescope, LSST, here on the ground, is doing surveys of lensing and supernovae and things like that. So many different complementary efforts. The Euclid satellite, all trying to, among other things, learn something about dark energy. And so that's an ongoing thing. Not to mention other smaller-scale things that have already given results, like the DESI collaboration, et cetera. Okay. So that's the experimental phenomenological side of things. What about the theoretical side of things? Is there any motivation really for thinking about this sort of sector of physics? So as I said, a big motivation was, even though the idea of the cosmological constant fits the data perfectly well, it leaves us with these puzzles, the cosmological constant problem, the coincidence problem, et cetera. Maybe by expanding our horizons and looking at dynamical models of dark energy, we could solve some of these problems or at least learn something that points us in the direction of a solution to these problems.

0:29:32.7 SC: So, roughly speaking, it hasn't worked, by the way. It never was gonna work really for the cosmological constant problem. The cosmological constant problem is not improved by saying that what is making the universe accelerate is not the cosmological constant. Because the cosmological constant problem is still a problem even if the vacuum energy is zero, right? That's still out there. So the puzzle, why is the vacuum energy so much smaller than its natural value? That's a puzzle whether or not the thing that is making the universe accelerate is vacuum energy or whether it's something else, okay? So we're not even trying that hard. You never had any expectations that making the dark energy dynamical would somehow help you with the cosmological constant problem. That's always there lurking in the background. In fact, what's gonna happen is we're gonna add more problems by inventing dynamical dark energy. But there might have been a hope to help with the coincidence problem. Why is it today that the dark energy seems to be important? And again, I don't think it quite worked, but there was a lot of excitement in the early days. There was an early paper by Robert Caldwell, Rahul Dave or Davé, I don't know how to pronounce his last name, and Paul Steinhardt, where they dubbed the idea quintessence, the idea of dynamical dark energy. In particular, they had a scalar field, which I'll talk about that in a second, but a scalar field model of what dynamical dark energy is, and they called it quintessence. That's the fifth element in ancient Greek physics, right? Earth, air, fire, water. Quintessence was the heavenly element.

0:31:13.3 SC: And they had this hope, I think it was in that paper, but certainly in follow-up papers, of developing what they called tracker models, or what people called tracker models. And the idea of a tracker model was maybe you could explain the coincidence problem by making it not be a coincidence anymore. Maybe you could have... You could take advantage of the dynamics of the scalar field, of the dark energy, so that it would sort of track the total amount of energy density of matter at all times. Or maybe it tracked at some era in the history of the universe and then stopped tracking later on, or some other dynamical story you could tell that would help explain why it's only now that the dark energy is becoming important. Now, roughly speaking, it didn't really work, or it didn't become convincing or something like that. I mean, when you have these aspirations to explain more things, then your theories are gonna have certain goals and therefore certain constraints, like you have to fulfill the goal. And that means they're not infinitely flexible anymore. And my impression, I haven't followed it very carefully, but my impression is that that hope of getting tracker like behavior doesn't actually fit the data very well. So it's a simple ruled out by an ugly fact kind of situation. But it was a perfectly good try, right? I think again and again in this story, you're gonna see people making legitimate college tries at getting a better understanding of what the dark energy might be and how it's working, and those college tries not quite panning out, okay? So that was a big aspiration, a big motivation, I should say, for dynamical models of dark energy. Could you help explain the coincidence problem in some way without just being anthropic, without just saying it's because there's a multiverse or something like that? Okay.

0:33:06.8 SC: So let's talk about the actual models that you would think about to develop. I already mentioned the scalar field, and that's absolutely the first thing that you're going to think about. So this is something that theoretical physicists were already good at thinking about because they had done this before, 20 years before, in the context of the inflationary universe scenario. Remember I mentioned inflation? This is a model of the early universe based on scalar fields, quantum fields that don't pick out a direction in space, they just have a value. And a scalar field can have a potential energy that is an amount of energy that is just sort of packed into the value of the scalar field. And you literally imagine in your head a picture of a ball rolling on a hill. And you probably have seen plots like this, maybe you haven't, I don't know. But if you hang out in the same circles that I hang out in, you see plots of potential energy as a function of value of the scalar field, okay? And inflation, interestingly, is a very similar idea in spirit to the idea of dark energy being dynamical today. What you want to make inflation work is something that makes the universe accelerate at an enormous amount, enormously fast rate in the very, very early universe. It's that acceleration that smooths everything out, makes everything homogeneous and isotropic, just like we see today. So theoretical cosmologists had a lot of practice writing down scalar fields with potential energies and asking what properties they needed to have to make the universe accelerate. And roughly speaking, what you want is a slowly rolling scalar field. Now, that's actually a technical term in inflationary cosmology, and that's not what I'm referring to here. I'm referring to the informal idea that if you have a scalar field, that scalar field has kinetic energy from its change over time and also its potential energy. And what you want is the potential energy is approximately constant, the kinetic energy is approximately zero. Then the potential energy in your scalar field is acting almost kind of like a cosmological constant. It's acting like dark energy.

0:35:21.2 SC: So this was the idea of what is called new inflationary cosmology. In old inflation, in Guth's first idea, the scalar field literally sat at the minimum of a potential called the false vacuum, and it did quantum tunneling to get out of the false vacuum, and that idea never really worked. In Guth's original paper, he admitted it never really worked. What other people realized, Albrecht and Steinhardt and also Andrei Linde, is that there is basically friction in the early universe. If you look at the equations of motion for these scalar fields, they're being pushed by the slope of the potential. Again, it's just like a ball rolling down a hill. So if the potential is steep, the field wants to roll down quickly. If the potential is almost flat, it wants to move slowly. But there's also friction from the expansion of the universe, that's literally called Hubble friction. The bigger the Hubble parameter is, the Hubble parameter telling you how fast the universe is expanding, the more friction there is. So if you have a potential energy function which is more or less flat, then the Hubble friction teams up with the flatness of the potential to keep the scalar field not moving very quickly, slowly rolling. And its energy density will be approximately constant, and it will make inflation happen in the early universe.

0:36:39.4 SC: And it would also, analogously, a similar thing would make the universe accelerate today. Now, the numbers are very, very different. The energy density that you need for inflation is hilariously high, much higher than any particle physics scale we've actually probed experimentally, whereas the energy you need for making the universe accelerate today is hilariously low. It's the average energy density of the universe, which is not very much. The universe is mostly empty, okay? But the basic equations look almost exactly the same. In fact, there are two ways... People sort of forget this because once you know the answer, you forget about the controversies, but the discovery that the universe is accelerating was hugely good news for the inflationary universe scenario for two separate reasons. The obvious reason is, the big prediction inflation had made is that the universe should be spatially flat, that the density of energy in the universe should be the critical density. And in the 1990s, that was not coming true. We were measuring the energy density, we were not getting there. We were only getting a third of the way there. And so the vacuum energy, or the cosmological constant, or the dark energy provided the extra amount of energy density you needed to explain that, that was beautiful news for inflation. It made a prediction that came true.

0:37:57.5 SC: The other thing, which is a little bit more subtle is, remember we always had the cosmological constant problem, right? That was known about since the '60s. And we didn't know the answer to that, so there was somehow always the possibility that something about vacuum energy made it not gravitate. In the previous episode, I talked about self-tuning solutions to the cosmological constant problem, and that's basically what they were doing. They were inventing a clever way that the particular exact thing called vacuum energy would have zero gravitational effect by replacing rho, the energy density, with rho plus p, energy density plus pressure. And for vacuum energy, p equals minus rho. So if that had been true, if that had been the correct solution to why the cosmological constant was so small, you could imagine mechanisms... Again, imagining in the space of all theories we haven't invented yet. But you could imagine mechanisms that would kill off the cosmological constant and at the same time kill off the possibility of inflation. Because inflation and the cosmological constant, or modern-day dark energy, act very similarly from the equation standpoint. So there might be something that made it impossible for the universe to accelerate, and that would have made inflation not work. So the discovery that our universe actually is accelerating was very good news for inflation because it said, yes, your prediction came true, and it is possible for the universe to accelerate. We know that because it's doing it now. Okay, so that was a big boost for inflation overall. And inflation returned the favor by saying, hey, I have this idea of a scalar field that is slowly rolling that could make the universe accelerate. Why don't you go play with it. Okay.

