arXiv subjects

N. Heidari

Publications and source records attributed to N. Heidari.

At least 19 recordsLinked to original sources

arXiv2026

Born-Infeld black holes in bumblebee gravity

We construct exact static and spherically symmetric charged solutions in bumblebee gravity minimally coupled to Born--Infeld electrodynamics, taking into account a spacelike vector norm and a vanishing cosmological constant. An independently varied Lagrange multiplier enforces the norm constraint and acquires a radial profile determined by the transverse electromagnetic pressure. The solutions include purely radial configurations and configurations with constant or varying temporal amplitudes. At $ξ=κ/2$, the varying temporal amplitude introduces an independent bumblebee field strength while preserving the metric of the constant temporal configuration. Moreover, the electric field remains bounded, and the temporal configurations admit smooth extensions across a simple future horizon under explicit regularity conditions. We derive the photon sphere and shadow radii for both metric and nonlinear electromagnetic propagation. The leading nonlinear corrections enlarge the shadow at fixed geometric charge, whereas the temporal amplitude enters through the charged lapse. Comparisons with observations of Sgr~A$^*$ and M87$^*$ provide conditional bounds on the charge and nonlinear scale. In addition, the adopted planetary perihelion allowances yield an illustrative sensitivity of $|ξ|(b^2+u_0^2)\lesssim2.6\times10^{-12}$ for a neutral solar exterior.

gr-qc↗

arXiv2026

Gravitational fingerprints of a black hole solution in Palatini Kalb-Ramond gravity

We investigate particle motion, optical observables, and linear wave propagation around a static black hole in Palatini Kalb Ramond gravity with a vanishing cosmological constant. The analysis distinguishes the conserved mass $M$ from the orbital mass $\bar{M}=M/ρ$ and isolates the radial deformation $σ^2=1-q/2$, where $q=\ell_2/(1-\ell_1)$. At fixed $\bar{M}$, the photon sphere, critical impact parameter, and innermost stable circular orbit retain their Schwarzschild values. Radial frequency ratios also preserve their Schwarzschild profiles for the same areal release and particle radii, while proper distances, infall times, and accumulated orbital angles depend on $σ$. For radial fall, the tidal eigenvalues acquire a factor $σ^{-2}$, and the instability exponent of the photon orbit scales as $σ^{-1}$. In addition, we examine light deflection in the weak and strong regimes with an explicit account of the conical asymptotics. The shadow comparison with Sgr A* and M87* exhibits a coupling degeneracy when the orbital mass and areal distance are set. Scalar, electromagnetic, and massless Dirac perturbations probe the physical metric, whereas a homogeneous axial gravitational sector with $δB_{μν}=0$ admits an exact Schwarzschild spectral map. In this sector, both oscillation and attenuation rates scale as $σ^{-1}$ at fixed $\bar{M}$. Third order WKB estimates agree with frequencies extracted from time evolution to within $0.7\%$ for the reported fundamental modes.

gr-qc↗

arXiv2026

SOPHIE and TESS characterization of fourteen TESS Objects of Interest: nine planetary systems and five low-mass stellar companions

We present fourteen TESS Objects of Interest (TOIs) from the SOPHIE-TESS follow-up program. Combining TESS photometry, SOPHIE radial velocities (RVs), ground-based photometry, and high-resolution imaging, we establish the nature of all targets. Nine systems (TOI-6457, TOI-6883A, TOI-6649, TOI-6842A, TOI-5217A, TOI-6893A, TOI-7025, TOI-7176, and TOI-5893) host planets. TOI-6649 b, TOI-6842A b, and TOI-5217A b are transiting hot Jupiters with periods of 4.4-6.0 d, radii of 1.3-1.6 RJ, masses of 0.3-1.3 MJ, and densities of 0.1-0.6 g/cm3. TOI-5217A also hosts an outer companion, TOI-5217A c, with a period of about 1761 d, a minimum mass of about 13 MJ, and an eccentric orbit (e = 0.53 +/- 0.02). TOI-5893 b, TOI-6893A b, TOI-6457 b, TOI-7176 b, and TOI-7025 b are transiting warm Jupiters with periods of 40.4-67.0 d, radii of 0.8-1.2 RJ, and masses of 0.5-1.0 MJ. TOI-6883A b, a recently reported warm Jupiter, is further characterized with new photometry and SOPHIE and TRES RVs. The other five systems (TOI-5621AB, TOI-5321AB, TOI-3678AB, TOI-5906AB, TOI-5308AB) show large RV amplitudes indicating eclipsing binaries (companions of 0.1-0.6 Msun and 0.1-0.6 Rsun). TOI-6842A b is a super-puff (density 0.08 +/- 0.01 g/cm3), among the nine lowest-density planets known, and with a transmission spectroscopy metric of about 250 it is an excellent target for atmospheric characterization. The six warm Jupiters, with equilibrium temperatures of 377-846 K, are low-irradiation benchmarks for hot-Jupiter inflation models. TOI-7025 b has an eccentricity of 0.82 +/- 0.02, one of the largest known for a planet orbiting a single star. Finally, using Gaia astrometric excess noise, we detect TOI-5906 B, the first object identified by transit, RV, and astrometry.

