Let's Find Out - The Universe in a Single Graph | ASMR

Episode Date: August 9, 2023

Tonight we're drawing and explaining a spacetime diagram of the entire universe. The universe is far larger than 14 billion light years, and expanding way faster than the speed of light. Let's find out the true nature of the universe. ▸ Want to leave a tip or connect?: https://linktr.ee/letsfindoutasmr See youtube video description for full credits.

Most of us have heard that the universe is about 14 billion years old. But this graph right here explains to us how it's not just 14 or 28 billion light years in diameter, but is instead 93 billion light years across and will eventually actually reach over 120 billion light years across. From our position in the universe, universe based on experiments here on earth and observations in the cosmos that's how we got this number they're in agreement that the earth is uh you know five billion years old ish and the universe itself the atoms in the universe are about 14 billion years old that isotopes radioactive isotopes like uranium and potassium decay predictably at predictable rates

which then, once we measure the abundances of the amounts of lead and argon that those decay into in rocks here on Earth, we can plug them into mathematical formulas that will tell us roughly how long they've been decaying. And similarly, in the cosmos, in stars measuring the emission lines, and we can measure, find some of the oldest stars that tell us how long those, radioactive isotopes have been in existence in the universe and how long they've been decaying since their creation in supernovae and the very earliest supernovae in the cosmos and then on top of that the cosmic microwave background

also tells us the age roughly of the universe and about when that redshifted light was emitted from a surface of last scattering, billions of light years away, and roughly about 14 billion light years ago. But as Terrence McKenna often likes to say, he likes to quote this biologist JBS Haldane, I think JBS, yeah,

that the universe is not only stranger than we suppose, but it's stranger than we can suppose. And that means he was of course referring to more psychedelic aspects of the universe unrevealing its strangeness to us. I don't think it's any less valid to say that all domains of inquiry and branches of science. If you dive deep enough down the rabbit hole of any particular topic reveals that strangeness to the same degree. But, although there might be layers of the onion that will never be able to wrap our minds around without the help of AI at least, I think it's a fundamental character of human nature to still search. And it's almost the most meaningful thing we can do is to mutually explore the universe and make sense of the strangest phenomena that we encounter.

This graph here is We're going really analog Tonight I went ahead and printed this out Is This graph is a huge Indicator I love how much information is packed in

To this graph here It shows the varying horizons Related to the expansion of space And the weird Distortions of space In time and light In an expanding universe A finite age

but possibly infinite size. It explains the trajectories of matter and light, non-relativistic and relativistic matter, as cosmologists call it. It shows the expansion of space. And when I came across this a few months ago, I just found it like compelling. I found it the most useful, intriguing picture

that is so information dense and packs in so many concepts about cosmology into one single picture. It's the entire universe in a single graph. And how cool is that? Just like Einstein's equation, emc squared equals MC squared is so revered because of its conciseness and elegance. I think the more information that can be packed into the most simple, you know, image or idea is the most useful in learning about those topics.

So although there's quite a wealth of knowledge that I do not understand hiding in this graph, we're going to explore tonight some of the more surface-level ideas and concepts behind these horizons. These three graphs right here are going to shed a lot of light. on all our questions about the size, the age, the expansion,

and, well, tons of other very strange concepts and features that we've been able to glean simply from the light pouring in from the most distant regions of the cosmos. And speaking of pouring in, I want to thank all of you guys for all the love that you guys always show and pour in to the channel. through surprisingly constructive positive, just genuine feedback and love you guys show in the comment section. And then on top of that, I want to thank my Patreon supporters, PayPal Donators, Coffee Donations, Super Chats.

You guys are a huge boost, a huge encouragement to what I do on this channel. So thanks. We're going to start drawing this graph. And we're going to be tying everything that we do. into this variable known as the scale factor that itself is derived from the Hubble constant measured at different times in the universe so locally it's a constant

turns out that it's actually better described as a parameter because at different ages of the universe it had different values so the better we can that's why the web telescope WMAP, Planck, all these different instruments and telescopes, one of their

main goals is to measure all the properties of the universe that these equations allow us to derive from the Hubble constant, the Hubble parameter at different values in time.

Currently it's somewhere between 68 and 72 kilometers per second per megaparsec. And once we know that we can relate that cross-reference that with observed red shifts and be able to interpret the different scales of the universe and all the characteristics of the universe such as time and distances and different types of distances of the objects that were observing whose light we're observing and it allows us to interpret the

evolution and dynamics of the universe which in turn allow us to interpret and check are models of the universe for accuracy. The Big Bang model, currently known as the Friedman, La Matro Robertson-Walker, Big Bang model of the universe. And it's the constant attempt to measure the values of the Hubble constant

and relate it to all these other factors that defines much of cosmology today. So this is a very important relation here between the Red Ship, and the scale of the universe at different epics and this is called cosmological redshift which itself is distinct from Doppler's redshift the relativistic Doppler redshift and even another aspect of general relativity called gravitational redshift this cosmological redshift is unique in that it is

from the expansion of the entire universe as a whole whole in the creation of space essentially the expansion of space itself which don't ask me what that even means believe me i've tried to understand which i don't think you can do unless you put in the work to understand the math which i unfortunately don't have time or probably the uh probably the skill to be able to pull off so we're going to put this aside and we are to draw this graph guys. We're going to get our graph paper out. So we're going to start off here by drawing the, what's called the proper distance graph.

This is where the distance in space includes the expansion of space itself, includes the expanding space and therefore the increased distances between very distant clusters. We're going to start here. This represents... So in a space-time diagram, we have the vertical axis representing space, the horizontal access. In two dimensions here, we're going to need to eliminate two dimensions of space to be able to have one dimension of space, one of time. So this represents literally just looking out in one exact direction in space. Think about like this, looking straight out, just along one line of sight, just to be.

one single dimension which is okay because we just want to know the dynamics of how light travels towards us or away from us as we're going to find out we're going to see just how the distance how the scale of the universe and how it's evolved since the big bang here this uh represents time zero we're going to go up until our present time and out to the boundary of the observable universe and we want to see how galaxies have traveled away or towards us with respect as a function of time we're going to see how light has traveled away and towards and at different rates towards us over the age of the universe and we're going to see i think the best way to start is to simply get a gauge of how far things are at the present uh at our present now

our present time since the birth of the universe. Each of these two lines here, so each grid is going to represent half a billion years, or about 500 million years. So each other grid, we're going to tick to represent a billion year increments, spacing them out evenly here, all the way up. A lot of these graphs actually go all the way up into infinity

because they use logarithmic or other scales, or they go a little bit beyond this, but just for this dimension of paper, this works best for us. And we're going to go all the way out in distance now. And for the sake, we're going to label our, for the sake of simplicity, we're going to label our own age as 14 billion years.

So let's see if I can manage this without it slipping here. A little bit off kilter there, but it's okay. This here represents the, now from zero to 14 billion years and it's not just the vertical axis but this actually represents our world line so from our perspective we're not 14 billion years old but if we were we would have stayed in the exact same space spatial coordinate this is a an arbitrary reference frame but it's the only one we that makes most sense to us for now This is that, in other words, we don't move with respect to the rest of the universe.

We're representing, we are watching the universe expand from our perspective, spirically outward. And it's expanding and it's been expanding since the initial, again, the birth of the universe, initiated the expansion. And we think that there is an inflation. and then even beyond that inflation, the universe kept expanding. And then after a certain point, about 7 billion years into the universe, six or 7 billion years ago,

they think that the universe has started accelerating. So it initially inflated, was accelerating, and actually decelerating due to gravity, kind of pulling things back together. So it was slowing down, and then it kept expanding. So at 14 billion years here, expansion happened about seven or seven billion years the big bang here down here at time zero and at time zero and uh at a point at which all other matter collapsed upon itself if we look at the universe in reverse so that means that everything that we see as far as distances away from us is going to expand out from this single origin point here.

