# Discovering intermediate-mass black hole lenses through gravitational wave lensing

## Abstract

Intermediate-mass black holes are the missing link that connects supermassive and stellar-mass black holes and are key to understanding galaxy evolution. Gravitational waves, like photons, can be lensed by these objects. In the diffraction limit a gravitational wave can self interfere when bent by a mass of Schwarzschild radius comparable to its wavelength, modifying the observed waveform. The required point mass scale is for significant wave effect to occur for compact binaries in the LIGO band, corresponding to intermediate-mass black hole deflectors. We perform a mock data study using lensed gravitational waves to investigate detectability of these intermediate-mass black hole deflectors in the LIGO-Virgo detector network. In particular, we simulate gravitational waves with different source distributions lensed by an astrophysical population of intermediate-mass black holes and use standard LIGO tools to infer the properties of these lenses. We show that one can discover intermediate-mass black holes through their lensing effects on gravitational waves in the LIGO-Virgo detector network. Moreover, we demonstrate that one can detect intermediate-mass black holes at 98% confidence level in of the lensed cases, subject to the astrophysical lens population.

gw short = GW , long = gravitational wave , short-plural = s, \DeclareAcronymbh short = BH , long = black hole , short-plural = s, \DeclareAcronymimbh short = IMBH , long = intermediate-mass black hole , short-plural = s, \DeclareAcronymsmbh short = SMBH , long = Supermassive Black Hole , short-plural = s , \DeclareAcronymdm short = DM , long = dark matter , short-plural = , \DeclareAcronymwimp short = WIMP , long = Weakly Interacting Massive Particle , short-plural = , \DeclareAcronympsd short = PSD , long = power spectral density , short-plural = , \DeclareAcronymsnr short = SNR , long = signal-to-noise , short-plural = , \DeclareAcronymligo short = LIGO , long = Laser Interferometer Gravitational-Wave Observatory , short-plural = , \DeclareAcronymlalsuite short = LALSuite , long = LSC Algorithm Library Suite , short-plural = , \DeclareAcronymsis short = SIS , long = singular isothermal sphere , short-plural = ,

## I Introduction

The existence of stellar-mass and supermassive black holes has become widely accepted thanks to X-ray observations of X-ray binary systems ([1]; [2]) and measurements of the orbits of stars in the center of the Milky Way ([3]; [4]; [5]). While the existence of \acpsmbh is widely accepted, their formation is a mystery due to a \acbh mass gap in the range (). Black holes in this mass range are called \acpimbh. In this range, we have yet to observe \acpbh but expect to see a transition from stellar-mass to supermassive \acpbh ([6]; [7]). Finding this link is crucial to understanding the formation of \acpsmbh and galaxies.

Only indirect evidence for \acpimbh exists ([8]), but there are multiple active detection efforts. A recent study focusing on mapping the potential of the globular cluster 47 Tucanae through pulsar timing in combination with N-body simulations casts indirect evidence towards an \acimbh in the center of the cluster ([9]). However, the potential for this cluster was derived from N-body simulations subject to a degree of model uncertainty (see [10], for a review). Other forms of searches involve locating X-ray and radio emissions from accretion onto \acpimbh, finding tidal disruption events, looking for \acimbh imprints in molecular clouds and microlensing experiments ([11]); for a review, see ([8]). Despite the many efforts to detect \acpimbh, the evidence is still inconclusive.

Gravitational lensing is the bending of light, waves or particles near concentrated mass distributions. Lensing events probe the \acimbh’s potential, opening a promising avenue to detect them. On 14 September 2015, the first gravitational wave event was observed with \acligo ([12]), opening a new window to the Universe. Similarly to light, \acpgw can be influenced by gravitational lensing ([13]; [14]; [15]; [16]; [17]; [18]; [19]). When the wavelength of \acpgw is comparable to the Schwarzschild radius of the lens, diffraction effects become relevant to the treatment of the lens effect ([18]). In the \acligo band, these wave effects happen in the \acimbh mass range.

Cao et al. 2014 ([20]) investigated the effect of lensing on \acgw parameter estimation using Markov-Chain Monte Carlo to study the lens degeneracy between lens parameters in the \acligo framework. However, only three different point mass lens scenarios (, and lens) and a non-spinning, inspiral-only waveform marginalized over time and phase were used; the waveform did not encode the merger of two \acpbh or spin information. In this work, we consider the full problem using realistic inspiral merger ringdown waveform (IMRPhenomPv2 ([21])) which includes the spin of the binary and is utilized in real \acligo searches. The merger-ringdown of the binary merger changes the lensing results for the waveform at higher frequencies where half of the \acsnr contribution comes from, for stellar-mass binaries. The morphology of the signal at higher frequencies is also more complicated (e.g. [22]), and by the inclusion of spin in our waveform model we account for the possibility that spin precession of the binary (see [23]) could mimic lensing. In addition, we infer the lens parameters using LALInference ([24]), a parameter inference software developed for the purpose of \acligo searches ([24]), and include \acligo and Virgo detection network at design sensitivity in our analysis ([25]; [26]). Furthermore, we study the parameter constraints in realistic lensing scenarios by including a wide range of lens masses (redshifted lens mass ) and the effect of diffraction in our lens model. Instead of a single lensing scenario, we inject a wide range of lensed signals drawn from an astrophysical distribution of binaries and lenses. We also investigate the ability to discriminate between finite and point size lens models, and comment on the effect of a potential host galaxy on the results.

