A&A 393, 25-32 (2002)
DOI: 10.1051/0004-6361:20020987
A. Goobar^{1} - E. Mörtsell^{1} - R. Amanullah^{1} - P. Nugent^{2}
1 - Department of Physics, Stockholm University,
SCFAB, 106 91 Stockholm, Sweden
2 -
Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA
Received 15 May 2002 / Accepted 1 July 2002
Abstract
We investigate the possibility of measuring the Hubble constant, the
fractional energy density components and the equation of state
parameter of the "dark energy'' using lensed multiple images of
high-redshift supernovae. With future instruments, such as the SNAP
and NGST satellites, it will become possible to observe several
hundred lensed core-collapse supernovae with multiple images. Accurate
measurements of the image separation, flux-ratio, time-delay and
lensing foreground galaxy will provide complementary information to
the cosmological tests based on, e.g., the magnitude-redshift relation
of type Ia supernovae, especially with regards to the Hubble parameter
that could be measured with a statistical uncertainty at the one
percent level. Assuming a flat universe, the statistical uncertainty
on the mass density is found to be
.
However, systematic effects from the uncertainty of the lens
modeling are likely to dominate. E.g., if the lensing galaxies are
extremely compact but are (erroneously) modeled as singular isothermal
spheres, the mass density is biased by
.
We argue that wide-field near-IR instruments such as the one proposed for the SNAP mission are critical for collecting large statistics of lensed supernovae.
Key words: gravitational lensing - cosmology: cosmological parameters
In this note we investigate how to use multiply imaged high-redshift supernovae (SNe) to constrain the Hubble parameter, the mass and dark energy density of the universe, and , as well as the equation of state parameter, , which we assume is constant for . SNe are well suited for this technique because of the expected high rate at high-redshifts and most importantly because of their well known lightcurves. In particular, the (rest-frame) optical lightcurves of core-collapse (CC) SNe show fast rise-times, typically about 1 week long. Thus, the time difference between images can be measured to better than one day's precision.
Another possibility is the use of the UV shock-breakout. As has been seen in SN 1987A and modeled in Ensman & Burrows (1992) the shock breakout can serve as a time stamp with a precision of just minutes. The entire UV flash occurs over a period of minutes to several hours in the rest-frame of the SN depending on the nature of the progenitor.
Wide field optical and NIR deep surveys, such as the planned SNAP satellite (Perlmutter et al. 2001), have the potential to discover CC SNe. While these SNe are sometimes regarded as a "background'' for the primary cosmology program based on Type Ia SNe, we argue that lensed SNe of any kind may also provide useful information on cosmological parameters.
In Holz (2001) the number of multiply imaged CC SNe up to z < 2 were estimated by simply scaling the Type Ia rate by a factor of 5. In this work we extend the considered redshift up to z=5 using a SN rate calculation derived from the star formation history. Further, we take into account the NIR wide field instrumentation in the current design of the SNAP satellite.
Using simple toy-models, we investigate the accuracy of the strong lensing technique to improve our knowledge of cosmological parameters. Our observables are the source redshift, the redshift of the lensing galaxy, the image-separation, , the time-delay, , and the flux-ratio, r. We derive the relation of our observables and cosmological parameters for different matter distributions in order to investigate the sensitivity to the choice of lens model. We speculate, that if the measurement is done using a very large sample of lensed systems, useful bounds on the cosmological parameters can be found in spite of large uncertainties in the lens model.
We have used the SNOC Monte-Carlo simulation package (Goobar et al. 2002b) to estimate the rate and measurable quantities of multiple image SNe, e.g. the distribution of time-delays, image-separations and flux-ratios. We also simulate extinction by dust both in the host galaxy of the SN and in the foreground lensing galaxy.
While about half of the known lensed systems exhibit more images (see e.g. Keeton et al. 1998), this work is limited to spherically symmetric lensing systems producing only two or ring-like images. Systems with more lensed images are potentially very interesting as they provide more measurables which can be used to constrain the lens model.
