Cosmology · The evidence motivating dark matter

Testing ordinary-matter explanations

Use a fixed nuclear inventory to separate changes in visibility and spatial distribution from an increase in the amount of ordinary matter.

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

Distinguish faint ordinary matter from nonbaryonic matter.
Check how redistribution changes enclosed amounts while preserving a supplied total.
Explain what additional physics and observations a real baryon-budget test needs.

A faint component still belongs to a physical account

Ordinary matter can occupy stars, remnants and gas in several phases. Changing the environment or charge state changes how it is traced; a component can be difficult to see without being nonbaryonic.

A gravitational mass inference and a baryon inventory ask different questions. Motion or lensing responds to gravity, while a baryon-sensitive analysis relates its observations to nuclei, electrons, radiation and their interactions.

Early-universe light-element and CMB analyses constrain ordinary matter through nuclear, thermal and perturbation models. Their conclusions require measurements, calibration, model assumptions and uncertainty treatment.

A local redistribution proposal should be tested against that broader account. The following finite example illustrates the bookkeeping condition only; it does not calculate a nucleosynthesis yield, a CMB likelihood or an empirical cosmic baryon fraction.

Count nucleons and define the proxy

Supply one finite comoving accounting region containing 64 hydrogen nuclei and four helium-4 nuclei. It has 68 nuclei but 64+4×4 = 80 nucleons, each counted as one baryon.

Choose an arbitrary mass proxy m_* per nucleon, so the total ordinary-matter proxy is 80m_*. This deliberately omits electron rest mass, nuclear binding differences, thermal energy and radiation.

During the comparison, allow spatial rearrangement but no nuclear reactions or boundary exchange. The conserved quantity here is the declared nuclear count and its proxy, not an exact full mass-energy ledger.

Divide the region into three ordered shells with no empirical radii assigned. Initially place proxy amounts 20, 30 and 30 in them. These partitions are supplied mathematical choices, not an observed hidden-gas census.

Redistribution can repair an inner amount

Move 20 proxy units from the outer shells into the innermost shell. One supplied rearrangement has shell amounts 40, 20 and 20, still summing to 80.

The initial cumulative amounts at the three boundaries are 20, 50 and 80. After rearrangement they are 40, 60 and 80. The inner profile changes while the whole-region amount stays fixed.

For comparison, stipulate cumulative total-gravitating proxy targets 40, 120 and 200 under one common gravitational interpretation. Their positive shell increments are 40, 80 and 80.

The rearrangement reaches the first target, but not the next two. These targets are not observations or inferred cosmological amounts; they are chosen numbers for testing a closed finite account.

One supplied closed 80-unit nuclear proxy; all amounts are in m_* and shell boundaries have no assigned physical radius
Shell boundary	Initial shell amount	Rearranged shell amount	Initial cumulative	Rearranged cumulative	Stipulated total-gravity target
1	20	40	20	40	40
2	30	20	50	60	120
3	30	20	80	80	200

An enclosed part cannot exceed the supplied total

Any rearrangement with nonnegative shell amounts has an enclosed ordinary-matter proxy at most 80m_*. This follows from the finite inventory, independently of which shell receives the material.

The target 120m_* exceeds that maximum by 40m_*, and the outer target 200m_* exceeds it by 120m_*. Moving the same 80 units cannot fill either difference.

Making the gas neutral, ionized, hotter, colder or fainter does not create nucleons in this stipulated closed model. It can change observational response, which a real inventory must model rather than assume away.

A proposed explanation could instead challenge the supplied budget, its uncertainty or the gravitational interpretation. That would be a different comparison with additional predictions and evidence, not a redistribution of the fixed account.

