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 strongest conclusion 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 compares these assumptions and limitations.

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.