Cosmological Distances Research Articles

Hubble constant and nuclear equation of state from kilonova spectro-photometric light curves

The merger of two compact objects of which at least one is a neutron star is signalled by transient electromagnetic emission in a kilonova (KN). This event is accompanied by gravitational waves and possibly other radiation messengers such as neutrinos or cosmic rays. The electromagnetic emission arises from the radioactive decay of heavy r-process elements synthesized in the material ejected during and after the merger. In this paper we show that the analysis of KNe light curves can provide cosmological distance measurements and constrain the properties of the ejecta. In this respect, MAAT, the new Integral Field Unit in the OSIRIS spectrograph on the 10.4 m Gran Telescopio CANARIAS (GTC), is well suited for the study of KNe by performing absolute spectro-photometry over the entire 3600 − 10 000 Å spectral range. Here, we study the most representative cases regarding the scientific interest of KNe from binary neutron stars, and we evaluate the observational prospects and performance of MAAT on the GTC to do the following: (a) study the impact of the equation of state on the KN light curve, and determine to what extent bounds on neutron star (NS) radii or compactness deriving from KN peak magnitudes can be identified and (b) measure the Hubble constant, H0, with precision improved by up to 40%, when both gravitational wave data and photometric-light curves are used. In this context we discuss how the equation of state, the viewing angle, and the distance affect the precision and estimated value of H0.

Search for Coincident Gravitational-wave and Fast Radio Burst Events from 4-OGC and the First CHIME/FRB Catalog

Advanced LIGO and Virgo have reported 90 confident gravitational-wave (GW) observations from compact-binary coalescences from their three observation runs. In addition, numerous subthreshold GW candidates have been identified. Binary neutron star (BNS) mergers can produce GWs and short-gamma-ray bursts, as confirmed by GW170817/GRB 170817A. There may be electromagnetic counterparts recorded in archival observations associated with subthreshold GW candidates. The CHIME/FRB Collaboration has reported the first large sample of fast radio bursts (FRBs), millisecond radio transients detected up to cosmological distances; a fraction of these may be associated with BNS mergers. This work searches for coincident GWs and FRBs from BNS mergers using candidates from the fourth Open Gravitational-wave Catalog and the first CHIME/FRB catalog. We use a ranking statistic for GW/FRB association that combines the GW detection statistic with the odds of temporal and spatial association. We analyze GW candidates and nonrepeating FRBs from 2019 April 1 to 2019 July 1, when both the Advanced LIGO/Virgo GW detectors and the CHIME radio telescope were observing. The most significant coincident candidate has a false alarm rate of 0.29 per observation time, which is consistent with a null observation. The null results imply, at most, – of FRBs are produced promptly from the BNS mergers.

Constraining Einstein-dilaton models using joint gravitational-wave and electromagnetic observations

With the advent of gravitational-wave astronomy it has now been possible to constrain modified theories of gravity that were invoked to explain the dark energy. In a class of dilaton models, distances to cosmic sources inferred from electromagnetic and gravitational wave observations would differ due to the presence of a friction term. In such theories, the ratio of the Newton's constant to the fine structure constant varies with time. In this paper we explore the degree to which it will be possible to test such models. If collocated sources (e.g. supernovae and binary neutron star mergers), but not necessarily multimessengers, can be identified by electromagnetic telescopes and gravitational-wave detectors one can probe if light and gravitational radiation are subject to the same laws of propagation over cosmological distances. This helps in constraining the variation of Newton's constant relative to fine-structure constant. The next generation of gravitational wave detectors, such as the Cosmic Explorer and Einstein Telescope, in tandem with the Vera Rubin Observatory and gamma ray observatories such as the Fermi Space Observatory will be able to detect or constrain such variations at the level of a few parts in 100. We apply this method to GW170817 with distances inferred by the LIGO and Virgo detectors and the observed Kilonova.

