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All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

Multi-scale VLBI observations of the candidate host galaxy of GRB 200716C

We present the discovery and the subsequent follow up of radio emission from SDSS J130402.36+293840.6 (J1304+2938), the candidate host galaxy of the gamma-ray burst (GRB) GRB 200716C. The galaxy is detected in the RACS (0.89 GHz), the NVSS, the Apertif imaging survey, and the FIRST (1.4 GHz), the VLASS (3 GHz), and in public LOFAR (130–170 MHz), WISE (3.4–22 μm), and SDSS (z, i, r, g, u filters) data. The luminosity inferred at 1.4 GHz is (5.1 ± 0.2) × 10 30 erg s -1 Hz -1 . To characterise the emission and distinguish between different components within the galaxy, we performed dedicated, high-sensitivity and high-resolution observations with the European VLBI Network (EVN) + e-MERLIN at 1.6 and 5 GHz. We did not detect any emission from a compact core, suggesting that the presence of a radio-loud active galactic nucleus (AGN) is unlikely, and therefore we ascribe the emission observed in the public surveys to star-forming regions within the galaxy. We confirm and refine the redshift estimate, z = 0.341 ± 0.004, with a dedicated Telescopio Nazionale Galileo (TNG) spectroscopic observation. Finally, we compiled a list of all the known hosts of GRB afterglows detected in radio and computed the corresponding radio luminosity: if GRB 200716C belongs to J1304+2938, this is the third most radio-luminous host of a GRB, implying one of the highest star-formation rates (SFRs) currently known, namely SFR ~ 324±61 M ⊙ yr -1 . On the other hand, through the analysis of the prompt emission light curve, recent works suggest that GRB 200716C might be a short-duration GRB located beyond J1304+2938 and gravitationally lensed by an intermediate-mass black hole (IMBH) hosted by the galaxy. Neither the public data nor our Very Long Baseline Interferometry (VLBI) observations can confirm or rule out the presence of an IMBH acting as a (milli-)lens hosted by the galaxy, a scenario still compatible with the set of radio observations presented in this work.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

Active galactic nucleus feedback in NGC 3982

The energetic feedback from supermassive black holes can influence star formation at the centres of galaxies. Observational evidence for active galactic nucleus (AGN) impact on star formation can be searched for in galaxies by combining ultraviolet imaging and optical integral field unit data. The ultraviolet flux directly traces recent star formation, and the integral field unit data can reveal dust attenuation, gas ionisation mechanisms, and gas kinematics from the central regions of the galaxy disk. A pilot study on NGC 3982 shows star formation suppression in the central regions of the galaxy, likely due to negative AGN feedback, and enhanced star formation in the outer regions. The case of NGC 3982 could be observational evidence of AGN feedback operating in a Seyfert galaxy.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

The SPAN512 mid-latitude pulsar survey at the Nançay Radio Telescope

The large number of ongoing surveys for pulsars and transients at various radio observatories is motivated by the science obtained from these sources. Timing and polarisation analysis of relativistic binaries can place strong constraints on theories of gravity. The observation of a growing number of millisecond pulsars (MSPs) spread over the celestial sphere may allow the detection of a stochastic gravitational wave background arising from supermassive black hole binaries. A more complete sample of young pulsars improves our knowledge of neutron star birth and evolution. Transients such as fast radio bursts can serve to probe the intergalactic medium. The SPAN512 pulsar survey covers intermediate Galactic latitudes using the L-band receiver of the Nançay Radio Telescope (NRT). The survey covers 224 sq. deg. of the sky for a total exposure time of 2200 h. Population syntheses predict the discovery of 3 to 19 new normal pulsars and a few MSPs. We present detailed modelling of the NRT beam with its L-band receiver and its sensitivity which we used to precisely assess the expected survey yield. We used the flexible Pulsar Arecibo L-band Feed Array data processing pipeline to search the 47 TB of SPAN512 data for pulsars and transients. The SPAN512 survey discovered two new MSPs and one new middle-aged pulsar. We focus on the analysis of the 2.4-ms spin period pulsar J2205+6012 for which we also report the detection of gamma-ray pulsations. Its narrow pulse width (35µs at an observing frequency of 2.55 GHz) allows for sub-microsecond timing precision over 8 yr, with exciting prospects for pulsar timing array programs.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

