Papers

The systematic generalisation of the well-established $S$-matrix positivity and bootstrap bounds from $2 \to 2$ scattering processes to higher-point amplitudes has, to date, remained elusive. To address this, we consider a large class of tree-level UV-completions of a single-scalar EFT. Specifically, we consider the most general renormalisable $N$-scalar theory in $D=4$. By explicitly computing tree-level amplitudes for four-, five- and six-point scattering and expanding at low energies, we can search for positivity relations among the expansion coefficients. We find that quite generally positivity bounds are unavailable due to contamination from a specific class of Feynman diagrams that are linear in sign-indefinite cubic couplings. However, by isolating a sector of the full crossing-symmetric low-energy amplitude that is unaffected by these sign-indefinite cubic couplings, we find that selected six-, five- and four-point Wilson coefficients organise into positive semi-definite matrices. While a violation of these positivity statements only implies that a given EFT is inconsistent with our class of UV completions, the result helps to guide towards what more general positivity bounds for higher-point interactions could, and could not, look like.

We investigate the extremization of Perlick's proper time in non-integrable Weyl geometry. We derive the corresponding generalized Euler–Lagrange equations and show that the resulting extremals do not, in general, coincide with the autoparallels of the Weyl connection, their difference being governed by the Weyl length curvature and vanishing in the integrable case. A distinctive feature of the extremal equation is its nonlocal character: in the Weyl proper-time parametrization, the acceleration depends explicitly on the remaining proper time to the endpoint of the variational interval. We illustrate this behavior for a weak constant Weyl field and show that a local Lorentz-force dynamics emerges in a double-scaling limit in which the Weyl field vanishes and the terminal proper time diverges while their product remains finite. These results uncover a nontrivial relation between proper-time extremization, Weyl non-integrability, and local force dynamics.

We study the quasinormal-mode and quasibound-state spectra of a massless scalar field in weakly magnetized Ernst–Schwarzschild spacetime. In the separated $O(B_0^2)$ regime, the Klein–Gordon equation reduces to a radial confluent-Heun equation. We construct an endpoint-selected spectral Wronskian directly from the canonical confluent-Heun recurrence and its regular-to-irregular connection limit. Quasinormal modes and quasibound states are then obtained as zeros of the same analytic spectral function on the radiative and bound sheets, respectively, providing a unified treatment of the two spectral sectors within the separated weak-field problem. The resulting QNM and QBS frequencies agree with independent weak-field Ernst and massive-Schwarzschild benchmarks available in the literature. We further show, for representative cases, that the frequencies selected by the local Heun-series termination relation examined here do not satisfy the endpoint-selected quasibound-state condition. The construction therefore provides a direct horizon-to-infinity spectral condition for both radiative and bound states within the weak-field Ernst problem.

We construct exact hairy black holes in Lovelock–Horndeski gravity and establish a relation between horizon topology, scalar hair and covariant charges. Rooted in the dimensional reduction of higher-curvature gravity, this framework connects scalar–tensor dynamics to the low-energy description of heterotic strings. In five dimensions, a single fixed theory admits a continuous planar family and two isolated hyperbolic black holes. The hyperbolic solutions share a locally AdS metric with distinct scalar dressings, while the planar family is Weyl curved and has a freely varying horizon scale. Both branches carry scalar hair regular on the future horizon. The hyperbolic construction extends above four dimensions; compatibility of the displayed planar profile selects five and seven dimensions. Time translations of the scalar act through a global shift, which enters the complete covariant charge balance. Along the planar family, a changing bulk shift-charge density supplies this balance despite zero radial flux and vanishing horizon charge variation. The hyperbolic roots admit no horizon-scale variation at fixed couplings.

