Temperature effects on white dwarfs in modified gravity

Abstract: In this article we analyze the effects of a finite temperature equation of state on the equilibrium structure of white dwarfs in massive Brans-Dicke theory as well as the symmetron and dilaton screening mechanisms. We compute and present the numerically obtained mass-radius relation, effective gravitational constant as well as radial profiles of the scalar field, pressure and metric within the star. We show that assuming a non-zero temperature effectively results in a larger radius while leaving the total mass of the star essentially unchanged, and discuss the interplay between the effective gravitational constant, central density, and radius of the star.

• The paper combines a finite-temperature Chandrasekhar equation of state with stellar structure equations for massive Brans–Dicke, symmetron, and dilaton gravity, finding that temperatures above approximately 10^6 K mainly increase white-dwarf radii at fixed mass.
• Thermal pressure creates a partial degeneracy with modified-gravity effects in mass–radius observations, while the central effective gravitational constant is suppressed by a few percent in massive Brans–Dicke theory and by roughly 10^-6 in symmetron models.
• The results support temperature-dependent white-dwarf cooling models but caution that predictions near 10^8 K are unreliable because the Chandrasekhar equation of state neglects non-degenerate envelopes, neutrino losses, and other relevant microphysics.

This paper by Vidal, Wojnar, and Järv investigates how replacing the standard zero-temperature Chandrasekhar equation of state (EoS) with a finite-temperature version modifies the equilibrium structure of white dwarfs (WDs) in scalar–tensor theories of gravity (STT), specifically massive Brans–Dicke theory and the symmetron and dilaton screening mechanisms (2608.12992). The motivation is twofold: most existing studies of WDs in modified gravity use cold EoSs, and finite-temperature effects may introduce a degeneracy between thermal contributions and genuine signatures of modified gravity, analogous to the known degeneracy between EoS choice and deviations from general relativity (GR).

Finite-temperature Chandrasekhar equation of state

The stellar matter is modeled as a degenerate relativistic Fermi gas of electrons with a carbon–oxygen core, treated via Fermi–Dirac integrals. The electron number density and pressure are expressed through relativistic Fermi–Dirac integrals $F_k(\eta,\beta)$ with degeneracy parameters $\eta = \mu/(k_B T)$ and $\beta = k_B T/(m_e c^2)$. In the $T \to 0$ limit the distribution reduces to a sharp Fermi step and the familiar zero-temperature Chandrasekhar pressure in terms of the dimensionless Fermi momentum $x = p_F/m_e c$ is recovered. The EoS is evaluated for $T \in [0, 10^8]$ K, with the zero-temperature results indistinguishable from those at $T = 10^4$ K.

Field equations and gravity models

The authors work in the Einstein frame of a general conformal STT with action characterized by a conformal factor $A(\tilde\varphi)$ and self-interaction potential $V(\tilde\varphi)$, related to the Jordan frame via $g_{\mu\nu} = A^2(\tilde\varphi)\tilde g_{\mu\nu}$. For static, spherically symmetric configurations, the reduced field equations together with the modified hydrostatic equilibrium condition $\eta = \mu/(k_B T)$0 form a closed system, supplemented by the EoS. The scalar coupling strength is $\eta = \mu/(k_B T)$1.

Three models are considered:

• Massive Brans–Dicke theory: constant coupling $\eta = \mu/(k_B T)$2 with quadratic potential; the field mass suppresses the scalar force beyond its Compton wavelength. The parameters used are $\eta = \mu/(k_B T)$3 and $\eta = \mu/(k_B T)$4 eV, well within observational constraints that only apply to the massless case ($\eta = \mu/(k_B T)$5).
• Symmetron screening: coupling $\eta = \mu/(k_B T)$6 with a symmetry-breaking potential; the field is driven to zero in high-density regions and to $\eta = \mu/(k_B T)$7 at low density. Parameters follow earlier work to ensure typical WDs are screened, though the authors note observational bounds would prefer $\eta = \mu/(k_B T)$8 and $\eta = \mu/(k_B T)$9.
• Dilaton screening: coupling $\beta = k_B T/(m_e c^2)$0 with an exponential runaway potential; the field settles at the coupling minimum in dense environments.

Notably, the authors report an unresolved discrepancy when reproducing prior dilaton results: the reported mass–radius relations are recovered only for potential parameters smaller by a factor of $\beta = k_B T/(m_e c^2)$1 than the theoretically converted values, and independent checks did not identify the source.

Numerical method

The boundary value problem (central conditions and spatial asymptotics) is solved via a shooting method on the scalar field, the only shooting parameter since the temporal metric function enters only through its derivative. The stellar surface is defined by $\beta = k_B T/(m_e c^2)$2, which the authors verify is insensitive to the threshold. Central densities span $\beta = k_B T/(m_e c^2)$3 g cm$\beta = k_B T/(m_e c^2)$4.

