The study of a wide peaking structure in the $B_c^{+}\gamma$ mass spectrum is reported using a data sample of proton-proton collisions collected by the LHCb detector at center-of-mass energies of 7, 8, and 13 TeV, corresponding to an integrated luminosity of $9{\mathrm{fb}}^{-1}$. The observed structure is consistent with the lowest excited $P$-wave $B_c^{+}$ states and exhibits a statistical significance exceeding 7 standard deviations relative to the background-only hypothesis. A two-peak model serves as an effective description of the data, with various theory-constrained models further explored to provide physical interpretation. Based on the predictions for the $B_c(1P{)}^{+}$ spectrum, the relative production cross section of the overall $B_c(1P{)}^{+}$ states with respect to the $B_c^{+}$ ground state with the transverse momentum $p_{\mathrm{T}}$ and rapidity $y$ of $B_c^{+}$ mesons in the regions $p_{\mathrm{T}}<20\mathrm{GeV}/c$ and $2.0<y<4.5$ at $\sqrt{s}=13\mathrm{TeV}$ is measured to be $0.20\pm 0.03\pm 0.02\pm 0.03$, where the uncertainty terms represent statistical, systematic, and uncertainties related to the choice of theoretical models, respectively. The results provide a test of theoretical models and deepen our understanding of quantum chromodynamics.

In this work, we study the dynamics of an extreme mass-ratio inspiral (EMRI) embedded within a scalar cloud populated around the massive black hole. This cloud may be generated through the black hole superradiant process if the wavelength of the scalar particle is comparable to the size of the massive black hole. The EMRI motion perturbs the cloud, producing scalar radiation toward infinity and into the black hole horizon. In addition, the backreaction of the scalar radiation onto the orbit modifies the motion of the EMRI and induces an observable gravitational-wave phase shift for a range of system parameters. We quantify the scalar flux and the induced phase shift, as one of the examples of exactly solvable, environmental effects of EMRIs.

In this article, we generalize the recursion methods based on cut equations [N. Arkani-Hamed, H. Frost, and G. Salvatori, arXiv:2412.21027.], originally developed for scalar theories, to gluons in pure Yang-Mills theory. In gauge theories, planar loop integrands are subtle to defined and obtained due to the existence of scaleless integrals. A critical challenge arises when constructing higher-loop integrands via recursion methods: integrands with or without all scaleless terms both are incompatible with the reconstruction formalism in Arkani-Hamed et al. To address this, we introduce a refined integrand by systematically removing specific scaleless contributions, and develop an algorithmic implementation of the recursion to all-loop level. We explicitly demonstrate the framework by three steps and obtain the recursion formula in pure Yang-Mills theory. In the Supplemental Material, we provide the results up to the two-loop five-point integrand and the simplified result in the large-$D$ limit for the three-loop four-point case.

We propose new bounds on black hole entropy in terms of the specific heat at fixed charges. For special spherically symmetric and static black holes ($g_{tt}{g}_{rr}=-1$), we prove the bounds with suitable sufficient energy conditions. For more general static and rotating black holes, we test the bounds with a variety of examples. This work extends the previously known algebraic inequalities of the black hole thermodynamic variables such as the Penrose inequality, to include their derivatives.

If primordial black holes (PBHs) are present in the early Universe, their contribution to the energy budget grows relative to that of radiation and generically becomes dominant unless the initial abundance is exponentially small. This black hole domination scenario is largely unconstrained for PBHs with masses $\lesssim {10}^9\mathrm{g}$, which evaporate prior to big bang nucleosynthesis. However, if the era of PBH domination is sufficiently long, the PBHs form clusters and can merge appreciably within these objects. We calculate the population statistics of these clusters within the Press-Schechter formalism and find that, for a wide range of PBH masses and Hubble rates at the onset of PBH domination, the mergers within PBH clusters can exhibit runaway behavior, where the majority of the cluster will eventually form a single black hole with a mass much greater than the original PBH mass. These mergers can dramatically alter the PBH mass distribution and leave behind merged relic black holes that evaporate after big bang nucleosynthesis and yield various observational signatures, excluding parameter choices previously thought to be viable.

