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DOE OSTI2023

Advancements in High-Q development for novel medium-velocity 644 MHz 5-cell elliptical superconducting RF cavities for continuous-wave operation in heavy-ion linacs

The Facility for Rare Isotope Beams is a world-leading center for experimental nuclear physics research, and relies on a first-in-kind superconducting RF driver linac to supply 400 kW beams of a uniquely large range of particles, from protons through the heaviest uranium ions. The FRIB400 project proposes to double the end energy of the current superconducting FRIB linac for the heaviest uranium ions from 200 MeV per nucleon (MeV/u) to 400 MeV/u, which equates to approximately 1 GeV for protons. The increased rare isotope production from higher-energy drive beams would deliver new and exciting capabilities to FRIB users, which include extending the facility's reach along the neutron drip-line, greatly increasing the facility's rare isotope yield, and significantly improving the precision of parameter measurements for the nuclear matter equation of state, which is a particularly timely complement to the recent advent of multi-messenger astronomy, and the study of neutron star mergers. The unique set of design parameters defined by the 400 MeV/u (uranium) energy goal, the physical size, continuous-wave operation, and cryogenic capacity of the extant facilities at FRIB, requires development of a first-in-kind beta = v/c = 0.65, 644 MHz 5-cell elliptical superconducting RF (SRF) cavity, with field-leading efficiency, measured by the cavity's intrinsic quality factor, Q_0. The field of SRF has recently experienced a very active period of development, which demonstrated the potential to raise the Q_0 of 1.3 GHz cavities by factors of two to three or more. In addition to uncovering fascinating new properties of the fundamental physics of superconducting RF phenomena, these improvements suggest substantial improvements could be made over the minimum Q_0 of the proposed FRIB400 upgrade SRF cavities, if these new techniques can be adapted to the beta = 0.65, 644 MHz regime. This work presents the first validation of the the novel beta = 0.65, 644 MHz 5-cell elliptical superconducting RF cavity design, and the first results of both conventional and advanced RF surface preparation techniques. In particular, this work focuses on two recently-developed high-Q_0 recipes: N-doping, and furnace baking, setting the current world-record Q_0 for this type of resonant cavity at 3.8 x 10^10 at the FRIB400 operating gradient of 17.5 MV/m. We also find evidence that these results may be further improved upon, and make the case for further RF surface preparation recipe optimization. We then pair these high-power RF investigations with techniques borrowed from material science to try to better understand the ways in which specific properties of the niobium material, such as grain structure and superconducting flux pinning force, can be related to its superconducting RF performance. These experiments better elucidate the effects of high-temperature annealing on niobium samples, which suggest methods of further optimizing cavity flux expulsion and thus SRF performance.Read

McGee, Kellen E.↗

DOE OSTI2022

Constraining ΛCDM cosmological parameters with Einstein Telescope mock data

ABSTRACT We investigate the capability of Einstein Telescope to constrain the cosmological parameters of the non-flat ΛCDM cosmological model. Two types of mock data sets are considered depending on whether or not a short gamma-ray burst is detected, and associated with the gravitational wave emitted by binary neutron stars merger, using the THESEUS satellite. Depending on the mock data set, two statistical estimators are applied: one assumes that the redshift is known, while the other marginalizes over it assuming a specific redshift prior distribution. We demonstrate that (i) using mock catalogues collecting gravitational wave signals emitted by binary neutron stars systems to which a short gamma-ray burst has been associated, Einstein Telescope may achieve an accuracy on the cosmological parameters of $\sigma _{H_0}\approx 0.40$ km s−1 Mpc−1, $\sigma _{\Omega _{k,0}}\approx 0.09$, and $\sigma _{\Omega _{\Lambda ,0}}\approx 0.07$; while (ii) using mock catalogues collecting all gravitational wave signals emitted by binary neutron stars systems for which an electromagnetic counterpart has not been detected, Einstein Telescope may achieve an accuracy on the cosmological parameters of $\sigma _{H_0}\approx 0.04$ km s−1 Mpc−1, $\sigma _{\Omega _{k,0}}\approx 0.01$, and $\sigma _{\Omega _{\Lambda ,0}}\approx 0.01$, once the redshift probability distribution of GW events is known from from population synthesis simulations and/or the measure of the tidal deformability parameter. These results show an improvement of a factor 2–75 with respect to earlier results using complementary data sets.

