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Sandbox Physics
The Lyman-alpha forest

The Lyman-alpha forest · L2

From a patch of hydrogen to an absorption spectrum

Sandbox Physics ·

Turning gas into a spectral dip has three steps: find its neutral hydrogen, determine which relative velocities those atoms absorb, and add the contributions along the actual light path. Keeping the units explicit leads to a dimensionless optical depth and a transmitted fraction.

Count atoms along a light path

Put the observer on the negative side and let light travel from the positive side. The comoving coordinate xx labels position on a grid that separates out the overall expansion. Begin with a short proper length ds\mathrm ds. Neutral-hydrogen density nHIn_{\mathrm{HI}} counts neutral atoms per cubic centimetre.

A cross section σ\sigma, measured in square centimetres, describes one atom’s interaction with the selected light. This effective area depends on wavelength; it describes an interaction probability per incident photon. Density times cross section times path length has no units: it is that segment’s optical depth.

For uniform neutral density, density times length is the column density NHIN_{\mathrm{HI}}, in inverse square centimetres. As a counting example, take 10−1010^{-10} neutral atoms per cubic centimetre over two proper megaparsecs, about 6.17×1024 cm6.17\times10^{24}\,\mathrm{cm}. The column is about 6.17×1014 cm−26.17\times10^{14}\,\mathrm{cm^{-2}}. No velocity distribution has been specified yet.

dτ=nHIσ dsNHI=∫nHI ds\begin{aligned}\mathrm d\tau&=n_{\mathrm{HI}}\sigma\,\mathrm ds\\N_{\mathrm{HI}}&=\int n_{\mathrm{HI}}\,\mathrm ds\end{aligned}
The integral adds the contributions of short path segments. A nonuniform gas needs the local density inside that sum.

Reading: Tepper García, Voigt profile fitting to quasar absorption lines, arXiv:astro-ph/0602124v1 · Independent hydrogen and sightline calculation materials

Find the part that still has electrons

The input gas density is usually total hydrogen nHn_{\mathrm H}. Its neutral fraction fHIf_{\mathrm{HI}} gives the neutral density when multiplied by the total. Assume pure hydrogen, a uniform radiation background and local ionization equilibrium. Each neutral atom is ionized at rate Γphoto\Gamma_{\mathrm{photo}}, while electrons and protons recombine with coefficient α(T)\alpha(T).

In pure hydrogen, electron and proton densities are equal: each is the total density times the ionized fraction. Substituting them into ionization balance gives the dimensionless expression below. RR measures recombination relative to photoionization.

When the neutral fraction is small, the ionized fraction is almost one and the neutral fraction is approximately RR. Doubling total density at fixed temperature and background then produces about four times as much neutral hydrogen, because both partners in a recombination become more abundant. Near neutrality, solve the complete equation instead of extending that square-law estimate.

The calculation uses the temperature-dependent total hydrogen recombination coefficient of Verner and Ferland. The radiation background is prescribed within this finite screen. Ionizing radiative transfer and self-shielding are not solved. The total case-A coefficient counts recombinations to all levels; reabsorption of their emitted photons is not followed.

fHI=R(1−fHI)2R=α(T)nHΓphoto\begin{aligned}f_{\mathrm{HI}}&=R(1-f_{\mathrm{HI}})^2\\R&=\frac{\alpha(T)n_{\mathrm H}}{\Gamma_{\mathrm{photo}}}\end{aligned}
The recombination coefficient is in cubic centimetres per second and the photoionization rate in inverse seconds, so RR is dimensionless. Choose the physical root between zero and one.

Reading: Verner & Ferland, Atomic Data for Astrophysics: Radiative Recombination Rates, arXiv:astro-ph/9509083v1 · Independent hydrogen and sightline calculation materials

One population absorbs a range of velocities

Atoms move thermally even when their gas has no bulk motion. The thermal Doppler width bb grows as the square root of temperature. Here the proton mass approximates the hydrogen-atom mass. The velocity standard deviation is smaller than bb by a factor of square root two.

The excited state’s finite lifetime gives a natural width. Thermal velocities supply an approximately Gaussian distribution. Their convolution, a weighted sum over atomic velocities, gives a Voigt profile. Its natural wings do not abruptly vanish at a selected velocity.

