Cosmology · Origins, primordial tests and the limits of knowledge

Bounce and quantum-gravity proposals

Solve a supplied leading effective stiff-fluid bounce background and distinguish its exact algebra from a complete quantum-gravity or observational claim.

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

State the effective equations, density convention and chosen smooth branch.
Verify conservation and the change from contraction to expansion.
Separate a background solution from a complete origin model and its tests.

A bounce needs equations as well as a smooth drawing

A bounce replaces a contracting phase with an expanding one through a nonzero minimum scale factor. Whether that transition is physically allowed depends on the equations, matter content and domain of the proposal.

Loop quantum cosmology provides specific symmetry-reduced quantum models in which a bounce can occur. The Ashtekar–Pawlowski–Singh improved-dynamics calculation studies a flat isotropic model with a massless scalar and also discusses a leading effective background equation.

An effective background is a controlled claim within stated assumptions. It is smaller than a complete quantum-gravity theory, a perturbation calculation or an observational detection of a pre-expansion phase.

Use an explicitly supplied effective stiff-fluid system below. The numerical reference density is left arbitrary; no value in Planck units and no empirical bounce date or density is adopted.

Keep the density convention and branch explicit

Take a flat homogeneous isotropic background with mass-equivalent density ρ≥0 in kg/m³, energy density ε = ρc² and stiff-fluid pressure P = ρc². Let the positive parameter ρ_B have the same density units.

Supply H² = (8πG/3)ρ(1−ρ/ρ_B), together with separate conservation ρ̇ = −6Hρ. These are the declared leading effective background equations, rather than the ordinary classical Friedmann system.

Define t_B = (24πGρ_B)^(−1/2) in seconds, s = t/t_B and normalize the minimum scale factor to a_B = 1. Choose the smooth contracting-to-expanding solution through s = 0.

The squared expansion equation alone does not select the sign of H or the continuation at its zero. The smooth solution below supplies that choice explicitly; no quantum-state selection is inferred from taking a square root.

Verify the solution directly

The supplied branch is a(s) = (1+s²)^(1/6), ρ/ρ_B = 1/(1+s²) and Ht_B = s/[3(1+s²)]. These expressions are defined for every finite real s.

Differentiate ln a with respect to cosmic time to recover H. Also ρa⁶ = ρ_B is constant, which gives the stated stiff-fluid conservation law.

Writing r = ρ/ρ_B, the expansion equation becomes (Ht_B)² = r(1−r)/9. Substitution of r = 1/(1+s²) gives the same s²/[9(1+s²)²] on both sides.

At s = 0, the minimum a is 1, density reaches ρ_B and H vanishes. The exact derivative is Ḣt_B² = (1−s²)/[3(1+s²)²], positive and equal to 1/3 at the bounce.

Exact branch values; the density and expansion rate are expressed in the stated reference units
s	a	ρ/ρ_B	Ht_B
−2	5^(1/6) ≈ 1.307660	1/5	−2/15
−1	2^(1/6) ≈ 1.122462	1/2	−1/6
0	1	1	0
1	2^(1/6) ≈ 1.122462	1/2	1/6
2	5^(1/6) ≈ 1.307660	1/5	2/15

The correction changes the high-density evolution

For negative s, H<0 and the branch contracts toward its minimum. For positive s, H>0 and it expands away from that minimum. Density is nonnegative and never exceeds ρ_B on this solution.

The largest magnitude of Ht_B occurs at s = ±1 and is 1/6. At the maximum density, H is zero rather than maximal, reflecting the supplied factor 1−ρ/ρ_B.

When |s| is large, ρ/ρ_B is small and the correction factor approaches 1. The scale factor behaves as |s|^(1/3), the large-|s| form associated with the stated classical stiff-fluid background.

This limiting check concerns the supplied homogeneous equations. The normalization a_B = 1 is a scale-factor convention, and the calculation supplies no thermal plasma, ordinary matter inventory or measured cosmic history.

