Note to fellow CONErs: I'm writing this at a sort of general-public level, or only slightly more technical than that, because I need practice doing that and I don't want to accidentally get caught up in a lot of jargon. Hopefully it's still helpful!
There have been a few big transitional moments in the history of the Universe in which something really fundamental about the cosmic environment changed. Some of these happened really early on -- the onset and end of cosmic inflation, reheating, the QCD phase transition, big bang nucleosynthesis... these all had to do with the nature of spacetime or the kinds of particles existing in the Universe going through a sudden change, and maybe we can write about all those sometime if we get a chance. But there have been other, somewhat more recent transitions in the Universe in which all the building blocks of matter were already present, but something big changed in the way they interacted.
The first of these was the epoch of recombination, also referred to as photon decoupling or last-scattering. (These are all subtly different things, but often discussed as the same event -- I will explain in a moment.) Recombination is probably the most inaccurately named event in the history of the Universe, on account of the fact that there was no "combination" before it.* In the very early universe, there was the all-encompassing energy-matter-plasma-fireball (which we sometimes refer to as the "hot big bang"), the product of the first cosmic explosion, which was rapidly expanding outward in all directions. This fireball was formed of protons and electrons (and other particles), all of which were hot and unbound and bouncing off photons and being energetic. (Terminology note: Protons and electrons or other charged particles that aren't bound together are called ions, so the fireball was made of ionized gas, also called plasma.) In the fireball, the particles making up the matter and the photons making up the radiation were coupled together, in the sense that they were all mixed up and bouncing around and a big indistinguishable mess. But the fireball expanded as spacetime did and the fireball got cooler, and the particles lost some of their frenetic energy. Eventually, there came a time when the fireball was cool and diffuse enough that protons and electrons could get close to each other and actually become bound atoms, instead of just careening around pinball-style. The photons were still there, still bouncing around, but now instead of just ricocheting off the ions, they could get absorbed by the atoms, or even sail right by them in the newly abundant spaces between. Sometimes the photons would be energetic enough to ionize the atom, but it would recombine and send the photon out again, and the Universe was becoming diffuse enough that atoms spent more time bound than not. *"Recombination" is a sort of technical term in physics, and in this case it refers to the joining of an electron and a photon, without regard to whether that particular electron and photon had made up an atom before. Even at very high temperatures, hydrogen atoms would form, but they'd be broken up immediately by energetic photons. The name "recombination" refers to the time when the hydrogen atoms that formed could stay bound for an appreciable amount of time.
And so, at the epoch of recombination, around 300,000 years after the big bang (at a redshift of about 1100), the gas went from being ionized to neutral. Recombination also reduced the amount of interaction between matter and radiation, setting off the era of decoupling. Decoupling is also known as last-scattering, because it was the last moment when photons would immediately be scattered off matter as they flew around -- after decoupling, the photons are free to sail around unimpeded and travel for long distances. Which is where the cosmic microwave background (CMB) comes from -- the newly decoupled photons free-streaming through the Universe out of the great primordial fireball.
The next phase of the Universe is distinctly unexciting. It's called the dark ages. During the dark ages, the Universe is full of cooling neutral gas (mostly hydrogen), and that gas is very very slowly coming together into clumps via gravity. At decoupling, the overdensities seeding these clumps were more dense than their surroundings by about one part in 10^5 -- these tiny overdensities, which we see as fluctuations in the CMB, were enough to tip the scales to draw more matter together into bigger and bigger clumps. But it took a while for anything particularly interesting to happen. Sometime between 100 and 500 million years after the big bang (z~20-14), one of these little clumps became dense enough to form the first star, and that defined what we sometimes call first light of the universe. (Of course it wasn't the first light -- the fireball made plenty of light, and we still see it as the CMB -- but it was the first starlight.)
Question: If we had a big enough telescope, could we look far enough back into the Universe to see that first star?
Answer: No. It turns out the dark ages were dark for two reasons. One was that there wasn't any (visible) light being produced (before the first star at least). The other is that neutral hydrogen is actually pretty opaque to starlight. During the dark ages, any photon energetic enough to be absorbed by a hydrogen atom very likely would be. Radio waves or other low-energy photons could get through, but neutral hydrogen is happy to accept photons of optical or infrared light and use them to knock its electrons into higher energy levels. Those atoms will then release the photons eventually, in a different direction, and through repeated scatterings and a little bit of Universe-expansion, the photons are unable to get very far.
