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# Limiting the abundance of LIGO/Virgo black holes with microlensing
observations of quasars of finite size

A. ESTEBAN-GUTIÉRREZ  Instituto de Astrofísica de Canarias, Vía Láctea S/N, La
Laguna 38205, Tenerife, Spain  Departamento de Astrofísica, Universidad de la
Laguna, La Laguna 38206, Tenerife, Spain  E. MEDIAVILLA  Instituto de
Astrofísica de Canarias, Vía Láctea S/N, La Laguna 38205, Tenerife, Spain
Departamento de Astrofísica, Universidad de la Laguna, La Laguna 38206,
Tenerife, Spain  J. JIMÉNEZ-VICENTE  Departamento de Física Teórica y del
Cosmos, Universidad de Granada, Campus de Fuentenueva, 18071 Granada, Spain
Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada,
18071 Granada, Spain  N. AGÜES-PASZKOWSKY  Instituto de Astrofísica de
Canarias, Vía Láctea S/N, La Laguna 38205, Tenerife, Spain  Departamento de
Astrofísica, Universidad de la Laguna, La Laguna 38206, Tenerife, Spain  J. A.
MUÑOZ  Departamento de Astronomía y Astrofísica, Universidad de Valencia,
46100 Burjassot, Valencia, Spain.  Observatorio Astronómico, Universidad de
Valencia, E-46980 Paterna, Valencia, Spain  S. HEYDENREICH  Argelander-
Institut für Astronomie, Auf dem Hügel 71, 53121, Bonn, Germany

###### Abstract

We present a simple but general argument that strongly limits the abundance of
Primordial Black Holes (PBHs) (or other unknown population of compact objects)
with masses similar to those determined by LIGO/Virgo from BH binary mergers.
We show that quasar microlensing can be very sensitive to the mass of the
lenses, and that it is able to distinguish between stars and BHs of high mass,
when the finite size of the source is taken into account. A significant
presence of massive BHs would produce frequent high flux magnifications
(except for unrealistically large sources) which have been very rarely
observed. On the contrary, a typical stellar population would induce flux
magnifications consistent with the observations. This result excludes PBHs (or
any type of compact object) in the mass range determined by LIGO/Virgo as the
main dark matter constituents in the lens galaxies.

(Primordial Black Holes — gravitational lensing: micro)

##  1 Introduction

The discovery of gravitational waves from binary black hole (BBH) mergers by
the LIGO/Virgo collaboration (cf. GWTC-1 and GWTC-2 by Abbott et al. 2019a and
Abbott et al. 2020a) with masses higher than previously expected for BHs of
stellar origin (but also the low effective spins of the components) renewed in
the last years the interest on the possibility that some of these BHs were of
primordial origin, and even that these primordial black holes of intermediate
mass (20$-$200 $M_{\odot}$), not excluded by galactic microlensing, could
constitute a significant fraction of the dark matter in the Universe (Carr &
Kühnel, 2020).

Quasar microlensing (Chang & Refsdal 1979; Wambsganss 2006) provides an
alternative path to study the abundance, not only of BHs, but also of any type
of compact object. The description of this phenomenon is simple: an
intervening galaxy (the lens) deflects the light from a distant quasar forming
several images. These images are usually seen through the lens galaxy and, as
far as the matter distribution in the galaxy is not smooth but granulated in
compact objects (stars, BHs, etc.), the light beams can suffer new secondary
deflections producing several microimages (that can not be resolved by
telescopes). The primary observational effect of this image splitting is a
change in the flux of the images (microlensing flux magnification). The
relevant question we will address here is: Can the amplitude and frequency of
these microlensing flux magnifications inform us about the mass of the
microlenses? Or, more specifically: can microlensing observations
unequivocally reveal the presence of LIGO/Virgo BHs?

Previous studies based on quasar microlensing (Mediavilla et al. 2017,
Esteban-Gutiérrez et al. 2020) do not support the existence of a significant
population of intermediate mass BH. However, neither these negative results
from microlensing, nor the existence of new proposed paths to explain the
stellar formation of BH of intermediate mass (see, e.g., Abbott et al. 2020b),
seem to have had a major impact in stopping the speculation about the
existence of a population of PBHs that could account for the dark matter. This
fact may be related to the indirect approach and rather complex statistical
modelling involved in those microlensing studies which we intend to avoid
here.

