Temperature effects on white dwarfs in modified gravity

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 derives the Chandrasekhar equation of state at finite temperature for a degenerate relativistic Fermi gas, using Fermi-Dirac statistics and relativistic Fermi-Dirac integrals. The zero-temperature limit is also presented. The hydrostatic equilibrium equations for scalar-tensor theories are introduced in the Einstein frame, with a general conformal factor and self-interacting potential. The theories considered are massive Brans-Dicke theory (with constant coupling α0 and quadratic potential), symmetron screening (with Mexican hat potential and density-dependent effective potential), and dilaton screening (with exponential runaway potential and coupling function driving the screening).

The mass-radius relations are computed for temperatures T ∈ [104, 108] K and central densities ρc ∈ [105, 1012] g cm−3. For all theories, deviations from zero temperature results remain very small up to ∼ 106 K, but become significant for T > 106 K. At higher temperatures, the radius increases for the same mass while the central density remains mostly unchanged, due to increasing thermal pressure. This leads to partial degeneracies between different sets of theory parameters, making it harder to distinguish between modified gravity effects and thermal effects. At T = 108 K, the mass-radius curves for all theories seem to converge at large radii and smaller masses, hinting that temperature effects may dominate over modified gravity effects in this regime, though the equation of state may be losing validity there.

The effective gravitational constant Geff = A2(ϕ̃)GN is computed. For all theories, the effective gravitation is weaker at the center of the star compared to its value at infinity. The largest deviations are seen in massive Brans-Dicke theory, of order O(10−1), while the symmetron shows deviations of O(10−6) and the dilaton shows intermediate values. For massive Brans-Dicke theory, the only visible temperature effect occurs at large central densities and only for T = 108 K, where the difference seems to saturate. The symmetron shows a clear plateau at high central densities due to the density-dependent effective potential driving the field to zero. The dilaton shows a drop-off towards lower densities as the star becomes less screened, and a saturation at high densities for T = 108 K.

Radial profiles of the scalar field, its derivative, pressure, and radial metric function are presented for massive Brans-Dicke theory with central pressure Pc ≈ 5.55 × 1024 dyn. The scalar field is strongest at the center and decreases towards infinity, with higher field masses leading to stronger and faster suppression. The pressure profile shows slight deviations at T = 107 K and more significant changes at T = 108 K. The radial metric function only deviates from the zero temperature case at T = 108 K. Overall, stellar properties are mostly unchanged with increasing temperature except for very high temperatures near T = 108 K.

The main conclusion is that the primary effect of temperature is an increase in radius with rising temperature, while other physical quantities retain their qualitative behavior except at extremely high temperatures. The authors caution that results at T = 108 K must be taken with care as the equation of state approaches its validity limit, and a more realistic model would require consideration of non-degenerate matter in outer layers, neutrino emission, or residual nuclear burning.

Improvement 1: Temperature-Aware Stellar Structure Prediction

The AI system can now incorporate finite-temperature Fermi-Dirac statistics into its stellar evolution models, enabling it to predict mass-radius relations for white dwarfs across a wide range of temperatures (104–108 K) and central densities (105–1012 g cm−3). This allows the system to generate accurate synthetic stellar profiles that account for thermal pressure effects, which is critical for modeling late-stage stellar remnants in high-temperature environments (e.g., accreting white dwarfs in binary systems).

Improvement 2: Modified Gravity Screening Discriminator

The system can now distinguish between thermal effects and modified gravity signatures (massive Brans-Dicke, symmetron, dilaton) in white dwarf observations. By leveraging the computed effective gravitational constant (G eff) profiles—which show characteristic deviations (O(10−1) for Brans-Dicke, O(10−6) for symmetron, intermediate for dilaton)—the AI can classify which theory best fits a given mass-radius dataset, even when temperature-induced radius inflation creates degeneracies. This improves the system’s ability to test alternative gravity theories using astrophysical data.

Improvement 3: High-Temperature Validity-Aware Inference

The AI system now includes a built-in reliability flag for equation-of-state (EOS) validity. Based on the paper’s caution about T = 108 K approaching EOS limits, the system can automatically down-weight or flag predictions in this regime, preventing overconfident conclusions. It can also suggest when to switch to more realistic models (e.g., including non-degenerate outer layers, neutrino losses, or residual nuclear burning) for extreme-temperature scenarios, improving robustness in stellar astrophysics pipelines.

Improved AI System Capabilities:

• Simulation Accelerator: Generates mass-radius curves and radial profiles (scalar field, pressure, metric) for white dwarfs under scalar-tensor gravity with thermal effects, reducing the need for costly numerical integrations.

• Observational Fitter: Given a white dwarf’s mass, radius, and temperature estimate, the system can infer the most likely gravity theory and scalar field parameters (e.g., coupling α0, field mass) while quantifying thermal contamination.

• Uncertainty Quantifier: Provides error bars on derived quantities (e.g., central density, G eff) that explicitly account for temperature-induced degeneracies, aiding in mission planning for telescopes like JWST or future X-ray observatories.

• EOS Boundary Detector: Automatically identifies when input conditions (T, ρ c) fall outside the regime where the Chandrasekhar EOS is valid, triggering alternative model selection or alerting the user to potential systematic errors.

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.