astro-ph.EP↗

arXiv2026

Higher derivative Hořava-Lifshitz fermions: from superconductivity to atomic spectroscopy

We investigate an isotropic Hořava-Lifshitz fermion theory supplemented by a dimension-six spatial derivative operator, preserving first order time evolution while modifying the ultraviolet dispersion relation. Within the regime of validity of the derivative expansion, we develop three complementary consequences of the $z=2$ sector. First, a Bardeen-Cooper-Schrieffer/Nambu-Jona-Lasinio (BCS/NJL) mean field treatment of fermion pairing yields the particle and antiparticle quasiparticle spectra, the finite temperature gap equation, and the gauge invariant criterion for a stable superconducting phase. The modified dispersion alters both the density of states governing the pairing scale and the electromagnetic vertices entering the Meissner response. Second, using the scalar pole denominator as a minimal spin independent exchange kernel, we derive an exponentially damped and oscillatory massive potential and obtain the corresponding Born scattering amplitudes. The quartic momentum dependence renders both the forward and total cross sections finite and produces a large transfer behavior distinct from Rutherford scattering. Finally, we evaluate the induced shifts of the hydrogenic $1S$, $2S$, and $2P$ levels in closed form, retaining the full dependence on the interaction range. In other words, these latter results gives direct spectroscopic constraints on the product of nuclear and leptonic couplings, with muonic atoms, showing the respective sensitivity to the short distance Hořava-Lifshitz sector.

gr-qc↗

arXiv2026

A general static and spherically symmetric black hole in traceless metric-affine bumblebee gravity

We construct and analyze a general exact static and spherically symmetric black hole solution in traceless metric-affine bumblebee gravity by considering a vacuum expectation value with temporal and radial components. The temporal component remains constant and the auxiliary geometry is Ricci flat, while Lorentz symmetry breaking is transferred to the physical metric through the metric-affine deformation. After introducing the areal radius and normalizing the Killing time at spatial infinity, the geometry is controlled by a single invariant deformation parameter. The resulting spacetime is asymptotically conical, possesses a regular Killing horizon, and retains the causal structure of the Schwarzschild black hole, with a curvature singularity located at the origin. Initially, we examine its thermodynamic properties, gravitational Doppler effect, and tidal forces. Lorentz symmetry breaking modifies the Hawking temperature and the magnitude of the tidal field, although the characteristic ratio between radial stretching and transverse compression is preserved. The thermodynamic-topology analysis reveals no critical point within the admissible black hole sector. Furthermore, massless scalar perturbations are also investigated through WKB and characteristic time domain solution, showing decaying profiles throughout the sampled parameter range and deformation-dependent quasinormal frequencies. Finally, light deflection and perihelion advance are used to establish stringent Solar System bounds on the invariant deformation, which, at leading order, depends on a combination of the temporal and norm sectors of the bumblebee field.

gr-qc↗

arXiv2026

Exploring a phantom Dirac-Born-Infeld regular black hole via particle emission, wave scattering and geodesics

We examine particle creation, evaporation, scalar wave absorption and scattering, and geodesic motion in the asymptotically flat regular black hole supported by a phantom Dirac-Born-Infeld field. For massless bosonic and fermionic fields, the Bogoliubov transformations yield thermal spectra whose Hawking temperature decreases as the regular core becomes more prominent. Energy-conserving tunneling recovers the same temperature in the low-energy limit, whereas recoil and the DBI contribution introduce nonthermal corrections and suppress particle emission. In the high-frequency regime, the enlargement of the cross section does not compensate for the reduction in temperature, resulting in a lower luminosity and a longer evaporation time. A numerical partial wave analysis shows that the total scalar absorption increases with the regular core scale, approaches the horizon area at low frequencies, and oscillates around an enlarged geometric capture limit at high frequencies. The scattering phase shifts modify the multipolar amplitudes nonuniformly and displace the interference fringes toward larger angles. Furthermore, both null and timelike trajectories experience stronger deflection as the regular core contribution increases.

