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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Support of the Development of EPP Plus SMT Design Method (FY2020 Test Results)

Experiments in support of the development of the integrated Elastic–Perfectly Plastic (EPP) analysis and Simplified Model Test (SMT) design methodology continued in FY 2020. The previously developed single-bar SMT (SBSMT) technique was successfully extended to tubular shaped Alloy 617 specimens for evaluation of the effect of primary pressure load on SMT creep-fatigue (CF) properties at 950°C. A series of pressurized SMT tests were performed with various elastic follow-up factors and strain ranges. The results show that the primary pressure load reduces the SMT CF cycles. The reduction of SMT CF life due to primary load was found to be dependent on strain ranges, elastic follow up and test temperatures. These results are crucial for verification of the EPP strain range analysis in the EPP+SMT evaluation procedure. Additionally, hold time effect on the SMT CF life cycles is a determining factor for generating the EPP+SMT CF design curves. Lack of test data in the high cycle and low strain range region results in major uncertainty in developing the design curves. In FY2020, experiments were designed, and testing started on Alloy 617 at 950°C to fill the gap in experimental data. The on-going CF and additional SMT CF with elastic follow-up at low strain ranges will generate the information needed to verify the methods for extrapolation to long hold time and finalize the EPP+SMT design method.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
NASA NTRS1977

Single-axis attitude determination accuracy

This paper extends the analysis of attitude determination accuracy introduced at the 1975 American Astronautical Society/American Institute of Aeronautics and Astronautics (AAS/AIAA) Astrodynamics Conference. It contains more generalized expressions, new geometrical relations, and additional applications. A complete set of attitude accuracy equations for both arc-length (portion of a great circle) and rotation-angle (dihedral angle) measurements are presented. These expressions can then be applied to determine the geometrical conditions under which a specified accuracy can be achieved by using either the attitude or a selected reference-vector direction as the independent variable. Representative applications to attitude determination accuracy studies and launch window analyses are discussed. The clear physical interpretation and straightforward graphical procedures greatly simplify mission analysis, maneuver planning, hardware configuration studies, and interpretation of results.

Chen, L. C.↗
NASA NTRSDate not supplied

Traffic Flow Analysis for Package Delivery Drones using a Queueing Model

A key component of the small unmanned aircraft systems traffic management ecosystem is the design of scalable algorithms for strategic deconfliction of drones prior to takeoff. In this work, we focus on efficient flow management of drones on a network of intersecting edges subject to two kinds of spacing constraints: 1) between any two adjacent vehicles on an edge and 2) between any two vehicles on two different edges arriving one after the other at an intersection. The spacing is designed to enable non-intersection of operational volumes corresponding to two different vehicles thereby properly separating the vehicles inside each volume. For simplicity, we assume a constant ground speed for the drones and fixed dimensions for the operational volume blocks. The deconfliction is managed by adjusting the takeoff time of the drones, thereby regulating their arrival time at various crossing waypoints in the network. This framework allows us to study the maximum flow (throughput) of vehicles on a network of edges connecting depots to drop off sites subject to the temporal spacing constraints. The departure scheduling of individual drones results in a combinatorial optimization problem. To alleviate this, we solve a max-flow formulation and use queueing theory to simplify the analysis and provide upper bounds to the underlying optimization problem for individual drone departure scheduling. Our results indicate that throughput drops rapidly after the density of drones in the network passes the max-flow limits.

