Lensing Time Delays and Cosmological Complementarity
aa r X i v : . [ a s t r o - ph . C O ] D ec Lensing Time Delays and Cosmological Complementarity
Eric V. Linder
Berkeley Lab & University of California, Berkeley, CA 94720, USA andInstitute for the Early Universe WCU, Ewha Womans University, Seoul, Korea (Dated: September 26, 2018)Time delays in strong gravitational lensing systems possess significant complementarity with dis-tance measurements to determine the dark energy equation of state, as well as the matter densityand Hubble constant. Time delays are most useful when observations permit detailed lens modelingand variability studies, requiring high resolution imaging, long time monitoring, and rapid cadence.We quantify the constraints possible between a sample of 150 such time delay lenses and a near termsupernova program, such as might become available from an Antarctic telescope such as KDUSTand the Dark Energy Survey. Adding time delay data to supernovae plus cosmic microwave back-ground information can improve the dark energy figure of merit by almost a factor 5 and determinethe matter density Ω m to 0.004, Hubble constant h to 0.7%, and dark energy equation of state timevariation w a to 0.26, systematics permitting. I. INTRODUCTION
Complementarity between cosmological probes in-creases their leverage on the cosmological model parame-ters, crosschecks results through differing systematic un-certainties, and breaks degeneracies. These all play im-portant roles in elucidating the nature of our universe:the energy densities in matter and dark energy, the scaleof the universe through the Hubble constant, and thecharacteristics of the dark energy behind the current cos-mic acceleration.Most probes, however, have substantial similarity intheir parameter dependencies, involving the same combi-nations of ingredients entering into the Hubble parameteras a function of redshift, H ( z ). Distances (and volumes)in particular, are essentially equivalent, and growth ofstructure also depends similarly on H ( z ). Looking for ahigh degree of complementarity, especially to determinethe dark energy equation of state value and time varia-tion, [1] investigated the use of distance ratios present instrong gravitational lensing as a means of breaking thisdegeneracy.While lensing distance ratios involve the mass struc-ture of the lens, and so are not purely geometric, therehas been impressive progress in modeling the mass distri-butions in lensing systems (e.g. [2–4]) and so it is worthconsidering strong lensing distances in more detail as acosmological probe, in particular for its complementarity.Here we revisit [1] with several important distinctions: 1)we concentrate on time delays, due to the recent obser-vational successes [4, 5] and modeling advances; 2) weconsider a more realistic range of future observationalprospects, involving projects starting to get underway,which will have important implications for complemen-tarity; and 3) we carry out studies of the science reachas a function of redshift range, and in the presence ofspatial curvature.Several authors (e.g. [6–12]) have addressed the sta-tistical power of strong lensing time delays from furtherfuture surveys such as LSST, calculating the numbers oflenses found and with measured time delays, and project- ing possible Hubble constant or cosmological constraints.These, however, treat the lensing systems as an ensembleto average over, and in fact identify the mass modelingas a major uncertainty capable of degrading constraintssubstantially. Here we concentrate on what are some-times called “golden lenses”, although now the meaning isnot systems with some special symmetry but rather oneswhere the survey design has specifically provided dataenabling detailed construction of the lens mass model.The number of such systems will be much less but wefind they can have significant scientific leverage.In Section II we discuss the cosmological impact oftime delay measurements and their complementaritywith other probes. Section III considers reasonable pos-sibilities for survey data sets in terms of number and red-shift range of time delay systems, and analyzes their con-straints in conjunction with a mid term supernova surveyand cosmic microwave background data. Survey require-ment issues with respect to imaging resolution, time sam-pling, etc. for time delay measurement, lens modeling,and systematic error control are outlined in Section IV,with specific reference to the Antarctic optical/infraredtelescope program. II. TIME DELAYS AS COSMIC PROBE
Strong gravitational lensing causes multiple images ofdistant sources, with the light rays from the images tak-ing different amounts of time to propagate to the ob-server. The time delay involves two parts: a geometricdelay from different path lengths and a gravitational timedelay from traversing different values of the gravitationalpotential of the lens. Thus both the image positions andthe lens mass model must be accounted for. Time de-lays are observed by looking for coordinated variationsin the flux from the images, e.g. of time varying quasars,which requires long time, well sampled monitoring. Typ-ical galaxy lens induced delays are ∼
