On the power spectrum of inflationary cosmologies dual to a deformed CFT

TL;DR

The paper establishes a concrete holographic computation of the inflationary power spectrum by relating it to the spectral density of the trace of the stress tensor in a three-dimensional CFT deformed by a nearly marginal operator. Using RG-improved conformal perturbation theory, it derives a controlled expansion for the spectral function and expresses the cosmological scalar power spectrum in terms of the dual QFT β-function and its derivatives, achieving exact agreement with the standard second-order slow-roll result. The approach requires a careful calibration of the QFT renormalization scheme to map the running coupling to the inflaton at horizon crossing, connecting UV-CFT data (A0,A1,A2) from AdS/CFT to cosmological observables. Overall, the work provides a nontrivial consistency check of holographic cosmology and outlines avenues for extending the analysis to non-Gaussianities, multi-field dynamics, and more general RG flows.

Abstract

We analyse slow-roll inflationary cosmologies that are holographically dual to a three-dimensional conformal field theory deformed by a nearly marginal scalar operator. We show the cosmological power spectrum is inversely proportional to the spectral density associated with the 2-point function of the trace of the stress tensor in the deformed CFT. Computing this quantity using second-order conformal perturbation theory, we obtain a holographic power spectrum in exact agreement with the expected inflationary power spectrum to second order in slow roll.

Figures (4)

• Figure 1: To derive the dispersion relation \ref{['disp2']} we integrate $B(-\rho^2)/(q^2+\rho^2)^3$ over the contour in the complex $\rho^2$ plane shown above. The contribution from the circular arc at infinity vanishes, since in the case of interest, $B(\rho^2) \rightarrow \rho^{3-2\lambda}$ as $\rho\rightarrow \infty$ with $0<\lambda\ll 1$. The discontinuity across the branch cut is then related to the residue at $\rho^2=-q^2$.
• Figure 2: For $b_2$ is positive and of order unity, the $\beta$-function has a perturbative IR fixed point located at $g_{IR}\sim\lambda$.
• Figure 3: The inflaton potential comprises a hilltop (corresponding to the IR fixed point at $\varphi_{IR}\sim \lambda$) with a nearby local minimum at the origin (corresponding to the UV fixed point). The decrease in height of the potential is of order $\lambda^3$.
• Figure 4: Witten diagrams contributing to $A_2$.