First let's look at some of the basic notions, both theoretical and observational, to set the stage for what cosmology really does. And the inflating balloon analogy is commonly used to describe expansion of the universe. Well, it all really began with general theory of relativity. That, you can say, would be start of modern cosmology, which Einstein published in 1915. This is the actual paper that was in the Annalen der Physik. And as I already probably mentioned, the important things about relativity, aside from being a fundamentally different way of looking at space and time and matter and energy, is that it connects geometry of space with presence of matter or energy in it. And the box here is two line summary of what general relativity is all about. So there has to be a consistent behavior if space does something, expands or contracts, then it has to follow whatever laws, Einstein's laws of gravity say. Now from the cosmology viewpoint, the important part is that this is the theory of gravity, nothing else. That's what general relativity is. And since gravity is the only interaction that's important at large scales, the subatomic ones are limited to subatomic particles. Electromagnetism usually doesn't matter very much because charges are well mixed. So if you want to have a theory of universe as a whole and dynamical behavior, you need theory of gravity. So, this was it. Now, observationally, in the early part of 20th century, astronomers started measuring velocities of galaxies. This is even before they knew that galaxies were galaxies, they were just nebulae. But, thanks to Hubble's discovery of variable stars in Andromeda in 1920s, it became clear that indeed galaxies are other island universes. And so the study of extragalactic world began in earnest. And in particular, one neglected pioneer is astronomer called Vesto Melvin Slipher who worked at Lowell Observatory, and he got a lot of first velocities of galaxies. Hubble used his data in his famous paper without necessarily giving him much credit. So Hubble plotted estimated distances to galaxies, which he just based on their brightness I think, versus their velocities. And so there is a linear trend. And this now known as Hubble's Law. That the velocity of a galaxy receding from us is directly proportional to the distance to it. And the constant of proportionality is call Hubble constant, and the plot's called Hubble diagram. Turns out that this still plays a very major role in modern cosmology, we began this way. So it was instantly recognized that this really corresponds to the uniform expansion of space. And since there was already some notion about it from general relativity, this was accepted right from the get-go. Now, where does Hubble law come from? So imagine space fill the galaxies, and the volumes stretch uniformly. So this is the most multiplicative change, all right? Every volume gets multiplied by some factor at one point, whatever, over a period of time. So that means that the increment over any given time interval is going to be also proportional to the length that you're looking at. So the increment of distance is proportional to the distance itself. And since the same delta t for all galaxies you look at, then velocities will be also proportional to the distance, and this is what Hubble's law says. In fact, it's equivalent to the statement that if the space is uniform, homogeneous, isotropic. If you see this effect, that means the space expands and vice versa. Now, there was another piece of well, assumption that one had to make. And that is what we call the Cosmological Principle. It's related to the Copernican Principle, which stated that Earth is not in a privileged place in the universe. And, so the galaxy itself is not in privileged space in the universe. And that at any given time, the things are pretty much same in all locations, in all directions. And this is what makes cosmology possible. This may not be the case in principle, but it just happens to be correct. And so that simplifies this problem enormously, as I'll show you in a moment. Now that can be generalized further to the so-called Perfect Cosmological Principle, which says that also should be same at all times. But how can you have that if the space is expanding? Well, you can have it if you keep creating matter so that it fills up whatever's left, new space, stay the same density. And that was the principle behind so called steady state cosmology. Universe expands, but new matter appears out of nowhere. And so it's always the same. And this was a serious contender for cosmological model until it was disproven largely by the cosmic micro background but also other things like counts of radio sources. So just to clarify what these symmetry assumptions mean. Here are the three cases. The first one, there is a preferred direction. So it's same everywhere, so homogeneous, but there is preferred direction, so not all directions are equal. So it's homogeneous, but not isotropic. The other one in the middle, all directions are equal, but that's a privileged place. And so it's isotropic from that point, but it's not homogeneous. And finally, if you just have random blobs going on forever, that's homogeneous and isotropic except around the blobs themselves where they defined local direction. And so the question is, how true is this? Well, it's actually pretty good. And so if you look at scales larger than about 100 megaparsecs, that's really cosmological scales, which is what matters here. We find out that, yeah, universe seems to be pretty much homogeneous isotropic. The map on the upper right is distribution on the sky of radio sources from particular survey. The missing pieces or parts our telescope could not observe. And it's pretty uniform, right, so it seems the universe at large scales really is homogeneous. The ellipse on the lower left is symbolic representation what picture of cosmic micro-background would look like. Now you've seen those garish blue-green yellow pictures of cosmic macrobackground fluctuations. That's with the contrast turned up by a factor of a million. If you just look at any lesser contrast, it's pretty damned uniform. It's equal in all directions to a few parts in a million. Well, that's a very large scale. Now, as we studied large scale structure, we know that this is not true at scales about 100 Mpc or less. There are all these blobs and filaments and clusters and voids, and so on. And