---
title: Cosmic Inflation
draft: false
module: The Hot Big Bang
moduleNumber: 12
lessonNumber: 5
order: 1205
summary: >
  The hot Big Bang leaves three initial-condition puzzles unexplained: why causally
  disconnected patches share a temperature, why the geometry is so nearly flat, and
  why no magnetic monopoles are seen. A brief epoch of accelerated expansion driven
  by a slowly rolling scalar field solves all three by stretching a small causal
  patch across the observable universe. The same accelerated expansion freezes
  quantum fluctuations into a near-scale-invariant spectrum of density perturbations,
  seeding all later structure.
topics: [The Hot Big Bang]
sources:
  - book: Ryden
    ref: "Ch. 10 — Inflation and the Very Early Universe"
  - book: Carroll & Ostlie
    ref: "Ch. 30 — The Early Universe; §30.4"
  - book: Guth 1981
    ref: "A. Guth, Inflationary universe, Phys. Rev. D 23, 347 (1981)"
---

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,

$$
d_H(t) = a(t)\int_0^t \frac{c\,\d t'}{a(t')} .
$$

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 $10^5$.

$$
% caption: At last scattering the causal horizon subtended only about one degree,
% so patches of the CMB separated by more than that were never in contact; the hot
% Big Bang offers no reason for their temperatures to match.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% observer
\fill[black!70] (0,0) circle (2pt);
\node[black!70, anchor=north] at (0,-0.15) {observer};
% last-scattering arc — radius = observer-to-patch distance (2.55*sqrt2 ~ 3.606),
% so the arc passes through the CENTRE of each patch circle
\draw[very thick] (-3.123,1.803) arc (150:30:3.6062);
\node[anchor=south] at (0,3.75) {surface of last scattering};
% two patches — their centres lie on the arc
\fill[acc!22] (-2.55,2.55) circle (0.35);
\fill[acc!22] (2.55,2.55) circle (0.35);
% labels pushed outside the dashed horizon circles (radius 0.6) to avoid overlap
\node[black!70, anchor=east] at (-3.4,2.55) {patch A};
\node[black!70, anchor=west] at (3.4,2.55) {patch B};
% sightlines
\draw[black] (0,0) -- (-2.55,2.55);
\draw[black] (0,0) -- (2.55,2.55);
% little horizon circles at each patch
\draw[black, dashed] (-2.55,2.55) circle (0.6);
\draw[black, dashed] (2.55,2.55) circle (0.6);
\node[black, anchor=north] at (0,-0.7) {patches never in causal contact};
\end{tikzpicture}
$$

## The flatness and monopole problems

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

$$
\Omega(t) - 1 = \frac{kc^2}{a^2 H^2} = \frac{kc^2}{\dot a^2} .
$$

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,

$$
\frac{\ddot a}{a} = -\frac{4\pi G}{3}\left(\rho + \frac{3P}{c^2}\right),
$$

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

$$
a(t) \propto e^{Ht}, \qquad H = \text{const},
$$

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

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

$$
N \gtrsim 60
$$

e-folds of expansion. This is the standard benchmark for a viable inflationary
model.[^guth]

> **Worked example.** Track the flatness parameter through inflation. During a de
> Sitter phase $H$ is constant and $a \propto e^{Ht}$, so
> $|\Omega - 1| = kc^2/(a^2 H^2) \propto e^{-2Ht} = e^{-2N}$ after $N$ e-folds. To
> reach the required $|\Omega - 1| \lesssim 10^{-60}$ at the end of inflation
> starting from an order-unity initial curvature,
> $$
> e^{-2N} \lesssim 10^{-60}
> \quad\Longrightarrow\quad
> N \gtrsim 30\ln 10 \approx 69.
> $$
> About seventy e-folds — a linear expansion factor of $e^{70} \approx 10^{30}$ —
> drives any initial curvature below the observable threshold and simultaneously
> stretches a sub-horizon patch across the present Hubble volume. The horizon and
> flatness problems require essentially the same number of e-folds, which is why a
> single episode solves both.

$$
% caption: Exponential expansion during inflation flattens the geometry and
% dilutes relics; a small curved, causally connected patch is magnified until its
% curvature is undetectable and it fills the observable universe.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% small curved patch
\draw[acc, very thick] (0.4,1.0) arc (210:330:1.2);
\node[acc, anchor=north] at (1.4,0.55) {small curved patch};
% magnification arrow
\draw[->, black, very thick] (3.0,1.4) -- (4.6,1.4);
\node[black, anchor=south] at (3.8,1.45) {expansion};
% large nearly flat patch
\draw[acc, very thick] (5.0,1.4) arc (255:285:9.0);
\node[acc, anchor=north] at (7.0,0.85) {stretched smooth};
\node[black, anchor=south] at (7.0,1.7) {curvature driven to zero};
\end{tikzpicture}
$$

## The inflaton and slow roll

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

$$
\rho c^2 = \tfrac{1}{2}\dot\phi^2 + V(\phi),
\qquad
P = \tfrac{1}{2}\dot\phi^2 - V(\phi).
$$

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

$$
\ddot\phi + 3H\dot\phi + \frac{\d V}{\d\phi} = 0,
$$

resembles a ball rolling down the potential $V(\phi)$ with a friction term
$3H\dot\phi$ 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 $\ddot\phi$ is negligible
and the field equation reduces to $3H\dot\phi \approx -\d V/\d\phi$.

