The Biggest Ideas in the Universe | 22. Cosmology

hello everyone welcome to the biggest ideas in the universe i'm your host sean carroll today we're on idea number 22 which is cosmology now when i say that you might have some conflicting reactions on the one hand of course i was going to talk about cosmology at some point it's an important part of our understanding of the physical universe it's something that i'm supposed to know something about on the other hand does it really count as an idea so much as a subject matter or field of study right but i think it does count as an idea because the real idea i want to get at is the fact that we can know so much about the universe that we can model the universe in a very very simple way you know we could imagine universes that were much harder to get to intellectually they were much harder to get a grasp of but our universe is actually something we can know a lot about it's the ultimate spherical cow we can make very nice assumptions that make us go very far that's the idea that i think is worth thinking about so the idea in other words the sort of in slightly more technical terms the idea is that the universe is more or less the same everywhere more or less uniform okay so there are technical ways of talking about this and so let me put make this specific the universe capital u's universe is more or less uniform now there was this long-standing uh sort of tradition of elevating this into a principle okay this was called the cosmological principle that if you averaged over local uh differences from place to place on very large scales the universe was two things number one it was homogeneous in space so it's the same at every location but also it was isotropic so if you looked around in different directions at every location you see the same kind of thing and this was developed into the cosmological principle that the universe was uniform and the same everywhere i don't think it should be a principle i mean maybe it's a principle is something that you think you know has to be right right you know it from the principle flow other things the uniformity of the universe i take it at this point in the history of cosmology is just an empirical fact we just observe the universe and it is that way it didn't have to be that way right we're so used to it by now that we sort of think of it as automatic but you could imagine what you used to be called the island universe hypothesis that you know something like the milky way galaxy was all by itself in otherwise empty universe in a lot of ways in many many ways that would have been a simpler universe if there'd just been one collection of stuff otherwise in an empty void that would have made a lot more sense more speculatively you could have imagined universes where conditions were just wildly different from place to place i mean maybe the density of stars or something like that or even the laws of physics could be wildly different from place to place now these days cosmologists do imagine such a thing but they put the regions where things could be very very different outside of our observable universe so really today in this biggest idea of video we're sticking to the observable universe where things look more or less the same everywhere now you know footnote not exactly the same everywhere things are not the same here on earth as they are in the desolate cold of interstellar space so we're doing some averaging we're mentioning that statistically on sufficiently large scales things look more and more uniform everywhere and you can go to you know when i was a graduate student you know late 80s early 90s there were still people who thought that the universe was sort of fractal on large scales it was kind of lumpy in some places very very under dense in other regions again our data has improved and that idea has fallen out of favor if you average over bigger and bigger regions our universe looks uniform that's an empirical fact so how do we get there um it was hubble edwin hubble in the 1920s he had helped and another you know another footnote here is that hubble worked with a lot of other people schleifer and humison and so forth um and he used ideas from a lot of other people like lemetra um but he gets and deserves a lot of credit for doing what he did so this is not a history lesson by all means check out the history it's fascinating and important you know that what one of the things hubble discovered was the universe is expanding but before that he found out that the universe was big okay that spiral nebulae these things you see in the sky uh if you're an astronomer these sort of fuzzy patches there's a whole bunch of fuzzy patches in the sky uh all called nebulae and they didn't know what they were in the early days and what hubble figured out is that some of them with these pretty spiral shapes are distinct galaxies just like the milky way so before that it was perfectly plausible to believe and many people did believe that we did live in an island universe okay there were a bunch of nebulae but they were just clouds of gas and dust within our galaxy and it was really hubble that showed that some of these clouds of gas and dust were very far away and given their size and their brightness they had to be separate galaxies all by themselves and then that's what really let people believe in a respectable way that maybe the universe was uniform hubble didn't prove that by any means but he sort of let you believe it um how could he do that there was a breakthrough by henrietta levitt even earlier 1912 i think which was the period luminosity relation for cepheid variable stars i'm just going to write period luminosity relation sometimes known as levitz law now and what this is there's a certain kind of star cepheid variables and again there's footnotes here too because astronomy is a god-awful mess if you get into the details of stars and galaxies so there's different kinds of cepheid variables they didn't know about that at the time but what levitt discovered was that there was uh these variable stars are pulsating okay they get bigger and they get smaller they get bigger and their brightness goes along with them what she discovered was that uh there was a relationship you know it was known that different stars had different periods different vibrational periods and what she was able to do was to use parallax to figure out the the distance to some nearby cepheid variables and she saw that their luminosity their brightness was correlated with their periods and the reason why this is crucially important is because you know i mentioned parallax so parallax is just the idea if here we are here's the sun and here's the earth going around it okay so you're here on earth and you're looking out at some star over here and there are even further away stars over here when you're looking at the star here at one time of the year you will see it in this background and then at the six months later you're going to see it behind a slightly different background so far away stars remain fixed nearby stars seem to move a little bit just like you know things seem to move if you just move your head a little bit okay that's parallax it only it works as a way of measuring the distance the parallax is bigger the closer the star is so it only works when the stars are pretty darn close right if they're very very far away then you can't measure the parallax at least you couldn't in 1912. now if you listen to my podcast we talked to lena nasib who talked about this gaia satellite project that we have now that can measure the parallax to a billion stars and you're figuring out where every star is in the milky way and that's great but back in the day you could only do this for very very nearby stars so for these nearby stars you find this relationship so what that means is that you could you measure nearby stars using parallax you find that some of them have this period luminosity relation and then if you find similar stars with the same period far away you know how bright they are and therefore you know how far away they are this is the fundamental challenge for observational cosmology is figuring out how far away things are if you walk out in a dark night and see stars in the sky they all look like you know bright spots in the sky they don't look any different from each other some are brighter some are dimmer if we lived in a wonderful world where every star had the same intrinsic brightness you'd instantly know how far away they are those will be called standard candles standard candles are things where you know precisely how bright they are intrinsically therefore how bright they appear to be tells you how far away they are stars are not standard candles real stars have intrinsic low luminosity high luminosity etc levitt showed that for these specific cepheid variables you could figure out what their intrinsic luminosity was by measuring their period what hubble was able to do was to locate you know he and his collaborators located cepheid variable stars in other galaxies stars inside spiral nebulae and what they found was that the they were so dim when you observe them they must be super duper far away which corresponds to being separate galaxies all by themselves okay so then you know that was what i said like i said this means that we do not live in an island universe there's a bunch of galaxies scattered throughout the cosmos by itself they didn't have nearly good enough data to show that they were uniformly scattered but it's a natural assumption and that's what people started assuming i mean people had guessed that already even before um okay so what would the world be like what happens to a universe in a universe with uniform density well this would be a really good and difficult problem if isaac newton had been correct about gravity right i mean newton said that space and time were fixed things are pulling on each other so we already talked about this puzzle before if you just scattered galaxies or stars or whatever uniformly through space they're all pulling on each other so do they all cancel out or somehow does everything move in toward each other that's a puzzle in newtonian gravity but by the time we were asking this question observationally we had general relativity right einstein had come along and it's no longer a puzzle there's a very clear answer the answer is a universe like this should either be expanding or contracting it can't be standing still so there is something called that's the well you can talk about the relative size of the universe and that will either be increasing or decreasing over time if it's just stuck there it will start decreasing okay now einstein was the first to figure this out around 1917 very soon you know kind of remarkably soon after he invented general relativity he plugged his theory into this and he realized you cannot have a static space-time with uniform matter all through it so this bothered him so you're thinking 1917 that's before hubble had discovered that there were spiral galaxies so there's i think i already mentioned this in another video but there is uh this uh story that is incorrect that goes around says that einstein was philosophically wedded to the idea of a static universe but that's not true he wasn't philosophically wedded to it he asked the astronomers what is the universe is it static or is it expanding or contracting in 1917 they said well it's static we don't see any any evidence that the universe is expanding or contracting so what einstein did is said well that that sounds like so what he could have done was be super bold right he could have said look it's 1917. observational cosmology has not advanced that far i'm going to disbelieve my experimental friends who say that the universe is static and i'm going to make a prediction that once cosmology gets to be more advanced we will realize that either the universe is expanding or it's contracting that is not actually what he did what he did was he believed his observational friends and he said look um i need to change my theory my theory general relativity seems to be in conflict with the data so i'm going to try to fix it this is a very respectable scientific thing to do it's not you know it has nothing to do with being hamstrung by an incorrect philosophical orientation so what he said was you know we have einstein's equation and this is the equation that he used to derive the fact that the universe must either be expanding or contracting r mu nu minus one half r g mu nu equals eight pi g i'm not gonna i'm gonna write a room here right [Music] 8 pi g t mu so we talked about this equation before when we talked about gravity don't didn't go into all the details but it's an equation between two different four by four matrices on the left-hand side the r mu minus one f r g minou that's a way of characterizing the curvature of space-time the right-hand side eight by g t nu that's the way characterizing the stuff the energy the momentum the heat the pressure all that stuff that is causing space time to curve and what einstein said is look i'm allowed to add a constant call it lambda the greek letter lambda times the metric tensor g mu nu to the left hand side of my equation and then i can actually what i find is that if that value lambda which he called the cosmological constant if lambda is positive it pushes space apart if it's negative it pulls things together so ordinary matter and radiation tends to pull together okay whereas positive cosmological constant pushes apart so he says i can pick a value of lambda that's exactly precisely right to balance the tendency of matter to pull toward each other and have the universe contract therefore i can make a static universe and there's he even solved