And we'll see the other graphs don't actually have that. And we'll talk about why that is. But I think the most intuitive way to start is to look at it from this perspective of the, what's called the proper distance measurement of distance between galaxies that includes the expansion of space. because when we talk about the universe being 93 billion light years across, that does include the expansion of space. And if we didn't include the expansion of space,

the universe would be considered much smaller. But in reality, light that we see has to travel across expanding space. So although they don't think new matter is being created, so therefore the matter itself at the coordinates, that they were at the beginning of time has stayed. Matter doesn't travel at the speed of light in its local reference frame. They think they roughly stayed bound to local gravitational cores, essentially,

that create the nodes to all these filaments of the universe. Like we can see here, this is a good visualization. So this is the Milky Way. This is the Milky Way in Andromeda and Triangulum right here. this is only about 2 million light years across. The whole local group is maybe at most 10 million light years across. Then we scale up. And now we see where's our local group?

Our local group right here is maybe these two are the Milky Way in Andromeda. And we're actually, we're not actually part of a really particularly dense group either, which is pretty interesting when you think about it. what repercussions that might have on the the hospitality to life flourishing maybe less you know you're less exposed to supernovae from a more than you otherwise would be in a more densely densely packed region of of the super cluster that we're a part of

so this is about on the order of see 10 million lighters away this is us about 100 million light years across 100 to 200 million and this is our local super cluster so our local group

is part of our local cluster and then our local cluster is part of our local super cluster the Virgo Supercluster and that scales up even more and to the point where

you can see the universe starts becoming a little bit more homogenous instead of really patchy with large voids they think at the scale of 100 to 2 to 300 billion light years across or sorry 200 million light years across the universe starts becoming more homogenous we have surrounding super clusters

which we together with those form filaments things called great walls there's a great wall of forming almost a string of filament and then these voids Gannis major void boots

void right here which is hundreds of millions of light years across of empty space and so beyond this beyond about a billion light years across our universe then becomes

completely homogenous and isotropic. Expansion happens radially outward at about the same rate no matter where you look. And this is the defining feature of our universe that allows us the simple allows us to actually model without being overwhelmed with the complexities of mathematics deriving out of Einstein's field equations, it allows us to model the actual universe that we're describing, which is a feature incredible in and of itself, that the simplest possible model, geometric model, for an expanding homogenous isotropic sphere essentially of the universe that allows for actual solutions to be solved or gained from Einstein's concept. complex field equations is actually the case in our actual universe.

If there were more dense regions over here, you know, they clumped up like waves or matter tended to just flow along one direction. The field equations would not be able to give us a solution. It would be far too complex to model a universe that was in, in homogenous and non-isotropic on the largest scales. And so it's really weird that it kind of smooths out into this cosmic foam web where the filaments are roughly equal in size and in direction, no matter where we look. It's really bizarre.

And as we label our spacetime axis, it's remarkable to think that each of these grids is a billion light years across the way up to 50 exactly 50 so our real universe goes out to about 63 billion light years but this is enough for us right here this dimension here is these are in billion light years or in other words again giga light years the time axis here it's a little narrow so I'll write it here is in giga years billion years each of these down here represent a billion light years so much of the universe fits in this small little grid the first grid here everything we've done up until 30 years ago was within this boundary here then we started expanding bit by bit

Hubble's discovery, his linear, the furthest galaxies he saw to be able to identify his linear, Hubble's law, relation between velocity and the distance of galaxies couldn't even be intelligibly depicted. My pencil is just not sharp enough. It's so close to where we are. This is a thousand million light years. Hubble's furthest measured galaxies initially was only about 6 million light years away.

It really is unable to be registered on here. Now, it's only within about 10 to 100 million light years that we're able to even register the cosmic expansion or the cosmological redshift. So any distances... any distances less than about a hundred million light years away from us, which is a tenth of the size of this square right here.

Most of those velocities are, in fact, due to the peculiar velocities of just the galaxies actually moving through real space, local space. So those are representative of their actual velocities with respect to each other. but as we go out here be on about halfway

to this this first grid here then the cosmological expansion starts to set in and become increasingly more dominant in terms of what the apparent velocity of galaxies

look like from any vantage point and it's about 1.8 billion so just under two squares that Hubble's constant stops being a constant. That's far enough out. Once you look a billion light years out, you're at minimum, at least looking a billion light years into the past.

And we'll get into that relationship soon. So the local group is 10 million light years across. That's 100th of this width of the square. So not even intelligible. it's essentially, we're essentially right on the origin. That's the local group. We have, when we span out again into the Virgo supercluster, it's about a hundred million, so maybe a tenth, just a smidgen further away. We have to sharpen my pencil for this one.

We have the Virgo Supercluster local groups about 10 million light years across. Virgo supercluster is about 100 to 500 million light years across and then we have the distance at which space starts becoming more homogenous say the kind of the distance uniformity or homogeneity 300 and you see how much I'm trying to crowd in here then the the lanakia linaika supercluster is about 500 million light years

so it's about halfway out down here that's the larger supercluster the Virgo supercluster sits within and then we go out to about a billion light years now is the local

it's the point at which we are no longer in any significant way gravitationally bound to the other matter in the universe. We are still in causal contact with their light because we are able to receive their light, but in no real meaningful way. And this is called the Pisces. Pisces Cetus supercluster complex. And this is about a billion light years out,

or a hundred or a thousand million light years our entire local universe is within this first square and it might be useful to interpret these as red shifts as well and there's a very well-known cosmological calculator that you can translate between red shifts and distance and all these other features but what's most important for us is and then you can see with redshift as you go back in space and distance and time you'll notice so mega parsecs a parsec is is 3.25 light years and that translates to a mega parsecs or a million parsecs being 3.25 million light years and that translates to a mega parsecs or a million light years. And so whenever we see you know a thousand million parsecs

it's really ripping out there. That means it's 3.25 billion light years. I got Ernie in here he's probably he's probably going to start getting real scared. So you might hear a little whining puppy out behind me. 10 million light years out. So 100 to 300 million light years is going to be about roughly 30 to 100 parsecs so that is and that's one probably 30 30 mega parsecs or about 100 million light years is about 0.007 we could say so probably right about there that point is that's the point at which cosmic expansion can start is going to start overriding the local peculiar velocity velocities and the redshifts due to those local motions and Hubble's furthest red shifts that he ever measured were only about six million light years

initially he eventually went out to almost maybe 200 million light years which is about here but he initially only measured maybe to 6 million light years which if we look here that corresponds to barely a thousandth on the redshift spectrum I think it was, I wrote it down, it was Z equals 0.00s is 0,002.

So 2,000th. One key to mention that a lot of cosmologists get irritated that aren't reported on in the news when we say we found the new for this galaxy is the red shift. They always say the distance, but they never give

accurate distances because they always want to say the distance in terms of just how far back in a how far back in terms of how far light could could have traveled in that time how old it is so if it's 13 billion years old they quote it as saying it's 13 billion light years away which is really inaccurate but the redshift is always accurate

and we've had web is so far until early 2020 the largest redshift is about 13 before that a Hubble I believe got the largest redshift and I had the record for almost a decade Hubble's redshift was 11 and these it turns out redshift is a really logarithmic progression so if we look at the furthest will ever probably ever be able to see with even infrared telescopes is a out to maybe 25 possibly even 30 right now we're at 13 but you could see scale this up pull pull the camera back a little bit it see the scale here it starts starting from zero the the distances away from us are pretty

linear not not exactly but fairly linear relative to the redshift going all the way out to just a redshift of one brings us all the way out three and a half billion light years away and then increasing beyond that with with each increase in you know integer increase in the redshift our distance I guess less and less distance is covered per red unit redshift So if one is three and a half, you would expect two to be seven, but in fact two is closer to five and a half. And if it's three and a half at one redshift of one, you would expect a redshift of ten to be about 30. But it's only, instead of 30, it's, you know, we're at about nine and a half. The distance represented by the redshift integer, each integer increase, increment of a redshift is successively. smaller and smaller and smaller. So when we hear these distances quoted,

the furthest galaxy initially in the very famous Hubble deep field, for instance, when we think about how iconic the Hubble Deepfield is from the mid to late 90s, it pierced through the local group, the Virgo and Lanakia superclusters, and even beyond the Pisces Cetus supercluster complex, that's a mouthful. and today we know that I think it's about 33 billion light years is what JWST observed at Z-13, 33 billion light years.

