Image credit: NASA/WMAP

Image credit: NASA/WMAP

Were the dark ages really dark? What is a perturbation, and how did they grow in the early universe? When the first stars awoke, what happened? I discuss these questions and more in today’s Ask a Spaceman!

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Music by Jason Grady and Nick Bain. Thanks to WCBE Radio for hosting the recording session, Greg Mobius for producing, and Cathy Rinella for editing.

Hosted by Paul M. Sutter, astrophysicist at The Ohio State University, Chief Scientist at COSI Science Center, and the one and only Agent to the Stars (http://www.pmsutter.com).

 

EPISODE TRANSCRIPTION (AUTO-GENERATED)

So I've repeated topics before. Like bringing up little things in a nugget of an episode that I've discussed fully in another episode, especially rich complicated topics like the big bang or black holes and so on and so forth. But this is the first time that I've rebooted an entire show. I first talked about this topic in episode six over three years ago, practically the dark ages and in about ten minutes you'll find out that's a really clever joke. And I want to talk about the first stars again.

And I'm going to take a different tack than I did oh so long ago because I'm teeing up an episode, someday don't don't hold your breath, about the formation of the cosmic web and large scale structure in our universe. And that's gonna be a big deal because it's a topic that I'm personally really fascinated by, a topic that intersects my own research, and so, yeah, I want I want to spend a lot of time on that. This is like the origin story for the large scale structure. This is what happens before the main trilogy, the main events, the main action, the big set piece blockbuster. This is this is what happens before that.

So that when we get to that big episode, I can say, hey, listen to that cosmic dawn episode again. You're like, oh, yeah. I remember I remember we did all that cool stuff. We know all the characters and their motivations and why they're doing what they're doing. We can jump right into the action.

In this this episode centers around a very very basic question. I love basic questions in science because basic simple questions are how we illuminate some really powerful stuff in nature. And in this case, it's when did the first stars appear? The short answer is we're not entirely sure. The long answer is like the rest of the episode.

And the reason it's a long answer is that there is a huge gap in our cosmological knowledge here. We do know, and this is a crazy statement to make, we do know that there was a time when no stars existed. And we know that there is a time when stars do exist. We know based on the existence of the Cosmic Microwave Background, that there was a time when stars don't exist. You remember the CMB Cosmic Microwave Background?

Couple episodes on this, pretty awesome topic. This is the leftover light from the early universe. Ions and the electrons were always constantly ripped apart from each other. They couldn't settle down and form neutral stable atoms. Then as the universe expanded, a switch flipped when it cooled off just enough that those electrons could attach themselves to the ions, become fully fledged and neutral atoms.

All the radiation that was bouncing around in that crazy, hot, messy plasma was released, had nothing left to intersect with because all the stuff is now neutral, and could just fly away free. And that radiation persisted, completely soaked the entire universe. And we can see that relic radiation today, of course, now thirteen point eight billion years later, it is cooled and it is redshift all the way down into the infrared, all the way down past that, all the way down into the microwave. And hence, it is cosmic because it comes from space, microwave because it's in microwaves, and background because it's light behind everything else. That was not a state that could generate stars.

So if you look at what the universe was capable of doing thirteen point eight billion years ago and say, oh, wow, it makes a cosmic microwave background. We can even predict its existence before it's observed, and that is not the kind of universe that manufactures stars. It's too hot. It's too dense. Density differences are too small.

We see bumps and wiggles in the cosmic microwave background, one part in a hundred thousand. Alright. Stars are more dense than that. More dense than the vacuum of space compared to that. More than one part in 100,000.

But then we look around at our present day universe, like, right nearby, you know, literally at the sun in the middle of the day, and we see that stars exist. So something had to happen. I'm not just being cheeky here because, well, I am being cheeky, but but I'm also trying to make a point. This is a big deal. One of the huge revelations of modern cosmology that nobody was expecting was that the universe changes, that the universe was different in the past.

It evolves. It changes character. It changes state on a global scale. I'm not just talking about changing like, oh, stars move around, or sometimes a nebula collapses, or sometimes there's a shockwave, or galaxies rotate, or sometimes a star blows up. Okay, that's small stuff here.

