Memory effects from symmetries for a vacuum gravitational plane wave

• Jibril BEN ACHOUR (ENS de Lyon / Arnold Sommerfeld Center (Munich))

Room: Auditorium Memory effects are persistent modifications of relatives observables (relative distance, relative velocity, etc ...) between test particles induced by the passage of a gravitational wave. These radiative effects stand as ones of the last predications of general relativity yet to be confirmed. On the one hand, modeling these memory effects is crucial for developing accurate waveform templates to be confronted with future observation. On the other hand, these radiatives effects are intimately related to the asymptotic symmetries of asymptotically flat spacetimes, thus revealing the fine structure of the infrared regime of gravity. In this talk, I will discuss the realization of memory effects in a non-asymptotically flat spacetime which corresponds to a pp-wave. I will review the recent classification of the different memory effects (displacement, velocity, etc...) induced by this simple exact non-linear radiative solution of GR. I will also review key theorems relating these memories to the explicit and hidden symmetries (associated to Killing tensors) of the spacetime geometry. This will provide a pedagogical example where memories can be treated fully analytically, the methods discussed here being applicable beyond this framework. This talk is based on the recent published article: https://inspirehep.net/literature/2796995

Relativity of the event: examples in JT gravity

• Alexander TASKOV (CPT)

Room: Auditorium Gravitational gauge constraints imply that local, bulk, measurable operators must be dressed to a fixed reference surface with the worldlines of some observers. Since observers live in their own inertial coordinate systems, they are unable to distinguish between different branches of the wavefunction when the metric is in superposition, and will identify the outcome of their measurement, whatever it is, as a single event occurring at a particular location. In the inertial frame of another set of observers, however, the event of this measurement will be identified as occurring in different locations according to the state of the metric. An event at a definite location for one set of observers will therefore be in a superposition of locations for another set of observers. We quantify an example of this effect in JT gravity, where certain observers see the black hole horizon as "smeared".

The operational meaning of total energy in general relativity

• Simone Speziale (CPT Marseille, CNRS)

Room: Auditorium The ADM and Bondi energies are key examples of diffeomorphism-invariant observables. They have played a crucial role to understand energy in general relativity, and by Noether's theorem, to understand (boundary) symmetries. In spite of their conceptual importance, we don't know how to directly measure them: their reconstructed values from gravitational waveforms are only inferred quantities. In this talk I will present thought experiments that provide an operational meaning to both ADM and Bondi energies.

Black holes in theories beyond general relativity

• Jacopo MAZZA (IJCLab)

Room: Auditorium Black holes beyond general relativity can be different from their general-relativistic counterparts—but in what ways and to what extent? I will explore this question discussing the particular example of Einstein–aether theory, an alternative to general relativity that displays a rich phenomenology and admits, among other things, faster-than-light signals. I will explain how the existence of such signals puts the very notion of black hole into question, but also describe how the serendipitous discovery of the so-called ‘universal horizons’ could, perhaps, salvage black holes from certain demise.

How to create a horizon in the lab and the route to measure entanglement in experiments

• Maxime Jacquet (Laboratoire Kastler Brossel, CNRS France)

Room: Auditorium Quantum field theory (QFT) in curved spacetimes predicts the amplification of field excitations and the occurrence of classical and quantum correlations, as in the Hawking effect for example. This raises the interest for experiments in which the curvature of spacetime can be controlled and correlations measured. Such analogue simulations are typically done with fluids accelerating from sub- to supersonic speeds: acoustic excitations are dragged by the supersonic flow, effectively trapped inside an acoustic horizon. Quantum fluctuations of the acoustic field are predicted to yield entangled emission across the horizon, as in black holes. Here we introduce such a QFT simulator in a one-dimensional polaritonic fluid of light. We demonstrate the unique tunability of our system by engineering smooth and steep horizons, which respectively have quasi-thermal, but weak, and strong Hawking radiation. We exploit the driven-dissipative nature of polaritons with a recently developed coherent probe spectroscopy method to measure the spectrum on either side of the horizon and evidence the excitation of negative energy waves for the first time. Notably, we explicitly show that, beyond phononic excitations as in other systems, our simulator also supports excitations with a massive, relativistic dispersion. This benchmarks and thereby establishes a QFT simulator of a new class. In the future, quantum optics techniques offer the possibility to measure entanglement in unexplored regimes, giving insight in this outstanding prediction of relativistic QFT.

