Constraining cosmic curvature by using age of galaxies and gravitational lenses Deepak Jain Deen Dayal Upadhyaya College University of Delhi INDIA Collaborators: Akshay Rana (DU), Shobhit Mahajan (DU), Amitabha Mukherjee (DU) Conference on Shedding Light on the Dark Universe with Extremely Large Telescopes 2-6 July 2018
Outl tline ine Ou Outline ne 1. Introduction 2. Test of FLRW metric : Using cosmic chronometers 3. Test of curvature : Using the mean image separation of strong gravitational lenses 4. Discussion
Int ntroduction oduction ▪ The spatial curvature is one of the most fundamental issue of modern cosmology ▪ Estimation of curvature of the Universe ( Ω 𝑙0 ) is directly linked with ▪ The validity of FLRW metric, ▪ Degeneracy with dark energy equation of state parameter ▪ Cosmic inflation and ▪ The ultimate fate of the Universe. ▪ The recent constraint on curvature (| Ω 𝑙0 |< 0.005) was obtained by the newest Planck 2015 observations.
Tes est t of FLRW RW me metri tric ▪ FLRW metric represents the homogeneous and isotropic Universe at sufficiently large scales 𝑒𝑠 2 𝑒𝑡 2 = −𝑑 2 𝑒𝑢 2 + 𝑏 2 𝑢 1 − 𝑙𝑠 2 + 𝑠 2 𝑒𝜄 2 + 𝑠 2 𝑡𝑗𝑜 2 𝜄𝑒𝜒 2 where k = 1, 0, − 1 for closed, flat and open geometry of the space. ▪ In FLRW Universe, the transverse comoving distance can be written as 𝑨 𝑒𝑨 −𝑙𝑑 2 𝑑 𝐸 𝑑 (𝑨) = 𝑇 |Ω 𝑙0 | න 𝑥ℎ𝑓𝑠𝑓 Ω 𝑙0 = 2 𝑏 02 & 𝐼 𝑨 = 𝐼 0 𝐹(𝑨) 𝐹(𝑨) 𝐼 0 𝐼 0 |Ω 𝑙0 | 0 ▪ By taking the first derivative of comoving distance, we can redefine 𝐼(𝑨) 2 𝐸 𝑑 ′2 −𝑑 2 Ω 𝑙0 = Clarkson et.al. (2008) 𝐼 02 𝐸 𝑑2 ▪ For FLRW metric to hold, estimate of present curvature density derived using observables at different redshift must remain constant.
Tes est t of FLRW RW met etri ric ▪ To check the consistency of this relation, we need an Independent datasets of ▪ comoving distance 𝐸 𝑑 ▪ its first derivative 𝐸 𝑑 ′ ▪ And Hubble parameter 𝐼 𝑨 ▪ Calculation of transverse comoving distance −1 −1 𝑒𝑢 𝐼 𝑨 = (1 + 𝑨) 𝑒𝑨 we obtain, 𝑑 0 (1 + 𝑨 ′ ) 𝑒𝑢 𝑒𝑨 ′ 𝑒𝑨 ′ 𝑡𝑗𝑜ℎ 𝐼 0 |Ω 𝑙0 | 𝑔𝑝𝑠 Ω 𝑙0 > 0 𝑨 𝐼 0 |Ω 𝑙0 | 0 (1 + 𝑨 ′ ) 𝑒𝑢 𝑒𝑨 ′ 𝑒𝑨 ′ 𝐸 𝑑 = 𝑑 𝑔𝑝𝑠 Ω 𝑙0 = 0 𝑨 𝑑 0 (1 + 𝑨 ′ ) 𝑒𝑢 𝑒𝑨 ′ 𝑒𝑨 ′ 𝑡𝑗𝑜 𝐼 0 |Ω 𝑙0 | 𝑔𝑝𝑠 Ω 𝑙0 < 0 𝑨 𝐼 0 |Ω 𝑙0 |
Tes est t of Homog mogeneit eneity: : Da Data taset ets ▪ Age of galaxies dataset 32 old and passive galaxies (0.11 < z < 1.84). Incubation time 𝑢 𝑗𝑜𝑑 = 1.50 ± 0.45 𝐻𝑧𝑠 (Wei et al.2015) Present age of universe, 𝑢 0 = 13.799 ± 0.021 𝐻𝑧𝑠 (Planck 2015) Polynomial fit ▪ 𝑢 = 𝐵 + 𝐶𝑨 + 𝐷𝑨 2 𝑒𝑢 𝑒𝑨 = 𝐶 + 2𝐷𝑨 ▪ For a flat universe 𝑨 2 2𝑨 3 − 𝐷(𝑨 2 + 𝐸 𝑑 = 𝑑 −𝐶 𝑨 + 3 ) 2 𝜓 2 𝑠𝑓𝑒𝑣𝑑𝑓𝑒 = 0.61 ′ = 𝑑 −𝐶 1 + 𝑨 − 2𝐷(𝑨 + 𝑨 2 ) 𝐸 𝑑
Tes est t of Homog mogeneity eneity: : Da Data tase set ▪ Hubble dataset • H(z) data consisting of 38 data points in the redshift range (0.07 < z < 2.36).