Discrepancies between distance measurements and $\mathrm{\Lambda }\mathrm{CDM}$ predictions reveal notable features in the distance-redshift relation, possibly suggesting the presence of an evolving dark energy component. Given the central role of the Friedmann-Lemaître-Robertson-Walker (FLRW) metric in modeling cosmological distances, we investigate here whether these features instead point to a possible departure from the fundamental FLRW symmetries. Exploiting the transverse and line-of-sight distances provided by baryonic acoustic oscillations (BAO) observations, we demonstrate that observed distances hint at a slight but systematic preference for an anisotropic expansion rate emerging regardless of the dark energy model considered. Leveraging this non-FLRW feature, we investigate an inhomogeneous extension of the $\mathrm{\Lambda }\mathrm{CDM}$ model that naturally provides an anisotropic expansion rate. Our analysis demonstrates that models featuring spherical overdensities can explain BAO, supernova, and cosmic microwave background data, providing fits statistically indistinguishable from those obtained with a phantom dark energy scenario. When $\mathrm{Pantheon}+\mathrm{data}$ is considered, our analysis challenges the FLRW framework at $2.8\sigma$ and yields scenarios that can be interpreted as subtle but non-negligible deviations from the FLRW metric. When DESY5 supernovae are considered instead, deviations are notably more significant, yielding scenarios that mildly violate the Copernican principle and exclude the FLRW assumption at $5.2\sigma$. Overall, our results motivate a more in-depth investigation of whether the perfectly homogeneous and isotropic FLRW paradigm can still be assumed to accurately predict cosmological distances in the era of precision cosmology.

We discuss a new source of gravitational waves (GWs) from first-order phase transitions. The collisions of bubbles of the new phase can efficiently produce particles that couple to the background field undergoing the transition, thereby transferring a significant fraction of the released vacuum energy into a distribution of inhomogeneous and dynamic particle populations that persist long after the bubbles have disappeared. We study the GWs produced by such particle distributions, showing that GWs arise from the quadrupolar anisotropy in the radiation emitted from the bubble collisions, and present a semianalytical calculation of the two-point correlation function for the associated energy distributions. We find that this new contribution can qualitatively modify the overall GW signal from such phase transitions, creating a distinct shift in the spectral slope at low frequencies that could be observed by future GW experiments. It is therefore important to take this new contribution into account for any transition where the background field has significant self-coupling or couplings to other fields that could lead to efficient particle production at bubble collision.

The high quiescent x-ray luminosity observed in some magnetars is widely attributed to the decay and evolution of their ultrastrong magnetic fields. Several dissipation mechanisms have been proposed, each operating with different efficiencies depending on the region of the star. In this context, ambipolar diffusion, i.e., the relative motion of charged particles with respect to neutrons in the neutron star core, has been proposed as a promising candidate due to its strong dependence on magnetic field strength and its capacity to convert magnetic energy into heat. We perform axisymmetric magnetohydrodynamic simulations to study the long-term magnetic evolution of a neutron star core composed of normal (non-Cooper-paired) matter under the influence of ambipolar diffusion. The core is modeled as a two-fluid system consisting of neutrons and a charged-particle fluid (protons and electrons), coupled to the magnetic field. Simulations are performed both at constant and variable temperatures. In the latter case, a strategy that decouples the magnetic and thermal evolution is employed, enabling efficient thermal modeling across a range of initial magnetic field strengths. At constant temperature, we obtain the expected result where neutrons reach diffusive equilibrium, the Lorentz force is balanced by chemical potential gradients of charged particles, and the magnetic field satisfies a nonlinear Grad-Shafranov equation. When thermal evolution is included, fields $B\gtrsim 5\times {10}^{15}\mathrm{G}$ can balance ambipolar heating and neutrino cooling, delaying the evolution over $\sim {10}^3[B/(5\times {10}^{15}\mathrm{G}){]}^{-6/5}\mathrm{yr}$. Although the surface luminosity is enhanced compared with passive cooling, the heating from ambipolar diffusion alone is insufficient to fully explain the persistent x-ray emission observed in magnetars.

Recently, ultracompact objects have been found to be susceptible to a new nonlinear instability, known as the light-ring instability, triggered by stable light rings. This discovery raises concerns about the viability of these objects as alternatives to black holes. In this work, we investigate the presence of the light-ring instability in scalarized Reissner-Nordström black holes, which have been previously shown to admit stable light rings. We employ fully nonlinear numerical evolutions of both scalarized black holes with and without stable light rings, perturbing them initially with spherically symmetric scalar perturbations. Our simulations demonstrate the long-term stability of these scalarized black holes, suggesting that the presence of a stable light ring may not necessarily induce the light-ring instability.

