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Register a new userRecursion relations for gravitational lensing
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Ben David Normann Mr.
Publication date
2019
fields
Physics
and research's language is
English
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The weak gravitational lensing formalism can be extended to the strong lensing regime by integrating a nonlinear version of the geodesic deviation equation. The resulting roulette expansion generalises the notion of convergence, shear and flexion to arbitrary order. The independent coefficients of this expansion are screen space gradients of the optical tidal tensor which approximates to the usual lensing potential in the weak field limit. From lensed images, knowledge of the roulette coefficients can in principle be inverted to reconstruct the mass distribution of a lens. In this paper, we simplify the roulette expansion and derive a family of recursion relations between the various coefficients, generalising the Kaiser-Squires relations beyond the weak-lensing regime.
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We discuss the prospects of gravitational lensing of gravitational waves (GWs) coming from core-collapse supernovae (CCSN). As the CCSN GW signal can only be detected from within our own Galaxy and the local group by current and upcoming ground-based GW detectors, we focus on microlensing. We introduce a new technique based on analysis of the power spectrum and association of peaks of the power spectrum with the peaks of the amplification factor to identify lensed signals. We validate our method by applying it on the CCSN-like mock signals lensed by a point mass lens. We find that the lensed and unlensed signal can be differentiated using the association of peaks by more than one sigma for lens masses larger than 150 solar masses. We also study the correlation integral between the power spectra and corresponding amplification factor. This statistical approach is able to differentiate between unlensed and lensed signals for lenses as small as 15 solar masses. Further, we demonstrate that this method can be used to estimate the mass of a lens in case the signal is lensed. The power spectrum based analysis is general and can be applied to any broad band signal and is especially useful for incoherent signals.
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Gravitational waves at suitable frequencies can resonantly interact with a binary system, inducing changes to its orbit. A stochastic gravitational-wave background causes the orbital elements of the binary to execute a classic random walk, with the variance of orbital elements growing with time. The lack of such a random walk in binaries that have been monitored with high precision over long time-scales can thus be used to place an upper bound on the gravitational-wave background. Using periastron time data from the Hulse-Taylor binary pulsar spanning ~30 years, we obtain a bound of h_c < 7.9*10^(-14) at ~10^(-4) Hz, where h_c is the strain amplitude per logarithmic frequency interval. Our constraint complements those from pulsar timing arrays, which probe much lower frequencies, and ground-based gravitational-wave observations, which probe much higher frequencies. Interesting sources in our frequency band, which overlaps the lower sensitive frequencies of proposed space-based observatories, include white-dwarf/supermassive black-hole binaries in the early/late stages of inspiral, and TeV scale preheating or phase transitions. The bound improves as (time span)^(-2) and (sampling rate)^(-1/2). The Hulse-Taylor constraint can be improved to ~3.8*10^(-15) with a suitable observational campaign over the next decade. Our approach can also be applied to other binaries, including (with suitable care) the Earth-Moon system, to obtain constraints at different frequencies. The observation of additional binary pulsars with the SKA could reach a sensitivity of h_c ~ 3*10^(-17).
Recently, the LIGO-Virgo Collaboration (LVC) concluded that there is no evidence for lensed gravitational waves (GW) in the first half of the O3 run, claiming We find the observation of lensed events to be unlikely, with the fractional rate at $mu>2$ being $3.3times 10^{-4}$. While we agree that the chance of an individual GW event being lensed at $mu>2$ is smaller than $10^{-3}$, the number of observed events depends on the product of this small probability times the rate of mergers at high redshift. Observational constraints from the stochastic GW background indicate that the rate of conventional mass BBH mergers (8 < M (M$_{odot}$) < 15) in the redshift range 1<z< 2 could be as high as O($10^7$) events per year, more than sufficient to compensate for the intrinsically low probability of lensing. To reach the LVC trigger threshold these events require high magnification, but would still produce up to 10 to 30 LVC observable events per year. Thus, all the LVC observed ordinary stellar mass BBH mergers from this epoch must be strongly lensed. By adopting low-rates at high redshift, LVC assumes that lensed events can not be taking place, thus incorrectly assigning them a closer distance and higher masses by a factor of a few (typically 2 to 5). The LVC adopted priors on time delay are in tension with the distribution of observed time delays in lensed quasars. Pairs of events like GW190421-GW190910 and GW190424-GW190910, which are directly assigned a probability of zero by LVC, should be instead considered as prime candidates to be strongly lensed GW pairs, since their separation in time is consistent with observations of time delays in lensed quasars. Correcting for the LVC wrong Bayesian priors, maximum merger rate of conventional mass BBH in 1<z<2, and gravitational lensing time-delay model, reverses the LVC conclusions and supports the strong gravitational lensing hypothesis.
We present a framework for studying gravitational lensing in spherically symmetric spacetimes using 1+1+2 covariant methods. A general formula for the deflection angle is derived and we show how this can be used to recover the standard result for the Schwarzschild spacetime.
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