Cosmology · Simulations, surveys and validation

Gas, feedback and unresolved physical processes

Compare two prescribed mass-transfer ledgers that match one final calibration target while retaining different gas and ejected reservoirs.

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 an unresolved-process prescription from a resolved physical calculation.
Follow a closed mass-proxy ledger through two stages.
Explain why matching one calibration target does not determine all predictions.

A numerical model cannot resolve every physical scale

A cosmological calculation represents a finite range of spatial and temporal scales. Gas cooling, star formation and energy or momentum injection can involve processes below its resolved scales.

A subgrid prescription states how unresolved processes affect the resolved variables. Its parameters and calibration targets are part of the model and need to be identified.

A calibration adjusts or chooses a prescription using specified information. Matching those targets does not by itself establish agreement with every other observable or resolve the underlying small-scale physics.

The historical EAGLE methods paper illustrates these distinctions. The following original ledger uses stipulated mass proxies and does not implement its feedback, hydrodynamics or empirical calibration.

Keep ejected material in the accounting

Let M_* be a positive mass unit in kilograms. Start with gas proxy G = 100, locked stellar mass proxy S = 0 and tracked ejected proxy E = 0, all measured in M_*. S denotes stellar mass here, not entropy.

At each of two arbitrary stages, form proxy mass F = fG. Immediately return fraction r = 1/4 of F to gas and lock the remaining (1−r)F in S.

Also move ηF from the gas into E. The ejected material remains within this bookkeeping ledger, rather than being destroyed or omitted from its total.

Supply model A with f = 1/4 and η = 1/4, and model B with f = 3/10 and η = 19/18. These coefficients are chosen inputs, and a stage is not a measured time or formation rate.

Two different prescriptions can reach the same locked mass

Each update gives ΔS = (1−r)fG, ΔE = ηfG and G_next = [1−((1−r)+η)f]G. Therefore G+S+E remains 100 at every stage.

Nonnegative reservoirs require nonnegative f and η with [(1−r)+η]f ≤ 1. Both supplied models satisfy this constraint. An η larger than one is allowed because it is ejected gas per newly formed mass.

For A, the first stage gives G = 75, S = 18.75 and E = 6.25. Its second stage gives G = 56.25, S = 32.8125 and E = 10.9375.

For B, the second stage also gives S = 525/16 = 32.8125, but G = 3025/144 ≈ 21.006944 and E = 3325/72 ≈ 46.180556. The equal locked mass was matched by construction.

Closed 100-unit mass-proxy inventory; all reservoir values are in M_* and stages have no assigned duration
Model	Stage	Gas G	Locked stars S	Tracked ejected E	Total
A	0	100	0	0	100
A	1	75	18.75	6.25	100
A	2	56.25	32.8125	10.9375	100
B	0	100	0	0	100
B	1	275/6 ≈ 45.833333	22.5	95/3 ≈ 31.666667	100
B	2	3025/144 ≈ 21.006944	32.8125	3325/72 ≈ 46.180556	100

Two panels show model A and model B at stages zero, one and two. Every horizontal stacked bar uses the same zero-to-one-hundred mass-proxy scale. Each actual segment is directly marked G, S or E; S has diagonal hatching and E a dot pattern. Ejected material remains in the inventory. Both models end at locked S = 525/16 = 32.8125 M*, but A ends with gas 56.25 and ejected 10.9375, whereas B ends with gas 3025/144 and ejected 3325/72. Every inventory totals 100 M*. The table preserves all six stages. This is supplied finite mass bookkeeping, not a physical time sequence, an energy-conservation test, an EAGLE simulation or a numerical convergence test. Closed 100-unit mass-proxy inventory; all reservoir values are in M_* and stages have no assigned duration. Columns: Model; Stage; Gas G; Locked stars S; Tracked ejected E; Total. Rows: A; 0; 100; 0; 0; 100 / A; 1; 75; 18.75; 6.25; 100 / A; 2; 56.25; 32.8125; 10.9375; 100 / B; 0; 100; 0; 0; 100 / B; 1; 275/6 ≈ 45.833333; 22.5; 95/3 ≈ 31.666667; 100 / B; 2; 3025/144 ≈ 21.006944; 32.8125; 3325/72 ≈ 46.180556; 100.

Optional mathematical path: sum the stage contributions

Define c = 1−[(1−r)+η]f. With constant coefficients, the gas recurrence gives G_n = 100cⁿ.

The accumulated locked mass is S_n = 100(1−r)f Σⱼ₌₀ⁿ⁻¹cʲ, and the ejected reservoir is E_n = 100ηf Σⱼ₌₀ⁿ⁻¹cʲ.

For A, c = 3/4. For B, c = 11/24. Inserting n = 2 gives the displayed common S₂ = 525/16 even though the coefficients and the two gas histories differ.

This equality is a calibration degeneracy within the supplied bookkeeping family. It does not show that arbitrary prescriptions are equivalent or that additional measurements could never distinguish them.

A calibration target is one part of model assessment

A matched final S does not determine G or E. Information about another reservoir could discriminate between these two prescriptions, provided its relation to the model and its observational uncertainties were specified.

Summing G, S and E checks only this mass-proxy ledger. No energies, cooling, temperature, radiation, momentum, metallicity or resolved outflow are supplied, so energy conservation has not been demonstrated.

No numerical resolution changed in the experiment. It therefore demonstrates neither fixed-model convergence nor weak convergence after recalibration; those require separate resolution comparisons.

Real feedback assessment needs a stated physical and numerical model, declared calibration information and additional relevant tests. Matching one endpoint is useful evidence about that target, without uniquely establishing the mechanism.

Worked reasoning

Compare the common calibration endpoint

Do models A and B have the same second-stage inventories because both reach S = 32.8125?

A has G = 56.25 and E = 10.9375, alongside S = 32.8125.
B has G = 3025/144 and E = 3325/72, alongside the same S = 525/16.
Both totals are 100, but their remaining gas and tracked ejected material differ substantially.

No. They match one target and make different additional predictions within the ledger.

Try the reasoning

Does a common final S establish the same feedback prescription?
Show the answer and reasoning
No.
The prescribed transfer coefficients differ and leave different G and E.
Has weak numerical convergence been demonstrated?
Show the answer and reasoning
No; no resolution was changed.
The experiment compares calibration-degenerate ledgers, rather than resolution changes with recalibration.
Does conservation of G+S+E establish energy conservation?
Show the answer and reasoning
No.
The ledger contains nonrelativistic mass proxies and supplies no energy or thermal evolution.

What to carry forward

Unresolved-process prescriptions and calibration choices are model inputs.
A conserved ledger can retain very different reservoir histories.
Matching one target does not uniquely determine a mechanism or establish convergence and physical validation.

References

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

The EAGLE project: Simulating the evolution and assembly of galaxies and their environmentsJoop Schaye; Robert A. Crain; Richard G. Bower; Michelle Furlong; Matthieu Schaller; Tom Theuns; Claudio Dalla Vecchia; Carlos S. Frenk; I. G. McCarthy; John C. Helly; Adrian Jenkins; Y. M. Rosas-Guevara; Simon D. M. White; Maarten Baes; C. M. Booth; Peter Camps; Julio F. Navarro; Yan Qu; Alireza Rahmati; Till Sawala; Peter A. Thomas; James TrayfordResearch paperarxiv.orgRead source