Sean Carroll on Dark Energy, the Cosmological Constant, and the Accelerating Universe

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. I know that in the past I've been happy to just go on at great length in individual episodes, but I do think this one naturally breaks into two different parts. And you don't have to listen to one part to get a lot out of the other. So it made sense to do it this way. The overall topic is dark energy and the accelerating universe: theories of how dark energy works, what it might possibly be. I planned out the episode and realized I needed to give background about vacuum energy and the cosmological constant, Einstein's old idea. That led to talking about the cosmological constant problem and the particle physics of it all, and that made its own episode — the previous one. Where we left off was this: 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, roughly speaking, are the cosmological constant problem — why the value is so much smaller than you might expect — and the coincidence problem. On the cosmological constant problem: if you think about effective field theory, an enormously successful paradigm for thinking about 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. The actual number is smaller than the expectation by something like 10 to the 122. That's bad, and it might be a clue to some interesting physics underlying what's going on. It certainly wasn't what particle physicists expected.

The other problem: if you think you've measured the vacuum energy — the energy density in empty space itself — and that's making the universe accelerate, 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 — 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, as a density, dilutes away to zero as the universe expands. So if they're approximately the same order of magnitude today, in the past there was way more matter density than vacuum energy, and in the future there'll be way more vacuum energy than matter. Why did we get so lucky as to be born at just the right time? It almost suggests some kind of anthropic, human-centred explanation. Maybe there is one. Maybe there isn't, and we can do better with new physical explanations.

Sean Carroll

When the idea of the accelerating universe became established in the late 1990s and early 2000s, physicists immediately said: we can't simply say it's the cosmological constant and stop there. Maybe it's something else. We should be open-minded — we were surprised once, we could be surprised again. That led people to the idea of dynamical dark energy: something that's not quite the cosmological constant but looks that way as a good approximation, and which we can look for experimentally as well as model theoretically. That's what we're getting to today — dynamical theories of what the dark energy could be. As I said in the previous episode, I'm not doing a historically fair and balanced treatment of this subject. I was involved in writing papers and thinking about this for a long time, so I'm telling you what I was thinking at the time and how that intersects with what other people were thinking. It covers a decent amount of ground for the different possibilities of what could be going on — because we still don't know what is actually going on out there. In some sense, I'm preparing you to think about new ideas as they get noticed and explained, or maybe even to come up with some new ideas of your own.

I really do want to stress that back in the late 1990s, early 2000s, when people first started thinking about dynamical dark energy, there was essentially zero empirical reason to do so. There was no data saying the cosmological constant doesn't fit very well — from the start, it fit quite well. There's always a little bit of error bars, and therefore tensions with predictions, but nothing that worried people. Today, as I'll talk about briefly — because I'm not an expert, nobody is an expert, because it's in flux — there are tiny bits of empirical evidence that maybe something is changing and we do need dynamical dark energy. But I'd say it's premature to get excited about right now. Back in the day, twenty-five years ago, people were 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. I think this was a major motivation: if you have just a number, the vacuum energy, there's really nothing you can do with it — it just sits there. You can try to come up with a deep explanation for it, but you're not getting any extra data. It's not flexible, not surprising you in some dynamical, time-dependent way. Whereas if you have some new vibrant thing, something dynamical, something that can change with time, then maybe it can help explain some of these puzzles — the cosmological constant problem, the coincidence problem, and so on. I think, roughly speaking, that ambition did not pan out. It's absolutely worth being ambitious, and people were. But I think, roughly speaking, it didn't really work. Maybe we're just not clever enough. Maybe we haven't come up with the right idea yet.

Sean Carroll

So what do you need? Let's recap the observational situation. There's something called the critical density of the universe — a theoretical number. Given Einstein's equation, given the way we relate the expansion rate of the universe to the stuff inside it, there is a certain energy density the universe could have, as a function of its expansion rate, its Hubble constant, that would leave it exactly balanced between negative and positive curvature — a geometrically flat universe. If you have less than the critical density, you're in a negatively curved universe; more, and you're in a positively curved one. Notice those words say nothing about what kind of energy density it is — cosmological constant, matter, radiation, something new. But the observations — grouping together everything measured between 1998 and 2005 or so, from the supernova measurements of the Hubble diagram (velocity versus distance for far-away supernovae in different galaxies), the microwave background, and other observations from large-scale structure — told us we were at the critical density.

That critical density is not simply one component. It divides up, in the simplest model, into about 70% cosmological constant and about 30% matter. Matter, to cosmologists, means particles moving slowly compared to the speed of light. Why does that matter? Because if particles are moving close to the speed of light — like photons — they lose energy as the universe expands, through redshift. But matter doesn't lose energy per particle: energy per particle is E = mc², 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 moves slowly. That's what matter is to cosmologists, and about 30% of the total density of the universe today is matter — about 25% dark matter, 5% ordinary matter. I'm not going to talk about dark matter, because it's a completely different kind of thing from dark energy — an interesting, important story, but not relevant to the cosmological constant story. Just because the word "dark" is in both names doesn't mean there's a necessary connection. Maybe there is — that's something to speculate about — but we have no reason to suspect it's true.

So: 70% cosmological constant, or something like it. To be like it, you need something almost constantly spread throughout space — because if this dark energy were clumping into galaxies and clusters, you'd 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 exactly what we missed before the supernovae came along. The real reason supernovae were such a good way to measure the cosmological constant — and the total density of the universe — was that they were responsive to the whole amount of energy in the universe, not just an individual galaxy or cluster you'd then extrapolate from. That was the hope of the Supernova Cosmology Project and the High-z Supernova team, and it was borne out. So you need dark energy to be almost smooth throughout space and almost constant throughout time. If you go back to the Friedmann equation: an energy density that is constant leads to a constant Hubble parameter, which is seen as an accelerating universe. All those sentences still go through if you replace "constant" with "almost constant" — an almost constant energy density gives an almost constant Hubble parameter and still an accelerating universe. Not quite exponential — sub-exponential, but still accelerating. The scale factor, telling you how big the relative distances between galaxies are as a function of time, is still curved upwards like a smile, not curved downward like a frown. That's what it means for the universe to accelerate. Of course, the cosmological constant, the vacuum energy, is perfectly good at this, and there's no experimental reason to go beyond it. But let's broaden our horizons a little.

