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Milisavljevic, Danny

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Milisavljevic

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Danny

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Milisavljevic, Danny

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    Publication
    Slowly Fading Super-Luminous Supernovae That Are Not Pair-Instability Explosions
    (Nature Publishing Group, 2013) Nicholl, M.; Smartt, S. J.; Jerkstrand, A.; Inserra, C.; McCrum, M.; Kotak, R.; Fraser, M.; Wright, D.; Chen, T.-W.; Smith, K.; Young, D. R.; Sim, S. A.; Valenti, S.; Howell, D. A.; Bresolin, F.; Kudritzki, R. P.; Tonry, J. L.; Huber, M. E.; Rest, A.; Pastorello, A.; Tomasella, L.; Cappellaro, E.; Benetti, S.; Mattila, S.; Kankare, E.; Kangas, T.; Leloudas, G.; Sollerman, J.; Taddia, F.; Berger, E.; Chornock, R; Narayan, Gautham; Stubbs, Christopher; Foley, R. J.; Lunnan, R; Soderberg, Alicia; Sanders, Nathan Edward; Milisavljevic, Danny; Margutti, Raffaella; Kirshner, Robert; Elias-Rosa, N.; Morales-Garoffolo, A.; Taubenberger, S.; Botticella, M. T.; Gezari, S.; Urata, Y.; Rodney, S.; Riess, A. G.; Scolnic, D.; Wood-Vasey, W. M.; Burgett, W. S.; Chambers, K.; Flewelling, H. A.; Magnier, E. A.; Kaiser, N.; Metcalfe, N.; Morgan, J.; Price, P. A.; Sweeney, W.; Waters, C.
    Super-luminous supernovae that radiate more than 10\(^{44}\) ergs per second at their peak luminosity have recently been discovered in faint galaxies at redshifts of 0.1–4. Some evolve slowly, resembling models of ‘pair-instability’ supernovae. Such models involve stars with original masses 140–260 times that of the Sun that now have carbon–oxygen cores of 65–130 solar masses. In these stars, the photons that prevent gravitational collapse are converted to electron–positron pairs, causing rapid contraction and thermonuclear explosions. Many solar masses of \(^{56}\)Ni are synthesized; this isotope decays to \(^{56}\)Fe via \(^{56}\)Co, powering bright light curves. Such massive progenitors are expected to have formed from metal-poor gas in the early Universe. Recently, supernova 2007bi in a galaxy at redshift 0.127 (about 12 billion years after the Big Bang) with a metallicity one-third that of the Sun was observed to look like a fading pair-instability supernova. Here we report observations of two slow-to-fade super-luminous supernovae that show relatively fast rise times and blue colours, which are incompatible with pair-instability models. Their late-time light-curve and spectral similarities to supernova 2007bi call the nature of that event into question. Our early spectra closely resemble typical fast-declining super-luminous supernovae, which are not powered by radioactivity. Modelling our observations with 10–16 solar masses of magnetar-energized ejecta demonstrates the possibility of a common explosion mechanism. The lack of unambiguous nearby pair-instability events suggests that their local rate of occurrence is less than 6 × 10\(^{−6}\) times that of the core-collapse rate.
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    PS1-10afx at z = 1.388: Pan-STARRS1 Discovery of a New Type of Superluminous Supernova
    (American Astronomical Society, 2013) Chornock, R; Berger, Edo; Rest, A.; Milisavljevic, Danny; Lunnan, R; Foley, R. J.; Soderberg, Alicia; Smartt, S. J.; Burgasser, A. J.; Challis, Peter; Chomiuk, L.; Czekala, Ian; Drout, Maria Rebecca; Fong, W; Huber, M. E.; Kirshner, Robert; Leibler, C.; McLeod, Brian; Marion, G. H.; Narayan, Gautham; Riess, A. G.; Roth, K. C.; Sanders, Nathan Edward; Scolnic, D.; Smith, K.; Stubbs, Christopher; Tonry, J. L.; Valenti, S.; Burgett, W. S.; Chambers, K. C.; Hodapp, K. W.; Kaiser, N.; Kudritzki, R.-P.; Magnier, E. A.; Price, P. A.