Now between that in the furthest possible galaxies where we are local group and furthest galaxies, we can observe a lot of time and a lot of space. Exists and a lot of evolution is apparent in that gap For instance the acceleration of the universe is detected at about redshift of point five to one Which corresponds here to about two billion parsecs or about somewhere between six and seven billion light years. All right, this part here at about red shift to 0.5, all the way to 0.1, which is about 11, so about here.

From this gap, 6 to roughly 11 billion light years away, we are able to detect the acceleration, the lambda acceleration. And that's the dark energy and a crucial aspect of space time. Is that light takes time to travel. And what that means is that as we, look back, we look at even our own planets, our nearby planets in the solar system, there are tens of minutes to hours

away at the speed of light. That means that we're only ever seeing them in the past as they were 20 minutes from Mars, roughly on this on average, because we do change distances between them, but a couple hours for the gas giants, and then

four years to the nearest star, Proxima Centauri. and we're only seeing them as they were that amount of time ago. So the nearest stars, we're still seeing them only as they were a couple years ago. And our Unar galaxy itself is 100 to 200,000 light years across. That means that there are stars in our own galaxy, whose light we haven't even received, or were now receiving, that was emitted before or right around the time humans as a species came on the map.

And a little bit of a sobering fact is that there could be supernovae exploding in our galaxy whose light and energy hasn't even reached us yet. And so we are none the wiser about it. So the concept is that space and time being intimately living, linked, means that there's no way we can ever have information about any object at any distance from us in the universe without that information being delayed by exactly the time it takes light to travel to us. And so as we look out, we're only seeing our own galactic core tens of thousands of years ago. We're seeing the opposite stars, opposite, on the opposite side

of the galaxy as us 100,000 years ago whose starlight was only emitted when before humans even left Africa. Now extending that, extrapolating out even further into our local group

we're seeing Andromeda as it was 2 million years ago and beyond that further out we look the further into time we're seeing into the past. what that means for the universe at the largest scales is that we're watching different ages and eras of the universe itself

the further we look into the past there's the acceleration at the redshift of 0.5 to about 0.1 this era here this is the furthest distance you know we can think of time as being increasingly less current. The further out we go, we're looking at the last billion years, then we're going out and looking at the last five billion years. And it's about, it's about at 5 to 10 billion light years away.

That space has increased enough, and the Hubble flow is significant enough that we're able to detect features specifically, supernova whose light is dilated and slowed down. These equations, based on the red shifts that we're noticing out to distances this far, it predicts not only how much dimmer the light should be based on not only how far away it is, but it's also shifted, and these equations tell us how much its time has dilated. something as as quick

cosmologically speaking as a supernova something as abrupt they only last 20 to 40 days usually which is instantaneous on geological and cosmological scales so that's something that we can actually measure time dilation of

and beyond certain distances it's so noticeable that the acceleration parameter is able to be gleaned out of the information that the time dilation tells us. So supernovae, and I'll have to look at this, bust out the big orange book again, they actually slow down and become dimmer.

They know roughly scientists know, and they have an idea of how certain types of supernovae, particularly supernovae type 1A, how their light curves should increase, peak, and then decrease. and over about roughly how long that should last. And what they actually find out is that the further away the supernovae are, the more the light has been redshifted, the dimmer their brightness curve is, and the longer the actual explosion takes.

So if it takes 20 days nearby, and they know kind of the stellar dynamics that's going on in there to cause that, they know they observe that it's taking almost twice as long in some instances I think I got to go to chapter 29 here it's in cosmology chapter and it's just so nuts to think about the fact that it's not it's not like a lens or mirror effect where it's like just

distorting the light in some superficial way it's literally the expansion the general relativistic cosmological expansion of space itself that's causing these supernovae light curves these the explosions themselves to appear in slow motion to us and it says here the so in the late 90s we all we had was these equations here and the Hubble's equations telling us that the universe was expanding at roughly a uniform rate and very few models predicted and accelerated expansion.

It was only in the last 20 years, starting in the late 90s, with a team of high-Z-supernova researchers. They were looking specifically for supernovae at high redshifts, from about 0.5 to about 1, maybe 2, redshift of 2. And they were trying to see, based on the fact that supernovae have these very human scale dynamics. They explode and dissipate. Their light curves go from, you know, before the explosion to nothing, all the way to a ramping up curve at a peak brightness over about 10 days, and then over the following 10 days, it diminishes and drops off back to nothing

as the light of the explosion dissipates and peters out. What they were noticing was that at about Redshift of .5, which is 6.5, 6.5 billion light years. The supernovae were about a quarter magnitude dimmer than expected for a universe with a particular model here. And that just represents the model of the curvature and the matter energy density, they assume. And a lambda of zero. So no dark energy. no cosmological constant there that's something i'll have to get into it in a different time but the supernovae turned out because of that they were further away than they would be in this

otherwise decelerating universe that had initiated with a big explosion and then was slowly slowing down due to the effects of gravity and they measured this their instruments are so precise that a deviation of a quarter of magnitude expected is huge. It's extremely significant. So they said that the possibility of an accelerating universe immediately leapt to their minds. And this was important because up until the late 90s, most of the models of an expanding universe that were perfectly in alignment with Friedman's in the Robertson-Walker metric predicted and indicated that the universe was had expanded and due to gravity was slowly slowing down but this was saying the exact opposite so we see that the supernovae here z.5 is about there and they measure it all the way out to over one the redshift of

one as they measured that the well I'll just stick to this here the deviation from the graph meant that the deviation, this upward bulge right here, this upward bulge as you go from you know, point one to one redshift, redshift to one is the signature of cosmic acceleration.

But then there's a turnover. This is the deviation from the expected the Hubble, you know, diagram, the law of velocity versus distance and it rapidly drops off. Because as you go beyond a redshift of one, you're going well beyond five, you know, the local universe in space and time. And at that point, we're seeing so far into the past in time that the universe is scaled down significantly to the point where there isn't nearly as much space. and therefore the acceleration due to the volumes of vacuum energy from the

just excessive amounts of empty space creating an outward pressure counteracting gravity just didn't exist there wasn't enough space in existence at that time and so the deceleration of the universe that they initially were looking for was in fact still dominating that era there were so far back in time at z equals one it was 10 about 10 to 11 billion light years distant which meant it was about this 13 is the current age of the universe so z equals one the universe was about 6 billion years old or almost you know 7 and a half or 8 billion years ago so as you look at further back the higher the redshift further back in the actual distance the further back in time we're looking until you're only looking at 200 million

years at a redshift of 20 and this means that there were a different set of dynamics controlling how the universe was expanding and there was a dominant deceleration before the dark energy could have its effect shown and instead of the supernovae being dimmer from the accelerated expansion of space and an increase in unexpected surplus in the actual space across which the light had to travel as you go further and further back in time that light actually extended from a smaller universe a universe that was decelerating and therefore its light was actually brighter than expected because that meant that the universe was in fact at a smaller scale and expanding at a smaller rate it was in fact decelerating so it was expanding outward but at a slower and slower rate that the light was actually brighter than you would expect from a simple linear expansion rate as defined by Hubble's law and so it says that

z 1.7 so we said one was about the age of the universe, well, if you go all the way out to Z, a redshift of 1.7, what you'll find is that you're going almost a third or two-thirds of the way back in time into the absolute age of the universe. You're going almost 10 billion years into the past. An age at which the universe was much smaller, its scale, was that, was such that everything was much closer together, much less space existed between galaxies and even galaxy clusters. And so for me it's just so, it's so fascinating that anything beyond a redshift of 0.76 here

