11institutetext:  Kapteyn Astronomical Institute, University of Groningen, PO
Box 800, 9700 AV Groningen, The Netherlands
11email<EMAIL_ADDRESS>22institutetext: Bernoulli Institute for Mathematics,
Computer Science and Artificial Intelligence, University of Groningen, 9700AK
Groningen, The Netherlands 33institutetext: Department of Astrophysics,
American Museum of Natural History, Central Park West and 79th Street, NY
10024-5192, USA 44institutetext: Department of Physics and Astronomy, Lehman
College of the CUNY, Bronx, NY 10468, USA 55institutetext: Institute of
Physics, Laboratory of Astrophysique, École Polytechnique Fédérale de Lausanne
(EPFL), Observatoire de Sauverny, 1290 Versoix, Switzerland 66institutetext:
Kavli Institute for Particle Astrophysics and Cosmology and Department of
Physics, Stanford University, Stanford, CA 94305, USA

# Probing compact dark matter objects with microlensing in gravitationally
lensed quasars††thanks: Light curves presented in this paper are only
available in electronic form at the CDS via anonymous ftp to
cdsarc.cds.unistra.fr (130.79.128.5) or via https://cdsarc.cds.unistra.fr/cgi-
bin/qcat?J/A+A/

P. Awad Probing compact dark matter objects with microlensing in
gravitationally lensed quasars††thanks: Light curves presented in this paper
are only available in electronic form at the CDS via anonymous ftp to
cdsarc.cds.unistra.fr (130.79.128.5) or via https://cdsarc.cds.unistra.fr/cgi-
bin/qcat?J/A+A/Probing compact dark matter objects with microlensing in
gravitationally lensed quasars††thanks: Light curves presented in this paper
are only available in electronic form at the CDS via anonymous ftp to
cdsarc.cds.unistra.fr (130.79.128.5) or via https://cdsarc.cds.unistra.fr/cgi-
bin/qcat?J/A+A/Probing compact dark matter objects with microlensing in
gravitationally lensed quasars††thanks: Light curves presented in this paper
are only available in electronic form at the CDS via anonymous ftp to
cdsarc.cds.unistra.fr (130.79.128.5) or via https://cdsarc.cds.unistra.fr/cgi-
bin/qcat?J/A+A/Probing compact dark matter objects with microlensing in
gravitationally lensed quasars††thanks: Light curves presented in this paper
are only available in electronic form at the CDS via anonymous ftp to
cdsarc.cds.unistra.fr (130.79.128.5) or via https://cdsarc.cds.unistra.fr/cgi-
bin/qcat?J/A+A/ J. H. H. Chan Probing compact dark matter objects with
microlensing in gravitationally lensed quasars††thanks: Light curves presented
in this paper are only available in electronic form at the CDS via anonymous
ftp to cdsarc.cds.unistra.fr (130.79.128.5) or via
https://cdsarc.cds.unistra.fr/cgi-bin/qcat?J/A+A/Probing compact dark matter
objects with microlensing in gravitationally lensed quasars††thanks: Light
curves presented in this paper are only available in electronic form at the
CDS via anonymous ftp to cdsarc.cds.unistra.fr (130.79.128.5) or via
https://cdsarc.cds.unistra.fr/cgi-bin/qcat?J/A+A/Probing compact dark matter
objects with microlensing in gravitationally lensed quasars††thanks: Light
curves presented in this paper are only available in electronic form at the
CDS via anonymous ftp to cdsarc.cds.unistra.fr (130.79.128.5) or via
https://cdsarc.cds.unistra.fr/cgi-bin/qcat?J/A+A/Probing compact dark matter
objects with microlensing in gravitationally lensed quasars††thanks: Light
curves presented in this paper are only available in electronic form at the
CDS via anonymous ftp to cdsarc.cds.unistra.fr (130.79.128.5) or via
https://cdsarc.cds.unistra.fr/cgi-bin/qcat?J/A+A/ M. Millon Probing compact
dark matter objects with microlensing in gravitationally lensed
quasars††thanks: Light curves presented in this paper are only available in
electronic form at the CDS via anonymous ftp to cdsarc.cds.unistra.fr
(130.79.128.5) or via https://cdsarc.cds.unistra.fr/cgi-bin/qcat?J/A+A/Probing
compact dark matter objects with microlensing in gravitationally lensed
quasars††thanks: Light curves presented in this paper are only available in
electronic form at the CDS via anonymous ftp to cdsarc.cds.unistra.fr
(130.79.128.5) or via https://cdsarc.cds.unistra.fr/cgi-bin/qcat?J/A+A/Probing
compact dark matter objects with microlensing in gravitationally lensed
quasars††thanks: Light curves presented in this paper are only available in
electronic form at the CDS via anonymous ftp to cdsarc.cds.unistra.fr
(130.79.128.5) or via https://cdsarc.cds.unistra.fr/cgi-bin/qcat?J/A+A/Probing
compact dark matter objects with microlensing in gravitationally lensed
quasars††thanks: Light curves presented in this paper are only available in
electronic form at the CDS via anonymous ftp to cdsarc.cds.unistra.fr
(130.79.128.5) or via https://cdsarc.cds.unistra.fr/cgi-bin/qcat?J/A+A/ F.
Courbin Probing compact dark matter objects with microlensing in
gravitationally lensed quasars††thanks: Light curves presented in this paper
are only available in electronic form at the CDS via anonymous ftp to
cdsarc.cds.unistra.fr (130.79.128.5) or via https://cdsarc.cds.unistra.fr/cgi-
bin/qcat?J/A+A/Probing compact dark matter objects with microlensing in
gravitationally lensed quasars††thanks: Light curves presented in this paper
are only available in electronic form at the CDS via anonymous ftp to
cdsarc.cds.unistra.fr (130.79.128.5) or via https://cdsarc.cds.unistra.fr/cgi-
bin/qcat?J/A+A/ E. Paic Probing compact dark matter objects with microlensing
in gravitationally lensed quasars††thanks: Light curves presented in this
paper are only available in electronic form at the CDS via anonymous ftp to
cdsarc.cds.unistra.fr (130.79.128.5) or via https://cdsarc.cds.unistra.fr/cgi-
bin/qcat?J/A+A/Probing compact dark matter objects with microlensing in
gravitationally lensed quasars††thanks: Light curves presented in this paper
are only available in electronic form at the CDS via anonymous ftp to
cdsarc.cds.unistra.fr (130.79.128.5) or via https://cdsarc.cds.unistra.fr/cgi-
bin/qcat?J/A+A/