Gaia -supported re-discovery of a remarkable weak line quasar from a variability and proper motion survey

We demonstrate that VPMS J170850.95+433223.7 is a weak line quasar (WLQ) which is remarkable in several respects. It was already classified as a probable quasar two decades ago, but with considerable uncertainty. The non-significant proper motion and parallax from the Gaia Early Data Release 3 have solidified this assumption. Based on previously unpublished spectra, we show that VPMS J170850.95+433223.7 is a WLQ at z = 2.345 with immeasurably faint broad emission lines in the rest-frame ultraviolet. A preliminary estimate suggests that it hosts a supermassive black hole of ~10 9 M ⊙ accreting close to the Eddington limit, perhaps at the super-Eddington level. We identify two absorber systems with blueward velocity offsets of 0.05 c and 0.1 c , which could represent high-velocity outflows, which are perhaps related to the high accretion state of the quasar.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

Classifying the full SDSS-IV MaNGA Survey using optical diagnostic diagrams: Presentation of AGN catalogs in flexible apertures

Accurate active galactic nucleus (AGN) identifications in large galaxy samples are crucial for the assessment of the role of AGN and AGN feedback in the co-evolution of galaxies and their central supermassive black holes. Emission-line flux-ratio diagnostics are commonly used to identify AGN in optical spectra. New large samples of integral field unit observations allow exploration of the role of aperture size in the classification process. In this paper, we present galaxy classifications for all 10010 galaxies observed within the Mapping Nearby Galaxies at Apache Point Observatory (MaNGA) survey. We use Baldwin-Philips-Terlevich line flux-ratio diagnostics combined with an Hα equivalent threshold in 60 apertures of varying size for the classification, and provide the corresponding catalogs. MaNGA-selected AGN primarily lie below the main sequence of star-forming galaxies, and reside in massive galaxies with stellar masses of ~10 11 M ⊙ and a median Hα-derived star formation rate of ~1.44M ⊙ yr -1 . We find that the number of “fake” AGN increases significantly beyond selection apertures of >1.0 R eff because of increased contamination from diffuse ionized gas (DIG). A comparison with previous works shows that the treatment of the underlying stellar continuum and flux measurements can significantly impact galaxy classification. Our work provides the community with AGN catalogs and galaxy classifications for the full MaNGA survey.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

The impact of environmental effects on active galactic nuclei: A decline in the incidence of ionized outflows

Active galactic nuclei (AGNs) have generally been considered to be less frequent in denser environments due to a lower number of galaxy-galaxy interactions and/or the removal of their gas-rich reservoirs by the dense intergalactic medium. However, recent observational and theoretical works suggest that the effect of ram-pressure stripping acting on galaxies in dense environments might reduce the angular momentum of their gas, causing it to infall towards the super massive black hole at their centre, activating the AGN phase. In this work we explore the connection between environment and nuclear activity by evaluating the variation in the incidence of ionized outflows, a common phenomenon associated with nuclear activity, in AGNs across different environments. We select a sample of approximately 3300 optical AGNs from the Sloan Digital Sky Survey Data Release 13, which we match with a group catalogue to identify galaxies in isolation or residing in groups. We further probe their environments through the projected distance to the central galaxy of the group or cluster and the projected surface density to the fifth neighbour (δ 5 ). The presence of ionized outflows is determined through the modelling of the [O III] λ5007 emission line. We find that at lower masses (< 10 10.3 M ⊙ ), the fraction of ionized outflows is significantly lower in satellite AGNs (~7%) than in isolated (~22%) AGNs, probably due to their different AGN luminosity, L [O III] , in this stellar mass range. The fraction of outflows decreases at distances closer to the central galaxy of the group or cluster for all satellite AGNs; however, only the lower-mass ones (10 9 - 10 10.3 M ⊙ ) display a significant decline with δ 5 . Although this study does not include AGNs in the densest regions of galaxy clusters, our findings suggest that AGNs in dense environments accrete less gas than those in the field, potentially due to the removal of the gas reservoirs via stripping or starvation, leading to a negative connection between environment and AGN activity. Based on our results, we propose that the observed change in the incidence of outflows when moving towards denser regions of groups and clusters could contribute to the higher gas metallicities of cluster galaxies compared to field galaxies, especially at lower masses.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