Supply a flat homogeneous leading effective background with mass-equivalent density ρ in kg/m³, energy density ε = ρc² and stiff pressure P = ρc². Let arbitrary ρ_B > 0, H² = (8πG/3)ρ(1−ρ/ρ_B) and ρdot = −6Hρ. Define t_B = (24πGρ_B)^(-1/2) in seconds, s = t/t_B and a at the bounce equal to 1. The explicitly chosen smooth contracting-to-expanding branch has a = (1+s²)^(1/6), r = ρ/ρ_B = 1/(1+s²) and h = Ht_B = s/[3(1+s²)]. Three aligned panels use the same linear s domain −3 to 3; vertical dotted guides mark s = 0. Panel A is solid blue with hollow circles and linear a ordinate 0.9 to 1.5, explicitly nonzero based. Panel B is dashed orange with hollow squares and zero-based linear r ordinate 0 to 1.1. Panel C is dotted green with hollow diamonds and signed linear h ordinate −0.2 to 0.2. All five marked s values are −2,−1,0,1,2. At these points a is approximately 1.307660,1.122462,1,1.122462,1.307660; r is 1/5,1/2,1,1/2,1/5; h is −2/15,−1/6,0,1/6,2/15. On the full drawn interval and entire chosen real-s branch, ρa⁶ = ρ_B and h² = r(1−r)/9. The density reaches its maximum while H is zero; t_B² dH/dt = (1−s²)/[3(1+s²)²] equals positive 1/3 at the bounce. Negative H at s < 0 is contraction, positive H at s > 0 is expansion. The squared expansion equation does not select a sign or unique smooth continuation by itself. These are exact checks only of the supplied leading effective homogeneous system; arbitrary ρ_B is not a numerical Planck density, measured bounce or full quantum-gravity solution. No quantum state, thermal history, perturbation stability/transfer, geodesic completeness or observational viability follows. Named separate panels, distinct patterns, marker shapes, explicit scales and exact point annotations provide noncolor access. Original Ark teaching model.

A background bounce leaves further physical questions

The solution is exact for the explicitly supplied effective system. It is not asserted to be an exact trajectory of a full quantum theory or a proof that every quantum-gravity proposal removes singularities.

A physical completion must specify its state, approximations and additional degrees of freedom, and assess perturbations and stability. A smooth a(t) alone does not determine primordial fluctuations or how they become observable signals.

Connecting a proposal to the hot early history also requires a mechanism and conditions for the relevant later matter and radiation. Distinguishing models requires their linked predictions to survive transfer, calibration and foreground uncertainties.

The useful conclusion is conditional: these equations admit a conservation-consistent smooth bounce with a classical low-density limit. Establishing whether such a mechanism describes our Universe requires additional theory and independent evidence.

Worked reasoning

Check the density and expansion equation at s = 1

Does the supplied solution satisfy the effective expansion equation at s = 1?

The solution gives r = ρ/ρ_B = 1/2 and Ht_B = 1/6.
The left-hand side is (Ht_B)² = 1/36.
The right-hand side is r(1−r)/9 = (1/2)(1/2)/9 = 1/36.

Yes; both sides equal 1/36 at this point.

Try the reasoning

What is Ht_B at s = 2?
Show the answer and reasoning
2/15.
Substitute s = 2 into s/[3(1+s²)].
What is ρa⁶ along this branch?
Show the answer and reasoning
ρ_B.
The factors 1/(1+s²) and a⁶ = 1+s² cancel exactly.
Does this background calculation determine observable primordial perturbations?
Show the answer and reasoning
No.
Their dynamics, state and propagation are additional ingredients that were not supplied.

What to carry forward

A bounce claim requires explicit dynamics, density conventions and a branch choice.
Conservation and a low-density limit can be checked without claiming empirical confirmation.
An effective background solution still needs a complete physical account and independent tests.

References

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