Constraining linear form of $f(\mathcal{R,G,T})$ gravity from astrophysical observations of the Pulsar U1724

TL;DR

The paper investigates a linear extension of gravity in the form $f(\mathcal{R},\mathcal{G},\mathcal{T}) = \mathcal{R} + \alpha\mathcal{G} + \beta\mathcal{T}$ and applies it to static, anisotropic compact stars. An exact analytic interior solution is derived using a KB-like metric ansatz, with all physical quantities expressed through dimensionless couplings $\alpha_1 = \alpha/R^{2}$ and $\beta_1 = \beta/\kappa^{2}$ and the compactness $C$. By matching to the pulsar U1724’s mass and radius, the parameters are constrained to $\alpha_1 = \pm 0.023$ and $\beta_1 = \pm 0.001$, and the radial sound speed satisfies $c_s^2 < c^2/3$, signaling a conformal bound on the core. The analysis shows consistent stability, energy conditions, and TOV-like hydrostatic balance, with MR relations that can reach several solar masses and compactness approaching the black-hole limit in this MG context. These results demonstrate that linear $f(\mathcal{R},\mathcal{G},\mathcal{T})$ gravity can considerably extend the viable neutron-star phenomenology beyond GR while remaining compatible with current observational constraints.

Abstract

In this work we examine the internal structure of compact stars within an extended gravitational framework described by the function $f(\mathcal{R},\mathcal{G},\mathcal{T})$. Throughout this work, the quantity $\mathcal{R}$ refers to the curvature scalar formed from the Ricci tensor. The term $\mathcal{G}$ denotes the Gauss--Bonnet curvature invariant, while $\mathcal{T}$ corresponds to the trace obtained by contracting the matter energy-momentum tensor. Our analysis is directed toward massive radio pulsars with masses above $1.8\,M_{\odot}$, which provide an exceptional testing ground for gravity under conditions inaccessible to laboratory experiments. Adopting the linear form $f(\mathcal{R},\mathcal{G},\mathcal{T})=\mathcal{R}+α\,\mathcal{G}+β\,\mathcal{T}$ where $α$ and $β$ are parameters of suitable dimensionality,\footnote{$α$ has dimensions of $[L^{2}]$ and $β$ carries units of $[N^{-1}]$.} we obtain an exact analytic solution for static anisotropic stellar matter in hydrostatic equilibrium. This solution allows all physical quantities to be expressed in terms of the dimensionless parameters $ α_{1}=α/R^{2},\qquad β_{1}=β/κ^{2}$ together with the compactness $C=2GM/(Rc^{2})$. We constraint the two parameters $α$ and $β$ by matching the model with the mass and radius of pulsar \textit{U1724} requires restricting these parameters to $α_{1}=\pm0.023$ and $β_{1}=\pm0.001$, where $κ^{2}=8πG/c^{4}$ is the standard Einstein coupling. The resulting stellar configuration satisfies the causal bound on the radial sound speed, $c_{s}^{2}<c^{2}/3$, distinguishing it from the corresponding behaviour in general relativity.

Figures (9)

• Figure 1: Matter distribution and anisotropy profiles for the pulsar U1724. Panels \ref{['fig:density']}--\ref{['fig:tangpressure']} display $\rho$ together with $p_r$ and $p_t$ profiles obtained from Eq. \ref{['sol']}, for the parameter values $\alpha_{1}=\pm 0.023$ and $\beta_{1}=\pm 0.001$. The curves demonstrate that both the density and pressure components remain finite throughout the stellar interior and decrease smoothly toward the stellar surface. Panel \ref{['fig:anisotf']} shows the variation of $\Delta(r)$ inside the star for $\alpha_{1}=0,\pm 0.023$ and $\beta_{1}=0,\pm 0.001$.
• Figure 2: Variation of the sound velocity within the matter of pulsar U1724 for $\epsilon_1 = 0,~ \pm 0.03$. Subfigures \ref{['fig:vr']} and \ref{['fig:vt']} depict how $v_r$ and $v_t$ propagates as described by Eq. \ref{['eq:sound_speed']}. Subfigure \ref{['fig:vt-vr']} indicates that the solution satisfies the stability requirement of the highly anisotropic regime, given by $(v_t^2 - v_r^2)/c^2 < 0$.
• Figure 3: The mass given by Eq. \ref{['Mf3']} associated to pulsar ${\textit{U}1724}$.
• Figure 4: The spacetime structure associated with the pulsar U1724. \ref{['fig:Junction']} illustrates the interior metric components $g_{tt}$ and $g_{rr}$, modeled through the KB prescription, together with the external Schwarzschild vacuum geometry. The figure confirms that both potentials remain regular throughout the star and transition smoothly to the outer spacetime at the boundary. \ref{['fig:redshift']} presents the corresponding gravitational redshift profile, derived from Eq. \ref{['eq:redshift']}, for the parameter sets $\alpha_1 = \pm 0.023$ and $\beta_1 = \pm 0.001$. The central value of the redshift is about $Z_s \simeq 0.59$, gradually declining to approximately $0.33$ at the stellar surface in all examined cases.
• Figure 5: All energy conditions associated with the effective matter tensor $\bar{\mathfrak{T}}_{\mu\nu}$, defined in Subsection \ref{['Sec:Energy-conditions']}, are verified to hold for the pulsar model J0740+6620, as illustrated in the plots.