Solution

Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 66 / 3 / iv / Solution

Proximity effect (astrophysics)

Proximity effect

Redshift evolution of the Lyman-alpha forest

Temperature-density relation of photoionized intergalactic gas

{c}
{wiki}

= Lyman forest
{c}
{synonym}

The <Lyman-alpha forest> is the multitude of foreground <Lyman-alpha absorption> features in a distant <quasar>'s spectrum, at <wavelengths> shorter than its own redshifted <Lyman-alpha emission>. The <quasar> supplies a bright background; intervening neutral <hydrogen atoms> remove <photons> near their local <Lyman-alpha line> resonance. Many dips can therefore come from one atomic transition encountered at different <redshifts>, rather than from many different <chemical elements>. Some features are instead transitions of other ions, so a dip alone does not identify <Lyman-alpha absorption>. This makes faint foreground gas detectable through the <light> it removes, even when its own emission would be difficult to see. https://ned.ipac.caltech.edu/level5/Sept01/Rauch/Rauch1.html[Rauch (1998), The Lyman Alpha Forest in the Spectra of QSOs, section 1]{ref}.

Ignoring <peculiar velocity> for the moment, an absorber at <cosmological redshift> $z_a$ puts its <Lyman-alpha line> at
$$
\lambda_{\mathrm{obs}}=\lambda_\alpha(1+z_a),\qquad z_a=\frac{\lambda_{\mathrm{obs}}}{\lambda_\alpha}-1,\qquad \lambda_\alpha=1215.67\,\text{Å}.
$$
The rest <wavelength> is in the <ultraviolet>; at $z_a=2$ it appears at $3647.01\,\text{Å}$ in the <optical spectrum>. In a toy sightline to a <quasar> at $z_Q=3$, absorbers at $z_a=2,\ 2.35,\ 2.7$ appear at $3647.01,\ 4072.4945,\ 4497.979\,\text{Å}$, all shortward of the <quasar>'s <Lyman-alpha emission> position $4862.68\,\text{Å}$. The absorbing atoms see the selected <photons> at their own resonance; the observed dip's <redshift> belongs to the foreground gas. <Peculiar velocity> adds a <Doppler shift>, so the inferred $z_a$ is not a pure distance coordinate. The rest <wavelength> and resonance interpretation are given in https://ned.ipac.caltech.edu/level5/Sept01/Rauch/Rauch1.html[Rauch (1998), section 1].

\Image[/lyman-alpha-forest-sightline.png]
{title=Synthetic foreground geometry and spectra: one Lyman-alpha transition at many redshifts}
{id=lyman-alpha-forest-sightline}

The diagram uses invented <optical depths> and 26 Doppler cores; three selected foreground regions are marked. Their drawn positions and sizes are schematic. The dashed comparison imposes a uniform <optical depth> $\tau=2$, giving transmission $e^{-2}\simeq0.135$ throughout the chosen foreground band. It is an idealized smooth-absorption model, not a measured spectrum or a prediction of the actual <intergalactic medium>.

The smooth comparison explains the Gunn-Peterson effect: neutral <hydrogen> distributed throughout an expanding <intergalactic medium> can absorb continuously over a <wavelength> interval, because different parts of the path resonate with different observed <wavelengths>. The resolved <Lyman-alpha forest> instead has strong transmission fluctuations and distinguishable <absorption lines>. These need not represent isolated compact clouds: diffuse sheets and filaments in the <cosmic web> can generate peaks in <optical depth> along one <line of sight>. A continuously varying gas distribution can look like a collection of lines. https://ned.ipac.caltech.edu/level5/Sept01/Rauch/Rauch1.html[Rauch (1998), section 1: smooth absorption]{ref}. https://ned.ipac.caltech.edu/level5/Sept01/Rauch/Rauch5_1.html[Rauch (1998), section 5.1: synthetic sightlines and extended structures]{ref}.

An early observation makes the identification concrete. Lynds's 1971 paper analyzed six spectrograms of the <quasar> 4C 05.34, with emission <redshift> $2.877$, taken in spring 1970 using an image-tube <spectrograph> on Kitt Peak's 84-inch telescope. Five covered $3200$–$7500\,\text{Å}$ with approximately $10\,\text{Å}$ resolution; the most informative covered about $3500$–$6000\,\text{Å}$ at $5\,\text{Å}$ resolution. Its reproduced spectrogram shows the strong <Lyman-alpha emission> flanked mostly on the short-<wavelength> side by numerous <absorption lines>. Lynds tabulated their <wavelengths> and estimated strengths, identified five absorption <redshift> systems, and proposed that many remaining unidentified lines were the same <Lyman-alpha line> at additional <redshifts>. He explicitly warned about blending, an incomplete census of weak lines, and <wavelength> errors around $0.5$–$1\,\text{Å}$. The result established a striking spectral pattern; the paper's proposed interpretation did not measure a separate physical cloud for every dip. https://adsabs.harvard.edu/pdf/1971ApJ...164L..73L[Lynds (1971), The Absorption-Line Spectrum of 4C 05.34, Figure 1, Table 1 and discussion]{ref}.

