arXiv:1907.02693v2 [astro-ph.CO] 25 Jan 2020 n aeams h aepa bouemgiue( magnitude explosions absolute luminous peak extremely same the are almost Ia have and SNe therefore up. is set Cosmography better the surveys. with large dramatically of development increased have (SNe) supernovae h aueo h opno tro h ht dwarf. white the of star for companion possibilities the many of are nature there for- the supernovae, the of For of luminosity [8]. mation absolute environments on the ac- dependent in the supernovae resulting in roles process, dif- different cretion in mag- have galaxies absolute stages Host the evolutionary example, to ferent For fac- related supernovae. new be of many to nitude theory, found Secondly, SN been have the [7]. tors of ignored also development be are the cannot with distances a that Cepheid and factors on contamination uncertain law photometric of of extinction the- slope effects changing different and The in controversial zero-point highly ories. the period- is both relationship the on ef- this The on metallicity [6]. of depends variables fect this Cepheid mainly in stud- of it uncertainties relationship cosmological some luminosity Firstly, in are ladder there results However, distance approach. good This 5]. to [4, ies [3]. led periods has with method luminosi- related whose are Universe local ties at stars variable Cepheid ladders. distance which local parameter, the free by calibrated a be as to brightness) considered needs intrinsic and (i.e., unknown magnitude is absolute of peak value as the the taken However, and therefore distances. extra-galactic luminosity are determining cosmological in They candles magnitudes standard [2]. absolute ideal peak dispersion For their small Ia, [1]. have mechanism SNe physical ” normal reasonable “ a most of basis the on vrteps eae,tenme fosre yeIa Type observed of number the decades, past the Over npatc,tewyt airt h N ai through is Ia SNe the calibrate to way the practice, In env S)osrain n opoeteitiscproper intrinsic the probe to magnitude and observations (SN) pernova steclbainsadrs h airtd1 calibrated the standards, calibration the as nftr bevto odtos o h iedlydistan delay time the For cosmolog bot for T conditions. at observation uncertainties Ia future 5% galaxies. SNe on with lens the systems the lensing anchor to high-quality th model-independently provide (ADD) can can Distances quasars ments Diameter lensed strongly Angular The the era. (LSST) scope ASnumbers: PACS supernovae. resu calibrating The for respectively. tool mag, promising 0.06 a be and mag 0.31 approximately .3mg epciey eie,w locnie nevolvi an consider also we Besides, respectively. mag, 0.03 ytecsi pct.I hscs,te1 the case, this In opacity. cosmic the by epooeanwmdlidpnetsrtg oclbaeth calibrate to strategy model-independent new a propose We INTRODUCTION 1 airtn h tnadcnlswt toglensing strong with candles standard the Calibrating M colo cec,WhnUiest fTcnlg,Whn43 Wuhan Technology, of University Wuhan Science, of School B aigo toglnigosrain nteucmn Larg upcoming the in observations lensing strong on basing , uogWen Xudong Dtd aur 8 2020) 28, January (Dated: σ M netite of uncertainties σ B ) 1 netite of uncertainties n a Liao Kai and atce 1] h omcoaiycudmk h N Ia SNe the make making could equivalently opacity other dimmer, to cosmic pho- photons the the transfer [12], redshifts. of that particles high mechanisms scattering of other absorption, or case potential tons the extrapo- the in this to effective Due whether is checked increases. method be redshift lation cannot the as it red- change Therefore, proper- may low the supernovae a between of than relations ties some massive [11], redshift more supernova high and shift extrapo- a redder that be is Considering to supernova redshifts. needs high calibration to mea- the are lated stars and variable locally, Cepheid the pro- sured Thirdly, are Ia the counterparts [10]. SNe typical in their duced subluminous But than path, dimmer [9]. are double-degenerate ex- Ia that a SNe in of the successful case of been observations the has plaining path single-degenerate The osat( constant redshift. ehi aibesasadSeI tlclUies has Universe local at Ia SNe 4 and stars variable Cepheid oe n aaeeso N a o example, high for Ia, at SNe Ia of the in parameters SNe parameters and the the model use fitting calibrate