## The CMB Energy Distribution: the Fingerprint of the Big Bang

In this post we’ll look at one of strongest pillars of evidence for the hot Big Bang. (For an introduction to the Big Bang, see this post.)

The decisive evidence for this extremely hot and dense origin of the universe as we know it is the thermal distribution of the cosmic microwave background (CMB). In a nutshell, the background radiation (light particles, or photons) filling the universe have exactly the same distribution of energies as particles in thermal equilibrium – that is, particles scattering off each other frequently enough to share a common temperature. This suggests that although they are not now in equilibrium, they were long ago – and this could only be the case in a much smaller, denser, and hotter universe.

If you took a large number of particles (say photons, electrons, and protons) and put them in a box, particles would scatter off each other, with more energetic particles sharing energy with other particles. Eventually the system would come to a state of thermal equilibrium, in which the distribution of energies clustered around an average energy, determining the temperature of the gas of particles. (Only a system in a state of equilibrium can be described with a temperature.) This distribution of energies is a thermal distribution:

$P(E)\sim\frac{E^2}{e^{E/T}-1}$.

$P(E)$ is the probability for a particle to have energy E. T is the temperature (a hotter gas means typical particles have more energy). The $\sim$ sign means “proportional to” – we need to scale everything so probabilities add up to one, but that’s not important here. (The “-1” in the denominator is a quantum mechanical effect that appears for photons, but is different for electrons or protons). For large energies, though, the amount of photons drops exponentially with the energy per photon.

No matter how you initially position the particles in the box – whether you cluster them all in one corner and give half the particles all the energy, or you position them randomly but with the exactly the same energy for each – given enough time they will come to equilibrium: their distribution of energies will approach a thermal distribution.

(For nonrelativistic particles, with speeds much less than the speed of light, the equilibrium energy distribution translates into a distribution of speeds. For photons, on the other hand, which are massless and travel at the speed of light, the distribution of energies translates into a distribution of frequency or wavelength, instead of speed.)

Well, guess how the energies (or frequencies) of CMB photons are distributed…in exactly this way. The CMB energy distribution – first measured by the COBE satellite – falls exactly on the thermal distribution described above (see the plot below), and is measured so precisely that the error bars shown are $400\sigma$ – this means that the odds of the actual value falling outside the experimental error bars is an unfathomably tiny number.

Two points to note about the plot above: First, this shows the distribution of intensity or, roughly speaking, the distribution of energy, which is proportional to the probability distribution given above multiplied by the photon energy. (More energetic photons obviously contribute more to the total energy, giving more weight to the high-energy part of the thermal distribution.) On the x-axis we have frequency, or equivalently the inverse of wavelength. Second, the theoretical curve is matched to the data points by adjusting the overall temperature; it is precisely through the CMB that we measure the temperature of the universe. But the shape of the distribution is the same for all temperatures, and it is this very definite prediction that the data matches precisely.

So CMB photons look like they’re in thermal equilibrium, but they can’t be in equilibrium today. Photons only affect each other by exchanging energy via other particles, such as electrons or protons. But all of these particles are clustered in galaxies and other bound objects. Empty space is far too dilute for particles to indirectly “pass energy” between photons scattering off them.

The natural explanation is that CMB photons were in equilibrium long ago, and have carried their equilibrium distribution of energies till today. (After the expansion of the universe made space too dilute for photons to continue colliding with other particles, they simply free-streamed through space, virtually untouched for billions of years. It’s true that all photons have lost energy due to redshift, but this just changes the overall temperature, leaving the thermal distribution intact.) But what conditions would put photons in equilibrium? Well, equilibrium is maintained through particles interacting and sharing energy, and the rate of interactions is determined by the density of particles and their speed. Faster particles forced into a smaller volume will scatter and share energy with each other more quickly. Photons, of course, travel at the speed of light, so they can’t have been moving faster in the early universe (although they did have more energy). So the only way they could have been in equilibrium would be by occupying a smaller volume, along with electrons and other particles via which they could share energy.

How much smaller a volume? It turns out, a little more than a thousand times smaller, with a temperature of over 5000 degrees Fahrenheit, over 13 billion years ago. This is hot, but not quite the hot Big Bang itself. If we rewind the clock even further, using relativity, we reach a moment a few hundreds of thousands of years beforehand where the universe is so hot and dense that atoms cannot even hold together! They would be bombarded apart into protons and neutrons. As the universe expands and cools after this point, atoms and elements start to form. When we apply what we know about atomic and nuclear physics, we’re able to predict how much Deuterium, Helium, and other elements should be created in this primordial epoch. And astronomical observations confirm the expectation, providing strong evidence for a history of the universe that extends back to nuclear temperatures and densities! But that is a story for another blog post.