As we will discuss, the key parameter that determines the differences in
amplitude and frequency of microlensing magnifications corresponding to
microlenses populations of different masses is the size of the lensed object,
which in our case is the size of the quasar accretion disk at (rest frame) UV
wavelengths. The motivation of this work is, then, to show using broadly
applicable arguments, that quasar microlensing is very sensitive to the mass
of BHs in the range of masses of LIGO/Virgo detections. Specifically, we will
show that there is a strong difference in predicted microlensing
magnifications for stars and LIGO/Virgo BHs when a source of finite size is
considered. We leave a thorough statistical analysis with quantitative
estimates of the limits in the abundance of BHs for an accompanying paper.

The letter is organized as follows. In §2 we analyze the impact of the size of
the source in the probability distribution of observing a given microlensing
magnification and compare the results with available microlensing
observations. In §3 we discuss the differences in microlensing of quasars by
stars or by LIGO/Virgo BHs. Finally, in this same section (§3), we summarize
the conclusions.

##  2 Results

We want to study the impact of the mass of the microlenses in both the
amplitude and frequency of magnifications, focusing on to what extent a
population of LIGO/Virgo type BHs can account for the unrevealed dark matter
in the lens galaxies. To do that, we are going to calculate the probability of
measuring a given microlensing flux magnification for two different
populations of microlenses, both with a 20% of matter in stars (with
$m=0.2M_{\odot}$, typical of old stellar populations), but one with the
remaining 80% of matter in smooth dark matter111This is the typical baseline
scenario (Schechter et al. 2014, Jiménez-Vicente et al. 2015a, 2015b) in which
the population of microlenses contributes to 20% to the total mass density
(including dark matter), and the remaining 80% is in the form of a smooth
distribution of matter., and the other in typical LIGO/Virgo BHs with
$m=30M_{\odot}$. Following the standard procedure, we obtain the Probability
Density Function (PDF) of the flux magnification from the histogram of the
magnification maps222Obtained by tracing light rays backwards from the image
to the source through the random distribution of microlenses (Kayser et al.,
1986). In these calculations, we have considered a typical lens system333We
take $\kappa=\gamma=0.45$, $z_{l}=0.5$, and $z_{s}=2$. with a mean flux
magnification of 10. We use a pixel size of 0.2 light days for the
magnification maps, very much smaller than typical quasar sizes. The PDFs of
measuring a certain flux magnification for both mass distributions are shown
in Figure 1 (left). Both PDFs match very well in the negative magnitude region
(corresponding to microlensing flux increase), although they show clear
differences in the positive wing (corresponding to flux decrease).

Figure 1: Probability distributions of microlensing flux magnifications for a
population of 20% of stars of $0.2M_{\odot}$ plus 80% of smooth dark matter
(orange curve) and a population of 20% of stars of $0.2M_{\odot}$ plus 80% of
LIGO/Virgo type BHs of $30M_{\odot}$ (blue curve). Left panel for a source of
0.2 light days. Right panel for a source of 10 light days. Scaling has been
chosen to give unit area under the PDF curves. The shadowed blue marks the
region containing $\sim$70% of observed microlensing magnifications (Pooley et
al. 2007) i.e., with the ordinate chosen to enclose 70% of the probability
(see text).

It is worth wondering, at this point, how well do these probability
distributions compare to actual observations. The optical fluxes (normalized
to their expected model values) reported by Pooley et al. (2007) (their Table
5) for nine444We have eliminated Q2237+0305 from their sample in this
comparison, as this system is produced by a nearby lens, and has nearly 100%
of the mass density in form of compact objects. quadruple lens systems are
very well suited for a quick comparison. While according to Pooley et al.
(2007) a $\sim$70% of the observed magnifications fall in the (-0.3,0.3)
magnitudes range (region shadowed in blue in Figure 1) our simulations for an
infinitesimal source predict only a $\sim$30% of measurements in this range.
As we will see below, this strong discrepancy is originated by the fact that
real quasars have sizes much larger than the pixel size used in our
calculations. To take this fact into account, we can calculate the
corresponding probability distributions for an extended source. We have chosen
here to model the source with a Gaussian brightness profile, but this is known
to have little effect on the microlensing magnification probability
distributions (Mortonson et al. 2005, Muñoz et al. 2016). We consider a source
size of 10 light-days, which approximates the Einstein radius (the natural
length scale of lensing) of the $m=0.2M_{\odot}$ stars, yet it is much smaller
than the Einstein radius of the BHs ($\sim$112 lt-days). Nevertheless,
conclusions remain essentially unchanged for any reasonable source size
compatible with observations.