That's small. Who cares? Lots of people care, but who cares cosmologically? For a very very long time, we assume that the universe was static, that small scale changes could happen all the time, but the universe has always been the universe and always will be the universe and it's never changed. Well, that viewpoint changed about a hundred years ago when we realized that no, the universe of the past is different than the universe of the present and will be different than the universe of the future.

There was a time before stars. There was a time when there wasn't a single star in the universe. Now we have a universe full of stars. What gives? What made that transition possible?

In the universe I'm about to describe, the universe of 13 something something billion years ago is radically different than our own. This is very exciting because it provides a lot of interesting theoretical challenges, AKA job security. Because we don't really understand it, because we don't have a lot of good observational hooks here, there's tremendous opportunity and there are tremendous challenges. For one, we have to explain how stars exist when they used to not exist. Why is there a huge gap in our knowledge?

I mean, haven't we been doing this whole cosmology thing for a really long time? Well, on one hand, we have galaxy surveys and maps and more maps and even more maps of precise distances and locations of galaxies all across the universe. And the further we go back in distance, the further we go back in time. So the more distant the galaxy we see, that galaxy existed in an earlier epoch of the universe. So you would think if you we wanted to spot when the first stars formed, it would just be a matter of building bigger and bigger telescopes so you can see further and further, so you can see a younger and younger universe, and eventually you'll come upon the first star ever, the first galaxy ever, or, you know, the first generation.

But there's a couple problems with that. One is stuff this far away, like the distances and timescales we're talking about, is really far away and really hard to see. You need a big telescope to see it. And you don't know exactly where to look. You know, it's it's just the universe is gonna be different back then.

So how do you know if you're seeing the first like, there's there are challenges. There are challenges. And also the universe at one point was neutral. Right? With the formation of the Cosmic Microwave Background, that neutralized the Universe.

Neutral Hydrogen, neutral Helium, neutral Lithium, neutral Gas floating around doesn't exactly give off a lot of light. So it's hard to see the epoch after the formation of the Cosmic Microwave Background until we get a whole bunch of stars and galaxies that are really easy to see because they're blazing hot furnaces of light. So we can see the bookends. We can see the beginning of this epoch and we can see the end of the epoch or our present day epoch, but it's hard to see the in between bits because there's not a lot of stuff giving off light. Overall, it's a tough problem.

The irony is that we can see the cosmic microwave background really, really well. It's 99.999 whatever percent of all the light in the universe, and it's relatively easy to build microwave telescopes. So it it's weird. So it's really ironic because it's it's it's actually easy to see, relatively easy to see the state of the universe when it was only, say, 300,000 years old. But it's very difficult for us to see the state of the universe when it was only 500,000,000 years old or a billion years old.

It's this weird gap in our knowledge that causes us so much trouble. That we see the beginning part, we see the end part, we don't see the middle parts, and the middle parts take up a good chunk of time. We're talking hundreds of millions of years of some really serious important action like, I don't know, the formation of the first stars. That seems like a pretty critical event. And at the present time in astronomy, it's a poorly understood event.

So we have to wait for the observations to catch up, and I'll get to at the end of this episode, I'll talk about how we are trying to tackle this epoch observationally. Until then, until we actually get some light, get some pictures, get a sense of what's going on, that means the theorists are in charge. We get to run the show and call the shots, tell those stupid observers what to do with those silly little telescope. Sorry, I got carried away. This is all hypothetical.

It's hypothetical. It's based on really solid physics, so it's not like I'm just making stuff up, but it is theoretical. It is hypothetical where there's a lot of models, there's a lot of question marks, there's a lot of uncertainties, and we have to wait for the observers to catch up and tell us what we're getting right and what we're getting wrong. And there are a lot of unanswered questions around this epoch. The epoch I'm talking about is the first billion years or so of the existence of the universe.

Here's what we do know. We do know that the Cosmic Microwave Background happened. It happened when the universe was about three hundred thousand years in. That means the universe was about a thousand times smaller in radius or about a billion times smaller in volume. And that's about it.

Like, what happens after the CMB? You know, after that big exciting release of all that radiation that we needed a two parter episode just to get through, What happens next? Well, back then back then, at the release of that radiation, it wasn't the CMB. It was actually the CVB. It wasn't microwave back then.

It was literally white hot. It was visible at a temperature of a few thousand degrees. And so, initially, after the release of that cosmic background, you have white hot light flooding the universe, but it quickly fades into obscurity. It goes red and then infrared, so you can't see it anymore. It just becomes generally warm.