The microlensing signal in the light curves of gravitationally lensed quasars
can shed light on the dark matter (DM) composition in their lensing galaxies.
Here, we investigate a sample of six lensed quasars from the most recent and
best COSMOGRAIL observations: HE 1104$-$1805, HE 0435$-$1223, RX J1131$-$1231,
WFI 2033$-$4723, PG 1115$+$080, and J1206$+$4332, yielding a total of eight
microlensing light curves, when combining independent image pairs and
typically spanning ten years. We explore the microlensing signals to determine
whether the standard assumptions on the stellar populations are sufficient to
account for the amplitudes of the measured signals or whether additional
microlenses are needed. We use the most detailed lens models to date from the
H0LiCOW/TDCOSMO collaboration to derive the microlensing parameters, such as
the convergence, shear, and stellar/dark matter mass fraction at the position
of the quasar images. We use these parameters to generate simulated
microlensing light curves. Finally, we propose a methodology based on the
Kolmogorov-Smirnov test to verify whether the observed microlensing amplitudes
in our data are compatible with the most standard scenario, whereby galaxies
are composed of stars as compact bodies and smoothly distributed DM. Given our
current sample, we show that the standard scenario cannot be rejected, in
contrast with previous results by Hawkins (2020a), claiming that a population
of stellar mass primordial black holes (PBHs) is necessary to explain the
observed amplitude of the microlensing signal in lensed quasar light curves.
We further estimate the number of microlensing light curves needed to
effectively distinguish between the standard scenario with stellar
microlensing and a scenario that describes that all the DM contained in
galaxies is in the form of compact objects such as PBHs, with a mean mass of
$0.2M_{\odot}$. We find that about 900 microlensing curves from the Rubin
Observatory will be sufficient to discriminate between the two extreme
scenarios at a 95% confidence level.

###### Key Words.:

gravitational lensing: micro – quasars: general – dark matter

##  1 Introduction

Dark matter (DM) is a hypothetical form of matter that is thought to account
for $\sim 80\%$ of mass in a galaxy. Although we generally see a broad
consensus regarding the gravitational effects of DM, its nature remains
largely unknown. Overall, DM is theorized to either take the form of a
particle (for a review, see e.g., Boyarsky et al., 2019) that is smoothly
distributed or the form of compact objects such as massive compact halo
objects (MACHOs; Alcock et al., 1992; Aubourg et al., 1994). The search for
the true form that would accurately describe the nature of DM is essential, as
this query lies at the foundation of efforts to build cosmological models of
the Universe.

One of the phenomena that has been used as a means to study the composition of
galaxies is microlensing. In strongly lensed quasars, the gravitational
lensing effect produces multiple images of the quasar source. Subsequently,
microlensing is induced by compact objects such as stars in the lensing
galaxies (Chang & Refsdal, 1979; Gott, 1981; Irwin et al., 1989): the light
from the source passing by these compact objects is further split to produce
”micro-images” that are separated from each other by a few $\mu$-arcseconds
(see a review in Wambsganss, 2006). These resulting micro-images cannot be
resolved, but they can still be detected through the anomalous fluxes in the
strongly lensed images. Microlensing induces variations in the quasar
brightness on timescales of several months to years (Mosquera & Kochanek,
2011), enabling us to study the properties of lensing galaxies, such as their
dark matter fractions (Schechter & Wambsganss, 2004), stellar mass functions
(Jiménez-Vicente & Mediavilla, 2019), and the properties of quasar accretion
disks (Morgan et al., 2008; Eigenbrod et al., 2008b; Morgan et al., 2010;
Jiménez-Vicente et al., 2014; Cornachione et al., 2020).

One aspect that has been generally assumed for the origin of microlensing
signals relates to the stellar populations inhabiting the halos of the lensing
galaxies (Schild, 1990; Falco et al., 1991; Kundic & Wambsganss, 1993).
However, in observed light curves of strongly lensed quasars, the stellar
distributions are not always sufficient for explaining microlensing flux
variations (Mediavilla et al., 2009; Pooley et al., 2012). This resulting
issue is, in fact, tied to the nature of DM particles. It has been argued that
compact bodies in the form of primordial black holes (PBHs) can offer a better
explanation for the microlensing events in the light curves of lensed quasars
than stellar populations (Hawkins, 2020a, b). These works have been reinforced
with the recent detection of gravitational waves from a black hole merger
(Abbott et al., 2016), which lends support to the idea that PBHs may make up a
significant fraction of the DM in galaxy halos (Bird et al., 2016; Sasaki et
al., 2016; Byrnes et al., 2018). In other works, galactic microlensing has
been used to set the limits of PBH abundances for different ranges of masses
(e.g., Alcock et al., 2001; Tisserand et al., 2007; Wyrzykowski et al., 2011;
Blaineau et al., 2022). Accordingly, microlensing is a prominent technique for
the exploration of PBH abundance and possibly provides an insight into the
nature of DM.

In this work, we build on what has been presented in Hawkins (2020a, b), with
the intention of exploring whether stars in lensing galaxies of lensed quasars
can indeed account for their microlensing signals. If the stellar
distributions are proven to be insufficient, then a fraction of another form
of compact bodies is needed to explain the variability in the observed
microlensing signals of lensed quasars. Compared to the study of Hawkins
(2020a, b), here we take a larger sample of lensed quasar systems that
contains light curves with long baselines measured by
COSMOGRAIL111http://www.cosmograil.org (Courbin et al., 2005; Millon et al.,
2020a). To explore the effect of stellar distributions on the microlensing
curves of these lens systems, we utilized the latest lensing model parameters
provided by the H0LiCOW222http://h0licow.org/TDCOSMO333http://www.tdcosmo.org
collaboration (Suyu et al., 2017; Millon et al., 2020b). Using the simulated
microlensing signals, we compared the amplitude to that seen in their
observational counterparts.

The paper is structured as follows. In Section 2, we describe the data used in
this work, including the chosen sample of lensed quasars, followed by the
extracted microlensing signals in observational and simulated light curves.
The description of the methodology for our statistical analysis is then
provided in Section 3. We present our results in Section 4 with the proposal
for future applications using data from the Rubin observatory in Section 5. In
Section 6, we discuss the assumptions in our analysis as well as the obtained
results. Our conclusions are given in Section 7. Throughout this work, we use
a flat-$\Lambda$CDM cosmology, with $H_{0}=70\leavevmode\nobreak\ {\rm
km\leavevmode\nobreak\ s^{-1}\leavevmode\nobreak\ Mpc^{-1}}$,
$\Omega_{m}=0.3,$ and $\Omega_{\Lambda}=0.7$.

##  2 Data

Our goal is to investigate the nature of the compact objects in lensing
galaxies that are responsible for the presence of microlensing events in the
light curves of lensed quasars. The sample of lensed quasar systems is
described in Section 2.1, followed by the methods used to extract the
microlensing signals of the observational data in Section 2.2. We then
describe in Section 2.3 the simulation methodology and setup that we use for
our comparison with the observational data.

###  2.1 Lensed quasar sample

To build a sample of lensed quasars, we looked through the most recent light
curve measurements of COSMOGRAIL (Millon et al., 2020a) and selected the
systems that have light curve measurements for longer than five years, as well
as those with lensing parameters modeled by the H0LiCOW and TDCOSMO
collaborations. In Table 4, we list the lensing parameters at the position of
each lensed quasar image, including the total convergence, $\kappa$, the
stellar convergence, $\kappa_{*}$, and the total shear, $\gamma$. Based on
this criteria, our sample consists of five lens systems, namely: HE
0435$-$1223, RX J1131$-$1231, WFI 2033$-$4723, PG 1115$+$080, and J1206$+$4332
(hereafter, HE0435, RXJ1131, WFI2033, PG1115, and J1206, respectively). To
this sample, we added the system HE 1104$-$1805 (hereafter, HE1104) to allow
for a comparison with the work of Hawkins (2020a). The derivation of the
lensing parameters for the latter system is explained in Section 2.3, which
follows similar procedures as all the other systems modeled by TDCOSMO.

