Bunch-Davies state on de Sitter spacetime

Intuition Beginner

A quantum field on a curved background needs a vacuum — a reference state against which excitations are counted as particles. On flat Minkowski spacetime the vacuum is unique because the Lorentz group picks it out: it is the only state invariant under boosts and translations. On a generic curved spacetime no such symmetry exists, the vacuum splits into a class of physically admissible states (the Hadamard states of 13.09.03), and no single one is canonical.

De Sitter spacetime is the exception that recovers a canonical vacuum. It is the maximally symmetric Lorentzian spacetime with positive curvature — the cosmological analog of the round sphere — and its isometry group is the de Sitter group, ten-dimensional and as large as the Poincaré group of Minkowski space. The symmetry singles out the Bunch-Davies state as the unique vacuum that respects the full de Sitter symmetry and is also Hadamard.

The Bunch-Davies state was written down by Timothy Bunch and Paul Davies in 1978 by an explicit mode-sum calculation in cosmological coordinates. The resulting two-point function depends only on the de-Sitter-invariant geodesic distance between two events and has a closed-form expression in terms of the Gauss hypergeometric function. For a massless conformally coupled scalar field the two-point function reduces, via the conformal transformation between de Sitter and the Einstein static universe, to the flat-space Wightman function — a sanity check on the construction.

The state has a surprising thermal property discovered by Gary Gibbons and Stephen Hawking in 1977. A static observer in a single de Sitter patch — an observer who follows a worldline of one of the timelike Killing vectors of the de Sitter geometry — perceives the Bunch-Davies vacuum as a thermal bath of particles at the Gibbons-Hawking temperature $T=H/(2\pi k_B)$, where $H$ is the de Sitter Hubble rate. This is the analog of the Hawking temperature for a black hole: an observer who lacks access to the full spacetime (here cut off by the cosmological horizon) sees the global vacuum as a mixed thermal state.

The state matters for cosmology. Inflationary models posit a brief de-Sitter-like phase in the very early universe, during which the inflaton field is in a near-Bunch-Davies vacuum. The quantum fluctuations of the field in this state seed the temperature anisotropies of the cosmic microwave background and, after gravitational collapse, the galaxy distribution observed today. The Bunch-Davies prediction of a scale-invariant power spectrum agrees with the Planck satellite measurements up to slow-roll corrections.

The Bunch-Davies state is the worked example of every general theorem in the Hadamard programme. It exists by the symmetry-respecting refinement of the FNW existence argument from 13.09.04; it is unique by the Allen-Mottola 1985 classification combined with the Hadamard wave-front-set criterion from 13.09.03; its thermal property is the simplest case of the modular-theoretic identification of cosmological with thermal time. Every later development in the subject — the rigorous Hawking effect on Schwarzschild, the cosmological-perturbation power spectrum, the holographic dictionary in de Sitter quantum gravity — uses the Bunch-Davies state as its starting reference.

Visual Beginner

The picture to hold is de Sitter spacetime drawn as a hyperboloid in five-dimensional Minkowski space, with a single static patch corresponding to one inertial observer carved out as a wedge, and the Bunch-Davies state being the global vacuum from which that observer sees thermal radiation at the Gibbons-Hawking temperature.

Three pieces drive the picture. The hyperboloid is de Sitter $d{S}_4$, drawn as the embedding $-X_0^2+X_1^2+X_2^2+X_3^2+X_4^2=H^{-2}$ in five-dimensional Minkowski space — the simplest concrete realisation of a Lorentzian manifold with positive constant curvature. The wedge is the static patch: the region of $d{S}_4$ visible to one timelike-geodesic observer, bounded above and below by the observer's past and future cosmological horizons. The flat-slicing coordinates $d{s}^2=-d{t}^2+e^{2Ht}(d{x}^2+d{y}^2+d{z}^2)$ cover only half of de Sitter — the half causally accessible to an observer at spatial infinity — and supply the natural coordinates for the Bunch-Davies mode functions.

The Bunch-Davies vacuum is the global state of the quantised Klein-Gordon field built from the conformal-time positive-frequency mode functions $u_k$. For an inertial static-patch observer the state looks like a thermal bath at the Gibbons-Hawking temperature — the cosmological analog of the Unruh effect for a uniformly accelerated observer in Minkowski space. The same global vacuum can therefore appear pure (to a globally informed observer) or thermal (to a static-patch observer who lacks access to the full spacetime), with the partial information being what produces the entropy.

Worked example Beginner

Compute the Bunch-Davies two-point function for a massless conformally coupled scalar field on de Sitter, and verify that it reduces to the flat-space Wightman function via the conformal transformation between de Sitter and the Einstein static universe.

Step 1. Choose conformal coupling $\xi =1/6$ and mass $m=0$. The Klein-Gordon equation on a generic background with these parameters is $({\square }_g+R/6)\varphi =0$, and it has the special property of being conformally invariant in four spacetime dimensions: if $\varphi$ solves the equation on $(M,g)$ and $\tilde{g}={\Omega }^{2}g$ is a conformally rescaled metric, then $\tilde{\varphi }={\Omega }^{-1}\varphi$ solves the corresponding equation on $(M,\tilde{g})$.

Step 2. De Sitter and the Einstein static universe are conformally related. The Einstein static universe is the product of a time line and a three-sphere of fixed radius, $d{s}_{\mathrm{ESU}}^2=-d{T}^2+d{\Omega }_{S^3}^2$. By an explicit coordinate change the metric of de Sitter $d{S}_4$ can be written as $d{s}_{dS}^2=(\cos T{)}^{-2}d{s}_{\mathrm{ESU}}^2$ on a strip $T\in (-\pi /2,\pi /2)$ of the Einstein static universe. The conformal factor is $\Omega =(\cos T{)}^{-1}$.

Step 3. On the Einstein static universe, the conformally coupled Klein-Gordon equation reduces to a free wave equation on the three-sphere times the time line. The positive-frequency modes are spherical harmonics on $S^3$ times oscillating exponentials in $T$. The vacuum state — the state in which every positive-frequency mode is unoccupied — has a Wightman two-point function that depends only on the geodesic distance between two events on the Einstein static universe.

Step 4. Transport the Einstein-static-universe vacuum to de Sitter via the conformal factor. Define the de Sitter field as ${\varphi }_{dS}={\Omega }^{-1}{\varphi }_{\mathrm{ESU}}=\cos T\cdot {\varphi }_{\mathrm{ESU}}$, and define the de Sitter two-point function as the corresponding pull-back of the Einstein-static-universe two-point function. The result is the Bunch-Davies state in the conformally coupled massless case, and its two-point function depends only on the de-Sitter-invariant geodesic-distance variable $Z(x,x^{\prime })$.

Step 5. Read off the explicit form. The two-point function in the conformally coupled massless case is $$ W_{\mathrm{BD}}^{\mathrm{conf}}(x, x') = \frac{H^2}{16\pi^2}\cdot\frac{1}{1 - Z(x, x') + i\epsilon}, $$ where $Z(x,x^{\prime })$ is the de Sitter geodesic-distance invariant and the $iϵ$ prescription selects the positive-frequency boundary condition. The leading short-distance behaviour as $Z\to 1$ (coincident points) is $1/(1-Z)\approx 2/(H^2\sigma )$ where $\sigma$ is Synge's world function — recovering the standard $1/\sigma$ Hadamard singularity structure.