is the point at which objects, specifically supernovae, those are the key targets of interest at these cosmological scales, they start to look brighter than they would otherwise if the universe has expanded at a constant rate. And they tell us that the universe is dynamic. It was initially decelerating and is now in the last few billion years starting to accelerate at an increasing rate. So here we're going so far out, we notice acceleration. By the time we get out here, we notice the deceleration is dominant. acceleration at about greater than 11 billion light years we get the universe is so old the light that we're observing is from an age so ancient that the universe was still

decelerating and was much smaller now a good example is uh I'm somewhere I got most of this information from is a username pulsar the physics staff exchange and breaks down an example between two galaxies in the universe and it shows you a good way to visualize how light expands away from a galaxy from which it is emitted and how it and the path okay ten years later I got an exceptionally slow internet today so I don't know what kind of time dilation is going on here but anyways this graph is super cool and we're going to use it to start off our our graph here. So remember that when we consider this example of us in a galaxy

and how its light travels across expanding space, how those, the light of that galaxy and the galaxy itself evolve and travel in time and space, we have to consider that we aren't just existing here in space and time. this is our world line which means that now we exist at 14 billion years here but also we've been traveling as we travel in time we've been traveling through space however this graph is this graph is set up so that we are the center the origin of our own reference frame so we have not traveled, deviated, any distance, because this spatially is centered on our coordinate in the universe.

Everything else that is expanding in any appreciable way from us is going to be really apparent when we start drawing this out. But if we are looking at ourselves at a certain time in the past, we have to change our position in our time. all the way back to, let's see, 2.5 billion years old in this example here. So at 2.5 billion years, our galaxy is now here. If we look at a galaxy that was 5 billion light years, 5.5 billion light years away from us at that time. So we'll go over here, about 5.

Its position is out here. So at that time, 2.5 billion years after the Big Bang into the age of the universe, we were separated by a distance of 5.5 billion light years. Now since then, we've traveled along our spacetime world line to the present. To the present. Unchanged coordinate in space, but definitely changed coordinate in time. traveled up, you know, over 10, 11.5 billion light years. What has this galaxy done in the meantime, and how has its light approached us throughout the intervening 10 and a half, 11.5 billion light years, billion years.

Now we want to know how this light has reached us. After traveling through the universe, its light has been redshifted by some amount, and that red shift is going to tell us everything we need to know about this galaxy's displacement in its own past since then where it's gone how it's expanded away from us since the light was emitted and the path that the light has taken from it since he was emitted at two and a half billion years after the age of the universe to get to us so we're going to know based on the red shift it's distance it's emitted distance from us when it was emitted, its present distance from us, how that's increased. We're going to know how old the universe was when that light was emitted.

We're going to know how far the light has traveled, how much expanding space, how much this universe has expanded since the emission time of that galaxy's light, and, yeah, how much time has elapsed since then. in now. So if the universe wouldn't be expanding,

then the light would have only needed it's 5.5 billion years away. So it would have only needed 5.5 billion years to reach us. So 2.5 plus 5.5 is 8. So it would have reached us at about 8 billion years. But we know that light is, or the universe is expanding. And it's expanding.

at variable rate too. It expands initially at a really accelerated rate, but that acceleration is slowly decelerating, and then it begins to accelerate again around 7 billion light years. What that means is that there's a certain distance, I don't know if I got to this earlier, but at which the Hubble constant tells us, there's a certain distance at which,

like I kind of hinted at earlier, that the velocity, the recessional velocity indicated by the Hubble velocity distance relation, the Hubble Law, the Hubble-Lumatra Law. There's a certain distance, I'm messed up there, at which the recession velocity equals the speed of light, or goes beyond it.

It was greater than or equal to the speed of light. And if we set that velocity in the Hubble's equation here to C and then we solve for whatever that distance might be to satisfy this equation then we have that C divided by the Hubble constant the Hubble distance

which is the distance in space at which space itself and anything around that region and beyond it with respect to us is expanding away from us at greater than the speed of light and for us it's about 14 billion light years

currently. So, right here, this, the recessional velocity, is equal to or greater than speed of light. That means things at that distance currently expanding away from us

faster than light. But that doesn't mean that they can't eventually they can't now emit light that will, won't eventually catch up to us and that we won't eventually be able to observe. This distance at which recessional velocities exceed the speed of light is actually called

the Hubble sphere because in three-dimensional space we're only working with one spatial dimension here. It is a spherical wavefront. And if we track that all the way back to the beginning of time, there was a point at which that sphere is ever decreasing. it expands with the expansion of space it expands because of the expansion of space but it also contracts and its rate decreases with the deceleration of the universe or its distance increases with the deceleration of the universe

and its distance contracts with the acceleration of the universe that sounds counterintuitive but when we think about it the current distance here. We said that it's about 14 billion light years. Right that? Well, the Hubble constant we said earlier is actually more of a

parameter that changes as a function of time, in other words, age of the universe. So this value, if this value changes, that means this value is going to change. So that distance is actually smaller in the past.

They think that the Hubble constant has increased, here we have the Hubble. As you go into the past, the Hubble constant gets larger and larger and larger. And so as that increases, it's going to decrease the size here. But it also decreases, not in a linear fashion, but in a directly proportional to the rate of change of the expansion of space. In other words, the rate of change of the scale factor, which we remember the Hubble parameter is the ratio of the rate of change of the universe scale factor to the current scale factor

so if the rate of change is increasing that means the Hubble parameter is going to increase and therefore the Hubble distance is going to get smaller so let's see what that looks like remember this is billion years down here so in the first billion years everything was right next to each other so the universe initially had a rapid expansion but that expansion

was always decelerating and what that meant was although space was expanding the distance at which space was expanding away at the speed of light grew aeon by eon

So at one billion years, it was about one and a half billion light years away. Two billion, it was about three billion light years away. It was kind of linear at first. And then it kind of starts turning. So around the 7 billion-year-old mark, the universe started accelerating. But before that, the deceleration meant that the distance, how far out you had to go before the expanding space reached the speed of light increased, which meant that the universe was, in fact, still expanding,

but it was slowing down in its expansion. Then at about, so I got to connect these two here, but in more of a curved sense, at about 7 billion years when the universe started accelerating, The distance to reach where the Hubble flow began to reach the speed of light decreased. Millennia after millennia. So as the universe is expanding at an accelerated rate, that means that the distance you have to go out, eon by eon, the distance you have to go out to reach the velocities approaching. the speed of light become shorter and shorter so they become closer and closer to us the faster

space is expanding around us the nearer you don't have to go out as far to reach speeds approaching the speed of light so it makes sense that this curve is going to slowly start approaching us here and it in fact keeps going there as the universe accelerates space closer and closer to us starts being reaching the speed of light and we have to go less and less further out and what that means for our galaxy here is that initially at five and a half billion light years away D at the time the distance at the time of emission was 5.5 light years and so you can see, it's actually at a position outside of the Hubble radius, the Hubble sphere, which means

that this galaxy, at the time this galaxy's light was emitted from it that we're now seeing, that light, including the galaxy and the light it emitted, was traveling away from us, greater than the speed of light. So what that actually meant was if we track the progress of its light, even the light pointing facing us. As it comes toward us, it initially, it initially traveled away from us. This is crazy to think about because that means that for the first few billion years of its journey towards us, it was actually receding from us, although it was oriented in our direction. Because that, because of the expansion of the universe, it was getting pulled away. It was not, it was unable to overcome the faster than light expansion of the space it was

sitting within. So we have to wonder how could it ever possibly reach us then. Well this has to do with the deceleration of the universe. Initially so our photon path without any expansion would just go straight in a linear with each billion year increment it will have progressed a billion light years and it goes over and over until it reaches us after exactly the amount of years it takes for light to travel five and a half billion light years but at this distance even if it was exactly on the Hubble sphere a beam of light traveling on the Hubble sphere is going to essentially be at a standstill relative to us. Now it's going to be moving through its local vicinity at the speed of light,

but that whole entire region of space time is itself going to be carried away with the expansion of us, of the universe, from us at the speed of light. So relative to us, the amount of space being created in that expanding universe, every second that photon is trying to travel towards us is going to simply, it's going to negate any progress that that they're, that photon made towards us. But the key here is the deceleration of the universe allows the Hubble distance, that Hubble sphere, to expand.