0:39:41.0 SC: So the simplest model for dark energy, for something dynamical that sort of mimics a cosmological constant, is a slowly rolling scalar field, also called quintessence. But I say that the numbers are different, the energy scales are different. Let me just emphasize how super different they are, okay? The mass of a proton, particle physicists like to measure masses in electron volts for various reasons. An electron volt is the amount of energy it takes to move an electron across one volt of voltage. I don't know why that has anything relevant about particle physics, but there you go. Since we set the speed of light equal to one, mc squared is just m, and energy and mass are interchangeable, so we use this energy unit to discuss the masses of elementary particles. So the mass of a proton, for example, is of order one billion electron volts. The mass of an electron is of order half a million electron volts. So it's about 1800th the size of a proton. The mass of a neutrino, we don't know. They're not zero mass, the neutrinos, but the order of magnitude that we're talking about for neutrinos is like a hundredth of an electron volt or something like that. So you see that the energy scales of particle physics, or mass scales if you wanna call them that, span quite a range, right? A billion electron volts for the proton, a hundredth, one percent of an electron volt for the neutrino, and of course they keep going up to places we don't know about yet. The Higgs boson is over a 100 times more massive than a proton, and these energy scales that we're talking about for inflation and things like that are like a quadrillion times the mass of a proton, a quadrillion electron volts, way, way higher than we can actually reach here in experiments done here on Earth.

0:41:31.2 SC: So what if you wanted to make a dark energy theory that applied to the universe today? So you want the scalar field to be rolling very slowly. So by slowly rolling, we mean this is a scalar field that is rolling down its potential and has been rolling down its potential for the entire history of the universe, 14 billion years, and it hasn't gotten very far. [laughter] We want the scalar field to not roll all the way to the bottom and start rocking back and forth. That would not be dark energy anymore. That would not be approximately constant energy density because those oscillations back and forth would be damped by the expansion of the universe. That energy density would go away. The way to get approximately constant energy is to have the scalar field be approximately not rolling at all, and then its potential energy just remains constant. And so you can kind of roughly... And here's some hand-waving going on, but you can parameterize the slope of the potential by thinking about the mass of the scalar field. It's not an exact fit because the slope is the first derivative and the mass is the second derivative, but okay, you're gonna go along with me, hopefully, for these purposes. You can work the observational facts about our universe today into, roughly speaking, an idea of what the mass of the scalar field that is the dark energy might have to be.

0:42:58.6 SC: And the answer, remember proton is a billion electron volts, electron is half a million, neutrino is 0.01, and the quintessence field has to have a mass of about 10 to the power minus 33 electron volts. So it's hugely tiny compared to any known particle physics scale. That 10 to the minus 33 electron volts is not pulled out of nowhere. It's just the Hubble constant in energy units. So it's the single parameter that tells you roughly the size and age and scope of the present-day universe. So it's not surprising that it's that number. So already you're kind of proposing something weird, right? You're saying, okay, I have a scalar field that has energy. That energy has recently taken over the total energy density of the universe, and the mass of this scalar field is 10 to the minus 33 electron volts, which is really, really tiny. And why? Why should it be so tiny? So let me put it this way. Again, it's a sort of naturalness question from a particle physics perspective. The Higgs boson, which I said is about 100 times the mass of the proton, so that's about 100 billion electron volts. Famously, there's a puzzle called the hierarchy problem. And you can phrase that puzzle by saying, why is the mass of the Higgs boson so small? By which we mean it's very small compared to things like the Grand Unification scale or the Planck scale or these ultra-high-energy particle physics scales. It's 100 times bigger than the mass of the proton and 200,000 times bigger than the mass of the electron or whatever, but those are things that we know symmetries that protect their masses. There's reasons that we have in particle physics as to why the mass of the mass of the proton, the mass of the electron, the mass of the neutrino are so incredibly tiny, not to mention the mass of the photon, which is zero. All these are very tiny compared to the Grand Unification scale or the Planck scale or whatever, but there are good reasons why. For the Higgs boson, there's no good reason why. That's the hierarchy problem. Why should the mass be so small, even though it's still 100 times the mass of the proton? The mass of our new quintessence boson is 10 to the minus 33 electron volts. That's just incredibly tiny, so that's a new puzzle to solve.

0:45:24.2 SC: Okay, so that's one thing to keep in mind. We have these scalar fields, we have some unnaturalness in their values. So can we possibly explain this? And this is sort of where I came into the game, because two things are going on. This is 1998, 1999, right? So what's important for the world is that we're discovering dark energy and beginning to think about it. What's important for me, Sean, is that I'm looking for a job, because I am a postdoc at the ITP, Institute for Theoretical Physics at UC Santa Barbara, and it's about time for me to get a faculty job if I'm ever gonna do it. It's my second postdoc. And I had realized recently that I was writing papers, I was productive, but the papers weren't that especially interesting to the rest of the world. I thought they were fun, but other people were working on other things, and they weren't that intersecting with what other people thought was interesting, and that's why I was not especially attractive on the job market. So I realized that I had to do something that was both interesting to me but also interesting to the rest of the world and make an impact there if I was going to get any faculty jobs.

0:46:38.5 SC: And happily, my friends discovered the acceleration of the universe, and I just happened to be the world's expert in that because I had written this review article with Bill Press and I was an expert on general relativity and all that stuff. So I said, okay, good. I'm gonna think about the acceleration of the universe and what we can do about it. And there had already been these papers by Steinhardt and Caldwell and Dave and other people. There were earlier papers by Jim Peebles and Bharat Ratra and others saying, oh, maybe there's a scalar field that is making the universe accelerate. So that had been done. But all those papers really bugged me because I knew enough particle physics to say, this is all really unnatural, right? The Higgs field is already anomalously low mass. This field is just crazy low mass. And it's actually worse than that. Okay? It's not just that the mass is small, that's one number that has to be small, but the danger when you go from constant vacuum energy to a dynamical field is that the field can do things. Not only can it do things by itself, like evolve with time, it can interact with other fields. That's generally what happens in particle physics. This is again part of the effective field theory paradigm. Even if nothing else, your new field that you're proposing will interact with gravity, you know that. And gravity interacts with other fields. So in the infrared effective theory, your new field should be interacting with other fields. And maybe you can say, well, okay, I'm just gonna make those interactions very, very small. Fine, but there is once again a set of expectations for the sizes of these interactions.

0:48:21.5 SC: And that means that your new quintessence field should lead to fifth forces of nature. If I have a pendulum, a torsion pendulum, so they're called, in a laboratory like they have at the University of Washington, you can look for tiny, really, really weak forces between ordinary matter over and above the gravitational force, the electromagnetic force, et cetera. And this new quintessence boson, which is essentially massless on the scale of laboratory experiments, right? It's easy to make, it's easy to source it, and therefore you can roughly predict order of magnitude, like should you have seen the fifth force due to quintessence in your laboratory experiments? And I worked out, back of the envelope kind of thing, yes, you certainly should have seen this already. And that's not all, because you not only have a new fifth force you could see in the laboratory, but this scalar field is supposed to be slowly changing with time. So what that means is it's slowly changing the values of everything else. There's feedback mechanisms that you would expect to exist if this field is interacting with other fields. So things like the fine structure constant, which tell you the strength of electromagnetism, or the masses of other particles, the mass of the Higgs boson, the mass of the electron, and things like that. All these should be slightly time-dependent over cosmological scales. You can invent new scalar fields and call them the dark energy, but those scalar fields should be talking to all the other fields in the world, and those should be leading to observable effects, and you can even estimate the sizes of the observable effects. And so I did that, and I realized, basically, in order to hide from the experiments and to be empirically, phenomenologically viable, this new scalar field has to be... Have all of its couplings suppressed by something like a factor of 10 to the minus five. Small, but not crazily small, but still pretty darn small.