60 days and thedesired measurement accuracy is a couple of days or bet-ter.To translate the image angular positions into spatialpositions, for computing both the path length and thegravitational potential effects, one needs the (conformal)distance to the lens, r l , to the source, r s , and betweenthe source and lens, r ls . Only in flat space is r ls = r s − r l .The particular combination of distances central to timedelays is T ≡ r l r s r ls . (1)Specifically, following [5], the time delay of an imageat position ~θ on the sky relative to an unlensed source atposition ~β is∆ t ( ~θ, ~β ) = r l r s r ls (1 + z l ) φ ( ~θ, ~β ) , (2)where the distance ratio is T , containing the key cosmo-logical information, and φ is the Fermat or time delaypotential given by φ ( ~θ, ~β ) = ( ~θ − ~β ) − ψ ( ~θ ) . (3)We see that the first term on the right hand side is thegeometric delay and the second term is the lensing poten-tial delay with ∇ ψ = 2 κ for κ the dimensionless lensingprojected surface mass density.The time delay potential φ connecting T to the ob-served time delays therefore depends on the lens massdistribution, the model for which is built up from infor-mation on image positions and flux ratios and perhapssurface brightness morphologies, ideally from many im-ages including arcs. See [4, 5, 10] for details on the mod-eling process and calculation of the potential factor. Allthe cosmological information comes from T (cf. the ap-proach of [13]), with the uncertainties in the potentialfactor entering into the error propagation, together withmeasurement uncertainties.The time delay probe T has interesting properties withregards to the cosmological parameter leverages. Asnoted by [1], the sensitivities to dark energy parameters w and w a , where the dark energy equation of state is wellfit by w ( z ) = w + w a z/ (1 + z ) [14], are actually posi-tively correlated in contrast to standard distance mea-surements such as from Type Ia supernovae. This offershope for complementarity with such probes. Further-more, the sensitivity to the dimensionless matter den-sity Ω m at low redshift is remarkably low compared tosolo distances, leading to the possibility of breaking theusual degeneracy between matter density and dark en-ergy equation of state. Finally, the ratio depends linearlyon the Hubble scale H − , and since [15] researchers havesought to use lensing time delays to measure the Hubbleconstant.In Figure 1 we highlight these special properties. Thederivatives ∂ ln T /∂p give the sensitivities for each pa-rameter p and are exactly what enters into a Fisher ma-trix analysis for cosmological parameter estimation. Raw (unmarginalized) sensitivities can be read directly, e.g. ∂ ln T /∂p/ .
01 = 10 means that a 1% measurement of T delivers an uncertainty σ ( p ) = 0 .
1. This must be foldedin with the covariances between parameters: sensitivitycurves having the same (reflected) shape indicate highlyanticorrelated (correlated) parameters. For the time de-lay probe, the curve shapes are not very similar – a goodsign for breaking degeneracies – and we see the unusualpositive correlation between w and w a over the wholerange z l = 0–0.6 (where for simplicity we have assumed z s = 2 z l ). FIG. 1. Sensitivity of the time delay distance combination T = r l r s /r ls to the cosmological parameters is plotted vslens redshift. Curves with opposite signs at the same redshiftindicate positive correlations between those parameters – veryunusual for the dark energy equation of state variables w and w a . To take advantage of the odd correlation propertiesof T to give strong complementarity in probing cosmol-ogy, we include Type Ia supernova distances as anotherprobe, having very different degeneracies. The super-nova distance-redshift relation has no sensitivity to theHubble constant h = H / (100 km/s/Mpc) however, thisbeing convolved with the unknown supernova absoluteluminosity. To profit from the time delay dependence on h , therefore, we also use cosmic microwave background(CMB) information, which determines the physical mat-ter density combination Ω m h very well (but not par-ticularly Ω m by itself). This further offsets the weakdependence of T alone on the matter density, and so theweakness of each is turned into strength in complemen-tarity. III. COSMOLOGICAL LEVERAGE
Another interesting property of the time delay probeis that its useful and unusual correlation properties oc-cur at low redshift, for z l = 0–0.6. Detailed observa-tions of lensing systems will be easier there, where thelens galaxy and source images will not be as faint as athigher redshift. We therefore take as our baseline a sur-vey producing time delay measurements at z l = 0 . z l = 0 . z s = 2 z l ; although there will be a distributionof source redshifts this has little impact on the cosmologyestimation (see, e.g., § z s = 4 z l affects the dark en-ergy figure of merit (uncertainty area) result by less than1%. In most of this section we assume a spatially flatuniverse, studying the effect of an additional parameterfor curvature in Section III B. A. Cosmological Parameter Constraints
To the time delay measurements we add supernova dis-tance (SN) and CMB information and carry out a Fishermatrix analysis to estimate the cosmological parameterconstraints. For the supernovae, we take a mid termsample reasonable for the next five years, consisting of150 SN at z = 0 . z = 0 . z = 1–1.7 asfrom Hubble Space Telescope observations such as theCLASH [18] and CANDELS [19] surveys. This seems likea reasonable estimate for a mid term, well characterizedsupernova sample. Each supernova is given a 0.15 mag(7% in distance) statistical uncertainty and each redshiftbin of 0.1 has a systematic floor at dm sys = 0 .
