Cosmic Inflation

The hot Big Bang model, run forward from nucleosynthesis, matches the CMB and the light-element abundances in quantitative detail. Run backward past those epochs, it requires initial conditions that are finely tuned and, within the standard expansion, causally inexplicable. Three of these — the uniform temperature of regions that were never in contact, the near-perfect spatial flatness, and the absence of the magnetic monopoles that grand unified theories predict — are resolved together by a single mechanism: a brief phase of accelerated expansion in the very early universe. Inflation also predicts the spectrum of primordial density perturbations, connecting the microphysics of a scalar field to the large-scale structure of the universe. This lesson states the three problems, derives the inflationary solution and its slow-roll dynamics, and traces the generation of the perturbation spectrum.

The horizon problem

The particle horizon is the maximum comoving distance light could have traveled since the beginning,

In a radiation- or matter-dominated universe this integral converges, so at any finite time the horizon is finite: only regions within $d_H$ of each other could ever have exchanged a signal, and hence come to a common temperature.

The CMB presents a difficulty. Photons arriving from opposite directions on the sky last scattered on the surface of last scattering, at a comoving distance equal to the present horizon. But at the time of last scattering the horizon subtended only about $1^{\circ }$ on the sky. Two CMB patches separated by more than a degree were outside each other's horizon at recombination — they had never been in causal contact — yet they share the same temperature to one part in $10^5$.

The flatness and monopole problems

Flatness. The Friedmann equation gives the deviation of the total density from critical in terms of the curvature,

The curvature $k$ is a fixed constant, so the entire time-dependence of $|\Omega -1|$ lives in the denominator ${\dot{a}}^2=(aH{)}^2$: as this shrinks, $|\Omega -1|$ grows in exact proportion. And in a decelerating universe $\dot{a}$ does shrink — during radiation domination $(aH{)}^2\propto a^{-2}$ and during matter domination $(aH{)}^2\propto a^{-1}$, so in both eras the denominator falls and $|\Omega -1|$ is driven upward. Spatial flatness ($\Omega =1$, i.e. $k=0$) is therefore an unstable fixed point: the exactly-flat case stays flat, but any nonzero curvature is amplified away from unity as the universe expands.

Read backward, this makes flatness today a fine-tuning puzzle. For $\Omega$ to lie within $0.005$ of unity now, it had to be tuned to within about $10^{-16}$ of unity at nucleosynthesis and $10^{-60}$ at the Planck time. The hot Big Bang alone offers no reason for a starting point so precisely balanced; a universe born with any generic curvature would have recollapsed, or diluted to emptiness, long before reaching an age of billions of years.

Monopoles. Grand unified theories predict that the symmetry breaking at $k_{B}T\sim 10^{16}\text{GeV}$ produces topological defects, including magnetic monopoles, at roughly one per horizon volume. These are superheavy, $\sim 10^{16}\text{GeV}/c^2$, and would dilute only as $a^{-3}$; their predicted abundance would dominate the energy density of the universe by many orders of magnitude. No monopole has ever been observed. The standard cosmology has no mechanism to remove them.

The three problems share a structure: each is a statement about initial conditions that the standard expansion preserves or worsens rather than explains.

The inflationary solution

All three problems are solved by an epoch of accelerated expansion, $\ddot{a}>0$, in the very early universe. The acceleration equation,

gives $\ddot{a}>0$ when $P<-\frac{1}{3}\rho c^2$, a fluid with strongly negative pressure. A cosmological-constant-like component, $P=-\rho c^2$, drives exponential (de Sitter) expansion,

for as long as it dominates. The number of e-folds of expansion is $N=\int H\mathbf{d}t=\ln (a_{\text{end}}/a_{\text{start}})$.

Each problem dissolves:

• Horizon: exponential growth stretches a tiny, causally connected patch to a size larger than the entire observable universe. The whole CMB sky descends from one pre-inflationary patch that had reached thermal equilibrium, so its uniformity is inherited, not coincidental.
• Flatness: during inflation $a^2{H}^2=a^2{H}_{\text{infl}}^2$ grows exponentially, so $|\Omega -1|\propto (aH{)}^{-2}$ is driven toward zero. Whatever the initial curvature, inflation flattens it; a small patch of any curved surface looks flat when magnified enormously.
• Monopoles: the exponential expansion dilutes any pre-existing monopoles to a density far below one per observable volume. Provided reheating after inflation does not exceed the GUT temperature, no new monopoles are produced.

