# Research

We present FANTASY (Finally A Numerical Trajectory Algorithm both Straightforward and sYmplectic), a user-friendly, open-source symplectic geodesic integrator written in Python. FANTASY is designed to work "out-of-the-box" and does not require anything from the user aside from the metric and the initial conditions for the geodesics. FANTASY efficiently computes derivatives up to machine precision using automatic differentiation, allowing the integration of geodesics in arbitrary space(times) without the need for the user to manually input Christoffel symbols or any other metric derivatives. Further, FANTASY utilizes a Hamiltonian integration scheme that doubles the phase space, where two copies of the particle phase space are evolved together. This technique allows for an integration scheme that is both explicit and symplectic, even when the Hamiltonian is not separable. FANTASY comes prebuilt with second and fourth order schemes, and is easily extendible to higher order schemes. FANTASY also includes an automatic Jacobian calculator that allows for coordinate transformations to be done automatically.

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The 2017 Event Horizon Telescope (EHT) observations of the central source in M87 have led to the first measurement of the size of a black-hole shadow. This observation offers a new and clean gravitational test of the black-hole metric in the strong-field regime. We show analytically that spacetimes that deviate from the Kerr metric but satisfy weak-field tests can lead to large deviations in the predicted black-hole shadows that are inconsistent with even the current EHT measurements. We use numerical calculations of regular, parametric, non-Kerr metrics to identify the common characteristic among these different parametrizations that control the predicted shadow size. We show that the shadow-size measurements place significant constraints on deviation parameters that control the second post-Newtonian and higher orders of each metric and are, therefore, inaccessible to weak-field tests. The new constraints are complementary to those imposed by observations of gravitational waves from stellar-mass sources.

**Detecting Black Hole Occultations by Stars with Space Interferometric Telescopes**

We show that the occultation of Sagittarius A* by stars can be detected with space-based or space-ground very-long-baseline-interferometers (SVLBIs), with an expected event rate that is high due to relativistic precession. We compute the tell-tale signal of an occultation event, and describe methods to flag non-occultation events that can masquerade as the signal.

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Closure phases are critical in astronomical interferometry. However, their uncertainties are difficult to compute numerically. We provide a method to efficiently compute interferometric closure phase distributions in terms of an approximate distribution that is valid in the low signal-to-noise ratio regime. This is done by first showing that the true phase distribution is well approximated by the von Mises distribution, then performing a convolution of three von Mises distributions. The resulting approximation is superior than the normal distribution for all signal-to-noise ratios and, being fully analytic, allow for fast computations in statistical algorithms.

**First M87 Event Horizon Telescope Results**** **

I am a member of the Event Horizon Telescope Collaboration, a world-wide effort to capture resolved images of black holes. Recently, we announced our first result: a "photograph" of the black hole at the center of M87 resolved at event horizon scales!

For this project, the collaboration won the 2020 Breakthrough Prize for Fundamental Physics!

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We investigate the ability of ground based gravitational wave observatories to detect gravitational wave lensing events caused by stellar mass lenses. We show that LIGO and Virgo possess the sensitivities required to detect lenses with masses as small as 30 Solar Masses provided that the gravitational wave is observed with a signal-to-noise ratio of ~30. Third generation observatories will allow detection of gravitational wave lenses with masses of ~1 Solar Mass. Finally, we discuss the possibility of lensing by multiple stars, as is the case if the gravitational radiation is passing through galactic nucleus or a dense star cluster.

**Evolution of the Black Hole Mass Function in Star Clusters from Multiple Mergers**

We investigate the effects of black hole mergers in star clusters on the black hole mass function. As black holes are not produced in pair-instability supernovae, it is suggested that there is a dearth of high mass stellar black holes. This dearth generates a gap in the upper end of the black hole mass function. Meanwhile, parameter fitting of X-ray binaries suggests the existence of a gap in the mass function under 5 solar masses. We show, through evolving a coagulation equation, that black hole mergers can appreciably fill the upper mass gap, and that the lower mass gap generates potentially observable features at larger mass scales. We also explore the importance of ejections in such systems and whether dynamical clusters can be formation sites of intermediate mass black hole seeds.