Alignments in the orientation of radio jets from AGN and their host galaxies

It is well established that active galactic nuclei (AGNs) play an important role in the evolution of galaxies. These AGNs can be linked to the accretion processes onto massive black holes and past merger events in their host galaxies, which may lead to different alignments of the jets with respect to the host galaxies. This paper presents a study of the position angle (PA) differences between radio and optical images of radio-AGNs based on the second data release (DR2) of the Low Frequency Array (LOFAR) Two-Meter Sky Survey (LoTSS), theKarl G. JanskyVery Large Array Faint Images of the Radio Sky at Twenty-Centimeters Survey (FIRST), the Dark Energy Spectroscopic Instrument (DESI) Legacy Imaging Surveys, and the Sloan Digital Sky Survey (SDSS). We assessed PA measurement biases in the data and classified the radio-AGNs based on the radio luminosity and infrared colour from the Wide-field Infrared Survey Explorer (WISE). This resulted in the largest sample of radio AGNs with reliable radio and optical PA measurements published to date, with a total of 3682 AGNs. The PA difference (dPA) distributions for the radio-AGN sample show a prominent minor-axis alignment tendency. Based on some simple assumptions, we simulated the projection effect to estimate the intrinsic jet-galaxy alignment. The observed dPA distribution can be well described by a two-component jet-alignment model in which one component is more aligned with the minor axis of the host galaxy than the other. The fitting results indicate that the jet alignment is dependent on radio luminosity and the shape of the host galaxies, with the jets being more likely to be aligned with the minor axis of the galaxy for lower radio luminosity and for optically more elongated radio-AGNs. The minor-axis alignment of the entire sample may suggest a coherent accretion model present in most AGN host galaxies, while a considerable number of luminous radio-AGNs with massive host galaxies might have undergone accretion – according to the chaotic model – or past merger events.

Astronomy & Astrophysics↗
DOE OSTI2024

A horseshoe-shaped ring of diffuse emission detected at 1.4 GHz

We identify a horseshoe-shaped ring (HSR) of diffuse emission in J1407+0453 from the Faint Images of Radio Sky at Twenty-cm (FIRST) survey using the Very Large Array telescope at 1.4 GHz. An optical galaxy, SDSS J140709.01+045302.1, is present near the limb of the HSR of J1407+0453, with a spectroscopic redshift ofz = 0.13360. The total extent of the source, including the diffuse emission of J1407+0453, is 65 arcsec (with a physical extent of 160 kpc), whereas the diameter of the HSR is approximately 10 arcsec (25 kpc). The flux density of the HSR is ∼47 mJy at 1400 MHz, whereas the flux densities of the whole diffuse emission of J1407+0453 at 1400 MHz and 150 MHz are 172 mJy and 763 mJy, respectively. We measure the radio luminosity of HSR J1407+0453 as 1.94 × 10 24 W Hz −1 , with a spectral index,α 150 1400 = −0.67. The black hole mass of J1407+0453 is 5.8 × 10 8 M ⊙ . We compare the HSR of diffuse emission of J1407+0453 with other discovered diffuse circular sources. The possible formation scenarios for J1407+0453 are discussed, so as to understand the nature of the source. We present a spectral index map of source J147+0453 to study the spectral properties of the source.

