# How to Break the Mass Sheet Degeneracy with the Lightcurves of Microlensed
Type Ia Supernovae

Luke Weisenbach,1, Thomas Collett,1 Ana Sainz de Murieta,1 Coleman Krawczyk,1
Georgios Vernardos,2,3 Wolfgang Enzi,1 Andrew Lundgren1
1Institute of Cosmology and Gravitation, University of Portsmouth, Burnaby
Road, Portsmouth, PO1 3FX, UK
2Department of Physics and Astronomy, Lehman College of the City University of
New York, Bronx, NY 10468, USA
3Department of Astrophysics, American Museum of Natural History, Central Park
West and 79th Street, NY 10024, USA E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The standardizable nature of gravitationally lensed Type Ia supernovae (glSNe
Ia) makes them an attractive target for time delay cosmography, since a source
with known luminosity breaks the mass sheet degeneracy. It is known that
microlensing by stars in the lensing galaxy can add significant stochastic
uncertainty to the unlensed luminosity which is often much larger than the
intrinsic scatter of the Ia population. In this work, we show how the temporal
microlensing variations as the supernova disc expands can be used to improve
the standardization of glSNe Ia. We find that SNe are standardizable if they
do not cross caustics as they expand. We estimate that this will be the case
for $\approx$6 doubly imaged systems and $\approx$0.3 quadruply imaged systems
per year in LSST. At the end of the ten year LSST survey, these systems should
enable us to test for systematics in $H_{0}$ due to the mass sheet degeneracy
at the $1.00^{+0.07}_{-0.06}$% level, or $1.8\pm 0.2$% if we can only extract
time delays from the third of systems with counter images brighter than $i=24$
mag.

###### keywords:

gravitational lensing: micro – gravitational lensing: strong – transients:
supernovae

††pubyear: 2023††pagerange: How to Break the Mass Sheet Degeneracy with the
Lightcurves of Microlensed Type Ia Supernovae–D

##  1 Introduction

Gravitational lensing is a powerful probe for understanding astrophysics and
cosmology. Lensing is of particular use for constraining the expansion history
of the Universe, since it is a geometric probe of the angular diameter
distances between observer, lens and source (Narayan, 1991). When a time
variable source is multiply imaged by a gravitational lens, the time delays
between images are inversely proportional to the Hubble constant $H_{0}$ (Treu
& Marshall, 2016). Some measurements of lensed quasars (Wong et al., 2020)
have found $H_{0}=73.3^{+1.7}_{-1.8}$ km s-1 Mpc-1, in agreement with local
measurements from Type Ia supernovae (Riess et al., 2022) but at odds with CMB
measurements from Planck (Planck Collaboration et al., 2020). However,
measurements are dependent on the mass distribution of the lens. Relaxing some
assumptions and including non-lensing constraints has also lead to results
which are consistent with those of Planck (Birrer et al., 2020). While
gravitational lensing has the potential to direct some light on the existing
tension between $H_{0}$ measurements, a thorough understanding of systematics
and degeneracies of lens modelling is necessary.

Historically time delays have been measured from lensed quasars (Millon et
al., 2020), though supernovae were originally proposed (Refsdal, 1964). Lensed
supernovae (glSNe) offer several benefits over quasars: they evolve over much
shorter timescales requiring months of monitoring as opposed to years, and
they fade away enabling follow-up observations of the lensed host. Type Ia
glSNe are even better: they have well understood lightcurves, and are
‘standardizable’ candles, with a scatter of $\approx$ 0.15 mags (Richardson et
al., 2014; Brout et al., 2022). The fact that Type Ia supernovae have a known
intrinsic luminosity means that their magnification can be directly measured,
helping to break lens modelling degeneracies.

In order to infer the value of $H_{0}$ from a lensed time variable source, one
needs to know the mass distribution of the lensing galaxy. The problem is that
lensing observables alone cannot uniquely tell us about that mass
distribution. A lensing model that predicts the positions, relative
magnifications, and time delays between observed images can be rescaled with a
sheet of mass in such a way that none of the observables change except for the
time delays (Falco et al., 1985; Schneider & Sluse, 2014). This mass sheet
degeneracy (MSD) must be broken to extract the Hubble constant from time delay
measurements, and this is one of the main systematic uncertainties in modern
time delay cosmography (Birrer et al., 2020). The MSD is broken if you know
the absolute magnification of the background source, which is possible with a
lensed standard candle such as a glSNe Ia.

Unfortunately, there is one main barrier to using glSNe Ia to break the MSD:
microlensing by stars. The image we observe is split into unresolvable
microimages that are (de)magnified by the stars in the lensing galaxy (Young,
1981; Paczynski, 1986; Vernardos et al., 2024). We can only observe the total
flux of these unresolved images, which can be (de)magnified from the macro-
model. We need to know the macrolensing magnification to break the MSD, but
the presence of microlensing adds significant stochastic noise (Dobler &
Keeton, 2006). In the worst case scenario this can introduce up to a magnitude
of scatter (Weisenbach et al., 2021), which would make microlensed glSNe Ia no
longer standardizable candles (Yahalomi et al., 2017).

Foxley-Marrable et al. (2018) were the first to examine if some lensed images
might suffer sufficiently small amounts of microlensing that they remain
standardizable. By analyzing the microlensing of a uniform disk, they found
that regions of low magnification and low stellar density have microlensing
scatter comparable to the intrinsic scatter in the luminosity of an unlensed
SN Ia111Throughout this paper we will refer to a glSNe Ia as standardizable if
the microlensing scatter is comparable to or less than the intrinsic scatter
in the luminosity of an unlensed SN Ia (0.15 mags). The microlensing scatter
is defined as the half-width of the inner 68% of the microlensing
magnification probability distribution. Microlensing magnification
distributions are non-Gaussian and skewed with a high-magnification tail (see,
e.g., Figure 4 of Vernardos & Fluke, 2013), hence this definition.. However,
they did not use temporal information in their inference. As the supernova
expands, it averages over different length scales of microlensing
fluctuations. The glSNe never becomes sufficiently large to completely average
out to the macro-model magnification, but the shape of the lightcurve can
teach us about the nature of the microimages and hence inform us about the
amplitude of the absolute microlensing magnification.