So that although the photon is initially... See, I think the inflection point is around 6 billion years. Something like that. So while the photon is initially, it's initially traveling away from us because the expansion of space is great. greater than the speed of light. So the expansion of space at that distance,

5.5 billion light years, at 2.5 billion years ago, subtracting the speed of light is a positive value. It's a positive value, meaning that light's absolute magnitude is going to be away from us. And it does so for quite a while. until we hit a point. Oh, it's more, uh, I got the right time, but the wrong shape here.

At about four and a half billion years, so about two billion years later, and I guess I probably should have, uh, should have shortened my time, my time, uh, axis so that this was a little more easy to visualize. But, um, there's an initial velocity away. for about 2 billion years it's receding away from us and again drawing the entire universe

on a single graph is kind of hard when you're dealing with anything less than a few billion light years away from us but we can see this initial distance was away from us the total movement including the

expansion of the universe was making negative progress towards us It hit a reflection point when it crossed the Hubble sphere. And it was only able to cross that Hubble sphere when the Hubble sphere itself, over 2 billion years, finally expanded to the point so the universe decelerated to a point where you had to go further and further away from us

to hit an expansion recession velocity of the speed of light. As the universe is decelerating, that Hubble sphere caught up to the region of space that this galaxy's light, that this galaxy was sitting within. And so at that point, once it crosses the Hubble sphere, this boundary allows it to start making progress towards us. And at first it's slow progress, but then it increases at a steady rate. And now it doesn't increase, you would think it might make an ever-increasing, the closer it gets to us, you would think it would be approaching us at an increased velocity. Well, the dark energy or the lambda parameter that tells us how much the universe has started, the expansion has started to accelerate again, switch from decelerating to accelerating, from slowing down, from slowing down, to quickening. That's about the threshold for this example

at which this galaxy was initially sitting. And so any time after it gets close enough to that, it's going to, by the time it hits, you know, got about, made about half a billion light years' progress, the universe had started accelerating. It was at about this stage that the universe began to expand. at an ever-increasing rate, which then contracts the Hubble sphere. So that means that as the universe is expanding, the Hubble sphere,

the radius at which you have to go out away from us to reach the speed of light is getting smaller and closer and closer to us. And so this light's trajectory from nearly 12 billion years ago was starting to be affected by the expansion of space but once you're already inside the Humble Sphere that's a

it is some type of horizon in the sense that any light that is within the Hubble sphere's boundary necessarily will make progress because the expansion of space is at that point the very moment it crosses the threshold into inside the Hubble

that lights speed of light is now greater in the direction towards us than the expansion of space is pushing it away so it is able to make steady progress now now anything outside that although it can eventually cross the threshold if it is in a position in the universe at which the Hubble sphere will eventually overtake it because it is even though the universe from seven billion years on has been accelerating and contracting this Hubble sphere. The Hubble sphere you can see is still expanding outward. So any region of space that this sphere eventually subsumes or consumes,

that light from that region will eventually reach us at least some point in the future. On into infinity. So let's put some numbers to this right now. We have the velocity, the emission velocity was greater than the speed of light. The time here, time of emission was two and a half. So we have our light path without expansion. We have our light path with expansion. What this amounts to, the light path without expansion would take five and a half,

half billion years traveling five and a half billion light years path with the expansion it's going to take about 11 years to reach us to travel what was initially a distance of 5.5 billion light years so it took about twice as long as it otherwise would have in a in our expanding universe here to get to us and then in the meantime now this galaxy has tried traveled. So on this graph, we see that the universe, this graph represents the universe expanding outward. This galaxy is ever increasing its distance from us. Its light emitted at this time 2.5 billion years, 9.5 billion years ago, or 11.5. Has finally made it to us after 11.5 billion years.

But this is the most recent version of this galaxy we see. And this is because this path of the light of the photon perfectly defines our past light cone. And it's this

trajectory here. And remember, this is only one dimension of space. So this is just a either forward or away or towards us direction. This is the path that all photons in the universe

to get to our point in space and time, our spacetime coordinate. This is the null geodesic. This is the past light cone. The null geodesic. It's the Robertson-Walker metric where the interval between, or the space-time interval, is zero. That means that the light travel time is exactly equal to the distance between events

multiplying by the scale factor. And this is a, I'm really distorting this equation here, but something of that form where the speed of light times the time between the events, the time, the time between this galaxy's emission of light and our observation of it is exactly equal to the distance between our two galaxies at the emission time, multiply by the scale factor. And this is a really cool... This is a really cool aspect here of our universe and our past light cone is that

everything we're currently seeing on this scale that is now reaching us perfectly lies on this past light gone so that means now at one billion years distant we're seeing

about one and a half billion years into the past and remember this is a little distorted because I just rounded up to 14 billion years for the current age of the universe but it is it's fairly accurate so at one two three billion light years away. We're seeing three or four billion light years into the past,

which is a roughly linear relationship, but as you go further and further into the past, what you're seeing, what we're seeing here is that, you know, at this point, let's just say we'll go five billion light years, we're seeing about seven billion years into the past. Further, we're observing the ratio of, distance in the past, distance and time to the distance in space is actually increasing.

We're seeing further into the past than objects are actually distant from us because the universe was shrinking. The universe had a smaller scale factor than it did than it does now. And a lot of these graphs, the scale factor is about one and a half billion years. The scale of the universe was 20% its original, its size it is now, at about seven and a half, so one, two, three, four, five, six, seven, and a half. It was about 60%. So that means that this distance here has now scaled up to be about double the distance in the current universe. this means that everything we see the further back we go actually starts to converge on this origin here

so that even though it's light even though it's light takes this long to reach us it takes 12 billion years to reach us it was much closer than 12 billion light years away it was only 5 billion and the further back you go the higher the redshift of the galaxies the closer they were when that red-shifted light was emitted. And this is that strange phenomenon

where if we have two similar-sized objects and the distant one looks clearly much smaller than the close one. But as you go back in time, the universe was actually smaller in scale. And so the distant objects beyond about this threshold from you know four to seven billion years

the universe was not only decelerating but it was increasingly smaller and smaller and smaller to the point that the more distant objects actually start to appear larger in the sky because the light that they emitted

was from a position at which they were much closer to us and this is so bizarre This is so weird to think about. Looking at an object like 2 billion years, or let's see, where's the graph? There's objects beyond which, a distance beyond which the objects are closer. They are larger in the sky because they were closer when that light was emitted than the objects that are physically in between us and them.

All right. Here we can see this is great. graph right here shows as you go further back after about a redshift of 1.5 the galaxies the galaxy start appearing larger on the sky and a galaxy at redshift 3 is actually appears larger than any galaxy between a redshift 1 and 3 so then these galaxies and then it just increases more and more. By the time you get out to a redshift of five, you appear as though you're larger than a redshift of maybe 0.6 or 0.7.

