Using the effective field theory framework for extended objects we describe the evolution of spinning compact objects in the late inspiral of the coalescence of a binary, before the plunge and merger, by including leading order corrections due to spin, tides, dissipation and gravitational wave radiation. Our implementation is of particular relevance for probing the stellar structure of compact objects with gravitational wave observations. A spinning compact object in the effective field theory framework is described as a spinning point particle, with its finite size effects encoded in higher order operators in the effective action, operators which have coefficients that encapsulates the internal structure of the star. For the inspiral regime described by non-relativistic general relativity, post-Newtonian corrections to each term of the action can be obtained in a diagrammatic approach, including gravitational radiation effects. Taking into account the aforementioned ingredients of the effective theory, we solve for the dynamics of the inspiral of binary systems using an algorithm for point particle simulations. We extract the gravitational wave as a function of the orbital frequency, input that is generated numerically and then evaluated in the analytic function of the waveform. By performing illustrative numerical experiments of systems that the LIGO-Virgo observatories have already detected, we show the role of the stellar structure and its coefficients in the phase evolution of the waveform, as well as the order in which they arise and the sensitivity required for the gravitational wave observatories to measure them. If these coefficients are to be measured, tight constraints on fundamental physics can be placed.
The observation of the inspiral and merger of compact binaries by the LIGO/Virgo collaboration ushered in a new era in the study of strong-field gravity. We review current and future tests of strong gravity and of the Kerr paradigm with gravitational-wave interferometers, both within a theory-agnostic framework (the parametrized post-Einsteinian formalism) and in the context of specific modified theories of gravity (scalar-tensor, Einstein-dilaton-Gauss-Bonnet, dynamical Chern-Simons, Lorentz-violating, and extra dimensional theories). In this contribution we focus on (i) the information carri…
Under the framework of gravitational effective field theory, we propose a theory agnostic strategy of searching for higher-spin particles with gravitational waves from compact binary inspirals. Using this strategy, we analyze gravitational wave signals from the binary black hole merger events GW151226 and GW170608, as well as the binary neutron star merger event GW170817. We find that the existence of higher-spin particles with mass ranged from $10^{-12} {\rm eV}$ to $10^{-11} {\rm eV}$ is strongly disfavored by these events unless the particles precisely combine within a supersymmetric superm…
We describe the search for gravitational waves from inspiraling neutron star binary systems, using data from the first Scientific Run of the LIGO Science Collaboration.
The INSPIRAL program is the LIGO Scientific Collaboration's computational engine for the search for gravitational waves from binary neutron stars and sub-solar mass black holes. We describe how this program, which makes use of the FINDCHIRP algorithm (discussed in a companion paper), is integrated into a sophisticated data analysis pipeline that was used in the search for low-mass binary inspirals in data taken during the second LIGO science run.
Contents: I. Introduction; II. Summary of optimal signal filtering; III. Newtonian binary polarization waveforms; IV. Newtonian orbital phase evolution; V. Post-Newtonian wave-generation; VI. Inspiral binary waveform.
We study the potential of gravitational wave astronomy to observe the quantum aspects of black holes. According to Bekenstein's quantization, we find that black hole area discretization can have observable imprints on the gravitational wave signal from an inspiraling binary black hole. We study the impact of quantization on tidal heating. We model the absorption lines and compute gravitational wave flux due to tidal heating in such a case. By including the quantization we find the dephasing of the gravitational wave, to our knowledge it has never been done before. We discuss the observability …
To enable detection and maximise the physics output of gravitational wave observations from compact binary systems, it is crucial the availability of accurate waveform models. The present work aims at giving an overview for non-experts of the (inspiral) waveforms used in the gravitational wave data analysis for compact binary coalescence. We first provide the essential elements of gravitational radiation physics within a simple Newtonian orbital dynamics and the linearized gravity theory, describing the adiabatic approximation applied to binary systems: the key element to construct the theoret…
The properties of neutron star matter above nuclear density are not precisely known. Gravitational waves emitted from binary neutron stars during their late stages of inspiral and merger contain imprints of the neutron-star equation of state. Measuring departures from the point-particle limit of the late inspiral waveform allows one to measure properties of the equation of state via gravitational wave observations. This and a companion talk by J. S. Read reports a comparison of numerical waveforms from simulations of inspiraling neutron-star binaries, computed for equations of state with varyi…
Modeling of gravitational waves (GWs) from binary black hole inspiral brings together early post-Newtonian waveforms and late quasinormal ringing waveforms. Attempts to bridge the two limits without recourse to numerical relativity involve predicting the time of the peak GW amplitude. This prediction will require solving the question of why the peak of the "source," i.e., the peak of the binary angular velocity, does not correspond to the peak of the GW amplitude. We show here that this offset can be understood as due to the existence two distinct components of the radiation: the "direct" radi…
Using two recent techniques giving non-perturbative re-summed estimates of the damping and of the conservative part of the dynamics of two-body systems, we describe the transition between adiabatic inspiral and plunge in binary non-spinning black holes moving along quasi-circular orbits.
