Lens modelling of the strongly lensed Type Ia supernova iPTF16geu

Improved observational constraints on the strongly lensed Type Ia supernova iPTF16geu, including the time delay between images, are used to decrease uncertainties in the lens model by a factor $\sim 7$ and to investigate the dependence on the universal expansion rate $H_0$. We constrain a combination of the dimensionless Hubble constant, $h\equiv H_0/(100\,{\rm km/s/Mpc})$, and the slope of the projected surface density of the lens galaxy, $\Sigma\propto r^{\alpha-2}$, at $r\sim 1\,{\rm kpc}$, to $h\gtrsim 2(1-2\alpha/3)$. This implies $\alpha\gtrsim 1$ using our current knowledge of the expansion rate, corresponding to a flatter surface density than an isothermal halo for which $\alpha=1$. Regardless of the slope, a smooth lens density fails to explain the iPTF16geu image fluxes, and additional sub-structure lensing is needed. Taking advantage of the standard candle nature of the source and including stellar microlensing, we show that the probability to obtain the observed fluxes is maximized for $\alpha=1.3$, confirming that $\alpha\gtrsim 1$. For $\alpha=1.3$, images 1 needs an additional $magnification$ from microlensing of $\Delta m_1\sim -0.7$, whereas images 3 and 4 require a $demagnification$ of $\Delta m_3\sim 0.8$ and $\Delta m_4\sim 1.0$, the total probability for which is $p_{\rm tot}\sim 26\,\%$. We conclude that the iPTF16geu flux"anomalies"are well within stellar microlensing predictions.


INTRODUCTION
The expansion history of the Universe can be constrained by measuring redshifts and distances of standard candles such as Type Ia supernovae (SNe Ia) (Goobar & Leibundgut 2011). The redshift gives the growth since the time when the light was emitted. This time is measured, together with the spatial curvature of the Universe, by the distance to the SNe Ia as inferred from its apparent magnitude.
Weak gravitational lensing from inhomogeneities in the matter distribution will cause a scatter in the distance measurements, possibly degrading the accuracy of the cosmological parameters derived from the expansion history. The first tentative detection of the gravitational magnification of SNe Ia was made in Jönsson et al. (2007), see also Mörtsell et al. (2001b); Jönsson et al. (2006); Nordin et al. (2014); Rodney et al. (2015); Rubin et al. (2018). In principle, the effect can be corrected for by cross-correlating the SNe Ia observations with data on the foreground galaxies responsible for the lensing effect (Amanullah et al. 2003;Jönsson et al. 2006Jönsson et al. , 2008Jönsson et al. , 2009). The induced scatter can also be used to measure the masses of the foreground galaxies (Jönsson et al. 2010) and to constrain the fraction of matter inhomogeneities in compact objects (Rauch 1991;Metcalf & Silk 1999;Seljak & Holz 1999;Goliath & Mörtsell 2000;Mörtsell et al. 2001a;Mörtsell 2002;Zumalacarregui & Seljak 2018), see also Dhawan et al. (2018).
In this paper, we study the first resolved strongly lensed SNe Ia, It is well known that the time delay between images in strong gravitational lensing systems can be used to constrain the Hubble constant, H 0 , (Refsdal 1964), see also Goobar et al. (2002); Mörtsell et al. (2005); Mörtsell & Sunesson (2006). SNe Ia are especially useful in this respect since (Kolatt & Bartelmann 1998;Oguri & Kawano 2003) (i) the time delay can potentially be measured with high accuracy, (ii) their standard candle nature can partly break the mass and source sheet degeneracy, (iii) their transient nature allows for accurate reference imaging.
Given the persistent tension [currently at the 4.4 σ-level (Riess et al. 2019)] between H 0 inferred from local distance indicators and the Cosmic Microwave Background (CMB) (Aghanim et al. 2016), independent measurements of the current expansion rate will shed light on possible explanations (e.g., ).
The first observed SN Ia with expected multiple images is PS1-10afx at redshift z = 1.388 and a flux magnification µ ∼ 30 (Chornock et al. 2013;Quimby et al. 2013). However, the strong lensing nature of the system was not verified by high spatial resolution imaging. Multiple images of the core-collapse supernova (SN) Refsdal at z = 1.49 have been observed (Kelly et al. 2016a,b), but could not be used to measure the lensing magnification directly [see also Amanullah et al. (2011)].
The first strongly lensed SN Ia, iPTF16geu at redshift z = 0.409, was identified by its high magnification µ > 50 (Goobar et al. 2017). Subsequent high spatial resolution imaging confirmed the multiple images of the SNe Ia. In Goobar et al. (2017), the positions of the SN images with respect to the lensing galaxy were used to construct a lensing model, an isothermal ellipsoid galaxy (Kassiola & Kovner 1993;Kormann et al. 1994) with ellipticity e = 0.15 ± 0.07 and mass M = (1.70±0.06)·10 10 M within a radius of ∼ 1 kpc. The total magnification of the SN images was not well constrained by the model, but the adopted smooth lens halo predicted brightness differences between the SN images in disagreement with observations, providing evidence for substructures in the lensing galaxy. The time delays between images were predicted to be shorter than 35 hours at 99.9 % confidence level. The system was also studied in More et al. (2017), where the anomalous image flux ratios were confirmed, see also Yahalomi et al. (2017).
Substantial effort in obtaining follow-up data, including reference imaging after iPTF16geu had faded, has resulted in smaller uncertainties in SN image and lens galaxy positions, the iPTF16geu host morphology, as well as the first observational constraints on the time delay between SN images, as presented in an accompanying paper (Dhawan et al., in preparation).
In this paper, we take advantage of the improved observational constraints on the system to improve the lens model and investigate the dependence on the expansion rate H 0 . We also quantify to what degree lens galaxy stellar microlensing can explain the observed image flux anomalies.
We use geometrized units for which c = G = 1 and express the dimensionless Hubble constant, h, in units of 100 km/s/Mpc, h ≡ H 0 100 km/s/Mpc . (1)