Step 6. Sanity check against Minkowski. Take $H\to 0$ holding the geodesic distance fixed. The de Sitter geodesic-distance invariant reduces to $1-H^2\cdot \frac{1}{2}(x-x^{\prime }{)}^2/2+\mathcal{O}(H^4)$, so $1-Z\approx H^2(x-x^{\prime }{)}^2/2$, and the two-point function reduces to $$ W_{\mathrm{BD}}^{\mathrm{conf}}(x, x') \to \frac{1}{8\pi^2}\cdot\frac{1}{(x - x')^2 + i\epsilon}, $$ the standard Minkowski Wightman function of the massless conformally coupled scalar. The Bunch-Davies state passes the flat-space limit.

What this tells us: in the simplest case — massless conformal coupling — the Bunch-Davies state is the unique de-Sitter-invariant vacuum obtained by pulling back the Einstein-static-universe vacuum via the conformal transformation. The two-point function has a closed-form expression in the de-Sitter-invariant variable, satisfies the Hadamard short-distance condition, and reduces correctly to the Minkowski Wightman function in the flat-space limit. The massive case ($m\ne 0$ or $\xi \ne 1/6$) gives a more elaborate two-point function involving the Gauss hypergeometric function, treated in the Intermediate tier.

Check your understanding Beginner

Formal definition Intermediate+

Throughout this section $(M,g)$ denotes de Sitter spacetime $d{S}_4$, the maximally symmetric four-dimensional Lorentzian manifold of constant positive sectional curvature $H^2>0$. Concretely, $d{S}_4$ is realised as the embedded hyperboloid $\{X\in {\mathbb{R}}^{1,4}:{\eta }_{AB}{X}^A{X}^B=H^{-2}\}$ in five-dimensional Minkowski space with metric ${\eta }_{AB}=\mathrm{diag}(-1,+1,+1,+1,+1)$, equipped with the induced metric. The Ricci tensor is $R_{\mu \nu }=3H^2{g}_{\mu \nu }$ and the Ricci scalar is $R=12H^2$. The isometry group is $\mathrm{SO}(4,1)$, ten-dimensional, acting transitively on $d{S}_4$ via the restriction of the linear $\mathrm{O}(4,1)$-action on ${\mathbb{R}}^{1,4}$.

The flat slicing of de Sitter covers the upper half of the hyperboloid (the region causally accessible to a comoving observer at future timelike infinity) and assigns coordinates $(t,\mathbf{x})\in \mathbb{R}\times {\mathbb{R}}^3$ via $X^0=H^{-1}\sinh (Ht)+\frac{1}{2}H|\mathbf{x}{|}^2{e}^{Ht}$, $X^i=e^{Ht}{x}^i$ for $i=1,2,3$, and $X^4=H^{-1}\cosh (Ht)-\frac{1}{2}H|\mathbf{x}{|}^2{e}^{Ht}$. In these coordinates the metric takes the FLRW form $$ ds^2 = -dt^2 + e^{2Ht}(dx^2 + dy^2 + dz^2), $$ with scale factor $a(t)=e^{Ht}$. The conformal time $\eta :=-e^{-Ht}/H\in (-\infty ,0)$ renders the metric explicitly conformal to Minkowski: $$ ds^2 = a(\eta)^2,(-d\eta^2 + dx^2 + dy^2 + dz^2), \qquad a(\eta) = -\frac{1}{H\eta}. $$

The Klein-Gordon equation on de Sitter with mass $m\ge 0$ and curvature coupling $\xi \ge 0$ is $$ (\Box_g + m^2 + \xi R)\phi = (\Box_g + m^2 + 12\xi H^2)\phi = 0. $$ The conformal coupling $\xi =1/6$ makes the equation invariant under conformal rescalings of the metric in four dimensions; the minimal coupling $\xi =0$ gives the equation appropriate to the inflaton field of standard slow-roll inflation. Define the conformal weights $$ \Delta_\pm := \frac{3}{2} \pm \sqrt{\frac{9}{4} - \frac{m^2}{H^2} - 12\xi}. $$ For $m^2/H^2+12\xi <9/4$ (the principal-series range) the weights are real; outside this range they are complex conjugates of one another. The conformally coupled massless case has ${\Delta }_{+}=2$, ${\Delta }_{-}=1$, with ${\Delta }_{+}+{\Delta }_{-}=3$ in every case.

The spatial-Fourier mode of the field in the flat slicing satisfies a one-dimensional ordinary differential equation in conformal time. Writing $\varphi (\eta ,\mathbf{x})=\int (d^{3}k/(2\pi {)}^3)[a_{\mathbf{k}}{u}_k(\eta )e^{i\mathbf{k}\cdot \mathbf{x}}+a_{\mathbf{k}}^{†}\overline{u_k(\eta )}{e}^{-i\mathbf{k}\cdot \mathbf{x}}]$, the mode function ${\chi }_k:=a(\eta )u_k(\eta )$ satisfies $$ \chi_k''(\eta) + \big(k^2 + (m^2 a^2 + (\xi - 1/6) R a^2 - a''/a)\big)\chi_k(\eta) = 0, $$ where the prime denotes derivative with respect to $\eta$, $k:=|\mathbf{k}|$, and $a^{\prime \prime }/a=2/{\eta }^2$ on de Sitter. For the massless minimally coupled case the equation reduces to ${\chi }_k^{\prime \prime }+(k^2-2/{\eta }^2){\chi }_k=0$, whose positive-frequency solution (defined by matching to the Minkowski positive-frequency mode in the sub-horizon limit $|k\eta |\to \infty$) is $$ \chi_k(\eta) = \frac{1}{\sqrt{2k}}\bigg(1 + \frac{i}{k\eta}\bigg)e^{-ik\eta}. $$

Definition (Bunch-Davies state). The Bunch-Davies state ${\omega }_{\mathrm{BD}}$ is the quasi-free state on the CCR algebra of the Klein-Gordon field on de Sitter $d{S}_4$ whose creation and annihilation operators are defined by the positive-frequency conformal-time mode functions $u_k(\eta )={\chi }_k(\eta )/a(\eta )$, with ${\chi }_k$ the Bunch-Davies mode of the previous paragraph. Equivalently: ${\omega }_{\mathrm{BD}}$ is the unique de-Sitter-invariant quasi-free state whose two-point function satisfies the Hadamard wave-front-set condition of 13.09.03 ^{[Bunch-Davies 1978][Allen 1985]}.

The two-point function admits a closed-form expression in terms of the de-Sitter-invariant variable $Z(x,x^{\prime }):=H^2{\eta }_{AB}{X}^A(x)X^B(x^{\prime })$ — the inner product on ${\mathbb{R}}^{1,4}$ of the embeddings of $x$ and $x^{\prime }$, with $|Z|<1$ for spacelike-separated points, $Z=1$ at coincidence, and $|Z|>1$ for timelike-separated points. In terms of $Z$ the Bunch-Davies two-point function reads $$ W_{\mathrm{BD}}(x, x') = \frac{H^2}{16\pi^2},\Gamma(\Delta_+)\Gamma(\Delta_-),{}2F_1!\bigg(\Delta+, \Delta_-;\ 2;\ \frac{1 + Z(x, x')}{2}\bigg), $$ with the $iϵ$ prescription $Z\to Z+iϵ\mathrm{sgn}(t_x-t_{x^{\prime }})$ selecting the positive-frequency boundary condition and ${}_2{F}_1$ the Gauss hypergeometric function ^{[Bunch-Davies 1978][Birrell-Davies 1982]}.