02 (1 + z )added in quadrature to the statistical error. Thus thesupernova sample is systematics limited out to z = 1.For CMB data, we take Planck quality information con-sisting of determination of the geometric shift parame-ter R to 0.2% and the physical matter density Ω m h to0.9%, roughly corresponding to constraints from the lo-cation and amplitude, respectively, of the temperaturepower spectrum acoustic peaks. The parameter set is { Ω m , w , w a , h, M} , where M is the convolution of thesupernova absolute luminosity and the Hubble constant.Current measurements can deliver the time delayprobe T to ∼
5% for a lensing system, dominated bysystematic uncertainties for individual systems. With asurvey designed to find many strong lensing images andcharacterize them accurately, it may be possible to con-sider 1% measurements of T in each redshift bin of 0.1from z = 0 . z = 0 . w – w a tightens by a factor 4.8 over thatfrom SN+CMB alone. All the cosmological parametersare better determined by factors of 2.6–3.1. Time delaystherefore have great complementarity with the supernovaand CMB probes, and such a strong lensing survey wouldbe highly valuable scientifically. FIG. 2. 68% confidence level constraints on the dark en-ergy equation of state parameters w and w a using mid termsupernova distances and CMB information, and with (solidcurve) or without (dashed curve) time delay measurements.The time delay probe demonstrates strong complementarity,tightening the area of uncertainty by a factor 4.8. The absolute level of the constraints with time delays isimpressive as well. The Hubble constant is determined to0.0051, or 0.7%; the matter density Ω m to 0.0044 (1.6%),and the present value of the dark energy equation of state w to 0.077 and its time variation w a to 0.26. Whilefalling short of the results from a space survey of su-pernovae (with CMB), such a mid term program coulddeliver important insights into the nature of cosmic ac-celeration and the cosmological model.The baseline time delay sample adopted seems plausi-ble, but let us consider variations to see how the cosmo-logical constraints depend on the survey characteristics.It may be difficult to find enough strong lens systemsat the lowest redshifts, due to the limited volume. Notehowever that the SLACS survey has been successful indetecting lenses [20], if not necessarily measuring timedelays, at z l ≈ .
