Cosmology · The cosmological constant and other dark-energy models

A cosmological constant in expansion models

Follow a constant density through an expanding volume and distinguish a geometric model term from a microscopic explanation.

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

Describe two equivalent ways to include a cosmological constant.
Compare density, pressure and energy in a specified growing volume.
Determine acceleration from the total stress-energy rather than a component label.

Choose one consistent description

A cosmological constant, written Λ, is a constant term in Einstein’s gravitational field equation. In a homogeneous expansion model, it can be kept on the geometric side of the equation or represented as an equivalent uniform fluid contribution.

These are two descriptions of the same term. If its equivalent density has already been included in the total fluid, adding Λ again on the geometric side counts its influence twice.

Use ε for energy per volume, in joules per cubic metre, and P for pressure, in pascals. These units are equivalent. The equivalent positive-Λ contribution has ε_Λ = Λc⁴/(8πG) and P_Λ = −ε_Λ.

This pressure-density relation defines its homogeneous gravitational behaviour. Calling the contribution vacuum-like does not, by itself, identify a quantum field mechanism or explain its magnitude.

Constant density can accompany increasing energy

Take a specified comoving region: its boundary follows the homogeneous expansion. If its volume at the chosen reference a = 1 is V_*, its volume at scale factor a is V = V_*a³.

Supply a constant positive density ε_* for the Λ contribution. Its energy within this region is E_Λ = ε_*V_*a³. When a doubles from 1 to 2, the region’s volume and this energy each increase by a factor of eight.

Local energy conservation for a homogeneous fluid has the pressure-work form d(εV) = −P dV. With P_Λ = −ε_*, the right side is +ε_*dV, exactly the increase of E_Λ.

This is a local conservation relation in an evolving spacetime. It does not establish a globally conserved Newtonian energy for the whole Universe, or require a particle-creation story for the changing chosen-volume energy.

A supplied matter-plus-Λ comparison

Choose a spatially flat model containing separately conserved pressureless matter and a positive cosmological constant, with radiation omitted. Set ε_m = ε_*a⁻³ and ε_Λ = ε_*. The equality at a = 1 is supplied, rather than inferred from observations.

Define H_*² = 8πGε_*/(3c²). This reference rate has units of inverse seconds squared. The Friedmann equation becomes H²/H_*² = a⁻³+1 when Λ is included through its equivalent fluid.

The acceleration equation gives (ä/a)/H_*² = 1−a⁻³/2. It depends on the total combination ε+3P: positive matter decelerates this model, while the constant negative-pressure contribution accelerates it.

The matter energy in the same chosen volume is ε_mV = ε_*V_*, independent of a. Its density decreases while its pressure is zero; the Λ density stays constant while its chosen-volume energy increases.

Supplied flat matter-plus-Λ model; energies use ε_*V_* and a = 1 is an arbitrary equality reference
a	ε_m/ε_*	ε_Λ/ε_*	H/H_*	(ä/a)/H_*²	E_m/(ε_V_)	E_Λ/(ε_V_)
0.5	8	1	3	−3	1	0.125
1	1	1	1.414214	0.5	1	1
2	0.125	1	1.060660	0.9375	1	8

A positive Λ does not guarantee acceleration at every epoch

At a = 0.5, matter density is eight times ε_*. The normalized acceleration is 1−8/2 = −3. Expansion can therefore be slowing even though Λ is positive.

Acceleration changes sign when a⁻³ = 2, or a = 2⁻¹ᐟ³ ≈ 0.793701. Density equality occurs later, at a = 1. These are different conditions because pressure enters the acceleration equation.

At a = 1, H/H_* = √2 and the normalized acceleration is +1/2. H need not increase during this acceleration: Ḣ = ä/a−H² = −3H_*²a⁻³/2 remains negative in the supplied model.

The table predicts a conditional history once the ingredients and reference densities have been chosen. It neither estimates today’s cosmological parameters nor shows that these ingredients are the unique explanation of measured distances.

