Tuesday December 2
The study of Cosmology is the study of the Universe as a whole. This means there is a fundamental problem with the subject: We have only one Universe to observe. Thus, cosmology differs from any other scientific pursuit, in that we do not have, and cannot have, an ensemble of objects to study.
This means we have to be very careful about distinguishing between the things we observe and the things we assume.
The Cosmological Principle states that the Universe is homogeneous and isotropic on large scales. Thus, it is a restatement of the first two points above.
The Copernican Principle states that we do not live in a preferred location of the Universe. And, more strongly, there IS no preferred location in the Universe. Thus, it combines the assumptions of homogeneity and universality.
Now, let me ask a question: Why is the night sky dark?
This seems pretty simple, but it isn't. It opens up a problem known as Olbers's paradox. Stated simply, "If the Universe is static and infinite, then the sky should be as bright as the surface of the Sun." This is because any line of sight would eventually intersect a star.
Heinrich Olbers wasn't the first person to realize this. The understanding that this was a problem goes back at least as far as Kepler. And it remained a problem until well into the 20th century. The resolution of the problem begins with another of our small store of cosmologically relevant observations: The Hubble Law.
Recall the form of Hubble's Law:
The obvious (but wrong) interpretation of this is that we are in the center of the Universe, and everything is expanding away from us. But any observer, in any galaxy, would see the same thing.
A classic analogy to demonstrate this is that of a loaf of raisin bread (unbaked). As the yeast produces gas, the loaf expands. If an observer, sitting on any given raisin, measured the recessional velocity of the other raisins, that observer would recover the Hubble Law (ignore the edges of the loaf).
Another classic analogy is that of an expanding balloon. This gets into some issues of geometry that many students find perplexing. But consider this: As the balloon expands, the spots on the surface of the balloon move apart from one another. If you were a two-dimensional observer, existing on the 2-D surface of that balloon, you would recover the Hubble Law for the spot recessional velocity ON THE SURFACE. Remember that you are a 2-D critter, and you have no physical perception of any third spatial dimension. Now, the "center" of the expansion is not part of the surface that is expanding. In the same sense, the "center" of the expansion of our Universe isn't part of the 3-D thing that is expanding. Some people like to consider the "center" of the expansion to be the point where t (time) = 0.
Another often troubling point is this: SPACE ITSELF IS WHAT IS EXPANDING. This is not a case of stuff expanding away from some center into a Euclidean space. The actual space is expanding (just like the surface of the balloon).
Now, this has a really remarkable consequence. We see the Universe expanding. If you run that movie backward, then there must have been a point at which t=0, when the volume of the Universe was V=0, and the density was infinite. The expansion of the Universe from this initial state is THE BIG BANG.
Now, let's go back to the Hubble Law, and do some algebra:
The Hubble constant has dimensions of 1/time, so the inverse of the Hubble constant is a time-scale for the age of the Universe. It isn't exactly the age of the Universe, because the expansion rate can change with time (due to gravity, say). But it's a useful scale. Our current best estimate for Ho implies a "Hubble time" of about 13 Gyr.
It's time to return to Olbers's paradox. A central assumption of the paradox was that the Universe is infinite. We do not know if the Universe is infinite or finite, but we do know that it appears to have a finite age. This being the case, Olbers's paradox is resolved because we can only see objects that are closer to us than the distance light has travelled since the Big Bang. So, if the Universe if 13 billion years old, then we cannot see anything more distant from us than 13 billion light years (even if the Universe is infinite).
Another part of the resolution of Olbers's paradox is the redshift-distance relation (Hubble's Law). More distant objects have higher redshift. Thus, we observe the photons from them at a much lower energy than they were emitted at.
This last point has another implication: The Big Bang was HOT. One can reach this conclusion from a number of angles. Perhaps the simplest one is to consider the case of gas collapsing under its own gravity. As it collapses, its density goes up, and it gets hot. If we run the Universal Movie backward, we see the same thing happening.