the equations uh this equation exactly and something that we now call guess what the einstein static universe okay sadly for einstein um he missed a chance because he should have stuck by his guns uh well maybe we'll we'll talk about the the subtleties here in a second hubble came along again and in 1927 the universe is expanding he never said this by the way it's interesting uh hubble hubble was actually you know he was an extraordinarily talented observational cosmologist he was not a theorist and he didn't pretend to be a theorist and all this gender relativity stuff was you know he was hands off about that he was non-judgmental so what he found was the hubble law or the hubble relation which is that when you measure the distance to a galaxy you can also measure its apparent velocity right this is something we can do like like distances are hard to measure but doppler shifts are easy to measure stars give off light and in certain regimes they give off very very well-known frequencies of light or wavelengths of light if the star is coming toward you that light will be blue shifted if it's moving away from you it's redshifted so there's an apparent velocity that stars and galaxies have and those are easy to measure that's what astronomers are good at and what hubble found is that there is a relationship it was a mess in his original plot velocity versus distance but it's more or less a straight line with a slope so velocity is a certain constant which we now call the hubble constant h times the distance okay so we can interpret that as saying that the universe is expanding hub will never use those words but if you think about it what this is saying this is the universe is isotropic the same in every direction so in that direction galaxies are moving away from us in that direction galaxies are moving away from us and the further they are away from us the faster they're moving away from us that's what hubble's law says so if you think about what this looks like to different observers you know here's an observer they see this galaxy moving away and they see this galaxy moving away and closer galaxies are moving away but more slowly right so there's this velocity field grows in size the further you get away and if you're worried about the fact that i can pick distances such the velocity is greater than the speed of light because the hubble constant is just a constant and the distance can be anything well hubble's not looking nearly that far away all these galaxies are moving much more slowly than the speed of light but what you notice here is that what if you take the point of view of this galaxy right there what if you're living there well then you see this galaxy moving away from you slowly you see us moving away from you slowly and you see the other ones moving away from you even faster so even though it seems to us like everyone's moving away from us okay as if like we're doing something bad and they're trying to get away and we're at the center of some evacuation this particular law velocity equals some constant times distance is exactly what you would expect not if we're at the center and everyone's moving away from us but if everything is moving away from everything else uniformly and that's exactly just what general relativity predicts so the unifor the universe is completely homogeneous it's the same everywhere there's no center there's no central point so all hubble said was that there's this relationship between velocity and distance but the whole story hangs together with this idea of a uniform universe expanding so we can think about that a little more quantitatively okay so expanding universe within the context of general relativity now by the way when hubble discovered this einstein was was purportedly uh you know reportedly uh upset that he missed the chance to predict the expansion or the contraction of the universe and he said all right we don't need the cosmos constant anymore that was a waste of time but he didn't actually throw it away he kept thinking about it of course nowadays we've discovered it so he was right even einstein's biggest mistakes are our wonderful new ideas um so here's an expanding universe space-time diagram time space okay as usual and space is getting bigger so what does that mean space is getting bigger that's that's the important question here okay so if you remember back we did a whole biggest ideas video on geometry where we emphasize the fact that from gauss and riemann and so forth we can think of geometry intrinsically not as some shape embedded in a bigger space but just the geometry of the space itself and in this case the space is space-time four-dimensional space-time so you have some galaxies okay there's a galaxy another galaxy another galaxy and they're expanding away from each other and what that means is the distance between these galaxies is getting bigger over time so if you say that this is a distance d1 and this is d2 the distance of those two galaxies at two different times t1 t2 then we can define something called the scale factor which says the relative distance between different galaxies okay a of t and how this is defined is you know the relationship distance two divided by distance one the relative size is the scale factor at time t two divided by the scale factor at time t one and the reason why this is worth going into and being explicit about is because you know people wonder about a bunch of things you know how can the universe expand if it's infinitely big you know this this space could go on forever how does an infinitely big universe get bigger well the answer is the relative distance between different objects grows bigger if you think on the number line think of the integers right minus 2 minus 1 0 1 2 3 i can multiply every integer by 2. so it becomes minus 4 minus 2 0 plus 2 plus 4 right so i'm expanding everything by a factor of 2. the integers got further apart but i didn't make any more integers right i didn't make any more numbers i just moved them apart from each other that's what is going on here they're an infant number of integers before and after there's an infinitely big space before and after that's completely compatible with space expanding and that expansion is intrinsic so things to note about this picture of the universe there are many things to know this is cosmology right one it's not expanding into anything that's not what's going on so people are constantly asking you know what is the universe expanding into and that's because we're used to visualizing things inside three-dimensional space number one and number two even cosmologists tend to give examples that are embedded in three-dimensional space so they say you know imagine a loaf of bread with raisins and you put in the oven and it expands or imagine a balloon with little dots and you expand it i don't like any of these analogies because they're all asking you to visualize something that's embedded in a bigger space and that leads you to ask what is the universe expanding into but it's not expanding into anything if you want to visualize the expansion of the universe visualize it's being inside the universe standing outside on a dark night visualize having perfect vision so you can see all the galaxies in the sky and they're all moving away from you they're all shrinking in size and getting dimmer because they're moving away from you that's what it means to say that the universe is expanding uh number two as we've said there's no center okay everything is moving away from everything else so d1 and d2 are increasing but then this is also increasing nothing is picked out as a preferred location in the middle or anything like that and also you know we said that space is uniform that's part of the uniformity of the universe so what that means in terms of the geometry of space is that space has a constant curvature so remember back again again when we talked about geometry there was loboshevsky in boye who invented negatively curved geometry and that is indeed a kind of non-euclidean geometry just like positively curved geometry on the sphere is but i quickly emphasized that these were just very very special cases real geometries tend to be lumpy not you know this perfectly smooth thing that boy and lobozhevsky were talking about or a sphere is so in you know the geometry of space-time in the solar system is not anything simple like positively curved negatively curved zero right it's it's something much more complicated than that but for cosmology now it works now since cut since the universe is such a simple place the geometry of our three-dimensional space on the largest scales on cosmological scales really is either positively curved like a sphere positive curvature it could be flat which is zero curvature right or it could be negatively curved like a saddle shape which is the sort of hyperbolic geometry the pul uh bulyai and lobashevsky i almost said bolashevsky bojaya and lebashevsky were originally talked about so the three-dimensional versions i've drawn little cartoons of two-dimensional spaces but the intrinsic geometry is dealing with questions like if i set off two light light beams that are parallel to each other do they stay parallel same distance apart forever maybe they diverge or they eventually come together that's what's asking whether you're positively curved negatively curved or flat and indeed empirically we seem to be here this is the real world the measure we've measured now years later we've measured the curvature of space to good accuracy to a percent or something like that and it's within the accuracy it is flat the universe seems to be flat could have been positively curved or negatively curved seems to be flat as a matter of fact so if you want to know does space go on forever if space were uniformly curved and the curvature was positive it would not go on forever right a sphere it would close in on itself but for flat or negatively curved for both of those cases you don't know the fact that the curvature is uniform is not enough to tell you whether space is infinitely big or not clearly a flat universe could go on forever could just be infinite in all directions likewise the negatively curved universe same thing you can have a saddle that extends out forever but there are more complicated topologies that are finite okay so a a taurus a three taurus which means a three-dimensional torus is a way of saying let's take a cube okay flat space inside but we're going to identify opposite sides so top and bottom get identified right and left get identified uh forward and backward get identified by identify we mean if you're walking here you are and you walk toward that wall you eventually come out the other side okay they're identified this is a three-dimensional torus you can't draw the whole thing without these identifications but it's spatially flat and finite and the same thing with negatively curved spaces there are negatively curved but finite spaces so unless we someday discover that the universe is positively curved we are not ever going to know empirically whether the universe goes on infinitely or whether it's finite we may come to a conclusion we may decide that it's overwhelmingly likely to be either infinite or finite based on some theoretical considerations that we don't yet have available to us but as of right now we don't know it's hard to see how we will ever know okay so that's a tough question to answer um one that we have more purchase on is how does the universe expand in other words we have this scale factor as a function of t here's the scale factor there is time we know it's expanding so a is increasing right now but what's it going to do in the future and what did it do in the past right that's what we're trying to ask what is the shape of this curve the scale factor the size of the universe as a function of time so actually this is exactly what einstein's equation is set up to tell us this is what general relativity gives us the ability to answer these questions quantitatively or at least to propose answers to them and then to test those ideas rigorously so let's do some definitions um the hubble constant remember velocity equals hubble constant times distance right you know you heard that so the hubble constant which is an observational thing um that's the way that we characterize how fast the universe is expanding at any one time the faster the universe is expanding the bigger the relationship between velocity and distance so we want to relate the hubble constant to the scale factor so you might guess that the hubble constant is just the slope right d a dt the velocity of the scale factor over time but that can't be right because the hubble constant is some physical thing you can measure it's a direct relationship between two observable quantities whereas a the scale factor only has meaning you know from up here if you change a by a constant factor if you multiply a times two it disappears from this equation right here so the overall value of a is completely meaningless that's bad if you want to think that it d a dt because if i change a by a factor of 2 here the hubble constant would change the way i fix that is i divide by a so sometimes this is called just so you know a dot that's d a dt is sometimes called a dot over a and that is the definition of the hubble constant so with that definition in mind you can take this idea of of a flat let's just go flat for the geometry of the universe henceforth because it's pretty close observationally so flat universe plug it into einstein's equation ask what the relationship between curvature and energy is and the answer is something called the friedman equation alexander friedman derived an equation for the expansion of the universe and