Mass–radius relations

The GR results with the temperature-dependent EoS agree with prior literature. Across all three modified gravity theories the pattern is qualitatively identical: deviations from the cold configuration are negligible up to $\beta = k_B T/(m_e c^2)$5 K, while above that, hotter WDs settle at larger radii for the same mass, with essentially unchanged total mass and central density — thermal pressure drives expansion. Two consequences follow directly. First, for WD cooling calculations (conventionally defined as cooling from $\beta = k_B T/(m_e c^2)$6 to $\beta = k_B T/(m_e c^2)$7 K), assuming a constant radius during cooling is evidently unrealistic. Second, finite-temperature effects produce partial degeneracies among different parameter sets within each theory, complicating the constraining of theory parameters from observations.

At $\beta = k_B T/(m_e c^2)$8 K the mass–radius curves for all theories converge toward larger radii and a mass of roughly $\beta = k_B T/(m_e c^2)$9, suggesting temperature effects dominate over gravity modifications in this regime. The authors caution, however, that at these temperatures the gas becomes less degenerate and the finite-temperature Chandrasekhar EoS approaches its validity limit; the observed lower Chandrasekhar mass and sharp drop-off for the symmetron and dilaton at $T \to 0$0 K are attributed to this non-applicability rather than to genuine physical effects.

Effective gravitational constant

The Jordan-frame effective gravitational constant $T \to 0$1 is evaluated at the stellar center and at infinity. For all three theories the scalar field weakens gravity at the center relative to infinity, which counterintuitively yields smaller radii at fixed mass: the positive $T \to 0$2 term steepens the pressure gradient, so pressure drops faster outward. Same-mass stars also require higher central pressure and density.

The magnitude of the effect varies strongly by model:

Model	Deviation $T \to 0$3	Temperature sensitivity
Massive Brans–Dicke	order a few percent ($T \to 0$4 relative)	only at $T \to 0$5 K (saturation)
Symmetron	$T \to 0$6, plateau $T \to 0$7 at high $T \to 0$8	none observed
Dilaton	rises with $T \to 0$9, drops to zero at low density	only at $x = p_F/m_e c$0 K (saturation)

Massive Brans–Dicke exhibits the largest deviations, consistent with the largest mass–radius departures. For the symmetron, at sufficiently high central density the field sits at zero, the density-independent potential minimum, so $x = p_F/m_e c$1 and the theory becomes locally equivalent to GR — visible as convergence of the mass–radius curves toward GR at small radii and high masses. The saturation of $x = p_F/m_e c$2 at high central densities and $x = p_F/m_e c$3 K for Brans–Dicke and selected dilaton parameters is acknowledged to be either an EoS-validity artifact or a numerical issue.

Interior profiles

Radial profiles of the scalar field, its derivative, pressure, and the metric function $x = p_F/m_e c$4 are presented for massive Brans–Dicke at $x = p_F/m_e c$5 dyn; the qualitative behavior is shared by all theories. The scalar field is maximal at the center and suppressed more strongly and rapidly for heavier fields. With radius normalized to the stellar surface, temperature effects are absent except at $x = p_F/m_e c$6 K, where the field suppression intensifies, the pressure declines faster and changes profile shape, and $x = p_F/m_e c$7 rises and decays more steeply. Below $x = p_F/m_e c$8 K the profiles are effectively temperature independent; a slight deviation appears at $x = p_F/m_e c$9 K in the pressure profile.

Limitations and open questions

The central limitation, conceded repeatedly by the authors, is the EoS itself: at $T \in [0, 10^8]$0 K the gas is no longer fully degenerate, the finite-temperature Chandrasekhar EoS loses validity, and additional microphysics — neutrino emission, residual or shell nuclear burning, and non-degenerate outer envelope layers — becomes relevant. The convergence of the $T \in [0, 10^8]$1 K mass–radius curves should therefore be interpreted with care. The unresolved dilaton parameter discrepancy of a factor $T \in [0, 10^8]$2 relative to theoretical conversion also remains unexplained. Finally, whether the thermal–modified-gravity degeneracy can be broken observationally — for example, by combining mass–radius data with the effective gravitational strength — is left open, as is the extension to fully evolutionary cooling models with realistic envelopes.

Conclusion

The paper establishes that for WDs in massive Brans–Dicke, symmetron, and dilaton gravity, finite temperature acts primarily through an increase in stellar radius at fixed mass, leaving mass and central density essentially unchanged, with significant deviations appearing only above $T \in [0, 10^8]$3 K. The effective gravitational constant is largely temperature independent except at the EoS validity boundary. The main quantitative results — percent-level $T \in [0, 10^8]$4 suppression for Brans–Dicke, $T \in [0, 10^8]$5 for the symmetron with a high-density plateau, and the thermal–parameter degeneracy of mass–radius curves — carry direct implications for WD cooling calculations and for observational constraints on scalar–tensor parameters, provided the high-temperature results are treated with the caution the authors themselves advise.