Archive for: February, 2012

The Minimum Size of the Whole Universe

The Observable Universe

When we talk about our Universe, we make a distinction between "the Universe" and "the Observable Universe". The latter includes only what we can see. By "can see", I don't mean what we have the technology to detect. Rather, I mean all objects out there from which light has had time to reach us given the age of the Universe, the speed of light, and the history and future of the expansion of the Universe. The age of the Unvierse is 13.8 billion years. Because the speed of light is finite, we can't see anything that is so far away that light would have taken longer than that from us to reach us. This isn't a technological limitation; this is a limitation on whether or not there is light, even in principle, for us to see given as much technological prowess as you could want.

Indeed, as we look towards the edge of the Observable Universe, we're looking back in time. If light took us 13.7 billion years to reach us, then we're seeing the Universe out there as it was 13.7 billion years ago, not as it is now.

The Universe as a whole, however, is probably infinite. This is easy enough to say, but it's a rather difficult concept to wrap your brain around when you really start thinking about it. One solution is not to think too hard about it. If you find yourself asking questions like "if it's already infinite, how can it expand?", you're not thinking properly about infinity. Infinity is a concept, not a number.

However, the Universe doesn't have to be infinite. According to General Relativity, there are other possibilities. I'm going to lump those possibilities into two categories, but only really talk about the latter.

Interesting Topologies

It's possible that our Universe has an interesting topology. Topology is different from geometry. Geometry includes things like lengths of lines, radii of curvature, sums of angles inside polygons, and so forth. Topology talks about how different parts of space are connected.

As an example, consider the classic video game asteroids:


This game takes place in a (very small) two-dimensional universe. The geometry of the Asteroids Universe is Euclidean— that is, parallel lines will never cross, the ratio of the circumference to the diameter of a circle is π, the sum of the three interior angles of a triangle is 180°, and so forth. However, if you ever played this game, you know that if you go off of the left of the screen, you come back on the right side of the screen. Likewise, if you go off of the top of the screen, you come back on the top. This universe is unbounded; you never hit a boundary, or an edge. However, it is also finite. Its topology is toroidal; it has the same topology as the surface doughnut, although it does not have the same geometry of a doughnut. (The surface of a doughnut has curvature.)

It's possible that our Universe is similar. It may have a flat geometry, but a topology that means that if you kept going in one direction, you came back where you started. If it does have this topology, it's on spatial scales larger than the Observable Universe. Otherwise, we would have seen the signature of this topology (i.e., the fact that parts of space are effectively repeats of each other if you keep going far enough in one direction) on the Cosmic Microwave Background.

For the rest of this post, I shall assume that the Universe does not have any interesting topologies like this. Either it's just infinite space, or it's finite space that is the 3d equivalent of the surface of a sphere.

Possible Geometries of the Universe

The geometry of our Universe doesn't have to be Euclidean. Depending on the total energy density (including the density of regular matter, dark matter, and dark energy), there are three possibilities for the curvature of our Universe.

[Possible Shapes of the Universe
Two-dimensional visualizations of the possible shape of the Universe. Our Universe would be the three-dimensional equivalent of one of these, depending on the total energy density of the Universe.

The parameter Ω is a convenient way of talking about the density of the Universe. There is a critical density, which depends on the current expansion rate of the Universe. That critical density is about 9×10-30 g/cm. That doesn't sound like a lot, but remember that the Universe is mostly empty space! Where we live, on Earth, is an extremely high-density place compared to most of the Universe. The parameter Ω is defined as the ratio of the density of the Universe to the critical density. If Ω=1, then the Universe has a flat geometry. Note that "flat" here doesn't mean "two-dimensional", the way you may used to be talking about flat. Rather, it means that the geometry of space is Euclidean, just like the geometry you probably learned about in high school.

On the other hand, if Ω>1, the Universe has a "closed" geometry. In this case, the geometry of the Universe is the same as the three-dimensional surface of a four-dimensional hypersphere. If that sounds like gobbledygook, think of it as the 3d equivalent to the surface of a sphere. Note that there doesn't need to really be a fourth spatial dimension or a 4d hypersphere out there. It's just that the geometry of the Universe— how parallel lines will behave when extended, how the angles of triangles will add up, what the ratio of the circumference to the diameter of a circle will be— is the same as the geometry of the surface of a sphere. It's possible to describe the mathematics of this geometry entirely using only three spatial dimensions, so there's no need for a higher spatial dimension in which to embed our Universe. However, for purposes of our visualization, it's worth thinking about the surface of a sphere, as that helps us get some idea about what sorts of things would be true in such a universe. The surface of a sphere is a two-dimensional closed universe. Remember, that the universe is the surface. There is no center to this universe, not within the universe— for everything within the universe is on the surface of the sphere, and no point there is any different from any other point.

