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At least 361 records · Page 20

Vapor shielding models and the energy absorbed by divertor targets during transient events

The erosion of divertor targets caused by high heat fluxes during transients is a serious threat to ITER operation, as it is going to be the main factor determining the divertor lifetime. Under the influence of extreme heat fluxes, the surface temperature of plasma facing components can reach some certain threshold, leading to an onset of intense material evaporation. The latter results in formation of cold dense vapor and secondary plasma cloud. This layer effectively absorbs the energy of the incident plasma flow, turning it into its own kinetic and internal energy and radiating it. This so called vapor shielding is a phenomenon that may help mitigating the erosion during transient events. In particular, the vapor shielding results in saturation of energy (per unit surface area) accumulated by the target during single pulse of heat load at some level E{sub max}. Matching this value is one of the possible tests to verify complicated numerical codes, developed to calculate the erosion rate during abnormal events in tokamaks. The paper presents three very different models of vapor shielding, demonstrating that E{sub max} depends strongly on the heat pulse duration, thermodynamic properties, and evaporation energy of the irradiated target material. While its dependence on the other shielding details such as radiation capabilities of material and dynamics of the vapor cloud is logarithmically weak. The reason for this is a strong (exponential) dependence of the target material evaporation rate, and therefore the “strength” of vapor shield on the target surface temperature. As a result, the influence of the vapor shielding phenomena details, such as radiation transport in the vapor cloud and evaporated material dynamics, on the E{sub max} is virtually completely masked by the strong dependence of the evaporation rate on the target surface temperature. However, the very same details define the amount of evaporated particles, needed to provide an effective shielding to the target, and, therefore, strongly influence resulting erosion rate. Thus, E{sub max} cannot be used for validation of shielding models and codes, aimed at the target material erosion calculations.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Mean-velocity profile of smooth channel flow explained by a cospectral budget model with wall-blockage

A series of recent studies has shown that a model of the turbulent vertical velocity variance spectrum (Fvv) combined with a simplified cospectral budget can reproduce many macroscopic flow properties of turbulent wall-bounded flows, including various features of the mean-velocity profile (MVP), i.e., the “law of the wall”. While the approach reasonably models the MVP’s logarithmic layer, the buffer layer displays insufficient curvature compared to measurements. The assumptions are re-examined here using a direct numerical simulation (DNS) dataset at moderate Reynolds number that includes all the requisite spectral and co-spectral information. Starting with several hypotheses for the cause of the “missing” curvature in the buffer layer, it is shown that the curvature deficit is mainly due to mismatches between (i) the modelled and DNS-observed pressure-strain terms in the cospectral budget and (ii) the DNS-observed Fvv and the idealized form used in previous models. By replacing the current parameterization for the pressure-strain term with an expansive version that directly accounts for wall-blocking effects, the modelled and DNS reported pressure-strain profiles match each other in the buffer and logarithmic layers. Forcing the new model with DNS-reported Fvv rather than the idealized form previously used reproduces the missing buffer layer curvature to high fidelity thereby confirming the “spectral link” between Fvv and the MVP across the full profile. A broad implication of this work is that much of the macroscopic properties of the flow (such as the MVP) may be derived from the energy distribution in turbulent eddies (i.e., Fvv) representing the microstate of the flow, provided the link between them accounts for wall-blocking.

54 ENVIRONMENTAL SCIENCES↗

Explicit formula for the Holevo bound for two-parameter qubit-state estimation problem

The main contribution of this paper is to derive an explicit expression for the fundamental precision bound, the Holevo bound, for estimating any two-parameter family of qubit mixed-states in terms of quantum versions of Fisher information. The obtained formula depends solely on the symmetric logarithmic derivative (SLD), the right logarithmic derivative (RLD) Fisher information, and a given weight matrix. This result immediately provides necessary and sufficient conditions for the following two important classes of quantum statistical models; the Holevo bound coincides with the SLD Cramér-Rao bound and it does with the RLD Cramér-Rao bound. One of the important results of this paper is that a general model other than these two special cases exhibits an unexpected property: the structure of the Holevo bound changes smoothly when the weight matrix varies. In particular, it always coincides with the RLD Cramér-Rao bound for a certain choice of the weight matrix. Several examples illustrate these findings.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