Algebraically special quadratic Schwarzschild perturbations

• Hugo Roussille (École Normale Supérieure de Lyon)

Room: Auditorium The equations describing linear perturbations around a Schwarzschild black hole admit analytical solutions that describe waves of specific wavelengths propagating outside the black hole. While perturbations around a Schwarzschild black hole are generally of Petrov type I, these analytical solutions describe spacetimes of Petrov type II, and are thus dubbed 'algebraically special modes'. The existence of these modes is linked to the isospectrality theorem for Schwarzschild. In this work, I go beyond the linear approximation and construct algebraically special perturbations around a Schwarzschild black hole at the quadratic order, making use of a family of exact twisting vacuum radiative solutions of General Relativity. These quadratic perturbations can still be expressed analytically, similarly to their linear sources. I study their properties and show in particular how static quadratic perturbations deform the Schwarzschild black hole.

Spinfoam models for quantum gravity

• Etera Livine (ENS-Lyon)

Room: Auditorium

Numerical Relativity in effective field theories of gravity

• Aaron Held

Room: Auditorium The age of gravitational-wave astronomy is now in full swing: For the first time, we gain observational access to the highly dynamical strong-field regime of the gravitational interaction. Constraining potential deviations from General Relativity (GR) requires reliable waveform predictions, not just in GR, but also when higher curvature corrections contribute to the dynamics. I will present an overview of recent progress on (i) mathematical well-posedness, (ii) physical time evolution in the presence of ghosts, and (iii) resulting numerical nonlinear waveforms. In combination, the above constitutes a feasible pathway to use current and future gravitational-wave observations to constrain effective field theories of gravity. The same methods may also be applied to address higher-curvature corrections in the early universe.

Vanishing of Quadratic Love Numbers of Schwarzschild Black Holes

• Nikola Savic (IPhT Paris)

Room: Auditorium The static tidal response of compact objects is characterized by **tidal Love numbers**, which provide insights into the black hole horizon or the internal structure of compact objects. These numbers can be directly extracted from gravitational wave measurements of compact binaries. It is well-known that asymptotically flat **Schwarzschild black holes in general relativity, in four spacetime dimensions, exhibit a vanishing static induced linear tidal response**. In this talk, I will extend this result to the quadratic response under an arbitrary static tidal field. By matching the second-order black hole perturbation calculations to point-particle effective theory, **we demonstrate that the nonlinear Love numbers describing the quadratic response also vanish**. I will discuss the implications of this result for black hole perturbation theory and gravitational wave phenomenology.

Carroll and flat holography

• Marios Petropoulos (CPhT Palaiseau)

Room: Auditorium The theme of asymptotically flat spacetimes has come back recently to the fore, fueled by the discovery of gravitational waves and the growing interest in what flat-space holography could be. In this quest, Carrollian geometry plays a central role. After outlining a synoptic survey of Carrollian geometric structures, I will show how Ricci-flat spacetimes are generally reached as a limit of Einstein geometries and how they are in fact constructed by means of an infinite set of data data defined on the conformal Carrollian boundary that is null infinity, and emerging as the coefficients of the Laurent expansion of the energy-momentum tensor in powers of the cosmological constant. Comparison with the celestial approaches is part of the agenda.

Mock modularity of Calabi-Yau threefolds

• Khalil Bendriss (L2C)

Room: Auditorium Generating functions h(τ ) of D4-D2-D0 BPS indices, appearing in Calabi-Yau compactifications of type IIA string theory and identical to rank 0 Donaldson-Thomas invariants,are known to be higher depth mock modular forms satisfying a specific modular anomaly, with depth dtermined by the D4-brane charge r. We develop a mmethod to solve the modular anomaly equation for arbitrary charges, in terms of indefinite theta series. This allows to find the generating functions up to modular forms that can be fixed by computing just a finite number of Fourier coefficients of h.

Bohmian mechanics and bouncing universes

• Gilbert Moultaka (CNRS - LUPM (Montpellier))

Room: Auditorium

Quintessence: analytical results

• David Andriot (LAPTh, CNRS)

Room: Auditorium I will present an analytical description of realistic quintessence solutions and their features, in a model independent manner. I will discuss consequences for cosmology and string theory model building, as well as related observational targets.