0:50:28.1 SC: So this added up to this feeling that, look, you can invent these new scalar fields, but they're very, very unnatural from a particle physics perspective, and we have a perfectly good theory, the cosmological constant, that fits all the data already. Let's just stick with that, okay? So I started, I sort of was debating back and forth, should I... Is this the kind of insight that is worth writing a paper about, or should I just tell people about it, right? Like, it's kind of just bringing a bit of knowledge from one subfield into another subfield. That's not necessarily worth writing a paper about just to inform your colleagues, right? But I was invited to give a talk at Fermilab. A whole workshop was going on on dark energy and the accelerating universe and thinking about all these things. I'm not sure if the phrase dark energy had even been coined yet at that time. And on the, I think it was on the plane ride to the conference, while I was working on my talk, as one does, I realized, okay, so you have this unnaturalness of the values of the mass and all the coupling constants for the scalar field, but that's something we've seen before in particle physics, unnaturally small coupling constants. And there are strategies for dealing with that. Basically, you can imagine that there is a symmetry that prohibits the existence of these coupling constants and even prohibits the existence of the mass.

0:51:52.9 SC: And so this goes back to a discourse that especially was associated with Gerard 't Hooft, famous Nobel Prize-winning physicist in the 1970s. And 't Hooft proposed that anomalously small coupling constants can be what is called technically natural. And technically natural means that if the coupling constants were exactly zero, there would be some symmetry that had been slightly broken that you have now restored. So if these coupling constants are breaking a symmetry, they're allowed to break it by just a little bit, and that would still count as technically natural. And then you can argue for why that's true using renormalization group and things like that. So could there be a symmetry that allows for squelching all of these couplings between the quintessence field and the fifth forces and things like that, and the fine structure constant and the masses of all the other particles? And the answer is yes. You just have a very simple thing, you imagine that there is a shift symmetry, phi, the scalar field, phi goes to phi plus a constant.

0:53:03.6 SC: You just move the value of the scalar field. Now you can't do this as an exact symmetry because that would mean there's no potential energy, right? The whole point of your quintessence field is you're trying to make the universe accelerate. You're trying to have a potential that does change as a function of phi and that phi is slowly rolling down it. But that's okay in this case because you're not saying that the symmetry phi goes to phi plus a constant is exact, you're saying it's approximate. And indeed, that's exactly what you need to explain why the mass of the scalar field is so small. This symmetry is protecting the mass. It's still a puzzle why it's so small. It's like really, really, really small, but it is technically natural for this kind of scalar field to be small, for the mass of this kind of scalar field to be small. And if you do that, that actually immediately gives you an analogous smallness to all the other coupling constants. So basically, implementing this idea of an approximate shift symmetry for the quintessence field helps you explain both the masses and the couplings, and everything's now good. And indeed, I was beginning to think that I knew what was going on here because this idea of a scalar field with an approximate shift symmetry, the technical term here is a pseudo-Nambu-Goldstone boson, or PNGB. Yoichiro Nambu, a famous physicist from the University of Chicago, Jeffrey Goldstone at MIT, did pioneering work on symmetry breaking all the way back in the '60s. And they showed that you could have these bosons that would come up if you spontaneously broke a symmetry, to both spontaneously break the symmetry and then explicitly break it by a little bit gives you a pseudo-Nambu-Goldstone boson.

0:54:55.2 SC: Okay, so this was a known thing. And in fact, it was known that you could make dark energy out of this. There was a paper by Josh Frieman, a cosmologist at Fermilab and UChicago, and other people, I think even before we discovered the accelerating universe, they pointed out that if you wanted to have energy in the... Again, people were skeptical or surprised to find the universe was accelerating, but they were still thinking about the possibility even before it was actually discovered. It was just sort of low-level thinking. It was not very big deal thinking. So Josh and his friends wrote a paper pointing out that pseudo-Nambu-Goldstone boson quintessence, as we would now call it, was technically natural and provided a nice way of explaining why the masses should be so small. They weren't worried about the coupling constants, but I could help with that, and it certainly followed immediately. And this also connected in my brain to work that I had done earlier, and that you can... I've talked about it a lot, and you can find it on previous podcast episodes where I talk about birefringence and the screwy universe. Okay, so I'll just give you the very quick version right now. If you have a pseudoscalar field, and this is a different word, different definition of the prefix pseudo. Pseudo, as in pseudo-Nambu-Goldstone boson, means you've broken a symmetry by a little bit. Pseudo as in pseudoscalar field, means it is a negative parity field. By parity, we mean what happens to the field when you change the orientation of space. So you change the X, Y, and Z axes. So you're like looking in a mirror, right? Parity, P-A-R-I-T-Y. So a scalar field has positive parity. You change the orientation of the axes, nothing happens to it. A pseudoscalar field has negative parity. You change the orientation of your axes, it goes to minus itself. Phi goes to minus phi. Okay? And so with George Field and Roman Jackiw, and in different various combinations, we had pointed out that a slowly rolling pseudoscalar field would couple to electromagnetism in a very particular way.

0:57:12.4 SC: Basically, what we're pointing out is this symmetry in this slightly tilted pseudoscalar field that eliminates almost all of the couplings to ordinary matter leaves one coupling untouched. And that is a coupling to a pseudoscalar electromagnetic coupling that you can make. It's basically the electric field dot magnetic field. So for those of you who are a little bit physics-y inclined here, the electric field is a vector, the magnetic field is a pseudovector. So the electric field has positive parity, the magnetic field has negative parity. When you take E dot B, take the dot product of those two electric and magnetic fields, you get a negative parity thing. When you multiply that by phi, the pseudoscalar, you get a positive parity thing. So phi E dot B is allowed. The symmetry does not rule it out in any way. And there's a longer reason I can explain why that's true, having to do with total derivatives and things like that. But the point is this pseudo-Nambu-Goldstone boson idea, so Josh and his friends just said it makes a good potential energy function. What I realized is it also helps you get rid of all the fifth forces, but it leaves one interaction allowed. And that interaction makes a prediction. It makes a prediction. It can happen. And as the scalar field is changing its value, the pseudoscalar field, it will rotate the plane of polarization of light from distant galaxies and the microwave background. So you have distant sources giving off photons and they're polarized, and in ordinary electromagnetism, that direction of polarization just stays fixed as the photon travels across empty space. But in the presence of this scalar field, it would slightly push the polarization angle of these photons, and in principle, that's detectable.

0:59:13.7 SC: And you could even, and this is me on a plane in 1998 or 1999, I forget exactly when, figuring out that you could again estimate what is a natural value for the amount of rotation. And it's small because there's factors of one over four pi and the fine structure constant and things like that, but it's not that small. It's about one degree is your general prediction. One degree of rotation between us and very, very distant galaxies. And then you can look at the data that people already had to constrain this effect, and at the time, the data said, well, we know that the rotation is less than five degrees. Okay? So one degree is just perfect. This never happens in science, or it very rarely happens in the course of a long scientific career, that the numbers come out this nicely. When you make an experimental prediction, what you most strongly want is a number that has not yet been ruled out but could be ruled out in your lifetime. So if the current limit is five degrees or less and you're predicting a feature of one degree, that's perfect for you. And so basically, I ended up giving a talk at Fermilab and I said, look, we're all talking about rolling scalar fields. This is what we do now as theoretical physicists faced with the accelerating universe. They're all hilariously unnatural from a particle physics perspective. Maybe that's okay because we were wrong about the cosmological constant. That's unnatural, so we can just live with that. But we can also try to fix it. And here's a way to fix it, impose this symmetry, and it gets... It both helps explain the low mass of the quintessence field and explains why the coupling constants are small, and it leaves you with one prediction which hasn't yet been tested, which is kind of pretty cool.