And if we go to our redshift little conversion table over here, what this means is that, so a galaxy, a redshift of 5, 8, so about 2, 24 billion light years away, about 25 billion light years away, that galaxy is going to start to look larger than anything between about this and its size here. So anything beyond about 9 billion light years away from us, these galaxies at 25 billion light years away are going to look larger and therefore closer in the sky than anything in between us and them that's uh that's out further than about nine or ten billion light years so it's this

really weird concept and and maybe looking at it like on on this scale here might be a little a little have a little more clarity so 10 billion light years is right here this is 30 billion light years so anything beyond here

beyond this distance from us is going to look larger than anything in this region right here so as you go out beyond a redshift of about one because remember this is about a redshift of one right here

anything beyond that is going to look smaller and then redshift of 1.5-ish from here on the galaxy start to look larger and therefore closer they start to look weirdly like they're they're actually getting closer to us although their light is dimmer more red shifted um and that's how we're able to indicate that they are indeed further away but the universe itself is acting like a gravitation lens and it's not only that it's that the light itself was emitted from an actual distance much closer than the distance of that galaxy at the present age and if we continue this galaxy's trajectory based on what we know now about based on what the equations tell us we find that it's

actually I think it's roughly about the same distance as the light path away so it's about 20 billion years and it's kind of done like a it's done an arcing trajectory where kind of started to slowly slope outward like that

and into the future the increase in the acceleration of the universe that by dark energy is going to increase its distance away from us even further and there's lines that roughly follow the same exact pack of

colored pencils here and I want to just color in the Hubble sphere so that we can easily spot it because I want to show you guys I want to show you guys the other galaxies, example galaxies for instance the

distances to the supernovae that were measured to detect the both acceleration and deceleration of the universe at different epics, different ancient epics in the past So this Hubble sphere is going to keep on going There

Everything inside it is traveling slower away from us than the speed of light Do some light shading Bras here One of the appeals of the graph was that it did have Nice colors to it because it was able to really You know jump out at your eyes And uh

You're able to gauge you able to easily decipher Kind of what was going on once you understood concepts like the Hubble sphere here and this tells us that everything inside this is traveling towards us and not away from us the speed of light inside the sphere once it hits it is able to travel along that path so this orange path here is the past light cone our past light cone for the Milky Way galaxy and really just honestly our entire super cluster really but the local group and this extends all the way back to the

beginning of the universe this too this just like our world line we were talking about earlier this too represents everything we've been in causal contact with everything that's influenced us been in causal contact with us and this here it tells us the the past light cone subsumes the I guess I like saying they were today don't I the it encompasses all other past light cones and other past light cones would look something like this following almost the exact same shape same teardrop shape and this graph is it's been peeking in and out of frame but this is based on this this graph right here so

it is a teardrop shape when it's extended in both directions but it's just a lot more drawing on my part so you can see here the different light cones in the past you know so at at about 10 billion light years or 10 billion years old the age of the universe our light cone was smaller and smaller still

the further back you go and I like thinking of about what it means. So let's just make a really, it's gonna be roughly, let's make a smaller one. I think it would be more like that. So this is an ancient light cone here. At a point if we could go back in time

seven billion years ago to half the age of the universe, everything inside this, so along this line is where we would have seen all objects in the universe at their respective, evolutionary stages that means that so so if we have see right here this object here though this about this represents redshift point five now out here to about let's see 16 billion light years this here represents about 1.25 out here was

so you can see how it jumps pretty dramatically. And 10 would be around here or so. And that means that this galaxy over here is that redshift here isn't the light that we're seeing from galaxies right now because their light takes time to reach us. These red shifts represent essentially the expanded space, how expansion of the universe

that the light has traveled through has redshifted the light since it was emitted and by tracing its world line we can trace exactly where on our past light cone these positions

these galaxies these coordinates in space crossed how old they were when they crossed so at what age of the universe we're seeing them sorry to some hand farts going on there as it's dragging across the desk but um

so what this means is that I tighten that up a little bit this galaxy crosses are so this galaxy at redshift to 1.251 in a quarter that represents a current distance of 16 light years, 16 billion light years away.

Point five, you can see it crosses it much closer to us that represents its current distance now being about 6.5 billion light years from us, which it was much closer still. Looks like it was about only 4 billion light years from us when its light was emitted. The further back in time we go, we see the closer, after about our four, five or four billion light year mark, the closer and closer the galaxies were to us. This is our global sphere.

Pretty important, so I'm going to box that in. Now I want to talk about some other horizons. If we talk about this, so these galaxies here, and it's almost at a certain point. maybe I could have maybe I could have dragged this down this past light cone a little bit further like that to encompass

almost let subsume slip out again to encompass the the actual other the world lines of more and more distant galaxies here so we could see maybe if I erase that

this colored pencil after all can I recent? Not the end of the world. Not the end of the world, just the world line. So everything kind of comes out of this origin point here. So we see as we go all the way out here now at our furthest reaches. So this is z.5 and it takes us all the way out till here to get to z one and a quarter. And then it's a pretty quick jump to we're getting to z11 and z13. And so continuing that logarithmic scale Z of infinity

due to the cosmological expansion is really only, you know, out here or so. So 46.5, this point right here is our theoretical Z equals infinity.

So our redshift, everything is not only redshifted to infinity. I think a whole aspect I forgot to. mention was that because we were able to have a cosmological redshift to scale factor relation we also have a time dilation redshift equation so in the same way that the redshift let me just mark that off there the redshift plus one is proportional to the scale of the current universe observe you know the um

age at which we're observing the light, which is now, which is just equal to one, relative to the emitted scale of the universe, that's exactly proportional to the wavelengths of light in how much they've been stretched, which means the frequencies have been stretched, which means the time between wave crests or the periods have also been stretched, which then corresponds to the actual time dilation, the time dilation factor of the universe, So it just means literally the change in time observed. The time it takes for any event to occur. If it, when it's emitted, say the supernova takes, you know, 20 days.

This time is going to be how long, how much longer. Or in other words, how much it's stretched. So the Z plus 1 is literally the time dilation factor. So whenever you have, so if you have another words, Z of 0.5, it's going to be 1.5, Z plus 1, is going to be the factor by which that supernova is stretched, and they've actually measured this. That's the craziest part about this, is that they've seen supernovae. If we were local with that supernova, we would observe it. It was right next to us before it wiped us out.

We would observe it to, as this redshift number increases, the event, the duration of that supernova, increases by that exact same factor. So if we made this nice neat number two, you know, I did point five there, but if it's redshift of one, this supernovae would take twice as long. It would be shifted by a factor of two.

It would take twice as long to explode. In other words, it would be exploding in slow motion. in that that's hard to wrap my head around so as you go out further to higher and higher red shifts you have factors of time dilation

for our furthest galaxy that galaxy anything any supernovae if we were able to observe it would be stretched by a factor of 14 it would be occurring 14 times slower at that redshift, that galaxy,

at a redshift of 13. And we go out and extend that all the way to Z, or redshift of infinity here. And that's our current theoretically

furthest observable position in the universe. Because at about a Z of 1,100, We have the cosmic microwave background, and that is actually, it's opaque to light, so we can't see beyond that. But a Z of 1100 means anything going on in the cosmic microwave background is essentially stationary. It's moving a thousand times slower from our perspective. Its time is dilated by a thousand times.

I made this graph way too. It's a little getting a little unwieldy for me here. I actually try to draw somewhat accurate curves here. Right, so, and what this really means, even though, you know, my drawing isn't really precise, is that it really is, things are expanding out in these regions of the universe. They're expanding from us well beyond the speed of light.

In the period of a billion years here, if we go all the way up here, in a billion years, it expanded, It shows on my scale here, but really it's like way off. But this world line here expanded shows 10 billion light years in the span of a billion years. This was the region that I started the video off with that little teaser. That was expanding at 58 times the speed of light. Its trajectory, its world line had so much space between us because it was the, and it is,

it's the current particle horizons, the current furthest distance will, ever see. And so I just realized that the particle horizon isn't able to really be understood unless we break out this graph, which I'm just not going to have time to draw now. So I'm going to show it to you guys. This one here is the exact same graph. We've got the scale factors, the time to now, but you see the units aren't simple linearly incrementing. It starts off at 100th or 10 million years, goes to 100 million years, 500 million years, then a billion. And so it increasingly gets each successive linear step encompasses more and more units of time

until it literally goes all the way to infinity right there. And so that's why this graph has different trajectories, different null geodesics, light paths. This is what I referenced earlier. This graph is oriented not only differently in time, but it's also different in space. The only points that correspond to our graph here are the points that represent the current cosmic time. And this might seem, as a side note, to violate the relativity principle where there is no absolute time, but cosmic time allows for proper distance. and that is the relation of the scale factor.

