The Biggest Ideas in the Universe | Q&A 22 - Cosmology

hello everyone welcome to the biggest ideas in the universe i'm your host sean carroll today we're doing the q a video for idea number 22 which was cosmology and miraculously somehow this week nothing went terribly wrong previous week i had broken the blog i didn't break the blog actually wordpress keeps updating itself so it made it impossible to leave comments uh those have all been restored the blog is working out we got good comments i think you know the microphones and lighting and everything are good in case you don't know i wrote a blog post um on how i make these videos so if you're regular watchers viewers you remember that in the q a video i think number three which is on force energy in action at the end i did a little video tutorial about how i was making these videos but uh number one i've gotten better at it and my techniques and and programs have changed a little bit and number two it's actually ironically easier to learn what i do from a text blog post than from a video because i can say you know i'm using screenflow and you can click and then go to what that is so check out the blog preposterous universe.com blog if you want to know more about the behind the scenes picture of what's going on here or make the biggest ideas videos otherwise uh it's pretty straightforward we got good questions so let's get to them um one question which is not a question it was a uh enduratum i made a boo boo and i was talking about henrietta levitt who showed the period luminosity relation uh for cepheid variable stars i went on and on about parallax but levitt was not directly using parallax after all i just sort of jumbled that into my memories of the history of astronomy um what in fact she just did was look at cepheid variables that are in either the large or small magellanic clouds these are small satellite galaxies which people more or less had an idea that that was true and she sort of uh assumed for good reasons and correctly that if you have a bunch of cephe variables that are in one little sub galaxy then for all intents and purposes you can treat them as the same distance away so even if you don't know how far away they are you know that they're basically the same size so there's a relationship between their apparent luminosity and their actual uh distance so that was sorry thank you for the i forget who said it but thank you for the correction there um in terms of actual questions you know let me i think a lot of them are pretty i was happy to see a lot of them are pretty straightforward expansion of the universe kind of questions how we talk about the expansion of the universe how we derive it uh the predictions for it and so it's good because we live in our everyday lives in you know rooms and spaces that are not expanding so understanding what it means to say the universe is expanding can be a little bit tricky so one question is uh the big bang which actually let me let me just talk about the big bang because i didn't uh say as much as i could about it it was a pretty long video that we did so we have this thing where over time you have the scale factor remember the scale factor as a function of time tells you the relative distances between things and putting aside speculative ideas like inflation and so forth the basic curve looks like this okay i'm not going to set t equals 0 at the beginning because who knows when the beginning was but it's expanding and in the future we start accelerating so it looks like that but okay we're accelerating now we've recently started to accelerate so the point is there is a moment when a equals zero when the scale factor was zero okay and this is a prediction of classical general relativity right einstein in 1915-16 came up with this theory of gravity you can check out previous biggest ideas videos to see what i'm talking about general theory of relativity and as we said you plug in this idea of a homogeneous and isotropic universe a smooth universe that is uniform on large scales into the equations of general relativity you get some predictions and if the universe is made of ordinary stuff right matter and radiation okay so i said we're forgetting about inflation and stuff like that then there's a very strong prediction that at some point in the past the scale factor was zero a equals zero that's the big bang and it's a singularity much like the singularity at the center of black holes okay shouldn't say the center remember we talked about in the in the gravity video uh the singularity inside black holes is in the future once you're there the big bang singularity is in the past so it is very similar but also different in that that that one fact is a very crucial difference if people ever ask you you know could the universe be a black hole it's like the opposite of a black hole it's like a white hole there's a singularity in our past with the big bang not in our future okay and um stephen hawking actually his first major contribution to science was showing that it doesn't matter that we made all these approximations about the universe being perfectly smooth there is still going to be a singularity in the past in classical general relativity if the universe is made of ordinary stuff and what he did actually it was roger penrose who did you know the foundational work doing uh what are now called singularity theorems for black holes showing that if you crunch into the black hole you will hit a singularity in the future hawking basically ran the clock backwards and showed that the same thing is true for cosmology and the singularity is a place where the riemann curvature tensor or other ways of measuring the curvature of the universe blow up become infinitely big the curvature of space-time itself is infinite okay so what do we make of that what we make of that is as people have long said general relativity is not correct it is not correct at the big bang basically the prediction of general relativity is that general relativity itself breaks down and this is incredibly non-surprising from the point of view of anyone who cares about the real world rather than the particular theory if you care about general relativity then you're a little bit upset by this but if you care about the real world you know the general relativity is not the final answer we have quantum mechanics and general relativity is a classical theory so when general relativity is breaking down near a singularity what it's telling you is the approximation that we can treat everything classically and just do classical general relativity is no longer a good approximation so we should do quantum gravity or something like that we don't have the right theory of quantum gravity to plug in so we don't know what to do so when people say you know many of my fellow cosmologists like to say things like the big bang is you know a moment before which there are no other moments it is the beginning of the universe it didn't pop into existence from any pre-existing nothingness it is just the first moment asking what is before the big bang is like asking what is north of the north pole okay all that might be true but we don't know it might be false also because we don't have a theory that number one we trust and number two is applicable to that moment all we can say for confidence is that unless there's some exotic form of matter that gets you out of this like some repulsive manner that is important in the early universe all we can say for confidence is that general relativity doesn't say what happens general relativity predicts a singularity but general relativity is wrong so don't trust that okay so maybe the big bang is the beginning maybe it's not that's one thing i wanted to say about the big bang but the other thing is okay let's take that under consideration when we care about you know what the real world is doing but still let's figure out what general relativity is saying in these circumstances so that we have a picture in our minds of what some of the possible things are that could be going on so remember we said that for space it could be positively curved negatively curved or flat so space could be positively curved like a sphere flat like a table top or negatively curved like a saddle or a pringle plus curvature zero curvature negative curvature and if it's flat or negatively curved space could go on forever in any one of these three cases space could be finite and compact in the positively curved case space has to be compact one good question that was asked was how do we know that space has to be compact if it's positively curved so parenthetically you know i'm trying to fill in all the details here that we glossed over last time it's very hard or if not impossible to know something like that because what we know is that the size of the universe we know even this we don't know 100 but to pretty good confidence we know that the physical size of the universe is larger than the size of the observable universe okay so when we say the universe is compact under this theory we're really extending the theory that we have beyond what we observed so it might not be true but if space is truly uniform and spherically uh positively curved then we know it is indeed compact and that's not just because we draw this picture or we embed a sphere in three-dimensional space in our minds and look at it and say it's compact there are theorems in mathematics which say if you have a spatial geometry that has a certain curvature and that curvature is uniform what are the possible topologies that that space can have and so when i say if it's flat or negatively curved it can go on forever or it could be compact whereas if it's positively curved it needs to be compact that is a theorem of uh differential topology and uh we can actually sort of get a intuitive feeling for why that's true because remember you're not supposed to think about any of these as embedded in a bigger space you're supposed to think about living in them and when we say positively curved flat or