Astronomy & Astrophysics↗
DOE OSTI2024

Evidence of a toroidal magnetic field in the core of 3C 84

The spatial scales of relativistic radio jets, probed by relativistic magneto-hydrodynamic (RMHD) jet launching simulations and by most very long baseline interferometry (VLBI) observations differ by an order of magnitude. Bridging the gap between these RMHD simulations and VLBI observations requires selecting nearby active galactic nuclei (AGN), the parsec-scale region of which can be resolved. The radio source 3C 84 is a nearby bright AGN fulfilling the necessary requirements: it is launching a powerful, relativistic jet powered by a central supermassive black hole, while also being very bright. Using 22 GHz globe-spanning VLBI measurements of 3C 84 we studied its sub-parsec region in both total intensity and linear polarisation to explore the properties of this jet, with a linear resolution of ~0.1 parsec. We tested different simulation set-ups by altering the bulk Lorentz factor Γ of the jet, as well as the magnetic field configuration (toroidal, poloidal, helical). We confirm the persistence of a limb brightened structure, which reaches deep into the sub-parsec region. The corresponding electric vector position angles (EVPAs) follow the bulk jet flow inside but tend to be orthogonal to it near the edges. Our state-of-the-art RMHD simulations show that this geometry is consistent with a spine-sheath model, associated with a mildly relativistic flow and a toroidal magnetic field configuration.

3C 84 (NGC 1275)↗
DOE OSTI2024

CIRCLEZ : Reliable photometric redshifts for active galactic nuclei computed solely using photometry from Legacy Survey Imaging for DESI

Photometric redshifts for galaxies hosting an accreting supermassive black hole in their center, known as active galactic nuclei (AGNs), are notoriously challenging. At present, they are most optimally computed via spectral energy distribution (SED) fittings, assuming that deep photometry for many wavelengths is available. However, for AGNs detected from all-sky surveys, the photometry is limited and provided by a range of instruments and studies. This makes the task of homogenizing the data challenging, presenting a dramatic drawback for the millions of AGNs that wide surveys such as SRG/eROSITA are poised to detect. This work aims to compute reliable photometric redshifts for X-ray-detected AGNs using only one dataset that covers a large area: the tenth data release of the Imaging Legacy Survey (LS10) for DESI. LS10 provides deep grizW1-W4 forced photometry within various apertures over the footprint of the eROSITA-DE survey, which avoids issues related to the cross-calibration of surveys. We present the results from CIRCLEZ, a machine-learning algorithm based on a fully connected neural network. CIRCLEZ is built on a training sample of 14 000 X-ray-detected AGNs and utilizes multi-aperture photometry, mapping the light distribution of the sources. The accuracy (σNMAD) and the fraction of outliers (η) reached in a test sample of 2913 AGNs are equal to 0.067 and 11.6%, respectively. The results are comparable to (or even better than) what was previously obtained for the same field, but with much less effort in this instance. We further tested the stability of the results by computing the photometric redshifts for the sources detected in CSC2 and Chandra-COSMOS Legacy, reaching a comparable accuracy as in eFEDS when limiting the magnitude of the counterparts to the depth of LS10. The method can be applied to fainter samples of AGNs using deeper optical data from future surveys (for example, LSST, Euclid), granting LS10-like information on the light distribution beyond the morphological type. Along with this paper, we have released an updated version of the photometric redshifts (including errors and probability distribution functions) for eROSITA/eFEDS.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

The advanced evolution of massive stars

The nuclear rates for reactions involving 12 C and 16 O are key to computing the energy release and nucleosynthesis evolution of massive stars during their advanced burning phases. Ultimately, these burning rates shape the stellar structure and evolution and influence the nature of the compact objects produced at the end of the stellar life. We explore the implications of new nuclear reaction rates from both experimental and theoretical studies for 12 C(α, γ) 16 ​O, 12 C+ 12​ C, 12 C+ 16 ​O, and 16 O+ 16 ​O reactions for massive stars. Our goal is to investigate how the chemical structure and nucleosynthesis evolve from the He-exhaustion stage to the O-burning phase and how these processes influence the ultimate stellar fate. We computed rotating and non-rotating models for stars of different masses at solar metallicity. We used the stellar evolution code GENEC, which includes a large network of nuclear reactions and isotopes involved in advanced phases, as well as updated rates for 12 C(α,γ) 16 O. For the three fusion reactions involving 12 C and 16 O, we considered new rates following a data-driven fusion suppression scenario (hereafter HIN(RES)) and new theoretical rates obtained with time-dependent Hartree-Fock (TDHF) calculations. The updated 12 C(α, γ) 16 ​O rates mainly impact the chemical structure evolution changing the 12 C/ 16 O ratio at He-exhaustion and have little effect on the CO core mass. This variation in the 12 C/ 16 O ratio is in some cases critical for predicting the final fate of the model, which is very sensitive to 12 C abundance, and in particular the 20 M ⊙ remnant may change from a black hole to a neutron star. The He-burning (C-burning) lifetime is also decreased (increased) by about −2% (+15%). The combined new rates for 12 C+ 12 ​C and 16 O+ 16 ​O fusion reactions according to the HIN(RES) model lead to shorter C- and O-burning lifetimes by ≈ − 10%, and −50%, respectively, and shift the ignition conditions to higher temperatures and densities. In contrast, the theoretical TDHF rates primarily affect C-burning, increasing its duration by about 30% and lowering the ignition temperature. These changes modify the chemical structure of the core, the size and duration of C-burning shells, and hence their compactness. They also impact the central and shell nucleosynthesis (by ±1 dex and by factors of ±2–10, respectively), while 12 C+ 16 ​O reaction rates variations remain the least important.