1:01:02.9 SC: And so people got excited about that, and that's basically what got me faculty jobs a year later. That's just how the game works. Suddenly I was saying something interesting. I didn't become any smarter, but I was putting my smarts to use in ways that the rest of the world thought were interesting. And to skip ahead to the present day, to 25 years later or so, we still haven't gotten the limit on the rotation of polarization down to where we want it, but we're getting very, very close. And there have been a couple of claims in the literature recently, especially by Eiichiro Komatsu, who is a well-known cosmologist, who is analyzing data from the cosmic microwave background, and he says that in the data there is a sign that it looks like the polarizations are rotating by a little bit, a tiny bit. And it's not quite statistically significant yet, but it's getting there. These things don't have momentum, so the fact that it's getting there is not very definitive, but it's a suggestion, and we're gonna absolutely be super duper interested in following up with data from better data sets that are more targeted exactly at looking for this. It turns out it's really, really hard to measure rotations of polarizations because that's usually not what the telescopes are built to do. You have to sort of trick them into doing it. But it's so interesting that people are now building telescopes with detectors, I should say, for telescopes, radio telescopes, that are specifically trying to look for this. And so if they find it, that would be direct detection of dark energy. Or probably they would come up with a 100 other explanations for it also, but it's at least plausibly a direct detection of dark energy, which is kind of fun.

1:02:51.3 SC: Okay, so that was, I thought, an interesting set of ideas bandied back and forth by the theoretical physics community, quintessence, is it natural, can you make it natural by adding symmetries and things like that? And then there was still a good five, ten years of people doing exactly that kind of thing, and I was somewhat involved with that. So what else can you do? Of course, you can just build models, like write down specific values of the potential. That's fine. But can you say something more lasting, more model-independent, more robust, something like that? Well, there is this question of w, the equation of state parameter. So remember that for the cosmological constant, w is minus one. If you impose the energy conditions of general relativity, it's slightly less, or it has to be less than minus one, less than or equal to minus one, so minus 0.9 or something like that. And indeed, if you have something physically reasonable, like a scalar field slowly rolling down its potential... And by the way, sometimes the word scalar field means as opposed to a pseudoscalar field, that is to say, a positive parity field. Sometimes scalar field just means either positive or negative parity. In fact, usually it means that. So when I just say the quintessence scalar field, that's what I mean. I'm not gonna... I'm not picking whether or not it's positive parity or not. Either possibility counts.

1:04:17.2 SC: Okay, so if you just have an ordinary scalar field with an ordinary kinetic energy rolling down an ordinary potential, you will predict that w, the equation of state parameter, is indeed less than or equal to... Sorry, I just said it wrong. Greater than or equal to minus one. Okay, something between zero and minus one are the allowed values for the equation of state parameter of a scalar field, so it's a good candidate to be dark energy. But of course, the whole thing is like free employment for theoretical physicists, so you can play other games. What if w is just less than minus one? And again, you can play the game phenomenologically. Like, forget about scalar fields. What happens if you just set w to be minus 1.1? And you don't tell me why, you just say that's what it is. And again, this is Robert Caldwell thought about this along with Marc Kamionkowski, and they called it phantom energy. I think that Rob Caldwell was the first to call it that in a solo paper. It was near the time when The Phantom Menace was being released, so you can sort of date all of the physics progress papers by what popular culture references they're making at the time. So he called phantom energy something that would have the feature that its energy density would increase over time. And so Caldwell and Kamionkowski and others said, okay, just again, forget about the actual underlying model, just say w is less than minus one and let it stay that way. What's gonna happen?

1:05:56.6 SC: Well, you're saying that not only is the energy density increasing with time, but it's increasing at a sort of a constant rate. So you might expect that sort of makes things go crazy, right? If the energy density is just constant, if you have the cosmological constant, then the energy density... Then, sorry, then the expansion rate is a constant, and that leads to an exponential growth in the scale factor, the size of the universe, A of t. A is the letter often given to the scale factor. A of t goes as e to the Ht, where H is the constant Hubble parameter, t is time, and exponential growth is pretty fast growth, okay? So if you have w less than minus one, the energy density is increasing in its own right, and so the scale factor of the universe is gonna increase even faster than exponential. And in fact, what they showed is, it hits a singularity at a finite time in the future. It goes, if you imagine in your mind a plot of the graph 1/x as a function of x. So at positive values of x, it trails off at large values of x, 1/x. But 1/x blows up, becomes infinity as x goes to zero. So sort of take that kind of behavior, but flip it around and have it go into the future. The scale factor of the universe would approach infinity at a finite time, which is a kind of singularity. And they called this the Big Rip, as opposed to the Big Bang or the Big Crunch.

1:07:30.3 SC: So this was a great marketing strategy for them. They came up with a very, very clever name for their cosmological scenario. And it's... Look, it's important to do that. This is good work. And it's good work because it's important to understand the space of possibilities. So we can have a dialogue, a discussion about how likely different possibilities are. And I would say the Big Rip is very, very unlikely. But it's one of the possibilities. It's not often that in your career you get to invent a whole new possibility for what the universe could do, right? So the invention of the Big Rip, I thought, was a very, very good idea. But it's... So it's a sufficiently good idea that it's worth taking it seriously and thinking, like, could you do this? Could there actually be an underlying particle physics model that would make the energy density of empty space increase with time? As I said way back in my paper with Garnavich et al., it would violate the energy conditions of general relativity. But that's not an absolute ruling out factor, that's just sort of a warning sign, like, dragons ahead, you should be careful here. So I became interested in this, and I started working with Mark Trodden, who is an old friend of mine from postdoc days, who was at the time a professor of physics at Syracuse. He has since moved to the University of Pennsylvania. And we wrote lots of papers together over the years. And so he and I and Mark Hoffman, who was my first ever graduate student at the University of Chicago, we wrote a paper that analyzed this question.

1:09:09.9 SC: Could the dark energy equation of state parameter w be less than minus one? And by that we meant, could you invent a good particle physics model, or what would go wrong if you did? Now, I think way back in Rob Caldwell's early papers, he had an attempt at a particle physics model, which is just the following, when you have a scalar field and you have... You wanna write down its equations of motion, right, so you wanna say, what does the scalar field do as a function of time? Well, you know, if you've read 'Quanta and Fields' or whatever, basically you have a kinetic energy and you have a potential energy, right? And we've already said that. The kinetic energy gives you the energy in the motion of the field. The potential energy just gives you the amount of energy that the field has just because it has a certain value. And the potential energy, in the usual way physicists think about things, potential energy can be positive or negative or zero. Kinetic energy for an ordinary scalar field is going to be minus... Sorry. Is going to be plus one half phi dot squared. Phi dot is the velocity of phi. Phi dot is just the derivative of phi with respect to t. So phi dot squared is positive or zero, and the kinetic energy is one half phi dot squared. So by the ordinary construction, the kinetic energy is always positive. Potential energy can be positive, negative, or zero. What Rob Caldwell realized is that in order to get w less than minus one, all you have to do is put a minus sign in front of the kinetic energy.

1:10:44.8 SC: That's what he suggested should be called phantom energy. And in fact, I think maybe he knew this, maybe he didn't, he must have known this, it's often particles with negative kinetic energies are often called ghost particles in quantum field theory, ghost degrees of freedom. So maybe that's where the phantom came from also. I'm sure he knew that. So it took me a while to remember it. I knew it when I was a grad student, then I forgot, and then I learned it again later. So, okay, so negative kinetic energy. Well, that's fine. And what a cosmologist does, is you instantly specialize to the case where the field is not fluctuating, right? It's just constant throughout space. It's only changing as a function of time. So if I have no spatial gradients from place to place, I only have dependence on t, the time parameter. And I put down my equations of motion with a negative kinetic energy and a positive potential. In an ordinary scalar field theory, you say, well, I have a field in the potential, and the field rolls down its potential, just like a ball rolling down a hill. But now I've changed the sign of the kinetic energy, and what that means is the ball rolls up the hill. So it rolls up to greater and greater amounts of energy, which is exactly what you want if you want phantom energy, if you want the Big Rip, okay?