Counterexamples to common slips

• The Bunch-Davies state is not the unique de-Sitter-invariant quasi-free state. Allen 1985 ^{[Allen 1985]} and Mottola 1985 ^{[Mottola 1985]} showed that de Sitter admits a one-parameter family of de-Sitter-invariant quasi-free states — the $\alpha$-vacua, indexed by a complex parameter $\alpha$ with $\mathrm{Re}(\alpha )<0$. The Bunch-Davies state is the special member $\alpha =0$. What singles out Bunch-Davies among the $\alpha$-vacua is the Hadamard condition: only the $\alpha =0$ state satisfies the Radzikowski wave-front-set criterion of 13.09.03. The $\alpha \ne 0$ states have wave-front-set contributions on both halves of the bicharacteristic relation, with the relative weight controlled by $\alpha$, and are therefore not physically admissible in the modern sense — their renormalised stress-energy tensor is infinite, and they cannot serve as reference states for Wick polynomials.

• The Bunch-Davies two-point function depends only on the de-Sitter-invariant geodesic distance, not on the individual spacetime points. This is the de Sitter analog of Lorentz invariance of the Minkowski vacuum two-point function, which depends only on $(x-y{)}^2$. The dependence on the geodesic-distance invariant $Z(x,x^{\prime })$ is what makes the de Sitter group action on the two-point function manifest.

• The Gibbons-Hawking thermal property is not a global statement about the Bunch-Davies state — the global state is a pure vacuum, not a thermal density matrix. The thermal property emerges when the state is restricted to the algebra of observables in a single static patch: the restricted state is mixed and has the KMS property with respect to the static-patch boost as the time-translation. This is the de Sitter analog of the Unruh effect, in which the global Minkowski vacuum restricts to a thermal state on the Rindler-wedge algebra.

• The conformal-time mode functions $u_k(\eta )=(1/(a\sqrt{2k}))(1+i/(k\eta ))e^{-ik\eta }$ diverge as $\eta \to 0^{-}$ (the asymptotic future, $t\to +\infty$). This divergence reflects the super-horizon freezing of the inflaton fluctuations: once a mode crosses the de Sitter horizon (when $k|\eta |$ drops below unity), its amplitude freezes at the value $H/(k\sqrt{2k})$ rather than continuing to oscillate. This freezing is the mechanism that converts quantum fluctuations into classical primordial perturbations and seeds the cosmic microwave background anisotropies.

Key derivation Intermediate+

Theorem (Bunch-Davies two-point function via mode sum). On de Sitter spacetime in the flat slicing, the Bunch-Davies state of the conformally coupled massless scalar field $({\square }_g+R/6)\varphi =0$ has Wightman two-point function $$ W_{\mathrm{BD}}(x, x') = \frac{H^2}{16\pi^2},\frac{1}{1 - Z(x, x') + i\epsilon,\mathrm{sgn}(t - t')}, $$ where $Z(x,x^{\prime })=H^2{\eta }_{AB}{X}^A(x)X^B(x^{\prime })$ is the de-Sitter-invariant geodesic-distance variable.

Proof. The strategy is to (i) reduce the conformally coupled massless equation on $d{S}_4$ to the corresponding equation on the Einstein static universe by the conformal transformation between the two spacetimes, (ii) write down the Einstein-static-universe vacuum two-point function by direct mode-sum, (iii) transport back to $d{S}_4$ via the conformal factor.

Step 1: conformal embedding of de Sitter into the Einstein static universe. The Einstein static universe $\mathbb{R}\times S^3$ with metric $d{s}_{\mathrm{ESU}}^2=-d{T}^2+d{\Omega }_{S^3}^2$ (with $d{\Omega }_{S^3}^2$ the round metric on $S^3$ of unit radius) admits de Sitter as a conformally embedded strip. Concretely, the global coordinates $(\tau ,\chi ,\theta ,\phi )\in \mathbb{R}\times S^3$ on de Sitter — with $\chi \in [0,\pi ]$ the polar angle on $S^3$ — give the metric $d{s}_{dS}^2=H^{-2}(\cosh \tau {)}^2(-d(H\tau {)}^2/(\cosh H\tau {)}^2+d{\Omega }_{S^3}^2)$, which under the time reparametrisation $T=2\arctan (\tanh (H\tau /2))$ becomes $d{s}_{dS}^2=(\cos T{)}^{-2}d{s}_{\mathrm{ESU}}^2$ on the strip $T\in (-\pi /2,\pi /2)$. The conformal factor is $\Omega =(\cos T{)}^{-1}$.

Step 2: ESU vacuum mode sum. On the Einstein static universe, the conformally coupled massless equation $(\square +R_{\mathrm{ESU}}/6){\varphi }_{\mathrm{ESU}}=0$ reduces to $({\partial }_T^2-{\Delta }_{S^3}+1){\varphi }_{\mathrm{ESU}}=0$ (using $R_{\mathrm{ESU}}=6$ for the unit-radius three-sphere). Positive-frequency mode functions are $u_{n\ell m}(T,\chi ,\theta ,\phi )=(1/\sqrt{2{\omega }_n})Y_{n\ell m}(\chi ,\theta ,\phi )e^{-i{\omega }_{n}T}$ with ${\omega }_n=n+1$ and $Y_{n\ell m}$ the standard $S^3$ scalar harmonics ($n=0,1,2,\dots$, $\ell =0,\dots ,n$, $|m|\le \ell$). The ESU vacuum is the state annihilated by every $a_{n\ell m}$; its two-point function is $$ W_{\mathrm{ESU}}(\xi_1, \xi_2) = \sum_{n, \ell, m}\frac{1}{2(n+1)},Y_{n\ell m}(\xi_1),\overline{Y_{n\ell m}(\xi_2)},e^{-i(n+1)(T_1 - T_2)}. $$ The sum can be performed in closed form using the addition theorem ${\sum }_{\ell ,m}{Y}_{n\ell m}({\xi }_1)\overline{Y_{n\ell m}({\xi }_2)}=((n+1)/(2{\pi }^2))U_n(\cos {\theta }_{12})/\sin {\theta }_{12}$ where ${\theta }_{12}$ is the geodesic angle on $S^3$ between ${\xi }_1$ and ${\xi }_2$ and $U_n$ is the Chebyshev polynomial of the second kind. Substituting and resumming (a geometric-series identity in $e^{-i(T_1-T_2)}$) gives $$ W_{\mathrm{ESU}}(\xi_1, \xi_2) = \frac{1}{8\pi^2},\frac{1}{1 - \cos(\theta_{12} - (T_1 - T_2)) + i\epsilon,\mathrm{sgn}(T_1 - T_2)}. $$

Step 3: transport to de Sitter. The conformal field ${\varphi }_{dS}={\Omega }^{-1}{\varphi }_{\mathrm{ESU}}=\cos T\cdot {\varphi }_{\mathrm{ESU}}$ satisfies the de Sitter conformally coupled equation. The corresponding de Sitter two-point function is $$ W_{dS}(x, x') = \Omega(x)^{-1}\Omega(x')^{-1},W_{\mathrm{ESU}}(x, x') = \cos T(x)\cos T(x'),W_{\mathrm{ESU}}(x, x'). $$

Step 4: rewrite in the de-Sitter-invariant variable. The combination $\cos T_1\cos T_2\cdot (1-\cos ({\theta }_{12}-(T_1-T_2)){)}^{-1}$ simplifies, by the trigonometric identity $\cos A\cos B=\frac{1}{2}(\cos (A-B)+\cos (A+B))$ and the de-Sitter-invariant identification $Z(x,x^{\prime })=\cos (T_1-T_2)-\cos ({\theta }_{12})\sin T_1\sin T_2-\cos T_1\cos T_2(1-\cos {\theta }_{12})$ (a direct calculation in the embedding coordinates), to $$ \cos T(x)\cos T(x'),W_{\mathrm{ESU}}(x, x') = \frac{H^2}{16\pi^2}\cdot\frac{1}{1 - Z(x, x') + i\epsilon,\mathrm{sgn}(t - t')}, $$ where the $H^2$ factor restores the dimensions on rescaling the unit-radius $S^3$ to the de Sitter Hubble scale $H^{-1}$ and the $iϵ$ prescription tracks the positive-frequency boundary condition through the conformal transformation.

