The Geometry of the Universe

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Astronomers sometimes say that an open universe has the geometry of a sphere, and a closed universe has the geometry of a saddle. What do they mean by that?
The sphere and the saddle are models to show that plane geometry (with the circumference of a circle exactly equal to the diameter of the circle times the number π (pi), and parallel lines never crossing) is not the only geometry that a universe might have, and that you can figure out information about the "shape" of the universe in a higher dimension than you can observe.

For our Universe, the sphere and the saddle don't relate to the three-dimensional shape of the Universe but to the four-dimensional shape.
Imagine creatures that live in only two dimensions, with length and width but no height. They can see their two-dimensional universe, but nothing outside of that. If the universe in which those creatures live is flat if you could see it in three dimensions (like a sheet of paper), then the circumference of a circle in their universe will always be equal to the diameter of that circle times the number π, and the sum of the three angles in a triangle will always be equal to 180 degrees. The geometry that applies in that case is plane geometry.
If their universe weren't flat but has the shape of a sphere when seen in three dimensions, then straight lines that are parallel somewhere can yet cross each other, and the circumference of a circle in that universe will always be less than π times the diameter, and the sum of the three angles in a triangle will always be more than 180 degrees.

Things work the same way on the two-dimensional surface of the Earth. The equator of the Earth is a circle that is everywhere equally far from the North Pole, so as seen in the two-dimensional surface of the Earth the North Pole is the center of the equator, and the equator is a circle around the North Pole.
To go from the equator along a straight line via the North Pole to the equator on the other side, you must travel half of the circumference of the Earth, so the diameter of the equator, measured on the two-dimensional surface of the Earth, is equal to half of the circumference. The length of the equator is equal to the circumference, so the circumference of the equator is equal to two times the diameter, which is less than π times the diameter (because π is approximately 3.14).
The ratio is closer to π for circles that are closer aroudn the North Pole than the equator is, and the ratio is even smaller than 2 for circles that are further away from the North Pole than the equator is.

If you walk from the North Pole to the equator along a meridian, then turn 90 degrees to the right, walk some distance along the equator, turn 90 degrees to the right again, and walk along a meridian again, then you end up at the North Pole again and have traveled along a triangle on the spherical surface of the Earth. To get back to the meridian that you started on at the beginning of your journey, you need to rotate through an angle that is related to how far you walked along the equator. The sum of the three angles in the triangle is then equal to 90 degrees + 90 degrees + the size of the stretch along the equator, so that is always more than 180 degrees. For smaller triangles, the sum of the angles will be closer to 180 degrees.
The geometry that goes with this is the spherical geometry. That there is less circumference (for a given diameter) in spherical geometry than in plane geometry is also the reason for what you see when you squash something that is spherical (such as the part of an egg carton that holds one egg): then the squashed thing gets tears that get wider the further you go from the center. Those tears occur because circles in that thing need less circumference than circles on a table do.

If their universe as seen from three dimensions does not have the shape of a sphere but of a saddle, then the circumference of a circle is always more than π times the diameter, and the sum of the three angles in a triangle is always less than 180 degrees. If you draw a circle around the center of such a saddle then that circle goes up a bit over here and down a bit over there, so it takes a longer path than a flat circle would, so it has a greater circumference than a flat circle has. If you squash such a saddle then it doesn't get torn but rather gets folds because there is more material than is needed to cover a flat area. The geometry that goes with this is the hyperbolic geometry.
If their two-dimensional universe as seen from three dimensions has the shape of a sheet of paper, then the circumference of a circle in that universe is equal to π times the diameter, and the rules of plane geometry hold. You can lay such a sheet flat without getting tears or folds, but you can also bend such a sheet until it looks like a letter "U" when seen from the side. That makes no difference for the kind of geometry that holds inside the sheet: that remains flat. That also means that two-dimensional creatures in a two-dimensional universe cannot figure out from measurements or observations inside that universe whether their universe as seen in three dimensions is flat or bent in the shape of a letter "U".

