# Dynamical Constraints on RG Flows and Cosmology

## TL;DR

This paper explores the constraints on cosmological and inflationary theories by deriving sum rules through a duality with anti-de Sitter spacetime, linking bulk effective field theories to boundary quantum field theories.

## Contribution

It introduces a new sum rule in AdS/CFT that constrains bulk couplings and relates them to boundary operator spreading, providing insights into inflationary constraints.

## Key findings

- Sum rule constrains bulk coupling constants.
- Spreading speed approaches light speed when the theory becomes free.
- Analogous sum rule discussed for inflationary fluctuations.

## Abstract

Sum rules connecting low-energy observables to high-energy physics are an interesting way to probe the mechanism of inflation and its ultraviolet origin. Unfortunately, such sum rules have proven difficult to study in a cosmological setting. Motivated by this problem, we investigate a precise analogue of inflation in anti-de Sitter spacetime, where it becomes dual to a slow renormalization group flow in the boundary quantum field theory. This dual description provides a firm footing for exploring the constraints of unitarity, analyticity, and causality on the bulk effective field theory. We derive a sum rule that constrains the bulk coupling constants in this theory. In the bulk, the sum rule is related to the speed of radial propagation, while on the boundary, it governs the spreading of nonlocal operators. When the spreading speed approaches the speed of light, the sum rule is saturated, suggesting that the theory becomes free in this limit. We also discuss whether similar results apply to inflation, where an analogous sum rule exists for the propagation speed of inflationary fluctuations.