Califano, Matteo↗

DOE OSTI2021

Composite anyons on a torus

An adiabatic approach put forward by Greiter and Wilczek interpolates between the integer quantum Hall effects of electrons and composite fermions by varying the statistical flux bound to electrons continuously from zero to an even integer number of flux quanta, such that the intermediate states represent anyons in an external magnetic field with the same “effective” integer filling factor. We consider such anyons on a torus, and construct representative wave functions for their ground as well as excited states. These wave functions involve higher Landau levels in general, but can be explicitly projected into the lowest Landau level for many parameters. We calculate the variational energy gap between the first excited state and ground state and find that it remains open as the statistical phase is varied. Lastly, we obtain from these wave functions, both analytically and numerically, various topological quantities, such as ground-state degeneracy, the Chern number, and the Hall viscosity.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2022

Effects of Coulomb collisions on lower hybrid drift waves inside a laboratory reconnection current sheet

We have developed a local, linear theoretical model for lower hybrid drift waves that can be used for plasmas in the weakly collisional regime. Two cases with typical plasma and field parameters for the current sheet of the magnetic reconnection experiment have been studied. For a case with a low electron beta (β e =0.25, high guide field case), the quasi-electrostatic lower hybrid drift wave is unstable, while the electromagnetic lower hybrid drift wave has a positive growth rate for a high-β e case (β e =8.9, low guide field case). For both cases, including the effects of Coulomb collisions reduces the growth rate but collisional impacts on the dispersion and growth rate are limited (≲20%).

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2025

Pair wave function symmetry in UTe2 from zero-energy surface state visualization

Although nodal spin-triplet topological superconductivity appears probable in uranium ditelluride (UTe 2 ), its superconductive order parameter Δ k remains unestablished. In theory, a distinctive identifier would be the existence of a superconductive topological surface band, which could facilitate zero-energy Andreev tunneling to an s-wave superconductor and also distinguish a chiral from a nonchiral Δ k through enhanced s-wave proximity. In this study, we used s-wave superconductive scan tips and detected intense zero-energy Andreev conductance at the UTe 2 (0-11) termination surface. Imaging revealed subgap quasiparticle scattering interference signatures with a-axis orientation. The observed zero-energy Andreev peak splitting with enhanced s-wave proximity signifies that Δ k of UTe 2 is a nonchiral state: B 1u , B 2u , or B 3u . However, if the quasiparticle scattering along the a axis is internodal, then a nonchiral B 3u state is the most consistent for UTe 2 .

Gu, Qiangqiang [Cornell Univ., Ithaca, NY (United ↗

DOE OSTI2023

New insights into the pole parameters of the Λ(1380), the Λ(1405) and the Σ(1385)

A coupled-channel S- and P-wave next-to-leading order chiral-unitary approach for strangeness S = –1 meson-baryon scattering is extended to include the new data from the KLOE and AMADEUS experiments as well as the Λπ mass distribution of the Σ(1385). The positions of the poles on the second Riemann sheet corresponding to the Σ(1385) pole and the Λ(1380) and Λ(1405) poles as well as the couplings of these states to various channels are calculated. We find that the resonance positions and branching ratios are on average determined with about 20% higher precision when including the KLOE and AMADEUS data. Additionally, for the first time, the correlations between the parameters of the poles are investigated and shown to be relevant. We also find that the Σ(1385) has negligible influence on the properties of the Λ states given the available data. Still, we identify isospin-1 cusp structures in the present solution in light of new measurements of π ± Λ line-shapes by the Belle collaboration.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2022

Unitarity bounds on effective field theories at the LHC

Effective Field Theory (EFT) extensions of the Standard Model are tools to compute observables (e.g. cross sections with partonic center-of-mass energy √s^) as a systematically improvable expansion suppressed by a new physics scale M. If one is interested in EFT predictions in the parameter space where M < √s^, concerns of self-consistency emerge, which can manifest as a violation of perturbative partial-wave unitarity. However, when we search for the effects of an EFT at a hadron collider with center-of-mass energy √s using an inclusive strategy, we typically do not have access to the event-by-event value of √s^. This motivates the need for a formalism that incorporates parton distribution functions into the perturbative partial-wave unitarity analysis. Developing such a framework and initiating an exploration of its implications is the goal of this work. Our approach opens up a potentially valid region of the EFT parameter space where M << √s. We provide evidence that there exist valid EFTs in this parameter space. The perturbative unitarity bounds are sensitive to the details of a given search, an effect we investigate by varying kinematic cuts.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

