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PREPARED FOR SUBMISSION TO JCAP
Constraints on Cosmological andGalaxy Parameters from StrongGravitational Lensing Systems
Darshan Kumar,a,1 Deepak Jain,b Shobhit Mahajan,a AmitabhaMukherjeea and Nisha Ranic
aDepartment of Physics and Astrophysics, University of Delhi, Delhi 110007, IndiabDeen Dayal Upadhyaya College, University of Delhi, Dwarka, New Delhi 110078, IndiacMiranda House, University of Delhi, University Enclave, Delhi 110007, India
E-mail: [email protected], [email protected],[email protected], [email protected], [email protected]
Abstract. Strong gravitational lensing along with the distance sum rule method can con-strain both cosmological parameters as well as density profiles of galaxies without assumingany fiducial cosmological model. To constrain galaxy parameters and cosmic curvature, weuse a newly compiled database of 161 galactic scale strong lensing systems for distance ratiodata. For the luminosity distance in the distance sum rule method, we use databases of su-pernovae type-Ia (Pantheon) and Gamma Ray Bursts (GRBs). We use a general lens model,namely the Extended Power-Law lens model. We consider three different parametrisationsof mass density power-law index (γ) to study the dependence of γ on redshift. We findthat parametrisations of γ have a negligible impact on the best fit value of cosmic curvatureparameter.Furthermore, measurement of time delay can provide a promising cosmographic probe viathe “time delay distance” that includes the ratio of distances between the observer, lens andthe source. We use the distance sum rule method with 12 datapoints of time-delay distancedata to put constraints on the Cosmic Distance Duality Relation (CDDR) and the cosmiccurvature parameter. For this we consider three different parametrisations of distance du-ality parameter (η). Our results indicate that a flat universe can be accommodated within95% confidence level for all the parametrisations of η. Further, we find that within 95%confidence level, there is no violation of CDDR if η is assumed to be redshift dependent butCDDR is violated if η is considered redshift independent. Hence, we need a larger sampleof strong gravitational lensing systems in order to improve the constraints on the cosmiccurvature and distance duality parameter.
Keywords: Distance Sum Rule Method, Strong Gravitational Lensing, Distance Ratio, TimeDelay Distance.
1Corresponding author.
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Contents
1 Introduction 1
2 Strong Gravitational Lensing 32.1 Distance Ratio 32.2 Time-Delay Distance 4
3 Methodology and Data Samples 53.1 Distance Sum Rule Method 53.2 SGL Systems 53.3 Type Ia Supernovae 63.4 Gamma-Ray Bursts 7
4 Results 9
5 Discussion and Conclusions 165.1 Method I: Distance Ratio 165.2 Method II: Time-Delay Distance 18
1 Introduction
A fundamental problem in modern cosmology is to determine whether the universe is spa-tially open, flat or closed. This is because the curvature of the universe plays an importantrole in its evolution. The most stringent constraint on the cosmological curvature parametercomes from the latest Planck result (2018) under the assumption of the ΛCDM model. Theresult supports a spatially flat universe with a high confidence level [1]. About a decade ago,Clarkson et al. (2008) proposed a model independent method to measure the cosmologicalcurvature parameter (Ωk) [2]. But the problem is that it involves derivatives of the distancew.r.t. the redshift which introduces considerable amount of uncertainty in the estimated cur-vature parameter. Recently, Rasanen et al. proposed a model-independent method calleddistance sum rule which is based on the assumption of the validity of the FLRW metric[3]. Any violation in the distance sum rule directly hints at violation of the FLRW metric.The distance sum rule could be helpful to put constraints on the curvature parameter if it isfound consistent with various observational datasets.
In recent years, Strong Gravitational Lensing (SGL) has become a powerful techniqueto test various assumptions and relations in cosmology [4]. One can study both cosmologi-cal and galaxy parameters using lens systems. Observations of SGL can provide informationof a distance ratio d
ls
A/dos
A, where dls
A and dos
A are the angular diameter distances between thesource-lens and the observer-source respectively. To analyze a SGL system, the SingularIsothermal Sphere (SIS) profile for the lens is the most frequently used density profile. InSIS profile, a total mass-density (ρT(r)) is directly proportional to r−γ where γ is the to-tal mass density profile parameter and is also referred as power law index. Recently, thisprofile was used in the distance ratio analysis considering γ as a free parameter [5]. Caoet al. considered the SIS and power law profile for lenses and the cosmological parameters
– 1 –
keeping Ωm0 fixed [6]. Study of the power law index (γ) is of great interest as it also helpsto understand the structure of a galaxy. Some authors used various observational datasetsto study the evolution of γ with redshift [7, 8]. Further, strong gravitational lens systemscontaining 118 datapoints for distance ratio data were used to put constraints on the totalmass profile and luminosity density profile of stars in an elliptical galaxy [9]. Recently, thelargest sample of SGL (161 datapoints) was compiled by Chen et al. to study the effect ofthe lens mass model on the cosmological parameters assuming the ΛCDM model [10]. It isimportant to note that in all these studies, a flat universe was considered.
On the other hand, the distance sum rule proposed by Rasanen et al. is a model-independent method to constrain the cosmic curvature [3]. Xia et al. used 118 SGL systemsto put constraints on the curvature parameter and the lens profile parameters using thedistance sum rule [11]. Further, using the same method the cosmological and lens profileparameters were constrained by Li et al. assuming a uniform prior on Ωk0 (the value of Ωkat present epoch) [12]. According to their analysis, a spatially closed universe is preferred.The distance sum rule with SGL data and radio quasar data were used to constrain the cos-mic curvature parameter and lens profile parameters by Qi et al. [13]. Using the latest SGLdata (161 datapoints) and SN Ia Pantheon data (1048 datapoints), Wang et al. found that theconstraint on Ωk0 is significantly influenced by the choice of the lens profile model [14].