Table 1: Properties of the chosen lensed quasar systems along with the
references for their lensing parameters.

System |  $z_{s}$ |  $z_{l}$ |  $R_{E}$ [$10^{16}$cm] |  $R_{1/2}$ [$10^{16}$cm] |  Image |  $\quad\kappa$ |  $\quad\kappa_{*}$ |  $\quad\gamma$ |  $\mathcal{M}$ [mag] |  Reference
---|---|---|---|---|---|---|---|---|---|---
HE1104 |  2.32 |  0.73 |  1.933 |  0.668 |  A |  0.648 |  0.125 |  0.605 |  $-1.540$ |  The current work
B |  0.332 |  0.019 |  0.290 |  $-1.103$
HE0435 |  1.69 |  0.46 |  2.406 |  0.885 |  A |  0.473 |  0.164 |  0.358 |  $-2.063$ |
B |  0.630 |  0.227 |  0.540 |  $-2.026$ |  Chen et al. (2019)
C |  0.494 |  0.165 |  0.327 |  $-2.066$
D |  0.686 |  0.260 |  0.575 |  $-1.586$ |
RXJ1131 |  0.66 |  0.29 |  2.081 |  1.685 |  A |  0.526 |  0.226 |  0.410 |  $-3.118$ |
B |  0.459 |  0.199 |  0.412 |  $-1.926$ |  Chen et al. (2019)
C |  0.487 |  0.201 |  0.306 |  $-2.276$
D |  0.894 |  0.519 |  0.807 |  $-0.493$ |
WFI2033 |  1.66 |  0.66 |  1.939 |  0.898 |  B |  0.445 |  0.145 |  0.208 |  $-1.443$ |  Rusu et al. (2020)
C |  0.792 |  0.396 |  0.538 |  $-1.522$
PG1115 |  1.72 |  0.31 |  2.959 |  0.872 |  B |  0.502 |  0.166 |  0.811 |  $-0.968$ |  Chen et al. (2019)
C |  0.356 |  0.072 |  0.315 |  $-1.252$
J1206 |  1.79 |  0.85 |  1.671 |  0.843 |  A |  0.656 |  0.095 |  0.685 |  $-1.137$ |  Birrer et al. (2019)
B |  0.401 |  0.020 |  0.364 |  $-1.613$

444From left to right, we give: the name of the system in the sample, source
redshift ,$z_{s}$, lens redshift, $z_{l}$, Einstein radius, $R_{E}$, half-
light radius of the lensed quasar disk, $R_{1/2}$, as computed in Eq. 5,
chosen images for each system, total convergence $\kappa$, stellar
convergence, $\kappa_{*}$, total shear, $\gamma$, macro-magnification,
$\mathcal{M,}$ using Eq. 2, and the reference for the lens models. The
Einstein radius of the microlenses $R_{E}$ is calculated following Eq. 4, with
$\langle M_{*}\rangle=0.2M_{\odot}$ and assuming a Salpeter IMF.

The observed light curves contain not only the signal of microlensing in
lensing galaxies, but also the intrinsic variability of quasar sources. In
order to extract the microlensing signals, we removed the intrinsic variation
of the quasar by shifting the curves of a pair of lensed images with their
time delay and subtracting them (see Section 2.2 for more). We refer to the
resulting difference light curve the ”microlensing curve” in the rest of this
paper. We used only independent pairs of light curves from each system to
avoid accounting twice for the same microlensing event, which would then
appear in all microlensing curves computed relative to a given lensed image.
In the cases of doubly lensed quasars (doubles): HE1104 and J1206, there is
only one pair available, so we obtained only one microlensing curve. The
quadruply lensed quasars (quads) have two independent pairs of images, thus
providing us with two independent microlensing curves. In the case of WFI2033
and PG1115, which are both quads in fold configuration, the two closest images
are not resolved in the monitoring data. Their image As are actually a blend
of two bright images, which makes the microlensing signals of the unresolved
image intractable. Thus, we only considered images B and C for these two
systems. Finally, we first considered images A and B for HE0435, along with
images A and C for RXJ1131, in order to match the choice of Hawkins (2020a).
Since there are an additional two images, we included the image pairs of C and
D for HE0435 and of B and D for RXJ1131, respectively. In total, our sample
consists of eight independent pairs of lensed images that made up the basis of
our subsequent analysis. The considered image pairs are listed in Table 2 and
their microlensing curves are shown in Fig. 1.

Figure 1:  Microlensing curves obtained from the COSMOGRAIL observations. The macro-magnifications computed from the lens models have been subtracted. The curves are half-yearly binned with the weighted mean and the standard deviation as the error bars, shown as black points. The original unaveraged measurements are shown in blue.  Table 2: Microlensing amplitudes and p-values, $p$ (obtained in this work) and $p_{\rm H20}$ obtained by Hawkins (2020a). System |  pair |  $\Delta m_{\rm obs}$ [mag] |  $p$ |  $p_{\rm H20}$
---|---|---|---|---
HE1104 |  A – B |  0.195 |  0.762 |  0.046
HE0435 |  A – B |  0.826 |  0.377 |  0.050
HE0435 |  C – D |  0.340 |  0.378 |  –
RXJ1131 |  A – C |  1.231 |  0.190 |  0.000
RXJ1131 |  B – D |  1.103 |  0.143 |  –
WFI2033 |  B – C |  0.161 |  0.787 |  0.131
PG1115 |  B – C |  0.053 |  0.610 |  –
J1206 |  A – B |  0.245 |  0.283 |  –

Figure 2:  Microlensing simulation for HE 0435$-$1223\. Magnification maps of
images A (top left) and B (middle left) are generated using microlenses with
$\langle M_{*}\rangle=0.2$ following the Salpeter initial mass function within
the mass range of $0.06M_{\odot}$ and $6.46M_{\odot}$. The convolved maps of
images A (top right) and B (middle right) are obtained with a Gaussian kernel
for the source half-light radius as listed in Table 4. Each map has
$57.3\leavevmode\nobreak\ \times\leavevmode\nobreak\ R_{E}$ on a side (or 2000
pixels). A few trajectories (white lines) are randomly drawn with a transverse
velocity of 600 km/s and a length equal to the observational duration (5338
days). A pair of simulated light curves (red circles) is shown on the bottom
panel. The corresponding difference light curve $\delta m$ yields the
microlensing amplitude $\Delta m\approx 0.75$ mag, following our definition in
Sect. 3.