Transdimensional Bayesian inference is becoming a key ingredient in gravitational-wave astronomy. In many relevant applications, the number of components needed to describe the data is not known a priori and must be inferred jointly with their parameters. We present a transdimensional Sequential Monte Carlo (SMC) framework designed to exploit massive parallelism. The method evolves a population of particles through a sequence of tempered distributions and uses the No-U-Turn Sampler for the exploration of fixed-dimensional parameter space while reversible-jump birth and death moves allow the number of components to vary. The same SMC construction also enables posterior samples obtained from a shorter data segment to be updated as additional data become available, enhancing the overall efficiency of the analysis. We validate the approach on two proof-of-concept problems: the recovery of a sequence of Gaussian pulses and a simplified LISA Galactic-binary inference problem. In both cases, the method recovers the injected number of components and produces consistent posterior estimates. These results indicate that the proposed method is a promising avenue toward scalable and parallel Bayesian transdimensional inference for current and future gravitational-wave astronomy.

Pulsar timing arrays are a powerful tool for probing low-frequency gravitational phenomena and physics beyond the Standard Model. We establish a general, gauge-invariant framework for the timing residual by identifying the measurable quantity as the proper time interval between consecutive pulse arrivals at Earth. In linearized general relativity, we derive a general expression for this proper time shift and show that it decomposes into Doppler, Shapiro, and Einstein delay contributions whose sum is gauge invariant, similar to frameworks that have been developed for other gravitational-wave detectors such as laser and atom interferometers. We also provide a recipe for computing the resulting timing residual for any specified perturbation, and we illustrate the method with several physically well-motivated sources, recovering known results where they exist in the literature. This recipe can be directly applied to search for new physics from real pulsar timing data using existing numerical software.

The origin of the stellar-mass binary black holes (BBHs) observed in gravitational wave (GW) events by the LIGO-Virgo-KAGRA (LVK) detectors remains uncertain. One promising formation pathway is the active galactic nucleus (AGN) channel, in which BBHs embedded in AGN accretion discs experience enhanced merger rates and may produce rapidly spinning black holes in the upper mass gap. Previous constraints on the AGN contribution to the LVK merger population have relied either on spatial correlations between GW localisations and AGN catalogues or on modelling the intrinsic properties of GW sources. We develop a hierarchical Bayesian framework that unifies these approaches within a single likelihood for the fraction of AGN-hosted GW events, $f_{\rm agn}$. This framework, combined with the latest gravitational wave transient catalogue GWTC-5.0 and the electromagnetic Quaia AGN catalogue, cannot exclude an AGN contribution to the observed merger rate. For unobscured AGN in the optical and infrared bands with bolometric luminosities $L_{\rm bol}\geq\{10^{44.5},10^{45.5},10^{46.5}\}\,\mathrm{erg\,s^{-1}}$, we exclude $f_{\rm agn} \geq \{0.52,0.41, 0.61\}$ at the 95 percent credible level under our fiducial model. The methodology and results presented in this work lay down the pathway for combining the full GW posterior with electromagnetic observations to infer the properties of GW populations and AGN discs.

Noether charges in gravity are always defined with respect to a reference metric, but the near-horizon extremal Kerr (NHEK) solution does not come with an obvious choice of reference metric (it is not asymptotically flat, for example). We discuss a one parameter generalization of the NHEK solution due to Kinnersley and Kelley which interpolates between NHEK and an analytically continued version of Minkowski spacetime. We use this interpolating family of solutions to compute the conformal energy of NHEK above the Minkowski reference metric. We find an interesting order of limits issue when computing the conformal energy of the Minkowski vacuum and we discuss a few natural prescriptions for fixing it, including one motivated by the Kerr/CFT correspondence.

