HOLISMOKES -- I. Highly Optimised Lensing Investigations of Supernovae, Microlensing Objects, and Kinematics of Ellipticals and Spirals

We present the HOLISMOKES project on strong gravitational lensing of supernovae (SNe) as a probe of SN physics and cosmology. We investigate the effects of microlensing on early-phase SN Ia spectra, and find that within 10 rest-frame days after SN explosion, distortions of SN Ia spectra due to microlensing are typically negligible (<1% distortion within the 68% credible region, and ~10% distortion within the 95% credible region). This shows great prospects of using lensed SNe Ia to obtain intrinsic early-phase SN spectra for deciphering SN Ia progenitors. As a demonstration of the usefulness of lensed SNe Ia for cosmology, we simulate a sample of mock lensed SN Ia systems that are expected to have accurate and precise time-delay measurements in the era of the Legacy Survey of Space and Time (LSST). Adopting realistic yet conservative uncertainties on their time-delay distances and lens angular diameter distances (of 6.6% and 5%, respectively), we find that a sample of 20 lensed SNe Ia would allow a constraint on the Hubble constant ($H_0$) with 1.3% uncertainty in the flat $\Lambda$CDM cosmology. We find a similar constraint on $H_0$ in an open $\Lambda$CDM cosmology, while the constraint degrades to 3% in a flat wCDM cosmology. We anticipate lensed SNe to be an independent and powerful probe of SN physics and cosmology in the upcoming LSST era.


Introduction
In the past few years, strongly lensed supernovae (SNe) have transformed from a theoretical fantasy to reality. First envisaged by Refsdal (1964) as a cosmological probe, a strongly lensed SN occurs when a massive object (e.g., a galaxy) by chance lies between the observer and the SN; the gravitational field of the massive foreground object acts like a lens and bends light from the background SN, so that multiple images of the SN appear around the foreground lensing object. The arrival times of the light rays of the multiple images are different, given the difference in their light paths. The time delays between the multiple SN images are typically days/weeks for galaxy-scale foreground lenses, and years for galaxy-cluster-scale foreground lenses. A strongly lensed SN is thus nature's orchestrated cosmic fireworks with the same SN explosion appearing multiple times one after another. Refsdal (1964) showed that the time delays between the multiple SN images provide a way to measure the expansion rate of the Universe.
The first strongly lensed SN system with multiple resolved images of the SN was discovered by Kelly et al. (2015), half a century after the prescient Refsdal (1964). The SN was named "Supernova Refsdal", and its spectroscopy revealed that it was a core-collapse SN. It was first detected serendipitously when it appeared in the galaxy cluster MACSJ1149.6+2223 in the Hubble Space Telescope (HST) imaging taken as part of the Grism Lens-Amplified Survey from Space (GLASS; PI: T. Treu) and the Hubble Frontier Field (PI: J. Lotz) programs. While this was the first system that showed spatially-resolved multiple SN images, Quimby et al. (2013Quimby et al. ( , 2014 had previously detected a SN in the PanSTARRS 1 survey (Kaiser et al. 2010;Chambers et al. 2016) that was magnified by a factor of ∼30 by a foreground intervening galaxy, although the multiple images of the SN could not be resolved in the imaging. Two years after the SN Refsdal event was the first discovery of a strongly lensed Type Ia SN by Goobar et al. (2017) in the intermediate Palomar Transient Factory (Law et al. 2009), namely the iPTF16geu system. This is particularly exciting given the standardisable nature of Type Ia SNe for cosmological studies.
With strongly lensed SNe being discovered, we have new opportunities of using such systems to study SN physics, particularly SN progenitors. Strongly lensed SNe allow one to observe a SN explosion right from the beginning, which was impossible to do in the past given the time lag to arrange follow-up observations after a SN is detected. By exploiting the time delay between the multiple SN images, the lens system can be detected based on the first SN image and follow-up (especially spectroscopic) observations can be carried out on the next appearing SN image from its beginning. Early-phase observations are crucial for understanding the progenitors of SNe, especially Type Ia SNe whose progenitors are still a puzzle after decades of debate -are they single-degenerate (SD) systems with a white dwarf (WD) accreting mass from a nondegenerate companion and exploding when reaching the Chandrasekhar mass limit (e.g., Whelan & Iben 1973), or double-degenerate (DD) systems with two WDs merging (e.g., Tutukov & Yungelson 1981;Iben & Tutukov 1984), a mix of the two, or other mechanisms? A few SNe Ia now have extremely early light-curve coverage and a UV excess is observed in some of them (e.g., Dimitriadis et al. 2019), but there are no rest-frame UV spectra at such early phases to constrain the origin of the UV emission. A continuum-dominated UV flux would hint at shocks and interaction of the ejecta with a companion star or circumstellar matter, favouring the SD scenario (Kasen 2010). A line-dominated early UV spectrum, on the other hand, would probe radioactive material close to the surface of the SN, as predicted by some DD models (Maeda et al. 2018).
