Edwin Hubble discovered that Andromeda is not a part of the Milky Way, and expanded our universe exponentially. He classified galaxy types, and neither of those is what he's really known for. What he's really known for had to do with measuring distances to galaxies. Again like he did to Andromeda, but to galaxies at a larger distances. so we're starting to measure distances at which finding cepheides becomes difficult. With todays' technology we can measure cepheides out to, as I say Maybe 10 million parsecs or 20 million parsecs. But at the time, this was still difficult. Astronomers needed ways to measure distances to galaxies. Galaxies are easier to see than Cepheids, they're more luminous. What you'd like is a period luminosity relation for galaxies. And indeed, there are such, at least approximate relations. For spiral galaxies the Tully-Fisher relation relates the rotation velocity in a galaxy to it's luminosity. and there's a Tully-Fisher relation for every kind of, every type of galaxy. So S B galaxies have a different Tully-Fisher relation than SA. Galaxies and so on. And roughly, the idea is that, larger galaxies are more luminous and have larger rotational velocities. And you see here the accuracy with which this is relation holds. It's not a very accurate distance measurement, but it tells you that if you can measure with Doppler shifts, the rotational velocity of a galaxy You can classify its type from its shape. You can now figure out the luminosity of the galaxy. You have galaxies as standard candles. This are very brilliant standard candles indeed you can see them out, to very large distances and you can measure distances because if you have luminosity and you measure brightness, you can figure out the distance to something. There's an analogous relationship for elliptical galaxies which allows the distance to ellipticals, to be measured, today at larger distances yet, we can measure with type 1a supernovae. But remember, that if you want to measure the distance to a galaxy in which alas, there is not a type 1a supernovae that has been observed. Then you have been reduced to variable stars or these kind of methods and refinements thereof. So, using all these methods Hubble goes around and he measures the distances to many of the galaxies and, of course, try to understand how they work and measures their spectra, and this spectra take a while to identify because galaxies are moving at rather high velocities relative to each other and therefore some are approaching us at high velocity. Some are receding at high velocity, and you have a case of big red shifts or big blue shifts, and you need to figure out which spectral lines are which. Astronomers, as I said, are very good at spectrometry. And what Hubble finds when he starts working through these spectra Is that is, it looks farther and farther out into the universe, and galaxies that are more distant, less and less of them appear to be having negative radio velocity. Moving towards the Milky Way, more and more of them appear to receding and furthermore, the farther they get, the faster they appear to be receding, and so, here's a nice plate with an example of the The shifted spectra that Doppler is, that Hubble is measuring, so here are four images of four galaxies they become smaller, appear smaller as they get more distance in these negatives. And here are their spectra, compared against some standard spectrum above and below. And we're looking for some particular line of potassium here. And what we see is that the farther the galaxy is, the farther to the red this line has shifted. And the bottom image gives you a sense of how good these guys were with spectra graphs because it's not totally obvious that this little gap right here is a spectra line and furthermore that it is the potassium line. And it turns out that it was. And so, with these Hubble can relate that known wavelength of the sode, of the potasium line to the wavelength that which he measures it. He can use the relativistic version of the doppler shift formula or the Newton approximation. Non-realistic expression. Both of those are things we are familiar by now. And ye can convert the observed redshift into an observed recessional velocity. And the statement that the redshift is increasing with distant is the statement that the farther the galaxy is, the faster he observes it moving away from us. And so, here is Hubble's collection of data from the paper. that he published, and what we see here the horizontal axis is distance. The vertical axis is recessional velocity, radial velocity. We see that here's a galaxy that's actually approaching the Milky Way. That is fine, but that the farther You get the more the galaxies are all receding, and Hubble very bravely draws a straight line through this bit of data, and declares that he's found a relation between distance and relational velocity. Notice the Hubble Law, this is