= Solution
<cosmic inflation> is an expanding epoch with $\ddot a>0$, equivalently a shrinking <comoving Hubble radius> $(aH)^{-1}=1/\dot a$. In <general relativity> it requires $\rho+3P/c^2<0$ for the total effective fluid. A <scalar field> whose <potential energy> dominates its <kinetic energy> has $P\simeq-\rho c^2$ and can produce approximately exponential expansion. <slow-roll inflation> is one realization; rapid expansion alone without acceleration does not have the required effect.
The <horizon problem> is that widely separated regions of the <cosmic microwave background> have nearly the same <temperature> despite lying outside one another's causal domains in a purely decelerating hot <Big Bang> extrapolation. Sufficient inflation stretches a previously small causally connected region across our observable universe; relevant perturbation <wavelengths> can begin inside the <Hubble radius>, leave it during inflation and re-enter later. This does not require local signals to outrun <light>. Inflation supplies a possible earlier causal history, not a proof that every possible initial state becomes homogeneous.
For the <flatness problem>, the <Friedmann equation> gives $\Omega_{\rm tot}-1=kc^2/(aH)^2$ when $a$ is length-valued. Since $aH$ grows during inflation,
$$
\boxed{|\Omega_{\rm tot}-1|\ \hbox{decreases during inflation}}.
$$
Pre-existing unwanted massive relics such as <magnetic monopoles> are also diluted: conserved <number density> falls as $a^{-3}$. Enough inflation, followed by <reheating> that does not recreate the relic problem, can make their abundance negligible. These were central motivations of the original proposal, whose first phase-transition implementation also faced a graceful-exit problem.<Inflationary universe: A possible solution to the horizon and flatness problems by Alan H. Guth>[Abstract: horizon, flatness, monopole suppression and limitations of the original scenario]{ref}
<quantum fluctuations> of the inflating field and metric supply seeds for <linear cosmological density perturbations>. Simple single-field slow-roll models predict a dominant <adiabatic mode> with nearly Gaussian, nearly scale-invariant <scalar cosmological perturbations>, consistent with the <cosmic microwave background>; <tensor cosmological perturbations> are another possible test. These predictions are model-dependent, and consistency is not unique proof of inflation. <reheating> must connect the accelerated phase to the hot expanding universe. \b[Inflation addresses causal reach, small <cosmological spatial curvature> and unwanted relic abundance, while providing a mechanism for primordial structure.]
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