Strongly lensed SNe with time-delay measurements also provide a direct and independent method to measure the expansion rate of the Universe, or the Hubble constant (H 0 ), as first pointed out by Refsdal (1964). There is currently an intriguing tension in the measurements of H 0 from independent probes, particularly between the measurement from observations of the Cosmic Microwave Background (CMB) by the Planck Collaboration (2018) and the local measurement from Cepheids distance ladder by the "Supernovae, H 0 , for the Equation of State of Dark Energy" (SH0ES) program ). This tension, if not due to any unaccounted-for measurement uncertainties, has great implications for cosmology as it would require new physics beyond our current standard "flat ΛCDM" cosmological model. The latest H 0 measurement from the Megamaser Cosmology Project by Pesce et al. (2020), which is independent of the CMB and SH0ES, corroborates the measurement of SH0ES, although it is within 3σ of the Planck measurement. On the other hand, Freedman et al. (2019) measured H 0 that is right in between the values from Planck Collaboration (2018) and Riess et al. (2019) through the Carnegie-Chicago Hubble Program (CCHP; Beaton et al. 2016) using a separate distance calibrator, the tip of the red giants, instead of Cepheids. There is ongoing debate about the method (e.g., Yuan et al. 2019) and the results from CCHP and SH0ES are not fully independent due to calibrating sources/data that are common among the two distance ladders. Strong-lensing time delays are therefore highly valuable for providing a direct H 0 measurement, completely independent of the CMB, the distance ladder, and megamasers (Riess 2019).
Given the rarity of lensed SNe, the method of time-delay cosmography has matured in the past two decades using lensed quasars that are more abundant. The H0LiCOW ) and COSMOGRAIL  collaborations have greatly refined this technique using high-quality data and state-of-the-art analyses of lensed quasars. The latest H0LiCOW H 0 measurement by Wong et al. (2019) from the analyses of 6 lens systems (Suyu et al. 2010(Suyu et al. , 2014Wong et al. 2017;Birrer et al. 2019;Rusu et al. 2019;Jee et al. 2019;Chen et al. 2019), which include 3 systems analysed jointly with the SHARP collaboration , is consistent with the results from SH0ES and is >3σ higher than the value from the Planck Collaboration (2018), strengthening the argument for new physics. Analysis of new lensed quasars is underway (e.g., Shajib et al. 2019, from the STRIDES collaboration), and a detailed account of systematic uncertainties in such measurements is presented by Millon et al. (2019) under the new TDCOSMO organisation. With time-delay cosmography maturing through lensed quasar, lensed SNe are expected to be a powerful cosmological probe.