certainly there are preferred directions wherever local acceleration pulls you. There is more matter in some places than others, but if you average this over hundreds of megaparsecs, it's pretty good. And so, on the global scale, this doesn't really matter very much. So homogeneity and isotropy are good, we'll use them in a moment, but what about the expansion of the space? This is something that people tend to get a little confused about, and the essential point is that there are actually two kinds of coordinates in general relativity. There are coordinates that expand with the expanding space, or contract, and there are those that really stay constant. Otherwise, how could you tell if everything was expanding at the same rate? You just couldn't tell, so there has to be a comparison. So it turns out that things that they're not gravitationally bound, like galaxies far apart, or massless quanta. Or any relativistic particles actually, stretch with the expanding space or move along with Big Bang space. But physical systems that they're bound by any forces, gravitational, electrostatic and so on, do not participate in that expansion. So sizes of atoms or planets or solar system or even galaxy doesn't change. The space kind of expands underneath, and so as long as the system is bound, it stays of the same physical proper size. So this is how we can tell. We can ask, well, why not the other way around? Why aren't proper coordinates shrinking and space is staying constant? Well, at some mathematical level it's equivalent, but on the other hand it will require inventing a whole new physics completely. And there is really no reason to do that. So the first question is okay, space is expanding, but into what? And the best thing that we can offer is two-dimensional analogy embedded in three-dimensional space. So you can have a surface of a sphere, which is finite. And the sphere can grow or shrink in time in the third dimension. So what's the equivalent of that in four-dimensional space then? Well, you can think of time perhaps as a radius, if you will, and the three-dimensional universe expands. Or take just coordinate grid, infinite coordinate grid, squares all the way to infinity, you could start expanding it, and there is no edge. Just expands to infinity, but it expands nevertheless. So there are two possibilities. Either universe is of a finite volume, so-called closed models, or infinite volume, so-called open model, or some clever combination of the two which comes with multiverse and so on. But it expands into itself if you will. And so there are no edges ever, and there is no center. All places are equally good. That's homogeneity and isotropy. So what we call cosmological redshift is stretching of the photons due to the expansion of the space. And again, if you take inflating balloon, you can glue little galaxy spots on it like coins or something. The balloon will inflate, but they'll stay the same size. But if you draw wiggly lines on the balloon that correspond, say, to photons, they will stretch as the balloon stretches. And since longer wavelengths correspond to redder shift, that's called a redshift. And this is related to the doppler shift that you are familiar with. This is formula for the special relativistic version of doppler effect. Redshift is ratio of velocity to the speed of light. And that corresponds to relative change in wavelength or frequency for that matter. So that's the familiar formula. Now, turns out that's actually completely equivalent to thinking that, for some reason, whatever is in galaxies is moving away from us. And so they have to have doppler shift due to that motion that's caused by the expansion of the universe. And that turns out to be exactly the same factor as if you just consider expansion of the space. So if you take any two points apart, they're not bound. And just let go, and then universe will kind of, space will carry them apart. And we can choose any two in comoving coordinates and call it R. It's a function of time. And so the wavelengths, or photons, will behave exactly the same way. And so the stretch factor of the universe is 1 + redshift. So if we measure a redshift of one for some distant galaxy, that means that universe is exactly one-half the present size at that time. Or rather things are closer by a factor of two. And the two are completely equivalent. So, the important concept here is that the space itself expands and carries galaxies and other stuff with it. And there are two different kinds of coordinates, those that do expand and those that do not expand. And thanks to the expansion, we see light coming from further away, meaning faster away from us, stretched more, and that's what we call redshift. So you can use, you can substitute measurements of redshift which are relatively straightforward by taking spectra for measurements of how much universe has inflated since then. And now here is the important part. I'm going to now tell you that the holy cow of physics that you've been told is absolutely true, that energy is conserved, is actually not true. Energy is not conserved in an expanding or contracting space. Locally, expansion is completely negligible. This is why locally you would see that energy is conserved just fine. But those photons that are getting stretched, that means their energies are getting lower. So where does that energy go? Doesn't go anywhere, they're just, it's not conserved. It's not exchange of energy like kinetic to potential or whatever. Likewise, take two galaxies that are some distance apart. There's some binding energy between them, even though they're not completely bound, but there's still potential energy. Space carries them apart, that potential energy will change. Again, there is no conversion between kinetic, potential, and whatnot. Energy is just not conserved in an expanding space. There is a deeper reason for this, and it has to do with symmetries of nature, the so-called Emmy Noether theorem. And that relates conservation of quantities like energy, momentum, angular momentum, to symmetries in space and time. And so for example, conservation of momentum is related to homogeneity of space. And conservation of angular momentum is connected to isotropy of space. And conservation of energy is related to homogeneity of time. But if the universe just keeps expanding, that then means not all times are equal. Things are changing because of that, energy is not conserved.
