Introduction

The scalar field $\phi$ in de Sitter space satisfies:

$$\Box \phi - m^2 \phi = 0$$

where the d'Alembertian in FRW coordinates takes the form:

$$\Box = -\frac{1}{\sqrt{-g}} \partial_\mu \left( \sqrt{-g} \, g^{\mu\nu} \partial_\nu \right)$$

Mode Expansion

We expand in terms of creation and annihilation operators:

$$\phi(x) = \int \frac{d^3k}{(2\pi)^3} \left[ a_k u_k(x) + a_k^\dagger u_k^*(x) \right]$$

The Bunch-Davies vacuum $|0\rangle$ satisfies $a_k |0\rangle = 0$.

Power Spectrum

The power spectrum of fluctuations is given by:

$$P(k) = \frac{k^3}{2\pi^2} |u_k|^2$$

For a massless field in de Sitter space:

$$P(k) = \left( \frac{H}{2\pi} \right)^2$$

This scale-invariant spectrum is a key prediction of inflation.

Numerical Example

import numpy as np

def hubble_parameter(H0, Omega_m, Omega_Lambda, z):
    """Compute H(z) for flat ΛCDM."""
    return H0 * np.sqrt(Omega_m * (1 + z)**3 + Omega_Lambda)

def power_spectrum(H, k):
    """Scale-invariant power spectrum."""
    return (H / (2 * np.pi))**2 * np.ones_like(k)

References

1. Birrell & Davies, Quantum Fields in Curved Space (1982)
2. Mukhanov, Physical Foundations of Cosmology (2005)