it says h squared equals some constant which happens to be 8 pi g over 3 times rho where rho is the energy density of space well not of space in space the density of stuff matter radiation dark energy whatever you want to have okay so the energy answers all you need to know because you hypothesized that space is uniform so we're glossing over all the lumpiness of galaxies and stars and things like that we're taking the average energy density and at every point there's an energy density it's the same at every other point at the same time and it's going to change with time as the universe expands and this is a very very simple relation between how fast the universe is expanding and how much stuff there is in it the friedman equation so what you want to do is tell me what kind of energy density is in the universe and then i can solve this equation right because if i look at the definition of h here i can rewrite the friedman equation take the square root of both sides multiply by a and what i get is that d a dt the sort of velocity of the scale factor equals the square root of 8 pi g over 3 a squared rho and then rho is going to be a function of a because as the universe expands it will dilute the density the cubics the amount of energy per cubic centimeter will change and i can solve this equation to get a as a function of time solve it do calculus and get a as a function of time there you go so i can give you some examples so this is the big question what is row of a what kind of stuff what is rho and how does it depend on the scale factor what is the amount of energy in the universe and how does it change as the universe expands so i'll give you a couple of uh again cosmology is very simple science it's the best one to go into for people who have short attention spans the universe is a simple place there are three kinds of energy density that really do more than you need for uh not more than you but as much as you need for a lot of modern cosmology so one kind of energy density is called matter remember when we talked about uh i guess is when we talked we had a biggest ideas video on matter and we talked about the fact that the word matter in the q a we talk about the fact the word matter is used differently by different kinds of scientists cosmologists use it in the most straightforward way matter is particles that move slowly i.e the velocity is much much less than the speed of light speed of light is the only universal speed so that's what you compare it to when you say it moves slowly why do you care how fast the particles are moving well because then the energy of the particle remember the energy of a particle has a contribution from its mass m c squared plus other contributions from its kinetic energy etc but by the hypothesis that the particles are moving slowly says that the kinetic energy is not going to be very important compared to the rest energy and we're going to set c equals one so this is just m right and then the energy density rho which is what we're after here that's just m the mass of the particle the energy per particle times n where n is the number density which is a fancy way of saying particles per volume whether volume is cubic centimeters or cubic megaparsecs it doesn't really matter so you have some density you have some region of space and it has some particles in it there are the particles every particle has an energy and that energy is fixed it's m c squared per and for whatever the m is of that particle and as the universe expands the number of particles stays the same roughly speaking there's going to be some some counter examples to that but roughly speaking the number of particles stays the same and so it's the number density times the mass therefore the energy per particle that gives you the energy density and we know what that does you know because space is three-dimensional um each one of these dimensions here of the box is going to expand along with a the scale factor so the volume goes up as a cubed that's what volume does and therefore the density if the number of particles stays the same the density goes down like one over a cubed so n is proportional to one over a cubed and therefore rho is proportional to let's just call it a to the minus three so rho matter this is matter this is what we mean by matter as a cosmologist a bunch of particles moving slowly compared to the speed of light so this is what cosmologists think of as matter uh some energy density that dilutes away as the universe expands and the way it dilutes away is that it's proportional to a to the minus three that is in contrast with what cosmologists call radiation now you think you know what radiation is electromagnetic radiation gravitational radiation it sounds like massless particles right but in fact two cosmologists radiation are particles moving at the speed of light okay so massless particles count right photons gravitons but also any particle that is traveling very very fast compared to the speed of light count or sorry near the speed of light any particle that is almost moving the speed of light to a cosmologist that counts as radiation why because what we care about is this relationship between energy density and the scale factor and the energy density in fast moving particles will evolve the same way whether those particles are moving at exactly the speed of light or just really really close okay and the reason why this is true is because of that redshift uh the redshift which we already mentioned hubble used it to discover the expansion of the universe the wavelength of a particle is proportional to the wavelength of a single photon for example is proportional to the scale factor this is just saying that as the universe expands so here's time the universe is expanding and i have a little photon its vibrations stretch along with the universe and you might say well how do i know that well you have to solve the equations you don't know that a priori right you have to actually say well what equations does this solve and you find out that it does in fact expand along with the universe and in other words the frequency of the particle which is proportional to i'm calling it a particle okay it's a wave but you could observe it and see that it's a particle frequencies are proportional to one over the wavelength and therefore to a to the minus one and we know back when we did quantum mechanics the energy of a single photon is h times the frequency where h is uh planck's constant and um therefore that is also proportional to a to the minus one per radiation particle okay therefore the energy density in radiation which is the energy per particle per particle p for particle uh times the number density that's still true is proportional to a to the minus one times a to the minus three therefore the energy density and radiation is proportional to a to the minus four there we go so conceptually what happens is the number density of radiation particles photons or whatever it could be electrons if they're moving at the speed of light for the electron the momentum will also slow down for massive particles the momentum in an expanding universe decreases like one over the scale factor so they will lose energy in exactly this way so for radiation particles the number density goes down like the volume goes up but also the energy per particle is going down and therefore the whole effect is that the energy density of radiation scales is a to the minus four you know i paused when i was thinking about the little h being for planck's constant because we've already talked about capital h being the hubble constant and it it just brings home this thing that i like to point out that the first half of the 20th century really the first 30 or 40 years um man it was the most exciting period in the history of physics like so many things were discovered relativity quantum mechanics those are the famous ones but also the big bang cosmology radiation quantum field theory you know so many things both special and general relativity all in the first half of the 20th century we were spoiled by that like physics now is like a lot more normal in some sense than it was then that was a very very unique time in the history of physics okay last thing to mention um well let me go let me say a little bit more about the the particles so so for massive particles by massive i mean not zero mass i mentioned this but let me actually draw a graph to make it clear um the velocity the the momentum of a particle p is proportional to 1 over a the scale factor so if i draw the velocity of a massive particle as the scale factor is increasing as the universe is expanding it it has a maximum value at one right it can't go faster than the speed of light it has a minimum value at zero and so roughly speaking it will always look like this so if you want to know the reason why i thought this was important to to uh say is that that's not really worth doing this but why do cosmologists get away with these two limiting cases where particles have zero velocity or particles are going at the speed of light and they're either matter or their radiation well this is the answer because particles are going to go through their life most of the time they'll either be radiation or they'll be matter there's only a relatively short period of their life when their velocity is somewhere interestingly in between one and zero it quickly goes from one to zero as the universe expands roughly speaking that's why these two categories are very very useful final category is vacuum we've already talked about this empty space cosmological constant rho lambda for the vacuum is just constant so that's a to the zero it doesn't evolve at all right it's the same density and this reminds us that the thing that you would normally call energy in a region of space that is expanding is not conserved in an expanding universe you might have thought it was if you thought that the whole universe were made of matter right because the energy per particle stays the same and the number of particles stays the same but either for radiation or for vacuum energy is not conserved in a co-moving region so this is what we call if you draw these little you know boxes and you let the box get bigger over time region of space that has a certain volume this is a co-moving volume it expands along with the expansion of the universe and the energy in the co-moving volume does not stay the same if the universe were nothing but radiation the energy in that box would decrease with time if the if the universe is mostly vacuum energy then the energy in that box increases with time either way it's not conserved decreasing with time is just as much not conserved as increasing with time that's not to say all hell has broken loose there's still a relationship between what happens to the energy and how space time expands but the energy is not the simple one that you would have gotten in flat space time one way of thinking about this is we talked about symmetries and nerd's theorem you see why all these previous ideas videos are relevant right we did we found the fact from the symmetry video that invariance over time leads to energy conservation but the universe is expanding then the the sort of the world in which particles live is not invariant over time the universe is expanding so you wouldn't expect energy to be conserved in the usual way and given those three options for what can be the stuff in the universe we can go and solve the friedman equation remember the freedom equation is h squared which is a dot over a squared is 8 pi g over 3 rho which now we know for different options we could have rho as a function of adding right and so for matter domination matter domination means most of the energy density in the universe comes from matter well then rho goes like row matter goes like a to the minus three and the freedom equation we can plug in and solve and we get that a is proportional to t to the two thirds you can check this at home then you know a dot proportional t to the minus one third one third and you can check at home that it solves the freedom equation likewise for radiation domination uh rho radiation proportional to a to the minus four and you get that a goes like t to the one half and for vacuum domination rho lambda is proportional a to the zero and then it's a little bit trickier but a is proportional to e to the t the only thing is that t needs to be multiplied by a constant to be dimensionless because you exponentiate dimensionless numbers and the thing you multiply by is basically the hubble constant or some particular hubble constant which i'm calling h star some particular hubble constant corresponding to that vacuum energy so that's it that's exactly how easy it is to solve for the expansion of the universe in cosmology and you can even check yourself but okay not done yet all i've done is given you three options for what could be in the universe so let's get to reality now what really is in the universe and what really was and what will be in the universe will the universe be matter dominate radiation dominate vacuum dominate or whatever so let's let's think let's use our brains before we start uh making guesses here um the past of the universe well what do we know we know that right now uh we live in the universe and there's stuff in it um particles galaxies stars but there's also light right you know emitted from stars and and other kinds of light also and we know that as we expand the universe the light stretches it goes from shorter wavelengths to longer wavelengths and what that means is that in some sense the universe is cooling down right a colder black body radiation has longer wavelengths than hotter black body radiation so what that means in the past