abundances↗
DOE OSTI2026

Astrometric microlensing probes of the isolated neutron star population with Roman

Notoriously hard to detect and study, isolated neutron stars (NSs) might provide valuable answers to fundamental questions about stellar evolution and explosion physics. With the upcoming Roman Space Telescope, scheduled for launch in 2026, a new and powerful channel for their detection will become available: astrometric microlensing. We set out to create a realistic sample of simulated gravitational microlensing events as observed by Roman with the Galactic Bulge Time Domain Survey. We focus in particular on the population of NS lenses, which has until now been largely understudied. We used dedicated Galactic models tailored for application to microlensing by compact objects. In addition to populations of stars, white dwarfs, and black holes, we simulated four different NS populations with Maxwellian natal kick distributions: v = (150, 250, 350, 450) km/s. For each simulation, we applied projected Roman precision, cadence, and detectability criteria. We found that the parameter space log10 t E –log 10 θ E , which will be accessible to Roman observations, is efficient for the classification of stellar remnants. We found a feature in this space that is characteristic of NSs; using this feature, optimal samples of NS candidates can be constructed from Roman -like datasets. We describe the dependence of the observable parameter distributions on the assumed mean kick velocities. As the effects of natal kicks are very complex and mutually counteracting, we suggest that more detailed studies focused on the dynamics of NSs are needed in anticipation of Roman and future surveys. We estimate that Roman will observe approximately 11 000 microlensing events, including ~100 with NS lenses, whose photometric and astrometric signals are detectable; the event yield decreases by 38% when gap-filling low-cadence observations are not included. We make all simulated microlensing event datasets publicly available in preparation for Roman data.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

Wee partons in QCD and gravity: Double copy and universality

We discuss a quantitative “double copy” between radiation from shockwave collisions in Einstein gravity and in QCD. The correspondence extends to 2 → N amplitudes in Regge asymptotics. The classicalization and unitarization of these amplitudes at maximal occupancy, corresponding to black hole and Color Glass Condensate (CGC) states respectively, are described by the emergent Goldstone dynamics of wee partons. We outline some consequences of the universal dynamics on both sides of the correspondence.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2023

Advanced data analysis in inertial confinement fusion and high energy density physics

Bayesian analysis enables flexible and rigorous definition of statistical model assumptions with well-characterized propagation of uncertainties and resulting inferences for single-shot, repeated, or even cross-platform data. This approach has a strong history of application to a variety of problems in physical sciences ranging from inference of particle mass from multi-source high-energy particle data to analysis of black-hole characteristics from gravitational wave observations. The recent adoption of Bayesian statistics for analysis and design of high-energy density physics (HEDP) and inertial confinement fusion (ICF) experiments has provided invaluable gains in expert understanding and experiment performance. In this Review, we discuss the basic theory and practical application of the Bayesian statistics framework. We highlight a variety of studies from the HEDP and ICF literature, demonstrating the power of this technique. Due to the computational complexity of multi-physics models needed to analyze HEDP and ICF experiments, Bayesian inference is often not computationally tractable. Two sections are devoted to a review of statistical approximations, efficient inference algorithms, and data-driven methods, such as deep-learning and dimensionality reduction, which play a significant role in enabling use of the Bayesian framework. We provide additional discussion of various applications of Bayesian and machine learning methods that appear to be sparse in the HEDP and ICF literature constituting possible next steps for the community. We conclude by highlighting community needs, the resolution of which will improve trust in data-driven methods that have proven critical for accelerating the design and discovery cycle in many application areas.

