Gravitational Lensing by a Massive Object in a Dark Matter Halo. I. Critical Curves and Caustics

We study the gravitational lensing properties of a massive object in a dark matter halo, concentrating on the critical curves and caustics of the combined lens. We model the system in the simplest approximation by a point mass embedded in a spherical Navarro–Frenk–White density profile. The low number of parameters of such a model permits a systematic exploration of its parameter space. We present galleries of critical curves and caustics for different masses and positions of the point in the halo. We demonstrate the existence of a critical mass, above which the gravitational influence of the centrally positioned point is strong enough to eliminate the radial critical curve and caustic of the halo. In the point-mass parameter space we identify the boundaries at which critical-curve transitions and corresponding caustic metamorphoses occur. The number of transitions as a function of the position of the point is surprisingly high, ranging from three for higher masses to as many as eight for lower masses. On the caustics we identify the occurrence of six different types of caustic metamorphoses. We illustrate the peculiar properties of the single radial critical curve and caustic appearing in an additional unusual nonlocal metamorphosis for a critical mass positioned at the halo center. Although we construct the model primarily to study the lensing influence of individual galaxies in a galaxy cluster, it can also be used to study the lensing by dwarf satellite galaxies in the halo of a host galaxy, as well as (super)massive black holes at a general position in a galactic halo.

Assimilating near-real-time mass balance stake readings into a model ensemble using a particle filter

Short-term glacier variations can be important for water supplies or hydropower production, and glaciers are important indicators of climate change. This is why the interest in near-real-time mass balance nowcasting is considerable. Here, we address this interest and provide an evaluation of continuous observations of point mass balance based on online cameras transmitting images every 20 min. The cameras were installed on three Swiss glaciers during summer 2019, provided 352 near-real-time point mass balances in total, and revealed melt rates of up to 0.12 m water equivalent per day (mw.e.d-1) and of more than 5 mw.e. in 81 d. By means of a particle filter, these observations are assimilated into an ensemble of three TI (temperature index) and one simplified energy-balance mass balance models. State augmentation with model parameters is used to assign temporally varying weights to individual models. We analyze model performance over the observation period and find that the probability for a given model to be preferred by our procedure is 39 % for an enhanced TI model, 24 % for a simple TI model, 23 %, for a simplified energy balance model, and 14 % for a model employing both air temperature and potential solar irradiation. When compared to reference forecasts produced with both mean model parameters and parameters tuned on single mass balance observations, the particle filter performs about equally well on the daily scale but outperforms predictions of cumulative mass balance by 95 %–96 %. A leave-one-out cross-validation on the individual glaciers shows that the particle filter is also able to reproduce point observations at locations not used for model calibration. Indeed, the predicted mass balances is always within 9 % of the observations. A comparison with glacier-wide annual mass balances involving additional measurements distributed over the entire glacier mostly shows very good agreement, with deviations of 0.02, 0.07, and 0.24 mw.e.

Imprints of Gravitational Millilensing on the Light Curve of Gamma-Ray Bursts

In this work, we search for signatures of gravitational millilensing in gamma-ray bursts (GRBs) in which the source−lens−observer geometry produces two images that manifest in the GRB light curve as superimposed peaks with identical temporal variability (or echoes), separated by the time delay between the two images. According to the sensitivity of our detection method, we consider millilensing events due to point-mass lenses in the range of 105 − 107 M ⊙ at lens redshift about half that of the GRB, with a time delay on the order of 10 s. Current GRB observatories are capable of resolving and constraining this lensing scenario if the above conditions are met. We investigated the Fermi/GBM GRB archive from the year 2008 to 2020 using the autocorrelation technique and found one millilensed GRB candidate out of 2137 GRBs searched, which we use to estimate the optical depth of millilensed GRBs by performing a Monte Carlo simulation to find the efficiency of our detection method. Considering a point-mass model for the gravitational lens, where the lens is a supermassive black hole, we show that the density parameter of black holes (ΩBH) with mass ≈ 106 M ⊙ is about 0.007 ± 0.004. Our result is one order of magnitude larger compared to previous work in the lower mass range of 102 − 103 M ⊙, which gave a density parameter ΩBH ≈ 5 × 10−4, and recent work in the mass range of 102 − 107 M ⊙, which reported ΩBH ≈ 4.6 × 10−4. The mass fraction of black holes in this mass range to the total mass of the universe would be f = ΩBH/Ω M ≈ 0.027 ± 0.016.

Quantification of a concentrated point mass by Haar wavelets and machine learning

The inverse problem of determining location and mass ratio of a concentrated point mass attached to the homogeneous Euler – Bernoulli beam was considered in this article. Under the assumption that the size of the point mass was small compared to the total mass of the beam, it was shown that the problem could be solved in terms of point-mass-induced changes in the natural frequencies or mode shapes. Predictions of the point mass location and its mass ratio were made by the artificial neural networks or the random forests. The dimensionless natural frequency parameters or the first mode shape transformed into the Haar wavelet coefficients were used at the inputs of the machine learning methods. The simulation studies indicated that the combined approach of the natural frequencies, Haar wavelets and neural networks produced accurate predictions. The results presented in this article could help in understanding the behaviour of more complex structures under similar conditions and provide apparent influence on design of beams.

DEPENDENCE OF THE FREQUENCY SPECTRUM OF MICRO- AND NANO- RESONATORS ON PRESSURE AND ATTACHED MASS

An elastic rod of circular or rectangular section is rigidly fixed on both ends. The applicability of classical equations for the deformation of thin elements like rods, plates and shells to describe the stated problem is assessed using such integral characteristics, as eigenfrequencies. The assembly pressure is uniform, specifically atmospheric, and acts also on the areas of strip edges. It is assumed that there are no strains in this case. Excess pressures act only on the strip’s surface. The self-weight of the strip is neglected. Accounting for the attached mass of the surrounding medium and radiation penetrating into it shows that pressures in the upper and lower parts of the rod differ. But these factors are not taken into account, which can be justified in case of light gases. Since the relative axial lengthening at the boundaries equals zero in case of rigid clamping, it will also equal zero along the entire length in the absence of external axial forces. Frequency equations have been derived in case of the action of the surrounding pressure and also uniformly distributed and attached point masses. The influence of the excess pressure of the surrounding medium on the frequency spectrum of the rod oscillations is determined by the non-dimensional parameter that increases with an increase in pressure and the rod length and decreases with an increase of bending rigidity. At the negative excess pressure (vacuuming) this parameter reverses its sign, and the frequencies become lower. With an increase in both distributed and attached point mass the eigenfrequencies of oscillations decrease due to the rod invariable bending rigidity. The displacement of the point mass towards the center results in a decrease in odd eigenfrequencies, while even eigenfrequencies remain the same. Using the first frequency measured we can determine the excess pressure acting on the rod’s surface. Using two frequencies of bending oscillations we can determine the attached point mass and its coordinate. These results can be used when simulating the performance of resonators, including micro and nano ones.

Inverse problems and rigidity questions in billiard dynamics

A Birkhoff billiard is a system describing the inertial motion of a point mass inside a strictly convex planar domain, with elastic reflections at the boundary. The study of the associated dynamics is profoundly intertwined with the geometric properties of the domain: while it is evident how the shape determines the dynamics, a more subtle and difficult question is the extent to which the knowledge of the dynamics allows one to reconstruct the shape of the domain. This translates into many intriguing inverse problems and unanswered rigidity questions, which have been the focus of very active research in recent decades. In this paper we describe some of these questions, along with their connection to other problems in analysis and geometry, with particular emphasis on recent results obtained by the authors and their collaborators.