the universe was smaller the scale factor was smaller um and therefore the radiation was shorter wavelength and therefore it was hotter so in the past the universe was hotter in some sense okay um in fact you can you can show that in the right circumstances where new particles are not being created or destroyed temperature is proportional to one over the scale factor so we often do is we plot a log log plot so the logarithm of the temperature as a function of the logarithm of the scale factor to be perfectly honest i forgot i've forgotten whether i've explained what a log is i must have explained what a log is right the logarithm of e to the x equals x logarithm is the operation that undoes exponentiation so when you have two variables that are related by one is the other one to some power like t is a to the power -1 then on a log log plot you get a straight line and indeed here's the straight line so as the universe expands the temperature goes down okay so what we can do is we can think about the thermal history of the universe in fact this is so this relationship is so nice that very often if i if i tell you you know something was happening when the scale factor was 10 to the minus nine of what it is now you don't really have a feeling for what that means okay scale factor is a billionth of what it is now i'm not really sure what's going on then but if i tell you what is happening when the temperature was let's say a million electron volts then you can relate that in your brain since you've listened to the other videos about particles and matter and atoms right scales and other videos so you know what all these particles have is their masses and so forth and you go oh yeah okay so that's heavier than a neutrino it's about the mass of an electron so you know you can instantly know which particles are heavy compared to the temperature and therefore moving slowly which particles are light compared to the temperature and therefore moving rapidly right so we'll we'll talk about the thermal history we'll think about the history of the universe as a function of temperature but you know that's a proxy for thinking of it as the function of the scale factor and so what happens as you heat things up you know the universe gets hotter and hotter let's imagine that you you know take a box with let's imagine we simplify our lives you have a box that has just photons in it and you make it smaller right and you might say well okay the photons get more and more energetic they get smaller and smaller wavelength in a contracting universe right but that's not all that happens because you know that photons can convert into other particles by interacting with each other so you can have you know two photons gamma plus gamma two photons convert into an electron and a positron so here's a feynman diagram that makes that happen gamma gamma e minus e plus uh let's draw a little fermion lines voila a conversion from electrons to photons and you can go the other way to an electron and a positron can annihilate into two photons so this is a conversion back and forth and what happens is if the typical wavelength of the photons or the typical energy of the photons is much less than the mass of the electrons this is never going to happen right because the photons don't have enough energy to make two real electrons or an electron and a positron so when t the temperature which is another way of thinking about the typical energy of the particles is greater than let's say two times the mass of the electron this happens in fact let's say it happens rapidly and it happens rapidly in both directions so what i'm saying is you start with a nothing but photons and you contract the universe those photons are going to be photons and get more and more energetic once they're energetic enough to pair create electron positron pairs then you're going to go from a box of just photons to a box of a combination of photons and electrons and positrons okay and you say well the electrons and positrons are going to annihilate away yes they will into photons which will then create more electrons and positrons so you get an equilibrium where this where this reaction goes back and forth very very rapidly and in fact if you're at temperatures way higher than the mass of the electron there are approximately equal numbers of photons and electrons and positrons roughly speaking so that's the trick to all of the thermal history of the universe once the temperature goes above the mass of a particle then that particle can be created very very efficiently and therefore that particle becomes part of the radiation gas the particle will also be relativistic as we say it moves close to the speed of light because you're at a temperature much much greater than the mass of that particle so its energy would be greater than its mass therefore its moving near the speed of light therefore its radiation uh not just matter so another way of saying this is uh if you were to plot so what we mean is so what we're saying here let's let's let's back up i was going to rush ahead a little bit um what this is saying is that it seems like the energy density of the universe should usually be in the form of radiation why because you know there's some radiation there and if the temperature is below the mass of a certain particle that particle and its antiparticle will annihilate away into radiation if you if the temperature is higher than that particle then you can then the mass of that particle then you can make it through this pair creation process but when you make it because the temperature is higher than its mass it's relativistic and it is also radiation so the photons which are sort of literal radiation make fast-moving electrons which are sort of figurative radiation but to cosmologists it's all just radiation so you might think that the universe is almost always all radiation and in fact so let's run it forward now let's go increasing uh let the universe expand and ask what happens to the energy density so if it's here's energy density again log log plot i'm not going to say that every single time or even write it but energy density versus scale factor and for photons you get something like this so the energy density and photons which are always relativistic they always move at the speed of light that's proportional to a to the minus four so that's a certain straight line on this graph but what about you know for electrons well when the temperature is higher than the mass then the electrons are very similar in density and they evolve in a very similar way but then when you hit the point where the temperature is of order the mass of the electrons they annihilate but they don't get recreated so the number density of electrons plummets okay the electrons just annihilate away and then there's always going to be some remnants right because you know these numbers are never exact but it's very very small so this is rho electron it's radiation-like for a while and then it becomes goes away quickly and this is when the temperature is of order the mass of the electron which is of order 1 mev or 10 to the sixth electron volts okay so this is what leads you to believe that you know maybe the universe should just be full of radiation all the time how could you get anything else well there are three exceptions to this rule one of them you probably already guessed uh namely that this only makes sense if there are an equal number of electrons and positrons okay so one exception is an asymmetry between particles and anti-particles so that's what actually happens that's what you and i are made of right and we talked about this uh the idea of bariogenesis there were a number of quarks and baryons which are the particles that quarks make up um and there was an imbalance in the early universe between the number of particles number of quarks the number of antiquarks and so almost all one all but one in a billion quarks annihilated away because of exactly a process like this we believe uh but there was a little remnant left over because now you know the atoms the protons and neutrons that are around us there aren't any anti-protons and anti-neutrons hanging around for them to annihilate you might ask you know is it possible like other galaxies are made of antimatter but the universe is not quite as empty as you might think even though interstellar space is pretty darn empty there's still some gas and dust there and so if another galaxy was made of antimatter there's some region in between the two galaxies where there would constantly be particles and anti-particles bumping into each other annihilating and giving off high-intensity radiation so that is not observed so no anti-particles and other galaxies are not made of antimatter they're all made of matter so this is one way to get you know more particles than antiparticles and therefore matter and in this case in the real world case it's called bariogenesis and we mentioned why is it variogenesis not leptogenesis or electron genesis we don't know if the number of leptons is bigger or smaller than the number of anti-leptons because there are neutrinos in the universe and we cannot count how many neutrinos there are versus anti-neutrinos whereas we can count the number of baryons there you go another possible exception is just that the interactions the annihilations are weak you know if the particles don't interact with each other that much you know the particles in order to interact in order to annihilate they have to hit each other and annihilate right for electrons and positrons this isn't so hard they have long range electromagnetic forces they will find each other and annihilate but maybe you have some hypothetical particle like maybe a weakly interacting massive particle that interacts weakly and it's massive so they at some point the universe becomes sufficiently dilute that these particles no longer annihilate away in other words for wimps the universe might very well have an equal number of wimps and anti-wimps it's just they never find each other to annihilate the weak interaction that they use to annihilate is not long range remember it's very very short range so the two wimps need to come really really close to each other in order to annihilate so it doesn't happen that often so you just get what is called a thermal relic because these are particles that were once in equilibrium at high temperature they were popping in and out of existence but then as the universe expanded and cooled they just stopped annihilating because the density went down they couldn't find each other anymore finally there's sort of a grab bag for other things um non-thermal relics so if you have particles that are massive but that we're never in thermal equilibrium so let's imagine particles that interact with each other so weakly that they never were either created or destroyed by all these photons bumping into each other but there could be some other mechanism for making them so i'm not going to go into details but axions are another dark matter candidate which people talk about very popular second most popular dark matter candidate after wimps and axions are created by various non-thermal production mechanisms so i'm not going to go into t as like i said you can google the word axion you can find out all about them and you'll you'll realize that there are different ways to have particles left over so this very naive story in which the universe is always just radiation because particles that become heavy annihilate away there are very very important exceptions to this that's why there can be matter in the universe okay so because of that therefore thus the generic thing we expect for the evolution of the energy density of the universe rho versus the scale factor is at super high temperatures uh the universe will indeed be dominated by radiation there are photons and and gravitons and so forth but there are also every other particle being created and destroyed and all moving close to the speed of light because we're at super high temperatures and the temperature is much higher than the masses of those particles so there's a component of radiation in the universe and it decays away quickly as a to the minus four but then there's also some leftover heavy particles like protons and neutrons or maybe like wimp or other dark matter particles and those their energy density goes away dilutes away more slowly right it dilutes away like a to the minus three so r is proportional to a to the minus four but matter it'll start subdominant but it will eventually catch up so here is matter proportional a to the minus three okay so we expect is that early times the universe is radiation dominated but if you have some mechanism that gives you leftover particles eventually leftover massive particles eventually they will dominate and the universe will become matter dominated even if there's very very few of them if you wait long enough because they dilute away more slowly in energy density eventually they're going to win then of course finally if you have any vacuum energy at all eventually eventually that's going to win because vacuum energy is just constant on this plot right so vacuum is just a to the zero so that will always win so this is the generic expectation for how the universe evolves it's radiation dominated early times then it's matter dominated then it's vacuum dominated and i'm not gonna i'm not gonna sort of reveal that this is all terribly mistaken this is right we think that this is exactly correct in broad strokes there can be details uh that change here um good there's i guess there's one thing that is worth saying here what is that little symbol is that just a badly drawn a yeah one thing worth saying here which is that what about this