The two known lensed SN systems, iPTF16geu and SN Refsdal, do not have early-phase SN observations for progenitor studies, and have yet to yield H 0 measurements. The time delays between the four SN images in iPTF16geu are short, 1 day (More et al. 2017;Dhawan et al. 2019), and all four SN images were past the early phase when the system was discovered by Goobar et al. (2017). The short delays also make it difficult to obtain precise H 0 from this system, since the relative uncertainties in the delays (which are 50%; Dhawan et al. 2019) sets the lower limit on the relative uncertainty on H 0 . On the other hand, SN Refsdal has one long time delay between the SN images (∼1 year; Grillo et al. 2016;Kawamata et al. 2016), in addition to shorter delay pairs (Rodney et al. 2016). The measurement of the long delay using multiple techniques is forthcoming (P. Kelly, priv. comm.), and this spectacular cluster lens system with multiple sources at different redshifts could yield the first H 0 measurement from a lensed SN (e.g., Grillo et al. 2018Grillo et al. , 2020. Even though lensed SNe are very rare, their numbers will increase dramatically in the coming years thanks to dedicated wide-field cadenced imaging surveys. In particular, Goldstein et al. (2019) forecasted about a dozen lensed SNe from the ongoing Zwicky Transient Facility (ZTF; Bellm et al. 2019;Masci et al. 2019); most of these lensed SNe will be systems with short time delays (days) and high magnifications, given the bright flux limit of the ZTF survey. The upcoming Legacy Survey of Space and Time (LSST; Ivezić et al. 2019) 2 at the Vera C. Rubin Observatory that will image the entire southern sky repeatedly for 10 years will yield hundreds of lensed SNe (e.g., Oguri & Marshall 2010;Goldstein et al. 2019;Wojtak et al. 2019). The efficiency of detecting these systems and measuring their time delays depends significantly on the observing cadence strategy. Huber et al. (2019) have carried out the first investigations of detecting lensed SNe Ia and measuring their delays in the presence of microlensing, with results that favour long cumulative season length and higher cadence.
With the upcoming boom in strongly lensed SNe, we initiate the HOLISMOKES project: Highly Optimised Lensing Investigations of Supernovae, Microlensing Objects, and Kinematics of Ellipticals and Spirals. We are developing ways to find lensed SNe (Cañameras et al., in prep.) in current/future cadenced surveys and to model the lens systems rapidly for scheduling observational follow-up (Schuldt et al., in prep.). We are also exploring in more detail the microlensing of lensed SNe Ia (Huber et al., in prep.) for measuring the time delays, following the works of Goldstein et al. (2018) and Huber et al. (2019).
In this first paper of the HOLISMOKES series, we study and forecast our ability to achieve two scientific goals with a sample of lenses from the upcoming LSST: constrain SN Ia progenitors through early-phase observations, and probe cosmology through lensing time delays. In Section 2, we investigate microlensing effects on SNe Ia to determine whether it is feasible to extract the intrinsic early-phase SN spectra that are crucial for revealing SN Ia progenitors. In Section 3, we forecast the cosmological constraints based on an expected sample of lensed SNe from LSST. We summarize in Section 4.

Microlensing of SNe Ia in their early phases
Early-phase spectra carry valuable information to distinguish between different SN Ia progenitors (e.g., Kasen 2010;Rabinak & Waxman 2011;Piro & Nakar 2013Piro & Morozova 2016;Noebauer et al. 2017). Problems arise when SNe are significantly influenced by microlensing (Yahalomi et al. 2017;Goldstein et al. 2018;Foxley-Marrable et al. 2018;Bonvin et al. 2019b;Huber et al. 2019), which distorts light curves and spectra, and therefore makes them hard to use as a probe for SN Ia progenitors. However, investigations by Goldstein et al. (2018) and Huber et al. (2019) show that microlensing of lensed SNe Ia is stronger in late phases than shortly after explosion. These results raise the hope to use lensed SNe Ia for the progenitor problem and motivates further investigation of the influence of microlensing on early-phase spectra.
To probe the effect of microlensing on SNe Ia, we need the time, wavelength, and spatial dependency of the SN radiation. For this, we use the theoretical 1D W7 model (Nomoto et al. 1984), where synthetic observables have been calculated via ARTIS (Kromer & Sim 2009). We assume that microlensing maps and positions in the map do not vary over typical time scales of a SN Ia and the microlensing effect is therefore just related to the spatial expansion of the SN. This approach is motivated by the work of Goldstein et al. (2018) and Huber et al. (2019). We follow closely the formalism described in Huber et al. (2019) to compute microlensing effects on a SN Ia, and briefly summarise the procedure. The observed microlensed flux of a SN at redshift z s and luminosity distance D L can be determined via where the emitted specific intensity I λ,e (t, x, y), is multiplied with the microlensing magnification map 3 µ(x, y) from GERLUMPH (Vernardos et al. 2015, J. H. H. Chan et al., in prep.) and integrated over the whole size of the projected SN Ia. The specific intensity I λ,e (t, x, y) depends on the time since explosion t, the wavelength λ, and the radial coordinate on the source plane p = x 2 + y 2 , given the 1D nature of the model 4 . The specific intensity profiles for different times after explosion are shown in Appendix A. Equation (1)  For this work we just focus on spectra, particularly at early phases. We investigate 30 different magnification maps. These maps depend on three main parameters: the lensing convergence κ, the shear γ, and the smooth matter fraction s = κ smooth /κ total . In our analysis we probe (κ, γ) = (0.29, 0.27), (0.36, 0.35), (0.43, 0.43), (0.57, 0.58), (0.70, 0.70), (0.93, 0.93), where we test for each combination of κ and γ the smooth matter fractions of s = 0.1, 0.3, 0.5, 0.7, 0.9. Six of these magnification maps are shown in Appendix B, where we explain also further inputs for producing the magnification maps. The values for the convergence and shear are calculated from the mock lens catalog of Oguri & Marshall (2010, hereafter OM10), taking into account 416 lensed SNe Ia that adopted a singular isothermal ellipsoid (Kormann et al. 1994) as lens mass model. The two pairs (κ, γ) = (0.36, 0.35) and (0.70, 0.70) correspond to the median values for type I lensing images (time-delay minimum) and type II images (time-delay saddle), respectively. The other (κ, γ) pairs are the 16th and 84th percentiles of the OM10 sample, taken separately for κ and γ.