In what follows, we provide a discussion on the theoretical reasoning behind
why certain regions of microlensing parameter space are or are not
standardizable in Section 2. We discuss simulations for a simple model in
Section 3. We discuss in Section 4 various methods and implementations for
either selecting microlensing lightcurves that are standardizable or inferring
a posterior of the microlensing magnification given an observed lightcurve. We
extend from our single point to the entirety of useful microlensing parameter
space in Section 5. In Section 6 we provide an estimate on the number of
standardizable glSNe Ia expected to be discovered in the next decade from a
forecasted LSST population, while discussing some of the limitations of our
work in Section 7. We provide our conclusions in Section 8.

##  2 Standardizable regions of microlensing parameter space

Figure 1: Top: Microlensing histograms for systems that are (from left to
right) standardizable, intermediate, and unstandardizable. The vertical dashed
lines mark the inner 68% of the histogram. The subhistograms are for those
regions completely outside the caustics ($N=1$), inside one caustic ($N=2$),
or deeper in the caustic network ($N\geq 3$). Middle: Sample caustic patterns
that produce the microlensing histograms. Bottom: Zoom of the caustics for the
indicated central square of the larger maps. The solid, dashed, and dotted
circles denote the size of a fiducial supernovae 0, 50, and 100 days after
peak luminosity respectively.

In this section, we investigate why the time evolution of a microlensed
lightcurve should be sensitive to the absolute microlensing magnification, and
hence inform us about which glSNe images are standardizable. This section
focuses on the theoretical underpinnings of why it should be possible to
select a subset of standardizable microlensed images, while Section 4 will
show how to do so with realistic observables.

###  2.1 The importance of the number of microimages

Microlensing fluctuations are highly dependent on where the image forms
relative to the lensing galaxy. There are three main parameters of importance:
the lens macro-model convergence and shear at the macroimage location,
$\kappa$ and $\gamma$, and the ratio of stellar to total matter density at the
location of the image, $1-s$ (where $s$ is the smooth matter fraction, which
is predominantly the dark matter fraction but also includes any other smooth
baryonic components).

Foxley-Marrable et al. (2018) and Weisenbach et al. (2021) have already shown
that microlensed systems with low macro-magnification and low stellar density
are standardizable; see for example Figure 5 of Suyu et al. (2024). The
physical reason that systems with low magnification and stellar density are
standardizable is that the caustic network is sparse – the majority of the
source plane is dominated by regions where the source does not lie within any
microcaustics. Fundamentally, this is related to the idea that the mean number
of microminima of the time delay surface (Wambsganss et al., 1992; Granot et
al., 2003) for such systems is small. This is opposed to regions of parameter
space with higher magnification or higher stellar density where the caustic
network is more dense (see, e.g., Figure 12 of Vernardos et al., 2024),
raising the expected number of microimages. Regions of the source plane can be
indexed by how many caustics a source lies within, which we will denote with
$N=1,2,3,...$222More precisely, $N$ denotes the number of microminima (Granot
et al., 2003). $N\geq 1$ for macrominima. Macrosaddles have an additional
$N=0$ region; see Appendix A..

Figure 1 shows three microlensing systems: one that would be standardizable
with the Foxley-Marrable et al. (2018) method, an intermediate scenario, and
one that is completely unstandardizable. Each of the microlensing histograms
is bimodal, and an examination of the magnification maps and caustic
structures reveals why: the histogram can be decomposed into subhistograms for
each value of $N$ (Rauch et al., 1992; Granot et al., 2003; Saha & Williams,
2011). These subhistograms are offset from each other, since increasing $N$ by
1 means the creation of another pair of microimages which introduces a jump in
the minimum allowable magnification. The relative importance of the
subhistograms is set by the density of the caustic network.

  1. 1.

For the standardizable system, the majority of the source plane consists of
regions with $N=1$, with a small fraction consisting of $N=2$. The scatter is
dominated by the $N=1$ region, so the width of the microlensing histogram is
small.

  2. 2.

For the intermediate system, the relative fraction of $N=1$ is lower and $N=2$
is higher, making the bimodality more prominent and increasing the scatter.

  3. 3.

The completely unstandardizable system has approximately equal fractions for
$N=1$ and $N=2$. Since the subhistograms are offset, the total microlensing
histogram is very broad.

The statement that a microlensed system is standardizable is really a
statement that the likelihood of lying inside a microcaustic is low. If we
could determine the value of $N$, many glSNe would be standardizable.
Unfortunately resolving the microimages is currently impossible. But there is
a key piece of data that is observable, namely the lightcurve of the expanding
supernova. The temporal variations as the supernova expands contain
information about where it lies within the caustic network.

###  2.2 Constraining the number of microimages

Figure 2: Top: Sample microlensing lightcurves for a source that i) crosses a
caustic inside to outside (blue), ii) crosses a caustic outside to inside
(orange), or iii) does not cross a caustic (green), within the time period
examined. Bottom: Sample noisy lightcurves, assuming a 2 day cadence in the
source rest frame and 0.05 mag Gaussian noise.

Since supernovae expand overtime, a source that lies deep in the microcaustic
network must eventually grow to the point that it crosses a caustic. The exact
timescales involved and the rate of change of the lightcurve depend on the
expansion velocity, the sizes of the caustics, and the mean spacing of the
caustics; the latter two depend on the mass of the microlenses, $\kappa$,
$\gamma$, and $s$. Looking at Figure 1, it is fairly easy to convince oneself
that sources starting in regions with $N\gtrsim 3$ will typically expand to
cross a caustic more quickly than sources located elsewhere. Since caustics
represent lines of extremely high magnification, caustic crossings induce
substantial changes in the microlensed supernova lightcurve. There are three
basic lightcurve scenarios possible as the source expands which we illustrate
in Figure 2:

  1. 1.

The source lies near and inside (on the more magnified side of) one of the
microcaustics. The micro magnification rises as the expansion approaches the
caustic and then falls when parts of the disc fall outside of the
microcaustic.

  2. 2.

The source lies near and outside (on the less magnified side of) one of the
microcaustics. The micro magnification sharply rises as parts of the disk
cross the caustic.

  3. 3.