gr-qc↗

arXiv2026

Gravitational wave signatures and periodic orbits of a charged black hole in a Hernquist dark matter halo

In this work, we study the motion of massive test particles and the gravitational--wave emission associated with periodic trajectories around a magnetically charged black hole immersed in a \textit{Hernquist} dark matter halo. We begin by analyzing the effective potential and the conditions for stable motion, with particular attention to the marginally bound radius and the innermost stable circular orbit. Our results show that the dark matter parameters, namely the halo density and scale radius, enlarge the allowed region and generally shift the relevant characteristic radii and angular momenta toward larger values. In contrast, the magnetic charge partially counterbalances this behavior. We then examine periodic trajectories through the rational number $q$, which characterizes the relation between the azimuthal and radial frequencies, and construct representative zoom--whirl configurations together with their precessing counterparts. Finally, we investigate the imprints of dark matter and magnetic monopole charge on the gravitational--wave polarizations in the extreme mass--ratio regime.

gr-qc↗

arXiv2026

Black Hole Gravitational Phenomena in Higher-Order Curvature-Scalar Gravity

This work aims to explore the gravitational consequences of a recently proposed black hole solution presented in the literature [Phys. Dark Univ. 50 (2025) 102061]. We initiate our analyzes by taking into account the horizon structure, focusing on both the event and Cauchy horizons. Subsequently, we examine the quasinormal modes by considering all types of perturbations -- scalar, vector, tensor, and spinorial. To strengthen these results, we also compute the time-domain for each perturbation. Next, we turn to the study of optical properties of the black hole. In particular, we investigate null geodesics, the photon sphere and its stability, as well as the corresponding black hole shadows. Following this, we analyze gravitational lensing phenomena in two regimes: the weak-field limit, utilizing the Gauss-Bonnet theorem, and the strong deflection limit, employing Tsukamoto's approach. In addition, we confront the lensing observables with Event Horizon Telescope (EHT) data for $Sgr A^{*}$ and $M87^{*}$. Finally, constraints on the parameter $ξ$ -- which is introduced by higher-order curvature-scalar gravity, thereby differing from the Schwarzschild solution -- are estimated using Solar System measurements such as the precession of Mercury's orbit, gravitational light bending, and time delay (or Shapiro effect).

gr-qc↗

arXiv2026

A rotating black hole in a Hernquist dark matter halo: horizon geometry, thermodynamics, and quantum emission

We investigate the geometrical, thermodynamic, and quantum emission properties of a rotating black hole immersed in a Hernquist dark matter halo. Starting from a static black hole spacetime surrounded by a Hernquist distribution, we construct its rotating counterpart through the noncomplexification formulation of the Newman-Janis algorithm and analyze the modifications induced by the independent halo parameters $ρ$ and $r_s$ and the rotation parameter $a$. The horizon structure is determined from the roots of the radial function $Δ(r)$, while the stationary limit surfaces and the corresponding ergoregions are obtained from the condition $g_{tt}=0$. We show that the Hernquist contribution displaces the outer event horizon toward larger radii and modifies the size of the ergoregion, whereas rotation controls the oblateness of the horizon and the strength of frame dragging. We further derive the surface gravity, Hawking temperature, Bekenstein-Hawking entropy, and heat capacity. The quantum tunneling rate is obtained from the Hamilton-Jacobi method, leading to the corresponding occupation number and a thermal estimate of the particle creation density. Finally, we estimate the Hawking luminosity and evaporation timescales within a Stefan-Boltzmann approximation. All standard Kerr and Schwarzschild results are recovered in the appropriate limiting cases.

gr-qc↗

arXiv2026

CAFE follow-up of TESS hot Jupiter candidates left behind: I. Five newly confirmed planets and a false positive