1, and this depends on the source pop-ulation targeted. Nevertheless, if we cut the time de-lay information to the range z = 0 . σ ( h ) degrading by55%. This then propagates into the Ω m constraint, whichweakens by 41%. These can be somewhat ameliorated ifwe have some information from z = 0 . z l = 1, still at the 1%accuracy per 0.1 redshift bin. Then the figure of meritimproves by 40%, though the constraints on Ω m and h only gain by 6%. Detailed characterization of the lens-ing systems at such high redshift could be problematic,however, due to lower fluxes and signal to noise. The red-shifts of well characterized time delay systems is slowlybeing pushed out toward z l = 1 [21, 22]. B. Including Curvature
Spatial curvature enters together with the Hubble pa-rameter into either the angular or luminosity distancebetween observer and source. Degeneracy between thecurvature density Ω k = 1 − Ω m − Ω de and dark energyequation of state can be severe; for example see Fig. 6 of[23] for effects on w , w a or [24] for general w ( z ). This canbe broken by using a wide redshift range of distances; inparticular high and low redshift distance measurementscan separate the curvature density from other compo-nents. Another possibility is direct measurement of theHubble parameter as well as distances (e.g. from the ra-dial baryon acoustic oscillation scale), or distance ratiosappearing in gravitational lenses or large scale structure(see, e.g., [25–27]). This has the advantage of not neces-sarily requiring high redshift measurements.We now examine the role that time delay measure-ments can play in breaking the curvature degeneracy, ifthe universe is not assumed to be spatially flat. Figure 3shows the results when we allow for curvature in the cos-mology fitting, using time delays, supernovae distances,and CMB information.The dark energy equation of state uncertainty indeeddegrades, with the area figure of merit declining by a fac-tor 4.1. This shows the degeneracy is not fully broken,but should be contrasted with the factor 20.2 degrada-tion from including curvature with only the SN+CMBdata for constraint. Thus the time delay probe is a use-ful tool even/especially when allowing for spatial curva-ture. Most of the covariance affects the time variation w a , with its uncertainty doubling. The present value w is only determined 12% worse, and the errors on Ω m and h increase by 29% and 27%. The curvature itself is esti-mated to σ (Ω k ) = 0 . FIG. 3. 68% confidence level constraints on the dark energyequation of state parameters w and w a using time delay,mid term supernovae, and CMB information, assuming spa-tial flatness (solid curve) or allowing curvature (dashed curve).The time delay probe moderates the curvature degeneracy,restricting the area degradation to a factor 4, rather than 20without time delay data. IV. SURVEY CHARACTERISTICS
In order to use time delays as a cosmological probe inthe individual lensing system approach, the survey mustdeliver detections and accurate measurements of the timedelays, detailed modeling of the lens systems, and con-trol of other systematic uncertainties. Systematics in-clude microlensing that induces variability, differentiallyaltering the images’ light curves, and projected mass nottruly part of the lens, altering the mass modeling.To detect a large sample of time delay systems, a widefield survey is needed, but to characterize them throughaccurate image positions, splittings, and flux variationsrequires high resolution imaging. Interestingly, a tele-scope at an excellent seeing site such as Dome A, Antarc-tica [28] could fulfill both roles. The Kunlun Dark Uni-verse Telescope (KDUST: [29]), a 2.5 meter telescopeplanned for Dome A would be situated above the lowground layer and possibly have 0 . ′′ median seeing in theoptical, 0 . ′′ in the low background noise infrared.The advantages of high resolution imaging for stronglensing are crucial and manifold [9, 30]. Such seeing alsohelps to separate the images from contamination by thelens galaxy light. To take advantage of this excellent see-ing for strong lensing, the point spread function (PSF)would need to be stable, or algorithms developed to fitsimultaneously the PSF and lens mass model. The stablewinter weather, with low winds and large isoplanatic an-gle, at Dome A could be advantageous. KDUST surveyswould overlap with Dark Energy Survey fields, as well asthose of the South Pole Telescope and LSST. DES couldsupply much of the supernova sample, although super-nova programs, at either low or very high redshift, arealso being studied for KDUST [31].Measuring time delays accurately from detected stronglensing systems requires a long time baseline, since thetime delay distribution of interest is in the range of ∼ z = 0 . V. CONCLUSIONS