A light ray which passes by a point-mass M at a minimum distance
,
is deflected by the "Einstein angle''
(2) |
(3) |
(4) |
y=x-1/x, | (5) |
(7) |
(8) |
(9) |
(10) |
y=r^{1/4}-r^{-1/4}. | (11) |
(12) |
(13) |
(14) |
(15) |
(17) |
(18) |
(23) |
(24) |
Since we in principle only are able to follow infinitesimal light-beams in SNOC, we have to use some approximations when trying to get information on multiple image systems. The main approximation is that we assume that in cases of strong lensing, the effects from one close encounter is dominant, i.e., the one-lens approximation. With this simplification, we can use the information from the magnification to derive quantities for systems with finite separations. In order to do this, we need to be able to derive analytical relations between the magnification and the image-separation and so forth. Here we show how this is done for the case of SIS lenses.
First, we concentrate on primary images. Studying
Eqs. (20) and (22), we see that multiple
imaging occurs whenever
and that the magnification of the
second image is given by
(26) |
(27) |
(28) |
(30) |
(31) |
(34) | |||
(35) |
(36) |
(37) |
A higher value of v_{*} would result in a larger number of wide separation lenses since the cross-section for multiple imaging scales as (see Eq. (29)) and the image separation as (Eq. (21)). Varying the galaxy mass distribution will also have an effect on the characteristics of the lensed events. However, since the mass and velocity dispersion are not variables in the fitting of cosmological parameters (see Sect. 5), any changes in these distributions will only have a marginal effect on the results obtained in this paper.
We have simulated CC SNe in the redshift range using a SIS model for the galaxy density profiles. The redshift dependence of the SN rates, the relative fraction among the various types of CC SNe (Ib/c, IIL, IIn, IIP, 87a-like), peak magnitudes and intrinsic dispersion were simulated following the prescriptions in Dahlén & Fransson (1999) and Dahlén & Goobar (2002). Using SN rate predictions derived from the star formation history (Chugai et al. 1999; Sullivan et al. 2000), our simulated sample corresponds approximately to the predicted number of CC SN explosions () in a period of 3-years in a 20 square degree field, i.e., 5.1 SNe year^{-1} arcmin^{-2}. In the simulations we have assumed a flat cosmology with and , and a Hubble constant H_{0} = 65 km s^{-1} Mpc^{-1}. We view these rates on the conservatively low side given the recent work by Lanzetta et al. (2002) which finds that the SFR plausibly increases monotonically with redshift through the highest redshifts observed.
Out of the simulated CC SNe, 2613 were multiply lensed. Figure 1 shows the number of detectable SNe as a function of the peak-brightness threshold in I and J-bands. The threshold refers to the faintest of the two images.
Figure 1: Number of multiple image SNe in the simulated sample vs magnitude threshold for the faintest image in I and J-band. | |
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In the simulations we have considered the possibility that one or both images suffer extinction. We assumed differential extinction properties as in Cardelli et al. (1989), with R_{V}=3.1 and a mean-free path of 1 kpc for V-band photons in the host and lensing galaxy. Details of the simulation procedure can be found in Goobar et al. (2002a,b).
In Figs. 2-5, we show the distribution of observables for the multiple-image SNe; source redshifts, image flux-ratios, image separations and time-delay between the images. The dashed lines show the subset of the data fulfilling case A SNe while the dotted curves correspond to case B.
The salient features of the events are: images of the same source with average redshift and the lensing galaxy typically at . The SN images are separated by 0.5 arcsec and a few weeks apart. With the imposed minimal brightness criteria, the images differ typically by less than one astronomical magnitude. Clearly, a very clean signature to search for, especially with space observations free of atmospheric blurring.
