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Harrison--Zeldovich spectrum from conformal invariance

V. A. Rubakov

TL;DR

The paper addresses generating a Harrison–Zeldovich (scale-invariant) spectrum without inflation by leveraging conformal invariance in a multi-component scalar field with a negative quartic potential and a global U(1) symmetry. It derives how entropy perturbations arising in the orthogonal direction to the rolling field become flat and then outlines a curvaton-like mechanism where a pseudo-Goldstone field converts these entropy perturbations into adiabatic perturbations with the correct amplitude. The work provides explicit dynamics, amplitude estimates, and robustness arguments against non-linear radial perturbations, highlighting a potential non-inflationary route to the observed cosmic perturbations. It also discusses caveats such as possible spectral tilt, non-Gaussianity, and lack of tensor modes, marking avenues for further investigation.

Abstract

We show that flat spectrum of small perturbations of field(s) is generated in a simple way in a theory of multi-component scalar field provided this theory is conformally invariant, it has some global symmetry and the quartic potential is negative. We suggest a mechanism of converting these field perturbations into adiabatic scalar perturbations with flat spectrum.

Harrison--Zeldovich spectrum from conformal invariance

TL;DR

The paper addresses generating a Harrison–Zeldovich (scale-invariant) spectrum without inflation by leveraging conformal invariance in a multi-component scalar field with a negative quartic potential and a global U(1) symmetry. It derives how entropy perturbations arising in the orthogonal direction to the rolling field become flat and then outlines a curvaton-like mechanism where a pseudo-Goldstone field converts these entropy perturbations into adiabatic perturbations with the correct amplitude. The work provides explicit dynamics, amplitude estimates, and robustness arguments against non-linear radial perturbations, highlighting a potential non-inflationary route to the observed cosmic perturbations. It also discusses caveats such as possible spectral tilt, non-Gaussianity, and lack of tensor modes, marking avenues for further investigation.

Abstract

We show that flat spectrum of small perturbations of field(s) is generated in a simple way in a theory of multi-component scalar field provided this theory is conformally invariant, it has some global symmetry and the quartic potential is negative. We suggest a mechanism of converting these field perturbations into adiabatic scalar perturbations with flat spectrum.

Paper Structure

This paper contains 5 sections, 43 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Getting flat spectrum
  3. Implementing the mechanism
  4. Generalizing to non-linear radial field perturbations
  5. Conclusions

Figures (1)

  • Figure 1: The scalar potential is negative quartic and respects global symmetry ($U(1)$ here) at relatively small $\phi$, and has one or more minima at large $\phi$. Dashed line shows the evolution of the field (dot with arrow) at the rolling stage. At that stage the perturbations (shown by double arrow) of the pseudo-Goldstone field are developed. At the hot Big Bang epoch, the pseudo-Goldstone field transfers its energy to hot matter, and its perturbations are converted into adiabatic perturbations.