Sean Carroll

Let me introduce a bit of jargon cosmologists use to discuss this: the equation of state parameter. The cosmological constant, interpreted as vacuum energy, has a pressure that is negative — in fact, minus the energy density of the vacuum. I shouldn't say the pressure is negative just because it's the cosmological constant; you can imagine cosmological constants that are themselves negative, giving a positive pressure — that's what you get in anti-de Sitter space, relevant if you're interested in the AdS/CFT correspondence. That doesn't seem relevant to our world, so let's put it aside and talk about a positive energy density, a negative pressure. For the strict cosmological constant, p equals minus rho — pressure is minus the energy density. For something not quite the cosmological constant, we can say p equals w times rho, where w is a number that would be minus one if it were exactly the cosmological constant, and not quite minus one — something close — if it were close to the cosmological constant but not quite. w is this new measurable quantity we can go out and ask about: we know there's dark energy, and it fits the data to have w be minus one, but is it exactly minus one? Data always has error bars — maybe it's minus 0.9 or something.

That might not sound tangible — how do you measure the pressure of dark energy separately from its energy density? But there's a nice, immediate connection between w and the rate of change of the dark energy's density as the universe expands. If w is exactly minus one, the energy density stays precisely constant — that's the cosmological constant. If w is greater than minus one — say minus 0.9, minus 0.8, numbers greater than minus one because minus one is more negative — then the dark energy density gradually fades away, decreasing over time. You can remember this because w equals zero means no pressure at all, which is just matter, and matter fades away quite quickly. For radiation, a gas of photons, w would be 1/3 — p equals a third of rho. So: for vacuum energy, w is minus one; for something not quite vacuum energy, maybe minus 0.9, minus 0.8, minus two-thirds — depends on the details. You could also imagine w being less than minus one — minus 1.2, say — which would mean the dark energy density is growing with time. Not just the total amount of dark energy growing because the universe expands, which happens with any dark energy model — energy isn't generally conserved in general relativity. If the universe is full of matter and nothing else, energy is conserved by coincidence, because it's just E = mc²: the number of particles per cubic centimetre goes down as the universe expands, but the energy per particle stays the same, and the number of cubic centimetres goes up — it exactly cancels. If the universe is full of radiation, energy is not conserved; it goes down, because radiation redshifts away some of its energy. If it's dark energy, the total energy goes up. For w less than minus one, we're saying something more dramatic: the energy density itself goes up. That might bother you — maybe it should, a little. We'll get back to whether it really should, but it's certainly allowed in the equations.

Sean Carroll

One of the first things people did after the acceleration of the universe was discovered was ask whether we could constrain this parameter w. This is a phenomenological approach to physics — we're not saying "here is my model of dark energy and here's what it predicts." We're saying: I don't have a theory, I'm parameterising the possibilities of my ignorance. There's some theory out there, not quite the cosmological constant — how close or how far is it? You add a new parameter to your model fitting, and that gives you more room to play, which is exactly the kind of situation scientists are good at.

I actually got to be a co-author on one of the first papers to do this — not the very first; I think there was a paper by Seljak and White, perhaps, that did it, but one of the first from the supernova groups. 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'd fix the equation of state at exactly minus one, but now you let it vary. As 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. Peter Garnavich, another friend of ours, had become a professor at Notre Dame and was a member of the team; he led the effort to write a paper on this. The group had a question: could the equation of state parameter be less than minus one? There seemed to be no problem putting it in their equations and plots, 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 I came up with a weasel-worded answer, a couple of paragraphs added to the paper. Roughly speaking, this was around 1999, maybe 2000, and I said that w less than minus one, from a general relativity point of view, violates energy conditions — a traditional thing general relativists do, because they care about the left-hand side of Einstein's equation, the curvature of spacetime, which is fun and interesting. The right-hand side, energy and pressure, is harder to understand, because it depends on your model of what the matter is. So general relativists invent energy conditions — energy density should be positive, pressure shouldn't be bigger than the energy density in absolute magnitude, and so on. There's no law of physics that says these conditions must be true, but they roughly guarantee that gravity isn't repulsive and things are stable. I explained that w less than minus one would violate energy conditions — not a guarantee it can't be done, but a license to say we're not going to consider that possibility. In fact, in Garnavich et al. 1999, we cut off the values of w we were looking at, at minus one. I might have said that if you want to look for w less than minus one you should be allowed to — you shouldn't be too blinded by theoretical prejudice — but I'm not sure it made it into the paper.

There's a funny anecdote about that paper's author list. Astronomers at the time were still learning their way around large collaborations — there weren't as many big collaborations in astronomy as in particle physics, where everyone's happy putting the author list alphabetically, because with a thousand or five thousand authors you can't keep track of who did what. Even in theoretical physics, with only a handful of authors, we just do alphabetical order — usually benefits me, sometimes I lose out, that's life. The High-z Supernova team's strategy was: whoever was the "boss" of a given paper, carrying out the analysis and checking everything, went first, and the rest of the team followed alphabetically. Peter Garnavich was the boss of this paper. As it would have turned out, if we'd done straight alphabetical order, Garnavich would have come first and I would have been second — but I was clearly not a real member of the High-z Supernova team; I hadn't done the work to earn my bones the way you do to be an author on those papers and eventually go to the Nobel Prize ceremony. So the strategy for our paper was: Garnavich first, the rest of the team alphabetically, and then me last — just to let everyone know I'd done a little bit of work, but not too much. I thought that was entirely fair. What we found, and what subsequent papers have found too, is that w equals minus one — the real cosmological constant — fits great, no problems, a perfect fit. But there's some wiggle room, and back in 1999 there was a good amount of it: you could increase the energy density in the dark energy a little and compensate by having it fade away a little, and still fit the data pretty well. One of the games in this active subset of cosmology is deciding what data counts — supernova data, but also large-scale structure, the cosmic microwave background, baryon acoustic oscillations, lensing statistics — different analyses put different data sets in, and it gets complicated to say what any given paper is actually claiming.

