GLOBAL KNOWLEDGE ARK

Cosmology · The cosmic web, voids and intergalactic matter

Absorption forests and line-of-sight sampling

Follow supplied optical-depth components through transmitted flux, instrumental convolution and a deliberately wrong continuum normalization.

Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.

Back to the chapter

After reading, you can

  • Explain how a background spectrum samples intervening gas along a line of sight.
  • Distinguish adding optical depths from smoothing transmitted flux.
  • Keep continuum, coordinate-area and physical equivalent-width assumptions explicit.
01

Many absorption features need a physical mapping

A background source can illuminate intervening gas whose resonant absorption appears at different observed wavelengths. A collection of such features is often called an absorption forest.

Relating a feature to gas location and composition requires a transition identification, redshift and velocity model, ionization conditions and an instrument response. A fitted component need not represent one distinct uniform cloud.

Construct a synthetic example with a background source assigned redshift 2.5 and features near redshift 2. These are chosen labels, not an actual quasar spectrum or observational redshift record.

The example supplies optical-depth profiles directly. It infers no gas density, neutral column, temperature or physical Doppler width from them.

02

Optical depths add before transmission is calculated

Let λ_α be the positive rest wavelength of an identified transition and choose λ_ref = 3λ_α. Define u = ln(λ/λ_ref)/w with w = 0.001, so λ/λ_ref = exp(wu). All wavelengths remain positive.

Ignoring peculiar velocities for these labels gives 1+z = 3exp(wu). Centres u = −1, 0 and 1.2 then correspond to redshifts about 1.9970015, 2 and 2.0036022, all in front of the supplied source.

Supply τ(u) = Σ_j τ₀,j exp[−((u−u_j)/b_j)²] with triples (u_j,b_j,τ₀,j) equal to (−1,0.35,0.8), (0,0.5,1.5) and (1.2,0.3,0.6). The widths b_j are dimensionless in u, not velocities or measured thermal widths.

With unit continuum, complete coverage and no added line emission into the beam, transmitted flux is F = exp(−τ). The analytic profiles extend over real u for the mathematical integrals; that extension is not an absorber catalogue beyond the source. Display only −3 ≤ u ≤ 3.

03

A response averages light rather than optical depth

Choose a normalized Gaussian response R_σ(t) = exp[−t²/(2σ²)]/(√(2π)σ) with σ = 0.25 in u units. Its convolved normalized flux is F_R = R_σ * F.

Exponentiation is nonlinear, so convolving optical depth and then exponentiating generally gives a different result. At u = 0, F is about 0.223079, F_R about 0.303943 and exp[−R_σ*τ] about 0.290799.

A response can raise flux near one trough and lower it near an adjacent high point. At u = −0.5, for example, the supplied intrinsic flux is about 0.519049 while the convolved value is about 0.449535.

The table contains exact-profile numerical evaluations, not noisy pixels, a fitted spectrograph or a likelihood. Gaussian optical-depth components also omit natural damping wings and a full gas-dynamical line calculation.

Entirely synthetic normalized spectrum and supplied log-coordinate response; the last column deliberately applies the operations in the wrong order
uTotal τIntrinsic FCorrect R_σ * FIncorrect exp(−R_σ * τ)
−3≈01.0000001.0000001.000000
−10.8274730.4371520.5339300.523132
−0.50.6557570.5190490.4495350.434406
01.5002280.2230790.3039430.290799
0.50.5544110.5744100.5500520.512419
1.20.6047270.5462240.6700740.660656
20.0004897620.9995100.9821150.980384
3≈01.0000001.0000001.000000
04

A conserved log-coordinate area is not a wavelength equivalent width

Define A_u = ∫(1−F)du over the complete real coordinate line. A normalized translation-invariant response preserves this area because the response redistributes the integrable flux deficit without changing its total.

For these supplied profiles, both full intrinsic and full convolved A_u are about 1.44711072. The finite displayed response window has area about 1.4471107103; its excluded tails are small but not erased or renormalized.

Physical wavelength equivalent width instead uses W_λ = λ_ref w∫exp(wu)(1−F)du, since dλ = λ_ref wexp(wu)du. A_u is dimensionless, while W_λ has wavelength units.

Conservation of A_u under this response in log wavelength does not assert conservation of the differently weighted wavelength integral or of every finite-window area. The integration variable and boundaries must travel with an area claim.

05

A continuum error can imitate a broad deficit

Now deliberately divide the convolved flux by a continuum estimate 1.05 instead of the true value one. Far from the features, normalized flux tends to 1/1.05 ≈ 0.952381 instead of one.

