Inflation is a period of accelerated expansion that helps explain the horizon, flatness, and monopole problems, while providing a mechanism for generating nearly scale-invariant primordial perturbations.

For a canonical scalar field $\phi$ with potential $V(\phi)$, the energy density and pressure are $$\rho_\phi = \frac{1}{2}\dot\phi^2 + V(\phi), \qquad p_\phi = \frac{1}{2}\dot\phi^2 - V(\phi).$$ Slow-roll occurs when $$\epsilon_V \equiv \frac{M_{\rm Pl}^2}{2}\left(\frac{V'}{V}\right)^2 \ll 1, \qquad \eta_V \equiv M_{\rm Pl}^2\frac{V''}{V} \ll 1.$$

Scalar and tensor perturbations are quantized on the inflationary background. Fluctuation modes begin as vacuum fluctuations, are stretched beyond the Hubble radius, and later re-enter to seed anisotropy and structure. The approximate scalar power spectrum is $$\mathcal P_{\mathcal R}(k) \simeq \frac{1}{8\pi^2 M_{\rm Pl}^2}\frac{H_*^2}{\epsilon_*}.$$

Research-grade inflation studies consider multi-field models, non-canonical kinetic terms, reheating dynamics, isocurvature perturbations, non-Gaussianity, and consistency relations. Observational constraints on $n_s$, $r$, and primordial non-Gaussianity strongly discriminate among model classes.