Cosmology · Primordial nuclei and light-element tests

How primordial abundances are inferred from later observations

Use a specified gas-mixture account to show why a present deuterium ratio does not uniquely reveal its unprocessed value without additional information.

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

Describe the inference steps between later spectra and primordial nuclear abundances.
Calculate a chosen two-component mixture and its inverse.
Keep a final hydrogen contribution distinct from a mass fraction or a complete chemical-evolution history.

A primordial abundance is inferred through a physical model

Primordial light-element abundances are not obtained by taking a sample directly during nucleosynthesis. They are inferred from later astronomical records, selected environments and models of how radiation and material have changed.

Deuterium studies can use gas absorption systems with relatively limited stellar processing. Extracting neutral-deuterium and neutral-hydrogen column densities requires line, blend, continuum, velocity and thermal modelling, followed by appropriate ionization assumptions.

Helium inference can use emission-line ratios with temperature, density, absorption, reddening, optical-depth and ionization modelling. Stellar lithium inference uses atmosphere and spectral-line models and must consider whether a surface abundance preserves the original material.

Low metallicity can help identify promising environments, but it does not logically certify zero processing, zero dust or an exact universal ionization correction. Different elements also have different observational tracers and later histories.

A supplied mixture can expose an inference ambiguity

Consider separate homogeneous gas mixtures, each with a final count N_H1 = 1,000,000 ordinary hydrogen-1 nuclei. Choose an unprocessed component with R_unprocessed = N_D/N_H1 = 3 × 10⁻⁵ and a processed component containing no deuterium.

Define f as the fraction of the final hydrogen-1 count contributed by that deuterium-free component. It is not a mass fraction, the fraction of all baryons processed through stars or a measured chemical-evolution parameter.

The unprocessed component contributes (1 − f)N_H1 hydrogen-1 nuclei and all of the final deuterium. Consequently R_final = (1 − f)R_unprocessed and N_D = R_final N_H1.

These are specified composition mixtures with a common final denominator. They are not successive stages of a closed gas parcel, a stellar-return model or observed absorption systems. The ratio inputs are supplied directly, without a synthetic spectrum.

A lower final ratio can reflect a changed composition

When f = 0, the chosen mixture contains 30 deuterium nuclei per million hydrogen-1 nuclei. When f = 0.2, the unprocessed hydrogen contribution is 800,000 and the deuterium count is 24. When f = 0.5, the deuterium count is 15.

The three final D/H1 ratios are 3 × 10⁻⁵, 2.4 × 10⁻⁵ and 1.5 × 10⁻⁵. They share the same chosen unprocessed ratio but differ because their final hydrogen contributions differ.

This demonstrates a possible missing-input ambiguity within the declared two-component account. It does not establish how much processing occurred in an actual cloud or supply a correction that can be applied to every observed system.

The ordinary-hydrogen denominator is explicit. Deuterium nuclei contribute to baryon count too, but the table holds H1 count fixed rather than claiming the total baryon count or total mass is identical across mixtures.

Chosen separate mixtures, each with one million final H1 nuclei; the unprocessed ratio 3 × 10⁻⁵ and deuterium-free processed component are inputs
Processed share f of final H1	Unprocessed H1 count	Processed H1 count	D count	Final D/H1
0	1,000,000	0	30	3 × 10⁻⁵
0.2	800,000	200,000	24	2.4 × 10⁻⁵
0.5	500,000	500,000	15	1.5 × 10⁻⁵

The same final ratio can imply different unprocessed ratios

If the two-component assumptions and f are supplied, inversion gives R_unprocessed = R_final/(1 − f), for 0 ≤ f < 1. Without f, the final ratio alone does not select a unique value.

For the same supplied R_final = 2.4 × 10⁻⁵, assuming f = 0 gives R_unprocessed = 2.4 × 10⁻⁵. Assuming f = 0.2 gives 3 × 10⁻⁵; assuming f = 0.4 gives 4 × 10⁻⁵.