Very little is known about the cosmological history from after the end of inflation until big bang nucleosynthesis. Various well-motivated models predict that the universe could have undergone a period of matter domination in this early epoch. We demonstrate that if the particles causing matter domination have self-interactions, they can form halos that undergo a gravothermal collapse. We thus propose a novel scenario for the formation of primordial black holes, which in particular can lie within the asteroid-mass range. We also find that it is not only black holes that can form in the aftermath of a gravothermal evolution. We show that number-changing annihilations of the particles can create sufficient heat to halt the gravothermal evolution, thus forming a “cannibal star.” Likewise, the pressure from the particle’s repulsive self-interactions can form a boson star during a gravothermal evolution. These stars can also eventually collapse into black holes. Thus, our study highlights that structure formation in the early universe can have a rich phenomenology.

The tidal deformation of a neutron star in a binary inspiral driven by the emission of gravitational waves affects the orbital dynamics and produces a measurable modulation of the waves. Late in the inspiral, the external timescale associated with changes in the tidal field becomes comparable to internal timescales associated with hydrodynamical processes within the star, and a regime of dynamical tides takes over from a prior regime of static tides. A recent analysis by Yu et al. [Mon. Not. R. Astron. Soc. 519, 4325 (2022)] reveals that nonlinear aspects of the tidal interaction are important during the regime of dynamical tides. Their theoretical framework is grounded in Newtonian gravity and fluid mechanics and relies on a representation of the tidal deformation in terms of the star’s normal modes of vibration; the nonlinearities are incorporated through a coupling of the modes and a nonlinear driving by the external tidal field. We confirm their observation in a general relativistic treatment of the tidal deformation of a neutron star, without relying on a mode representation of this deformation. The starting point of our description is a simultaneous time-derivative and nonlinear expansion of the tidal deformation, expressed in terms of three encapsulating constants, the static $k_2$, dynamic ${\ddot{k}}_2$, and nonlinear $p_2$ tidal constants. We describe the neutron star’s deformation in terms of a well-defined quadrupole moment tensor ${\mathcal{Q}}_{ab}$, which is related to the tidal quadrupole moment ${\mathcal{E}}_{ab}$ through a frequency-domain response function ${\tilde{k}}_2(\omega )$. In a pragmatic extension of our simultaneous time-derivative and nonlinear expansion, we express this in a form proportional to $(1-{\omega }^2/{\omega }_{*}^2{)}^{-1}$, the characteristic response of a harmonic oscillator subjected to a driving force of frequency $\omega$, with a natural-frequency parameter ${\omega }_{*}$ constructed from the tidal constants. We compute these for polytropic stellar models and show that the nonlinear constant $p_2$ lowers the frequency parameter by as much as 15% relative to an estimation based on a purely linear treatment of the tidal deformation. This means that the approach to resonance is triggered at an earlier frequency during an inspiral, which causes a significant enhancement of the star’s tidal response.

Building on recent approaches, we develop an effective field theory for the interaction of spinning particles modeling Kerr black holes within the gravitational self-force expansion. To incorporate dimensional regularization into this framework, we analyze the higher-dimensional metric arising from the minimal coupling solution, comparing it against the Myers-Perry black hole and its particle description. We then derive the 1SF self-force effective action up to quadratic order in the spin expansion, identifying a new type of spinning recoil term that arises from integrating out the heavy dynamics. Next, we study the 1SF metric perturbation both from the traditional self-force perspective and through the diagrammatic background field expansion, making contact with the radiative waveform. This leads us to consider a novel recursion relation for the curved space 1SF Compton amplitude, which we study up to one-loop in the wave regime and compare with the flat space one-loop Compton for Kerr up to quadratic order in spin. Finally, we investigate the 1SF spinning Compton amplitude in the eikonal regime, clarifying how the strong-field effect—such as the location of the separatrix—emerges from the resummation of the perturbative weak-field expansion.