###  2.2 Microlensing light curves from observations

We can obtain microlensing signals by shifting a pair of observed light curves
with its respective time delay. For example, given a pair of lensed images
labeled $i$ and $j$, assuming that image $j$ occurs with a time delay, $\Delta
t$, we can describe the light curve for each image as a function of time, $t,$
in the unit of magnitude:

$\displaystyle S_{i}(t)$ $\displaystyle=I(t)+\mathcal{M}_{i}+m_{i}(t),$ (1)
$\displaystyle S_{j}(t)$ $\displaystyle=I(t-\Delta
t)+\mathcal{M}_{j}+m_{j}(t-\Delta t),$

where $I(t)$ is the quasar intrinsic variation, $m(t)$ is the microlensing
variations, and $\mathcal{M}$ is the macro-magnification induced by the
lensing setup. We can express the macro-magnification factor in the unit of
magnitude as the inverse of the Jacobian of the lensing matrix (Schneider et
al., 1992):

$\mathcal{M}=-2.5\log_{10}\Bigg{(}\Big{\lvert}{\frac{1}{(1-\kappa)^{2}-\gamma^{2}}}\Big{\rvert}\Bigg{)}.$
(2)

After shifting the light curve of image $j$ by the time delay, $\Delta t$, we
subtract the induced macro-magnification, $\mathcal{M,}$ in each image and
compute the difference between the signals, $S_{i}$ and $S_{j}$. Since the
intrinsic signal of the source is the same in both lensed images, the
corresponding term will cancel out upon taking the difference between the two
signals of the pair, so that we are able to retain the difference between the
microlensing contributions from each image. In other words, from Eq. 1, we can
derive a microlensing curve, $\delta m,$ using a pair of lensed images:

$\displaystyle\delta m(t)$ $\displaystyle=S_{i}(t)-S_{j}(t+\Delta
t)-\left(\mathcal{M}_{i}-\mathcal{M}_{j}\right)$ (3)
$\displaystyle=m_{i}(t)-m_{j}(t).$

We employed PyCS3555https://cosmograil.gitlab.io/PyCS3/ (Tewes et al., 2013;
Millon et al., 2020c), a curve-shifting python package, to fit, shift, and
subtract the light curves. In this work, we are only interested in the long-
term microlensing variations attributed to stars passing in front of the
quasar images and modulating the microlensing magnification. These variations
occur on a time scale of months to years. Hence, we re-binned our data half-
yearly to reduce the photometric noise within the fluctuating signals. We
calculated the weighted mean of each bin, taking the weights on both axes as
the inverse of the error on the magnitude measurements for each point in a
given bin. The error for each bin represents the standard deviation of all the
photometric measurements within that bin. Our original and binned
observational microlensing curves are illustrated in Fig. 1 by the blue and
black data points, respectively.

###  2.3 Microlensing light curves from simulations

In order to test if stars in lensing galaxies can explain the observed
microlensing signal, we simulated light curves drawn randomly from
microlensing magnification maps for each quasar image. We generated the
magnification maps using GPU-D, an inverse ray-shooting code on the GPU
(Vernardos & Fluke, 2013) that efficiently computes networks of lensing
microcaustics in the source plane. This requires an estimate of the total
projected mass, $\kappa$, shear, $\gamma$, and projected fraction of mass
under the stellar form, $\kappa_{*}$, at the position of the quasar images.

For the quadruply imaged quasars, we considered a composite mass model for the
lens to infer $\kappa$, $\kappa_{*}$, and $\gamma$, following Suyu et al.
(2014) and Chen et al. (2019). This is also the methodology adopted by the
H0LiCOW (Wong et al., 2020) and TDCOSMO collaborations (e.g., Millon et al.,
2020b). In these works, the lens mass has two components. The first represents
the stellar mass, namely, a fit to the 2D projected light distribution of the
lens, scaled by a radially constant mass-to-light ratio. The second is a
standard Navarro-Frenk-White (NFW) profile (Navarro et al., 1996). The stellar
and dark mass components are fitted jointly and the mass-to-light ratio of the
stellar component is a free parameter during this fit. The resulting values
for the models are given in Table 4, along with the reference of the papers
that specifically studied these lenses.

For HE1104, which only has two lensed images, a composite model has too many
degrees of freedom. We instead used a simpler power-law elliptical mass
distribution with shear (PEMD + shear), as implemented in the Lenstronomy
package (Birrer & Amara, 2018). In this model, the stellar and dark mass
components cannot be treated separately. We therefore followed the approach of
Auger et al. (2009), who modeled a large sample of (lensing) early-type
galaxies (ETG) with power-law models and found that the mean fraction of
stellar mass within half of the effective radius, that is, where the lensed
images fall, is $f_{*}=0.7$. To find $\kappa_{*}$ we first integrate the lens
light in our HST images in the F160W band, in the same aperture as Auger et
al. (2009). We then assume that $f_{*}=0.7$ is constant across the galaxy and
integrate the convergence of the mass model in the same aperture as for the
light and compute the normalization factor to apply on the lens light at the
image position to obtain $\kappa_{*}$. The values for HE1104 are listed in
Table 4 along with the values for the other systems.

It is also possible to calculate the lens models by analyzing the microlensing
effects of flux measurements for individual lensed quasars (Mediavilla et al.,
2009; Pooley et al., 2012; Jiménez-Vicente et al., 2015; Esteban-Gutiérrez et
al., 2022). This method, however, can be unreliable as it depends on several
assumptions that can lead to underestimating the stellar mass fraction. Some
of these assumptions include presuming the source to be infinitely compact and
attributing flux ratio anomalies entirely to microlensing. Such anomalies,
however, can also arise from millilensing by substructures in the main
deflector or along the line of sight. Correcting for these assumptions would
lead to higher estimates of stellar fractions to explain measured flux ratios.
We therefore rely on the lens modeling performed by the TDCOSMO collaboration
for detailed modeling of each lensed system.

The mass under the form of stars is distributed following a Salpeter initial
mass function (IMF), with the mean mass of $\langle M_{*}\rangle=0.2M_{\odot}$
and the mass ratio between the heaviest and lightest microlenses fixed to be
$100$ in the mass range $0.06M_{\odot}$ and $6.46M_{\odot}$ (Chan et al.,
2021). Other IMFs have not been explored since they do not produce a
noteworthy effect on the results (Kochanek et al., 2007; Wyithe et al., 2000).

The size of a magnification map is scaled to the Einstein radius of
microlenses, $R_{E}$, on the source plane, described as:

$R_{E}=D_{\mathrm{S}}\times\sqrt{\frac{4G\langle
M_{*}\rangle}{c^{2}}\frac{D_{\rm LS}}{D_{\rm L}D_{\rm S}}},$ (4)

which depends on the angular diameter distances from the observer to the lens,
$D_{\rm L}$, from the observer to the source, $D_{\rm S}$, and between the
lens and the source, $D_{\rm LS}$. The Einstein radius for each system is
provided in the fourth column of Table 4, assuming $\langle
M_{*}\rangle=0.2M_{\odot}$ and the source and lens redshifts listed in the
second and third columns of the same table.