Probing Our Universe’s Past Using Earth’s Geological and Climatological History and Shadows of Galactic Black Holes

In this short review, we discuss how Earth’s climatological and geological history and also how the shadows of galactic black holes might reveal our Universe’s past evolution. Specifically we point out that a pressure singularity that occurred in our Universe’s past might have left its imprint on Earth’s geological and climatological history and on the shadows of cosmological black holes. Our approach is based on the fact that the H0 tension problem may be resolved if some sort of abrupt physics change occurred in our Universe 70–150 Myrs ago, an abrupt change that deeply affected the Cepheid parameters. We review how such an abrupt physics change might have been caused in our Universe by a smooth passage of it through a pressure finite-time singularity. Such finite-time singularities might occur in modified gravity and specifically in F(R) gravity, so we show how modified gravity might drive this type of evolution, without resorting to peculiar cosmic fluids or scalar fields. The presence of such a pressure singularity can distort the elliptic trajectories of bound objects in the Universe, causing possible geological and climatological changes on Earth, if its elliptic trajectory around the Sun might have changed. Also, such a pressure singularity affects directly the circular photon orbits around supermassive galactic black holes existing at cosmological redshift distances, thus the shadows of some cosmological black holes at redshifts z≤0.01, might look different in shape, compared with the SgrA* and M87* supermassive black holes. This feature however can be checked experimentally in the very far future.

Gravitational wave memory and its tail in cosmology

We study gravitational wave memory effect in the FRW cosmological model with matter and cosmological constant. Since the background is curved, gravitational radiation develops a tail part arriving after the main signal that travels along the past light cone of the observer. First we discuss first order gravitational wave sourced by a binary system, and find that the tail only gives a negligible memory, in accord with previous results. Then we study the nonlinear memory effect coming from induced gravitational radiation sourced by first order gravitational radiation propagating over cosmological distances. In the light cone part of the induced gravitational wave we find a novel term missed in previous studies of the cosmological memory effect. Furthermore, we show that the induced gravitational wave has a tail part that slowly accumulates after the light cone part has passed and grows to a sizeable magnitude over a cosmological timescale. This tail part of the memory effect will be a new component in the stochastic gravitational wave background.

Fast algorithm for simulating light curves of stars at extreme magnification affected by microlensing

We present a fast algorithm aimed at reproducing the light curves of distant stars undergoing microlensing near critical curves. The need for these type of algorithms has been motivated by recent observations of microlensing events of distant stars at high redshift and involving extreme magnification factors. The algorithm relies on a low-resolution computation of the deflection field due to an ensemble of microlenses near critical curves and takes advantage of the gradually varying nature of the deflection field to infer the magnification of the unresolved images. The algorithm is capable of resolving microlenses at cosmological distances with planet-sized masses, as well as typical background luminous stars of a few solar radii. Using this algorithm, light curves covering decades of relative motion between the source and the web of microcaustics, at nanoarcsec resolution, can be reproduced within a matter of minutes on a basic laptop. Classic inverse ray tracing simulations at the same resolution would take days or weeks to produce, consuming massive computational resources.

The power spectrum of cosmological number densities

This paper studies the cosmological power spectrum (PS) of the differential and integral galaxy volume number densities, respectively, \(\gamma _i\) and \(\gamma _i^{*}\), constructed with the cosmological distances \(d_i\) \((i={\scriptstyle A},{\scriptstyle G},{\scriptstyle L},{\scriptstyle Z})\), where \(d_{\scriptscriptstyle A}\) is the angular diameter distance, \(d_{\scriptscriptstyle G}\) is the galaxy area distance, \(d_{\scriptscriptstyle L}\) is the luminosity distance and \(d_z\) is the redshift distance. Theoretical and observational quantities were obtained in the Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime with a non-vanishing cosmological constant. The radial correlation \(\Xi _i\), a quantity defined in the context of these densities, is also discussed in the wave number domain. All observational quantities were computed using luminosity function (LF) data obtained from the FORS Deep Field (FDF) galaxy survey. The theoretical and observational power spectra of \(\gamma _i\), \(\gamma _i^{*}\), \(\Xi _i\) and the ratio \(\gamma _i / \gamma _i^*\) were calculated by performing Fourier transforms on values of these densities which were previously derived by Iribarrem (Astron. Astrophys. 539:A112, 2012) from the observational values \([\gamma ]_{{\textstyle {\scriptscriptstyle \mathrm {obs}}}}\) and \([\gamma ^*]_{{\textstyle {\scriptscriptstyle \mathrm {obs}}}}\) obtained by using the galactic absolute magnitudes and Schechter’s parameters of the galaxy LF presented in Gabasch (Astron. Astrophys. 421:41–58, 2004; Astron. Astrophys. 448:101, 2006). These parameters were evaluated from a I-band selected dataset of the FDF in the redshift range \(0.5 \le z \le 5.0\) for its blue bands and \(0.75 \le z \le 3.0\) for its red ones. The results show similar behavior of the power spectra obtained from \(\gamma \) and \(\gamma ^{*}\) using \(d_{\scriptscriptstyle L}\), \(d_z\) and \(d_{\scriptscriptstyle G}\) as distance measures. The PS of the densities defined with \(d_{\scriptscriptstyle A}\) have a different and inconclusive behavior because this cosmological distance reaches a maximum at \(z\approx 1.6\) in the adopted cosmological model. For the other distances, our results suggest that the PS of \([\gamma _i]_{{\textstyle {\scriptscriptstyle \mathrm {obs}}}}\), \([\gamma ^*_i]_{{\textstyle {\scriptscriptstyle \mathrm {obs}}}}\) and \([\gamma _i / \gamma ^{*}_i]_{{\textstyle {\scriptscriptstyle \mathrm {obs}}}}\) has a general behavior approximately similar to the power spectra obtained with the galaxy two-point correlation function and, by being sample size independent, they may be considered as alternative analytical tools to study the galaxy distribution.