We present baryon acoustic oscillation (BAO) measurements from more than 14 million galaxies and quasars drawn from the Dark Energy Spectroscopic Instrument (DESI) Data Release 2 (DR2), based on three years of operation. For cosmology inference, these galaxy measurements are combined with DESI Lyman-$\alpha$ forest BAO results presented in a companion paper (M. Abdul-Karim et al., companion paper, Phys. Rev. D 112, 083514 2025.). The DR2 BAO results are consistent with DESI DR1 and the Sloan Digital Sky Survey, and their distance-redshift relationship matches those from recent compilations of supernovae (SNe) over the same redshift range. The results are well described by a flat $\mathrm{\Lambda }$ cold dark matter ($\mathrm{\Lambda }\mathrm{CDM}$) model, but the parameters preferred by BAO are in mild, $2.3\sigma$ tension with those determined from the cosmic microwave background (CMB), although the DESI results are consistent with the acoustic angular scale ${\theta }_{*}$ that is well measured by Planck. This tension is alleviated by dark energy with a time-evolving equation of state parametrized by $w_0$ and $w_a$, which provides a better fit to the data, with a favored solution in the quadrant with $w_0>-1$ and $w_a<0$. This solution is preferred over $\mathrm{\Lambda }\mathrm{CDM}$ at $3.1\sigma$ for the combination of DESI BAO and CMB data. When also including SNe, the preference for a dynamical dark energy model over $\mathrm{\Lambda }\mathrm{CDM}$ ranges from $2.8-4.2\sigma$ depending on which SNe sample is used. We present evidence from other data combinations which also favor the same behavior at high significance. From the combination of DESI and CMB we derive 95% upper limits on the sum of neutrino masses, finding $\sum m_{\nu }<0.064\mathrm{eV}$ assuming $\mathrm{\Lambda }\mathrm{CDM}$ and $\sum m_{\nu }<0.16\mathrm{eV}$ in the $w_0{w}_a$ model. Unless there is an unknown systematic error associated with one or more datasets, it is clear that $\mathrm{\Lambda }\mathrm{CDM}$ is being challenged by the combination of DESI BAO with other measurements and that dynamical dark energy offers a possible solution.

We present the baryon acoustic oscillation (BAO) measurements with the Lyman-$\alpha$ ($\mathrm{Ly}\alpha$) forest from the second data release (DR2) of the Dark Energy Spectroscopic Instrument (DESI) survey. Our BAO measurements include both the autocorrelation of the $\mathrm{Ly}\alpha$ forest absorption observed in the spectra of high-redshift quasars and the cross-correlation of the absorption with the quasar positions. The total sample size is approximately a factor of 2 larger than the DR1 dataset, with forest measurements in over 820,000 quasar spectra and the positions of over 1.2 million quasars. We describe several significant improvements to our analysis in this paper, and two supporting papers describe improvements to the synthetic datasets that we use for validation and how we identify damped $\mathrm{Ly}\alpha$ absorbers. Our main result is that we have measured the BAO scale with a statistical precision of 1.1% along and 1.3% transverse to the line of sight, for a combined precision of 0.65% on the isotropic BAO scale at $z_{\mathrm{eff}}=2.33$. This excellent precision, combined with recent theoretical studies of the BAO shift due to nonlinear growth, motivated us to include a systematic error term in $\mathrm{Ly}\alpha$ BAO analysis for the first time. We measure the ratios $D_H(z_{\mathrm{eff}})/r_d=8.632\pm 0.098\pm 0.026$ and $D_M(z_{\mathrm{eff}})/r_d=38.99\pm 0.52\pm 0.12$, where $D_H=c/H(z)$ is the Hubble distance, $D_M$ is the transverse comoving distance, $r_d$ is the sound horizon at the drag epoch, and we quote both the statistical and the theoretical systematic uncertainty. The companion paper presents the BAO measurements at lower redshifts from the same dataset and the cosmological interpretation.