Write the Voigt function as H(a,w)H(a,w). The dimensionless offset ww is relative velocity divided by thermal width, and aa compares natural and thermal broadening. The area convention is ∫H(a,w) dw=π\int H(a,w)\,\mathrm dw=\sqrt{\pi}. Dividing the integrated line strength SvS_v by thermal width then gives the normalized cross section.

The atomic decay rate Γline\Gamma_{\mathrm{line}} belongs to the excited state. The environmental rate Γphoto\Gamma_{\mathrm{photo}} in the preceding section ionizes neutral atoms. Increasing the radiation background changes how many absorbers remain; it does not change this transition’s lifetime.

b=2kBTmpa=Γlineλ04πbσv(Δv)=SvπbH ⁣(a,Δvb)\begin{aligned}b&=\sqrt{\frac{2k_{\mathrm B}T}{m_p}}\\a&=\frac{\Gamma_{\mathrm{line}}\lambda_0}{4\pi b}\\\sigma_v(\Delta v)&=\frac{S_v}{\sqrt{\pi}b}H\!\left(a,\frac{\Delta v}{b}\right)\end{aligned}
The constants are Boltzmann’s constant, proton mass and rest wavelength. All velocities must use one unit. SvS_v has units of cross section times velocity.

Reading: Tepper García, Voigt profile fitting to quasar absorption lines, arXiv:astro-ph/0602124v1 · Independent hydrogen and sightline calculation materials

Convert the expanding grid to a proper path

Comoving coordinates follow the expanding grid, while the number density counts atoms per proper volume. The scale factor acos=1/(1+z)a_{\mathrm{cos}}=1/(1+z) connects the two. Its subscript distinguishes it from the Voigt damping parameter.

At redshift three, the scale factor is one quarter. Eight comoving megaparsecs span two proper megaparsecs. Substituting the comoving length directly into the absorption integral would overcount the path by a factor of four.

Each location also needs its recession velocity u(x)u(x). Expansion plus peculiar motion determines which spectral velocity vv its atoms absorb most strongly. The next chapter examines this mapping; for now, treat it as an input for every gas segment.

The integral sums positive spatial lengths. Light travels from positive toward negative position; that orders the cumulative dimming, while every absorption contribution stays positive.

ds=acos dxτ(v)=∫nHI(x) σv[v−u(x)] acos dx\begin{aligned}\mathrm ds&=a_{\mathrm{cos}}\,\mathrm dx\\\tau(v)&=\int n_{\mathrm{HI}}(x)\,\sigma_v[v-u(x)]\,a_{\mathrm{cos}}\,\mathrm dx\end{aligned}
Convert positions in megaparsecs into centimetres before using centimetre-based cross sections. One redshift and scale factor describe this thin slice.

Reading: Peeples et al., Pressure Support versus Thermal Broadening in the Lyman-alpha Forest, arXiv:0910.0256v3 · Independent hydrogen and sightline calculation materials

Add absorption before calculating the remaining light

For every selected spectral velocity, traverse the same physical path. Each segment contributes optical depth. Add those depths and only then exponentiate to obtain transmission. Averaging the separate transmissions would describe parallel sightlines rather than successive segments.

If two segments have optical depths of 0.3 and 0.7 at one velocity, the total is one and about 36.8 percent of the background passes through. Reversing their order preserves this pure-attenuation result, although the positions at which the light dims are reversed.

A deep Lyman-alpha line does not require the screen to block all ionizing light. The line-centre cross section can greatly exceed the ionization-threshold continuum cross section. A small neutral abundance can therefore make a deep narrow line while the screen remains comparatively transparent to ionizing radiation. The declared ionizing-depth bound applies to this finite region.

