What does ionization of neutral Hydrogen have to do with "transparency"? Most accounts of the early history of the Universe make some reference to (re)ionization as being the reason that the Universe becomes transparent after a period of opacity caused by the absence of ionized hydrogen. These accounts suggest that neutral H is opaque, wile ionized H is transparent. But I don't follow the reasoning.
I understand that, while neutral H is primarily absorbing at a single relevant wavelength, those wavelengths are "smeared out" by expansion, as seen by an observer, so that a dense and nearly continuous neutral H at a large range of redshifts would result in absorption at a corresponding wide range of wavelengths (i.e. "opacity" at those wavelengths, as seen by an observer "on the other side" of the absorbing intervening H). But I'm not clear on why ionization of the same H, alone, would result in clearing up this opacity.
 A: There is a lot of confusion on this issue, and indeed plenty of textbooks have got their terminology mixed up. The brief story of the opaqueness of the universe is as follows:


*

*In the beginning, everything was a plasma. The photons coupled to the free protons and electrons, unable to travel far before scattering.

*As the universe expanded and cooled, neutral hydrogen formed, decoupling matter and radiation. This is known as recombination.1 Using redshift to demarcate time, this occurred at $z = 1091.64\pm0.47$.2,3 Photons could now travel very far indeed.

*Eventually, matter started to collapse into gravitationally bound structures. The first stars, so-called Population III stars, formed. They were massive and emitted a great deal of ultraviolet radiation capable of ionizing neutral hydrogen. Thus at a redshift of $z = 10.1\pm1.0$2 the universe underwent reionization. This naturally reduced the mean free path of the photons.

*However, at the time of reionization, the scale factor of the universe was a hundred times larger than at the time of recombination. The universe was diffuse enough that this decrease in mean free path was not enough to make it opaque again.

*We continue to live in a universe with a largely ionized interstellar medium. It is nonetheless diffuse enough to not interfere with photons, and it is growing thinner with the passage of time.


Putting in some numbers, the present baryon density (read: proton density) of the universe2 is $n_0 = 0.2497\pm0.0037~\mathrm{m}^{-3}$. We can also make some simplifying assumptions:


*

*Photons are not energetic enough to worry about relativistic corrections, even when scattering off electons.

*Photons thus scatter off electrons with Thompson cross section $\sigma = 6.652\times10^{-29}~\mathrm{m}^2$.

*Photon-proton scattering is negligible, being reduced by a factor of the mass ratio squared.

*Recombination and reionization are instantaneous.

*When ionized, the ionization fraction $\chi$ is $100\%$.

*When neutral, the ionization fraction is $10^{-5}$.

*Photons only scatter off charged particles - everything else is transparent.


With these assumptions, the number density of electrons is
$$ n(z) = \frac{n_0}{a(z)^3} \chi(z), $$
where $a(z) = 1/(1+z)$ is the scale factor of the universe. The mean free path is then
$$ \lambda(z) = \frac{1}{n(z)\sigma}. $$
A quick plot of both $\lambda$ (violet) and $\lambda/a$ (blue; in some sense a more "fair" measurement normalized by the size of the universe) as functions of $z$ is shown below.

Time goes from right to left. As you can see, the jump caused by reionization is countered by the increase that occurs between recombination and reionization. Once the first stars reionized the universe, the mean free path was still a gigaparsec ($10$ comoving gigaparsecs), enough for the universe to be considered transparent.

