Table of Contents
- Introduction
- Problems in Standard Cosmology
- Motivation for Inflation
- The Inflationary Epoch
- Scalar Field Dynamics: The Inflaton
- Slow-Roll Conditions
- Quantum Fluctuations During Inflation
- Generation of Perturbations
- Scalar and Tensor Perturbations
- Horizon Crossing and Freezing
- Power Spectrum of Scalar Modes
- Scale Invariance and Tilt
- Tensor Power Spectrum
- Quantum Origin of Structure
- Quantum-to-Classical Transition
- Role of Decoherence
- Stochastic Inflation
- Eternal Inflation
- Reheating and End of Inflation
- Observational Signatures in CMB
- Non-Gaussianities and Higher-Order Effects
- Primordial Gravitational Waves
- Constraints from Planck and Other Experiments
- Open Problems in Inflationary Cosmology
- Conclusion
1. Introduction
Inflation is a period of accelerated expansion in the early universe, proposed to resolve several shortcomings of the standard Big Bang model. During inflation, quantum fluctuations in the inflaton field seeded the large-scale structure of the universe we observe today.
2. Problems in Standard Cosmology
The traditional Big Bang model faces several challenges:
- Horizon problem: CMB regions were never causally connected
- Flatness problem: Why is the universe spatially flat?
- Monopole problem: No relics predicted by GUTs are observed
3. Motivation for Inflation
Inflation solves these problems by introducing a phase of exponential expansion:
\[
a(t) \propto e^{Ht}
\]
This stretches space and smoothens out any inhomogeneities or curvature.
4. The Inflationary Epoch
Inflation typically occurs between \( 10^{-36} \) s and \( 10^{-32} \) s after the Big Bang. The universe expands by a factor of at least \( e^{60} \), setting the stage for the hot Big Bang.
5. Scalar Field Dynamics: The Inflaton
Inflation is driven by a scalar field \( \phi \) called the inflaton, with potential \( V(\phi) \). The dynamics are governed by:
\[
\ddot{\phi} + 3H\dot{\phi} + V'(\phi) = 0
\]
\[
H^2 = \frac{8\pi G}{3} \left( \frac{1}{2}\dot{\phi}^2 + V(\phi) \right)
\]
6. Slow-Roll Conditions
Inflation requires the potential energy to dominate over kinetic energy:
- \( \epsilon = \frac{M_{\text{Pl}}^2}{2} \left( \frac{V’}{V} \right)^2 \ll 1 \)
- \( \eta = M_{\text{Pl}}^2 \left( \frac{V”}{V} \right) \ll 1 \)
These ensure slow evolution and prolonged inflation.
7. Quantum Fluctuations During Inflation
Quantum fluctuations of \( \phi \) and the metric get stretched to macroscopic scales. These become classical density perturbations after horizon exit and re-entry.
8. Generation of Perturbations
Scalar perturbations arise from inflaton fluctuations \( \delta \phi \). These perturb spacetime via the Einstein equations, producing curvature perturbations \( \zeta \) on superhorizon scales.
9. Scalar and Tensor Perturbations
Two key modes:
- Scalar perturbations: curvature (density) perturbations
- Tensor perturbations: primordial gravitational waves
Both originate from vacuum fluctuations of fields during inflation.
10. Horizon Crossing and Freezing
Perturbations evolve inside the horizon as quantum oscillators. When they exit the Hubble radius \( k = aH \), they “freeze”, retaining their amplitude until re-entry.
11. Power Spectrum of Scalar Modes
The dimensionless power spectrum:
\[
\mathcal{P}_\zeta(k) = \left( \frac{H^2}{2\pi \dot{\phi}} \right)^2
\]
evaluated at horizon crossing. Nearly scale-invariant if \( H \) and \( \dot{\phi} \) vary slowly.
12. Scale Invariance and Tilt
Perfect scale invariance means equal power at all \( k \). Inflation predicts a tilted spectrum:
\[
n_s – 1 = -6\epsilon + 2\eta
\]
with observations giving \( n_s \approx 0.96 \), a slight red tilt.
13. Tensor Power Spectrum
Tensor mode power:
Characterized by tensor-to-scalar ratio:
14. Quantum Origin of Structure
Inflation explains how quantum vacuum fluctuations lead to the observed anisotropies in the CMB and formation of galaxies, clusters, and voids.
15. Quantum-to-Classical Transition
Mechanisms include:
- Squeezing: suppresses phase space uncertainty
- Decoherence: interaction with environment
- Classicalization: dominance of growing mode
These explain the emergence of classical density perturbations.
16. Role of Decoherence
Decoherence suppresses interference between different fluctuation modes, making them behave like classical stochastic variables — essential for understanding the classical universe.
17. Stochastic Inflation
Treats long-wavelength modes as a stochastic process influenced by short-wavelength quantum noise. Useful for modeling eternal inflation and landscape dynamics.
18. Eternal Inflation
In regions where quantum kicks dominate over classical roll, inflation never ends — leading to a multiverse of eternally inflating patches.
19. Reheating and End of Inflation
Inflation ends when \( \epsilon \sim 1 \). The inflaton decays into standard particles, reheating the universe and initiating the radiation-dominated era.
20. Observational Signatures in CMB
Inflation predicts:
- Gaussianity
- Nearly scale-invariant spectrum
- Flat geometry
- Tensor modes (yet undetected)
CMB observations strongly support these.
21. Non-Gaussianities and Higher-Order Effects
Non-Gaussianity probes interaction strength during inflation. Most models predict small levels (e.g., \( f_{\text{NL}} \ll 1 \)), consistent with observations.
22. Primordial Gravitational Waves
Predicted by inflation. Detected via B-mode polarization in the CMB. Detection would directly probe inflationary energy scale.
23. Constraints from Planck and Other Experiments
Planck data constrains:
- \( n_s \approx 0.9649 \)
- \( r < 0.07 \)
- Gaussianity consistent with zero
Future experiments (e.g., CMB-S4, LiteBIRD) aim to improve constraints.
24. Open Problems in Inflationary Cosmology
- Initial conditions for inflation
- Embedding in fundamental theory
- Alternatives to inflation
- Understanding the landscape and multiverse
25. Conclusion
Inflation provides a compelling framework for the early universe, explaining the smoothness, flatness, and structure we observe today. The quantum fluctuations during inflation act as seeds for cosmic structure, bridging quantum mechanics and cosmology. While many questions remain, inflationary cosmology continues to be refined by theory and experiment, offering deep insights into the origin of the universe.