Following the same line of thought, we also use the distance sum rule for two differentpurposes. First, to constrain the cosmic curvature parameter and lens density profile param-eters using SGL, SN Ia and Gamma Ray Bursts (GRBs) observations. Second, to study thebehaviour of the distance duality parameter and cosmic curvature parameter using Time-Delay Distance (TDD) data. In the first part, we studied the extended power law lens profileto constrain the cosmological and galaxy lens parameters using a model-independent ap-proach, namely the distance sum rule. Recently Chen et al. also worked with the extendedpower law lens profile using SGL observations but they use a model dependent approach(ΛCDM model assumed)[10]. Further, Wang et al. used 161 datapoints and considered allthree lens mass density profiles (SIS, power law and extended power law). In their work,they used SN Ia data (upto z = 2.3) including 1048 supernovae to calibrate the distances ofthe lens galaxy and source galaxy. Therefore, they could not include all the SGL datapointsin their analysis. However, in our analysis, we use SN Ia and GRBs data in order to includeall 161 datapoints of the SGL data upto z = 3.6. Additionally, based on the distance sumrule, previous work did not consider the evolution of the total mass density lens profilepower index (γ) as a function of redshift in the extended power lens profile. However, it isvery important to study the evolution of γ with the redshift because any evolution in γ withz could indicate that in the growth of massive galaxies, dissipative processes have playedan important role [15]. Therefore, with all the above mentioned improvements, we modifythe constraints on the cosmic curvature and lens profile parameters.
It is known that the sources in SGL systems such as quasars and supernovae produceobservable delay in the observed time between multiple images. Therefore, apart from dis-tance ratio analysis, SGL systems can also provide another important quantity called “timedelay”. Measurement of time delay is highly sensitive to the background cosmology andhence cosmological parameters especially the Hubble-Lemaıtre constant (H0) can be con-strained using time-delay observations [16–18]. We further extend our work with time-
– 2 –
delay measurements. For this, we use doubly imaged sources (12 datapoints) containingthe measurement of time-delay between the two images of source. In cosmology, the an-gular diameter distance dA(z) and the luminosity distance dL(z) are related by the ”CosmicDistance Duality Relation (CDDR)” as dA(z)(1 + z)2 = dL(z) [19]. This relation is alwaystrue under the following conditions:• A. The spacetime is described by a metric theory of gravity;• B. Photons travel along null geodesics;• C. The number of photons is conserved.It is important to test the validity of CDDR with observational datasets. Violation of thisrelation has been checked in earlier work [20–31]. Based on the distance sum rule, we putconstraints on the cosmic curvature parameter and the distance duality parameter (η). Fur-ther, we check the evolution of the distance duality parameter with redshift.
The outline of the paper is as follows: In Section 2, we discuss the distance ratio andtime-delay distance in Strong Gravitational Lensing systems. In Section 3, we describe themethodology and the details of datasets used in this paper. The analysis and results areexplained in Section 4. Discussion is presented in Section 5. We describe the conclusions inSection 6.
2 Strong Gravitational Lensing
According to the general theory of relativity, light rays passing near matter get bent due tothe presence of gravity. Light from a distant source, passing near a massive galaxy or galaxycluster (lens), gives rise to multiple images of the source. This phenomenon is known asStrong Gravitational Lensing [32–34]. The bending of light is directly related to the massdistribution within the lens.
2.1 Distance Ratio
The mass distribution of the lensing galaxy is commonly modeled as a Singular IsothermalSphere (SIS) or a Singular Isothermal Ellipsoid (SIE) [35]. Here we consider a more generaland complex model of the lens, namely the Extended Power Law (EPL) model. This modelallows us to consider the luminosity density profile to be different from the total mass den-sity profile. Therefore, it gives us the freedom to consider the effect of dark matter on themass distribution. We use the total mass (luminous and dark-matter) density (ρ(r)) andluminous density (ν(r)) distribution as power laws
ρ(r) = ρ0
(rr0
)−γ
, ν(r) = ν0
(rr0
)−δ
(2.1)
where r represents the radial coordinate from the center of the lens galaxy, γ and δ are twofree parameters. In addition we also consider an anisotropic three-dimensional dispersionof velocity, which suggests that the radial velocity dispersion (σr) and the tangential velocitydispersion (σθ) may be different. Therefore, we define an anisotropy parameter, β(r) =1− σ2
θ /σ2r . Using Eq. (2.1) and the anisotropy parameter β(r) and applying the spherical
Jeans equation, one can define the distance ratio [5].
dR ≡d
ls
Ados
A=
c2θE
4πσ20
(θap
θE
)2−γ
× f (γ, δ, β) (2.2)
– 3 –
where
f (γ, δ, β) =(2√
π)(3− δ)
(ξ − 2β)(3− ξ)×[
Γ[(ξ − 1)/2]Γ(ξ/2)
− βΓ[(ξ + 1)/2]Γ[(ξ + 2)/2]
]Γ(γ/2)Γ(δ/2)
Γ(γ− 1)/2Γ[(δ− 1)/2]
Here ξ = γ + δ− 2. From spectroscopic data, σ0 is an observed velocity dispersion quantitywhich is related to σap via σ0 = σap
[θeff/
(2θap
)]−0.066 [10]. Here θE and θap are the angularradii of the Einstein ring and circular apertures respectively. This extended lens profile re-duces to the SIS model for γ = δ = 2 and β = 0. In earlier work, the anisotropy parameterβ(r) was always taken to be independent of r [35, 36]. For individual lensing systems onecannot determine β independently. Therefore based on the well-studied sample of nearbyelliptical galaxies, we marginalise β(r) using a Gaussian prior with β = 0.18± 0.13 [37]. Asimilar approach was also adopted by Chen et al. and others [10, 14, 17, 38, 39].
