 When we look out into the universe, far enough, we see on average the same distribution of galaxies in all directions. In other words, there is no preferred direction in space, it's isotropic. But being isotropic for us doesn't necessarily mean it is isotropic for viewers in far-off locations. It is possible to imagine galaxy configurations where things look the same in all directions from one point of view and not from another. But this is unlikely. Our assumption is that all observers will see the same isotropic distribution of galaxies. In other words, there is no preferred location in the universe. It's homogeneous. Galaxy densities are the same everywhere as long as we are viewing large enough distances and volumes. This is the cosmological principle. It simply states that the universe is both isotropic with no preferred direction and homogeneous with no preferred place. All directions are equal and all places are equal as far as the laws of physics are concerned. But we know that galaxies clump up into galaxy clusters, and that these galaxy clusters clump up into superclusters. So to get to this cosmological principle, we need to be talking about distances of many billions of light-years. For comparison, it's like saying the surface of the earth is a sphere, but up close we see mountains and valleys galore. It isn't clear that we have a sphere until we get far enough away for the nearby structures to average out. We have to go out 45,000 kilometers to actually see the sphere. For thousands of years, it was taken for granted that the universe was static. It always was, as we see it now, and it always will be. This was the case when Newton developed his gravitational equations, and it was the case when Einstein developed his general theory of relativity. Then in 1929, Edwin Hubble published his studies of galactic velocities. He found that, except for a few nearby galaxies, all the spectra shifts were to the red. All of them were moving away from us. Here's what we see from galaxies in our Virgo supercluster out to 100 million light-years. He discovered that the universe was expanding away from us. His Hubble law says that the further away a galaxy is, the faster it is receding away from us. The relationship is linear, a straight line, so the equation is simple. The receding velocity of a galaxy is equal to its distance times a constant, now called the Hubble constant. This constant has been refined over time, and the distances used have increased by orders of magnitude using tools like the Hubble Space Telescope to analyze Type 1a supernova out to billions of light-years. The box at the lower left shows the region that Edwin Hubble probed. The current best value for the Hubble constant using this approach is 20.86 km per second per million light-year. That is, the receding velocity of a galaxy goes up by almost 21 km per second for each additional million light-years away from us. This slow and steady movement of galaxies away from us is called the Hubble flow. Here are galaxy, galaxy A, separated by a large distance from galaxy B. In galaxy A's frame of reference, it is at rest. Galaxy B is moving away. Its distance from the Milky Way will continue to increase as time goes on. It follows that going backwards in time, galaxy B was getting closer to the Milky Way. We see that, at some point in the distant past, they would have been extremely close to each other, assuming for now that the velocity is constant. We can divide it into the current distance between the two galaxies to see how long it took them to get this far apart. That's just one over the Hubble constant. So without actually knowing the distance between them or their separation velocity, we find the two galaxies would have taken 14 billion years to reach their current separation. When we combine the cosmological principle with our view that all distant galaxies are moving away from us, we conclude that all galaxies are moving away from each other and that the further away a galaxy is from any other galaxy, the faster it is moving away. It follows that going back in time, all galaxies in the universe were once extremely close together, if not actually in the exact same place at the exact same time, 14 billion years ago. The flow of all galaxies away from each other, with faster velocities the further away from each other they are, cannot happen in a fixed volume because in a fixed volume some reference frames would have to have distant objects heading towards them for others to have them moving away. It can only be explained if the space that these galaxies exist in is itself expanding. This is the base assumption for the Big Bang Theory. Here's a one dimensional example to illustrate why this is the case. Consider an 8 meter circle with marks 1 meter apart. If we are at the top mark and all the other marks are moving away from us, then from other points of view marks are getting closer. The system is not homogeneous, but if the apparent motion is due to the amount of space expanding we get a different picture. Here the marks hold their position on the line, but the line grows. Let's say each meter on the line expands to 2 meters over the course of a minute. We see that the distance between adjacent marks goes up 1 meter and their apparent velocity as seen by each other is 1 meter per minute. But more distant marks have increased their distance and velocity by more than that. And the further away any two marks are the more their distance and velocity have increased. And most importantly, this will be the same no matter which mark is used for the reference frame. In order to illustrate the point, this example uses an expansion rate that is 74,000 trillion times greater than the actual expansion rate as determined by the Hubble constant. The real expansion is very slow. If we take a look at what the expansion does to 1 meter, we see that it would take a million years