Our results indicate that lensed \acpgw can be used to infer the mass of \acpimbh, providing a novel avenue to detect \acpimbh. In particular, if a \acgw is lensed through a potential induced by an \acimbh in our parameter range, we can claim detection with confidence in of the cases. Moreover, we show that we can distinguish astrophysical fine structures larger than , while structures smaller than this can be effectively treated as point mass lenses. Finally, we discuss the implications of our results on detection of \acpimbh.

## Ii Point mass lens

Consider a system composed of a source emitting \acpgw, a lens, and a distant observer. The source is close (sub-parsec scale) to the line-of-sight between the lens and the observer for lensing to occur; we denote this distance with . The angular diameter distances along the line-of-sight between source-lens, source-observer, and lens-observer, are denoted as , , and , respectively. \Acpimbh can be approximated as point mass lenses ([19]). Given that we ignore the near horizon contribution to the lensing effect, the lensed waveform is ([18]; [19])

(1) |

(2) | ||||

where is the waveform without lensing, is complex gamma function, is confluent hypergeometric function of the first kind, is dimensionless frequency, is the redshifted lens mass, is the source position, is a normalization constant (Einstein radius for point mass lens), and and are the lens mass and redshift, respectively. The magnification function includes the information of the time delay and is not to be confused with its geometric optics counterpart. To calculate the magnification function , we construct a lookup table, and retrieve its values by bilinear interpolation; the error between the table and the exact solution is less than 0.1%. For the \acgw waveform, we use IMRPhenomPv2 model, which includes the whole binary inspiral-merger-ringdown phase ([27]). This assumes an isolated point lens, but we also discuss the effect of external shear and host galaxy.

## Iii Simulations, nested sampling, and post-processing

We inject \acgw signals from an astrophysical population of binary sources lensed by \acpimbh and infer the properties of the \acimbh lens using nested sampling. Following ([28]), the astrophysical distribution of the binary source is uniform in component masses, dimensionless spin magnitude, and volume; isotropic in spin directions and in sky location. We assume an isolated lens, so that the source distribution of lenses follows ([19]) and our lens population is uniform in source parameter squared and redshifted lens mass , chosen to include the lower \acimbh mass range. If the lens is no longer isolated, then the distribution requires corrections; however, investigating such corrections requires numerical simulations and is outside the scope of this work. \Acligo requires an \acsnr of 8 for claiming a detection, and signals with \acsnr greater than 32 are rare ([29]). Therefore, we choose our unlensed , distributed uniformly. Source masses are fixed to investigate the effect of mass ratio on parameter inference.

We infer the lens mass using nested sampling algorithm (LALInference) ([30]). The lens mass and redshift are fully degenerate with each other. However, we obtain a lower bound for the lens mass since \acligo events are detectable at redshifts ([31]). Therefore, we choose the probability to indicate a successful detection of \acimbh. We show an example redshifted lens mass posterior distribution recovered from an injected \acgw that passes through a lens of mass (Fig. 1). The posterior peaks around the injected value and all samples are above the \acimbh mass limit. In our analysis, this posterior is classified as detection.

## Iv Lens Parameter Constraints

We find detections over a wide range of lens masses (), but find no clear rising or lowering trend in detections with higher or lower lens mass (Fig. 2). Of these, we find around % of detected \acpimbh with relatively small redshifted lens masses (; Fig. 2). Approximately 20% of lenses are detectable in our parameter range. However, there are two false alarms with masses lower than , which is statistically acceptable at 98 % confidence level, given that we have over 100 detections; these are no longer detections at confidence level.

In addition to redshifted lens mass, we characterize the effect of source position on the detectability of \acpimbh. The source position is proportional to the horizontal distance from the line-of-sight. Because smaller source positions correspond to larger lens effects, we expect to put better constraints on the \acimbh masses at small . Indeed, we detect a more substantial number of \acpimbh at low source positions, where more than 55% of them are in the range for all source masses (Fig. 2). Meanwhile, we find that there are also detections at relatively large source positions () but the number decreases for increasing position. The source position at can be translated back to the displacement from the line-of-sight. Assuming typical lens-to-source distance , lens distance and source distance , we have . Hence, the line-of-sight distance where we detect \acpimbh is likely sub-parsec.