We have quantified the significant complementarityas cosmological probes that strong gravitational lensingtime delays, involving distance ratios, have with solo dis-tance measurements such as from Type Ia supernovae.A well designed time delay survey can add to practical,near term supernova and CMB data to provide surpris-ingly incisive constraints on the dark energy equation ofstate, the Hubble constant, and the matter density. Theimprovement in equation of state area uncertainty (figureof merit) is almost a factor 5 over the data sets withouttime delays.Time delays also significantly ameliorate the degenera-cies in parameter determination caused by allowing forspatial curvature, again improving the area uncertaintyby a factor 5. Determination of the Hubble constant to 0.7% as well would be valuable for several astrophysicaland cosmological applications.We have focused on what seem to be near term, rea-sonable data sets. An exciting possibility for achievingthese is telescopes being developed at promising Antarc-tic astronomical sites, such as KDUST at Dome A. Ifthese truly deliver high resolution, stable seeing muchbetter than conventional ground based optical conditions(if not quite space quality), the baseline time delay surveyconsidered here to deliver one order of magnitude timeslarger sample of well characterized time delay systems ap-pears practical. Another advantage is the synergy withother southern surveys, such as the Dark Energy Sur-vey in the near term. (While we have intentionally notextrapolated to long term developments, synergy withLSST is clear as well.)Systematic uncertainties would be ameliorated by thehigh resolution imaging, whether single epoch to char-acterize in detail the lens model and separate the hostgalaxy light, or multiepoch to finely measure the fluxvariations and measure clean and accurate time delays.The redshift range for the survey could be modest, z l ≈ . ACKNOWLEDGMENTS
I gratefully acknowledge Sherry Suyu for helpfuldiscussions, and thank the Niels Bohr InternationalAcademy and Dark Cosmology Centre, University ofCopenhagen for hospitality during the workshop inspir-ing this research. This work has been supported in partby the Director, Office of Science, Office of High EnergyPhysics, of the U.S. Department of Energy under Con-tract No. DE-AC02-05CH11231 and by World Class Uni-versity grant R32-2009-000-10130-0 through the NationalResearch Foundation, Ministry of Education, Science andTechnology of Korea. [1] E.V. Linder, Phys. Rev. D 70, 043534 (2004)[arXiv:astro-ph/0401433][2] S.H. Suyu, P.J. Marshall, R.D. Blandford, C.D. Fass-nacht, L.V.E. Koopmans, J.P. McKean, T. Treu, Ap. J.691, 277 (2009) [arXiv:0804.2827][3] F. Courbin et al, Astron. Astrophys. 536, A53 (2011)[arXiv:1009.1473][4] R. Fadely, C.R. Keeton, R. Nakajima, G.M. Bernstein,Ap. J. 711, 246 (2010) [arXiv:0909.1807][5] S.H. Suyu, P.J. Marshall, M.W. Auger, S. Hilbert, R.D.Blandford, L.V.E. Koopmans, C.D. Fassnacht, T. Treu,Ap. J. 711, 201 (2010) [arXiv:0910.2773][6] M. Oguri & P.J. Marshall, MNRAS 405, 2579 (2010)[arXiv:1001.2037][7] D. Coe & L. Moustakas, Ap. J. 706, 45 (2009)[arXiv:0906.4108][8] B.M. Dobke, L.J. King, C.D. Fassnacht, M.W. Auger,MNRAS 397, 311 (2009) [arXiv:0904.1437][9] L.V.E. Koopmans et al, arXiv:0902.3186[10] M. Oguri, Ap. J. 660, 1 (2007) [arXiv:astro-ph/0609694][11] G.F. Lewis & R.A. Ibata, MNRAS 337, 26 (2002)[arXiv:astro-ph/0206425][12] K. Yamamoto, Y. Kadoya, T. Murata, T. Fu-tamase, Prog. Theor. Phys. 106, 917 (2001)[arXiv:astro-ph/0110595][13] D. Paraficz & J. Hjorth, Astron. Astrophys. 507, L49(2009) [arXiv:0910.5823][14] E.V. Linder, Phys. Rev. Lett. 90, 091301 (2003)[arXiv:astro-ph/0208512][15] S. Refsdal, MNRAS 128, 307 (1964)[16] http://snfactory.lbl.gov ; G. Aldering et al, SPIE4836, 61 (2002)[17] [18] M. Postman et al, arXiv:1106.3328[19] N.A. Grogin et al, Astrophys. J. Suppl. 197, 35 (2011)[arXiv:1105.3753][20] A.S. Bolton, S. Burles, L.V.E. Koopmans, T. Treu, R.Gavazzi, L.A. Moustakas, R. Wayth, D.J. Schlegel, Ap.J. 682, 964 (2008) [arXiv:0805.1931][21] S. Suyu, private communication[22] F. Courbin, private communication; M. Tewes, in prepa-ration[23] E.V. Linder, Astropart. Phys. 24, 391 (2005)[arXiv:astro-ph/0508333][24] A. Shafieloo & E.V. Linder, Phys. Rev. D 84, 063519(2011) [arXiv:1107.1033][25] L. Knox, Phys. Rev. D 73, 023503 (2006)[arXiv:astro-ph/0503405][26] G. Bernstein, Ap. J. 637, 598 (2006)[arXiv:astro-ph/0503276][27] E.V. Linder, Phys. Rev. D 68, 083503 (2003)[arXiv:astro-ph/0212301][28] W. Saunders et al, PASP 121, 976 (2009)[arXiv:0905.4156]; J.S. Lawrence et al, PASA 26,379 (2009) [arXiv:0905.4432][29]