For each of the 30 magnification maps, we draw 10,000 random positions in the map to quantify the effect of microlensing on the SN spectra. For each position we calculate the microlensed flux F micro via Equation (1) and compare it to the case without microlensing F no micro (µ = 1). From this, we can calculate the deviation ∆ λ (t) from the macro magnification as: where the last term corresponds to the mean value of the ratio over the wavelength, which is the magnification one would observe in a bolometric light curve. The deviation quantifies distortions in the spectra of a microlensed SN relative to the intrinsic SN without microlensing, i.e., a deviation of 0 across all wavelengths implies no microlensing distortion on the intrinsic SN spectra. We refer readers to Figures A.1 and A.3 of Huber et al. (2019) for examples of F micro F no micro . From the 30 × 10,000 random configurations, we determine the median deviation of ∆ λ (t) with the 1σ range (68% credible region) and 2σ range (95% credible region) shown in Figure 1 for different times after explosion. We find that at early times (within ∼10 rest-frame days after explosion), the 2σ spread of ∆ λ (t) for most wavelength regions is at the ∼10% level. The 1σ spread is almost indistinguishable (within 1%) from the median around zero. Therefore, microlensing would not distort the spectra of SNe beyond 1% at any wavelength in 68% of all strongly lensed SNe Ia at early phases. At later times, the influence of microlensing becomes substantially larger, as also visible in the increased 1σ spread, but the 1σ spread is still mostly below the 10% level.
In total, we have investigated 20 different time bins covering rest-frame day 4.0 to 39.8 after explosion. The results are summarised in Figure 2, where the median of the 1σ and 2σ spread of ∆ λ (t) over different wavelength bins is shown. We find that up to 10 rest-frame days after SN explosion, the median 2σ spread is around 10%. The slightly higher values for day 4 are related to worse statistics of the ARTIS simulations for very early times. The overall trend shows that the median 2σ spread is increasing over time. This can be explained because SNe Ia are expanding over time and therefore it is much more likely to cross a micro caustic at later times. The second reason is that the specific intensity profiles for different filters deviate more strongly from each other at later phases Huber et al. 2019, and Appendix A), which leads to higher deviations in the spectrum between different wavelength regions. The median 1σ spread is increasing as well but the values are always below a few percent, and therefore far less problematic.
To summarise, at early times ( 10 rest-frame days) we have very good prospects to collect good quality spectra with negligible distortions from microlensing, which is necessary to address the SN Ia progenitor problem. Nevertheless, we would like to point out that there are extreme cases where microlensing can significantly influence even very early spectra. These extreme microlensing cases could potentially allow one to probe the specific intensity distribution of SNe. A comparison showing the dependency of microlensing effects on different parameters, such as s, will be presented in Huber et al. (in prep.), but especially for high magnification cases with both values of κ and γ close to 0.5, we find that microlensing is more likely. This can be understood by looking at the magnification maps shown in Appendix B, where more caustics and higher gradients exist for (κ, γ) = (0.43, 0.43) and (κ, γ) = (0.57, 0.58) in comparison to the other cases. Fortunately, in practice we can always estimate for a given lensed SN Ia image the likelihood of it being microlensed, to determine whether it is suitable for obtaining a "clean" SN spectrum that has little distortion from microlensing.