First-principles investigations on ionization and thermal conductivity of polystyrene for inertial confinement fusion applications

Using quantum molecular-dynamics (QMD) methods based on the density functional theory, we have performed first-principles investigations of the ionization and thermal conductivity of polystyrene (CH) over a wide range of plasma conditions (ρ = 0.5 to 100 g/cm{sup 3} and T = 15 625 to 500 000 K). The ionization data from orbital-free molecular-dynamics calculations have been fitted with a “Saha-type” model as a function of the CH plasma density and temperature, which gives an increasing ionization as the CH density increases even at low temperatures (T < 50 eV). The orbital-free molecular dynamics method is only used to gauge the average ionization behavior of CH under the average-atom model in conjunction with the pressure-matching mixing rule. The thermal conductivities (κ{sub QMD}) of CH, derived directly from the Kohn–Sham molecular-dynamics calculations, are then analytically fitted with a generalized Coulomb logarithm [(lnΛ){sub QMD}] over a wide range of plasma conditions. When compared with the traditional ionization and thermal conductivity models used in radiation–hydrodynamics codes for inertial confinement fusion simulations, the QMD results show a large difference in the low-temperature regime in which strong coupling and electron degeneracy play an essential role in determining plasma properties. Hydrodynamic simulations of cryogenic deuterium–tritium targets with CH ablators on OMEGA and the National Ignition Facility using the QMD-derived ionization and thermal conductivity of CH have predicted ∼20% variation in target performance in terms of hot-spot pressure and neutron yield (gain) with respect to traditional model simulations.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Non-additive non-interacting kinetic energy of rare gas dimers

We show approximations of the non-additive non-interacting kinetic energy (NAKE) as an explicit functional of the density are the basis of several electronic structure methods that provide improved computational efficiency over standard Kohn-Sham calculations. However, within most fragment-based formalisms, there is no unique exact NAKE, making it difficult to develop general, robust approximations for it. When adjustments are made to the embedding formalisms to guarantee uniqueness, approximate functionals may be more meaningfully compared to the exact unique NAKE. We use numerically accurate inversions to study the exact NAKE of several rare-gas dimers within partition density functional theory, a method that provides the uniqueness for the exact NAKE. We find that the NAKE decreases nearly exponentially with atomic separation for the rare-gas dimers. We compute the logarithmic derivative of the NAKE with respect to the bond length for our numerically accurate inversions as well as for several approximate NAKE functionals. We show that standard approximate NAKE functionals do not reproduce the correct behavior for this logarithmic derivative and propose two new NAKE functionals that do. The first of these is based on a re-parametrization of a conjoint Perdew-Burke-Ernzerhof (PBE) functional. The second is a simple, physically motivated non-decomposable NAKE functional that matches the asymptotic decay constant without fitting.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Cross-verification of neoclassical transport solutions from XGCa against NEO

Radial neoclassical transport from the global gyrokinetic code XGC has been cross-verified against the local drift-kinetic neoclassical code NEO in NEO's local regime using a Grad-Shafranov equilibrium with a circular boundary. XGC shows excellent agreement with NEO, except for a difference of approximately 10% in the ion heat flux. Close investigation reveals that this difference is due to the simplification in NEO to use the same Coulomb logarithm ln Λ to represent all particle species, while XGC uses different formulas for collisions between different species. When XGC uses NEO's Coulomb logarithm, excellent agreement in the ion heat flux is recovered. Here, the present work not only crossverifies the equation solvers in two codes, but also verifies that the gyrokinetic and the drift-kinetic equations agree and that the linearized and nonlinear Coulomb collision operators agree in the mild gradient, local regime.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Quantitative metrics for evaluating thermonuclear design codes and physics models applied to the National Ignition Campaign