Late-time cosmic acceleration is conventionally ascribed to a cosmological constant, though $\Lambda$CDM continues to face several theoretical difficulties that keep alternative gravity models under active consideration. This work examines the Hu-Sawicki $f(R)$ model in a spatially non-flat background, with the modified-gravity parameter $b$ and the curvature density $\Omega_k$ both treated as free parameters, constrained using DESI-DR2 BAO, BBN, and four Type Ia supernova compilations – PantheonPlus, PantheonPlus+SH0ES, Union3, and DESY5yr. Across all four combinations, $H_0$ and $\Omega_m$ stay close to their $\Lambda$CDM values, with a noticeable shift in $H_0$ appearing only for the SH0ES-calibrated dataset. The parameter $b$ departs from zero at better than $2\sigma$ in three of the four fits, most prominently for DESY5yr, whereas the SH0ES-calibrated combination alone prefers $b<0$. A strong positive correlation among $b$, $H_0$, and $\Omega_k$ underlies this shift, suggesting that curvature signatures obtained under $\Lambda$CDM can be reabsorbed into the modified-gravity sector once this additional freedom is allowed. Statistical model comparison via AIC and BIC gives a mixed picture: AIC leans toward the Hu-Sawicki model in three of the four combinations, while BIC's heavier penalty on the extra parameter favors $\Lambda$CDM in most cases, and only DESY5yr is preferred under both criteria. These findings show that the Hu-Sawicki $f(R)$ scenario with unconstrained curvature is nonetheless a statistically feasible, if not obviously preferred, alternative to $\Lambda$CDM, and that curvature restrictions generated inside $\Lambda$CDM cannot be viewed as independent of the underlying gravity model.

We present a complete analytic expression for the $N$-body Hamiltonian of nonspinning point particles at second post-Newtonian (2PN) order in the Arnowitt-Deser-Misner (ADM) gauge. Recent work reduced the general $N$-body Hamiltonian to a form containing a single unresolved integral term, which prevented a fully explicit closed-form representation. Here, we evaluate this integral analytically and thereby obtain a fully explicit closed-form expression for the general 2PN $N$-body Hamiltonian. We outline the main steps of the derivation and validate the analytic expression through direct numerical comparisons with the original integral representation.

We study quasinormal modes governed by the Teukolsky equation in Kerr-Newman-de Sitter spacetime, formally treating the spin weight and field charge as arbitrary parameters. We use exact Heun connection formulae expressed through classical conformal blocks. We analytically derive the quasinormal mode spectra in the small cosmological constant, near-extremal, and near-Nariai limits. In the small cosmological constant limit, we obtain the leading charged de Sitter spectrum together with its subleading corrections. In both the small cosmological constant and near-extremal limits, solving the full connection condition beyond the leading pole approximation gives rise to generically noninteger-power corrections to the quasinormal mode frequencies, whereas no analogous correction appears in the near-Nariai limit. We also analyze the angular eigenvalue problem as an illustration of the same connection-formula method.

Scalar-Tensor theories extend General Relativity (GR) by including both a scalar field and a metric tensor in describing gravity. Although the matter sector is minimally coupled to the metric in GR, the scalar field coupling to matter may be adjusted by means of Disformal Transformations. One of the most general of these transformations, formulated by Takahashi, Motohashi, and Minamitsuji, include arbitrarily high-order derivatives of the scalar field and are designed to map between two ghost-free Higher-order Scalar-Tensor theories. We used these transformations to determine the most general conditions for the disformal form-invariance of the Klein-Gordon equation for the scalar field. In the second-order case, we identified a family of disformally-related metrics that result in the same propagation solutions. Additionally, higher-order cases could comply with theoretical and phenomenological constraints alongside KG-invariance. Finally, we investigated the notion of disformal transformations as “deformations” of the spacetime manifold.

The high-overtone quasinormal-mode spectrum encodes the near-singularity Kasner scaling of black-hole interiors. We show that distinct structures in this spectrum independently determine the temporal and spatial Kasner grades, allowing the corresponding metric exponents to be reconstructed without imposing Kasner constraints or invoking holography. We demonstrate this independent reconstruction at the fixed Kasner point of five-dimensional asymptotically flat Schwarzschild and planar Schwarzschild–AdS$_5$, and then extend it to a continuously varying Gibbons–Maeda black-hole family. The reconstructed exponents satisfy the Kasner relation a posteriori, providing a nontrivial geometric consistency check. More generally, the high-overtone spectrum separates into global spectral data and local algebraic corrections whose fractional powers are fixed by the near-singularity Kasner geometry. These fractional corrections therefore provide a direct spectral probe of the local geometry behind the horizon.

