Horizon and flatness questions motivating inflation

A successful thermal history still needs initial conditions

A model can explain how a nearly uniform hot background evolves while leaving a further question: why were its starting conditions suitable for that explanation? Inflationary proposals address aspects of that initial-condition question by adding an early phase of accelerated expansion.

The horizon question concerns whether regions whose later properties are compared could have exchanged signals in a specified earlier history. Answering it requires an account of propagation over time, together with the beginning and end of the phases being considered.

The flatness question concerns the evolution of spatial-curvature importance relative to the expansion rate. A small curvature contribution at one epoch can require a much smaller starting contribution in a purely decelerating history. This is a model-dependent sensitivity question, not a mathematical contradiction in the expansion equations.

Motivation does not establish a unique mechanism. To assess an inflationary proposal, specify its dynamics and duration, how it ends, and how its predictions connect to later evidence. The finite example here isolates two background relationships rather than settling that full account.

A length at an instant is different from an integral

Let a be a dimensionless scale factor, H the expansion rate in inverse seconds, and N = ln(a/a_i) the number of e-folds since a chosen reference. Define the comoving Hubble length R_c = c/(aH), measured in metres in this convention.

For a prescribed constant ε_H = −d ln H/dN, H/H_i = exp(−ε_H N). The instantaneous length therefore obeys R_c/R_ci = exp[(ε_H − 1)N], where R_ci = c/(a_i H_i). It decreases when ε_H < 1.

A radial light signal accumulates comoving travel Δχ = ∫ c dt/a. Since dN = H dt, the contribution during this phase is Δχ/R_ci = ∫₀ᴺ exp[(ε_H − 1)n] dn. Its integrand is positive, even when that integrand decreases.

The Hubble length is thus a useful background scale, while the integral measures the travel accumulated over the stated interval. Neither expression by itself supplies a particle horizon from the full past or an event horizon depending on the whole future.

Shrinking Hubble length can accompany growing light reach

Choose two kinematic histories over 0 ≤ N ≤ 4: ε_H = 0.1 and ε_H = 2. They share their initial a_i and H_i. These prescribed functions let us compare lengths without assigning a physical starting temperature, age or observed parameter fit.

For ε_H ≠ 1, the finite-phase integral is Δχ/R_ci = {exp[(ε_H − 1)N] − 1}/(ε_H − 1). The ε_H = 1 limit is N, obtained directly from the integral, rather than by dividing zero by zero.

At N = 4 in the ε_H = 0.1 history, R_c/R_ci is about 0.0273237, while Δχ/R_ci is about 1.08075. The instantaneous length has become small, but the accumulated travel has increased from zero. There is no conflict between these statements.

At the same N in the ε_H = 2 history, the two ratios are about 54.5982 and 53.5982. Their different numerical behavior follows from the supplied histories. It does not make the larger integral a complete causal history or prove that interactions established equilibrium.

The curvature ratio follows a separate identity

For this calculation use ds² = −c²dt² + a²[dr²/(1 − Kr²) + r²dΩ²], with r in metres and K in inverse square metres. The curvature-density parameter is Ω_k = −Kc²/(a²H²). Its sign follows the chosen spatial curvature convention.

For fixed nonzero K, divide by the initial value to obtain Ω_k/Ω_ki = (R_c/R_ci)². The ratio is positive for either sign of nonzero K, and describes how the magnitude changes. At K = 0 both parameters vanish, so this particular ratio is undefined.

The ε_H = 0.1 history suppresses that magnitude by exp(−7.2) ≈ 0.000746586 after four e-folds. The ε_H = 2 history instead multiplies it by exp(8) ≈ 2980.96. This demonstrates sensitivity to expansion history without assigning an initial curvature magnitude.

Here H(N) is prescribed kinematically. In particular, ε_H = 2 with nonzero K is not being presented as the exact solution of a separately specified radiation-plus-curvature matter model. A complete dynamical proposal must also satisfy its energy and gravitational equations.

Turning a motivation into a testable proposal takes more work

The example shows why an accelerating phase with decreasing c/(aH) can reduce the relative importance of spatial curvature. It also shows why language about a shrinking horizon needs care: the accumulated causal travel during that phase still grows.

Whether later-observed regions share an earlier causal domain depends on the earlier history, the phase duration, the transition out of it and the subsequent expansion. Four chosen e-folds in a comparison table are not a calculation of the duration required by the real Universe.

Signal contact also differs from physical equilibration. Exchange of light establishes a possible causal connection under the assumed geometry; the actual interactions and their time scales determine whether populations or temperatures become equilibrated.

The next lesson supplies an exact scalar-field background as one possible source of acceleration. Subsequent lessons distinguish reheating assumptions, statistical initial patterns and observational tests. Agreement among some predictions still leaves the task of comparing alternative mechanisms.

Worked reasoning

Try the reasoning

1. Can Ω_k/Ω_ki be used at exactly K = 0?

   Show the answer and reasoning

   No.

   Both numerator and denominator then vanish. The squared-length ratio comparison was stated for fixed nonzero curvature.

2. Does possible signal exchange prove that a plasma reached thermal equilibrium?

   Show the answer and reasoning

   No.

   The interaction processes and their rates must also be effective during the available time.

3. Do four e-folds in this table solve the observed horizon question?

   Show the answer and reasoning

   The table does not establish that.

   The complete relevant causal history and its connection to later scales have not been specified.

What to carry forward

• Inflationary motivations concern the initial conditions and their evolution in a specified history.
• An instantaneous comoving Hubble length and accumulated causal reach are different quantities.
• For fixed nonzero curvature, its density-parameter ratio follows the square of the comoving Hubble-length ratio.