The important point is that from measurements and observations only inside your own universe you can figure out what kind of geometry holds in that universe: spherical, plane, or hyperbolic. Two-dimensional creatures can do that on their sphere or saddle, and us three-dimensional creatures can do that in our universe. We can (in principle) measure whether our universe is flat, but not if it is bent in higher dimensions like a two-dimensional sheet of paper can be bent in three dimensions.
That the geometry near a particular location looks like that of a sphere, a saddle, or a sheet of paper does not mean that the geometry has to be like that everywhere. It might even be that the geometry of our universe looks like that of a sphere in some places, and looks like that of a saddle in other places.
And, for all kinds of geometries, if the circle is sufficiently small (compared to the typical scale of the universe), then the circumference of a circle will be so close to π times the diameter (as for plane geometry) that you cannot detect the difference, and then the sum of the angles in a triangle will be so close to 180 degrees that you cannot detect the difference. At sufficiently small scales, you can use the rules of plane geometry.
How much the geometry deviates from plane geometry is indicated by the curvature or radius of curvature of the space. If the curvature of the two-dimensional space is like that of a sphere, then the radius of the sphere that fits the best at that location is the radius of curvature of space at that location. (That radius can be different in different directions, but let's forget about that for now.) The curvature is one divided by the radius of curvature, because the smaller the sphere, the greater the curvature.

For plane geometry, the curvature is equal to zero and the radius of curvature is infinitely great. How much the circumference of a circle differs from π times the diameter, and how much the sum of the three angles in a triangle differs from 180 degrees, depends on how large that circle is compared to the radius of curvature of space. Turning this argument around, you can figure out the curvature of space from the diameter and circumference of circles, and from the size and the sum of angles of a triangle. For plane geometry the curvature is equal to zero, for spherical geometry (like on a sphere) it is positive, and for hyperbolic curvature (like on a saddle) it is negative. You can figure out the curvature in three or more dimensions in the same way.
Why is an open universe like a saddle and a closed universe like a sphere? With the preceding explanation, this means that in an open universe the curvature of space is negative and the geometry is that of a saddle (hyperbolic), and in a closed universe the curvature of space is positive and the geometry is that of a sphere (spherical). You can take that for a definition: a universe is called open if it has hyperbolic geometry, and closed if it has spherical geometry. I guess that a universe with the geometry of a sphere is called closed because a sphere is closed, and a universe with the geometry of a saddle is called open because a saddle surface is not closed.

One might invent other definitions of "open" and "closed", for example that a universe with a finite volume is closed, and a universe with an infinite volume is open. I don't know if a universe that is open according to one of these definitions of "open" must also be open according to the other one, and likewise for "closed".
In our universe the curvature of space is very small, and the curvature radius therefore very large ― greater than the size of the visible universe itself. The curvature is so close to zero (and the geometry so close to plane geometry) that we cannot yet be certain if the curvature is positive or negative.




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The Fate of the Universe

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 If the mass in the Universe is distributed densely enough, then the combined gravity of it eventually reverses the expansion and makes it collapse on itself. If the average mass density is too small, then the combined gravity of the mass cannot stop the expansion. The boundary density between these two cases is called the critical density, which goes down as the Universe expands.

Until recently, it was assumed that only the force of gravity was important for the ultimate fate of the Universe, and the mass density was too close to the critical density to call, so if the Universe was going to collapse on itself again, then it was not going to be for many, many billions of years.

Recently, astronomers have discovered that there is in fact an additional force that is important for the ultimate fate of the Universe. This force is repulsive and becomes important only when the mass density has become small enough (in an absolute sense), which it did a few thousand million years ago. The expansion of the Universe is therefore not slowing down but rather seems to be accelerating. It seems, then, that the Universe will not collapse on itself at all.



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The Big Bang

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Astronomers have found a lot of evidence that says that the Universe that we know today started about 13.7 thousand million years ago in a so-called Big Bang, and that the Universe has gotten ever larger and colder since that time.

When the Big Bang came the whole Universe was squeezed together in a very small space, that was very dense and hot. It wasn't that all matter and energy was squeezed together and the rest of space was empty, but that there was so little space that all matter and energy only fit if it were pressed together incredibly tightly everywhere.