The source lies sufficiently far from the microcaustics for the timescale of
interest. The micro magnification is broadly insensitive to the size of the SN
disk.

There can certainly be more complicated behaviour, for example, if the source
lies closer to a cusp or overlapping caustics, but these are much rarer.

Table 1: Microlensing magnification scatter for an expanding source which has not crossed a caustic up to some time for the intermediate configuration ($\kappa=\gamma=0.4,s=0.75$). Values were found by convolving one large scale map. Fractions indicate what percentage of locations on the source plane have not crossed a caustic by the indicated time. The first row gives results for all of the source plane under consideration, while subsequent rows show the decomposition into regions with various values of $N$. | Point source | Peak | 50 days past peak
---|---|---|---
| Fraction | Scatter | Fraction | Scatter | Fraction | Scatter
All | 1 | 0.420 | 0.840 | 0.335 | 0.564 | 0.191
$N=1$ | 0.799 | 0.187 | 0.722 | 0.178 | 0.539 | 0.167
$N=2$ | 0.186 | 0.338 | 0.112 | 0.170 | 0.025 | 0.126
$N\geq 3$ | 0.015 | 0.389 | 0.005 | 0.108 | - | -
Figure 3: Left: Microlensing magnification distributions for the intermediate
scenario of Figure 1. The vertical dashed lines mark the inner 68% of the
histograms. While the point source histogram is for the entire source plane,
the subsequent histograms are only for those regions of the source plane where
an expanding source will not cross a caustic by some given time. Right:
Visualization of the ruled out regions in the source plane. The solid and
dashed white circles in the magnification maps denote the size of a fiducial
supernova at each time period.

The shape of the lightcurve provides information about where the source lies
within the caustic network. Clearly microlensing lightcurves that are flat
originate far from caustics and so are not likely to be highly magnified. One
would expect that this subpopulation will show much less micromagnification
scatter, and hence be more standardizable.

We can quantify this by tracking whether or not a source placed within the
microlensing map for the intermediate system ($\kappa=\gamma=0.4,s=0.75$)
crosses a caustic in a given time period. A visualization of how this affects
the microlensing magnification distribution is shown in Figure 3. Results are
shown in Table 1. The high magnification tail of the distribution comes from
regions of the source plane that are either i) near the inside of fold
caustics, ii) near the outside of cusps, or iii) deep inside the caustics.
These regions are quickly ruled out by the absence of caustic crossings. The
magnification distribution for non-crossing events shrinks and becomes much
more standardizable.

A problem with this approach is that configurations with dense caustic
networks — high macro-magnification images or images forming in regions of
high stellar density — will have very few source positions that do not cross a
caustic as the SN expands. This approach also does not work for saddlepoint
images as they have a low magnification tail which cannot be ruled out by the
absence of a caustic crossing (see Appendix A). However, we expect
intermediate systems to be standardizable under this approach and for it to be
a substantial improvement upon previous analyses.

##  3 Simulations and Data

In this section, we discuss the assumptions used for our simulations, the
simulated data that we will use throughout the paper, and sources of noise
that are relevant for the lightcurves.

###  3.1 Assumptions

We will use a simple model for a microlensed supernova: we take the supernova
to be a uniformly luminous disk that is expanding in radius constantly with
time. While more complicated supernova models such as those used by Goldstein
et al. (2018) and Huber et al. (2019) may technically be more accurate,
Mortonson et al. (2005) and Vernardos & Tsagkatakis (2019) have shown that
microlensing fluctuations are most sensitive to the half-light radius of the
source as opposed to its luminosity profile. We furthermore use the following
assumptions throughout this work:

  1. 1.

Our lens is at a redshift $z=0.5$ and our source is at a redshift $z=0.8$, in
a flat $\Lambda$CDM universe using the parameters of Planck Collaboration et
al. (2020). Results loosely depend on the redshifts.

  2. 2.

The microlenses are all of mass $0.3M_{\odot}$, which determines the Einstein
radius $\theta_{E}$. Conversions to other masses can be made using the fact
that $\theta_{E}\propto\sqrt{M}$.

  3. 3.

The supernova has a constant expansion rate of $10^{4}$ km/s (Pan, 2020). The
main purpose of this is to convert from size to approximate times where
necessary.

  4. 4.

The supernova peaks at 20 days in its rest frame. Days listed are days in the
rest frame of the supernova unless otherwise specified.

  5. 5.

We will assume that we can separate the effects of microlensing from the
intrinsic variations of the SN lightcurve, up to a constant uncertainty of
0.05 mag. This is comparable to the uncertainties in the model fitting of type
Ia SNe (e.g. SALT3, Kenworthy et al., 2021, see Figure 13), which will set a
fundamental floor on our ability to separate out the time evolution of the
microlensing333Unless it is possible to use the multiple microlensed images of
the glSNe to improve upon classical Ia template fitting.. Figure 2 shows some
example noisy microlensing lightcurves.

###  3.2 Simulations

We generate 100,000 microlensed lightcurves for the intermediate system
parameters ($\kappa=\gamma=0.4$, $s=0.75$) to serve as our dataset. We use the
inverse ray shooting method (Kayser et al., 1986; Wambsganss, 1990) to create
magnification maps that are $\approx 10\theta_{E}\times 10\theta_{E}$ and
$\approx$10,000 x 10,000 pixels with a pixel scale of $0.001\theta_{E}$, or
$\approx$ a quarter of a day of supernova expansion. We use enough stars to
capture the bulk of the magnification (Katz et al., 1986), and distribute them
in a rectangular region to reduce computational complexity while still
accounting for the average microlensing deflection (Zheng et al., 2022). We
can fit 10 x 10 = 100 expanding uniform disks onto a single map with no
overlap after $\approx$ 100 days, requiring 1000 maps. We generate the maps
and perform the convolutions on GPUs . Since we are only interested in
lightcurves from sources that do not overlap on the source plane (not
correlated), we do not need to convolve the entirety of the maps, greatly
reducing the computation time needed. In addition to creating the lightcurves
for each expanding source, we use a GPU version of Hans Witt’s method (Witt,
1990) to find the caustics of each star field in order to track whether the
expanding disk crosses any caustics throughout the entirety of the lightcurve.