τtotal(v)=∑jΔτj(v)F(v)=exp⁡[−τtotal(v)]\begin{aligned}\tau_{\mathrm{total}}(v)&=\sum_j\Delta\tau_j(v)\\F(v)&=\exp[-\tau_{\mathrm{total}}(v)]\end{aligned}
Sum before taking the exponential. Emission and scattered light entering the selected narrow beam are neglected.
Saved prescribed neutral-hydrogen density, integrated optical depth and transmitted fraction on the default sightline
Original finite gas screen at redshift three. Density uses spatial position; optical depth and transmission use spectral velocity. The last two panels derive from the same full Voigt ray integral, with no observed-quasar fit. Open the full figure

Reading: Tepper García, Voigt profile fitting to quasar absorption lines, arXiv:astro-ph/0602124v1 · Independent hydrogen and sightline calculation materials

Finish with a limit you can integrate yourself

Take a uniform neutral density with no peculiar motion, expanding only with the Hubble flow. If the medium extends infinitely along the sightline, each spectral velocity encounters a complete copy of the same line profile. Changing the integration coordinate from proper distance to Hubble velocity leaves just the total cross-section area.

The resulting uniform Gunn–Peterson limit depends on neutral density, integrated line strength and Hubble rate. Here HcosH_{\mathrm{cos}} is the cosmic expansion rate in inverse seconds, distinct from the Voigt function H(a,w)H(a,w).

A finite path omits the wings contributed by more distant gas. First compare it with an independent finite integral, then quantify the difference from the infinite-medium limit. Increasing the spatial quadrature resolution reduces discretization error; it does not restore gas excluded by the boundary.

τGP=nHISvHcosSv=∫−∞∞σv(v) dv\begin{aligned}\tau_{\mathrm{GP}}&=\frac{n_{\mathrm{HI}}S_v}{H_{\mathrm{cos}}}\\S_v&=\int_{-\infty}^{\infty}\sigma_v(v)\,\mathrm dv\end{aligned}
Density and distance are proper quantities. The uniform infinite medium is an analytic reference for units and normalization.

Reading: Peeples et al., Pressure Support versus Thermal Broadening in the Lyman-alpha Forest, arXiv:0910.0256v3 · Independent hydrogen and sightline calculation materials

Exercises and explained answers

  1. At fixed temperature and radiation background, does doubling total hydrogen always double neutral hydrogen?

    Read the explanation

    For a small neutral fraction it approximately quadruples the neutral density: both recombination partners become more abundant. Near neutrality, use the complete balance equation.

  2. What happens if eight comoving megaparsecs at redshift three are used directly as the proper path?

    Read the explanation

    With other inputs fixed, the path and optical depth are overcounted by four. The proper length is one quarter of the comoving length.

  3. Why might doubling quadrature resolution still leave a difference from the infinite Gunn–Peterson limit?

    Read the explanation

    Gas beyond the finite boundary is absent from the model. Refinement improves integration over the retained path, not the omitted line-wing contribution.

Try the prediction

Read neutral abundance together with width

  1. Keep the ray fixed. Predict what stronger photoionization will do, then change it.
  2. Increase only temperature and explain why column and thermal width both change.
  3. Find a deep line centre and compare how readily its centre and wings change.

Sources and calculation scope

  1. Verner & Ferland, Atomic Data for Astrophysics: Radiative Recombination Rates, arXiv:astro-ph/9509083v1

    Targeted reading of equation 4, the hydrogen row of the fitted-parameter table, and the rate definitions. Pure-hydrogen case-A radiative recombination is used over 5,000–30,000 K. No helium, collisional ionization, self-shielding or complete ion network is solved. This is not a full reading of every ion in the paper.

  2. Tepper García, Voigt profile fitting to quasar absorption lines, arXiv:astro-ph/0602124v1

    Targeted equations 1–8 establish the complete Lorentz/Maxwell convolution and normalization. Our calculation uses the full Voigt function, not the paper’s small-damping approximation. The atomic scalars are explicitly rounded assumptions. Hydrogen mass is approximated by proton mass.

  3. Peeples et al., Pressure Support versus Thermal Broadening in the Lyman-alpha Forest, arXiv:0910.0256v3

    Targeted abstract, sections 2–3 and equations 6–7: photoionization equilibrium, uniform Gunn–Peterson opacity, and distinctions between thermal broadening and pressure history. This reading does not reproduce the hydrodynamic runs or claim a full-paper review.

  4. Independent hydrogen and sightline calculation materials

    Original prescribed periodic gas screen and linear-parent peculiar velocity; fixed redshift three, no hydrodynamic pressure evolution. Full Voigt integration is checked against independent SciPy and finite-grid refinements. The offline Python and own synthetic examples are separate from the protected browser Engine.

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