A contrasting observation demonstrates smooth, very strong absorption. Becker and collaborators (2001) used the Echelle Spectrograph and Imager on Keck II to observe the <quasar> SDSSp J103027.10+052455.0 at $z_Q=6.28$. Sky lines gave an instrumental full width at half maximum of about $1.8\,\text{Å}$ near $8500\,\text{Å}$, or <spectral resolving power> $R\simeq4700$. After sky subtraction and flux calibration, the mean continuum-normalized transmission over $8450$–$8710\,\text{Å}$ was $0.0038\pm0.0026$, consistent with zero; this corresponds to absorption <redshifts> approximately $5.95$–$6.16$. They identified a Gunn-Peterson trough. Crucially, resonant <Lyman-alpha absorption> is so sensitive that even a small neutral fraction can make transmission undetectable: the trough alone does not establish a mostly neutral <intergalactic medium> or uniquely date <reionization>. https://arxiv.org/pdf/astro-ph/0108097[Becker et al. (2001), sections 2–4: instrument response, transmitted flux and interpretation]{ref}.

What a spectrum constrains is a <radiative transfer> problem. For an ideal foreground screen with negligible emission into the observed beam, the intrinsic continuum-normalized transmission is
$$
\mathcal T(\lambda)=\frac{F(\lambda)}{F_{\mathrm{cont}}(\lambda)}=e^{-\tau(\lambda)},\qquad N_{\mathrm{HI}}=\int n_{\mathrm{HI}}\,d\ell.
$$
Here $F$ is spectral <radiative flux>, $F_{\mathrm{cont}}$ is its unabsorbed continuum, $\tau$ is <optical depth>, $n_{\mathrm{HI}}$ is neutral <hydrogen> <number density>, and $N_{\mathrm{HI}}$ is neutral <hydrogen> <column density> along proper path length $\ell$. A <Voigt profile> model constrains <redshift>, <column density> and a velocity width together. A nearly black line has lost much of its sensitivity to further increases in <optical depth>; blending and an uncertain continuum can also admit different fitted components. Finite <spectral resolving power> and noise limit which widths and weak features can be measured. https://ned.ipac.caltech.edu/level5/Sept01/Rauch/Rauch2.html[Rauch (1998), section 2: resolution, noise and profile analysis]{ref}. https://arxiv.org/pdf/astro-ph/0101005[Kim, Cristiani and D'Odorico (2001), section 3: fitted parameters and nonunique profile fits]{ref}.

For a purely thermal Doppler core, use the velocity convention $\tau(v)\propto\exp[-(v-v_0)^2/b^2]$. The <Maxwell-Boltzmann velocity distribution> then gives
$$
b_{\mathrm{th}}=\sqrt{\frac{2k_BT}{m_{\mathrm H}}}\simeq12.85\sqrt{\frac{T}{10^4\,\mathrm K}}\,\mathrm{km\,s^{-1}},
$$
where $k_B$ is the <Boltzmann constant>, $T$ is <temperature> and $m_{\mathrm H}$ is a <hydrogen atom>'s mass. This $b$ is $\sqrt2$ times the <Gaussian distribution>'s standard deviation. Higher <temperature> spreads the same <column density> over more velocities. An instrument also broadens a line: locally, for a nearly flat continuum, the recorded transmission is approximately the <convolution> of $\mathcal T$ with the normalized instrumental response, not the exponential of a convolved <optical depth>. A Gaussian response with full width at half maximum $c/R$ has standard deviation $c/[R\sqrt{8\log2}]$, where $c$ is the <speed of light> and $R$ the <spectral resolving power>.

\Image[/lyman-alpha-forest-broadening.png]
{title=Synthetic absorption profiles: thermal broadening, instrumental smoothing and unresolved blends}
{id=lyman-alpha-forest-broadening}

The upper model fixes the integral $\int\tau(v)\,dv$, hence the neutral <hydrogen> <column density> for this transition, while changing <temperature>. A cooler line blurred by an instrument resembles a warmer intrinsic line. The lower model has two equal Doppler cores separated by $24\,\mathrm{km\,s^{-1}}$: reducing <spectral resolving power> merges them into one apparent feature. Both panels use invented profiles, Gaussian instrumental responses, and no noise or damping wings. They illustrate an <inverse problem>, rather than measurements of <temperature> or a universal relation between a dip and a cloud.