to expand by just 7 millionths of a meter. That's way too slow to ever notice or even measure in a lab in a lifetime. And it is why it's so easy to overcome it with local gravity out to the Andromeda galaxy. It should be noted that this expansion of space itself does not pull apart objects that exist in that space. A meter stick does not expand. It will measure 2 meters where there once was only one. That's because the size of the meter stick is determined by the forces that hold it together and these forces are not changing. Expanding space has significant implications for measuring distance. Consider a racetrack that's 100 kilometers long at the time the race car crosses the starting line traveling 200 kilometers per hour. If we set its odometer to zero when it leaves the starting line, it will read 100 kilometers when it reaches the finish line in half an hour. Now suppose the track is expanding at 50 kilometers per hour. How long will it take to reach the finish line and how far will it have traveled? It will be more than a half hour and further than the 100 kilometers that the track started with but it will be less than the length of the track when the car crosses the finish line because some of the expansion will have happened to space the car had already traveled through. A little algebra gives us the exact numbers. Now suppose we didn't know the track's distance at the start but we did know how fast the car travels and the car's odometer tells us how far it traveled. And furthermore, suppose we found a way to figure out that space was expanding at 50 kilometers per hour. With that, we can figure the original track size and we can figure out how far apart they were once the car reached the finish line. The principles are the same for light traveling to us from distant galaxies. Here we are zooming into GNZ11, the most distant object ever found. The galaxy's redshift, combined with Hubble's law, gives us the distance the light traveled 13.4 billion light years. And we know the speed of light so the time traveled was 13.4 billion years. We normally say that the galaxy is therefore 13.4 billion light years away, but during its long travel time, space expanded considerably. In fact, GNZ11 was less than 2.7 billion light years away from us when the light started its journey and the galaxy is now over 30 billion light years away. In order to calculate these distances, we need to know how the universe expanded during the light's journey. It's not like the simple constant we used in the car racetrack example. In fact, we'll be spending the remainder of this video on just how and why our universe's expansion behaves the way it does. And we'll return to GNZ11 along the way. Note that if a galaxy is far enough away, its apparent velocity will be faster than the speed of light. And its light would never reach us. It would be beyond the physical visible horizon for the universe. It's not that it is moving through space that fast, it's just that more space is being created per second between us and them, then light can traverse in one second. Plugging in the numbers, we find that all galaxies beyond 14 billion light years could never be seen here. GNZ11 is now 32 billion light years away, so the light that is leaving GNZ11 now will never reach us. We now turn our attention to the forces that are causing the universe to expand. We'll start with Newton's gravitational equations. Back in the 1700s, Newton proved a theorem now named after him. It has two parts. The first is that an isotropic, spherically symmetric body affects external objects gravitationally, as though all of its mass were concentrated at its center. For part two of the theorem, we'll drill 30 kilometers into the planet. From here, we are on the Earth's mantle and under the Earth's crust. Counting the oceans, around 1.05% of the Earth's mass is now further from the center than our 1 kilogram object. We see that, although the object is closer to the center, the overall force is just a little bit less due to the smaller mass. But what about all the matter in the Earth's crust that is now further away from the center than the object? The force on the object from the shell of matter will come from the sum of all the forces produced by all the molecules in the shell. And what would the total be if the object were elsewhere inside the sphere? Newton had to actually develop calculus to solve this problem. The remarkable result is that the sum total of all these forces is equal to zero. The shell mass has no impact on the object whatsoever. So now, we have the second part of Newton's shell theorem. It states that a homogeneous, spherically symmetric shell exerts no gravitational force on objects within the shell. This is a key idea for our study of cosmology, and it is not readily understood just why we would get a zero result anywhere inside the shell. A quick look at the geometry involved helps. If we put our object at the center and build a cone that intersects with the shell in two opposite directions, we can analyze the force on the object as it moves around inside the shell. At the center, all the forces cancel out. Now move the object towards one side and away from the other. On the right side, the number of molecules exerting a force is increasing by the square of the distance. At the same time, the force from each molecule is decreasing by the square of the distance. The reverse is happening on the left side. The forces continue to cancel each other out. The total force remains equal to zero. The gravitational acceleration of an object on the surface is always towards the center. But it is possible that the velocity of the object is away from the center. This happens when there is an initial velocity, such as when a projectile is shot from a cannon. To understand how the projectile will progress, we need to use the conservation of energy principle together with the total