Cosmology · Matter–antimatter asymmetry and hypothetical relics

Early phase transitions as conditional scenarios

Analyze a supplied free-energy landscape to distinguish coexistence, metastability, loss of local stability and the additional dynamics needed for a cosmic transition.

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

Find competing extrema and their stability in a stated scalar model.
Calculate coexistence, a free-energy barrier and a latent-heat difference.
Separate equilibrium phase structure from nucleation and empirical history.

Cooling alone does not specify a first-order transition

A proposed early phase transition depends on the physical theory, its parameters and the thermal state. A first-order transition, a continuous transition and a smooth crossover are distinct possibilities.

An equilibrium phase diagram identifies favorable states and possible barriers. It does not by itself determine whether bubbles nucleate, how quickly a transition completes or what relics survive.

Finite-system numerical features also need careful interpretation. A sharp susceptibility peak alone need not establish a thermodynamic transition; scaling and the validity of the effective description matter.

The following scalar polynomial is a supplied phenomenological free-energy density. It is not a gauge-invariant Standard Model calculation, a lattice result or evidence that a particular electroweak transition occurred.

Define the order parameter and units

Choose positive constants T₀ in kelvin, φ₀ for the field scale and f₀ in J/m³. Define t = T/T₀, q = φ/φ₀≥0 and f(q,T) = f₀u(q,t).

Supply u = (t²−1)q²/2−tq³/3+q⁴/4 over the displayed temperature interval 1≤t≤1.25. An omitted common background free energy cancels when comparing phases.

The stationary equation is ∂u/∂q = q[q²−tq+(t²−1)] = 0. Its nonzero stationary points are q_± = [t±√(4−3t²)]/2 when the square root is real.

For 1<t<2/√3, the origin is a local minimum, q_− is a barrier and q_+ is another local minimum. Comparing their free energies determines the global preference; the existence of a local minimum alone does not.

Equal minima can remain separated by a barrier

Coexistence requires u(q_c,t_c) = u(0,t_c) as well as stationarity. The supplied polynomial gives t_c = 3/√7 and q_c = 2/√7.

At this temperature it factorizes exactly as u(q,t_c) = q²(q−q_c)²/4. The minima q = 0 and q = q_c both have zero relative free energy.

The intervening barrier is at q = q_c/2 = 1/√7, where u = 1/196. Thus equal free energies at the minima do not make all intermediate states equally favorable.

For 1<t<t_c, the nonzero minimum is global and the origin metastable. For t_c<t<2/√3, the origin is global and the nonzero minimum metastable; metastability refers to a local minimum that is not globally favored.

Landscape checkpoints for the supplied equilibrium model
T/T₀	Stationary structure	q_−	q_+	Relevant interpretation
1	Origin has zero curvature	0	1	Origin loses local stability; not a separate barrier
1.10	Two minima with a barrier	0.245862	0.854138	Nonzero phase favored
3/√7 ≈ 1.133893	Coexisting minima	1/√7	2/√7	Barrier f₀/196 remains
2/√3 ≈ 1.154701	Nonzero extrema merge	1/√3	1/√3	Merged stationary inflection, not a distinct minimum
1.25	Only the origin minimum	Absent	Absent	No nonzero stationary points

The supplied scalar free-energy density is f = f0 u, with q >= 0 and u = (t²−1)q²/2−tq³/3+q⁴/4 for t = T/T0. The upper panel at coexistence t_c = 3/sqrt(7) shows the exact factorization u = q²(q−q_c)²/4, with q_c = 2/sqrt(7). The equal minima at 0 and q_c have u = 0, while the intervening barrier at 1/sqrt(7) has u = 1/196. The lower panel shows stationary branches over the finite temperature interval 1 <= t <= 1.25. The q_plus local minimum and q_minus barrier exist separately only for 1 < t < 2/sqrt(3), and merge into one stationary inflection at the upper endpoint. The origin is a local minimum for t > 1, but loses local stability at t = 1. Coexistence and loss of local stability are distinct. This is an equilibrium polynomial, not a nucleation calculation, gauge-theory phase diagram or established cosmic transition. Exact relative free energies at coexistence; barrier height is f₀/196. Columns: Stationary point; q; u; Interpretation. Rows: Origin; 0; 0; Minimum / Intermediate; 1/√7; 1/196; Barrier / Nonzero; 2/√7; 0; Minimum.

Latent heat follows from a temperature derivative

For a stationary phase, its entropy density is s = −∂f/∂T evaluated at the minimum. The implicit derivative through the minimizing q contributes no extra term because ∂f/∂q = 0 there.

At coexistence, ∂u/∂t = tq²−q³/3 gives s_broken−s_origin = −4f₀/(3√7 T₀). This is an entropy-density difference in J/(m³ K), not an absolute entropy assignment.

The corresponding released latent heat per volume is L = T_c(s_origin−s_broken) = 4f₀/7, positive for the specified transition direction. The omitted common background does not affect this phase difference.

This thermodynamic s differs from the entropy-divided-by-k_B number-density convention used in the annihilation example. The units and normalization must be stated before using either quantity.

Coexistence does not determine a completed cosmic event

At t = 1, the origin’s curvature becomes zero and the positive-q cubic term makes the nearby free energy decrease. This loss of local stability is distinct from the earlier coexistence temperature.

At t = 2/√3, the nonzero stationary points merge. Counting that merged point as both a stable minimum and a separate barrier would misread the landscape.

No nucleation rate, surface tension, bubble-wall velocity or completion time was supplied. A metastable phase can persist beyond coexistence, and computing its decay requires additional dynamics.

Connecting a proposed transition to baryogenesis, defects or gravitational radiation requires a consistent field theory and evolution. The useful results here are conditional properties of the stated polynomial, not a measured cosmic heat release or established early event.

Worked reasoning

Calculate the barrier at coexistence

What barrier separates the two equal minima?

Use t_c = 3/√7 and q_c = 2/√7 to write u = q²(q−q_c)²/4.
The intermediate stationary point is q = q_c/2 = 1/√7.
Substitution gives u_barrier = (1/7)(1/7)/4 = 1/196.

The barrier height relative to either minimum is f₀/196.

Try the reasoning

At coexistence, are all q values between the two minima equally favorable?
Show the answer and reasoning
No.
The two endpoints have equal free energy, but the intervening barrier remains positive.
Does T_c fix the temperature at which bubbles actually nucleate?
Show the answer and reasoning
No.
Coexistence is an equilibrium condition; nucleation and possible supercooling require barrier-crossing dynamics.
Does this scalar polynomial establish an actual first-order electroweak transition?
Show the answer and reasoning
No.
Its coefficients are supplied teaching inputs; the relevant gauge theory, couplings and phase structure need their own calculation.

What to carry forward

Equilibrium phase preference, metastability and dynamical transition rates are different questions.
Equal free-energy minima can remain separated by a finite barrier.
A proposed cosmic transition needs a physical theory and evolution beyond a plotted potential.

References

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