Forecasted cosmological constraints from strongly lensed SNe
Each lensed SN provides an opportunity to measure two distances: the time-delay distance D ∆t and the angular diameter distance to the deflector/lens D d (e.g., Refsdal 1964;Suyu et al. 2010;Paraficz & Hjorth 2009;Birrer et al. 2016;Jee et al. 2019).
The time-delay distance is defined by Suyu et al. (2010) as where D ds and D s are angular diameter distances to the source from the deflector and from the observer, respectively. Measuring D ∆t requires three ingredients: (1) time delays, (2) strong lens mass model, and (3)  Lensed SNe have several advantages over lensed quasars: (1) the time delays are easier to measure with simple and sharply varying light-curve shapes that are less prone to strong microlensing effects, (2) the lens mass distribution is easier to model without strong contamination by quasar light that typically outshines everything else in the lens system (SNe are bright as well, but they fade in months, revealing their host galaxy and lens galaxy light clearly), (3) some SNe are standardisable candles and their intrinsic luminosities could mitigate lens model degeneracies in cases when microlensing effects are negligible, and (4) the effect of microlensing time delay, pointed out by Tie & Kochanek (2018) for lensed quasars, is negligible for typical lensed SNe (Bonvin et al. 2019b).
We create a mock sample of lensed SNe Ia expected from the upcoming LSST, with simulated D ∆t and D d measurements in Section 3.1, and forecast the resulting cosmological constraints based on the sample in Section 3.2.

Mock distance measurements from lensed SNe Ia
We focus on a sample of lensed SNe Ia that would have "good" time-delay measurements even in the presence of microlens-ing, i.e., those systems with accuracy better than 1% and precision better than 5% in their time-delay measurements. From the investigations of Huber et al. (2019), the expected number of spatially-resolved lensed SNe Ia is ∼75 for 10 years of LSST survey with baseline-like LSST cadence strategies. Accounting for the effects of microlensing, lensed SN Ia systems that have delays longer than 20 days could yield accuracy better than 1%, whereas shorter delays could suffer from inaccuracy (see Figure  13 of Huber et al. 2019). SNe Ia at lower redshifts, z s < 0.7 are brighter and would yield good delays (i.e., delays with accuracy and precision within the target), whereas for SNe Ia at z s > 0.7, only about half of the systems could yield good delays with deep follow-up imaging (see Figure 15 of Huber et al. 2019). Using these results, we start with the mock sample of lensed SNe Ia expected for LSST from OM10 (Oguri & Marshall 2010), and select the fraction of lensed SN systems with delays longer than 20 days, resulting in 30 lensed SNe Ia systems. 5 Of these 30 systems, 10 have z s < 0.7 which we keep, whereas 20 have z s > 0.7 and we randomly select half of them. This leads to a final sample of N SNIa = 20 mock lensed SNe Ia that we expect to have good delays. Figure 3 shows the redshift distributions of these mock lens systems.
To estimate the precision for D ∆t measurements, we conservatively adopt 5% for the time-delay uncertainties (given that the precision of the delays would be better than 5%), 3% for the lens mass modelling uncertainties, and 3% for the lens environment uncertainties, which are realistic given current lensed quasar constraints (e.g., Suyu et al. 2010;Greene et al. 2013;Collett et al. 2013;Suyu et al. 2014;Rusu et al. 2017;Wong et al. 2017;Tihhonova et al. 2018;Bonvin et al. 2019a;Chen et al. 2019). Adding these in quadrature, we assign 6.6% uncertainty to D ∆t from each lensed SN Ia system. For the precision on D d , we consider the scenario of having spatially-resolved kinematics of the foreground lens (e.g., Czoske et al. 2008;Barnabè et al. 2009Barnabè et al. , 2011, such that we can measure D d with its uncertainty essentially dominated by the time-delay uncertainty (Yıldırım et al. 2019). Spatially-resolved kinematic observations of the lens systems would be relatively straightforward to obtain after all the multiple SN images fade away in 1 year. We thus adopt 5% uncertainties on D d for each lensed SN Ia system.