In Fig. 2, we show the magnification maps of images A and B of HE0435 as an
example. The observed flux at a particular source position can be computed by
convolving the quasar’s accretion disk’s light profile with the micro-caustics
pattern. This, of course, depends on the size of the accretion disk. In this
work, we estimate the accretion disk sizes of $15$ quasars with continuum
reverberation mapping from Mudd et al. (2018), which avoids any circular
arguments caused by choosing disk sizes measured with previous assumptions on
the stellar populations of the lensing galaxies (e.g., Morgan et al., 2010,
2018; Cornachione et al., 2020). We further discuss the choice of disk sizes
in Section 6.

In order to estimate the disk size for each lens system, we started from the
estimates of Mudd et al. (2018) for the mean black hole mass ($5.5\times
10^{8}\leavevmode\nobreak\ M_{\odot}$) and for the flux-weighted mean disk
size at the rest wavelength $\lambda=2500\leavevmode\nobreak\ \AA$ ($7\times
10^{15}$ cm). We then scaled the disk sizes with the mean black hole mass of
our sample, $\overline{M}_{\rm BH}=8.35\times 10^{8}\leavevmode\nobreak\
M_{\odot}$, and we corrected for the redshift of each individual lensed
quasars to obtain an estimate of the disk size at the rest-frame wavelength
$\lambda_{\rm rest}=\lambda_{\rm obs}/(1+z_{s})$. The COSMOGRAIL observations
are taken in the R band, which corresponds to $\lambda_{\rm
obs}=6500\leavevmode\nobreak\ \AA$. We chose to scale the disk sizes by the
mean – and not by the individual black hole mass, since the black hole masses
of lensed quasars may not all be well constrained. We therefore used the mean
mass of our sample to avoid any biasing that could occur in the case of one or
more of the mass estimates being highly inaccurate, although this choice does
not affect our analysis significantly. The half-light radii of the disks can
be expressed as:

$R_{1/2}=7\times 10^{15}\leavevmode\nobreak\ {\rm cm}\
\Big{(}\frac{\lambda_{\rm
rest}}{2500\AA}\Big{)}^{4/3}\Big{(}\frac{\overline{M}_{\rm BH}}{5.5\times
10^{8}\leavevmode\nobreak\ M_{\odot}}\Big{)}^{2/3},$ (5)

and they are listed in Table 4 for each of the six systems of our sample. We
note that this scaling relation follows the thin-disk model (Shakura &
Sunyaev, 1973), which is supported by microlensing observations (e.g.,
Eigenbrod et al., 2008a). The mean half-light radius of our sample is then
$\overline{R_{1/2}}\sim 10^{16}$ cm, which corresponds to about four light-
days and is comparable with the choice of Hawkins (2020a). The disk light
distribution is also assumed to be Gaussian to match the light profile used in
Hawkins (2020a). The choice of disk models has been reported to have only a
minor effect on the expected amplitude of the microlensing signal (Mortonson
et al., 2005). We generated the magnification maps with the size of
$72\leavevmode\nobreak\ \times\leavevmode\nobreak\ R_{E}$ in $2512$ pixels on
each side and the disk light distribution with the size of
$14.7\leavevmode\nobreak\ \times\leavevmode\nobreak\ R_{E}$ in $512$ pixels.
As the convolution can slightly reduce the map size, the final effective map
size is $57.3\leavevmode\nobreak\ \times\leavevmode\nobreak\ R_{E}$, with a
resolution of $2000$ pixels on a side.

After we obtained the convolved maps, we drew random trajectories on each map
to simulate the light curves. In this work, we assume (conventionally) a
transverse velocity of $600$ km/s for the microlenses. More precise estimates
are available (Neira et al., 2020) but our choice of velocity also matches
that of Hawkins (2020a). The time duration of each simulated light curve is
the same as the observed one (shown in Fig. 1). Since the observed
microlensing signal is the outcome of the difference between a pair of lensed
images, we also computed the difference between the simulated light curves
from two maps. Given the light curves of a lensed image pair in flux units
$F_{i}(t)$ and $F_{j}(t)$, the simulated difference curve in magnitude can be
expressed as:

$\displaystyle\delta m(t)$
$\displaystyle=-2.5\log_{10}\Bigg{(}\frac{F_{i}(t)}{F_{j}(t)}\Bigg{)}-\left(\mathcal{M}_{i}-\mathcal{M}_{j}\right)$
(6) $\displaystyle=m_{i}(t)-m_{j}(t),$

where $\mathcal{M}$ is the macro-magnification obtained from the lens macro
model, equal to that adopted for the observed light curves, as in Eq. 3. We
illustrate a few simulated light curves, $m(t)$, using tracks in the
microlensing caustics. Examples of such tracks are indicated as white lines in
Fig. 2. We note that $m=0$ corresponds to no microlensing. An example of a
pair of curves (red circles) is highlighted on the bottom panel of the figure.

##  3 Method

Figure 3: Probability distributions of magnification amplitude, $\Delta m$.
The black histogram is generated from $\Delta m_{\rm sim}$ in the simulated
light curves, with the median (dashed line), along with the 84th and 16th
percentile indicated as grey shaded areas. The red vertical lines, $\Delta
m_{\rm obs}$, are measured in the observed light curves and given in Table 2.
We also display the quadratic sum of the photometric uncertainties as a red
shaded line.

We went on to compare the amplitude of the microlensing signals in
observations and simulations. We adopted the difference between the maximum
and the minimum magnitudes as the definition for the amplitude of each curve:

$\Delta m=\max(\delta m(t))-\min(\delta m(t)),$ (7)

which follows the choice of Hawkins (2020b). Importantly, this definition is
different from Hawkins (2020a), where $\Delta m$ is defined as the maximum
deviation from the difference in macro-magnification
($\mathcal{M}_{i}-\mathcal{M}_{j}$), dubbed ”zero points.” The ”zero point”
can be estimated from the lens modeling or from the flux ratios if the
observing wavelength corresponds to an emission region that is sufficiently
large not to be affected by microlensing (e.g., the broad line regions or the
extended radio emission regions). This definition, however, is prone to
significant biases, as the amplitude is directly affected by the level of
accuracy of either the lens models or the observational measurements. Instead,
our definition in Eq. 7 is purely empirical. It does not rely on the precision
of the lens models since its respective zero points are canceled out when
calculating the difference between the maximum and minimum of a difference
curve. This choice is therefore more robust since it allows us to study the
microlensing variations over several years without using any information about
the absolute amplitude of the microlensing magnification. We list in Table 2
our measurements of $\Delta m$ on the observed difference light curves in Fig.
1.

