= Solution
<Cosmic inflation> is accelerated expansion with decreasing comoving Hubble radius, $\ddot R>0$ or $\epsilon_H=-\dot H/H^2<1$. It can make the observed region causally connected before inflation and dilute spatial curvature or unwanted relics. A slowly rolling scalar has energy and pressure $\rho_\phi=\dot\phi^2/2+V$, $P_\phi=\dot\phi^2/2-V$ in natural units. Potential domination gives near-exponential expansion; ending inflation and reheating must create the hot radiation era.
For a canonical single scalar with reduced Planck mass $M_{\rm Pl}$,
$$
\epsilon_V=\frac{M_{\rm Pl}^2}{2}(V'/V)^2\ll1,\qquad|\eta_V|=|M_{\rm Pl}^2V''/V|\ll1,
\qquad n_s-1\simeq-6\epsilon_V+2\eta_V,\quad r\simeq16\epsilon_V.
$$
Quantum fluctuations stretched to cosmological scales seed density perturbations. Tests include the scalar spectral tilt, acoustic coherence, spatial curvature, non-Gaussianity and tensor-induced CMB B-mode polarization. Foregrounds and lensing must be separated from primordial tensors. A tensor non-detection does not exclude all inflation; tilt and tensor consistency predictions depend on the model. Approximate flatness and near-scale-invariant fluctuations support the framework but do not uniquely determine a potential or prove an exact de Sitter phase.
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