simultaneously can to by one redshifts model calibration, cosmological Cepheid the a besides model. that cosmological Note standard im- the mea- or beyond physics distance measurements new ply observations local CMB in and errors surements systematic man- either unknown would contradiction ifest This [13]. model standard ΛCDM the in observations (CMB) Background crowave hszdhr htteeaea es he eet to benefits three least at em- are be there should It that here cross-check). phasized the cosmological achieve model- at to new (even distances explore methods to calibration necessary independent is it model above, cosmological standard mentioned the and observations in rors . 4 eety h omnt spzldb h Hubble the by puzzled is community the Recently, hrfr,det h susaotbt ytmtcer- systematic both about issues the to due Therefore, σ imthwt htcntandfo omcMi- Cosmic from constrained that with mismatch gdsac eain o xml,caused example, for relation, distance ng D n D esrmnsbasing measurements ADD and TDD h M e n h nua imtrdistances diameter angular the and ces iso N a seilyteabsolute the especially Ia, SNe of ties B M t hwta h D ehdwill method ADD the that show lts ieDlyDsacs(D)and (TDD) Distances Delay Time e H airtdwt D n D are ADD and TDD with calibrated 1, B itnerlto nTp asu- Ia Type in relation distance e 0 cldsacs esmltd55 simulated We distances. ical ∗ eeaslt itnemeasure- distance absolute hese eso su.The issue. tension ) r prxmtl .4mgand mag 0.24 approximately are yotcSre Tele- Survey Synoptic e 00 China 0070, M B pert vlewith evolve to appear H 0 esrdfrom measured M B [14]. 2 do this: 1) understand the properties of SNe Ia them- Sect. we introduce the angular diameter distance and selves at any redshifts directly and cosmological-model- time delay distance, respectively. We also introduce the independently; 2) provide new ways to anchor SNe Ia and mock catalog of the strong lensing systems. In Sect. We then apply them in cosmological studies; 3) the newly cal- introduce the method for calibrating SNe Ia with or with- ibrated SNe Ia may shed light on the H0 tension issue. out the consideration of cosmic opacity. In Sect. we In the literature, the effective absolute magnitude present our analysis and results. Finally, we summarize M(z) was calibrated by using the Etherington’s distance- our work in Sect. . duality relation and the angular baryonic oscillation (BAO) scale observed at any redshifts [16]. The disad- vantage of this method is that it produces the quantity DISTANCES FROM STRONG LENSING that cannot be directly compared with SN simulations. The inverse distance ladder technique can not only de- Thousands of lensed quasars will be detected by the termine H0 but also calibrate supernovae. Some articles upcoming wide-field synoptic surveys. In particular, the use this method to calibrate the intrinsic magnitude of Large Synoptic Survey Telescope (LSST) [28] will find supernovae by combining supernovae and BAO [17, 18]. more than 8000 lensed quasars, of which a considerable However, BAO relies on the scale of the sound horizon at part have well-measured time delays [29]. With ancillary recombination rs to convert angular measurements into data consisting of high-quality imaging from next genera- angular-diameter distance [19]. This means that once tion space telescope, the central velocity dispersion of the the rs is fixed the H0 has already been determined. Re- lens galaxies and the line-of-sight (LOS) measurements, cently, the value of H0 was determined by using the in- we can measure the TDD and ADD. We introduce both verse distance ladder method in combination with super- of them in the following. To make it clearer, we take the novae and gravitational lenses [20], though it depends Singular Isothermal Sphere (SIS) [30] as the model of the on a specified cosmological model. Another study used lens for example, although realistic lenses are much more three time-delay lenses to calibrate the distance ladder complicated. at low-redshifts, combined them with relative distances Firstly, The time delay between two images of the from SNe Ia and BAO, leaving rs completely free [19]. lensed AGN can be expressed by an equation contain- This method calibrates supernovae from a new perspec- ing TDD as: tive and is promising in the future. Moreover, the dis- D ∆t = ∆t ∆φ, (1) covery of a coalescing gravitational wave (GW) signal of c a compact binary system and its electromagnetic coun- terpart provides a new method for calibrating supernova where c is the light speed. ∆φ is the difference between absolute magnitudes [21, 22]. It is expected that the Fermat potentials at different image positions, which can third generation of gravitational wave detectors will pro- be inferred by high resolution imaging observations of 2 − Einstein ring (or arcs). In the SIS model, ∆φ = (θi vide more abundant data in the future. 