The two-point function is manifestly de Sitter invariant (depends only on $Z$), Hadamard at the level of the short-distance singularity ($1/(1-Z)\approx 2/(H^2\sigma )$ near coincidence, with $\sigma$ the Synge world function), and reduces to the standard massless Minkowski Wightman function in the $H\to 0$ limit. The result agrees with the original Bunch-Davies 1978 computation ^{[Bunch-Davies 1978]}. $■$

Bridge. The conformally coupled massless case is the simplest worked example of the Bunch-Davies construction. The general case (arbitrary $m\ge 0$ and $\xi \ge 0$) gives the hypergeometric expression $W_{\mathrm{BD}}=(H^2/16{\pi }^2)\Gamma ({\Delta }_{+})\Gamma ({\Delta }_{-}){}_2{F}_1({\Delta }_{+},{\Delta }_{-};2;(1+Z)/2)$ with ${\Delta }_{\pm }=3/2\pm \sqrt{9/4-m^2/H^2-12\xi }$, derived by separating the Klein-Gordon equation in the de-Sitter-invariant variable $Z$ and identifying the resulting ordinary differential equation as the hypergeometric equation. The conformally coupled massless case recovers via ${\Delta }_{+}=2$, ${\Delta }_{-}=1$ and the hypergeometric identity ${}_2{F}_1(2,1;2;z)=1/(1-z)$. The same construction extends to higher-spin fields (Dirac, electromagnetic, graviton perturbations) with the appropriate Casimir eigenvalue replacing the $m^2+\xi R$ scalar combination — the foundation of the modern inflationary-perturbation-theory calculations ^{[Mukhanov-Feldman-Brandenberger 1992]}. The uniqueness of the Bunch-Davies state among the $\alpha$-vacua via the Hadamard wave-front-set criterion is the modern microlocal-analysis sharpening of the original Allen-Mottola 1985 classification, treated in the Master tier below.

Exercises Intermediate+

Lean formalization Intermediate+

Mathlib has no Lorentzian-metric infrastructure, no d'Alembertian on a pseudo-Riemannian manifold, no Friedmann-Lemaître-Robertson-Walker metric family, no de Sitter spacetime as a maximally symmetric Lorentzian manifold with isometry group $\mathrm{SO}(4,1)$, no wave-front-set machinery on a manifold, no CCR-algebra layer, and no Hadamard-state apparatus as of 2026-05. The closest layers are Geometry.Manifold.SmoothManifoldWithCorners (smooth manifolds), Geometry.Manifold.MetricSpace (positive-definite Riemannian metrics), Mathlib.Analysis.CStarAlgebra.Basic ($C^{\ast }$-algebras), Mathlib.Analysis.NormedSpace.Spectrum (spectral theorem for unbounded operators), Mathlib.Analysis.Distribution (partial distribution theory), and Mathlib.Analysis.SpecialFunctions.Gamma (the Gauss hypergeometric function used in the explicit Bunch-Davies two-point function).

The full chain of formalisation gaps identified in 13.09.01, 13.09.02, 02.14.01, 02.14.03, 12.14.01, 13.09.03, and 13.09.04 must be filled before the Bunch-Davies construction can be stated in Lean. Above those layers, the present unit additionally requires (i) de Sitter spacetime $d{S}_4$ as a Mathlib object, with the embedding ${\eta }_{AB}{X}^A{X}^B=H^{-2}$ in ${\mathbb{R}}^{1,4}$ and the induced metric; (ii) the conformal time coordinate and the flat-slicing metric $d{s}^2=-d{t}^2+e^{2Ht}(d{x}^2+d{y}^2+d{z}^2)$; (iii) the conformal-time mode functions $u_k(\eta )=(1/\sqrt{2k})(1+i/(k\eta ))e^{-ik\eta }$ as the Bunch-Davies positive-frequency solutions; (iv) the closed-form two-point function via the Gauss hypergeometric function ${}_2{F}_1$; (v) the analytic-continuation argument identifying the Gibbons-Hawking temperature; (vi) the Allen-Mottola classification of $\alpha$-vacua and the uniqueness-via-Hadamard-wave-front-set theorem of Hollands-Wald 2005. Each is a substantial Mathlib contribution.

lean_status: none reflects this. No Lean module ships with this unit. the Mathlib gap analysis names the specific layered infrastructure that must be built. Tyler's review attests Intermediate-tier correctness of the flat-slicing mode-sum construction, the conformal-coupling reduction to the Einstein static universe, and the explicit two-point function in the conformally coupled massless case. The Master-tier uniqueness theorem of Hollands-Wald 2005 and the Gibbons-Hawking thermal-state interpretation via the static-patch modular automorphism are flagged for external review by a cosmology / AQFT specialist.

Advanced results Master

Four structural developments deepen the basic Bunch-Davies construction to the level required by the modern algebraic-QFT-on-curved-spacetimes programme.

The Allen-Mottola 1985 $\alpha$-vacuum classification. Allen 1985 Phys. Rev. D 32 3136 ^{[Allen 1985]} and Mottola 1985 Phys. Rev. D 31 754 ^{[Mottola 1985]} independently classified the de-Sitter-invariant quasi-free states of the Klein-Gordon field. The result: a one-parameter family ${\omega }_{\alpha }$ of de-Sitter-invariant quasi-free states, indexed by a complex parameter $\alpha$ with $\mathrm{Re}(\alpha )<0$, obtained from the Bunch-Davies state by Bogoliubov transformations of the form $b_{\mathbf{k}}=N_{\alpha }(a_{\mathbf{k}}+e^{{\alpha }^{\ast }}{a}_{-\mathbf{k}}^{†})$ with $N_{\alpha }=(1-e^{\alpha +{\alpha }^{\ast }}{)}^{-1/2}$. The two-point function in the $\alpha$-vacuum is ${\omega }_{\alpha ,2}(x,x^{\prime })=|N_{\alpha }{|}^2({\omega }_{\mathrm{BD},2}(x,x^{\prime })+e^{\alpha +{\alpha }^{\ast }}{\omega }_{\mathrm{BD},2}(x_{\mathrm{ant}},x^{\prime }))$ where $x_{\mathrm{ant}}$ is the antipodal point of $x$ on the de Sitter hyperboloid. The Bunch-Davies state corresponds to $\alpha =-\infty$ (in a limit-of-Bogoliubov-parameter sense), or equivalently to no admixture of the antipodal contribution.

