Decoherence in Primordial Fluctuations

• Decoherence in primordial fluctuations is the process by which quantum inhomogeneities become classical stochastic fields through interactions with environmental modes.
• An open-system approach using a Lindblad-type master equation models the evolution of Fourier modes, capturing the rapid suppression of quantum interference.
• Efficient decoherence driven by gravitational and field interactions generates entropy that preserves classical observables like the CMB acoustic peaks.

Primordial fluctuations generated during inflation are fundamentally quantum, yet the observed cosmological structures—CMB anisotropies and the large-scale distribution of matter—are described by classical stochastic fields. Decoherence processes provide the rigorous framework for understanding how quantum primordial inhomogeneities lose their quantum coherence and become effectively classical stochastic variables. The decoherence of cosmological perturbations is driven by interactions—either with explicit environmental degrees of freedom or through unavoidable gravitational self-interactions—that entangle long-wavelength ("system") modes with short-wavelength or unobserved ("environment") modes, resulting in the rapid suppression of quantum interference and a rise in entropy. Decoherence not only underpins the quantum-to-classical transition but also constrains the potential observability of genuine quantum features in primordial fluctuations.

1. Open-System Approach and Master Equation Formalism

The contemporary theoretical framework models each Fourier mode of the inflaton fluctuation, typically rescaled as $y_k \equiv a\,\delta\varphi_k$, as an open quantum system interacting with an environment. The total density matrix $\rho_\text{tot}(t)$ evolves unitarily under a full Hamiltonian that includes system, environment, and interaction parts: $$H_\text{tot} = H_\text{sys} + H_\text{env} + H_\text{int}\,, \qquad \frac{d\rho_\text{tot}}{dt} = -i\,[H_\text{tot},\,\rho_\text{tot}]$$ The reduced density matrix for the system,

$$\rho_\text{red}(t) = \operatorname{Tr}_\text{env}\,\rho_\text{tot}(t)\,,$$

generically evolves into a mixed state due to tracing over environmental degrees of freedom (0810.0087).

Under the standard Born-Markov and rotating-wave approximations, $\rho_\text{red}$ obeys a Lindblad-type master equation: $$\frac{d\rho_\text{red}}{dt} = -\,i\,[H_\text{sys},\,\rho_\text{red}] + L\rho_\text{red}L^\dagger - \frac{1}{2}\{L^\dagger L,\,\rho_\text{red}\}$$ with $L$ the Lindblad operator. For super-Hubble inflationary modes, $L \propto \sqrt{\gamma_k}\,y_k$, since the relevant system-environment coupling is to the field amplitude $y_k$ rather than its conjugate momentum. This structure captures the environment-induced decoherence responsible for the transition from quantum to classical fluctuations (0810.0087, Burgess et al., 2014).

2. Pointer Basis, Decoherence Rate, and Time Scale

The field-amplitude basis $\{|y_k\rangle\}$ is dynamically selected as the "pointer basis" because for super-Hubble modes, $[y_k, H_\text{sys}]\approx 0$ ("freezing" of the growing mode) and $H_\text{int}$ couples directly to $y_k$. In the $y_k$-representation, the reduced density matrix has the schematic form: $$\rho_\text{red}(y, y'; t) = \rho_0(y, y'; t)\, \exp\left[-D_k(t)\,(y - y')^2\right]$$ where $\rho_0$ is the evolution without decoherence, and $D_k(t) = \int_0^t ds\,\xi_k(s)$ is the integrated decoherence rate, with $\xi_k$ determined by the environmental fluctuation spectrum (0810.0087).

The characteristic decoherence time for a mode $k$ during inflation is

$t_\text{dec}(k) \simeq H_I^{-1} \ln(H_I t_0^{-1})$

where $H_I$ is the inflationary Hubble parameter and $t_0$ represents the correlation timescale of the environment. This time is typically a few Hubble times, indicating that decoherence is extremely efficient, rapidly suppressing off-diagonal elements and rendering quantum superpositions unobservable (0810.0087, Burgess et al., 2014, Kiefer, 24 Mar 2025).

