Cosmology · Conditional futures of the Universe

The future of locally bound structures

Compare local gravity with a constant cosmological term and distinguish radial force balance from stable orbital motion.

Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.

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After reading, you can

Explain why comoving expansion is not a rule for every local separation.
Calculate a force-balance scale in a specified weak-field model.
Distinguish a circular orbit from a stable circular orbit.

Do not expand every internal distance by a

A comoving separation follows the homogeneous cosmological expansion by definition. A locally bound system instead has internal gravitational and other dynamics that must be solved in their own regime.

A star orbiting a central mass is therefore not assigned an ever-growing orbital radius merely because the cosmic scale factor increases. Conversely, being bound at one time is not a guarantee of unchanged structure forever.

Consider a passive test mass moving around a fixed spherical mass M, with a positive constant Λ. Use local weak-field mechanics, and omit mass loss, encounters, radiation reaction and changes of the central potential.

Define H_Λ² = Λc²/3 and the length r_s = (GM/H_Λ²)¹ᐟ³. This is a supplied local model; H_Λ is the scale of the constant term, rather than the total measured Hubble rate of a matter-filled Universe.

An opposing force can balance gravity without stabilizing it

With no angular momentum, the local radial acceleration is r̈ = −GM/r²+H_Λ²r. The first term points inward, while the second points outward in this approximation.

Set x = r/r_s. In units H_Λ²r_s, the inward force magnitude is x⁻² and the outward magnitude is x. They are equal at x = 1.

That nonrotating balance is unstable: a small outward displacement increases the relative outward acceleration, while a small inward displacement favours inward motion. Force balance alone is not a stable bound configuration.

Supply GM/(c²r_s) = 10⁻⁶. Then r_s/(c/H_Λ) = 10⁻³, so the displayed radii 0.25 ≤ x ≤ 1 are both weak-field and small compared with the constant-term horizon scale. No real object’s mass or radius is inferred.

Circular motion adds a separate condition

Let ℓ be conserved angular momentum per unit test mass, in square metres per second, and define j² = ℓ²/(GMr_s). Circular motion requires the radial gravity, constant-term and centrifugal contributions to balance.

The circular-orbit condition is j² = x−x⁴. Its angular-frequency relation is Ω²/H_Λ² = x⁻³−1. Positive j² permits circular motion inside x = 1, but does not yet establish stability.

To test radial stability, perturb the radius while holding angular momentum fixed. The squared radial oscillation frequency is κ²/H_Λ² = x⁻³−4 in this local approximation.

Positive κ² gives a restoring radial response; negative κ² gives instability. The stable branch therefore requires x < 4⁻¹ᐟ³ ≈ 0.629961, a stricter condition than x < 1.

Supplied weak-field constant-M, constant-Λ model; stability is tested at fixed angular momentum
x	Inward magnitude x⁻²	Outward magnitude x	Circular j²	Ω²/H_Λ²	κ²/H_Λ²	Classification
0.25	16	0.25	0.246094	63	60	Stable circular
0.5	4	0.5	0.4375	7	4	Stable circular
0.75	1.777778	0.75	0.433594	1.370370	−1.629630	Unstable circular
1	1	1	0	0	−3	Unstable nonrotating balance

Optional mathematical path: hold ℓ fixed

The effective potential per unit test mass, divided by GM/r_s, is U_eff/(GM/r_s) = j²/(2x²)−1/x−x²/2. The first term is the angular-momentum contribution.

A stationary point satisfies −j²/x³+1/x²−x = 0, giving j² = x−x⁴. This derivative tests radial balance at the chosen angular momentum.

The second derivative at fixed j is 3j²/x⁴−2/x³−1. Substituting the circular-orbit value gives x⁻³−4, the normalized radial stability criterion.

Differentiating along a sequence in which j changes with x would answer a different question. Stability concerns a perturbation of one orbit with its conserved angular momentum held fixed.

A supplied passive test mass moves around fixed spherical mass M with constant positive Λ in local Newtonian weak-field mechanics. Define H_Λ² = Λc²/3, r_s = (GM/H_Λ²)^(1/3) and radius x = r/r_s, shown from 0.25 to 1. Panel A has a zero-based ordinate from 0 to 17 for acceleration magnitudes in units H_Λ²r_s. Inward gravity x^−2 is a solid blue line with circles; the outward constant term x is dashed orange with squares. They balance at x = 1, which is an unstable nonrotating equilibrium. Panel B separately shows signed squared radial frequency κ²/H_Λ² = x^−3−4, with an ordinate from −5 to 65. Its dotted green diamonds at x = 0.25, 0.5, 0.75 and 1 give 60, 4, −44/27 and −3. A violet square marks the zero at x = 4^(−1/3) ≈ 0.629961. Positive values below that radius give restoring radial responses; negative values do not. Circular motion alone needs j² = x−x⁴, where j² = ℓ²/(GMr_s) and ℓ has units m²/s. Stability perturbs one orbit at fixed angular momentum, rather than changing j with x. Supplied GM/(c²r_s) = 10^−6 implies r_s/(c/H_Λ) = 10^−3. No exact GR stability, virial radius, observed size or permanent survival theorem is asserted. Mass loss, encounters and radiation reaction are omitted. Original Ark teaching model.

A local bound orbit is not an eternal-survival theorem

This weak-field calculation extends a spherical radial-force model by including angular momentum. It is not an exact general-relativistic orbital-stability result, a virial-radius definition or a universal observational bound for nonspherical objects.

Real bound systems can change through stellar mass loss, encounters, mergers, gas dynamics and dissipative or radiative processes. Those effects were deliberately absent from the supplied fixed-M orbit.

Changing the assumed future stress or gravity would also change the local comparison. A result derived for a constant Λ cannot be silently transferred to every evolving dark-energy proposal.

The calculation establishes a distinction: homogeneous expansion, radial force balance and stable local motion are different dynamical questions. Assessing a structure’s future requires the appropriate conditions for each.

Worked reasoning

Test the orbit at x = 1/2

Does the circular orbit have a restoring radial response?

The required j² is 1/2−1/16 = 7/16.
Its circular angular-frequency square is Ω²/H_Λ² = 8−1 = 7.
The radial-frequency square is κ²/H_Λ² = 8−4 = 4 > 0.

Yes: it is radially stable in the stated fixed-angular-momentum weak-field model.

Try the reasoning

Is the circular orbit at x = 3/4 radially stable?
Show the answer and reasoning
No: κ²/H_Λ² = 64/27−4 = −44/27.
Positive circular j² permits an orbit, but the negative radial curvature makes that orbit unstable.
Does equal inward and outward radial force at x = 1 imply stability?
Show the answer and reasoning
No.
A small radial displacement moves the nonrotating test mass away from the balance point.
Does this fixed-M calculation guarantee that a real bound structure survives unchanged forever?
Show the answer and reasoning
No.
Mass evolution, interactions and additional physics are outside the model.

What to carry forward

Bound internal separations require local dynamics rather than automatic comoving scaling.
Radial force balance is distinct from stable circular motion.
A finite weak-field stability result remains conditional on the specified forces and conserved quantities.

References

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