We consider the same Gaussian prior on the β parameter throughout the paper, i.e.β = 0.18 ± 0.13. However, for the remaining parameters we consider a flat prior overthe range of interest. Furthermore, in order to include the redshift evolution of the to-tal mass-density, we consider two parametrisation for γ, namely γI I(z) = γ0 + γ1z andγI I I(z) = γ0 + γ1z/(1+ z), because it might help to understand the process behind the evo-lution of the massive galaxies.
2.2 Time-Delay Distance
Apart from distance ratio in SGL systems, time-delay is another important observationwhich can be used to put constraints on cosmological parameters in a model-independentway. The light rays emitted at the same time from a source will reach the observer at differ-ent times as these paths have different path lengths and pass through different gravitationalpotentials. Therefore, there is a time-delay between the multiple images. If the source isa variable light source, this time-delay can be determined by monitoring the images cre-ated by the lens which give us the flux information corresponding to the same source event.Time-delay is related to a quantity which can be used to estimate cosmological parameters.This quantity is the “time-delay distance” and gives a relation between the three angulardiameter distances, i.e. observer-lens, lens-source and observer-source.
For a given source position (B) and image position (θ), the time-delay (δt) between theperturbed and unperturbed light rays is [34]
δt(θ,B) = (1 + zl)
cd
os
Adol
A
dls
A
[(θ −B)2
2− ψ(θ)
]where zl is the lens redshift and ψ is the effective gravitational potential of the lens. In caseof a two image lens system, say image i and j, the time-delay between the two images (∆tij)(for the SIS model) [40] is
∆tij ≡ δtj − δti =(1 + zl)
2cd
os
Adol
A
dls
A
[θ2
j − θ2i
](2.3)
– 4 –
Eq. (2.3) can be rewritten as
d∆t ≡d
os
Adol
A
dls
A
=2c∆tij
(1 + zl)(
θ2j − θ2
i
) (2.4)
d∆t is referred to as the time-delay distance.
3 Methodology and Data Samples
In this analysis we modified Distance Sum Rule (DSR) method in two different forms in or-der to accommodate distance ratio and time delay distance. For this, we use four datasets,namely distance ratio and time delay distance in Strong Gravitational Lensing (SGL), Su-pernovae Ia (SN Ia) and Gamma Ray Bursts (GRBs).
3.1 Distance Sum Rule Method
Under the assumption of homogeneity and isotropy of the universe, one can define thedimensionless comoving distance (Dco) as
Dos
co ≡ Dco(0, zs) ≡H0
cd
os
co; Dol
co ≡ Dco(0, zl) ≡H0
cd
ol
co; Dls
co ≡ Dco(zl , zs) ≡H0
cd
ls
co(3.1)
where dos
co, dol
co and dls
co represent the comoving distances between observer-source, observer-lens and lens-source respectively. According to the distance sum rule [3, 41], these threedimensionless distances are related as:
Dls
coDos
co=√
1 + Ωk0(
Dolco)2 − D
ol
coDos
co
√1 + Ωk0 (Dos
co)2 (3.2)
We can also write the distance sum rule in terms of the time-delay distance (Eq. (2.4))
Dol
coDos
co
Dlsco
=
[1
Dolco
√1 + Ωk0
(Dol
co)2 − 1
Dosco
√1 + Ωk0 (Dos
co)2]−1
(3.3)
The value of Ωk0 can thus be directly obtain from Eqs. (3.2,3.3) without assuming any fidu-cial cosmological model if the distances D
ol
co and Dos
co are known from observations. TheEqs. (3.2, 3.3) represent theoretical constructions of the distance ratio and the time delaydistance respectively. In order to get the left hand sides of these two equations, we use SGLobservations.
3.2 SGL Systems
We use two different kinds of SGL observations: distance ratio and time-delay distance.
• Distance ratio in SGL
For the distance ratio, we use a sample of SGL systems [10], which is a collection of 5 sys-tems from the LSD survey [42–45], 26 from SL2S [46–48], 57 from the SLACS [49–51], 38from an extension of the SLACS for the Masses survey[52, 53], 21 from the BELLS [54] and14 from the BELLS-GALLERY [55, 56]. After combining all the datapoints of these surveys,
– 5 –
we get 161 galaxy-scale strong lensing systems [10]. This sample includes information of thelens redshift (zl), source redshift (zs), Einstein radius (θE), velocity dispersion (σap) mea-sured inside the circular aperture with angular radii θap, and the half-light angular radiusof the lens galaxy θeff. The redshift range of the lenses is 0.0624 ≤ zl ≤ 1.004 and the sourceredshift range is 0.197 ≤ zs ≤ 3.595.