GRAVITATIONAL LENSING
When light from a source at angular position ì β and redshift z s , passes a mass at z l along its path, it is deflected and observed at an angle ì θ. The time delay, ∆t, compared to an unlensed image is given by Here, D l , D s and D ls are angular distances to the lens, source and between the lens and source, respectively, and Ψ( ì θ) is the scaled, projected Newtonian potential, Φ, of the lens integrated along the path of the light ray. It is related to the surface mass density of the lens, or convergence, through the Poisson equation, giving where Using Fermat's principle that light rays traverse paths of stationary time with respect to variations of the path, we obtain the lens equation where ì α( ì θ) = ∇Ψ( ì θ). The magnification, µ( ì θ), is given by where is the Jacobian matrix of the lens mapping.

Mass sheet degeneracy
If we rescale Ψ using the scalars ξ and u, and the vector ì s as the deflection angle ì α = ∇Ψ rescales as and the convergence κ = ∇ 2 Ψ/2 according to If we also rescale the (unobserved) source position ì β = ξ ì β−ì s, the image positions will be unchanged. This is the mass sheet degeneracy; given only the observed image positions, we are free to rescale the projected mass 1 . Magnification and time delay predictions, however, will change according to µ = µ ξ 2 and ∆t = ξ∆t.
From equation 2, the inferred h is proportional to the predicted ∆t from the lens model, h = ξ h. Error propagation gives For a SN Ia, |∆m| ≈ 0.1 and the fractional uncertainty on h from the mass sheet degeneracy is of order 5 %.

LENS MODEL
For a cored isothermal ellipsoid, the convergence is given by (Kassiola & Kovner 1993;Kormann et al. 1994) where b is a mass normalization corresponding to the Einstein radius for s = 0, s is the core radius and is related to the minor and major axis ratio q as In terms of the ellipticity, e , and eccentricity, e, We can generalize equation 14 as In the core-less circularly symmetric case ( = s = 0), the parameter α relates to the slope, n, of a three-dimensional density profile ρ ∝ r −n , where r 2 = θ 2 1 +θ 2 2 , for which κ ∝ r −n/2 as α = 2 − n/2, or equivalently n = 2(2 − α). For a singular isothermal sphere (SIS), = s = 0 and α = 1 → n = 2. We refer to its elliptic generalization 0 as a singular isothermal ellipsoid (SIE). Note that for an SIE lens, the image magnification and convergence are related by µ SIE = (1 − 2κ SIE ) −1 .
In terms of the dimensionless τ, the time delay between two lensed images, I and II, of a single source is For a SIS halo, Ψ = ì θ · ì α and Converting to polar coordinates, for two images on opposite sides of the lens, we can gain some analytical insight by Taylor expanding in (Mörtsell et al. 2005) Note that the time delay only depends on the slope of the projected surface density at radii between the images (Falco et al. 1985). The main observational uncertainty affecting the precision of ∆τ, and thus h, is the location of the lens centre (from its impact on ∆t SIS ). The main theoretical assumption is the slope of the surface mass density, 2 − α, in the annulus between the images.

