Slow-Roll Inflation
The inflaton equation of motion, slow-roll parameters, scalar and tensor perturbation spectra, the spectral index, and confrontation with Planck data
Overview
The simplest realisation of inflation uses a single scalar field \(\phi\) — the inflaton — slowly rolling down a flat potential \(V(\phi)\). The near-constancy of the potential energy drives quasi-exponential expansion. Quantum fluctuations of \(\phi\) generate an almost scale-invariant spectrum of curvature perturbations, while tensor fluctuations of the metric produce primordial gravitational waves. Both predictions are testable and have been spectacularly confirmed (scalars) or constrained (tensors) by the Planck satellite.
1. Inflaton Dynamics
The energy density and pressure of a homogeneous scalar field are:
$$\rho_\phi = \frac{1}{2}\dot{\phi}^2 + V(\phi), \qquad p_\phi = \frac{1}{2}\dot{\phi}^2 - V(\phi)$$
When the potential energy dominates (\(\dot{\phi}^2 \ll V\)), we have\(p \approx -\rho\), giving accelerated expansion. The equation of motion follows from the Klein-Gordon equation in the FRW background:
Inflaton Equation of Motion
$$\boxed{\ddot{\phi} + 3H\dot{\phi} + V'(\phi) = 0}$$
The Hubble friction term \(3H\dot{\phi}\) acts as a drag force, slowing the field as it rolls. Here \(V' = dV/d\phi\).
Combined with the Friedmann equation \(3H^2 M_{\text{Pl}}^2 = \rho_\phi\), this forms a closed system for \(\phi(t)\) and \(a(t)\).
2. The Slow-Roll Approximation
Inflation requires the field to roll slowly: \(|\ddot{\phi}| \ll |3H\dot{\phi}|\)and \(\dot{\phi}^2 \ll V\). These conditions are quantified by the slow-roll parameters:
Slow-Roll Parameters (Potential)
$$\boxed{\epsilon = \frac{M_{\text{Pl}}^2}{2}\left(\frac{V'}{V}\right)^2, \qquad \eta = M_{\text{Pl}}^2\,\frac{V''}{V}}$$
Slow-roll requires \(\epsilon \ll 1\) and \(|\eta| \ll 1\). Inflation ends when \(\epsilon(\phi_{\text{end}}) = 1\).
Under slow-roll, the equations simplify dramatically:
$$3H\dot{\phi} \approx -V', \qquad 3H^2 \approx \frac{V}{M_{\text{Pl}}^2}$$
3. Number of e-Folds
The number of e-folds of inflation between field values \(\phi_*\) (horizon crossing of the pivot scale) and \(\phi_{\text{end}}\) is:
$$N_* = \int_{\phi_{\text{end}}}^{\phi_*} \frac{H}{\dot{\phi}}\,d\phi \approx \frac{1}{M_{\text{Pl}}^2}\int_{\phi_{\text{end}}}^{\phi_*} \frac{V}{V'}\,d\phi$$
Solving the horizon and flatness problems requires \(N_* \sim 50\text{--}60\).
4. Scalar Perturbation Spectrum
Quantum fluctuations of the inflaton generate curvature perturbations \(\mathcal{R}\). At horizon crossing (\(k = aH\)), the power spectrum of these perturbations is:
Scalar Power Spectrum
$$\boxed{\Delta_s^2 = \frac{H^2}{8\pi^2 \epsilon\, M_{\text{Pl}}^2}\bigg|_{k=aH}}$$
The scalar spectral index, which measures the departure from exact scale invariance, is:
Spectral Index
$$\boxed{n_s - 1 = -6\epsilon + 2\eta}$$
Since \(\epsilon > 0\) during slow-roll, we generically predict \(n_s < 1\): a red-tilted spectrum with slightly more power on large scales.
5. Tensor Perturbation Spectrum
Quantum fluctuations of the metric tensor produce primordial gravitational waves with power spectrum:
Tensor Power Spectrum
$$\Delta_t^2 = \frac{2H^2}{\pi^2 M_{\text{Pl}}^2}\bigg|_{k=aH}$$
The tensor-to-scalar ratio is a direct measure of the energy scale of inflation:
Tensor-to-Scalar Ratio
$$\boxed{r = \frac{\Delta_t^2}{\Delta_s^2} = 16\epsilon}$$
Together with the consistency relation \(n_t = -r/8\), this provides a powerful test of single-field slow-roll inflation.
6. Confrontation with Planck Data
The Planck 2018 results provide precision measurements:
Planck 2018 Constraints
$$n_s = 0.9649 \pm 0.0042 \quad (68\%\;\text{C.L.})$$
$$r < 0.056 \quad (95\%\;\text{C.L.})$$
$$\ln(10^{10} A_s) = 3.044 \pm 0.014$$
The red tilt \(n_s < 1\) is detected at \(> 8\sigma\), confirming a key prediction of slow-roll inflation. The bound on \(r\)rules out monomial potentials \(V \propto \phi^n\) for \(n \geq 2\)and favours plateau-like models such as Starobinsky \(R^2\) inflation.
For the quadratic potential \(V = \frac{1}{2}m^2\phi^2\):\(n_s = 1 - 2/N_* \approx 0.967\) and \(r = 8/N_* \approx 0.13\), which is now ruled out by the Planck/BICEP bound. Starobinsky inflation gives\(n_s = 1 - 2/N_*\) and \(r = 12/N_*^2 \approx 0.004\), well within current limits.
7. The Lyth Bound and the Energy Scale of Inflation
The tensor-to-scalar ratio directly determines the inflaton field excursion during inflation. The Lyth bound relates \(r\) to the total field displacement:
Lyth Bound
$$\frac{\Delta\phi}{M_{\text{Pl}}} \approx \mathcal{O}(1)\times\left(\frac{r}{0.01}\right)^{1/2}$$
If \(r \gtrsim 0.01\), the inflaton traverses super-Planckian distances in field space (\(\Delta\phi > M_{\text{Pl}}\)), requiring UV-complete control over the inflaton potential — a major challenge for embedding inflation in string theory.
The energy scale of inflation is directly related to \(r\):
$$V^{1/4} = 1.06 \times 10^{16}\;\text{GeV}\left(\frac{r}{0.01}\right)^{1/4}$$
Detection of primordial gravitational waves would thus probe physics at the GUT scale — far beyond the reach of any terrestrial particle accelerator.