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

$$
\epsilon = \frac{M_{\text{Pl}}^2}{2}\left(\frac{V'}{V}\right)^2,
\qquad
\eta_V = M_{\text{Pl}}^2\,\frac{V''}{V},
$$

with $M_{\text{Pl}} = (\hbar c/8\pi G)^{1/2}$ the reduced Planck mass and primes
denoting $\d/\d\phi$. Inflation requires $\epsilon \ll 1$ and $|\eta_V| \ll 1$: a
flat potential. It ends when the field reaches the steep part of the potential and
$\epsilon \to 1$, 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.

$$
% caption: A slow-roll inflaton potential. The field rolls slowly across the flat
% plateau, driving inflation with nearly constant energy density, then falls into
% the minimum where reheating converts its energy into a hot particle bath.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (9.4,0) node[right, black!70] {scalar coordinate};
\draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {potential};
% flat plateau then drop into minimum then rise
\draw[acc, very thick] (0.5,3.6) .. controls (2.6,3.55) and (3.6,3.45) .. (4.6,3.1)
  .. controls (5.6,2.6) and (6.0,0.6) .. (6.8,0.5)
  .. controls (7.5,0.45) and (8.0,1.3) .. (8.6,2.2);
% ball on the plateau
\fill[black!70] (2.4,3.55) circle (3pt);
\draw[->, black, very thick] (2.6,3.7) -- (3.6,3.55);
\node[black!70, anchor=south] at (2.0,3.7) {slow roll};
\node[acc, anchor=south] at (1.2,3.65) {level plateau};
% minimum
\draw[black, densely dotted] (6.8,0) node[below, black!70] {reheating} -- (6.8,0.5);
\node[black, anchor=west] at (6.9,1.2) {oscillate and reheat};
\end{tikzpicture}
$$

## 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 $\delta\phi$ 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, $\delta\phi \sim H/2\pi$. 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,

$$
\mathcal{P}_{\mathcal{R}}(k) = A_s\left(\frac{k}{k_0}\right)^{n_s - 1},
\qquad
n_s - 1 = 2\eta_V - 6\epsilon .
$$

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,

$$
\mathcal{R} \sim \frac{H\,\delta\phi}{\dot\phi} \sim \frac{H^2}{\dot\phi},
$$

evaluated at horizon exit. The measured amplitude $A_s \approx 2 \times 10^{-9}$
therefore constrains the combination $H^2/\dot\phi$ 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 $10^{16}\ \text{GeV}$.

$$
% caption: A comoving mode is stretched by inflation until its wavelength exceeds
% the Hubble radius; its amplitude then freezes, and the near-constant expansion
% rate imprints nearly equal power on every scale.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (9.4,0) node[right, black!70] {time};
\draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {comoving scale (log)};
% Hubble radius: shrinks during inflation (comoving)
\draw[black, very thick, dashed] (0.4,3.8) .. controls (2.5,2.4) and (4.0,1.2) .. (6.0,0.6);
\node[black, anchor=west] at (4.4,0.9) {Hubble radius};
% two modes: horizontal lines (fixed comoving scale) that cross the Hubble radius
\draw[acc, very thick] (0.4,2.6) -- (8.8,2.6);
\draw[black, very thick] (0.4,1.5) -- (8.8,1.5);
% exit markers
\fill[black!70] (2.1,2.6) circle (2pt);
\fill[black!70] (3.7,1.5) circle (2pt);
\node[black!70, anchor=south] at (2.1,2.65) {exit};
\node[black!70, anchor=south] at (3.7,1.55) {exit};
\node[acc, anchor=west] at (6.4,2.75) {frozen mode};
\end{tikzpicture}
$$

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** $r = 16\epsilon$. 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](/astrophysics-cosmology/the-hot-big-bang/cmb-anisotropies-and-cosmological-parameters);
the frozen perturbations are the initial conditions for the [growth of
structure](/astrophysics-cosmology/the-hot-big-bang/structure-formation-and-the-growth-of-perturbations)
treated next.

[^guth]: A. Guth, "The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems," Phys. Rev. D 23, 347 (1981) — the original statement of the horizon, flatness, and monopole problems and their resolution by a de Sitter phase. https://doi.org/10.1103/PhysRevD.23.347