If Ω<1, the Universe has an "open" geometry. This is harder to visualize. It turns out that you can't embed a slice of an open 3d universe into three dimensions to visualize it, the way you can with a closed universe (in which case you get the surface of a sphere, as described above). However, the closest two-dimensional equivalent would be a saddle or a potato chip (each of which is a hyperboloid or hyperbolic paraboloid). This is an unbounded and infinite universe. It keeps on going forever. However, it's also clearly not flat, and so will have an interesting geomoetry.

The Geometry of Our Universe

You can figure out the geometry of your universe several ways. One way is to create a triangle by drawing three straight lines through space. Then, measure the angle between each of those pairs of lines. If the three interior angles add up to 180°, you're in a flat universe. If they're more than 180°, then you're in a closed universe; if they're less than 180°, you're in an open universe. The problem is the precision needed for this measurement. In order to be able to tell whether or not the angles add up to 180°, you either need to measure them mind-bogglingly precisely, or you need to draw huge triangles, such that the length of one side of the triangle approaches the radius of curvature of your universe. (How close it approaches it depends on how precisely you can measure angles.)

Effectively, we have done this. Measurements of the Cosmic Microwave Background (CMB) give us triangles. One leg of the triangle is given by the characteristic size of fluctuations in the CMB. We know the physical size of those fluctuations. The other legs of the triangle are given by the path of light travelling from either side of one of those fluctuations. By measuring the angle between the light coming from either side of a fluctuation, we can figure out what the geometry of this isosceles triangle is. We did this. The answer: our Universe is flat. However, as with all physical measurements, there is uncertainty on this measurement. The latest form of this measurement tells us that Ω must be between 0.9916 and 1.0133, to 95% confidence (see "Reference" at the end for the source of these numbers). That means that there still is the possibility that our Universe is either infinite (in the case of Ω≤1) or finite (in the case of Ω>1).

The Minimum Size of Our Universe

The Universe is big. You just won't believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it's a long way down the road to the chemist's, but that's just peanuts to the Universe.

With all due apologies to Douglas Adams, let's quantify how big our Universe is.

First, the age of the Universe is 13.8 billion years. That is a long time compared to you and I, but as compared to the age of the Universe, it's just about right. The edge of the observable Universe, right now, is 48 billion light-years away. "Wait!" you may cry. "How can light from something 48 billion light-years away have reached us in a mere 13.8 billion years!" Remember that while that light was working its way towards us, the Universe was expanding. The light, in a sense, had to try to "catch up" with that expansion. This is imperfect language, and indeed if you know Special Relativity you should object to it. However, it does (sort of) make sense in the context of General Relativity.

How does this size compare to the size of the Universe as a whole? If we make the assumption that Ω=1.0133— the highest total energy density consistent with our current data, and thus the smallest closed universe consistent with our data— it's possible to calculate how big the Universe is. The result looks something like the following:

[Observable Universe in the Minimum Total Universe]
Click for a bigger version

In this picture, the surface of the sphere is meant to represent the whole Ω=1.0133 Universe. The parts that are "greyed out" are outside our Observable Universe; the patch at the top that you fully see is the Observable Universe. The radius of curvature of this universe is 120 billion light years. Its circumference is 760 billion light years. That means that the diameter of our Observable Universe is just 1/8 of the full length of a line you'd have to draw through space if you wanted it to connect back go yourself. The volume of the whole Universe is about 100 times the volume of our observable Universe. (If you object to the fact that 83 is not equal to 100, remember that we're not talking Euclidean space here, so your intuition for how radii and volumes of spheres relate doesn't entirely apply.)

Remember, though, that this is the minimum size of the Universe given our current data. Most of us suspect that the Universe is really a whole hell of a lot bigger than that, and indeed may well be infinite.

Size and Fate Are Separate

If you read almost any cosmology book written more than 12 or so years ago, and some written since, you will probably read something about a closed universe being one that recollapses, and an open universe one that expands forever. This is true only if the dark energy density of the universe is zero! Implicitly, those texts assumed that our Universe was matter dominated, and as such the geometry and fate of the Universe were linked. In a universe such as ours, where there is dark energy, the fate and geometry are not so tightly linked. Dark matter and dark energy both affect both the shape of the Universe and its ultimate fate, but they affect it differently. Exactly what will happen to our Universe depends on the details of what dark energy really turns out to be. However, for what most of us consider to be the most likely versions of dark energy, the Universe will keep expanding forever, with clusters of galaxies getting ever more and more separated. This is true whether the geometry of the Universe is flat, open, or closed.


