Cosmic curvature in astronomical measurements

Cosmic curvature in astronomical measurements Krzysztof Bolejko The University of Sydney 21 September 2017, Astrofizyka Cząstek

Outline Historical overview Evidence for spatial flatness of our Universe Emerging spatial curvature

Über das zulässige Krümmungsmaß des Raumes, Vierteljahrschrift d. Astronom. Gesellschaft. 35, 337-47 (1900) As has recently been shown by Professor Seeliger, the most rational view of the arrangement of stars that one can build on the basis of current observations, is that all visible stars, whose number can be estimated to be no greater than 40 million, can be considered to lie within a space of a few hundred AU, and that outside this is a relative void. Even if this view is reassuring, by offering us a significant step in our understanding of the Universe through a complete investigation of this limited stellar system, this reassurance, so satisfying to reason, would be experienced to an even greater degree if we could Karl Schwarzschild conceive of space as being closed and finite and filled, more or less completely by this stellar system. Class. Quantum Grav., 15 2539 (1998)

G ab =T ab Annalen der Physik, 354, 769 (1916) original manuscript available at http://new.alberteinstein.info Albert Einstein

G ab Λ g ab =T ab Königlich Preußischen Akademie der Wissenschaften, 142 (1917) Albert Einstein

Willem de Sitter Proceedings of the Royal Netherlands Academy of Arts and Sciences, 20 I, 229 (1918)

Über die Krümmung des Raumes Zeitschrift für Physik, 10, 377 (1922) original manuscript available at https://www.lorentz.leidenuniv.nl Alexander Friedman Gen. Rel. Grav. 31, 1991, 1999

Über die Moglichkeit einer Welt mit konstanter negativer Krümmung des Raumes Zeitschrift für Physik, 21, 326 (1924) Alexander Friedmann Gen. Rel. Grav. 31, 2001, 1999

Edwin Hubble

Georges Lemaître K >0 Annals of the Scientific Society of Brussels, 47A, 41 (1927)

Edwin Hubble Hubble, PNAS, 15, 168 (1929)

Hubble, PNAS, 15, 168 (1929)

Hubble, PNAS, 15, 168 (1929) Λ=0 K =0

Fred Hoyle K =0

Fl at Un ive rs e

Emerging spatial curvature Buchert, Carfora, Class. Quant. Grav. 25, 195001 (2008)

Millennium universe Springel, Frenk & White, Nature, 440, 1137 (2006)

G ab Λ g ab=t ab

G ab Λ g ab =T ab T ab =ρua ub + p hab +π ab +q a ub +ua q b

G ab Λ g ab =T ab conservation equation T ab ;b =0

G ab Λ g ab =T ab conservation equation ab ρ +Θ(ρ+ p)+σ π ab +q a a ;a +q A a =0

G ab Λ g ab =T ab conservation equation ab ρ +Θ(ρ+ p)+σ π ab +q Ricci identities a a ;a +q A a =0 u a ; d ; c ua ; c ; d =R abcd u b

G ab Λ g ab =T ab conservation equation ab ρ +Θ(ρ+ p)+σ π ab +q a a ;a +q A a =0 Ricci identities Θ = 13 Θ2 12 (ρ+3 p) 2(σ 2 ω2 )+ D a A a + A a A a +Λ σ ab = 32 Θ σ ab σ c a σ c b ω a ωb + D a A b + A a A b E ab + 12 π ab 2 1 ω a = 3 Θ ω a 2 curl A a +σ ab ωb

G ab Λ g ab =T ab conservation equation ab ρ +Θ(ρ+ p)+σ π ab +q a a ;a +q A a =0 Ricci identities Θ = 13 Θ2 12 (ρ+3 p) 2(σ 2 ω2 )+ D a A a + A a A a +Λ σ ab = 32 Θ σ ab σ c a σ c b ω a ωb + D a A b + A a A b E ab + 12 π ab 2 1 ω a = 3 Θ ω a 2 curl A a +σ ab ωb Bianchi identities R ab[cd ; e]=0

G ab Λ g ab =T ab conservation equation ab ρ +Θ(ρ+ p)+σ π ab +q a a ;a +q A a =0 Ricci identities Θ = 13 Θ2 12 (ρ+3 p) 2(σ 2 ω2 )+ D a A a + A a A a +Λ σ ab = 32 Θ σ ab σ c a σ c b ω a ωb + D a A b + A a A b E ab + 12 π ab 2 1 ω a = 3 Θ ω a 2 curl A a +σ ab ωb Bianchi identities E ab = Θ E ab 12 (ρ+ p)σ ab +curl H ab 12 π ab 16 Θ π ab + 3 σ c a ( E b c 16 π b c ) +ϵcd a 2 A c H bd ωc ( E b d + 12 π db ) [ ] H ab = Θ H ab curl E ab + 12 curl π ab +3 σ c a H b c ϵcd a ( 2 A c E db +ω c H bd )