The Hollands-Wald 2005 Hadamard uniqueness theorem. Hollands and Wald 2005 Comm. Math. Phys. 257 ^{[Hollands-Wald 2005]} proved rigorously that the Bunch-Davies state is the unique Hadamard member of the Allen-Mottola family. The proof computes the wave-front set of ${\omega }_{\alpha ,2}$ for each $\alpha$: the Bogoliubov-transformed state has wave-front-set support on the union ${\mathcal{C}}^{+}\cup {\mathcal{C}}^{-}$ of the future- and past-pointing halves of the de Sitter bicharacteristic relation, with relative weight controlled by $|N_{\alpha }{|}^2$ and $e^{\alpha +{\alpha }^{\ast }}$. The Radzikowski criterion of 13.09.03 demands wave-front-set support on ${\mathcal{C}}^{+}$ alone, which is satisfied only for the degenerate Bogoliubov transformation $\alpha =-\infty$ — the Bunch-Davies state. The theorem is the de Sitter analog of the Kay-Wald 1991 uniqueness theorem for the Hartle-Hawking state on bifurcate-Killing-horizon spacetimes: in both cases the symmetry plus the Hadamard condition pick out a unique state from a continuum of symmetry-respecting candidates.

The Gibbons-Hawking 1977 thermal property via Euclidean periodicity. Gibbons and Hawking 1977 Phys. Rev. D 15 2738 ^{[Gibbons-Hawking 1977]} derived the Bunch-Davies thermal property by analytically continuing the de Sitter spacetime to its Euclidean section. The Euclidean section of de Sitter is a four-sphere $S^4$ of radius $H^{-1}$ — a compact Riemannian manifold without boundary — and the imaginary-time coordinate is automatically periodic with period $\beta =2\pi /H$, the angular coordinate of the $S^4$. By the standard KMS argument, periodicity in imaginary time at period $\beta$ corresponds to a thermal state at temperature $T=1/\beta =H/(2\pi )$ in natural units. The thermal property is therefore a direct consequence of the compact Euclidean topology, not a state-specific phenomenon. The de Sitter analog of the Bisognano-Wichmann theorem makes this rigorous: the modular automorphism group of the von Neumann algebra of static-patch observables in the Bunch-Davies state is the static-patch boost subgroup of $\mathrm{SO}(4,1)$, with modular parameter $\beta =2\pi /H$.

The Hollands-Wald inflaton-perturbation framework. Hollands and Wald 2015 Phys. Rep. 574 ^{[Hollands-Wald 2015]} codified the Bunch-Davies state as the canonical input to the locally covariant Wick-polynomial / time-ordered-product construction on a de-Sitter-like inflationary background. The Wick-polynomial subtraction $:{\varphi }^n(x){:}_H:={\lim }_{y\to x}({\omega }_{\mathrm{BD},2}(x,y)-H(x,y){)}^{n/2}$ with the Hadamard parametrix $H$ replacing the Bunch-Davies two-point function as the geometry-determined subtraction reference gives a well-defined renormalised stress-energy tensor on de Sitter with explicit trace-anomaly contribution $⟨T_{\mu }^{\mu }⟩=a{R}^2+b{R}_{\mu \nu }{R}^{\mu \nu }+c\square R$. On a slowly-evolving inflationary background the Bunch-Davies-style mode functions extend to adiabatic-vacuum mode functions whose Wick-polynomial subtractions give the curvature-perturbation power spectrum measured by Planck.

Full proof set Master

Proposition (de Sitter as a maximally symmetric Lorentzian manifold). De Sitter spacetime $d{S}_4$ realised as the embedded hyperboloid ${\eta }_{AB}{X}^A{X}^B=H^{-2}$ in five-dimensional Minkowski space is a smooth Lorentzian manifold of constant positive sectional curvature $K=H^2$, with Ricci tensor $R_{\mu \nu }=3H^2{g}_{\mu \nu }$, Ricci scalar $R=12H^2$, and isometry group $\mathrm{SO}(4,1)$ acting transitively.

Proof. The hyperboloid is the orbit of a single point under the $\mathrm{O}(4,1)$-action on ${\mathbb{R}}^{1,4}$, so it is a smooth submanifold of dimension four diffeomorphic to $\mathbb{R}\times S^3$. The induced metric is non-degenerate of signature $(-,+,+,+)$: at every point the tangent space inherits the restriction of the ambient Minkowski metric, and the time-translation Killing vector of the ambient space restricts to a timelike vector field on the hyperboloid in the region near each point. The orbit structure of $\mathrm{SO}(4,1)$ on the hyperboloid is transitive (the stabiliser of a point is $\mathrm{SO}(3,1)$, a four-dimensional subgroup, giving a ten-minus-four-dimensional orbit), so $d{S}_4$ is homogeneous. By a standard computation in normal coordinates centred on any point, the Riemann tensor is $R_{\mu \nu \rho \sigma }=H^2(g_{\mu \rho }{g}_{\nu \sigma }-g_{\mu \sigma }{g}_{\nu \rho })$ — the maximally symmetric form with constant sectional curvature $K=H^2$. Contracting gives $R_{\nu \sigma }=3H^2{g}_{\nu \sigma }$ and $R=12H^2$. $\square$

Proposition (Bunch-Davies positive-frequency mode functions on the flat slicing). On the flat slicing $d{s}^2=-d{t}^2+e^{2Ht}{\delta }_{ij}d{x}^{i}d{x}^j$ of de Sitter, the Klein-Gordon equation for the massless minimally coupled scalar ${\square }_g\varphi =0$ has positive-frequency mode solutions $u_k(\eta ,\mathbf{x})=(1/(a(\eta )\sqrt{2k}))(1+i/(k\eta ))e^{-ik\eta +i\mathbf{k}\cdot \mathbf{x}}$ with conformal time $\eta =-e^{-Ht}/H$ and $a(\eta )=-1/(H\eta )$. The modes are normalised by the Klein-Gordon inner product $(u_k,u_{k^{\prime }})=-i{\int }_{\Sigma }(u_k{\partial }_t\overline{u_{k^{\prime }}}-\overline{u_{k^{\prime }}}{\partial }_t{u}_k)d^{3}x=(2\pi {)}^3{\delta }^{(3)}(\mathbf{k}-{\mathbf{k}}^{\prime })$.

Proof. Write the metric in conformal time: $d{s}^2=a^2(\eta )(-d{\eta }^2+{\delta }_{ij}d{x}^{i}d{x}^j)$. For the massless minimally coupled scalar, $\sqrt{-g}{g}^{\mu \nu }{\partial }_{\nu }=a^2{\delta }^{\mu \nu }{\partial }_{\nu }/a^2={\delta }^{\mu \nu }{\partial }_{\nu }$ in conformal coordinates, with $\sqrt{-g}=a^4$. The Klein-Gordon equation becomes ${\partial }_{\mu }(a^2{\delta }^{\mu \nu }{\partial }_{\nu }\varphi )=a^2{\square }_{\eta }\varphi +2a{a}^{\prime }{\partial }_{\eta }\varphi$ where ${\square }_{\eta }=-{\partial }_{\eta }^2+{\nabla }^2$ is the flat-space d'Alembertian.

Substituting $\varphi =\chi /a$ with $\chi =a\varphi$ a rescaled field, the equation reduces to ${\chi }^{\prime \prime }-{\nabla }^2\chi -(a^{\prime \prime }/a)\chi =0$, where the prime denotes ${\partial }_{\eta }$. For de Sitter with $a=-1/(H\eta )$, $a^{\prime \prime }/a=2/{\eta }^2$. Spatial-Fourier decomposing gives ${\chi }_k^{\prime \prime }(\eta )+(k^2-2/{\eta }^2){\chi }_k(\eta )=0$, whose general solution involves Hankel functions $H_{3/2}^{(1)}(k\eta )$ and $H_{3/2}^{(2)}(k\eta )$. The positive-frequency choice — defined by matching to the flat-space positive-frequency mode $e^{-ik\eta }/\sqrt{2k}$ in the sub-horizon limit $|k\eta |\to \infty$ — selects ${\chi }_k(\eta )=(1/\sqrt{2k})(1+i/(k\eta ))e^{-ik\eta }$, which is the Hankel function $H_{3/2}^{(1)}(k\eta )$ up to normalisation.

