1. Cosmology with gravitational lensing
Title Page Template 2
Probing the surface mass density of the intervening lens system.
Presented by: Marick Manrho

2. Why use weak gravitational lensing techniques?
Introduction
Why use weak gravitational lensing techniques?
Figure 1. Images of the merging ‘bullet’ cluster 1E
Unbiased for type of matter
Source: Clowe, D., Bradac, M., Gonzalez, A. H., et al. 2006, A Direct Empirical Proof of the Existence of Dark Matter, ApJ, 648, L109

3. OUTLINE Introduction to gravitational lensing
Cosmology with gravitational lensing
OUTLINE
Introduction to gravitational lensing
2D mass surface density reconstruction
Measurment problems
Results
Future developments
Conclusions

4. Introduction to gravitational lensing
Cosmology with gravitational lensing
Introduction to gravitational lensing
2D mass surface density reconstruction
Measurement problems
Results
Future developments
Conclusions

5. Geometry and lens potential
1. Introduction to gravitational lensing
Cosmology with gravitational lensing
Geometry and lens potential
Thin lens approximation
𝜙 𝜃 = 1 𝜋 Σ 𝑐𝑟𝑖𝑡 𝑑 2 𝜃 ′ Σ 𝜃 ′ ln 𝜃− 𝜃 ′
Σ 𝜃 =𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝑚𝑎𝑠𝑠 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑙𝑒𝑛𝑠
Σ 𝑐𝑟𝑖𝑡 = 𝑐 2 𝐷 𝑠 4𝜋𝐺 𝐷 𝐿 𝐷 𝐿𝑆
Figure 2. Geometry of a lens system
Source: Krishnavedala (Own work) [CC0], via Wikimedia Commons

6. Amplification of source image
1. Introduction to gravitational lensing
Cosmology with gravitational lensing
Amplification of source image
Gravitational lensing amplifies source image
Amplification is split into two contributions
Magnification 𝜅
Shear 𝛾
𝐴= 1 1−𝜅 2 − 𝛾 2
Convergence 𝜅
Shear (ellipticity) 𝛾
𝜅= 1 2 𝛻 2 𝜙= Σ 𝜃 Σ 𝑐𝑟𝑖𝑡
𝛾= 𝛾 1 +𝑖 𝛾 2
𝜙=𝑃𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝑓𝑖𝑒𝑙𝑑
Σ 𝜃 =𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝑚𝑎𝑠𝑠 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑙𝑒𝑛𝑠

7. Shear (ellipticity) 1. Introduction to gravitational lensing
Cosmology with gravitational lensing
Shear (ellipticity)
𝐼𝑚 𝛾
𝛾= 𝛾 1 +𝑖 𝛾 2
𝛾 1 = 𝜕𝜙 𝜕 𝜃 1 𝜕 𝜃 1 − 𝜕𝜙 𝜕 𝜃 2 𝜕 𝜃 2 = 𝐷 1 𝜙
𝛾 2 = 𝜕𝜙 𝜕 𝜃 1 𝜕 𝜃 2 = 𝐷 2 𝜙
𝑅𝑒(𝛾)
Figure 3. The effect of shear distortions on a circular source. 𝑒 𝑡 and 𝑒 𝑟 are the real and imaginary parts of the ellipticity (or shear).
Source: van Waerbeke L., Mellier Y., Gravitational Lensing by Large Scale Structures: A Review. Proceedings of Aussois Winter School, astroph/ (2003)

8. 2D mass surface density reconstruction
Cosmology with gravitational lensing
Introduction to gravitational lensing
2D mass surface density reconstruction
Measurement problems
Results
Future developments
Conclusions

9. Relating convergence and shear
2. 2D mass surface density reconstruction
Cosmology with gravitational lensing
Relating convergence and shear
𝜅 𝜃 − 𝜅 0 ∝ 1 𝑛 𝛼 𝛾 1 +𝛽 𝛾 2 𝜃 2
Up to a constant 𝜅 0 (mass sheet degeneracy)
Applying directly gives much noise – smoothing of field is needed!
Several smoothing options excists. For example smoothing the shear field does the trick.

10. Measuring shear 2. 2D mass surface density reconstruction
Cosmology with gravitational lensing
Measuring shear
Measured ellipticity depends on source ellipticity
𝑒= 𝑒 𝑠 +𝛾 1+ 𝛾 ∗ 𝑒 𝑠
𝑒 𝑠 =𝑆𝑜𝑢𝑟𝑐𝑒 𝑒𝑙𝑙𝑖𝑝𝑡𝑖𝑐𝑖𝑡𝑦
Assuming the sources are randomly oriented
𝑒 ≅𝛾
Galaxies randomly
distributed
Slight alignment
Figure 4.
Source: E Grocutt, IfA, Edinburgh

11. E- and B-mode shear 2. 2D mass surface density reconstruction
Cosmology with gravitational lensing
E- and B-mode shear
Assumption that galaxies are randomly oriented might not be true
Only E-mode shear is predominantly produced by weak gravitational lensing.
By making lensing potential complex one can filter out high B-mode sources.
Figure 5. Seperation of shear in E- and B-mode contributions.
Source: van Waerbeke L., Mellier Y., Gravitational Lensing by Large Scale Structures: A Review. Proceedings of Aussois Winter School, astroph/ (2003)