2 Strong gravitational lensing has become an effective θj )/2 [26]. The TDD is defined by: tool in astrophysics and cosmology [23]. When light from A A Dl Ds a distant object passes through an elliptical lens galaxy, D∆t = (1+ zl) , (2) DA multiple images of AGN can be observed and time de- ls lays exist among them due to the geometric and Shapiro which is the combination of three different angular diam- A A A effects for different paths. Distances can be obtained by eter distances [31]. Dl ,Ds ,Dls are the angular diameter analyzing the imaging and time delays. There are two distances between observer and lens, observer and source, methods to extract distance information. One is to mea- and lens and source, respectively. zl is the lens redshift. sure the “ time delay distance ” (TDD) consisting of three Therefore, if we obtain the time delay through monitor- angular diameters distance [24]. The other is to measure ing the light curves and model the potential of the lens, the angular diameter distance (ADD) of the lenses, which we can get the TDD. can be obtained by measuring the time delays and the ve- Secondly,the random motion of stellar in an elliptical locity dispersion of the lens galaxy [25, 26]. The current galaxy produces Doppler shift on the spectra correspond- and upcoming large surveys are bringing us a large num- ing to each stellar, and the velocity dispersion σ can be ber of lensed quasars, making time delay measurements obtained by observing the integrated spectrum of the of strong lensing systems very promising for cosmology. whole galaxy. From the Virial theorem, σ is related to 2 In this work, we propose to use two kinds of lensing the mass Mσ in radius R, σ ∝ Mσ/R [25]. In a grav- distances for calibrating the SNe Ia at cosmological dis- itational lens system, the relationship between Einstein tances. It should be noted that the lensing observations angle θE and mass MθE is as follows: are angular separation and spectroscopy measurements, A thus the distances measured should be free of cosmic 4GMθE Dls θE = 2 A A , (3) opacity [27]. The structure of the paper is as follows. In s c Dl Ds 3 where the radius of the Einstein ring can be expressed A 2 ∝ as R = Dl θE. Therefore, it can be deduced that: σ A Ds DA θE. In the SIS model, velocity dispersion is given ls by [26] 2 A 2 c Ds σ = θE . (4) A 4π Dls
A A Ds Dl Considering that ∆t is proportional to A and the Dls A 2 Ds s velocity dispersion σ is proportional to A , the angular z Dls A diameter distance Dl to the lens can be obtained by the ratio ∆t/σ2 [25]. In a SIS lens, angular diameter distance A Dl can be written as c3 ∆t A Dl = 2 . (5) 4π σ (1 + zl) The Time Delay Challenge (TDC) program tested the zl accuracy of current algorithms [46]. And with the first challenge (TDC1), the average precision of the time delay measurement was approximately ∼ 3%, which was com- FIG. 1: The lens and source redshift distributions of the lens systems with well-measured time delay light curves observed parable to the uncertainty of current lens modeling [48]. by LSST plus excellent auxiliary data such that the measured Considering that the metric efficiency was about 20%, distances have 5% precision. For visualization, we show it in TDC1 gave at least 400 well-measured time delay sys- the form of probability density tems [46]. Since the TDD and ADD are sensitive to the mass distribution of the lens, auxiliary data such as high-resolution imaging and stellar velocity disper- by the SDSS-II and SNLS collaborations. The catalog sion observations are required for accurate lens modeling, includes several low-redshift samples, three seasons from so that reasonably accurate distance information can be SDSS-II 0.05
1 10 M , if M < 10 M⊙. M = B stellar (7) METHODOLOGY B 1 (MB + ∆M , otherwise.