N-body simulations as a tool for investigating the nature of dark energy and gravity at cosmological scales.

• Yann RASERA (LUTH/Obs. de Paris/Univ. Paris Cité/IUF)

Room: Auditorium N-body simulations are a powerful tool for investigating the nature of dark energy and gravity at cosmological scale. In this talk, I will present new progress made in this field in two directions. Firstly, by launching billions of light rays inside light cones extracted from simulation volumes, it is now possible to use them as laboratories for studying general relativistic effects at cosmological scale. Secondly, by running thousands of simulations in modified gravity, it is possible to build emulators that allow fast and accurate predictions for the distribution of large-scale structures beyond general relativity.

Covariant Cosmography with the Expansion Rate Fluctuations Field

• Christian Marinoni (Centre de Physique Théorique, Aix-Marseille Univ.)

Room: Auditorium Is it possible to construct a detailed model of local spacetime in a completely model-independent and non-perturbative manner? Specifically, this would involve developing an observationally viable and physically meaningful method for identifying and classifying angular distortions present in the distance-redshift relation, without relying on the cosmological principle or the notion of peculiar velocities. The first difficulty along this path is the meaningful generalization of the notion of the cosmic expansion rate at an arbitrary point P in a generic spacetime. This involves defining covariant cosmographic parameters, which are a set of line-of-sight dependent functions, with finite degrees of freedom, characterizing deviations from isotropy and critically dependent on the observer's state of motion. The second challenge revolves around the optimal estimation of anisotropies in the expansion rate using observational data. To this end, we define the expansion rate fluctuation field η, an observable designed to maximize measurement accuracy while minimizing potential biases. Using analytical models, we analyze the virtues and limitations of this formalism. Current results and future survey prospects for constraining the shape of the expansion rate field in the local universe will also be presented.

The formation of the first nonlinear dark matter structures in the universe

• Stéphane Colombi

Room: Auditorium In the standard picture of cosmic structure formation, the first dark matter objects to form are expected to be microhalos of roughly Earth mass and solar system size. In this framework I will discuss how pancakes seed these primordial structures and how dark matter microhalos emerge through violent relaxation and the appearance of a power-law prompt cusp, then grow and relax to the so-called Navarro Frenk and White profile through subsequent infall and mergers.

A self-similar approach to Dark Matter halo dynamics in 2D Vlasov simulations

• Abineet Parichha (Institut d'Astrophysique de Paris - Sorbonne Université)

Room: Auditorium Understanding dark matter halo dynamics can be pivotal to unravelling the nature of dark matter particles. Analytical treatment of the multistream flows inside the turn-around region of a collapsed CDM (cold dark matter) halo using various self-similar approaches already exist. In this work, we adapt the Fillmore and Goldreich self-similar solutions assuming cylindrical symmetry to data from 2D Vlasov-Poisson (ColdICE package) simulations of isolated halos seeded with sin-wave initial conditions. We measure trajectories in position and phase-space, mass and density profiles and compare these to predictions from the self-similar model, with an aim to establish the limits of self-similar behaviour and the factors leading to deviations from it.

The variance of the Hellings-Downs curve

• Giulia Cusin (IAP)

Room: Auditorium Recently different Pulsar Timing Array collaborations have found evidence of a gravitational wave background, consistent with an astrophysical background from black hole binaries in the inspirling phase. To claim a detection, one has to measure an almost-quadrupolar correlation between the time of arrival of pulses from an array of pulsars: the Hellings-Downs curve. I will address the question: assuming noise-free measurements, how well do we expect data to follow this curve? I will review different sources of "theoretical" variance affecting this curve, focusing in particular on the effect of source clustering on the pulsar correlations.

Glory and fate of post-newtonian approaches to general relativity and beyond

• Laura Bernard (LUTH - CNRS / Observatoire de Paris)

Room: Auditorium

Small-scale clustering of Primordial Black Holes

• Pierre Auclair (UCLouvain)

Room: Auditorium In this talk, based on arXiv:2402.00600, we revisit the initial spatial clustering of Primordial Black Holes (PBHs) originating from the Hubble reentry of large Gaussian density fluctuations in the early Universe. Using an excursion-set approach, we derive the two-point correlation functions of PBHs, properly accounting for the “cloud-in-cloud” mechanism. Our expressions naturally and intrinsically correlate the formation of pairs of PBHs, which is a key difference with the Poisson model of clustering. Our approach effectively includes short-range exclusion effects and clarifies the clustering behaviors at small scale: PBHs are anticorrelated at short distances. Using a scale-independent collapse threshold, we derive explicit expressions for the excess probability to find pairs of PBHs, as well as the excess probability to find pairs with asymmetric mass ratio. Our framework is model independent by construction, and we discuss possible other applications.