The amplitude, $\Delta m,$ from the simulation was calculated in the exact
same way as for the observational curves. We indicate the amplitudes from
simulations as $\Delta m_{\rm sim}$ and those from observations as $\Delta
m_{\rm obs}$. For each image pair, we drew $10^{3}$ light curves $m(t)$
randomly oriented on the corresponding magnification maps and subtracted all
pairs of light curves using Eq. 6, resulting in $10^{6}$ simulated difference
curves $\delta m(t)$. The probability distribution of $\Delta m$ from
simulated light curves, $P(\Delta m)$, is shown in Fig. 3 for each system. The
observed amplitudes, $\Delta m_{\rm obs}$, are labeled as red lines with
observational errors corresponding to the sum in quadrature of the the maximum
and the minimum photometric uncertainties for each observational difference
curve. By counting the frequency with which the amplitude of the simulated
light curve, $\Delta m_{\rm sim}$, exceeds that of the observed one, $\Delta
m_{\rm obs}$, we can estimate the probability of observing a microlensing
event with amplitude larger than $\Delta m$. This corresponds to an empirical
measurement of the p-value, which can be expressed as

$p=P(\Delta m_{\rm sim}>\Delta m_{\rm obs})=\frac{N(\Delta m_{\rm sim}>\Delta
m_{\rm obs})}{10^{6}}.$ (8)

We list the p-values ($p$) of the microlensing amplitudes in Table 2. The
p-values obtained by Hawkins (2020a) are also provided for comparison. The
difference between our results and the work of Hawkins (2020a) is further
discussed in Section 6.

Figure 4: Cumulative distribution functions (CDF) of the p-values. The CDF of
a uniform distribution represented by the solid grey line is what is expected
when the hypothesis is true. Three tests represented by the dotted black,
solid blue, and orange lines produce $p_{\rm KS}>0.05$, showing that the
hypotheses of microlenses composed of compact objects only and of stars+smooth
DM cannot be rejected due to the small sample size. Figure 5: Probability
distributions of magnification amplitude, $\Delta m,$ using the LSST 10-year
light curves, under the scenarios of stellar microlenses (black) and of
stellar+DM microlenses (blue). The probabilities of two scenarios are employed
to predict the number of difference light curves needed to distinguish between
the two scenarios (see also Fig. 6).

##  4 Results

The probability distributions, $P(\Delta m)$, shown in Fig. 3, lead us to the
following remarks. For all systems, $\Delta m_{\rm obs}$ falls within one or
two standard deviations around the median of the corresponding distribution.
The largest difference from the mean can be seen in the case of RXJ1131, where
$\Delta m_{\rm obs}$ is within two standard deviations from the median of the
simulated distribution. This is because RXJ1131 is the system with the highest
magnification amplitude (see Fig. 1). Such high magnifications are rare, but
can still occur in case of caustic crossings. This might well be the case of
RXJ1131. Thus, even though the errors on the light curve measurements might be
large in some cases (as seen in Fig. 1), there is no obvious indication that
any of our observations for any individual object significantly deviate from
expectations when assuming purely stellar microlensing.

We can go beyond the above phenomenological statements and quantify the
agreement between our observations and microlensing model using statistical
hypothesis testing. Traditionally, a popular choice is to use Pearson’s
$\chi^{2}$ test (Snedecor & Cohran, 1989) to compare expected outcomes with
measurement distributions. However, the $\chi^{2}$ test requires a normality
condition on the distributions we are comparing with and since our simulated
amplitudes are not normally distributed as can be seen from Fig. 3, a
Pearson’s $\chi^{2}$ test is not a viable option to use.

Alternatively, we examine the uniformity of the p-value measurements to test
our hypothesis using the Kolmogorov-Smirnov (KS) test (Chakravarti et al.,
1967). When a null hypothesis is true, the corresponding p-values appear to be
distributed between 0 and 1 uniformly666The proof can be seen from
https://statproofbook.github.io/P/pval-h0. Here, we define the null hypothesis
$H_{0}$ as our conventional stellar microlenses model. Under this hypothesis,
we test whether the probability distribution of $\Delta m_{\rm sim}$ matches
the observational microlensing amplitude, $\Delta m_{\rm obs}$, using the
p-values defined in Eq. 8. We then used the KS test to examine whether the
eight p-values of our systems are indeed statistically compatible with a
uniform probability distribution. Specifically, the KS test quantifies the
distance between the cumulative distribution functions (CDFs) of the measured
p-values and the uniform distribution. This distance can be converted to a
single p-value $p_{\rm KS}$ (ranging from 0 to 1), showing how close these two
CDFs are. We emphasize that $p_{\rm KS}$ is derived from the KS test, which is
different from the p-value in Eq. 8. The resulting value for $p_{\rm KS}$ can
be compared to any desired level of significance, $\alpha$. For example, if
$p_{\rm KS}<\alpha=0.05$, we can conclude that our data rejects the null
hypothesis $H_{0}$ at $95\%$ confidence level.

In Fig. 4, the orange line shows the CDF constructed for the p-values obtained
in this work. We note that the CDF of the uniform probability distribution is
the identity line $y=x$ (also shown in the figure). Performing the KS test to
compare between these two CDFs yields $p_{\rm KS}=0.63\gg\alpha=0.05$; we are
therefore not able to reject $H_{0}$. We then display the sample of Hawkins
(2020a) with the dotted line, which yields $p_{\rm KS}=0.87$. The null
hypothesis in this case also cannot be rejected, which contradicts the finding
in Hawkins (2020a). In Section 6, we propose several possible reasons to
explain such a difference.

The sample size in this work is still small, even though it has been expanded
by a factor of two compared to the sample in Hawkins (2020a). Supposing that
we define an alternative hypothesis, $H_{1}$, stating that all DM behaves as
compact objects, such as PBHs, we consider whether our microlensing curves
would then be able to reject $H_{1}$. Assuming that the mass distribution of
PBHs follows the stellar distribution as a toy model, we generated the
magnification maps by enforcing $\kappa_{*}=\kappa$, which means that all the
mass of the galaxies is now in the form of compact objects with a mean mass of
$0.2M_{\odot}$. We then repeated our analysis with this alternative model. The
CDF of the new p-values resulting from this experiment is illustrated by the
blue line in Fig. 4, yielding $p_{\rm KS}=0.89\gg\alpha=0.05$. Because of the
small sample size, we conclude that it is unachievable to argue whether DM in
a lensing galaxy is smooth or in compact form. However, the Rubin
Observatory/LSST will provide thousands of lenses in the future, which may
offer an opportunity to distinguish the hypotheses, $H_{0}$ and $H_{1}$, as
presented in the next section.

##  5 Predictions with ten years of LSST Data

The Rubin Observatory Legacy Survey of Space and Time (LSST; Ivezić et al.,
2019) will have an effective aperture of $6.4$-m with field of view of
$9.6\leavevmode\nobreak\ \deg^{2}$. The LSST Camera will be devoted to a 10-yr
imaging survey over $20\,000\leavevmode\nobreak\ \deg^{2}$. There will be
$\approx 3000$ lensed quasars (consisting of 2400 doubles and 600 quads)
expected to be found in the LSST (Oguri & Marshall, 2010), yielding $\approx
3600$ independent difference microlensing light curves.

Given that the total convergence, $\kappa,$ of a lensed system is the sum of
convergence created due to both the fraction of stars and DM (which could be
smooth or compact as in PBHs), we test the ability of the upcoming
observational data from the Rubin Observatory to discriminate between: 1) a
fiducial scenario, where a percentage of the total convergence is in a form of
compact bodies (such as stars), while the rest is still in a form of smooth
matter; and 2) an alternative scenario: all the convergence is contributed by
a form of compact bodies, namely, $\kappa_{*}/\kappa=1$.