energy of the system at the time the projectile leaves the barrel of the cannon. The initial velocity that enables the object to escape the Earth's gravitational pull is called the escape velocity. It is the velocity that makes the system's total energy equal to zero and the object will slowly approach a velocity of zero but never quite reach it and it will never return to the Earth. In this example, the initial velocity is less than the escape velocity. So the total energy is negative and the object will fall back to Earth. If the initial velocity is greater than the escape velocity, the total energy is positive and the object will go on with a non-zero velocity forever and never return to the Earth. It has escaped. If we scale up our previous Earth-bound bullet example to galaxies in the universe, we can apply Newton's equations for force, gravity, acceleration, and his shell theorem to gauge how the universe might evolve over time. Picture a wide enough expanse of the universe so that the cosmological principle holds true. We'll center our example on ourselves here in the Milky Way. We are at rest in our own reference frame. Consider a galaxy like IC 1101 a billion light-years away. Our question comes down to how will this galaxy move with respect to us? Given the galaxies are made up of electrically neutral molecules, the only force at work here is gravity. If we build a sphere with us at the center and IC 1101 at the surface, we can calculate the gravitational force on an acceleration of IC 1101 from all the matter, from all the galaxies within the sphere. And we can use Newton's shell theorem to cancel out all the other gravitational forces in the universe. We find that the mass density inside the sphere is all that matters. If we take a look at this equation for a minute, you can see its implications for cosmology. If the acceleration, r double dot, is zero, then the mass density of the universe, rho, would have to be zero. In other words, the universe cannot be static unless it's empty. The existence of matter in the universe means that galaxies must not only be moving, they must be accelerating. The other thing to note is that the acceleration is always negative, meaning that it is in the direction of contraction. But we know from our examination of escape velocity that initial conditions can have the universe expanding, even as that expansion is slowing down. Looking at it from an energy point of view, Alexander Friedman, a Russian mathematician and physicist using Newton's model, developed an equation, now named after him, which showed how the universe would behave under various initial conditions. The constant u in the equation represents the total energy per unit mass at the surface of the expanding sphere. There are three possibilities for this constant. It will be zero if the kinetic energy is equal to the gravitational binding energy. In this case, an early rapid expansion will continue to slow as it approaches a steady volume, but never reaches it. This is like the bullet example having the exact escape velocity. It will be a positive number if the kinetic energy is large enough to overcome the gravitational binding energy. In this case, the universe will expand forever. This is like the bullet examples having exceeded the escape velocity. And it will be a negative number if the kinetic energy is insufficient to overcome the gravitational binding energy. In this case, the universe will eventually collapse. This is like the bullet examples having less than the escape velocity. Note that r dot over r is velocity over distance. This is the Hubble constant. We see that it can vary with time. This means that the Hubble constant is not really constant. We call it the Hubble parameter. The value we've been measuring is designated H0 and represents the value of the Hubble parameter at the current time. In order to more precisely analyze our expanding universe, modern cosmologists place a grid over three-dimensional space. We treat the distance between two galaxies, r, as a constant. Then we set the grid's scale factor, a, equal to one at the present time, and vary it to account for changes in distance over time instead of changing r. Now consider a cube enclosing a volume of space containing some number of galaxies. With our scale factor approach, the amount of matter inside the volume remains the same as the volume increases or decreases. But the matter density goes down when the scale factor increases, and it goes up when the scale factor decreases. Sticking with Newton's model and incorporating the cosmic scale factor, we can rewrite the Friedman equation. We see that the scale factor, a, is the only variable. In other words, the history of the universe comes down to the history of the scale factor. And the history of the scale factor depends completely on the contents of the universe and how that content affects the space it exists in. Understanding the evolution of the universe is what cosmology is all about. Up to this point, we've been using the Newtonian equations. For a full picture, we need to use Einstein's general theory of relativity that includes mass energy and pressure. Plus, we need to consider the curvature of spacetime given its mass energy contents. Here's the equation Friedman developed from this starting point. It is quite similar to the Newtonian version with two key differences. First, the mass density is replaced by the energy density, epsilon. And second, the total energy is replaced by the radius of curvature and the curvature constant that equals minus 1, 0, or plus 1, that tells us which metric to use depending on the nature of the curvature. For an isotropic homogeneous universe, this says we must exist in one of these three possible universes. If it's flat, the universe will expand forever at an ever-decreasing rate. If it's spherical, it is closed and will eventually collapse back in a