Indirect dark matter searches (PNHE-TH)

• Silvia Manconi (LAPTh)

Room: Auditorium

Constraints on the equation of state of dense matter (PNHE-TH)

• Anthea FANTINA ({CNRS}UPR3266)

Room: Auditorium

Compact objects beyond general relativity as seen in the EHT images (PNHE-TH)

• Irene Urso (LESIA, Paris Observatory)

Room: Auditorium

Inflation at the crossroads

• Christophe Ringeval (CURL, UCLouvain)

Room: Auditorium mini-review

Strong Mixing at the Cosmological Collider

• Arthur Poisson (IAP)

Room: Auditorium Apart from its manifest interest in the understanding of the first moments of the universe, the framework of cosmic inflation is also the best way we know to probe fundamental physics at very high energies. In particular, the spontaneous production of massive particles due to the expanding background can leave potentially visible imprints in cosmological correlation functions known as the cosmological collider signal. Within the effective field theory of inflation (EFTI), it is possible to treat these exchange processes in a model-independent way, and explicit computations taking advantage of the conformal invariance of late-time observables have been carried out using various techniques such as the cosmological bootstrap. More recently, the full parameter space allowed by the EFTI has been explored allowing for boost-breaking setups leading to more striking phenomenological signatures, and the recently developed cosmological flow approach numerically gives us access to any correlation function. In this talk, I will expose a treatment of a parameter space region that remains analytically unknown: the strong mixing regime where the inflaton field and the massive particle can experience an infinite number of flavor transformations during the process. I will describe ongoing efforts to describe this regime based on extensions of standard single-field effective field theory techniques.

Clocking the End of Inflation

• Baptiste Blachier (CURL, UCLouvain and LPENS)

Room: Auditorium Making observable predictions for cosmic inflation requires determining when the wavenumbers of astrophysical interest today exited the Hubble radius during the inflationary epoch. These instants are commonly evaluated using the slow-roll approximation and measured in e-folds $\Delta N=N−N_{\mathrm{end}}$​, in reference to the e-fold $N_{\mathrm{end}}$​ at which inflation ended. Slow roll being necessarily violated towards the end of inflation, both the approximated trajectory and $N_{\mathrm{end}}$​ are determined at, typically, one or two e-folds precision. Up to now, such an uncertainty has been innocuous, but this will no longer be the case with the forthcoming cosmological measurements. In this work, we introduce a new and simple analytical method, on top of the usual slow-roll approximation, that reduces uncertainties on $\Delta N$ to less than a tenth of an e-fold.

Revisiting the stochastic QCD axion window: departure from equilibrium during inflation

• Vadim Briaud (LPENS)

Room: Auditorium If dark matter is made of QCD axions, its abundance is determined by the vacuum expectation value acquired by the axion field during inflation. The axion is usually assumed to follow the equilibrium distribution arising from quantum diffusion during inflation. This leads to the so-called stochastic window under which the QCD axion can make up all the dark matter. However, in realistic inflationary potentials, I will show that the axion never reaches the equilibrium distribution at the end of inflation. This is because the relaxation time of the axion is much larger than the typical time scale over which $H$ varies during inflation. As a consequence, the axion acquires a quasi-flat distribution as long as it remains light during inflation. This leads to a reassessment of the stochastic axion window.

Line intensity mapping in cosmology

• Azadeh Moradinezhad (Laboratoire d'Annecy-le-Vieux de Physique Théorique (LAPTh))

Room: Auditorium mini-review

Effective theory of the large scale structure of the universe

• Pierre Zhang (ETH-Zurich)

Room: Auditorium

Optimal constraints on Primordial non-Gaussianity with the eBOSS DR16 quasars in Fourier space

• Marina Cagliari (LAPTh)