Better observations provide the inputs to that <inverse problem>. Kim, Cristiani and D'Odorico (2001) analyzed three <quasars> with the VLT's UVES <spectrograph>. For HE2217–2818, Table 1 records $z_Q=2.413$ and $16200\,\mathrm s$ of exposure in the $3050$–$3860\,\text{Å}$ setup in October 1999. Isolated thorium-argon lamp lines measured $R\simeq45000$; the data were resampled into $0.05\,\text{Å}$ bins, with signal-to-noise ratios about $45$–$50$ per pixel over $3550$–$4050\,\text{Å}$. They fitted a local polynomial continuum, fitted <Voigt profiles>, and compared statistics of the transmitted flux. Their analysis excluded the region near the <quasar> to avoid the <quasar proximity effect> and used the interval between its Lyman-beta and <Lyman-alpha emission> to reduce confusion with other Lyman-series absorption. Features below $N_{\mathrm{HI}}=10^{12.5}\,\mathrm{cm^{-2}}$ became confused with noise, setting their adopted lower <column density> limit. Figure 1 shows the measured spectrum, fitted profiles and residuals. This was a calibrated absorption measurement, with explicit selection limits. https://arxiv.org/pdf/astro-ph/0101005[Kim, Cristiani and D'Odorico (2001), UVES observations, sections 2–3, Table 1 and Figure 1]{ref}.

Even after instrumental correction, a width need not be purely <thermal Doppler broadening>. <Hubble flow> across extended gas, <peculiar velocity> structure and unresolved blends can contribute. If an isolated line's intrinsic width obeys $b^2=b_{\mathrm{th}}^2+b_{\mathrm{nt}}^2$ with an independent Gaussian nonthermal contribution $b_{\mathrm{nt}}^2\geq0$, then $T\leq m_{\mathrm H}b^2/(2k_B)$; $b=25\,\mathrm{km\,s^{-1}}$ gives a computed upper bound about $3.79\times10^4\,\mathrm K$. A complex blend need not satisfy this single-component model. Noise can also bias the fitted widths of weak lines. This is why the <thermal cutoff of Lyman-alpha line widths>, calibrated against gas models, is more informative than assigning a <temperature> to every dip. https://ned.ipac.caltech.edu/level5/Sept01/Rauch/Rauch3_3.html[Rauch (1998), section 3.3: width measurements, noise bias and bulk motions]{ref}.

Finally, neutral <hydrogen> <column density> is not total gas <mass density>. In a pure <hydrogen> approximation, with gas optically thin to ionizing radiation and nearly fully ionized in <photoionization equilibrium>, the <Lyman-alpha forest photoionization scaling> gives
$$
n_{\mathrm{HI}}\simeq\frac{\alpha(T)n_{\mathrm H}^2}{\Gamma},\qquad x_{\mathrm{HI}}\simeq\frac{\alpha(T)n_{\mathrm H}}{\Gamma},
$$
where $n_{\mathrm H}$ counts all <hydrogen> nuclei, $\Gamma$ is the <photoionization rate>, $\alpha(T)$ is the radiative recombination coefficient and $x_{\mathrm{HI}}=n_{\mathrm{HI}}/n_{\mathrm H}$ is the neutral fraction. At fixed <temperature> and $\Gamma$, doubling $n_{\mathrm H}$ quadruples $n_{\mathrm{HI}}$; doubling $\Gamma$ halves it. These computed examples explain both the sensitivity to <density contrast> and the ambiguity: stronger absorption can mean denser gas, weaker ionizing radiation, a different <temperature>, or more path length. The <Jeans-length ionization correction for the Lyman-alpha forest> adds a size model rather than an independently measured depth. Optically thick <Lyman limit systems> and <damped Lyman-alpha systems> require different treatment. https://ned.ipac.caltech.edu/level5/Sept01/Rauch/Rauch5.html[Rauch (1998), section 5: neutral fraction and ionization correction]{ref}. https://arxiv.org/pdf/astro-ph/0104272[Schaye (2001), section 2 and Appendix equations A6–A7: ionization balance and the Jeans-length model]{ref}.

The beauty of the <Lyman-alpha forest> is that a spectrum converts otherwise faint foreground gas into a sequence of markers through cosmic time. With a model of <radiative transfer>, velocities, ionizing radiation and the <temperature-density relation of photoionized intergalactic gas>, its transmission statistics probe the <cosmic web> and gas thermal history. A single <line of sight> remains one narrow sample, not a unique three-dimensional density map. https://ned.ipac.caltech.edu/level5/Sept01/Rauch/Rauch5_2.html[Rauch (1998), section 5.2: the forest as a record of fluctuations]{ref}.

Codex @codex  0

 Incoming links: Lyman-alpha forest