In calculations of the inspiral of binary black holes an intermediate approximation is needed that can bridge the post-Newtonian methods of the early inspiral and the numerical relativity computations of the final plunge. We describe here the periodic standing wave approximation: A numerical solution is found to the problem of a periodic rotating binary with helically symmetric standing wave fields, and from this solution an approximation is extracted for the physically relevant problem of inspiral with outgoing waves. The approximation underlying this approach has been recently confirmed with…
Using the NRGR effective field theory formalism we calculate the remaining source multipole moments necessary to obtain the spin contributions to the gravitational wave amplitude to 2.5 Post-Newtonian (PN) order. We also reproduce the tail contribution to the waveform linear in spin at 2.5PN arising from the nonlinear interaction between the current quadrupole and the mass monopole.
Binary black holes are the most promising candidate sources for the first generation of earth-based interferometric gravitational-wave detectors. We summarize and discuss the state-of-the-art analytic techniques developed during the last years to better describe the late dynamical evolution of binary black holes of comparable masses.
Recent work on the expected event rate of neutron star inspiral signals in the LIGO detector is summarized. The observed signals will be from inspirals at cosmological distances, and the important cosmological effects on the event rate and spectrum are discussed. This paper is a contribution to the proceedings of the 17th Texas Symposium held in Munich, 12-17 December 1994.
Gravitational radiation from a binary neutron star or black hole system leads to orbital decay and the eventual coalescence of the binary's components. During the last several minutes before the binary components coalesce, the radiation will enter the bandwidth of the United States Laser Inteferometer Gravitational-wave Observatory (LIGO) and the French/Italian VIRGO gravitational radiation detector. The combination of detector sensitivity, signal strength, and source density and distribution all point to binary inspiral as the most likely candidate for observation among all the anticipated so…
Now that LIGO has revealed the existence of a large number of binary black holes, identifying their origin becomes an important challenge. They might originate in more isolated regions of the galaxy or alternatively they might reside in dense environments such as galactic centers or globular clusters. In the latter case, their center of mass motion as well as their orbital parameters should lead to observable changes in the waveforms, which would reflect their gravitational interactions with the surrounding matter. This would be reflected in the gravitational wave signal by a net phase change …
We generalize the world line EFT formalism to account for parity violating finite size effects. Results are presented for potentials and radiating moments of a binary inspiral for the parity conserving sector, and agreement is found with, previous calculations. Furthermore, we generate new results in this sector, calculating the current quadrupole moment induced by finite size gravitomagnetic effects. We also present novel results for parity violating sources, which might be due to beyond standard model physics, and show that they generate GW signals with the unique signature that the current-…
We perform long-term simulations of spinning binary neutron stars, with our highest dimensionless spin being $χ\sim 0.32$. To assess the importance of spin during the inspiral we vary the spin, and also use two equations of state, one that consists of plain nuclear matter and produces compact stars (SLy), and a hybrid one that contains both nuclear and quark matter and leads to larger stars (ALF2). Using high resolution that has grid spacing $Δx\sim 98$ m on the finest refinement level, we find that the effects of spin in the phase evolution of a binary system can be larger than the one that c…
Presented is a description of a Markov chain Monte Carlo (MCMC) parameter estimation routine for use with interferometric gravitational radiational data in searches for binary neutron star inspiral signals. Five parameters associated with the inspiral can be estimated, and summary statistics are produced. Advanced MCMC methods were implemented, including importance resampling and prior distributions based on detection probability, in order to increase the efficiency of the code. An example is presented from an application using realistic, albeit fictitious, data.