big crunch. And if it's hyperbolic, it is open and will expand forever at an increasing rate. To take into consideration the impact of pressure, the other key component of Einstein's field equations, we need to consider the relationship between pressure and energy density. When we change the amount of energy in a box, the pressure on the walls changes. In a slowly changing volume, the constant of proportionality is called W and varies depending on the nature of the contents of the box. In physics, this is called the equation of state. For matter that is not actually moving inside the box, the pressure is zero, so W is zero. For radiation, it can be shown that W equals one-third, where the number three comes from the number of spatial dimensions. This gives us the same relationship between energy density and the scale factor that we had with Newton's version. With this and the Friedman equation, we can now calculate the history of the scale factor and thereby the history of the universe if we can determine its radiation energy density, matter energy density, and curvature. We define the critical density of the universe as the density that would give us flat space. Any more than this and we have a closed universe. Any less than this and we have an open universe. The flat matter dominated version is called the Einstein Decider Universe after the scientists that developed the model. This critical mass density comes to around five protons per cubic meter. The actual density of interstellar space is on the average of about one proton per cubic centimeter. That's a million times denser than the critical density, but much of the universe is made up of vast voids with far less than this. So five protons per cubic meter could be the number we actually have. Cosmologists like to work mostly with ratios. In this case, we have the ratio of the energy density over the critical density called the density parameter omega. Current measurements have it at very close to one. It is the sum of all forms of energy that fill the universe. At this point in our analysis, we have three components that add up to one, radiation, mass, and curvature. When we observe light from distant galaxies, we are seeing the light from the stars in those galaxies and that light has absorption lines. The same lines measured in a lab give us the wavelength of the light at the time it was emitted. What we observe is the wavelength stretched over the time it took to get here. We define redshift z as the difference between the two divided by the wavelength emitted. In this hypothetical example, we have an object with a redshift equal to six. Once a model for the change in the cosmic scale factor over time is specified, redshift gives us a great deal of information. For now, we'll assume a flat matter-dominated Einstein-Desider universe. This will only get us part of the way to the actual numbers, but it helps illustrate the key role redshift plays in cosmology. First, redshift gives us an object's receding velocity. With our model, we have the object moving away at six times the speed of light. Redshift also gives us the actual cosmic scale factor at the time the light was emitted. It gives us the age of the universe at the time the light was emitted. And it gives us the amount of time the light was traveling. Redshift gives us the distance to the object at the current time and it gives us the distance to the object at the time the light was emitted. You can see why astronomers rely so heavily on redshift measurements. Next, we'll use it extensively to count galaxies. We now turn our attention to some cosmological observations. The way cosmologists judge any model for the universe is to compare the model's predicted outcomes with what we actually observe. You may recall from our video book on general relativity that curved space has different volume implications than we have for Euclidean flat space. So, one way to determine if the universe is flat, spherical, or hyperbolic is to count galaxies at different distances, i.e. different redshifts. If we look out into a flat universe, we would see the number of galaxies vary with the volume. If we look out into a spherical space, we would see the number of galaxies increase more slowly than for flat space, reach a maximum, and then come back down. And if we look out into a hyperbolic space, we would see the number of galaxies increase dramatically faster than for flat space. With our modern technologies, we can see galaxy populations out to around 10 billion light-years. And as far as we can see, the number of galaxies increases according to the flat space model. Observations of galaxy diameters and luminosity distances also show a flat universe. Large-scale space looks completely flat. But if the universe is large enough, say with a radius of curvature at around 200 billion light-years, it is possible for it to look flat to us, examining such a small part of it. So it is still possible that we live in a three-dimensional sphere with a huge radius of curvature. But for the rest of our study of the benchmark model, we'll assume we exist in flat space. By the early 1960s, we had a consistent theory for how the universe scaled over time, with three basic curvature models. Observation of galaxy counts at different distances indicated that the flat model was the best fit. And Type 1A supernova studies gave us a good reading on the value of the Hubble parameter at the current time. But the Big Bang Theory was challenged by a steady-state cosmology that held that the cosmological principle was true for all observers and for all time. Galaxy Redshift was explained as photons losing energy to space rather than space expanding. Plus, the supporters argued that a Big Bang would have left a trace that should be detectable. But nothing had been found. That all changed when the cosmic microwave background radiation was discovered in the mid-60s.