what he's famous for, the recessional velocity of the galaxy is given by a constant, Hubble's Constant, times its distance from the Milky Way or the Sun, it's the same thing. And Hubble's Constant is therefore, conversion of distance to velocity, so 1 way to express its units, because that's the way we measure it, is a kilometer per second recession. Per megapersec distance, and Hubble has an estimate for the value. he this estimate has been refined over the years, we have a lot more data. but this consant is of such importance that long before its exact value was known Astronomers needed to use it in their expressions and so they developed a method of sort or parameterizing their ignorance. The Hubble constant is characteristically writen and as 100 times h kilometer per second per megaparsec. In other words if someone were to measure the Hubble constant and its value were to be discovered to be 100 kilometers per second per megaparsec, we'd say h is 1. And we'd go around into all the formulas, and all the books and papers that have various powers of H in them, and set H to 1 and get an answer, but at the moment, until we're absolutely certain of the value of the Hubble constant, people use this little h to parameterize. The dependence on that measurement because so much depends on this, as we shall see, but the value of h is not too far from one, We now have far better measurements that hubble had, we have measured much more statistics, the most recent measurements suggest a value of H on the order of point seventy one, and as you see we know it quite accurately, We have here a plot of recent measurements in the vein of Hubble of recessional velocities, a function of distance out to 1.6 billion light years away, so 500 megaparsecs. We see that Hubble's relation, linear relationship Holds up very, very well what's exciting about this plot is that little rectangle at the lower left, that represents the data that Hubble actually had. So he was predicting the straight line velocity on the base, the straight line plot, and it's slope on the basis of a very small amount of data and was gloriously vindicated. Now typically in astronomy we write the Doppler shift as lamda is 1 + z * lamda 0. And we call z the red shift, so z is a positive number. And assuming something is red shifted and z = 0 is a red shift of 0, in other words lamda = lamda 0. And so clearly z is related to velocity. The expression is written down here, lets figure it out. So, lamda / by lamda 0 is 1 + z. And lamda / by lamda 0, remember we had lamda / by lamda 0 is the square root of 1 + v / c. This is the completely Relativistic formula, and this is supposed to be 1+z. So, the first thing I do, of course, is I, I want to figure out how z determines v, and so I want to solve this for v over c, and the first thing I do, of course, is I square both sides. And I write that, 1+z^2=1+vC/1-vC. Then I multiply both sides by 1-v/C, and I find 1-v/C. *(1+z)²=1+V/C and then, I move all the V/C over here, so I have V/C * (1+(1+z) ^ 2) and then everything without V/C goes over here and that (1+z)²-1. So, I solve for V/C and I get that V/C = (1+z)²-1 /(1 + z)² + 1 and again if z is small, if the red-shift is small, you can use Newton's approximation to show that this is approximately z. Which is indeed when this is approximately 1+v over c, lambda 0, then z is approximately v over c, but this is the exact relativistic expression. So, astronomers will rarely tell you, unless they're talking to, you know, the Hoi polloi like us, that an object is at a distance of 2.8 billion parsecs, they will tell you that it has a redshift of .3. The reason is that we'll see next week that distances are ill defined but certainly redshifts are measured, observable quantities and The larger H redshift and so we basically can speak about redshift as a surrogate for distance and indeed the Hubble Law tells us, remember we solve for v over c as a function of z here, but the Hubble Law tells us that v over c is the same as H0 D over c. So if you know z you can figure out D as well as v, so. Knowing the red shift tells you not only the speed obviously with which the galaxy is receding, but also its distance. So we have to digest this. In whatever direction we look, galaxies are receding from us. At the rate of recession does not depend on the direction of the galaxy. Galaxies at the same distant at all directions are receding with a equal velocity, but the farther they are the faster they are receding. Now, depending on your taste, you could either conclude that were not popular galaxy, or more technically that we're in the center of some cosmic expansion, everything is moving away from us. It turns out that this is not exactly correct. Your intuition is in this case actually misleading you. Imagine this, collection of galaxies here they are all going to be moving away from each other. The center of expansion could be the