Slow-Roll Inflation Dynamics

About Inflation and Slow-Roll

Cosmic inflation is a period of exponential expansion in the very early universe, driven by a scalar field φ (the inflaton) slowly rolling down a potential V(φ). This elegant mechanism solves several cosmological puzzles: the horizon problem (why distant regions have the same temperature), the flatness problem (why Ω is so close to 1), and generates the primordial density perturbations that seed all cosmic structure.

The Slow-Roll Approximation

Field Dynamics

The inflaton evolves according to the equation of motion: φ̈ + 3Hφ̇ + V'(φ) = 0, where H is the Hubble parameter and V'(φ) is the potential's derivative. During slow-roll, the field velocity is small (φ̇² ≪ V), and the acceleration is negligible (φ̈ ≪ 3Hφ̇), so we have the approximate solution: 3Hφ̇ ≈ -V'(φ). The field "slowly rolls" down the potential with friction from cosmic expansion.

Slow-Roll Parameters

We quantify how slowly the field rolls using two parameters: ε = (Mₚₗ²/2)(V'/V)² and η = Mₚₗ²(V''/V). When ε ≪ 1 and |η| ≪ 1, slow-roll inflation occurs. Inflation ends when ε ≈ 1 (the field accelerates down the potential). The number of e-folds N = ln(a_end/a_start) measures the total expansion: N ≈ (1/Mₚₗ²) ∫ V/V' dφ. We need N ≳ 60 to solve the horizon and flatness problems.

Scale Factor Growth

During inflation, the Friedmann equation gives H² ≈ V(φ)/(3Mₚₗ²). Since V is roughly constant during slow-roll, H is nearly constant, and the scale factor grows exponentially: a(t) ∝ exp(Ht). This exponential expansion is what "inflates" the universe, stretching quantum fluctuations to cosmological scales and smoothing out initial inhomogeneities.

Observable Predictions

Scalar Spectral Index nₛ

Inflation predicts a nearly scale-invariant spectrum of density perturbations with spectral index nₛ = 1 - 6ε + 2η (evaluated at horizon crossing). The deviation from exact scale invariance (nₛ = 1) is a smoking gun for inflation. Planck satellite measurements give nₛ = 0.965 ± 0.004, confirming the prediction. Different potentials predict different values of nₛ.

Tensor-to-Scalar Ratio r

Inflation also produces gravitational waves with amplitude parameterized by r = 16ε. This is the ratio of tensor (gravitational wave) to scalar (density) perturbations. Large-field models (like quadratic) predict large r ≳ 0.01, while small-field models (like natural inflation) predict small r ≪ 0.01. Current upper limit is r < 0.036, ruling out some simple models. Detecting r via B-mode polarization in the CMB would be a triumph.

Types of Inflation Potentials

• Quadratic (m²φ²/2): The simplest chaotic inflation model. Predicts nₛ ≈ 0.97, r ≈ 0.13 for N=60, which is now disfavored by data (r too large). Historically important but ruled out in its simplest form.
• Quartic (λφ⁴/4): Another large-field model with even larger r ≈ 0.27, strongly disfavored. Shows how observational data constrains inflationary models—not all potentials are viable.
• Natural (Λ⁴(1-cos(φ/f))): Motivated by axion physics, this periodic potential has small r for sub-Planckian f. Fits current data well with nₛ ≈ 0.965, r ≪ 0.01. The field scale f controls the amplitude.
• Starobinsky-like (R² gravity): Arising from modified gravity, this gives nₛ ≈ 0.965, r ≈ 0.003, consistent with all observations. Currently a leading model. Shows inflation can emerge from quantum gravity corrections.

Things to Explore

1. Chaotic Inflation: Select "Quadratic" potential with initial field φᵢ = 15 Mₚₗ. Watch the field roll down from large values. Notice ε increases as the field descends—inflation ends when ε = 1. Check the predicted r ≈ 0.13, which is too large compared to observations.
2. Natural Inflation: Switch to "Natural" potential. This periodic potential has a gentler slope, giving smaller ε and thus smaller r. Adjust the field scale f—smaller f means tighter curvature and larger ε. This model can fit nₛ and r constraints.
3. Number of e-folds: Count the e-folds from the evolution plot. We need N ≳ 60 to solve the horizon problem. Try different initial field values— larger φᵢ gives more e-folds. This is why "large-field" inflation is attractive: plenty of expansion.
4. Slow-roll violation: Watch the slow-roll parameters ε and η. While ε ≪ 1, inflation continues. When ε approaches 1, slow-roll breaks down and inflation ends. The field then oscillates around the minimum, reheating the universe.
5. Potential shape matters: Compare quadratic vs quartic. Steeper potentials (quartic) have larger V'/V, hence larger ε and r. Flatter potentials (Starobinsky) have smaller ε and r. The data prefers flat potentials.

Why Inflation Matters

Inflation is our leading explanation for the universe's large-scale structure. Without it, we have no explanation for:

• CMB uniformity: Regions separated by 100+ degrees on the sky have the same temperature to 1 part in 10⁵, despite never being in causal contact without inflation.
• Spatial flatness: Ω_total = 1.000 ± 0.001. Without inflation, this requires incredibly fine-tuned initial conditions. Inflation dynamically drives any universe toward flatness.
• Density perturbations: Galaxies, clusters, and the CMB power spectrum all trace back to quantum fluctuations of the inflaton, stretched to cosmic scales by exponential expansion. This is the only mechanism we know that generates the observed nearly scale-invariant perturbations.
• Absence of monopoles: Grand unified theories predict magnetic monopoles with mass ~10¹⁶ GeV. We see none. Inflation dilutes them to unobservable densities by expanding the universe by a factor of e⁶⁰ or more.

Yet inflation remains incomplete. We don't know what the inflaton is—it could be a fundamental scalar field, a pseudo-Goldstone boson (axion), or an emergent degree of freedom from quantum gravity. We don't know what ended inflation (reheating). And we don't know how to embed inflation in a fundamental theory like string theory (the "eta problem"). Despite these mysteries, inflation has passed every observational test for 40 years. It is the cornerstone of modern cosmology, connecting the very small (quantum mechanics) to the very large (the observable universe).