negatively curved think about two light rays that initially start off parallel and ask what happens to them in a flat geometry they stay parallel forever negatively curve they go away positive curve they come together okay so the point is if you send out a bunch of test particles that are moving parallel to each other or even a little bit away from each other in a positively curved geometry if you just if you don't let it expand if you just like keep it fixed in size they will eventually come back right they will eventually hit each other that's not a proof but that is an intuitive demonstration of how we do think about the fact that positively curved uh manifolds have to close in on each other and be compact okay what i wanted to do was um that was a footnote what i wanted to do was talk about a couple of things the difference between what i really want to talk about is can the universe be infinitely big at the big bang okay so let's just talk about that i can we can come back to where we wanted to go um so look this is saying if you just take at face value everything that is literally on this uh tablet right now okay on the one hand space can be infinite if it's flat or negatively curved on the other hand the scale factor will hit zero at the big bang so people will say that the universe was small near the big bang but here you're telling me that space is infinitely big and space being infinitely big doesn't change as a qualitative fact even if the scale factor is smaller right if space is infinitely big and you shrink it by half it's still infinitely big so one thing to emphasize is that you know you have to keep in mind what when people are talking about the observable part of the universe and when they're talking about the whole universe extended beyond what we can observe so when people say you know resolutely and with confidence that the universe was smaller near the big bang that's okay they're not lying but what they mean is our observable part of the universe okay so remember we talked about co-moving volumes if you take all the matter and radiation universe today and just follow its evolution into the past it forms a volume and you can ask how big that was in the past the answer was it was very small like smaller than a centimeter right our whole observable universe today so outside our universe was the universe small near the big bang outside our observable universe uh no not necessarily and that's that's the important point so the in the big bang theory right in the big bang theory not the tv show but the cosmological model um what you can have if if curvature equals zero so a spatially flat cosmology what you're saying is that space time looks like this here's time here's space and there is an earliest moment okay there's an earliest moment of time t equals zero by the way let me emphasize let me go back to this discussion of uh the universe popping into existence because i was trying to say we don't know whether the big bang was the earliest moment of time but it could have been right that's a that's a possible thing so the other thing i want to say was there's a vocabulary problem when the universe has an earliest moment in time because of course none of us are familiar with that as a feature of the physical world we all came into existence when time already existed and will go out when time is still existing so we not we don't have the vocabulary to talk in ordinary plain spoken language about the big bang we are tempted to say that the big bang came from nothing right but that's misleading that's wrong i would go so far as to say it's just wrong because it makes it sound like there's something called nothing that transformed into the big bang okay but that's not the idea the idea which again they may not be true but the idea is that existence itself has a boundary at the big bang so that's why a careful cosmologist will say the big bang represents the earliest moment of time the moment of time before which there were no other moments and remember it's a moment of time it's not a point in space everything is uniform there's no location at which the big bang happened it's a moment and it didn't come from anything there wasn't anything before the big bang there was not space or time or existence or anything like that it's just saying that the universe has a boundary in time it has an earliest moment so the coming from nothing vocabulary is seductive and people use it but it's not what is being described at least according to classical general relativity now in quantum gravity maybe things are different it's very plausible that there is a meaningfulness you could attach to that phrase but we don't know what it is exactly right because we don't understand quantum gravity well all right there you go um here we're doing classical general relativity and saying look space can i don't know how to draw this how do you draw an infinitely big space the point is that there is space at some early time it expands and what we mean by expanding is the scale factor gets bigger but the scale factor is just the relative size of the universe at any one moment of time so if the universe is infinite infinite in all directions as freeman dyson said then it always was in the big bang model okay it's not true this is a very long-winded way of getting to this point if according if we live in an open universe a universe that is spatially infinite in extent then according to the big bang model there was never a moment of time in which the universe was small things were closer together things were the scale factor was small the scale factor went all the way to zero the density goes to infinity the curvature goes to infinity but the universe was always infinitely big and this is why to bring it back home it sort of is not smart to use a vocabulary of in the big bang theory the universe came into existence because you need to posit an infinitely big universe coming into existence okay now if you're really technical you could say wait a minute i shouldn't even be defining the size of the universe at the big bang because it's curved and the curvature is infinite and therefore the metric tensor which we talked about long ago doesn't make sense that's perfectly valid if you want to say that at the big bang itself i shouldn't be talking about sizes or anything like that that's fine i will totally let you get away with that but if you give me any instant of time after the big bang planck time you know one quadrillionth of a plank time whatever it is in classical general relativity the universe is infinitely big right away and if you don't like that well then you just don't like general relativity right okay it's probably if you don't like it because you're thinking that the universe should pop into existence somehow but that's not what it is the universe has a beginning it has a first moment that's what's going on that's what i wanted to say okay um good then there's a sort of slightly related thing when we talked about inflation i drew a picture that looked kind of like this so there was the [Music] expansion of the universe there's t there's the scale factor and then i said you know here's the traditional thing it goes like this big bang then inflation it would go like this quasi-exponential expansion so someone says wait a minute it looks from this picture like during inflation the yellow thing the universe was expanding more slowly than it was without inflation right like the slope here is less than the slope there but i thought that during inflation the universe expanded super super fast so how how is this compatible that's a very good question and there's a there's a few things going on here that are worth emphasizing so remember the friedman equation the friedman equation is uh alexander friedman's invention soviet physicist who said you can take this idea of and expand the universe plug it into einstein's equations get an equation for it it says that h squared equals eight pi g over three rho and i'm going to add a piece to this i didn't tell you about last time because last time i talked about just if the universe is flat let's add a little piece just so you know what it is uh oops it's i got to get it right otherwise i want to lose my cosmologist license k over a squared where k equals curvature of space so k is a number plus positive negative or zero depending on whether space is flat zero or uh is possibly curved flat or negatively curved so k equals zero the universe is flat that's why we could ignore it that's perfectly valid in fact let's ignore it for a little bit um and then remember what h is h is 1 over a d a dt okay so for one thing h is not the derivative of a h of the derivative of a divided by a okay so even though in this little picture here it looks like the the slope is very small for d a d t at early times uh the value of a is also small so during inflation let me make it in yellow here h is approximately constant that's the idea of exponential expansion right a goes as e to the ht where h is approximately constant so um even though d a d t even though the slope is small as you go to negative values of t or whatever the hubble constant is not any less in fact let's look at this equation let's set k equals zero okay um so this is in other words a o a dot which is d a d t over a a dot is defined to be d a dt the slope of a so you have to remember that it's that ratio that gives you h so when k equals zero let's make our lives easy then h squared equals eight pi g over three rho and rho is the energy density the you know we're taking to be uniform everywhere through space and this is a number greater than zero like strictly greater than zero it's never equal to zero without curvature okay um curvature or the cosmological constant are the two things that can make h be zero on the right hand side of the friedman equation but without that if you're just in a flat universe this is what you get okay so h squared is never um zero if k equals zero and and also lambda equals zero that's the cosmological constant