= Solution
The <Lyman-alpha forest> is the multitude of <Lyman-alpha absorption> lines blueward of the broad emission line in a distant <quasar> spectrum. Neutral hydrogen along the line of sight absorbs at its rest wavelength $\lambda_\alpha\simeq1216\,\mathrm{\mathring A}$, so each absorber appears at $\lambda_{\rm obs}=(1+z_c)\lambda_\alpha$. Much of the ordinary forest traces modest overdensities in the mostly ionized <intergalactic medium>, rather than compact objects made primarily of neutral gas. Strong self-shielded systems form a distinct absorption regime. Line strengths constrain neutral-hydrogen <column density>, while line widths contain thermal and velocity information.
The <redshift evolution of the Lyman-alpha forest> is normally described, for a fixed line selection, by an incidence $d\mathcal N/dz=A(1+z)^\gamma$. At redshifts of a few the line incidence and mean absorption increase rapidly with redshift; typical selected populations have $\gamma$ of order $2$–$3$, not a universal exponent. Low-redshift evolution is appreciably flatter. Geometry alone cannot explain the full change: for a fixed comoving abundance and fixed proper cross section one obtains $d\mathcal N/dz\propto(1+z)^2/H(z)$, only $(1+z)^{1/2}$ in matter domination. Density evolution, thermal evolution and the <cosmic ionizing background> change how many structures exceed the chosen absorption threshold. In optically thin <photoionization equilibrium>,
$$
n_{\rm HI}\Gamma_{\rm HI}=n_en_p\alpha(T),
$$
so for roughly fixed overdensity the rising cosmic density alone makes $n_{\rm HI}$ scale approximately as $\alpha(T)(1+z)^6/\Gamma_{\rm HI}$. At low redshift, the declining ionizing radiation partly offsets the decrease in gas density. Near the end of <reionization> the forest becomes very opaque and the simple individual-line description ceases to suffice.
The <quasar proximity effect> is the statistically reduced absorption near the quasar's own redshift, contrary to the general increase of absorption towards higher redshift. It results from additional ionizing radiation from the nearby quasar. At fixed density and temperature the local <photoionization rate> becomes $\Gamma_{\rm bg}+\Gamma_Q$. Accordingly
$$
N_{\rm HI}=\frac{N_{\rm HI}^{(0)}}{1+\omega},\qquad\omega=\frac{\Gamma_Q}{\Gamma_{\rm bg}},
$$
where $N_{\rm HI}^{(0)}$ is the value with the background alone. Lines weaken and some fall below the detection threshold; the absorbers need not have been physically removed. This gives the <proximity-effect estimator of the ionizing background>.
For an approximately isotropic quasar and a nearby cloud at proper distance $d$, calculate its flux from $F_{\nu,Q}\simeq L_{\nu,Q}/(4\pi d^2)$, allowing for absorption and relative redshift if necessary. The quasar and isotropic background ionization rates are
$$
\Gamma_Q=\int_{\nu_0}^\infty\frac{F_{\nu,Q}\sigma_{\rm HI}(\nu)}{h_P\nu}\,d\nu,\qquad
\Gamma_{\rm bg}=4\pi\int_{\nu_0}^\infty\frac{J_\nu\sigma_{\rm HI}(\nu)}{h_P\nu}\,d\nu,
$$
where $\nu_0$ is the hydrogen ionization threshold, $h_P$ is the <Planck constant> and $J_\nu$ is the background <specific intensity>. Establish the ordinary absorption statistics far from quasars, then fit their reduction as a function of distance and known quasar luminosity. The scale at which $\omega\sim1$ gives the background rate; an assumed spectral shape converts this to an intensity near the ionization threshold. For matching spectral shapes, $\omega\simeq L_{\nu_0,Q}/(16\pi^2d^2J_{\nu_0})$. A column-density distribution proportional to $N^{-p}$ gives a line-incidence suppression $(1+\omega)^{1-p}$ above a fixed threshold, by shifting that threshold to $(1+\omega)N_{\min}$ in the unperturbed distribution. This is the physical measurement principle established in https://adsabs.harvard.edu/pdf/1988ApJ...327..570B .
The inference is statistical. Quasars live in overdense environments, which enhance absorption and can mask extra ionization. Emission-redshift errors smear the effect, as part (iii) demonstrates. Finite source lifetimes, anisotropic emission, variability and departures from ionization equilibrium can also alter the radial profile. These effects must be modeled rather than attributed entirely to the <cosmic ionizing background>.
The background is produced mainly by ionizing radiation from accretion-powered <quasars> and hot massive stars in star-forming <galaxies>. Their relative importance changes with redshift. Quasars have hard spectra and can ionize helium as well as hydrogen; stellar photons depend strongly on their <ionizing photon escape fraction> from galaxies. Integrating source luminosities over the <luminosity function> and propagating radiation through absorbing gas gives the background. At high redshift the observed quasar population alone need not supply the required hydrogen ionization, so star-forming galaxies can be essential, as discussed in https://arxiv.org/abs/0807.4177 . Diffuse recombination radiation can contribute, but a residual primordial radiation field is not an adequate ionizing source.
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