High-energy and ultra-high-energy neutrinos: A Snowmass white paper

Astrophysical neutrinos are excellent probes of astroparticle physics and high-energy physics. With energies far beyond solar, supernovae, atmospheric, and accelerator neutrinos, high-energy and ultra-high-energy neutrinos probe fundamental physics from the TeV scale to the EeV scale and beyond. They are sensitive to physics both within and beyond the Standard Model through their production mechanisms and in their propagation over cosmological distances. They carry unique information about their extreme non-thermal sources by giving insight into regions that are opaque to electromagnetic radiation. This white paper describes the opportunities astrophysical neutrino observations offer for astrophysics and high-energy physics, today and in coming years.

The power of locality: primordial non-Gaussianity at the map level

Primordial non-Gaussianity is a sensitive probe of the inflationary era, with a number of important theoretical targets living an order of magnitude beyond the reach of current CMB constraints. Maps of the large-scale structure of the universe, in principle, have the raw statistical power to reach these targets, but the complications of nonlinear evolution are thought to present serious, if not insurmountable, obstacles to reaching these goals. In this paper, we will argue that the challenge presented by nonlinear structure formation has been overstated. The information encoded in primordial non-Gaussianity resides in nonlocal correlations of the density field at three or more points separated by cosmological distances. In contrast, nonlinear evolution only alters the density field locally and cannot create or destroy these long-range correlations. This locality property of the late-time non-Gaussianity is obscured in Fourier space and in the standard bispectrum searches for primordial non-Gaussianity. We therefore propose to measure non-Gaussianity in the position space maps of the large-scale structure. As a proof of concept, we study the case of equilateral non-Gaussianity, for which the degeneracy with late-time nonlinearities is the most severe. We show that a map-level analysis is capable of breaking this degeneracy and thereby significantly improve the constraining power over previous estimates.

The spectral geometry of de Sitter space in asymptotic safety

Within the functional renormalization group approach to Background Independent quantum gravity, we explore the scale dependent effective geometry of the de Sitter solution dS4. The investigation employs a novel approach whose essential ingredient is a modified spectral flow of the metric dependent d’Alembertian, or of similar hyperbolic kinetic operators. The corresponding one-parameter family of spectra and eigenfunctions encodes information about the nonperturbative backreaction of the dynamically gravitating vacuum fluctuations on the mean field geometry of the quantum spacetime. Used as a diagnostic tool, the power of the spectral flow method resides in its ability to identify the scale dependent subsets of field modes that supply the degrees of freedom which participate in the effective field theory description of the respective scale. A central result is that the ultraviolet of Quantum Einstein Gravity comprises far less effective degrees of freedom than predicted (incorrectly) by background dependent reasoning. The Lorentzian signature of dS4 is taken into account by selecting a class of renormalization group trajectories which are known to apply to both the Euclidean and a Lorentzian version of the approach. Exploring the quantum spacetime’s spatial geometry carried by physical fields, we find that 3-dimensional space disintegrates into a collection of coherent patches which individually can, but in their entirety cannot be described by one of the effective average actions occurring along the renormalization group trajectory. A natural concept of an entropy is introduced in order to quantify this fragmentation effect. Tentatively applied to the real Universe, surprising analogies to properties of the observed cosmic microwave background are uncovered. Furthermore, a set of distinguished field modes is found which, in principle, has the ability to transport information about the asymptotic fixed point regime from the ultraviolet, across almost the entire “scale history”, to cosmological distances in the observed Universe.