DATA
For the iPTF16geu  The observed image positions are listed in table 1. The lens galaxy is observed to have ellipticity 0.127 < e < 0.252 and orientation (from North to East) 10 • < RA < 20 • . Expressing positions in arc seconds [arcsec, "], τ is given in units of The arrival time of the images can be observationally constrained to the values given in table 2 with the corresponding time delays in table 3 (Dhawan et al., in preparation). The fact that data is consistent with zero time delay between images means that we will not be able to put an upper limit on h.  Table 5. Observed magnification ratios defined as r i j ≡ µ i /µ j . Images The observed image magnifications are given in table 4 with the corresponding magnification ratios in table 5 (Dhawan et al., in preparation).

METHODS AND RESULTS
We use the lensmodel software (Keeton 2001a,b), to fit the parameters of the lens model, including b, , the slope α, the core size s, the orientation RA of the major axis, the position of the centre of mass of the lensing galaxy, and the corresponding iPTF16geu source position.
First, we check how the quality of the fits depends on the allowed freedom in the lens model and the observational constraints included. Next, we present constraints on the mass distribution of the lens and the dimensionless Hubble constant h.
A note on the number of degrees of freedom (dof) for the fits: The lens and source position represent 2 + 2 dof, the mass, ellipticity and orientation of the lens 1 + 1 + 1 dof, in total 7 dof. Fitting also the slope and core size gives an additional 2 dof. The host galaxy is described by its position, ellipticity s and orientation (4 dof).
The image positions represent 8 data points, the flux and time delays 3 each (since the source luminosity and explosion time are not fixed). The host ring radius has 60 data points, and the extremal points along this curve an additional 8 data points.

iPTF16geu image positions
Restricting to iPTF16geu data (i.e., not fitting for the host galaxy image), we can in principle include the observed image positions and fluxes. Since the fluxes can be subject to systematics effects such as lensing by substructure and dust extinction not properly corrected for, in table 6, we show results when fitting the lens model to the iPTF16geu image positions only. The quality of the fit does not improve when allowing for a varying slope, α, and core size, s, as compared to the SIE model. However, although their best fit values are close to the corresponding SIE values α = 1 and s = 0, they are not well-constrained by the iPTF16geu positions, since varying α and/or s can be compensated for by changing b in a way that keeps the total mass within the images constant.
In table 7, we show best fit parameters when fitting   Figure 1 shows the critical curves, corresponding to image positions with infinite magnification, i.e., det A( ì θ) = 0, and caustics, the corresponding source positions, for the best fit SIE lens model.
Given the high degree of circular symmetry of the lens system, the ellipticity of the lensing galaxy is highly degenerate with a possible external shear component for light deflections close to the critical curve. In fact, the system can be equally well fitted with a spherically symmetric lens galaxy and an external shear component with amplitude γ = 0.040 ± 0.002 and direction θ γ = 65.29 ± 0.12. If the external shear is caused by a SIS galaxy, the shear magnitude is given by where b ext is the Einstein radius of the external perturber, r the projected distance between the lensing galaxies and σ v the velocity dispersion of the perturber. The largest expected contribution from galaxies in a square field with side length 100 arcsec centred on the iPTF16geu system is from a galaxy with r ≈ 58 arcsec at φ ≈ 40 • and an SDSS photometric redshift estimate of z = 0.336 ± 0.0365 (Blanton et al. 2017). From its g-band magnitude of m g = 20.77, we can estimate its velocity dispersion to σ v ≈ 187 km/s (Mitchell et al. 2005;Jönsson et al. 2008). This corresponds to an external shear contribution of γ = 1.4 · 10 −4 for iPTF16geu. Other individual galaxies in the field contribute at most 20 % of this value and we conclude that the major part of the shear is caused by the ellipticity of the lensing galaxy itself.