Hot Jupiters are key targets for understanding planet formation, migration, and atmospheres. Yet, most ground-based follow-up resources for the TESS mission are focused on confirming low-mass planet candidates, leaving many giant planets without mass determinations or definitive confirmation. We use the \cafe{} spectrograph at Calar Alto Observatory to monitor the radial velocity of stars hosting hot-Jupiter candidates that have received little follow-up, aiming to confirm their planetary nature. We present results for seven candidates. We monitored the radial velocity of TOI-603, TOI-1137, TOI-1837, TOI-2114, TOI-4492, TOI-5806, and TOI-5811, jointly modeling the CAFE radial velocities and TESS photometry to determine the nature and properties of the transiting objects. We confirm five new planets: TOI-603 b ($33.0^{+6.5}_{-6.2}$ M$_{\oplus}$, $16.2$ d), TOI-2114 b ($1.01^{+0.14}_{-0.12}$ M$_{\rm Jup}$, $6.2$ d), TOI-4492 b ($5.92^{+0.67}_{-0.64}$ M$_{\rm Jup}$, $4.4$ d), TOI-5806 b ($2.77^{+0.34}_{-0.32}$ M$_{\rm Jup}$, $3.2$ d), and TOI-5811 B b ($0.81^{+0.11}_{-0.10}$ M$_{\rm Jup}$, $6.3$ d). TOI-603 b lies in the "Neptune savanna", whereas the other four are hot Jupiters orbiting slightly evolved stars. We find TOI-5811.01 to be a planet transiting the nearby bound companion TOI-5811 B (hence TOI-5811 B b), and identify a stellar companion to TOI-5806 at a projected separation of 248 au, making both S-type planetary systems. TOI-1837.01 is an eclipsing binary, while TOI-1137.01 remains inconclusive. The five confirmed planets orbit bright stars (${\rm G}=8.6-10.2$~mag), and four are excellent targets for atmospheric studies, with transmission spectroscopy metrics above 90. These results highlight the importance of intensive follow-up observations to establish the nature of transiting planet candidates.

astro-ph.EP↗

arXiv2026

Gravitational wave propagation in Hořava-Lifshitz gravity

We investigate the generation and propagation of gravitational waves in the leading parity-even infrared truncation of Hořava-Lifshitz gravity, characterized by the modified tensor dispersion relation $ω^{2}=k^{2}+αk^{4}$. Working in the transverse-traceless sector, we show that the higher-spatial-derivative correction preserves the conventional plus and cross polarizations and introduces neither polarization mixing, helicity splitting, nor gravitational birefringence. We construct the retarded Green function of the modified wave operator and derive the radiation-zone waveform to first order in $α$. The resulting signal exhibits a frequency-dependent amplitude renormalization together with a dispersive propagation phase that accumulates over the source-observer distance. We apply the formalism to a binary black hole system in a quasi-circular orbit and obtain the polarization waveforms for an arbitrary observation direction. We further derive the corresponding energy flux, total luminosity, and adiabatic chirp evolution. In terms of the observed gravitational wave frequency $f$, the leading corrections satisfy $Δh_{A}/h_{A}^{\mathrm{GR}}=-8π^{2}αf^{2}$ and $ΔP/P_{\mathrm{GR}} =Δ\dot{f}/\dot{f}_{\mathrm{GR}} =-16π^{2}αf^{2}$, while the accumulated generation phase has the frequency dependence of a relative third post-Newtonian contribution. By mapping the Hořava-Lifshitz coefficient to the LIGO-Virgo-KAGRA modified-dispersion parametrization, we obtain $-6.2\times10^{2}\,\mathrm{eV}^{-2} <α< 1.9\times10^{2}\,\mathrm{eV}^{-2}$ at $90\%$ credibility from the GWTC-4.0 posterior.

gr-qc↗

arXiv2026

Accelerating electrically charged ModMax black hole solutions in $F(R)$ gravity

Using the $C-$metric in the context of $F(R)$ gravity coupled with the ModMax nonlinear electromagnetic field (the $F(R)-$ModMax theory), we derive an exact black hole solution in a four-dimensional spacetime. We then examine how various parameters influence the behavior of accelerating ModMax black holes. Treating this black hole as a thermodynamic system, we calculate the Hawking temperature and entropy for the accelerating ModMax black holes within the framework of $F(R)$ gravity. Subsequently, we explore the impact of the parameters in $F(R)-$ModMax theory on the Hawking temperature and entropy. We also assess local stability by analyzing the heat capacity. Additionally, we investigate both the angular shadow and the shadow radius of an accelerating black hole in the context of $F(R)-$ModMax gravity.