Sean Carroll

This idea — constraining the properties of dark energy — became a huge deal in cosmology. Many experiments, satellites and ground-based, were motivated by "we are going to probe the dark energy, we're going to learn about it." I have mixed feelings about that, to be honest. On the one hand, of course you should probe it, try to learn something — that's good, useful science. On the other hand, if it is the cosmological constant, then we're done probing it: we're just going to measure this one number to increasing precision, and the precision doesn't tell us much about the underlying physics. It's still just a number we have no explanation for. So this program of experimentally constraining the dark energy is a bit overblown if it is the cosmological constant, because there's not that much to constrain. But if it's not, you've got to do it. And I think this is the killer argument: maybe we learn something completely different. Even though we're motivating these experiments by testing theories of dark energy, we're still doing cosmology — still collecting data, learning about supernovae, structure, early galaxies. That's not how observational cosmology generally works, targeting one single thing.

So these days there's a lot of effort: satellites like the Nancy Grace Roman Space Telescope, recently up there; the Vera Rubin Observatory, which used to be the Large Synoptic Survey Telescope, doing surveys of lensing and supernovae; the Euclid satellite — all trying, among other things, to learn something about dark energy. And that's an ongoing thing, not to mention smaller-scale efforts that have already given results, like the DESI collaboration. So that's the experimental, phenomenological side. What about the theoretical side — is there any real motivation for thinking about this sector of physics? A big motivation was that even though the cosmological constant fits the data perfectly well, it leaves us with these puzzles — the cosmological constant problem, the coincidence problem. Maybe by looking at dynamical models of dark energy we could solve some of these, or at least learn something pointing toward a solution. Roughly speaking, it hasn't worked. It never was going to work for the cosmological constant problem — that problem is still there even if the vacuum energy making the universe accelerate is zero. Why is the vacuum energy so much smaller than its natural value? That puzzle exists whether or not the thing making the universe accelerate is vacuum energy or something else. You're not even trying that hard there; you never had any expectation that dynamical dark energy would help with the cosmological constant problem — that's always lurking in the background, and in fact we're going to add more problems by inventing dynamical dark energy. But there might have been hope for the coincidence problem.

Sean Carroll

Why is it, today, that the dark energy seems to be important? 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, and Paul Steinhardt, where they dubbed the idea quintessence — dynamical dark energy, in particular a scalar field model of it. Quintessence is the fifth element in ancient Greek physics — earth, air, fire, water, and quintessence, the heavenly element. They had a hope — certainly in follow-up papers — of developing what they called tracker models. The idea was that maybe you could explain the coincidence problem by making it not a coincidence any more: take advantage of the dynamics of the scalar field so it tracks the total energy density of matter at all times, or tracks it for some era and then stops, or some other dynamical story that would explain why it's only now the dark energy is becoming important. Roughly speaking, it didn't really become convincing — when you have these aspirations to explain more, your theories have certain goals to fulfil, so they're not infinitely flexible any more, and my impression is that hope of tracker-like behaviour doesn't fit the data very well. Simply ruled out by an ugly fact. But it was a perfectly good try, and you'll see this pattern again and again in this story — legitimate college tries at understanding the dark energy that didn't quite pan out.

So let's talk about the actual models. I already mentioned a scalar field — the first thing you'd think about, and something theoretical physicists were already good at, because they'd done this before, twenty years earlier, in the inflationary universe scenario. Inflation was a model of the early universe based on scalar fields — quantum fields that don't pick out a direction in space, that just have a value, and can have a potential energy packed into that value. You imagine a ball rolling on a hill — plots of potential energy as a function of the value of the scalar field. Inflation is a very similar idea in spirit to dark energy being dynamical today: what you want to make inflation work is something that makes the universe accelerate at an enormously fast rate in the very early universe — that acceleration smooths everything out, makes it homogeneous and isotropic, just as 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 make the universe accelerate.

Sean Carroll

Roughly speaking, what you want is a slowly rolling scalar field. That's a technical term in inflationary cosmology, but I'm using it informally here: if you have a scalar field, it has kinetic energy from its change over time and potential energy from its value. What you want is the potential energy approximately constant, the kinetic energy approximately zero — then the potential energy acts almost like a cosmological constant, like dark energy. This was the idea of new inflationary cosmology. In old inflation — Alan Guth's original idea — the scalar field sat at the minimum of a potential called the false vacuum and did quantum tunnelling to get out; that idea never really worked, and Guth admitted as much in his original paper. What Andreas Albrecht and Paul Steinhardt, and also Andrei Linde, realised is that there's basically friction in the early universe. If you look at the equations of motion for these scalar fields, they're pushed by the slope of the potential — again, a ball rolling down a hill: steep potential, the field rolls quickly; almost flat, it moves slowly. But there's also friction from the expansion of the universe, literally called Hubble friction — the bigger the Hubble parameter, the more friction. So if the potential energy function is more or less flat, Hubble friction teams up with the flatness of the potential to keep the scalar field rolling slowly, its energy density approximately constant, making inflation happen in the early universe, and analogously making the universe accelerate today. Now, the numbers are very different: the energy density you need for inflation is hilariously high, much higher than any particle physics scale we've probed experimentally, whereas the energy needed to make the universe accelerate today is hilariously low — the average energy density of the universe, which is mostly empty. But the basic equations look almost exactly the same.

In fact — people forget this once they know the answer — the discovery that the universe is accelerating was hugely good news for the inflationary universe scenario, for two separate reasons. The obvious one: inflation's big prediction was that the universe should be spatially flat, that its energy density should be the critical density, and in the 1990s that wasn't coming true — we were only getting a third of the way there. The vacuum energy, the cosmological constant, the dark energy, provided the extra energy density needed — beautiful news for inflation, a prediction that came true. The other reason is more subtle: we'd always had the cosmological constant problem, 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, replacing rho, the energy density, with rho plus p — and for vacuum energy, p equals minus rho. If that had been the correct solution to why the cosmological constant was so small, 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 dark energy act very similarly from the equations' standpoint. So the discovery that our universe actually is accelerating was good news for inflation, because it showed the prediction came true and that it is possible for the universe to accelerate — we know that, because it's doing it now. Inflation returned the favour by offering its idea of a slowly rolling scalar field for people to go play with. So the simplest model for something dynamical that mimics a cosmological constant is a slowly rolling scalar field — also called quintessence.

