Lyman-alpha forest
By Bruce A. Peterson (The Australian National University). May, 2013.
Edit by Yuan-Sen Ting (Harvard University).
The hot early-Universe cools as it expands, and after about half a million years, the temperature has dropped enough for protons and electrons to combine to form hydrogen atoms. At this point the Universe is transparent to radiation with wavelengths longer than the Lyman-limit at 912 Å, except at wavelengths that correspond to ground state hydrogen absorption lines.
Radiation with wavelengths shorter than the Lyman-limit will ionize an atom of neutral hydrogen and be absorbed. Radiation shorter than the Lyman-alpha line at 1215 Å travels outward from a quasar and is seen redshifted along its path.
The optical depth is given by \(p\), the probability of scattering of a photon in a proper length interval \(\mathrm{d}l = c\, \mathrm{d}t\) at cosmic time \(t\), which is \begin{equation} \mathrm{d}p = n(t) \sigma(\nu_s) c \, \mathrm{d} t, \end{equation} where \(c\) is the speed of light in vacuum, \(n(t)\) is the number density of neutral hydrogen atoms at time \(t\), and \(\sigma(\nu)\) is the radiative excitation cross-section for the Lyman-alpha transition. This has the form \begin{equation} \sigma(\nu) = \frac{\pi e^2}{m_e c} f \phi (\nu - \nu_\alpha), \end{equation} where \(e\) is the electric charge, \(m_e\) is the electron mass, \(f\) is the oscillator strength (here \(f =\) 0.416) and \(\phi\) is the line profile function.
In the Friedman-Lemaître Universe, the derivative of the scale factor, \(a(t)\), is determined by \begin{equation} \frac{\dot{a}^2}{a^2} = \frac{8\pi G \rho}{3} - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3}. \end{equation} where \(\rho\) is the mass density of the Universe, \(G\) is the gravitational constant, \(\Lambda\) is the cosmological constant. We define the dimensionless cosmological parameters \begin{equation} \Omega = \frac{8\pi G \rho}{3H^2}, \qquad \lambda = \frac{\Lambda c^2}{3 H^2}, \qquad \kappa = \frac{kc^2}{a^2 H^2}, \end{equation} These parameters are subject to the constraint \begin{equation} \Omega_0 + \lambda_0 = \kappa_0 + 1. \end{equation}
The Hubble parameter evolves as: \begin{equation} H(z) = H_0 (1+z) \sqrt{\Omega_0 (1+z) - \kappa_0 + \lambda_0/(1+z)^2} \end{equation} where \(H_0\) is also known as the Hubble constant. The total optical depth becomes: \begin{equation} p = \frac{4.14 \times 10^{10} n_s}{h_0(1+z) \sqrt{\Omega_0 (1+z) - \kappa_0 + \lambda_0/(1+z)^2}} \end{equation} where \(h_0 = H_0/\)100 km s-1 Mpc-1 and \(n_s\) is the number density of neutral hydrogen atoms in the scattering region. The flux is reduced by the factor \(e^{-p}\).
With \(\Omega_0 =\) 0.3, \(\lambda_0 =\) 0.7 and \(h_0 =\) 0.7, the mass density of the Universe is given by \begin{equation} \rho = \frac{3 H_0^2 \Omega_0}{8 \pi G} (1+z)^3 = 2.76 \times 10^{-30} (1+z)^3 \; [\mathrm{g/cm^3}]. \end{equation} Most of this is dark matter. The baryonic density is only about 10 per cent of the total matter density. The expected number density of intergalactic hydrogen is \begin{equation} n_{\small \mathrm{H}} = 0.1 \times 0.75 \times 0.2 \times \rho/m_{\small \mathrm{H}} = 2.5 \times 10^{-8} (1+z)^3. \end{equation} Over the redshift range 2 \(< z_Q <\) 6, less than 1 in 104 hydrogen atoms in the neutral ground state will produce an optical depth greater than one.
The ground based spectra of quasars with redshifts in the range 2 \(< z_Q <\) 6 have large numbers of absorption lines at wavelengths \(\lambda_o\) shorter than \((1+z_Q)\lambda_\alpha\). The number density of absorption lines in a quasar spectrum increases rapidly with the redshift of the quasar. At a redshift of \(z_Q \sim\) 6, the absorption lines have merged into a continuous absorption trough. This phenomenon provides clear evidence for the reionization of the clumpy Universe, starting at a redshift somewhat beyond six.