For SN Ia data, we use a catalog of direct SN Ia obser- For robustness and simplicity, we only consider the sta- vations: a joint analysis of SN Ia observations obtained tistical uncertainties. The error of the distance modulus 4
µ can be expressed as: estimate of the precision level based on data simulated by future observation conditions. In the actual data, the 2 2 2 2 2 bias is assessed by the non-Gaussian effect, which has to σµ = σmB + α σx + β σc , (8) q start from the original observations, that is, the pixel val- where σmB , σx, and σc are the errors of the peak magni- ues of the host imaging, the velocity dispersion, the AGN tude mB and light curve parameters (x, c) of the SNe Ia, positions, and the time delays taken as Gaussians [31, 35]. respectively. However, the details of the observational uncertainty set The luminosity distance of SN Ia in Mpc can be ob- up for the LSST lens are not yet known. The current tained by published data shows the inference of D∆t and H0 ap- proximation follow the Gaussian distribution [34]. There- L µ/5−5 DSN = 10 . (9) fore, in this paper, we only consider normal distribution for mock data. This assumption would not affect our To compare distances of SNe Ia with lensing distances, we main conclusion. More detailed discussions can be found need to use the corresponding angular diameter distances in [35, 36]. DA from SNe Ia which can be easily obtained through SN We now give the statistics for constraining MB, α and the Distance Duality Relation (DDR) [54]: β in the two methods, respectively. In the ADD method, the statistical quantity can be expressed by using χ2: DL DA = SN , (10) SN (1 + z)2 2 N DA(i) − DA(i) χ2 = SN,l GL,l , (15) where the error of DA can be expressed as σi SN i=1 " DA # X l A σ A = (ln 10/5)D σµ. (11) DSN SN A(i) where DSN,l is the ith ADD term obtained from the SNe A(i) By using Eq. 10, the angular diameter distances from Ia data. DGL,l is the ith data of ADD obtained from the A the observer to the lens (DSN,l) and from the observer to strong lensing observations. N is the number of matched A the source (DSN,s) can be obtained, respectively. Consid- pairs that meet the screening criteria. The ith total error i ering that in the flat universe case, the comoving distance σ A can be written as Dl r = (1+ z)DA between the lens and the source can be − written as rls = rs rl [33]. Thus, the angular distance i 2 2 A σDA = σ A(i) + σ A(i) , (16) from the lens to the source DSN,ls is given by: l DGL,l DSN,l r 1+ z A A − l A where σ A(i) and σ A(i) are ADD errors from gravita- DSN,ls = DSN,s DSN,l. (12) DGL,l DSN,l 1+ zs tional lensing and SNe Ia, respectively. Note that the Then we can construct the TDD from SN Ia observations latter depends on the parameters (MB,α,β). In the TDD method, the statistical quantity can be A A DSN,lDSN,s written as D∆t,SN = A , (13) DSN,ls 2 N D(i) − D(i) χ2 = ∆t,SN ∆t,GL , (17) with the corresponding error being σi i=1 " ∆t # X 2 2 (i) ∂D∆t,SN ∂D∆t,SN 2 2 where D∆t,SN is the ith time delay distance term calcu- σD∆t,SN = σ A + σ A . v DSN,s A DSN,l A u ∂DSN,s ! ∂DSN,l ! lated from the supernova data according to Eq. 13 and u (i) t (14) D∆t,GL is the ith data of time delay distance obtained In order to perform the calibration, in principle, the through observing gravitational lensing. N is the num- distances from two kinds of data should correspond to ber of matched pairs that meet the screening criteria. i the same redshift. However, their redshift cannot always The ith total error σ∆t can be written as be matched perfectly. One solution is to select nearby data pair whose redshift difference is small enough to i 2 2 σ = σ i + σ i , (18) ∆t D( ) D( ) be considered the same. In this paper, we use redshift ∆t,GL ∆t,SN r difference ∆z < 0.005 as the screening criterion [27]. If where σ (i) and σ (i) are the time delay distance there are more than one point in the range of screening D∆t,GL D∆t,SN criterion, the one with the smallest ∆z