Cosmic Inflation

Introduction

Cosmic inflation is a period of exponential expansion in the very early universe ($t \sim 10^{-36}$ to $10^{-32}$ s), proposed to solve fundamental problems with the standard Big Bang model and explain the origin of cosmic structure.

During inflation, the scale factor grows as $a(t) \propto e^{Ht}$, increasing by at least 60 e-folds:$N = \ln(a_{\text{end}}/a_{\text{start}}) \gtrsim 60$.

1. Problems Solved by Inflation

Horizon Problem

The CMB is isotropic to $\sim 10^{-5}$, yet causally disconnected regions at recombination have the same temperature. The particle horizon size at recombination:

$$d_H(t_{rec}) = a(t_{rec})\int_0^{t_{rec}} \frac{cdt}{a(t)} \sim 200 \text{ Mpc}$$

corresponds to only ~2° on the sky today. Inflation solves this by making the observable universe originate from a causally connected patch.

Flatness Problem

The density parameter evolves as:

$$\Omega(t) - 1 = \frac{\Omega_0 - 1}{\Omega_0}a^{-2}(t)$$

For $\Omega_0 \approx 1$ today, we need $|\Omega(t_P) - 1| \sim 10^{-60}$ at the Planck time—extreme fine-tuning! Inflation drives $\Omega \to 1$ exponentially fast.

Monopole Problem

Grand Unified Theories predict massive magnetic monopoles formed during phase transitions. Their relic density would be:

$$\frac{n_M}{n_\gamma} \sim 10^{-10}$$

Inflation dilutes monopoles to unobservable levels by expanding the universe exponentially.

Structure Formation

Quantum fluctuations during inflation are stretched to cosmic scales, seeding all structure in the universe. This provides a natural, causal mechanism for the origin of galaxies, clusters, and the CMB anisotropies.

2. Inflaton Field Dynamics

Action and Equation of Motion

The inflaton is a scalar field $\phi$ with action:

$$S = \int d^4x\sqrt{-g}\left[\frac{1}{2}g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi - V(\phi)\right]$$

In a flat FLRW spacetime, this gives the Klein-Gordon equation:

$$\ddot{\phi} + 3H\dot{\phi} + V'(\phi) = 0$$

The $3H\dot{\phi}$ term acts as Hubble friction, slowing the field's evolution.

Energy-Momentum Tensor

The stress-energy tensor components:

$$\rho_\phi = \frac{1}{2}\dot{\phi}^2 + V(\phi)$$
$$p_\phi = \frac{1}{2}\dot{\phi}^2 - V(\phi)$$

The equation of state parameter:

$$w_\phi = \frac{p_\phi}{\rho_\phi} = \frac{\dot{\phi}^2/2 - V(\phi)}{\dot{\phi}^2/2 + V(\phi)}$$

Inflation requires $w < -1/3$, achieved when potential energy dominates: $V(\phi) \gg \dot{\phi}^2/2$.

3. Slow-Roll Inflation

Slow-Roll Parameters

The slow-roll parameters quantify the flatness of the potential:

$$\epsilon_V = \frac{M_P^2}{2}\left(\frac{V'}{V}\right)^2$$
$$\eta_V = M_P^2\frac{V''}{V}$$

where $M_P = \sqrt{\hbar c/G} \approx 1.22 \times 10^{19}$ GeV is the Planck mass.

Slow-roll conditions: $\epsilon_V \ll 1$ and $|\eta_V| \ll 1$.

Slow-Roll Approximation

Under slow-roll, we approximate:

$$3H\dot{\phi} \approx -V'(\phi)$$
$$H^2 \approx \frac{8\pi G}{3}V(\phi) = \frac{V(\phi)}{3M_P^2}$$

Number of e-folds

The number of e-folds from field value $\phi$ to the end of inflation:

$$N(\phi) = \int_{t}^{t_{end}} H dt = \int_{\phi}^{\phi_{end}} \frac{H}{\dot{\phi}} d\phi \approx \frac{1}{M_P^2}\int_{\phi_{end}}^{\phi} \frac{V}{V'} d\phi$$

Observable scales left the horizon around $N \sim 50-60$ e-folds before the end of inflation.

4. Primordial Perturbations

Scalar Perturbations (Curvature)

Quantum fluctuations $\delta\phi$ generate curvature perturbations $\mathcal{R}$. The power spectrum:

$$\mathcal{P}_\mathcal{R}(k) = \left.\frac{1}{8\pi^2M_P^2}\frac{V}{\epsilon_V}\right|_{k=aH}$$

The amplitude is conventionally written:

$$\mathcal{P}_\mathcal{R}(k) = A_s\left(\frac{k}{k_*}\right)^{n_s - 1}$$

where the scalar spectral index is:

$$n_s - 1 = -6\epsilon_V + 2\eta_V$$

Planck 2018: $n_s = 0.9649 \pm 0.0042$ (slight red tilt, favoring $\eta_V < 0$).