NASA NTRS1973

Ion streaming instabilities with application to collisionless shock wave structure

The electromagnetic dispersion relation for two counterstreaming ion beams of arbitrary relative strength flowing parallel to a dc magnetic field is derived. The beams flow through a stationary electron background and the dispersion relation in the fluid approximation is unaffected by the electron thermal pressure. Magnetic effects on the ion beams are included, but the electrons are treated as a magnetized fluid. The dispersion relation is solved with a zero net current condition applied and the regions of instability in the k-U space (U is the relative velocity between the two ion beams) are presented. These results are extensions of Kovner's analysis for weak beams. The parameters are then chosen to be applicable for parallel shocks. It is found that unstable waves with zero group velocity in the shock frame can exist near the leading edge of the shock for upstream Alfven Mach numbers greater than 5.5.

Golden, K. I.↗

NASA NTRS1995

Wavenumber selection and hysteresis in nonlinear baroclinic flow

Wavenumber transition and hysteresis in a highly unstable baroclinic flow are investigated using a high-resolution spectral numerical model. As the flow becomes more supercritical, the dominant wave gradually shifts from the most unstable wave predicted by the linear theory to a longer wave with a larger time-averaged amplitude, while the rectified mean flow attains a stronger shear at the center of the channel. The numerical results display a complex hysteresis behavior, which occurs not only between the states of different dominant wavenumbers, but also between the states of identical dominant wavenumber but of different dynamic characteristics. In a certain parameter range three stable states, each with different dominant wavenumber, are possible, and in another parameter range four stable states are possible, among them three stable states with an identical dominant wave. The numerical results suggest that a multiple weather regime exists even without external forcing in which the flow aperiodically varies between two distinct behaviors. The effects of stable higher harmonics are assessed and it is found that their presence contributes not only to the better approximation of the model solutions but also to the selection of the final equilibrium state, due to the chaotic nature of the initial transient period.

Chou, Shih-Hung↗

NASA NTRS1983

Microprocessor-Based Neural-Pulse-Wave Analyzer

Microprocessor-based system analyzes amplitudes and rise times of neural waveforms. Displaying histograms of measured parameters helps researchers determine how many nerves contribute to signal and specify waveform characteristics of each. Results are improved noise rejection, full or partial separation of overlapping peaks, and isolation and identification of related peaks in different histograms. 2

Kojima, G. K.↗

DOE OSTI2024

Multi-GeV wakefield acceleration in a plasma-modulated plasma accelerator

We investigate the accelerator stage of a plasma-modulated plasma accelerator (P-MoPA) [Jakobsson , ] using both the paraxial wave equation and particle-in-cell (PIC) simulations. We show that adjusting the laser and plasma parameters of the modulator stage of a P-MoPA allows the temporal profile of pulses within the pulse train to be controlled, which in turn allows the wake amplitude in the accelerator stage to be as much as 72 % larger than that generated by a plasma beat-wave accelerator with the same total drive laser energy. Our analysis shows that Rosenbluth-Liu detuning is unimportant in a P-MoPA if the number of pulses in the train is less than ∼ 30 , and that this detuning is also partially counteracted by increased red-shifting, and hence increased pulse spacing, towards the back of the train. An analysis of transverse mode oscillations of the driving pulse train is found to be in good agreement with 2D (Cartesian) PIC simulations. PIC simulations demonstrating energy gains of ∼ 1.5 GeV ( ∼ 2.5 GeV ) for drive pulse energies of 2.4 J ( 5.0 J ) are presented. Our results suggest that P-MoPAs driven by few-joule, picosecond pulses, such as those provided by high-repetition-rate thin-disk lasers, could accelerate electron bunches to multi-GeV energies at pulse repetition rates in the kilohertz range. Published by the American Physical Society 2024

van de Wetering, J. J. (ORCID:0000000303292355)↗

DOE OSTI2020

A ``Tail" of Two Cities: Fatality-based Modeling of COVID-19 Evolution in New York City and Cook County, IL