##  4 Selecting standardizable lightcurves

In this section, we discuss how, in practice, to infer the posterior of the
microlensing magnification given an observed lightcurve. We start by
presenting two simple criteria for picking out lightcurves which do not cross
caustics and discuss some of the difficulties that might arise due to noise
when using these criteria. We then discuss how a bank of lightcurves can be
used to estimate the amount of microlensing (de)magnification. Next, we
examine how neural network regression can be used to predict the microlensing
(de)magnification. Finally, we use a neural network to classify the
lightcurves into two categories: did or did not cross a caustic.

Throughout this section, we assume glSNe lightcurves are observed from 5 days
before peak up to 50 days after peak in the SN rest frame, with a 2 day
cadence in the rest frame.

###  4.1 Simple criteria

We consider perhaps the simplest metric: measuring the standard deviation of
our simulated lightcurves $\sigma_{\text{lightcurve}}$. We can then determine
the fraction of simulated lightcurves that have $\sigma_{\text{lightcurve}}$
less than some cutoff value, and what the microlensing scatter at peak is for
those lightcurves. Results are shown in Figure 4. By selecting lightcurves
which have small standard deviations, we select a subset with low scatter. A
given amount of noise in the data sets a lower limit for the cutoff however,
limiting the utility of this metric. In practice this noise is likely to come
from imperfect knowledge of the unlensed SN lightcurve, rather than
observational noise. If the standard deviation can only be recovered with 0.05
mag precision the improvement is marginal - 70% of microlensing lightcurves
will be consistent with flat and only the most extreme caustic crossings can
be excluded. If a precision of 0.01 mag is achievable (which is likely the
case if SALT3 mismatches correlate with time) then half of the lightcurves
will be consistent with flat and the standardizability for this half of the
dataset improves to approximately 0.15 magnitudes.

Figure 4: Given the standard deviation $\sigma_{\text{lightcurve}}$ of the
lightcurve data points, we can select a subset (fraction) of lightcurves that
have $\sigma_{\text{lightcurve}}$ less than some desired cutoff. Lowering the
cutoff reduces the fraction of lightcurves selected and their scatter. The red
“+” marks where the cutoff for $\sigma_{\text{lightcurve}}=0.05$, while the
red “x” marks where the cutoff for $\sigma_{\text{lightcurve}}=0.01$.

###  4.2 A bank of lightcurves

Since we are able to simulate microlensing curves, the statistically rigorous
way to infer microlensing magnifications is to compare observations with
similar simulated data (Kochanek, 2004).

We assume that an observed lightcurve can be parameterized by a single value,
the amount of microlensing (de)magnification at peak, $\mu$. We then have that

$P(\mu|D)\propto P(D|\mu)P(\mu)$ (1)

There is additionally a nuisance parameter, the position $\mathbf{y}$ of the
source within the microcaustics, that must be marginalized over. The
marginalization is approximated as a sum

$P(\mu|D)\approx\sum_{\mathbf{y}}P\left(D|\mu(\mathbf{y})\right)P(\mathbf{y})$
(2)

over a finite number of source positions, i.e. a finite number of simulated
lightcurves. The likelihood $P\left(D|\mu(\mathbf{y})\right)$ is given by a
$\chi^{2}$ statistic of the difference between the data and the simulations.
Thus, $P(\mu)$ is determined by summing up the finite collection (bank) of
lightcurves, where each lightcurve in the bank is weighted by how well its
shape matches the observed lightcurve.

We separate our 100,000 lightcurves into two sets: 90,000 lightcurves with no
noise to serve as perfect members of the bank, and 10,000 lightcurves with
noise to serve as our mock observed data. The lightcurves are all shifted in
magnitudes such that their means are each 0. This way, there is only relative
knowledge of how their shapes evolve over time.

We use Equation 2 to calculate the microlensing magnification posterior for
each of the 10,000 mock observed lightcurves. We then calculate the scatter of
the posterior for each lightcurve. The majority of mock observed lightcurves
have microlensing scatters less than 0.2 mags. This is roughly a factor of 2
improvement from the point source microlensing histogram, indicating that
information about the lightcurve shape can narrow down the predicted
microlensing (de)magnification. Figure 5 shows these scatters as a function of
how many bank lightcurves are similar.

Figure 5: Inferred microlensing scatter at peak luminosity for our 10,000 mock
observed lightcurves, versus the number of bank lightcurves that make up the
inner 95% of their posteriors. The majority of mock observed lightcurves have
scatter less than 0.2 mags – 33% fall in the yellow bin, with a scatter of
$\approx$0.12 mags. However, some of the lightcurves are matched only by a
handful of bank lightcurves and so their posteriors are likely spurious.

The majority of the bank and the dataset consists of lightcurves with no
distinguishing features (flat) and these are easily standardized as seen in
Figure 5. In contrast, there are few lightcurves which have greater time
variability, i.e. caustic crossings. The lack of similar curves in the bank
makes the inferred posteriors unreliable: the approximation in Equation 2 only
works if the bank is sufficiently well sampled. One could simulate more
lightcurves to give a reliable posterior for everything (see, e.g., Kochanek,
2004, where the order of $10^{8}$ lightcurves are used), but since the bulk of
the lightcurves are flat, and these are standardizable, we leave this for
future work.

###  4.3 Machine learning – regression

We train a neural network to predict the (de)magnification due to
microlensing. The network is a fully connected network with 2 hidden layers –
simple, but sufficient for our purposes. The size of the input layer is
determined by the observation length and cadence of the lightcurve, while the
two hidden layers each have half as many neurons as the input layer to avoid
overfitting. We take our set of 100,000 noisy lightcurves and set aside 80,000
as as training data, 10,000 as validation data, and 10,000 as test data. The
label for each lightcurve in the training and validation sets is the amount of
(de)magnification at peak supernova luminosity. Training is stopped when the
training loss (mean squared error) on the validation data stops decreasing.
The network is then applied to the 10,000 test lightcurves.

Figure 6: Predicted versus true microlensing (de)magnification from the
regression neural network. The white solid and dashed lines mark the median
and inner 68% of the distribution of true values for lightcurves within the
same bin of predictions. The black dotted line marks where prediction = truth.

