Hartle–Hawking No-Boundary Proposal

• Hartle–Hawking No-Boundary Proposal is a quantum cosmology framework that defines the universe’s wave function using a gravitational path integral over smooth, compact, boundary-less Euclidean geometries.
• It employs semiclassical approximations and minisuperspace saddle-point methods to derive dominant instanton solutions, leading to predictions for the cosmological power spectrum and non-Gaussian signatures.
• Generalizations with additional Dirichlet boundaries create excited states, extending the holographic dictionary in de Sitter space and influencing late-time cosmological correlators.

The Hartle–Hawking no-boundary proposal is a foundational framework in quantum cosmology defining the ground state wave function of the universe through a gravitational path integral over smooth, compact, Euclidean geometries without an initial boundary. The proposal specifies both the quantum state and initial conditions for cosmological evolution in theories with a positive cosmological constant Λ. Recent generalizations extend the construction to families of excited states, especially in the context of de Sitter (dS) holography, by introducing extra Euclidean boundaries with Dirichlet conditions. This framework directly impacts the structure of late-time cosmological correlators and their non-Gaussianities, with implications for the cosmological power spectrum and holographic dictionaries in dS quantum gravity (Botta-Cantcheff et al., 20 Jun 2025).

1. Definition of the No-Boundary Proposal

The Hartle–Hawking no-boundary wave functional, Ψ_HH, is defined as a gravitational path integral: $$\Psi_{\mathrm{HH}}[h_{ij}, \varphi] = \int_{(g, \Phi)|_{\Sigma}=(h, \varphi)} \mathcal{D}g\,\mathcal{D}\Phi\, \exp \big[ -S_E[g, \Phi] \big]$$ where $S_E$ is the Euclidean action,

$$S_E[g, \Phi] = -\frac{1}{16\pi G} \int_{M_E} d^{d+1}x \sqrt{g}\, (R - 2\Lambda) + \int_{M_E} d^{d+1}x \sqrt{g}\, [\frac{1}{2} g^{\mu\nu}\partial_\mu\Phi\partial_\nu\Phi + V(\Phi)]$$

and the integration is over all smooth, compact Euclidean $(d+1)$-dimensional geometries $M_E$ that end on a Cauchy surface $\Sigma$ with prescribed three-metric $h_{ij}$ and matter profile $\varphi$. Crucially, no further boundary or singularity is permitted in the past (“no-boundary” condition) (Botta-Cantcheff et al., 20 Jun 2025).

2. Semiclassical Evaluation and Minisuperspace Saddle

In the semiclassical (WKB) approximation, path integrals localize on dominant saddle-point (instanton) solutions:

• In $d$ spatial dimensions and positive Λ, the dominant Euclidean solution is

$$ds_E^2 = d\chi^2 + a^2(\chi) d\Omega_d^2, \quad a(\chi) = \ell\sin(\chi/\ell), \quad \ell^2 = \frac{d(d-1)}{2\Lambda}$$

on the interval $S_E$0. The on-shell action is

$S_E$1

with $S_E$2 the volume of the $S_E$3-sphere. The corresponding wave function is approximately

$S_E$4

For a free scalar, expanding around this background leads to a Gaussian wavefunction in φ: $S_E$5 where $S_E$6 is the boundary-to-boundary Green function (Botta-Cantcheff et al., 20 Jun 2025).

3. Generalization to Excited States via Additional Dirichlet Boundary

To define a family of excited quantum states over the Hartle–Hawking ground state, one introduces an additional (inner) Dirichlet boundary $S_E$7 at Euclidean time $S_E$8 in the Euclidean region, fixing fields $S_E$9 at that location. The wavefunctional for the excited state is

$$S_E[g, \Phi] = -\frac{1}{16\pi G} \int_{M_E} d^{d+1}x \sqrt{g}\, (R - 2\Lambda) + \int_{M_E} d^{d+1}x \sqrt{g}\, [\frac{1}{2} g^{\mu\nu}\partial_\mu\Phi\partial_\nu\Phi + V(\Phi)]$$0

with $$S_E[g, \Phi] = -\frac{1}{16\pi G} \int_{M_E} d^{d+1}x \sqrt{g}\, (R - 2\Lambda) + \int_{M_E} d^{d+1}x \sqrt{g}\, [\frac{1}{2} g^{\mu\nu}\partial_\mu\Phi\partial_\nu\Phi + V(\Phi)]$$1 at $$S_E[g, \Phi] = -\frac{1}{16\pi G} \int_{M_E} d^{d+1}x \sqrt{g}\, (R - 2\Lambda) + \int_{M_E} d^{d+1}x \sqrt{g}\, [\frac{1}{2} g^{\mu\nu}\partial_\mu\Phi\partial_\nu\Phi + V(\Phi)]$$2, joined to Lorentzian evolution. In the semiclassical limit, the on-shell action generalizes to

$$S_E[g, \Phi] = -\frac{1}{16\pi G} \int_{M_E} d^{d+1}x \sqrt{g}\, (R - 2\Lambda) + \int_{M_E} d^{d+1}x \sqrt{g}\, [\frac{1}{2} g^{\mu\nu}\partial_\mu\Phi\partial_\nu\Phi + V(\Phi)]$$3