I describe $SIR$ modeling of the COVID-19 pandemic in two U.S. urban environments, New York City (NYC) and Cook County, IL, from onset through the month of June, 2020. Since testing was not widespread early in the pandemic in the U.S., I do not use data on confirmed cases and rely solely on public fatality data to estimate model parameters. Fits to the first 20 days of data determine a degenerate combination of the basic reproduction number, $R_0$, and the mean time to removal from the infectious population, $\gamma^{-1}$, with $\gamma (R_0-1) = 0.25 (0.21)$ inverse days for NYC (Cook County). Equivalently, the initial doubling time was $t_d = 2.8 (3.4)$ days for NYC (Cook). The early fatality data suggest that both locations had infections in early February. I model the mitigation measures implemented in mid-March in both locations (distancing, quarantine, isolation, etc) via a time-dependent reproduction number $R_t$ that declines monotonically from $R_0$ to a sma ller asymptotic value, with a parameterized functional form. The timing (mid-March) and duration (several days) of the transitions in $R_t$ appear well determined by the data. However, the fatality data determine only a degenerate combination of the parameters $R_0$, the percentage reduction in social contact due to mitigation measures, $X$, and the infection fatality rate (IFR), $f$. With flat priors, based on simulations the NYC model parameters have 95.45\% credible intervals of $R_0 = 3.0-5.4$, $X= 80-99.9\%$ and $f=2-6\%$, with $5-13\%$ of the population asymptotically infected. A strong external prior indicating a lower value of $f$ or of $\gamma^{-1}$ would imply lower values of $R_0$ and $X$ and higher percentage infection of the population. For Cook County, the evolution was qualitatively different: after mitigation measures were implemented, the daily fatality counts reached a plateau for about a month before tailing off. This is consistent with an $SIR$ model that exhibits ``critical slowing-down", in which $R_t$ plateaus ! at a value just above unity. For Cook County, the 95.45\% credible intervals for the model parameters are much broader and shifted downward, $R_0=1.4-4.7$, $X=26-54\%$, and $f=0.1-0.6\%$ with $15-88\%$ of the population asymptotically infected. Despite the apparently lower efficacy of its social contact reduction measures, Cook County has had significantly fewer fatalities per population than NYC, $D_\infty/N=100$ vs. 270 per 100,000. In the model, this is attributed to the lower inferred IFR for Cook; an external prior pointing to similar values of the IFR for the two locations would instead chalk up the difference in $D/N$ to differences in the relative growth rate of the disease. I derive a model-dependent threshold, $X^{crit}_e$, for `safe' re-opening, that is, for easing of contact reduction that would not trigger a second wave; for NYC, the models predict that increasing social contact by more than 20\% from post-mitigation levels will lead to renewed spread, while for Cook County the threshold value is very uncertain, given the parameter degeneracies. The timing of 2nd-wave growth will depend on the amplitude of contact increase relative to $X^{crit}_e$ and on the asymptotic growth rate, and the impact in terms of fatalities will depend on the parameter $f$.

Frieman, Joshua↗

DOE OSTI2022

Probabilistic inversion of circular phase spectra: application to two-station phase-velocity dispersion estimation in western Canada

SUMMARY Periodic directional and temporal measurements are common in seismology, and necessitate specific statistical analyses that are appropriate for circular quantities. In this work, we explore the use of a von Mises distribution as a representation of errors on circular seismological observations. Specifically, we automate the estimation of surface-wave phase-velocity dispersion for the teleseismic two-station method, which generally suffers from a 2π phase ambiguity. The use of Bayesian inverse techniques, which aim to rigorously quantify model parameter uncertainty, have become widespread throughout seismology over the last decade. Here, we apply Bayesian inversion to measurements of surface-wave phase spectra in order to estimate 1-D, path-averaged Earth structure between station pairs. The dispersion curve and associated uncertainties are additional results of the inversion, which can then be used as input for subsequent analyses (e.g. tomography). We demonstrate this technique through application to surface-wave recordings from long-running seismic stations throughout western Canada. Our results for over 10 000 station pairs reveal first-order tectonic features consistent with previous studies, which provides confidence in our approach as well as an initial step towards resolving a full 3-D seismic velocity model for the region. This work also presents a foundation for the inversion of surface-wave phase spectra to estimate 3-D Earth structure directly. Finally, the ideas presented in this work are not limited to the inversion of surface-wave phase spectra, but can also be considered for Bayesian geophysical inversion of any circular quantities.