In this study, physics models and design codes for hot dense plasmas undergoing thermonuclear burn are evaluated objectively using statistical metrics that compare the difference between calculations and data relative to the experimental uncertainties. The analysis is applied to the National Ignition Campaign (NIC) because it is relevant, comprehensive, and well documented. The statistics confirm that a key process afflicting NIC performance is mix driven by hydrodynamic instabilities as approximated here using the KL model [G. Dimonte and R. Tipton, Phys. Fluids 18, 085101 (2006)]. New physics models are also presented for instability-driven magnetic fields [B. Srinivasan et al., Phys. Rev. Lett. 108, 165002 (2012)] and the Coulomb logarithm for electron–ion thermal relaxation [G. Dimonte and J. Daligault, Phys. Rev. Lett. 101, 135001 (2008)]. The plasma-generated magnetic fields improve the agreement between code and data in a statistically significant manner by reducing the electron thermal conduction in the hot-spot via the Hall term. The Coulomb logarithm is presented in a numerically practical form that incorporates recent theoretical advances. However, it does not improve the agreement between the code and data because the thermal relaxation is so fast in non-ignited NIC plasmas that the typical 20% error does not change the tight coupling between electrons and ions. Even with these improved physics models, the one-dimensional simulations presented here are not able to describe the high-convergence NIC implosions in a statistically acceptable manner. Of the heroic multi-dimensional simulations of Clark et al. [Phys. Plasmas 23, 056302 (2016)], only those in three-dimensions satisfy the statistical acceptance criterion.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Reconciling calculations and measurements of inverse bremsstrahlung absorption

It was recently shown that the use of Coulomb logarithms appropriate for bremsstrahlung radiation (rather than transport processes) along with corrections for the Langdon effect and ion screening reproduced measurements of collisional absorption in well-characterized underdense plasmas [D. Turnbull et al., Phys. Rev. Lett. 130, 145103 (2023)]. However, it was recognized at the time that the use of the standard absorption-reduction factor from Langdon's seminal paper was inconsistent with the use of Coulomb logarithms that are thermally averaged over a Maxwellian electron-velocity distribution function. A more accurate approach would be to average over the expected super-Gaussian distribution function while accounting for the Gaunt factor's velocity dependence, which somewhat mitigates the Langdon effect; however, at that time, this theory matched the data less well. This conflict is now eliminated with the additional insight that the ionization state of our mid-Z ion species (when present) was lower than had been assumed, as evidenced by the Thomson-scattering data and time-dependent Cretin simulations. We are now able to show that an improved treatment of the Langdon effect provides the best match to data. Otherwise, the prior conclusions remain unchanged. We also show an example of the substantial expected impact to the absorption rate in calculations of indirect-drive hohlraums.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

THE CENTRAL SLOPE OF DARK MATTER CORES IN DWARF GALAXIES: SIMULATIONS VERSUS THINGS

We make a direct comparison of the derived dark matter (DM) distributions between hydrodynamical simulations of dwarf galaxies assuming a {Lambda}CDM cosmology and the observed dwarf galaxies sample from the THINGS survey in terms of (1) the rotation curve shape and (2) the logarithmic inner density slope {alpha} of mass density profiles. The simulations, which include the effect of baryonic feedback processes, such as gas cooling, star formation, cosmic UV background heating, and most importantly, physically motivated gas outflows driven by supernovae, form bulgeless galaxies with DM cores. We show that the stellar and baryonic mass is similar to that inferred from photometric and kinematic methods for galaxies of similar circular velocity. Analyzing the simulations in exactly the same way as the observational sample allows us to address directly the so-called cusp/core problem in the {Lambda}CDM model. We show that the rotation curves of the simulated dwarf galaxies rise less steeply than cold dark matter rotation curves and are consistent with those of the THINGS dwarf galaxies. The mean value of the logarithmic inner density slopes {alpha} of the simulated galaxies' DM density profiles is {approx}-0.4 {+-} 0.1, which shows good agreement with {alpha} = -0.29 {+-} 0.07 of the THINGS dwarf galaxies. The effect of non-circular motions is not significant enough to affect the results. This confirms that the baryonic feedback processes included in the simulations are efficiently able to make the initial cusps with {alpha} {approx}-1.0 to -1.5 predicted by DM-only simulations shallower and induce DM halos with a central mass distribution similar to that observed in nearby dwarf galaxies.