We prove a purely quasi-local positivity theorem for the Wang–Yau mass for a class of initial data whose Jang deformation, after a boundary-preserving conformal reduction to zero scalar curvature, lies in a sufficiently small transverse–traceless (TT) perturbative neighborhood of a strictly convex Euclidean fill-in.We explicitly construct a nontrivial class of physical initial data whose admissible Jang reductions realize this TT-generated sector. The argument reduces the Wang–Yau energy to the Brown–York mass of the resulting scalar-flat compact metric, together with nonnegative bulk terms determined by the Jang deformation, and establishes strict positivity by computing the second variation of the Brown–York functional at the Euclidean metric in transverse–traceless directions. The proof is entirely confined to the compact fill-in and uses neither an asymptotically flat extension nor the positive mass theorem. This gives a partial answer to a question of R. Schoen concerning a genuinely quasi-local proof of positivity for quasi-local mass.

Quantum imaginarity, as a critical resource metric for quantifying intrinsic nonreal coherence encoded in quantum states, exhibits nontrivial evolutionary behaviors in curved spacetime backgrounds. This work focuses on bipartite Dirac field reduced states in the quantum atmosphere of a static Schwarzschild black hole, aiming to explore the modulation of three mainstream imaginarity measures by Hawking thermal radiation. We reveal that the relative-entropy imaginarity, the geometric imaginarity and robustness of imaginarity, the fully accessible state presents a valley-shaped radial profile with a local minimum inside the quantum atmosphere, while the cross-coupling and fully inaccessible states follow opposite peak-shaped trends. Further analysis demonstrates that the Hartle-Hawking constant significantly strengthens the redistribution effect of imaginarity across both regions, whereas an increase in the event horizon radius weakens such redistribution and flattens the extremum features. These findings offer a new perspective for decoding the information structure of black hole quantum atmospheres and the intrinsic quantum nature of Hawking radiation.

We numerically investigate the gravitational collapse of a charged scalar field coupled to the Einstein and Maxwell equations in spherical symmetry, in 4+1 spacetime dimensions. At the threshold of collapse we find type-II critical phenomena very similar to what is seen in 3+1 dimensions, including self-similarity of the critical solution and power-law scaling of the black hole mass and charge. Well inside the collapse region, we also identify initial data that collapse to an extremal black hole. By varying all parameters of the initial data by up to $\sim0.5\%$, we give evidence that there is a small neighbourhood in the space of initial data where an exactly extremal charged black hole is formed in the evolution of a codimension-1 set of initial data. We believe this is the first such evidence for scalar field collapse from regular initial data.

The conformal boundary of asymptotically flat spacetime is null and therefore carries a Carrollian rather than Lorentzian geometry. This changes the relation between symmetry, variational principles and evolution equations in an essential way: unlike a Lorentzian Levi-Civita connection, a torsion-free Carrollian connection compatible with the conformal structure contains genuine independent data. We develop a metric-affine formulation of conformal dynamics that keeps track of these connection degrees of freedom through hypermomenta. Local Carroll boosts, rotations, Weyl transformations and diffeomorphisms then imply a set of covariant evolution equations. We apply this framework to Einstein gravity in four-dimensional asymptotically flat spacetimes. Penrose's conformal compactification identifies null infinity as a conformal Carrollian manifold; the undetermined transverse symmetric part of its compatible connection is the asymptotic Bondi shear. After fixing a Bondi-van der Burg-Metzner-Sachs (BMS) frame and identifying the Carrollian momenta and hypermomenta with the Bondi data, the general Carrollian evolution equations reproduce the standard evolution equations for the mass and angular-momentum aspects. The result provides a boundary-intrinsic derivation of the gravitational evolution equations and a geometric realization of radiative degrees of freedom in flat-space holography. This is a proceedings contribution to the "Athens Workshop in Theoretical Physics: 10th Anniversary", held at the National and Kapodistrian University of Athens on December 17-19 2025.