    We present the Pan-STARRS1 discovery of PS1-10afx, a unique hydrogen-deficient superluminous supernova (SLSN) at redshift z = 1.388. The light curve peaked at z P1 = 21.7 mag, making PS1-10afx comparable to the most luminous known SNe, with Mu = –22.3 mag. Our extensive optical and near-infrared observations indicate that the bolometric light curve of PS1-10afx rose on the unusually fast timescale of ~12 days to the extraordinary peak luminosity of 4.1 × 1044 erg s–1 (M bol = –22.8 mag) and subsequently faded rapidly. Equally important, the spectral energy distribution is unusually red for an SLSN, with a color temperature of ~6800 K near maximum light, in contrast to previous hydrogen-poor SLSNe, which are bright in the ultraviolet (UV). The spectra more closely resemble those of a normal SN Ic than any known SLSN, with a photospheric velocity of ~11, 000 km s–1 and evidence for line blanketing in the rest-frame UV. Despite the fast rise, these parameters imply a very large emitting radius (gsim 5 × 1015 cm). We demonstrate that no existing theoretical model can satisfactorily explain this combination of properties: (1) a nickel-powered light curve cannot match the combination of high peak luminosity with the fast timescale; (2) models powered by the spindown energy of a rapidly rotating magnetar predict significantly hotter and faster ejecta; and (3) models invoking shock breakout through a dense circumstellar medium cannot explain the observed spectra or color evolution. The host galaxy is well detected in pre-explosion imaging with a luminosity near L*, a star formation rate of ~15 M ☉ yr–1, and is fairly massive (~2 × 1010 M ☉), with a stellar population age of ~108 yr, also in contrast to the young dwarf hosts of known hydrogen-poor SLSNe. PS1-10afx is distinct from known examples of SLSNe in its spectra, colors, light-curve shape, and host galaxy properties, suggesting that it resulted from a different channel than other hydrogen-poor SLSNe.
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    Rapidly-Evolving and Luminous Transients From Pan-Starrs1
    (IOP Publishing, 2014) Drout, M. R.; Chornock, R.; Soderberg, Alicia; Sanders, Nathan Edward; McKinnon, R.; Rest, A.; Foley, R. J.; Milisavljevic, Danny; Margutti, R.; Berger, Edo; Calkins, M.; Fong, W.; Gezari, S.; Huber, M. E.; Kankare, E.; Kirshner, R. P.; Leibler, C.; Lunnan, R.; Mattila, S.; Marion, G. H.; Narayan, G.; Riess, A. G.; Roth, K. C.; Scolnic, D.; Smartt, S. J.; Tonry, J. L.; Burgett, W. S.; Chambers, K. C.; Hodapp, K. W.; Jedicke, R.; Kaiser, N.; Magnier, E. A.; Metcalfe, N.; Morgan, J. S.; Price, P. A.; Waters, C.
    In the past decade, several rapidly-evolving transients have been discovered whose timescales and luminosities are not easily explained by traditional supernovae (SN) models. The sample size of these objects has remained small due, at least in part, to the challenges of detecting short timescale transients with traditional survey cadences. Here we present the results from a search within the Pan-STARRS1 Medium Deep Survey (PS1-MDS) for rapidly-evolving and luminous transients. We identify 10 new transients with a time above half-maximum (t1/2) of less than 12 days and −16.5 > M > −20 mag. This increases the number of known events in this region of SN phase space by roughly a factor of three. The median redshift of the PS1-MDS sample is z=0.275 and they all exploded in star forming galaxies. In general, the transients possess faster rise than decline timescale and blue colors at maximum light (gP1 − rP1 . −0.2). Best fit blackbodies reveal photospheric temperatures/radii that expand/cool with time and explosion spectra taken near maximum light are dominated by a blue continuum, consistent with a hot, optically thick, ejecta. We find it difficult to reconcile the short timescale, high peak luminosity (L > 1043 erg s−1), and lack of UV line blanketing observed in many of these transients with an explosion powered mainly by the radioactive decay of 56Ni. Rather, we find that many are consistent with either (1) cooling envelope emission from the explosion of a star with a low-mass extended envelope which ejected very little (<0.03 M⊙) radioactive material, or (2) a shock breakout within a dense, optically thick, wind surrounding the progenitor star. After calculating the detection efficiency for objects with rapid timescales in the PS1-MDS we find a volumetric rate of 4800 − 8000 events yr−1 Gpc−3 (4 − 7% of the core-collapse SN rate at z=0.2).