It's anywhere in the universe, doesn't matter how far away it is, as long as that is oriented such that the universe was at the same scale and therefore the same age after since the Big Bang or the same age of the universe. But everything orients back to the red shifts, which tell us the scale, which from that we can derive the age since the beginning of time, age since the Big Bang, and all these horizons and the distances between objects. Now we have the concept of co-moving distance here,

and our graph that we've been referencing is called the proper distance. The proper distance includes the expansion of space. So everything on a proper, the reason I chose to draw this one instead of this one, even though you can see a lot of, you can get a lot more information from certain angles from this graph. I just thought it was more intuitive to draw this graph. This graph shows us exactly what we would expect from a universe starting out at a single.

singularity expanding from there. The distances from us to galaxies and between galaxies at the current now time is the only distance set of distances on both of these graphs that correspond to each other. So whatever galaxy, whatever distance it is now from us, the co-moving distance and the proper distance is exactly equal. It's only when you go into the past or the future away from our current time that the proper distance converges or all converge to a single point on the proper distance graph going into the past.

Going into the future, they all diverge infinitely far away from each other. And I thought that this best represents, again, it best represents the actual universe because it is the, it does represent the expansion of space. and light has to travel that expansion of space that's apparently what makes light redshifted and although it's not a simple as that because we as we reference with Einstein's field equations

the geometry involved in the curvature of space and how so much information is packed in Einstein's general theory of relativity in his field equations it's very complex but this sense of expansion to me was most intuitive in this graph. This graph here tells us that everything was equally far apart, no matter how far into the past or the future you go.

That's called co-moving coordinates. The co-moving distance of any galaxy from us is exactly the same at any point in time. So the fundamental difference between them is the lack of expansion in co-moving coordinate system. And what this means is that it allows for Einstein's field equations to work fundamentally is what it breaks down into. And if the universe would not appear to be homogeneous and isotropic, the equations of defining the scale of the universe and everything we know about it would be incredibly complex, if not impossible to solve.

So what we do is actually we find the co-moving distance between galaxies, and then we multiply it by a scale factor to get the proper distance, which would be less, because you're multiplying by a fraction in the past. It would be less in the past, and you're multiplying by multiples of the current scale factor, which is one, as you go into the future. So by 40 billion years from now, we're roughly a factor of four more distant between any given coordinate in the universe. But if we are in a coordinate system, which was adjusted for the scale factor to allow the light cone trajectories to be perfectly straight lines, and there's another one that has a linear scale factor here, a linear incremental scale factor.

here, a linear incremental scale factor where we could see that the light cone looks a little different. The lines of light, even though the distances between galaxies stay the same, the light cones change if the universe,

if time is at constant intervals. Versus here, you can see it's at exponentially increasing in the the value of time in between units here. So time is getting crunched. More and more time is being crunched

in the more and less and less space as this one progresses vertically up the time axis. Where this here only goes from 0 to 25 in a linear, equally in equal increments per unit as you move up that time axis there. So that tells you that that's why it's cut off here.

But here in this one, it all comes to a point because that's at T equals infinity, infinitely far into the future. These distances, although, you know, this obviously wasn't 40 billion, 46 billion light years away at the beginning of time when everything was on top

of each other. But what it represents is the co-moving coordinate that is now that far away. And therefore, it's easy to see that although you can multiply, you know, at the beginning of time, if just an increment, just a small amount above it, you'd be at 1,000th, which is about the cosmic microwave background, you'd be about 1,000th the size of the current scale factor. So here, multiply that 46 divided by 1,000, you'd be at about 46 million light years away, which is more accurate. So you

At here Everything is kind of compacted Just like our Proper distance graph So here this one Everything at the bottom is kind of like you untie it And it all expands out Here

And it's just an easy way to visualize From how far Your light cone actually goes Because the You know anything beyond Less than a billion light years Yeah that takes takes up almost half the space between the beginning of time and now is at about a billion

light years or maybe a third. Whereas in this graph, a billion light years cuts off down here. So all of this, well really everything in this section of the graph is contained in this tiny section right here which is really, really hard to decipher. So I kind of made a calculated risk where I'm losing a time. ton of information about the universe in this region right here. And that's why I'm going to show you this. Right here, it really does help to explain the furthest reaches of our past light comb

and how that light cone will eventually converge onto what's called the event horizon, which is essentially our past light cone infinitely far, at some point infinitely far in our future. So if we go into, you know, a trillion light years from, or a trillion years from now, our past light cone will essentially be this event horizon right here. So this is the future trajectory of where our past light cone will eventually become. And the particle horizon, the particle horizon is just simply any point on the particle horizon is for any given time. So, for instance, now at about 13.8 billion years, the particle horizon, that point right there is 46 billion light years, roughly. And you could see that marks our current past light cone that converges right on our current time in the universe.

That tells us the boundary of our past light cone, the furthest in time, and therefore the furthest in space. possibly see is 46 billion light years in co-moving coordinates, which was about only 40 billion light years in proper distance in our graph. Now as we go in the past, if we go down to 6 million light years on this graph here, our light cone actually, and I like this graph because it's so easy to just add things to it, because everything is just a straight line instead of me trying to do those curves. Our light cone is just a shrinking triangle. It just shrinks like that.

I think I found it easy to use another piece of paper. So it literally just shrinks like that. So at 6 billion light years, we can see our past light cone goes to about here. You know, it's just under 40. So it would be 40 million light years. And as we go in the past, about 1 billion light years, or sorry, at 1 billion years old, our past light cone was this small here and so we could see at one billion years old our past light cone only extended out to a co-moving coordinate of 20 billion light years or about 20 million light years away from us and it's just an exponential decrease we go back to 500 million a hundred million years into the past or a hundred million years after the Big Bang.

Our light cone gets smaller and smaller until it's only 10 million. But you can see that, you know, at 100 million years after the Big Bang, our universe is roughly, it's over 100 times older than the span of time from the Big Bang to 100 million years after it.

So the amount of time covers, or the amount of space that our particle horizon covered, how far back we could see at 100 million years after the Big Bang, was 10, you know, 10 million light years and proper distance, which includes expanding space, or in our co-moving coordinates, it's 10 billion light years away. So it's a rapid expansion at the beginning of time. Within the first billion light years,

we already covered a half of the total. distance we have been able to see roughly. So that shows us how fast the universe was truly expanding. In the first billion years alone, light had been able to cover a span a co-moving distance of 20 billion light years, but in reality, that scale factor here of about 0.15, if we multiply that by 20 billion light years, 20 times 0.15, 10% would be 0.2 or sorry 2 so I'm guessing about 3 light years Kiga light years so light it was still able to cover distances greater than the speed of light

in a static universe but it wasn't 20 billion light years in just 1 billion years the comics so that's a way that the this that's kind of deceptive with the coordinate system there but it does allow us to see a lot of what's going on here. And now here, this right here allows us to see that these green spheres, these shells here, are the Hubble spheres, the Hubble distances, where the first one is the actual Hubble distance, where the recession velocity is the speed of light, and then the second, third, and fourths are the respective twice, third, and four times the speed of light.

So that means that at the beginning, if you extend that out in the way it's kind of going, that means, you know, here, our edge of the universe was traveling, who knows, you know, that's where we get the 58 times the speed of light away from us figure. So it was initially light that we're just now seeing pop into our field of view, almost infinitely redshifted, moving incredibly slow due to the incredible. time dilation due to the red shift and the expansion of space it's having to overcome. It was traveling away at 60, you know, close to 60 times the speed of light. Now, to take this example here, this galaxy represents the GNZ11 galaxy.

That was at the time this graph was drawn, that was the pre-James Webb. so it was the furthest. It's about 30 to 32 billion light years away. On these graphs, it's a nice, neat, straight world line because co-moving distances don't change with time. In reality, I actually drew it here. Let's see, where did I go? Yeah, this one right here.

So its line actually looks more like this. And we could see it goes away. And our world line, it crosses it all the way down here. So at about a billion years old, it crossed our past light cone, which that light was the last light it emitted that has been able to travel along our, that null geodesic, our past light comb, to reach our current position on our world line. And so since then, it's traveled. It was only about two or three billion light years away. from us in actual distance, proper distance, which is why I like this graph, because it represents to me more intuitively real distances between objects.