So we have that radiation. We have a bunch of neutral hydrogen, a whole bunch, like a universe full of it. Just boring neutral hydrogen in helium and lithium, but mostly hydrogen, of course. And how do you go? Here's the big question.

How do you go from a universe with a bath of background radiation being a bunch of neutral hydrogen just hanging out to stars and galaxies and magnetic fields and podcasts? I'm skipping a few bits. But how do you how do you do that? The answer is perturbations. I'm going to use this word a lot.

Go ahead and count the number of times I say it. So it's important to describe what I mean in the physical sense of the word perturbation. Have you ever been perturbed? You know, annoyed, but just a little bit not enough to go do something about it, but enough to to mention it? Like, oh, look at the way that lady parked that car over the lines.

I'm so perturbed. Or, man, look at this guy is taking forever at the self checkout line. I'm perturbed. Man, I hope Paul doesn't do a Patreon ad right now. Patreon.com/pmsutter is how you support the show.

It is your contributions, really, that keep all of my education and outreach initiatives alive and going and growing. I can't thank you enough. Even a dollar a month is enough to get you started, and there's goodies and prizes and behind the scenes stuff that you get if you join Patreon. Patreon dot com slash p m center, and go to astrotours.co because we have lots of cool trips around the world. Have you ever been perturbed?

Yes. It doesn't change your global state. You're still a human. You're still alive. You're not, like, viciously ill because of this perturbation, but it irritates you at some small level.

The universe after the formation of the CMB was perturbed. It was irritated at small scales. It was still expanding, still doing its global cosmological evolution thing, but funny stuff was happening at small scales. These perturbations are just density differences. They're tiny, tiny, tiny differences in density.

Some places, some pockets of the universe are slightly more dense than others. These perturbations were laid down way in the distant past. I'm talking about the first second of the big bang in an event we call inflation, which is a totally different episode. Plenty of people have already asked about it. Feel free to add your name to the list, and maybe one of these days I'll get around to it.

We saw in the episodes on the cosmic microwave background how these tiny perturbations grew to be from microscopic to just small, and the density differences were like one part in a hundred thousand. And they gave all sorts of bumps and wiggles to the cosmic microwave background itself, which we can see. But after the cosmic microwave background was released, when the universe was just three hundred thousand years in, these bumps and wiggles were still there. There were still tiny density differences in the neutral hydrogen and they grew. Because if you have a universe with density differences and you have a universe with gravity, which we do, then the rich get richer and the poor get poorer.

If you have just a slightly higher amount of density, just a bare, like just an inch, just a tiny bit more, just a speck more, you have slightly more gravity than your neighbors, so you will exert a gravitational pull on your neighbors, and they will start to empty out. And then you you have a little bit more mass, which means you have an even stronger gravitational pull, so you will pull on more stuff and more stuff and more stuff. It's this runaway effect where certain pockets completely empty out and then certain small regions get incredibly high densities. And it's just simple gravity, folks. Simple gravity is all it takes.

It does all the work of enhancing the bumps and wiggles, the differences, the perturbations, all while the whole global expansion of the universe is happening at these small scales. You're getting these enhanced density differences. The richer getting richer and the poorer getting poorer in the early universe. So imagine you have this small pocket of neutral hydrogen that's slightly denser than average, grows and grows and grows and grows, pulls material onto itself. It's happening simultaneously in all sorts of pockets all across the universe, but this is a universe that isn't really emitting light.

It's called the dark ages. Oh, wait. Let me make it more dramatic. It's called The dark ages of the universe. It was before the first new light.

The gas itself, this neutral hydrogen, is still pretty warm. It's heated from the leftover cosmic microwave background. And from gravitational collapse, as these clumps are growing and growing, there's a lot of friction involved that's releasing heat. So there is it's a bit of a misnomer, just like the real dark ages is a bit of a misnomer. It's there is light being generated in this epoch, but it's hard to get dramatic with the slightly glowing and kinda warm ages, so we just kinda stick to dark ages, you know, to make it a little bit more fancy.