because the reason for that is ordinary energy densities rho right what is the number of ergs per cubic centimeter in photons or protons and neutrons or in dark matter whatever real energy densities are never less than zero okay but the cosmological constant the vacuum energy can be either positive or negative so the two ways to get a negative number on the right hand side of the friedman equation are either to have a cosmological constant be negative or to have positive spatial curvature if k is greater than zero since it comes in with a minus sign you can get a negative term on the right hand side so h is never zero in these in this simple case which means that the universe has to be expanding and i mean this is obvious i won't even write it down but h is always bigger when rho is bigger right so in the past um when rho the energy density in space the density of stuff was bigger h was always bigger in fact if you look really closely you can convince yourself h monotonically decreases so if we plot the hubble constant versus time or versus the scale factor clean that up a bit it never goes up in this kind of cosmology it can only go down then the details of how it goes down depend on what's going on and in inflation this is still true we have ordinary matter and radiation so i'm going to go into a little bit more detail about what's going on in inflation i'm going to draw this pictures that look exactly the same but mean completely different things over and over again so this was the hubble constant versus time let me plot the thing that drives inflation is some scalar field again we don't know if inflation is true but we invent a field called the inflaton which serves as the source of energy density in the early universe we call it phi cleverly enough actually the vertical axis here is v of phi and this is phi so we have some potential for phi okay so phi is a scalar field it has some potential energy the energy density in phi is one half times phi dot squared the time derivative of i i'll be nice to you if you're not familiar with this notation and write it out d phi dt i love how now you are familiar with that notation because you learn calculus in uh the second biggest ideas video plus vfi so there's an energy density that the field has because it's rolling right it's moving down the potential and there's an energy density that gets just from having a potential right there's some value of the potential energy and so rho for the inflaton is the sum of two terms but they're both positive right v of phi is uh the potential energy is positive and one half d phi dt squared is manifestly bigger than or equal to zero so the energy density during inflation is uh greater than zero and in fact you can do a little bit more work to show that the energy density only ever decreases so even during inflation uh the hubble constant only ever decreases so inflation is not something where you start with a non-expanding universe and you start it going right that you could invent a cosmology like that and people have but that's not the usual inflationary story the usual inflationary stories of the universe for some reason starts like this in other words it starts with an energy density a super duper high energy density in this scalar field which slowly decreases and if you want to ask why did the universe start that way with that energy density well good you should be asking that question nobody knows is the answer maybe stephen hawking knew or thought he knew um but then the energy density goes down and what why inflation works is you can say well if d phi dt if the rate at which the scalar field is rolling down the hill is small then phi the scalar field itself is approximately constant d phi dt squared is approximately zero v of phi is therefore approximately constant because it's just a function of phi and phi isn't changing therefore rho is approximately constant because i don't know what happened there because rho is the sum of zero this one is zero and this one is a constant and then you get a constant energy density therefore h is approximately constant because h is proportional to rho here it is right there you might say well what if there is curvature or something like that but this curvature term k over a squared if you start with some constant energy density in rho and also k over a squared well as a increases k over a squared goes down so even if you had some k over a squared whether k was positive or negative doesn't matter k over a squared is going to diminish away compared to the constant rho if you have a row which is approximately constant and in inflation you do okay so even though the point of all this i have these very long-winded ways of saying very simple things the point of all this is in inflation inflation does not ignite the expansion of the universe from zero to expanding to beat the band inflation imagines that the universe always had a high energy density or at least at some beginning point which is ill-defined had a high energy density and the energy density always ever decreases during inflation i didn't prove this to you but if you go through the equations of motion for the inflaton field um this combination of energy uh d phi dt squared plus v of phi never increases in an expanding universe in a contracting universe it could increase but as the universe is expanding the energy density of the inflaton always goes down therefore h always goes down even during inflation and therefore uh you know the the last thing to say is if you plug in rho equals constant h equals constant h equals a dot over a we actually inferred this uh last time where we mentioned it if h is approximately constant then a is proportional to e to the ht and you get this exponential expansion okay so just trying to clear up some misconceptions here inflation is not the beginning of the expansion of the universe and we don't really know what happened at the beginning of inflation or what put the inflaton up there on his potential um should i yeah so that i have more to say about that should i say it now is the question why don't i say it now um why do we talk about inflation at all wow i'm getting very far away from my outline but that's okay we're all friends by now we've spent hours of our lives together uh you'll put up with me so the thing about inflation is it was invented to it to explain two puzzles that uh well three puzzles in fact that alan guth identified all of them had been identified by other people before but guth uh actually proposed a solution to them for the first time so he he popularized them the history of inflation is very interesting there were other people who had similar ideas other than goods but they didn't quite pinpoint the usefulness of them or the mechanism to make them happen in a convincing way so his paper was really the one that got everyone paying attention to the idea and the puzzles that were being solved hadn't really been bothering people that much it was one of those things where ah you could see what these puzzles are and go on your way because they were not you know in some sense they were not like direct conflicts between theory and experiment or theory and observation right they were sort of fine-tuning naturalness problems and these days decades later uh that's all we have so we take those very very seriously but you know in the 1970s and 80s guth publishes paper in 80 81 i think um there we had like real experimental puzzles to deal with so these fine-tuning puzzles were not as as taken seriously until he proposed the answer to them in which case it was so beautiful and elegant that people suddenly said oh yes these are very very serious problems anyway the puzzles of conventional inflationary cosmology one was the monopole problem the monopole problem was a very straightforward thing we talked earlier about um topological defects remember we had a whole video on topology and geometry i say this knowing that not all of you watch the video because i see the viewership statistics but you should watch that one and what you find is that uh depending on how gauge theories get broken we also talked about this in the gauge theory video so we talked about things in the topology video which we then put to use in the gauge theory video where we said look there could be symmetry breaking and the symmetry breaking can lead to topological defects in ordinary physics as we know it electric charges come in either pluses or minuses so they're up there's there's two different electric charges they're completely independent of each other but in magnetism every magnet has both a north pole and a south pole so every magnet in the real world that we know about in the world of experiment is bipolar in some sense dipolar yeah dipolar i guess would be a technical term bipolar means something else but then you can hypothesize could there be a magnetic monopole could there be just a north pole if you take an ordinary magnet which is a north pole and a south pole so ordinarily here's a magnet north pole south pole you break it in two so you do that well then this becomes a south pole and that becomes a north pole and so you don't actually get let me do that prettier sorry a little quick there um if you break it into two pieces the north pole and the south pole stay where they were but this end becomes a south pole of its own this end becomes a north pole and there's sort of magnetic field lines stretching between them so you never get in ordinary everyday physics a magnetic field a magnetic monopole which is just north or just south so nevertheless there are theories like grand unified theories that predict that there should be magnetic monopoles because of the symmetry breaking pattern and the topology of the gauge fields and a bunch of people including my caltech colleague john prescott and his youth uh noticed that not only you know we talked about in the cosmology video and the one this is q a for we talked about relics that are created out of thermal equilibrium okay like axion particles or