Cosmology · Cosmic dark ages, first light and reionisation

Timing, duration and patchiness in reionisation models

Construct different finite ionization histories with the same partial Thomson optical depth, then separate a spatial mean from its patchiness.

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

Identify the electron-density weighting in a Thomson optical-depth integral.
Compare two supplied different histories with equal partial optical depth.
Explain why neither a single integrated quantity nor a mean ionized fraction uniquely determines the full history and spatial pattern.

Optical depth weights free electrons along propagation

Thomson optical depth accumulates scattering opportunity through proper free-electron density along a light path. Its differential contribution is dτ = cσ_T n_e |dt|, where σ_T is the Thomson cross section.

Inferring reionization timing from an integrated depth therefore requires a relation between electron density, cosmic time and the proposed history. Helium, density variations and ionization-density correlations can matter in a fuller account.

An integral can constrain a history without recovering every point of that history uniquely. Two different time dependences may have the same weighted total, even though additional observables can distinguish them.

The following finite mathematical comparison isolates that loss of information. Its partial depths are not measurements of the full CMB optical depth or solutions of source formation, recombination and radiative transfer.

A supplied matter-only history gives an explicit integral

Use y = 1 + z on 6 ≤ y ≤ 12, corresponding to 5 ≤ z ≤ 11. Supply a matter-only expanding background H = H_ref y^(3/2), with homogeneous proper total hydrogen density n_H = n_ref y³.

Take pure hydrogen, define x_e = n_e/n_H and exclude helium and density-ionization correlations. Since |dt/dy| = 1/(yH), the interval contribution becomes Δτ = K∫₆¹² x_e(y)√y dy, where K = cσ_T n_ref/H_ref.

Choose the dimensionless normalization K = 0.001. No numerical H_ref or n_ref is adopted from data, and behavior outside the finite interval is unspecified. Later time in the expanding background corresponds to decreasing y and redshift.

These definitions make the weight explicit. In a spatially varying medium, the electron-density average entering the optical depth is not generally the same as an unweighted volume average of an ionized fraction.

History A supplies x_e = 1/2 throughout the interval. History B supplies x_e = 1 for 6 ≤ y < y_c and x_e = 0 for y_c < y ≤ 12. A single boundary value does not change either integral.

Choose y_c = [(6^(3/2) + 12^(3/2))/2]^(2/3) ≈ 9.250066, or z_c ≈ 8.250066. This choice makes the integral of √y over B’s ionized interval equal to half the integral over the full interval.

Both histories then give Δτ ≈ 0.008957427 with the supplied K. They differ point by point, and their plateau or transition behavior differs, but this one partial scattering-depth number is equal.

The step is an ideal mathematical comparison, not a literal instantaneous ionization front or an observationally established chronology. A physical history must satisfy its source, transfer and gas equations; matching one integral does not establish that viability.

Two supplied pure-hydrogen histories over y = 1 + z from 6 to 12; K = 0.001 and no contribution outside the interval is specified
History	Supplied x_e(y)	Partial Thomson Δτ	What remains unspecified
A	1/2 throughout	0.008957427	Earlier/later behavior and source dynamics
B	1 below y_c ≈ 9.250066; 0 above	0.008957427	A physical transition mechanism and outside behavior

A separate four-cell account distinguishes two patterns

Now consider a separate same-epoch illustration with four equal-volume, equal-total-hydrogen-density cells. It is not a discretization of the two temporal histories and assigns no physical cell separation, bubble size or spatial topology.

In pattern U, every cell has ionized atomic fraction 1/2. In pattern P, the fractions are 1, 1, 0 and 0. Both have equal-density mean 1/2, and therefore the same mean electron density under the stated pure-hydrogen conditions.

Their population variance of the four fractions differs. Pattern U has variance zero. In pattern P, each squared departure from the mean is (1/2)², giving variance 1/4. The common mean has not fixed the distribution among cells.

Real spatial morphology requires more information about source positions, density, recombinations and radiation transport. These four supplied values demonstrate a difference in moments, rather than predicting a bubble-size distribution or a 21-centimetre power spectrum.

