A barotropic alternative to Early Dark Energy for alleviating the $H_0$ tension

We propose a cosmological scenario in which, beyond matter and radiation, an additional barotropic fluid with positive equation of state $\omega_s$ contributes to the cosmic energy budget, in contrast to Early Dark Energy (EDE). We investigate the theoretical implications of this framework, here dubbed the $\Lambda_{\omega_s}$CDM model, at both the background and perturbative levels, exploring its impact on the expansion history and structure formation. We show that, while remaining subdominant at late times and therefore consistent with current observational bounds, the additional fluid modifies the early-time expansion rate, leading to a higher inferred value of the Hubble constant. Thus, we perform a full Bayesian analysis using a modified version of the \texttt{CLASS} Boltzmann code interfaced with \texttt{MontePython}, considering combinations of \textit{Planck} 2018 Cosmic Microwave Background (CMB) data, DESI DR2 Baryon Acoustic Oscillations (BAO) measurements, Pantheon Type Ia supernovae (SNe Ia), and SH0ES determinations of $H_0$. We find that the inclusion of the SH0ES prior, $H_0 = 73.04 \pm 1.04\,\mathrm{km/s/Mpc}$, leads to a preference for a nonvanishing barotropic fluid. In particular, we obtain $\omega_s = 0.290^{+0.017(0.021)}_{-0.007(0.028)}$ and density $10^5\Omega_s = 1.47^{+0.35(1.14)}_{-0.62(0.94)}$ for the dataset combination CMB + BAO + Pantheon + SH0ES, and $\omega_s = 0.302^{+0.024(0.034)}_{-0.013(0.038)}$ and $10^5\Omega_s = 1.21^{+0.31(1.10)}_{-0.65(0.86)}$ when BAO data are excluded. We further compare our scenario with the EDE framework and show that, statistically, no strong evidence is found against the $\Lambda_{\omega_s}$CDM model. Finally, we provide a physical interpretation of our fluid in terms of matter with pressure, indicating that the standard cosmological model may be incomplete in its current minimal formulation.