In other words, in the fiducial scenario, $\kappa_{*}$ comes from the stellar
contribution, while in the alternative scenario, we enforce
$\kappa_{*}=\kappa$. The new set of light curves in the simulation is also
extended to ten years to match the duration of LSST measurements. The aim of
this experiment is to estimate the number of light curves needed to
discriminate between these two scenarios with a confidence level of 95%.

Following the simulations, the amplitude distributions for both scenarios are
shown in Fig. 5, where the histograms in blue show the results for the
alternative setup, while the histograms in black show the distributions for
the fiducial setup where we used the macro-model parameters listed in Table 4.
We note that the black histograms in Figs. 3 and 5 are slightly different due
to the ten-year extension in the simulation.

A microlensing light curve provides an amplitude of $\Delta m_{\rm obs}$. To
mimic the observations, we randomly drew $\Delta m_{\rm obs}$ from the
distribution in the alternative scenario ($\kappa_{*}=\kappa$) as a reference
amplitude and calculated the p-value of its corresponding distribution in the
fiducial scenario using Eq. 8. Once we obtained a set of p-values from a
certain number of light curves, we performed the KS test to see if $p_{\rm
KS}<\alpha$. When the number of light curves increases, $p_{\rm KS}$ decreases
so that $H_{0}$ can be rejected with a confidence level of 95% (as shown in
the blue curve of Fig. 6). Given a certain number of light curves, we further
repeated this experiment 100 times in order to obtain the $p_{\rm KS}$
distribution, and its median is illustrated as the solid line with the shaded
area bounded between the 16th and the 84th percentiles. The number of
samplings of reference amplitudes is increased by picking more events randomly
from the eight distributions of the COSMOGRAIL sample, since we have precisely
measured lensing parameters ($\kappa$, $\gamma$, $\kappa_{*}$) only for these
eight systems. Here, we make the assumption that the COSMOGRAIL sample is a
representative sample of all LSST lensed quasars. We show that the median
$p_{\rm KS}$ goes below $\alpha=0.05$ for about 900 curves. This means that it
is possible to distinguish the fiducial and alternative scenarios with a
confidence level of 95%, provided that 900 image pairs are available. As we
would expect, when sampling both $\Delta m_{\rm obs}$ and $\Delta m_{\rm sim}$
from the distributions in the fiducial scenario, the median $p_{\rm KS}$
oscillates around $0.5$, regardless of the number of light curves considered.
This result is illustrated as the solid orange line in Fig. 6, showing that
$H_{0}$ can no longer to be rejected, since $\Delta m_{\rm obs}$ is drawn from
the same distribution as $\Delta m_{\rm sim}$.

Figure 6:  P-value from the KS test $p_{\rm KS}$ against number of
microlensing light curves. The solid line indicate the median of $p_{\rm KS}$
distribution with the shade region bounded between the 84th and the 16th
percentiles. The fiducial scenario adopts the conventional stellar
distribution as microlenses, while the alternative scenario assumes that all
the mass in the galaxy is in the form of compact objects, namely,
$\kappa_{*}/\kappa=1$ (see Fig. 5). Given that $\Delta m_{\rm obs}$ and
$\Delta m_{\rm sim}$ come from the same distribution, $p_{\rm KS}$ is expected
to be 0.5 on average (orange line), regardless of the number of light curves
used for this experiment. When $\Delta m_{\rm obs}$ and $\Delta m_{\rm sim}$
come from different distributions, the median value of $p_{\rm KS}$ decreases
steadily with the sample size, enabling us to distinguish between two
scenarios. The horizontal dashed line marks $\alpha=0.05$, corresponding to
the threshold where the null hypothesis $H_{0}$ is rejected with a confidence
level of $95\%$. The blue and green lines represent our results when using
disk sizes from continuum reverberation mapping and from the thin disk model,
respectively.

Given that the methodology of this work is dependent upon the source sizes
(see the discussion in Section 6), we re-performed our analysis using a
different set of disk sizes for the chosen sample of lensed quasars, following
the thin disk model (Morgan et al., 2010; Mosquera & Kochanek, 2011):

$R_{s}=9.7\times 10^{15}\leavevmode\nobreak\ {\rm cm}\
\Big{(}\frac{\lambda_{\rm rest}}{\mu{\rm m}}\Big{)}^{4/3}\Big{(}\frac{M_{\rm
BH}}{10^{9}M_{\odot}}\Big{)}^{2/3}\Big{(}\frac{L}{\eta L_{E}}\Big{)}^{1/3},$
(9)

where the black hole mass of each system $M_{\rm BH}$ is obtained in Table 1
of Mosquera & Kochanek (2011). We conventionally assume that the quasar
accretion efficiency $\eta=0.1$ and the luminosity in units of the Eddington
luminosity $L/L_{E}=1/3$ (Kollmeier et al., 2006; Hopkins & Hernquist, 2009;
Schulze, A. & Wisotzki, L., 2010). The second set of sizes is then adopted as
$R_{1/2}=2.44\ R_{s}$. We illustrate the results of the thin disk model as the
solid green line in Fig. 6. We can now observe that the number of curves
needed to distinguish between the two scenarios drops to $\sim 300$, as a
result of the smaller estimates of the thin disk radii, which have the mean
radius of $\sim 4\times 10^{15}$ cm ($\sim 1.5$ light-days). The smaller the
source size, the less blurred the caustic pattern. This then makes it easier
to distinguish between the distribution of microlenses in either scenario.
Given the span of disk size between 1.5 and 4 light-days, thus, the number of
microlensing curves that are needed to test the validity of the compact DM
scenario lies within a range from 300 to 900. Obtaining such a number of
systems will be difficult but feasible given the capabilities of LSST. We thus
propose, as an extension of this work, to apply this methodology to the future
observations provided by LSST when the data becomes available.

##  6 Discussion

In this section, we discuss the implications of several assumptions made in
our work. Concretely, we review the choices made for the source size and
transverse velocity. We also discuss the definition for the amplitude of the
microlensing signals and the methodology developed for the calculation of
their corresponding p-values.

Since the convolution of magnification maps with a source brightness
distribution smooths the caustic patterns, we expect the amplitude of
microlensing variations to become smaller with increasing source size (Refsdal
& Stabell, 1991; Witt & Mao, 1994). Therefore, we expect our results to vary
as a function of the adopted disk sizes for our simulations, due to the
uncertainties in the size measurements. In this work, we follow a physically
motivated approach and we assume, for each system, its own source size with a
half-light radius as calculated in the relevant literature. Although some
microlensing studies can also provide disk size measurements (e.g. Morgan et
al., 2018; Cornachione et al., 2020), we avoided using them, as they already
included assumptions on the stellar microlenses in a halo of the lensing
galaxy. Using their reported disk sizes might lead to the conclusion that
stars are the only components needed to produce sufficient amplitudes in the
lensed quasar microlensing curves, as this is the underlying hypothesis of
their work. To overcome this circular argument, we instead chose the disk
sizes reported in recent continuum reverberation mapping studies, such as the
work of Mudd et al. (2018), which is a technique that is independent of the
assumption on the stellar populations (see Eq. 5).We note that disk sizes from
reverberation mapping are usually associated with large error bars. However,
our method relies on a statistical treatment of the microlensing signal and
not on individual measurements; thus, even though individual measurements may
not be very precise, they serve well as an unbiased average measurement of the
disk sizes. It is also possible to scale the source size with the black hole
mass under the thin disk model (see Eq. 9), but since the thin disk size has
frequently reported much smaller measurements than those from microlensing and
reverberation mapping studies, we avoided choosing this scaling. As for our
prediction in Section 5, we present the results given two different sets of
source sizes to highlight the possible range for the number of future
microlensing curves needed to distinguish between the smooth DM regime and the
compact DM regime.