Scientists don't know for certain what, if anything, came before the Big Bang. It was so mindbogglingly hot and dense just after the Big Bang that anything you can find in the Universe today or that we've been able to do in our laboratories is like a freezing vacuum of space by comparison. If you were to be transported back in time to that period, then you'd instantly be vaporized so thoroughly that not even a single atom from your body would stay intact. Because we haven't been able to study anything like the conditions that existed then, we can't be sure that the laws of physics that we've discovered so far in our current, old Universe were important in the first instants after the Big Bang, or before the Big Bang (if there was anything then).

We can imagine that there may have been stars and galaxies before the Big Bang, but it seems impossible that we could ever find proof of this. If there was something before the Big Bang, then it would have been crushed and vaporized beyond any recognition during the Big Bang. We may as well consider the Big Bang to be the beginning of time, because no information from earlier times could have gotten to us through the Big Bang (if there was in fact something before the Big Bang).

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Which World View is the Best One?

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Any geocentric world view is not by definition worse than any heliocentric world view. Ptolemy, Copernicus, and Tycho assumed motion along circles, and if you add a sufficient number of circles in just the right way to any one of their models, then you can get results at any desired accuracy, regardless of whether you assume that the Sun revolves around the Earth or the Earth around the Sun, and of whether you assume that the planets revolve around the Sun or around the Earth.

The key difference between the models of Ptolemy/Copernicus/Tycho and that of Kepler is that the older models assume motion along circles, and Kepler assumed motion along ellipses.
Tycho's model, if suitable corrections for elliptical orbits are applied, can work as well as Kepler's model, but the same holds for Copernicus or even Ptolemy. Moreover, "suitable corrections" means, if you want to keep to circular motions, that you have to keep adding more and more circles until the desired accuracy is reached.
For the results it does not matter which point you take to be the origin of the coordinate system (the Sun or the Earth), or (with certain restrictions) what shapes you want to use to describe the orbits, but such choices do affect the length of the calculations and the form of the formulas and how much time it takes to explain it all to someone else.

If you use a model with motion along circles, then you need all kinds of circles on top of circles with each their own period of revolution, diameter, and orientation, without any clear understanding (if you don't know what the real situation is) of why a certain circle needs to have just that period, diameter, and orientation. Besides, certain periods keep coming back in the motion of the same or other planets, without it being clear why this should be so. All in all, a complete description of such a system is very large, with many free parameters (such as the period, diameter, and orientation), that all have certain values that cannot be predicted from the theory and therefore have to be measured separately.
In addition, until the end of the Middle Ages, these kinds of models were not presented as descriptions of the true situation, but merely as mathematical tricks to allow prediction of certain things. People saw no fundamental problem if one model was used to predict the longitude but a completely different model to predict the latitude. I seem to recall having read that Galileo used the excuse that "it was merely a mathematical convenience" to get rid of the Inquisition, though he probably did not believe in that excuse himself.

In general, if there are two models that can each provide predictions to the same accuracy, then the one that has the smaller number of free parameter is preferred. This number is smaller in a heliocentric model than in a geocentric model, because the motion of the Earth around the Sun (from the heliocentric model) is reflected (in a geocentric model) in the apparent motion of all other planets around the Earth (for example, in the occasional retrograde motion of the outer planets), so in a geocentric model some additional free parameters are needed including one for the size of the retrograde loop) of which we now know that they really reflect the motion of the Earth around the Sun. If you don't know that really all planets revolve around the Sun, then you don't know how these additional free parameters are related to one another, so then you have to determine their values each separately from observations.
The switch from the geocentric (Ptolemy) to the heliocentric model (Copernicus) meant that a number of these previously free parameters were no longer free, because they could (if desired) be predicted from a transformation from the heliocentric models. I cannot say how this worked for the model of Tycho, because it depends a lot on whether he assumed that some of these parameters were free, or whether he recognized that they really weren't free at all.