Tes est t of Homog mogeneit eneity: : Da Data taset et ▪ Hubble dataset • H(z) data consisting of 38 data points in the redshift range (0.07 < z < 2.36). • To avoid the extrapolation of age function, we restrict analysis till z< 1.84 . (3 points removed)
Tes est t of Homog mogeneity eneity: : Da Data tase set ▪ Hubble dataset ▪ H(z) data consisting of 38 data points in the redshift range (0.07 < z < 2.36). ▪ To avoid the extrapolation of age function, we restrict analysis till z< 1.84 . (3 points removed) ▪ Further after removing H(z) points derived from Age datasets, we finally left with 27 data points. (8 points removed)
Tes est t of Homog mogeneity eneity: : Da Data tase set ▪ Hubble dataset ▪ H(z) data consisting of 38 data points in the redshift range (0.07 < z < 2.36). ▪ To avoid the extrapolation of age function, we restrict analysis till z< 1.84. (3 points removed) ▪ Further after neglecting H(z) points derived from Age datasets, we finally left with 27 data points. (8 points removed) (0.07 < z < 1.37).
Tes est t of Homog mogeneit eneity: : Res esult Error propagation in Ω 𝑙0 ▪ 2 2 2 2 2 2 𝜏 𝐸𝑑′ 𝜏 𝐸𝑑 𝜏 𝐼0 𝑑 𝜏 𝐼 2 = 4 (Ω 𝑙0 ) 2 + 4 Ω 𝑙0 + 𝜏 Ω 𝑙0 + + 𝐸 𝑑′ 𝐸 𝑑 𝐼 0 𝐼 0 𝐸 𝑑 𝐼 ′ respectively Where 𝜏 𝐼 0 , 𝜏 𝐼 , 𝜏 𝐸 𝑑 and 𝜏 𝐸 𝑑′ are the error in 𝐼 0 , H(z), 𝐸 𝑑 and 𝐸 𝑑 ▪ On applying model independent non-parametric smoothing technique Gaussian Process
Tes est t of Homog mogeneit eneity: : Res esult 𝐼 0 ( 𝑙𝑛 𝑡𝑓𝑑 −1 𝑁𝑞𝑑 −1 ) Ω 𝑙0 73.24 ± 1.74 0.025 ± 0.57 68 ± 2.8 0.036 ± 0.62 ▪ The reconstructed plot completely encloses Ω 𝑙0 =0 within 1 𝜏 confidence level and remains constant along z ▪ This shows consistency with the assumption of homogeneity of the universe and also concordance with the FLRW metric
Tes est t of Cu Curvatur ture: e: Usi sing g St Strong ng Gravi vita tational tional Len enses ses Strong gravitational lensing (SGL) is a phenomenon in which light ▪ coming from a distant source get distorted to form multiple images of source in the presence of a foreground galaxy. ▪ Statistical istical prop oper erties ties of SGL GL Let us assume that lensing galaxies are non evolving with • comoving density 𝑜 0 • Let effective Einstein radius of the lens is given by 𝑏 𝑑𝑠 . The differential