The ALMA REBELS Survey: The Cosmic H i Gas Mass Density in Galaxies at z ≈ 7

The neutral atomic gas content of individual galaxies at large cosmological distances has until recently been difficult to measure due to the weakness of the hyperfine H i 21 cm transition. Here we estimate the H i gas mass of a sample of main-sequence star-forming galaxies at z ∼ 6.5–7.8 surveyed for [C ii] 158 μm emission as part of the Reionization Era Bright Emission Line Survey (REBELS), using a recent calibration of the [C ii]-to-H i conversion factor. We find that the H i gas mass excess in galaxies increases as a function of redshift, with an average of M Hi /M ⋆ ≈ 10, corresponding to H i gas mass fractions of f Hi = M Hi /(M ⋆ + M Hi ) = 90%, at z ≈ 7. Based on the [C ii] 158 μm luminosity function (LF) derived from the same sample of galaxies, we further place constraints on the cosmic H i gas mass density in galaxies (ρ Hi ) at this redshift, which we measure to be . This estimate is substantially lower by a factor of ≈10 than that inferred from an extrapolation of damped Lyα absorber (DLA) measurements and largely depends on the exact [C ii] LF adopted. However, we find this decrease in ρ Hi to be consistent with recent simulations and argue that this apparent discrepancy is likely a consequence of the DLA sight lines predominantly probing the substantial fraction of H i gas in high-z galactic halos, whereas [C ii] traces the H i in the ISM associated with star formation. We make predictions for this buildup of neutral gas in galaxies as a function of redshift, showing that at z ≳ 5, only ≈10% of the cosmic H i gas content is confined in galaxies and associated with the star-forming ISM.

A Probabilistic Autoencoder for Type Ia Supernova Spectral Time Series

We construct a physically parameterized probabilistic autoencoder (PAE) to learn the intrinsic diversity of Type Ia supernovae (SNe Ia) from a sparse set of spectral time series. The PAE is a two-stage generative model, composed of an autoencoder that is interpreted probabilistically after training using a normalizing flow. We demonstrate that the PAE learns a low-dimensional latent space that captures the nonlinear range of features that exists within the population and can accurately model the spectral evolution of SNe Ia across the full range of wavelength and observation times directly from the data. By introducing a correlation penalty term and multistage training setup alongside our physically parameterized network, we show that intrinsic and extrinsic modes of variability can be separated during training, removing the need for the additional models to perform magnitude standardization. We then use our PAE in a number of downstream tasks on SNe Ia for increasingly precise cosmological analyses, including the automatic detection of SN outliers, the generation of samples consistent with the data distribution, and solving the inverse problem in the presence of noisy and incomplete data to constrain cosmological distance measurements. We find that the optimal number of intrinsic model parameters appears to be three, in line with previous studies, and show that we can standardize our test sample of SNe Ia with an rms of 0.091 ± 0.010 mag, which corresponds to 0.074 ± 0.010 mag if peculiar velocity contributions are removed. Trained models and codes are released at https://github.com/georgestein/suPAErnova.

A galaxy-driven model of type Ia supernova luminosity variations

ABSTRACT Type Ia supernovae (SNe Ia) are used as standardizable candles to measure cosmological distances, but differences remain in their corrected luminosities which display a magnitude step as a function of host galaxy properties such as stellar mass and rest-frame U−R colour. Identifying the cause of these steps is key to cosmological analyses and provides insight into SN physics. Here we investigate the effects of SN progenitor ages on their light-curve properties using a galaxy-based forward model that we compare to the Dark Energy Survey 5-yr SN Ia sample. We trace SN Ia progenitors through time and draw their light-curve width parameters from a bimodal distribution according to their age. We find that an intrinsic luminosity difference between SNe of different ages cannot explain the observed trend between step size and SN colour. The data split by stellar mass are better reproduced by following recent work implementing a step in total-to-selective dust extinction ratio (RV) between low- and high-mass hosts, although an additional intrinsic luminosity step is still required to explain the data split by host galaxy U−R. Modelling the RV step as a function of galaxy age provides a better match overall. Additional age versus luminosity steps marginally improve the match to the data, although most of the step is absorbed by the width versus luminosity coefficient α. Furthermore, we find no evidence that α varies with SN age.