Sean Carroll

But the energy scales are wildly different. Particle physicists measure masses in electron volts — the amount of energy it takes to move an electron across one volt of voltage. Since we set the speed of light to one, mc² is just m, and energy and mass are interchangeable, so we use this energy unit for the masses of elementary particles. The mass of a proton is of order one billion electron volts. The mass of an electron is of order half a million electron volts — about 1/1800th the size of a proton. Neutrinos aren't zero mass, but the order of magnitude we're talking about is something like a hundredth of an electron volt. The Higgs boson is over a hundred times more massive than a proton, and the energy scales for inflation are like a quadrillion times the mass of a proton — way higher than we can reach in experiments on Earth. So if you want a dark energy theory for the universe today, you want the scalar field rolling very slowly — rolling down its potential for the entire history of the universe, fourteen billion years, and it hasn't got very far. We don't want it to roll all the way to the bottom and start rocking back and forth — that wouldn't be dark energy any more, because those oscillations would be damped by the expansion of the universe and the energy density would go away. The way to get approximately constant energy is to have the scalar field approximately not rolling at all, so its potential energy just remains constant.

You can roughly parameterise the slope of the potential by thinking about the mass of the scalar field — not an exact fit, because the slope is the first derivative and the mass the second, but roughly. Working the observational facts about our universe into an idea of what that mass would have to be, the answer is about 10 to the minus 33 electron volts — hugely tiny compared to any known particle physics scale. That number isn't pulled from nowhere; it's just the Hubble constant in energy units, the single parameter that tells you roughly the size, age, and scope of the present-day universe, so it's not surprising it's that number. But it's still a naturalness question. The Higgs boson, about a hundred billion electron volts, gives rise to the famous hierarchy problem: why is its mass so small compared to the grand unification scale or the Planck scale? For the proton, the electron, the neutrino — even the massless photon — we have symmetries that protect their masses, good reasons why they're so tiny compared to those ultra-high scales. For the Higgs, there's no good reason — that's the hierarchy problem. And the mass of our new quintessence boson is 10 to the minus 33 electron volts — incredibly, incomparably tinier still. A new puzzle to solve.

This is roughly where I came into the story. Two things were going on at once. What was important for the world was that we were discovering dark energy and beginning to think about it. What was important for me was that I was looking for a job — I was a postdoc at the ITP, the Institute for Theoretical Physics at UC Santa Barbara, on my second postdoc, and it was about time to get a faculty job if I was ever going to. I'd realised that my papers, while fun, weren't especially interesting to the rest of the world, which is why I wasn't attractive on the job market. I needed something interesting both to me and to other people. Happily, my friends had discovered the acceleration of the universe, and I happened to be something of an expert, having written a review article with Bill Press and knowing general relativity well. There'd already been papers by Steinhardt, Caldwell, and Dave, and earlier ones by Jim Peebles and Bharat Ratra, proposing a scalar field making the universe accelerate. But all of those papers bugged me, because I knew enough particle physics to see this was all really unnatural — the Higgs field is already anomalously low mass, and this quintessence field is just crazy low mass. And it's worse than that: it's not just the mass that has to be small. The danger in going from a constant vacuum energy to a dynamical field is that the field can do things — not just evolve with time by itself, but interact with other fields, which is generally what happens in particle physics, part of the effective field theory paradigm. At the very least, your new field will interact with gravity, and gravity interacts with other fields, so in the infrared effective theory your new field should be interacting with other fields too. You can try to make those interactions very small, but there's a set of expectations for how small, and your new quintessence field should lead to fifth forces of nature — detectable, in principle, with a torsion pendulum in a laboratory, like the one at the University of Washington, looking for tiny forces between ordinary matter beyond gravity and electromagnetism. This quintessence boson is essentially massless on laboratory scales, easy to source, so you can roughly predict, order of magnitude, whether you should have already seen a fifth force due to quintessence — and the back-of-the-envelope answer is yes, certainly. And that's not all: because this scalar field is slowly changing with time, that means it's slowly changing the values of everything else — the fine structure constant, the strength of electromagnetism, the masses of other particles — all of which should be slightly time-dependent over cosmological scales, and observable, at least in principle. I estimated the sizes of these effects and realised that to hide from experiments and stay phenomenologically viable, this new scalar field's couplings have to be suppressed by something like a factor of 10 to the minus 5 — small, but not crazily small. So this added up to the feeling that these new scalar fields are very unnatural from a particle physics perspective, and we already have a perfectly good theory — the cosmological constant — that fits all the data. Let's just stick with that.

Sean Carroll

I debated whether this insight was worth a paper, or just worth telling people about. But I was invited to give a talk at Fermilab — a whole workshop on dark energy and the accelerating universe. I'm not sure the phrase "dark energy" had even been coined yet. On the plane ride to the conference, working on my talk, I realised: this unnaturalness of the masses and coupling constants for a scalar field is something we'd seen before in particle physics — anomalously small coupling constants — and there are strategies for dealing with it. You can imagine a symmetry that prohibits the existence of these couplings and the mass. This goes back to a discourse associated with Gerard 't Hooft, the Nobel Prize-winning physicist, in the 1970s, who proposed that anomalously small coupling constants can be what's called technically natural: if the coupling constants were exactly zero, there would be some symmetry, slightly broken, that you've now restored. If these couplings are breaking a symmetry, they're allowed to break it by just a little and still count as technically natural — you can argue for that using renormalisation group methods and so on.