Gosselin, Jeremy M. (ORCID:0000000203754102)↗

DOE OSTI2022

Role of electromagnetic gauge-field fluctuations in the selection between chiral and nematic superconductivity

Motivated by the observation of nematic superconductivity in several systems, we revisit the problem of the leading pairing instability of two-component unconventional superconductors on the triangular lattice—such as (p x , p y )-wave and (d x 2 –y 2 , d xy )-wave superconductors. Such a system has two possible superconducting states: the chiral state (e.g., p + ip or d + id), which breaks time-reversal symmetry, and the nematic state (e.g., p + p or d + d), which breaks the threefold rotational symmetry of the lattice. Weak-coupling calculations generally favor the chiral over the nematic superconducting state, raising the question of what mechanism can stabilize the latter. Here, we show that the electromagnetic field fluctuations can play a crucial role in selecting between these two states. Specifically, we derive and analyze the effective free energy for the two-component superconducting order parameter after integrating out the gauge-field fluctuations, which is formally justified if the spatial order parameter fluctuations can be neglected. A nonanalytic cubic term arises, as in the case of a conventional s -wave superconductor. However, unlike the latter, the cubic term depends on the relative phase and on the relative amplitudes between the two order parameter components, in such a way that it generally favors the nematic state. This result is a direct consequence of the fact that the stiffness of the superconducting order parameter is not isotropic. Competition with the quartic term, which favors the chiral state, leads to a renormalized phase diagram in which the nematic state displaces the chiral state over a wide region in the parameter space. Here, we analyze the stability of the fluctuation-induced nematic phase, generalize our results to tetragonal lattices, and discuss their applicability to candidate nematic superconductors, including twisted bilayer graphene.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

NASA NTRS1983

The average solar wind in the inner heliosphere: Structures and slow variations

Measurements from the HELIOS solar probes indicated that apart from solar activity related disturbances there exist two states of the solar wind which might result from basic differences in the acceleration process: the fast solar wind (v 600 kms(-)1) emanating from magnetically open regions in the solar corona and the "slow" solar wind (v 400 kms(-)1) correlated with the more active regions and its mainly closed magnetic structures. In a comprehensive study using all HELIOS data taken between 1974 and 1982 the average behavior of the basic plasma parameters were analyzed as functions of the solar wind speed. The long term variations of the solar wind parameters along the solar cycle were also determined and numerical estimates given. These modulations appear to be distinct though only minor. In agreement with earlier studies it was concluded that the major modulations are in the number and size of high speed streams and in the number of interplanetary shock waves caused by coronal transients. The latter ones usually cause huge deviations from the averages of all parameters.