Host galaxy
We next include host galaxy image data when constraining the lens mass distribution, following the method outlined in Kochanek et al. (2001). Assuming the host has a surface brightness monotonically decreasing with radius, we can predict the position of the image peak surface brightness in the radial direction around the lens galaxy, as well as the tangential position of the surface brightness maxima and minima along this curve. Maxima correspond to the image position of the host centre, and minima to points where the image crosses the critical curve of the lens model. Apart from the lens galaxy parameters, the additional model parameters are the position of the host centre, the host ellipticity, s , and orientation, φ s . Since the method relies on high spatial resolution data, preferably at long wavelengths in order to minimize sensitivity to dust extinction, we use 1st epoch k s -data for this analysis.
In figure 2, the flux along the curve of maximal surface brightness of the host image is shown. We estimate the maxima to be at at θ max /(•) = [40, 160, 210, 300] with uncertainties σ θ /(•) = [10, 10, 10 3 , 10], setting a large uncertainty for entry 3 to effectively not include it in the fit. The minima are set to θ min /(•) = [0,110,190,230] with uncertainties σ θ /(•) = [20, 10, 10 3 , 10], again setting a large uncertainty for entry 3 to exclude it from the fit. We have also indicated (black dotted lines) the predicted maxima given the best fit host position obtained using the iPTF16geu positions and the radial position of the host image. Due to the proximity of the source to the inner caustic, the positions of the extremal points are very sensitive to the exact source position. The yellow dashed lines corresponds to the host centre being shifted 0.01 (corresponding to a physical scale of 56 parsec) to the East and the red dotted-dashed lines 0.01 to the West. The projected distance from the SN to the host galaxy centre is 0.02 arcsec, corresponding to 0.1 kpc. However, since this represents a projected quantity, the physical distance can be larger.
Fitting the host galaxy together with iPTF16geu positions yield b/( ) = 0.2863 ± 0.0003 compared to b/( ) = 0.2863 ± 0.0009 when fitting only SN data, see table 8. However, the model fails to fit the extremal point correctly. Given  their susceptibility to dust extinction, sensitivity to the exact location of host galaxy centre, and difficulty in constraining their angular position unambiguously, we defer from using the extremal points as input to the lens modelling in the following.
Fitting the radial position of the host peak surface brightness, excluding the tangential position of maxima and minima along this curve, together with iPTF16geu positions yield b/( ) = 0.2863 ± 0.0009, i.e., the same as for iPTF16geu positions only. Given that constraints on the lens model changes very little when including also host data, in the remainder of the paper, in the interest of minimizing possible systematic effects, we will only use iPTF16geu position data.

iPTF16geu flux
When fitting image positions and (relative; here we do not fit the source luminosity) fluxes with a lens model without substructure, we generally obtain χ 2 min > 400. The badness of the fit is completely dominated by the fact that we are not able to fit the relative image fluxes. We conclude that it is not possible to successfully reproduce the observed fluxes with the employed lens model, even when allowing for varying slope and core size, α and s. Using only the SN positions as input, the predicted image magnifications are listed in table 9, and the corresponding magnification ratios in table 10, for three different values of the slope parameter α. When varying the slope of the lens, flux ratios vary little, but the total magnification a lot. We can thus get a wide range of magnifications, even without invoking the mass sheet degeneracy. In the remainder of the paper, since the effects of changing the slope and adding a mass sheet are highly degenerate, we focus on the former by investigating different α. Since fluxes are well constrained, they can be used to study additional magnification (or demagnification) from substructure in the lens, see section 6.
In figure 3, we plot the predicted magnitude shifts, defined as ∆m ≡ 2.5 log µ, for the four images, together with the observed magnitude shifts indicated by the horizontal bands (1 σ). The fact that the predicted flux ratios are consistently off from the observed values, shows that we need to have additional magnification and/or demagnification for at least two of the images.