probability d𝜐 of a beam of light to interact with • uniformly distributed lenses at redshift z would be Image credits: (Credit: ESA / NASA /JPL-Caltech / Keck / SMA) 𝑒𝜐 = 𝑜 0 1 + 𝑨 𝑀 3 𝜏 𝑑𝑒𝑢 where 𝜏 = 𝜌𝑏 𝑑𝑠2 𝑒𝑨 𝑀 𝑒𝑨 𝑀 𝑑𝑒𝑢 𝑑 1 𝑒𝑨 𝑀 = and Ω 𝑛 (1+𝑨 𝑀 ) 3 +Ω Λ +(1−Ω 𝑛 −Ω Λ )(1+𝑨 𝑀 ) 2 𝐼 0 (1+𝑨 𝑀 ) • The total probability of interaction 𝑨 𝑡 𝑒𝜐 𝜐 = 𝑒𝑨 𝑀 𝑒𝑨 𝑀 0
Tes est t of Cu Curvatur ture: e: Using g St Strong ng Gravita tational tional Len enses es On integrating over the full range of 𝑨 𝑀 we get the overall mean image ▪ separation 𝑨 𝑡 Δ𝜄 1 𝑒𝜐 2𝑏 𝑑𝑠 < Δ𝜄 >= 𝜐 𝑒𝑨 𝑀 𝑒𝑨 𝑀 where Δ𝜄(𝑨 𝑀 ) = 0 𝐸 𝑃𝑀 For SIS mass profile of lens galaxy with velocity dispersion ( 𝑤 ) ▪ 2 𝐸 𝑃𝑀 𝐸 𝑀𝑇 𝑏 𝑑𝑠 = 4𝜌 𝑤 Credits: http://www.jb.man.ac.uk/distance/frontiers/glens/section2.html 𝑑 𝐸 𝑃𝑇 On solving ▪ 3 𝐸 𝑃𝑀 2 (1 + 𝑨 𝑀 ) 2 𝑨 𝑡 𝐸 𝑀𝑇 𝑒𝑨 𝑀 3 𝐹(𝑨) < Δ𝜄(𝑨 𝑡 ) > 0 𝐸 𝑃𝑇 = 2 𝐸 𝑃𝑀 2 (1 + 𝑨 𝑀 ) 2 Δ𝜄 0 𝑨 𝑡 𝐸 𝑀𝑇 𝑒𝑨 𝑀 2 𝐹(𝑨) 0 𝐸 𝑃𝑇 2 𝑤 Where, Δ𝜄 0 = 8𝜌 𝑑 ▪ For the singular isothermal sphere (SIS) lensing galaxies, the mean image separation is completely independent of the source redshift for the all FLRW based cosmological models having flat Universe i.e. Ω 𝑙0 = 0
Tes est t of Cu Curvatur ture: e: Da Data taset et ▪ We use the final statistical sample of lensed quasars from the SDSS Quasar Lens Search (SQLS). The SDSS DR7 quasar catalog consists of 26 lenses in a well-defined statistical sample and 36 additional lenses identified by various techniques Select ection ion Criteri terion on and proced cedur ure ▪ ➢ Limits the number of source images to two. ➢ Maximum image separation between two images should be less than 4′′. ➢ After applying the selection criteria, out of 62 lensing systems, we are finally left with 44 galaxy lenses. ➢ For calculating the mean image separation first we divide this dataset in a redshift bin-size of 0.3 and 0.5 respectively and determine the mean value of Δ𝜄 in each interval.
Tes est t of Cu Curvatur ture: e: Res esul ults ts
Tes est t of Cu Curvatur ture: e: Res esul ults ts ➢ Statistical tests to check the correlation between image separation (Δ𝜄) and source redshift, 𝑨 𝑡 . Statistical tests Values Spearman's rank coefficient 𝜍 0.22 ± 0.09
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