Pseudospectral analysis provides a powerful way to probe the spectral stability of non-self-adjoint operators and has been widely used in black hole physics, but its application to braneworld scenarios has not yet been explored. In this work, we apply this method to tensor gravitational perturbations in a representative scalar-field-generated thick brane background. To the best of our knowledge, we provide the first hyperboloidal formulation of braneworld perturbations and propose a height-function gauge adapted to the warped geometry. This construction converts the outgoing boundary conditions of quasinormal modes into regularity conditions at finite compactified boundaries and recasts the perturbation equation as a first-order system generated by a non-self-adjoint hyperboloidal evolution operator. With the corresponding energy norm, we compute the condition numbers and pseudospectra of the localized graviton zero mode and the quasinormal-mode spectrum. We find that the condition numbers grow rapidly along the overtone sequence and that the corresponding pseudospectral contours develop broad, connected structures in the high-overtone region. These results show a strongly mode-dependent spectral sensitivity: among the damped modes analyzed, the higher overtones are less robust than the fundamental mode. The zero mode also has a larger condition number than the fundamental mode, indicating stronger local first-order sensitivity. These diagnostics characterize sensitivity to generic norm-bounded operator perturbations. Relating that sensitivity to a specific braneworld deformation requires the corresponding self-consistent perturbation constraints.

Gravitational-wave (GW) microlensing occurs when an intervening compact object causes diffraction of the GW, imprinting characteristic frequency-dependent modulations on the observed signal. Current searches for microlensing signatures in the detected GW events, including those used to constrain the fraction of dark matter in the form of compact objects, assume an isolated point mass lens (PML) model. However, astrophysical compact objects are typically embedded within larger structures, such as galaxies, whose potentials introduce additional convergence and tidal shear, thereby perturbing the lensed signal. In this work, we study the impact of the background galaxy in the search for microlensing by compact objects using a PML model. We distribute the compact objects inside a galaxy following the Navarro-Frenk-White density profile. We model the microlensing effects using the potential of a PML embedded in a constant convergence and shear, which are, in turn, calculated from a singular isothermal sphere model of the galaxy lens. We demonstrate that neglecting the galaxy's effects can systematically reduce sensitivity in microlensing searches, particularly for high microlens masses. Thus, future searches may need to incorporate the additional effect of the lens galaxy, particularly those aimed at placing robust constraints on the abundance of compact dark matter.

A central question in bottom-up holography is whether a given bulk effective theory admits a consistent holographic dual. In this work, we explore whether the near-singularity Kasner scaling of planar AdS black hole interiors can serve as a useful diagnostic based on interior-sensitive holographic probes. By examining the metric combinations that control interior-sensitive observables, including Complexity=Volume (CV), the bulk contribution to Complexity=Action (CA), Hartman-Maldacena (HM) entropy, and the thermal $a$-function, we derive algebraic bounds on the Kasner exponents $(p_t,p_s)$ for the relevant semi-classical regimes: finite-radius late-time branches of CV and HM, finite CA complexity, and a finite near-singularity thermal $a$-function. The resulting inequalities delineate the corresponding regions of Kasner space in which the terminal scaling alone is sufficient to realize these behaviors. Furthermore, we demonstrate that classical energy conditions, specifically the null and dominant energy conditions, provide simple sufficient criteria ensuring that the Kasner exponents fall within these holographically allowed regions. These results establish a direct connection between classical bulk energy conditions and the interior geometry selected by holographic probes, and suggest that Kasner scaling can provide a complementary diagnostic in bottom-up holography.