Could there be a symmetry that allows for squelching all these couplings — to the fifth forces, the fine structure constant, the masses of other particles? Yes: a shift symmetry, phi (the scalar field) goes to phi plus a constant — you just move the value of the scalar field. You can't make this an exact symmetry, because that would mean no potential energy, and the whole point of your quintessence field is a potential that changes as a function of phi, slowly rolling down it. But that's fine, because you're saying the symmetry is approximate, not exact — and that's exactly what you need to explain why the mass of the scalar field is so small. This symmetry protects the mass — still a puzzle why it's so small, but technically natural. And it immediately gives you an analogous smallness for all the other coupling constants. Implementing this approximate shift symmetry for the quintessence field explains both the masses and the couplings, and everything is good.

This is a scalar field with an approximate shift symmetry, technically called a pseudo-Nambu-Goldstone boson, or PNGB. Yoichiro Nambu, of the University of Chicago, and Jeffrey Goldstone, of MIT, did pioneering work on symmetry breaking back in the 1960s, showing you could get these bosons if you spontaneously broke a symmetry and then explicitly broke it a little more. This was a known thing, and it was known that you could make dark energy out of it — there was a paper by Josh Frieman, a cosmologist at Fermilab and Chicago, and others, pointing out — even before the accelerating universe was discovered — that if you wanted energy of this kind, pseudo-Nambu-Goldstone-boson quintessence, as we'd 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 followed immediately.

Sean Carroll

This also connected to work I'd done earlier — I've talked about it a lot on previous podcast episodes — on cosmic birefringence. The quick version: if you have a pseudoscalar field — a different use of the prefix "pseudo" from the pseudo-Nambu-Goldstone case. Pseudo, as in pseudoscalar, means it is a negative-parity field. Parity refers to what happens to a field when you change the orientation of the axes of space — like looking in a mirror. A scalar field has positive parity: change the orientation of the axes, nothing happens to it. A pseudoscalar field has negative parity: phi goes to minus phi. With George Field and Roman Jackiw, in various combinations, we'd pointed out that a slowly rolling pseudoscalar field would couple to electromagnetism in a very particular way. This symmetry, this slightly tilted pseudoscalar field, eliminates almost all the couplings to ordinary matter, but leaves one untouched: a coupling of the pseudoscalar to the electric field dotted with the magnetic field. The electric field is a vector, positive parity; the magnetic field is a pseudovector, negative parity. Taking the dot product of the two gives a negative-parity quantity; multiplying that by phi, the pseudoscalar, gives a positive-parity quantity — so this term is allowed, not ruled out by the symmetry.

What I realised on that plane is that this same pseudo-Nambu-Goldstone-boson idea, which gives you a good potential energy function and gets rid of all the fifth forces, leaves this one interaction allowed — and it makes a prediction. As the pseudoscalar field changes value, it will rotate the plane of polarisation of light from distant galaxies and the microwave background. Ordinarily, in electromagnetism, the direction of polarisation stays fixed as a photon travels across empty space; in the presence of this scalar field, it would slightly push the polarisation angle. In principle, that's detectable. I estimated a natural value for the amount of rotation — small, because of factors of one over four pi and the fine structure constant, but not that small: about one degree of rotation between us and very distant galaxies. And the data people already had at the time constrained the rotation to less than five degrees. One degree is just perfect — this rarely happens in science. What you most want, making an experimental prediction, is a number not yet ruled out but that could be ruled out within your lifetime. So I gave a talk at Fermilab saying: we're all talking about rolling scalar fields, hilariously unnatural from a particle physics perspective — maybe that's okay, because we were wrong about the cosmological constant too — but we can also try to fix it, and here's a way, which leaves one untested prediction. People got excited about that, and that's basically what got me faculty jobs a year later. I didn't get any smarter — I was just putting my smarts to use in ways the rest of the world thought were interesting.

Skipping ahead twenty-five years or so: we still haven't got the limit on the rotation of polarisation down to where we want it, but we're getting close. There have been a couple of recent claims in the literature, especially by Eiichiro Komatsu, a well-known cosmologist analysing cosmic microwave background data, that there's a sign in the data that polarisations are rotating by a tiny bit — not quite statistically significant yet, but getting there. These things don't have momentum, so that's not very definitive, but it's a suggestion, and we're going to be very interested in following up with better, more targeted data sets. It's really hard to measure rotations of polarisation, because that's usually not what telescopes are built to do — you have to trick them into it. But people are building detectors, radio telescopes, specifically to look for this. If they find it, that would be — plausibly, though people would come up with a hundred other explanations too — a direct detection of dark energy, which would be kind of fun.

Sean Carroll

So that was an interesting set of ideas bandied back and forth in the theoretical physics community — quintessence, is it natural, can you make it natural with symmetries — and there was a good five or ten years of people doing exactly that. I was somewhat involved. What else can you do? You can build models — write down specific potentials — but can you say something more lasting, more model-independent? Well, there's still the question of w. For the cosmological constant, w is minus one; if you impose the energy conditions of general relativity, it has to be less than or equal to minus one. And if you have something physically reasonable, like an ordinary scalar field with an ordinary kinetic energy rolling down an ordinary potential, you predict w is greater than or equal to minus one — something between zero and minus one are the allowed values. So it's a good candidate for dark energy.

But the whole thing is free employment for theoretical physicists, so you can play other games. What if w is just less than minus one? Forget about scalar fields — what if you just set w to minus 1.1 and don't explain why? Robert Caldwell thought about this, along with Marc Kamionkowski, and called it phantom energy — I think Caldwell was first to use the name, in a solo paper, around the time The Phantom Menace was released; you can date physics papers by what popular culture references they make. Phantom energy has the feature that its energy density increases over time, at a roughly constant rate — so you might expect things to go crazy. If the energy density is constant, the cosmological constant case, the expansion rate is constant and that leads to exponential growth in the scale factor: a of t goes as e to the Ht, where H is the Hubble parameter and t is time. If w is less than minus one, the energy density is increasing in its own right, so the scale factor increases even faster than exponential — in fact, it hits a singularity at a finite time in the future. Think of a plot of 1/x: it trails off for positive x but blows up to infinity as x goes to zero — flip that around into the future, and the scale factor approaches infinity at a finite time. That's a kind of singularity, and Caldwell and Kamionkowski called it the Big Rip, as opposed to the Big Bang or Big Crunch — a great marketing move, and important work, because it's important to understand the space of possibilities so we can discuss how likely different ones are. I'd say the Big Rip is very unlikely, but it's one of the possibilities, and it's not often you get to invent a whole new possibility for what the universe could do.