something like that all very hypothetical well magnetic monopoles could be produced in the early universe and some of them would still survive to today and you might say okay well that's probably true but they're probably rare and hard to find no in fact what you prove very pretty quickly is that uh the universe should be almost all monopoles in ordinary grand unified theories the energy density of monopoles should be orders of magnitude higher than the energy density we observe in the universe today so there has to be something going wrong with that now there's an obvious solution to this problem namely grand unified theories are not correct and grand unified theories remember made the prediction that protons should decay they should not live forever and people looked for proton decay and didn't find it so very plausibly that's the actual correct solution to this problem so this is not a problem with cosmology this the monopole problem is a problem that is at the intersection of cosmology and a particular set of ideas for fundamental particle physics namely grand unified theories that's what alan guth was working on as a postdoc at stanford when he was thinking about this he was thinking about gage theories and magnetic monopoles and he was a latecomer to cosmology so this is actually what motivated him and then he went to a talk i think it was by peebles but uh or maybe by bob dickey yeah by bob dickey so bob dickey and jim peebles two very famous cosmologists both of whom were at princeton and they collaborated on you know uh a paper about conundrums and puzzles in cosmology and so they point out these other two problems one is called the flatness problem and uh guth actually sat in the audience for a lecture that dickie gave at cornell where guth was also a postdoc prior to stanford and he heard about this problem and that sort of settled in the back of his mind it's a good lesson that you can actually learn things by going to lectures there you go so what is the flatness problem think about the friedman equation h squared equals 8 pi g over 3 rho minus k over a squared okay and i said you know if you have if rho is constant then the k over a squared term dilutes away and becomes unimportant but for ordinary matter and radiation so you know think about what people were thinking of in 1977 all right so they weren't thinking about scalar fields filling the universe they were not thinking about the cosmological constant they were thinking about particles and photons matter and radiation so for matter so we can rewrite this as 8 pi g over 3. rho matter has some value when a equals one and then it goes like a to the minus three okay so i'm writing the energy entity in matter which is dependent on the scale factor as a constant times a to the minus three because the number of particles just dilutes away as the scale factor gets bigger and the volume goes as a cubed radiation we talked about rho radiation starts at some fiducial value and then goes as a to the minus 4 because the energy density and radiation is the number density goes down as a cubed and then the energy per particle goes down as as a because the wavelength gets stretched so then minus k a to the minus two okay so this is what you would have been thinking of as a cosmologist in the mid 70s you would think there's matter there's radiation there could be curvature we don't know and so there's no term in here that is constant and so if you plot these puppies as a function of the scale factor rho i for matter radiation and call this call this term row curvature rho sub k it's not an energy density but it enters the friedman equation in a way very similarly to an energy density what you find is of course there's a to the minus three a to the minus four a to the minus two so a to the minus four goes down very quickly that's radiation matter goes down somewhat more slowly and curvature goes down the least slow at all so it's the opposite story the story that we told for inflation was if you have a constant energy density curvature gets diluted away if you don't have a constant energy density if you have curvature matter and radiation it's matter and radiation that get diluted away and curvature always wins okay so if you didn't know if you didn't know what the initial curvature of the universe was if you thought for whatever reason and maybe this for whatever reason should be interrogated more carefully but if you thought you know i don't know there's some matter there's some radiation there's some curvature just start at some random values and let them go what you would predict because the scale factor has increased by many orders of magnitude what you would not maybe not predict but expect naturally is that the universe should be all curvature it should be wildly curved the curvature of space should be big much bigger the matter density or the radiation density in the friedman equation okay um it's not right even in the 70s we knew the universe was not really really curved euclidean geometry works pretty well on your tabletop right i mean the amount of curvature we're talking here would be noticeable in your everyday life and that's not true so something went wrong there so the in the conventional cosmology um the answer to this well the only way to answer this is to say you just start the universe with k really really really really small so that even if it grows faster than matter and radiation it still hasn't caught up to them today or only just started catching up to them today that's the flatness problem the finest problem is our universe looks pretty flat you'd expect it to look really curved why was the curvature so small at early times and then there is the uh the horizon problem and the horizon problem comes from the fact that you know you just look at the universe with the cosmic microwave background in the 1970s they had they detected the cosmic ray background in the 60s they had not yet detected the temperature fluctuations from place to place so as far as they knew the microwave background looked perfectly smooth everyone knew it wasn't perfectly smooth but the precision of the experiments to the day so it said that as far as they could tell it looked smooth and that's an issue because here's another space-time diagram here's time going up okay and here we are right now here's us as observers okay here's the past let's say that this is the big bang so this is well this is space and this is t equals zero the big bang let's say here is the cosmic microwave background right remember 380 000 years after the big bang and this is a space-time diagram so i should be consistent about the dashed lines so we can draw light cones backward in time and what you notice here is that even though space can be infinitely big you're only looking at a certain part of space at any one time okay there's only a certain part of the universe that can possibly influence what happens to you so an event let's put this in colors an event here trying to draw a star and failing an event that happens there i don't know what kind of event happened before recombination but maybe it would so the causal influence can certainly propagate upward in the universe and affect you but something that happens out here it just can't get you even if it moves at the speed of light it is outside your light cone okay so nothing that happens outside your light cone can possibly affect you and the thing that is new about cosmology is that there's a beginning right in flat space in minkowski space your light cones traced into the past just get bigger and bigger and bigger you get more and more of the past of the universe could possibly influence you but here in cosmology they are cut off by the fact that the universe had a beginning so if you look at the points where your vision your looking out at the universe and looking at the cosmic microwave background so the points where your past light cone intersects the moment of recombination the moment where the microwave background was formed in some sense when the universe became transparent we can ask well what about the light cones of those points so look at the sky see the cosmic microwave background ask what it would be like to be living at that point in the microwave when the microwave background was formed when the universe was becoming transparent when the electrons were recombining with protons and helium nuclei well they have a horizon also well they have a past light cone okay so you can call this distance here the horizon for this point right there whereas this whole thing is the horizon for us right there so the horizon in this case is take your location in space time trace your light cones back to the big bang okay that defines a region of space at or very very very very close to the big bang call that your horizon and what you see is if you call this point a and point b right two different points that you're looking at in the cosmic microwave background the horizons of a and b so this is horizon sub a this is horizon sub b they're non-overlapping there's nothing in literally nothing in the universe that affects what happens at point a and also affects what happens at point b their past light cones are non-overlapping again in minkowski space in a non-expanding universe you would just keep tracing those light cones back and eventually they would intersect each other but here that's not true the big bang is the beginning it's the boundary to space and time okay under this theory we haven't put inflation yet that was the idea the horizon problem is a problem inflation's supposed to solve it so the the specific problem is on the one hand when you look at point a and b these are two points in the cosmic microwave background they're the same temperature right they are 2.7 degrees kelvin and that temperature is set by the initial conditions of that part of the universe its density its expansion rate number of photons number of electrons all that stuff that's true for both point a and point b they have the same temperature but in