vacuum dominated regime so vacuum dominance it's worth saying a few more words i said um that a is proportional to e to some specific hubble constant value times time and you might ask well you know you can even check this right d a dt is then equal to h star times e to the h star t this is the nice thing is why e the specific constant is so nice because the derivative of e to the x is just e to the x the derivative of e to the some constant times x brings down the constant and then you get e to the x again so this is equal to h star times a and you can plug in the friedman equation and you can see that it works so this is another way of saying that a fixed amount of energy vacuum energy cosmological constant imparts a perpetual impulse that expands the universe at a constant rate and what you see is the scale factor going exponentially right an exponential looks like it expands very quickly exponentially fast as it were um and so this is it's clear why this counts as an accelerating universe right because if you think about the hubble hubble law velocity equals h times d uh the distance to something is increasing and this is the the thing about this particular vacuum domination thing is that this constant h star is actually truly constant so this is why i wanted to get the nomenclature right we talk about the hubble constant and when we talk about the hubble constant we're referring to the value of the hubble of the hubble parameter today right so in general relativity we define one over a times da dt as the hubble parameter and that's different at different eras in the history of the universe in the early universe the density was much higher and the hubble parameter was much larger than it is today so the hubble constant in other words isn't really a constant if you define it or calculate it at different moments in time but if the universe has nothing in it but vacuum nothing in it but a cosmological constant then there's a constant energy density and remember from the friedman equation remember friedman says h squared is eight pi g over three rho and if rho is constant all these other things are constant so h is constant so that's h star so h star squared is the vacuum energy density times 8 pi g over 3. and so that gets a little surprising because you're saying well wait a minute i told you before that the hubble parameter is the way that we characterize the expansion rate of the universe and you're telling me now that expansion rate is constant so why is that accelerating why is a constant expansion rate accelerating well it's because if this expansion rate is constant the velocity that you would observe of any distant galaxy will be increasing over time right so if the distance is increasing if this is constant then the velocity that we observe will be increasing so that's really observationally why we call it an accelerating universe because distant galaxies accelerate away from us so this is a feature of non-euclidean geometry that we have in the expanding universe in general relativity that on the one hand the expansion rate is constant if you were to calculate the riemann tensor the curvature of space-time in the universe like this it would be constant it would not be changing over time but we say the universe is accelerating at the same time the expansion rate is constant and the universe is accelerating why well because that's how the geometry works it's just yet to go a little bit beyond your conventional intuition a little bit but you should be able to do that okay so what i want to do is to sort of close up here well there's a lot to say you know there's a lot of cosmology out there so you're going to indulge me with a lot of a lot of knowledge we got to drop in this particular biggest ideas video we talked about the fact that the friedman equation tells you how fast the universe is expanding we talked about the fact that there's matter radiation in vacuum as different examples we even talked about the fact that electrons would be relativistic and part of the radiation at high temperatures or they'll be matter at low temperatures let's go through the actual history of the universe a little bit about that thermal history that we talked about for the real world and so this sort of brings things together you know even though cosmology is fundamentally simple it involves little bits and pieces of all the physics that is out there so that's why i keep mentioning previous installments in the biggest ideas series because cosmology really brings together a lot of different ideas from a lot of different areas of physics so particle physics in particular is going to be relevant here so let's start the universe at high temperatures and then let it expand and cool and see what happens so we need to ask you know how high is high what counts as high temperature uh by this way of counting the philosophy we have here at the biggest ideas is that we're trying to tell you ideas we think are true that is to say we have very very good reason to think that they're correct um we're trying you know i'm happy to speculate occasionally but that's not the point so a lot of discussions of cosmology go instantly to you know the big bang inflation and the multiverse and stuff like that but i'm trying to tell you what actually happens in the world and then we can see what we'll speculate about in uh in other contexts but what do we know about the thermal history depends on what we know about the particles that the universe is made of right and we get a long bunch of conversations about the standard model of particle physics and we have the higgs boson the top quark and things like that but beyond that at energies higher than 100 or a couple hundred billion electron volts we don't know what is in the universe so let's call that our cutoff at temperature is higher than a hundred billion electron volts that is to say 10 to the 11 electron volts we don't know what's going on the universe is a stew of particles and relativistic particles relativistic again means just moving close to the speed of light and that includes standard model and others maybe unknown particles because we don't know grain unification super symmetry extra dimensions who knows what's up there some set of relativistic degrees of freedom as we would call them okay we begin to know what's going on at temperatures around so this is radiation in case that's not clear uh temperatures around 10 to the 11 electron volts this is you know 100 gev 100 billion electron volts is where the standard model kicks in um and the first thing that happens is electro weak symmetry breaking so what that means is let me write rather than talking we talked about this when we talked about symmetries and matter and so forth um gauge theories remember gage theories have different phases and one of the phases of gage theory can be in is when the symmetry that gives rise to the gauge theory is spontaneously broken in particular the higgs boson breaks the electroweak symmetry of the standard model of particle physics so this is what the higgs is doing and what it's doing is you know it has a potential v of phi as a function of phi and the potential looks like this this mexican hat form right and at higher temperatures than 10 to the 11 gev it's there basically because the higgs is being buffeted around right so you might say well why does it fall to the bottom well it's very hot so the field itself is sort of vibrating and on average it sits at zero at the middle of its potential and then once the universe cools down to 10 to the 11 gev then the higgs can fall down into the brim of the hat and the symmetry breaking takes place so above 100 gev electrons and quarks are all massless and they're identical to neutrinos electrons and neutrinos are identical up and down quarks are identical et cetera the symmetry has not yet been broken after that now electrons and quarks have masses their masses are still small compared to the temperature so they're still relativistic but now they're officially massive then there's a series of big events series of very fortunate events so at temperatures of around 10 to the 8 electron volts that's about 100 mev you'll sometimes hear it said this is where um qcd quantum chromodynamics the strong nuclear force becomes strong you might remember or you might not in that gauge theory video uh spontaneous symmetry breaking in the higgs phase was one thing we talked about another thing we talked about was the coulomb phase where the force lines of a gauge theory can extend out infinitely far you get a coulomb like force the one over r squared newtonian kind of potential uh or force i should say and there's also a confining phase where the field where the gauge fields interact with each other so strongly that they just bundle up into little particles rather than stretching out at temperatures and we also said i forget whether we said this i hope i said this qcd has the very special feature that is called asymptotic freedom when we talk about renormalization we talked about the fact that the coupling constants of theories could have different values at different energies it's also true at different temperatures because temperature and energy are kind of the same thing at high temperatures the strong nuclear force is weak it is not very strong it goes to zero strength at infinite energies or infinite temperatures at low temperatures it's strong so this is the cut-off this is the dividing line at temperatures above 10 to the 8th electron volts qcd is weak and therefore quarks and gluons are not confined into hadrons they are freely floating at lower temperatures at temperatures below 10 to the 8th gev now quarks and gluons confine into hadrons so quarks and gluons confined higher temperatures they were separately moving particles lower temperatures they're bound into protons and neutrons and mesons and things like that that's the next big event um what's next at temperatures of around 10 to the sixth electron volts there's a there's a series of things also going on here as particles go from being uh relativistic to non-relativistic they decay away and there's some or they annihilate away and sort of the last big event of that form was near one mev where electrons annihilate away so e plus e minus annihilation and this is basically the last time in the history of the universe that a whole bunch of particles annihilated away okay electrons are the lightest particles that can still annihilate away neutrinos are lighter but they don't annihilate away because they interact too weakly so there's still a gas of neutrinos all around you there's a cosmic neutrino background but electrons and positrons find each other and they do an island away at around their mass which is around a million electron volts so after that now radiation by radiation to cosmologists we now mean photons and neutrinos at higher temperatures we meant all sorts of other particles but at temperatures below an mev it's photons neutrinos you might ask why not gravitons you know shouldn't gravitons like photons count the answer is because remember we mentioned if particles interact weakly enough then they might never have been in thermal equilibrium they might never have been produced or they might never have annihilated into other particles gravitons count gravitons interact so weakly there's no reason for them to ever have been in equilibrium so the actual particles that are there in the background are mostly photons and neutrinos as far as we know there could be some gravitons there that we don't know about but the dominant sources are photons and neutrinos to be said another way you know those electrons and positrons just like up quarks and down quarks and taus and so forth these all these particles existed and when they annihilate they annihilate the photons they don't violate the gravitons so even if there were a lot of gravitons in the very early universe by the time you've been through all these stages you're left with photons and neutrinos that's what that's what you're actually left with and you can calculate exactly how many and this is one of the things you learn to do if you're a cosmology graduate student you calculate the ratio of neutrinos to photons in the universe and then there's a crucially important phase at ten to the five electron volts okay uh which is big bang nucleosynthesis big bang nucleus bag big bang nucleosynthesis i think i have that temperature right something like that i should actually check that um so or you should check it there you go homework check that is the right temperature but the point is it's in the right order anyway big bang nuclear synthesis is when protons and neutrons can come together they could always come together you know and you know you can make a proton neutron make a deuteron but at temperatures higher than this all the photons hitting them would just tear that apart right away so they wouldn't be stable so after this the temperature is lower than this protons and neutrons can come together to make deuterium helium and lithium okay now you might say so there's another question here um well good these are like the lightest things you could have made out of protons and neutrons why did you stop there why don't you make carbon and oxygen and so forth so this is all this theory is worked out by george gamows ralph alpher and i think it's robert herman 1950s and 60s they're really the pioneers of the big bang by the way there's a shameful story about nobel prizes here you know edwin hubble never won the nobel prize for discovering the expansion of the universe gamov and his collaborators never won the nobel prize for the big bang theory uh even though they should have it's kind of a weird thing that i don't quite