Cosmology · Spacetime, curvature and cosmological models

Critical density, curvature and model parameters

Use a common critical-density denominator to compare three hypothetical reference-epoch accounts, distinguish energy shares from density parameters and keep the curvature sign convention explicit.

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 critical energy density and a component density parameter with consistent units.
Apply the stated GR/FLRW reference-epoch closure relation including vacuum once.
Distinguish a ratio to critical density from a fraction of total physical energy density.

Critical density belongs to a rate and a model

A critical density is a useful reference quantity in a specified cosmological model. Here use general relativity with an FLRW background, a reference scale factor a_ref = 1 and a chosen positive reference expansion rate H_ref. No measured present-day values are supplied.

Write critical energy density as ε_crit = 3c²H_ref²/(8πG), in J/m³. It is the total energy density corresponding to zero spatial curvature at that rate under the stated Friedmann equation and component convention.

A mass-density version is ρ_crit = 3H_ref²/(8πG), in kg/m³. Multiplying that mass-density quantity by c² gives ε_crit. The two formulas use different density units; placing kilograms per cubic metre into the energy-density formula without conversion would be an error.

The reference quantity changes if the chosen epoch and its H change. It is not a fixed universal material density that is identical throughout every possible history. A quoted parameter must therefore identify its reference epoch and model.

Density parameters need not sum to one by themselves

Define each physical component parameter as Ω_i = ε_i/ε_crit at this reference epoch. All components use the same critical-density denominator. Let Ω_total be their sum, including vacuum energy if it is present in this convention.

Treat the cosmological constant as an equivalent vacuum component already included in ε_total. Do not also add a separate explicit Λ term. With dimensionless a and K = k/R_ref² in m⁻², the reference Friedmann equation becomes 1 = Ω_total − Kc²/H_ref².

Define Ω_k = −Kc²/H_ref² at a_ref = 1. Then Ω_total + Ω_k = 1. Positive Ω_k corresponds to negative spatial curvature, and negative Ω_k corresponds to positive spatial curvature. The minus sign is part of the definition, not a statement that curvature is a negative amount of matter.

Ω_k is an effective geometric term in this algebraic account. It is not an identified physical energy species whose particles must be added to the physical component total. The relation uses all included physical components, not only matter.

Keep the reference rate fixed and vary the contents

Compare three separate hypothetical reference accounts at the same positive H_ref. Set Ω_m = 2/5 in each, and choose vacuum parameters Ω_vac = 2/5, 3/5 or 4/5. Radiation and other components are omitted from these chosen examples, not declared absent from our actual Universe.

The total physical critical-density ratios are respectively 4/5, 1 and 6/5. The closure relation gives Ω_k = +1/5, 0 and −1/5. Under the stated conventions their spatial curvatures are negative, zero and positive.

Matter alone is below the critical benchmark in all three examples. Comparing Ω_m with 1 therefore cannot distinguish their curvature signs. The other physical components contribute to the density appearing in the equation.

These are choices illustrating the equation, not observational fits or probabilities assigned to curvature. Their common denominator makes a bar comparison meaningful. Renormalizing every physical total to one before plotting would conceal the very difference being examined.

Three hypothetical reference-epoch GR/FLRW accounts, with a_ref = 1, the same positive H_ref and vacuum counted as a physical component
Ω_m	Ω_vac	Ω_total	Ω_k = 1 − Ω_total	Spatial curvature
2/5	2/5	4/5	+1/5	Negative
2/5	3/5	1	0	Zero
2/5	4/5	6/5	−1/5	Positive

A component share asks a different question

In the third account, Ω_m = 2/5 means the matter energy density is 2/5 of critical energy density. It does not mean matter is 2/5 of the physical total, because that total is 6/5 of critical density.

The matter share of the two-component physical total is ε_m/ε_total = Ω_m/Ω_total = (2/5)/(6/5) = 1/3. The vacuum share is (4/5)/(6/5) = 2/3. Those two shares sum to one, while the two critical-density ratios sum to 6/5.

In the flat second account, Ω_total happens to equal one, so component critical-density ratios and physical shares have the same numerical values. That agreement is a consequence of the stated total, not the definition of every Ω_i in every model.

When reading a percentage or density parameter, ask for its denominator. Is it a share of total energy density, a ratio to critical density, or something else? Identical component names do not make those quantities interchangeable.

Three separate hypothetical reference-epoch accounts use a_ref = 1, the same chosen positive H_ref, and the same critical energy density ε_crit = 3c²H_ref²/(8πG). Horizontal stacked bar lengths use one common dimensionless energy-density/critical-density scale; they are not normalized shares. Blue matter is 2/5 in every account. Orange vacuum is 2/5, 3/5 and 4/5 in accounts A, B and C, giving total ratios 4/5, 1 and 6/5. A dashed vertical guide marks critical ratio 1. Under the stated GR/FLRW closure Ω_total + Ω_k = 1, the geometric terms are +1/5, 0 and −1/5, corresponding respectively to negative, zero and positive spatial curvature. Vacuum is included in the total once; no separate explicit Lambda term is added, and radiation is omitted only in this chosen comparison. In account C the actual matter share of the included two-component total is (2/5)/(6/5) = 1/3, and the vacuum share is 2/3. Thus critical-density ratios and total-energy shares have different denominators. Ω_k is a geometric term, not a physical energy component. No measured cosmic composition, topology or future fate is inferred. Original Ark teaching model.

Geometry still needs its own scope

At a general epoch the corresponding geometric parameter is Ω_k(t) = −Kc²/[a(t)²H(t)²]. It can change with time even when K is fixed. The reference values in the table should not be treated as constants at every epoch without evolving the model.

An optional length check uses the instantaneous nonflat curvature-radius magnitude R_curv = a/√|K|. At the chosen reference epoch this is (c/H_ref)/√|Ω_k|. For either nonflat example, |Ω_k| = 1/5, so R_curv = √5 times c/H_ref. The flat example has no finite curvature radius from this expression.

That curvature-radius calculation does not supply global topology or total spatial volume. The previous flat periodic construction already shows why local curvature alone cannot establish infinite extent.

Nor does a curvature sign alone settle future fate when contents and pressure behaviour are not fully specified. Use the expansion dynamics and conditions for that question. Critical-density closure is a precisely defined relation within the stated model, rather than a replacement for the whole history.

Worked reasoning

Try the reasoning

May Ω_m alone determine the curvature sign when vacuum or other components contribute?
Show the answer and reasoning
No.
The stated Friedmann closure relation uses the total included physical energy density.
If vacuum is included in Ω_total, should its explicit Λ contribution be added again?
Show the answer and reasoning
No.
That would count the same contribution twice.
Do all physical Ω_i necessarily sum to one in a nonflat model?
Show the answer and reasoning
No.
They use critical density as the denominator; Ω_total + Ω_k = 1 in the stated convention. Physical shares instead divide by ε_total.

What to carry forward

Critical density uses an epoch’s expansion rate and a stated gravitational/component convention.
Density parameters and physical energy shares have different denominators.
The geometric curvature term needs its sign convention and does not alone fix topology or future fate.

References

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