Dark Energy Stars in Rastall-Rainbow Gravity: Structure, Stability and Observational Constraints

TL;DR

This work investigates dark energy stars in the two-parameter Rastall–Rainbow gravity, combining an energy‑dependent spacetime with nonconserved matter–geometry coupling. The authors derive the modified field equations, recast them into an Einstein‑like form, and solve the generalized TOV system for a modified Chaplygin gas interior, analyzing how the Rastall parameter $\\eta$ and the rainbow function $\\Sigma$ alter the mass–radius relations. Their results show that stable, causal DES configurations exist within broad regions of parameter space and can be compatible with observations from massive pulsars ($\\sim$2$-$$2.4\\,M_\\odot$) and gravitational‑wave events (GW170817, GW190814), with rainbow effects generally increasing maximum mass and Rastall effects shifting the GR baseline. The findings highlight Rastall–Rainbow gravity as a viable framework for strong‑field stellar structure, offering potentially observable signatures in tidal deformability and surface redshift that can be tested with future measurements.

Abstract

In this work, we investigate static configurations of dark energy stars within the framework of Rastall-Rainbow (R-R) gravity, which combines an energy-dependent deformation of spacetime with a nonminimal coupling between matter and geometry. We begin by deriving the modified field equations corresponding to R-R gravity and subsequently reformulate the stellar structure equations to describe hydrostatic equilibrium. The generalized Tolman-Oppenheimer-Volkoff (TOV) equations are then solved numerically by adopting the modified Chaplygin equation of state to model the interior matter distribution. The R-R parameters, along with fluid constants, are shown to influence the maximum mass, radii, and stiffness of the star sequences compared to the baseline set by general relativity. We apply observational benchmarks from high-mass pulsars and binary-merger events (e.g., GW170817 and GW190814) to appraise viability within the explored parameter space. The results collectively suggest that stable, causal configurations arise from physically meaningful parameter selections, with deviations from general relativity leading to systematic changes in structural characteristics while adhering to theoretical limits. These findings illustrate that Rastall-Rainbow gravity can support stable, observationally consistent dark energy stars, providing verifiable signatures in strong gravitational fields.

Figures (6)

• Figure 1: Mass-radius (left panel) and mass-compactness (right panel) relations for dark energy star configurations in Rastall-Rainbow gravity. The calculations are obtained with the parameter choice $G=1$, $A=\sqrt{0.4}$, $B = 0.23 \times 10^{-3}\,{\rm km}^{-2}$, and $\Sigma = 0.9$. The curves correspond to different values of the rainbow parameter $\eta = 0.8, 0.9, 1.0$ (Rainbow gravity limit), $1.1$, and $1.2$. Observational constraints are overlaid for PSR J0952-0607 Romani:2022jhd (yellow band), PSR J0740+6620 Fonseca:2021wxt (red contours), PSR J0348+0432 Antoniadis:2013pzd (green band), and HESS J1731-347 Doroshenko:2022nwp (magenta shaded region), together with the GW170817 LIGOScientific:2018cki bounds (blue contours). Filled circles mark the maximum mass points along each sequence.
• Figure 2: Mass–radius (top panel) and mass–compactness (bottom panel) relations for dark-energy star configurations in Rastall–Rainbow gravity. The calculations use the parameter set $G=1$, $A=\sqrt{0.4}$, $B=0.23\times10^{-3}\,\mathrm{km^{-2}}$, with fixed $\eta=0.8$. The curves correspond to different values of the rainbow parameter $\Sigma=\{0.75,0.80,0.85,0.90,0.95\}$. Observational constraints are overlaid for PSR J0952–0607 Romani:2022jhd (yellow band), PSR J0740+6620 Fonseca:2021wxt (red contours), PSR J0348+0432 Antoniadis:2013pzd (green band), the GW190814 secondary mass range LIGOScientific:2020zkf (cyan band), GW170817 component masses LIGOScientific:2018cki (blue contours), and HESS J1731–347 Doroshenko:2022nwp (magenta shaded region). Filled circles mark the maximum-mass points along each sequence.
• Figure 3: Mass–radius (left panel) and mass–compactness (right panel) relations for dark-energy star configurations in Rastall–Rainbow gravity. Calculations use the parameter set $G=1$, $A=\sqrt{0.4}$, and $B=0.23\times10^{-3}\,\mathrm{km^{-2}}$. Five sequences are shown for the pairs $(\eta,\Sigma)=\{(0.80,0.90),(0.90,0.95),(1.0,1.0),(1.1,1.05),(1.2,1.1)\}$; the black dashed curve denotes the GR baseline $(\eta=1,\Sigma=1)$. Observational constraints are overlaid for PSR J0952--0607 Romani:2022jhd (yellow band), PSR J0740+6620 Fonseca:2021wxt (red band), PSR J0348+0432 Antoniadis:2013pzd (green band), the GW190814 secondary-mass interval LIGOScientific:2020zkf (cyan band), GW170817 component masses LIGOScientific:2018cki (blue contours), and HESS J1731--347 Doroshenko:2022nwp(magenta shaded region). Filled circles mark the maximum-mass (radial stability turning-point) configurations on each sequence, and the same points are repeated in the compactness plane.
• Figure 4: Profiles of the mass--central energy density relation $M(\rho_c)$ for dark energy stars in Rastall--Rainbow gravity. Left: variation with the Rastall parameter $\eta$ (Table \ref{['table1']}). Middle: variation with the rainbow parameter $\Sigma$ (Table \ref{['table2']}). Right: joint variations around the GR baseline, $(\eta,\Sigma)$, as summarized in Table \ref{['table3']}. The dashed black curve denotes the GR case $(\eta=1,\Sigma=1)$. Blue markers indicate the turning point (maximum mass) along each sequence; these configurations correspond to the entries reported in the respective tables. Unless stated otherwise, all remaining model constants are fixed as in the main text.
• Figure 5: Variation of the adiabatic index $\gamma(r)$ for dark energy stars in Rastall--Rainbow gravity. Left: dependence on the Rastall parameter $\eta$ (Table \ref{['table1']}). Middle: dependence on the rainbow parameter $\Sigma$ (Table \ref{['table2']}). Right: joint variations around the GR baseline $(\eta,\Sigma)$, as summarized in Table \ref{['table3']}. The horizontal dashed line marks the dynamical–stability threshold $\gamma=4/3$. For the parameter choices shown, $\gamma(r)$ remains above $4/3$ throughout the stellar interior, thereby satisfying the local stability criterion. Unless stated otherwise, all remaining model constants are fixed as in the main text.