The Dark Energy Spectroscopic Instrument (DESI) Collaboration has obtained robust measurements of baryon acoustic oscillations in the redshift range $0.1<z<4.2$, based on the Lyman-$\alpha$ forest and galaxies from data release 2. We combine these measurements with cosmic microwave background (CMB) data from Planck and the Atacama Cosmology Telescope to place our tightest constraints yet on the sum of neutrino masses. Assuming the cosmological $\mathrm{\Lambda }\mathrm{CDM}$ model and three degenerate neutrino states, we find $\sum m_{\nu }<0.0642\mathrm{eV}$ (95%) with a marginalized error of $\sigma (\sum m_{\nu })=0.020\mathrm{eV}$. We also constrain the effective number of neutrino species, finding $N_{\mathrm{eff}}=3.23_{-0.34}^{+0.35}$ (95%), in line with the Standard Model prediction. When accounting for neutrino oscillation constraints, we find a preference for the normal mass ordering and an upper limit on the lightest neutrino mass of $m_l<0.023\mathrm{eV}$ (95%). However, we determine using frequentist and Bayesian methods that our constraints are in tension with the lower limits derived from neutrino oscillations. Correcting for the physical boundary at zero mass, we report a 95% Feldman-Cousins upper limit of $\sum m_{\nu }<0.053\mathrm{eV}$, breaching the lower limit from neutrino oscillations. Considering a more general Bayesian analysis with an effective cosmological neutrino mass parameter, $\sum m_{\nu ,\mathrm{eff}}$, that allows for negative energy densities and removes unsatisfactory prior weight effects, we derive constraints that are in $3\sigma$ tension with the same oscillation limit, while the error rises to $\sigma (\sum m_{\nu ,\mathrm{eff}})=0.053\mathrm{eV}$. In the absence of unknown systematics, this finding could be interpreted as a hint of new physics not necessarily related to neutrinos. The preference of DESI and CMB data for an evolving dark energy model offers one possible solution. In the $w_0{w}_a\mathrm{CDM}$ model, we find $\sum m_{\nu }<0.163\mathrm{eV}$ (95%), relaxing the neutrino tension. These constraints all rely on the effects of neutrinos on the cosmic expansion history. Using full-shape power spectrum measurements of data release 1 galaxies, we place complementary constraints that rely on neutrino free streaming. Our strongest such limit in $\mathrm{\Lambda }\mathrm{CDM}$, using selected CMB priors, is $\sum m_{\nu }<0.193\mathrm{eV}$ (95%).

The Dark Energy Spectroscopic Instrument (DESI) Data Release 2 (DR2) galaxy and quasar clustering data represents a significant expansion of data from Data Release 1 (DR1), providing improved statistical precision in baryon acoustic oscillation (BAO) constraints across multiple tracers, including bright galaxies, luminous red galaxies, emission line galaxies, and quasars. In this paper, we validate the BAO analysis of DR2. We present the results of robustness tests on the blinded DR2 data and, after unblinding, consistency checks on the unblinded DR2 data. All results are compared with those obtained from a suite of mock catalogs that replicate the selection and clustering properties of the DR2 sample. We confirm the consistency of DR2 BAO measurements with DR1 while achieving a reduction in statistical uncertainties due to the increased survey volume and completeness. The combined BAO precision, including both statistical and systematic errors, improves from $\sim 0.52\%$ in DR1 to 0.30% in DR2—a factor of 1.7 gain. We assess the impact of analysis choices, including different data vectors (correlation function vs power spectrum), modeling approaches and systematics treatments, and an assumption of the Gaussian likelihood, finding that our BAO constraints are stable across these variations and assumptions with a few minor refinements to the baseline setup of the DR1 BAO analysis. We summarize a series of pre-unblinding tests that confirmed the readiness of our analysis pipeline, the final systematic errors, and the DR2 BAO analysis baseline. The successful completion of these tests led to the unblinding of the DR2 BAO measurements, ultimately leading to the DESI DR2 cosmological analysis, with their implications for the expansion history of the Universe and the nature of dark energy presented in the DESI key paper (companion paper).

We conduct an extended analysis of dark energy constraints, in support of the findings of the Dark Energy Spectroscopic Instrument (DESI) second data release cosmology key paper, including DESI data, Planck cosmic microwave background observations, and three different supernova compilations. Using a broad range of parametric and nonparametric methods, we explore the dark energy phenomenology and find consistent trends across all approaches, in good agreement with the $w_0{w}_a\mathrm{CDM}$ (cold dark matter) key paper results. Even with the additional flexibility introduced by nonparametric approaches, such as binning and Gaussian processes, we find that extending $\mathrm{\Lambda }\mathrm{CDM}$ to include a two-parameter $w(z)$ is sufficient to capture the trends present in the data. Finally, we examine three dark energy classes with distinct dynamics, including quintessence scenarios satisfying $w\ge -1$, to explore what underlying physics can explain such deviations. The current data indicate a clear preference for models that feature a phantom crossing; although alternatives lacking this feature are disfavored, they cannot yet be ruled out. Our analysis confirms that the evidence for dynamical dark energy, particularly at low redshift ($z\lesssim 0.3$), is robust and stable under different modeling choices.

