Late-Time Wavefunctions and Boundary Data

The cosmological wavefunction is an amplitude for a specified late-time field configuration, prepared from a specified initial state and complex-time contour. Its logarithm separates local, often divergent phases from nonlocal coefficient functions. Those coefficients are useful candidate holographic data, but symmetry does not select the state, and a wavefunction coefficient is neither a probability nor an in-in correlator without the explicit Born-rule conversion.

Required background. Wavefunction and Correlator Object Dictionary fixes coefficient conventions; Bunch–Davies, Euclidean, and Alpha-State Diagnostics fixes the state.

Helpful background. Contours, iε, and Initial-Boundary Terms supplies the contour; Euclidean Preparation and Lorentzian State Dictionaries separates preparation from Lorentzian evolution.

Definition and coefficient expansion

At a cutoff slice $\eta_c<0$, define

$$\Psi[\varphi,\eta_c] =\int_{\phi(\eta_c)=\varphi}^{\text{initial state}}\!\mathcal D\phi\, e^{iS[\phi]},$$

where the lower end of the contour implements the state. A common expansion is

$$\log\Psi=-\sum_{n\ge2}\frac1{n!} \int_{\mathbf k_1\cdots\mathbf k_n} (2\pi)^d\delta\!\left(\sum_i\mathbf k_i\right) \psi_n(\mathbf k_1,\ldots,\mathbf k_n;\eta_c) \prod_i\varphi_{\mathbf k_i}.$$

Local counterterms shift polynomial or analytic pieces of $\psi_n$. Nonlocal momentum dependence, factorization singularities, and discontinuities can be more invariant, but remain state- and contour-dependent.

First application: the Gaussian Bunch–Davies wavefunction

For a free scalar in four-dimensional de Sitter,

$$S=\frac12\int d\eta\,d^3x\,a^2 \left[(\phi')^2-(\nabla\phi)^2-a^2m^2\phi^2\right], \qquad a=-\frac1{H\eta}.$$

The Bunch–Davies mode is proportional to $(-\eta)^{3/2}H_\nu^{(1)}(-k\eta)$ with

$$\nu=\sqrt{\frac94-\frac{m^2}{H^2}}, \qquad \Delta_\pm=\frac32\pm\nu.$$

Evaluating the on-shell action gives a Gaussian

$$\Psi[\varphi,\eta_c] =\mathcal N(\eta_c)\exp\!\left[-\frac12\int_{\mathbf k} \Omega_k(\eta_c)\varphi_{\mathbf k}\varphi_{-\mathbf k}\right], \qquad \Omega_k=-i a^2\frac{u_k^{*\prime}}{u_k^*},$$

where the conjugate mode implements the decaying Euclidean contour in this convention. For $m=0$,

$$u_k^*(\eta)\propto(1-ik\eta)e^{ik\eta}, \qquad \Omega_k(\eta_c)=\frac{k^3}{H^2}-i\frac{k^2}{H^2\eta_c}+O(\eta_c).$$

The imaginary $k^2/\eta_c$ term is a divergent local phase removable or shiftable by a local boundary counterterm. The real nonlocal $k^3/H^2$ term controls the Gaussian probability and produces

$$\langle\varphi_{\mathbf k}\varphi_{-\mathbf k}\rangle =\frac{H^2}{2k^3}$$

under the convention above. For general mass the two falloffs carry weights $\Delta_\pm$; principal-series masses give complex-conjugate weights rather than dimensions of an ordinary reflection-positive Euclidean CFT.

Interactions generate $\psi_{n>2}$. Tree coefficients inherit total-energy and factorization singularities and can be related to analytically continued AdS diagrams, while loops require the full contour, normalization, and counterterms. Maldacena’s inflationary calculation is a canonical example in which a wavefunctional computation must still be converted to late-time correlators Maldacena 2003, §§ 3–4.

Adversarial control: change the initial state

Replace $u_k$ by a Bogoliubov mode $A_k u_k+B_k u_k^*$ with $|A_k|^2-|B_k|^2=1$. de Sitter covariance may be retained in formal alpha-state choices, but $\Omega_k$, its nonlocal coefficient, and its singularity structure change. A dictionary inferred from symmetry alone therefore fails to select a unique wavefunction. Add a local phase $e^{iF_{\mathrm{loc}}[\varphi]}$ as a second control: coefficient functions shift although equal-time probabilities do not.

The approximation requires a specified initial state, perturbative interactions, an IR prescription, and $G_NH^2\ll1$ for neglecting gravitational loops. It supplies no automatic $g_s$, $\alpha'H^2$, Kaluza–Klein, or metastable-vacuum control. The evidence ceiling is a well-defined late-time functional and robust nonlocal coefficient data within that regime. A probability, in-in expectation value, inner product, and complete holographic boundary theory require additional maps handled on the following pages.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Maldacena, J. (2003). “Non-Gaussian Features of Primordial Fluctuations in Single Field Inflationary Models.” Journal of High Energy Physics 2003(5), 013. DOI.