Cosmology · Primordial nuclei and light-element tests

Lithium and the test of joint consistency

Separate the real lithium inference problem from an invented multi-output example showing why matching one quantity alone does not establish a successful explanation.

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 why stellar lithium is an inference requiring atmosphere and processing models.
Test a chosen mathematical adjustment against several outputs together.
Distinguish an illustrative survival factor from a physically established explanation of a real discrepancy.

Lithium tests several linked parts of the inference

Lithium provides a demanding consistency question for primordial nucleosynthesis and the interpretation of old metal-poor stars. A stellar surface abundance is inferred through spectral-line and atmosphere models rather than counted as an untouched sample from the early Universe.

Temperature, gravity, line formation and departures from local thermodynamic equilibrium can affect the inferred abundance. Stellar mixing or depletion can also change a surface composition after the star forms.

The inspected August 2025 specialist review discusses the lithium discrepancy and favours stellar depletion among candidate explanations. That dated assessment does not establish one universal survival factor for every star or justify presenting all proposed mechanisms as equally supported.

The lesson supplies no current discrepancy factor, measured abundance or statistical significance. Its numerical example below is deliberately invented to teach joint consistency; its functions are not BBN fitting formulas or approximations to actual isotope sensitivities.

A mathematical control problem can isolate the logic

Choose a dimensionless control b from 0.5 to 1.5 and define three response proxies d(b) = 1/b, h(b) = 1 + (b − 1)/20 and l(b) = b². Their labels are deuterium, helium and lithium proxies only; they are not actual D/H, helium fractions or Li/H.

Set exact chosen reference conditions d = 1, h = 1 and lithium surface proxy = 1/4. These are mathematical conditions, not observed points, fitted best values or uncertainty intervals.

Initially take the surface proxy equal to l, corresponding to a separate survival factor s = 1 in l_surface = s l. Later, a supplied 0 < s ≤ 1 can illustrate a distinct processing account by construction.

The control b is not an inferred baryon density, and the response slopes and factor of four are not measurements of the real lithium problem. Their purpose is to make one-output matching and multi-output consistency visibly different questions.

Fixing one proxy can spoil the others

At b = 1, the d and h proxies both equal their chosen conditions, while the birth and unprocessed surface lithium proxy is 1, above its chosen condition 1/4.

Changing b to 1/2 makes l = 1/4 and therefore matches that one condition when s = 1. But it also gives d = 2 and h = 0.975, which fail the other exact chosen conditions.

This is an algebraic demonstration that one-output matching does not imply joint consistency. No statistical rejection level follows, since the example has supplied exact conditions rather than a likelihood or measured covariance.

A real proposed change to nuclear inputs, cosmic conditions or new physics must similarly be tested across its full set of consequences, using actual observations and uncertainties rather than these invented proxy functions.

Invented response proxies with separate normalizations; these functions and values are not empirical nucleosynthesis fits or measured abundances
Chosen control b	d proxy = 1/b	h proxy	Birth l proxy = b²	Surface proxy if s = 1/4
0.5	2	0.975	0.25	0.0625
0.75	4/3	0.9875	0.5625	0.140625
1	1	1	1	0.25
1.25	0.8	1.0125	1.5625	0.390625
1.5	2/3	1.025	2.25	0.5625

A tunable correction needs independent physical support

Alternatively keep b = 1 and choose s = 1/4. Then the surface proxy becomes 1/4 while d and h remain 1, because this invented survival control was defined to act only on the lithium surface proxy.

That algebra can reproduce the chosen condition but does not explain why a real star would have that survival factor. A physical account must predict or constrain processing consistently with stellar structure, spectra, populations and other evidence.

A parameter added simply to match one output can hide an unresolved physical question. Its allowed range, dependence on stellar properties and effects on other observables need tests beyond the original discrepancy.

The example’s chosen survival factor is therefore neither an observed depletion estimate nor a solution of the actual lithium problem. It illustrates what an additional inference input can do before any claim about its physical support.

