= Solution
<cosmic inflation> is an early interval of accelerated expansion, $\ddot R>0$. For expansion this means a shrinking <comoving Hubble radius>, $d[(RH)^{-1}]/dt<0$. A nearly constant positive <scalar potential> can supply it: in $c=1$ units,
$$
\rho_\phi=\frac{\dot\phi^2}{2}+V(\phi),\qquad
P_\phi=\frac{\dot\phi^2}{2}-V(\phi).
$$
Potential domination gives $\rho_\phi+3P_\phi<0$ and approximately exponential expansion. Sustaining the interval requires sufficiently slow evolution, not just setting the initial field velocity to zero on a steep potential.
The <horizon problem> is the near equality of conditions in widely separated microwave-background regions that could not equilibrate in the standard decelerating hot <Big Bang> history. <cosmic inflation> permits the currently observed patch to descend from an earlier region small enough to have been causally connected, before that patch is stretched to enormous size. The claim concerns a causal history extended before the conventional hot phase; merely accelerating a collection of already disconnected regions would not explain their common <temperature>.
For the <flatness problem>, the Friedmann constraint gives
$$
|\Omega_{\rm tot}-1|=\frac{|k|c^2}{R^2H^2}.
$$
A growing $RH$ drives this curvature fraction toward zero, whereas ordinary matter- or radiation-dominated deceleration makes it grow. <cosmic inflation> therefore suppresses observable curvature without requiring an equally extreme initial tuning within the later hot evolution.
Pre-existing monopoles and other unwanted relics are also diluted by the volume increase: a conserved particle number <density> falls as $e^{-3N}$ over $N$ e-folds. The transition to the hot universe must reheat efficiently, and its <temperature> must not regenerate an excluded relic population. The required number of e-folds is model and <reheating> dependent; “about sixty” is a useful scale for many scenarios rather than a universal exact requirement.
<cosmic inflation> also supplies a mechanism for perturbations. <quantum fluctuations> are stretched beyond the Hubble <radius>, and nearly scale-invariant <comoving curvature perturbations> can later seed cosmic structure. Simple single-field slow-roll models commonly predict approximately adiabatic, nearly Gaussian fluctuations, with a small spectral tilt; tensor perturbations would provide an additional test. Acoustic coherence and the observed scalar spectrum are compatible with this account, but they do not single out one <inflaton> potential. \b[<cosmic inflation> addresses causal homogeneity, curvature and relic dilution while motivating the initial perturbations; it does not automatically solve the <cosmological constant> problem or determine <reheating>.]
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