Sean Carroll

Given that it's a sufficiently good idea, is it worth asking whether there's an underlying particle physics model that would make the energy density of empty space increase with time? As I said, it would violate the energy conditions of general relativity, but that's not an absolute ruling-out — more a warning sign, dragons ahead, be careful. I became interested in this and started working with Mark Trodden, an old friend from postdoc days, at the time a professor at Syracuse, since moved to the University of Pennsylvania — we wrote lots of papers together over the years — and Mark Hoffman, my first ever graduate student at the University of Chicago. We wrote a paper analysing whether the dark energy equation of state parameter could be less than minus one — could you invent a good particle physics model, or what would go wrong if you tried? Rob Caldwell's early papers had an attempt: when you write down the equations of motion for a scalar field, you have kinetic energy and potential energy. Potential energy can be positive, negative, or zero. Kinetic energy for an ordinary scalar field is plus one-half phi-dot squared — phi-dot being the velocity, the derivative of phi with respect to t — so phi-dot squared is positive or zero, and the kinetic energy is positive. What Caldwell realised is that to get w less than minus one, all you have to do is put a minus sign in front of the kinetic energy — negative kinetic energy, which he suggested should be called phantom energy. Particles with negative kinetic energy are often called ghost particles in quantum field theory — maybe that's where "phantom" came from too.

Negative kinetic energy is fine as long as a cosmologist instantly specialises to the case where the field is constant throughout space, changing only in time. But in the real world, there'll be at least a little spatial fluctuation, which means quantising the theory and getting particles. Crack open the old quantum field theory textbook, replace plus signs with minus signs, and see what happens with a negative energy density. A lot of particle physicists didn't do this correctly at the time — there's a way of handling ghost particles where you say this corresponds to a negative norm in Hilbert space and you can project them out, because they're only arising as virtual particles in Feynman diagrams, not as real particles making the universe accelerate. So you had to relearn what you'd been taught, and the particle excitations of this field turn out to have negative mass — negative kinetic energy, even without changing the potential, turns into a negative mass for the particle. We realised that if you have a negative-mass particle in the world of Feynman diagrams — where energy is conserved, so a heavy particle can decay into a light one (a neutron decays into a proton by spitting off an electron and a neutrino, the extra mass turning into kinetic energy) but a proton can't decay into a neutron, because a proton is lighter and there's not enough energy — well, now there exist particles with negative mass somewhere in the universe. That allows the proton to decay into a neutron by spitting off a positron, a neutrino, and a bunch of negative-mass particles, still conserving energy, turning into a heavier particle. You might think this must be very rare, that you could make the coupling constant tiny to suppress it. No — as it turns out, it's infinitely likely, because if you sum 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, there's only a finite number of ways to distribute the extra energy between the electron and the neutrino; but with a negative-energy particle, you can make as many as you want, so phase space becomes infinitely big, and something you might have thought unlikely becomes infinitely likely. So our paper said: no, w cannot be less than minus one, because it would be catastrophically unstable. Forget protons decaying into neutrons — empty space, with zero energy, could decay into a bunch of positive-mass and negative-mass particles. The vacuum would be catastrophically unstable.

Of course, the whole history of particle physics is a back-and-forth — someone comes up with a good idea, someone points out a problem, someone else comes up with a fix. After our paper, there was work by people like Nima Arkani-Hamed and others on something called ghost condensation — ghost particles, negative-mass particles that could settle into a stable equilibrium, ways to get negative-mass particles hidden in the details of a theory that still wouldn't lead to a Big Rip. So I'm still pretty much on the side of thinking w is not going to be less than minus one — I think it's very hard to make a theoretically respectable model where that happens. But I'm open to someone coming up with a clever way.

Sean Carroll

There's more to life than scalar fields and negative kinetic energies. We followed up our paper — Mark, Mark, and I — with one asking: can you be tricked into thinking w is less than minus one? There's another game you can play with scalar fields: let them affect the value of Newton's constant of gravity — an old idea going back to the 1960s, with Carl Brans and Robert Dicke and others, scalar-tensor theories of gravity. When your scalar field moves around, it makes the effective value of Newton's constant change, which can lead to crazy behaviour in the strength of gravity, and therefore the expansion rate of the universe. We looked at whether, even without true phantom energy and its instabilities, you could have a model of cosmology with a time-dependent Newton's constant where, plugging in the data, you'd think you were seeing w less than minus one. This is just a feature of how physics goes: the constraints you get out depend on the theoretical possibilities you allow when you start the analysis. We found that if you didn't consider the possibility that Newton's constant was changing, and naively fit supernova and large-scale structure data, you could be tricked into a best-fit value of w a little less than minus one. To be fair, our numerical simulations showed it's hard to make that work — hard to thread the needle and stay compatible with all the various limits and observational constraints — but it's possible.

I bring this up because people have recently found hints in the data that the dark energy is not constant, and might even have w less than minus one. As soon as those hints appeared, I remained sceptical that it's really phantom energy, but I'm open to the possibility that we could be tricked into thinking w is less than minus one by constraining an incomplete theory. Looking at the papers, I think their implication that w is less than minus one isn't that believable — I think they did too quick a job fitting the parameters, and it's pretty easy to find models that fit the data even with w not less than minus one. But it is interesting: more than one hint, over the last five years, that w is not exactly minus one. If you ask me right now, I'd still say the smart money is that w is minus one — just the cosmological constant. There are theoretical virtues there, but we need to be open-minded.