Alexey A Munishkin↗
NASA NTRS1961

Theory of Electrostatic Fields in the Ionosphere at Equatorial Latitudes

The properties of the elongated electrostatic fields that are required to provide the coupling mechanism in the dynamo-motor concept of the E and F regions of the ionosphere are examined theoretically for the conditions that prevail in equatorial latitudes. The analysis is developed for an electrostatic field of arbitrary horizontal scale in a horizontally stratified partly ionized gas subject to an imposed magnetic field having the form of a parabolic arch over the equator. The anisotropic character and continuous variation with height of the conductivity are retained throughout , and numerical solutions are determined for the attenuation of the electric field with distance along the field line. The results are similar qualitatively to those found previously upon analysis of the corresponding problem for middle latitudes, but the attenuation of the electrostatic field with height is considerably greater. It develops, in particular, that the coupling between E and F regions is very small for fields having horizontal wavelengths of a few kilometers. It follows that the dynamo-motor concept could not be used to account for the presence of irregularities of this scale. On the other hand, it is found that almost all the attenuation occurs at heights near that of the E region. If an electrostatic field having a horizontal wavelength of a few kilometers could be produced at heights of, say 200 km or greater, it follows that it would be very elongated and extend with little change in amplitude from hemisphere to hemisphere. The results also indicate that effective coupling could be achieved for fields having horizontal wavelengths of the order of tens of kilometers, but the assumptions introduced to simplify the analysis may impair the quantitative reliability of the results for fields of this scale.

Briggs, Benjamin R.↗
NASA NTRS2003

Torque Limit for Bolted Joint for Composites: TTTC Properties of Laminated Composites - Part A

The existing design code for torque limit of bolted joints for composites at Marshall Space Flight Center is MSFC-STD-486B, which was originally developed in 1960s for metallic materials. The theoretical basis for this code was a simplified mechanics analysis, which takes into account only the bolt, nut and washers, but not the structural members to be connected. The assumption was that metallic materials would not fail due to the bearing stress at the contact area between washer and the mechanical member. This is true for metallic materials; but for composite materials the results could be completely different. Unlike most metallic materials, laminated composite materials have superior mechanical properties (such as modulus and strength) in the in-plane direction, but not in the out-of-plane, or through-the-thickness (TTT) direction. During the torquing, TTT properties (particularly compressive modulus and compressive strength) play a dominant role in composite failure. Because of this concern, structural design engineers at Marshall are currently using a compromised empirical approach: using 50% of the torque value for composite members. Companies like Boeing is using a similar approach. An initial study was conducted last summer on this topic to develop theoretical model(s) that takes into consideration of composite members. Two simplified models were developed based on stress failure criterion and strain failure criterion, respective. However, these models could not be used to predict the torque limit because of the unavailability of material data, specifically, through-the-thickness compression (TTTC) modulus and strength. Therefore, the task for this summer is to experimentally determine the TTTC properties. Due to the time limitation, only one material has been tested: IM7/8552 with [0 degrees,plus or minus 45 degrees, 90 degree ] configuration. This report focuses the test results and their significance, while the experimentation will be described in a separate report by Mr. Kris Kostreva.

Zhao, Yi↗
DOE OSTI2023

An Experimental Investigation of Students? Learning Effects When Using a Simplified Nuclear Simulator

This study focuses on investigating students' learning effects and performance trends over a certain period using the Rancor Microworld Simulator. Specifically, it aims to determine the training required to collect HRA data from non-experts (i.e., students) using Rancor Microworld and the differences in human performance measures between students and professional operators. A longitudinal experiment is conducted with sixteen undergraduate students, using four Rancor Microworld scenarios in each of the four experiment trials. The study considers four human performance measurements workload, situation awareness, time, and error. Finally, the trend of students' performance is compared with operator data collected from the previous experiment. Overall, this research complements previous studies by providing insights into how much training is required to collect HRA data from non-experts and the differences in human performance measures between students and professional operators.