The switch from circles (Ptolemy, Copernicus, Tycho) to ellipses (Kepler) reduced the number of free parameters even more, because instead of a large number of joined circles with each its own position and diameter and orientation (and hence many free parameters), now you needed only a single ellipse. If you know that really an ellipse is appropriate, then you can take the small number of free parameters of the ellipse and calculate from those the positions and diameters and orientations of all of those circles that you can use to make the same predictions, but if you don't know that you're really dealing with an ellipse, then you have to separately deduce each of the characteristics of each of the circles from observations, so then they count as free parameters.
Kepler's ellipses gave a preferred position to the Sun, because the Sun (and not the Earth) must be in one of the foci of each of those elliptical orbits. Kepler also found (in his Harmonic Law) a relationship between the periods of revolution of the planets and their distances from the Sun (and not the Earth), which got rid of some more free parameters (because now you could calculate the period from the distance, or vice versa). These were two good reasons to take the Sun and not the Earth as the center of the Solar System, and hence to reject the model of Tycho.

Today we know not just how but also why the planets move in elliptical orbits, namely because of the Law of Gravity of Isaac Newton. There are now only six free parameters for each planet (the so-called orbital elements), if you ignore the disturbance of one planet's orbit by another planet (which the geocentric models did as well): one for the size of the orbit, one for the shape of the orbit, three for the orientation of the orbit, and one for the position of the planet in the orbit at a given time. This number, six, is far smaller than the number of free parameters that is needed for models with only circular orbits, because each additional circle requires four extra free parameters (one for the size, two for the orientation, and one for the position along the circle).

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The World View of Tycho Brahe

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Tycho Brahe proposed a world view in which the Sun revolves around the Earth (i.e., geocentric), but all other planets revolve around the Sun (i.e., heliocentric). His world view lay somewhere between that of Ptolemy (all planets geocentric) and that of Copernicus (all planets heliocentric).

The model of Copernicus was immediately more popular than that of Tycho, and it has remained like that ever since. However, in principle, Tycho's model explains the movement of the planets equally well as the model of Copernicus does, so that provides no argument to prefer one of the models over the other one. None of the models (of Tycho, Copernicus, or even Ptolemy) as they were applied in the 17th century gave much more accurate predictions for the positions of planets than the other models, so that did not help to select the best one, either.

The choice for a particular model was therefore based on other arguments, such as opinions about "elegance" or "common sense", or about how well each model seemed to fit with particular interpretations of passages from the Bible, but people did not all hold the same opinions about these things, so both models had some fans.

Copernicus published his model in 1543. Tycho published his model over 40 years later, in 1588. In 1627, Kepler published the "Rudolfinian Tables" that were based on his new model of the Solar System with planets in elliptical orbits around the Sun, and these tables provided more accurate predictions of the positions of the planets than the models of Copernicus and Tycho did. The improved accuracy of the results gave a lot of support for the heliocentric model, and hence also (after the fact) to the model of Copernicus but not that of Tycho.



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The Geocentric World View

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The key assumption of the Geocentric Model was that the Earth was at rest in the center of the Universe and that all other celestial bodies revolved around the Earth. This model or theory was written down by the famous Greek philosopher Aristotle before the beginning of our era. It was used by the astronomer Ptolemy in his book the Almagest, which was the standard book of astronomy for about 1500 years. The Geocentric Model was adopted by the Catholic Church.
To be able to get reasonable predictions for the motion of the planets, the Geocentric Model requires the specification of the size and rotation rate of many circles, of which it was not clear why the sizes and rotation rates should be just those values.

As the science of astronomy progressed, it became clear that the Earth is in fact not the center of the Universe. The first blow to the Geocentric Model came when Nicholas Copernicus (1473 - 1543) showed that puzzling aspects of the motion of the planets followed naturally if you assumed that the planets and the Earth revolved around the Sun and that the Earth rotated around its own axis. That model, with the Sun rather than the Earth in the center of the Solar System, is called the Heliocentric Model.

Johan Kepler (1571 - 1630) deduced from accurate observations that the orbits of the planets are ellipses around the Sun rather than combinations of circles, and managed to link the period ("year") of each planet to its distance from the Sun in a simple formula.

Galileo Galilei (1564 - 1642) discovered the four great moons of Jupiter, using a telescope that he built himself. These moons proved that the Earth is not the center of all motion in the Universe. Galileo's support of the Heliocentric Model of Copernicus got him into trouble with the Catholic Church which had much power in Italy at the time and which held that the Bible supported the Geocentric Model. Galilei was put under house arrest for the rest of his life.

