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Register a new userMultipole representation of a generic gravitational lens
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We consider gravitational lensing by a generic extended mass distribution. We represent the static external gravitational field of the lens as a potential via an infinite set of symmetric trace free (STF) moments. We discuss the possibility of determining the physical characteristics of the lens including its shape, orientation and composition via gravitational lensing. To do that, we consider STF multipole moments for several well-known solids with uniform density. We discuss the caustics formed by the point spread function (PSF) of such lenses, and also the view seen by an imaging telescope placed in the strong interference region of the lens. We show that at each STF order, all the bodies produce similar caustics that are different only by their magnitudes and orientations. Furthermore, there is ambiguity in determining the shape of the lens and its mass distribution if only a limited number of moments are used in the model. This result justifies the development of more comprehensive lens models that contain a greater number of multipole moments. At the same time, inclusion of higher multipole moments leads to somewhat limited improvements as their contributions are suppressed by corresponding powers of the small parameter $(R/b)^ell$, where $R$ characterizes the bodys physical size and $b$ is the impact parameter, resulting in a weaker signature from those multipole moments in the PSF. Thus, in realistic observations there will always be some ambiguity in the optical properties of a generic lens, unless the properties of the lens can be determined independently, as in the case of the solar gravitational lens (SGL). Our results are novel and offer new insight into gravitational lensing by realistic astrophysical systems.
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We report on the initial results obtained with an image convolution/deconvolution computer code that we developed and used to study the image formation capabilities of the solar gravitational lens (SGL). Although the SGL of a spherical Sun creates a greatly blurred image, knowledge of the SGLs point-spread function (PSF) makes it possible to reconstruct the original image and remove the blur by way of deconvolution. We discuss the deconvolution process, which can be implemented either with direct matrix inversion or with the Fourier quotient method. We observe that the process introduces a ``penalty in the form of a reduction in the signal-to-noise ratio (SNR) of a recovered image, compared to the SNR at which the blurred image data is collected. We estimate the magnitude of this penalty using an analytical approach and confirm the results with a series of numerical simulations. We find that the penalty is substantially reduced when the spacing between image samples is large compared to the telescope aperture. The penalty can be further reduced with suitable noise filtering, which can yield ${cal O}(10)$ or better improvement for low-quality imaging data. Our results confirm that it is possible to use the SGL for imaging purposes. We offer insights on the data collection and image processing strategies that could yield a detailed image of an exoplanet within image data collection times that are consistent with the duration of a realistic space mission.
We study gravitational lensing by a generic extended mass distribution. For that, we consider the diffraction of electromagnetic (EM) waves by an extended, weakly aspherical, gravitating object. We account for the static gravitational field of this lens by representing its exterior potential in the most generic form, expressed via an infinite set of symmetric trace free (STF) tensor multipole mass moments. This yields the most general form of the gravitational phase shift, which allows for a comprehensive description of the optical properties of a generic gravitational lens. We found that at each order of the STF moments, the gravitational phase shift is characterized by only two parameters: a magnitude and a rotation angle that characterize the corresponding caustics, which form in the point spread function (PSF) of the lens. Both of these parameters are uniquely expressed in terms of the transverse-trace free (TT) projections of the multipole moments on the lens plane. Not only does this result simplifies the development of physically consistent models of realistic lenses, it also drastically reduces the number of required parameters in the ultimate model. To gain physical insight and to help with the interpretation of the results obtained, we established the correspondence of the gravitational phase shift expressed via the TT-projected STF multipole mass moments and its representation via spherical harmonics. For axisymmetric mass distributions, the new results are consistent with those that we obtained in previous studies. For arbitrary mass distributions, our results are novel and offer new insight into gravitational lensing by realistic astrophysical systems. These findings are discussed in the context of ongoing astrophysical gravitational lensing investigations as well as observations that are planned with the solar gravitational lens (SGL).
We study image formation with the solar gravitational lens (SGL). We consider a point source that is positioned at a large but finite distance from the Sun. We assume that an optical telescope is positioned in the image plane, in the focal region of the SGL. We model the telescope as a convex lens and evaluate the intensity distribution produced by the electromagnetic field that forms the image in the focal plane of the convex lens. We first investigate the case when the telescope is located on the optical axis of the SGL or in its immediate vicinity. This is the region of strong interference where the SGL forms an image of a distant source, which is our primary interest. We derive analytic expressions that describe the progression of the image from an Einstein ring corresponding to an on-axis telescope position, to the case of two bright spots when the telescope is positioned some distance away from the optical axis. At greater distances from the optical axis, in the region of weak interference and that of geometric optics, we recover expressions that are familiar from models of gravitational microlensing, but developed here using a wave-optical treatment. We discuss applications of the results for imaging and spectroscopy of exoplanets with the SGL.
We compute the effect of scattering gravitational radiation off the static background curvature, up to second order in Newton constant, known in literature as tail and tail-of-tail processes, for generic electric and magnetic multipoles. Starting from the multipole expansion of composite compact objects, and as expected due to the known electric quadrupole case, both long- and short-distance (UV) divergences are encountered. The former disappears from properly defined observables, the latter are renormalized and their associated logarithms give rise to a classical renormalization group flow. UV divergences alert for incompleteness of the multipolar description of the composite source, and are expected not to be present in a UV-complete theory, as explicitly derived in literature for the case of conservative dynamics. Logarithmic terms from tail-of-tail processes associated to generic magnetic multipoles are computed in this work for the first time.
We propose a novel method to test the consistency of the multipole moments of compact binary systems with the predictions of General Relativity (GR). The multipole moments of a compact binary system, known in terms of symmetric and trace-free tensors, are used to calculate the gravitational waveforms from compact binaries within the post-Newtonian (PN) formalism. For nonspinning compact binaries, we derive the gravitational wave phasing formula, in the frequency domain, parametrizing each PN order term in terms of the multipole moments which contribute to that order. Using GW observations, this {it{parametrized multipolar phasing}} would allow us to derive the bounds on possible departures from the multipole structure of GR and hence constrain the parameter space of alternative theories of gravity. We compute the projected accuracies with which the second generation ground-based detectors, such as Advanced Laser Interferometer Gravitational-wave Observatory (LIGO), the third generation detectors such as Einstein Telescope and Cosmic Explorer, as well as space-based detector Laser Interferometer Space Antenna (LISA) will be able to measure these multipole parameters. We find that while Advanced LIGO can measure the first two or three multipole coefficients with good accuracy, Cosmic Explorer and Einstein Telescope may be able to measure the first four multipole coefficients which enter the phasing formula. Intermediate mass ratio inspirals, with mass ratio of several tens, in the frequency band of planned space-based LISA mission should be able to measure all the seven multipole coefficients which appear in the 3.5PN phasing formula. Our finding highlights the importance of this class of sources for probing the strong-field gravity regime. The proposed test will facilitate the first probe of the multipolar structure of Einsteins general relativity.
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