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    Systematic Uncertainties Associated with the Cosmological Analysis of the First Pan-STARRS1 Type Ia Supernova Sample
    (IOP Publishing, 2014) Scolnic, D.; Rest, A.; Riess, A.; Huber, M. E.; Foley, R. J.; Brout, D.; Chornock, R.; Narayan, G.; Tonry, J. L.; Berger, Edo; Soderberg, Alicia; Stubbs, Christopher; Kirshner, Robert; Rodney, S.; Smartt, S. J.; Schlafly, E.; Botticella, M. T.; Challis, P.; Czekala, I.; Drout, M.; Hudson, M. J.; Kotak, R.; Leibler, C.; Lunnan, R.; Marion, G. H.; McCrum, M.; Milisavljevic, Danny; Pastorello, A.; Sanders, Nathan Edward; Smith, K.; Stafford, E.; Thilker, D.; Valenti, S.; Wood-Vasey, W. M.; Zheng, Z.; Burgett, W. S.; Chambers, K. C.; Denneau, L.; Draper, P. W.; Flewelling, H.; Hodapp, K. W.; Kaiser, N.; Kudritzki, R.-P.; Magnier, E. A.; Metcalfe, N.; Price, P. A.; Sweeney, W.; Wainscoat, R.; Waters, C.
    We probe the systematic uncertainties from the 113 Type Ia supernovae (SN Ia) in the Pan-STARRS1 (PS1) sample along with 197 SN Ia from a combination of low-redshift surveys. The companion paper by Rest et al. (2013) describes the photometric measurements and cosmological inferences from the PS1 sample. The largest systematic uncertainty stems from the photometric calibration of the PS1 and low-z samples. We increase the sample of observed Calspec standards from 7 to 10 used to define the PS1 calibration system. The PS1 and SDSS-II calibration systems are compared and discrepancies up to ∼ 0.02 mag are recovered. We find uncertainties in the proper way to treat intrinsic colors and reddening produce differences in the recovered value of w up to 3%. We estimate masses of host galaxies of PS1 supernovae and detect an insignificant difference in distance residuals of the full sample of 0.037±0.031 mag for host galaxies with high and low masses. Assuming flatness and including systematic uncertainties in our analysis of only SNe measurements, we find w =−1.120+0.360 −0.206(Stat)+0.269 −0.291(Sys). With additional constraints from BAO, CMB (Planck) and H0 measurements, we find w = −1.166+0.072 −0.069 and Ωm = 0.280+0.013 −0.012 (statistical and systematic errors added in quadrature). Significance of the inconsistency with w = −1 depends on whether we use Planck or WMAP measurements of the CMB: wBAO+H0+SN+WMAP = −1.124+0.083−0.065.
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    Rapidly evolving and luminous transients from Pan-STARRS1
    (IOP Publishing, 2014) Drout, Maria Rebecca; Chornock, R; Soderberg, Alicia; Sanders, Nathan Edward; McKinnon, R.; Rest, Armin; Foley, R. J.; Milisavljevic, Danny; Margutti, Raffaella; Berger, Edo; Calkins, Michael; Fong, W; Gezari, S.; Huber, M. E.; Kankare, E.; Kirshner, Robert; Leibler, C.; Lunnan, R; Mattila, S.; Marion, G. H.; Narayan, Gautham; Riess, A. G.; Roth, K. C.; Scolnic, D.; Smartt, S. J.; Tonry, J. L.; Burgett, W. S.; Chambers, K. C.; Hodapp, K. W.; Jedicke, R.; Kaiser, N.; Magnier, E. A.; Metcalfe, N.; Morgan, J. S.; Price, P. A.; Waters, C.
