Asymptotically Friedmann self-similar scalar field solutions with potential

TL;DR

This paper explores self-similar scalar field solutions with exponential potential that are asymptotic to Friedmann universes, revealing two distinct solution families and conditions for their existence in accelerating universes.

Contribution

It identifies two types of asymptotic solutions in scalar field cosmologies and clarifies conditions under which they occur, especially in accelerating universes with quintessence.

Findings

Two families of asymptotic solutions: proper Friedmann and quasi-Friedmann.

Proper Friedmann solutions exist only in accelerating or negative potential scenarios.

Density perturbations decay differently depending on the solution type.

Abstract

We investigate self-similar solutions which are asymptotic to the Friedmann universe at spatial infinity and contain a scalar field with potential. The potential is required to be exponential by self-similarity. It is found that there are two distinct one-parameter families of asymptotic solutions,one is asymptotic to the proper Friedmann universe, while the other is asymptotic to the quasi-Friedmann universe, i.e., the Friedmann universe with anomalous solid angle. The asymptotically proper Friedmann solution is possible only if the universe is accelerated or the potential is negative. If the potential is positive, the density perturbation in the asymptotically proper Friedmann solution rapidly falls off at spatial infinity, while the mass perturbation is compensated. In the asymptotically quasi-Friedmann solution, the density perturbation falls off only in proportion to the inverse square of the areal radius and the relative mass perturbation approaches a nonzero constant at spatial infinity. The present result shows that a necessary condition holds in order that a self-gravitating body grows self-similarly due to the constant accretion of quintessence in an accelerating universe.