There's more to the story of dynamical models than rolling scalar fields on their own. People were thinking about what we now call dark energy even before the acceleration of the universe was discovered, and I was one of them — I wrote a paper around 1996 with Greg Anderson, a friend from my postdoc days at MIT, on what we called variable-mass particles, or VAMPs. The idea: what if you have a scalar field that can roll down a potential with no minimum — not phi-squared, but something like one over phi, or e to the minus phi — so the field can just roll off to phi equals infinity forever? That's not impossible in supersymmetric theories and string theory. Greg's original interest was whether you could control these runaway potentials. What he realised is that these scalar fields, much like the Higgs mechanism, can have a feedback where the value of the field feeds into the masses of the particles around it. That wouldn't work for electrons and protons, which have very constant masses — we know from big bang nucleosynthesis and other observations that it's very hard to make known particle masses change with time. But what if dark matter got its mass from a scalar field the way electrons get theirs from the Higgs boson? As the field rolls toward infinity, it gives mass to the ambient dark matter particles, and that costs energy — an effective contribution to the potential proportional to the number density of dark matter particles. So you could stabilise the field at a value balancing its own potential against that effective contribution. As the universe expands, the number density of dark matter goes down, so the effective contribution shifts, and the scalar field keeps adjusting its minimum — variable-mass particles, a candidate for both dark matter and dark energy at once. It doesn't quite work in its naive form, because dark matter isn't perfectly homogeneous, and there's an instability where those inhomogeneities grow with time. But it helped inspire other people — something called chameleon fields, where Justin Khoury and Amanda Weltman suggested that if a scalar field gets an effective contribution to its potential from coupling to ordinary matter, that could pin it in place near matter and stop it giving rise to fifth forces, while letting it roll freely elsewhere. The interesting thing isn't any one model in detail — it's that there exist whole different worlds to play in, with exciting possibilities. Data will guide us if we eventually figure something out, but in the meantime theorists keep coming up with fun ideas, and you can't just wait for experiments, because sometimes a theoretical idea tells you what data to go and look for. If we hadn't discovered the acceleration of the universe, no one would be trying to measure its equation of state. There's a constant give and take between theory and experiment in physics.

Sean Carroll

The final idea I want to put on the table: what if the acceleration of the universe is not due to dark energy at all? I don't mean we've made a mistake in the data — I think the universe is accelerating, that's what the data tell us. But what if that acceleration is due to modified gravity instead? One thing dark matter and dark energy have in common is that, to date, we haven't detected either directly — we've inferred their existence from the behaviour of spacetime, gravitational fields in galaxies and clusters, and the expansion rate of the universe. So it's a very old idea that gravity might be different from what we think on cosmological scales, tricking us into inferring dark matter and dark energy. For dark matter, the most famous version of this is MOND, developed into further theories in various ways. You can try to play the same game with dark energy: modify Einstein's equation of general relativity to make the universe accelerate without dark energy.

I was thinking about this as a young assistant professor. Nowadays modifying gravity to get rid of dark matter just doesn't work — we have data from the cosmic microwave background, and the predictions MOND and other modifications made for the microwave background anisotropies came out false, ruled out by the data. Some people stubbornly keep thinking about it — it's a free world — but to a very good approximation the data have ruled it out. In the early 2000s, that wasn't yet true. I started from a motivation for MOND that's a numerical coincidence. Mordehai Milgrom's idea of modified Newtonian dynamics starts from this: if you look at a spiral galaxy like the Milky Way, there's evidence for dark matter because the rotation curve — the velocity of things orbiting as a function of distance from the centre — matches Newtonian gravity near the centre, where you can count the stars and matter. But as you go further out, you'd expect the rotation velocity to keep getting smaller, the way Uranus and Neptune move more slowly than Mercury and Venus because the gravitational field is weaker further out. What you actually see, going back to data collected by Vera Rubin and others, is that rotation curves flatten out rather than diminishing — more gravitational force at the fringes of the galaxy than expected. This is evidence for dark matter: cold dark matter, non-interacting, doesn't bump into itself and radiate away energy the way ordinary matter does, so it doesn't fall into a dense centre the way ordinary matter does; the galactic halo has a much lower density contrast between the middle and the outskirts than ordinary matter does. So the natural prediction, with dark matter, is that the ratio of ordinary matter to dark matter should be high in the centre of a galaxy and low far away — which is exactly what you see.

But Milgrom noticed something else: a puzzling numerical coincidence. There's always some radius, in a given spiral galaxy, where you cross over from not needing dark matter to needing it. That radius isn't the same in every galaxy, but if you calculate the acceleration due to gravity — a legitimate concept in the Newtonian limit, even if it makes a general relativist wince — at that crossover point, it's roughly the same number across many different galaxies, even though the galaxies themselves are very different. And if you plug in the right factors of the speed of light, that acceleration turns out to be numerically about the same as the Hubble parameter today — which makes no obvious sense, because the Hubble parameter is a statement about the overall expansion rate of the universe, and this crossover acceleration is about the local dynamics of an individual galaxy. Maybe secretly there's a connection — the galaxy formed somehow, from somewhere, and that might be relevant — but at first blush it's surprising. That universality, and its rough numerical equality with the Hubble constant, was Milgrom's motivation for proposing MOND. It doesn't work for clusters of galaxies, where dark matter doesn't appear at that same acceleration — which makes perfect sense for dark matter but not for MOND.

Sean Carroll

So there I was, in the early 2000s, noticing that not only is MOND motivated by this universality, and that acceleration numerically about the size of the Hubble parameter, but there's another numerical coincidence — the coincidence problem for dark energy, where the dark energy density needed to fit the data is approximately equal to the matter density today, parameterised by the Hubble parameter's value today. Two similar numerical coincidences, one in the dark matter sector, one in the dark energy sector — very juicy and provocative to a theoretical physicist. Could you unify these? Dark matter and dark energy seem very different, with no obvious connection, but if you modify gravity, maybe you connect the two phenomena — maybe there's no dark matter, no dark energy at all, just one number, the Hubble constant today, that you don't explain but just put in. Seductive — and you have to be able to be seduced by an idea like that while also being able to give it up if it doesn't work.