This is one stipulated finite comoving nuclear accounting region, not an observed baryon census or inferred cosmic fraction. It contains 64 hydrogen nuclei and four helium-4 nuclei: 68 nuclei but 64 + 4×4 = 80 nucleons counted as baryons. Assign an arbitrary positive proxy m_* per nucleon, giving total 80 m_*. Electron rest mass, nuclear binding differences, thermal energy and radiation are excluded; this is not a full mass-energy conservation claim. During the comparison there are no nuclear reactions or boundary exchange. Panel A draws two different shell assignments of the same total, on a zero-based linear mass-proxy axis from zero to 80 m_*. Initially shells one, two and three have 20, 30 and 30 m_*; after the supplied redistribution they have 40, 20 and 20. Each actual segment has a centered circle for shell one, square for shell two or diamond for shell three, with direct shell/value captions, so identification is not color dependent. Twenty units move inward in aggregate; neither panel adds them to the total. Segment widths are mass amounts, not physical radial widths. Panel B uses a linear ordered boundary index from zero to three and zero-based cumulative proxy from zero to 200 m_*. Dotted blue circles show initial cumulative amounts 20, 50, 80 at boundaries one, two, three. Dashed orange squares show rearranged 40, 60, 80. Solid violet diamonds show stipulated total-gravitating targets 40, 120, 200, with positive shell increments 40, 80, 80. These targets are supplied mathematical numbers, not data or measured gravitational masses. The horizontal long-dashed green line at 80 marks the entire available nuclear proxy. The step placement is only discrete bookkeeping: zero represents an empty partial account before the first supplied shell, and nothing specifies a physical density between boundaries or empirical shell radii. The rearranged first amount and target coincide at 40; both baryon accounts coincide at the final 80, with no artificial visual offsets. The new inner account meets the first target, but no nonnegative rearrangement of the available 80 can reach the cumulative targets 120 or 200. Those exceed the whole budget by 40 and 120 m_* respectively. Making the same matter neutral, ionized, hotter, colder or fainter can alter observational response but does not create nucleons under the declared closed conditions. A different budget, its uncertainty or a different gravitational interpretation would be a new test requiring forward predictions. No nucleosynthesis yield, CMB likelihood, empirical hidden-gas amount, observed nonbaryonic fraction or universal rejection of modified gravity follows from this ledger. Original Ark teaching model.

Real alternatives must satisfy the relevant independent responses

To use this reasoning in cosmology, the ordinary-matter budget must be inferred rather than stipulated. Nuclear reaction physics, radiation history, perturbation response and actual observations establish the conditional comparison.

A local faint-matter proposal must also predict its emission, absorption, electron column, dynamics or other relevant signals. A change in tracer conversion can affect the estimated inventory and should be propagated through its uncertainties.

Modified gravity changes how the observations are converted into gravitating mass. It requires a common account of motion, lensing and cosmology, rather than treating the chosen target ledger as a universal refutation.

The finite example shows why spatial or phase rearrangement alone does not enlarge an independently fixed nuclear total. It supplies no empirical amount of nonbaryonic matter and excludes no entire class of theories by itself.

Worked reasoning

Track what the rearrangement accomplishes

Can the supplied rearrangement match the inner target and the outer 200-unit target?

The initial shell total is 20+30+30 = 80, and the rearranged total is 40+20+20 = 80.
The first rearranged cumulative amount is 40, matching the first supplied target.
Even putting every available unit inside the outer target leaves only 80, which is 120 below 200.

It can match the inner target but cannot reach the outer target under the fixed proxy and closed inventory.

Try the reasoning

Do 64 hydrogen nuclei and four helium-4 nuclei contain 68 or 80 baryons?
Show the answer and reasoning
80 baryons.
There are 68 nuclei, while each helium-4 nucleus contains four nucleons.
Can any redistribution of this nonnegative 80-unit inventory reach an enclosed 120 units?
Show the answer and reasoning
No.
An enclosed part cannot exceed the whole supplied account.
Does the example derive a measured cosmic baryon fraction?
Show the answer and reasoning
No.
Both the 80-unit budget and the gravitational targets were stipulated, without observations or their inference model.

What to carry forward

Faintness and nonbaryonic composition are different properties.
Redistribution can change an enclosed profile while preserving the supplied nuclear total.
A real ordinary-matter explanation must fit the broader physical and observational account.

References

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