Holographic No-Boundary Measure

TL;DR

The paper shows that the complex saddle points of the no-boundary wave function with a positive cosmological constant and a positive scalar potential admit a representation in which the interior geometry is a regular Euclidean AdS domain wall smoothly transitioning to a Lorentzian inflationary de Sitter universe. The AdS portion regularizes volume divergences via counterterms and supplies a characteristic phase, enabling a holographic reformulation in which the semiclassical NBWF is encoded by the partition function of a dual field theory on the final boundary (a relevant deformation of an AdS/CFT CFT). This leads to a concrete dS/CFT-like duality: the NBWF amplitudes for coarse-grained classical configurations are given by $e^{I^{reg}_{DW}}/Z_{QFT}$ with a boundary phase, and bulk time evolution corresponds to inverse RG flow in the boundary theory. The authors conjecture that this duality extends beyond leading order, offering a pathway to compute string and quantum corrections to the cosmological measure from the dual QFT, and to test holographic cosmology in a controlled setting.

Abstract

We show that the complex saddle points of the no-boundary wave function with a positive cosmological constant and a positive scalar potential have a representation in which the geometry consists of a regular Euclidean AdS domain wall that makes a smooth transition to a Lorentzian, inflationary universe that is asymptotically deSitter. The transition region between AdS and dS regulates the volume divergences of the AdS action and accounts for the phases that explain the classical behavior of the final configuration. This leads to a dual formulation in which the semiclassical no-boundary measure is given in terms of the partition function of field theories on the final boundary that are certain relevant deformations of the CFTs that occur in AdS/CFT. We conjecture that the resulting dS/CFT duality holds also beyond the leading order approximation.

Figures (4)

• Figure 1: Left panel: The contour $C_D$ in the complex $\tau$-plane. The horizontal part is the geometry of half a Euclidean three-sphere. The vertical part is Lorentzian deSitter space (cf Fig \ref{['iconic']}). Right panel: The contour $C_A$. The part (a) along the imaginary axis $x=0$ is AdS. The part (d) along the $x=\pi/2H$ line is Lorentzian deSitter space. The part (h) is a complex geometry that transitions between them.
• Figure 2: Two embedding diagrams showing the different geometries representing the same deSitter saddle point. The generally complex geometries are embedded using the real metrics defined by the modulus of the scale factor $|a|$. The two figures show the representation of a two dimensional slice of the same saddle point corresponding to an equator of the three-sphere in terms of the contours $C_D$ and $C_A$ in Figure \ref{['contours']}. The slices along the vertical parts of these contours are embedded in a flat Lorentz signatured three-dimensional space. They are shaded in red. The slices along the horizontal parts of the contours are embedded in a Euclidean three dimensional flat space and shaded blue. The left figure is the NBWF deSitter saddle point as half a Euclidean three sphere joined to half a Lorentzian deSitter space across an equator. The next figure corresponds to the contour $C_A$ and consists of Euclidean AdS space joined (moving upwards) to the geometry of the horizontal branch, and then to deSitter space. Both representations give the same action and, in the more general case discussed below, the same predictions for the ensemble of classical histories.
• Figure 3: The contour $C_A$ when matter is included. The scale factor $a$ and field $\phi$ are real along an asymptotically vertical contour (d) but one which is displaced from $x=\pi/2H$ to $x=x_r$. The vertical part (a) of the contour along which the saddle point is asymptotically AdS is shifted by an amount $\pi/2H$ in $x$ from that. The horizontal part of the contour (h) connects these two vertical parts --- relating AdS to dS.
• Figure 4: The saddle point representation that serves as a guide in the derivation of a dual formulation of the NBWF in terms of a field theory defined on the conformal boundary geometry at $\upsilon$.