Ignacy Sawicki CEICO, Institute of Physics, Prague Cosmology in the Time of Dark Energy Lecture 2 Ignacy Sawicki CEICO, Institute of Physics, Prague ceico

Previously on… Acceleration is there under weak assumptions Image credit: NASA and ESA Acceleration is there under weak assumptions C.c. or not c.c.? Degeneracy between DM and DE 𝑤 not observable, only 𝐻 𝑧 physical Today: inhomogeities tell you about the theory

The background is the playground SDSS/BOSS Planck (2015)

Linear Theory: Superposition How to Test Gravity? Newtonian gravity Continuous media: ∇ 2 Φ 𝑁 =−4𝜋 𝐺 𝑁 𝜌 Φ N 𝑥 =∫ 𝑑 3 𝑥 ′ 𝜌 𝑥 ′ 𝑥− 𝑥 ′ Spherical symmetry: Φ 𝑁 =− 𝐺𝑀 𝑟 𝑟 =−∇ Φ 𝑁 Closed Orbits Linear Theory: Superposition

How to Test Gravity? General Relativity: Schwarzschild unique* spherical vacuum solution Weak field ( Φ 𝑁 ≪1): 𝑔 00 ≈− 1−2 Φ 𝑁 +2 Φ 𝑁 2 𝑔 𝑖𝑗 = 1+2 Φ 𝑁 𝛿 𝑖𝑗 Geodesic motion Massive: 𝑢 𝜈 ∇ 𝜈 𝑢 𝜇 =0 (1,0,0,0) accelerated by 𝑔 00 Null: 𝑘 𝜈 ∇ 𝜈 𝑘 𝜇 =0 (1,1,0,0) accelerated by 𝑔 00 , 𝑔 𝑖𝑗

How to Test Gravity? PPN Extension Weak field ( Φ 𝑁 ≪1): Will (1971), Norvedt (1972) How to Test Gravity? Weak field ( Φ 𝑁 ≪1): 𝑔 00 =− 1−2 Φ 𝑁 +2𝛽 Φ 𝑁 2 𝑔 𝑖𝑗 = 1+2 𝛾Φ 𝑁 𝛿 𝑖𝑗 PPN Extension Lensing of light Δ𝜃=2 1+𝛾 𝐺𝑀 𝑏 Orbits 𝐺𝑀 𝑟 𝛾−1< 2.1±2.3 ⋅ 10 −5

What do we know about gravity? Baker, Psaltis, Skordis (2015) Chart shows curvature and potential All our tests are in coloured regions. Have tested gravity to 1 in 10^5 there (PPN formalism of Will etc) Know nothing much about extremely low curvatures: cosmology allows us to probe these PPN was for isolated and static spherical bodies. Now have FRW: time dependent. Need to redo

Inhomogeneities 10 metric components Origin of inhomogeneities d 𝑠 2 =− 1+2Ψ 𝑥,𝑡 d 𝑡 2 + 𝐵 𝑖 𝑥,𝑡 d𝑡d 𝑥 𝑖 + + 𝑎 2 𝑡 1−2Φ(𝑥,𝑡) 𝛿 𝑖𝑗 + ℎ 𝑖𝑗 (𝑥,𝑡) d 𝑥 𝑖 d 𝑥 𝑗 10 metric components 4 removed by diffeomorphism invariance (gauge choice; here Newtonian) ℎ 𝜇𝜈 → ℎ 𝜇𝜈 + 𝜕 (𝜇 𝜉 𝜈) 2 tensors ℎ 𝑖𝑗 – real dynamical d.o.f. – gravitational waves 2 vector polarisations 𝐵 𝑖 – frame dragging: Einstein eqs constraints, decay without source (e.g. cosmic strings) 2 scalars Φ and Ψ: Einstein eqs constraints. Dynamics from EMT Origin of inhomogeneities Inflaton creates FRW background which can carry fluctuations (scalar) Exponential expansion leads to universal solution for scalar and tensor modes CMB implies amplitude of scalar fluctuations √ Δ 𝜁 2 ∼ 10 −5 , tensors much less Evolution then driven by theory of gravity

The Perturbed FLRW Lightcone 𝜒 𝑧 𝑡

GR/ΛCDM: Scalars sourced by matter 3 𝐻 2 =8𝜋𝐺𝜌 Einstein: 𝑘 2 Φ=4𝜋𝐺𝜌 𝛿+3𝐻𝜌𝑣 Ψ−Φ=8𝜋𝐺𝜎 Matter Conservation 𝛿 +2𝐻𝛿= 𝑘 2 Ψ≈ 3 2 𝐻 2 Ω 𝑚 𝛿 𝑘 2 Ψ+Φ =3 𝐻 2 Ω 𝑚 𝛿

Matter Power Spectrum 𝜁𝜁 ∼ Δ 𝜁 2 Adiabatic Transfer Fn: 𝛿(𝑎) DES (2015) 𝜁𝜁 ∼ Δ 𝜁 2 Adiabatic Transfer Fn: 𝛿(𝑎) Δ 2 = 𝑘 3 𝑃(𝑘) 𝛿(𝑥)𝛿(𝑥) 8 = 𝜎 8 2 𝑘 2 Φ= 𝑘 2 Ψ= 𝐻 2 Ω 𝑚 𝛿