79 ASTRONOMY AND ASTROPHYSICS↗

DARK ENERGY FROM THE LOG-TRANSFORMED CONVERGENCE FIELD

A logarithmic transform of the convergence field improves 'the information content', i.e., the overall precision associated with the measurement of the amplitude of the convergence power spectrum, by improving the covariance matrix properties. The translation of this improvement in the information content to that in cosmological parameters, such as those associated with dark energy, requires knowing the sensitivity of the log-transformed field to those cosmological parameters. In this paper, we use N-body simulations with ray tracing to generate convergence fields at multiple source redshifts as a function of cosmology. The gain in information associated with the log-transformed field does lead to tighter constraints on dark energy parameters, but only if shape noise is neglected. The presence of shape noise quickly diminishes the advantage of the log-mapping, more quickly than we would expect based on the information content. With or without shape noise, using a larger pixel size allows for a more efficient log-transformation.

79 ASTRONOMY AND ASTROPHYSICS↗

STELLAR POPULATIONS OF LYMAN BREAK GALAXIES AT z {approx_equal} 1-3 IN THE HST/WFC3 EARLY RELEASE SCIENCE OBSERVATIONS

We analyze the spectral energy distributions (SEDs) of Lyman break galaxies (LBGs) at z {approx_equal} 1-3 selected using the Hubble Space Telescope (HST) Wide Field Camera 3 (WFC3) UVIS channel filters. These HST/WFC3 observations cover about 50 arcmin{sup 2} in the GOODS-South field as a part of the WFC3 Early Release Science program. These LBGs at z {approx_equal} 1-3 are selected using dropout selection criteria similar to high-redshift LBGs. The deep multi-band photometry in this field is used to identify best-fit SED models, from which we infer the following results: (1) the photometric redshift estimate of these dropout-selected LBGs is accurate to within few percent; (2) the UV spectral slope {beta} is redder than at high redshift (z > 3), where LBGs are less dusty; (3) on average, LBGs at z {approx_equal} 1-3 are massive, dustier, and more highly star forming, compared to LBGs at higher redshifts with similar luminosities (0.1L* {approx}< L {approx}< 2.5L*), though their median values are similar within 1{sigma} uncertainties. This could imply that identical dropout selection technique, at all redshifts, finds physically similar galaxies; and (4) the stellar masses of these LBGs are directly proportional to their UV luminosities with a logarithmic slope of {approx}0.46, and star formation rates are proportional to their stellar masses with a logarithmic slope of {approx}0.90. These relations hold true-within luminosities probed in this study-for LBGs from z {approx_equal} 1.5 to 5. The star-forming galaxies selected using other color-based techniques show similar correlations at z {approx_equal} 2, but to avoid any selection biases, and for direct comparison with LBGs at z > 3, a true Lyman break selection at z {approx_equal} 2 is essential. The future HST UV surveys, both wider and deeper, covering a large luminosity range are important to better understand LBG properties and their evolution.

79 ASTRONOMY AND ASTROPHYSICS↗

Dwarf galaxy dark matter density profiles inferred from stellar and gas kinematics

We present new constraints on the density profiles of dark matter (DM) halos in seven nearby dwarf galaxies from measurements of their integrated stellar light and gas kinematics. The gas kinematics of low-mass galaxies frequently suggest that they contain constant density DM cores, while N-body simulations instead predict a cuspy profile. We present a data set of high-resolution integral-field spectroscopy on seven galaxies and measure the stellar and gas kinematics simultaneously. Using Jeans modeling on our full sample, we examine whether gas kinematics in general produce shallower density profiles than are derived from the stars. Although two of the seven galaxies show some localized differences in their rotation curves between the two tracers, estimates of the central logarithmic slope of the DM density profile, γ, are generally robust. The mean and standard deviation of the logarithmic slope for the population are γ = 0.67 ± 0.10 when measured in the stars and γ = 0.58 ± 0.24 when measured in the gas. We also find that the halos are not under-concentrated at the radii of half their maximum velocities. Finally, we search for correlations of the DM density profile with stellar velocity anisotropy and other baryonic properties. Two popular mechanisms to explain cored DM halos are an exotic DM component or feedback models that strongly couple the energy of supernovae into repeatedly driving out gas and dynamically heating the DM halos. While such models do not yet have falsifiable predictions that we can measure, we investigate correlations that may eventually be used to test models. We do not find a secondary parameter that strongly correlates with the central DM density slope, but we do find some weak correlations. The central DM density slope weakly correlates with the abundance of α elements in the stellar population, anti-correlates with H I fraction, and anti-correlates with vertical orbital anisotropy. We expect, if anything, the opposite of these three trends for feedback models. Determining the importance of these correlations will require further model developments and larger observational samples.