We examine the exact propagation of light in a McVittie spacetime to quantify the observational bias introduced on the inferred Hubble constant $H_0$ by a localized central mass embedded in an expanding asymptotically de Sitter background. By integrating the Sachs optical equations for radial null geodesics, we derive linearized analytical expressions for the areal and luminosity distances to first order in the dimensionless Hawking-Hayward coupling parameter $\mu = H_{\text{dS}} m_H$. We evaluate the magnitude of this local gravitational field effect across realistic astrophysical scales, ranging from supermassive black holes ($\text{Sgr A}^*$) to galaxy cluster halos. We show that while a McVittie central mass induces a coordinate-invariant path-focusing effect on incoming photons, the quantitative correction for local galactic parameters ($\mu \sim 10^{-16}$) is negligible ($\sim 10^{-15}\%$), proving that a localized galactic mass alone cannot resolve the $\sim 8\%$ Hubble tension. We establish strict upper bounds on the extent to which local spacetime inhomogeneity can bias low-redshift distance ladder measurements.

We survey recent developments on the null energy condition (NEC) from the perspective of optimal transport, with emphasis on non-smooth Lorentzian geometry. After recalling the role of classical energy conditions in general relativity, we discuss the relation between Ricci curvature and entropy convexity in optimal transport. The main focus is the authors' optimal-transport characterization of the NEC. In the smooth setting, the NEC is equivalent to displacement concavity of the Shannon entropy power along suitable null-geodesic transport plans supported on null hypersurfaces and measured with respect to rigged measures. We review the geometric ingredients of this characterization and applications, including a weighted Hawking area theorem. We then turn to the synthetic theory introduced in a second paper of the authors. Synthetic null hypersurfaces are defined using achronal boundaries, gauge functions, and reference measures in topological causal spaces. In this framework, the displacement-concavity characterization becomes a synthetic curvature condition, $\mathsf{NC}^{e}(N)$, meaningful without differentiable structure. We discuss compatibility with the smooth theory, invariance under natural changes of gauge and reference measure, and stability under convergence. Finally, we present a synthetic version of Penrose's singularity theorem for continuous Lorentzian metrics. These results indicate that geometric and causal consequences of the NEC persist beyond the smooth setting and highlight optimal transport as a framework connecting curvature, causality, and singularity formation in Lorentzian geometry.

Strongly lensed gravitational waves (GWs) are commonly searched for using population priors, including time-delay and magnification-ratio distributions, to suppress false alarms. Current LIGO, Virgo, and KAGRA (LVK) searches mainly adopt priors calibrated with galaxy-scale lenses; however, these may penalize the longer time delays produced by galaxy clusters, which could contribute 10%–20% of lensed events. Cluster member galaxies may also perturb the macro-lens potential and alter image magnifications. We investigate the time-delay and magnification-ratio distributions of binary black hole GWs lensed by galaxy clusters, together with microlensing systematics. In the full lensed sample, the median time delay is about 76 days for cluster lenses, compared with 6.9 days for galaxy lenses. After imposing an SNR threshold of $\rho>8$ at LVK O4 sensitivity, the medians are 15.4 and 0.24 days. The detected cluster sample retains a long-delay tail, with 9.1% of image pairs delayed by more than 365 days and 3.3% by more than 1000 days, so galaxy-based time-delay priors can suppress cluster-lensed pairs. By contrast, although cluster member galaxies broaden the magnification-ratio distribution of the full cluster sample, the detected cluster- and galaxy-lensing samples are very similar at O4 sensitivity, suggesting little bias in candidate selection. Cluster lenses also generally produce weak microlensing distortions: only about 0.8% of single-image mismatches exceed ${\cal M}=0.03$, while 4.8% exceed the parameter-estimation threshold $1.35/\rho^2$. Thus, microlensing is subdominant for most cluster-lensed events, mainly because images often form far from the brightest cluster galaxy, where the stellar density is low and dominated by intracluster light, although nearby satellites can locally enhance it. These results motivate a dedicated cluster-lensing pipeline for GW lensing searches.