The resulting probability distributions for this larger source are shown in
Figure 1 (right). In this Figure, it can be clearly seen that while the PDF of
the stars becomes significantly narrower (therefore predicting less extreme
magnifications in better agreement with observations) the PDF of the BHs has
changed much less (still predicting over $\sim$60% of magnifications outside
the (-0.3,0.3) magnitude range). The reason for this different behavior is
indeed rather simple: for a finite source size, the gradients in the
magnification maps produced by lenses with an Einstein radius comparable to
the source size get blurred/averaged, while this effect is minimized if the
source has a negligible size compared to the Einstein radius of the lens.

We would like to stress again that although we have chosen here some values of
the parameters to illustrate the principle, this result is, nevertheless, very
general, and has little dependence on the specific choice as long as it is in
reasonable agreement with observations.

##  3 Discussion and Conclusions

An immediate result of the previous section is that we need to consider sizes
of quasars comparable to the Einstein radius of the lenses in order to narrow
the predicted PDF to approximate (even roughly) the experimental histogram of
microlensing magnifications. If BHs were to account for a significant fraction
of the mass in the lens galaxies, this would imply source sizes of $\gtrsim
100$ light-days, which is absolutely discarded by reverberation mapping
estimates of the size of the accretion disks of quasars (see e.g. Edelson et
al. 2015, Fausnaugh et al. 2016, Jiang et al. 2017, Cackett et al. 2018,
Homayouni et al. 2019 and Yu et al. 2020). On the contrary, for typical
estimated sizes of quasar accretion disks at these wavelengths of a few light
days, the PDF predicted by a population of stars is reasonably consistent with
the observations. Therefore, if the dark matter of the lens galaxies were in
the form of BHs of the masses detected by LIGO/Virgo, much larger microlensing
magnifications should have been regularly observed. As this is not the case,
an explanation to dark matter based on this kind of objects can safely be
discarded.

On the other hand, although we have used here for comparison the estimates of
microlensing magnifications from Pooley et al. (2007) (because the reported
observed fluxes normalized to model predictions provide directly the flux
magnifications), the rarity of high flux magnifications is also extensively
confirmed by the statistics of differential flux magnifications between images
(Mediavilla et al. 2009, Fian et al. 2016, Fian et al. 2018) and by the
microlensing induced variability observed in the light curves of lensed quasar
images (see, e.g. Mediavilla et al. 2016 and references therein).

Regarding the possibility of a partial explanation of dark matter in terms of
LIGO/Virgo BHs, we have also calculated the PDFs of populations including less
than an 80% of BHs, just to a 10%, confirming the significant over-prediction
of non observed large microlensing magnifications, even for this small
abundance of BHs. In an accompanying paper we discuss the likelihood of a
population of BHs in all the range of BHs abundances.

From the above analysis, we can reach the following conclusions:

  1. 1.

Observed microlensing magnification statistics in optical observations of
lensed quasars can only be explained if the source (the quasar accretion disk)
size is comparable to the Einstein radius of the microlenses.

  2. 2.

For reasonable values of the size of the observed quasars, if the dark matter
were in form of compact objects with masses in the range of the BHs detected
by LIGO/Virgo, much larger magnifications should have been frequently
observed. The absence of such observations is therefore a strong proof that
the dark matter in lens galaxies is not formed by these objects.

The above conclusions are based on very general arguments which do not depend
on specific models. Notwithstanding, a quantitative full Bayesian statistical
modelling of a mixed populations of stars and BHs setting quantitative limits
to the abundance of the latter is included in an accompanying paper.

We thank the anonymous referee for ideas and comments, which greatly
contribute to enhance the scope of our paper. This research was supported by
the Spanish MINECO with the grants AYA2016-79104-C3-1-P and
AYA2016-79104-C3-3-P. J.J.V. is supported by the project AYA2017-84897-P
financed by the Spanish Ministerio de Economía y Competividad and by the Fondo
Europeo de Desarrollo Regional (FEDER), and by project FQM-108 financed by
Junta de Andalucía. AEG thanks the support from grant FPI-SO from the Spanish
Ministry of Economy and Competitiveness (MINECO) (research project
SEV-2015-0548-17-4 and predoctoral contract BES-2017-082319).

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