Room: Auditorium The statistical properties of the primordial curvature perturbations are a key ingredient of the success of the LCDM model in explaining the Universe as we observe it today. In simplest models of inflation initial fluctuations are Gaussian for all practical purposes, and measurements of the CMB bispectrum by the Planck satellite constrain any deviation from the Gaussian regime in a part in ten thousand. On the other hand, the theoretical target for the amplitude of Primordial Non-Gaussianities (PNG) in the initial perturbations is roughly an order of magnitude smaller than what Planck has measured. We have almost saturated the information content in the CMB, and any further improvement will likely come from the late-time distribution of galaxies or any other tracers of the Large Scale Structure (LSS) of the Universe. I will present constraints on the amplitude of local PNG, $f_{\rm NL}$, using the quasar sample in the Sloan Digital Sky Survey IV extended Baryon Oscillation Spectroscopic Survey (eBOSS) Data Release 16 (DR16) from https://arxiv.org/abs/2309.15814. We analyze the power spectrum monopole, testing for the presence of scale-dependent galaxy bias induced by local PNG. Our analysis makes use of optimal redshift weights that maximize the response of the quasar sample to the possible presence of non-zero PNG. We find $-4<f_{\rm NL}<27$ at $68\%$ confidence level, which is among the strongest bounds with LSS data. The optimal analysis reduces the error bar by about $10\%$ compared to the standard one, but this improvement is lower than the one expected from previous forecasts. In addition, the larger volume of this dataset, when compared to previous releases of the eBOSS quasar catalog, does not always correspond to a reduction of the final uncertainty on local PNG. This could suggest the presence of still unknown systematic effects in the data. If the quasars have a lower response to local PNG, our optimal constraint becomes $-23<f_{\rm NL}<21$ at $68\%$, with an improvement of $30\%$ over standard analyses. We also show how to use the optimal weights to put data-driven priors on the sample's response to local PNG.

Breaking the “distance duality relation” to explain the Hubble tension

• Elsa Teixeira (LUPM Montpellier)

Room: Auditorium The standard Lambda Cold Dark Matter (ΛCDM) cosmological model has proven remarkably successful in describing a broad range of observational data, ranging from the cosmic microwave background (CMB) radiation to the large-scale structure of the Universe. However, recent advances in precision cosmology have revealed persistent statistical discrepancies between independent data sets and observational methods. One prominent example is the "Hubble tension," which refers to the irreconcilable predictions of the present expansion rate of the Universe when inferred from early-Universe measurements (such as the CMB) compared to local observations. Low-redshift observables like Baryon Acoustic Oscillations (BAO) and Type Ia Supernovae (SN1a) are used to build the cosmological distance ladder, which relies on calibrations using either early- or late-Universe data. Therefore, the Hubble tension is also reflected in the incompatibility between these distances and how they are calibrated. However, this comparison assumes that the distance-duality relationship (DDR) holds and can be used to compare measurements of the luminosity and angular diameter distances. In this talk, we will examine the implications of relaxing this assumption to more general relations, its implications to the current cosmic tensions and how it could potentially explain the apparent need for the introduction of new physics to address current cosmic tensions.

A post-LHC view of particle dark matter

• Genevieve Belanger (LAPTH)

Room: Auditorium Review talk on particle dark matter.

Searches for ultralight dark matter: from the lab to space

• Aurelien Hees (SYRTE - Observatoire de Paris)

Room: Auditorium UltraLight Dark Matter (ULDM) is a class of bosonic Dark Matter candidates whose mass is below the electronVolt. Such Dark matter candidates have become very popular in the last decade, in particular because of the lack of direct detection of WIMPs with particles accelerators. The most studied ULDM candidates are the axion (pseudo-scalar particle), the dilaton (a scalar particle) and the hidden photon (a spin 1 particle). The phenomenology of ULDM is very rich and in particular, it can be searched for using atomic sensors like atomic clocks, cavities, atom interferometry, etc... In this talk, we will review the basic properties of such dark matter candidates and discuss some of their theoretical properties. We will then focus on experimental results and also present some new proposals of experiments specifically designed to search for such new fields.

Ghosts without runaway instabilities

• Cédric Deffayet (CNRS-LPENS)

Room: Auditorium Ghosts, i.e. degrees of freedom with negative kinetic energies, have been used in cosmology to tackle various issues such as dark energy or non standard dynamics of the universe. On the other hand, field theory models with ghosts are usually considered as suffering from a deadly instability. This instability is often thought to appear already at classical level and in mechanical models with just a ghost oscillator coupled to a positive energy (standard) one. We show that it is not the case, in a large class of such mechanical models where the motion can be shown analytically to be fully stable thanks to integrability. This is based on a work in collaboration with Aaron Held, Shinji Mukohyama and Alex Vikman.