1:12:08.0 SC: And that's fine, and that's... And they wrote papers about that. But in the real world, you will have at least a little bit of fluctuation in space as well. So what is that going to do? You're gonna quantize this theory, you're gonna get particles, et cetera. In fact, you just crank through and you crack open your old quantum field theory textbook, this is literally what we did, and you replace a bunch of plus signs with minus signs, and you see, so what happens if I have a negative energy density? And I think a lot of particle physicists were... Did not do this correctly at the time because there's a way of dealing with ghost particles in quantum field theory where you say, oh, this corresponds to a negative norm in Hilbert space, and I can project them out. There's a whole set of things you're doing. But that's all because these particles are only arising as virtual particles in Feynman diagrams, not because they're real particles out there making the universe accelerate. So you had to sort of relearn what you'd been taught in your quantum field theory class, replacing a bunch of plus signs with minus signs. And the particle excitations of this field turn out to have a negative mass. Negative kinetic energy, even if you don't change the potential, turns into a negative mass for the particle. And people hadn't really talked about this that much, or at least we didn't know when we were writing our paper of any work on it that had been done before. And so we thought about what's going on, and we realized that there was a kind of a bad thing that could happen, which is that if you have a negative mass particle...

1:13:40.0 SC: When you have particle physics and you have Feynman diagrams telling you what the different possibilities are, energy is conserved in those Feynman diagrams. So a heavy particle can decay into a light one. The neutron can decay into a proton by spitting off an electron and a neutrino, and the total mass afterward is less than the mass before, and that extra energy gets turned into the kinetic energy of the particles. But the proton can't decay into a neutron, right? Because a proton is lighter. There's not enough energy for the proton to decay into a neutron. Except now you're telling me there exists somewhere in the universe particles with negative mass. So that allows for the proton to decay into a neutron by spitting off a positron, a neutrino, and a bunch of negative mass particles. Okay? And it can turn into a heavier particle while still conserving energy. And you might say, well, okay, but that must be very, very rare. I can make the coupling constant, such that that's a very unlikely process. No, as it turns out. This is what we did, we checked, in fact, it's infinitely likely. If you sort of sum up over all the different possibilities, you've made phase space, the space of possible energies for the particles you're making, infinitely big. In an ordinary particle physics calculation, if you have the neutron decaying into a proton, there's only a finite number of ways you can distribute the extra energy between the electron and the neutrino, et cetera. But now you can have this negative energy particle, and you can make as many of them as you want, these negative mass particles. So phase space becomes infinitely big, and something that you might have thought was unlikely becomes super-duper infinitely likely. And maybe you can try to cut it off with an effective field theory, blah, blah, blah.

1:15:29.5 SC: So basically, the paper that we wrote said, no, w cannot be less than minus one, because it would be catastrophically unstable. Forget about protons decaying into neutrons, empty space with zero energy can decay into a bunch of positive mass particles and negative mass particles. The vacuum would be catastrophically unstable in this way of doing things. So we suggested that the Big Rip and phantom energy were not as plausible as people had thought. Now, of course, the whole history of particle physics is a back and forth, right? So you should go back and forth, the whole history of physics. Like someone comes up with a good idea, someone points out a problem with it, someone else comes up with a way of fixing the problem. And so after our paper came out, there was work by people like Nima Arkani-Hamed and others that talked about something called ghost condensation. So these were ghost particles, negative mass particles that could settle into a stable equilibrium. So there were ways to get negative mass particles hidden in the details of your theory. It still wouldn't lead to a Big Rip or anything like that.

1:16:41.3 SC: So I'm still pretty much on the side of thinking that w is not gonna be less than minus one. I think that it's very, very hard to make a theoretically legitimate, respectable model in which something like that happens. So, of course, I'm open to the possibility that someone comes up with a clever way of doing it. There's more to life than scalar fields and negative kinetic energies, right? In fact, we followed up our paper, Mark and... The two Marks and I, with a paper called, Can you be tricked into thinking that w is less than minus one? So there's another game you can play with scalar fields, which is you can let them affect the value of Newton's constant of gravity. This is an old, old idea going back at least to the '60s with Brans and Dicke and people like that, where you have scalar-tensor theories of gravity. So when your scalar field moves around, it makes the effective value of Newton's constant change. And that can lead to all sorts of crazy behavior for things like the strength of gravity and therefore the expansion rate of the universe. So we looked into the question, like, even without true phantom energy and all the instabilities that it implied, could you have a model of cosmology, perhaps with a time-dependent Newton's constant, where you would plug in the data and think that you were seeing that w is less than minus one? This is just a feature of how physics goes. When you're saying, I have some data and I wanna constrain my theory, the constraints you get out depends on the theoretical possibilities that you allow for, that you put in when you start your analysis.

1:18:22.9 SC: So we were saying that if you didn't take into consideration the possibility that Newton's constant was changing and just naively said, I have some data from supernovae and large-scale structure and whatever, what would I infer were the values of the energy density of the dark energy and its equation of state parameter? You could be tricked into getting a best-fit value of w that was a little bit less than minus one. Now, to be fair, what we found by doing the numerical simulations and things like that was, it's hard to make that work. It's very, very hard to thread the needle and be compatible with all of the various limits and observational constraints we already have and still open yourself to the possibility of getting w less than minus one. But it's possible. And I bring this up, of course, because people have recently found possibilities, found hints in the data that, number one, the dark energy is not constant, and number two, it might even have w less than minus one. So as soon as people got these hints from the data that w might be less than minus one... I'm still skeptical that it really is something like phantom energy, but I am open to the possibility that we could be tricked into thinking that w is less than minus one by constraining an incomplete theory. But then I looked at the papers and I think that their implication that w is less than minus one is actually not that believable. I think they did too quick a job in fitting the parameters, and it's pretty easy to find models that would fit the data even with w not being less than minus one. But it is interesting because there's more than one little bit of a hint in the data in the last five years that w is not exactly minus one.

1:20:17.3 SC: If you ask me right now, I would still say that the smart money is that w is minus one, that it is just the cosmological constant. There's a lot of theoretical virtues there, but we need to be open-minded. And let me sort of round this out by... Well, actually, there's two things, there's two more things to talk about. One is, you can get more complicated, right? These scalar fields rolling down potentials are just the simplest things that you can do. And in fact, like I said, people were thinking about what we now call dark energy even before the acceleration of the universe was discovered. And I was one of those people. I wrote a paper in something like 1996 with Greg Anderson, a friend of mine who had been a fellow postdoc at MIT, we wrote a paper on what we called variable mass particles, or VAMPs. So here is the idea for that. What if you have a scalar field, and what if it can roll down a potential, and what if the potential doesn't have a minimum, right? This is something that these days is very common in dark energy investigations. What if the potential energy function is not like phi squared, but something like one over phi or e to the minus phi or something like that? So the scalar field can just roll off to phi equals infinity forever. And in fact, things like this are not impossible in supersymmetric theories, in string theory, and things like that. I think that was part of Greg's original idea, like is there a way to control these potential runaways of the scalar field that wants to run off to infinity? Because we didn't have data then that there was an acceleration of the universe.

1:21:59.7 SC: And what he knew about, he didn't know about the cosmology aspects of things, but what he realized is that these scalar fields, very much like the Higgs mechanism, can have a feedback where the value of the scalar field feeds into the masses of the particles around it. So you have... It wouldn't work phenomenologically or experimentally, I should say, with things like electrons and protons, because electrons and protons have pretty darn constant masses in the universe since early times. We know that from experiments, from observations, from looking at the fine structure constant early on, from looking at Big Bang nucleosynthesis, et cetera. It's just very, very hard to make the masses and couplings of the known particles change with time. But what if you had dark matter? Okay, which you do. You do have dark matter. What if the dark matter was a field that got its mass just like electrons get their mass from the Higgs boson, what if dark matter gets its mass from a scalar field? And what if that scalar field doesn't have a minimum to its potential? So it wants to run off to infinity, just wants to roll off. It's a ball rolling down a hill, and the hill just rolls on forever. But here's the problem. As the scalar field rolls, it gives mass to the particles around it, to all the dark matter particles, okay? And that costs energy.