99 GENERAL AND MISCELLANEOUS↗
NASA NTRS2011

Z-Pinch Magneto-Inertial Fusion Propulsion Engine Design Concept

Fusion-based nuclear propulsion has the potential to enable fast interplanetary transportation. Due to the great distances between the planets of our solar system and the harmful radiation environment of interplanetary space, high specific impulse (Isp) propulsion in vehicles with high payload mass fractions must be developed to provide practical and safe vehicles for human spaceflight missions. Magneto-Inertial Fusion (MIF) is an approach which has been shown to potentially lead to a low cost, small fusion reactor/engine assembly (1). The Z-Pinch dense plasma focus method is an MIF concept in which a column of gas is compressed to thermonuclear conditions by an estimated axial current of approximately 100 MA. Recent advancements in experiments and the theoretical understanding of this concept suggest favorable scaling of fusion power output yield as I(sup 4) (2). The magnetic field resulting from the large current compresses the plasma to fusion conditions, and this is repeated over short timescales (10(exp -6) sec). This plasma formation is widely used in the field of Nuclear Weapons Effects (NWE) testing in the defense industry, as well as in fusion energy research. There is a wealth of literature characterizing Z-Pinch physics and existing models (3-5). In order to be useful in engineering analysis, a simplified Z-Pinch fusion thermodynamic model was developed to determine the quantity of plasma, plasma temperature, rate of expansion, energy production, etc. to calculate the parameters that characterize a propulsion system. The amount of nuclear fuel per pulse, mixture ratio of the D-T and nozzle liner propellant, and assumptions about the efficiency of the engine, enabled the sizing of the propulsion system and resulted in an estimate of the thrust and Isp of a Z-Pinch fusion propulsion system for the concept vehicle. MIF requires a magnetic nozzle to contain and direct the nuclear pulses, as well as a robust structure and radiation shielding. The structure, configuration, and materials of the nozzle must meet many severe requirements. The configuration would focus, in a conical manner, the Deuterium-Tritium (D-T) fuel and Lithium-6/7 liner fluid to meet at a specific point that acts as a cathode so the Li-6 can serve as a current return path to complete the circuit. In addition to serving as a current return path, the Li liner also serves as a radiation shield. The advantage to this configuration is the reaction between neutrons and Li-6 results in the production of additional Tritium, thus adding further fuel to the fusion reaction and boosting the energy output. To understand the applicability of Z-Pinch propulsion to interplanetary travel, it is necessary to design a concept vehicle that uses it. The propulsion system significantly impacts the design of the electrical, thermal control, avionics, radiation shielding, and structural subsystems of a vehicle. The design reference mission is the transport of crew and cargo to Mars and back, with the intention that the vehicle be reused for other missions. Several aspects of this vehicle are based on a previous crewed fusion vehicle study called Human Outer Planet Exploration (HOPE), which employed a Magnetized Target Fusion (MTF) propulsion concept. Analysis of this propulsion system concludes that a 40-fold increase of Isp over chemical propulsion is predicted. This along with a greater than 30% predicted payload mass fraction certainly warrants further development of enabling technologies. The vehicle is designed for multiple interplanetary missions and conceivably may be suited for an automated one-way interstellar voyage.

Miernik, Janie H.↗
DOE OSTI2021

Up‐And‐Coming Advances in Optical and Microwave Nonreciprocity: From Classical to Quantum Realm

Reciprocity is a fundamental physical principle that roots in the time‐reversal symmetry of physical laws. It allows making predictions on any arbitrary complex system's response and operation and hence simplifies the analysis. However, there are many practical situations in which it is advantageous to break reciprocity, e.g., isolators preventing wave scattering back to lasers and generators, full‐duplex systems for multiplexing transmission and receiving in the same channel, nonreciprocal cavity excitation, and protection of fragile states of superconductor quantum computers from thermal noise. The most widespread approach to time‐reversal symmetry breaking and nonreciprocity based on magnetic field biasing suffers from bulkiness, cost ineffectiveness, and loss, motivating researchers and engineers to search for more practical approaches. Herein, the up‐and‐coming advances in optical nonreciprocity, including new materials (Weyl semimetals, topological insulators, metasurfaces), active structures, time‐modulation, parity‐time (PT)‐symmetry breaking, nonlinearity combined with a structural asymmetry, quantum nonlinearity, unidirectional gain and loss, chiral quantum states and valley polarization are overviewed. A general description of nonreciprocal systems is provided and the pros and cons of the mentioned approaches to nonreciprocity are discussed.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

Simulation-based anomaly detection for multileptons at the LHC

Decays of Higgs boson-like particles into multileptons is a well-motivated process for investigating physics beyond the Standard Model (SM). A unique feature of this final state is the precision with which the SM is known. As a result, simulations are used directly to estimate the background. Current searches consider specific models and typically focus on those with a single free parameter to simplify the analysis and interpretation. In this paper, we explore recent proposals for signal model agnostic searches using machine learning in the multilepton final state. These tools can be used to simultaneously search for many models, some of which have no dedicated search at the Large Hadron Collider. We find that the machine learning methods offer broad coverage across parameter space beyond where current searches are sensitive, with a necessary loss of performance compared to dedicated searches by only about one order of magnitude.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2025