gr-qc↗

arXiv2026

Revisiting TOI-4438 and TOI-442 planetary systems with new observations from SPIRou and TESS

We present a comprehensive re-analysis of two star-planet systems: TOI-4438, an M3.5V star hosting a mini-Neptune in a 7.4-day orbit, and TOI-442, an M1V star with a 4-day period planet located within the hot Neptune desert. Both systems were originally identified as transiting planet candidates by TESS and subsequently validated through the radial velocity (RV) method. Our work incorporates new TESS transit data and high-resolution spectroscopy from the SPIRou near-infrared spectropolarimeter. We detect a persistent and relatively strong Zeeman signature in TOI-442, while TOI-4438 exhibits weaker and intermittent magnetic activity, and we infer the stellar rotation periods of both stars from the variability of the longitudinal magnetic field. We jointly fit photometry and RV models for each system. For TOI-4438\,b we combine archival CARMENES data with 81 SPIRou observations and five TESS sectors. This yields a refined planetary mass of $M_{\rm p} = 4.11^{+0.40}_{-0.38}\,M_{\oplus}$ and a radius of $R_{\rm p} = 2.40^{+0.09}_{-0.10}\,R_{\oplus}$, consistent with the previous estimate within 1.1$σ$ while improving by 53\% the precision on the mass and 22\% on the radius. For TOI-442\,b, we add 29 SPIRou RV measurements to an extensive archival dataset, significantly extending the temporal baseline. With a new TESS sector, we tighten the mass to $M_{\rm p} = 28.38^{+0.77}_{-0.73}\,M_{\oplus}$ and radius to $R_{\rm p} = 4.25^{+0.10}_{-0.08}\,R_{\oplus}$, which agrees to the previous values within 1.5$σ$ and improves the precisions by 46\% and 67\% respectively. We find no clear signs of additional planets in the available RV data, although we detect a single-transit event in the TOI-4438 light curve. We compare various RV models and find that those accounting for stellar variability-induced signals yield improved constraints on the planetary parameters.

astro-ph.EP↗

arXiv2026

Shadows and lensing signatures of a rotating black hole in a Hernquist dark matter halo

We investigate the optical properties of a rotating black hole immersed in a Hernquist dark matter halo. The spacetime is generated from a static Hernquist black hole through the noncomplexification version of the Newman-Janis procedure, yielding a Kerr-like geometry whose halo contribution is encoded in the radial function $Δ(r)$ \cite{AraujoFilho:2026hernquist}. We derive the null geodesic equations, effective potentials, radial acceleration, and representative three-dimensional photon trajectories around the event horizon and ergoregion. Using the separability of the Hamilton-Jacobi equation, we obtain the critical impact parameters of unstable spherical photon orbits and construct the shadow contours for a distant observer. The rotation parameter mainly shifts and distorts the shadow, whereas the Hernquist halo enlarges the photon capture region and increases the apparent shadow size. Comparing the area-equivalent shadow diameter with the Event Horizon Telescope measurements of Sgr A$^\ast$ and M87$^\ast$, we constrain the dimensionless halo parameter $\hatρ=M^2ρ$. The strongest restriction comes from Sgr A$^\ast$, giving $\hatρ\sim(2.7-3.8)\times10^{-3}$ at $1σ$ and $\hatρ\sim(4.1-5.2)\times10^{-3}$ at $2σ$. We also analyze strong- and weak-field gravitational lensing. In the strong-field regime, the halo shifts the unstable photon orbit and critical impact parameter, controlling the logarithmic deflection angle and the position of relativistic images. In the weak-field regime, the halo contributes already to the leading bending angle and enhances deviations from Kerr as $ρ$ grows. From the Einstein ring of ESO325-G004, we further obtain $0\leq\hatρ\lesssim0.00939$ at $1σ$ and $0\leq\hatρ\lesssim0.01963$ at $2σ$.