Schwenn, R.↗

NASA NTRS2013

Silent and Efficient Supersonic Bi-Directional Flying Wing

The supersonic bi-directional (SBiDir) flying wing (FW) concept has a great potential to achieve low sonic boom with high supersonic aerodynamic performance due to removal of performance conflict between high speed and low speed by rotating goo in flight. This NIAC Phase 1 research has achieved three objectives: 2) prove the concept based on simulation that it can achieve very low boom with smooth Sine wave ground over-pressure signature and excellent aerodynamic efficiency; 3) conduct trade study to correlate the geometric parameters with sonic boom and aerodynamic performance for further automated design optimization in Phase II. The design methodology developed in Phase I includes three parts: 1) an advanced geometry model, which can vary airfoil meanline angle distribution to control the expansion and shock waves on the airplane surface to mitigate sonic boom and improve aerodynamic efficiency. 2) a validated CFD procedure to resolve near field flow with accurate shock strength. The sonic boom propagation from near field to far field ground is simulated by NASA NF Boom code. The surface friction drag prediction is based on fiat plate correlation adopted by Seebass and supported by the experimental study of Winter and Smith, which is on the conservative side and is more reliable than CFD RANS simulation. 3) a mission analysis tool based on Corke's model that provides design requirements and constraints of supersonic airplanes for range, payload, volume, size, weight, etc. The design mission target is a supersonic transport with cruise Mach number 1.6, 100 passengers, and 4000nm range. The trade study has several very important findings: 1) The far field ground sonic boom signature is directly related to the smoothness of the flow on the airplane surface. The meanline angle distribution is a very effective control methodology to mitigate surface shock and expansion wave strength, and mitigating compression wave coalescing by achieving smooth loading distribution chord-wise. Compared with a linear meanline angle distribution, a design using nonlinear and non-monotonic meanline angle distribution is able to reduce the sonic boom ground loudness by over 20dBP1. The design achieves sonic boom ground loudness less than 70dBP1 and aerodynamic dynamic efficiency 1/D of 8.4. 2) Decreasing sweep angle within the Mach cone will increase 1/D as well as sonic boom. A design with variable sweep from 84° at the very leading edge to 68° at the tip achieves an extraordinarily high 1/D of 10.4 at Mach number 1.6 due to the low wave drag. If no sonic boom constraint is attached, SBiDir-FW concept still has a lot of room to increase the 1/D. 3) The round leading edge and trailing edge under high sweep angle are beneficial to improve aerodynamic performance, sonic boom, and to increase volume of the airplane. 4) Subsonic performance is benefited greatly from the high slenderness of supersonic configuration after rotating goo. A design with excellent supersonic aspect ratio of 0.44, 1/D of 8.g, gives an extraordinary subsonic aspect ration of 10 and 1/D of 1g.7. Two configurations are designed in details to install internal seats, landing gears, and engine installation to demonstrate the feasibility of SBiDir-FW configuration to accommodate all the required volume for realistic airplane. Here we emphasize that the qualitative findings in Phase I are very encouraging, more important than the quantitative results. Qualitative findings give the understanding of physics and provide the path to achieve the ultimate high performance design. The promising quantitative results achieved in Phase I need to be confirmed by wind tunnel testing in Phase II and ultimately proved by flight test. The other important step forward will be made to study the rotation transition from both CFD unsteady simulation and wind tunnel testing.

Zha, Gecheng↗

DOE OSTI2021

Adaptive hyperparameter updating for training restricted Boltzmann machines on quantum annealers

Restricted Boltzmann Machines (RBMs) have been proposed for developing neural networks for a variety of unsupervised machine learning applications such as image recognition, drug discovery, and materials design. The Boltzmann probability distribution is used as a model to identify network parameters by optimizing the likelihood of predicting an output given hidden states trained on available data. Training such networks often requires sampling over a large probability space that must be approximated during gradient based optimization. Quantum annealing has been proposed as a means to search this space more efficiently which has been experimentally investigated on D-Wave hardware. D-Wave implementation requires selection of an effective inverse temperature or hyperparameter (β) within the Boltzmann distribution which can strongly influence optimization. Here, we show how this parameter can be estimated as a hyperparameter applied to D-Wave hardware during neural network training by maximizing the likelihood or minimizing the Shannon entropy. We find both methods improve training RBMs based upon D-Wave hardware experimental validation on an image recognition problem. Neural network image reconstruction errors are evaluated using Bayesian uncertainty analysis which illustrate more than an order magnitude lower image reconstruction error using the maximum likelihood over manually optimizing the hyperparameter. The maximum likelihood method is also shown to out-perform minimizing the Shannon entropy for image reconstruction.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Gauge invariants of linearized gravity with a general background metric

In linearized gravity with distributed matter, the background metric has no generic symmetries, and decomposition of the metric perturbation into global normal modes is generally impractical. This complicates the identification of the gauge-invariant part of the perturbation, which is a concern, for example, in the theory of dispersive gravitational waves (GWs) whose energy–momentum must be gauge-invariant. Here, we propose how to identify the gauge-invariant part of the metric perturbation and the six independent gauge invariants per se for an arbitrary background metric. For the Minkowski background, the operator that projects the metric perturbation on the invariant subspace is proportional to the well-known dispersion operator of linear GWs in vacuum. For a general background, this operator is expressed in terms of the Green’s operator of the vacuum wave equation. Further, if the background is smooth, it can be found asymptotically using the inverse scale of the background metric as a small parameter.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