It gets confusing, actually, because people talk about two kinds of "opacity" and "scattering" in the early universe. The primordial fireball is opaque to the high-energy photons produced in the big bang, because the fireball is very dense and there's a lot of scattering, especially for those high-energy photons. But around the time of recombination, the cosmic microwave background photons have been redshifted by the Universe's expansion to be on the long end of optical wavelengths, moving over to the infrared. These photons aren't energetic enough to really do much to neutral hydrogen, so they can pass through the newly neutral IGM unimpeded. So after recombination, the universe is no longer opaque to the cosmic microwave background, which is the only light there is at that point.
But that same neutral gas is, later, opaque to visible and ultraviolet light -- the light produced by stars. So if you talk to an optical or infrared astronomer (because the visible/UV light produced by the first stars would be infrared by the time it got to us), they'll say the dark ages are when the Universe is opaque, because the atoms are neutral.
Anyway, once stars are forming, astrophysics can really get going, and fun things start to happen. The vast majority of the gas in the Universe (the intergalactic medium, IGM) was still neutral at this point -- mostly hydrogen not doing much -- but each star or galaxy that formed would heat the gas around it and make a bubble of ionized gas. As more and more of these bubbles formed, the IGM had a bit of a swiss-cheese nature, with bubbles of ionized gas growing and coming together in a background of neutral gas.
Once there are enough stars and galaxies to ionize a significant fraction of the IGM, you get the overlap phase and then reionization: the (more aptly named) epoch when the IGM goes from being neutral to being fully ionized again. And this time, the universe is much less dense and the starlight can easily pass through the ionized gas, so the universe is transparent to starlight. This transition occurs somewhere between redshift 11 and 6 (around a billion years after the big bang) -- and probably gradually and clumpily and at different times in different places.
Here's a handy graphic George Djorgovski made about reionization: http://www.haystack.mit.edu/ast/science/epoch/pictures/reionexpl.jpg
And here's an artsier version, from an article in Scientific American by Avi Loeb: http://reionization.org/tt12b.jpg
And here's a nicely annotated one from a Nature article: http://www.nature.com/nature/journal/v468/n7320/carousel/nature09527-f1.2.jpg
So, how do we know when reionization happened? And why does it matter? Second question first: it matters because understanding reionization means understanding how the first sources of light in the Universe formed and how the IGM turned into the large-scale structure we see today. Also, it's a major milestone in the Universe's history, and a phase transition of the entire IGM, so it seems important.
Back to the other question: there are a few ways to about it. One way is to look for the onset of the opacity mentioned above. If you can find a quasar at a high enough redshift that it's around the epoch of reionization, you can use the way the IGM partially absorbs its light to get evidence of neutrality. There are two things you need to know about quasars, first. One is that they have a lot of emission in Lyman-alpha, which is the 2->1 transition of neutral hydrogen. So if you look at a quasar spectrum, the big peak in it is the Lyman-alpha line. The other thing you need to know is that they also have continuum emission, so there's a lot of light put out at higher and lower wavelengths.
Here are a couple of quasar spectra, at two different redshifts, but shifted to the emitted wavelengths: http://www.astro.ucla.edu/~wright/Lya-forest-60.gif The big peak is the Lyman-alpha emission, at 1216 Angstroms. To the left of that on this plot is "blue-ward" (higher photon energies) and to the right is "red-ward" (lower photon energies).
The most obvious difference between these spectra are the "bar code" absorption lines in the higher-redshift spectrum. This is called the Lyman-alpha forest. Here's a movie of how it works. Basically, the neutral hydrogen really likes to absorb Lyman-alpha-wavelength light, and has trouble effectively absorbing light at longer wavelengths. So as photons from the quasar on the blue-ward side of the spectrum travel toward us, redshifting along the way, they will at some point be at the Lyman-alpha wavelength and be immediately eaten up by whatever neutral hydrogen is around. After reionization, neutral hydrogen only exists in little clumps, so a lot of those bluer photons just pass right through the nice transparent IGM and they don't hit anything troubling. But if the light does encounter a neutral region, whatever part of the emitted spectrum is at the Lyman-alpha wavelength at that point will get absorbed, and you'll get a narrow absorption line in the spectrum, as you can see in the z=3.62 spectrum linked to above. Once in a while, the light will pass through a large dense neutral region (like a piece of a galactic disk, maybe) and the damping wings will take a much bigger bite out of the spectrum, and that's called a damped Lyman-alpha system (DLA). Here's an animation of how the Lyman-alpha forest happens and what makes a DLA, the big v-shaped gap in the spectrum: http://www.cosmocrunch.co.uk/media/dla_credited.mov As you can see in the movie, there are other lines that are important too, such as Lyman-beta and some metal lines. But Lyman-alpha is really the main one.