79 ASTRONOMY AND ASTROPHYSICS↗

Mass and environment as drivers of galaxy evolution. III. The constancy of the faint-end slope and the merging of galaxies

Using our continuity approach, we explore the underlying connections between the evolution of the faint-end slope α{sub s} of the stellar mass function of star-forming galaxies, the logarithmic slope β of the specific star formation rate (sSFR)-mass relation, and the merging of galaxies. We derive analytically the consequences of the observed constancy of α{sub s} since redshifts of at least z ∼ 2. If the logarithmic slope β of the sSFR-mass relation is negative, then the faint-end slope α{sub s} should quickly diverge due to the differential mass increase of galaxies on the star-forming main sequence, and this will also quickly destroy the Schechter form of the mass function. This problem can be solved by removing low-mass galaxies by merging them into more massive galaxies. We quantify this process by introducing the specific merger mass rate (sMMR) as the specific rate of mass added to a given galaxy through mergers. For a modest negative value of β ∼ –0.1, an average sMMR ∼ 0.1 sSFR across the population is required to keep α{sub s} constant with epoch, as observed. This in turn implies a merger rate of ∼0.2 sSFR for major mergers, which is consistent with the available observational estimates. More negative values of β require higher sMMR and higher merger rates, and the steepening of the mass function becomes impossible to control for β < –(α{sub s} + 2). The close link that is required between the in situ sSFR and the sMMR probably arises because both are closely linked to the buildup of dark matter halos. These new findings further develop the formalism for the evolving galaxy population that we introduced earlier and show how striking symmetries in the galaxy population can emerge as the result of deep links between the physical processes involved.

79 ASTRONOMY AND ASTROPHYSICS↗

GENERATION OF MAGNETOHYDRODYNAMIC WAVES IN LOW SOLAR ATMOSPHERIC FLUX TUBES BY PHOTOSPHERIC MOTIONS

Recent ground- and space-based observations reveal the presence of small-scale motions between convection cells in the solar photosphere. In these regions, small-scale magnetic flux tubes are generated via the interaction of granulation motion and the background magnetic field. This paper studies the effects of these motions on magnetohydrodynamic (MHD) wave excitation from broadband photospheric drivers. Numerical experiments of linear MHD wave propagation in a magnetic flux tube embedded in a realistic gravitationally stratified solar atmosphere between the photosphere and the low choromosphere (above β = 1) are performed. Horizontal and vertical velocity field drivers mimic granular buffeting and solar global oscillations. A uniform torsional driver as well as Archimedean and logarithmic spiral drivers mimic observed torsional motions in the solar photosphere. The results are analyzed using a novel method for extracting the parallel, perpendicular, and azimuthal components of the perturbations, which caters to both the linear and non-linear cases. Employing this method yields the identification of the wave modes excited in the numerical simulations and enables a comparison of excited modes via velocity perturbations and wave energy flux. The wave energy flux distribution is calculated to enable the quantification of the relative strengths of excited modes. The torsional drivers primarily excite Alfvén modes (≈60% of the total flux) with small contributions from the slow kink mode, and, for the logarithmic spiral driver, small amounts of slow sausage mode. The horizontal and vertical drivers primarily excite slow kink or fast sausage modes, respectively, with small variations dependent upon flux surface radius.