    In the past decade, several rapidly evolving transients have been discovered whose timescales and luminosities are not easily explained by traditional supernovae (SNe) models. The sample size of these objects has remained small due, at least in part, to the challenges of detecting short timescale transients with traditional survey cadences. Here we present the results from a search within the Pan-STARRS1 Medium Deep Survey (PS1-MDS) for rapidly evolving and luminous transients. We identify 10 new transients with a time above half-maximum (t 1/2) of less than 12 days and –16.5 > M > –20 mag. This increases the number of known events in this region of SN phase space by roughly a factor of three. The median redshift of the PS1-MDS sample is z = 0.275 and they all exploded in star-forming galaxies. In general, the transients possess faster rise than decline timescale and blue colors at maximum light (g P1 – r P1 lesssim –0.2). Best-fit blackbodies reveal photospheric temperatures/radii that expand/cool with time and explosion spectra taken near maximum light are dominated by a blue continuum, consistent with a hot, optically thick, ejecta. We find it difficult to reconcile the short timescale, high peak luminosity (L > 1043 erg s–1), and lack of UV line blanketing observed in many of these transients with an explosion powered mainly by the radioactive decay of 56Ni. Rather, we find that many are consistent with either (1) cooling envelope emission from the explosion of a star with a low-mass extended envelope that ejected very little (<0.03 M ☉) radioactive material, or (2) a shock breakout within a dense, optically thick, wind surrounding the progenitor star. After calculating the detection efficiency for objects with rapid timescales in the PS1-MDS we find a volumetric rate of 4800-8000 events yr–1 Gpc–3 (4%-7% of the core-collapse SN rate at z = 0.2).
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    Zooming in on the progenitors of superluminous supernovae with the HST
    (IOP Publishing, 2015) Lunnan, R; Chornock, R; Berger, Edo; Rest, A.; Fong, W; Scolnic, D.; Jones, D. O.; Soderberg, Alicia; Challis, Peter; Drout, Maria Rebecca; Foley, R. J.; Huber, M. E.; Kirshner, Robert; Leibler, C.; Marion, G. H.; McCrum, M.; Milisavljevic, Danny; Narayan, Gautham; Sanders, Nathan Edward; Smartt, S. J.; Smith, K. W.; Tonry, J. L.; Burgett, W. S.; Chambers, K. C.; Flewelling, H.; Kudritzki, R.-P.; Wainscoat, R. J.; Waters, C.
    We present Hubble Space Telescope (HST) rest-frame ultraviolet imaging of the host galaxies of 16 hydrogen-poor superluminous supernovae (SLSNe), including 11 events from the Pan-STARRS Medium Deep Survey. Taking advantage of the superb angular resolution of HST, we characterize the galaxies' morphological properties, sizes, and star formation rate (SFR) densities. We determine the supernova (SN) locations within the host galaxies through precise astrometric matching and measure physical and host-normalized offsets as well as the SN positions within the cumulative distribution of UV light pixel brightness. We find that the host galaxies of H-poor SLSNe are irregular, compact dwarf galaxies, with a median half-light radius of just 0.9 kpc. The UV-derived SFR densities are high ($\langle {{{\Sigma }}_{{\rm SFR}}}\rangle \simeq 0.1{{M}_{\odot }}\;{\rm y}{{{\rm r}}^{-1}}\;{\rm kp}{{{\rm c}}^{-2}}$), suggesting that SLSNe form in overdense environments. Their locations trace the UV light of their host galaxies, with a distribution intermediate between that of long-duration gamma-ray bursts (LGRBs; which are strongly clustered on the brightest regions of their hosts) and a uniform distribution (characteristic of normal core-collapse SNe), though cannot be statistically distinguished from either with the current sample size. Taken together, this strengthens the picture that SLSN progenitors require different conditions than those of ordinary core-collapse SNe to form and that they explode in broadly similar galaxies as do LGRBs. If the tendency for SLSNe to be less clustered on the brightest regions than are LGRBs is confirmed by a larger sample, this would indicate a different, potentially lower-mass progenitor for SLSNe than LRGBs.
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    Toward Characterization of the Type IIP Supernova Progenitor Population: A Statistical Sample of Light Curves from Pan-STARRS1
    (IOP Publishing, 2015) Sanders, Nathan Edward; Soderberg, Alicia; Gezari, S.; Betancourt, M.; Chornock, R; Berger, Edo; Foley, R. J.; Challis, Peter; Drout, Maria Rebecca; Kirshner, Robert; Lunnan, R; Marion, G. H.; Margutti, Raffaella; McKinnon, R.; Milisavljevic, Danny; Narayan, G.; Rest, A.; Kankare, E.; Mattila, S.; Smartt, S. J.; Huber, M. E.; Burgett, W. S.; Draper, P.W.; Hodapp, K. W.; Kaiser, N.; Kudritzki, R. P.; Magnier, E. A.; Metcalfe, N.; Morgan, J. S.; Price, P. A.; Tonry, J. L.; Wainscoat, R. J.; Waters, C.
