Table of Contents
- Introduction
- Motivation for Quantum Cosmology
- Classical Cosmology and General Relativity
- Singularities and the Big Bang
- Quantum Gravity and Early Universe
- Wheeler–DeWitt Equation in Cosmology
- Minisuperspace Approximation
- Canonical Quantization of Cosmological Models
- Quantum States of the Universe
- Boundary Conditions: No-Boundary and Tunneling Proposals
- Hartle–Hawking No-Boundary Proposal
- Vilenkin’s Tunneling Proposal
- Quantum Fluctuations and Inflation
- Quantum-to-Classical Transition
- Decoherence in the Early Universe
- Quantum Initial Conditions
- Loop Quantum Cosmology (LQC)
- The Big Bounce Scenario
- Discrete Quantum Geometry in LQC
- Effective Dynamics and Phenomenology
- Observational Consequences and CMB
- Singularity Resolution in Quantum Cosmology
- Multiverse and Quantum Cosmology
- Open Problems and Interpretations
- 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.
