6
1 Introduction and Summary
we introduce the mathematical structures upon which our formalism is built. These
can be organized in three categories, each one of them being associated with a given
manifold. More precisely, we discuss the fields, equations and symmetries associated
with the space-time manifold in Sect. 3.2, those associated with the line manifold on
which geodesics are defined in Sect. 3.3 and those associated with the phase space
on which particle distributions are defined in Sect. 3.4. In Chap. 4 we then present
the observer space-time formalism for observables from localized sources, which is
constructed using the structures introduced in Sect. 3.3, and conclude by building the
distance, weak lensing and number count maps in Sect. 4.7. Finally, in Chap. 5 we
provide the relevant derivations and discussions for general-relativistic matrix kinetic
theory, based on the structures introduced in Sect. 3.4. In Sect. 5.10 we provide, as a
concrete example, the lowest-order collision term for a fluid of photons, electrons and
protons, including the full spin/polarization information, and in Sect. 5.11 we define
the CMB observable maps using the spectral observer sky construction of Sect. 4.3.
We use natural units 8πG = c = = 1 and a space-time signature of mostly pluses.
References
1. E. Komatsu, K. M. Smith, J. Dunkley, C. L. Bennett, B. Gold, G. Hinshaw et al., Seven-year
Wilkinson Microwave Anisotropy Probe (WMAP) observations: cosmological interpretation.
Astrophys. J. Suppl. Ser. 192 , 18 (2011). arXiv:1001.4538
2. Planck Collaboration et al., Planck 2018 results. VI. Cosmological parameters (2018).
arXiv:1807.06209
3. Planck Collaboration et al., Planck 2018 results. I. Overview and the cosmological legacy of
Planck (2018). arXiv:1807.06205
4. SDSS collaboration, Cosmological constraints from the SDSS luminous red galaxies. Phys.
Rev. D74, 123507 (2006). https://doi.org/10.1103/PhysRevD.74.123507, arXiv:0608632
5. M. Levi, C. Bebek, T. Beers, R. Blum, R. Cahn, D. Eisenstein et al., The DESI Experiment, a
whitepaper for Snowmass 2013 (2013). arXiv:1308.0847
6. C.W. Stubbs, D. Sweeney, J.A. Tyson, LSST Collaboration, An overview of the large synoptic
survey telescope (LSST) system, in American Astronomical Society Meeting Abstracts, vol. 36
of Bull. Am. Astron. Soc. (2004), p. 108.02
7. J. Green, P. Schechter, C. Baltay, R. Bean, D. Bennett, R. Brown et al., Wide-Field InfraRed
Survey Telescope (WFIRST) final report. arXiv:1208.4012
8. P.E. Dewdney, P.J. Hall, R.T. Schilizzi, T.J. Lazio, The square kilometre array. IEEE Proc. 97,
1482 (2009)
9. R. Laureijs, J. Amiaux, S. Arduini, J. Auguères, J. Brinchmann, R. Cole et al., Euclid definition
study report (2011). arXiv:1110.3193
10. A. Challinor, A. Lewis, Lensed CMB power spectra from all-sky correlation functions. Phys. Rev. D71, 103010 (2005). https://doi.org/10.1103/PhysRevD.71.103010,