A possible core and central image
Allowing for a lens galaxy core, there is an almost perfect degeneracy between the core size s and the mass normalization parameter b, keeping the mass inside the images constant. However, at least in principle, the size of the core can be constrained by the fact that we do not observe a central image of iPTF16geu. Close to the centre of the lens, the image magnification is given by If we can observationally constrain µ central < µ max , then (for κ > 1) where Σ crit ≈ 10.50 kg/m 2 for iPTF16geu. Since we expect the central image to be subject to large dust extinction not accounted for in the analysis, equation 26 represents the most optimistic limit obtainable for the lens galaxy central surface density. Assuming a Hsiao template (Hsiao et al. 2007), with a Bessell B band magnitude of -19.3, reddened by A V = 0.23 magnitudes in the Milky Way and A V = 0.58 magnitudes in the host galaxy, the 3 σ upper limit on a possible fifth central image corresponds to a demagnification of 1.2 magnitudes, or µ max = 0.33, implying Σ central > 29 kg/m 2 .
In terms of the size of a possible core of the lens galaxy,  Images 1.88 ± 0.10 0.76 ± 0.03 0.32 ± 0.01 ∆t 13 −0.23 ± 0.01 −0.12 ± 0.01 −0.054 ± 0.004 ∆t 14 0.58 ± 0.03 0.27 ± 0.01 0.12 ± 0.01 figure 4 shows predictions for the central image magnification, µ central , for three different values of the slope parameter α. Here, s is the size of the core in arcsec (1 arcsec corresponds to 5.6 kpc at the source redshift). For α = [0.8, 1, 1.2], we derive a maximum core size of s max /( ) = [0.08, 0.07, 0.05], respectively. Note again that dust in the lens galaxy degrade constraints on the core size. The expected time between the fifth central image and the second image, which is predicted to arrive first in the models, is of order ∼ 1 day.

Time delays and h
In table 11, we list the predicted time delays between images for different α, assuming a Planck cosmology. Comparing with the measured time delays in table 2 give constraints on h as depicted in figure 5. Since the observed delays are consistent with being zero at 1 σ, we can only (weakly) constrain h from below. We can approximate the combined constraints on h and α as (see equation 21), or, equivalently, h 2(n−1)/3 at ∼ 1 σ. If we set h = 0.7±0.05, and fit the observed image positions and time delays, we can constrain α > 1 at 1 σ, corresponding to a slope n < 2, see figure 6. This is in agreement with independent indications from the observed iPTF16geu fluxes in section 6.  . Constraint on α assuming h = 0.7 ± 0.05. The solid line indicates the 1 σ limit, indicating that α > 1, corresponding to n < 2.

Substructure mass constraints
The fact that we can not resolve any multiple sub-images for any of the four iPTF16geu images, allows us to constrain the mass, M, of possible substructures responsible for the anomalous image fluxes. Assuming the substructure is in the form of compact objects, the image splitting can be calculated perturbing the gravitational field of a point mass with that of the lens galaxy, a so called Chang-Refsdal lens (Chang & Refsdal 1979;Schneider et al. 1992). The image splitting, ∆θ, can be approximated by putting the substructure lens at the position of the image in absence of the sub-structure, giving where µ th is the magnification of the image in absence of the substructure. The lack of resolved sub-images gives an upper constraint on the image splitting of ∆θ max ≈ 0.05 arcsec, and For α = 1, the magnification in absence of the lens for image 1 and 2 is µ th,1 ≈ µ th.2 ≈ 6 (see table 9), and M max ≈ 2 · 10 7 M .