G ab Λ g ab =T ab conservation equation ab ρ +Θ(ρ+ p)+σ π ab +q a a ;a +q A a =0 Ricci identities Θ = 13 Θ2 12 (ρ+3 p) 2(σ 2 ω2 )+ D a A a + A a A a +Λ ω ab =0 A a =0 qa= 0 1 2 c σ ab = 3 Θ σ ab σ c a σ b ω a ωb + D a A b + A a A b E ab + 2 π ab 2 1 ω a = 3 Θ ω a 2 curl A a0 +σ ab ωbπ ab =0 p= H ab =0 Bianchi identities E ab = Θ E ab 12 (ρ+ p)σ ab +curl H ab 12 π ab 16 Θ π ab + 3 σ c a ( E b c 16 π b c ) +ϵcd a 2 A c H bd ωc ( E b d + 12 π db ) [ ] H ab = Θ H ab curl E ab + 12 curl π ab +3 σ c a H b c ϵcd a ( 2 A c E db +ω c H bd )

Silent Cosmology ρ = Θ ρ 1 2 1 2 Θ = 3 Θ 2 ρ 6 Σ + Λ 2 2 Σ = 3 Θ Σ+ Σ W 1 W = Θ W 2 ρ Σ 3 Σ W Bruni, Matarrese, Pantano, Astroph. J. 445, 958 (1995) van Elst, Uggla, Lesame, Ellis, Maartens, Class. Q. Grav. 14, 1151 (1997)

FLRW Cosmology ρ = Θ ρ 1 2 1 Θ = 3 Θ 2 ρ+ Λ Σ 0, shear free W 0, conformally flat

Millennium universe Springel, Frenk & White, Nature, 440, 1137 (2006)

Silent Cosmology ρ = Θ ρ 1 2 1 2 Θ = Θ ρ 6 Σ + Λ 3 2 2 2 Σ = Θ Σ + Σ W 3 1 W = Θ W ρ Σ 3 Σ W 2

Silent Cosmology ρi = ρ + Δ ρ= ρ (1+ δi ) 1 Θi = Θ + Δ Θ= Θ (1 3 δi ) 1 1 Σi = 3 Δ Θ= 9 Θ δi 1 W i = 6 ρ δi Bolejko, arxiv:1708.0940 (2017)

Silent Universe Simulation Bolejko, arxiv:1708.0940 (2017)

Evolution of the volume Millennium Millennium Millennium Bolejko, arxiv:1708.0940 (2017)

Evolution of the cosmic parameters ium n n lle i M Mi lle nn ium Millennium Bolejko, arxiv:1708.0940 (2017)

Low-redshift constraints on curvature Rasanen, Bolejko, Finoguenov, Phys. Rev. Lett. 115, 101301 (2015)

Low-redshift constraints on curvature Future data: 10,000 supernovea and 10,000 lensed galaxies Current data: 580 supernovea and 30 lensed galaxies Rasanen, Bolejko, Finoguenov, Phys. Rev. Lett. 115, 101301 (2015)

Measuring the curvature Bolejko, arxiv:1707.01800 (2017)

JLA Supernova constraints Fla t Un ive rs e Betoule, Kessler, Guy, et al. Astron. & Astroph. 568, A22, (2014)

Hubble constant Planck (2015) Riess (2016) 60 65 70 75 80 85 Riess, Macri, Hoffmann et al, Astrophys. J. 826, 56 (2016)

H0 tension resolved Planck high-redshift H 0 =67.81 ±0.92 km s 1 Mpc 1 Riess et al. low-redshift H 0 =73.24 ±1.74 km s 1 Mpc 1

H0 tension resolved Planck high-redshift Planck low-redshift Riess et al. low-redshift H 0 =72.38 ±1.74 km s 1 Mpc 1 H 0 =73.24 ±1.74 km s 1 Mpc 1

Summary

Silent Cosmology ρ = Θ ρ 1 2 1 2 Θ = Θ ρ 6 Σ + Λ 3 2 2 2 Σ = Θ Σ + Σ W 3 1 W = Θ W ρ Σ 3 Σ W 2 FLRW limit: Σ 0, W 0

Bolejko, arxiv:1708.0940 (2017)

Brief history of spatial curvature -300: Euclid's Elements 1813: beginning of non-euclidean geometry 1900: astronomical constraints on spatial curvature 1917: cosmological models with K > 0 1923: extragalactic astronomy 1929: expanding Universe 1932-1990s: Λ = 0, K = 0 1997-2010s: Λ > 0, K = 0 2008-2017: theoretical suggestions for Λ > 0, K < 0 2022-2025: Euclid measures K, K < 0 enters standard cosmology