This figure is an invented multi-output mathematical example, not measured or fitted D/H, helium or Li/H. The dimensionless control b from 0.5 to 1.5 is not an inferred baryon density. The three separately named response proxies are d=1/b, h=1+(b−1)/20 and birth l=b², with their own definitions and normalisations. Solid blue circles show d, dashed green squares h and dotted orange diamonds l on a common linear display with zero baseline. At b=0.5, 0.75, 1, 1.25, 1.5, the d proxies are 2, 4/3, 1, 0.8, 2/3; h proxies are 0.975, 0.9875, 1, 1.0125, 1.025; birth l proxies are 0.25, 0.5625, 1, 1.5625, 2.25. Chosen exact conditions are d=1, h=1 and surface l=1/4; they are mathematical targets, not observed points or uncertainty bands. With the default survival s=1, b=1 satisfies d and h but leaves surface l=1. Retuning b to 1/2 gives l=1/4 but also d=2 and h=0.975, missing the other exact conditions. The lower bars show a different supplied survival account at b=1 with l_surface=s l_birth. On the shared linear zero-baseline surface-proxy scale, s=1 gives 1 and s=1/4 gives 1/4. d and h remain 1 only because this additional control was defined to act on the lithium surface proxy alone. Across the upper five b values, quarter-survival surface proxies would be 0.0625, 0.140625, 0.25, 0.390625, 0.5625. No proxy slope, factor of four or survival amount is a real BBN sensitivity, measured lithium discrepancy, universal depletion law or solved stellar mechanism. Matching one chosen output does not imply joint consistency, and adding a tunable survival factor requires separate physical evidence. Exact conditions supply no statistical significance, confidence interval or likelihood. The actual dated specialist assessment belongs to the prose and is not represented as numerical data in this proxy figure. Original Ark teaching model.

Evaluate an explanation across observations and mechanisms

A discrepancy can motivate scrutiny of observational inference, stellar processing, reaction inputs and cosmological assumptions. Its existence alone does not identify which part is responsible or prove that new particles are required.

Conversely, a successful prediction for some light elements does not make every abundance automatically agree or remove the need to investigate a persistent inconsistency. Each comparison has its own observation and theory dependencies.

Useful proposals state which inputs change, why that change is physically justified and what additional predictions follow. A change affecting more than one element must carry those responses and their correlated uncertainties into the joint test.

The core lesson is a method: distinguish a measured record, its inference model, a predicted abundance and an illustrative proxy. Then ask whether an explanation improves the complete physical account, rather than only matching one selected number.

Worked reasoning

Test a single-target adjustment against the other conditions

In the invented functions, does setting b = 1/2 to obtain l = 1/4 also preserve d = h = 1?

At b = 1/2, l = b² = 1/4 matches the one chosen lithium condition for s = 1.
But d = 1/b = 2 and h = 1 + (b − 1)/20 = 0.975, so the other conditions fail.
Keeping b = 1 and choosing s = 1/4 can instead match the surface proxy by construction, but needs separate physical justification before any real interpretation.

No. One-output matching is not joint consistency, and neither proxy adjustment solves the real lithium problem.

Try the reasoning

Are the factor of four and proxy response slopes measurements of the real lithium discrepancy?
Show the answer and reasoning
No.
They are deliberately chosen mathematical inputs with no empirical fit attached.
Does a discrepancy alone prove new particles or invalidate every successful BBN prediction?
Show the answer and reasoning
No.
Test observational inference, stellar processing, nuclear inputs and the wider predictions with their actual uncertainties and dependencies.
Does selecting s = 1/4 establish a universal stellar depletion mechanism?
Show the answer and reasoning
No.
It changes the invented surface proxy by definition; real stellar processing requires independent evidence and a physical model.

What to carry forward

Stellar lithium inference includes spectral and processing assumptions.
Matching one output does not establish joint consistency.
An adjustable survival proxy is not a measured depletion factor or an explanation of a real discrepancy.

References

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