## Relativity, Cosmic Expansion, and the Big Bang

(Continuing from “The Expanding Universe.”) The expansion of the universe was also a vindication of Einstein’s theory of gravity, general relativity. Although it wasn’t understood at first, general relativity predicted either an expanding or contracting universe. A static universe of constant size, neither growing nor shrinking, would be unstable, like a carefully balanced upside-down pendulum. Why? According to general relativity, the dynamics of space (e.g. its expansion or contraction) is controlled by the matter and energy moving around in space. Slightly too much matter, and the universe would start to contract. Slightly too little, and it would begin to grow.

In general, Einstein’s theory gives a relationship between the matter and energy at a given point in space and time to the shape or “curvature” of space and time itself. When we zoom out on large enough scales, the universe looks the same everywhere, in terms of the distribution of matter. Since space is the same everywhere (“homogeneous”), and is nearly flat, the curvature of spacetime in this case simply means the expansion of the universe in time. (When we think of space and time together as a single entity, expansion in time can be thought of as curving the shape of spacetime.)

So for a homogeneous universe, the equations of Einstein’s theory relate its expansion to the matter within it, and they show that the amount of matter must be very delicately balanced with the cosmological constant (roughly, the amount of energy per volume of empty space itself) in order to keep the universe static. Too much matter, and the universe collapses in a crunch. Too little, and the pressure from the cosmological constant dominates, pushing space apart and driving expansion – precisely what is happening in our universe. Without a cosmological constant, a static universe is even less stable. Even if it isn’t expanding or contracting initially, it will immediately begin to contract if there is any matter at all present. (Those familiar with the math might be interested in this page.)

In light of the instability of a static universe, we can say that general relativity predicted an expanding universe. The expanding solution to Einstein’s gravity, along with the soon-to-be-measured Hubble parameter or expansion rate, was first pointed out by Georges Lemaitre, a physicist and Catholic priest, in 1927. Lemaitre also introduced the idea of a Big Bang, another prediction of Einstein’s theory.

As mentioned above, general relativity predicts a relationship between the expansion history of the universe and the matter and energy filling it. Both for the normal matter we’re familiar with (particles or collections of particles moving much less than the speed of light) and for radiation (anything moving at speeds close to the speed of light) the size of the universe is related to the passage of time in a simple way. If we know the amount of matter and radiation in the universe, general relativity allows us to rewind the clock back to moments when the universe was extremely small and hot, with particles scattering at extraordinary energies. In the colorful language of physicists, the energy of particles “blows up’’ – becomes limitlessly large – at a finite time in the past. General relativity allows us to infer this time – the age of the universe as we know it – in terms of measured amounts of matter and radiation. It turns out to be roughly 13.8 billion years.

What exactly does general relativity predict about the Big Bang? Not a whole lot – the reason is that when energies become too spectacularly high, we expect general relativity not to apply anymore. But that’s another story, one involving quantum mechanics. What general relativity does say is that if we trace the history of the universe back in time, we reach energies so high that we don’t know what particles or physical processes occurred (see this post). In this sense, the “Big Bang” is the point in the history of the universe where things are hot, dense, and energetic beyond what established physics can currently describe. As we’ll see in future blog posts, the paradigm of inflation extends the story back even further.

## The Expanding Universe

The universe is expanding.

As explained here, this makes the past history of the universe a window into unknown high-energy physics. In this post I’ll address two questions: What does it mean to say that the universe is expanding? And how do we know that it is?

Let’s start with the first question. Consider a balloon being blown up. We can describe its expansion by saying that someone looking on sees it getting bigger – it’s volume and surface area are increasing. But we can also talk about its expansion purely in terms of the surface of the balloon. If we were two-dimensional creatures living on a balloon, we would only be able to look along its surface, not perpendicular to it. The expansion of the balloon could be discovered by measuring the increasing distance between two points, or, by measuring the increasing area of the balloon. A very small balloon observer would not see the curvature of the balloon, but could notice its expansion: everything nearby would simply be moving away, with more distant objects moving away more rapidly.

Now, the universe is three-dimensional, not two-dimensional. But everything about the balloon said above applies. The shortcoming of this balloon analogy is that the universe does not (necessarily) exist in a higher dimensional space, as the two-dimensional balloon does in three-dimensional space. But since that extra dimension isn’t needed as far as understanding the expansion goes, we can simply forget about it.