1:23:27.3 SC: So basically we pointed out you could stabilize the potential energy of the scalar field because there's effectively a contribution from the fact that it costs energy to give mass to all of the ambient dark matter particles all around it. So instead of just the scalar field having a potential energy one over phi or something like that, in the presence of some dense collection of dark matter particles, some number density of dark matter particles, there's also an effective term in the scalar field that is proportional to phi and is positive. The value of phi is increasing the masses of the scalar field... Sorry. The masses of the dark matter particles, and the energy density in the dark matter particles is the number of dark matter particles times their masses. And so that's an effective contribution to the potential for the scalar field, and therefore you could stabilize it. You could have a value of phi that was constant because of this ambient collection of dark matter particles with an energy density. And then you say, you keep going and you say, but the universe expands. So the universe expands, and so the density, the number density of the dark matter particles, changes with time. It goes down. And as a result of that, the effective contribution to the potential energy for the scalar field shifts, because it's proportional to the number density of the dark matter particles, and that's going down as the universe is expanding. So basically the scalar field keeps adjusting its minimum to sit at a happy place where it's a balance of its potential energy one over phi and its effective contribution, which is proportional to the number density of dark matter particles. And that changes with time as the universe expands. So we get variable mass particles, or VAMPs, as we called them.

1:25:17.7 SC: And this was supposed to be a candidate for both dark matter and dark energy all at once. It doesn't quite work in the naive form because, again, you should allow the dark matter particles to be not completely homogeneous, right? They have some inhomogeneities, and there's an instability where those inhomogeneities grow with time, which can be bad. But this helped inspire other people. There was something called chameleon fields, where, as I said a while ago, there's a problem with fifth force experiments when you have all these light scalar fields going around. And so what if you could give those scalar fields a temporary mass when they were in the vicinity of ordinary matter, right? So if the scalar field gets an effective contribution to its potential from its coupling to ordinary matter, that could pin it and stop it from giving rise to fifth forces. This is called the chameleon mechanism, and Amanda Weltman and Justin Khoury suggested this idea. And so I'm not trying to... I'm sorry if I'm breezing through this too quickly, but I'm not trying to go into all the details because the details of no one model are all that interesting. The idea is... The interesting thing is that there exist models, that there exist whole different worlds to play in where the possibilities are very exciting and interesting. So the data will help guide us if we eventually do figure something out here, but in the meantime, the theorists are coming up with all these fun ideas. And you can't just sit around and wait for the data, because sometimes coming up with a theoretical idea suggests to you that you should go look for certain pieces of data, right? Like if we hadn't discovered the acceleration of the universe, no one would be trying to measure the equation of state parameter of the dark energy. So you can't just wait for the experiments to come in. There's a constant give and take, an eternal interplay between the theory and the experiments in physics, and this is all part of that.

1:27:20.2 SC: Okay, the final idea I wanna get on the table is, what if there isn't dark energy? And by that I mean not that we've made a mistake in the data. I think the universe is accelerating. I think that's what the data are telling us. The universe is accelerating. What if the acceleration of the universe is not due to dark energy? That's something that you absolutely are allowed to think about. The alternative, of course, is that it's due to modified gravity. So one thing that is in common between dark matter and dark energy is that, in so far to date, we haven't detected either one directly. What we've done is inferred their existence by looking at the behavior of spacetime, by looking at the gravitational fields in galaxies and clusters and the expansion rate of the universe. And so it's a very old idea that what if gravity is different than we think it is on cosmological scales? That could trick us into thinking that there is dark matter and/or dark energy. In the case of dark matter, this goes... The most famous version of this is MOND, which has been developed into more theories in various ways. But you can try to play that same game with dark energy. Modify Einstein's equation of general relativity to make the universe accelerate even without dark energy. That's a game that you could imagine playing. And in fact, so I was thinking about this, again, as a young assistant professor. And the way that I thought about it, it's again a funny story. Like, the history behind these things is always kind of amusing and sometimes a little bit embarrassing because you made mistakes. So I was thinking along the following lines. There is something that we know about dark matter that motivates modifying gravity. So, by the way, nowadays it doesn't work. Modifying gravity to get rid of dark matter just doesn't work.

1:29:15.9 SC: And the reason why, as opposed to 25 years ago, is we have data from the cosmic microwave background and the idea of MOND and other modifications of gravity made predictions for the microwave background anisotropies and they came out false, they have been ruled out by the data. I know there's some people who sort of stubbornly insist that they're gonna keep thinking about it, it's a free world you can keep thinking about it but the data rule it out. Now that's never a 100% statement, right, because well maybe I can change my theory a little bit to hide from it but to a very very good approximation, the data have ruled out this idea. But in the year 2000, that was not true or the year 2003 or whatever, I don't know exactly when it was. So I was thinking about the following idea, right, I started saying a motivation for MOND is a numerical coincidence, so MOND Milgrom's idea of Modified Newtonian Dynamics is the following idea, he noticed in the data, that if you look at something like a spiral galaxy like the Milky Way or other spiral galaxies, famously there's evidence for dark matter because the rotation curves of the galaxies which is to say, the speed... The velocity of rotation of things going around the galaxy as a function of the distance from the center, if you look near the center, it's exactly what you would predict like you count the stars, you count the amount of matter in there you do Newtonian gravity you figure out the gravitational field the rotation is exactly what you expect. But what you also expect is that as you go further and further out into the fringes of the galaxy, the rotation curve should get lower and lower. The velocity should be smaller and smaller because the gravitational field is weaker. Uranus and Neptune move much more slowly than Mercury and Venus because the gravitational field is much stronger near Mercury and Venus. But what you see going back to the data collected by Vera Rubin and others, is that the rotation curves do not diminish to nothing.

1:31:17.9 SC: They more or less flatten out. So there is more gravitational force at the distant fringes of the galaxy than you would expect from ordinary physics. This is evidence for dark matter. If you have dark matter, dark matter, if it's just cold dark matter, non-interacting, the simplest kind of model, dark matter doesn't bump into other dark matter. It doesn't dissipate. The reason why in a galaxy there's so much density of matter near the center is ordinary matter, like made of atoms with electrons, and they can bump into each other and radiate away energy. And by losing energy they fall into the center of the galaxy and the centers of the galaxies become very dense. Dark matter doesn't do that, dark matter doesn't bump into itself and radiate away energy, it just moves under the force of gravity so it becomes slightly more dense in the center but not that much really the galactic halo of dark matter as we talk about it, has a much lower density contrast between the middle and far away, than the density contrast in ordinary matter. So the 100% natural prediction, the expectation in a theory with dark matter, is that the ratio of ordinary matter to dark matter should be very high in the center of a galaxy and very low far away. And that's exactly what you see in the data. The numerics, you can try to work out exactly where the crossover should be, et cetera. That's fun to do and people do it. But the general qualitative expectation is exactly what you see. But what Milgrom realized is there is a weird and legitimately puzzling numerical coincidence, that if you look at different spiral galaxies and there's always a crossover point where you go from, you don't need dark matter near the middle of the galaxy to, you do need dark matter near the outskirts, right?