We develop an inverse lensing formalism for reconstructing the metric function of a static, spherically symmetric spacetime directly from the gravitational deflection angle of light. By formulating the inverse problem through an Abel transformation, we derive an integro-differential relation connecting the observable deflection profile to the underlying spacetime geometry. As a consistency check, we consider the case of vanishing deflection, \(\alpha(b)=0\), and recover \(A(r)=1\), corresponding to flat Minkowski spacetime. We then apply the formalism to the gravitational deflection of light by the Sun using the observationally motivated leading-order expression $\alpha(b)=\frac{2(1+\gamma)GM_{\odot}}{c^{2}b}.$The resulting metric function is shown to recover the Schwarzschild form in the weak-field limit \(r\gg M\). We further consider the higher-order correction to the solar deflection angle and reconstruct the corresponding metric beyond the leading-order approximation. Interestingly, the \((M/r)^2\) term vanishes in the resulting weak-field expansion, yielding an improved approximation compared with the metric reconstructed from the leading-order deflection angle. Our results demonstrate that gravitational lensing observations can provide a direct route to reconstructing the underlying spacetime geometry without assuming a specific metric a priori. This formalism therefore, offers a novel framework for connecting observational light propagation with spacetime geometry.

Stationary-axisymmetric black holes are the standard theoretical framework for astrophysical black holes, yet their interiors remain largely inaccessible: the absence of a hypersurface-orthogonal timelike Killing vector renders the metric non-diagonal, and frame dragging obstructs the usual techniques for probing the interior. We prove that any stationary-axisymmetric black hole satisfying the classical energy conditions contains at most one nondegenerate inner Killing horizon per connected interior branch, irrespective of its matter content. For compact horizon sections, we apply the Raychaudhuri equation to the congruence normal to constant-time hypersurfaces: the strong energy condition makes the squared lapse subharmonic, and the maximum principle excludes a second inner horizon. For noncompact planar and hyperbolic sections central to holography, the null energy condition instead yields a monotonicity obstruction through a geometric flow we introduce here, the “inverse lapse flow,” whose weak continuation via viscosity solutions we also construct. The attractive nature of gravity, encoded in these energy conditions, thus both drives horizon formation and caps the number of inner horizons, revealing a striking dual role of the same focusing mechanism.

A dark matter density profile alone does not determine the spacetime around a black hole; the radial stress must be prescribed separately and governs the strong-field orbital response. We study a Bardeen black hole in a Hernquist halo through three spherical models sharing the same density, mass function, and cutoff but differing in radial stress: a halo with $p_r^{\mathrm{DM}}=-\rho_{\mathrm{DM}}$, its truncated form, and a truncated Einstein cluster with $p_r^{\mathrm{DM}}=0$. Treating the halo as a small perturbation, we derive a first-order criterion for the leading innermost stable circular orbit shift, $C_\lambda=C_\rho+\lambda C_p$. The two truncated closures shift this orbit in opposite directions for the Hernquist profile, and unexpanded calculations for five density profiles confirm the predicted signs. Checks using a Hayward background and a published Dehnen-halo result show that the criterion is not restricted to the Bardeen-Hernquist system. A continuous stress interpolation identifies a critical closure at which the leading shift vanishes. For a representative four-year extreme-mass-ratio inspiral, changing the radial stress at fixed density and cutoff produces a phase difference of several radians in a leading-order adiabatic treatment.

Thermodynamic rewritings of horizon equations depend on which relations are varied and which have already been imposed. We formulate a differential rank criterion that makes these choices explicit. The construction uses a stationary metric family together with an independently marked radial surface, physical charges determined by the metric parameters, and specified extensions of the horizon thermodynamic quantities. A first-law covector is pointwise representable by selected constraints precisely when it lies in the span of their differentials. We distinguish this algebraic condition from a smooth identity in a neighborhood. If regular constraints already define the complete physical family, pointwise representability follows from the first law itself; a neighborhood identity additionally depends on the chosen extensions. With explicit prescriptions, Kerr–Newman, rotating BTZ black holes in new massive gravity, and the new-type NMG black hole all exhibit exact alignment with their horizon constraints. By contrast, substituting an exact metric family into a field-equation component can annihilate that residual before any horizon condition is tested. We also give an explicit entropy–volume comparison and specify the role of fixed ensemble data in free-energy variations. The framework separates consequences of the established first law from additional claims about selected constraints and their extensions.