the conventional big bang model they have non-overlapping pasts nothing was uh causally influencing both of them now you could say that i have a theory of cosmology where you know stephen hawking or god or the wave function of the universe sets up initial conditions at the universe at the beginning of the universe all over space instantaneously okay because remember it could be infinitely big um well you know good for you that's great good glad you have a theory like that but we have no idea why that should be the case or whether that theory is true so the horizon problem the point of the horizon problem is that in the absence of a specific predictive theory of what was happening at the big bang we have no reason no understanding of why the temperature in two very distant regions of the cosmic microwave background should be the same how do they know to be the same even though they have no causal influence over each other and in fact you can ask you how far apart on the sky do you have to look to get two points which are causally disconnected in this way and the answer about one degree on the sky so you can go 360 times around and get causally non-overlapping regions of the cosmic microwave sky that's the horizon problem okay and so inflation solves all of these and i'm not going to go into detail well sorry inflation says it solves all of these um we kind of already know how it solves the flatness problem because it dilutes away the curvature right if you start with a constant energy density that energy density remains constant while the curvature goes away k over a squared degree decreases while the energy density remains constant and then at the end of inflation what happens we're supposed to imagine is there's sort of a drop off here at the end of inflation and at that point all the energy density in the inflaton converts into ordinary matter and radiation it does not convert into curvature the inflaton is a field made of stuff and that stuff decays into other stuff that we know and love okay so you create big matter and radiation densities you do not create big curvature and that explains why the universe is flat for the horizon problem roughly speaking what inflation does is say well actually the universe had been expanding a lot longer than that and i say roughly speaking because it's not in time that it had been uh spending a lot longer than that but it is expanded by a lot more than that so these light cones that you were tracing backward are you know deserve to be traced backward a lot more and in fact they do have an overlapping past so i hope no professional cosmologists are listening here because i'm going to air the dirty laundry of cosmology both the horizon problem and the flatness problem are entirely bs all right i know that everyone loves them and i used to think they were great great in the sense that they are good motivations for inflation if you read my general relativity book there they are presented with all the earnest sincerity that a good cosmologist can muster but they're entirely bogus there are problems and there are problems that inflation addresses they're just not these problems okay and the reason why very very briefly is the flatness problem is is entirely bogus because look you said you started with this vague statement about you know maybe the matter density the radiation density the curvature maybe they're all roughly equal to each other and then what you need is that in fact the curvature is much much much smaller in order to be consistent with the data today and then the implicit step is how likely is that seems very very unlikely that the curvature of the universe would be very very small right the thing is we have a measure we talked about probability and measures a little bit there is a measure defined by general relativity on the space of initial conditions and in fact in that measure which maybe you don't want to use but it's literally the only one that is predicted by the theory in that measure the universe is overwhelmingly likely to be flat it is overwhelmingly likely to have an infinitesimally tiny curvature in other words if you pick a universe randomly out of a box probably that was not how our universe was made but if you imagine doing something like that you would not imagine getting equal amounts of matter radiation in curvature you would imagine getting much more matter radiation than curvature so the flatness problem isn't a problem at all it's just the most natural thing in the world for the universe to be flat the horizon problem is a little bit trickier because in the horizon problem also has has a step in it that is a bit of a cheat okay the horizon problem says look there are these two points uh they're causally disconnected from each other there is no reason uh that there's no thing that ever happened in history the universe that set them to synchronize in some way so why should they be at the same temperature and then if you have inflation then they do have an overlapping causal past so they can be set to the same temperature right you saw what happened there i didn't say they are set to the same temperature i just said they can be and the point is that again i'm not saying that inflation doesn't solve problems i'm just saying this isn't the way to say what the problems are if you didn't have inflation if you just for some reason let the universe expand very very slowly for a long time you could also expand these light cones so they would overlap in the past but you wouldn't expect these different parts of the universe to equilibrate the universe should become lumpier we talked about this when we're talking about entropy right the universe should grow under gravitational instability its differences in matter density from place to place so we have an intuition from like an ordinary box of gas where if you start a box of gas with cold on one side hot on one side they will even out right but that's not what happens in cosmology they don't even out because the gravity is important so they become even more lumpier so there's a sleight of hand being pulled here in the horizon problem of course inflation is not like a box of gas either so once you start inflation then there is a reason why the two different regions of the universe become similar to each other and the reason why that cosmic micro background is homogeneous so inflation does what it purports to do it makes the universe spatially flat homogeneous approximately the same everywhere it predicts the kind of universe we actually live in but this motivational pep talk about the horizon and flatness problem much less the monopole problem is a little bit off base the just to drive that home a little bit think about entropy okay we said that in the entropy video we said that there is this issue of the past hypothesis okay why is the entropy of the early universe so small so if you think about our universe today so here's today and think about some earlier time let's think about um i don't know near the think about near the cmp near the recombination okay the universe was smaller but the reason why that's a perfectly good place to think about is because there were no black holes or anything like that i mean the universe was pretty much smooth near the cmb the the variations in density from place to place were one part in a hundred thousand ten to the minus five very very tiny so this is a co-moving volume okay this is a region of space that has moved that is its definition of its boundaries is moving along with the matter in the universe and the entropy at this time cmb this means not the entropy of the cmb but the entropy of our observable co-moving patch of universe at the time the cmb was formed is about 10 to the 88th we know that because it's a box of gas it really is just a bunch of particles and we know from the 1800s formulas for calculating the entropy of bunches of particles today the entropy of the universe is much higher why because gravitational instability has had time to work it's made black holes uh i said that the black hole the center of our galaxy all by itself has an entropy of something like 10 to the 90th bigger than the whole universe at the cmb time so the entropy has gone up um i forget actually what the number is today but it's something like 10 to the 100 and well it's over 10 to the 100 let's let's put it that way it's gone up by a lot okay 10 to 103 maybe i'm forgetting the number i used to know this number off the top of my head but anyway it's gone up that's what matters and the point is if you trace this let's say this is what we know and then you trace backwards okay into what we don't know well if you just did the conventional big bang right so s ordinary big bang obb ordinary big bang it's still about 10 to the 88th right and it's still just a box of gas with a bunch of particles in it maybe slightly fewer particles maybe it's 10 to 87 or something like that in fact actually it is smaller than that maybe 10 to 86 there are fewer particles is that right yeah something like that entropy's only going to go up but um okay entropy went up and there's still a long way for it to go you know the entropy that you could get if you took all the matter in our universe and put in a black hole as roger penrose has emphasized is over 10 to 120. so 10 120 is much bigger than 10 to the 103 but 10 to 103 is much bigger than 10 to the 86 so there's a huge problem why was the entropy of the early universe so small and inflation is supposed to be a theory that tells you that makes the initial conditions of the universe natural but inflation says well the reason why the entropy of the universe was small then is because it started with a little inflationary patch with an entropy of order 10 or something like that you know 1 or 10 to the 10 to the 3 or something like that it's an incredibly tiny number that's what you need to have this little remember we talked about what happens when you start inflation you have a scalar field and it has energy density which is its time derivative and