79 ASTRONOMY AND ASTROPHYSICS↗

RELATIVISTIC EQUATION OF STATE FOR CORE-COLLAPSE SUPERNOVA SIMULATIONS

We construct the equation of state (EOS) of dense matter covering a wide range of temperature, proton fraction, and density for the use of core-collapse supernova simulations. The study is based on the relativistic mean-field (RMF) theory, which can provide an excellent description of nuclear matter and finite nuclei. The Thomas-Fermi approximation in combination with assumed nucleon distribution functions and a free energy minimization is adopted to describe the non-uniform matter, which is composed of a lattice of heavy nuclei. We treat the uniform matter and non-uniform matter consistently using the same RMF theory. We present two sets of EOS tables, namely EOS2 and EOS3. EOS2 is an update of our earlier work published in 1998 (EOS1), where only the nucleon degree of freedom is taken into account. EOS3 includes additional contributions from {Lambda} hyperons. The effect of {Lambda} hyperons on the EOS is negligible in the low-temperature and low-density region, whereas it tends to soften the EOS at high density. In comparison with EOS1, EOS2 and EOS3 have an improved design of ranges and grids, which covers the temperature range T = 0.1-10{sup 2.6} MeV with the logarithmic grid spacing {Delta}log{sub 10}(T/[MeV]) = 0.04 (92 points including T = 0), the proton fraction range Y{sub p} = 0-0.65 with the linear grid spacing {Delta}Y{sub p} = 0.01 (66 points), and the density range {rho}{sub B} 10{sup 5.1}-10{sup 16} g cm{sup -3} with the logarithmic grid spacing {Delta}log{sub 10}({rho}{sub B}/[g cm{sup -3}]) = 0.1 (110 points).

79 ASTRONOMY AND ASTROPHYSICS↗

Brane-antibrane backreaction in axion monodromy inflation

We calculate the interaction potential between D5 and D-bar 5 branes wrapping distant but homologous 2-cycles. The interaction potential is logarithmic in the separation radius and does not decouple at infinity. We show that logarithmic backreaction is generic for 5-branes wrapping distant but homologous 2-cycles, and we argue that this destabilises models of axion monodromy inflation involving NS5 brane-antibrane pairs in separate warped throats towards an uncontrolled region.

79 ASTRONOMY AND ASTROPHYSICS↗

Reconciling induced-gravity inflation in supergravity with the Planck 2013 & BICEP2 results

We generalize the embedding of induced-gravity inflation beyond the no-scale Supergravity presented in ref. http://dx.doi.org/10.1088/1475-7516/2014/08/057 employing two gauge singlet chiral superfields, a superpotential uniquely determined by applying a continuous R and a discrete ℤ{sub n} symmetries, and a logarithmic Kähler potential including all the allowed terms up to fourth order in powers of the various fields. We show that, increasing slightly the prefactor (−3) encountered in the adopted Kähler potential, an efficient enhancement of the resulting tensor-to-scalar ratio can be achieved rendering the predictions of the model consistent with the recent BICEP2 results, even with subplanckian excursions of the original inflaton field. The remaining inflationary observables can become compatible with the data by mildly tuning the coefficient involved in the fourth order term of the Kähler potential which mixes the inflaton with the accompanying non-inflaton field. The inflaton mass is predicted to be close to 10{sup 14} GeV.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

Gravity waves from non-minimal quadratic inflation

We discuss non-minimal quadratic inflation in supersymmetric (SUSY) and non-SUSY models which entails a linear coupling of the inflaton to gravity. Imposing a lower bound on the parameter c{sub R}, involved in the coupling between the inflaton and the Ricci scalar curvature, inflation can be attained even for subplanckian values of the inflaton while the corresponding effective theory respects the perturbative unitarity up to the Planck scale. Working in the non-SUSY context we also consider radiative corrections to the inflationary potential due to a possible coupling of the inflaton to bosons or fermions. We find ranges of the parameters, depending mildly on the renormalization scale, with adjustable values of the spectral index n{sub s}, tensor-to-scalar ratio r≃(2−4)⋅10{sup −3}, and an inflaton mass close to 3⋅10{sup 13} GeV. In the SUSY framework we employ two gauge singlet chiral superfields, a logarithmic Kähler potential including all the allowed terms up to fourth order in powers of the various fields, and determine uniquely the superpotential by applying a continuous R and a global U(1) symmetry. When the Kähler manifold exhibits a no-scale-type symmetry, the model predicts n{sub s}≃0.963 and r≃0.004. Beyond no-scale SUGRA, n{sub s} and r depend crucially on the coefficient involved in the fourth order term, which mixes the inflaton with the accompanying non-inflaton field in the Kähler potential, and the prefactor encountered in it. Increasing slightly the latter above (−3), an efficient enhancement of the resulting r can be achieved putting it in the observable range. The inflaton mass in the last case is confined in the range (5−9)⋅10{sup 13} GeV.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