Across the finite window of width six in u, the apparent deficit area becomes A_app = 6(1−1/1.05) + A_R,window/1.05 ≈ 1.663915. The five percent offset is a chosen sensitivity test, not a measured continuum uncertainty.

Actual absorption inference must handle continuum placement, velocity blending, thermal structure, instrumental response and noise together. A deeper feature does not by itself identify a unique separate cloud or total baryon column.

This synthetic sequence separates optical-depth addition, nonlinear transmission, instrumental averaging and normalization. Keeping those stages explicit makes line-of-sight evidence useful without mistaking a convenient profile model for a direct three-dimensional gas map.

Synthetic absorption flux, response order and a separate continuum sensitivity testThis is a supplied analytic spectrum, not quasar data or noisy pixels. A background source is assigned z =2.5 and three components near z =2. Let u = ln(λ/λ_ref)/w, with w =0.001 and positive λ_ref =3λ_α for one identified transition. Then 1+z =3exp(wu) when peculiar shifts are ignored. Only −3≤u≤3 is displayed. Optical depths add as τ =Στ₀ exp[−((u−u₀)/b)²] with triples(−1, 0.35, 0.8), (0, 0.5, 1.5), (1.2, 0.3, 0.6). These are dimensionless input widths, not physical Doppler temperatures or velocities. With unit continuum, full coverage and no added line emission, intrinsic transmitted flux F =exp(−τ). A normalized Gaussian response with σ =0.25 in u gives correct response flux R*F. Panel A has linear u from −3 to 3 and continuum-normalized flux from zero to one. Solid blue circles are intrinsic F; dashed orange squares are correct R*F. All eight manuscript u samples are marked. At u =−1, −0.5, 0, 0.5, 1.2, 2, intrinsic fluxes are about 0.437152, 0.519049, 0.223079, 0.574410, 0.546224, 0.999510; response values are 0.533930, 0.449535, 0.303943, 0.550052, 0.670074, 0.982115. Thus averaging can raise one trough and lower an adjacent high point. Exponentiating smoothed optical depth is different: at u =zero exp(−R*τ)≈0.290799. Panel B uses the same u but a separate signed-difference scale from −0.08 to zero. Dotted violet diamonds show incorrect exp(−R*τ) minus correct R*F. Dashed green squares show incorrectly normalized R*F/1.05 minus correct R*F. These negative differences are not negative physical flux. Dividing by the supplied high continuum 1.05 makes the unabsorbed limit 1/1.05≈0.952381. Full-line dimensionless log-coordinate deficit A_u =∫(1−F)du is about 1.44711072 and is preserved by a normalized translation-invariant response. The finite convolved display window area is about 1.4471107103; excluded tails are small but are not set to zero or renormalized. In that window the wrong-continuum apparent area is 6(1−1/1.05)+A_R, window/1.05≈1.66391496. Physical wavelength equivalent width instead is W_λ =λ_ref w∫exp(wu)(1−F)du and has wavelength units; log-coordinate area conservation does not imply this differently weighted integral or every finite-window area is conserved. The analytic tails exist on the whole real mathematical coordinate line only for these integrals; they are not an absorber catalogue beyond the source. No gas-density, neutral-column, thermal history, fitted spectrograph, significance, three-dimensional map or one-component-per-cloud inference is supplied.A response averages transmitted fluxEntirely synthetic: chosen source z = 2.5; features near z = 2.u = ln(λ/λ_ref)/w; w = 0.001; λ_ref = 3λ_α.τ components (u₀, b, τ₀): (−1, 0.35, 0.8), (0, 0.5, 1.5),(1.2, 0.3, 0.6); widths are dimensionless, not velocities.intrinsic F = exp(−τ)correct R * FA · Continuum-normalized flux · dimensionless linear scale00.250.50.751-3-2-10123Log-wavelength coordinate u · finite display onlyGaussian response σ = 0.25 in u; unit area; no noisy pixels.At u = 0: F ≈ 0.223079; R * F ≈ 0.303943.exp(−R * τ) − R * FR * F / 1.05 − R * FB · Incorrect operation minus correct flux · signed difference-0.08-0.040-3-2-10123Same u · negative differences are not negative physical fluxFull-line A_u = ∫(1−F)du ≈ 1.44711072, conserved by R.Finite response window [−3, 3]: A ≈ 1.4471107103; tails remain.Wrong continuum 1.05 gives apparent area ≈ 1.66391496.W_λ = λ_ref w∫exp(wu)(1−F)du; this weighted area differs.No measured spectrum, Doppler temperature or separate-cloud count.
This is a supplied analytic spectrum, not quasar data or noisy pixels. A background source is assigned z =2.5 and three components near z =2. Let u = ln(λ/λ_ref)/w, with w =0.001 and positive λ_ref =3λ_α for one identified transition. Then 1+z =3exp(wu) when peculiar shifts are ignored. Only −3≤u≤3 is displayed. Optical depths add as τ =Στ₀ exp[−((u−u₀)/b)²] with triples(−1, 0.35, 0.8), (0, 0.5, 1.5), (1.2, 0.3, 0.6). These are dimensionless input widths, not physical Doppler temperatures or velocities. With unit continuum, full coverage and no added line emission, intrinsic transmitted flux F =exp(−τ). A normalized Gaussian response with σ =0.25 in u gives correct response flux R*F. Panel A has linear u from −3 to 3 and continuum-normalized flux from zero to one. Solid blue circles are intrinsic F; dashed orange squares are correct R*F. All eight manuscript u samples are marked. At u =−1, −0.5, 0, 0.5, 1.2, 2, intrinsic fluxes are about 0.437152, 0.519049, 0.223079, 0.574410, 0.546224, 0.999510; response values are 0.533930, 0.449535, 0.303943, 0.550052, 0.670074, 0.982115. Thus averaging can raise one trough and lower an adjacent high point. Exponentiating smoothed optical depth is different: at u =zero exp(−R*τ)≈0.290799. Panel B uses the same u but a separate signed-difference scale from −0.08 to zero. Dotted violet diamonds show incorrect exp(−R*τ) minus correct R*F. Dashed green squares show incorrectly normalized R*F/1.05 minus correct R*F. These negative differences are not negative physical flux. Dividing by the supplied high continuum 1.05 makes the unabsorbed limit 1/1.05≈0.952381. Full-line dimensionless log-coordinate deficit A_u =∫(1−F)du is about 1.44711072 and is preserved by a normalized translation-invariant response. The finite convolved display window area is about 1.4471107103; excluded tails are small but are not set to zero or renormalized. In that window the wrong-continuum apparent area is 6(1−1/1.05)+A_R, window/1.05≈1.66391496. Physical wavelength equivalent width instead is W_λ =λ_ref w∫exp(wu)(1−F)du and has wavelength units; log-coordinate area conservation does not imply this differently weighted integral or every finite-window area is conserved. The analytic tails exist on the whole real mathematical coordinate line only for these integrals; they are not an absorber catalogue beyond the source. No gas-density, neutral-column, thermal history, fitted spectrograph, significance, three-dimensional map or one-component-per-cloud inference is supplied. Original Ark teaching model.