We investigate the dynamic stability of inspiraling neutron stars by performing multiple-orbit numerical relativity simulations of the binary neutron star inspiral process. By introducing eccentricities in the orbits of the neutron stars, significant changes in orbital separation are obtained within orbital timescales. We find that as the binary system evolves from apastron to periastron (as the binary separation decreases), the central rest mass density of each star decreases, thus stabilizing the stars against individual prompt collapse. As the binary system evolves from periastron to apastr…
We report on a search for gravitational waves from binary black hole inspirals in the data from the second science run of the LIGO interferometers. The search focused on binary systems with component masses between 3 and 20 solar masses. Optimally oriented binaries with distances up to 1 Mpc could be detected with efficiency of at least 90%. We found no events that could be identified as gravitational waves in the 385.6 hours of data that we searched.
Gravitational waves emitted during the inspiral, plunge and merger of a black hole binary carry linear momentum. This results in an astrophysically important recoil to the final merged black hole, a ``kick'' that can eject it from the nucleus of a galaxy. In a previous paper we showed that the puzzling partial cancellation of an early kick by a late antikick, and the dependence of the cancellation on black hole spin, can be understood from the phenomenology of the linear momentum waveforms. Here we connect that phenomenology to its underlying cause, the spin-dependence of the inspiral trajecto…
The LIGO Scientific Collaboration is currently engaged in the first search for binary black hole inspiral signals in real data. We are using the data from the second LIGO science run and we focus on inspiral signals coming from binary systems with component masses between 3 and 20 solar masses. We describe the analysis methods used and report on preliminary estimates for the sensitivities of the LIGO instruments during the second science run.
Third-generation gravitational wave (GW) observatories such as Einstein Telescope (ET) and Cosmic Explorer (CE) will be ideal instruments to probe the structure of neutron stars through the GWs they emit when undergoing binary coalescence. In this work we make predictions about how well ET in particular will enable us to reconstruct the neutron star equation of state through observations of tens of binary neutron star coalescences with signal-to-noise ratios in the hundreds. We restrict ourselves to information that can be extracted from the inspiral, which includes tidal effects and possibly …
Tidal effects in gravitational-wave (GW) observations from binary neutron star mergers have the potential to probe ultra-dense matter and shed light on the unknown nuclear equation of state of neutron stars. Tidal effects in inspiralling neutron star binaries become relevant at GW frequencies of a few hundred Hz and require detectors with exquisite high-frequency sensitivity. Third generation GW detectors such as the Einstein Telescope or Cosmic Explorer will be particularly sensitive in this high-frequency regime, allowing us to probe neutron star tides beyond the adiabatic approximation. Her…
We explore how a recently developed analytical black-hole binary spacetime can be extended using numerical simulations to go beyond the slow-inspiral phase. The analytic spacetime solves the Einstein field equations approximately, with the approximation error becoming progressively smaller the more separated the binary. To continue the spacetime beyond the slow-inspiral phase, we need to transition. Such a transition was previously explored at smaller separations. Here, we perform this transition at a separation of D=20M (large enough that the analytical metric is expected to be accurate), and…
The early part of the gravitational wave signal of binary neutron star inspirals can potentially yield robust information on the nuclear equation of state. The influence of a star's internal structure on the waveform is characterized by a single parameter: the tidal deformability lambda, which measures the star's quadrupole deformation in response to the companion's perturbing tidal field. We calculate lambda for a wide range of equations of state and find that the value of lambda spans an order of magnitude for the range of equation of state models considered. An analysis of the feasibility o…
Emission of anisotropic gravitational radiation from compact binary system leads to a flux of linear momentum. This results in the recoil of the system. We investigate the rate of loss of Linear momentum flux in the far zone of the source using various mass type and current type multipole moments for inspiralling compact binary mergers in quasi-elliptical orbits at 2.5 Post Newtonian order. We compute the linear momentum flux accurate up to $\mathcal{O}(e_t)$ in harmonic coordinate. A 2.5 Post Newtonian Quasi-Keplarian representation of the parametric solution to the Post Newtonian equation of…