center of the screen, or if I take this and add to it a motion of everybody to the right, the center of expansion could be way over to our left. Each galaxy, as it observes each of the others will see that each of of the other galaxies is growing farther and farther away from it. So all that is going on is that as these galaxies grow farther and farther away, all of their distances are increasing and the fact that in this case. It looks like the expansion is about the point of the center of the screen is completely irrelevant. They are all moving away and the farther they are, the faster they are moving. There is no centrality involved, this is just the way it is. So we need not invoke some particular central location for ourselves. Everything is moving away from everything else, all distances are growing, every observer as he looks around him or she looks around her will see all other observers receding and the mathematical expression of this is that if you measure. Any distance in the universe at time t*0, by t*0 I mean now and then you measure it either later or earlier. If you measure it later, it will have grown by a factor which depends on time in this fashion. What this tells you is that over a time, t Distances grow, by an amount that is H*D, oh, yeah. The speed of recession is H*D. Now this is valid both into the future and of course this didn't just start happening now, so it's valid into the past. If T is less T0, if you're looking into the past, then this is a negative number, distance in the past were smaller. Than they are now. Fine. If everything is receding, that means everything used to be closer together. I should point out immediately that what we're seeing is the galaxies, all distances between all galaxies appear to be growing. This does not mean that the distance between say, the sun and the Earth is growing, first of all, or that my belly's growing That's not cosmic expansion. This is a description that is valid for objects too far away to be gravitationally bound. The Earth and the Sun and the Milky way are all moving as one object. The Milky is not in particular. Expanding this is the motion of distant galaxies, that are not gravitationally bound to each other. It's to that case the Hubble applies. The other point, Point is that of course, there are still peculiar motions, galaxies and are still orbiting various things and that they do on top of this global expansion. So everything is expanding but that doesn't necessarily the only motion that galaxies do so for example one could ask everything is expanding how do galaxies collide that is, That is a case where the parculiar, are larger than the Hubble expansion for the relatively small distance between them and that usually happens in cases where their gravitationally strongly interacting or bound. Okay. So, Hubble gives us this picture, where the entire universe is growing, every galaxy is receding from every other galaxy. What's it growing into? What is beyond the place where these galaxies are now? More of the universe, the galaxies are receding into the universe. We'll talk about the, sort of mathematical implications of this expansion a lot next week. But at the moment there need not be an edge to the universe in order for all distances to grow. All we have is a bunch of galaxies that are receding from each other, and beyond them are probably more galaxies that are receding even faster from where we sit, and no matter which galaxy you sit on, everything recedes at a rate proportional to its distance, because all distances grow by the same factor. This is the symmetric picture. Okay, so Hubble teaches us that all distances are increasing. Right from there, you have a big discovery. Look, this can't have been going on forever. Because, if in the future, distances are going to bigger, as I said in the past, they are going to be smaller. And, farther in the past they were smaller yet, And farther in the past they were smaller yet. At, oh, and at some point in the past you can solve the equation that says D(t) = 0. The distance is zero. Which distance? It doesn't really matter. If I make this factor zero, all distances are zero. All of the galaxies that we see and the ones that are too far for us to see. What is that time well solve the equation, one plus H times t minus t0 is zero, move the one over to there you find that H t minus Hto equal. To -1, and so what does that tell me? That tells me dividing by H, I call that H0 ^ -1 and cross out the H's over here and I find that t = t0 - H0 ^ -1. Wait, so H0 ^ -1 is a time? I thought H was something in a kilometer per second per megaparsec. Well, once you remember that a megaparsec and a kilometer are both distance, you realize that you can convert one to the other. And indeed, kilometer per second per megaparsec is one over seconds times some small number or megaparsecs per kilometer per second The units of Ho inverse are indeed the