Figure 2: Redshift distribution of lensed SNe up to z=5, solid line. The dotted line shows SNe for which the SN system satisfies the case B criteria. The dashed lines shows case A. A SIS lens profile was used. The lensing galaxy is typically at a redshift | |
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Figure 3: Flux-ratio of SN images. The dotted line shows SNe for which the SN system satisfies the case B criteria. The dashed line shows case A. A SIS lens profile was used. | |
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Figure 4: Image separation in arcseconds. The dotted line shows SNe for which the SN system satisfies the case B criteria. The dashed line shows case A. A SIS lens profile was used. | |
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Figure 5: Time delay between the SN images in days. The dotted line shows SNe for which the SN system satisfies the case B criteria. The dashed line shows case A. A SIS lens profile was used. | |
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Next, we estimate the target statistical uncertainty of a 3 year mission studying a 20 square degree patch of the sky down to 28.5 peak I-band (and J-band) magnitudes. We perform a maximal likelihood analysis of the multiply lensed systems with and as free parameters where h is the dimensionless Hubble parameter defined by 100 km s^{-1}Mpc^{-1}. With and f(r) being the r-dependent factors in Eqs. (16) and (25) for point mass lenses and SIS respectivelywe define the "experimental'' ( ) and cosmology dependent () quantities:
While we will consider more pessimistic scenarios later, in this section we assume that
the uncertainties
of the four observables are:
The image separation uncertainty corresponds to 0.1 pixel of the proposed SNAP instrument. This estimate could even improve considering that all of the SN images along the lightcurve can be co-added to get signals on the individually lensed SNe. E.g. in Anderson & King (2000) it is stated that with HST/WFPC one could reach 0.02 pixel precision on reasonable bright stars.
In estimating we have assumed that the differential extinction of the lensed SNe will not largely exceed what has been measured for a set of 23 gravitational lens galaxies in Falco et al. (1999). The median extinction for those systems ( ) was found to be mag.
Figure 6: 68% CL region in the projection of a three parameter fit . The 366 two-image events fulfilling criteria B. The dark (green) region shows the 68% CL region that would result if h would be exactly known from independent measurements. The line shows , i.e. a flat universe. The diamond shows the value that was used in the simulation. | |
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With large statistics, such as the case in our simulations, it is
feasible to constrain the possible values of
if a flat universe
is assumed. Figure 7 shows the 68% CL-region of the
plane for case B. The dark shaded region shows the systematic
effect introduced by fitting the SIS simulated data with the
assumption of point mass lenses. The fit in the light shaded region (solid
curve) of
Fig. 7 yields
h = 0.65^{+0.006}_{-0.003};
.
The uncertainties are 68% CL for a two
parameter fit, i.e., 1.51 .
Introducing the bias due to the
wrong lens model, we found the fitted central values to be h = 0.69and
.
To further test potential bias in the cosmological parameters
from systematic uncertainties in the lens model we have tested
adding an ad-hoc offset with a different dependence on the
image ratio:
It is interesting to note that if the Hubble parameter and the energy densities are known to good accuracy from other cosmological tests, it should be possible to put very useful constraints on the average lens properties.
Figure 7: The light (yellow) shaded region bounded by the solid line shows 68% CL region of fit from 366 two-image events where the peak brightness of the faintest image fulfills criteria B. A flat universe was assumed. The dark (green) shaded region shows the bias introduced if the fit is done (erroneously) assuming a point-mass lens model. The dotted and dashed curves show the effect of a systematic error in the lens model, according to Eq. (41). | |
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Next, we investigate the potential of this method in setting limits in the parameter plane. Figure 8 shows that upper limits on the equation of state parameter of dark energy ^{}, w_{0} < -0.8 may be derived from the considered data sample. Meaningful limits on the possibility of w_{0}<-1 could be derived, especially if an independent estimate of H_{0} is used as a prior in the fit.