Figure 6 shows the predicted microlensing magnifications compared to the true
values. The peak in the distribution at 0.5 mag is due to all of the flat
lines. The neural network has learned i) the average magnification of the flat
lines (which make up the bulk of the data), and ii) that it can can minimize
the error by assigning the mean value of the flat $N=1$ lines to all of them.
There are, however, flat lines that come from the $N=2$ region which therefore
have an incorrect prediction. This is the reason for a slight tail in the
distribution to the right of the peak, which cannot be removed with the simple
point estimator of this network. More complicated networks that return a full
posterior should be able to resolve this problem (e.g. Bayesian Neural
Networks, Jospin et al., 2022).

The neural network made some progress on the remaining data, which show a
roughly even amount of scatter around the predicted = truth line with widths
of 0.2-0.25 magnitudes. This is slightly greater than the assumed intrinsic
supernova scatter of 0.15 magnitudes. Furthermore, the number of lightcurves
with bright predicted magnifications is small, making it somewhat difficult to
be as confident in their scatter. While the small number of lightcurves with
bright magnifications is due to the particular point we picked in microlensing
parameter space, and could potentially be remedied with more simulations, the
improvements to the flat lightcurves are in line with Table 1 and our
expectations.

###  4.4 Machine learning – classification

We train a different neural network to classify the lightcurves into two
categories: did it cross a caustic or not? We use the same simple network
architecture as before but instead of training the network to learn the
underlying magnification at peak, the training set is tagged 1 if it did not
cross a caustic and 0 if it did cross. The output of the network then is no
longer the magnification at peak, but the probability of a lightcurve
belonging to either of the two categories. Figure 7 shows the ROC curve for
the network when applied to the test data, along with the fraction of
lightcurves selected as a function of the classification threshold and their
associated scatter. At a classification threshold of 75% (roughly the
threshold with the highest true positive rate and lowest false positive rate),
slightly less than 60% of the lightcurves are inferred not to have crossed a
caustic, and the scatter from those lightcurves is roughly 0.2 mags –
consistent with the expectations from Table 1.

Figure 7: Top: ROC curve for the classification neural network. Bottom:
Fraction of lightcurves selected as not crossing a caustic, and their scatter,
as a function of classification threshold. The red plus symbol marks the 75%
classification threshold in both figures.

###  4.5 Remarks

We have shown that there are a variety of viable methods for reducing the
scatter for the intermediate scenario microlensing parameters that we chose
($\kappa=\gamma=0.4,s=0.75$). Ultimately, each method gives approximately the
same results, with the same underlying physical reason: flat microlensing
lightcurves, which come from supernovae that do not cross a caustic as they
expand over a long enough timescale, are more standardizable. The performance
of the lightcurve bank and the regression neural network yield some further
improvement for lightcurves that do cross a caustic, but since these events
start off with much higher variance the improvement will not have much impact
on the inference of the Hubble constant from a population of glSNe. The simple
method of using the standard deviation of the lightcurve is sufficient to
quickly select out the standardizable microlensing lightcurves.

##  5 Covering microlensing parameter space

Figure 8: Contour plot showing the scatter over microlensing parameter space
for those lightcurves which the neural network predicts did not show any
caustic crossings. The lightcurves used assume we observe every 2 days from 5
days before peak up to 50 days after peak (in the rest frame of the supernova)
and have 0.05 mag noise on the data. The solid white line denotes the contour
of 0.15 mag microlensing scatter for the lightcurves. The dashed white line
denotes the contour of 0.15 mag scatter when considering only the point source
histograms, i.e. no time-series information (Foxley-Marrable et al., 2018).
The vertical black dotted line marks the boundary where the counter-image of a
singular isothermal sphere is demagnified (left) or magnified (right).

Whilst the previous section focused on a particular set of parameters
($\kappa=\gamma=0.4,s=0.75$), we expect that other regions of microlensing
parameter space with lower magnifications or lower stellar fractions can be
similarly improved. We turn now to covering the rest of the useful parameter
space. We opt to use the final neural network-based approach where lightcurves
are sorted into two categories.

Any triplet of model values $(\kappa,\gamma,s)$ can be transformed into a
doublet of $(\kappa=\gamma,s)$ (Paczynski, 1986; Kochanek, 2004; Vernardos et
al., 2014; Schechter et al., 2014), which is applicable for a singular
isothermal elliptical potential. We therefore cover the space from
$\kappa=\gamma=0.25$444Strong lensing does not occur below $\kappa=0.25$ in an
isothermal sphere, although it will if the system is more elliptical. to
$\kappa=\gamma=0.5$ and from $s=0$ to $s=1$. For each point sampled in this
space, we create 5,000 microlensing lightcurves using the same procedures
discussed in Section 3.

We take 4000 lightcurves from every point sampled in the parameter space to
create our training set and use 1000 lightcurves from each point as validation
to avoid overfitting555The final amount taken from each sample is technically
smaller, as we want equal number of lightcurves which do or do not cross a
caustic to train on. We ultimately end up with $\approx$300,000 lightcurves in
our training set and $\approx$75,000 as validation.. This is done for two
reasons: first, we would expect to get a fair number of flat lightcurves in
the standardizable regions and non-flat lightcurves in the unstandardizable
regions; second, by training the network on a sample of lightcurves that come
from everywhere in the space and therefore show potential complexities from
multiple caustic crossing events, we expect it to be more general and
applicable.

Once the training is completed, we generate 5000 lightcurves at each point in
parameter space to test the performance of the network. We calculate the
scatter on those lightcurves which the network says does not cross a caustic.
Figure 8 shows the results. The neural network selects only a fraction of the
lightcurves (shown in Figure 14). However, the standardizable region of
parameter space has been improved when compared to considering just the point-
source histogram (i.e. using no time-series information from the lightcurve,
Foxley-Marrable et al., 2018). More systems with magnified counter-images are
standardizable, and a large fraction of parameter space with demagnified
counter images are now standardizable as well.

