Cosmic Inflation
Slow-roll exponential expansion — fixing horizon, flatness, and seeding structure.
Standard hot Big Bang has three puzzles:
- Horizon problem: the CMB is uniform to $10^{-5}$ across regions that were never in causal contact at recombination.
- Flatness problem: $|\Omega - 1|$ grows in time; finding $|\Omega_0 - 1| \lesssim 0.01$ today requires absurd fine-tuning of initial conditions.
- Monopole problem: GUTs predict copious magnetic monopoles; none are observed.
Inflation (Guth 1981; Linde, Albrecht, Steinhardt): a brief epoch of accelerated expansion driven by a slowly-rolling scalar field $\phi$ (the inflaton). With potential $V(\phi)$ and slow-roll parameters $\epsilon, \eta \ll 1$,
$$3 H \dot\phi \approx -V'(\phi), \qquad H^2 \approx \frac{V(\phi)}{3 M_{Pl}^2}.$$For $\dot\phi^2 \ll V$, $a(t) \approx e^{Ht}$ — quasi-de-Sitter. ~60 e-folds dilute monopoles to unobservable density, blow a tiny patch into our observable universe (uniformity), and drive $\Omega$ arbitrarily close to 1.
Vacuum quantum fluctuations of $\phi$ become classical density perturbations at horizon crossing — seeding the CMB anisotropies measured by COBE, WMAP, Planck. Predicted scalar spectrum $P(k) \propto k^{n_s - 1}$ with $n_s$ slightly below 1; observed $n_s = 0.9649 \pm 0.0042$ (Planck 2018). Tensor-to-scalar ratio $r$ probes the energy scale of inflation; not yet detected.