In our analysis, we adopted $600$ km/s as a conventionally assumed transverse
velocity for all systems. Indeed, any increase or decrease in this parameter
value results in the respective lengthening/shortening of the trajectories
drawn on the magnification maps, which could affect microlensing amplitudes in
these simulated light curves. However, this bias is reduced with long-enough
curves which are ensured by the long monitoring baseline of COSMOGRAIL. In our
simulations, the trajectories are sufficiently long to explore a large span of
dynamical range in the generated maps. Unlike the disk size, we therefore
argue that the transverse velocity has marginal impact on our results.

This study is inspired by the works of Hawkins (2020a, b), who have suggested
a need for other forms of compact objects to explain their measurements of the
microlensing amplitudes. Our results, however, challenge these findings and
agree with other works, such as Esteban-Gutiérrez et al. (2022). We
investigate a few possible causes to explain the inconsistency. Firstly, a
difference lies between the macro-model parameters used: while the lensing
parameters from the TDCOSMO collaboration are considered to be more robust,
the lensing models in Hawkins (2020a) are obtained from a singular iso-thermal
sphere plus an external shear model, which can produce different micro-caustic
patterns and zero-points. Although Hawkins (2020a) estimated the zero-points
from the flux ratios of emission regions larger than the accretion disk, we
notice that the measurements can deviate from those using lens modeling.
Secondly, the metric for the microlensing amplitudes chosen in this work is
less subject to biases from macro-magnifications, millilensing, dust
absorption, and so on. Our empirical definition of $\Delta m$ does not require
any measurement of the zero-points, which make them insensitive to this highly
unreliable quantity. In order to understand how the lens models affect the
hypothesis test, we repeated our analysis using the lensing parameters in
Table 3 of Hawkins (2020a) but using our definition of $\Delta m$. We still
find that $H_{0}$ still cannot be rejected. Lastly, we think that the major
difference between our work and that of Hawkins (2020a) rests in the
statistical treatments of the p-values when combining all considered systems.
While we use the KS test to evaluate our stated hypothesis, the approach
followed by Hawkins (2020a) consists of simply taking the product of all
calculated p-values. Since p-values are, by definition, smaller than 1, the
latter statistical treatment will artificially disfavor the $H_{0}$ hypothesis
as more and more systems are added to the sample. Our methodology avoids this
biasing by following a statistical method for combining the information
provided by each p-value.

Additionally, we discuss the effectiveness of our method in discriminating
between the defined fiducial and the alternative scenarios, which is subject
to the disk sizes of lens systems. This is explained by the fact that larger
disk sizes yield smoother micro-caustic patterns on magnification maps,
blurring the distinctions between the two compared scenarios. As a result, one
would need more light-curves to reach a conclusion within $95\%$ confidence.
In this work, we adopt the disk sizes from the reverberation mapping, and show
that we need – at most – about 900 microlensing curves to make the distinction
between the two scenarios. We further explore the possibly smaller disks,
scaled with the black hole masses using the typical thin disk model, which is
about 2.5 times smaller than the reverberation mapping sizes. In this case, we
only require $\sim 300$ curves to test the validity of the compact DM
scenario. Since accretion disks are smaller and more affected by microlensing
at shorter wavelength, we emphasize that using bluer bands than the R band
considered here will also sensibly increase the constraining power of this
experiment. Although in this work, we test two extreme cases, it is possible
to apply the same methodology to compare different scenarios under a broader
range of assumptions; this exploration, however, will be left for a future
work. Finally, LSST observations will be ongoing for total of $10$ years,
consistently providing a greater number of realistic measurements of the
microlensing amplitudes and, thus, helping set better constraints on the
calculated p-values.

##  7 Conclusion

In this work, we study the origin of high-magnification events in microlensing
light curves of strongly lensed quasars. More precisely, we test whether stars
in the lensing galaxies of a population of lensed quasars are sufficient to
account for the observed microlensing signal. Our statistical analysis of the
data leads to the following conclusions:

  * $\bullet$

The considered population consists of the strongly-lensed quasar systems: HE
1104$-$1805, HE 0435$-$1223, RX J1131$-$1231, WFI 2033$-$4723, PG 1115$+$080,
and J0126$+$4332\. From these lenses, eight independent pairs of lensed images
are chosen and the microlensing curves for those pairs are used from the most
recent light curves and time-delay measurements of COSMOGRAIL.

  * $\bullet$

We defined a robust metric to evaluate the amplitude of the microlensing
signal and we computed, for each system, the p-value corresponding to the
probability that our predicted microlensing amplitudes would be greater than
the actual measurements.

  * $\bullet$

We proposed a coherent statistical approach to carry out a quantitative
comparison of our microlensing observations and simulations. Our method
utilizes the KS test to examine the uniformity of the distribution of p-values
of our sample. This allows us to statistically test the null hypothesis, which
posits that 1) galaxies are composed of stars and smoothly distributed DM
against 2) an alternative scenario where all the mass of the galaxy is in the
form of compact objects, namely, stars+PBH.

  * $\bullet$

On the basis of the current COSMOGRAIL sample, we demonstrated that our null
hypothesis cannot be rejected statistically.

  * $\bullet$

We explored an alternative hypothesis in which DM is completely constituted by
compact objects, such as PBHs, and also found that this hypothesis cannot be
rejected.

  * $\bullet$

Finally, we showed that $\sim 900$ microlensing curves are needed to ascertain
the validity of either of the assumed scenario, within a $95\%$ confidence
level.

Our current sample of light curves from the COSMOGRAIL collaboration is the
only and largest one to date poised to explore the possibility of using quasar
microlensing to discriminate between different assumptions on the nature of
microlenses. Although is is still too small, it lends considerable hope to the
possibility of making the method truly effective, when hundreds of light
curves become available from the Rubin Observatory/LSST in the coming years.

## Acknowledgements

We give our thanks to M. Hawkins, D. Sluse, G. Vernardos, E. Soutter, and M.
Auger for helpful discussions. This work is supported by the DSSC Doctoral
Training Programme of the University of Groningen. This research was made
possible by the generosity of Eric and Wendy Schmidt by recommendation of the
Schmidt Futures program. This work is also supported by the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
program (COSMICLENS: grant agreement No 787886) and by the Swiss National
Science Foundation (SNSF). M.M. acknowledges the support of the Swiss National
Science Foundation (SNSF) under grant P500PT_203114.

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