How does this help us find reionization? As we get to higher redshifts, there will be more and more neutral regions, and bigger parts of the spectrum will be taken out. Here's a graphic from the Djorgovski group, showing a spectrum they used that they claimed was the first hint of a quasar that was really in the epoch of reionization: http://www.astro.caltech.edu/~george/reion/discovexpl.jpg You can see that instead of a forest of dark absorption lines, now you have just a few little peaks of transmission where the light goes through ionized bubbles. Eventually, you'll have so much neutral gas in the IGM that pretty much all the light just blue-ward of the Lyman-alpha emission peak is absorbed, as you can see here: http://cmblenser.files.wordpress.com/2009/04/gp.png?w=450&h=507 In the highest-redshift quasar (bottom panel), you see a flat-line region to the left of the peak called the Gunn-Peterson trough. Fan et al. put together a nice compilation of a bunch of high-redshift quasars to show the Gunn-Peterson trough here: http://ned.ipac.caltech.edu/level5/Sept06/Loeb/Figures/figure60.jpg
That should be it, right? We found reionization? Unfortunately not. It turns out that neutral hydrogen absorbs Lyman-alpha so effectively that if only one part in 10^5 of the hydrogen is neutral, it absorbs enough light that the Gunn-Peterson trough is complete: zero transmission. So we can't use the Lyman-alpha forest to say that we've reached into the neutral IGM -- all we can say is that around redshift of 6, at least 10^(-5) of the hydrogen was neutral. It's a nice limit, but it can't really shed light on how reionization progressed, because once the Lyman-alpha is gone, it's gone. There isn't really any more information to be had there.
The CMB gives us another clue, in a slightly less direct way. Instead of being sensitive to the neutral gas (which, as I said before, is transparent to CMB photons), the CMB is sensitive to free electrons. There's a polarization primer on this on Wayne Hu's webpage, here: http://background.uchicago.edu/~whu/polar/webversion/node3.html#SECTION00021000000000000000 and a really nice explanation of the effect of reionization on the CMB in Section 2.1.2 here: http://ned.ipac.caltech.edu/level5/March06/Choudhury/Choudhury2.html The gist is that CMB photons scatter off free electrons, and the presence of free electrons and a density quadrupole alter the polarization of the CMB photons. The quadrupole is important here. It only happens if you have significant density contrasts, and the scale on which it happens is determined by the scale of fluctuations. In the early universe, around the time of decoupling, the scale is really small and the fluctuations are really small. There is a polarization signal induced from that, but it is tiny and only occurs at a multipole of about 100. Polarization signals at larger scales would have to have been generated later, which means they come from after reionization, when there are free electrons around again. If reionization happened all at once, the strongest signal would be at the horizon scale at that epoch, and you'd have a peak in the polarization signal at l ~ zr^(1/2) where zr is the redshift of instantaneous reionization. The amplitude of that peak is proportional to the TOTAL optical depth of free electrons -- which comes from the integrated sum of all the free electrons between the last-scattering surface and us. That's why you have to assume an instantaneous reionization to get some solid number out of it. In effect, that gives us an upper limit on the redshift of the completion of reionization, because an instantaneous reionization at zr would give the same optical depth as a gradual reionization finishing at some redshift z<zr.
The scattering of CMB photons off free electrons also damps the temperature signal, just because scattering moves some of the photons away from the path toward us, but that signal is highly degenerate with other signals. And the polarization is hard to detect directly. So the easiest thing to look for is the temperature-polarization cross-correlation -- specifically the TE cross-correlation power spectrum. It has a peak at low l (large scales) whose amplitude can give you the free-electron optical depth, also called the Thomson scattering optical depth. And that gives you a handle on the end of reionization. WMAP7 gives us the optical depth tau of 0.088 and an instantaneous reionization redshift of 10.5 (http://lambda.gsfc.nasa.gov/product/map/current/params/lcdm_sz_lens_wmap7.cfm). So we can say pretty confidently that the process of reionization started sometime after z=10.5 and ended sometime before z=6, but getting more precise than that is complicated. (There are some methods to constrain this a bit using things like Lyman-alpha damping wings, but I won't get into that right now.)