In this dark ages, you know, we're talking the first hundred or so million years of the universe. There's no new sources of light and but some pockets of gas go beyond mere perturbations and trigger to become catastrophic collapse where the densities reach a critical point where it just can't help itself, and it shoves down into an incredibly small volume, reaches insanely high densities, packing layers upon layers of material into a as small a volume as possible until the densities and temperatures reach a critical point and nuclear fusion ignites, where protons from those hydrogen nuclei can smash into each other and form helium atoms. That releases a little bit of energy. That releases a little bit of light. The first star is born.

At some point in the past, there were no stars. Through this process of continued gravitational collapse at small scales, the first star is born and then another and another and another and another. Before galaxies even begin to coalesce, before we get those larger structures, the first stars appear on the cosmic stage. This event is what we call, drum roll, please, the cosmic dawn. Sounds pretty cool.

Right? For once, astronomers give something a really cool and appropriate and useful name, cosmic dawn. For the first time since the cosmic microwave background was was released, we have a new source of light from nuclear fusion. For the first time since the first twenty minutes of the Big Bang itself, we have nuclear fusion. There has been no nuclear fusion for hundreds of millions of years in the universe up until this point.

From there, the groups of stars continue to coalesce into larger structures into what we will someday call galaxies. And by the way, this episode is thanks to Joyce s via email who asked what is the cosmic dawn. This is a mind blowing result. I mean, those stars and indeed all cosmic structures started off as microscopic bumps and wiggles, mere perturbations, just small. And then just gravity and time, gravity and time, gravity and time, over the course of these eons produce something spectacular.

In this case, nuclear fusion in the core of a ball of neutral hydrogen, AKA the first population of stars. It's also an incredibly unsatisfying picture because because that's the general picture. It's the broad brushes. It's our best guess based on evidence that we have of how the first stars formed. It's how we get from point a, the CMB at year 300,000, to point b, a whole bunch of stars and galaxies a billion years later.

And we're able to match the statistics of the bumps and wiggles in the cosmic microwave background to the statistics of later arrangements of stars and galaxies. So we know it's it's the legit path, but there's so many unanswered questions. And that's because that the picture I just painted of density differences growing, eventually gained to smaller scales, etcetera, etcetera, in the backdrop of an expanding universe, that's a picture from the lens of cosmology. Right? How the universe itself behaves.

Now we're confronted by astrophysics, the flows of gas, magnetic fields, accretion disk, nuclear fusion, radiation transport, all that juicy, juicy, juicy, and also sometimes disgusting stuff. Cosmology is simpler than astrophysics. It's easier to talk about the whole entire universe than what's happening at these small scales. And we when we dig down, it gets messy, really messy, but in a kind of fun way. For example, you may have noticed I haven't been very firm in any of my mentions of time.

Did it take ten years for the first stars to form? Probably, it took a little bit longer than ten years after the CMB. Did it take ten billion years for the first stars to form? Well, no. Because we see plenty of mature stars, you know, ten billion years in.

So it's somewhere between ten and ten billion years. Well, that narrows it down. Some new lines of evidence, though, show that the first stars formed kinda early, cosmologically early, astronomically early. Somewhere in the ballpark of the first few hundred million years, we get the first stars. The evidence we have for that is is actually pretty cool.

We have techniques to occasionally see some very, very distant galaxies. We can't if you just point a telescope at the sky, you can't see a very, very distant galaxy in this very young epoch. It's just too far away. It's too faint. You're never gonna get it.

Just give up. But through gravitational lensing, sometimes everything lines up just right and we get a massive object like a cluster of galaxies in between us and a distant, more distant galaxy. The light from that distant galaxy gets bent through gravitational lensing and that lens acts like a magnifying glass. It actually magnifies distant objects. We get more light than we ought to, and we are actually able to see.

So it's like almost like a gravitational slingshot y kind of thing where we can see farther than we normally would expect to be able to. So we do have some samplings of some relatively early galaxies from the early universe. And they're really mature, Surprisingly mature. I mean, these are galaxies with giant black holes, with plenty of heavier elements that can only have been fused over multiple life cycles, multiple generations of stars building up the rest of the periodic table. Full on formed, you know, spiral galaxies, the whole deal.

It's like it's like walking up to a three year old, and the three year old is delivering a lecture on quantum mechanics. They're like, wait a minute. How did you get so mature? Shouldn't you be playing with colored blocks or something? That's basically what it's saying.

