Spatial Phonons: A Phenomenological Viscous Dark Energy Model for DESI

TL;DR

This work proposes a viscoelastic description of space as an elastic brane with tension $T_s$ hosting a longitudinal phonon fluid described by scalar fields $\phi^I$ and invariant $b$, yielding a background energy density near a cosmological constant and a phonon bulk modulus controlled by $\varepsilon$ and $\kappa$ with a sound speed $c_s^2=\kappa/\varepsilon$. Dissipation is incorporated via a Maxwell-type bulk viscosity with a relaxation time $\tau(H)$, tied to a mass gap $m_\phi$ and horizon temperature $T_H=H/(2\pi)$, producing a transient phantom dip in the effective dark energy equation of state $w_{\rm eff}(H)$ around $H\tau(H)\sim1$ while asymptoting to $-1+\varepsilon$ at early and late times. The authors derive compact expressions for $w_{\rm eff}(H)$, show how DESI data motivate parameter choices $\varepsilon\approx\kappa\approx1/3$ and $H_*/H_0\approx2.1$, and demonstrate that the resulting $w_{\rm eff}(z)$ closely tracks a DESI-motivated CPL form over the redshift range most sensitive to BAO measurements. The framework provides a flexible, physically motivated mechanism for transient dark-energy dynamics without introducing new long-lived matter components, with clear avenues for deriving microphysical contact through perturbations and perturbative growth analyses.

Abstract

We explore a phenomenological model of dark energy in which space is treated as an elastic brane with uniform tension $T_s$ supporting a longitudinal phonon fluid. The brane tension gives a residual geometric contribution to the vacuum energy, while the phonon sector is described by three scalar fields $φ^I$ and an invariant $b = \sqrt{\det B_{IJ}}$ entering an effective action $F(b)$. At the background level this reproduces a perfect fluid with energy density, pressure and bulk modulus set by two dimensionless parameters $\varepsilon$ and $κ$. These parameters fix the enthalpy and bulk modulus in units of the space tension and determine the phonon sound speed through $c_s^2 = κ/\varepsilon$. Dissipative effects are modeled by a bulk viscous pressure obeying a Maxwell type relaxation law with a characteristic time scale $τ(H)$ that depends on the Hubble rate. Motivated by a Boltzmann suppressed scattering rate at a mass-gap scale $H_\star$, we adopt a simple ansatz for $Hτ(H)$ and obtain a compact expression for the effective dark energy equation of state $w_{\rm eff}(H)$. The viscous correction is transient, is most active around $H \sim H_\star$ and drives a temporary phantom deviation. At the same time, we also show that $w_{\rm eff}$ approaches $-1 + \varepsilon$ at very early and very late times. Using a flat $Λ$CDM background for $H(z)$, we scan the parameter space and identify a region with $κ\ \simeq \ \varepsilon \ \simeq \ 1/3$ and $H_\star/H_0 \ \simeq \ 2.1$. This choice gives an ultralight phonon with sound speed close to the causal limit and a viscous dark energy history $w_{\rm eff}(z)$ that closely tracks a DESI motivated Chevallier-Polarski-Linder parametrization over the redshift range most relevant for the DESI BAO measurements.

Figures (1)

• Figure 1: Comparison between the Chevallier--Polarski--Linder equation of state $w_{\rm CPL}(z)$ (solid line) with parameters $(w_0, w_a) = (-0.856, -0.53)$ and the effective equation of state $w_{\rm eff}(z)$ (dashed line) of the viscoelastic spatial phonon model for $\varepsilon = 0.335$, $\kappa = 0.33489$ and $H_\star/H_0 = 2.10$. The horizontal dotted line shows the cosmological constant value $w = - 1$. Over the DESI redshift range $0 \le z \lesssim 1.6$, the viscous dark energy model closely tracks the CPL fit.