The idea that the Big Bang was hot goes back to a series of papers by Ralph Alpher and collaborators, written in the late 1940s. But at that time there wasn't any way to test the model observationally. So the point languished for some 20 years. In the mid-1960s Arno Penzias and Robert Wilson, a pair of physicists working for Bell Labs, were studying low-power radio communications technology. They noticed that they had a persistent source of noise in their signal. It showed up no matter what direction they pointed their antenna . No matter what the conditions were. No matter what. Penzias and Wilson had accidentally discovered THE COSMIC MICROWAVE BACKGROUND RADIATION. It is wonderfully fit by a Blackbody curve, with a temperature of 2.73 K. The data in that plot come from the COBE satellite mission. Penzias and Wilson were able to assign about the correct temperature from their first observations, but the COBE fit is the current standard.
Now, if you think about it, this brings us back to Olbers's Paradox again. Another resolution of the paradox is that the sky IS bright. It's just bright in microwaves. Not is visible light.
The CMB is evidence that the early Universe was hot. How so? Recall the analogy of running the movie projector backwards. As we run the Universe backward from current conditions, the density and temperature of the Universe increase. At sufficiently high temperature, structure is unable to form (no galaxies, no stars, nothing but gas). When the gas was sufficiently hot, it would have been ionized. So the early Universe would have been filled with a plasma of protons, electrons, and photons. The Universe was fully ionized at this point, so the matter and the photons were coupled together. But as the Universe expanded and cooled, it eventually reached a temperature (about 3000 K) at which the gas could no longer remain ionized. At that point, neutral Hydrogen atoms formed, and the Universe became tranparent to the photons. This is known as The Epoch of Decoupling (because before that time, the photons and the matter were "coupled" together, and after that point they were not). It is also sometimes called "The Epoch of Recombination", although this is something of a mis-nomer (the electrons and protons are combining for the first time!).
Thursday December 4
The CMB was discovered in the mid-1960s, and for about 25 years there was no evidence that it was anything but completely smooth. The only structure we could detect is the dipole structure that arises due to the Doppler effect from our own motion with repect to the background. Any actual structure in the CMB is the seed structure out of which all the current structure in the Universe formed, so if we are to have any hope of understanding things like galaxy formation, we need to know what the structure in the CMB looks like. This structure was finally detected in the early 1990s by the COBE satellite. The CMB is smooth to about 1 part in 100,000 (that is, the Universe is VERY isotropic). The observed structure of the CMB tells us what the small-scale perturbations were that evolved into the large-scale structure we see today. They are thus one of our only inputs for any model of galaxy formation.
In the last 15 years, our ability to measure anisotropy on smaller angular scales has improved a great deal. The current state of the art is provided by the WMAP mission. The improvement in the angular resolution is important because the COBE maps showed structure only down to scales that are much larger than the structures we see in the current Universe. At WMAP resolution, we are beginning to be able to see anisotropies on spatical scales that have evolved into the structures (clusters of galaxies) that we see in the Universe today.
Now lets return to the exercise of running the Universal movie projector backwards. Starting with now. The current redshift is z=0, T~3 K, t~15 Gyr.
The most distant actual objects we have observed are galaxies and QSOs at a redshift of z~6. At z=5, T=15 K, and t~1 Gyr. So somehow, the very smooth Universe that we see in the CMB managed to develop enough structure to form the first galaxies and QSOs in something like 1 Gyr.
The next point where we have any information is the Epoch of Decoupling. Here z~1000, T=3000 K, and t~100,000 yrs.
Before decoupling, the Universe was ionized. At earlier times it was increasingly dense and hot. In order to study the Universe at times before decoupling, we must use indirect methods. Just before decoupling, the Universe was a fairly smooth plasma of radiation, electrons, protons, and a few ions of other light elements. As we run the movie backward, the temperature and density both become higher. When we reach the first 30 minutes or so after the Big Bang, the temperature and density of the Universe are high enough for fusion reactions to occur. Before these fusion reactions, the matter in the Universe was entirely in the form of protons, neutrons, and electrons.