We see the the few samples we have of the younger universe, the younger universe is actually pretty dang mature. And the only way to make that happen is if the first stars come in even earlier. If you're seeing a mature galaxy when the universe is only one or 2,000,000,000 years old, and you know you needed multiple generations of stars to make and build a mature galaxy like that, then, yeah, the first stars need to come pretty dang early. The other piece of evidence we have is, and I'm gonna toss out a fancy pants astronomy term here, optical depth. Optical depth.

Have you ever, been in, like, a smoggy or foggy or cloudy day where if the sky is perfectly crystal clear, you can see the sun really, really well. Don't look at the sun. But imagine looking at the sun. You can see it really well. And then if a cloud rolls in, it's harder to see the sun.

And then if a lot of clouds roll in or a lot of smog or a lot of fog roll in, it's harder and harder to see the sun. The clouds have a shorter optical depth than a clear blue sky. Light can go further in a clear sky than it can in a foggy or smoggy sky. And so the depth of the optics or optical depth is lower inside of a cloud. We can measure the this is crazy.

We can measure the optical depth of the universe. Because we have the cosmic microwave background, it's b for background, remember, that light filters through the entire universe before it reaches our telescopes. And we can use that to measure the optical depth of the whole entire universe. And those first generations of stars and quasars, they did something special to the universe. They re ionized it.

Remember when the Cosmic Microwave Background was released, the universe became neutral. But now we live in a universe not just with stars and galaxies, but all the gas is pretty much ionized again. All the electrons, if you look at random cloud of gas and dust in the universe, the electrons have been ripped off it. It's been re ionized, re plasma sized. The sun is a plasma.

The Orion Nebula is a plasma. Most of the Milky Way galaxy is a plasma. It's been reionized, and those first stars did it. Or at least it happened in the same epoch right after the cosmic dawn. This epoch of reionization that we call that shortly followed the cosmic dawn completely changed the character of the universe.

It's like Puerri, but a lot longer and involving a lot more radiation. Changed the character the phase, the state of the universe one last time, gave us the universe that we have today, And how long the universe has been neutral versus how long the universe has been ionized changes the optical depth of the universe from us to the CMB, so we can use the CMB to measure the optical depth, and that puts a limit on when the first stars had to show up. And that gives us these numbers of a few hundred million years. And a few hundred million years is basically nothing, especially considering the achingly slow force that is gravity. Gravity is so weak.

It takes a long time. You're starting with tiny perturbations of one part in a hundred thousand, which is super small. You're building from that to something like a star capable of nuclear fusion in a bath of background radiation using only neutral hydrogen. That's quite an accomplishment. Good job, gravity.

The universe grew up fast, reached its present day character in less than half a billion years, and has basically remained unchanged in the billions of years since then, and we don't know much about it. Here's another big mystery besides, you know, how long it took and when it happened. How big were these stars? That seems like a basic question. Initially, we thought these things would be huge.

They'd be monsters, easily one or 200 times the mass of the sun. Why? Well, you have a smaller universe with the same amount of stuff, and so as we crammed in a smaller volume, easy to build big megastructures like giant stars. And at this epoch, because it's all neutral hydrogen, there aren't a lot of, shall we say, efficient pathways to emit radiation and cool down to compress to make really, really tiny balls of, you know, like stars like our sun. It's it's just harder.

The the you don't have the right chemistry mixture that we do today to manufacture small stars that you did in the past. The universe just didn't have the technology because it was missing all those heavy elements. So you had to just keep piling stuff on more and more and more and more and just kind of brute force your way into nuclear fusion rather than doing it, say, elegantly and cleanly the way the universe does today. And we don't see any first stars today. The first generations of stars is all gone.

They're all dead, and we know that more massive stars live shorter lives. Of these first generations of stars that are missing all their heavy elements because heavy elements weren't a thing back then, is there like, if we if we look around, are there any stars that are almost completely pure hydrogen, helium, and no trace anything else? We don't see any of those stars anywhere, and we've really looked. So the only conclusion is that these stars must have died a long time ago, which means they had short lifespans, which means they must have been big. Some of the early simulations we've done of this epoch of these condensing and collapsing neutral hydrogen clouds supported this idea, but then more sophisticated simulations are showing maybe it's not all big monsters.

Maybe you can get a big monster, but then surrounded by fragmented balls of smaller stars. Maybe the stars are big, but not that big, like 50 times the mass of the sun. So over the past few years, it's become a much more complicated story. Where it used to be just like, yep. These are gonna be giant stars and that's it.