We present the Damped $\mathrm{Ly}\alpha$ Toolkit for automated detection and characterization of damped $\mathrm{Ly}\alpha$ absorbers (DLAs) in quasar spectra. Our method uses quasar spectral templates with and without absorption from intervening DLAs to reconstruct observed quasar forest regions. The best-fitting model determines whether a DLA is present while estimating the redshift and HI column density. With an optimized quality cut on detection significance ($\mathrm{\Delta }{\chi }_r^2>0.03$), the technique achieves an estimated 80% purity and 79% completeness when evaluated on simulated spectra with $\mathrm{S}/\mathrm{N}>2$ that are free of broad absorption lines (BALs). We provide a catalog containing candidate DLAs from the DLA Toolkit detected in DESI DR1 quasar spectra, of which 21 719 were found in $\mathrm{S}/\mathrm{N}>2$ spectra with predicted ${\log }_{10}(N_{\mathtt{HI}})>20.3$ and detection significance $\mathrm{\Delta }{\chi }_r^2>0.03$. We compare the Damped $\mathrm{Ly}\alpha$ Toolkit to two alternative DLA finders based on a convolutional neural network and Gaussian process models. We present a strategy for combining these three techniques to produce a high-fidelity DLA catalog from DESI DR2 for the $\mathrm{Ly}\alpha$ forest baryon acoustic oscillation measurement. The combined catalog contains 41 152 candidate DLAs with ${\log }_{10}(N_{\mathtt{HI}})>20.3$ from quasar spectra with $\mathrm{S}/\mathrm{N}>2$. We estimate this sample to be approximately 85% pure and 79% complete when BAL quasars are excluded.

Third-generation gravitational wave detectors such as Einstein Telescope and Cosmic Explorer will have significantly better sensitivities than current detectors, as well as a wider frequency bandwidth. This will increase the number and duration of the observed signals, leading to many signals overlapping in time. If not adequately accounted for, this can lead to biases in parameter estimation. In this work, we combine the joint parameter estimation method with relative binning to handle full parameter inference on overlapping signals from binary black holes, including precession effects and higher-order mode content. As this method is computationally more expensive than traditional single-signal parameter estimation, we test a prior-informed Fisher matrix and a time-frequency overlap method for estimating expected bias to help us decide when joint parameter estimation is necessary over the simpler methods. We improve upon previous Fisher matrix implementations by including the prior information and performing an optimization routine to better locate the maximum likelihood point, but we still find the method unreliable. The time-frequency method is accurate in 86% of close binary black hole mergers. We end by developing our own method of estimating bias due overlaps, where we reweight the single signal parameter estimation posterior to quantify how much the overlapping signals affect it. We show it has 99% accuracy for zero noise injections (98% in Gaussian noise), at the cost of one additional standard sampling run when joint parameter estimation proves to be necessary.

We present a novel and rigorous framework for determining the equilibrium configurations of uniformly rotating, self-gravitating fluid bodies. This work addresses the longstanding challenge of accurately modeling the rotational deformation of celestial objects such as stars and planets. By integrating classical Newtonian potential theory with modern mathematical techniques, we develop a unified formalism that significantly enhances both the precision and generality of shape modeling in astrophysical contexts. Our methodology employs Lie group theory and exponential mapping to characterize vector flows associated with rotational deformations. We derive functional equations governing perturbations in density and gravitational potential, which are analytically resolved using the shift operator and Neumann series summation. This approach extends Clairaut’s classical linear perturbation theory into the nonlinear regime. The resulting formulation yields an exact nonlinear differential equation for the shape function, describing hydrostatic equilibrium under rotation without relying on the assumption of slow rotation. This equation generalizes Clairaut’s classical result—which applies only in the linear, slow-rotation regime—by incorporating nonlinear effects and accommodating sufficiently large rotational speeds, the upper limit of which remains subject to further investigation. We refer to this as the generalized Clairaut equation to emphasize its role as a natural extension of Clairaut’s original theory. We validate this formulation by deriving exact solutions, including the Maclaurin spheroid, Jacobi ellipsoid, and the unit-index polytrope. Furthermore, we introduce spectral decomposition techniques to analyze the radial harmonics of the shape function and associated gravitational perturbations. Utilizing Wigner’s formalism for angular momentum addition, we compute higher-order spectral corrections, thereby enhancing both the accuracy and computational efficiency of existing models. Our framework also derives rigorously defined boundary conditions for the spectral radial harmonics, enabling the systematic computation of Love numbers and gravitational multipole moments with high accuracy. This work represents a substantial advancement in the mathematical theory of rotating fluid figures, offering a comprehensive, nonperturbative methodology for modeling rotational deformations in astrophysical and planetary systems.