The common feature of the dark matter phenomenon and the acceleration of the universe is that they both kick in when gravity becomes weak, when the curvature of spacetime becomes small — which flies in the face of effective field theory expectations: you expect things to go crazy in the ultraviolet, at high energies, but to stay normal in the infrared, not to become weird when spacetime is close to flat. A very famous theoretical physicist, whom I won't name, told me he should write a paper just explaining why none of this will ever work — and there probably was such a paper to be written, though it never was. It certainly flies in the face of effective field theory, but there are things we don't understand, so maybe you shouldn't be too wedded to those expectations. I took off my effective field theory hat and thought phenomenologically: how would you change gravity so the change only became noticeable when the gravitational field was weak, rather than strong? One way of thinking about general relativity is through the action, the principle of least action — a formula written down by David Hilbert around the time Einstein himself derived his equation. You look at all possible spacetime geometries subject to some boundary conditions and find the one with minimum action. Hilbert's formula is the simplest one you could write: a scalar quantity, no directionality, called the curvature scalar — boiling down the Riemann curvature tensor to a number, sometimes called the Ricci scalar, given the symbol capital R. The integral of R, plus the action for matter, gives you Einstein's equations. From an effective field theory point of view, you'd expect R to be just the first term, with R-squared and R-to-the-fourth and so on also present — but because R is a tiny number in the real world, those higher terms are negligible until you get to very strong gravitational fields, so in cosmology you can just go with R, and that gives you ordinary general relativity.

So the question became: could you modify the action for general relativity so it didn't change things at large curvatures but did at small curvatures — no theoretical motivation, just an attempt to fit the data, since both the dark matter data and the accelerating universe data say something weird happens when gravity is weak, when spacetime is close to flat, when R is close to zero. If the ordinary action is just R, the simplest thing to add is a constant — but that's already the cosmological constant. The next obvious thing is to add 1/R, the reciprocal of R — something that kicks in as R goes to zero. I wrote it down, worked out the equations of motion in special symmetric circumstances, and got disappointed: the Schwarzschild solution — the solution to Einstein's ordinary equations for a spherically symmetric distribution — turned out to be exactly a solution for this new theory too. Even though I'd modified the action, R doesn't capture everything, and R is zero for the Schwarzschild solution, so it remained a perfectly good solution to the new equation. That meant the Newtonian limit, far from the event horizon — which is what's going on in a galaxy — was unaffected: this 1/R theory made no new prediction for the rotation curves of spiral galaxies. It was of no help explaining what we'd attributed to dark matter, though it could help explain the acceleration of the universe. I later learned, by reading, that people had thought about theories of gravity based on functions of R other than R itself, and had shown you could do a clever change of variables to rewrite an f(R) theory as ordinary general relativity plus a scalar field doing something interesting. That's a feature of classical field theory worth knowing: tensors like the metric in general relativity can hide other degrees of freedom, so a theory that looks like it's just describing the metric can be secretly describing a scalar field too. Rewritten that way, you could show that an accelerating universe was a solution to the equations — early times looking like ordinary cosmology, late times accelerating. I later realised that solution is unstable — an issue — but in principle it could act like dynamical dark energy.

Here's where my cleverness ran out. I decided the theory didn't help at all with dark matter or the rotation curves of spiral galaxies, that it was just a scalar-tensor theory people already knew about, and therefore not interesting — and I left the files on my computer. A year or two later, in the space of a single week, two things happened. Vikram Duvvuri, a graduate student working with Michael Turner at the University of Chicago, knocked on my door and asked whether anyone had ever thought of adding 1/R to the Lagrangian. I told him the story, said it wasn't very interesting, didn't help with dark matter. He said, okay, I guess not. Then Mark Trodden called, or emailed, and asked the exact same question. I told him I'd thought about it and decided it wasn't interesting, but that he and a graduate student had both just suggested it independently, so maybe it was. So we wrote a paper — with Michael Turner too, four of us — with a goofy title along the lines of "Can cosmic speedup be explained by a new theory of gravity?" We were among the first, if not the first, to explicitly connect functions of R in the Lagrangian to the acceleration of the universe. I wasn't that excited, because I thought it worked but didn't have that many interesting new properties. What I didn't appreciate — and I keep making this mistake, though I'm trying not to as I get older — is that it's a new tool for other people to play with, and more clever people can find more clever ways to modify gravity and make progress. The whole program now called f(R) gravity became hugely popular; other groups, at Chicago and elsewhere, followed up on questions like whether you can fit the data carefully, avoid the experimental tests of modified gravity, whether 1/R is the best function to use, couplings of the scalar degree of freedom to matter, the chameleon mechanism. I did a couple of small things with it myself — one on the cosmological consequences, one on a variant we called modified source gravity, which avoided some problems but raised others. I still feel ambivalent about it: it's a good idea, I don't think it quite works, and it's popular enough that I don't want to disown it, but I can't be that excited about it myself.

If you want to ask me right now, in the middle of 2026, I still think 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, keeping some credence for a scalar field, for modified gravity, for other kinds of things we haven't thought of yet, and space in our credences for new data to update them. Knowing what new data to look for is sometimes driven by theoretical speculation, so even before the data comes in, we need theorists proposing new models and new tests. Science moves slowly in fundamental physics and cosmology, in part because a prediction made in 1999 — that a certain scalar field could cause photon polarisations to rotate across the universe — takes decades before telescopes good enough to test it exist. As I mentioned, there have been hints from the DESI collaboration that maybe it's not the cosmological constant — I'm not going into those hints in detail, because I don't want to rely on them or give the impression we've discovered that w isn't minus one and it's some dynamical dark energy we should now build models around. My point is: we don't know, and we should keep an open mind, collect data, nod approvingly when it comes in, and have the patience to wait and see whether a borderline-significant hint goes away or gets stronger.

Where I can finish is where I began, one and a half — two — podcasts ago: the discovery that the universe is accelerating is the single most surprising and profound discovery in fundamental physics in the time I've been doing this professionally. It's a big deal. Even the paper I recently wrote about a cyclic universe, though it isn't precisely about explaining dark energy, is a hundred percent motivated by the idea that there's a cosmological constant giving rise to a horizon, and a finite-dimensional Hilbert space, and we need to look at the implications of that. I'm still very much interested in figuring out the implications of the accelerating universe, and I will be until we've totally figured 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.