Observational constraints on interacting dark energy models with multiple measurements of quasars

Dark energy and dark matter, two subjects of basic physics, have received a lot of attention in the 21st century. From the observational point of view, the interaction between dark energy and dark matter can significantly affect cosmological distances. This gives rise to the possibility of indirectly detecting such interaction through high-redshift cosmological probes. Theoretically, the introduction of interaction between dark energy and dark matter can assist in alleviating the coincidence problem of the standard cosmological model ($\Lambda$CDM model). Furthermore, this can provide a new method of studying the properties of dark matter particles. In this paper, based on the latest observations of multiple measurements of quasars (X-ray+UV quasars acting as standard candles, compact radio quasars acting as standard rulers) covering the redshift range of $0.04~<~z~<~5.1$ and baryonic acoustic oscillation between ($0.38~<~z~<~2.34$), we investigate the observational constraints on a variety of interacting dark energy models ($\gamma_d~$IDE model, $\gamma_m~$IDE model) and other cosmological models ($\Lambda$CDM model, XCDM model). The results provide us with a quantitative analysis of the possible interaction between dark energy and dark matter, as well as the possible range of the mass of dark matter particles. The joint analysis shows that: (1) Multiple measurements of quasars can provide more stringent constraints on the interacting dark energy models, which can further strengthen the potential of quasars acting as effective cosmological standard probes at higher redshifts; (2) In the framework of both $\gamma_m$IDE model and $\gamma_d$IDE model, the quasar data supports possible conversion of dark energy into dark matter at high redshift, which alleviates the coincidence problem to some extent. We also found that the interaction term is of a small value, which demonstrates the negligible interaction between dark matter and dark energy; (3) In the framework of $\Lambda$CDM model, which has shown the best consistency with quasar data, the density parameter of matter in the Universe is constrained at $\Omega_~m=0.317^{+0.007}_{-0.007}$, with the best-fit Hubble constant $H_0=68.177^{+0.497}_{-0.505}$ at 68.3% confidence level. These findings are consistent with the recent microwave background radiation (CMB) measurements from the Planck satellite; (4) If dark matter in the Universe exists in the form of scalar-field dark matter with $Z_2$ symmetry, we obtain the range of the mass of dark matter particles as $56~{\rm~GeV}\lesssim~m_S\lesssim~63~{\rm~GeV}$ or $m_S\gtrsim450~{\rm~GeV}$, based on the dark energy-dark matter coupling term from multiple measurements of quasars. Such conclusions agree well with the latest experimental results aimed at the direct detection of dark matter particles.

Using Neutrino Oscillations to Measure H0?

Recently, the idea of using neutrino oscillations to measure the Hubble constant was introduced. We show that such a task is unfeasible because for typical energies of cosmic neutrinos, oscillations average out over cosmological distances and so the oscillation probability depends only on the mixing angles.

Luminosity distribution of fast radio bursts from CHIME/FRB Catalog 1 by means of the updated Macquart relation

Fast radio bursts (FRBs) are extremely strong radio flares lasting several micro- to milliseconds and come from unidentified objects at cosmological distances, most of which are only seen once. Based on recently published data in the CHIME/FRB Catalog 1 in the frequency bands 400-800 MHz, we analyze 125 apparently singular FRBs with low dispersion measure (DM) and find that the distribution of their luminosity follows a lognormal form according to statistical tests. In our luminosity measurement, the FRB distance is estimated by using the Macquart relation which was obtained for 8 localized FRBs, and we find it still applicable for 18 sources after adding the latest 10 new localized FRBs. In addition, we test the validity of the luminosity distribution up to the Macquart relation and find that the lognormal form feature decreases as the uncertainty increases. Moreover, we compare the luminosity of these apparent non-repeaters with that of the previously observed 10 repeating FRBs also at low DM, noting that they belong to different lognormal distributions with the mean luminosity of non-repeaters being two times greater than that of repeaters. Therefore, from the two different lognormal distributions, different mechanisms for FRBs can be implied.