47 OTHER INSTRUMENTATION↗
DOE OSTI2020

Generation of thermofield double states and critical ground states with a quantum computer

Significance Our experiment prepares two types of nontrivial quantum states on a trapped ion quantum computer: the thermofield double state of the transverse-field Ising model at arbitrary temperature and the quantum critical state of the zero-temperature model. We use techniques motivated by the quantum approximate optimization algorithm, and we implement a hybrid quantum–classical optimization loop to prepare the quantum critical state. Our results pave the way for exploring strongly correlated models at finite temperature and teleportation protocols inspired by black hole physics.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

Oscillon preheating in full general relativity

Abstract Oscillons are dense nonperturbative objects that may be copiously produced in the preheating period after inflation. Oscillon preheatings are usually simulated with nonlinear matter interactions but in a rigid FLRW background, without taking into account the gravitational backreactions. We investigate the oscillon preheating scenario in full general relativity with a full numerical relativity scheme, and compare the general-relativistic simulations with the traditional ones done in an FLRW background. We find that in certain parameter regions the general-relativistic corrections are significant, producing more and denser oscillons, and can be strong enough to collapse the oscillons to black holes.

Astronomy & Astrophysics↗
DOE OSTI2021

A world without pythons would be so simple

We show that bulk operators lying between the outermost extremal surface and the asymptotic boundary admit a simple boundary reconstruction in the classical limit. This is the converse of the Python's lunch conjecture, which proposes that operators with support between the minimal and outermost (quantum) extremal surfaces—e.g. the interior Hawking partners—are highly complex. Our procedure for reconstructing this 'simple wedge' is based on the HKLL construction, but uses causal bulk propagation of perturbed boundary conditions on Lorentzian timefolds to expand the causal wedge as far as the outermost extremal surface. As a corollary, we establish the Simple Entropy proposal for the holographic dual of the area of a marginally trapped surface as well as a similar holographic dual for the outermost extremal surface. Here, we find that the simple wedge is dual to a particular coarse-grained CFT state, obtained via averaging over all possible Python's lunches. An efficient quantum circuit converts this coarse-grained state into a 'simple state' that is indistinguishable in finite time from a state with a local modular Hamiltonian. Under certain circumstances, the simple state modular Hamiltonian generates an exactly local flow; we interpret this result as a holographic dual of black hole uniqueness.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Gravity of two photon decay and its quantum coherence

A linear analytical solution is derived for the gravitational shock wave produced by a particle of mass M that decays into a pair of null particles. The resulting space-time is shown to be unperturbed and isotropic, except for a discontinuous perturbation on a spherical null shell. Formulae are derived for the perturbation as a function of polar angle, as measured by an observer at the origin observing clocks on a sphere at distance R. The effect of the shock is interpreted physically as an instantaneous displacement in time and velocity when the shock passes the clocks. The time displacement is shown to be anisotropic, dominated by a quadrupole harmonic aligned with the particle-decay axis, with a magnitude δτ ~ GM/c 3 , independent of R. The velocity displacement is isotropic. The solution is used to derive the gravitational effect of a quantum state with a superposition of a large number of randomly oriented, statistically isotropic particle decays. This approach is shown to provide a well-controlled approximation to estimate the magnitude of gravitational fluctuations in systems composed of null point particles up to the Planck energy in a causal diamond of duration τ = 2R/c, as well as quantum-gravitational fluctuations of black holes and cosmological horizons. Coherent large-angle quantum distortions of macroscopic geometry from fluctuations up to the Planck scale are shown to grow linearly with the duration, with a variance $\langle$δτ 2 $\rangle$ ~ τtP much larger than that produced in models without causal quantum coherence.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