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Sandbox Physics
The Lyman-alpha forest

The Lyman-alpha forest · L3

When different places share one spectral velocity

Sandbox Physics ·

A spectrum arranges gas contributions by received wavelength. Different places can absorb together when their recession velocities are similar. Drawing the velocity map, then integrating along the original spatial path, explains this crowding without turning a fold in the map into infinite absorption.

Separate expansion from peculiar motion

Keep the observer on the negative side, with positive velocity directed away. Gas recedes with the overall expansion and can also move relative to that flow. The latter is peculiar velocity. In a thin slice at fixed redshift, the sum gives the spectral centre assigned to each comoving location.

Here HcosH_{\mathrm{cos}} is the Hubble rate, acosa_{\mathrm{cos}} converts comoving distance to proper distance, and vpec,xv_{\mathrm{pec},x} is the peculiar-velocity component along the sightline. Positive peculiar motion shifts absorption redward; a negative component shifts it blueward.

Without peculiar motion, the graph is a straight rising line. With it, the mapping can become shallower, steeper or locally turn back. Its horizontal axis still represents spatial position; its vertical axis is spectral velocity.

u(x)=acosHcosx+vpec,x(x)u(x)=a_{\mathrm{cos}}H_{\mathrm{cos}}x+v_{\mathrm{pec},x}(x)
Position is comoving and recession is positive. Convert the units before adding expansion and peculiar velocity.
Saved velocity maps and spectra for peculiar-velocity strengths zero and two
The same prescribed density, neutral column and temperature with two velocity strengths. Velocity crowding changes spectral overlap. These are saved forward calculations, not trajectories or independently sampled random fields. Open the full figure

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

A shallow map gathers more gas

Temporarily use proper distance ss and examine a monotonic piece of the curve. A narrow velocity interval corresponds to a short path where the curve is steep. The same interval gathers a longer path where it is shallow.

The slope du/ds\mathrm du/\mathrm ds measures how much recession velocity changes per distance travelled. The path contained in a small velocity interval is inversely proportional to the slope’s magnitude. At equal neutral density, the shallower region supplies more column to that interval.

To isolate this weighting, idealize the entire line profile as a spike of vanishing width but fixed area. Both thermal and natural widths are neglected in this mathematical limit. If the selected velocity has several separate spatial crossings with nonzero slopes, add all their contributions.

For a pure Hubble flow, the proper-distance slope is the Hubble rate. Uniform density then recovers the preceding chapter’s uniform Gunn–Peterson relation: the two arguments agree in units and weighting.

τnarrow(v)=Sv∑si: u(si)=vnHI(si)∣du/ds∣si\tau_{\mathrm{narrow}}(v)=S_v\sum_{s_i:\,u(s_i)=v}\frac{n_{\mathrm{HI}}(s_i)}{|\mathrm du/\mathrm ds|_{s_i}}
Sum over every location with the selected velocity. In centimetre-based units the denominator is in inverse seconds. Every crossing must have a nonzero slope.

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

Keep finite width at a fold

At a turning point, the narrow-line denominator approaches zero. The zero-width position-to-velocity weighting diverges there. Finite line width spreads the crowded absorption across a range of velocities. The atoms still have thermal velocities and a natural line width.

The actual calculation walks through space and adds the complete Voigt cross section of each segment. Neighbouring segments may share similar velocities, but they contain finite numbers of atoms and the cross-section peak is finite. Forward integration does not divide by the slope at a fold.

Every spatial location still has one prescribed bulk velocity. The folded graph maps positions into spectral velocities; it does not calculate several fluid streams at one location. The fold itself supplies no evidence for collisions, shocks or evolved gas dynamics.

τ(v)=∫nHI(s) σv[v−u(s)] ds\tau(v)=\int n_{\mathrm{HI}}(s)\,\sigma_v[v-u(s)]\,\mathrm ds
Keep the finite line width inside the integral. Position, density and velocity belong to the same spatial ray.

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

One column can give different mean transmission

Change only peculiar velocity, keeping the sightline, neutral abundance and temperature fixed. Column density stays fixed. Integrating the optical depth over all velocities also preserves its area: motion redistributes the contributions.

Transmission is an exponential of optical depth. Overlapping two contributions can further darken an already dark region while leaving another part of the spectrum free. Equal column therefore does not prescribe equal mean transmission.

For a counting example, use two ideal rectangular lines, each of width ww and optical depth one. Separated lines have two missing-light areas; overlapping them puts optical depth two inside one width. Their areas below differ even though the optical-depth area is identical. Rectangles are only this arithmetic example, not the hydrogen profile used in the calculation.

The fixed gas screen gives a concrete comparison. Between −500 and +500 kilometres per second, the mean of 601 samples is about 66.8 percent at zero peculiar-velocity strength and 72.6 percent at strength two. Both columns are about 4.65×1014 cm−24.65\times10^{14}\,\mathrm{cm^{-2}}. These are prescribed-model results, with a declared velocity window and sampling.

Wsep=2w(1−e−1)≃1.264wWov=w(1−e−2)≃0.865w\begin{aligned}W_{\mathrm{sep}}&=2w(1-e^{-1})\simeq1.264w\\W_{\mathrm{ov}}&=w(1-e^{-2})\simeq0.865w\end{aligned}
WW integrates the missing light over velocity. If ww is in kilometres per second, WW has that unit too. The labels sep and ov denote separated and fully overlapping lines.

Reading: Independent hydrogen and sightline calculation materials

What is missing when we infer locations from dips?

The forward problem has a clear order: specify gas in space and its motion, then calculate the spectrum. A single dip permits several explanations, including more neutral hydrogen, a shallower velocity map or overlapping contributions. Temperature changes the width of each contribution as well.

Neighbouring sightlines sample different density and velocity combinations. Multiple transitions, statistical distributions and simulations add further information. Each addition must be connected to the particular unknown that it constrains.

Rotating a graph on paper does not change the frequencies already received by a telescope. Fix the physical viewing direction first, then keep spatial position distinct from spectral velocity. Crowding becomes a well-defined physical question.

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. In a valid narrow-line, monotonic region of fixed neutral density, what if the velocity slope becomes one quarter as large?

    Read the explanation

    Optical depth at that velocity is approximately four times larger: the same velocity interval contains four times the path. Near a fold, restore the finite-width integral.

  2. Why can equal neutral columns give different mean transmission?

    Read the explanation

    Velocity can separate or overlap absorption. Exponential attenuation and saturation change the missing-light area. Keep the finite velocity window consistent when comparing.

  3. Does a folded velocity map establish a shock?

    Read the explanation

    No. It maps different positions into the same spectral velocity. A shock needs fluid dynamics and evidence for the corresponding jump conditions.

Try the prediction

Locate crowded absorption on the velocity graph

  1. Set peculiar-velocity strength to zero and predict the velocity graph before checking.
  2. Select crowded velocities and find a shallow or folded segment, then inspect the spectrum.
  3. Compare column and mean transmission and explain why one stays fixed while the other changes.

Sources and calculation scope

  1. 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.

  2. 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.

  3. 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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