Temperature-Dependent Mechanical Properties of Ni-Based Concentrated Alloys: Insights from First-Principles Calculations

The present work focuses on predicting temperature-dependent mechanical properties of Ni-based concentrated alloys Ni 18 Cr 10 Co 10 Fe 6 M 4 (abbreviated by X 44 M 4 , with M = Al, V, Mn, Fe, Nb, Mo, and W) using density functional theory (DFT). These predictions are based on shear (plastic) and elastic deformations, utilizing the special quasirandom structure (SQS), the phonon-based quasiharmonic approach (QHA), and the quasistatic approach. The resulting properties include coefficient of thermal expansion via QHA, ideal shear strength (τ IS ), and stable and unstable stacking fault energies (γ SF and γ US ) through pure alias shear deformation, and elastic constants (c ij ), bulk modulus (B 0 ), and shear modules (G 0 ) via elastic deformation. Notably, predicting accurate γ SF is challenging due to uncertainties that can exceed the γ SF values. τ IS and γ US exhibit a strong linear relationship, enabling the accurate prediction of γ US based on the precisely determined τ IS . All mechanical properties of X 44 M 4 decrease with increasing temperature, except for some γ SF cases such as X 44 M 4 with M = V, Mn, Fe, Mo, and W. Among the X 44 M 4 alloys, X 44 Nb 4 exhibits the lowest τ IS , γ US , and G 0 values, and the highest B 0 /G 0 ratio, while X 44 Mn 4 has the lowest B 0 and B 0 /G 0 ratio. We found that volume is a crucial descriptor for understanding and modeling mechanical properties (except B0 and maybe also γ SF ) affected by alloying elements and temperature. Ni-based dilute alloys (e.g., Ni 11 M 1 and Ni 31 M 1 ) and concentrated alloys (e.g., X 44 M 4 ) show similar trends in mechanical properties influenced by alloying elements and temperature, simplifying the analysis and design of Ni-based alloys.

Elastic properties↗
DOE OSTI2021

Empirical validation and comparison of methodologies to simulate micro and macro-encapsulated PCMs in the building envelope

Thermal Energy Storage (TES) has the potential to shift peak electricity demand. Passive TES is usually implemented in building envelope as micro and macro encapsulated phase change materials (PCM) to shift electric energy demand and therefore requires careful heat transfer analysis. Whole building energy modelling with simplified heat transfer analysis has become extremely important for designers, architects, engineers, and researchers to predict energy performance of buildings. It is important to validate PCM modelling algorithms used in building energy programs to quantify their error and prove their capacity to model different PCM encapsulation types. This study uses data from a microencapsulated PCM and two macroencapsulated PCMs (Bio based PCM and hydrate salts) tested in full-scale using the Advanced Multiscale Building Energy Research (AMBER) Lab located at the Colorado School of Mines and is used to validate a numerical algorithm written in MATLAB language. To approximate the heat transfer through a wall assembly with macroencapsulated PCM pouches, several modelling techniques that can reduce 3D heat transfer characteristics to 1D are explored in this research. A parallel path heat transfer modelling approach is found to give the closest agreement with the experimental data for the pouched PCMs in building envelope applications.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2024

Health and air pollutant emission impacts of net zero CO2 by 2050 scenarios from the energy modeling forum 37 study