its potential you notice we did not have a spatial derivative d phi dx okay where x is little vector there was no energy from the variations in the field from place to place in space that was an assumption that there wasn't any why well because if there were inflation wouldn't start that energy density would dilute away very very quickly and you not have this constant push to the expansion of the universe so you assume that there isn't that much as we say gradient energy at early times that assumption i'm skipping some steps but that assumption is equivalent to saying that the entropy of the early universe was super duper small so inflation explains the low entropy of the early universe by saying that there was an even lower entropy at earlier times which is obviously cheating that is not okay right um this is why i am personally ambivalent about inflation on the one hand it's good it solves a lot it explains let's put it this way forget about words like solves and explains it predicts that the universe should look like the universe we see that's why inflation is good that's one reason the other reason is to start inflation it needs to be very this is a very small amount of space you know the in ordinary big bang you know the early universe that corresponds to our observable universe today was i forget but it's you know millimeters or micrometers something like that which is a small size compared to the universe today but enormously big compared to particle physics scales whereas in inflation the universe could start as small as the planck scale and then expand so inflation provides in principle a bridge from a true theory of quantum gravity that explains what the universe was doing when it's at the planck scale to the observable universe with the microwave background and all that stuff and via that bridge it predicts the universe should look like the universe we see that's why i think it's an attractive theory i just don't think it actually solves any of the puzzles that it was purported to solve either because it doesn't or because those puzzles are not really puzzles right the real puzzle is the entropy puzzle and it totally boots that one it totally just assumes that the entropy of the universe was small what can i say all right that's my little rant about inflation for today let's see we have some other questions those are those are the major ones i want to get to but we have some other fun ones uh one was why does the why does general relativity predict that the universe is either expanding or contracting remember that's what got einstein hot and bothered when he realized oh no my theory is incompatible with the data so remember the friedman equation i'm going to write it again so i don't have to keep referring back 8 pi g over 3 rho minus k over a squared so we're putting the curvature in um well look the prediction that the universe is either expanding or contracting is just a prediction that h is not equal to zero right if you don't have the curvature term so let's say that you just erase this right if there's no curvature if you're in a flat universe then clearly the universe has to be expanding or contracting h is a non-zero number so h because it's h squared h could be positive or negative right so he could either be expanding or contracting but it's definitely not zero if you don't have the cosmological constant because he didn't invent it yet in 1917. so you could put back in the curvature term right and then you could say okay why don't i balance it why don't i say that k is greater than zero that's squared sorry k is greater than zero um rho is some number and they exactly cancel maybe that's what it is but then you have to say okay that's perfectly okay but then you have to look at the next there's another equation that we weren't looking at before okay the point is h is one over oops let's write it as a dot over a so that depends on the first derivative of a but there is another equation for the second derivative of a and what you find if you plug in i'm not showing you what that equation is it's a mess but if you set the universe to be exactly h equals zero by balancing rho and k the second derivative of the scale factor with time is not zero so if you truly have a universe that is not expanding so if you think that what you're getting is as a function of time here's the scale factor if you think you're getting this okay um that's not compatible because that would be a solution where the scale factor has 0 first derivative 0 second derivative zero third derivative it's literally not changing instead what einstein's equation by the friedman equation predicts is you are at the peak of a universe that is expanding and then contracting again okay so that's what you would actually live in so there's no universe in general relativity with the with the ingredients of curvature matter and radiation no cosmological constant that could just be constant and stay constant that's not an allowed solution um so that's why einstein had to invent the cosmological constant once he gets uh some once he gets a universe where there is a cosmological constant then you can balance everything out again i'm not going to go to the details but that becomes possible but that's where the prediction comes from which is related to the question of you know is there a relationship between the fate of the universe and the geometry of the universe there is but kind of it becomes messy and this is probably less important for me to explain now than it would have been 20 years ago but when i was your age we were taught that there was an ironclad relationship between the geometry of space positively curved flat or negatively curved and the future of the universe because if you have h squared equals 8 pi g over 3 it's getting sloppy here over 3 minus k over a squared if that's your equation oops rho sorry and there's no cosmological constant lambda equals zero then you can actually show it's pretty easy to show that as time goes on and the scale factor goes on there's two possibilities um well let's put it this way there is here's what would happen if the universe let's just say that rho is matter okay row matter is proportional to a to the minus three you can put in radiation but it doesn't really matter as long as we don't put in the cosmological constant so here is the universe where k equals zero then a is proportional to t to the two thirds this is my attempt to draw t to the two thirds if k is a negative number then the universe will expand forever also so t to the two thirds expands forever k is less than zero the universe also expands forever and whenever k is greater than 0 look at what happens if k is greater than 0 the matter dilutes away as the universe expands the curvature dilutes away also but more slowly right so if the curvature is small initially the matter is what is matter what matters as it were in the universe um that's good but eventually it will catch up so eventually this k over a squared term which is negative for k equals greater greater than zero this k over a squared term will exactly balance the matter term eventually if you wait long enough so positively curved universes look like this k greater than zero and so there is a relationship positively curved universes are finite in time negatively curved universes can last forever okay um sadly so that was a very nice thing because when people thought they were measuring the spatial geometry of the universe they said we're measuring the future of the universe we're going to tell you whether it expands forever or not um once they realized in 1980 1998 that the cosmos constant was greater than zero they had to rewrite all those statements because i mean they should have known all along and the best ones did know all along but the cosmological constant changes this set of predictions then if you a better way to plot it let me see if i can actually do this correctly [Music] the parameter space which we might want to draw is well i need to tell you what omega is don't i well okay i will here is omega matter here is omega lambda so omega is the density parameter of the universe and we're imagining once again a universe that is just matter and cosmological constant now no radiation doesn't change anything qualitatively omega i is 8 pi g over 3 rho over h squared so we take the friedman equation and we divide both sides by h squared okay so we get something that is normalized in some nice way and so h so in other words this with this definition we get 1 equals omega minus k over a squared h squared okay so you see that instantly you can get k equals a squared h squared times uh let me get it right omega minus 1. so there's instantly a relationship between the density density curvature parameter density parameter density parameter and the curvature namely if omega is greater than one then k is a positive number if omega is less than one then k is an is a negative number so omega even though it's a measure of density if you believe the friedman equation instantly tells you the curvature of space that's why it's called that's why it's a useful way of parameterizing the density and so if you have matter and the cosmological constant what we what we care about what you thought you cared about is is omega total greater than one right that tells you the spatial curvature so here's the line omega total equals omega matter plus omega lambda equals one right so here for omega greater than one k is greater than zero this way k is less than zero on the line k equals zero so the way to think about this is the the new thing that enters with the cosmological constant is go back to the k greater than zero case with no cosmological constant okay if you think about these two terms that are on the right hand side of the friedman equation what's going on the matter is always expanding the universe right i mean the way to think about this in terms of einstein's equation is energy density sources the curvature of space-time and the expansion of space-time the expansion of space