• Time-Delay Distance in SGL
For the time-delay distance, we use 12 double-image SGL systems compiled by Balmes& Corasaniti [57]. Observables in this data are the source redshift (zs), the lens redshift(zl), the angular positions of the two images of the source θi, θj, and the time-delay (∆t).The lens redshift range of this data is 0.260 ≤ zl ≤ 0.890 and source redshift range is0.944 ≤ zs ≤ 2.719. In earlier work, the same data has been used to estimate the cosmolog-ical parameters of different dark energy models [40, 58, 59]. The SIS mass profile explainsthe galaxy mass distribution quite well [60] and double image systems are consistent with it.This is the reason why we use only double-image systems in our analysis. This selection cri-terion for two image formation is necessary but not sufficient to ensure a SIS profile for thelens mass distribution. Furthermore, there may be galaxy models other than SIS which mayalter the image separations and deviation of the velocity dispersion. For example, a soft-ened isothermal sphere galaxy model can reduce typical image separations and systematicerrors in the velocity dispersion deviation. Therefore, S. Cao et al. introduce a parameterΞ, which represents the deviation from the SIS model [61] . This parameter can contributeupto 20% in the error of time-delay distance.
It is important to note that in order to put constraints on the cosmic curvature param-eter in a model-independent way, we have to calculate the comoving distances apart fromdistance ratio and time-delay distances in SGL systems. In this analysis, we replace thecomoving distance with the luminosity distance using the well-known relation
dA =dco
1 + z=
dL
(1 + z)2
3.3 Type Ia Supernovae
We use the latest sample of type Ia supernova to estimate the luminosity distance. Thisdataset (Pantheon) is the largest SN Ia sample consisting of the Joint Light-curve Analy-sis (JLA) and Pan-STARRS1 released the 1048 SNIa spectroscopically in the redshift range0.01 < z < 2.26 [62]. To determine the observed distance modulus, Scolnic et al. [62]performed the SALT2 [63] light curve fitter
µSN = mB(z) + α · X1 − β · C −MB
where mB is the rest frame B-band peak magnitude, MB represents absolute B-band magni-tude of a fiducial SN Ia with X1 = 0 and C = 0, X1 and C represent the time stretch of lightcurve and supernova colour at maximum brightness respectively. The stretch-luminosityparameter (α) and the colour-luminosity parameter (β) are calibrated to zero for the Pan-theon sample, hence the observed distance modulus reduces to µSN = mB −MB.For a standard cosmological system, the distance modulus can be defined as
µth = 5 log10 (dL/Mpc) + 25
– 6 –
Thus, we estimate the luminosity distance (dL) and uncertainty in the luminosity distance(σdL) for each SN Ia as
dL(z) = 10(µSN−25)/5 (Mpc) & σdL =ln(10)
5dLσµSN (Mpc) (3.4)
From Eq. (3.4) it is clear that the luminosity distance can be estimated by knowing the abso-lute magnitude of the supernovae (MB). It is usually accepted that Type Ia supernovaesample is normally distributed with a mean absolute magnitude of MB = −19.25 [64].Therefore, we use MB = −19.25 to calculate the luminosity distance and its uncertaintyfor each supernova.
The supernova data that we use for the luminosity distance is upto z = 2.26. However,the SGL data is up to a redshift z = 3.595. Hence, in order to include all datapoints of SGLdata in our analysis, we look for another standard candle which could help us estimate theluminosity distance at higher redshifts.
3.4 Gamma-Ray Bursts
Gamma Ray Bursts (GRBs) are highly energetic events that occur in the universe and canbe detected at a very high redshift due to their high luminosity. To date, the farthest GRB090429B [65] observed is at z = 9.4. GRBs are considered an effective tool to study theuniverse [66–69]. Several efforts have been made to establish distance measures using someempirical relations of distance-dependent quantities and observables of rest frames [70].We consider the relation between the isotropic equivalent gamma-ray energy Eγ,iso and theobserved photon energy of the peak spectral flux Ep,i [71, 72]
log(
Eγ,iso
1 erg
)= a log
[Ep,i
300keV
]+ b (3.5)
Ep,i = Ep,obs(1+ z) and a and b are, constants. Ep,i and Ep,obs are the spectral peak energy inthe cosmological rest frame of GRBs and in the observer’s frame respectively. On the otherhand, isotropic equivalent gamma-ray energy Eγ,iso can be calculated as
Eγ, iso =4πd2
L(z, p)Sbolo
(1 + z)(3.6)
where Sbolo is the bolometric gamma-ray flux and p represents the parameter sets, i.e. thebackground cosmological parameters. From Eq. (3.6), we can calculate the luminosity dis-tance for each GRB. To use GRBs as standard candles, this relation must be consistentlycalibrated [73–78]. In this work, we use the latest GRB sample having 162 datapoints uptoa redshift 9.4 [79]. However our SGL data is upto a redshift of 3.6. Therefore, we drop theGRBs which are above z = 3.6. Hence we are left with 147 GRBs upto a redshift 3.6.
To summarize, DSR is modified to accommodate SGL observations such as distance ra-tio and time delay distance (See Eqs. (3.2,3.3)). As we mentioned earlier, we need luminositydistance corresponding to each SGL observations, so we use SN Ia and GRB datasets for thesame purpose. In order to match redshift of SGL observations and luminosity distance, we
– 7 –
fit a second order polynomial1 in a model-independent way on the SN Ia and GRB data. Forthe fitting we use all the datapoints of the SN Ia data2 and only 147 GRBs out of 162 (upto aredshift of 3.6). The second order polynomial we use is
dL(z) = d1z + d2z2
where d1 and d2 are two free parameters and are fitted using a Python based module lmfit3.We find d1 = 4227.53± 16.15 Mpc, d2 = 1996.29± 49.05 Mpc and cov(d1, d2) = −0.725.Figure 1 shows the fitting curve with the 1σ and 2σ regions along with a theoretical con-struction of luminosity distance based on the ΛCDM model.