is chosen. errors obtained by gravitational lensing and the corre- We consider that this work does not attempt to give sponding supernovae data, respectively. an accurate constraint on parameters of SNe from the We also consider the case where the universe is opaque. realistic data, but to propose a new method and give an Such effect has been proposed to account for the dimming 5 through dust distribution in the Milky Way, host galax- ies, intervening galaxies and intergalactic media [37–41]. Cosmic opacity might result from some exotic mecha- nisms including the theory of gravitons [42], Kaluza- Klein modes associated with extra-dimensions [43], or a chameleon field [44, 45]. The flux received by the ob- server will be reduced by the opacity depth factor, and the observed luminosity distance can be expressed by the opaque depth [46]:
α DL = DL eτ(z)/2, (19) SN,obs SN,true − L where DSN,obs is the observed luminosity distance from L SN Ia, DSN,true is the true luminosity distance without the influence of opacity. In this paper, the opacity depth τ(z) is parameterized and can be written as τ(z)=2εz. (20) − − − α Note that the distance information of the gravitational lensing is obtained by measuring the angular separation, regardless of the absolute intensity. That is, the distance FIG. 2: In the case of no opacity, the 1-D and 2-D marginal- ized distributions and 1σ and 2σ constraint contours for SNe measured by gravitational lensing are not biased even in Ia nuisance parameters (α, β, MB ), respectively. The green the presence of opacity. Therefore, We also study the contours and red contours represent the constraint results for calibration by using ADD and TDD respectively under the ADD method and the TDD method, respectively. the influence of cosmic opacity.
SIMULATIONS AND RESULTS
To perform a study on the power of calibration, we take a flat ΛCDM universe with matter density ΩM =0.3 and −1 −1 Hubble constant H0 = 70kms Mpc as our fiducial model in the following simulations. α Since this work aims at giving a prediction of con- straints on α,β,MB,ε rather than using realistic data − to get a result, we give an unbiased analysis reflecting an average constraining power by the following steps. Firstly, on the basis of the distributions in JLA, we set the parameters (α,β,MB) and the theoretical observa- tional quantities (mB ,x,c) of SNe Ia such that the lu-
minosity distances of SNe Ia can be converted to the fiducial values. Secondly, we randomly select the red- − shifts of lenses and sources from Fig.1, then calculate − − − − α the corresponding lensing distances, note that the num- ber of matched pairs might be different for each selection. Thirdly, We perform the noise distribution by considering FIG. 3: The same as Fig.2 but in the case of considering the the uncertainty levels of supernovae data in JLA and 5% cosmic opacity. uncertainties for lensing distances to generate the mock data. Fourthly, we do minimizations to find the best-fits of parameters (α,β,MB, ε) by using the minimization The constraints of each parameter of supernovae are com- function in Python. Finally, we repeat the minimization posed of one-dimensional distributions corner plots and process for 50,000 times under different noise realizations. two-dimensional constraint for the combination of two We take all the best-fits from each minimization as the parameters, where the innermost contour and the out- expected distributions of the parameters. For both meth- ermost contour represent the 1σ and 2σ ranges, respec- ods, we show results that are not affected by opacity in tively. Fig.2, and that considering the effect of opacity in Fig.3. The simulation results in the ADD method show that 6
MB ε
FIG. 4: The comparison of constraint results on MB between FIG. 5: The comparison of constraint results on ǫ between Pantheon sample and JLA sample with 5% uncertainty of Pantheon sample and JLA sample with 5% uncertainty of ADD and TDD. ADD and TDD.