1 Not to be confused with matter-radiation equality, which occurred earlier and denotes the transition from when the gravitationally dominant components of the universe were relativistic to the dominance of non-relativistic matter in determining the evolution of the universe.
2 Value from WMAP nine-year data, Table 17.
3 Actually, photons decoupled from baryons slightly earlier than baryons decoupled from photons. And of course the process was not instantaneous.
A: http://en.wikipedia.org/wiki/Recombination_(cosmology)

The first phase change of hydrogen in the universe was recombination,
  which occurred at a redshift z = 1100 (400,000 years after the Big
  Bang), due to the cooling of the universe to the point where the rate
  of combination of an electron and proton to form neutral hydrogen was
  higher than the ionization rate of hydrogen. The universe was opaque
  before recombination because photons scatter off free electrons (and,
  to a significantly lesser extent, free protons), but it became
  transparent as more and more electrons and protons combined to form
  hydrogen atoms. While electrons in neutral hydrogen (or other atoms or
  molecules) can absorb photons of some wavelengths by going to an
  excited state, a universe full of neutral hydrogen will be relatively
  opaque only at those wavelengths, and transparent over most of the
  spectrum. The Dark Ages start at that point, because there are no
  light sources yet other than the gradually darkening cosmic background
  radiation.

The mean-free path of photons was always very low when temperatures were so high that they could barely travel anywhere before they scattered off free electrons (electrons off ionized atoms).  Once temperatures cooled down, though, the mean free path of the average photon increased to the point where it could expect to travel across the entire universe without getting scattered off by a stray free electron (and we got http://en.wikipedia.org/wiki/Free_stream).
There is one question I do have though: do non-ionized electrons occupy less "space" (on average) than ionized electrons? Hydrogen atoms only have 1 electron, after all, so you're not going to see electrons clumped around atoms. If this question has an answer of "yes", then the above explanation can be easily understood.
A: Before the time when the universe first became neutral (recombined) at z approximately 1100, the ion density was very high so that the mean free path of light photons was small enough that you could consider the universe as opaque. This opacity is what we see when we look at the Cosmic Microwave Background (CMB). After the recombination, the universe was transparent to visible light, but 'rapidly' (a million years or so) became dark as the CMB radiation was redshifted into the infrared. It stayed dark (in visible light) until the first stars formed. These stars also re-ionized the universe, a process that was completed by redshifts of 10-20, a few hundred million years after the Big Bang. But this time, the universe stayed essentially transparent because with the scale factor increasing by a factor of about 100, the density of the universe had decreased by a factor of a million, and the opacity varies as the square of the density.
So, the universe, with the exception of dense dust clouds, has been transparent to visible light ever since the recombination at the time of the CMB. But there is an important issue when we try to look back to that time in visible light or the near infrared (as with Hubble). That is, the neutral hydrogen which dominated from z = 1100 to z = 10-20 is strongly opaque to far ultraviolet light. First there are a series of absorption lines, called the "Lyman Series" (see wikipedia), starting at wavelengths 121.6 nm (nanometers) where the sole electron in a hydrogen atom is kicked from the ground state into various excited states, followed by the Lyman continuum, starting at 91.2 nm whereby the electron is kicked out completely, temporarily ionizing the atom. So, a universe of neutral hydrogen is opaque at wavelengths less than 91.2 nm. But look back to redshifts of 9, and that limit becomes 912 nm, well into the infrared. So Hubble becomes stymied at redshifts much greater than this.
Indeed, the Lyman series of absorption lines starting at 121.6 nm has long been, using ground-based telescopes, from remnants of neutral hydrogen clouds along the lines of sight from very distant quasars, and dubbed the 'Lyman forest' in these spectra.
So there is a second meaning to 'dark ages' when considering our present attempts to look back into the neutral era. The James Webb space telescope is designed with far infrared detectors to probe back into the 'dark ages' to see what was emitted from the earliest stars as visible light.
A: Ionized plasma is highly opaque, because there are so many electrons not bound to atoms and they can take on nearly energy level (and thus able to absorb photons of almost any energy). It is also typically highly luminous due to being at a very high temperature. The Cosmic Microwave Background represents viewing the super-heated plasma of the early Universe, specifically the boundary (in time) between that era and the era when the Universe cooled enough to cause most ionized atoms to combine back to neutral forms (which are more or less transparent). However, there were periods of small-scale re-ionization which add small perturbations to the CMB.