##  6 Standardizable LSST lensed Ia supernovae

The discussion up to this point has focused on the theoretical improvements
that could be made to standardizing microlensed Ia supernovae based on
temporal information from their lightcurves. In this section, we discuss the
practical difficulties of actually observing a glSN Ia which can be
standardized. We then estimate the number of standardizable glSNe Ia to be
discovered by LSST in the next decade. Finally, we examine how well we can
constrain systematics in measurements of $H_{0}$ from the mass sheet
degeneracy with a sample of standardizable glSNe Ia.

Given that the unstandardizable saddlepoint macroimages are the trailing
images, standardizing lensed supernovae heavily relies on discovering the
leading image (or two, if the system is a quad) before peak. For a doubly
imaged system, there is only one chance to standardize it. For a quad, there
can be a second chance, but only for the rare quads with two standardizable
images discovered early enough.

###  6.1 Generating (micro-)lensed supernovae samples

We follow Sainz de Murieta et al. (2023) to generate a population of glSNe Ia
and determine the fraction that will be observable, useful for time-delay
cosmography, and standardizable. While Sainz de Murieta et al. (2023) focused
on unresolved lightcurves, the methods can be used to create a catalogue of a
large population of glSNe Ia observed in the $i$ band.

Since we need to measure the microlensing in each image, the images need to be
resolvable; we limit ourselves to systems where the minimum image separation
is $0.8^{\prime\prime}$. In order to be a good candidate for time delay
cosmography, the supernova images also need to have an appreciable ($\gtrsim$
10 days) time delay, as time delays can be recovered to within a day or two
(Pierel et al., 2021; Huber et al., 2022). We additionally impose that the
system be discovered no later than 10 days before the peak of the first
(second) image for doubles (quads) in the observer frame. This allows a long
baseline to measure the evolution of the microlensing signal as the SN disk
grows as in Section 4.

We estimate the stellar fractions at the image locations by assuming an
elliptical de Vaucouleurs profile for the light (Vernardos, 2019) with
effective radii calculated from the scaling relation in Hyde & Bernardi
(2009). Following Dobler & Keeton (2006), we assume isothermal total density
profiles and normalise the stellar component such that the maximum stellar
fraction is 1. This will make our forecasts somewhat pessimistic since
observations prefer lower stellar fractions, but it depends on the stellar
initial mass function (Auger et al., 2010).

We furthermore limit ourselves to systems where microlensing cannot demagnify
the first image enough to be unobservable. This is done to ensure that any
observed image would be a fair draw from the microlensing magnification
distribution, rather than an unfair sampling from the brighter tail of the
distribution. This would naively restrict us to redshifts where the unlensed
supernova magnitude is greater than the detection limit of the survey; when
accounting for microlensing, the minimum magnification is
$\mu=(1-s\cdot\kappa)^{-2}$ which, depending on the macro-model and the
stellar fraction for the image, allows us to push to slightly farther
redshifts. We use 24th magnitude at this minimum magnification as our
detection limit, approximately the value appropriate for LSST.

###  6.2 Results

Figure 9: One realisation from the catalogue of glSNe systems expected after a
decade of LSST observations. The dashed and solid black lines denote the
boundaries where systems are standardizable either via the microlensing
magnification histogram or their lightcurves.

We can now estimate rates of standardizable glSNe and make forecasts for
breaking the mass sheet degeneracy with these systems.

####  6.2.1 Rate estimates

We start with a population of glSNe from Sainz de Murieta et al. (2023) with a
rate of 13 doubles (4 quads) per year which are discovered early enough for
standardization to be possible. We show one realization from the catalogue of
such systems in Figure 9. Of those, 8.3 (1.3) per year are cosmologically
golden, with time delays $>10$ days and minimum image separations of
$0.8^{\prime\prime}$. Of the golden systems, 6.0 doubles and 0.3 quads per
year have standardizable images with flat lightcurves.

####  6.2.2 Constraints on $H_{0}$

The MSD is governed by a parameter $\lambda$. The propagation of magnification
uncertainties onto $\lambda$ is given in Appendix D. If we assume the
intrinsic supernova scatter of 0.15 mag dominates the uncertainty on the
observed magnification and microlensing dominates the uncertainty on the model
magnification, we can infer how each standardizable system will constrain the
systematics from the mass sheet transformation. Combining the $\approx$60
expected cosmological golden standardizable doubles, we find that they should
constrain the population average $\langle\lambda\rangle$ to
$1.00^{+0.07}_{-0.06}$% fractional uncertainty, and therefore detect
systematics in $H_{0}$ from the MSD at the 1% level. We note that if the
intrinsic supernova scatter is lower (0.1 mag), this changes to a
$0.74^{+0.06}_{-0.05}$% fractional uncertainty on $\langle\lambda\rangle$ and
$H_{0}$.

###  6.3 Comparison to previous works

Our results are more pessimistic than those of Foxley-Marrable et al. (2018).
The main factor behind this is a decrease in the estimated number of systems
to be discovered with LSST; compare, e.g., Goldstein et al. (2019), Wojtak et
al. (2019), and Sainz de Murieta et al. (2023) (particularly their discussion
in Section 4.3). We additionally imposed more restrictions on what systems
might be considered cosmologically useful and standardizable than Foxley-
Marrable et al. (2018).

Arendse et al. (2023) estimate slightly higher rates of detection for LSST,
which in part come about due to considering alternative image detection
methods that do not rely solely on magnification (see also, e.g., Wojtak et
al., 2019). However, detection methods such as image multiplicity miss early
time information which could be key for standardizing microlensed lightcurves.

Comparing our results to those ignoring the time evolution of the lightcurve
(i.e., roughly following Foxley-Marrable et al., 2018, and using just the
point source microlensing histograms), we find that $\approx$20% more of the
cosmological golden systems will be standardizable. We were able to
substantially increase the size of the standardizable region of microlensing
parameter space in Figure 8, which gives a 30% improvement on the number of
standardizable systems in the whole population. However, the majority of
systems with long time delays tend to come from the regions that were already
standardizable under the considerations of Foxley-Marrable et al. (2018).