The numbers for the current expansion rate of the Universe (used to derive the critical density) and for the limits on the curvature of the Universe come from the cosmological implications of the WMAP 7-year data as described in Komatsu et al., 2011, ApJS, 192, 18. The image used to wrap the universe sphere is the Hubble Ultra-Deep Field.

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Charged Particles and Magnetic Fields

Magnetic Fields

You are probably familiar with magnets. Magnetized bits of iron can stick to metal things such as your refrigerator or paperclips. (It's always a fridge or paperclips, for some reason.) You may even have made an electromagnet by wrapping a wire many times around a nail (or some other iron thing) and running a current through that wire by attaching it to a battery.

When physicists want to describe the forces that result from magnetized objects like this, we talk about magnetic fields. Really, a magnetic field means the strength and direction of this abstract physical thing (called, er, the "magnetic field") in all regions of space around the spot we're talking about. We then know how to use the numbers associated with the strength and direction of this magnetic field to calculate how it will interact with other things, such as iron objects (that react to magnetic fields), other magnetic fields, and charged particles. We will often visualize these magnetic fields by drawing "magnetic field lines". These lines point along the direction of the magnetic field. Where magnetic field lines are closer together, it's a stronger magnetic field. Iron filings will tend to line themselves up along magnetic fields, allowing a pretty direct (and physical) visualization of small segments of magnetic field lines. The weakness of this visualization is that how "close together" the lines appear depends on where the iron filings fall as well as how strong the magnetic fields are.

[Iron filings lining up along a magnetic field]
Iron filings line up along the magnetic field lines of a bar magnet. Image from Black & Davis, 1913, Practical Physics, via Wikimedia Commons

Magnetic Fields... innnn... Spaaaaaace

There are magnetic fields all over the place in space. The Earth has a magnetic field associated with it. Indeed, the Earth's magnetic field is very similar to that of a huge bar magnet embedded in the Earth, with the south magnetic pole of the magnet pretty close to the North Pole of the Earth.

[Earth's magnetic field]
Earth's magnetic field. Image by Zureks, via Wikimedia Commons.

When a magnetic field is embedded in plasma— and plasmas are very common in space— the dynamics of the magnetic field and the dynamics of the plasma are closely linked. Charged particles interact with magnetic fields, and the motion of charged particles gives rise to magnetic fields. Wheres you can use simple [sic] fluid dynamics or hydrodynamics to describe the motion of a non-charged fluid (a liquid or a gas), when that fluid is made up of charged particles (and is therefore a plasma), you have to take into account the magnetic fields and start using what is called magnetohydrodynamics. The most important result of magnetohydrodynamics is that the magnetic field lines and the plasma will, for the most part, move together. If the plasma is dense enough, as it moves around, it will convect the magnetic field lines along with it. If the plasma is being compressed, or if it is twisting around, it can stretch out or squeeze magnetic field lines (which corresponds to a strengthening and weakening of the magnetic field), and it can also distort and twist up the magnetic fields. Just as there is energy in electric or gravitational fields, you can also store energy in magnetic fields, and as magnetic field lines get compressed and twisted up, a lot of energy can be stored in them.

Like the Earth, the Sun has a magnetic field. Unlike the Earth, the Sun is entirely gaseous, not solid. What's more, throughout much of the Sun the gas is ionized— that is, electrons are free, not trapped on atoms. That means that the particles in the gas are charged, and the gas is a plasma. While the physics of the Sun is by far dominated by the balance between its tremendous gravity and the tremendous pressure generated by the nuclear fusion at its core, magnetic fields are important, and give rise to things such as the Solar Corona (that wispy "crown" of extremely hot gas that you may have seen in pictures of Solar eclipses) and the various loops and prominences you see on the Sun, as well as solar flares and ejections of groups of charged particles from the surface of the Sun. The big loops you see sticking out on the Sun in some images such as the one below are where bundles of magnetic fields have blooped out of the surface of the Sun, and plasma has been carried along with them.

The Solar Corona during an eclipse. Image by NH53, via Wikimedia Commons

A solar prominence, imaged in ultraviolet light. Charged particles are streaming along a magnetic field; this particular image captures the light from ionized Helium. Image from the Solar Dynamics Observatory, via Wikimedia Commons.