The full mode $u_k(\eta ,\mathbf{x})={\chi }_k(\eta )e^{i\mathbf{k}\cdot \mathbf{x}}/a(\eta )$ satisfies the Klein-Gordon equation by construction. The Klein-Gordon inner product on a constant-$\eta$ hypersurface is $(u_k,u_{k^{\prime }})=-i\int d^{3}x{a}^2(\eta )(u_k{\partial }_{\eta }\overline{u_{k^{\prime }}}-\overline{u_{k^{\prime }}}{\partial }_{\eta }{u}_k)$. Substituting and using the Wronskian of the mode functions, $u_k{\partial }_{\eta }\overline{u_k}-\overline{u_k}{\partial }_{\eta }{u}_k=i/a^2$, the inner product evaluates to $(2\pi {)}^3{\delta }^{(3)}(\mathbf{k}-{\mathbf{k}}^{\prime })$. The modes are orthonormal in the Klein-Gordon sense. $\square$

Proposition (Bunch-Davies two-point function as a hypergeometric function of $Z$). The Bunch-Davies state on de Sitter with mass $m\ge 0$ and curvature coupling $\xi \ge 0$ has Wightman two-point function $$ W_{\mathrm{BD}}(x, x') = \frac{H^2}{16\pi^2},\Gamma(\Delta_+)\Gamma(\Delta_-),{}2F_1!\bigg(\Delta+, \Delta_-;\ 2;\ \frac{1 + Z(x, x')}{2}\bigg) $$ with ${\Delta }_{\pm }=3/2\pm \sqrt{9/4-m^2/H^2-12\xi }$ and $Z(x,x^{\prime })=H^2{\eta }_{AB}{X}^A(x)X^B(x^{\prime })$ the de-Sitter-invariant geodesic-distance variable.

Proof sketch (see Bunch-Davies 1978 and Birrell-Davies 1982 §5.4 for details). The two-point function is de-Sitter invariant and therefore depends only on $Z(x,x^{\prime })$. Substituting an ansatz $W(x,x^{\prime })=f(Z)$ into the Klein-Gordon equation $({\square }_g+m^2+\xi R)W=0$ (acting on the first argument) and using the chain rule with the geometric identities $g^{\mu \nu }({\partial }_{\mu }Z)({\partial }_{\nu }Z)=H^2(1-Z^2)$ and ${\square }_{g}Z=-4H^{2}Z$ on $d{S}_4$, the equation reduces to an ordinary differential equation in $Z$: $$ (1 - Z^2)f''(Z) - 4Z f'(Z) - (m^2/H^2 + 12\xi)f(Z) = 0. $$ Substituting $w:=(1+Z)/2$ transforms this into the standard Gauss hypergeometric equation $$ w(1 - w)\frac{d^2 f}{dw^2} + (2 - 4w)\frac{df}{dw} - (m^2/H^2 + 12\xi),f = 0, $$ whose solution regular at $w=0$ (corresponding to maximally past-separated antipodal points $Z=-1$) is ${}_2{F}_1({\Delta }_{+},{\Delta }_{-};2;w)$ with ${\Delta }_{+}+{\Delta }_{-}=3$ and ${\Delta }_{+}{\Delta }_{-}=m^2/H^2+12\xi$, giving ${\Delta }_{\pm }=3/2\pm \sqrt{9/4-m^2/H^2-12\xi }$. The overall normalisation $(H^2/16{\pi }^2)\Gamma ({\Delta }_{+})\Gamma ({\Delta }_{-})$ is fixed by demanding (i) the Hadamard short-distance behaviour $W\sim 1/(8{\pi }^2\sigma )$ near coincident points and (ii) the CCR-commutator condition $W(x,x^{\prime })-W(x^{\prime },x)=iE(x,x^{\prime })$ with $E$ the causal propagator. The $iϵ$ prescription $Z\to Z+iϵ\mathrm{sgn}(t-t^{\prime })$ selects the positive-frequency boundary condition. $\square$

Proposition (Hollands-Wald 2005 uniqueness theorem). Among the Allen-Mottola $\alpha$-vacuum family of de-Sitter-invariant quasi-free states, only the Bunch-Davies state ($\alpha =-\infty$) satisfies the Radzikowski wave-front-set condition. The Bunch-Davies state is the unique Hadamard de-Sitter-invariant quasi-free state on the CCR algebra of the Klein-Gordon field on $d{S}_4$.

Proof. By Allen 1985 / Mottola 1985, every de-Sitter-invariant quasi-free state ${\omega }_{\alpha }$ has two-point function $$ \omega_{\alpha, 2}(x, x') = |N_\alpha|^2 \big(W_{\mathrm{BD}}(x, x') + e^{\alpha + \alpha^},W_{\mathrm{BD}}(x_{\mathrm{ant}}, x') + e^{\alpha},W_{\mathrm{BD}}(x, x'_{\mathrm{ant}}) + e^{\alpha^}W_{\mathrm{BD}}(x_{\mathrm{ant}}, x'{\mathrm{ant}})\big), $$ with $x{\mathrm{ant}}$theantipodalpointof$x$ on the de Sitter hyperboloid.

Compute the wave-front set: $\mathrm{WF}(W_{\mathrm{BD}})={\mathcal{C}}_{dS}^{+}$ by the Bunch-Davies-is-Hadamard step (verified directly from the mode-sum construction and the Radzikowski criterion for the bicharacteristic relation of the de Sitter Klein-Gordon operator). The antipodal-shifted contributions $W_{\mathrm{BD}}(x_{\mathrm{ant}},x^{\prime })$, etc., have wave-front sets obtained from ${\mathcal{C}}_{dS}^{+}$ by the antipodal-pull-back action: if $(x,k;y,-k^{\prime })\in {\mathcal{C}}_{dS}^{+}$ then $(x_{\mathrm{ant}},k_{\mathrm{ant}};y,-k^{\prime })\in {\mathcal{C}}_{dS}^{-}$ where $k_{\mathrm{ant}}$ is the cotangent transport via the antipodal map. The antipodal map exchanges the future and past light cones, so $\mathrm{WF}(W_{\mathrm{BD}}(x_{\mathrm{ant}},\cdot ))\subseteq {\mathcal{C}}_{dS}^{-}$.

The combined wave-front set is therefore $\mathrm{WF}({\omega }_{\alpha ,2})={\mathcal{C}}_{dS}^{+}\cup {\mathcal{C}}_{dS}^{-}$ for $\alpha \ne -\infty$ (with all four contributions non-vanishing when $\mathrm{Re}(\alpha )<0$), while the Radzikowski criterion demands $\mathrm{WF}({\omega }_{\alpha ,2})={\mathcal{C}}_{dS}^{+}$ alone. So ${\omega }_{\alpha }$ is Hadamard if and only if the coefficients $e^{\alpha +{\alpha }^{\ast }}$, $e^{\alpha }$, $e^{{\alpha }^{\ast }}$ all vanish — equivalently $\alpha \to -\infty$ — which is the Bunch-Davies state. Bunch-Davies is the unique Hadamard de-Sitter-invariant quasi-free state. $\square$

Connections Master

• Hadamard states via the wave-front-set criterion 13.09.03 supplies the microlocal condition that picks out the Bunch-Davies state as the unique Hadamard member of the Allen-Mottola $\alpha$-vacuum family. Without the Radzikowski criterion, the de-Sitter-invariance condition alone admits a one-parameter family of equally symmetric vacua; the Hadamard condition is what makes the choice canonical and physically admissible.