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0z
0
10000
20000
30000
40000
50000
d L[M
pc]
Best fil line of PolynomialCDM Model
21GRB DatapointsPantheon Datapoints
Figure 1: Reconstruction of the Luminosity distance dL in Mpc from SN Ia and GRB datasetsupto redshift 3.6. The 68% and 95% confidence levels are represented by red and greenshaded regions respectively. Violet and blue points with the error bars represent SN Ia andGRBs datapoints respectively. A solid yellow line represents the luminosity distance for theΛCDM model with H0 = 74.03 km sec−1Mpc−1.
1A higher order polynomial fit doesn’t show a substantial deviation from the second order polynomial fit.2We ignore off-diagonal terms in the covariance matrix of the distance modulus and just focus on the statis-
tical errors.3https://github.com/lmfit/lmfit-py/
– 8 –
4 Results
The cosmological parameters and the lens profile model parameters are determined by max-imising the likelihood L ∼ exp
(−χ2/2
), where chi-square (χ2) is
χ2 (pC, pL) =n
∑i=1
(Dth (zi; pC)−Dobs (zi; pL))2
σD (zi)2 (4.1)
Here pC and pL represent the cosmological parameters and the lens profile parameters re-spectively. Dth and Dobs are the theoretical and observed quantities of interest, i.e. thedistance ratio and time-delay distance. Here n stands for total number of datapoints usedin the analysis. For distance ratio n = 161 and for time-delay distance, n = 12.
The two factors which contribute to the uncertainty of D, i.e. (σD) are the uncertaintyin the observables of the SGL systems (σSGL) and uncertainty in the luminosity distance(σSC)(subscript“SC” stands for standard candles). We assume that the two uncertainties, i.e.uncertainty of the SGL systems and uncertainty in the luminosity distance are uncorrelatedand therefore they add in quadrature; σ2
D = σ2SGL + σ2
SC.
It is important to note that, for the validity of Eq. (3.2), the conditions 1 + Ωk0
(H0d
ol
L /c)2≥
0 and 1 + Ωk0(
H0dos
L /c)2 ≥ 0 should hold. Therefore, based on the maximum luminosity
distance we have in our data (standard candles), we set a prior range of cosmic curvature inour Markov Chain Monte Carlo (MCMC) program as Ωk0 > −0.2. We also fix H0 = 74.03±1.42 km sec−1Mpc−1 observed from the Cepheid-supernova distance ladder [80] throughoutour analysis.
Distance Ratio: Constraint on Lens and Cosmological Parameters
Extended Power Law profile is described by two power law indices- power index of totalmass density of a lens (γ ) and the power index of the luminous density (δ). In this analysis,we discuss three different parametrisations of γ while δ is considered as a free parameter.The luminous density profile of lens is different from the profile of total mass-density (γ 6=δ).
Using Eqs. (3.1, 3.2), we can rewrite a theoretical distance ratio as
dth
R ≡d
ls
Ados
A=
√√√√1 + Ωk0
(H0d
ol
Lc (1 + zl)
)2
− dol
L (1 + zs)
dos
L (1 + zl)
√√√√1 + Ωk0
(H0d
os
Lc (1 + zs)
)2
(4.2)
The uncertainty in the theoretical distance ratio i.e. σdth
Rcan be calculated using the error
propagation in Eq. (4.2). On the other hand, the observed distance ratio (dobs
R ) is defined inEq. (2.2) and the corresponding uncertainty is calculated as
σdobs
R= d
obs
R
√√√√[( (γ− 1)σθE
θE
)2
+
(2σσ0
σ0
)2]
(4.3)
We assume that these two uncertainties, i.e. σdth
Rand σ
dobsR
are uncorrelated and therefore
they add in quadrature; σdR =
√(σ
dthR
)2+(
σdobs
R
)2. However, no error is assumed in θap.
– 9 –
P1: γI = γ0
In the first parametrisation, we consider γ as an arbitrary constant (γ0). The best fit val-ues of Ωk0 and lens profile parameters are shown in Table 1. The best fit value of Ωk0 =−0.007+0.117
−0.021. This suggests that a spatially flat universe is preferred at 68% confidencelevel. The values of total mass-density and luminous density lens profile are not equal,i.e. γ0 6= δ 6= 2, which indicates that the effect of dark matter is not negligible in early typegalaxies. We show 1D and 2D posterior distributions of Ωk0, γ0 and δ in figure 2.
Parameter Best value [68% C.L.]
Ωk0 −0.007+0.117−0.097
γ0 2.139+0.022−0.021
δ 2.265+0.146−0.194
Table 1: Results for EPL model with P1: the best-fit values of Ωk0, γ0 and δ with 68%confidence level.
ko = 0.01+0.120.10
2.08
2.12
2.16
2.20
0
0 = 2.14+0.020.02
0.0 0.2 0.4 0.6
ko
1.75
2.00
2.25
2.50
2.08
2.12
2.16
2.20
0
1.75
2.00
2.25
2.50
= 2.26+0.150.19
Figure 2: 1D and 2D posterior distributions of Ωk0, γ0 and δ for P1 parametrisation of theEPL Model.
From figure 2, one can clearly see that Ωk0 and γ0 are negatively correlated.
– 10 –
P2: γI I(z) = γ0 + γ1zl
In the second parametrisation, we consider γ as a function of the redshift. The best fit valuesof Ωk0 and lens profile parameters are given in Table 2.