We study whether binary black hole template banks can be used to search for the gravitational waves emitted by general binary coalescences. To recover binary signals from noisy data, matched-filtering techniques are typically required. This is especially true for low-mass systems, with total mass $M \lesssim 10 \, M_\odot$, which can inspiral in the LIGO and Virgo frequency bands for thousands of cycles. In this paper, we focus on the detectability of low-mass binary systems whose individual components can have large spin-induced quadrupole moments and small compactness. The quadrupole contrib…
Multiple approaches are required to study the evolution of black-hole binaries. While the post-Newtonian approximation is sufficient to describe the early inspiral (even from infinitely large orbital separation), only numerical relativity can capture the full complexity of the dynamics near merger. We combine multi-timescale post-Newtonian integrations with numerical-relativity surrogate models, thus mapping the entire history of the binary from its asymptotic configuration at past-time infinity to the post-merger remnant. This approach naturally allows us to assess the impact of the precessio…
We model the adiabatic inspiral of relativistic binary neutron stars in a quasi-equilibrium (QE) approximation, and compute the gravitational wavetrain from the late phase of the inspiral. We compare corotational and irrotational sequences and find a significant difference in the inspiral rate, which is almost entirely caused by differences in the binding energy. We also compare our results with those of a point-mass post-Newtonian calculation. We illustrate how the late inspiral wavetrain computed with our QE numerical scheme can be matched to the subsequent plunge and merger waveform calcula…
Thanks to recent measurements of tidal deformability and radius, the nuclear equation of state and structure of neutron stars are now better understood. Here, we show that through resonant tidal excitations in a binary inspiral, the neutron crust generically undergoes elastic-to-plastic transition, which leads to crust heating and eventually meltdown. This process could induce $\sim \mathcal{O}(0.1)$ phase shift in the gravitational waveform. Detecting the timing and induced phase shift of this crust meltdown will shed light on the crust structure, such as the core-crust transition density, wh…
We investigate Nordström's second theory of gravitation, with a focus on utilizing it as a testbed for developing techniques in numerical relativity. Numerical simulations of inspiraling compact star binaries are performed for this theory, and compared to the predictions of semi-analytic calculations (which are similar to Peters and Mathews' results for GR). The simulations are based on a co-rotating spherical coordinate system, where both finite difference and pseudo-spectral methods are used. We also adopt the "Hydro without Hydro" approximation, and the Weak Radiation Reaction approximation…
Extreme-mass-ratio inspiral (EMRI) is an important gravitational-wave (GW) source and it normally consists of one stellar-mass black hole (BH) whirling closely around a supermassive black hole (SMBH). In this Letter, we demonstrate that the small body, in fact, could be a BH binary (BHB). Previous numerical scatting experiments have shown that SMBHs can tidally capture BHBs to bound orbits. Here we investigate the subsequent long-term evolution. We find that those BHBs with a semi-major axis of $a\lesssim5\times10^{-3}$ AU can be captured to tightly-bound orbits such that they will successfull…
Combining recent techniques giving non-perturbative re-summed estimates of the damping and conservative parts of the two-body dynamics, we describe the transition between the adiabatic phase and the plunge, in coalescing binary black holes with comparable masses moving on quasi-circular orbits. We give initial dynamical data for numerical relativity investigations, with a fraction of an orbit left, and provide, for data analysis purposes, an estimate of the gravitational wave-form emitted throughout the inspiral, plunge and coalescence phases.
It is known that massive black holes (MBHs) of $10^{5-7}\,M_\odot$ could capture small compact objects to form extreme-mass-ratio inspirals (EMRIs). Such systems emit gravitational waves (GWs) in the band of the Laser Interferometer Space Antenna (LISA) and are ideal probes of the space-time geometry of MBHs. Recently, we have shown that MBHs could also capture stellar-mass binary black holes (about $10\,M_\odot$) to form binary-EMRIs (b-EMRIs) and, interestingly, a large fraction of the binaries coalesce due to the tidal perturbation by the MBHs. Here we further show that the coalescence coul…
We consider EGO as a possible third-generation ground-based gravitational wave detector and evaluate its capabilities for the detection and interpretation of compact binary inspiral signals. We identify areas of astrophysics and cosmology where EGO would have qualitative advantages, using Advanced LIGO as a benchmark for comparison.