3. Entropy Generation and Information Loss

Decoherence produces entropy in the system as its state evolves from pure to mixed. For a decohered mode, the von Neumann entropy is

$$S_k = -\operatorname{Tr}\left[ \rho_\text{red} \ln \rho_\text{red} \right]$$

For the reduced density matrix of Gaussian form,

$$\rho_\text{red}(y, y') \propto \exp[-R_k(y^2 + y'^2) + iI_k(y^2 - y'^2) - D_k(y - y')^2]$$

one finds (cf. [(0810.0087), eq. 53])

$$S_k = \ln\left(\frac{\sqrt{1 + x_k} + 1}{2}\right) - \frac{\sqrt{1 + x_k} - 1}{2} \ln\left(\frac{\sqrt{1 + x_k} - 1}{\sqrt{1 + x_k} + 1}\right)$$

where $x_k = D_k/(2R_k)$. In the strong decoherence limit $x_k\gg1$, $S_k \simeq \ln x_k$. However, the entropy is bounded above by the squeezing parameter $r_k$, i.e., $S_k < r_k$, ensuring survival of quantum coherence necessary for observed features such as the acoustic peaks in the CMB (0810.0087).

4. From Decoherence to Classical Observables

Once $t > t_\text{dec}(k)$, the reduced density matrix is effectively diagonal in field amplitude, and its diagonal elements define a classical probability distribution over $y_k$. The Wigner function of such a state becomes positive and highly squeezed along the classical trajectory in phase space. All quantum two-point functions $\langle y_k y_k' \rangle$, $\langle p_k p_k' \rangle$, $\langle y_k p_k' + p_k y_k' \rangle$ then coincide with classical stochastic averages over $P(y_k)$, reproducing the classical power spectrum and two-point statistics (0810.0087, Burgess et al., 2014, Kiefer, 24 Mar 2025).

This transition justifies the practical use of classical stochastic field theory for post-inflationary evolution, structure formation, and CMB phenomenology. Environment-induced decoherence thus accomplishes the quantum-to-classical transition: selection of the field-amplitude basis, suppression of quantum interference on super-Hubble scales, and production of an entropy controlled by the squeezing parameter while preserving the observables (0810.0087, Franco et al., 2011, Kiefer, 24 Mar 2025).

5. Model Dependence and the Role of the Environment

The precise rate and efficiency of decoherence depend on the system-environment coupling. In standard inflationary cosmology, plausible environmental candidates include:

• Sub-Hubble modes of the inflaton (inside-the-horizon regions),
• Other light (spectator) fields,
• Gravitational degrees of freedom (tensor modes),
• Short-wavelength sectors induced by non-linear gravitational constraints (0810.0087, Franco et al., 2011, Kiefer, 24 Mar 2025).

The scenario is robust: even the unavoidable gravitational self-interaction is sufficient to induce rapid decoherence. Specifically, scalar-tensor interactions result in a decoherence rate for the scalar sector that is vastly greater than the Hubble rate, with $\Gamma_k / H \sim 10^{40} - 10^{120}$ for typical parameter choices (Franco et al., 2011). This massive factor ensures that any residual quantum coherence in observable scalar perturbations is exponentially small, justifying classical stochastic treatment.

6. Entropy, Quantum Correlations, and Observational Implications

Decoherence during inflation leads to mixed states characterized by nonzero entropy, but the process is constrained to avoid "over-decoherence" (i.e., $S_k$ exceeding the squeezing parameter $r_k$), which would spoil observable phase correlations. As such, the classical stochastic description emerges with statistical properties matching those expected from the quantum calculation, but the state remains sufficiently pure to account for phenomena such as CMB acoustic oscillations (0810.0087).

Observationally, any experimental probe searching for quantum signatures (e.g., violations of Bell inequalities, non-Gaussian features due to quantum entanglement rather than classical nonlinearity) would be sensitive to deviations from perfect decoherence. Current estimates establish that for realistic inflationary scenarios—and certainly for the CMB observable range—such quantum residuals are unobservably small (0810.0087, Kiefer, 24 Mar 2025, Franco et al., 2011). However, in models or regimes of weak environmental coupling or for modes that exit the Hubble radius at the very end of inflation, some minimal coherence may persist, motivating precise studies of the associated entropy budget (0805.0424, 0805.0548).

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References:

• "Why do cosmological perturbations look classical to us?" (0810.0087)
• "Decoherence in the cosmic background radiation" (Franco et al., 2011)
• "EFT Beyond the Horizon: Stochastic Inflation and How Primordial Quantum Fluctuations Go Classical" (Burgess et al., 2014)
• "Decoherence and entropy of primordial fluctuations. II. The entropy budget" (0805.0424)
• "From a quantum world to our classical Universe" (Kiefer, 24 Mar 2025)
• "Decoherence and entropy of primordial fluctuations. I: Formalism and interpretation" (0805.0548)