Astronomy · Unseen components and unresolved questions

Comparing competing physical explanations

Fix two original acceleration predictions with the same anchor, compare an unused datum, and separate that narrow result from a verdict on whole physical theories.

Continue in the approved chapter order. Necessary vocabulary, quantities and assumptions are explained in the lesson.

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After reading, you can

Distinguish an empirical acceleration relation from a complete physical explanation.
Calibrate two supplied functions and compare their fixed predictions on an unused datum.
Identify additional observables and nuisance assumptions needed for a fair physical comparison.

A discrepancy opens several questions

A mismatch between motion and an ordinary-matter gravity calculation can motivate changes in the gravitating components, the gravitational description or the measurement and source model. These alternatives are not interchangeable verbal labels. Each needs to specify how it predicts the actual observations under the relevant conditions.

An empirical relation summarizes a pattern in data. A physical explanation must say why that pattern arises and what else should happen. McGaugh and colleagues’ radial-acceleration fitting function is a useful example of a descriptive constraint; fitting it alone does not supply a complete relativistic model of light deflection or cosmic structure.

Within the established cosmological comparison framework, general relativity with cold dark matter supplies a baseline for joint comparisons. Proposed modified-gravity descriptions require their own predictions and domain checks. A galaxy-scale phenomenology alone cannot be assumed to provide the same cluster, lensing or background-radiation predictions.

The Foundations model-comparison record points to those broader questions and to the distinction between an established baseline and proposed extensions. This lesson performs a narrower mathematical exercise. It does not re-fit Cosmology results or score every possible dark-matter and modified-gravity theory.

Agreement where a model was fixed is not a new test

Define two invented one-parameter acceleration functions. Function A is gA = gbar + K. Function B uses the empirical fitting form gB = gbar/[1 − exp(−√(gbar/gd))], with a positive acceleration scale gd. Both gbar and g have units m s⁻², so the exponential argument is dimensionless.

Supply one exact anchor: gbar = 4 × 10⁻¹¹ m s⁻² and g = 1 × 10⁻¹⁰ m s⁻². For A, K = g − gbar = 6 × 10⁻¹¹ m s⁻². This constant is a mathematical prediction term, not a fully specified halo-density model across all positions.

For B, the anchor requires 1 − exp(−√(gbar/gd)) = gbar/g = 0.40. Therefore exp(−√(gbar/gd)) = 0.60, √(gbar/gd) = −ln 0.60 and gd = gbar/(ln 0.60)² ≈ 1.532902891 × 10⁻¹⁰ m s⁻².

Both functions now reproduce the anchor exactly because it fixed their respective parameters. This is not independent evidence favouring one. The inferred gd here belongs only to these invented inputs; it is not the fitted acceleration scale quoted by the observational paper.

One anchor can leave different predictions elsewhere

Reserve a different input gbar = 1 × 10⁻¹¹ m s⁻² for a prediction check. Keep K and gd fixed at the values obtained above. Function A predicts 7 × 10⁻¹¹ m s⁻², while B predicts approximately 4.436491673 × 10⁻¹¹ m s⁻².

Now supply an independent held-out measurement y = 4.5 × 10⁻¹¹ m s⁻² with known Gaussian standard uncertainty σ = 3 × 10⁻¹² m s⁻². For this exercise, the input gbar and the anchor-derived parameters are treated as exact. No anchor uncertainty or cross-measurement covariance is represented.

Define each standardized residual as r = (y − prediction)/σ. A gives rA ≈ −8.333333333 and B gives rB ≈ 0.211694423. Their squared residual contributions are approximately 69.444444444 and 0.044814529 respectively.

Under this stipulated equal-σ Gaussian comparison, the held-out datum fits B’s fixed prediction much better. That conclusion concerns these two functions at this point under the supplied assumptions. A squared residual is neither a posterior probability nor a universal verdict on entire families of physical explanations.