units of time. So what is this time? This is the so called Hubble Time. It's very important. This is the time where if you imagine if this equation governs the universe, this is the time in the past of go, the time in the past in which everything was on top of everything else. So let's compute it, the 100 gets converted to .01. H converted to H inverse, this is the inverse of a kilometer per second per megaparsec. A megaparsec is how many kilometer's? Well a megaparsec is this many Au. And then Au is this many kilometer's, except, I also, while I was at it, Introduced the mega parsecs. So, there's 10 ^ 11, this is how many, a mega, a mega parsec is and the seconds come out. I plug in all the numbers, I get this number. If I allow multiply through, I get 9.8h inverse, plug in the most Recent value for H of .71 I get finally my result for the Hubble time, 13.8 billion years. What is this? Well this is the maximum amount of time the universe can exist because before that, at that time everything was on top of everything else and I can't go farther back. [UNKNOWN]. So we have a finite age for the universe, given Hubble's expansion. And this is our rough and ready estimage for how old the universe is. It turns out to be surprisingly good. We will need to improve it. It will turn out that this precise behavior is a Newton approximation to some More complicated time dependents of distances, that'll be the topic of next week. But first, we can get even more out this story. Remember that the red shift, z, determines distance, at least for small z, where the Newton Approximation holds, as, v/ c is 1 +, z/ c is z Remember that V/C was supposed to be Z and V/C is therefore H0/C, D/C and so D is ZC/H0. So if you know the red-shift you can figure out the number, the distance. This, allows you to figure out something else though. Someting called the look back time. If a galaxy is a distance d away, then, what I am seeing is the light that left that galaxy a while ago, and the farther the galaxy is, the longer the light has taken to get here. The Andromed galaxy is 2 1/2 million light years away. We see the Andromeda galaxy as it was 2 and a half million years ago. The farther an object is the farther back in time we see it. The look back time is simply the time that the light was emitted is t0 which is our notation for the present minus D over c, this is how long it takes light at the speed c to cover that distance. PLUGGING IN THE VALUE FOR D WE GET THAT THE TIME WE SEE A GALAXY ACT IS T ZERO MINUS Z TIMES EIGHT ZERO INVERSE. AGAIN Z IS THE FRACTION OF THE HUBBLE TIME IF YOU WANT A FRACTION OF THE AGE OF THE UNIVERSE, THAT HAS PASSED SINCE THE LIGHT WAS EMITTED FROM THIS GALAXY. [INAUDIBLE] us another exciting way to think about things. let's think about it this way. So, remember that I'm going to compute the wavelength at which light was emitted / the wavelength at which I'm going to observe it. This is kind of backwards to the usual calculation of the Redshift, the redshift is lambda observed divided by lambda emitted, so this is actually 1+z, inverse. it'll be clear why I'm doing it this way. Now I'm going to assume that I'm in the small And that, reasonably small redshift neighborhood of non relativistic motion so that this expression is correct, first of all and second of all that because I can use Newton's approximation because z is much less than 1, then I can use Newton's approximation to approximate 1 + z to the -1. As one minus z. Ok, and then from here i have an expression for minus z which says that this is equal to one minus to get minus z i take h zero times minus z is plus H 0 times T emission minus T 0. Oh, but I know what 1 plus H 0 times T minus T 0 is. Remember that our expression for the Hubble change in distances, was D of T, is D of T 0. Times 1 + H zero T - T zero. Oh good, so this is exactly D at the time of emission, for any distance / that distance at T zero. Okay. So what do we have? That, the wavelength at the time of omission, / by the light wave length that we observe, is the, which is 1 - Z approximately is the same as distance at the time of emission, and distance now. Which distance? Doesn't matter. They all scale by the same factor. Write that out cleanly. And now you see, that it's telling me something, What is this? This is a distance, the wave length. This is telling me, that the wavelength over the time between emission and observation or absorption has scaled the same way any other distance would. So the red, Hubble's redshift formula is basically telling me that the wave length of a light wave grows along with the distances between galaxies. That's a very strange way to think about it. We'll see next week that it's very natural, this is thinking of the redshift, of the redshift not as a Doppler shift but as a cosmological shift. The wave lengths of light stretch just like any other distance, we'll come back to this next week.