Problems with the Standard Hot Big Bang
The horizon problem, flatness problem, monopole problem, and the origin of perturbations: four puzzles that motivate cosmic inflation
Overview
The standard FLRW cosmology with radiation and matter is extraordinarily successful at explaining BBN, the CMB, and the Hubble expansion. However, it suffers from several fine-tuning and initial-condition puzzles that strongly suggest a period of accelerated expansion — inflation — preceded the hot Big Bang. We examine each problem quantitatively and show how inflation resolves all of them simultaneously.
1. The Horizon Problem
The CMB is isotropic to one part in \(10^5\) across the entire sky. Yet at the time of last scattering (\(z \approx 1100\)), the particle horizon subtended only about \(1^\circ\) on today's sky. Points separated by more than\(\sim 2^\circ\) were never in causal contact.
Quantitatively, the comoving particle horizon at decoupling is:
$$d_H(t_{\text{dec}}) = \int_0^{t_{\text{dec}}} \frac{dt}{a(t)} \approx \frac{2}{H_{\text{dec}}\,a_{\text{dec}}}$$
whereas the comoving distance to the last-scattering surface seen today is:
$$d_{\text{LSS}} = \int_{t_{\text{dec}}}^{t_0} \frac{dt}{a(t)} \approx \frac{3}{H_0\,a_0}$$
The ratio of the angular size of the horizon at decoupling to the full sky is:
Number of Causally Disconnected Patches
$$N_{\text{patches}} \sim \left(\frac{d_{\text{LSS}}}{d_H(t_{\text{dec}})}\right)^2 \sim 10^4$$
The CMB sky consists of roughly \(10^4\) causally disconnected regions, yet they all have the same temperature to within \(\Delta T/T \sim 10^{-5}\). Why?
2. The Flatness Problem
The first Friedmann equation can be rewritten as:
$$|\Omega(t) - 1| = \frac{|k|}{a^2 H^2}$$
During radiation domination \(aH \propto a^{-1}\) and during matter domination \(aH \propto a^{-1/2}\), so \(|\Omega - 1|\) grows with time. Working backward from the observed \(|\Omega_0 - 1| \lesssim 0.01\):
Fine-Tuning at the Planck Time
$$|\Omega(t_{\text{Pl}}) - 1| \lesssim 10^{-60}$$
The universe must have been flat to 60 decimal places at the Planck time. Without a dynamical mechanism, this is an extraordinary fine-tuning of initial conditions.
3. The Monopole Problem
Grand Unified Theories (GUTs) predict that topological defects — magnetic monopoles, cosmic strings, domain walls — form at phase transitions in the early universe. For monopoles produced at the GUT scale \(T_{\text{GUT}} \sim 10^{16}\) GeV:
$$n_M \sim \frac{T_{\text{GUT}}^3}{\xi^3} \sim T_{\text{GUT}}^3$$
where \(\xi\) is the correlation length, of order the horizon size at the transition. Each monopole has mass \(m_M \sim M_{\text{GUT}}/\alpha_{\text{GUT}} \sim 10^{17}\) GeV. The predicted monopole density today would vastly overclose the universe:
$$\Omega_M \sim 10^{11} \gg 1$$
No magnetic monopoles have ever been observed. Either GUTs are wrong, or something diluted the monopoles away.
4. The Origin of Primordial Perturbations
The standard hot Big Bang provides no mechanism for generating the nearly scale-invariant, Gaussian, adiabatic density perturbations observed in the CMB. These perturbations have a power spectrum:
$$\mathcal{P}_\mathcal{R}(k) = A_s \left(\frac{k}{k_*}\right)^{n_s - 1} \approx 2.1 \times 10^{-9}$$
with \(n_s \approx 0.965\), very close to (but not exactly) scale-invariant. These perturbations must have been laid down at very early times and on super-horizon scales. The hot Big Bang has no causal mechanism to produce them.
5. How Inflation Resolves All Four Problems
Inflation is a period of accelerated expansion, \(\ddot{a} > 0\), typically modeled as quasi-de Sitter: \(a \propto e^{Ht}\) with \(H \approx \text{const}\). During inflation the comoving Hubble radius shrinks:
Key Property of Inflation
$$\frac{d}{dt}\left(\frac{1}{aH}\right) < 0 \qquad \Longleftrightarrow \qquad \ddot{a} > 0$$
Horizon problem solved: Regions that appear causally disconnected today were in fact in causal contact before inflation stretched them apart. A minimum of \(N \sim 60\) e-folds of inflation is required:
$$N = \ln\frac{a_{\text{end}}}{a_{\text{start}}} \gtrsim 60$$
Flatness problem solved: During inflation\(|\Omega - 1| = |k|/(aH)^2\) decreases exponentially, driving the universe toward flatness regardless of initial conditions.
Monopole problem solved: Monopoles produced before or during inflation are diluted by the exponential expansion to negligible densities:\(n_M \propto a^{-3} \to 0\).
Perturbations explained: Quantum fluctuations of the inflaton field, stretched to macroscopic scales by expansion, become the seeds of all cosmic structure. We derive this in the next chapter.
6. The Necessary Conditions for Inflation
Accelerated expansion \(\ddot{a} > 0\) requires, from the second Friedmann equation:
$$\rho + 3p < 0 \quad \Longleftrightarrow \quad w < -\frac{1}{3}$$
No known form of matter or radiation satisfies this — ordinary matter has \(w = 0\)and radiation has \(w = 1/3\). A scalar field with potential energy dominating over kinetic energy naturally provides \(w \approx -1\). Equivalently, inflation requires the comoving Hubble radius to shrink, meaning modes that were once inside the horizon get pushed outside:
Shrinking Comoving Hubble Radius
$$\frac{d}{dt}\left(\frac{1}{aH}\right) = -\frac{\ddot{a}}{(aH)^2} < 0$$
This is the defining property of inflation: scales that are sub-Hubble before inflation become super-Hubble during inflation, freeze, and re-enter the horizon later during the standard decelerated expansion.