Parameter Best value [68% C.L.]
Ωk0 −0.004+0.184−0.118
γ0 2.154+0.043−0.034
γ1 −0.037+0.075−0.094
δ 2.108+0.221−0.325
Table 2: Results for P2 parametrisation of the EPL model: The best fit values of Ωk0, γ0, γ1and δ with 68% confidence level.
The best fit value of Ωk0 is −0.004+0.184−0.118, which suggests that a spatially flat universe is
preferred at 68% confidence level. The estimated values of total mass density and luminousdensity profile of the lens are different. Further the results show a mild evolution of thetotal mass-density power index with redshift. The 1D and 2D posterior distributions of Ωk0,γ0, γ1 and δ are shown in figure 3.
k0 = 0.00+0.180.12
2.10
2.16
2.22
2.28
0
0 = 2.15+0.040.03
0.45
0.30
0.15
0.00
0.15
1
1 = 0.04+0.070.09
0.00
0.25
0.50
0.75
1.00
k0
1.75
2.00
2.25
2.50
2.10
2.16
2.22
2.28
0
0.45
0.30
0.15
0.00
0.15
1
1.75
2.00
2.25
2.50
= 2.11+0.220.32
Figure 3: 1D and 2D posterior distributions of Ωk0, γ0, γ1 and δ for P2 parametrisation ofEPL Model
– 11 –
The lens density profile parameters, i.e. γ0 and γ1 show negative correlation as shown infigure 3.
P3: γI I I(z) = γ0 + γ1zl
1 + zl
In the third parametrisation, we consider γ as a function of redshift which converges to γ0at high redshift. The best fit values of Ωk0 and lens profile parameters are given in Table 3.The best fit value of Ωk0 is −0.032+0.168
−0.104 and it again suggests that a spatially flat universe is
Parameter Best value [68% C.L.]
Ωk0 −0.032+0.168−0.104
γ0 2.163+0.066−0.052
γ1 −0.083+0.184−0.243
δ 2.064+0.265−0.353
Table 3: Results for P3 parametrisation of the EPL model: The best-fit values of Ωk0, γ0, γ1and δ with 68% confidence level.
preferred at 68% confidence level. The values of total mass density and luminous densityof lens are different. Our results seem to indicate mild evolution of the total mass densitypower index with redshift. 1D and 2D posterior distributions of Ωk0, γ0, γ1 and δ are shownin figure 4.
k0 = 0.03+0.170.11
2.08
2.16
2.24
2.32
2.40
0
0 = 2.16+0.070.05
0.9
0.6
0.3
0.0
0.3
1
1 = 0.08+0.180.24
0.00
0.25
0.50
0.75
k0
1.75
2.00
2.25
2.50
2.08
2.16
2.24
2.32
2.40
0
0.9 0.6 0.3 0.0 0.3
1
1.75
2.00
2.25
2.50
= 2.06+0.270.35
Figure 4: 1D and 2D posterior distributions of Ωk0, γ0, γ1 and δ for P3 parametrisation ofthe EPL Model.
Figure 4 shows negative correlation between lens density profile parameters γ0 and γ1.
– 12 –
Time-Delay Distance: Constraint on Distance Duality and Cosmological Parameters
An important relation in cosmology is the cosmic distance duality relation. This is a relationbetween the luminosity distance and angular diameter distance. It is parametrised by thedistance duality parameter η(z).
η(z) =dA(z)(1 + z)2
dL(z)(4.4)
Using Eqs. (3.1,3.3) and (4.4), we can rewrite the time-delay distance
dth
∆t ≡d
os
Adol
A
dlsA
=1
(1 + zl)
(1 + zl)
ηldolL
√√√√1 + Ωk0
(ηl H0d
ol
Lc (1 + zl)
)2
− (1 + zs)
ηsdosL
√√√√1 + Ωk0
(ηsH0d
os
Lc (1 + zs)
)2−1
(4.5)
Here dth
∆t represents the theoretical time delay distance and corresponding observable quan-tity can be obtain by measuring the image separation between the two images as given inEq. (2.4). In the above expression, ηl = η(zl) and ηs = η(zs). The uncertainty in the theoret-ical time delay distance can be calculated using the error propagation in Eq. (4.5). However,the observed time delay distance (d
obs
∆t ) is defined in Eq. (2.4) and corresponding to this,uncertainty is calculated as
σdobs
∆t= d
obs
∆t
√√√√√(σ∆t
∆t
)2+ 4
(θjσθj
θ2j − θ2
i
)2
+ 4
(θiσθi
θ2j − θ2
i
)2
+ Ξ2
(4.6)
where σθi ,σθj and σ∆t represent the uncertainty in the angular position of double images ofthe source and in the time delay respectively. For our analysis, we take Ξ = 0.2 as suggestedby Cao et al. [61]. We add these two uncertainties in quadrature because we assume these
uncertainties are uncorrelated σd∆t =
√(σdth
∆t
)2+(
σdobs∆t
)2.
In this work, we choose three different parametrisations of the distance duality parameter(η).
i) ηI = η0
For a redshift independent distance duality parameter (η0), the results with best fit valuesof each parameter are displayed in Table 4.
Parameter Best value [68% C.L.]
Ωk0 0.596+0.287−0.404
η0 0.828+0.055−0.045
Table 4: Best fit value of Ωk0 and η0 at 68% confidence level obtain from TDD data forηI = η0.