Isaac Newton (1642 - 1727) showed that the orbits of the planets could be understood as a result of gravity between the planets and the Sun. The Sun turned out to be many times more massive than all other things in the Solar System combined. His theories also successfully explained the motion of comets. Still, it was generally assumed that the Sun was in the center of the Universe.

Just before 1840, several astronomers first measured the distances to some nearby stars and found them to be very, very large. If you know the distance of a star and its brightness in our sky then you can calculate its intrinsic ("real") brightness corrected for its distance. Measurements and calculations such as these showed that the Sun is but an ordinary star, and that there are many much larger, brighter, and more massive stars. Around 1930, it became clear that the Sun is not in the center of the Milky Way but in the outskirts, and that the Milky Way does not fill the Universe but is just one of a very large number of galaxies.

Today we know that the Earth is a planet of modest size, orbiting an ordinary star in the outskirts of an unremarkable galaxy near the edge of a common supercluster of galaxies of which there are very many more throughout the Universe.



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How Many Stars and Galaxies are there in the Universe?

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It is very difficult to give a reasonable estimate for the total number of stars and galaxies in the Universe, because we cannot always draw a well-defined line between stars and non-stars, or between different galaxies.

A star is usually defined as a ball of gas that is kept together by its own gravity and in which nuclear reactions take place or took place. Whether nuclear reactions are happening in a ball of gass depends mostly on the mass of the gas. A ball of gas can only become a star if it has enough mass. According to our theories, the dividing line is at a mass of about a tenth of that of the Sun. If a ball of gas has at least that much mass, then it can become a star. If the ball of gas has less mass, then it is a so-called brown dwarf, or a planet (such as Jupiter).

There are three problems with this definition of a star: Firstly, you cannot always tell from the outside whether nuclear reactions are happening on the inside, and how many. Secondly, nuclear reactions don't suddenly turn on at a particular mass, and at slightly smaller masses there are also already some nuclear reactions. Thirdly, there are very many more balls of gas with small masses than with large masses, so for the exact number of stars it is very important at what mass exactly you draw the line.

A galaxy is a collection of gas and dust and stars that orbit around a common center of mass and that are separated form other galaxies by vast expanses of empty space, but this definition is not sufficient, because it doesn't say how large or empty those expanses must be. Cases are known of multiple galaxies being in the process of coalescing into one, and then it is not clear when they should still be counted as separate galaxies, and when as just a single one. It is also not clear how small such a collection of gas and dust and stars may be to yet be called a galaxy. A star that escaped from a galaxy a long time ago and that now drifts alone through space between galaxies will probably not be called a galaxy by anyone, but perhaps a thousand stars orbiting around each other might be. And just as for balls of gas, there are very many more small galaxies than large ones, so for the precise count it is very important at what size exactly you draw the line.
All in all you can get very different answers, depending on the lower boundaries that you put on balls of gas and on collections of stars.

It is often said that a typical galaxy contains on the order of 100 thousand million stars, and that the number of galaxies in the visible Universe is about that same number. With these assumptions, the number of stars in the visible Universe would be about 1 × 10²². However, it is not clear which galaxy is typical, and whether the typical galaxy that contains 100 thousand million stars is also the typical galaxy of which there are 100 thousand million. According to http://www.space.com/scienceastronomy/star_count_030722.html there are about 7 × 10²² stars in the visible Univese that could be observed by our telescopes in principle, but that number probably does not include very many dim stars.

If my calculations are right, then the SKY-model of our own Milky Way Galaxy (Wainscoat et al, Astrophysical Journal Supplement Series, volume 83, pagina 111) says that in the disk of our Galaxy alone there are already about 2.2 million million (2.2 × 10¹²) stars, of which about a quarter are very dim dwarf stars of spectral type M (spectral classes greater than or equal to M5), about half are less dim M-dwarfs (spectral classes M0-4), and a quarter are brighter stars (earlier than M or brighter than V). This total count is about a factor of 20 greater than the typical value that is often mentioned, even though our Galaxy is often seen as fairly typical.