1:33:20.1 SC: So there's some radius, there's some distance from the center where you go from not needing dark matter to needing it to explain the rotation curves. And Milgrom noticed that this radius at which you suddenly need dark matter is not the same in every spiral galaxy. And it's not a... So it's not a constant distance, but if you calculate the acceleration due to gravity, okay, at that radius, so the acceleration due to gravity in just a Newtonian sense, like if you're an expert in general relativity, the phrase acceleration due to gravity bothers you because there is no such thing, but in the Newtonian limit, it's perfectly legitimate to talk about acceleration due to gravity, you calculate in many different spiral galaxies the acceleration due to gravity at the point where you're apparently crossing over from not needing dark matter to needing dark matter. And it's roughly speaking the same number in all these different galaxies, even though the galaxies themselves are very different from each other. And furthermore, if you do the plug in the numbers and plug in appropriate factors of the speed of light and things like that, that acceleration due to gravity is the Hubble parameter today. Numerically, it's approximately the same numeric value as the Hubble constant. And that makes no sense at all, or at least it... At first blush, it makes no sense because the Hubble parameter is a statement about the overall expansion rate of the universe, and this acceleration in a galaxy where suddenly you need dark matter to explain what's going on, has nothing to do with the overall expansion rate of the universe. It has to do with the local dynamics of the galaxy.

1:35:05.0 SC: Now, secretly, maybe it does have something to do with the Hubble parameter because that galaxy came from somewhere and it was formed somehow, and maybe that's relevant to this dynamics. But okay, at least, again, at first blush, that seems very surprising. So this was the motivation for Milgrom for proposing Modified Newtonian Dynamics, the universality of this phenomenology, of the fact that in all these many different spiral galaxies you saw the same number creeping up over and over again. And the fact that it's approximately equal to the Hubble constant is also kind of provocative. Now, again, it doesn't work, like if you go to clusters of galaxies and things like that, suddenly the dark matter doesn't appear at that acceleration. It's a different thing, which again makes perfect sense in the dark matter models, doesn't make sense in the MOND models, but okay, put that aside. Here I am thinking about this in the early 2000s, and I said, so not only is modified Newtonian dynamics motivated by this universality of the acceleration, but that acceleration is numerically about the same size as the Hubble parameter today. And we have another numerical coincidence, which is the coincidence problem for dark energy in the cosmological constant. Namely, the dark energy density needed to fit the data is approximately equal to the matter density needed to fit the data today, in the current universe. And today, the current universe is parameterized by a certain value for the Hubble parameter, right? Today's value.

1:36:37.1 SC: So when I say that the galaxies, according to the MOND phenomenology, have an acceleration radius that is equal to the Hubble constant, I mean equal to the Hubble constant today. The Hubble constant is not a constant, it changes over time. So there was this numerical coincidence just in the dark matter sector, and there's sort of a similar numerical coincidence in the dark energy sector. Like, this is just very juicy and provocative to a theoretical physicist, right? So I tried to say to myself, is there a way to unify this together? Is it possible that if it is... I mean, dark matter and dark energy are just very different seeming things. So there's no obvious connection between them. But if you modify gravity, maybe you can connect these two phenomena. Maybe there's no dark matter, no dark energy. You see how it's very seductive, right? And you have to be able to be seduced by it, but also be able to give it up if it doesn't work. So if you can modify gravity to explain both of these things away, then you only need one number. You need the Hubble constant today. Like, okay, we're not gonna explain that number, we're just gonna put it in, but maybe you could explain away both the rotation curves of galaxies and the acceleration of the universe. So, okay, it's easy to be inspired, it's hard to actually come up with the theory. So what is going on? What is the common feature that the dark matter phenomenon and the acceleration of the universe share? Well, the common feature is that they're both things, phenomena, that kick in when gravity becomes weak, when the curvature of spacetime becomes small, right? This is entirely incompatible with, or entirely flying in the face of your expectation as an effective field theory person. In effective field theories, you expect things to go crazy in the ultraviolet, at high energies. But you expect everything to just be normal and under control in the infrared. You don't expect things to become weird when spacetime becomes approximately flat, fields become weak, distances become large, any of those things.

1:38:56.3 SC: In fact, when I later was talking to a very famous theoretical physicist, who I won't name, about this idea, he said, I should write a paper just saying why none of this will ever work. And there probably was such a paper to be written. The paper was never written, but it certainly will never work in the sense that it's absolutely not what you expect from effective field theory. But there's things we don't understand, so maybe you shouldn't be too wedded to those possibilities. So I took off my effective field theory hat and just thought phenomenologically and said, could I fit the data? So how would you change gravity in such a way that the change would only become noticeable when the gravitational field was weak, rather than the gravitational field being strong? Being strong is easy to change, being weak is hard. So I knew about how to think about general relativity from the point of view of a field theorist, and I'll just get a little bit technical because it helps with understanding the vocabulary here. I mentioned in the previous podcast when we were talking about Euclidean quantum gravity, one way of thinking about general relativity is through the action, the principle of least action. There is a formula. It was actually written down by David Hilbert and used to derive Einstein's equation very close to the time when Einstein himself was first doing it. So there's a formula for the action associated with a spacetime geometry, and you can derive Einstein's equation just by saying, let's look at all the possible spacetime geometries subject to some boundary conditions and find the one that has the minimum action, right? The principle of least action. That's what's going on. And the formula that Hilbert wrote down is like the simplest, dumbest formula you can write down. So that's, again, always a sign that you're on the right track when the simplest formula actually works.

1:40:51.5 SC: There is a scalar quantity, that is to say a quantity that has no directionality in spacetime, it's just a number, called the curvature scalar. It's a way of taking the Riemann curvature, the curvature tensor that has a lot of directionalities, and boiling it down to a number, the curvature scalar, sometimes called the Ricci scalar, and it's given the name capital R. And that's it. That's Hilbert's action principle, the integral of capital R plus the action for whatever is matter and particles in your theory, that gives you Einstein's equations of general relativity. It's just that simple. So this is the simplest thing you could possibly do. From an effective field theory point of view, you would expect that that action, the integral R, so let me... Again, one more little vocabulary word here, and all this is explained in 'Quanta and Fields' if you're interested in. The action is an integral over all of spacetime of a quantity called the Lagrangian. If you've ever heard of Lagrangians, that's what they are. They're just quantities you make from the fields in your theory, like the metric and the electromagnetic field and whatever. You make different quantities, you add them together to make a Lagrangian, you integrate that over the spacetime manifold to make the action, and it's that action that you minimize to get the equations of motion.

1:42:12.8 SC: So Hilbert's Lagrangian is just R, the curvature scalar. There's some details in there about the measure over spacetime volume, but let's ignore that, okay? It's just R. It's the simplest thing you could write down. As an effective field theorist, you expect this term R to just be the first term in your effective field theory, and you should have plus R squared and R to the fourth and a whole bunch of other things. That's exactly what you would expect. But because R, the curvature scalar, is a very, very tiny number in the real world, R squared is even tinier, and R to the fourth is tinier than that, and so all these other things are completely negligible until you get to very, very strong gravitational fields, okay? So in cosmology, you can just go with R. You can just go with the simplest thing. Everything works fine. That gives you general relativity. So what we're asking is, okay, just because it fits the data or just because the data cries out for some help, not because there's any theoretical motivation for it, could we modify the action for general relativity in such a way that it didn't change things at large curvatures but changes things at small curvatures? Again, no motivation for this theoretically, but we wanna try to fit the data because both the dark matter data and the accelerating universe data are saying something weird is going on when gravity is weak, which means something weird is going on when spacetime is close to flat, which means something weird is going on when R, the curvature scalar, is close to zero.

1:43:50.3 SC: Well, if the ordinary action is just R, the simplest thing you can do is to add a constant, right? That would change things when R is small. But we've already done that. That's the cosmological constant. Adding a constant to the action for gravity is exactly just adding the possibility of vacuum energy. So that already existed. So the next obvious thing to do would be to add one over R, the reciprocal of R. No reason to do that except it's something that would begin to kick in when the gravitational field became weak, right? When R went close to zero. Now, the experts in general relativity are shaking their heads here because, even you can have a strong gravitational field with R close to zero, because R, the curvature scalar, is just part of the overall curvature. In fact, near a black hole, which in some sense has a strong gravitational field, R is still close to zero. But that's okay. We have to work with what we got, and what we have is an attempt to change the action of general relativity. So let's add the reciprocal, R plus one over R. So instead of just having the curvature scalar R, add R plus one over R. And then you hope, the idea is you hope that this can explain both the dark matter phenomenon and the dark energy phenomenon. So it explains the spiral... The rotation curves of spiral galaxies and the acceleration of the universe today. So I did that. I wrote it down, wrote down the equation, I worked out the equations of motion.