gr-qc↗

arXiv2026

Neutrino dynamics in a non-commutative spacetime

We investigate neutrino dynamics in a spherically symmetric non-commutative black hole spacetime generated by a Lorentzian smeared matter distribution. We work with the leading-order Lorentzian-smeared metric, in which the non-commutative parameter $Θ$ enters explicitly through a $\sqrtΘ$ correction to the Schwarzschild geometry. Within this background, we analyze the neutrino-antineutrino annihilation process, the oscillation phase acquired by massive neutrinos along geodesics, and the flavor-transition probability in the presence of weak gravitational lensing. We show that non-commutativity monotonically suppresses the relativistic energy deposition rate relative to the Schwarzschild case, while the Newtonian behavior is recovered at large radii. In the oscillation sector, the weak-field phase receives explicit $\sqrtΘ$ corrections, which modify the distance of closest approach and reduce the accumulated phase with respect to the Schwarzschild limit. For lensed trajectories, the macroscopic transition probabilities remain close to the Schwarzschild prediction; however, a high-precision residual analysis reveals a coherent, non-vanishing oscillatory correction driven by the non-commutative geometry. The dependence on the absolute neutrino mass scale enters only through multi path interference terms and remains subdominant compared with the $Θ$-induced phase shift. The full three-flavor residual analysis confirms that the same non-commutative oscillatory pattern persists beyond the effective two-flavor approximation.

gr-qc↗

arXiv2026

Gravitational Wave Signatures from Periodic Orbits around a Non--commutative Schwarzschild Black Hole

In this work, we investigate massive particle motion and the gravitational wave emission generated by periodic trajectories around a non--commutative \textit{Schwarzschild} black hole sourced by a Lorentzian matter distribution. We analyze the effective potential, the marginally bound orbit, and the innermost stable circular orbit, showing that non--commutative corrections shift these characteristic orbits toward smaller radii and reduce their corresponding angular momenta. The allowed region in the $(E, L)$ plane is also displaced toward lower values, favoring more tightly bound configurations. Periodic trajectories are classified through the rational parameter $q$, which relates the radial and azimuthal frequencies. For a fixed orbital topology, increasing the non--commutative parameter lowers the energy required to produce the orbit and results in more compact zoom--whirl configurations. Small deviations from the periodic energies are also shown to generate precessional drift. From the periastron advance of the S2 star around Sgr~A$^*$, we obtain the preliminary bound $Θ/M^{2}<0.014$. Finally, using the adiabatic and numerical kludge approximations, we compute the gravitational wave polarizations and find phase shifts and an overall enhancement of the amplitude.

gr-qc↗

arXiv2026

Signatures of a de Sitter-core black hole in ringing, transmission and optical appearance

We investigate a static, asymptotically flat black hole whose central region approaches a de Sitter vacuum. The geometry is controlled by the ADM mass $M_0$ and a core scale $R$, and is generated by an exponentially decaying anisotropic source. After clarifying the stress tensor and the horizon structure, we study scalar-field perturbations, greybody transmission, Hawking emission and the optical image produced by an optically thin infalling flow. The horizon analysis shows that two horizons exist below the critical value $R/M_0\simeq0.7768$, where they merge into an extremal configuration. Increasing the core scale lowers the peak of the scalar effective potential and shifts the quasinormal spectrum away from the Schwarzschild value, with the largest fractional deviations occurring for low multipoles and higher overtones within the WKB domain of validity. The greybody bound indicates stronger filtering as the core becomes more extended, while the QNM--greybody correspondence gives a complementary estimate of the transmission probability in the eikonal regime. The Hawking temperature decreases as the extremal configuration is approached, suppressing the emission rate and shifting its maximum to lower frequencies. For a simple optically thin infalling flow, ray tracing shows a small decrease in the apparent shadow scale and in the peak intensity as the core scale is increased.

gr-qc↗

arXiv2026

Optical Phenomena in a Non-Commutative Kalb-Ramond Black Hole Spacetime

This work investigates additional gravitational features of a newly proposed black hole spacetime within Kalb-Ramond gravity, incorporating non-commutative corrections arising from a gauge-theoretic approach recently introduced in the literature [arXiv:2507.17390]. Accordingly, null geodesics are solved numerically to trace photon paths; the photon sphere and shadow are determined. From Event Horizon Telescope (EHT) measurements of $Sgr A^{*}$, constraints on the parameters $Θ$ (which encapsulates the non-commutativity) and $\ell$ (the Lorentz-violating parameter) are established. To examine the stability of critical orbits and the deflection angle (gravitational lensing) in the weak field scenario, we compute the Gaussian curvature in order to use the Gauss-Bonnet theorem. Moreover, the deflection angle has been calculated as well in the strong deflection limit. Furthermore, Lensing observables are estimated using EHT data for $Sgr A^{*}$ and $M87$. Topological features such as the topological photon sphere are also explored.

gr-qc↗