Original fixed-prediction comparison. Both functions were calibrated to the same exact anchor; only the distinct held-out datum enters these residuals.
Quantity	Function A	Function B
Anchor prediction (m s⁻²)	1 × 10⁻¹⁰	1 × 10⁻¹⁰
Held-out prediction (m s⁻²)	7 × 10⁻¹¹	4.436491673 × 10⁻¹¹
Standardized residual, data minus prediction	−8.333333333	0.211694423
Squared standardized residual	69.444444444	0.044814529

Ask what each explanation predicts outside the fit

A physical added-component model needs a consistent density and gravitational potential, including their spatial variation and relation to ordinary matter. A constant extra acceleration over selected inputs does not establish such a model everywhere. Conversely, a successful empirical acceleration mapping alone does not specify light propagation or relativistic dynamics.

Lensing and cluster data can therefore add different constraints. Clowe and colleagues compare lensing structure with separated galaxies and X-ray gas in a merging cluster. Those observables provide a challenge beyond reproducing a disk’s rotation curve, while still requiring source, response and gravitational assumptions.

Angus and colleagues explicitly examine how convergence-to-matter reconstruction changes in a nonlinear modified-gravity description. Their analysis illustrates the need to compute the relevant mapping rather than assume the standard density conversion carries over unchanged. The studied cluster models also required an additional collisionless component; changing gravity was not an automatic removal of every unseen-matter requirement.

Those historical analyses do not certify every proposed model or establish current candidate-particle limits. Compare the actual specified theory, matter content, geometry and observations. A failure of one prediction rejects or constrains that prediction’s conditions; it does not logically test every alternative bearing a similar name.

Design a comparison that can change the conclusion

Real acceleration estimates share uncertain distance, inclination, mass-to-light conversion, gas calibration and geometric assumptions. Points along one rotation curve can inherit common nuisance errors. An honest comparison needs the joint data relationship and uncertainty in both the predicted and measured quantities.

Parameter flexibility matters too. If a model is retuned after inspecting every result, the residuals no longer represent the fixed-prediction exercise above. Parameter estimation, predictive checking and model comparison have different evidence requirements; priors and parameter volume matter when a posterior or Bayesian comparison is claimed.

Useful new observations should respond differently to the serious alternatives. Additional radii, source populations, lensing configurations or independent calibrations can help, depending on the competing mechanisms. Repeating the original anchor more precisely cannot distinguish functions that were constructed to agree there unless another dependency supplies new information.

Retain the exact hypotheses, their shared assumptions, the calibration data, the reserved tests and the scope of any failed prediction. The next lesson applies this approach to fast radio bursts: measured transient structure and a demonstrated source association constrain explanations without closing every question about the wider population.

Worked examples

Fix the empirical function with the supplied anchor

How is gd obtained when gbar/g = 0.40 at gbar = 4 × 10⁻¹¹ m s⁻²?

Rearrange the denominator to exp(−√(gbar/gd)) = 0.60.
Take logarithms to obtain √(gbar/gd) = −ln 0.60.
Square and solve gd = gbar/(ln 0.60)².

gd ≈ 1.532902891 × 10⁻¹⁰ m s⁻² for this invented anchor.

Compare two fixed predictions

Which fixed function has the smaller squared standardized residual for the supplied held-out datum?

Use y = 4.5 × 10⁻¹¹ and σ = 3 × 10⁻¹² m s⁻².
Calculate residuals from predictions 7 × 10⁻¹¹ and 4.436491673 × 10⁻¹¹ m s⁻².
Square the standardized differences to obtain approximately 69.4444 and 0.0448145.

Function B fits this reserved point better under the supplied Gaussian and exact-parameter assumptions.

Try the reasoning

Does reproducing the datum that fixed each parameter independently validate either function?
Show the answer and reasoning
No.
The anchor was calibration information, so agreement there was imposed.
Why is √(gbar/gd) allowed inside the exponential?
Show the answer and reasoning
It is dimensionless.
Both accelerations have the same units.
Does this held-out score settle all dark matter versus all modified gravity?
Show the answer and reasoning
No.
It compares two supplied functions under narrow assumptions, not complete physical theories.
Can points on one rotation curve share calibration or geometry uncertainty?
Show the answer and reasoning
Yes.
Common nuisance inputs can correlate their inferred accelerations and must be represented.

What to carry forward

An empirical regularity is a constraint to explain, rather than a complete physical theory.
A reserved datum can discriminate between explicit fixed predictions.
Physical comparisons require consistent observables, uncertainty, parameter treatment and tests outside the original fit.

References

Teaching resources and research supporting this lesson’s explanations and examples.