In the 1920s, Edwin Hubble observed that more distant galaxies are moving away from us more quickly: galaxies twice as far away were seen to be rushing away from us twice as fast. This is just as you would expect to see if the universe were expanding. The velocity of galaxies can be measured in terms of redshift: light from an object moving away from you will be measured to have a longer wavelength. (If it is visible light, it will be shifted towards the red end of the spectrum.) The distance to a given galaxy is trickier: One method of distance measurement uses supernova observations. Supernovae of a certain kind (“Type 1A”) always give off the same amount of light (they have the same luminosity), yet they will appear fainter or brighter to us based on their distance. Since the luminosity is fixed, the distance to supernovae (and their host galaxies) can be inferred from their brightness. Measurements of distance and redshift indicate that the speed at which galaxies are receding from us is indeed proportional to their distance.

How do we know that space itself is expanding, instead of some invisible force which is causing galaxies to repel and move away from each other? First, in the absence of expansion we have no reason to expect the distance-velocity relationship Hubble observed. No known force of nature could lead galaxies to repel each other in this way. Second, as we’ll see in the next post, Einstein’s (well-tested) theory of gravity – general relativity – leads us to expect an expanding universe, and provides a simple framework in which to understand the expansion.

## The Shape of the Universe (is there a boundary to the universe?)

•January 6, 2015 • 2 Comments

You may have wondered at one point, is there an edge to the universe?

On the one hand, the universe might simply be infinite in space, simply extending forever. Such a scenario is a perfectly good solution of Einstein’s equations of general relativity, and is consistent with all observations of the universe. There may be a legitimate philosophical objection to the existence of an actually infinite physical quantity, but I won’t get into that here.

Broadly speaking, an infinite universe is called open, and a finite universe compact or closed. And our universe could just as well be closed as open. A sphere, for example, is closed: you can go from every point to every other point, and there is no edge.

In fact, it’s possible that the shape of the universe is spherical – but a three-dimensional rather than two-dimensional sphere. Just as one can sail around the earth, it may be possible (in principle) for a spaceship to fly off in one direction and return to the same place coming from the other direction. This may seem bizarre, but it’s only confusing because our brains aren’t wired to visualize three-dimensional spheres as they can two-dimensional spheres. We wouldn’t necessarily notice the spherical shape of the universe, just as a person standing on earth cannot notice that they are standing on a sphere.

How would we know if the universe was shaped like a sphere? We might be able to measure its curvature. Think about the example of the earth again. Suppose I was standing on the South Pole, and wanted to test the hypothesis that the earth was flat. And suppose I knew of two very distant lighthouses – I knew their distance from each other, and from the South Pole. If the earth was flat, I could use simple geometry to find the angle I would expect to measure between the incoming light beams. But a sphere, that angle would be larger. (See Wayne Hu’s tutorial for an image.) This is because if you connect three points on a curved surface, the sum of the three angles formed is no longer 180 degrees.

So, by measuring the angles formed between incoming light signals from known sources, I could determine whether or not the earth was flat. The universe works in a similar way, except with one more dimension of space. The cosmic microwave background, background radiation filling space, gives us a temperature map of the early universe at a distance of almost 13.8 billion light-years (CMB light has been travelling for nearly the age of the universe). The physics of the CMB is well-understood, and we know the actual physical size of “hot spots” and “cold spots” in the CMB, so by measuring how large of an angle they take up on the sky, we can determine the curvature of the universe. As far as we can tell, it’s a flat universe. But this may only be because we lack the precision to distinguish curvature, just as a terrestrial observer would need precise instruments to discover the globe beneath them.

Now, if we did measure this kind of curvature in space, it would not necessarily mean that the universe was spherical. It could have a more complicated shape, in which our local region appears to be curved like a sphere, just as a nearly flat sheet with many tiny ripples could look curved when viewed in a very small region. In fact, this is exactly what the universe is like. Small hills and valleys in the gravitational field (which also lead to the temperature variation in the CMB) curve space locally, and a hill or valley stretching over the largest observable distances could look like a curved universe, and could obscure measurements of the global curvature just as mountainous terrain could screw up the earthly experiment described above. Fortunately, these variations in the gravitational field are tiny (one part in 100,000!) so only an equally miniscule global curvature could be hidden by them. In other words, a spherical universe would have to be enormously large to look so nearly flat from our vantage point.

We would have stronger evidence of a sphere-shaped universe if, for example, we observed the same galaxy cluster on opposite sides of the sky. In this case, we would be looking around the sphere on opposite sides, and seeing the same thing.