The Sun rotates, about once every 24 days or so. However, that rotation is not uniform; the poles rotate faster than the equator. This means that the magnetic fields in the Sun are always getting twisted up and distorted. Every so often the magnetic fields will "break", and relax to a less twisted state, releasing the energy that was stored in the magnetic fields. This is what causes so-called "coronal mass ejections" (CME). There is a constant stream of charged particles coming off of the Sun; when there's a CME, the sun briefly sends out charged particles at a much greater rate into the Solar System. If that mass ejection is pointed more or less in the direction of the Earth, then those particles will hit us.

Charged Particles Intersecting Magnetic Fields

Whereas an electric field points exactly along the direction that it will push a charged particle (or exactly opposite it, if the charge on that particle is negative, as is the case with an electron), when a charged particle comes across a magnetic field the force on it is perpendicular to both its direction of motion and to the magnetic field line. If you imagine that you're going straight, and then you feel a force to your left, you'll curve to the left. If you keep having a force to your left (your new left, now that you've curved), you'll eventually go around in circles. This is what happens when charged particles come into a region of magnetic fields. It depends on much energy the particles have— how fast they're moving— as well as the strength of the magnetic field. If they're moving very fast, they may just get deflected a little before going on their way. However, if the magnetic field is strong enough, and especially if the particle loses a little energy, say by colliding occasionally with other particles, it can get trapped by the magnetic field, at which point it will circle around it. In fact, particles will tend to spiral around magnetic field lines, moving along the lines as they orbit around them. That part of their velocity that is pointing along the magnetic field is not affected, but the part that's perpendicular circles around them.

A charged particle (red) spiraling around a magnetic field (blue).

Because the Sun is continually throwing charged particles at us, the Earth has an ample source of particles to collect in its magnetic fields. There are belts of radiation above the Earth, called the Van Allen Belts, where charged particles (mostly protons, or Hydrogen nuclei, and electrons) spiral around the Earth's magnetic field.

Look at the picture above of the Earth's magnetic field. Pick one of the field lines that is away from the surface of the Earth at the equator. If you follow that field line along, you'll see that as you get closer to the poles, eventually the field line plunges down into the Earth. If you trap charged particles on these field lines, as the charged particles move along them they will eventually intersect the atmosphere. As these charged particles plow into the atmosphere, they run into atoms in the atmosphere, giving them energy and exciting their electrons. When the atoms in the atmosphere release this energy, they glow. This is what we observe as the northern and southern aurorae. This also explains why you only tend to see the aurorae if you're fairly far north or fairly far south: you have to be there for the magnetic field lines that trap the charged particles to intersect the atmosphere. It also explains why you might expect more impressive aurorae after a big mass ejection from the Sun. When there are more charged particles coming off of the Sun, there's more basic material for the Earth's magnetic fields to capture and spiral down into the atmosphere.

The southern aurora, imaged from space. Image from NASA.

Acceleration of Cosmic Rays

When a supernova explodes, it sends a blast wave out into the circumstellar blast. This blast wave is driven by the outer layers of the star that get blown off in the supernova, and the blast wave then sweeps up the existing circumstellar gas into its expansion. Remember above that we talked about magnetic field lines being locked into the fluid. If you have gas crashing into other gas in a shockwave with this expansion, you'll have gas getting highly compressed right at the shock. That also means that you're going to be squeezing together magnetic field lines right at the shock. As the magnetic field lines squeeze together, some of the kinetic energy from the expanding gas goes into accelerating the particles spiraling around those magnetic field lines. The result is that they spiral faster and faster around the stronger and stronger magnetic fields.

Eventually, with all those particles there, some particles will run into other particles and be knocked free. By this time, however, they will have picked up a lot of energy because of these compressing and strengthening magnetic fields, and the from the process of them spiraling faster and faster around those magnetic fields. As a result, by the time the particles are kicked out they might have quite a lot of kinetic energy. This fast-moving charged particles then go flying through the galaxy, and they're what we observe as cosmic rays here on Earth.

There is a constant stream of charged particles coming off of the Sun. Cosmic rays tend to have higher energy than that, though, and there are a lot of them that are too energetic to have come off of the Sun. Indeed, there are some cosmic rays that are too energetic, we believe, even to have come from supernovae, and their origin is still something of a mystery (or at least was last time I checked). However, the vast majority of the higher-energy cosmic rays that show up here at Earth were accelerated long ago through this mechanism, called the "Fermi mechanism", in the magnetic fields in the shocks of supernovae. Ever since then, these charged particles have been wandering about the Galaxy, steered by the magnetic fields of the galaxy, until by chance some of them run into our planet.

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