    In recent years, wide-field sky surveys providing deep multiband imaging have presented a new path for indirectly characterizing the progenitor populations of core-collapse supernovae (SNe): systematic light-curve studies. We assemble a set of 76 grizy-band Type IIP SN light curves from Pan-STARRS1, obtained over a constant survey program of 4 yr and classified using both spectroscopy and machine-learning-based photometric techniques. We develop and apply a new Bayesian model for the full multiband evolution of each light curve in the sample. We find no evidence of a subpopulation of fast-declining explosions (historically referred to as "Type IIL" SNe). However, we identify a highly significant relation between the plateau phase decay rate and peak luminosity among our SNe IIP. These results argue in favor of a single parameter, likely determined by initial stellar mass, predominantly controlling the explosions of red supergiants. This relation could also be applied for SN cosmology, offering a standardizable candle good to an intrinsic scatter of lesssim 0.2 mag. We compare each light curve to physical models from hydrodynamic simulations to estimate progenitor initial masses and other properties of the Pan-STARRS1 Type IIP SN sample. We show that correction of systematic discrepancies between modeled and observed SN IIP light-curve properties and an expanded grid of progenitor properties are needed to enable robust progenitor inferences from multiband light-curve samples of this kind. This work will serve as a pathfinder for photometric studies of core-collapse SNe to be conducted through future wide-field transient searches.
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    An Ultraviolet Excess in the Superluminous Supernova Gaia16apd Reveals a Powerful Central Engine
    (American Astronomical Society, 2017) Nicholl, Matthew; Berger, Edo; Margutti, R; Blanchard, Peter; Milisavljevic, Danny; Challis, Peter; Metzger, B.D.; Chornock, R.
    Since the discovery of superluminous supernovae (SLSNe) in the last decade, it has been known that these events exhibit bluer spectral energy distributions than other supernova subtypes, with significant output in the ultraviolet. However, the event Gaia16apd seems to outshine even the other SLSNe at rest-frame wavelengths below ∼3000 \AA. Yan et al (2016) have recently presented HST UV spectra and attributed the UV flux to low metallicity and hence reduced line blanketing. Here we present UV and optical light curves over a longer baseline in time, revealing a rapid decline at UV wavelengths despite a typical optical evolution. Combining the published UV spectra with our own optical data, we demonstrate that Gaia16apd has a much hotter continuum than virtually any SLSN at maximum light, but it cools rapidly thereafter and is indistinguishable from the others by ∼10-15 days after peak. Comparing the equivalent widths of UV absorption lines with those of other events, we show that the excess UV continuum is a result of a more powerful central power source, rather than a lack of UV absorption relative to other SLSNe or an additional component from interaction with the surrounding medium. These findings strongly support the central-engine hypothesis for hydrogen-poor SLSNe. An explosion ejecting Mej=4(0.2/κ) M⊙, where κ is the opacity in cm2g−1, and forming a magnetar with spin period P=2 ms, and B=2×1014 G (lower than other SLSNe with comparable rise-times) can consistently explain the light curve evolution and high temperature at peak. The host metallicity, Z=0.18 Z⊙, is comparable to other SLSNe.
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    A Panchromatic View of the Restless Sn 2009ip Reveals the Explosive Ejection of a Massive Star Envelope
    (IOP Publishing, 2013) Margutti, R.; Milisavljevic, Danny; Soderberg, Alicia; Chornock, R; Zauderer, B; Murase, K.; Guidorzi, C.; Sanders, Nathan Edward; Kuin, P.; Fransson, C.; Levesque, E. M.; Chandra, P.; Berger, Edo; Bianco, F. B.; Brown, P. J.; Challis, P.; Chatzopoulos, E.; Cheung, C. C.; Choi, C.; Chomiuk, L.; Chugai, N.; Contreras, C.; Drout, M. R.; Fesen, R.; Foley, R. J.; Fong, W.; Friedman, A. S.; Gall, C.; Gehrels, N.; Hjorth, J.; Hsiao, E.; Kirshner, R.; Im, M.; Leloudas, G.; Lunnan, R.; Marion, G. H.; Martin, J.; Morrell, N.; Neugent, K. F.; Omodei, N.; Phillips, M. M.; Rest, A.; Silverman, J. M.; Strader, J.; Stritzinger, M. D.; Szalai, T.; Utterback, N. B.; Vinko, J.; Wheeler, J. C.; Arnett, D.; Campana, S.; Chevalier, R.; Ginsburg, A.; Kamble, A.; Roming, P. W. A.; Pritchard, T.; Stringfellow, G.