1:45:25.2 SC: I solved the equations of motion in certain extra special symmetric circumstances and I got disappointed. What I realized was that the Schwarzschild solution, which is the solution for Einstein's ordinary equations for a spherically symmetric distribution, is 100% exactly also a solution for this new theory with one over R added to the action. So even though I had modified, it's exactly this technicality that I was worried about before, R does not capture everything, okay? So even though R is zero for the Schwarzschild solution, that solution is still a perfectly good solution to this new equation. And what that means is that in the Newtonian limit, the Newtonian limit is just the Schwarzschild solution far away from the event horizon or the Schwarzschild radius. And that's what's going on in a galaxy, et cetera. So this theory, this one over R theory, doesn't make any new predictions for the rotation curves of spiral galaxies. It's of no help in explaining what we thought of as arising from dark matter. It could help explain the acceleration of the universe. So I realized by reading, not by actually thinking, that people had thought about theories of gravity based on functions of R that were not simply R. So R is Hilbert's idea. Any arbitrary function of R is something other people had thought of, and they realized there is a way to do a clever change of variables and rewrite this theory with a function of R as your Lagrangian into a theory with the ordinary general relativity Lagrangian plus a scalar field doing something interesting. And this is a feature of field theory, just of classical field theory, that is really sort of weird and worth exploring, that you write down these tensors like you have in general relativity, et cetera, and these tensors hide many different degrees of freedom in them. And so sometimes you can write a theory that you thought was just describing a certain tensor field like the metric, but it's actually describing other tensor fields as well.

1:47:43.4 SC: So that's why you can take an f of R theory and rewrite it as what is called a scalar tensor theory, just a theory with a scalar field as well as a tensor field. And from looking at that theory, you could look pretty obviously and show that an accelerating universe is a solution to the equations. At early times, it would look like an ordinary general relativistic cosmology, and at late times, it could accelerate. Now, I later realized that that solution is unstable, so that's an issue, but okay, in principle, it could accelerate. It could basically act like dynamical dark energy, basically. But here's where my not very cleverness comes in. I'd been clever up to that point, and then I became very non-clever. I said to myself, well, that's too bad. This theory seems to maybe work to make the universe accelerate, but it doesn't help at all with dark matter, the spiral curves of galaxies, the rotation curves of spiral galaxies, and it's just a scalar-tensor theory at the end of the day. People knew about them already. Therefore, it is not interesting. And so I took the files I had written up and just left them on my computer. And I think at least a year or probably two years passed after that. And within the course of a week, two things happened. One is that Vikram Duvvuri, who was a graduate student working with Michael Turner at the University of Chicago, knocked on my door and said, "Hey, has anyone ever thought of a theory of gravity where you add one over R to the Lagrangian?" And I said, "Well, yeah, I thought about that." And I told him the story, and I said, "It's not very interesting. It doesn't help with dark matter." And he goes, "Oh, okay. I guess not."

1:49:26.1 SC: And then I got a phone call, I think it was, maybe an email, from Mark Trodden, the guy who we'd worked together on w less than minus one. And Mark says to me, "Has anyone ever thought about a theory of gravity where they added one over R to the Lagrangian?" And I said, "Well, I thought about that, and I decided it wasn't interesting. But you and a graduate student just came up to me and suggested it, so maybe it is interesting. Maybe if people keep inventing it, we should write a paper about it." And we did. And so we wrote a paper also with Michael Turner, so there was four of us, and it has a goofy title, Can Cosmic Speed-Up Be Explained by New Theory of Gravity? Or something like that. And this became, like I said, the idea of modifying gravity by having a function of R in the Lagrangian was an older idea, but we were either the first or among the first to very, very explicitly say, this can be used to explain the acceleration of the universe. And again, I wasn't that excited about the paper because I thought, it's a model, it kind of works, but it doesn't have that many new interesting properties. But what I didn't get right, and I constantly make this mistake, I'm trying not to make it as I get older now, what I realize is it's a new tool for people to play with. Like, once you say, oh, maybe I can modify gravity in this way, maybe other more clever people than you can modify gravity in more clever ways and make progress. So the whole program of what is now called f of R gravity became super duper popular. And we were not the only paper. Like I said, we were one of the first, if not the first, but there were other papers near the same time saying somewhat similar things, so you can look up all the references. I don't wanna over claim the amount of credit.

1:51:15.8 SC: But there's a million questions that come up. Can you fit the data? Beyond just saying the universe accelerates, can you literally fit the data very carefully? Can you be avoiding all the various experimental tests of modified gravity that people have done? And is one over R the best thing to do? Maybe it's not. Maybe you want some other function of R. What about couplings of this scalar degree of freedom to matter? And oh, maybe you wanna impose the chameleon mechanism and things like that. So other people at Chicago and elsewhere followed up on this, and it became very popular. And I did a couple of small things with it, one looking at the cosmological consequences, one with a slightly different modified theory of gravity, which we called modified source gravity, which avoided some of the problems but then raised others. And it's fun. I feel always somewhat ambivalent about it because it's a good idea, I don't think it quite works, and it's really popular. And so I don't wanna disown it, but I can't be that excited about it myself. If you wanna ask me right now in 2026, the middle of the year, I still think that the vast amount of smart money is on the good old cosmological constant being the source of dark energy, making the universe accelerate. We don't know. We're trying to be good Bayesians. We're trying to say there's a credence for it being a scalar field or modified gravity or various other kinds of things we haven't thought of yet. We need to allow space in our credences for new data to come in and let us update our credences. Knowing what new data to look for is sometimes driven by theoretical speculation. So even before the data comes in, we need theorists to come up with new models and propose new tests of those models. Science moves slowly these days, and that's for different reasons, so different parts of science. Some science moves very quickly.

1:53:22.2 SC: Fundamental physics and cosmology move slowly in part because you make a prediction in 1999 that says, oh, a certain kind of scalar field can cause photon polarizations to rotate across the universe, and it takes decades to make telescopes that are good enough to test that kind of prediction. So it's not like we're not trying, right? We're trying our best to learn things. And as mentioned, there have been hints from the DESI collaboration and things like that that maybe it's not the cosmological constant. I'm not talking about those hints in detail here because I don't wanna rely on them. I'm not trying to give you the impression that we have discovered that w is not minus one and therefore it's not a cosmological constant, it's some dynamical dark energy, and therefore we should be building models that fit that. That's not my point. My point is, we don't know, and therefore we should keep an open mind, and we should collect data, and we should nod approvingly when the data comes in and keep it in mind, knowing that the data might be on the borderline of being statistically significant and maybe it'll go away, or maybe it gets stronger in the future. We have to have the patience to be able to wait and see what's happening.

1:54:36.0 SC: Where I can finish is where I began one and a half pod... Or two podcasts ago, with the idea that this discovery that the universe is accelerating is the single most surprising and profound discovery in fundamental physics in the time that I've been doing fundamental physics professionally. So it's important. It's a big deal. Even the paper I recently wrote about a cyclic universe, even though it's not precisely trying to explain the dark energy or whatever, it's 100% motivated by the idea that there's a cosmological constant and that gives rise to a horizon and that gives rise to a finite dimensional de Sitter space, and we need to look at the implications of that. So I'm still very much interested in figuring out what are the implications of the accelerating universe, and it'll remain until we totally figure it out one of the most important things for us to stretch our brains around. Hopefully that stretching gets us somewhere good and we find some new discoveries along the way. So thanks for listening. Thanks for putting up with a two-part solo podcast, and I will talk to you next time.

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