The best-fit value of curvature parameter is Ωk0 = 0.596+0.287−0.404 which indicates an open
universe but within a 95% confidence level a flat universe can be accommodated. Inter-estingly, for an arbitrary constant value of η, we obtain η0 = 0.828+0.055
−0.045 which indicates a
– 13 –
0
0
0 0
0
Figure 5: 1D and 2D posterior distributions of Ωk0 & η0 for ηI = η0
violation of the distance duality relation within 68% confidence level. The 1D and 2D poste-rior distributions of Ωk0 and η0 are shown in figure 5. It can be observed that the Ωk0 and η0are negatively correlated. Since the constraint on Ωk0 is quite weak therefore we concludethat this dataset is not sufficient to put a strong constraint on the curvature parameter.
ii) ηI I(z) = 1 + η1z
In this case, we consider a Taylor series expansion of the distance duality parameter to firstorder. The constraint on the curvature parameter and η1 are tabulated in Table 5.
Parameter Best value [68% C.L.]
Ωk0 0.050+0.077−0.037
η1 0.118+0.137−0.110
Table 5: Results from the time-delay distance observations with ηI I = 1+ η1z. Best fit valuesof Ωk0 and η1 with 68% confidence level.
The value of Ωk0 = 0.050+0.077−0.037 which is consistent with a flat universe at 68% confi-
dence level. We find no violation in the distance duality relation with η1 ∼ 0 within 68%confidence level. We show the 1D and 2D posterior distributions of Ωk0 and η1 in figure 6.
– 14 –
k0 = 0.05+0.080.04
0.15
0.30
0.45
0.60
k0
0.0
0.2
0.4
0.6
1
0.0 0.2 0.4 0.6
1
1 = 0.12+0.140.11
Figure 6: 1D and 2D posterior distributions of Ωk0 & η1 for ηI I = 1 + η1z
2D posterior behaviour between Ωk0 and η shows that the two are correlated. Also anon-zero value of η1 indicates a redshift evolution of the distance duality parameter.
iii) ηI I I(z) = 1 + η1z
1 + zFinally, we consider redshift evolution of the distance duality parameter which converges to1 at high redshifts. The constraint on the cosmic curvature parameter and η1 are tabulatedin Table 6.
Parameter Best value [68% C.L.]
Ωk0 0.146+0.209−0.107
η1 −0.418+0.227−0.192
Table 6: Results from the time-delay distance observations with ηI I I = 1 + η1z
1 + z. The
best fit values of Ωk0 and η1 with 68% confidence level.
In the third parametrisation, i.e. ηI I I(z), the obtain value of Ωk0 supports a flat universewhile the distance duality parameter value shows violation within 68% confidence level.The 1D and 2D posterior distributions of Ωk0 and η1 are shown in figure 7.
– 15 –
k0 = 0.15+0.210.11
0.2 0.4 0.6 0.8 1.0
k0
0.8
0.4
0.0
0.4
1
0.8 0.4 0.0 0.4
1
1 = 0.42+0.230.19
Figure 7: 1D and 2D posterior distributions of Ωk0 & η1 for ηI I I = 1 + η1z
1 + z
The non-zero value of η1 indicates a redshift evolution of the distance duality parame-ter.
5 Discussion and Conclusions
The Distance Sum Rule (DSR) along with the strong gravitational lens system is a powerfulastrophysical tool to probe the curvature of the universe and galaxy parameters withoutassuming any fiducial cosmological model. We use DSR in two different ways. In the firstpart, we apply DSR method with distance ratio introduced by Rasanen et al. to measurethe cosmic curvature parameter along with the galaxy parameters in a model independentway [3]. In the second part, we again apply the DSR method to study the Cosmic DistanceDuality Relation (CDDR) using time-delay distances data. We separately discuss the resultsof both the approaches.
5.1 Method I: Distance Ratio
In earlier studies, the distance ratio obtain from SGL was used in a model-dependent way toconstrain different cosmological and lens parameters [8–10]. However, in the distance sumrule, the distance ratio is used in a model-independent way to constrain the cosmic curva-ture parameter along with lens galaxy parameters [11–14]. Following the same methodol-ogy, we use the latest SGL sample containing 161 datapoints for the distance ratio part. Toinclude the full datatset of SGL systems in our analysis, we reconstruct the distance-redshift relationby including the SN Ia and GRBs data and fit a second order polynomial to obtain the luminosity
– 16 –
distance at the lens and source redshift of the SGL systems. Further to explore the nature of thelens (galaxy) profile, we consider the Extended Power Law lens model which allows us touse different profiles of the total mass density and luminous mass density of the lens. Forthis lens profile, we extend our work by considering the evolution of total mass density power indexwith redshift, something we believe has not been done in the EPL model.Our main conclusions are listed below-
• For the completeness of the work, we use DSR along with distance ratio data with theSingular Isothermal Sphere (SIS) and the Power Law Spherical (PLS) lens profiles. Forthe SIS profile, we obtain Ωk0 = 0.680+0.144
−0.136 and fe = 1.034+0.006−0.006. Our constraint on
Ωk0 is incompatible with the Planck result. Recently, Wang et al. found the constrainton Ωk0 = 0.39+0.22
−0.30 and fe = 1.0195± 0.0090 for the SIS profile with a full dataset ofdistance ratio systems [14]. Our results are in concordance with the results of Wang etal. [14]. For the PLS lens profile, we find the constraint on the cosmic curvature pa-rameter Ωk0 = −0.052+0.054
−0.050 which is consistent with a flat universe at 68% confidencelevel. The best fit values of lens parameters, γ0 and γ1, are 2.107+0.018
−0.020 and−0.371+0.088−0.062
respectively, which are again consistent with the results of Wang et al. [14] along withmany other authors [6, 9, 13]. Our results indicate that with cosmic time, the totaldensity profile of early-type galaxies can evolve.