A byproduct of our methods is a decrease in the scatter on the systems that
were already standardizable, at the cost of losing a small fraction of
lightcurves. However, we get similar constraints on $H_{0}$ for our simulated
systems if we ignore information about the shape of the lightcurves: the rate
of standardizable doubles decreases to 5.1 per year, which leads to a
$1.11^{+0.09}_{-0.07}$% fractional uncertainty on $H_{0}$ from the MSD. This
suggests that model macro-parameters are the main driving factors in
standardizability. This comes from the fact that whilst we have tightened
$P(\mu)$ for all macrominima, the intrinsic scatter in the Ia population sets
a floor on the utility of an individual system in breaking the MSD.

###  6.4 Observing the counter-images of standardizable images

Actually doing time-delay cosmography with these systems requires us to detect
the trailing saddlepoint image(s). Figure 10 shows the distribution of counter
image magnifications at peak, from the lens macro-model only. The distribution
peaks at 24.5 mags in the $i$ band for the doubles, 24 mags for image 4 of the
quads, and 22.5 mags for image 3 of the quads. However, this ignores the fact
that saddlepoint images are more susceptible to microlensing demagnifications
(Schechter & Wambsganss, 2002). Followup observations of the trailing image(s)
will be difficult, but possible.

When considering the number of standardizable systems with bright counter-
images, our lightcurve method represents a substantial improvement upon the
previous histogram method of Foxley-Marrable et al. (2018). If we consider
only the doubles which have a trailing image that peaks brighter than 25th
mag, our yearly rate of standardizable doubles drops from 6.0 to 3.9, compared
to the 3 per year expected when ignoring lightcurve shape. After ten years, we
should have 39 systems and a $1.28^{+0.12}_{-0.09}$% fractional uncertainty on
$H_{0}$ from the MSD. For counter-images brighter than 24th magnitude we
expect 2 standardizable systems per year and $1.8\pm 0.2$% fractional
uncertainty on $H_{0}$ from the MSD.

Figure 10: Distribution of counter image Sloan i band magnitudes under the
lens macro-model only for the standardizable systems.

##  7 Limitations

The detection of any microlensing signal is limited due to uncertainties in
the supernova model. We have pessimistically assumed 0.05 mag uncertainty on
each point in our lightcurves. Whilst this is comparable to the mismatch in
SALT3, we have neglected temporal or chromatic correlations, both of which
might be able to improve upon our assumed 0.05 mag uncertainty.

The standardizable lightcurves are typically demagnified from the macro-model
prediction. Although in our study we limited ourselves to simulated systems
where microlensing could never demagnify the first image below the detection
threshold of LSST, we must keep in mind that this will not always be the case.
Supernovae at higher redshifts $(z\gtrsim 1)$ begin to suffer from a
microlensing magnification bias in LSST, as microlensing can demagnify
(relative to the macro-model) the first image of systems below the detection
threshold of the survey. Incorporating such systems would require a careful
simulation of their Malmquist bias.

We have assumed that the lens macro-model parameters for the images are
perfectly known. In reality, uncertainties on $\kappa$, $\gamma$, and $s$
require properly marginalizing over a region of the parameter space in Figure
8, rather than taking the scatter at an individual point. If uncertainties on
the macro-model parameters are not large, the slow variations of the scatter
in the standardizable region suggests that there will be minimal changes to
results.

While our simulated systems have values of $s$ calculated with simple
assumptions, $s$ should properly be inferred from the mass to light ratio of
the lensing galaxy. If we were to use a Salpeter or Chabrier IMF, the smooth
matter fraction would increase (Foxley-Marrable et al., 2018), providing a
small boost in the number of standardizable systems. A rough estimation of
this effect by halving the stellar fractions used leads to little change in
our results: 7.6 standardizable doubles per year and 0.4 quads, with the
doubles constraining $H_{0}$ to $0.90^{+0.06}_{-0.05}$% fractional uncertainty
from the MSD.

Furthermore, there is an additional dependence of microlensing on the mass
function of the lenses (Schechter et al., 2004) that is usually ignored. This
dependence is difficult to manifest in microlensed quasars for physical
reasons (Lewis & Gil-Merino, 2006) but the implication for lensed supernovae
has not been investigated.

While we considered the simple model of a uniform expanding disc, real
supernovae will have a 2D intensity profile that must be convolved with the
microlensing magnification pattern (Goldstein et al., 2018; Huber et al.,
2019). We don’t expect any changes to our conclusions due to the fact that the
standardizable lightcurves have essentially constant microlensing
magnification.

##  8 Conclusions

In this work, we have investigated how the temporal information from
microlensing lightcurves can be used to reduce the uncertainty on the unlensed
SN magnitude. We have shown that it is possible to select a subset of glSNe Ia
images which can be used to break the mass sheet degeneracy. The main idea we
have put forward is that by selecting lightcurves of supernovae which do not
cross any microcaustics as they expand, you can restrict yourself to regions
of the source plane typically outside the microcaustics which have lower
amounts of scatter. Our method relies on having a long enough sequence of
observations to rule out caustic crossings.

Using a simulated sample population of lensed Type Ia supernovae with
resolvable ($>0.8^{\prime\prime}$ separation) images and long ($>10$ days)
time delays, we estimate the number of detectable, standardizable glSNe Ia
systems to be discovered by LSST in the next decade as $\approx$60 doubly
imaged systems and $\approx$3 quadruply imaged system. The doubles can
constrain systematics in $H_{0}$ from the MSD at a $1.00^{+0.07}_{-0.06}$%
precision level. While the first image for the majority of cosmologically
useful doubles should be able to break the mass sheet degeneracy, there will
be observational challenges in following up the second image to measure the
time delay. This is mostly driven by the faintness of the (typically
demagnified) counter-images, which have a median brightness of $\approx$25 mag
before accounting for additional microlensing (de)magnification. If we are
only able to followup counter-images brighter than 24th magnitude, only a
third of the double image systems are retained, and the fractional uncertainty
on the breaking of the MSD degrades to $1.8\pm 0.2$%.

Time delay measurements will already require high-quality observations up to
and after peak supernova luminosity (Pierel & Rodney, 2019; Huber et al.,
2019). In that sense, no additional data should be required to determine
whether the expanding supernova crosses a microcaustic or not. Given the
observational cost of followup, it should be focused on systems where the
first image is most likely to be standardizable. When the microlensing effect
does not vary with time we essentially know the intrinsic shape of the
unlensed SN lightcurve. By focusing on standardizable systems it will be
easier to infer time delays, and after 10 years of LSST allow us to test the
impact of the MSD on $H_{0}$ at $1\%$ precision.