$$S_E[g, \Phi] = -\frac{1}{16\pi G} \int_{M_E} d^{d+1}x \sqrt{g}\, (R - 2\Lambda) + \int_{M_E} d^{d+1}x \sqrt{g}\, [\frac{1}{2} g^{\mu\nu}\partial_\mu\Phi\partial_\nu\Phi + V(\Phi)]$$4 is the HH vacuum two-point kernel, $$S_E[g, \Phi] = -\frac{1}{16\pi G} \int_{M_E} d^{d+1}x \sqrt{g}\, (R - 2\Lambda) + \int_{M_E} d^{d+1}x \sqrt{g}\, [\frac{1}{2} g^{\mu\nu}\partial_\mu\Phi\partial_\nu\Phi + V(\Phi)]$$5 a mixed kernel coupling outer and inner boundaries, and $$S_E[g, \Phi] = -\frac{1}{16\pi G} \int_{M_E} d^{d+1}x \sqrt{g}\, (R - 2\Lambda) + \int_{M_E} d^{d+1}x \sqrt{g}\, [\frac{1}{2} g^{\mu\nu}\partial_\mu\Phi\partial_\nu\Phi + V(\Phi)]$$6 is supported only on $$S_E[g, \Phi] = -\frac{1}{16\pi G} \int_{M_E} d^{d+1}x \sqrt{g}\, (R - 2\Lambda) + \int_{M_E} d^{d+1}x \sqrt{g}\, [\frac{1}{2} g^{\mu\nu}\partial_\mu\Phi\partial_\nu\Phi + V(\Phi)]$$7 (Botta-Cantcheff et al., 20 Jun 2025).

4. Structure of the Wavefunctional and n-Point Functions

Expressing the excited-state wavefunctional in momentum space for a free scalar: $$S_E[g, \Phi] = -\frac{1}{16\pi G} \int_{M_E} d^{d+1}x \sqrt{g}\, (R - 2\Lambda) + \int_{M_E} d^{d+1}x \sqrt{g}\, [\frac{1}{2} g^{\mu\nu}\partial_\mu\Phi\partial_\nu\Phi + V(\Phi)]$$8 where

$$S_E[g, \Phi] = -\frac{1}{16\pi G} \int_{M_E} d^{d+1}x \sqrt{g}\, (R - 2\Lambda) + \int_{M_E} d^{d+1}x \sqrt{g}\, [\frac{1}{2} g^{\mu\nu}\partial_\mu\Phi\partial_\nu\Phi + V(\Phi)]$$9

$(d+1)$0

This identification extends: the n-point function $(d+1)$1 is interpreted as the connected n-point correlator in the presence of the source $(d+1)$2 dual to $(d+1)$3 per standard holographic logic. Thus, the state is fully characterized by the kernel structure determined by the imposition of Dirichlet data at both boundaries (Botta-Cantcheff et al., 20 Jun 2025).

5. Implications for Cosmological Observables

Expectation values of late-time boundary fields are computed from $(d+1)$4: $(d+1)$5

$(d+1)$6

Higher $(d+1)$7-point functions, in the presence of non-Gaussianities (e.g., interaction-generated $(d+1)$8), yield corrections. To linear order in cubic coupling $(d+1)$9: $M_E$0 Key observable impacts:

• The power spectrum shifts from $M_E$1 (vacuum) to $M_E$2.
• A non-zero one-point background arises: $M_E$3, i.e., classical inhomogeneity seeded by $M_E$4.
• The bispectrum and higher non-Gaussianity (e.g., $M_E$5) gain new contributions, distorting the template shapes away from the Bunch–Davies vacuum (Botta-Cantcheff et al., 20 Jun 2025).

6. Comparison with AdS/CFT and Holography

In AdS/CFT, quantum states—vacuum or excited—are specified by Dirichlet data at the asymptotic AdS boundary, and their properties are directly mapped to Euclidean AdS geometries. In de Sitter space, there is no asymptotic spatial boundary; the Hartle–Hawking no-boundary construction fills this role by defining the ground state as the path integral over smooth, boundary-less Euclidean geometries ending on a spatial Cauchy slice. The generalization to excited states by extra Euclidean boundaries (with arbitrary prescribed data) mimics the AdS/CFT mechanism, extending holographic dictionary prospects for dS quantum cosmology (Botta-Cantcheff et al., 20 Jun 2025).

7. Significance and Research Directions

The no-boundary proposal specifies initial quantum conditions for the emergence of classical spacetime. The key technical innovation of allowing extra Euclidean boundaries produces a continuous family of quantum states with distinguishable late-time correlators. These differences are encoded in both the power spectrum and the pattern of primordial non-Gaussianities, offering observational signatures of excited initial states.

This formalism clarifies the extension of quantum state constructions to dS holography and potentially provides a principled framework for implementing the “initial state” in cosmological correlator computations. The approach has further relevance for the construction of the dS holographic dictionary and the microphysical understanding of cosmological initial conditions (Botta-Cantcheff et al., 20 Jun 2025).