Worked reasoning

Calculate the effect of a supplied continuum offset

What apparent area results on −3 ≤ u ≤ 3 when F_R is divided by 1.05?

  1. The true convolved finite-window area is A_R,window ≈ 1.4471107103.
  2. Integrate 1−F_R/1.05: the constant offset contributes 6(1−1/1.05), and the true deficit contributes A_R,window/1.05.
  3. The result is about 1.66391496, rather than 1.44711071.

The chosen high continuum creates an extra finite-window deficit; it does not reveal additional gas in this construction.

Try the reasoning

  1. Should the supplied instrument response act on F or on τ before exponentiation?

    Show the answer and reasoning

    On transmitted flux F.

    The response averages arriving light; nonlinear exponentiation does not commute with that averaging.

  2. Is the conserved A_u the usual wavelength equivalent width?

    Show the answer and reasoning

    No.

    Wavelength integration includes λ_ref wexp(wu) and has wavelength units; A_u is a dimensionless log-coordinate area.

  3. Do three supplied optical-depth components prove three distinct physical clouds?

    Show the answer and reasoning

    No.

    Components describe a chosen profile decomposition; gas structure, velocity and ionization require further inference.

What to carry forward

  • Absorption forests record line-of-sight transmission under transition, velocity and ionization assumptions.
  • Optical depths add, transmission is exponential and instrumental convolution acts on flux.
  • Continuum choice, coordinate Jacobians and finite windows matter before spectral areas become physical claims.

References

Sources supporting the explanations, physical distinctions and examples in this lesson.

  1. The Physics of the Intergalactic MediumAvery A. MeiksinTeaching resourcearxiv.orgRead source