understand but anyway um we live in a world where in the universe where there's a whole bunch of different kinds of chemical elements right there is carbon and there's all the way up like iron and uranium and lead in in the universe here on earth where did they come from if the universe started with nothing but protons and neutrons it was very natural to think that they were produced in the big bang right because you know the particles protons neutrons come together and make heavier elements until eventually the universe cools down when you run the numbers it doesn't work that way because there's a competition yes protons and neutrons are coming together to fuse together to form heavier elements but the universe is also expanding and becoming less dense and all of these nuclei are positively charged right electromagnetically they're repelling each other not attracting so it just very quickly you go from a phase where protons and neutrons can't even stick together to a phase where they can't fuse together anymore in the first place and in that tiny window which is the big bang nucleosynthesis window when the universe was a few seconds or a few minutes old that's when you could make elements you end up making mostly helium a little bit of deuterium and lithium that's why steven weinberg's famous popular book on cosmology is called the first three minutes that's when nucleosynthesis was going on and this is worth a whole you know if i were more ambitious or didn't have other things to do big bang nucleosynthesis sounds pretty down to earth but oh sorry i didn't finish the story um therefore you didn't make all the heavier elements in the big bang you make them in stars so roughly speaking you know the universe is mostly hydrogen and helium that helium was produced in the big bang these trace amounts of deuterium and lithium were also produced in the big bang but everything else was produced in stars either exploding or just spitting out heavier elements so the interiors of stars or supernova explosions were necessary to produce heavier elements okay and this is worth a huge amount of exploration because we know enough physics to say more or less exactly what was happening at big bang nuclear synthesis so we're talking literally a second after the big bang or a few seconds after the big bang and this depends on the expansion rate of the universe which therefore depends on the energy density of the universe which therefore depends on photons and neutrinos right so you say well there are three families of neutrinos people asked in other q and a videos how do you know there are more families of elementary particles right we have three generations or three families how do you know there aren't four well if there were four light neutrinos the energy density of the universe would have been a little bit different during big bang nucleosynthesis because that fourth neutrino would also have contributed to the radiation energy density and that means the hubble parameter would have been a little bit different and that means the expansion of the universe would have been a little bit different and that means the relative abundance of hydrogen helium lithium and deuterium predicted by big bang nuclear synthesis would have been different and when you compare to the data the prediction with three neutrinos is right and a prediction with four neutrinos is wrong furthermore even better you say you know you think you know you have this dark matter stuff how do you know this dark matter stuff isn't just protons and neutrons that you haven't found maybe they're just dark well the ratio of the number of protons to the number of photons is crucially important for big bang nucleosynthesis if there were fewer photons they'd be less efficient at breaking apart the nuclei if there were more photons they'd be too efficient at breaking them apart we know how many photons there are in the universe because we can look at them they're mostly in the cosmic microwave background therefore if we know how many photons there are in the universe and we know that big bang nucleosynthesis works we know how many protons and neutrons there are in the universe we know the total amount of ordinary matter in the universe and it's much less than the total amount of matter that's one of the best reasons to think we need dark matter so this big band nucleosynthesis story is incredibly important and i encourage you if you're not already familiar with it read all about it it's really one of the triumphs of modern cosmology you know i remember again when i was in grad school cosmology was in the process of going from slightly disreputable to a really central part of modern physics you know really is as as recently as the 80s and 90s people were a little skeptical and i won't name names but you know i ran into famous particle physicists who were like you don't believe that big bang synthesis stuff do you like that that really constrains the number of neutrinos it really does it really did it got everything right you know uh david schramm and other people get a lot of credit for figuring that stuff out okay um then when the temperature is of order uh 10 electron volts between 1 and 10 electron volts finally nuclei mostly hydrogen and helium in fact mostly hydrogen and electrons can come together to form atoms and this is the process known as recombination some people make fun of the word recombination in this context in laboratory physics when you take an atom and you ionize it so you remove the electron and then you let the electron come back you call that recombination so therefore in cosmology when the electron comes back to the atom we also use the same word recombination so some people try to be clever and say shouldn't it just be combination since it's the first time it ever happened so if you ever meet one of those unfortunate people who are trying to be pedantic and annoying you can prove that you are even more pedantic and annoying by saying actually recombination is appropriate because a typical atom an electron or a typical atom let's say will gain an electron and then lose it and gain it and lose it many many many times before finally it gains its final electron in the history of cosmology so it is actually recombination uh but it's a crucially important stage in the history of the universe because when those electrons are flying freely the universe is opaque right photons keep bumping into those electrons so you wouldn't be able to see your hand in front of your face uh before recombination after recombination when the universe is all atoms now photons can travel freely so this is when the universe becomes transparent and what that means is all that light all those photons that we had running around all through space can suddenly stream freely through the universe and we call that the background radiation the cosmic microwave background and this you know helps us understand a little bit about people ask like why is the cosmic microwave background hitting us right now you know how does it know so let's just think about exactly what's going on here again we have our usual space time diagram for the universe okay space and the point is uh we have photons going in all directions at every point in space okay there's no special point in space no central point or anything like that as space expands and the universe cools photons are still there they're now slightly longer wavelengths same number of photons i have to draw the same number of arrows whatever that was uh one two three one three four there you go okay but they cool down the temperature goes down right so they were short wavelength you know they were 10 electron volts the temperature today 2.7 kelvin which is about 10 to the minus 4 electron volts here the temperature was between 1 and 10 electron volts and this this moment so now the universe is 14 let's say 13.8 i think billion years giga years after the big bang uh and here this moment of recombination is 380 000 years after the big bang so it's been a long time this is still relatively early in the universe's history and here we are you know today and we look back at our past light cone so our light cone intersects this recombination uh moment in some sphere and so it doesn't matter where we were if there were some other if there were some aliens who lived here in some other galaxy they would look back and their light cone would intersect the surface or this time of recombination and they would see a cosmic microwave background there's always some photons moving exactly along those light cones no matter what lycone you're talking about so it's not that the cosmic microwave background is literally a sphere it's just that the sky looks like a sphere to us everywhere in the universe the universe is suffused with these relic photons from the big bang that we call the cosmic microwave background okay it's the same overall features for the cosmic microwave background in all places it's a snapshot of what the universe looked like 380 000 years after the big bang so the two big events that we love in observational cosmology are big bang nuclear synthesis and recombination because those give us observational data about what the universe was doing at early times and the big nuclear synthesis is still a story of um a uniform universe you know that we the differences in density from place to place are not that large but the microwave background recombination now the tiny differences in density from place to place suddenly become important so that's why the cmb is like the growth industry like this is where well i mean it's already grown it's a huge industry if you want to be a modern theoretical cosmologist you got to know the cosmic microwave background in and out and this leads us directly into this uh the next feature of what we're saying here which is that so far we've taken very seriously the idea that the universe is perfectly smooth okay and that gets us very very far so to sort of in the last few last few topics here very quickly let's let the universe be a little bit lumpy right the universe is not perfectly smooth so the universe isn't perfectly smooth and we can characterize that quantitatively we can say that there's a difference in density from place to place call that delta rho okay so delta rho well you know the energy density itself which is rho that's changing as the universe expands likewise delta rho is changing it's hard to remember what it is so we take delta o we divide it by rho so this is the fractional difference in energy density from place to place in the universe so delta rho of x and roughly speaking on average in the early universe it's ten to the minus five okay so one part in a hundred thousand so if in one region of space there is a hundred thousand atoms then in a nearby region on average they'll be a hundred thousand one or ninety nine thousand nine hundred ninety nine okay they'll be fluctuations of order one part in a thousand overall and what we can do this is a very traditional physics-y move to analyze you know what this means is well sorry before i analyze it let me just mention something that should be obvious but you know maybe it's worth saying um these tiny perturbations this is literally a perturbation by which you mean a tiny change 10-5 is a small amount the tiny perturbations grow by uh gravitational instability which is a fancy way of saying that a slightly over dense region will pull itself together under gravity and also pull gas and dust and matter from regions around it so it will become even more over dense whereas an under dense region will lose its matter to nearby overdense regions so it becomes even more under dense so that's why it's an instability okay the tiny variations will grow with time okay and that they grow into galaxies and stars and planets and things like that so it's kind of a big deal the universe right now 14 billion years after the big bang is much lumpier than it was in the very early universe but i don't want to that's i just wanted to say that so you know what's going to eventually happen but i'm not going to go into a lot of details about that let's think about the early universe let's think about these tiny fluctuations in density in the early universe and let's analyze them that's why i used the word analyze before so how do we analyze it well we don't know what the density of the universe is at any one point right like that's just a uh it could be big could be small who knows how do i even know what point we're talking about what we do is same thing we do with wavelengths of light we do a fourier transform we look at modes of different wavelengths and so we analyze the fluctuations in density not from point to point in space but scale to scale we say on large distances what is the variation in density on small distances what is the variation density and so forth and at early times we think and we'll go back to why that's true at the end but we think that it was basically the same story at every scale there were scale-free perturbations in density so density perturbations are approximately not exactly and that becomes important but not for us we're not going to that much detail scale free which is a fancy way of saying the same amount of fluctuation at every distance that we might want to quantify it at okay so what you can do is say well let's analyze that by looking at the cosmic microwave background you could analyze it today by looking at galaxies right and you do believe me that's large-scale structure study is one of the things that you do as a working cosmologist but it's much harder it's much messier because by now the this gravitational instability has led to an enormous variation in density like the center of the sun is way