is one version of the curvature of space-time one way in which space-time can be curved the spatial curvature is another way it sort of divides up that way in cosmology so rho is positive therefore h squared kind of wants to be positive right but if k is greater than zero k over a squared goes away more slowly than k over then something over a cubed like the matter density does so as the universe expands this term will always start to dominate and this term wants to cancel off the push that the energy density of the universe is getting from that the expansion rate of the universe is getting from matter so eventually these two terms will be of the same size and they will say h equals zero and that traces out this path that's the zero point and then it will recollapse okay so the basic mechanism is that k over a squared the curvature always goes away more slowly than the matter but when you have a cosmological constant the cosmological constant if it's positive wants to make the universe expand positive energy density and i know this is a little bit counterintuitive positive matter let me let's put it this way matter like galaxies dark matter slows the expansion of the universe down but at the same time make sure the universe is still expanding right so that's why we talk about the deceleration of the universe if we didn't know about the cosmological constant the universe would expand but ever more slowly and you see that in the a goes as t to the two cubed two-thirds uh curve so there's these two effects going on okay rho slows down the expansion the universe because galaxies pull on each other but also it can't be zero if it's just matter because h squared is proportional to rho as long as there is energy density and we don't have a curvature or a negative energy density the universe has to expand a negative cosmological constant okay can make the universe recollapse a positive cosmological constant though wants to push it so matter goes away faster than the curvature so if matter is positive and curvature is negative they can cancel but if you have a positive cosmological constant that goes away even more slowly than matter than curvature does because it doesn't go away at all positive cause marshall constant will always win at the end of the day and so in this little diagram here what you get is let's get a different color for k less than zero i should i should have said this more explicitly before it's easy to know what happens when k is less than zero without the cosmological constant the universe wants to expand forever if there's a negative cosmological constant that will always eventually win because it doesn't dilute away so the inverse expands and expands gets rid of all the matter all the curvature and all you have is a negative cosmological constant which forces the universe to re-collapse so we can draw the dividing line like this so this is uh i can just draw keep drawing it this way so these are universes that expand forever and these are universes that re-collapse okay so for on this line where omega lambda equals zero then it's just is omega matter greater than or equal to one um when omega lambda is not zero when there's any cosmological constant a negative cosmological constant always makes the universe re-collapse what about when omega matter is greater than one that means that without the cosmos constant the universe would be positively curved and would recollapse okay well if it's just right there's a little window in there the universe gets big big enough that the you might think it's going to re-collapse but the cosmological constant becomes important so if there's a positive cosmological constant a positive cosmological constant wants the universe to expand forever the only way to stop that from happening is if the matter is big enough that it causes the universe to recollapse before the cosmological constant can take over so in this plot you see there's a region that kind of looks like this okay so in this little wedge down there even though there's a positive cosmological constant you don't have enough time for it to matter the universe recollapses so this is still the re-collapsing and this is still the expand forever so but notice that this to the top side of the diagonal line let's let's make the diagonal line different colors so it kind of stands out here's the diagonal line this is the spatial curvature line right so these are negatively curved universes below the line and some of them re-collapse but they will expand forever if they have a positive cosmological constant sorry that's obvious when curvature is negative they want to expand forever they will still do that if the cosmos constant is positive but if the hospital constant is negative even a negatively curved universe will recollapse whereas if the curvature is positive it depends on the details of exactly the relative amount of cosmological constant to matter so the point is the very long way away once again of saying that things become complicated once you allow there to be both matter and cosmos constant the simple relationship between the fate of the universe and its geometry its spatial geometry is no longer there okay um yes i want to talk about energy conservation because people did mention this and this is a thing uh i mentioned it but let me mention say one more thing about it entropy i always think about entropy energy conservation remember i said that if you have a box of universe here's space but i'm sort of making it look three-dimensional now it's going to expand as before i'm going to move this whole thing down to give me more room so it expands you have more space so if the universe has nothing in it but matter particles slowly moving particles then in that expanding region the energy density goes the number of particles that remains constant the energy remains constant but so rho let me put it this way um let's define the energy in the box let me move this box a little bit here there's something i can define which maybe might not be a good thing to define if you're a professional cosmologist but it gets intuitively to what's going on here i will define the energy inside the box as the integral over the volume of the box of rho the energy density right so i just add up inside the box the energy density and then as the universe expands uh for matter ebox equals constant for radiation the number of radiation particles remains the same but the energy per particle goes down so ebox decreases and for vacuum energy the energy per cubic centimeter remains the same the total number of cubic centimeters goes up so e-box increases and the question is um is there some way that we can describe this as the energy going into the gravitational field right can we can we somehow rescue conservation of energy by saying that really what we're doing here is talking about the energy matter and radiation uh and vacuum but there's also the energy of the gravitational field and the answer there is you know not really honestly like i wish there were but it's not quite like that if you follow the structure of the equations of general relativity there's no natural way to associate uh with anything that appears in those equations the energy of a box including the gravitational field and everything else so that it remains constant but i think that's okay like i don't think it should bother you people tried and you know in other circumstances it kind of works right you can sort of fudge your way into it but i think there's a different way of thinking about it the way of thinking about it is there's still an equation right the equation used to be the equation of energy conservation in non-expanding universes just says d e d t equals zero right where it you know it doesn't change over time the energy of a closed system so this is before gr came along but now we have an expanding universe and it's not that the equation just disappears it goes to d e dt is some function of the kinds of matter we have and what space-time is doing how the universe is expanding so general relativity predicts a very specific equation for how these things change and you you know that right as the universe expands the vacuum energy is proportional to the volume the radiation density goes down as the scale factor goes up it's exactly like that it's not like all craziness is broken loose there's something very very definite that happens so this is my particular way of doing it you know the reason why this is worth saying even though it's kind of a fuzzy conclusion because different cosmologists like to use different words for this um is that it's a lesson that different cosmologists like to use different words for this the words aren't what matter right all the cosmologists who will slightly disagree about is energy conserved or not in expanding universe they all agree on the equations they have zero disagreement about physically what is happening the only disagreement is as to what is most convenient to attach to equations and words which was the most convenient way to attach words to the equations okay that is something that people can disagree about and the lesson that i'm trying to get across is that's fine like who cares about that you know i don't even say that people who disagree about this particular vocabulary are wrong i just say they're just using different vocabulary words to express what is going on okay so that's a good lesson to end on here i think that uh you know there's many more things to say there are a lot of good questions a lot of good things to talk about oh wait yeah no there's one one thing i just got to talk about sorry you were hopeful that we're going to end but there's one more thing to talk about uh is brooklyn expanding this of course is uh a illusion reference to a scene in annie hall the woody hall woody allen movie where young uh woody allen is at the therapist's office and he's uh full of anxiety because the universe is expanding and his mother tells you what do you care the universe is expanding brooklyn is not