So, where does that leave us? Well, we know reionization happened sometime between z=10.5 and z=6, but we don't know how it happened and we can't see anything in the dark ages. This brings us to the next stage of reionization / dark age exploration -- using the 21cm transition of neutral hydrogen to look at the dark ages directly.
There are two things that are important/special about 21cm radiation. One is that it's a transition of neutral hydrogen -- specifically, the spin-flip hyperfine transition of the hydrogen ground state. The ground state has two energy states: one where the electron and proton have parallel spins, and one where they have opposite spins. (In technical terms: the lower state is the singlet state with total spin S=0, electron and proton anti-parallel, and the upper state is the triplet state with total spin S=1, electron and proton parallel. Here are some more details: http://www.ucolick.org/~krumholz/courses/spring10_ast230/notes7.pdf)
The other important thing about the 21cm transition is that it's in the radio -- wavelength about 21cm, frequency about 1420 MHz -- which means that it can travel unimpeded through neutral IGM and probe the dark ages directly. Also, because it's such a low-energy transition, it doesn't take a lot of energy to excite it, and even hydrogen that's mostly cold and boring can emit 21cm radiation that can get to us today. (You might ask why if the hydrogen can emit it, it doesn't immediately absorb it again -- the main reason is because the interaction cross section is fairly low and as soon as the photon is emitted, it starts to redshift with the expanding universe. So once it gets out of the immediate region in which it's created, it really can't get absorbed again by much of anything. There aren't a lot of transitions at lower energies than the 21cm transition.)
Of course, by the time 21cm emission gets to us, it won't be at 21cm anymore. It'll be at 21*(1+z) cm, where z is the redshift at which the photon is emitted. So if you want to look at reionization, you're looking at wavelengths of at least 21*(1+6) =147 cm -- almost a meter and a half! The upshot of that is that you need large telescopes, or dipoles, or interferometry arrays, to see the signals.
One of the downsides to working at those frequencies (we're talking about something like 100-200 MHz for the epoch of reionization and a bit of the dark ages) is that you're now smack in the middle of all sorts of terrestrial radio communication: FM radio, cell phones, satellite transmissions... it's a big mess. Also, at low frequencies, the ionosphere is highly refractive and can do all sorts of horrible things to your signals as they're coming through the atmosphere. If you go all the way down to 70 MHz, the ionosphere is actually opaque. So you have to find a place that's relatively radio-quiet (i.e., unpopulated) to do this sort of study, or you have to find a way to deal with the radio noise.
Another big challenge is the Galactic synchrotron emission. The Galaxy has a big magnetic field and it produces a lot of synchrotron radiation which is extremely bright at the low frequencies we're dealing with here. 21cm signals are typically about 10 mK in brightness temperature (a measure of the amount of radiation produced at the telescope) -- the synchrotron from the Galaxy is 100s of Kelvins. And it's spatially varying in weird and complex ways. Here's a map of the Galactic synchrotron at 408 MHz: http://healpix.jpl.nasa.gov/healpixSkymapsGalSynchEmssn.shtml It gets worse for lower frequencies.
Nonetheless, there's a lot of effort right now going into building the telescopes to see this signal, because it would allow us to actually get a picture of the IGM in the epoch of reionization. Ideally, we'd get pictures like this: http://reionization.org/eor.21cm.png where each square represents a different redshift, going down in redshift (or forward in time) as you move left->right and top->bottom in the figure. In the upper left-hand panel, the IGM is largely neutral. In the lower-right hand panel, it's ionized. The features in the other panels are ionized bubbles forming and growing. Each of these simulations represents a small patch of sky, but in theory you can imagine doing a full-sky map. Let's say you wanted to see the 21cm emission coming from redshift 9. You know that it's emitted at 21cm and by the time it gets here it's at 210 cm, or about 142 MHz. So if you tune your radio telescope to 142 MHz, and you can somehow take out the Galactic synchrotron and the terrestrial foregrounds and whatever else there might be, you have a map of all the neutral hydrogen at z=9. Tuning the telescope to a number of different frequencies corresponding to 21*(1+z) cm, you can get a sort of CMB-style map at each redshift. Here's a (rather slow) simulation of what a piece of that map might look like: http://gamow.ist.utl.pt/~msantos/fp-content/attachs/t21_mov.gif
But before we get too excited, I should reiterate that dealing with the foregrounds and instrumental calibration and stuff is hard. There are actually a number of intermediate steps (including getting an all-sky integrated signal, or getting a power spectrum) that would have to happen before real mapping (or "tomography"), but that's the ultimate goal.