more dense than the space in between the galaxies okay that becomes harder to analyze it's this small variation that you have at early times this i should say like early primordial small variation that's truly a perturbation anything that's just a small variation from simplicity is much easier to analyze than large deviations from simplicity so even though large-scale structure and galaxies are an important source of cosmological information when the micro background was formed the snapshot of the universe at early times is way easier to analyze and to predict and so that's why it's been such a treasure trove for cosmologists you know it was only 1992 uh that the kobe satellite actually measured slight deviations in temperature in the cosmic microwave background so uh the cmb causing microarray background has the property that these fluctuations in density from place to place translate into fluctuations in the temperature of the cmb we said you know 2.7 kelvin on average is the temperature that you get from the cmb but it's not exactly that for a long time since we discovered in the 60s up until 92 it was exactly that and finally we found small variations delta t over t of order ten to the minus five okay and you can look at them at different angles on the sky so basically you can ask this question about density perturbations as a function of distance but you can do it on the microwave background you look at points close to each other what is their variation in temperature look at points far away what is their variation temperature there's math involved spherical harmonics and things like that but roughly it's the same kind of story that you would do with fourier transforms or whatever so this becomes an industry predict delta t over t as a function of angular scale on the sky so we're going to do that we're going to predict it we're going to get it wrong then we're going to fix it okay so here is what you might guess very very naively i already said look perturbations are approximately scale free so that's density perturbations so therefore if i look at c and b temperatures delta t over t as a function of angular scale and so these are by the way the usual way that we plot this is large scale and small scale large scale on the left small scale on the right why well because there is a largest scale on the sky 180 degrees right whereas small scales can go infinitely small so it makes more it's just more convenient but large scale uh near the beginning of the graph what you might predict is this right that's your guess that you get the same amount of temperature variation at large scales intermediate scales small scales because the underlying density perturbations are the same right there's there was some mechanism i'm being intentionally mysterious about this there's some mechanism that created i guess density perturbations that were similar at all scales the reason i'm being mysterious is because most cosmologists at least many cosmologists think that inflation in the very early universe was the mechanism that made these perturbations we don't know that's a speculation that's something we don't know but as observational cosmologists we know that they were approximately scale-free and you might guess that those approximately scale-free primordial perturbations turn into a flat uh temperature variation as a function of scale when we look at them in telescopes on the sky but that's not true that's not exactly what we see it's a little bit slightly more subtle than that and the reason why is because perturbations perts evolve okay like we said if you have a slightly over dense region it will become more over dense gravitational instability so there turn out to be a number of different mechanisms by which the perturbations evolve one is just exactly that gravitational instability so [Music] over dense regions contract not literally contract in size but you know the matter in them contracts and becomes more dense so you have some region of space you have a region inside where there's more particles than average there are some particles outside but not as many and the over dense region under the force of gravity brings together even more particles into it okay that's one thing that happens uh another thing that happens is well it's the real world you know this is why cosmology gets messy because all the physics matters so if if some matter contracts it's kind of like you know you have a gas in a piston and you're you're compressing it right and you know when you have a gas in a piston you compress it it heats up and you also know that when gas heats up it pushes back on you there's pressure that resists right it doesn't just collapse to a black hole there's in a in a piston it will push back on you you'll become too weak to keep pushing it eventually so that will happen here in the real world when particles come together they will bump into each other they will heat up the radiation will start pushing back on them there will be a pressure restoring force so we're thinking in our minds here about times before the microwave background was formed okay so we're going to see what is the effect on the microwave background so we're talking about times in the history of the universe before electrons and protons and helium atoms came together so the universe is not transparent it's opaque so those electrons and protons are bunching together the photons keep buffeting them around so the second effect is contracting regions heat up and develop pressure which means that they bounce back in fact you might think that it sounds kind of like a sound wave or the ringing of a bell because it exactly is that's exactly what it is it's an acoustic oscillation a sound wave is just when you know here in the air in the room when you're listening to me i make slightly higher density of air molecules and that sort of pushes on the air molecules nearby and in any one region any one point in space the density goes up and down right it vibrates up and down when you have a sound wave uh or literally when you bang a gong the gong will vibrate and send out sound waves and they go up and down in density as as time goes on so that's what happens in cosmology so you have this plasma right all the electrons are freely moving they have electrons and you have protons protons just kind of go along for the ride because they're heavy the electrons and the photons are what are doing most of the work here and so the dense region increases in density the photons heat up and they push back and in fact it's so pronounced that the region that was over dense pushes back and becomes under dense now so it really is oscillating back and forth exactly like a sound wave because it is a sound wave finally effect number three those perturbations get damped because radiation flows between them in other words if i have some region where it's contracting the density is going up heating up and it's pushing back so now i have a difference in temperature from this region to the nearby region which means there are more photons here in the dense region but photons move at the speed of light you know it's true they keep bumping into electrons so that's fine but if they're not bumping into anything they leak out of that region the photons in some sense don't want to gravitationally contract photons want to be smoothly distributed so as long as photons are important and here they are um the photons are going to damp out these oscillations so when i said it's like banging a gong it really is because you know if you bang the gong it vibrates back and forth but the vibrations dampen out right you know the photons want to restore everything to a smooth configuration and the final thing to keep in mind so so what we actually get is this in the real world here's our plot of your small scales large scales again this means on the sky when we're looking at it here's delta t over t sorry making it this way uh here was our dumb naive prediction that it was completely flat the other thing to keep in mind is larger sized regions evolve more slowly okay because everything here is evolving you know pretty quickly like relative to the speed of light you know things are actually moving uh comparable in because photons are moving around and things like that it's not at the speed of light but that's that's a limiting factor and so large regions you know it just takes a long time for them to oscillate and bounce back and oscillate whereas a small region can go boom very quickly right so in the small scales what you see are regions where they've oscillated and damped a lot whereas on large scales they've barely moved and on intermediate scales maybe they've oscillated and damped once or twice or three times okay so here you have you know few or no oscillations really let's just say few oscillations here you have a yeah let's say no oscillations down here at the the very large scale here you have a few and here you have many oscillations okay so given these three effects that over dense regions contract they heat up and they bounce and they actually become rarefied and then those oscillations are damped what you would expect is instead of this straight line at very large scales you get the straight line because you just haven't had time to evolve these these scales that you're looking at the microwave background are literally larger than the size of the observable universe when the micro background was created okay the universe was big this at early times the observable universe at those early times was much smaller than it is today because we just haven't had time to look at it they they haven't had time to look at it so these large-scale perturbations we see are just whatever they were primordially they haven't evolved and as we go to smaller scales then you get to a region where you know if you look at the universe was 380 000 years old so once you look at regions which are approximately 380 000 light years across that's a region where you've had time to contract just barely right and then you haven't even had time to bounce back yet and that contraction increases the contrast increases the density perturbation increases the temperature perturbation so what you expect is something like this that it goes up okay but then it begins to relax it bounces back so you expect that it begins to go down again and then what you're going to see is it's going to oscillate back and forth but those oscillations will be damped so by the time you're down here you're down at tiny little wiggles at very very small scales because they've oscillated and reached some equilibrium and in between you get something that looks like this so you get a series of oscillations that are damped and this is what you guess this is what you so this is well we guessed before the straight line this is a much more sophisticated guess and you can actually calculate something like this and ask whether it's right or wrong then you go observe it right then you do the experiment this is what we do we're this is i'm not telling you a historically accurate story but this is a story that could be accurate if you had slightly different thoughts um you go observe it and you ask what you see in the sky and you have a very very specific prediction compared to what you see this is not precisely what you see in the real cosmic microwave background sky it's almost what you see but it's not quite let me show you i'm going to try my best to use my drawing skills here to show you what you actually see here's what you actually see let me make it a little bit thicker also just so it shows up here's what you actually see in the sky okay what you're supposed to get from this i hope that you know more or less was accurate um the first peak is a little bit higher than you expected the second peak is a little bit lower than you expected the third peak is a little bit higher the fourth peak is a little bit lower see i did it wrong already i did it wrong it's supposed to be lower what's supposed to be the case is that the odd numbered peaks are higher and the even numbered peaks are lower so what we what we actually observe is that odds are boosted and evens are suppressed now remember i'm not telling you a historically accurate story this is not how the story uh came to be but this is a good way of thinking about it if this is what you discovered uh you'd be puzzled like you're literally you have a oscillating wave going back and forth and it's being damped and you know that damping should be more or less uniform i mean how in the world does it know that odd oscillations and even numbered oscillations are different but if you think about it there is a difference because remember that first oscillation this big peak right here that's when an over dense region first collapsed and we said that since it really is an oscillation when it bounced that this second peak so you know this is let's pick us a certain mode well what what should we say here um let's talk about regions that were initially over dense they're also underdense regions and they do the complementary thing but this is a region that is over dense over and then the second peak is when it's rarified okay in other words it's under dense now the a certain region will collapse become overly dense and then it will actually empty out more than the average so it goes to being less dense than the average so there is a difference between the odd-numbered peaks and the even numbered peaks the odd-numbered peaks are ones where an initially over dense region is moreover dense even-numbered peaks are where initially over dense regions are now under dense okay so you ask yourself well what what difference could that make you know i mean it's bouncing but it might strike you well