Cosmology · Cosmic contents and background radiation

Dark matter and dark energy as different questions

Use an orbital mass estimate, gravitational lensing and supernova distances to distinguish missing gravitating matter from the inference of accelerated expansion.

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

Explain how motion or gravitational lensing can constrain mass without identifying its particles.
Distinguish the dark-matter question from the dark-energy question.
State the assumptions and limits of a simple circular-orbit calculation.

How can motion reveal mass?

Suppose a small tracer follows a circular orbit around an approximately spherical mass distribution. In a Newtonian model, the inward acceleration needed for the orbit is v²/r, where v is orbital speed and r is orbital radius. Gravity supplies GM(<r)/r², where G is the gravitational constant and M(<r) is the mass enclosed inside that radius.

Equating these gives v² = GM(<r)/r, or M(<r) = rv²/G. A speed measurement can therefore constrain an enclosed mass under the model’s assumptions. The units agree: r in metres and v² in square metres per square second, divided by G in cubic metres per kilogram per square second, produce kilograms.

This is a gravitational measurement, not a chemical analysis. It does not label the mass as gas, stars or a new particle. Astronomers compare such gravitational information with estimates of visible and otherwise detectable ordinary matter. A persistent mismatch under a specified analysis motivates an additional gravitating component.

Real galaxies are not all spherical, and tracers need not follow perfectly circular orbits. Gas motions, geometry, the stellar mass estimate and other uncertainties matter. The simple calculation below teaches the inference; it is not a complete galaxy fit.

A flat speed curve asks for more enclosed mass

Choose a reference radius r₀ and reference speed v₀, and define M₀ = r₀v₀²/G. Compare radii r₀, 2r₀ and 4r₀. These are chosen dimensionless examples, not measured galaxy data.

First suppose the enclosed mass is the same M₀ at all three radii, as for orbits outside an effectively concentrated spherical mass. Then v/v₀ = √(r₀/r). The speeds are 1, about 0.707 and 0.5 times v₀. Gravity from that fixed enclosed mass gives smaller circular speeds farther out.

Now choose the same speed v₀ at every radius: a flat speed curve. The inferred enclosed mass is then proportional to r, giving M/M₀ = 1, 2 and 4. Flat does not mean that gravity vanishes. In this model, keeping the speed up at larger r requires more mass inside the larger orbit.

The conclusion is conditional on the gravitational and orbital model. It demonstrates why an extended mass distribution can be inferred. It does not, by itself, prove a particular dark-matter particle or rule out every alternative gravitational explanation.

Two synthetic spherical circular-orbit models; all ratios are dimensionless
r/r₀	v/v₀ for fixed enclosed M₀	M(<r)/M₀ for chosen constant v₀
1	1	1
2	1/√2 ≈ 0.707	2
4	0.5	4

Two aligned analytic plots use radius r/r₀ from 1 to 4, with positive arbitrary reference radius r₀ and speed v₀ and M₀ = r₀v₀²/G. All axes are dimensionless and linear. Solid blue lines describe a fixed enclosed mass M₀: speed v/v₀ falls as (r/r₀)⁻¹ᐟ², taking values 1, 1/√2 ≈ 0.707 and 1/2 at radii 1, 2 and 4. Dashed orange lines impose speed v/v₀ = 1: the required mass M(<r)/M₀ then grows as r/r₀ to 1, 2 and 4. These are chosen Newtonian spherical circular-orbit models, not galaxy measurements or a fit. The comparison infers gravitating mass under its assumptions and does not identify dark-matter particles. Original Ark teaching model.

Lensing supplies a different gravitational probe

Gravity also changes the paths of light. A foreground mass distribution can distort images of more distant galaxies. Patterns in those distortions allow a gravitational-lensing reconstruction of the foreground mass, given the source distances, geometry and analysis assumptions.