The strong tidal forces that arise during the last stages of the life of a black hole-neutron star binary may severely distort, and possibly disrupt, the star. Both phenomena will imprint signatures about the stellar structure in the emitted gravitational radiation. The information from the disruption, however, is confined to very high frequencies, where detectors are not very sensitive. We thus assess whether the lack of tidal distortion corrections in data-analysis pipelines will affect the detection of the inspiral part of the signal and whether these may yield information on the equation o…
Compact binaries are an important class of gravitational-wave (GW) sources that can be detected by current and future GW observatories. They provide a testbed for general relativity (GR) in the highly dynamical strong-field regime. Here, we use GWs from inspiraling binary neutron stars and binary black holes to investigate dipolar gravitational radiation (DGR) and varying gravitational constant predicted by some alternative theories to GR, such as the scalar-tensor gravity. Within the parametrized post-Einsteinian framework, we introduce the parametrization of these two effects simultaneously …
We present the first fully relativistic calculations of the crustal strain induced in a neutron star by a binary companion at the late stages of inspiral, employing realistic equations of state for the fluid core and the solid crust. We show that while the deep crust is likely to fail only shortly before coalescence, there is a large variation in elastic strain, with the outermost layers failing relatively early on in the inspiral. We discuss the significance of the results for both electromagnetic and gravitational-wave astronomy.
Compact binaries inspiralling along quasi-circular orbits are the most plausible gravitational wave (GW) sources for the operational, planned and proposed laser interferometers. We provide new class of restricted post-Newtonian accurate GW templates for non-spinning compact binaries inspiralling along PN accurate quasi-circular orbits. Arguments based on data analysis, theoretical and astrophysical considerations are invoked to show why these time-domain Taylor approximants should be interesting to various GW data analysis communities.
Presented in this paper is a Markov chain Monte Carlo (MCMC) routine for conducting coherent parameter estimation for interferometric gravitational wave observations of an inspiral of binary compact objects using data from multiple detectors. The MCMC technique uses data from several interferometers and infers all nine of the parameters (ignoring spin) associated with the binary system, including the distance to the source, the masses, and the location on the sky. The Metropolis-algorithm utilises advanced MCMC techniques, such as importance resampling and parallel tempering. The data is compa…
Standard choices of quasi-circular orbit parameters for black-hole binary evolutions result in eccentric inspiral. We introduce a conceptually simple method, which is to integrate the post-Newtonian equations of motion through hundreds of orbits, and read off the values of the momenta at the separation at which we wish to start a fully general relativistic numerical evolution. For the particular case of non-spinning equal-mass inspiral with an initial coordinate separation of $D = 11M$ we show that this approach reduces the eccentricity by at least a factor of five to $e < 0.002$ as compared t…
We probe the ability of various types of post-Newtonian(PN)-accurate circular templates to capture inspiral gravitational-wave (GW) signals from compact binaries having tiny orbital eccentricities. The GW signals are constructed by adapting the phasing formalism, available in T. Damour, A. Gopakumar, and B. R. Iyer, [Phys. Rev. D 70, 064028 (2004)], employing the orbital energy and the time-eccentricity to describe the orbital evolution. Using the fitting factor estimates, relevant for the initial LIGO, we show that circular templates, based on the adiabatic TaylorT1, complete adiabatic Taylor…
Treating general relativity as an effective field theory, we compute the leading-order quantum corrections to the orbits and gravitational-wave emission of astrophysical compact binaries. These corrections are independent of the (unknown) nature of quantum gravity at high energies, and generate a phase shift and amplitude increase in the observed gravitational-wave signal. Unfortunately (but unsurprisingly), these corrections are undetectably small, even in the most optimistic observational scenarios.
Phasing formulas in a recent paper of ours' (Phys. Rev. D 63, 044023-1 (2001) are updated taking into account the recent 3.5PN results. Some misprints in the published version of the recent paper are also corrected.
The detection and characterization of gravitational wave signals from compact binary coalescence events relies on accurate waveform templates in the frequency domain. The stationary phase approximation (SPA) can be used to compute closed-form frequency-domain waveforms for non-precessing, quasi-circular binary inspirals. However, until now, no fast frequency-domain waveforms have existed for generic, spin-precessing quasi-circular compact binary inspirals. Templates for these systems have had to be computed via a discrete Fourier transform of finely sampled time-domain waveforms, which is far …
In the context of binary inspiral of mixed neutron star - black hole systems, we investigate the excitation of the neutron star oscillation modes by the orbital motion. We study generic eccentric orbits and show that tidal interaction can excite the f-mode oscillations of the star by computing the amount of energy and angular momentum deposited into the star by the orbital motion tidal forces via closed form analytic expressions. We study the f-mode oscillations of cold neutron stars using recent microscopic nuclear equations of state, and we compute their imprint into the emitted gravitationa…
We use the post-Newtonian-Affine model to assess the validity of the adiabatic approximation in modeling tidal effects in the phase evolution of compact binary systems. We compute the dynamical evolution of the tidal tensor, which we estimate at the 2 PN order, and of the quadrupole tensor, finding that their ratio, i.e. the tidal deformability, increases in the last phases of the inspiral. We derive the gravitational wave phase corrections due to this phenomenon and quantify how they affect gravitational wave detectability.
