 Solution

Cosmic inflation is accelerated expansion with decreasing comoving Hubble radius, $\ddot{R}>0$ or ${ϵ}_H=-\dot{H}/H^2<1$. 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 ${\rho }_{\varphi }={\dot{\varphi }}^2/2+V$, $P_{\varphi }={\dot{\varphi }}^2/2-V$ 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 $M_{\mathrm{Pl}}$,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.