This original supplied flat FLRW model contains separately conserved pressureless matter and one positive cosmological constant, with radiation omitted. The constant is counted once through its equivalent fluid, not again geometrically. The positive reference energy density ε_* is in J/m³, P_Λ = −ε_* in Pa, H_*² = 8πGε_*/(3c²) in s⁻² and ε_Λ = Λc⁴/(8πG). The scale factor a is dimensionless; a = 1 is an arbitrary equality of matter and Lambda densities, not a measured present epoch. Panel A uses linear a from 0.5 to 2 and zero-based normalized energy density from 0 to 8.5. Dashed blue square samples show ε_m/ε_* = a⁻³, with values 8, 1, 1/8 at a = 0.5, 1, 2. Solid orange circles show constant ε_Λ/ε_* = 1. The two samples at a = 1 coincide intentionally. Panel B has its own signed acceleration ordinate from −3.5 to 1.1. Dotted green diamonds show (ä/a)/H_*² = 1−a⁻³/2, giving −3, 1/2 and 15/16 at the same three a values. A solid zero reference and separately marked square identify the sign change at a = 2^(−1/3), approximately 0.793701. It occurs before density equality at a = 1. Distinct line styles, marker shapes, written formulas and panel headings preserve the meanings without color. A separate native table concerns energy in one chosen expanding comoving region, not an additional density plot. Its columns are a, V/V_*, E_m/(ε_*V_*) and E_Λ/(ε_*V_*). Rows are (0.5, 1/8, 1, 1/8), (1, 1, 1, 1) and (2, 8, 1, 8). V = V_*a³, matter energy remains ε_*V_*, and Lambda energy grows as ε_*V_*a³. Between a = 1 and 2, local pressure work −P_ΛΔV equals the energy increase +7ε_*V_*. This does not assert a globally conserved Newtonian energy or particle creation. H can decrease even while the expansion accelerates. The ingredients and reference are stipulated; no fitted current fractions, observed data, unique cosmic mechanism or microscopic vacuum origin is inferred. Original Ark teaching model.

Keep the model term and its origin as separate questions

A constant term is a compact hypothesis about the homogeneous stress-energy or geometry. Testing it requires measured expansion and structure observables together with their calibration, selection and uncertainty models.

A successful fit of a constant density would constrain its gravitational behaviour within that model. A microscopic account must additionally explain why an effective contribution takes that value and remains consistent with the wider theory.

Other stress-energy descriptions may have evolving pressure or density. They must be compared through their own conservation and gravitational equations, rather than by changing a label in this ledger.

The next lesson makes that comparison using pressure divided by energy density. Keep the distinction between energy density and energy in a selected volume: both are useful, but they answer different questions.

Worked reasoning

Follow one region from a = 1 to a = 2

How can the Λ energy increase while its density remains constant?

The selected region grows from V_* to 8V_*, so E_Λ grows from ε_*V_* to 8ε_*V_*.
The change is +7ε_*V_*; the pressure-work contribution is −P_ΛΔV = +7ε_*V_*.
The matter energy in that same region stays ε_*V_* because ε_m falls by eight.

The Λ energy increases by +7ε_*V_* in the growing region, consistently with local pressure work.

Try the reasoning

Should an equivalent Λ fluid and the same geometric Λ term both be added?
Show the answer and reasoning
No.
They are equivalent descriptions of one contribution; adding both counts it twice.
Is the supplied model accelerating at a = 0.5?
Show the answer and reasoning
No: (ä/a)/H_*² = −3.
The pressureless matter contribution is sufficiently large to make the total acceleration negative.
Does acceleration begin exactly at matter–Λ density equality?
Show the answer and reasoning
No: the transition is a = 2⁻¹ᐟ³, while equality is a = 1.
Acceleration depends on energy density plus three times pressure, rather than density equality alone.

What to carry forward

Include a cosmological constant once, geometrically or as its equivalent fluid.
Constant density and increasing energy in a growing comoving region are compatible.
Acceleration and a microscopic origin require distinct calculations and evidence.

References

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