The signal-to-noise ratios (SNRs) for quasi-circular binary black hole inspirals computed from restricted post-Newtonian waveforms are compared with those attained by more complete post-Newtonian signals, which are superpositions of amplitude-corrected harmonics of the orbital phase. It is shown that if one were to use the best available amplitude-corrected waveforms for detection templates, one should expect SNRs in actual searches to be significantly lower than those suggested by simulations based purely on restricted waveforms.
We study the coalescence of non-spinning binary black holes from near the innermost stable circular orbit down to the final single rotating black hole. We use a technique that combines the full numerical approach to solve Einstein equations, applied in the truly non-linear regime, and linearized perturbation theory around the final distorted single black hole at later times. We compute the plunge waveforms which present a non negligible signal lasting for $t\sim 100M$ showing early non-linear ringing, and we obtain estimates for the total gravitational energy and angular momentum radiated.
There has been remarkable progress in numerical relativity recently. This has led to the generation of gravitational waveform signals covering what has been traditionally termed the three phases of the coalescence of a compact binary - the inspiral, merger and ringdown. In this paper, we examine the usefulness of inspiral only templates for both detection and parameter estimation of the full coalescence waveforms generated by numerical relativity simulations. To this end, we deploy as search templates waveforms based on the effective one-body waveforms terminated at the light-ring as well as s…
Using effective field theory techniques we calculate the source multipole moments needed to obtain the spin contributions to the power radiated in gravitational waves from inspiralling compact binaries to third Post-Newtonian order (3PN). The multipoles depend linearly and quadratically on the spins and include both spin(1)spin(2) and spin(1)spin(1) components. The results in this paper provide the last missing ingredient required to determine the phase evolution to 3PN including all spin effects which we will report in a separate paper.
We construct a new global, fully analytic, approximate spacetime which accurately describes the dynamics of non-precessing, spinning black hole binaries during the inspiral phase of the relativistic merger process. This approximate solution of the vacuum Einstein's equations can be obtained by asymptotically matching perturbed Kerr solutions near the two black holes to a post-Newtonian metric valid far from the two black holes. This metric is then matched to a post-Minkowskian metric even farther out in the wave zone. The procedure of asymptotic matching is generalized to be valid on all spati…
The late stage of the inspiral of two black holes may have important non-Newtonian effects that are unrelated to radiation reaction. To understand these effects we approximate a slowly inspiralling binary by a stationary solution to Einstein's equations in which the holes orbit eternally. Radiation reaction is nullified by specifying a boundary condition at infinity containing equal amounts of ingoing and outgoing radiation. The computational problem is then converted from an evolution problem with initial data to a boundary value problem. In addition to providing an approximate inspiral wavef…
In this paper, we study gravitational waves generated by binary systems within an extension of General Relativity which is described by the addition of quadratic in curvature tensor terms to the Einstein-Hilbert action. Treating quadratic gravity as an effective theory valid in the low energy/curvature regime, we argue that reliable calculations can be performed in the early inspiral phase, and furthermore, no flux of additional massive waves can be detected. We then compute the massive dipole (-1PN) leading corrections to the post-Newtonian (PN) expansion of the standard waveform. By confront…
Using accurate and fully general-relativistic simulations we assess the effect that magnetic fields have on the gravitational-wave emission produced during the inspiral and merger of magnetized neutron stars. In particular, we show that magnetic fields have an impact after the merger, because they are amplified by a Kelvin-Helmholtz instability, but also during the inspiral, most likely because the magnetic tension reduces the stellar tidal deformation for extremely large initial magnetic fields, B_0>~10^{17}G. We quantify the influence of magnetic fields by computing the overlap, O, between t…
