Cosmic inflation is the idea that the very early Universe underwent a phase of extremely rapid, nearly exponential
expansion. Instead of growing by an ordinary power law for a brief moment, the scale factor increased roughly like
\(a(t) \propto e^{Ht}\), where \(H\) stayed approximately constant during the inflationary stage. This simple behavior
has enormous consequences: even a tiny fraction of a second of exponential expansion can stretch microscopic regions into
patches much larger than the observable Universe today.
Exponential expansion
The defining inflation relation is
Scale-factor law.
\[
\begin{aligned}
a(t) &= a_i\,e^{Ht}
\end{aligned}
\]
Here \(a_i\) is the scale factor at the start of inflation, \(H\) is the inflationary Hubble parameter, and \(t\) is
time measured during the inflationary era. Because the exponential grows so quickly, it is convenient to describe the
total amount of inflation with the number of e-folds, written \(N\). It is defined by
E-fold definition.
\[
\begin{aligned}
N &= \ln\!\left(\frac{a_f}{a_i}\right)
\end{aligned}
\]
If \(H\) is approximately constant, then this becomes
Constant-H approximation.
\[
\begin{aligned}
N &= H\,\Delta t
\end{aligned}
\]
where \(\Delta t\) is the duration of inflation. Rearranging gives the total growth factor directly:
Total growth factor.
\[
\begin{aligned}
\frac{a_f}{a_i} &= e^N
\end{aligned}
\]
For the famous benchmark \(N=60\),
Standard benchmark.
\[
\begin{aligned}
\frac{a_f}{a_i} &= e^{60} \\
&\approx 1.14\times 10^{26}
\end{aligned}
\]
This means the Universe can expand by about twenty-six orders of magnitude during the inflationary interval.
Why inflation was proposed
Inflation was introduced to address several problems of the older non-inflationary Big Bang picture. Two of the most
famous are the horizon problem and the flatness problem.
The horizon problem asks why the cosmic microwave background looks so uniform in temperature across regions of the sky
that seem too far apart to have exchanged signals in an ordinary expanding model without inflation. Inflation answers this
by proposing that those regions were once much closer together, inside a single causally connected patch, before they were
stretched enormously apart.
The flatness problem asks why the present Universe appears so close to spatial flatness, meaning why the density
parameter \(\Omega\) is so close to \(1\). During inflation, deviations from flatness are rapidly driven down. In a
simple qualitative treatment, one often writes
Flatness suppression.
\[
\begin{aligned}
\left|\Omega-1\right|_f &\approx \left|\Omega-1\right|_i\,e^{-2N}
\end{aligned}
\]
So even a moderate increase in \(N\) produces an enormous suppression. For \(N=60\),
Suppression for 60 e-folds.
\[
\begin{aligned}
\frac{\left|\Omega-1\right|_f}{\left|\Omega-1\right|_i}
&\approx e^{-120} \\
&\approx 7.67\times 10^{-53}
\end{aligned}
\]
This is why inflation is often described as flattening the Universe dynamically.
Comoving Hubble radius and the horizon problem
Another useful concept is the comoving Hubble radius \((aH)^{-1}\). In ordinary decelerating expansion it tends to grow,
but during inflation it shrinks. In the same qualitative constant-\(H\) approximation,
Comoving Hubble-radius shrinkage.
\[
\begin{aligned}
\frac{(aH)^{-1}_f}{(aH)^{-1}_i}
&\approx e^{-N}
\end{aligned}
\]
This means physical scales that were once well inside the causal horizon can be pushed far outside it during inflation,
then later re-enter during the radiation- and matter-dominated eras. That is the main geometric reason inflation can
explain the smoothness of the observable Universe.
Worked benchmark: \(N=60\)
If the number of e-folds is \(N=60\), then the most important benchmark quantities are
Growth factor.
\[
\begin{aligned}
\frac{a_f}{a_i} &= e^{60} \\
&\approx 1.14\times 10^{26}
\end{aligned}
\]
Base-10 growth measure.
\[
\begin{aligned}
\log_{10}\!\left(\frac{a_f}{a_i}\right)
&= \frac{60}{\ln 10} \\
&\approx 26.1
\end{aligned}
\]
Flatness suppression.
\[
\begin{aligned}
\frac{\left|\Omega-1\right|_f}{\left|\Omega-1\right|_i}
&\approx e^{-120} \\
&\approx 7.67\times 10^{-53}
\end{aligned}
\]
These values show why \(N\approx 60\) is so often quoted in introductory cosmology: it is the order of magnitude
typically needed for the standard qualitative solution of the horizon and flatness problems.
Advanced note
At university level, one goes beyond the constant-\(H\) approximation and studies inflation with a scalar field,
usually called the inflaton. The field rolls slowly in a potential \(V(\phi)\), and the details are described by
slow-roll parameters such as \(\epsilon\) and \(\eta\). In that more complete treatment, \(H\) is not
perfectly constant, the precise number of e-folds depends on the reheating history, and one can connect inflation to
primordial density fluctuations and the observed cosmic microwave background spectrum.
Still, the constant-\(H\), exponential-growth model is the right first step because it captures the essential physics:
inflation blows up tiny patches, suppresses curvature, and makes the observable Universe easier to understand as the
descendant of a once-causally connected region.
| Concept |
Main relation |
Meaning |
| Exponential growth |
\(a(t) = a_i e^{Ht}\) |
Approximate scale-factor behavior during inflation |
| E-folds |
\(N = \ln(a_f/a_i)\) |
Logarithmic measure of total inflationary expansion |
| Constant-H relation |
\(N = H\Delta t\) |
E-folds from rate and duration |
| Total growth |
\(a_f/a_i = e^N\) |
Total scale-factor increase |
| Flatness suppression |
\(|\Omega-1|_f \approx |\Omega-1|_i e^{-2N}\) |
Qualitative reduction of spatial-curvature deviations |
| Horizon-resolution factor |
\((aH)^{-1}_f/(aH)^{-1}_i \approx e^{-N}\) |
Shrinkage of the comoving Hubble radius during inflation |