Neutrons are unstable particles if they are not in atomic nuclei. They have mean lifetimes of about 15 minutes before they decay into protons and electrons. This places a contraint on how the Big Bang Nucleosynthesis worked in detail. In addition to this, the Universe was expanding and cooling while the nucleosynthesis was occuring.
The lifetime of free neutrons, and the density of the Universe during the period when the temperature allowed fusion reactions determine what mix of fusion products should have been produced. We can use observations of those elements and isotopes in the current Universe to determine what the mix actually was. Combining these two sorts of information tells us about the physical conditions of the early Universe.
The basic reasoning is that a high density at a given temperature will produce a lot of helium, but very little deuterium. Deuterium is easy to destroy, and to fuse into helium. Similar arguments can be imposed on the other light elements (isotopes of He, Li, and Be) to constrain the conditions when fusion was occuring.
Before the epoch of Big Bang Nucleosynthesis, the Universe was a sea of photons, electrons, protons, and neutrons. As we run the movie backward, we reach a point where the energy a a typical photon is high enough for Pair-Production of particles to occur from photons. This takes us to about one second before the Big Bang. Before this time, there was a thermal equilibrium between matter particles and photons. By this, I mean not only that they were at the same temperature, but that matter and energy could convert into each other as a matter of course. And that there was a nearly equal amount of matter and anti-matter in the Universe. Once the Universe cooled off enough that photons could no longer produce particle-antiparticle pairs, the pairs annhilated, leaving a small amount of matter, and a LOT of photons.
For some reason that is mysterious, there was a slight imbalance between the amount of matter and anti-matter at this time. There was about a part in a billion more matter than anti-matter. So, when all the anti-matter had annhialated, there was a small amount of normal matter left. That is what we see in the Universe today.
As we push the clock back further, conditions become increasingly more exotic. The Universe was dense enough to be opaque to neutrinos at times earlier than a second or so. At about a millionth of a second, the temperature became high enough that protons and neutrons would be unable to form. Before this, the Universe was filled with unbound quarks. Quarks are the particles that make up protons and neutrons in normal atoms.
Our ability to understand the early Universe becomes more and more compromised by our inability to reproduce analogous conditions in accelerators, and by the need to include both quantum mechanics and general relativity in the physical models.
The existence of the CMB, and the observed products of Big Bang Nucleosynthesis are strong arguments in favor of the Big Bang Model. But there are a number of problems that the Big Bang Model doesn't address. One of them is called The Horizon Problem. In essence, the problem is that the Universe IS isotropic. If we look in two opposite directions, we see the same CMB, and even the same scale of fluctuations in the CMB. This, despite the fact that those two directions are not yet in causal contact.
Another problem is The Flatness Problem. We observe the density of the Universe to be very close to the critical density, but NOT the critical density. But the Hubble expansion drives the ratio away from 1. Fluffing over a lot of UGLY mathematics, the basic problem is that for the current density to be as close to the critical density as it appears to be, the ratio of those two numbers must have differed from 1 by only a part in 10 to the 50th power or so. This is an example of a "fine-tuning problem".
An idea that tries to address these issues is The Inflationary Universe Model, proposed in the early 1980s. The idea is that the vacuum energy density of the Universe drove an episode of exponential expansion in the first 10E-24 of a second. Thus the size-scale of the Universe increased by 50 orders of magnitude or so. This means that the subsequent evolution of the horizon has simply brought back in more of the previously causally connected Universe. It also means the Universe was driven to a high degree of flatness by the inflation.
This is similar in some senses to asking "If you through a rock straight up, will it keep going, or will it eventually fall back to the ground?" The practical answer to that is "It will fall down", but that's because you can't throw it hard enough to have it escape Earth's gravity. So in principle, the answer is "that depends on how hard you through it, and how strong the local gravity is."
