3.2.1 Cosmological preliminaries Recent CMB experiments, in conjunction with new measurements of the large-scale structure in the present-day Universe, the SN Ia Hubble diagram, and other observations, have led to a substantial reduction in the uncertainties in the parameters describing the ACDM concordance cosmology. We appear to be living in a flat + ^a = 1) universe dominated by a cosmological constant and seeded with an approximately scale-invariant primordial spectrum of Gaussian density fluctuations. A ACDM cosmology with = 0.24, =0.76, = 0.042, h = H°/100kms—1 Mpc—1 =0.73, n = 0.95, and a8 = 0.75 is consistent with the best-fit parameters from the WMAP 3-year data release (Spergel et al. 2006). Here = pm/p°^ is the present-day matter density (including cold dark matter as well as a contribution from baryons) relative to the critical density p° = 3H°/(8nG), ^a is the vacuum energy contribution, H° is the Hubble constant today, n is the spectral index of the matter power spectrum at inflation, and a8 normalizes the power spectrum: it is the root-mean-square amplitude of mass fluctuations in a sphere of radius 8h—1 Mpc. The lower values of a8 and n compared with WMAP 1-year results (Spergel et al. 2003) have the effect of delaying structure formation and reducing small-scale power. In this cosmology, and during the epochs of interest here, the expansion rate H evolves according to the Friedmann equation

a dt where z is the redshift. Light emitted by a source at time t is observed today (at time t°) with a redshift z = 1/a(t) — 1, where a is the cosmic scale factor; a(t°) = 1. The age of the (flat) Universe today is dz' 2 ^f VHX +1\ 1 noc TT_1

(Ryden 2003). At high redshift, the Universe approaches the Einstein-de Sitter behaviour, a oc t2/3, and its age is given by t(z) «-(1 + z)-3/2. (3.3)

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The average baryon density today is nb = 2.5 x 10-7 cm-3, and the hydrogen density is nH = (1 — Yp)nb = 0.75nb, where Yp is the primordial mass fraction of helium. The photon density is nYf = (2.4/n2)[kBT0/(fi,c)]3 = 400 cm-3, where T0 = 2.728 ± 0.004K (Fixsen et al. 1996), and the cosmic baryon-to-photon ratio is then n = nb/ny = 6.5 x 10—10. The electron-scattering optical depth to redshift zr is given by fz' dt fz' ne(z)aTc . .

where ne is the electron density at redshift z and aT the Thomson cross-section. Assuming a constant electron fraction xe = ne/nH with redshift, and neglecting the vacuum energy contribution in H(z), this can be rewritten as

Te(zr) = XfH!T-C T dz VT+z = 0.0022[(1 + zr)3/2 — 1]. (3.5)

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t Hereafter densities measured at the present epoch will be indicated by either superscript or subscript "0."

Ignoring helium, the 3-year WMAP polarization data requiring Te = 0.09 ± 0.03 are consistent with a universe in which xe changes from essentially close to zero to unity at zr = 11 ± 2.5, and xe « 1 thereafter.

Given a population of objects with proper number density n(z) and geometric cross-section sigma £(z), the incremental probability dP that a line of sight will intersect one of the objects in the redshift interval dz at redshift z is dP = n(z)E(z)c £ dz = n(z)E(z)cd^(z). (3.6)

3.2.2 Physics of recombination Recombination marks the end of the plasma era, and the beginning of the era of neutral matter. Atomic hydrogen has an ionization potential of I = 13.6 eV: it takes 10.2 eV to raise an electron from the ground state to the first excited state, from which a further 3.4 eV will free it,

According to Boltzmann statistics, the number-density distribution of non-relativistic particles of mass mi in thermal equilibrium is given by mik—T\3/2 f ¡i — mic2

where gi and ¡ii are the statistical weight and chemical potential of the species. When photoionization equilibrium also holds, then ¡e + ¡p = ¡H. Recalling that ge = gp = 2 and gH = 4, one then obtains the Saha equation:

which can be rewritten as l^ Xe ^ = 52.4 - 1.5ln(1 + z) - 58000(1 + z)-1. (3.10)

The ionization fraction goes from 0.91 to 0.005 as the redshift decreases from 1500 to 1100, and the temperature drops from 4100 K to 3000 K. The time elapsed is less than 200 000 yr.

Although the Saha equation describes reasonably well the initial phases of the departure from complete ionization, the assumption of equilibrium rapidly ceases to be valid in an expanding universe, and recombination freezes out. The residual electron fraction can be estimated as follows. The rate at which electrons recombine with protons is dne np

dt ^rec where trec is the characteristic time for recombination and aB is the radiative recombination coefficient. This is the product of the electron-capture cross-section an and the electron velocity ve, averaged over a thermal distribution and summed over all excited states n > 2 of the hydrogen atom, aB = J2n(anve). The radiative-recombination coefficient is well approximated by the fitting formula aB = 6.8 x 10-13 T-0'8 cm3 s-1 = 1.85 x 10-10(1 + z)-0'8 cm3 s-1, (3.12)

Figure 3.1. Helium and hydrogen recombination for a WMAP 3-year cosmology. The step at earlier times in the left panel is due to the recombination of He iii into He ii. We used the code RECFAST (Seager et al. 1999) to compute the electron fraction xe. Note that the residual electron fraction (determined by the condition trecH = 1) scales with cosmological parameters as xe x VQm/(Obh).

where T3 = T/3000 K and in the second equality I used T = To(1 + z). When the recombination rate falls below the expansion rate, i.e. when trec > 1/H, the formation of neutral atoms ceases and the remaining electrons and protons have negligible probability of combining with each other:

where I assumed z = 1100 in the third equality.

The recombination of hydrogen in an expanding universe is actually delayed by a number of subtle effects that are not taken into account in the above formulation. An e-captured to the ground state of atomic hydrogen produces a photon that immediately ionizes another atom, leaving no net change. When an e- is instead captured to an excited state, the allowed decay to the ground state produces a resonant Lyman-series photon. These photons have large capture cross-sections and put atoms into a high energy state that is easily photoionized again, thereby annulling the effect. That leaves two main routes to the production of atomic hydrogen: (1) two-photon decay from the 2s level to the ground state; and (2) loss of Lya resonance photons by virtue of the cosmological redshift. The resulting recombination history was derived by Peebles (1968) and Zel'dovich et al. (1969).

In the redshift range 800 < z < 1200, the fractional ionization varies rapidly and is given approximately by the fitting formula

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