These alternatives are conditional composition accounts, not confidence intervals, measurements of primordial D/H or a fit to observed absorbers. They make the role of an additional physical assumption visible.

In actual work, selection of lightly processed systems and independent physical modelling help control such ambiguities. Their remaining uncertainties and correlations need to be carried into the inferred result.

The two panels are separate specified composition comparisons. Each forward mixture has one million final ordinary hydrogen-1 nuclei, an unprocessed component with supplied R_unprocessed=N_D/N_H1=3×10^(−5), and a deuterium-free processed component. f is narrowly the processed component’s share of that final H-1 count; it is neither a mass fraction nor all baryons ever processed through stars. The upper panel has linear f from 0 to 0.5 and linear R_final/R_unprocessed from zero, following 1−f. At f=0, 0.2, 0.5 the unprocessed H-1 counts are 1000000, 800000, 500000 and processed counts 0, 200000, 500000. Their D counts are 30, 24, 15 and final D/H-1 ratios 3×10^(−5), 2.4×10^(−5), 1.5×10^(−5). These are separate mixtures, not a closed parcel evolving through three times; fixed final H-1 does not fix the total nucleon count or mass when D changes. The lower panel instead holds one supplied R_final=2.4×10^(−5) fixed and uses R_unprocessed=R_final/(1−f). It shares the finite linear f range, but its own vertical axis displays R_unprocessed in units of 10^(−5) with a zero baseline. At assumed f=0, 0.2, 0.4 the conditional values are 2.4, 3, 4 in those units. Without f the final ratio alone does not choose one original value; the formula requires f<1. No actual spectrum, absorber measurement, ionization correction, chemical-evolution fit or confidence interval is invented. Real spectroscopic and later-processing inferences require their own physical and uncertainty models; the exact two-component rule is not transferred to helium or stellar lithium. Original Ark teaching model.

Observation, processing and theory enter different steps

The spectroscopic inference must first relate a detector record to the appropriate neutral or ionic columns or emission properties. Converting that result to a nuclear abundance may require a further ionization and physical-state model.

A primordial interpretation then considers later creation, destruction, transport or surface changes. Real gas evolution can include stellar returns, inflows, outflows and dust. The simple equation here does not include a full account of those processes.

Helium and lithium cannot be assigned the deuterium toy’s exact correction rule. Their nuclear production or depletion, tracers and observational systematics differ, even when they help test the same cosmological framework.

A credible comparison between nucleosynthesis and observations therefore states the abundance definition, selected objects, measurement model, processing assumptions and uncertainties. Agreement or disagreement is assessed across those connected steps rather than attributed automatically to early-Universe physics.

Worked reasoning

Recover a conditional unprocessed value

A supplied mixture has R_final = 2.4 × 10⁻⁵ and f = 0.2. What unprocessed ratio does this model require?

The unprocessed component supplies 1 − f = 0.8 of the final H1 count.
All deuterium comes from that component under the stated assumptions.
R_unprocessed = (2.4 × 10⁻⁵)/0.8 = 3 × 10⁻⁵. Assuming f = 0 instead would give a different answer.

3 × 10⁻⁵, conditional on the supplied two-component model and f = 0.2.

Try the reasoning

Are the table rows measured deuterium absorption systems?
Show the answer and reasoning
No.
They are exact forward accounts for chosen component mixtures.
Does low metallicity alone prove zero processing and a universal exact ionization correction?
Show the answer and reasoning
No.
It helps select candidates but does not replace the relevant physical and observational models.
Is f the fraction of all baryonic mass that passed through stars?
Show the answer and reasoning
No.
It is defined narrowly as the processed component’s share of the final ordinary-H1 count.

What to carry forward

Primordial abundance inference uses later records and explicit measurement and processing models.
A specified final ratio can correspond to different unprocessed ratios when an input is unknown.
Abundance definitions and element-specific histories must match before results are compared.

References

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