An Eikonal Approach to Gravitational Bremsstrahlung

• Carlo Heissenberg (IPhT)

Room: Auditorium In this talk, I will discuss recent developments in the calculation of the gravitational waveform emitted during a scattering of two compact objects, considering two complementary regimes. The first is the post-Minkowskian (PM) approximation, where one focuses on widely separated objects, i.e. scatterings at large impact parameters. In this setup, interactions are weak and can be treated perturbatively. A particularly natural approach to attack this problem is to exploit the connection with scattering amplitudes, for which the eikonal framework offers a systematic way to describe the classical limit. I will discuss in particular how the next-to-leading PM waveform can be extracted from a one-loop 2->3 amplitude. The second approximation consists in focusing on low-frequency emissions, which are governed by universal soft theorems. These are simple relations that dictate in particular the structure of leading log-enhanced pieces of the type $\omega^{n-1}(\log\omega)^n$ for $n=0,1,2...$ in the low-frequency expansion, as $\omega\to0$. I will present a recent proposal for a resummation of all such terms and discuss their contribution to the energy emission spectrum.

New post-Newtonian results in general relativity and scalar-tensor theory

• David Trestini (IAP et LUTH)

Room: Auditorium In the first part of the talk, I review the recent computation (in general relativity) of the radiation-reaction sector of the equations of motion at 4.5PN [1]. In particular, I will discuss novel hereditary terms arising from the passage to the center-of-mass frame. In the second part of the talk, I will discuss the study of gravitational waves generated by compact binaries on elliptic orbits in scalar-tensor theories. I will first review the quasi-Keplerian parametrization for scalar-tensor theories [2]. Then, I will review the computation of the flux at 1.5PN order [3], which includes: instantaneous terms, tails, memory, and the same novel hereditary terms that we found in GR. Finally, I obtain the secular evolution of the frequency and eccentricity, which is the main observable for gravitational wave observatories. [1] Blanchet, Faye and Trestini (2024), arXiv:2407.18295 [2] Trestini, Phys. Rev. D 109, 104003 (2024), arXiv:2401.06844 [3] Trestini (2024), arXiv:2410.12898

Tidal contributions to the gravitational waveform amplitude to the second-and-a-half post-newtonian order

• Eve Dones (LUTH - Observatoire de Paris)

Room: Auditorium The study of tidal effects between compact objects such as neutron stars is particularly promising to better understand their physics. Including these effects in our waveform models could allow us to probe their internal structure, but also possibly to distinguish signals coming from black holes, neutron stars or even more exotic objects. This will be of paramount importance when interpreting the multiple signals expected with the arrival of third-generation gravitational wave detectors. The tidal interaction affects both the dynamics and the gravitational wave emission processes of compact binaries resulting in a change in the orbital phase and the gravitational wave amplitude that are directly observable. In this talk, I will present how we completed the computation of gravitational waveform amplitude modes using the post-Newtonian-multipolar-post-Minkowskian formalism with an order of accuracy of 2.5PN and wrote them in form suitable for effective-one-body template building.

Second order gravitational waves: paving the way for a full calculation.

• Raphael Picard (Queen Mary University London)

Room: Auditorium Gravitational waves provide a new observational tool to study the universe. There has been extensive research on scalar induced gravitational waves (SIGWs), since they could be the counterpart signal of primordial black holes. SIGWs are sourced by terms quadratic in first order scalar fluctuations from inflation. In this talk, I will discuss the possibility and implication of including tensor fluctuations at first order in the source term. Furthermore, I will talk about the correlation of third order and first order gravitational waves and comment on the imprint this leaves on the spectral energy density. Finally, I will discuss implications for their detectability and observational constraints for models of inflation.

Solution to the binary black hole system with arbitrary spins, eccentricity, and masses at the second post-Newtonian order

• Sashwat Tanay (LUTH, Paris Observatory)

Room: Auditorium We present an analytical solution to the binary black hole system with arbitrary spins, eccentricity, and masses at the second post-Newtonian order. This solution owing to its analytical nature is expected to facilitate quick gravitational wave (GW) template generation and accelerate GW detection and parameter estimation.