Using the black hole merger rate inferred from LIGO, we calculate the abundance of tightly bound binary black holes in the Milky Way galaxy. Binaries with a small semimajor axis originate at larger separations through conventional formation mechanisms and evolve as a result of gravitational wave emission. We find that LISA could detect them in the Milky Way. We also identify possible X-ray signatures of such binaries.

One avenue for testing the no-hair theorem is obtained through timing a pulsar orbiting close to a black hole and fitting for quadrupolar effects on the time-of-arrival of pulses. If deviations from the Kerr quadrupole are measured, then the no-hair theorem is invalidated. To this end, we derive an expression for the light travel time delay for a pulsar orbiting in a black-hole spacetime described by the Butterworth-Ipser metric, which has an arbitrary spin and quadrupole moment. We consider terms up to the quadrupole order in the black-hole metric and derive the time-delay expression in a closed analytic form.

Newton’s theorem of revolving orbits states that one can multiply the angular speed of a Keplerian orbit by a factor k by applying a radial inverse cubed force proportional to (1 − k^{2}). In this paper we derive an extension of this theorem in general relativity, valid for the motion of massive particles in any static, spherically symmetric metrics. We verify the Newtonian limit of this extension and demonstrate that there is no such generalization for rotating metrics. Further we also extend the theory to the case of charged particles in the Einstein-Maxwell and Kaluza-Klein theories.

We show that an interferometer moving at a relativistic speed relative to a point source of light offers a sensitive probe of acceleration. Such an accelerometer contains no moving parts, and is thus more robust than conventional ”mass-on-a-spring” accelerometers. In an interstellar mission to AlphaCentauri, such an accelerometer could be used to measure the masses of exoplanets and their host stars as well as test theories of modified gravity.

**Mapping the Dynamics of Cold Gas around Sgr A* through 21 cm Absorption**

The presence of a circumnuclear stellar disk around Sgr A* and megamaser systems near other black holes indicates that dense neutral disks can be found in galactic nuclei. We show that depending on their inclination angle, optical depth, and spin temperature, these disks could be observed spectroscopically through 21 cm absorption. Related spectroscopic observations of Sgr A* can determine its HI disk parameters and the possible presence of gaps in the disk. Clumps of dense gas similar to the G2 could could also be detected in 21 cm absorption against Sgr A* radio emission.

**Probing the Spacetime Around Supermassive Black Holes with Ejected Plasma Blobs**

Millimeter-wavelength VLBI observations of the supermassive black holes in Sgr A* and M87 by the Event Horizon Telescope could potentially trace the dynamics of ejected plasma blobs in real time. We demonstrate that the trajectory and tidal stretching of these blobs can be used to test general relativity and set new constraints on the mass and spin of these black holes.

**Pulsar Timing Constraints on Cumulative and Individual Mass of Stars in the Galactic Center**

We consider the time derivatives of the period P of pulsars at the Galactic Center due to variations in their orbital Doppler shifts. We show that in conjunction with a measurement of a pulsar’s proper motion and its projected separation from the supermassive black hole, Sgr A*,measuring two of the period time derivatives sets a constraint that allows for the recovery of the complete six phase space coordinates of the pulsar’s orbit, as well as the enclosed mass within the orbit. Thus, one can use multiple pulsars at different distances from Sgr A* to determine the radial mass distribution of stars and stellar remnants at the Galactic center. Furthermore, we consider the effect of passing stars on the pulsar’s period derivatives and show how it can be exploited to measure the characteristic stellar mass in the Galactic Center.

**Measuring the X-ray background in the reionization era with first generation 21 cm experiments**

The X-ray background during the epoch of reionization is currently poorly constrained. We demonstrate that it is possible to use first generation 21 cm experiments to calibrate it. Using the semi-numerical simulation, 21cmFAST, we calculate the dependence of the 21 cm power spectrum on the X-ray background flux. Comparing the signal to the sensitivity of the Murchison Widefield Array (MWA) we find that in the redshift interval z =8-14 the 21 cm signal is detectable for certain values of the X-ray background. We show that there is no degeneracy between the X-ray production efficiency and the Lyα production efficiency and that the degeneracy with the ionization fraction of the intergalactic medium can be broken.