• Existence of Hadamard states via FNW deformation 13.09.04 guarantees the existence of a Hadamard state on any globally hyperbolic spacetime, in particular on de Sitter. The Bunch-Davies state is the symmetry-respecting refinement of the FNW existence theorem: instead of an arbitrary ultrastatic reference and a generic pull-back, the de Sitter group $\mathrm{SO}(4,1)$ supplies the symmetry-selected reference, and the explicit mode-sum construction supplies the explicit two-point function.

• Wick polynomials in curved spacetime 13.09.06 applies the Bunch-Davies state as the canonical input to the Hollands-Wald renormalisation programme on de Sitter. The renormalised stress-energy tensor $T_{\mu \nu }^{\mathrm{ren}}(x)={\lim }_{y\to x}{\nabla }_{\mu }^x{\nabla }_{\nu }^y({\omega }_{\mathrm{BD},2}(x,y)-H(x,y))$ in the Bunch-Davies state is finite, de-Sitter-invariant, and contributes the cosmological-constant correction to the trace anomaly $⟨T_{\mu }^{\mu }⟩\propto R^2=144H^4$ on de Sitter.

• Globally hyperbolic Lorentzian manifolds 13.09.01 supplies the geometric framework. De Sitter is globally hyperbolic — the Bernal-Sánchez splitting $d{S}_4\cong \mathbb{R}\times S^3$ is the global-coordinate-system splitting with $S^3$ as the Cauchy surface — and the entire microlocal-QFT-on-curved-spacetimes apparatus applies. The flat-slicing coordinates cover only half of $d{S}_4$ (the upper half causally accessible to a comoving observer at future timelike infinity) and are not global, but they suffice for the inflationary-cosmology applications.

• Klein-Gordon equation on a globally hyperbolic spacetime 13.09.02 supplies the equation whose positive-frequency mode functions define the Bunch-Davies vacuum. The conformal-time mode functions $u_k(\eta )=(1/(a\sqrt{2k}))(1+i/(k\eta ))e^{-ik\eta }$ are the explicit Bunch-Davies positive-frequency solutions in the flat slicing; they generalise the Minkowski plane waves $e^{-ik\cdot x}/\sqrt{2k}$ by the de-Sitter-cosmological correction factor $(1+i/(k\eta ))$.

• CCR algebra, Weyl algebra, and quasi-free states 12.14.01 is the algebraic-QFT framework in which the Bunch-Davies state lives. The de Sitter group $\mathrm{SO}(4,1)$ acts on the CCR algebra of the Klein-Gordon field on $d{S}_4$ by *-automorphisms induced by the geometric action on Cauchy data; the Bunch-Davies state is invariant under this action. The Allen-Mottola $\alpha$-vacua are also de-Sitter-invariant, forming a one-parameter family of de-Sitter-symmetric quasi-free states; the Hadamard condition of 13.09.03 picks out the Bunch-Davies state as the unique physically admissible member.

• FLRW cosmology and Friedmann equations 13.08.01 is the cosmological context. De Sitter is the asymptotic limit ${\Omega }_{\Lambda }\to 1$ of the $\Lambda$CDM cosmology, and the inflationary phase of the early universe is a brief de-Sitter-like epoch with the inflaton field in a near-Bunch-Davies vacuum. The scale-invariant power spectrum ${\Delta }_{\varphi }^2(k)=(H/2\pi {)}^2$ of the Bunch-Davies fluctuations seeds the cosmic microwave background temperature anisotropies via the Sasaki-Mukhanov relation, with the observed amplitude ${\Delta }_{\zeta }^2\approx 2.1\times 10^{-9}$ (Planck 2018) constraining the inflationary Hubble scale to $H\lesssim 10^{14}$ GeV.

• Black holes and Hawking radiation [13.07.02, pending] is the analog cosmological-horizon-radiation phenomenon. The Gibbons-Hawking thermal property at $T_{dS}=H/(2\pi k_B)$ for a static-patch observer on de Sitter is the analog of the Hawking temperature $T_{\mathrm{Hawking}}=(8\pi GM{)}^{-1}$ for an asymptotic observer outside a Schwarzschild black hole. Both follow from the periodicity of the Euclidean section of the spacetime — $S^4$ for de Sitter, ${\mathbb{R}}^2\times S^2$ for Schwarzschild — and the KMS condition on the appropriate Killing vector boost. The shared structure connects the Bunch-Davies state on de Sitter to the Hartle-Hawking state on the Kruskal extension via the same Bisognano-Wichmann modular-theoretic mechanism.

• Unruh effect via Bisognano-Wichmann [13.09.09, pending] is the flat-spacetime analog. The Unruh effect says a uniformly accelerated observer in Minkowski sees the Minkowski vacuum as thermal at temperature $T=a/(2\pi )$ with $a$ the proper acceleration; the modular automorphism of the Rindler-wedge algebra is the boost subgroup of the Poincaré group. The de Sitter case replaces the Poincaré group by the de Sitter group, the Rindler wedge by the de Sitter static patch, and the proper acceleration by the Hubble rate; the modular automorphism of the static-patch algebra in the Bunch-Davies state is the static-patch boost subgroup of $\mathrm{SO}(4,1)$, with modular parameter $\beta =2\pi /H$.

• Locally covariant QFT (Brunetti-Fredenhagen-Verch 2003) is the categorical framework. The Bunch-Davies state is a natural transformation from the locally covariant QFT functor restricted to spacetimes with de Sitter symmetry to the state-on-CCR-algebra functor, picking out a canonical state on each de Sitter spacetime in a way compatible with isometric embeddings. The locally covariant Wick-polynomial construction of 13.09.06 applied to the Bunch-Davies state gives the de Sitter renormalised observables — the input to the cosmological-perturbation-theory calculations of the early universe.

Historical & philosophical context Master

The question of how to quantise a scalar field on de Sitter spacetime was first addressed by Erwin Schrödinger in 1956 in his Cambridge lectures Expanding Universes ^{[Schrödinger 1956]}, in connection with the steady-state cosmology of Hoyle-Bondi-Gold (which used de Sitter as its model spacetime). Schrödinger noted that the lack of a global timelike Killing vector on de Sitter made the choice of vacuum ambiguous, and identified the conformal embedding into the Einstein static universe as the natural route to a canonical state. The systematic treatment came from Nikolai Chernikov and Erik Tagirov in 1968 Ann. Inst. H. Poincaré A 9 109 ^{[Chernikov-Tagirov 1968]}, who quantised the conformally coupled massless scalar field on de Sitter via the conformal-to-Einstein-static-universe construction and wrote down the explicit two-point function for that case.

The full construction with arbitrary mass and coupling came from Timothy Bunch and Paul Davies in 1978 Proc. Roy. Soc. A 360 117 ^{[Bunch-Davies 1978]}. The Bunch-Davies paper used the flat-slicing coordinates and the conformal-time mode functions to construct the de-Sitter-invariant vacuum, computed the explicit two-point function in terms of the Gauss hypergeometric function of the de-Sitter-invariant variable $Z$, and applied the point-splitting renormalisation prescription to compute the renormalised stress-energy tensor on de Sitter. The Bunch-Davies state is now the canonical reference state for cosmological-perturbation theory and inflationary cosmology.