Carbon dioxide and non-greenhouse gas air pollutants are emitted from many of the same sources. Decarbonization actions thus typically yield air pollutant emission reductions, resulting in significant air quality benefits. Although several studies have highlighted this connection, including in the context of net zero carbon emission targets, substantial uncertainty remains regarding how alternative technological pathways to this goal will affect the spatial distribution and magnitude of air pollutants. Comprehensive multi-model and multi-scenario analyzes are needed to explore the relative impacts of alternative pathways. Here, our study begins to address this gap by leveraging the results from the recent Energy Modeling Forum 37 inter-model comparison exercise on U.S. decarbonization pathways. Comparing the results of the six teams who submitted air pollutant emissions suggests that strategies that target net zero U.S. carbon emissions would yield significant reductions in many air pollutants, and that this finding is generally robust across pathways. However, some energy sources, such as biomass and fossil fuels with carbon capture, will emit air pollutants and can potentially influence the magnitude, spatial distribution, and even sign of localized air pollutant emission changes. In the second part of this analysis, a simplified air quality and health impacts screening model is used to evaluate the air quality impacts in 2035 of sectoral emission changes from the three models that provided sectoral detail. Relative to a reference scenario, a net zero pathway is estimated to reduce fine particulate matter concentrations across the contiguous U.S., with health benefits from reduced mortality ranging from $\$$65 billion to $\$$250 billion in 2035 alone (2023$\$$s). These benefits would be expected to grow over time as the net zero trajectory becomes more stringent. Both the magnitude of potential benefits and the substantial variation of the projections across models underscore the need for an EMF-like inter-model comparison exercise focused on air quality.

Air pollutants↗
DOE OSTI2020

Near-axis expansion of stellarator equilibrium at arbitrary order in the distance to the axis

A direct construction of equilibrium magnetic fields with toroidal topology at arbitrary order in the distance from the magnetic axis is carried out, yielding an analytical framework able to explore the landscape of possible magnetic flux surfaces in the vicinity of the axis. This framework can provide meaningful analytical insight into the character of high-aspect-ratio stellarator shapes, such as the dependence of the rotational transform and the plasma beta limit on geometrical properties of the resulting flux surfaces. The approach developed here is based on an asymptotic expansion on the inverse aspect ratio of the ideal magnetohydrodynamics equation. The analysis is simplified by using an orthogonal coordinate system relative to the Frenet–Serret frame at the magnetic axis. The magnetic field vector, the toroidal magnetic flux, the current density, the field line label and the rotational transform are derived at arbitrary order in the expansion parameter. Moreover, a comparison with a near-axis expansion formalism employing an inverse coordinate method based on Boozer coordinates (the so-called Garren–Boozer construction) is made, where both methods are shown to agree at lowest order. Finally, as a practical example, a numerical solution using a W7-X equilibrium is presented, and a comparison between the lowest-order solution and the W7-X magnetic field is performed.

Physics↗
DOE OSTI2021

Quasioptical modeling of wave beams with and without mode conversion. IV. Numerical simulations of waves in dissipative media

We report the first quasi-optical simulations of wave beams in a hot plasma using the quasi-optical code PARADE (PAraxial RAy DEscription) [K. Yanagihara, I. Y. Dodin, and S. Kubo, Phys. Plasmas 26, 072112 (2019)]. This code is unique in that it accounts for inhomogeneity of the dissipation-rate across the beam and mode conversion simultaneously. We show that the dissipation-rate inhomogeneity shifts beams relative to their trajectories in cold plasma and that the two electromagnetic modes are coupled via this process, an effect that was ignored in the past. Additionally, we propose a simplified approach to account for the dissipation-rate inhomogeneity. This approach is computationally inexpensive and simplifies the analysis of actual experiments.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

Thermo-Mechanical Distortion of Tungsten-Coated Steel During High Heat Flux Testing Using Plasma Arc Lamps

An experimental setup and a test section were designed and fabricated for high heat flux testing (HHFT) of neutron-irradiated specimens using water-wall plasma arc lamps. Because of the radiological considerations and limitations of reactor irradiation, the size of the test articles was limited to disks less than 10 mm in diameter. The specimen was clamped onto an actively cooled block, and clamping allowed the insertion of several thermocouples on the back surface of the specimen through a copper (Cu) block. Five vacuum plasma sprayed tungsten (W)–coated F82H steel specimens were subjected to HHFT. Surface profilometry measurements, which were conducted after HHFT, revealed central bowing of the top W surface. This type of residual distortion occurred for all of the specimens, and the larger the specimens were, the larger was the distortion.In an attempt to understand specimen distortion and address the science questions related to the testing of subsize specimens during HHFT, a simplified thermo-mechanical model was developed. By using a measured temperature in the Cu as an isothermal boundary condition, the model eliminated the need for coupling cooling fluid flow models with stress models, greatly simplifying the analysis. The main variable in the proposed model is hC, i.e., the thermal contact conductance between the F82H and the Cu washer. Inelastic properties, including hardening properties, were considered for F82H steel and Cu. Numerical simulation results demonstrated a buildup of residual deformation during HHFT and a very complex state of stress and deformation during typical heat flux (HF) cycling. Additionally, hoop stress evolution during a high heat flux cycle reveals that F82H at an interface with W would be mainly in compression during HF application and experienced a transition to a tension state during cooldown. Also, specimen distortion evolves during each HF cycle, as the specimen bows downward during HF application and upward during the cooldown period between HF cycles. The final specimen distortion, i.e., upward bowing of the specimen center, was qualitatively predicted for hC values of 4000 to 5000 W/(m 2 ·K). This hC range of values, for which bulging is obtained, is at the lower spectrum of the range of values for hC, consistent with the low thermal contact conductance expected from the unpolished F82H surface.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2021