Solving the horizon and flatness problems requires the observable universe to have been inside the horizon at the start of inflation, which needs

e-folds of expansion. This is the standard benchmark for a viable inflationary model.¹

The inflaton and slow roll

Inflation is driven by a scalar field $\varphi$, the inflaton, with a potential $V(\varphi )$. Its energy density and pressure are

When the kinetic term is small compared with the potential, ${\dot{\varphi }}^2\ll V(\varphi )$, the pressure approaches $P\approx -\rho c^2$, the equation of state that drives inflation. The field equation in an expanding universe,

resembles a ball rolling down the potential $V(\varphi )$ with a friction term $3H\dot{\varphi }$ set by the expansion (Hubble friction). For inflation to last many e-folds the field must roll slowly, so that $V$ stays nearly constant and $H$ nearly fixed. This is the slow-roll regime, in which $\ddot{\varphi }$ is negligible and the field equation reduces to $3H\dot{\varphi }\approx -\mathbf{d}V/\mathbf{d}\varphi$.

Slow roll is quantified by two dimensionless parameters built from the potential,

with $M_{\text{Pl}}=(\hslash c/8\pi G{)}^{1/2}$ the reduced Planck mass and primes denoting $\mathbf{d}/\mathbf{d}\varphi$. Inflation requires $ϵ\ll 1$ and $|{\eta }_V|\ll 1$: a flat potential. It ends when the field reaches the steep part of the potential and $ϵ\to 1$, at which point the kinetic energy becomes comparable to the potential, the accelerated expansion stops, and the inflaton's energy is transferred to a hot bath of particles in reheating, setting the initial conditions for the standard hot Big Bang.

Quantum fluctuations and the perturbation spectrum

Inflation does more than smooth the universe; it seeds the structure within it. During inflation the inflaton has quantum fluctuations $\delta \varphi$ on all scales. A given comoving wavelength starts inside the Hubble radius, where it oscillates as a normal quantum mode, and is then stretched by the exponential expansion until it exceeds the Hubble radius — the mode exits the horizon. Outside the horizon the fluctuation can no longer evolve causally; its amplitude freezes.

Because $H$ is nearly constant during inflation, every mode exits the horizon with approximately the same amplitude, $\delta \varphi \sim H/2\pi$. The fluctuations in the field translate into fluctuations in the time at which inflation ends locally, and hence into density perturbations when the modes later re-enter the horizon. The result is a near-scale-invariant spectrum of curvature perturbations,

Exact scale invariance, $n_s=1$ (the Harrison-Zel'dovich spectrum), would follow from strictly constant $H$; the small slow-roll corrections make $n_s$ slightly less than one. The measured value $n_s=0.965$ is the mild red tilt inflation predicts, and it is one of the observational successes of the framework.

The amplitude of the curvature perturbation follows from the field fluctuation divided by the rate at which the field crosses field values,

evaluated at horizon exit. The measured amplitude $A_s\approx 2\times 10^{-9}$ therefore constrains the combination $H^2/\dot{\varphi }$ during inflation, and through the slow-roll relations it ties the energy scale of inflation to the observed $10^{-5}$ CMB fluctuations. A larger $H$ (higher inflation scale) produces larger fluctuations, so the observed amplitude caps the inflationary energy scale below about $10^{16}\text{GeV}$.

Inflation also stretches quantum fluctuations of the spacetime metric itself into a background of primordial gravitational waves, with a spectrum characterized by the tensor-to-scalar ratio $r=16ϵ$. These tensor modes would imprint a curl (B-mode) pattern on the CMB polarization. No primordial B-modes have been detected; the current bound $r<0.06$ rules out the steepest inflaton potentials and is the target of ongoing polarization experiments.

The predictions that distinguish inflation from a merely assumed set of initial conditions are a spatially flat universe (${\Omega }_{\text{tot}}=1$), a near-scale-invariant spectrum ($n_s$ slightly below 1), and Gaussian, adiabatic perturbations correlated on scales larger than the last-scattering horizon. The first two are confirmed by the CMB acoustic peaks; the frozen perturbations are the initial conditions for the growth of structure treated next.