Note that in neither of these cases does the universe have an edge. General relativity – Einstein’s theory of gravity which describes the geometry of space and time, and has been confirmed with many experimental tests – does not describe space that way.

## Is the universe a biased sample?

Recently, at the conference for the American Scientific Affiliation, I gave a short talk on the question “Is our universe a biased sample?” You can listen to it and see a pdf of the slides, and see more info on this page.

The question here is, since we only observe a finite volume of the universe, how do we know we have a representative sample?  We hope that our measurements of galaxies, galaxy clusters, the cosmic microwave background, and other observables are representative of the larger universe. That way, we can directly compare predictions of models for the early universe, such as inflation in its many varieties, to what we observe. Of course, no model predicts exactly what structures will form and where they will form. Due to the uncertainty from quantum effects becoming dominant in the early universe, we can only predict probability distributions for observables.

So the question is, are the probability distributions which describe our finite observable universe representative, or are they biased by our cosmic environment, when compared to the probability distributions characterizing the larger universe? The short answer is: it depends on the physics at play in the early universe, which determines the initial conditions for structure formation. But if there is a systematic bias due to our location, there are observations we could carry out to constrain how large that bias could be.  See the talk for more, as well as a longer and more technical version at the Perimeter Institute here.

In the midst of other posts, I plan to highlight online videos of talks, interviews, lectures, etc. in an effort to collect some good resources on theoretical physics and cosmology.

Recently I have been working through a series of videos (starting here) from Stanford featuring a course on string theory, with Leonard Susskind. You can watch them yourself, although they’re quite long. Here I’ll just boil down some of the key ideas:

Strings, like fields, can be described as a collection of masses connected by springs – except, we’re taking the continuous limit where there are an infinite number of infinitely small masses and springs chained together. This is very similar to the description of a field in quantum field theory – in that case the masses fill all of space, and each is attached or “coupled” to its neighbors with a spring. The continuous limit gives us a field.

As with particles, fields, spacetime, and just about anything else in physics, we can talk about a classical string or a quantum string. A classical string has a definite shape at each moment in time: each of the masses oscillate back and forth and pull on their neighbors via the connecting springs. When we talk about a quantum string, each of the masses is treated as a quantum harmonic oscillator (again, this is just like the description of a quantum field, in which case there is an oscillator at each point in space). Because the constituent masses no longer have definite positions, the quantum string doesn’t have a definite shape. Instead, like anything quantum mechanical, it is described by a wave function, which tells you the likelihood (actually, the “amplitude”) for any particular shape – that is, how much different shapes contribute to the quantum state of the string.

(Funny fact: a string can pass through itself – so you would have trouble tying knots – because each point on it only interacts with its immediate neighbor points, and thus isn’t sensitive to some distant patch on the string.)

A string can be in its ground state (state of lowest energy), or in an excited state. Different particles, such as photons or gravitons, emerge as different excited states of the same string. Excited states can be described in terms of little waves propagating around the strings.

Strings can be open or closed – that is, with two free ends, or joined in a loop. Excited states of a closed string can act like the graviton (the particle which mediates the gravitational force), whereas excited states of an open string can act like the photon (the particle which mediates the electromagnetic force).

Strings can join together into one, or break into two – this is where scattering and decaying of particles comes from. A “coupling constant” or “interaction strength” (similar to the strength of an electric force, for example), tells you how common it is for strings to break or join.

If you’re interested in more, check out the video link above.

Why am I telling you all this? Aside from the fact that it’s fascinating as a theory in itself, regardless of any connection to the real world, string theory can lead to physics we know when you “zoom out” to scales where you can’t distinguish strings from point particles. So it may be the right way to describe the world at extremely small scales! Either way, it can give us new understanding of the mathematical structure of theories which do describe our world, such as quantum field theory and general relativity, which emerge from string theory in certain limiting regimes of the parameter space.

## What did the Big Bang look like?

If you want a quick answer, see the image at the bottom of the post. What follows is an explanation of what goes into that image.

By “Big Bang,” I mean the hot, dense state of the early universe, when particles were scattering at extremely high energies, and the universe expanding at an explosive rate. This state was not the beginning of the universe, although it is only a fraction of a second after times when the universe was so hot and dense that familiar physics (eg. the Standard Model) breaks down as a description of what is going on. So, the “Big Bang” is the beginning of the universe as we know it – but there is much to discover at even earlier times. Time itself may indeed reach its origin shortly before the Big Bang – perhaps at the beginning of inflation.