arXiv:astro-ph/0502425
11. A. Lewis, A. Challinor, Weak gravitational lensing of the CMB. Phys. Rept. 429, 1 (2006).
https://doi.org/10.1016/j.physrep.2006.03.002, arXiv:astro-ph/0601594
12. D. Hanson, A. Challinor, A. Lewis, Weak lensing of the CMB. Gen. Rel. Grav. 42, 2197 (2010).
https://doi.org/10.1007/s10714-010-1036-y, arXiv:0911.0612
13. A. Lewis, A. Challinor, D. Hanson, The shape of the CMB lensing bispectrum. JCAP 1103,
018 (2011). https://doi.org/10.1088/1475-7516/2011/03/018, arXiv:1101.2234
1 Introduction and Summary
we introduce the mathematical structures upon which our formalism is built. These
can be organized in three categories, each one of them being associated with a given
manifold. More precisely, we discuss the fields, equations and symmetries associated
with the space-time manifold in Sect. 3.2, those associated with the line manifold on
which geodesics are defined in Sect. 3.3 and those associated with the phase space
on which particle distributions are defined in Sect. 3.4. In Chap. 4 we then present
the observer space-time formalism for observables from localized sources, which is
constructed using the structures introduced in Sect. 3.3, and conclude by building the
distance, weak lensing and number count maps in Sect. 4.7. Finally, in Chap. 5 we
provide the relevant derivations and discussions for general-relativistic matrix kinetic
theory, based on the structures introduced in Sect. 3.4. In Sect. 5.10 we provide, as a
concrete example, the lowest-order collision term for a fluid of photons, electrons and
protons, including the full spin/polarization information, and in Sect. 5.11 we define
the CMB observable maps using the spectral observer sky construction of Sect. 4.3.
We use natural units 8πG = c = = 1 and a space-time signature of mostly pluses.
References
1. E. Komatsu, K. M. Smith, J. Dunkley, C. L. Bennett, B. Gold, G. Hinshaw et al., Seven-year
Wilkinson Microwave Anisotropy Probe (WMAP) observations: cosmological interpretation.
Astrophys. J. Suppl. Ser. 192 , 18 (2011). arXiv:1001.4538
2. Planck Collaboration et al., Planck 2018 results. VI. Cosmological parameters (2018).
arXiv:1807.06209
3. Planck Collaboration et al., Planck 2018 results. I. Overview and the cosmological legacy of
Planck (2018). arXiv:1807.06205
4. SDSS collaboration, Cosmological constraints from the SDSS luminous red galaxies. Phys.
Rev. D74, 123507 (2006). https://doi.org/10.1103/PhysRevD.74.123507, arXiv:0608632
5. M. Levi, C. Bebek, T. Beers, R. Blum, R. Cahn, D. Eisenstein et al., The DESI Experiment, a
whitepaper for Snowmass 2013 (2013). arXiv:1308.0847
6. C.W. Stubbs, D. Sweeney, J.A. Tyson, LSST Collaboration, An overview of the large synoptic
survey telescope (LSST) system, in American Astronomical Society Meeting Abstracts, vol. 36
of Bull. Am. Astron. Soc. (2004), p. 108.02
7. J. Green, P. Schechter, C. Baltay, R. Bean, D. Bennett, R. Brown et al., Wide-Field InfraRed
Survey Telescope (WFIRST) final report. arXiv:1208.4012
8. P.E. Dewdney, P.J. Hall, R.T. Schilizzi, T.J. Lazio, The square kilometre array. IEEE Proc. 97,
1482 (2009)
9. R. Laureijs, J. Amiaux, S. Arduini, J. Auguères, J. Brinchmann, R. Cole et al., Euclid definition
study report (2011). arXiv:1110.3193
10. A. Challinor, A. Lewis, Lensed CMB power spectra from all-sky correlation functions. Phys. Rev. D71, 103010 (2005). https://doi.org/10.1103/PhysRevD.71.103010,
arXiv:astro-ph/0502425
11. A. Lewis, A. Challinor, Weak gravitational lensing of the CMB. Phys. Rept. 429, 1 (2006).
https://doi.org/10.1016/j.physrep.2006.03.002, arXiv:astro-ph/0601594
12. D. Hanson, A. Challinor, A. Lewis, Weak lensing of the CMB. Gen. Rel. Grav. 42, 2197 (2010).
https://doi.org/10.1007/s10714-010-1036-y, arXiv:0911.0612
13. A. Lewis, A. Challinor, D. Hanson, The shape of the CMB lensing bispectrum. JCAP 1103,
018 (2011). https://doi.org/10.1088/1475-7516/2011/03/018, arXiv:1101.2234