    The double explosion of SN 2009ip in 2012 raises questions about our understanding of the late stages of massive star evolution. Here we present a comprehensive study of SN 2009ip during its remarkable rebrightenings. High-cadence photometric and spectroscopic observations from the GeV to the radio band obtained from a variety of ground-based and space facilities (including the Very Large Array, Swift, Fermi, Hubble Space Telescope, and XMM) constrain SN 2009ip to be a low energy (E ~ 1050 erg for an ejecta mass ~0.5 M ☉) and asymmetric explosion in a complex medium shaped by multiple eruptions of the restless progenitor star. Most of the energy is radiated as a result of the shock breaking out through a dense shell of material located at ~5 × 1014 cm with M ~ 0.1 M ☉, ejected by the precursor outburst ~40 days before the major explosion. We interpret the NIR excess of emission as signature of material located further out, the origin of which has to be connected with documented mass-loss episodes in the previous years. Our modeling predicts bright neutrino emission associated with the shock break-out if the cosmic-ray energy is comparable to the radiated energy. We connect this phenomenology with the explosive ejection of the outer layers of the massive progenitor star, which later interacted with material deposited in the surroundings by previous eruptions. Future observations will reveal if the massive luminous progenitor star survived. Irrespective of whether the explosion was terminal, SN 2009ip brought to light the existence of new channels for sustained episodic mass loss, the physical origin of which has yet to be identified.
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    Ultra-Luminous Supernovae as a New Probe of the Interstellar Medium in Distant Galaxies
    (IOP Publishing, 2012) Berger, Edo; Chornock, R; Lunnan, R.; Foley, R.; Czekala, Ian; Rest, A.; Leibler, C.; Soderberg, Alicia; Roth, K.; Narayan, G.; Huber, M. E.; Milisavljevic, Danny; Sanders, Nathan Edward; Drout, M.; Margutti, R.; Kirshner, Robert; Marion, G. H.; Challis, P. J.; Riess, A. G.; Smartt, S. J.; Burgett, W. S.; Hodapp, K. W.; Heasley, J. N.; Kaiser, N.; Kudritzki, R.-P.; Magnier, E. A.; McCrum, M.; Price, P. A.; Smith, K.; Tonry, J. L.; Wainscoat, R. J.
    We present the Pan-STARRS1 discovery and light curves, and follow-up MMT and Gemini spectroscopy of an ultra-luminous supernova (ULSN; dubbed PS1-11bam) at a redshift of z = 1.566 with a peak brightness of MUV ≈ −22.3 mag. PS1-11bam is one of the highest redshift spectroscopically-confirmed SNe known to date. The spectrum is characterized by broad absorption features typical of previous ULSNe (e.g., C II, Si III), and by strong and narrow Mg II and Fe II absorption lines from the interstellar medium (ISM) of the host galaxy, confirmed by an [O II]λ3727 emission line at the same redshift. The equivalent widths of the Fe IIλ2600 and Mg IIλ2803 lines are in the top quartile of the quasar intervening absorption system distribution, but are weaker than those of gamma-ray burst intrinsic absorbers (i.e., GRB host galaxies). We also detect the host galaxy in pre-explosion Pan-STARRS1 data and find that its UV spectral energy distribution is best fit with a young stellar population age of τ∗ ≈ 15 − 45 Myr and a stellar mass of M∗ ≈ (1.1 − 2.6) × 109 M⊙ (for Z = 0.05 − 1 Z⊙). The star formation rate inferred from the UV continuum and [O II]λ3727 emission line is ≈ 10 M⊙ yr−1, higher than in any previous ULSN host. PS1-11bam provides the first direct demonstration that ULSNe can serve as probes of the interstellar medium in distant galaxies. At the present, the depth and red sensitivity of PS1 are uniquely suited to finding such events at cosmologically interesting redshifts (z ∼ 1 − 2); the future combination of LSST and 30-m class telescopes promises to extend this technique to z ∼ 4.