To generate mock D ∆t and D d measurements for the N SNIa (= 20) lensed SN Ia systems, we adopt as input a flat ΛCDM cosmology with H 0 = 72 km s −1 Mpc −1 and Ω m = 1 − Ω Λ = 0.32. Given the deflector and SN source redshifts from the OM10 catalog, we compute the D true ∆t,i and D true d,i of lensed SN Ia system i, where i=1..N SNIa . Using the estimated 1σ uncertainty of 6.6% for D ∆t and 5% for D d , which we denote as σ ∆t,i and σ d,i , respectively, we draw random Gaussian deviates, δD ∆t,i and δD d,i , to obtain the mock measurements for lensed SN Ia system i as follows,  (2)) of SN Ia spectra due to microlensing for different times after explosion. The black dashed line represents the median, and the 1σ (68% credible region) and 2σ (95% credible region) spreads are shown in blue and red shades, respectively, for a sample of 30 different magnification maps with 10,000 random positions per map. The grey dot-dashed line indicates a deviation of 10% in the spectra relative to the intrinsic one without microlensing effects. The small zoomed in panels show a region of 150Å to illustrate the small extent of the 1σ spread especially at early times.
From this, we get the following mock distance measurements for our lensed SN Ia sample:

Cosmological constraints from the mock lensed SN Ia sample
To obtain the cosmological constraints, we sample the posterior distribution of the cosmological parameters π in the same way as we do for the analysis of lensed quasars Wong et al. 2019;Millon et al. 2019). We first describe our like- Fig. 3. Source redshift z s and lens redshift z d distribution of the mock sample of lensed SNe Ia from LSST with good time-delay measurements (accuracy better than 1% and precision better than 5%). The top and right panels show the histograms of the number of systems in each lens-redshift and source-redshift bin, respectively. lihoods and priors for the cosmological parameters that enter the posterior probability distribution function.
For lensed SN Ia system i, we assume Gaussian likelihoods for D mock ∆t,i and D mock d,i with their corresponding uncertainties σ ∆t,i and σ d,i as the Gaussian standard deviations. That is, the likelihood for (D mock ∆t,i , D mock d,i ) is Article number, page 5 of 9 A&A proofs: manuscript no. holismokesI where We then multiply the likelihoods of the individual mock lenses together to compute the joint likelihood for the sample, We adopt uniform priors on the cosmological parameters in the sampling. We consider three background cosmological models as listed in Table 1, and sample the cosmological parameters in the models. The first cosmological model is the flat ΛCDM model with two variable cosmological parameters H 0 and the matter density Ω m . The second model is open ΛCDM where the variable parameters are H 0 , Ω m and the curvature density Ω k (with the dark energy density Ω Λ = 1 − Ω m − Ω k > 0). The third model is flat wCDM with three variable parameters H 0 , Ω m and the dark energy equation-of-state parameter w (where w = −1 corresponds to the cosmological constant Λ for dark energy). The priors for these parameters are summarised in Table 1.
For each background cosmological model, we sample the cosmological parameters π by computing the posterior probability which is the joint likelihood P joint multiplied with the prior. Specifically, for a given set of π values, we can compute D ∆t,i and D d,i for system i of the mock lensed SN Ia sample to calculate P i in equation (6), and thus P joint via equation (9). Given our uniform priors on π, our posterior is, up to a constant factor, P joint . We then sample the posterior probability distribution using emcee (Foreman-Mackey et al. 2013) with 32 walkers and 40,000 samples. To compare the constraining power of the two distance measurements on the cosmological parameters, we also consider the constraints from only D ∆t and only D d measurements.
The results of the sampling in flat ΛCDM are shown in Figure 4 with the marginalised cosmological constraints listed in Table 1. The time-delay distances D ∆t provide tight constraints on H 0 but little information on Ω m (grey contours). Since D d has a different dependence on cosmological parameters from that of D ∆t (the orange contours from D d are tilted with respect to the grey contours from D ∆t ), the combination of the two distance constraints tightens slightly the constraint on H 0 and substantially the constraint on Ω m . The input cosmological parameter values (marked in black) are recovered within the marginalised 68% credible intervals. With the two distances from the forecasted sample of 20 mock lensed SN Ia systems, we expect to measure H 0 with uncertainties of 1.3%.
