# Exotic compact objects with soft hair

## TL;DR

This paper develops a general framework for modeling exotic compact objects with soft or hard hair, extending the Hartle-Thorne formalism, and analyzes their multipolar structure to aid future gravitational wave tests distinguishing them from black holes.

## Contribution

It introduces a perturbative, axisymmetric solution class for exotic compact objects with soft and hard hair, relaxing symmetry assumptions and including higher multipole moments.

## Key findings

- Soft exotic objects become less hairy as they become more compact.
- Multipole moments approach Kerr values logarithmically or linearly depending on their origin.
- Distinguishing black holes from exotic objects via multipolar structure may be feasible with advanced gravitational-wave detectors.

## Abstract

Motivated by the lack of a general parametrization for exotic compact objects, we construct a class of perturbative solutions valid for small (but otherwise generic) multipolar deviations from a Schwarzschild metric in general relativity. We introduce two classes of exotic compact objects, with "soft" and "hard" hair, for which the curvature at the surface is respectively comparable to or much larger than that at the corresponding black-hole horizon. We extend the Hartle-Thorne formalism to relax the assumption of equatorial symmetry and to include deformations induced by multipole moments higher than the spin, thus constructing the most general, axisymmetric quasi-Schwarzschild solution to Einstein's vacuum equations. We explicitly construct several particular solutions of objects with soft hair, which might be useful for tests of quasi-black-hole metrics, and also to study deformed neutron stars. We show that the more compact a soft exotic object is, the less hairy it will be. All its multipole moments can approach their corresponding Kerr values only in two ways as their compactness increases: either logarithmically (or faster) if the moments are spin-induced, or linearly (or faster) otherwise. Our results suggest that it is challenging (but possibly feasible with next-generation gravitational-wave detectors) to distinguish Kerr black holes from a large class of ultracompact exotic objects on the basis of their different multipolar structure.