## Acknowledgements

We would like to thank D’Arcy Kenworthy for his kind assistance in answering
questions about SALT3. We would also like to thank Nikki Arendse and Matt
O’Dowd for extremely helpful discussion.

Numerical computations were done on the Sciama High Performance Compute (HPC)
cluster which is supported by the ICG, SEPNet, and the University of
Portsmouth.

This work has received funding from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (LensEra:
grant agreement No. 945536). TC is funded by the Royal Society through a
University Research Fellowship. For the purpose of open access, the authors
have applied a Creative Commons Attribution (CC BY) license to any Author
Accepted Manuscript version arising.

## Data Availability

The simulated data from this work can be made available upon reasonable
request to the corresponding author.

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##  Appendix A Why macrosaddles cannot be standardized

We stated previously that macrosaddles cannot be standardized. It is well
known that saddlepoint images are more susceptible to microlensing
(de)magnifications (Schechter & Wambsganss, 2002), and the scatter for
saddlepoint images can be found in Weisenbach et al. (2021). The only reason
that macrominima can be standardized is that the dominant source of scatter
eventually comes from the $N=1$ region. For macrosaddles, there is an $N=0$
region – i.e. there are no microimages which are minima of the time delay
surface. Minima are important because they cannot have $\mu<1$; therefore, the
histograms and subhistograms must have hard boundaries for how much they can
be demagnified from the macro-model. One might notice in Figure 1 that
subhistograms for different values of $N$ take on roughly the same shape; this
has been mentioned in the literature but not greatly explored, and we reserve
such exploration for future work. Saddlepoint images have no such restriction
or lower bound on $\mu$ however, and can become arbitrarily demagnified.
Therefore, the subhistograms for $N=0$ regions can look slightly different and
have a greater width; see Figure 11 or, e.g., Figure 4 from Granot et al.
(2003) or Figure 3 from Saha & Williams (2011).

One might argue that the saddlepoint shown in Figure 11 is a bright one with
many subhistograms that have similar probabilities, which we have already said
makes images unstandardizable. However, the unstandardizability holds true for
faint saddlepoints as well, as shown in Figure 12 – even if we consider ruling
out regions where an expanding supernova would cross a caustic. In essence, we
can remove the high magnification tail from minima and saddlepoints, but
saddlepoints also have a low magnification tail which cannot be removed. This
results in saddlepoint macroimages being unstandardizable.

Figure 11: Microlensing magnification histogram for a bright macrosaddle
showing the $N=0$ subhistogram. Figure 12: Microlensing magnification
histogram for a faint macrosaddle showing the $N=0$ subhistogram. Even when
ruling out regions where an expanding supernova would cross a caustic, the low
magnification tail of the histogram prevents the system from being
standardizable.

##  Appendix B SALT3 model errors

Figure 13: Top: SALT3 supernovae templates with errors due to the model
covariances. Bottom: Just the errors as a function of supernova phase.

Figure 13 shows SALT3 supernova templates for the rest frame $g$ and $r$
bands, which will be typically redshifted to the $r$ and $i$ bands in the
observer frame. The errors associated with the templates are also shown. For
the majority of rest frame phases, the errors are on the order of 0.05 mags.

##  Appendix C Fraction of lightcurves selected

Figure 14 shows the fraction of lightcurves which the neural network is 75%
confident did not have any caustic crossings as a function of lens macro-model
parameters. For the majority of standardizable glSNe Ia, at least 70% of the
lightcurves should be useable (flat).

Figure 14: Contour plot showing the fraction of lightcurves which the neural
network predicts did not show any caustic crossings. The lightcurves used
assume we observe every 2 days from 5 days before peak up to 50 days after
peak (in the rest frame of the supernova) and have 0.05 mag noise on the
microlensing lightcurve. The solid white line denotes the contour of 0.15 mag
microlensing scatter for the lightcurves. The vertical black dotted line marks
the boundary where the counter-image of a singular isothermal sphere is
demagnified (left) or magnified (right).

##  Appendix D Propogation of uncertainties

Scaling the mass model by the mass sheet parameter $\lambda$ and introducing
the presence of a mass sheet of convergence $\kappa=1-\lambda$ causes the
magnification to transform as

$\mu\rightarrow\lambda^{-2}\mu.$ (3)

Given the known intrinsic brightness of a supernovae and its observed
brightness, we can infer the magnification. By comparison with the
magnification predicted by the model, we can constrain the value of $\lambda$.
In the absence of microlensing, a single standardizable glSN Ia can constrain
the value of $\lambda$ to within $\approx$ 5% fractional uncertainty (Mörtsell
et al., 2020; Birrer et al., 2022).

Using the fact that $\lambda$ is a ratio of model and observed magnifications,

$\lambda^{2}=\frac{\mu_{\text{model}}}{\mu_{\text{obs}}}$ (4)

and the change in magnitudes from lensing is

$\Delta m=-2.5\log\mu=\frac{-2.5}{\ln 10}\ln\mu$ (5)

the fractional uncertainty on the mass sheet parameter $\lambda$ from an
individual system is

$\displaystyle\Big{(}\frac{\sigma_{\lambda}}{\lambda}\Big{)}^{2}$
$\displaystyle=\Big{(}\frac{1}{2}\Big{)}^{2}\Big{[}\Big{(}\frac{\sigma_{\text{obs}}}{\mu_{\text{obs}}}\Big{)}^{2}+\Big{(}\frac{\sigma_{\text{model}}}{\mu_{\text{model}}}\Big{)}^{2}\Big{]}$
(6) $\displaystyle=\Big{(}\frac{\ln 10}{5}\Big{)}^{2}\Big{(}\sigma_{\Delta
m_{\text{obs}}}^{2}+\sigma_{\Delta m_{\text{model}}}^{2}\Big{)}$ (7)

where $\sigma_{\Delta m_{\text{obs}}}$ is the uncertainty on the observed
magnification of the supernova and $\sigma_{\Delta m_{\text{model}}}$ is the
uncertainty on the magnification of the model, in magnitudes.