Cosmology · The cosmological constant and other dark-energy models

Pressure, density and an equation of state

Use pressure work to compare four supplied homogeneous fluids and see which assumptions make a constant equation of state useful.

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

Define w using energy density and pressure in compatible units.
Derive conditional density and chosen-volume energy scalings.
Apply the acceleration threshold to the correct total fluid.

Pressure and energy density play different roles

For a homogeneous isotropic fluid with positive energy density ε, define its equation-of-state ratio as w = P/ε. Pressure P and energy density ε have the same units, so w is dimensionless.

Negative w means negative pressure when ε is positive. It does not mean the energy density is negative. Nor is w a speed: a background pressure-density ratio alone does not determine how perturbations propagate.

Consider four separate spatially flat single-fluid models with w = 0, −1/3, −2/3 and −1. Each has the same positive density ε_* at the chosen reference a = 1.

The four fluids are alternative supplied universes in this example. They are not four components being summed, and their reference normalization is not a measurement of the present Universe.

Follow the work done during expansion

For a separately conserved component in a comoving volume V = V_*a³, local conservation gives d(εV) = −P dV. Substituting P = wε connects the change of density to expansion.

When w is constant, the result is ε/ε_* = a⁻³⁽¹⁺ʷ⁾. Equivalently, the energy within the chosen volume is E/(ε_*V_*) = a⁻³ʷ.

For pressureless matter, w = 0: the density falls as a⁻³ while this region’s energy stays constant. For w = −1, density stays constant while the energy grows as a³.

These power laws require separate conservation and constant w over the interval. If w varies, conservation must be integrated using its dependence on a; exchanging energy with another component changes the equation as well.

One reference density, four different histories

At a = 1 all four densities equal ε_*. Away from that reference, pressure work distinguishes them. From a = 1 to a = 2, the w = −2/3 density halves, while the chosen-volume energy becomes four times larger.

Supply H_*² = 8πGε_*/(3c²). In each separate flat single-positive-fluid model, H/H_* = √(ε/ε_*). Equal reference density therefore means equal reference H, even though the accelerations differ.

The deceleration parameter is q = −äa/ȧ². Here q = (1+3w)/2, giving 1/2, 0, −1/2 and −1 for the four choices. Negative q denotes positive acceleration.

The table displays both density and energy so that a change in one is not mistaken for a change in the other. Its numbers follow the supplied conservation laws; none is an observed density or equation-of-state estimate.

Four separate constant-w single-fluid models; entries at the same a do not describe a mixture
w	ε/ε_* at a = 0.5	ε/ε_* at a = 1	ε/ε_* at a = 2	E/(ε_V_) at a = 2	q
0	8	1	0.125	1	0.5
−1/3	4	1	0.25	2	0
−2/3	2	1	0.5	4	−0.5
−1	1	1	1	8	−1

Apply the threshold to the stated model

For a single positive fluid, acceleration requires ε+3P < 0, so w < −1/3. The equality w = −1/3 gives q = 0 in this example: expansion is neither accelerating nor decelerating.

For a mixture, the relevant combination is the sum of ε_i+3P_i over components. A small negative-pressure component does not automatically overcome the other contributions.

If the total density is positive, an effective background w can be defined by total pressure divided by total density. It depends on the component proportions and need not remain constant when their densities evolve differently.

This explains why the previous matter-plus-Λ model changed from deceleration to acceleration despite each component having a fixed individual w. The mixture changed, rather than either component changing its own equation of state.

This original finite comparison contains four separate spatially flat homogeneous positive single-fluid FLRW models, not four components being added. Their constant equation-of-state ratios are w = 0, −1/3, −2/3 and −1, with pressure P = wε. All have the same positive supplied energy density ε_* at arbitrary reference a = 1. Each component is separately conserved on a from 0.5 to 2. Panel A has linear a and a zero-based normalized density ordinate from 0 to 8.5, showing ε/ε_* = a^(−3(1+w)). At a = 0.5, 1, 2 the four series are respectively (8, 1, 1/8), (4, 1, 1/4), (2, 1, 1/2) and (1, 1, 1). Panel B uses a separate ordinate from 0 to 8.5 for selected-region energy E/(ε_*V_*) = a^(−3w), with V = V_*a³. At a = 0.5, 1, 2 the four energy series are respectively (1, 1, 1), (1/2, 1, 2), (1/4, 1, 4) and (1/8, 1, 8). The reference samples all coincide at a = 1 by construction. Solid blue circles, dashed orange squares, dotted green diamonds and dash-dot violet triangles identify w = 0, −1/3, −2/3 and −1 in both panels without relying on color. The compact q row gives deceleration parameter q = −äa/ȧ² = (1+3w)/2 as 1/2, 0, −1/2, −1. These are separate single-fluid backgrounds: negative q means acceleration, and w = −1/3 is the boundary. A mixture instead depends on total energy density plus three times total pressure. Positive density with negative pressure does not mean negative energy. Pressure and energy density have compatible SI units Pa = J/m³; w, a and all displayed ratios are dimensionless. The power laws require constant w and separate conservation; changing either requires a different conservation calculation. A homogeneous w does not define perturbation propagation, sound speed, couplings or microscopic identity. No observational equation-of-state estimate, actual cosmic composition or preferred mechanism is supplied. Original Ark teaching model.

Ask what remains unspecified

A chosen w supplies a relation for a homogeneous background. It does not by itself specify a particle species, a field potential, couplings, sound speed or the evolution of inhomogeneities.

A dynamical field can have a changing balance between kinetic and potential energy and hence a changing w. Its evolution must satisfy the field equation and the gravitational equations together.

An inference of w from data is conditional on the expansion model, calibration and other parameters. A narrow parameter interval inside one family is not a direct measurement of every possible microscopic mechanism.

Use the power laws as a transparent first calculation. Then return to the assumptions: which component is conserved, is w constant, what else gravitates, and which observables test the proposed theory?

Worked reasoning

Double a in the w = −2/3 model

What changes between a = 1 and a = 2?

Density scales as a⁻¹, so ε becomes ε_*/2.
Volume grows by eight, giving E = (ε_*/2)(8V_*) = 4ε_*V_*.
The active combination ε+3P = −ε is negative, and q = −1/2 for this single-fluid universe.

The density halves, the selected-volume energy quadruples and the expansion accelerates under the stated assumptions.

Try the reasoning

Does w = −2/3 require negative ε?
Show the answer and reasoning
No.
With positive ε, this ratio specifies a negative pressure P = −2ε/3.
Does w = −1/3 produce acceleration in the supplied single-fluid model?
Show the answer and reasoning
No: q = 0.
The active combination ε+3P vanishes at the boundary.
Can a tiny component with w = −1 be assumed to accelerate any mixture?
Show the answer and reasoning
No.
The total sum of energy density plus three times pressure must be negative.

What to carry forward

Constant-w density scaling follows from separate conservation.
Energy in a chosen expanding region and local energy density have different scalings.
A background equation of state leaves substantial microphysics and perturbation physics unspecified.

References

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