Bunch-Davies Vacuum in de Sitter Space

• Bunch-Davies vacuum is the unique de Sitter-invariant quantum state whose mode functions match Minkowski positive-frequency solutions at short distances.
• It sets the initial conditions for inflation, providing analytic, IR-finite two-point functions essential for predicting primordial fluctuation spectra.
• Thermal properties emerge in static patches due to horizon entanglement, linking it to Bogoliubov-transformed α–vacua in quantum field theory.

The Bunch-Davies vacuum is a distinguished state for quantum fields in de Sitter space, widely recognized as the canonical choice for defining quantum fluctuations during inflation and cosmological scenarios. It is characterized by its de Sitter invariance, minimal energy properties, and the matching of its short-distance (UV) behavior to that of the Minkowski vacuum.

1. Definition and Construction

The Bunch-Davies vacuum is constructed as the unique quantum state in which mode functions of free fields behave as Minkowski positive-frequency solutions at early times or short distances. For a free scalar field, on the Poincaré patch of de Sitter space in $d+1$ dimensions, the positive-frequency mode functions are

$$u^{(\mathrm{BD})}_k(\eta, x) = \frac{e^{i k \cdot x}}{(2\pi)^{d/2}} \chi_k(\eta),$$

with

$$\chi_k(\eta) = \sqrt{\frac{\pi \ell}{4}}\ e^{i\pi(\nu/2 - (d+2)/4)} \left( -\frac{\eta}{\ell} \right)^{d/2} H_\nu^{(1)}(-k \eta),$$

where $H_\nu^{(1)}$ is the Hankel function of the first kind, $\ell$ is the de Sitter radius, and the order $\nu$ is determined by

$$\nu = \ell \sqrt{m_*^2 - m^2}, \quad m_* = \frac{d}{2\ell}.$$

This construction ensures that at early conformal times ($\eta \to -\infty$), the modes asymptotically coincide with plane waves of Minkowski space. The corresponding two-point (Wightman) function is maximally symmetric, depending only on the geodesic distance between points.

2. Bunch-Davies Vacuum and De Sitter Invariance

The Bunch-Davies vacuum is invariant under the full de Sitter group. Its selection is closely related to the "Euclidean" or Hartle–Hawking prescription, in which the quantum state is analytically continued from a regular solution of the field equation on the de Sitter sphere's Euclidean section. This regularity at the so-called "South Pole" of the Euclidean geometry uniquely fixes the state, so that after analytic continuation to Lorentzian signature, one obtains the Bunch-Davies vacuum (Chen et al., 23 Apr 2024).

In the compact instanton (no-boundary) scenario of the Euclidean path integral approach, regularity conditions on the geometry and matter fields at the shrinking point select precisely the (growing) Euclidean mode—after Wick rotation, this becomes the de Sitter–invariant Bunch-Davies vacuum. The analytic continuation ensures the modes behave as positive-frequency Minkowski modes in the ultraviolet (UV) limit.

3. Mathematical Properties and Mode Expansions

The mode expansion of a quantum field $\phi$ in the Bunch-Davies vacuum is given by

$$\phi(x) = \sum_{n} \left[ a_n u_n^{(+)}(x) + a_n^\dagger u_n^{(-)}(x) \right],$$

where $u_n^{(+)}(x)$ are positive-frequency Bunch-Davies modes. Any other de Sitter–invariant vacuum (such as an $\alpha$-vacuum) can be related to the Bunch-Davies vacuum through a Bogoliubov transformation: $$u_k^{(\alpha,\beta)} = \cosh\alpha\, u_k^{(\mathrm{BD})} + \sinh\alpha\, e^{i\beta}\, \overline{u_{-k}^{(\mathrm{BD})}}.$$

In various coordinate patches—global, Poincaré (cosmological), or static—these positive-frequency modes can be expressed using spherical harmonics, hypergeometric functions, or linear combinations of modes in different charts, preserving the global definition under analytic continuation (Higuchi et al., 2018).

The Bunch-Davies two-point function in the global patch (and in general dimension) takes the form

$$W_E(x, y) = \frac{\Gamma(h_+)\Gamma(h_-)}{4\pi \ell^2 \Gamma(D/2)}\, {}_2F_1\left(h_+, h_-, D/2; (1+Z(x,y)+i\epsilon\, \text{sign}(x^0 - y^0))/2 \right),$$

where $h_\pm = d/2 \pm \nu$ and $Z(x, y)$ is the invariant embedding variable determined by de Sitter geometry (Aslanbeigi et al., 2013).

4. Physical Implications and Applications

Quantum Fluctuations and Cosmological Initial Conditions

The Bunch-Davies vacuum sets the initial state for quantum fluctuations during inflation. Its UV behavior matches the Minkowski vacuum, providing a well-defined initial condition for the computation of primordial spectra. When seeded into numerical relativity simulations, the quantum Bunch-Davies state is implemented by generating classical stochastic fields whose statistics match the quantum two-point correlators, ensuring the correct initial statistical properties for cosmological inhomogeneities (Launay et al., 10 Feb 2025).