First generation. But now maybe it's maybe it's more nuanced than that, and we don't really have a way out until the observations come online. There are some crazy ideas. I don't wanna call it crazy ideas. Like, interesting ideas that are highly speculative.

Let's put it that way. Maybe the first stars formed from clumps of dark matter. Maybe they provided the seeds to get those first stars going. Maybe black holes formed first and provided the anchors for galaxies that allowed matter to accumulate, and then the stars formed around those, etcetera, etcetera. There there's some pretty cool ideas, and I'd love to dig in more.

Feel free to ask, and I can get into the more speculative stuff about the dark ages and the cosmic dawn. But this is just what happens when you let theorists run free and wild. They they tend to get a little crazy. You know, they get bored. They they solve the main problem.

Like, okay, yeah, here's the broad brush picture of how the first stars formed. But then if you don't feed them some, like, new data, they they start to get antsy and they start, you know, they got jobs, they got tenure, you know, they gotta keep writing papers, they gotta do something, so they they start writing some weird stuff. So we need some real data and observations to get those theorists back on track. And over the next few years, and if you're listening to this far into the future, then in your past, there have been two major efforts to explore the dark ages and the cosmic dawn. One is through neutral hydrogen itself.

It turns out, of all the coincidences that mother nature could provide, That neutral hydrogen with just one proton and one electron, usually the spins and remember the whole quantum spin thing? No. Okay. Fine. But particles have spin, and they the proton and the electron, they can either spin the same direction or they can spin opposite directions.

They prefer to spin opposite directions because of various quantum mechanical rules, but every once in a while you can get a random flip of the electron, so they're both spinning. The proton and the electron are spinning the same way. And then, of course, that's an unstable situation, so the electron will flip back because I know never mind. I I didn't I changed my mind. I didn't want it that way.

Anyway, it will flip back to normal, and it will give off a little bit of light. That light has a very precise wavelength, 21 centimeters. You know, hold your hands out yay far apart, and that's about 21 centimeters. Very, very specific wavelength of light. So if the universe if a neutral cloud of gas and we can see this light from our own, their pockets and neutral hydrogen still left in our own Milky Way galaxy.

We can use that to map it out. It's great. We can use this same radiation to probe the early universe, but the early universe was the early universe. Light gets red shifted in an expanding universe and instead of 21 centimeters now, it's now a couple meters, which is radio. So the irony here, the weird thing is you actually need a very sensitive radio telescope to probe the dark ages.

How cool is that? The challenge here is that there's, like, a bajillion other things in the universe, including humans, that are also very loud in the radio, so it's very, very hard to to pick out that signal from the very early universe. It's a very, very faint whisper, but there are dedicated telescope operations running right now trying to get at that signal. The second thing is the James Webb Space Telescope. The James Webb Space Telescope is designed to hunt for the first generations of galaxies.

Can't see the stars themselves, probably can't push back all the way to the epoch when the very, very first stars formed, but shortly after that, after the first stars formed and grouped themselves together into galaxies, and the galaxies had a little bit of time to, you know, get their feet underneath them, get up and running, figure out what it means to be an an adult mature galaxy, that's when the James Webb will be able to spot them. Again, light, they're emitting plenty of visible light just like galaxies do today, but it's in the distant past that light is red shifted. That's why James Webb is an infrared telescope because it's hunting for galaxies in the early universe. So the dark ages of the universe won't stay dark for long. Thanks to my top Patreon contributors this month, Robert R, Justin G, Kevin O, Justin R, Chris C, Helgeby, and Andrew p.

It's your contributions and the contributions of all the fine Patreon contributors that keep this show going. Patreon.com/pmsudder. And of course, go to astrotours.co. That's pretty awesome. Iceland, Costa Rica, Ireland, trips are coming mile a minute here.

It's a fabulous ride, and I would love to see you in any of these amazing destinations. Check out spaceradioshow.com if you wanna hear me live on the air. And of course, if you get a chance, put in a positive review on iTunes. Send me a question, hashtag ask a spaceman, ask a spaceman at gmail dot com or ask a spaceman dot com. Love those questions.

That how that's how the show keeps running. And, that's it. That's all I got. I'll see you next time for more complete knowledge of time and space.

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