This supplied pure-hydrogen homogeneous matter-only background uses y = 1+z from 6 to 12, H = H_ref y^(3/2), proper n_H = n_ref y³ and x_e = n_e/n_H. Since |dt/dy| = 1/(yH), the partial Thomson depth is Δτ = K integral x_e(y)√y dy on only this interval, with chosen dimensionless K = cσ_T n_ref/H_ref = 0.001. No data-derived H_ref, n_ref or outside history is supplied. Panel A has a linear fraction axis zero to one and y increasing to the right, so later cosmic time is toward the left. Solid blue history A is one half throughout. Dashed orange history B is one for 6 ≤ y < y_c and zero for y_c < y ≤ 12. The chosen boundary y_c = [(6^(3/2)+12^(3/2))/2]^(2/3) is approximately 9.250066, z_c approximately 8.250066. The separate horizontal B segments end at two square boundary markers; a dotted vertical guide identifies the discontinuity, not an intermediate physical trajectory. A single boundary value has zero integral measure. Both histories have partial depth approximately 0.008957427 by construction. They remain different point by point, and the ideal step does not claim a physical instantaneous front or viable source/transfer history. Panel B is a distinct same-epoch illustration, not time bins of A or B. Four equal-volume, equal-total-density cells have fractions U = (1/2, 1/2, 1/2, 1/2) or P = (1, 1, 0, 0). Each outlined column height represents fraction zero to one; its colored filled share is the supplied fraction, with the fraction printed below. Both mean one half and therefore have equal mean electron density under the stated conditions. Their population variances are zero and one quarter. Cell placement specifies no distance, adjacency correlation, bubble size or cosmic topology. Density–ionization correlations and helium are omitted. A general electron-density weighted depth is not an unweighted volume average. Equal partial depth does not give identical full CMB or 21-centimetre spectra, temperature or spatial information. No observed timing, duration, full optical depth, likelihood or global morphology is inferred. Original Ark teaching model.

An equal scalar depth is not equality of every observable

A complete CMB prediction uses more than one partial optical-depth integral. Different visibility histories can affect angular correlations differently, so equality of the scalar Δτ here does not prove equality of full CMB spectra.

Likewise, an equal mean ionized fraction does not fix the 21-centimetre signal. Spin temperature, neutral density, velocities and their spatial statistics introduce further information and modelling requirements.

Observations of sources, absorption and scattering can constrain complementary parts of a reionization account. Combining them requires consistent definitions, selection and uncertainty models rather than assigning one preferred timing from a single summary quantity.

When reporting a history, identify its finite or full time domain, hydrogen and helium treatment, density weighting, spatial assumptions and data constraints. Those choices explain how a timing or duration inference was obtained and which alternatives remain possible.

Worked reasoning

Find a step with the same partial optical depth

How is y_c chosen so that history B matches the half-ionized plateau’s partial Δτ?

For A, integrate half of √y over 6 ≤ y ≤ 12, using ∫√y dy = (2/3)y^(3/2).
For B, integrate √y only from 6 to y_c and equate the two results.
The equality gives y_c^(3/2) = [6^(3/2) + 12^(3/2)]/2, hence y_c ≈ 9.250066 and the shared Δτ ≈ 0.008957427.

The weighted integrals match by construction, although the histories have different pointwise ionization and transition behavior.

Try the reasoning

Is the supplied Δτ the full measured CMB optical depth?
Show the answer and reasoning
No.
It integrates only a chosen finite interval in a hydrogen-only mathematical model, with no outside history specified.
Does mean ionized fraction 1/2 require every cell to be half ionized?
Show the answer and reasoning
No.
The equal-density pattern 1, 1, 0, 0 shares that mean but has variance 1/4 rather than zero.
Does equality of this partial Δτ prove identical full CMB or 21-centimetre spectra?
Show the answer and reasoning
No.
The full history, transfer, temperatures and spatial statistics can carry information beyond this one scalar integral.

What to carry forward

Thomson optical depth integrates proper electron density with a history-dependent weight.
Different supplied histories can share one partial scattering-depth integral.
A spatial mean, patchiness and the complete observable predictions retain distinct information.

References

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