# Universal Formula for the Holographic Speed of Sound

## TL;DR

This paper derives a universal formula for the holographic speed of sound in five-dimensional black holes with scalar fields, applicable to various boundary conditions and dimensions, revealing cases where it exceeds the conformal value.

## Contribution

It introduces a universal formula for the holographic speed of sound applicable to scalar-metric theories within the Breitenlohner-Freedman window, including new solutions in type IIB supergravity.

## Key findings

- Speed of sound can exceed the conformal value in certain solutions.
- Universal formula applies to arbitrary dimensions within the Breitenlohner-Freedman window.
- Constructed new black hole solutions with scalar fields preserving specific symmetries.

## Abstract

We consider planar hairy black holes in five dimensions with a real scalar field in the Breitenlohner-Freedman window and show that is possible to derive a universal formula for the holographic speed of sound for any mixed boundary conditions of the scalar field. As an example, we locally construct the most general class of planar black holes coupled to a single scalar field in the consistent truncation of type IIB supergravity that preserves the $SO(3)\times SO(3)$ R-symmetry group of the gauge theory. We obtain the speed of sound for different values of the vacuum expectation value of a single trace operator when a double trace deformation is induced in the dual gauge theory. In this particular family of solutions, we find that the speed of sound exceeds the conformal value. Finally, we generalize the formula of the speed of sound to arbitrary dimensional scalar-metric theories whose parameters lie within the Breitenlohner-Freedman window.