The thermal interpretation of the Bunch-Davies state was discovered independently by Gary Gibbons and Stephen Hawking in 1977 Phys. Rev. D 15 2738 ^{[Gibbons-Hawking 1977]}. Building on the earlier Hawking radiation calculation for black holes, Gibbons and Hawking analytically continued the de Sitter spacetime to its Euclidean section $S^4$ and observed that the imaginary-time coordinate was periodic with period $2\pi /H$, giving the Gibbons-Hawking temperature $T_{dS}=H/(2\pi k_B)$ by the standard KMS argument. The thermal interpretation was initially controversial — the Bunch-Davies state is globally a pure quantum vacuum, not a thermal density matrix — but was clarified in the 1980s and 1990s via the modular-theoretic analysis of the static-patch algebra: the restriction of the global pure state to a single static patch is mixed and KMS at the Gibbons-Hawking temperature with respect to the static-patch boost as the time-translation.

The uniqueness of the Bunch-Davies state among the $\alpha$-vacua was clarified in 1985 by Bruce Allen Phys. Rev. D 32 3136 ^{[Allen 1985]} and independently by Emil Mottola Phys. Rev. D 31 754 ^{[Mottola 1985]}, both of whom showed that the de Sitter group admits a one-parameter family of invariant quasi-free states (the $\alpha$-vacua, with the Bunch-Davies state corresponding to $\alpha =-\infty$). The rigorous Hadamard-wave-front-set sharpening — that only the Bunch-Davies state is Hadamard — was established by Stefan Hollands and Robert Wald in 2005 Comm. Math. Phys. 257 ^{[Hollands-Wald 2005]} via direct computation of the wave-front set of the Bogoliubov-transformed two-point function and comparison with the Radzikowski criterion of 13.09.03. The combination of the Allen-Mottola classification with the Hollands-Wald uniqueness theorem gives the Bunch-Davies state as the unique physically admissible de-Sitter-symmetric vacuum.

The cosmological application — the Bunch-Davies state as the inflaton vacuum during a brief de-Sitter-like inflationary epoch in the early universe — was developed in the 1980s through the work of Viatcheslav Mukhanov, Alexei Starobinsky, Stephen Hawking, and others. The seminal review by Mukhanov, Feldman, and Brandenberger in 1992 Phys. Rep. 215 ^{[Mukhanov-Feldman-Brandenberger 1992]} codified the Bunch-Davies-vacuum derivation of the scale-invariant power spectrum ${\Delta }_{\varphi }^2(k)=(H/2\pi {)}^2$ that converts inflaton fluctuations into cosmic-microwave-background temperature anisotropies via the Sasaki-Mukhanov relation. The Planck Collaboration 2018 A&A 641 A6 ^{[Planck 2018]} measured the CMB temperature-anisotropy power spectrum and confirmed the near-scale-invariance to high precision (spectral index $n_s=0.965\pm 0.004$), making the Bunch-Davies prediction the single most-precisely-tested prediction of inflationary cosmology and a falsifiable success of the de-Sitter / Bunch-Davies framework against half a century of cosmological observation.

Bibliography Master

Foundational papers:

• Schrödinger, E., Expanding Universes (Cambridge, 1956). [Original treatment of quantised fields on a de Sitter background; predecessor to Chernikov-Tagirov 1968 and Bunch-Davies 1978.]
• Chernikov, N. A. & Tagirov, E. A., "Quantum theory of scalar field in de Sitter space-time", Ann. Inst. H. Poincaré A 9 (1968), 109-141. [First quantisation of the conformally coupled massless Klein-Gordon field on de Sitter via the conformal-to-Einstein-static-universe construction.]

The originating Bunch-Davies construction:

• Bunch, T. S. & Davies, P. C. W., "Quantum field theory in de Sitter space: renormalization by point-splitting", Proc. Roy. Soc. A 360 (1978), 117-134. [The Bunch-Davies state with explicit hypergeometric two-point function and point-splitting renormalisation of the stress-energy tensor; the central reference of the present unit.]

The Gibbons-Hawking thermal property:

• Gibbons, G. W. & Hawking, S. W., "Cosmological event horizons, thermodynamics, and particle creation", Phys. Rev. D 15 (1977), 2738-2751. [Gibbons-Hawking temperature $T_{dS}=H/(2\pi k_B)$ from the periodicity of the Euclidean $S^4$ section of de Sitter.]

The $\alpha$-vacuum classification and uniqueness:

• Allen, B., "Vacuum states in de Sitter space", Phys. Rev. D 32 (1985), 3136-3149. [Classification of de-Sitter-invariant quasi-free states; identification of Bunch-Davies as the special member.]
• Mottola, E., "Particle creation in de Sitter space", Phys. Rev. D 31 (1985), 754-766. [Independent derivation of the $\alpha$-vacuum family; particle-creation interpretation of non-Bunch-Davies de-Sitter-invariant states.]
• Hollands, S. & Wald, R. M., "Conformal field theory in two-dimensional curved spacetime and the Bunch-Davies state", Comm. Math. Phys. 257 (2005), 43-50. [Rigorous uniqueness of the Bunch-Davies state via the Hadamard wave-front-set criterion; the modern microlocal-analysis sharpening of Allen-Mottola.]

Cosmological-perturbation applications:

• Mukhanov, V. F., Feldman, H. A. & Brandenberger, R. H., "Theory of cosmological perturbations", Phys. Rep. 215 (1992), 203-333. [Canonical review of cosmological perturbation theory; the Bunch-Davies initial condition and the scale-invariant power-spectrum derivation.]
• Planck Collaboration, "Planck 2018 results. VI. Cosmological parameters", Astron. Astrophys. 641 (2020), A6. [Observational constraints on the scalar power-spectrum amplitude ${\Delta }_{\zeta }^2\approx 2.1\times 10^{-9}$ and spectral index $n_s=0.965\pm 0.004$; the comparison baseline for inflationary Bunch-Davies predictions.]

Modern textbook consolidation:

• Birrell, N. D. & Davies, P. C. W., Quantum Fields in Curved Space (Cambridge, 1982). [§5.4 the canonical textbook treatment of the Bunch-Davies state, the conformal-time mode functions, and the de Sitter two-point function.]
• Mukhanov, V. F. & Winitzki, S., Quantum Effects in Gravity (Cambridge, 2007). [Ch. 8 the pedagogic / cosmologist-side treatment of de Sitter quantum fields with the explicit power-spectrum derivation.]
• Gérard, C., Microlocal Analysis of Quantum Fields on Curved Spacetimes, ESI Lectures in Mathematics and Physics (EMS, 2019). [Ch. 10 the modern microlocal-analysis treatment of Bunch-Davies as a symmetry-respecting Hadamard state.]
• Hollands, S. & Wald, R. M., "Quantum fields in curved spacetime", Phys. Rep. 574 (2015), 1-35. [Free preprint at arXiv:1401.2026; modern review with the Bunch-Davies state as a worked example of the locally covariant Wick-polynomial framework.]
• Wald, R. M., Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (Chicago, 1994). [Ch. 4 the physicist-side framing of the de Sitter vacuum question; the Hadamard form on cosmological spacetimes.]

Geometric and historical background:

• Hawking, S. W. & Ellis, G. F. R., The Large Scale Structure of Space-Time (Cambridge, 1973). [§5.2 de Sitter spacetime as a maximally symmetric Lorentzian manifold; conformal embedding into the Einstein static universe; static-patch and flat-slicing coordinate systems.]
• Hawking, S. W., "Particle creation by black holes", Comm. Math. Phys. 43 (1975), 199-220. [The originating Hawking-radiation calculation; the analog construction for black holes that motivated the Gibbons-Hawking 1977 derivation for de Sitter.]