Thermodynamic Characteristics of a Fermi Gas with an Invariant Energy Scale and its Astrophysical Implications

TL;DR

The paper investigates how a modified dispersion relation with an invariant ultraviolet scale $\kappa$, within the Magueijo–Smolin Doubly Special Relativity framework, alters the thermodynamics of a relativistic Fermi gas. It develops analytic low-temperature results via a Sommerfeld expansion and computes high-temperature thermodynamics numerically from the exact grand potential, demonstrating convergence to standard relativistic behavior as $\kappa$ grows. By applying the low-temperature equation of state to white dwarfs and neutron stars through Lane-Emden and Tolman–Oppenheimer–Volkoff equations, it reveals that helium-core white dwarfs exhibit strong $\kappa$-sensitivity, while heavier-core white dwarfs show degeneracies with composition; neutron stars become smaller and lighter than nucleonic EOS predictions, challenging some observational constraints. Overall, the work shows that modified relativity theories imprint observable signatures on compact objects and motivates further exploration of density-dependent $\kappa$ and dynamical astrophysical scenarios to test MDR frameworks.

Abstract

We investigate the thermodynamics of a relativistic Fermi gas governed by a modified dispersion relation in the Magueijo Smolin (MS) formulation of Doubly Special Relativity (DSR), characterized by the presence of an invariant ultraviolet energy (deformation) scale. We study the system in two physically distinct regimes: the near degenerate low temperature limit, and the high temperature regime. In the low temperature regime, we derive the thermodynamic quantities using the standard Sommerfeld expansion. In the high temperature regime, we evaluate all thermodynamic quantities numerically from the exact grand canonical potential and demonstrate that the thermodynamics of the Fermi gas reduces to the standard relativistic ideal gas behavior. We apply the resulting low temperature equation of state to study compact astrophysical objects, namely, non rotating white dwarfs and neutron stars. Helium white dwarfs exhibit a strong dependence on the deformation scale, while white dwarfs composed of heavier elements are less affected. For neutron stars, the modified equation of state leads to configurations that are smaller in radius and lower in mass than is by nucleonic equations of state. Our results highlight how modified relativity theories can be probed by studying astrophysical objects.

Figures (4)

• Figure 1: Pressure (P), entropy (S), internal energy (U) and specific heat ($C_V$) versus particle number density (n) for $\kappa = 2\, \mathrm{GeV},\, 3\, \mathrm{GeV},\, 7\, \mathrm{GeV},\, 20\, \mathrm{GeV}$ DSRs in the low temperature limit. The red plots represent the thermodynamical quantities in the special relativistic case, i.e., in the limit $\kappa\to\infty$. The plots are done taking a low temperature of $8.62\times10^{-9} \,\mathrm{GeV}$, a rest mass of $0.939\, \mathrm{GeV}$ and a spherical volume with $10\, \mathrm{Km}$ as the radius.
• Figure 2: The pressure $(P)$, entropy $(S)$, internal energy $(U)$, and specific heat $(C_V)$ are plotted as functions of the temperature $(T)$ at fixed number density in the high temperature regime. The blue curves correspond to the deformed case with $\kappa = 300~\mathrm{GeV}$, while the red curves represent the standard relativistic limit, $\kappa \rightarrow \infty$. The plots are obtained by taking the rest mass to be $m_{0} = 0.939~\mathrm{GeV}$ and imposing the constraint $m \ll T \ll \kappa$.
• Figure 3: Left: Lane-Emden parameters $\theta$ v $\xi$ for combinations of $m_b$ and $\kappa$. Lower panel shows $\Delta\theta$ (w.r.t. $m_b=2$ and $\kappa=5000~\rm{GeV}$ curve) v $\xi$ to highlight differences in the $\theta-\xi$ curves. Right: White dwarf mass-radius produced by the DSR for a range of particle mass $m_b$ and $\kappa$.
• Figure 4: Left: Pressure v. energy density for DSR at supra-nuclear saturation densities. Nuclear saturation density ($n_0$) is $0.15 ~\rm{nucleons}~\rm{fm}^{-3}$, energy density $\sim 0.71~\rm{fm}^{-4}$. Right: Mass-Radius relations for neutron stars obtained for the DSR. Curves in the colors blue, orange, green, red, purple, brown, pink, gray, lime, and aqua are for $\kappa$ = 5000, 1000, 500, 100, 20, 7, 5, 3, 2.5, and 2 GeV respectively.