expanding but people wonder about that they say wait a minute um the universe is expanding but pretty slowly right you know the universe the expansion the universe is not a speed it's not a velocity any one galaxy has a velocity but hubble's constant is velocity equals distance times the hubble constant so for different distances galaxies move at different velocities the best way to think about the hubble constant is as a rate you can think about you know the hubble constant h in the real world there's a controversy over what the value of h is today h naught but roughly speaking it's approximately 70 in units of kilometers per second per megaparsec now these are silly units remember when we talked about units uh earlier what this means is of course we have velocity equals hubble constant times distance so this is measured in kilometers per second distance is measured in megaparsecs so the hubble constant converts from distance to velocity and that's why it's in these weird units but these units are distance divided by time divided by distance so this has units of one over time it does not have units of velocity that's crucially important so these are convenient units to use but the actual physical thing going on is one over time because the hubble constant is basically saying how much time does it take for the universe to double in size okay so we talk about the hubble time t h which is just 1 over the hubble constant i am setting all c equal one things like that but i think in this case c the speed of light doesn't even appear and this is about you know 10 billion years right 10 to the 10 years roughly speaking so what this is saying is it takes almost 10 billion years for the universe to double in size and so in one year or 10 years the universe doesn't expand by very much so you might very well ask [Music] is it possible that brooklyn is expanding or at least the solar system brooklyn is held together by the structural forces of the earth right and you know the ground and the buildings and so forth but less poetically is the solar system expanding because planets are not bound by material objects to the sun right is the earth slowly moving away from the sun because of the expansion of the universe and we just haven't noticed because the expansion of the universe is so small so the answer is no that's the answer the answer is not it's expanding very slowly it's not expanding at all or rather it may be doing all sorts of weird things you know the solar system is buffeted by the gravitational fields of other stars and gravitational waves passing through so there's always slight deviations in the distances between the planets and the sun but it is not overall either expanding or contracting at all okay it is hard for me to explain that i've two ways of explaining it neither one of which have historically been very effective but i'll give them both of you the first one is just in words um you can think of here not just in words i'll draw a picture so here's time once again space okay and this is the expansion of the universe but rather than drawing the parallelogram to be all space let me draw some particles okay and as the universe expands the particles move further apart from each other this is what it means to be an expanding universe right not the prettiest diagram ever but there you go and you can think of this as sort of just inertial motion given some initial conditions okay uh it is almost kind of halfway analogous to take a baseball and throw it into the sky throw it very very fast you know that if you throw an object away from the earth uh for realistic amounts of velocity that you can give the baseball it will come back after a while go up and then come down but if there is a escape velocity where if you throw it fast enough the thing will just take off into outer space and never come back okay so you can think of the expansion of the universe as kind of like that this picture that i drew here corresponds to a perfectly uniform initial condition which we think is is true but um but sorry it's almost true but it's not perfect right there are deviations there are slightly over dense and under dense regions so if you have some set of particles more nearby that are in over dense region they will pull together and they will sort of initially be expanding apart but then they will start collapsing together and we call this the formation of a galaxy or something like that right a galaxy or a star or whatever and so the first explanation is to why brooklyn is not expanding or the solar system's not expanding is that these bound structures have departed is to part of the right word um yeah okay they well the part is not the right word they have left the decoupled from the expansion of the universe what is affecting their dynamics is their local interactions not the expansion of the universe and people don't like that they say well look if i have you know if they go back to the analogy right like the rubber sheet is being pulled apart or the balloon is being blown up if you if you take a a small balloon and you put dots on it with a magic marker and then you blow it up it's not only that the dots get further apart the dots get bigger right and so they say like maybe there is a tiny effect or if the rubber sheet if i have you know if i draw pictures on the rubber sheet and i pull it apart then everything gets bigger that's just not a good analogy the analogy has broken down in that case uh it's much better if you think of the rubber sheet and you put pennies on the rubber sheet or coins on the rubber sheet you pull them apart right the coins get further apart as you pull the rubber sheet but they don't get bigger they might have infinitesimally incredibly tiny force exerted on them by the sheet below them but then there's a restoring force from the tension inside the coin so they don't get bigger at all um the raisins in the raisin bread if you put them in uh the oven another famous analogy for the expanding universe is raisin bread you put in the oven let it expand the raisins don't expand along with the bread because they're held together by the forces inside the raisin it's even more true here for the universe because there's just as many things pulling it together or more things pulling it together than pulling it apart so that's one explanation the other explanation is i can give you an exact solution to einstein's equation that is not exactly the real world but might be that resembles the real world in an evocative way so let me draw not space time but just space so here is space okay and i imagine that i fill space with stuff dense matter right so i do this correctly it will be exactly the same shade of gray everywhere representing the fact that the matter is perfectly smooth there we go okay and so in this case this represents a universe where it's exactly homogeneous everywhere exactly uniform and it would expand and continue to be exactly uniform so that is a solution to einstein's equations for cosmology but now let me take this and duplicate it let me give you another solution einstein's equation i can take a circular region let me take it this way i can take a circular region in this universe and i don't eliminate all the matter in it but i take all the matter that was in that region and i move it to the center okay so i put it all right in the middle and i can do that at various points so i can do that over here and it doesn't even matter the size i can do a bigger region oops i can't make it lumpy it needs to be perfectly spherical that's the rule of the game it's perfectly spherical then all the matter went there so you're imagining that the matter collapsed to that central point okay and i don't disturb the particles that are still left in between but i can do this i can make this hole called a vacuole and i can make a hole in the universe and contract all the matter to the center and as the universe expands it turns out this as long as the amount of matter that i put at the center of that region is the same that was in there before i evacuated it this is still an exact solution to einstein's equation so in other words if i'm living here at some point in the uniform distribution of matter i don't notice that i've changed from this picture to that picture the dynamics near me oops is exactly the same what even happened there okay so what that means is that the size of the region expands along with the universe and the life within this region so if we live here a little galaxy the metric of space-time in that region has zero expansion it is the short shield metric locally it is the metric of a single blob of matter with a spherically symmetric vacuum around it there is exactly zero uh influence of the fact that it's surrounded by an expanding universe so this is not a completely realistic version of the universe we live in but it is a an analogy a close relative of it to drive home the fact that we can have an exact solution einstein's equation in which the universe as a whole is expanding and in local regions where things have become over dense and collapsed there is precisely zero expansion it's not that there's just a little bit and we can solve the equations exactly and we can show that so the analogies that we use for the expansion of the universe are helpful in some ways and not helpful in other ways and this is a way in which uh they're the the balloon and the raisin bread and all these things are slightly less than perfect brooklyn's not expanding the solar system's not expanding nothing like that the universe is expanding though all right and this is you know like i said there are many more questions that i could have answered uh the cosmology it's a big field there's a lot going on i encourage you to you know hit up the internet and uh look for more information because you'll never stop learning more about how cool the universe is

Channel: Sean Carroll

Views: 36,573

Rating: 4.8884664 out of 5

Keywords:

Id: Qf_4vWO6UTk

Length: 86min 53sec (5213 seconds)

Published: Sun Aug 23 2020