the MB uncertainty range of the supernova can be de- TABLE I: termined at 0.03 under 1σ confidence level without the Methods MB α β ǫ influence of opacity. Under the influence of opacity, the − +0.03 +0.03 +0.40 ADD(5%) 19.10−0.03 0.1−0.03 2.7−0.36 \ simulation results can determine the uncertainty range − +0.24 +0.11 +0.99 TDD(5%) 19.10−0.24 0.1−0.10 2.7−1.13 \ of MB at 0.08 (1σ). We also consider these two cases − +0.04 +0.04 +0.54 ADD(10%) 19.10−0.04 0.1−0.04 2.7−0.50 \ with the TDD method. The uncertainty range of MB is − . +0.31 . +0.16 . +1.15 \ 0.26 (1σ) and 0.31 (1σ), respectively. For comparison, TDD(10%) 19 10−0.31 0 1−0.18 2 7−1.66 − +0.06 +0.03 +0.37 +0.06 we also consider the case when the lens data uncertain- ADD(5%) 19.10−0.06 0.1−0.03 2.7−0.36 0.0−0.06 − +0.31 +0.15 +1.20 +0.34 ties are 10%. Results from the two methods are shown TDD(5%) 19.10−0.31 0.1−0.15 2.7−1.18 0.0−0.39 − +0.11 +0.08 +1.89 +0.10 in table I. ADD(10%) 19.10−0.11 0.1−0.07 2.7−1.50 0.0−0.10 − +0.39 +0.17 +1.38 +0.38 We further consider the Pantheon sample[47] and com- TDD(10%) 19.10−0.37 0.1−0.17 2.7−1.76 0.0−0.42 pare the results with the JLA sample under the same con- Constraint results of supernova parameters for two methods ditions that the uncertainty of ADD and TDD are 5%. with different degrees of uncertainty for the JLA sample. For the Pantheon sample, there is no stretch-luminosity The top half of the table shows the results without opacity, parameter α and color-luminosity parameter β, so we and the bottom half considers results with opacity. only consider absolute magnitude MB and cosmic opac- ity ǫ. As shown in Fig.4 and Fig.5, both samples have almost the same results. CONCLUSION AND DISCUSSIONS The results show that ADD method is much more pow- erful than TDD method. There are two reasons for this. First, and the most important reason, this is because the SNe Ia play an important role in modern astronomy, screening criteria we set for the TDD method will ar- especially in measuring cosmological distances. However, tificially weaken the constraints. There are ∼ 53 pairs some theories and observations introduced controversies of valid data that meet our screening criteria under the about the absolute magnitude of supernovae and the dis- ADD method, but only ∼ 2 pairs for the TDD method tance relation from Cepheids. due to the high-redshifts (typically 2-3) of the sources in In this paper, we propose a strategy to calibrate the LSST. Second, TDD method contains two distance errors absolute magnitude of supernovae by using two kinds of from SNe Ia, while ADD method uses only one SN dis- lensing distances. The simulation is based on the high- tance. Previous studies [55] have shown that combining quality data available in the future LSST era. The re- ADD and TDD can improve the ability to constrain the sults show that gravitational lensing systems can con- parameters. However in this work, the power of TDD is strain the SN Ia parameters to a high precision. Com- too weak. Jointing TDD and ADD does not affect, unless pared with the TDD method, the ADD method is more the power of TDD is at the same level as ADD. There- powerful in constraining the parameters of supernovae. fore, we will not further study the effects of combining There are two main origins for the large uncertainty of ADD and TDD. the TDD method. On the one hand, it contains the error Our study shows ADD method is excellent for calibrat- of two distances, which significantly increases the uncer- ing SNe Ia. Nevertheless, we still incorporate the TDD tainty. On the other hand, it is necessary to match the method in this paper for completeness. redshifts of both lens and source, resulting in a smaller 7 amount of data that match successfully. 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