12. E- and B-mode shear 2. 2D mass surface density reconstruction
Cosmology with gravitational lensing
E- and B-mode shear
E-mode shear seems to follow predicted line.
Figure 6. E- and B-modes from an early analysis of CFHTLS data [7]. Top points are the E-modes and bottom the B-modes.
Source: Benjamin J., et al., MNRAS, 281, 792 (2007)

13. Measurement problems 3. Measurement problems
Cosmology with gravitational lensing
Introduction to gravitational lensing
2D mass surface density reconstruction
Measurement problems
Results
Future developments
Conclusions

14. Measurement problems 3. Measurement problems
Cosmology with gravitational lensing
Measurement problems
Point spread function (PSF)
Use unlensed stars to correct for PSF.
Figure 7. The true image is corrupted by many sources. Finding these sources is key for reconstructing the true image.
Source: H. Hoekstra, arXiv: , Weak gravitational lensing (2013)

15. Point spread function 3. Measurement problems
Cosmology with gravitational lensing
Point spread function
Lines represent the amount of ellipticity added by point spread function.
Figure 8. An example of the pattern of PSF anisotropy for MegaCam on CFHT
Source: Hoekstra H., Mellier Y., van Waerbeke L., Semboloni E., Fu L., Hudson M. J.,
Parker L. C., Tereno I. and Benabed K., ApJ, 647 (2006) 116.

16. Results 4. Results Cosmology with gravitational lensing
Introduction to gravitational lensing
2D mass surface density reconstruction
Measurement problems
Results
Future developments
Conclusions

17. Navarro-Frenk-White profile
4. Results
Cosmology with gravitational lensing
Navarro-Frenk-White profile
Is a model for cold dark matter (CDM)
Figure 9. Excess surface density from stacked galaxy clusters from the SDSS survey, with best-fitting NFW profiles. N200 is a measure of the richness of the clusters.
Source: Mandelbaum R., Seljak U., Hirata C.M., JCAP 8, 6 (2008)

18. Cosmological parameters
3. Measurement problems
Cosmology with gravitational lensing
Cosmological parameters
Ω 𝑚 = Mass density parameter
𝜎 8 = Amplitude of the matter power spectrum of an average sphere with a radius of 8 h −1 Mpc.
Figure 10. An example of the pattern of PSF anisotropy for MegaCam on CFHT
Source: Hoekstra H., Mellier Y., van Waerbeke L., Semboloni E., Fu L., Hudson M. J.,
Parker L. C., Tereno I. and Benabed K., ApJ, 647 (2006) 116.

19. Future developments 5. Future developments
Cosmology with gravitational lensing
Introduction to gravitational lensing
2D mass surface density reconstruction
Measurement problems
Results
Future developments
Conclusions

20. Future telescopes 5. Future developments
Cosmology with gravitational lensing
Future telescopes
2007
2016
2020
2023
CFHT (Canada France Hawai Telescope)
Pan-STARRS1
Euclid
LSST
Current source of data.
First data-set released december 2016 and the next is scheduled this month.
Space telescope specially designed to probe dark matter.
Large Synoptic Survey Telescope, promises excellent image quality.

21. Next 5. Future developments Cosmology with gravitational lensing
3D density reconstruction using precision redshift measurement of courses.
Lift degeneracy between 𝜎 8 and Ω 𝑚
Measure equation of state dark matter

22. Conclusions 6. Conclusions Cosmology with gravitational lensing
Introduction to gravitational lensing
2D mass surface density reconstruction
Measurement problems
Results
Future developments
Conclusions

23. Cosmology with gravitational lensing
6. Conclusions
Cosmology with gravitational lensing
Lensing amplifies source
Amplification due to convergence and shear
Determine shear
Average over multiple sources
Select E-mode shear
Sources might be intrinsically aligned
Reconstruct mass density
Which enables search for dark matter and provides estimates of cosmological parameters.

24. Relating convergence and shear
Additional slides
Cosmology with gravitational lensing
Relating convergence and shear
𝜅 𝜃 − 𝜅 0 ∝ 1 𝑛 𝑔 𝛾 1 𝜃 𝑔 cos 2 𝜃 𝑔 −𝜃 + 𝛾 2 𝜃 𝑔 sin 2 𝜃 𝑔 −𝜃 𝜃 𝑔 −𝜃 2
Up to a constant 𝜅 0 (mass sheet degeneracy)
Applying directly gives much noise – smoothing of field is needed!
Several smoothing options excists. For example smoothing the shear field does the trick.

25. Bibliography Additional slides Cosmology with gravitational lensing
Clowe, D., Bradac, M., Gonzalez, A. H., et al., ApJ, 648, L109 (2006)
van Waerbeke L., Mellier Y., Gravitational Lensing by Large Scale Structures: A Review. Proceedings of Aussois Winter School, astroph/ (2003)
Martin Kilbinger, Cosmology with cosmic shear observations: a review , arXiv: v2 (2015)
H. Hoekstra, Weak gravitational lensing, arXiv: v1 (2013)
Alan Heavens, Cosmology with gravitational lensing, arXiv: v1 (2011)
E Grocutt, IfA, Edinburgh (Figure 4)
Benjamin J., et al., MNRAS, 281, 792 (2007)
Mandelbaum R., Seljak U., Hirata C.M., JCAP 8, 6 (2008)
Hoekstra H., Mellier Y., van Waerbeke L., Semboloni E., Fu L., Hudson M. J.,
Parker L. C., Tereno I. and Benabed K., ApJ, 647, 116 (2006)