Cosmology · Matter–antimatter asymmetry and hypothetical relics

Hypothetical defects and other relic signatures

Use an exact finite phase-interpolation experiment to distinguish allowed topological winding from formation probability, field energy and a surviving cosmic relic.

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

Calculate signed winding using explicitly chosen phase interpolation.
Distinguish a finite phase law from a continuous phase law.
Separate topology from field dynamics and observed relic abundance.

Possible configurations are different from observed relics

A symmetry-breaking model can have several ways for its field to approach a vacuum state across space. The topology of its vacuum manifold helps identify configurations that cannot be smoothly unwound while remaining in that manifold.

In familiar three-dimensional examples, disconnected vacua can support walls, noncontractible loops can support strings, and noncontractible maps from a surrounding two-sphere can support monopoles. The physical setting and boundary conditions matter.

These possibilities do not determine how often configurations form, whether they survive or what signal they produce. Field dynamics, expansion and later evolution are additional parts of a relic prediction.

The following original phase experiment calculates a boundary winding and its exact probability under a small supplied law. It is not a string-network simulation or a measurement of a cosmic relic population.

The spatial loop and vacuum circle are different objects

Take a spatial triangle traversed counterclockwise through vertices A,B,C. Independently assign each vertex phase θ = 0,2π/3 or 4π/3 with equal probability 1/3, giving 27 labeled equiprobable assignments.

The vacuum manifold is still the continuous circle S¹. Restricting the sampled phase values to three choices does not replace that circle with three disconnected vacua.

On each spatial edge, interpolate along the unique shorter arc of the phase circle. No two allowed values are antipodal, so no tie-breaking rule is required.

Define the integer winding w as the sum of the three signed principal phase changes divided by 2π. Keep the arrows on the spatial traversal distinct from the direction and length of each phase-space arc.

A shorter arc can cross the zero-phase label

For phase indices (0,1,2), corresponding to turns (0,1/3,2/3), each step along the triangle contributes +1/3 turn. The last 2/3→0 step uses the shorter positive arc across one full turn.

The sum is +1. For indices (0,2,1), all three shorter steps are −1/3 turn and the winding is −1.

For indices (0,0,1), the changes are 0,+1/3,−1/3, giving zero. A repeated phase need not make every edge change zero, but the total is zero for these repeated-value assignments.

A nonzero boundary winding cannot extend continuously over a spatial disk while the field stays on S¹ everywhere. A physical configuration can leave the vacuum manifold, for example by vanishing at a core; its energy still needs a field solution.

Three exact assignments; phase indices label the allowed turns 0,1/3,2/3
Indices at A,B,C	AB step	BC step	CA step	w
(0,1,2)	+1/3	+1/3	+1/3	+1
(0,2,1)	−1/3	−1/3	−1/3	−1
(0,0,1)	0	+1/3	−1/3	0

The winding probability belongs to the chosen phase law

Exactly six assignments use all three distinct phases. Three cyclic orderings have w = +1 and three opposite orderings have w = −1; the remaining 21 assignments have w = 0.

Thus P(nonzero winding) = 6/27 = 2/9. The signed expectation E[w] is zero, while E[w²] = 2/9; cancellation of signs does not mean that every triangle is trivial.

For three independent continuous uniform phases with shortest-arc interpolation, the corresponding probability is 1/4. In that different law, the probability that all three phases fit within a semicircle is 3/4.

The difference between 2/9 and 1/4 is a change in the supplied phase distribution, not an experimental discrepancy. Neither number is a universal probability per cosmic volume or a present relic abundance.

Complete 27-state classification under the finite independent phase law
w	Assignment count	Probability
−1	3	1/9
0	21	7/9
+1	3	1/9
Any nonzero	6	2/9

Three spatial triangles illustrate the exact assignments (0,1,2), (0,2,1) and (0,0,1), where phase indices denote 0, 1/3 and 2/3 of a turn on the continuous vacuum circle S1. Each spatial loop is traversed counterclockwise A to B to C to A, as shown by its arrows. On each edge the phase uses the unique shorter arc, so the three example sums are +1, −1 and 0. A spatial arrow does not depict a phase-space arc or propagation speed. The lower probability bars classify all 27 labelled equiprobable independent three-choice assignments: counts 3,21,3 for winding −1,0,+1. Thus the nonzero probability and mean squared winding are 2/9 while the signed mean is zero. This differs from 1/4 for a separately supplied continuous uniform phase law; it is not an observed tension or a universal relic rate. No string-core energy, gauge-field solution or surviving cosmic network is calculated. Three exact shorter-arc sums; sampled phases do not disconnect the continuous circle. Columns: Indices A,B,C; AB turns; BC turns; CA turns; w. Rows: (0,1,2); +1/3; +1/3; +1/3; +1 / (0,2,1); −1/3; −1/3; −1/3; −1 / (0,0,1); 0; +1/3; −1/3; 0. Complete finite classification; Any nonzero is the sum of the two nonzero categories. Columns: w; Count; Probability. Rows: −1; 3; 1/9 / 0; 21; 7/9 / +1; 3; 1/9 / Any nonzero; 6; 2/9.

A field and its evolution connect winding to observables

The topology calculation constrains what a field configuration can do under particular continuity and vacuum conditions. It does not calculate a string core profile, tension, interaction rate or complete network evolution.

For local strings, the gauge field is important to a finite energy per unit length. A bare phase drawing alone does not supply that gauge-field solution or its physical energy.

A relic prediction must also specify formation, subsequent dilution or evolution and the observational response. Possible gravitational, particle or radiation signatures depend on that additional dynamics.

Topology is therefore a useful discriminator among conditional models, while a surviving population or observed signal remains a separate claim. Preserve that distinction when interpreting a possible defect or comparing it with other relic scenarios.

Worked reasoning

Calculate winding across the zero-phase label

What winding follows from phase indices (0,1,2)?

The first edge changes phase from 0 to 1/3 turn, contributing +1/3.
The second contributes +1/3; the shorter final arc from 2/3 to 0 also contributes +1/3.
Add the signed turns: w = 1/3+1/3+1/3 = 1.

The boundary winding is +1 under the declared traversal and shortest-arc rule.

Try the reasoning

Does E[w] = 0 mean that every triangle has zero winding?
Show the answer and reasoning
No.
Positive and negative winding cancel in the signed mean, but 6 of the 27 assignments have nonzero winding.
Why does 2/9 not contradict the continuous-phase value 1/4?
Show the answer and reasoning
They use different phase probability laws.
The finite experiment allows only three phase values; the continuous example samples the whole circle independently.
Does a winding drawing establish a finite-energy local cosmic string?
Show the answer and reasoning
No.
A consistent microscopic field and gauge configuration, plus formation and evolution assumptions, are still needed.

What to carry forward

Specify the vacuum manifold and interpolation rule before calculating winding.
A topological possibility and its formation probability are distinct.
A surviving relic signal requires field dynamics and an observable response.

References

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

Topology of cosmic domains and stringsT. W. B. KibbleResearch paperdoi.orgRead source