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

Why use weak gravitational lensing techniques? Introduction Why use weak gravitational lensing techniques? Figure 1. Images of the merging ‘bullet’ cluster 1E0657-558 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

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

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

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

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

Shear (ellipticity) 1. Introduction to gravitational lensing Cosmology with gravitational lensing Shear (ellipticity) 𝐼𝑚 𝛾 𝛾= 𝛾 1 +𝑖 𝛾 2 𝛾 1 = 1 2 𝜕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/0305089 (2003)

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

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.

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

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/0305089 (2003)

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)

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

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:1312.5981, Weak gravitational lensing (2013)

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.

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

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)

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.

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

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.

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

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

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.

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.

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/0305089 (2003) Martin Kilbinger, Cosmology with cosmic shear observations: a review , arXiv:1411.0115v2 (2015) H. Hoekstra, Weak gravitational lensing, arXiv:1312.5981v1 (2013) Alan Heavens, Cosmology with gravitational lensing, arXiv:1109.1121v1 (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)