The early universe, at the time of the hot Big Bang, was a soup of particles – at large scales, matter (electrons, protons, etc.) and radiation (photons) could be described as a fluid. But this was a fluid of extremely high density, with particles scattering off of each other at extraordinary energies. What did it look like? Can we visualize it? If we were to zoom in, we would see individual particles scattering off of each other (just as zooming on in water reveals the molecular structure of hydrogen and oxygen), but at larger distances we would simply see a homogeneous fluid – everything would look the same in every direction. Thus, the simplest picture of the early universe would be uniform:

This is a remarkable fact! We know of the uniformity of the early universe mainly from the uniformity of the CMB or cosmic microwave background (see the next post). Only because of this uniformity are we able to extrapolate measurements of the CMB and large scale structure into the distant past, and learn about the initial conditions at the Big Bang. Otherwise, things would be so complicated that we couldn’t extrapolate into the past any more than we can predict the weather in your home town in 5 years.

However, the CMB is not exactly uniform. There are tiny variations in the temperature of the background radiation in different directions in the sky, as measured very precisely last year by the Planck satellite:

(The oval shape here is a flattened map of the spherical sky, just like a map of the earth.)

These variations are the result of slight variations in the density of the matter/radiation fluid at very early times – in fact, there is a straightforward way to translate between variations in the density and temperature.

Now, each fluidy chunk of matter (or even single particle) contributing to the matter/radiation fluid exerts a gravitational pull, just like the earth. Consequently, there is a gravitational force field filling space. At any point in space, this field points in the direction in which gravity would pull if you put a particle there. For example, at a point close to a region of higher density, the force field will likely point towards that region. We can think of this force as coming from a gravitational landscape or “potential” in which particles roll around, as if on a landscape of little hills and valleys – little ripples in the geometry of space and time, which are distorted by the variation in matter density, and have the effect of causing particles to fall “downhill.” It would look something like this:

Two points: (1) these ripples are exaggerated – in reality, they would be extremely tiny: one part in 100,000 compared to the flat, featureless background space. And (2), this image only shows the “altitude” as a function of two spatial dimensions. In reality, there is a particular value of the gravitational potential at each point in three-dimensional space, but this is harder to visualize.

The force from gravity at any point is a “pull” in the downhill directions.  The entire fluid (which is made up of both matter and radiation particles) sloshes around in this landscape of hills and valleys, with a preference for moving to lower altitude, lower energy regions.

In equation form, the variation in density $\delta\rho$ and gravitational potential $\Phi$ are related by the Poisson equation,
$\nabla^2\Phi = 4\pi G \delta\rho.$
What this equation says in words is that if you are in a region where the density is larger than average (positive $\delta\rho$), this will create a corresponding valley in the landscape pictured above. ($\Phi$ can be thought of as the “altitude” on the landscape I described above, and positive $\nabla^2\Phi$ at some point means that if you move away from that point, you will on average move uphill, so you are in a valley.) The $4\pi G$ indicates the strength of gravity, which allows the variation in matter $\delta\rho$ to have an effect on the “landscape” ripples in spacetime (that is, increase $\Phi$), which in turn produces gravitational forces on the matter.

We can represent the “landscape” image of the potential shown above in terms of color variation, like the CMB map. If we looked at a two-dimensional slice through the three-dimensional landscape that maps correctly into the CMB, it would look something like this:

This is just a sample visual – representative, but not exact. And of course, the colors here should not be taken literally. As with the CMB map above, they are meant to be indicators of slight variation throughout space – in this case, variation of the gravitational potential $\Phi$.

So what happens to this nearly uniform gravitational landscape? And what happens to the nearly uniform fluid of matter and radiation? Well, particles will eventually fall into these tiny valleys due to the gravitational force towards “lower altitude” (=lower energy), which will in turn make the valleys bigger, and thus more effective in pulling more matter in. In this way, the tiny initial nonuniformities in matter ($\delta\rho$) and spacetime ($\Phi$) grow to be much larger, eventually even forming complex structures like galaxies. Note that larger structures take longer to form, and on super large scales the expansion of the universe is fast enough to prevent structures from ever forming (at least by today). If we smooth over the universe on these large scales, it still looks uniform and featureless.

A final teaser: the gravitational potential is the same at all scales – that is, if you zoom in or out on the landscape, it will look exactly the same (unlike a geological landscape). This is a crucial inference made from the CMB that ends up being related to inflation, as we’ll see in a future post.

###### Note: My use of the word “landscape” here has absolutely nothing to do with the multiverse “landscape” of string theory.

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