Uniqueness, Analyticity, and Infrared Behavior

The Bunch-Davies vacuum is unique among de Sitter–invariant vacua in being analytic and free from infrared divergences across the full spacetime (Takook et al., 2015). Its two-point functions obey the Hadamard condition, essential for renormalization and the consistent construction of quantum field theory and quantum linear gravity in de Sitter space:

• Analyticity ensures all correlation functions can be constructed via analytic continuation from the complexified de Sitter manifold.
• Infrared finiteness is maintained in the analytic two-point functions, contrasting with alternative vacua where zero modes can induce infrared pathologies.

Entanglement, Thermal Properties, and Static Patches

Due to the presence of cosmological horizons, the Bunch-Davies vacuum, when restricted to a static patch, appears as a thermal state at the Gibbons–Hawking temperature $T = H/(2\pi)$. This property is deeply associated with the entanglement between causally disconnected regions and is made manifest by expanding the state as an entangled ("thermofield double") state of modes inside and outside a given horizon (Higuchi et al., 2018, Choudhury et al., 2017): $$\rho = \mathrm{Tr}_{\text{unseen}}\, |\text{BD}\rangle\langle \text{BD}|,$$ with the density matrix corresponding to a thermal ensemble for an observer within a single static patch.

Moreover, measures such as entanglement entropy and entanglement negativity computed for causally separated regions in de Sitter space explicitly reflect the entanglement structure of the Bunch-Davies state (Choudhury, 2022).

5. Relationship to Other De Sitter–Invariant Vacua

The Bunch-Davies vacuum is a special limit within the broader family of de Sitter $\alpha$-vacua, obtained via Bogoliubov transformations (parameterized by $\alpha,\beta$), and is uniquely singled out by the requirement of regularity at the Euclidean South Pole or by connection to the no-boundary compact instanton (Chen et al., 23 Apr 2024). In contrast, Euclidean wormhole instanton geometries admit a superposition of growing and decaying modes—allowing for generic $\alpha$-vacua—with the geometry of the instanton providing a geometric underpinning for the full family of de Sitter–invariant vacua.

The Sorkin–Johnston (SJ) vacuum, an alternative covariant prescription for the ground state in arbitrarily curved spacetimes, coincides with the Bunch-Davies state for heavy fields ($m \geq m_*$) on the global patch in even spacetime dimensions. The SJ state differs in other regimes, particularly for light fields or restricted spacetime regions, but always belongs to the class of $\alpha$-vacua (Aslanbeigi et al., 2013).

6. Limitations, Instabilities, and Extensions

While the Bunch-Davies vacuum is privileged by analyticity and cosmic regularity, it is not always dynamically stable under perturbations. In global de Sitter space, even infinitesimal deviations can lead to a build-up of energy density via blueshifting in contracting phases, indicating an instability toward breaking de Sitter invariance (Anderson et al., 2013). The vacuum’s stability thus depends on the global temporal structure of the background spacetime.

When applied as the initial state for nonlinear, non-perturbative evolution—such as in numerical relativity simulations—the Bunch-Davies vacuum provides a natural prescription for initial fluctuations but does not guarantee perturbative behavior throughout evolution, especially in the presence of resonances or strong nonlinearity (Launay et al., 10 Feb 2025).

7. Summary Table: Key Features of the Bunch-Davies Vacuum

Feature	Property	Context/Comments
Invariance	Full de Sitter invariance	Unique up to $\alpha$-vacua ambiguity
Analyticity	Yes (Hadamard state, analytic two-point function)	Essential for well-defined QFT
Infrared	Free from IR divergences	Via analytic and covariant construction
Thermal Prop.	Thermal response in static patch: $T=H/(2\pi)$	Due to entanglement across horizons
Initial Data	Canonical inflationary initial state	Ensures correct primordial spectra
Stability	Unstable to infinitesimal perturbations globally	Blueshifting of modes, quantum anomaly
Relation to SJ	Coincides for heavy fields, even $D$ in global patch	Otherwise is a different $\alpha$-vacuum

References to Key Results

• Construction, de Sitter invariance, and uniqueness: (Aslanbeigi et al., 2013, Takook et al., 2015, Wrochna, 2017).
• Causal set and numerical implementation: (Aslanbeigi et al., 2013, Launay et al., 10 Feb 2025).
• Thermal and entanglement properties: (Higuchi et al., 2018, Saharian et al., 2021, Choudhury, 2022).
• Stability and global issues: (Anderson et al., 2013).
• Origin in path integral and geometric frameworks: (Chen et al., 23 Apr 2024).
• Relationship to $\alpha$-vacua and wormhole scenarios: (Chen et al., 23 Apr 2024).
• Quantum field implementation and analytic structure: (Takook et al., 2015, Wrochna, 2017).

The Bunch-Davies vacuum continues to serve as a foundational element for rigorous studies of quantum fields in curved space, inflationary cosmology, quantum gravity models, and the exploration of fundamental questions in vacuum selection, vacuum stability, and the role of quantum entanglement in the structure of spacetime.