Cosmology · Simulations, surveys and validation

Simulated universes and their stated assumptions

Track four labelled test tracers in a prescribed expanding background and distinguish an analytic code test from a model of observed structure.

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

Identify state, evolution, initial conditions and boundary assumptions.
Follow comoving positions and peculiar speeds through an expanding background.
Separate verification of an update rule from physical validation.

A simulation evolves a specified model

A numerical universe begins with equations, initial conditions, a background description and boundary choices. Numerical methods approximate how that declared system evolves.

A simulation tracer can represent a weighted sample of a matter distribution. Its numerical mass need not be the mass of a fundamental dark-matter particle.

Verification asks whether a calculation implements the stated mathematical model adequately. Physical validation asks whether the model describes the relevant real system well enough for its intended use.

Begin with a deliberately restricted analytic test. Four nonrelativistic test tracers have negligible self-gravity, no collisions and no gas; their background expansion is supplied externally.

Separate expansion from motion relative to the background

Let t_* be a positive time in seconds, s = t/t_* range from 1 to 8, and prescribe dimensionless scale factor a = s²ᐟ³. The four tracers do not determine this background.

Let x be a one-dimensional comoving coordinate in a periodic interval of length L metres. Initially, the labelled tracers A, B, C and D have x/L = 1/8,3/8,5/8,7/8.

Their common initial physical peculiar speed is v_* > 0, meaning motion relative to the supplied expansion. Choose v_*t_*/L = 1/8 and, if a speed hierarchy is desired, v_*/c = 0.001.

The periodic boundary identifies x = L with x = 0 for this calculation. It is a modeling choice and supplies no conclusion about the topology of the real Universe.

Wrapping a coordinate does not remove a tracer

With no peculiar force, a times the physical peculiar speed stays equal to v_*. Thus v_pec/v_* = s⁻²ᐟ³, decreasing to 1/4 at s = 8.

The unwrapped coordinate obeys d(x/L)/ds = (1/8)s⁻⁴ᐟ³. Its common displacement is Δx/L = (3/8)(1−s⁻¹ᐟ³).

At s = 27/8 the displacement is L/8. Tracer D reaches unwrapped x = L and is represented at periodic x = 0; it remains the same labelled tracer.

At s = 8 the displacement is 3L/16. D has unwrapped coordinate 17L/16 and periodic coordinate L/16. All four tracers remain present.

Exact supplied snapshots; position tuples follow the same labelled tracers A, B, C, D
s	a	Displacement/L	v_pec/v_*	Unwrapped x/L	Periodic x/L
1	1	0	1	(1/8,3/8,5/8,7/8)	(1/8,3/8,5/8,7/8)
27/8	9/4	1/8	4/9	(1/4,1/2,3/4,1)	(1/4,1/2,3/4,0)
8	4	3/16	1/4	(5/16,9/16,13/16,17/16)	(5/16,9/16,13/16,1/16)

Three exact snapshots show the same labelled tracers A through D on a common periodic coordinate axis. Initial positions are one eighth, three eighths, five eighths and seven eighths of L. At s = 27/8 the displacement is L/8 and D wraps from unwrapped L to periodic zero. At s = 8 the displacement is 3L/16, so D is at unwrapped 17L/16 and periodic L/16. Separate curves show declining normalized peculiar speed s to the power minus two thirds and normalized peculiar kinetic energy s to the power minus four thirds, both starting at one. At the three snapshots the speed ratios are 1,4/9,1/4 and energy ratios are 1,16/81,1/16. The background is supplied externally; no peculiar force or self-gravity is solved. Exact supplied snapshots; position tuples follow the same labelled tracers A, B, C, D. Columns: s; a; Displacement/L; v_pec/v_*; Unwrapped x/L; Periodic x/L. Rows: 1; 1; 0; 1; (1/8,3/8,5/8,7/8); (1/8,3/8,5/8,7/8) / 27/8; 9/4; 1/8; 4/9; (1/4,1/2,3/4,1); (1/4,1/2,3/4,0) / 8; 4; 3/16; 1/4; (5/16,9/16,13/16,17/16); (5/16,9/16,13/16,1/16).

Optional mathematical path: the conserved quantity is specified

For a tracer weight m in kilograms, canonical comoving momentum is p = ma²dx/dt = ma v_pec. It is constant for this force-free peculiar-motion test.

This does not mean v_pec is constant. The growing scale factor requires v_pec = v_*/a, while dx/dt = v_*/a².

Nonrelativistic peculiar kinetic energy scales as a⁻². Relative to its initial value it is 16/81 at s = 27/8 and 1/16 at s = 8.

A time-dependent prescribed background does not supply ordinary conservation of each tracer’s physical peculiar kinetic energy. The conserved canonical quantity and the changing physical speed must remain distinct.

An analytic test checks a restricted implementation

A numerical implementation can compare its computed displacements, wrapping and speed ratios with these exact results. A failure can reveal an update, unit or boundary error.

Passing this test verifies only the restricted conditions tested. It supplies no calculation of self-gravitating collapse, initial perturbations, gas dynamics or galaxy formation.

No peculiar force here does not imply an empty background Universe. The background law was imposed rather than solved from these four test tracers.

To interpret a cosmological simulation physically, state which ingredients were solved, approximated or supplied, then compare appropriate predictions with observations. Correctly following a small numerical world is one part of that larger task.

Worked reasoning

Follow tracer D to s = 8

Where is the tracer whose initial coordinate was 7L/8?

The common displacement is (3/8)(1−1/2)L = 3L/16.
Add 7L/8 to obtain unwrapped coordinate 17L/16.
Subtract one periodic length L to represent the same tracer at L/16.

Its unwrapped coordinate is 17L/16 and its periodic coordinate is L/16; it has not disappeared.

Try the reasoning

At s = 27/8, is tracer D lost when its unwrapped coordinate reaches L?
Show the answer and reasoning
No; its periodic coordinate is zero.
The periodic boundary identifies the two endpoints while preserving the tracer identity.
At s = 8, what is the physical peculiar speed?
Show the answer and reasoning
v_*/4.
The supplied scale factor is 4 and a v_pec = v_* is constant.
Does passing this analytic test establish a model of observed galaxies?
Show the answer and reasoning
No.
Self-gravity, gas, initial structure and observational comparisons are outside this restricted test.

What to carry forward

A simulation evolves declared equations and inputs.
Comoving coordinates, peculiar velocity and boundary wrapping have distinct meanings.
Verification against an analytic solution is different from validation against nature.

References

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