We provide an explicit rotating black-hole realization of the predicted \(\overline{W}^{1-}\) thermodynamic topological subclass by studying a slowly rotating, magnetically charged Kerr black hole in nonlinear electrodynamics (NLED). Using a reduced fixed-parameter off-shell potential, we derive the inverse-temperature curve analytically rather than relying solely on numerical plots. In AdS spacetime, this curve begins at a finite nonzero value, increases monotonically, and diverges at large horizon radius; it therefore contains exactly one defect with \(w=W=-1\) and has no stable large-black-hole branch. Positivity of the analytic coefficient establishes this result throughout the admissible slow-rotation parameter domain. Under the same reduced prescription, the Kerr-AdS benchmark instead has two branches and belongs to the \(W^{0-}\) class, establishing a clear topological distinction between the NLED-Kerr and classical rotating cases. In dS spacetime, the formally extended off-shell curve remains monotonic and any isolated zero has \(w=-1\); because the parameter-dependent horizon-admissible range is not fixed in the present reduced analysis, no physical dS asymptotic subclass is assigned.

Black holes in the Universe span nearly ten orders of magnitude in mass over a large cosmic timescale, dating back to at least the Universe's early history, when it was only a few hundred million years old. The formation and evolution of these objects remain a mystery because no direct probe can trace each black hole across the full mass range and cosmic time. This work shows, for the first time, a statistical mechanical description of black hole evolution. I show that though each black hole and its evolution are governed by stochastic processes, one can make a macroscopic description of the distribution function of black holes in the Universe, and its evolution over the entire mass range and cosmic time can be expressed in terms of the Kramers-Moyal expansion. This bridging between statistical mechanics and black hole evolution in terms of the Kramers-Moyal expansion enables the discovery of the underlying physical processes that play a role in the evolution of black holes. In the future, the application of this technique to data accessible from different messengers, such as gravitational waves and electromagnetic waves, which probe black holes in the Universe, can lead to a data-driven understanding of the underlying physical processes by this statistical mechanics technique despite the ignorance of the underlying microscopic stochastic processes.

The millihertz gravitational-wave sky is expected to be observed by a network of space-based detectors (LISA, Taiji, and TianQin) in the 2030s. A dominant noise in this band is the confusion foreground produced by O(10^7) unresolved Galactic binaries (GBs). Because the same population projects onto the time-delay interferometry (TDI) channels of the different detectors, the foreground is necessarily correlated across the network. We construct the full foreground noise covariance matrix for the LISA-Taiji-TianQin network from a catalogue of ~3 x 10^7 GBs through numerical simulation, and we derive an analytic model of the cross-detector foreground coherence that provides a cross-check and physical interpretation of the numerical results. We derive the overall sensitivity of the detector network based on this covariance matrix, and further characterize the frequency- and time-dependent cross-detector foreground correlations. Taking massive black hole binaries (MBHBs) and GBs as representative transient and continuous sources, we further compare the block-diagonal (i.e., neglecting the cross-detector foreground correlation) and full-covariance noise models in terms of their impacts on the signal-to-noise ratio (SNR), parameter uncertainties, and Bayesian posteriors. The impact is confined to specific signal regimes, affecting primarily high-mass (M_c ~ 10^7 M_sun) MBHBs and low-frequency (f_0 <~ 2 mHz) GBs, with SNR relative differences of up to ~30% and ~10%, respectively. No statistically significant parameter bias arises under either noise model, as verified by probability-probability tests. Parameter estimations for MBHBs and GBs are performed using the network analysis pipelines implemented in the Triangle-BBH and Triangle-GB codes. Both repositories, together with the simulated foreground data and network sensitivities, are publicly released for diverse scientific investigations.