STELLAR MICROLENSING
Lens galaxy stars can microlens strongly lensed compact sources, such as quasars or SNe, significantly altering their observed fluxes (Chang & Refsdal 1979). Although the principles of microlensing are the same as those of galaxy-scale strong gravitational lensing (see section 2), there are some important differences between the two scenarios. First, the significantly lower deflector masses involved in microlensing (m ∼ M ) lead to microimage time delays and image separations that are too small to be observed with current optical instrumentation (typically of order microseconds and microarcsec, respectively; Moore & Hewitt 1996). Second, individual deflectors are replaced by fields of lens stars, having complex caustic patterns that vary over spatial scales of microarcseconds. These patterns can magnify or demagnify compact sources by several magnitudes over the value expected from a smooth lens model (e.g., Kayser et al. 1986;Schneider & Weiss 1987;Wambsganss 1992). This effect has been observed in many lensed quasars (e.g., Irwin et al. 1989;Witt et al. 1995;Tewes et al. 2013), and simulations indicate that it should be ubiquitous in the multiple images of strongly SNe (Dobler & Keeton 2006;Goldstein et al. 2018). Consequently, microlensing by lens galaxy field stars is a natural explanation for the flux anomalies observed in iPTF16geu.
To determine if microlensing can explain the flux anomalies in iPTF16geu, we use a set of publicly available microlensing magnification patterns produced by the GER-LUMPH project (Vernardos et al. 2014). The maps were generated by the inverse ray-shooting method (Wambsganss 1999), in which a uniform surface density of rays is followed from the observer through a random field of point-mass deflectors in the lens galaxy, and collected in pixels on the source plane, The ray count in each pixel is proportional to the lensing magnification that a source at the position of the pixel would experience.
The statistical properties of the microlensing are determined by the macrolensing model parameters κ and γ, as well as the fraction of matter in stars, f * , at the location of each image. Since f * is unknown, we scan over its value. In the simulations, fields of stars were realized at the location of each image by assuming a uniform deflector mass m = M , (Vernardos et al. 2014 fields have established that m = 0.3M is a more representative value (Poindexter & Kochanek 2010), but other studies indicate that the deflector mass has a negligible effect on the microlensing magnification probability distributions (Wambsganss 1992;Lewis & Irwin 1995;Wyithe & Turner 2001;Schechter et al. 2004). In figure 7, we show the microlensing magnification probability density functions (PDFs) for each image, for three representative values of the slope parameter α = [1.0, 1.2, 1.4]. Here, µ tot is the total magnification of each image and µ th the smooth lens model magnification. Thanks to the standard candle nature of SNe Ia, the required microlensing magnification can be estimated (vertical dotted lines) and compared to the microlensing PDFs. In table 12, we list the maximum one-tailed p-value for each image for 1.0 ≤ α ≤ 1.5. The total probability, p tot , for each αscenario is obtained by combining the individual p-values using the Fisher's combined probability test (Fisher 1925). The macrolensing model in best agreement with the observed fluxes, when including stellar microlensing, has α = 1.3, with p tot = 26 %. For α = 1, the large microlensing magnification required for image 1 and 2 decreases the probability to p tot = 1 %. An equally low probability is obtained for α = 1.5 due to the large microlensing demagnification required for image 3 and 4.
We conclude that for 1.1 ≤ α ≤ 1.4, stellar microlensing can explain the observed flux anomalies in iPTF16geu. Note that the use of microlensing to place constraints on the macrolens model is only possible since the source is a precisely calibrated standard candle.

CONCLUSIONS
With the aid of follow-up observations of iPTF16geu, we derive improved constraints on the lens model, as well as the first constraint on the Hubble constant from observations of a strongly lensed SN Ia, following Refsdal's original proposal (Refsdal 1964).
The projected mass of the lens galaxy within the iPTF16geu images is measured with a precision of 0.3 %, the ellipticity of the surface mass density with 5 % and its orientation with 0.2 % precision, representing a factor of ∼ 7 improvement from earlier constraints.
The fact that iPTF16geu exploded very close to the inner caustic of the lens, makes the predicted images for a smooth lens to be very similar, in terms of radial position, magnification and arrival time. Since the majority of the data is post first maximum, the measurement of the time delays between images is challenging and the resulting fractional uncertainties are large. We can therefore only obtain Microlensing magnification PDFs for each image of iPTF16geu. Histograms show the distribution of differences, in magnitudes, between the total magnification of each image, µ tot , and the magnification predicted from a smooth model, µ th . The required microlensing magnifications as derived from the observed image fluxes are indicated by vertical dotted lines. f * denotes the fraction of matter in stars at the location of each image. a relatively weak lower limit on the current expansion rate h 2(1 − 2α/3), where α is connected to the slope of the projected surface mass density, Σ, of the lens in the annulus between the iPTF16geu images as Σ ∝ r α−2 . Given our current knowledge of the expansion rate where h ∼ 0.7, this implies α 1.
Observational limits on the image fluxes are better constrained, and show beyond doubt that substantial additional (de)magnification has to take place for at least two images. For example, adjusting the slope to α ∼ 1 (corresponding to an isothermal halo) to fit the flux of image 3 and 4, image 1 needs an increased flux of ∆m 1 ∼ −2, and image 2 ∆m 2 ∼ −1. For a flatter density profile with α ∼ 1.45, image 2, 3 and 4 need to be demagnified by ∆m 2 ∼ 0.5 and ∆m 3 ∼ ∆m 4 ∼ 1.5, respectively.
Using stellar microlensing magnification probabilities derived for the individual images of iPTF16geu, we compute the total probability for different macrolens models, parametrized by the slope parameter α, to explain the observed flux anomalies. For α = 1.3, the probability to obtain the observed fluxes is p tot = 26 %, again indicating a preferred value of α > 1 at the radial position of the iPTF16geu images, for which stellar microlensing can comfortably explain the observed flux anomalies. This is the first time that microlensing has been used in combination with a precisely calibrated standard candle to place constraints on the macrolens model in a strong lensing system.