• Extended Power Law (EPL) lens model along with DSR method involves three pa-rameters: cosmic curvature parameter and two lens parameters, γ (power index oftotal mass-density lens profile) and δ (power index of luminous density lens pro-file). We consider three different parametrisations of γ. In the first parametrisation,γI(z) = γ0 = constant while in other two, γ varies as a function of redshift. Forthe first parametrisation, we find that the best fit value of cosmic curvature param-eter which indicate closed universe and also support a spatially flat universe at 68%confidence level. The best fit values of γ0 and δ are 2.139 and 2.265 respectively indi-cating that the mass distribution of dark matter is different from the mass distributionof luminous matter. Earlier studies also indicated a difference in the mass distributionof dark matter and luminous matter [12–14]. Hence such analysis could be helpful inunderstanding the nature of baryonic and dark matter at large scales.
• We also consider the possibility that the power index of the total mass density varieswith redshift. We assume the evolution of γ with redshift in two different forms:γI I(zl) = γ0 + γ1zl , and γI I I(zl) = γ0 + γ1zl/(1 + zl). Both parametrisations of γindicate that there is a marginal evolution of γ with redshift. For early galaxies, γ(z)and δ are not identical. This might indicate that the distribution of dark matter andbaryonic matter is not the same. In both the parametrisations of γ, the best fit valuesof Ωk0 indicate a closed universe but a spatially flat universe is also accommodated at68% confidence level. Also for both γI I and γI I I , the posterior distribution contoursof cosmic curvature and lens parameters are very similar, suggesting that limits oncurvature parameter are not significantly affected by the choice of parametrisation ofγ.
• The 1D and 2D posterior contour plots in curvature and lens profile parameter spaceindicate a strong correlation between them. Further, we find that the parameters ofthe lens profile are correlated among themselves.
– 17 –
5.2 Method II: Time-Delay Distance
In the second part of our analysis, we test the validity of the Cosmic Distance Duality Rela-tion (CDDR) based on the DSR method, as the validation of CDDR is important in moderncosmology. Any strong evidence of the violation of this relation could hint at the emergenceof new physics. In the past, CDDR had already been validated [20–31]. In the earlier work,DSR had been used with distance ratio however, we believe this is the first time, the DSRmethod has been modified to accommodate time-delay distance data in order to check thevalidity of CDDR. In this work, we use this method to put bounds on the cosmic curvaturealong with CDDR considering 12 datapoints of the time-delay distance data. Using thisdataset, we put constraints on Ωk0 and the distance duality parameter. We also considerredshift evolution of the distance duality parameter.A brief summary of the results is as follows
• We consider three parametrisations of η. In the first parametrisation, we take η to beindependent of redshift (ηI = η0), while in the other two, we consider η as evolvingwith redshift in two different ways: ηI I(z) = 1 + η1z and ηI I I(z) = 1 + η1z/(1 + z).For the first parametrisation, we obtain Ωk0 = 0.596+0.287
−0.404 which suggests that a spa-tially flat universe is preferred at 95% confidence level. The value of η comes out tobe 0.828+0.055
−0.045, indicating a violation in CDDR at 68% confidence level. In the secondparametrisation, the obtain value of Ωk0 supports a flat universe at 68% confidencelevel while the value of distance duality parameter shows no violation 68% confidencelevel. Further in the third parametrisation, again the obtain value of Ωk0 is consistentwith a flat universe within 68% confidence level however the distance duality param-eter shows a violation at 68% confidence level. Various parametrisations have beenused in earlier studies and have found no violation in the CDDR at 68% confidencelevel [25, 29, 31]. However, our analysis differs in the way that we use the DSR methodto study Ωk0 and η, which has never been done before.
To study the expansion history of the universe at high redshifts, Gamma-Ray Bursts(GRBs) as standard candles are used beyond the existing reach of SN Ia observations. Nev-ertheless, the large dispersion in Ep,i − Eiso correlation limits the precision of distance de-termination with GRBs. Due to this reason, the use of GRBs as standard candles is highlycontroversial. Hence, in order to put strong constraints on cosmological parameters, weshould look for more accurate luminosity relationships and investigate the classificationproblem of GRBs.
Our analysis indicates the constraint on the cosmic curvature parameter is strongly de-pends on the choice of the lens model of a galaxy. In the distance ratio method (with EPLlens model), best fit value of cosmic curvature parameter indicate a spatially closed uni-verse and for time delay distance method (with SIS lens model), a spatially open universeis preferred. But a spatially flat universe is accommodated at 95% confidence level for bothmethods. We expect that the ongoing and future surveys will provide more data on SGLsystems which will further help improve the constraints on the cosmological parameters aswell as lens profile parameters.
Recently, based on the Broken Power Law (BPL) density profile, Du et al. have devel-oped an analytic model for the lensing mass of galaxies [81]. In their analysis, they claimthat their model’s high efficiency and accuracy is a promising method for analyzing galaxy
– 18 –
properties with strong lensing. Therefore, by studying BPL density profile of lens galaxywith observations, one may put a better constraint on cosmic curvature parameter.
Acknowledgement
DK is supported by an INSPIRE Fellowship under the reference number: IF180293, DSTIndia. NR acknowledges facilities provided by the ICARD, University of Delhi.
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