The Hubble Diagram: Jump from Supernovae to Gamma-ray Bursts

The Hubble diagram (HD) is a plot that contains a luminous distance modulus presented with respect to the redshift. The distance modulus–redshift relation of the most well-known “standard candles”, the type Ia supernovae (SN), is a crucial tool in cosmological model testing. In this work, we use the SN Ia data from the Pantheon catalogue to calibrate the Swift long gamma-ray bursts (LGRBs) as “standard candles” via the Amati relation. Thus, we expand the HD from supernovae to the area of the Swift LGRBs up to z∼8. To improve the quality of estimation of the parameters and their errors, we implement the Monte-Carlo uncertainty propagation method. We also compare the results of estimation of the Amati parameters calibrated by the SN Ia, and by the standard ΛCDM model and find no statistically significant distinction between them. Although the size of our LGRB sample is relatively small and the errors are high, we find this approach of expanding the cosmological distance scale promising for future cosmological tests.

Ultra Long Period Cepheids: Observation and Theory

Ultra Long Period Cepheids are becoming a very interesting and important topic thanks to the contribution that they can give to understanding the current tension existing between the early-universe and local Hubble constant measurements. These bright pulsating variables are observable up to cosmological distances (larger than 100 Mpc) allowing us, in principle, to measure the Hubble constant without the need for secondary indicators, thus reducing the possible systematic errors in the calibration of the extragalactic distance scale. The Ultra Long Period Cepheids also represent a useful tool for obtaining information on the star formation history of the host galaxy and a challenge for the evolutionary and pulsational models, particularly in the very metal poor regime. In this paper, the largest known ULP sample, consisting of 72 objects, including 10 new candidates, is analyzed to give an observational and theoretical overview of their role as distance indicators and of their evolutionary properties.

Optical and X-ray GRB Fundamental Planes as cosmological distance indicators

ABSTRACT Gamma-ray bursts (GRBs), can be employed as standardized candles, extending the distance ladder beyond Type Ia supernovae (SNe Ia, z = 2.26). We standardize GRBs using the three-dimensional (3D) Fundamental Plane relation (the Dainotti relation) among the rest-frame end time of the X-ray plateau emission, its corresponding luminosity, and the peak prompt luminosity. Combining SNe Ia and GRBs, we constrain ΩM = 0.299 ± 0.009 assuming a flat Λ cold dark matter (ΛCDM) cosmology with and without correcting GRBs for selection biases and redshift evolution. Using a 3D optical Dainotti correlation, we find this sample is as efficacious in the determination of ΩM as the X-ray sample. We trimmed our GRB samples to achieve tighter planes to simulate additional GRBs. We determined how many GRBs are needed as stand-alone probes to achieve a comparable precision on ΩM to the one obtained by SNe Ia only. We reach the same error measurements derived using SNe Ia in 2011 and 2014 with 142 and 284 simulated optical GRBs, respectively, considering the error bars on the variables halved. These error limits will be reached in 2038 and in 2047, respectively. Using a doubled sample (obtained by future machine learning approaches allowing a light-curve reconstruction and the estimates of GRB redshifts when z is unknown) compared to the current sample, with error bars halved we will reach the same precision as SNe Ia in 2011 and 2014, now and in 2026, respectively. If we consider the current SNe precision, this will be reached with 390 optical GRBs by 2054.

An efficient method for simulating light curves of cosmological microlensing and caustic crossing events

ABSTRACT A new window to observing individual stars and other small sources at cosmological distances was opened recently, with the detection of several caustic-crossing events in galaxy cluster fields. Many more such events are expected soon from dedicated campaigns with the Hubble Space Telescope and the James Webb Space Telescope. These events can not only teach us about the lensed sources themselves, such as individual high-redshift stars, star clusters, or accretion discs, but through their light curves they also hold information about the point-mass function of the lens, and thus, potentially, the composition of dark matter. We present here a simple method for simulating light curves of such events, i.e. the change in apparent magnitude of the source as it sweeps over the net of caustics generated by microlenses embedded around the critical region of the lens. The method is recursive and so any reasonably sized small source can be accommodated, down to sub-solar scales, in principle. We compare the method, which we dub Adaptive Boundary Method, with other common methods such as simple inverse ray shooting, and demonstrate that it is significantly more efficient and accurate in the small-source and high-magnification regime of interest. A python version of the code is made publicly available in an open-source fashion for simulating future events.