That means that it was rapidly receding. I think it was something like, yeah, I have it here, so. This is so cool. This is so cool to think about this galaxy that we've seen light from. On here, yeah, it shows it. It was receding faster than four times the speed of light. And so this graph helps us understand what's been going on. So it's light since about 400 million years after the Big Bang,

less than a billion years old into the universe. It emitted the last light that we're able to currently see 500, less than 500 million years after the Big Bang, which was pretty, pretty young into the universe. That light traveled along this trajectory, simple enough on this graph. And as this, so this world line here has been doing, you know, going straight and vertical, it hasn't changed. So the galaxy evolved out of the mists of the Big Bang. It started first coalescing as a smaller galaxy, again got bigger.

It emitted some light at about 500 million years old, but it's since gone on to keep evolving. But here's where the change in rates of expansion of the universe come into play. At 500 million years old, this galaxy was receding from us at four times the speed of light. But as the universe aged, and remember this graph showing exponential age here, So that's one billion years and that's two billion years and that's four billion years. It's not a linear aging. As the universe expanded and that Hubble distance, the distance to which the space traveling at the speed of light increased because the expansion was slowly decelerating, that light eventually went through regions of space traveling.

at a lower and lower velocity until about three and a half billion years later it finally entered a region inside and crossed the threshold past our Hubble sphere entering a region inside which space was expanding slower than light and at that point when it's a relativistic particle

like light it can no longer not reach us it's inevitable that it will eventually reach us here. And so we see that any time after that in this galaxy's coordinates, you know, what's currently about 32 billion light years from us, as it moved in space, we know again you've got to remember it was closer in the past, in proper distance. But any time after that 500 million years,

that galaxy was expanding away from us and has now itself entered slower and slower and slower recessional regions of the universe and to the point about seven years ago seven billion years ago that point seven billion years ago or it happens to just be seven billion years after the Big Bang

kind of another weird coincidence don't you think exactly half the age of the universe ago the universe started accelerating again and so it just pierced into the region that was just less than two times the speed of light in its recessional velocity but now as it stays in its

same world line trajectory at the same co-moving coordinates because the galaxy itself remember isn't moving through space at any velocity near the speed of light it's mostly these velocities we're talking about our velocities due to the expansion, the overall expansion of space. Accelerating expansion of space now causes closer and closer distances to us to be receding increasingly faster away. So therefore the threshold at which regions of space are receding at the speed of light

and twice the speed of light three and four times two are getting closer and closer and so therefore that galaxy slipped out of that less than sub, you know, 2C region. It is now in the region approaching three times the speed of light, which it will hit at about in about 6 billion years from now. And then by about 30 billion years after the Big Bang, it'll be receding at four times the speed of light again. And so due to the scale factor, we're able to determine that this galaxy, GnZ11, was only about 3.2 billion light years away, just a factor of a thousand less than it currently is. Or sorry, a factor of 10, 10% of its current distance based on this scale factor here.

So we know it was about, because its current distance or its co-moving coordinate is about 32 billion light years away, it was about 3.2. billion light years away, but it took 13.5 billion years to reach us, despite only being an initial distance, remember with the scale factor of 3.2 billion light years away. Since then, the universe, the galaxy has traveled to another, an additional, about 29 billion light years to hit its final 32 billion light year distance from us. So the light, covered, a majority of that distance in the expanding universe since it was emitted. And so I was a little bit off on this one here. So this distance, yeah, I think I mixed up the current distance here.

So it was in fact the same. Same. The time it took for this galaxy's emitted light at 2.5 billion years to overcome the expansion of space and reach us at our current spacetime coordinate it was exactly equal to the distance light can travel in that time so that's that was a mistake right there guys sorry about that so this galaxy now is the in light years so the particle horizon is at any given point it's not really a line in the sense that the past light cone is a line that the actual an actual path through space time.

It, well, I guess it is in the sense that if we emitted light there, it would be the path that light would take out into infinitely far into the future and infinitely far away. So in that sense, it would be any light that our location in space time emitted at the beginning of time will have traveled that far. But for our purposes, it's not. It's just a series of points. so that represent the furthest distance

we can see at any given time in the universe. So at 10 million light years, the particle horizon is about here or about 1, 2, 3, 4 billion light years away, and that exactly matches our past light cone. The furthest distance our past light cone reaches. So at 6 billion light years, years old, our past light cone.

Again, is the path light takes, the null geodesiccate takes through space time to reach us, overcome the expansion of space. It's already about 37 billion light years away. And if we track that up,

that's exactly where the particle horizon says we should have the most distant observable point in the universe to us at that point in time. So currently, the furthest light can ever have traveled from to reach our location in the universe.

Now at 13.8 billion years after the Big Bang is defined by this point, at this time, 13.8 billion years, this point along the particle horizon line, which is 46.5 billion light years away. Now the event horizon is an inverse, and I think the easiest way to describe it, it's like an inverse of the particle horizon. Because this is our past light cone. Got to remember this graph is logarithmic or exponential, depending on how you want to look at it, I guess. And so this, the very top, the furthest position along the y-axis is, At, or in other words, the time axis is at T equals infinity right there.

So T equals infinity right there. And that means at, you know, for our practical purposes, infinity is impossible to imagine. So we can just say even 100 billion years into the future is far enough where it would essentially be, it would converge on this light cone. So what that means is that, again, as we, we go forward into time, our past light cone, if we notice here down here, as you move forward into time, the furthest reaches of your past light cone extend further and further out in the space until you get to a point where they hit about 62, a little over 60 billion light years.

and that due to the expansion of space is the furthest co-moving coordinate we will ever observe this current region of space will have evolved to a point of about 6 million light years old that means that the current what we're currently seeing as the cosmic microwave background

in 100 billion years or even a trillion years due to time dilation meaning we observe it to evolve extremely slowly even in 100 or 500 billion years it will have only evolved that interval of time meaning about 5 to 600 million years which is wild to think about

to any other distance for instance when you think about galaxies actually instead of the CMB if we go to about 20 million or about another 6 billion light years in the future, we're only going to observe our galaxy evolving maybe another billion years because it is slowed down by a factor of between 10 and 20, or 10% and 20%, I guess.

So it's going to be moving between a 10th or a fifth as slow. Its evolution will appear if we have a telescope trained on it for 6 billion years. And we do a quick time lapse. We'll only see it evolve about 500 million to a billion years in time. Things are actually moving in slow motion at the furthest reaches of the cosmos. And it's going to look much larger on the sky than anything from here to here. It's going to look like it's up here. It's going to look like it's less than 20 billion light years from us,

when in reality it's going to start out at 32 billion light years from us. But we know from this graph here, it's going to expand in proper coordinates, probably out to well beyond 50, 60, 70 billion light years from us, which is just unreal. So despite this entire region, outside our past light cone being inaccessible to us entirely

inaccessible anything beyond this you know at two billion years old ago anything beyond roughly a billion light years away is completely inaccessible to us and we're going to simply have to wait for its light to reach us

and then anything out here is entirely the elsewhere It's a really stranger than fiction the universe. We live in really our existence So I hope you guys really weren't too confused and I hope you guys got a lot out of this because I really did too I really hope you guys enjoyed this I I had a good time making this video It was a lot of work to try to wrap my head around things that are Well beyond my depth, but it was extremely insightful

and really, really just not only humbling to have a taste of just the complexity of the mathematics that some people are able to handle, work with, but the genius of the physicists and cosmologists that were able to come up, Einstein first and foremost among them, come up with these mathematical models that predict so much about what we observe in the universe and it's encouraging to see where and what else we might we might discover about the universe and its and its most fundamental workings at the deepest levels in the future i want to thank you guys for watching and i really want to thank my patrons and everybody who supports me financially and even if you can't donate

all the moral support I get from you guys in the comments is hugely uplifting to me and it means a lot so I want to thank all you guys everybody who shows love and sends encouragement my way

really does mean a lot hope you guys enjoyed this one I'll see you next time