Quantum Cosmology

Table of Contents

1. Introduction
2. Motivation for Quantum Cosmology
3. Classical Cosmology and General Relativity
4. Singularities and the Big Bang
5. Quantum Gravity and Early Universe
6. Wheeler–DeWitt Equation in Cosmology
7. Minisuperspace Approximation
8. Canonical Quantization of Cosmological Models
9. Quantum States of the Universe
10. Boundary Conditions: No-Boundary and Tunneling Proposals
11. Hartle–Hawking No-Boundary Proposal
12. Vilenkin’s Tunneling Proposal
13. Quantum Fluctuations and Inflation
14. Quantum-to-Classical Transition
15. Decoherence in the Early Universe
16. Quantum Initial Conditions
17. Loop Quantum Cosmology (LQC)
18. The Big Bounce Scenario
19. Discrete Quantum Geometry in LQC
20. Effective Dynamics and Phenomenology
21. Observational Consequences and CMB
22. Singularity Resolution in Quantum Cosmology
23. Multiverse and Quantum Cosmology
24. Open Problems and Interpretations
25. Conclusion

1. Introduction

Quantum cosmology applies the principles of quantum mechanics to the universe as a whole, especially its earliest moments. It aims to understand the birth, evolution, and fundamental structure of the cosmos using quantum gravity.

2. Motivation for Quantum Cosmology

• General relativity predicts singularities, where physical quantities diverge.
• Quantum effects are expected to become significant at the Planck scale.
• A quantum treatment of spacetime is necessary to explain the origin of the universe and initial conditions.

3. Classical Cosmology and General Relativity

In classical cosmology, the universe is modeled using the Friedmann–Lemaître–Robertson–Walker (FLRW) metric:

\[
ds^2 = -dt^2 + a(t)^2 \left( \frac{dr^2}{1 – kr^2} + r^2 d\Omega^2 \right)
\]

The Friedmann equations govern the scale factor \( a(t) \):

\[
\left( \frac{\dot{a}}{a} \right)^2 = \frac{8\pi G}{3} \rho – \frac{k}{a^2}
\]

4. Singularities and the Big Bang

Classical solutions imply a singularity at \( a = 0 \), the Big Bang. Here, energy density, curvature, and temperature become infinite — indicating a breakdown of classical physics.

5. Quantum Gravity and Early Universe

Quantum cosmology seeks to resolve this using quantum gravity, incorporating quantum effects into the geometry and dynamics of spacetime at early times.

6. Wheeler–DeWitt Equation in Cosmology

Derived from canonical quantum gravity, the Wheeler–DeWitt equation:

\[
\hat{H} \Psi[h_{ij}, \phi] = 0
\]

describes the quantum state \( \Psi \) of the universe. In minisuperspace (homogeneous models), this reduces to a simpler partial differential equation in variables like \( a \) and \( \phi \) (scalar field).

7. Minisuperspace Approximation

To make the problem tractable, only a few degrees of freedom (e.g., \( a(t), \phi(t) \)) are quantized. The wavefunction \( \Psi(a, \phi) \) satisfies a Schrödinger-like equation with no external time parameter.

8. Canonical Quantization of Cosmological Models

Quantization proceeds by:

• Defining a Hamiltonian constraint
• Promoting variables to operators
• Imposing \( \hat{H} \Psi = 0 \)

This yields a timeless wavefunction — a hallmark of quantum cosmology.

9. Quantum States of the Universe

The solution \( \Psi(a, \phi) \) encodes all possible universes. Different interpretations (e.g., many-worlds, consistent histories) attempt to make sense of this quantum state.

10. Boundary Conditions: No-Boundary and Tunneling Proposals

To uniquely define \( \Psi \), boundary conditions must be specified. Two major proposals are:

• No-boundary (Hartle–Hawking)
• Tunneling (Vilenkin)

11. Hartle–Hawking No-Boundary Proposal

Suggests the universe “tunnels” from nothing, with Euclidean (imaginary time) geometry:

\[
\Psi(a) = \int \mathcal{D}[g] \, e^{-S_E[g]}
\]

This yields a smooth beginning without singularity — the universe has no initial boundary in time.

12. Vilenkin’s Tunneling Proposal

The universe originates via quantum tunneling from a “nothing” state. This selects an outgoing wavefunction that describes an expanding universe.

13. Quantum Fluctuations and Inflation

Quantum fluctuations of the inflaton field during inflation are amplified, seeding the cosmic microwave background (CMB) anisotropies and large-scale structure of the universe.

14. Quantum-to-Classical Transition

After inflation, these fluctuations become classical. The mechanism involves:

• Squeezing of quantum states
• Decoherence from interaction with the environment
• Emergence of classical perturbations

15. Decoherence in the Early Universe

Decoherence explains how superpositions collapse into definite outcomes. In cosmology, it helps explain why quantum fluctuations appear as classical density perturbations in the CMB.

16. Quantum Initial Conditions

Quantum cosmology provides natural candidates for initial conditions:

• Specific form of the wavefunction
• Predictive probabilities for inflation, curvature, etc.

17. Loop Quantum Cosmology (LQC)

A symmetry-reduced version of Loop Quantum Gravity:

• Replaces Big Bang with Big Bounce
• Discrete quantum geometry modifies Friedmann equations

18. The Big Bounce Scenario

Instead of a singularity, the universe contracts, reaches a minimum volume, and rebounds due to quantum repulsion — offering a non-singular origin.

19. Discrete Quantum Geometry in LQC

In LQC, geometry is quantized:

• Area and volume have discrete spectra
• Operators for curvature and energy density are bounded

20. Effective Dynamics and Phenomenology

Modified equations in LQC:

\[
\left( \frac{\dot{a}}{a} \right)^2 = \frac{8\pi G}{3} \rho \left(1 – \frac{\rho}{\rho_c} \right)
\]

where \( \rho_c \) is the critical density at which the bounce occurs.

21. Observational Consequences and CMB

Quantum cosmology could affect:

• Primordial power spectrum
• Non-Gaussianities
• Tensor modes in the CMB

Efforts are underway to constrain models using cosmological observations.

22. Singularity Resolution in Quantum Cosmology

One of the strongest results: quantum cosmology resolves classical singularities through either wavefunction regularity or quantum repulsion mechanisms.

23. Multiverse and Quantum Cosmology

The wavefunction \( \Psi \) may include many possible universes — a quantum multiverse. Measures are needed to extract physical probabilities from this landscape.

24. Open Problems and Interpretations

• What is the correct interpretation of \( \Psi \)?
• Can quantum cosmology make testable predictions?
• How does time emerge from a timeless equation?

25. Conclusion

Quantum cosmology offers a framework for understanding the universe’s origin, singularity resolution, and the quantum nature of spacetime. Through tools like the Wheeler–DeWitt equation, loop quantum cosmology, and boundary proposals, it bridges general relativity and quantum mechanics. As observations improve and quantum gravity progresses, quantum cosmology may illuminate the ultimate beginning of the cosmos.