what if it wasn't bouncing what if there was something in the universe that didn't bounce like that in other words some kind of particle some kind of matter that was contributing to the energy density uh that felt the force of gravity so it could shrink under the gravitational pull and it could you know this first thing gravitational instability happened but this second thing where contractions heat up and bounce didn't happen in other words some particle that was you know electrically neutral right a particle that did not interact with photons some kind of matter that didn't have electromagnetic interactions and therefore it couldn't heat up but didn't feel the pressure if you had particles like that let's call them dark matter particles because they don't interact with photons right if you had dark matter particles what would they do well they would just contract they don't bounce dark matter doesn't feel pressure it's pressureless so ordinary matter contracts and then bounces and then contracts again then bounces whereas the dark matter just contracts in that one region of space and that gives a difference to the odd-numbered peaks and the even-numbered peaks the difference is on the odd-numbered peaks the behavior of the ordinary matter is synchronized it's in phase with the behavior of the dark matter and in the even-numbered peaks they're out of phase and the even-numbered peaks the dark matter is combating the in homogeneities in the ordinary matter and in the odd-numbered peaks they're working together so it is a generic prediction of the idea of dark matter that you should see a difference in odd numbered peaks and even numbered peaks in the cosmic microwave background and that is exactly what you really observe in the data okay the reason why i'm telling you this story in this quixotic historically inaccurate way is because the first evidence we had for dark matter was in things like clusters and galaxies and stuff like that okay and it very naturally led people to say look you say that there must be more matter in the galaxy because the rotation curve or something like that there's more gravity there than we can account for using ordinary matter so the first thing they say is probably there's more ordinary matter than you think right nowadays we know that can't be right because of big bang nuclear synthesis the number density of protons and neutrons versus helium and hydrogen would be different versus helium and lithium and ethereum would be different so we know that it's not because we're missing ordinary matter so the second thing they say is well maybe gravity is wrong you don't understand gravity uh and so they invent theories like mond modification of newtonian dynamics that purport to explain the rotation curves of spiral galaxies now there's a lot more evidence for dark matter than the rotation curves of spiral galaxies there's gravitational lensing there's cluster dynamics there's the growth of structure et cetera et cetera but you could always try to wriggle around all those things and find a way to mess with gravity to duplicate them the cosmic microwave background is crystal clear evidence that matter is not ordinary matter protons neutrons and electrons is not the only source of gravity if it were you would see a uniform dampening of the oscillations in the microwave background if you have another source of gravity that does not oscillate then you'd expect to see this difference between odd and even numbered peaks which is exactly what you see therefore the idea that there is no dark matter in the universe is wrong it has been ruled out by the data it might be that gravity is modified also in addition to having dark matter but dark matter certainly exists or as certain as you can be in an experimental science like cosmology now people are still trying right so jacob beckenstein years ago i mean so i had to back up the problem with mond as a problem as a proposal for modifying gravity was it wasn't really a theory it was a phenomenological relationship but it wasn't like there was a lagrangian like you've all watched all these videos you know that if you want a new model in theoretical physics what you need is a lagrangian that you can get equations of motion from etc and mon never had a lagrangian so beckenstein uh in fact there was a joke there was something called bekenstein's theorem when the theorem was uh there can't be a lagrangian vermont because jacob beckenstein tried to find one and he failed and he was very smart so it's not possible to be done the problem with that joke was that beckenstein had not tried to find a lagrangian and when beckenstein heard about the joke he's like i never tried to do that and he tried and he succeeded because he was very smart to jacob beckenstein so he invented a theory called tensor vector scalar gravity which was an ungodly mess right it's like general relativity plus a whole bunch of other fields interacting in some very very specific ways and he was able to reproduce the predictions for spiral galaxies that you get from mond so it was a sort of respectable version of mond and it made predictions for the cosmic microwave background before we had measured the actual uh density perturbations and they were wrong because of course they were wrong cause there was no dark matter in the model but secretly there was dark matter because you know there's a whole bunch of other fields in his theory right and these fields are sort of mimicking up the effects of modified gravity but they are fields that have their own energy density that propagate independently from protons neutrons and electrons and that act as a source of gravity to anyone else we just say that's dark matter that's what it is so you may hear people claim oh i have a theory of modified gravity that gets the microwave background right the question to ask them is how much dark matter is in your theory and by dark matter i mean not just what you call dark matter what i mean is how many sources of energy density do you have in your theory because without them you're never going to get this right this is the reason not spiral galaxies or gravitational lensing this is the reason the microwave background is the reason why we think that what we're really seeing in cosmology is dark matter not to mention the fact there's a dozen other tests that all give the same predictions so dark matter is real by all by all means there could be other things going on there could also be modifications of gravity but dark matter is really there so the what we're left with is an inventory for the universe where the universe is uh five percent sometimes people say four percent but i think that gives a an impression of precision that isn't really there so five percent is a rounder number ordinary matter that's what you need to fit nucleosynthesis and direct measurements and things like that so by ordinary matter we need protons neutrons electrons um you know i guess there's radiation right the radiation is something like 10 to the minus 4 radiation today a tiny little sliver the radiation the photons and neutrons were photons and neutrinos were crucially important in the early universe they're less important today 25 roughly speaking of the energy density universe is dark matter and again we don't know what it is so it's perfectly you know good to wonder what that is but it's some particle that doesn't interact with photons doesn't feel the restoring pressure doesn't form atoms it's not noticeable in experiments yet the problem is that that's too broad a category it's too easy to come up with theories of what dark matter could be so we don't know exactly what it is yet and then 70 dark energy we call it dark energy um but it's almost certainly the cosmological constant you know as soon as we proposed we found in 1998 that indeed the universe is accelerating that's why we have evidence for dark energy and everyone said okay good the cosmos constant is not zero but you know most of us had thought including me absolutely that there were good naturalness arguments that if the cosmos constant was not huge remember we we talked about the cosmos constant problem in quantum field theory in effective field theory the cosmological constant should be huge and it's not so there's probably we said some symmetry that sets it equal to zero okay and we are wrong about that so because we actually found the cosmological constant people said look we were wrong before let's be cautious let's not assume that it's the cosmological constant so people including me made predictions or built models for dynamical theories of dark energy where the dark energy was almost constant over time but not quite and those theories made predictions and those predictions have so far turned out wrong that doesn't mean that they will continue to turn out wrong just like you know super symmetry or string theory or whatever sometimes the theories are just hiding from you but still as we talked about when we talked about probability and statistics um when you have a theory that you could have discovered evidence for and you didn't your credence should go down so i think that the chances that the dark energy is really something dynamical versus something absolutely constant or lower now than they were 20 years ago or 22 years ago when we first found the cosmodrome constant and i say that this is what we call in a court of law testimony against interest because i would love it if the dark energy were dynamical i would love it if gravity were modified instead of dark matter these are all ideas i spent time thinking about but when the data come in and i have to change my credences that's what i try to do so anyway that's cosmology as we understand it right now as i said we've left out the very very earliest times in the history of the universe you know we have this plot of scale factor versus time and we said well what does it do well the answer is a goes like t to the one half in the radiation dominated era then it goes like t to the two thirds in the matter dominated era so this is rd md and these days it has begun to go like this e to the ht in the vacuum dominated era okay as far as we know well can he do that forever right vacuum energy doesn't decay away it can if there are noticeable changes in the underlying fundamental physics so i recommend that you either go listen to the podcast i did with katie mack or buy her book she just had a book come out about the future of the universe and how it can end where we talk about the future but we also care about the past you know if you extrapolate backwards the simplest thing is you hit zero at some time and you call that the big bang and we don't know what happened at the big bang or we don't even know if it was there there's another possibility which is that i'll make a different color which is something called inflation right inflation is basically the idea that there was a temporary period of dark energy in the early universe and in that case let's blow it up here in that case there was this e to the h t behavior with a much larger value of h so inflation looks like this so inflation and then who knows what happens before then dragons there be dragons before that but inflation is the idea that there was this form of energy that dominated the universe made it expand exponentially and accelerate and then that energy converted into radiation and matter and that's inflation and it's a whole another complicated thing that you know i've written a lot about and you can read about i wrote about it from eternity to here for example but we don't know if it's true you know i think that personally 50 chance that inflation is true inflation is so good it's really compelling theoretically for various reasons but it it raises new questions we don't know the answer to you know like as as you know from the entropy video our big question about the early universe is why was it low entropy inflation by itself provides zero answer to that question in fact it makes it worse the state of the universe you need to start in to get inflation to go is even lower entropy than the conventional big bang model would have you predict so i even though i like inflation i don't know that it's true and it's certainly an extrapolation well beyond the physics we understand so i'm remaining agnostic about it but it has become the working hypothesis for the mo for the majority of contemporary cosmologists why because it predicts this curve bang on right this microwave background prediction starting from scale-free or almost scale-free remember the exact thing that inflation predicts is approximately scale-free density perturbations and that is exactly what we observe in the cosmic microwave background so that's a prediction that has been tested the problem is it's a simple prediction approximately scale-free density perturbations you could imagine that there are other mechanisms that also produce those so we don't know so i like to remain agnostic about it but you know progress is being made data-driven progress gravitational waves have now come in we're still measuring the migrate background we're still learning a lot about the early universe i am optimistic that this this mystery that we have why the universe is made to the stuff it is where it started from why it looks the way it does these are questions we're going to answer maybe not in the next 10 years maybe some young person who is watching this video will be inspired to think about these and come up with the right answer and then i hope you remember what inspired you to come up with those answers

Channel: Sean Carroll

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Id: tZQadPmTd84

Length: 119min 14sec (7154 seconds)

Published: Tue Aug 18 2020