Due to the different reshift distributions considered, the shape of
the CL-regions differ from what is expected from the Hubble diagram of
Type Ia SNe for the SNAP satellite (Goliath et al. 2001). In particular, if
the Hubble parameter is further constrained by some independent
method, the estimates of w_{0} from the strong lensing data becomes
comparable in precision the limits that may be derived from Type Ia
SNe, the dashed countour line in Fig. 8. The other 2D
projection, the one onto the w_{0}-h plane, is shown in
Fig. 9. If the wrong halo model is used in the
fitting procedure, i.e., a point-mass halo model instead of SIS,
a 5% bias is introduced in the estimate of w_{0}.
Figure 8: 68% CL region of fit from the 366 events fulfilling criteria B. A flat universe was assumed. The dark (green) region shows the smaller confidence region that would result if h would be exactly known from independent measurements. The dashed line shows the expected statistical uncertainty from a 3 year SNAP data sample of Type Ia SNe. | |
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Figure 9: 68% CL region of h-w_{0} fit from 366 two-image events in case B. A flat universe was assumed. The dark (green) region shows smaller confidence region that would result if would be exactly known from independent measurements. | |
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Lensed SNe are potentially interesting as they provide independent measurements of cosmological parameters, mainly H_{0}, but also the energy density fractions and the equation of state of dark energy. The results are independent of, and would therefore complement the Type Ia program. At the faint limits considered in this note, several quasars per square arcminute are expected (Conti et al. 1999). Thus, we expect that an instrument like SNAP would find several hundred multiply imaged QSOs, in addition to the strongly lensed SNe. Thus, the statistical uncertainty could become smaller than what we have considered here.
While the systematic uncertainties remain a source of concern, we show that the simplest spherically symmetric models introduce moderate biases ( ), at least as long as multiple images with similar fluxes are considered.
While projections of different SNe could in principle be interpreted as a lensed SN the two scenarios may be distinguished. The signatures of CC SNe are unique. Crudely, the lightcurves are a product of the progenitor mass, the mass loss during its evolution off the main sequence, the amount of radioactive Ni synthesized during the explosion and the kinetic energy imparted to the ejecta. In addition, the environments the SNe explode in (the density and structure of their local interstellar medium), often play a significant role in what we eventually see of the CC event. These differences have given rise to all the different classifications of these events we currently have; Type IIP, IIL, Ib, Ic, IIn, etc. Given all this diversity it makes it quite easy to distinguish one event from another and to not confuse a lensing event with a coincident CC event along the same line of sight.
In our simulations we find that extinction in the foreground galaxies does not severely affect the detectability of multiple lensed events nor the ability to derive the flux-ratio between the images. Clearly, multi-band observations of the SNe will be important in order to correct for the different amounts of extinction of the images. At the same time, the data could provide important results on the dust properties of the foreground lensing galaxies, similar to the studies done with multiple imaged quasars at (Falco et al. 1999). Unlike quasars, CC events have a very deliberate color evolution along their lightcurves. In general, from the moment of shock-breakout onwards, the atmospheres of CC SNe expand and become cooler and redder. This signature not only helps in the relative timing of these events, but also allows us to measure the differential extinction due to the varying amounts of dust along the lensed paths to the SN quite well. With 3 or more filters one could make a measurement of both the total extinction relative to the bluest event in addition to the ratio of the total to selective extinction due to differences in the dust properties. Furthermore, one could enhance this method by taking a spectrum of the SN (any of the lensed events would do) at a given epoch and through spectrum synthesis derive the true, unextinguished spectral energy distribution of the event (see e.g. Mitchell et al. 2002; Baron et al. 2002).
Acknowledgements
It is a pleasure to acknowledge E. Linder for careful reading of the manuscript and important suggestions. We thank G. Bernstein, A. Kim and the rest of the SNAP collaboration for useful discussions. AG is a Royal Swedish Academy Research Fellow supported by a grant from the Knut and Alice Wallenberg Foundation. This work was supported by a NASA LTSA grant to PEN and by resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098.