 
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Rotation in the Universe

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All things that rotate clockwise also rotate counterclockwise, and vice versa: just look at it from the other side. For example, the Earth rotates counterclockwise if you look at it from above its north pole, but clockwise if you look at it from above its south pole.

There are no indications that there is a preferred direction in the Universe, so the average rotation of all matter in the Universe is probably very close to zero.



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The Size of the Universe

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You can define the size of the Universe in different ways, because the Universe has kept expanding since its formation. A light ray from the formation of the Universe that has just now reached us has taken 14 thousand million years (the current age of the Universe) to reach us, but the point from where that ray came has in the meantime gotten much farther away from us because the Universe has kept growing, so that place of origin is now much farther away from us than 14 thousand million lighyears. And if we were to send a ray of light back to that point today, then that ray would have to travel even further still, because during its journey the Universe would expand yet more. Also, there may be much more Universe beyond the boundaries to which we have been able to see so far, but we don't know anything about those regions, and in particular we don't know how big they are. Which distance you can use best for the size of the Universe depends on what you want to use that distance for.

Cosmologists (astronomers who study the structure of the Universe as a whole) usually measure the size of the Universe in a relative fashion, with a scale factor that indicates the size of the Universe compared to its size at a reference time, for which usually today is chosen. They don't say "when the Universe was 7 thousand million lightyears in size", but "when the Universe was half as large as today".

Light coming from the furthest galaxies that we have detected took so long to get to us that it must have started travelling when the Universe was much younger and smaller than it is today. Even though the Universe was much smaller then, the light has still needed all of this time to get to us, because the Universe has kept expanding the whole time. Imagine an ant walking over a balloon that is being inflated. While the ant is walking, the distance to its target increases, so it has to walk a long way even though the target was not so far away in the beginning. The same happens to light rays traveling between galaxies in the Universe.
A source emits a signal in our direction at a certain moment. If the distance between the source and us decreases while the signal travels, then the signal reaches us sooner than if the distance stayed constant. If the distance increases while the signal travels, but not faster than the speed of the signal compared to the source, then the signal reaches us later than if the distance stayed the same. And if we travel faster away from the source than the signal does, then the signal will never reach us. It does not matter what the signal is; it could also be an object.

This holds whether we're talking of someone who tosses an apple to us from a moving train, or of a galaxy that sends a ray of light to us from a great distance in an expanding Universe.
So, if you were to ask how for 14 thousand million years we can stay ahead of a ray of light that was sent in our direction when the Universe was much smaller and we were only 2 thousand million lightyears from the source, then the answer is that the distance between the source and us has been increasing with nearly the speed oflight, so that the ray of light takes a very long time to catch up with us.

Now the question shifts to why the distance between the source and us increases with nearly the speed of light. The answer is that the source is so very far away. Hubble's Law says that the speed at which the distance between two points in the Universe increases is proportional to the starting distance between those points, so you can find any speed of separation that you want if you begin at a sufficiently large separation.
A large speed of separation does not mean that the source moves at that speed compared to its surroundings, but only that the distance between the source and us increases that fast, just like the distance between two marks on a balloon increases when you inflate the balloon, even though the marks do not move compared to their surroundings on the balloon.

The "edge" of the visible Universe is at those locations from where light has taken exactly the current age of the Universe to reach us now. The simplest models give galaxies on that edge a speed of recession (from us) equal to the speed of light, so then the edge travels with those galaxies. A galaxy that was on that edge a long time ago will then still be on the edge today, so then even people from the far future won't be able to receive information from galaxies that are today just beyond the edge. More complicated models (with acceleration or deceleration of the expansion) allow movement of the edge compared to the local galaxies, so then there would be a chance that people from the very far future might receive information about galaxies that are today still beyond the edge.

The simple model fits with the "ant-on-a-balloon" model. It works best if you increase the radius of the balloon at about 1/3 of the speed of light on the balloon. Then the "other side of the balloon" corresponds to the edge of the visible Universe. Faster is OK, too, and then the visible Universe covers a smaller part of the balloon.

The end conclusion is that a source of which light took (for example) 99 % of the age of the Universe to get to us now must lie at 99 % of the distance to the edge of the visible Universe.


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