From a quantum world to our classical Universe (2503.18499v1)

Abstract: Modern cosmological theories invoke the idea that all structure in the Universe originates from quantum fluctuations. Understanding the quantum-to-classical transition for these fluctuations is of central importance not only for the foundations of quantum theory, but also for observational astronomy. In my contribution, I review the essential features of this transition, emphasizing in particular the role of Alexei Starobinsky.

• The paper demonstrates how quantum fluctuations during inflation transition into classical states through the process of squeezing and decoherence.
• It elucidates the role of decoherence in selecting the field amplitude as the preferred variable, preserving correlations essential for the observed cosmic microwave background features.
• The study highlights how quantum field theory and gravitational models together explain the emergence of large-scale cosmic structures from initial quantum conditions.

This paper, "From a quantum world to our classical Universe" (2503.18499), reviews how the large-scale structure of the Universe, such as galaxies and galaxy clusters seen in images like Webb's First Deep Field, can originate from initial quantum fluctuations. The central theme is the transition of these primordial fluctuations from a quantum state to the classical, stochastic fluctuations needed to seed structure formation. The review emphasizes the role of cosmic inflation and the process of decoherence in this transition, highlighting the contributions of Alexei Starobinsky.

The observed structure in the Universe requires initial conditions beyond a perfectly homogeneous state. The prevailing cosmological theory for the early Universe is inflation, a period of quasi-exponential expansion. A key feature of inflation is that it provides a mechanism for generating initial density and gravitational wave fluctuations from quantum fluctuations of the spacetime metric and other fields (like the inflaton). These fluctuations, initially microscopic and quantum, are stretched to cosmological scales by the rapid expansion. The paper discusses the standard picture where modes of different wavelengths leave the Hubble scale during inflation and re-enter it much later in the radiation or matter-dominated era, providing the initial conditions for structure formation.

From a fundamental quantum gravity perspective, the early Universe can be described by a quantum state (e.g., a solution to the Wheeler-DeWitt equation in quantum geometrodynamics). In a semiclassical approximation, this leads to a Schrödinger equation for quantum fields on an evolving background. The primordial fluctuations, often described by a gauge-invariant variable (like the Mukhanov-Sasaki variable), can be treated as independent quantum modes, each evolving in time.

For a single mode, the dynamics during inflation, particularly the coupling to the evolving background (analogous to a time-dependent harmonic oscillator), causes the initial adiabatic vacuum state (a simple Gaussian) to become a squeezed state. In a squeezed state, the quantum uncertainty is reduced in one variable (like momentum) at the expense of increased uncertainty in its conjugate variable (like field amplitude or position). For cosmological fluctuations during inflation, the state becomes highly squeezed, with a very large variance in the field amplitude variable.

While in the limit of large squeezing, certain quantum expectation values can mimic those of a classical stochastic system, this alone is not sufficient for explaining the emergence of classical properties. The crucial missing ingredient is the interaction of these quantum modes with other degrees of freedom – their environment. This leads to the process of decoherence.

Decoherence, first rigorously studied by H.-Dieter Zeh, explains how a quantum system interacting with a complex environment loses its coherence (superposition and entanglement with other systems) when viewed in isolation. The system becomes entangled with the many inaccessible degrees of freedom of the environment. When one traces out the environment's degrees of freedom, the system's reduced density matrix rapidly becomes diagonal in a preferred basis, known as the "pointer basis." The system then appears to be in a statistical mixture of states from this pointer basis, mimicking classical behavior, although the global state of system-plus-environment remains pure and entangled.

Applying this concept to primordial fluctuations, the paper argues that the interaction of the highly squeezed modes with environmental degrees of freedom (e.g., non-linear self-interactions of the modes, or interactions with other fields) causes them to decohere. Because the squeezed state has a large variance in the field amplitude, interactions that are local in the field amplitude variable (typical in quantum field theory) are particularly effective at inducing decoherence.

The paper presents the form of the reduced density matrix for a mode undergoing decoherence, showing an exponential damping of off-diagonal terms in the field amplitude basis. This implies that the field amplitude itself constitutes the pointer basis for these fluctuations. The strength of decoherence is characterized by a parameter $\xi$. For decoherence to be effective, $\xi$ must be much larger than a characteristic parameter of the squeezed state ($\Omega_R$).

The timescale for decoherence during inflation is found to be approximately the Hubble time $H_I^{-1}$. The decoherence process alters the shape of the Wigner function for the mode: while the squeezing aligns the major axis along the field amplitude axis, decoherence broadens the state in the conjugate momentum direction, resulting in an elliptical contour where correlations between field amplitude and momentum are partially damped.

However, the decoherence cannot be arbitrarily strong. Observations of the cosmic microwave background (CMB) anisotropy spectrum (like those from the Planck satellite) show acoustic peaks, which are a signature of specific correlations between field amplitude and momentum in the fluctuations. If decoherence completely washed out these correlations, these peaks would not be present. The paper notes that the condition $\xi/k \ll e^{2r}$ (where $r$ is the squeezing parameter) must hold, meaning decoherence is significant but not total.

The degree of decoherence is also related to the entropy of the modes. For a pure squeezed state, the von Neumann entropy is zero. With decoherence, the modes are in a mixed state, and their entropy increases. For large squeezing and partial decoherence consistent with CMB observations, the entropy per mode is found to be limited to approximately $r$, which is half the maximal possible entropy for a state with the same variance. This limited entropy reflects the fact that decoherence selects the field amplitude as the classical variable, but correlations with the environment related to the conjugate momentum are preserved to some extent, allowing for the observed acoustic features.

The paper discusses how these concepts extend to decoherence after inflation, showing that the relevant correlations persist long enough to influence structure formation. It also draws a parallel to Hawking radiation from black holes, where similar squeezing occurs, but interactions select a different pointer basis (particle number), leading to a thermal (maximal entropy) state.

Ultimately, the paper positions decoherence as essential for understanding the emergence of classicality in cosmology, not just for the primordial fluctuations but also for the background spacetime itself. This process contributes to the arrow of time, as the formation of entanglement and the resulting decoherence are irreversible, leading to an increase in entropy. The selection of a preferred initial state, such as the adiabatic vacuum for primordial fluctuations, is argued to be key to this emergence of irreversibility, potentially linking to concepts like the Weyl curvature hypothesis.

In summary, the paper explains how the observable large-scale structure of the Universe likely originated from initial quantum fluctuations during inflation, which transitioned to classical stochastic behavior through the unavoidable process of decoherence driven by interactions with other degrees of freedom. This quantum-to-classical transition, primarily selecting the field amplitude as the classical variable, is consistent with cosmological observations and is seen as a crucial step in the emergence of classicality and the arrow of time in our Universe.