Tensor Perturbations (Gravitational Waves)

Quantum fluctuations of the metric generate primordial gravitational waves:

$$\mathcal{P}_t(k) = \left.\frac{2}{\pi^2M_P^2}H^2\right|_{k=aH}$$

The tensor-to-scalar ratio:

$$r = \frac{\mathcal{P}_t}{\mathcal{P}_\mathcal{R}} = 16\epsilon_V$$

Current constraint: $r < 0.032$ (95% CL), implying $\epsilon_V < 0.002$. Detection of $r > 0$ would be smoking-gun evidence for inflation!

Consistency Relation

Single-field slow-roll inflation predicts:

$$r = -8n_t$$

where $n_t = -2\epsilon_V$ is the tensor spectral index. Violation would indicate multiple fields or non-standard inflation.

5. Inflationary Models

Chaotic Inflation

Simple monomial potential (Linde 1983):

$$V(\phi) = \frac{1}{2}m^2\phi^2 \quad \text{or} \quad V(\phi) = \frac{\lambda}{4}\phi^4$$

Predictions for quadratic potential:

  • $n_s \approx 1 - 2/N \approx 0.967$
  • $r \approx 8/N \approx 0.13$

The large $r$ prediction is disfavored by current data.

Starobinsky Inflation (R² Gravity)

Modified gravity with $f(R) = R + R^2/(6M^2)$ is equivalent to a scalar field with:

$$V(\phi) = \frac{3M^2M_P^2}{4}\left(1 - e^{-\sqrt{2/3}\phi/M_P}\right)^2$$

Predictions:

  • $n_s \approx 1 - 2/N \approx 0.967$
  • $r \approx 12/N^2 \approx 0.004$

Excellent agreement with Planck! Small $r$ makes detection challenging.

Natural Inflation

Pseudo-Nambu-Goldstone boson with periodic potential:

$$V(\phi) = \Lambda^4\left[1 + \cos\left(\frac{\phi}{f}\right)\right]$$

Protected by shift symmetry. Predictions depend on decay constant $f$.

Higgs Inflation

The Standard Model Higgs field as inflaton with non-minimal coupling:

$$S = \int d^4x\sqrt{-g}\left[\frac{1}{2}(M_P^2 + \xi h^2)R + \frac{1}{2}(\partial h)^2 - V(h)\right]$$

Large $\xi \sim 10^4$ flattens the potential at high energies, enabling inflation.

6. Reheating

Inflaton Decay

After inflation, the inflaton oscillates around the minimum and decays to Standard Model particles. For perturbative decay with coupling $g$:

$$\Gamma_\phi \sim \frac{g^2m_\phi}{8\pi}$$

Reheating Temperature

The universe reheats to temperature:

$$T_{rh} \sim \left(\Gamma_\phi M_P\right)^{1/2}$$

Constraints from Big Bang Nucleosynthesis require $T_{rh} \gtrsim 4$ MeV.

Preheating

Non-perturbative particle production via parametric resonance can be very efficient, with exponential growth:$n_\chi \propto e^{\mu t}$. This "preheating" can complete reheating much faster than perturbative decay.

7. Observational Signatures

CMB Angular Power Spectrum

Inflationary predictions tested against CMB $C_\ell$ measurements:

  • Nearly scale-invariant spectrum ($n_s \approx 0.96$)
  • Adiabatic perturbations (same phase for all components)
  • Gaussian statistics (non-Gaussianity $f_{NL} \lesssim 10$)

B-mode Polarization

Primordial gravitational waves produce B-mode polarization in the CMB:

$$C_\ell^{BB} \propto r \, A_s$$

Future experiments (LiteBIRD, CMB-S4) aim to detect $r \sim 10^{-3}$.

Non-Gaussianity

The bispectrum quantifies deviations from Gaussianity:

$$\langle\zeta_{k_1}\zeta_{k_2}\zeta_{k_3}\rangle = (2\pi)^3\delta^3(k_1+k_2+k_3) f_{NL} F(k_1,k_2,k_3)$$

Single-field slow-roll inflation predicts $f_{NL} \sim \mathcal{O}(0.01)$. Current constraint: $f_{NL}^{\text{local}} = -0.9 \pm 5.1$.

8. Open Questions and Alternatives

Theoretical Challenges

  • Trans-Planckian problem: Observable scales have $k/a > M_P$ early in inflation
  • Initial conditions: How does inflation start? Eternal inflation?
  • UV completion: Embedding in quantum gravity (string theory)
  • Fine-tuning: Flatness of $V(\phi)$ requires protection mechanism

Alternative Scenarios

  • Ekpyrotic/Cyclic Universe: Contraction phase before bounce, scale-invariant from kinetic energy
  • String Gas Cosmology: Winding modes stabilize extra dimensions
  • Variable Speed of Light: $c(t)$ varies in early universe to solve horizon problem

Despite alternatives, inflation remains the most successful paradigm for explaining the large-scale structure of the universe and CMB properties, with multiple independent lines of evidence supporting it.