In the merging system commonly called the Bullet Cluster, the 2006 Clowe and colleagues study compared such a reconstruction with X-ray observations of hot gas. The dominant observed ordinary plasma was spatially separated from the principal lensing-inferred mass concentrations. This supplies evidence that the gravitational mass does not simply follow the gas that dominates the observed ordinary matter in that system.

A lensing mass map is a reconstruction with noise and limitations. It is not a photograph of dark particles. Motion and lensing are valuable together because they use different observations to ask where gravitating mass lies. The interpretation must still account for the ordinary matter and the assumptions of each probe.

Dark matter names the additional matter component inferred in such gravitational explanations, for which no identifying electromagnetic signal has established its underlying particle nature. It need not be a single known substance. The gravitational case and a laboratory identification are separate achievements.

Dark energy addresses the expansion history

Return to the scale factor a from the preceding chapter. Accelerated expansion means its absolute growth rate increases: ä > 0. This is a question about the large-scale expansion history. It is different from asking how much mass lies inside an orbit or a gravitational lens.

A major historical line of evidence came from Type Ia supernovae. Their light curves and colours can be used to standardize their luminosities, with calibration and astrophysical uncertainties. Comparing that luminosity estimate with apparent brightness yields a luminosity distance. Comparing these distances with redshift tests alternative expansion histories.

The 1998 Riess and colleagues analysis favoured accelerated expansion within its cosmological model comparison and treatment of systematic errors. A distance–redshift relation is evidence about a history; it is not a local photograph of an outward-pushing substance.

Within general-relativistic cosmological models, dark energy is the name used for a component invoked to account for accelerated expansion. A cosmological constant, usually written Λ, is one model: its energy density stays constant as the background expands. Other physical explanations are studied. The name does not establish a known force or identify a particle.

Keep the inference and the physical identity separate

The shared word dark records a difficulty of observation and interpretation; it does not make dark matter and dark energy the same component. Dark matter is used to explain additional gravitating matter and its spatial structure. Dark energy is used in modelling accelerated cosmic expansion. Their roles and expansion behaviour differ.

A useful reading habit is to ask three questions: What was observed? What did a stated model infer? What physical explanation remains unresolved? For the orbit example these are speed and radius, enclosed mass, and the mass’s composition. For the supernova example they are measured light and redshift, a distance–redshift history, and the cause of acceleration.

No current percentage is needed to make these distinctions. A numerical cosmic budget must additionally state its epoch, observations, model and uncertainty. The final lesson of this chapter returns to that accounting.

Worked reasoning

Compare two radii at the same speed

In the spherical circular-orbit model, a tracer at 4r₀ has the same speed v₀ as one at r₀. What enclosed mass is inferred at 4r₀?

Use M(<r) = rv²/G with the same v₀ at both radii.
At r₀, M₀ = r₀v₀²/G. At 4r₀, M = 4r₀v₀²/G = 4M₀.
This compares enclosed masses under the chosen geometry and gravity. It does not identify the material that supplies the extra mass.

The inferred mass inside 4r₀ is 4M₀.

Try the reasoning

For the fixed-enclosed-mass model, what is v/v₀ at 4r₀?
Show the answer and reasoning
0.5.
v/v₀ = √(r₀/r) = √(1/4). A flat curve would instead require more enclosed mass in this model.
Does a lensing reconstruction identify the particle species of dark matter?
Show the answer and reasoning
No. It constrains gravitational mass through distorted background images.
Geometry, distances and reconstruction assumptions enter the inference. Particle identification requires additional evidence.
Which question is directly addressed by comparing standardized supernova luminosity distances with redshift?
Show the answer and reasoning
The cosmic expansion history.
It is a test of distance–redshift predictions, distinct from an enclosed-mass estimate around a galaxy.

What to carry forward

Motion and lensing can reveal gravitational mass while leaving its physical composition unresolved.
The flat-speed example requires enclosed mass to grow with radius under its stated assumptions.
Dark matter and dark energy name different cosmological questions; neither label supplies a confirmed microscopic identity.

References

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