Spatial Phonons: A Phenomenological Viscous Dark Energy Model for DESI

We explore a phenomenological model of dark energy in which space is treated as an elastic brane with uniform tension $T_s$ and supports a longitudinal phonon sector described by three scalar fields $\phi^I$. At the background level the construction reproduces a perfect fluid whose enthalpy and bulk modulus are controlled by two dimensionless parameters $\varepsilon$ and $\kappa$, which encode the elastic and viscous response and determine both the effective equation of state $w_{\rm eff}(z)$ and the physical sound speed thriugh $c_s^2 = \kappa/\varepsilon$. Motivated by the consistency requirements of a phonon effective description, we focus on the stable regime $0 < \kappa < \varepsilon < 1$, which prevents tachyonic behavior and is natural given that $\varepsilon$ and $\kappa$ are defined as fractional response coefficients. To connect the model predictions to current observational summaries, we map $w_{\rm eff}(z)$ onto the Chevallier--Polarski--Linder form $w(z) = w_0 + w_a z/(1+z)$ over the redshift range relevant to DESI. We then use the public DESI DR1 compressed Gaussian likelihood in $(w_0,w_a)$ for BAO combined with Pantheon+ Type I a supernovae and Planck 2018, and scan the model parameter space to identify viable regions. We find that the model can reproduce the DESI compressed constraints at the background level, with representative best match values $(w_0,w_a) \simeq (-0.83,-0.74)$ and therefore, a near luminal sound speed $c_s^2 \simeq 0.9618$ as an output of the fit rather than a built in restriction. The covariance weighted distance is found to satisfy $\chi^2 \ll 1$ in the $(w_0,w_a)$ plane. The associated phonon mass scale $m_\phi = H_\star/(2\pi)$ is ultralight, corresponding to a Compton wavelength of order the cosmological horizon, consistent with interpreting the relevant excitations as infrared collective modes of the brane.