We can approach the problem by considering the current rate of expansion, and work out the density of matter required to halt that expansion due to gravity. So imagine an average sphere of the Universe, of some radius R, and containing some total mass M. Also, put a galaxy of mass m at the edge of this sphere. The question boils down to this: Does the galaxy (m) have enough kinetic energy due to the Hubble expansion to overcome the gravitational attraction of the total mass M? If the answer is exactly yes, then the kinetic energy of the galaxy will equal the gravitational energy that the total mass M exerts on the galaxy. Here is the algebra (that I skipped over in class):
The "test-mass" m drops out. This is a good thing, as we are looking for a global relationship. But notice the v. This is just the expansion velocity from Hubble's Law: v = Ho R. So we can do the following:
As I said, we are looking for a global relationship, so we really don't want those M's and R's in the equation. We are assuming we have a representative sphere of Universe, and that means that the density of the sphere in question is the same as the average density of the Universe. Also, we are examining the case where the density of matter is just enough for gravity to counteract the Hubble expansion. So we are solving for what is called The Critical Density. Density is just mass per unit volume, so
Where I have given the volume of the sphere in terms of its radius R. Now we take this expression for the mass M, and put it in the equation above:
Notice that all the R's drop out!
This is very good, as ALL of the local variables (M, m, R) have dropped out of the relationship, leaving only constants, and global quantitites. So, I now rearrange the above to solve for the critical density:
So the critical density depends only on Ho. Putting in our best estimates for these numbers, this means
Will the Universe continue to expand forever, or will it eventually recollapse? Put another way, "Is the Universe finite or infinite?" Put still another way, "Is the overall curvature of the Universe positive or negative?" If the Universe is positively curved, and finite, the Universe really is like a 3-D balloon, embedded in a 4-D spacetime. On the other hand, the Universe could be negatively curved, and infinite. Or it could be exactly between the two, a flat Universe.
Observational estimates of the matter density are only about 10% of this. And only about 10% of that is due to normal matter (the other 90% is Dark Matter). So it appears that the density of the Universe is less than the critical density. This would imply that the Universe will not collapse back on itself, but will instead expand forever.
Another way of trying to test this is to observe objects at large distances, try to determine their distance by some means other than the Hubble Law, and look for deviations from a strictly linear Hubble law.
Well, we think we have a way to do this. One needs to detect Type I Supernovae in distant galaxies. This turns out to be a doable thing, if you plan your observing carefully enough. Now, we know that SNIa's in the nearby Universe are pretty good standard candles (or standard bombs, if you will): They reach a fairly consistent maximum luminosity. These are the supernovae that we think are due to White Dwarfs accreting enough matter to go over the Chandrasekhar mass limit. There is no obvious reason why such objects should have behaved differently in the distant (and thus long-ago) Universe. And if they do not (that is, if they really are the same sort of beast as the ones we observe in the current Universe), then we can determine their distances just by measuring their maximum apparent brightnesses (and assuming they have intrinsic maximum luminosities equal to those we see nearby).
The results of this experiment have been nothing short of astonishing. The synopsis is that the expansion of the Universe appears to be accelerating with time. Not slowing down (as one would expect if the influence of gravity were all that was in play), but actually speeding up as time goes on. To understand how bizarre this is, imagine seeing someone throw a rock up into the air, and then seeing the rock accelerate upward after it left the hand of the person who threw it.
This is a supreme puzzle. It is the nature of scientific enquiry that when one reaches the boundaries of what we understand, one is left with a puzzle. This is particularly the case in a field such as cosmology. If it were 20 years ago, I would have left you with a different puzzle. Perhaps in 20 more years this puzzle will have been resolved and I will leave your followers with a new and deeper puzzle. But here is the mystery I leave for you.