We show in Figure 5 the results in open ΛCDM, with the marginalised constraints in Table 1. We see in the bottom-left panel of the figure that the parameter degeneracies between H 0 and Ω k are in different directions from the D ∆t and D d constraints, and the combination of the two helps to reduce the degeneracies. As a result, the inferred H 0 from both D ∆t and D d measurements is relatively insensitive to other cosmological parameters (the blue contours are nearly vertical in the left column of Figure 5). In fact, the marginalised H 0 constraint of 72.7 ± 1.0 is comparable in precision to that in flat ΛCDM (see Table 1), while the constraint on Ω m degrades substantially by a factor of 2 compared to that in flat ΛCDM.
For the background cosmological model of flat wCDM, the cosmological constraints are shown in Figure 6 and summarised in Table 1. When the dark energy equation-of-state parameter is allowed to vary, this substantially weakens the cosmological constraint on H 0 (to 3% uncertainty), given the strong parameter degeneracy between H 0 and w. Having D d measurements is important for constraining w and thus limit the possible range of H 0 values, as also previously shown by e.g. Jee et al. (2016). We see clearly here that while D ∆t is mainly sensitive to H 0 , it does depend on the assumed background cosmological model.  Huber et al. 2019). Lensed core-collapse SNe, not considered here, 6 provide additional D ∆t and D d measurements, and studies indicate more numerous lensed core-collapse SNe than lensed SNe Ia (e.g., Oguri & Marshall 2010;Goldstein et al. 2019;Wojtak et al. 2019). Therefore, we expect that a measure-  ment of H 0 with 1% uncertainty in flat ΛCDM from lensed SNe in the LSST era to be achievable.

Summary
We initiate the HOLISMOKES project to conduct Highly Optimised Lensing Investigations of Supernovae, Microlensing Objects, and Kinematics of Ellipticals and Spirals. In this first paper of the project, we investigate the feasibility of achieving two scientific goals in the LSST era: (1)  Panels and labels are in the same format as in Figure 4. The input values, marked in black, are recovered within the marginalised 68% credible intervals. When the dark energy equation-of-state parameter w is allowed to vary, significant parameter degeneracy between H 0 and w exists which weakens the constraint on H 0 .
for progenitor studies, and (2) cosmology through the time-delay method. We summarise as follows.
-The time delays between the multiple appearances of a lensed SN would allow us to observe the SN in its early phases. We find that microlensing distortions of early-phase SN Ia spectra (within ∼10 rest-frame days) are typically negligible, with distortions within 1% (68% credible region) and within ∼10% (95% credible region). This provides excellent prospects for acquiring intrinsic early-phase SN Ia spectra, free of microlensing distortions, to shed light on the progenitors systems of SNe Ia.
-We forecast the cosmological parameters constraints from a sample of 20 lensed SNe Ia in the LSST era. We assume that D ∆t and D d to these systems could be constrained with uncertainties of 6.6% and 5%, respectively. From this sample, we expect to measure H 0 in flat ΛCDM with a precision of 1.3% including (known) systematics, completely independent of any other cosmological probes. In an open ΛCDM cosmology, we find a similar constraint on H 0 , while in the flat wCDM cosmology, the constraint loosens to 3%. -Given the additional lensed core-collapse SNe, we expect a measurement of H 0 with 1% uncertainty in flat ΛCDM from lensed SNe to be achievable in the LSST era.
With ongoing wide-field cadence surveys like ZTF and the upcoming LSST, we are entering an exciting time of catching and watching SNe being strongly lensed. While the next systems from ZTF are likely to have short time delays ( 10 days), which could limit their use for cosmological and supernova studies, as the surveys like LSST image deeper, lensed SNe with longer time delays are expected to appear . Each one of these systems will provide an excellent opportunity for studying SN physics and cosmology. The cosmological analyses of lensed SNe will be complementary to the growing sample of lensed quasars, and the combination of the two types of lensed transients will be an even more powerful probe of cosmology. The challenges associated with lensed SNe will be to find these systems amongst the millions of daily transient alerts from LSST, and to analyse them quickly. Methods based on machine learning are being developed to overcome such challenges (e.g., Jacobs et al. 2019;Avestruz et al. 2019;Hezaveh et al. 2017;Perreault Levasseur et al. 2017;Pearson et al. 2019), and we are exploring these avenues in our forthcoming publications.