AC losses of Roebel and CORC ® cables at higher AC magnetic fields and ramp rates

We have measured ReBCO coated conductor-based conductor on round core (CORC ® ) and Roebel cables at 77 K in a spinning magnet calorimeter, which subjected the tapes in the samples to a radial magnetic field of 566 mT (peak) at frequencies up to 120 Hz (272 T s -1 , cyclic average) with an approximately sinusoidal waveform. The samples were oriented such that the field applied to the tapes within the cables was entirely radial, simplifying subsequent analysis. An expression for loss which included hysteretic, flux creep and eddy current losses was fit to both the CORC ® and the Roebel cables. This expression allowed easy comparison of the relative influence of eddy currents and flux creep (or power-law behavior) effects. The loss of both the CORC ® and Roebel cables measured here were seen to be essentially the sum of the hysteretic loss, flux creep effects, and the normal metal eddy current losses of the individual tapes. The losses of these cables were measured at high B × d B /d t with no coupling current loss observed under the present preparation conditions. The influence of flux creep effects on loss were not negligible. The losses of the CORC ® cable per meter of tape were seen to be reduced from the case of a flat tape because of the helical geometry of the tapes.

Physics↗
DOE OSTI2024

Not even 6 dB: Gaussian quantum illumination in thermal background

Abstract In analyses of target detection with Gaussian state transmitters in a thermal background, the thermal occupation is taken to depend on the target reflectivity in a way which simplifies the analysis of the symmetric quantum hypothesis testing problem. However, this assumption precludes comparison of target detection performance between an arbitrary transmitter and a vacuum state transmitter, i.e. ‘detection without illumination’, which is relevant in a bright thermal background because a target can be detected by its optical shadow or some other perturbation of the background. Using a target-agnostic thermal environment leads to the result that the oft-claimed 6 dB possible reduction in the quantum Chernoff exponent for a two-mode squeezed vacuum transmitter over a coherent state transmitter in high-occupation thermal background is an unachievable limiting value, only occurring in a limit in which the target detection problem is ill-posed. Further analyzing quantum illumination in a target-agnostic thermal environment shows that a weak single-mode squeezed transmitter performs worse than ‘no illumination’, which is explained by the noise-increasing property of reflected low-intensity squeezed light.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

Mixing bispectrum multipoles under geometric distortions

We derive general expressions for how the Alcock–Paczynski distortions affect the power spectrum and the bispectrum of cosmological fields. We compute explicit formulas for the mixing coefficients of bispectrum multipoles in the linear approximation. The leading-order effect for the bispectrum is the uniform dilation of all three wavevectors. The mixing coefficients depend on the shape of the bispectrum triplet. Our results for the bispectrum multipoles are framed in terms of the ‘natural’ basis of the lengths of three wavevectors but can be easily generalized for other bases and reduction schemes. Our validation tests confirm that the linear approximation is extremely accurate for all power spectrum multipoles. The linear approximation is accurate for the bispectrum monopole but results in sub-per cent level inaccuracies for the bispectrum quadrupole and fails for the bispectrum hexadecapole. Our results can be used to simplify the analysis of the bispectrum from galaxy surveys, especially the measurement of the baryon acoustic oscillation peak position. They can be used to replace numeric schemes with exact analytical formulae.

79 ASTRONOMY AND ASTROPHYSICS↗