We find that \acpimbh may be detected as long as the median inferred \acsnr is above and we detect across the \acsnr range , even at relatively small \acsnr thresholds. To put this into the context of the current \acligo detections, all of the confirmed detections have had a network inferred \acsnr higher than (see the first observing run summary ([32])).

In all four source mass realizations, we detect around 20% of the \acpimbh at a 98% confidence level. Among the detected signals, we also compare the Bayes factors between the lensed model and the unlensed model. The Bayes factors for the lensed hypothesis are significantly ( times) larger than the unlensed hypothesis for more than 70% of the signals. We have also analyzed the first \acgw event GW150914, finding no evidence of lensing (Bayes factors both being the same up to 4th significant digit for the lensed and unlensed case).

In conclusion, we find detections across , and , and find that higher lens masses, smaller source positions and higher \acpsnr are favored.

## V Discriminating between point and finite-size lens

Other small astrophysical lens objects could mimic \acimbh lenses. We study a finite-size \acsis model to test our ability to discriminate between finite and point lenses using \acpgw. If the size is small enough, the object will collapse into a \acbh. The \acsis model represents the approximate mass distribution of an extended astrophysical object and its magnification function ([19])

(3) | ||||

where , is the redshifted mass inside the Einstein radius , is a characteristic dispersion velocity of the model and is the normalized impact parameter. We expect the \acsis model to be indistinguishable from an \acimbh model due to mass screening effect when the Einstein radius is small.

In order to compare the \acsis and the point lens model, we compute the match ([33]; [34]) between two waveforms and maximized over time, phase and amplitude. For this comparison, we simulate \acpgw from a (30,30) source oriented in the overhead direction and compare the match between the signals lensed by an \acsis and a point lens.

We consider different pairs of , and maximize the by non-linear least squares fitting and classify as distinguishable in \acligo waveform (following [35]). We show that the \acsis lens and point lens can be discriminated when redshifted lens mass , shown as a match lower than 97% in Fig. 3. The source positions and show higher match for all redshifted lens masses because very small source positions cause only a total magnification of the signal, while very large cause only small lens effect.

The SIS model has an intrinsic length scale, which is the Einstein radius. Since astrophysical structures with diameters smaller than show high match (Fig. 3), they can not be discriminated from point lenses. Indeed, our results suggest we could distinguish an \acimbh from a globular cluster (half-mass radius at pc scale ([36])), but not smaller than AU structures.

## Vi Discussion and conclusions

We demonstrate that it is possible to discover \acpimbh in the LIGO-Virgo network by analyzing \acpgw lensed by these \acpbh even for relatively small lens masses (). We find that in of cases the effect of lensing is strong enough to discover an \acimbh with 98% confidence in our parameter range. Moreover, we find that we can discriminate between \acsis and point lens models when the Einstein radius of the \acsis is larger than . In particular, our results suggest that we may discriminate an \acimbh lens from an extended astrophysical object, but it is hard to distinguish between \acimbh lenses and compact objects of similar mass. However, there is currently no conclusive evidence of compact objects with masses greater than .

In our results, we do not account for shear effects by host galaxies. However, it is important to discuss its effect on the results, as compact objects are typically discovered as part of a galaxy. Such shear magnifies the \acgw signal and introduces a degeneracy between the inferred lens mass and shear magnification. In particular, external shear enlarges the point lens’ Einstein radius, stretching it along the deflection field of the host galaxy and changing the lens time delay (see [37]). Consequently, the effective mass of the lens becomes owing to its dependence on the Einstein radius. The stretching is modest when the magnification by galaxy is reasonably low (), and the new radius is larger by a factor of , with being the tangential magnification component. The lensing probability at high magnification goes as . As a consequence, typical magnifications are modest, between . Taking such typical shear, we would need to measure at least 300 lens to confirm the event as an \acimbh for typical magnifications. Meanwhile, the magnification in shear would boost the \acgw event rates.

Our results imply that we can detect \acpimbh within \acligo data. However, there is also an interesting prospect of detecting stellar mass \acpbh with \acpgw. \acligo may not be sensitive enough to constrain the properties of lenses and the event rate required for \acpgw lensed by lenses with high enough \acsnr may be too low, but there is an interesting prospect of detecting these \acpbh with future third-generation detectors such as the Telescope and Cosmic Explorer (see [38]; [39]; [40]; [41]); these prospects are discussed by ([42]).

In conclusion, we have shown that lensing of \acpgw by \acpimbh is detectable over a wide range of parameters and that a detection of a point mass lens of mass higher than in principle would warrants a discovery of \acpimbh. In the future, we will expand our study on the effect of different lensing models, and mixed models with \acpbh and surrounding matter; for example, it is essential to investigate lens models with globular clusters containing \acpimbh and lenses admixed in shear.

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