“Modify Gravity” Einstein: ∇ 2 Φ=−4𝜋𝐺 𝜇 𝜂 𝜌 𝛿+3𝐻𝜌𝑣 𝜂Ψ−Φ=8𝜋𝐺𝜎≈0 Matter Conservation 𝛿 +2𝐻𝛿= 𝑘 2 Ψ=4𝜋𝐺𝜇𝜌𝛿 𝑘 2 Ψ+Φ =4𝜋𝐺𝜌 1+𝜂 𝜇𝛿 c.f. 𝛾 PPN

Observables in the Late Universe

Massive Probes: Galaxy Surveys SDSS/BOSS Determine position and redshift of galaxies (and type) Pixelise and count deviations from average Δ 𝒏,𝑧 ≡ 𝑁 𝒏,𝑧 − 𝑁 𝑧 𝑁 𝑧 What does it mean? ~ 3 million galaxies

Correlation Function 2500 deg2 up to 𝑧=0.7 SDSS/BOSS SDSS/BOSS DR10 400000 thousand positions and spectra Bump: BAO same as CMB. Map out d_A Normalisation arbitrary: galaxy bias 2500 deg2 up to 𝑧=0.7

But bias 𝛿 𝑔 𝛿 𝑔 = 𝑏 2 𝛿 𝑚 𝛿 𝑚 +⋯

Galaxies Redistributed 𝜌 𝒙 obs d 𝑉 obs = d𝑁 gal =𝜌 𝒙 d𝑉 𝜌 𝑧 +𝛿𝜌 𝒏,𝑧 𝜌 𝑧 d 𝑉 obs = 𝜌 𝑧 +𝛿𝜌 𝒏 , 𝑧 𝜌 𝑧 d𝑉 𝜌 𝑧 +𝛿𝜌 𝒏,𝑧 𝜌 𝑧 − 𝜌 ,𝑧 𝛿𝑧 = 1+𝛿 𝒏 , 𝑧 d𝑉 d 𝑉 obs Δ 𝒏,𝑧 =𝛿 𝒏 , 𝑧 + d𝑉 d 𝑉 obs −1− 𝜌 ,𝑧 𝜌 𝑧 𝛿𝑧 Δ 𝒏,𝑧 =𝛿 𝒏 , 𝑧 + 𝛿𝑉 𝑉 − 3𝛿𝑧 1+ 𝑧

Effects of Metric Fluctuations d 𝑠 2 = 𝑎 2 (𝜂) −(1+2Ψ)d 𝜂 2 +(1−2Φ)(d 𝜒 2 + 𝜒 2 d Ω 2 ) 𝜕𝑉 𝜕 𝑉 obs =1+ 𝜕𝜒 𝜕 𝜒 obs + 𝜕Ω 𝜕 Ω obs 1+𝑧≡ 𝑢 𝜇 𝑘 𝜇 S 𝑢 𝜈 𝑘 𝜈 O 𝑢 𝜇 =(1−Φ, 𝒗 ) 𝜒 obs =𝜒 𝑧 ≃𝜒 𝑧 + 𝜒 ,𝑧 𝛿𝑧= =𝜒 𝑧 + 𝛿𝑧 𝐻 𝑧 =𝑎( 𝑡 O )/𝑎( 𝑡 S ) 𝛿𝑧= d𝜒 Φ + Ψ + + 𝒏 ⋅ 𝒗 ​ S + Φ S + Ψ S + − 𝒏 ⋅ 𝒗 ​ O − Φ O − Ψ O 𝜕 𝜒 obs 𝜕𝜒 ≃1+ 1 𝐻 𝜕 𝜒 𝒏 ⋅ 𝒗 + 𝐻 𝐻 2 𝒏 ⋅ 𝒗 Redshift perturbation, 𝛿𝑧 Volume perturbation, 𝛿𝑉

What Counting Measures Bonvin & Durrer (2011) Physical counts Δ g 𝒏,𝑧 = 𝛿 𝑔 𝒏 , 𝑧 −2Φ+Ψ+ + 1 𝐻 Φ − 𝜕 𝜒 𝒏 ⋅ 𝒗 + Redshift-space distortions + 𝐻 𝐻 2 + 2 𝜒 S 𝐻 Ψ S + 𝒗 ⋅ 𝒏 S + 0 𝜒 S d𝜒 Φ + Ψ Doppler + ISW Convergence 𝜅 + 1 𝜒 S 0 𝜒 S d𝜒 2− 𝜒 𝑆 −𝜒 𝜒 𝛻 ⊥ 2 (Φ+Ψ) Shapiro + Shear

Relative Importance Galaxy correlations ΛCDM, 𝑙=20 Bonvin & Durrer (2011) Galaxy correlations ΛCDM, 𝑙=20

Redshift-Space Distortions Kaiser (1985) 𝛿 𝑔 𝑧 = 𝛿 𝑔 − 1 𝐻 𝑛 𝑖 𝜕 𝑖 ( 𝑛 𝑗 𝑣 𝑗 ) 𝜃 𝑘 𝑛 𝛿 𝑔 𝑧 𝛿 𝑔 𝑧 = 𝑏 2 𝛿 𝑚 𝛿 𝑚 − 1 𝐻 𝜇 2 𝑏 𝛿 𝑚 𝜃 𝑔 + 1 𝐻 2 𝜇 4 𝜃 𝑔 𝜃 𝑔 𝜇≡ cos 𝜃 Real space Redshift space 𝜃= 𝜕 𝑖 𝑣 𝑖 𝛿 𝑚 ≡𝐻𝑓 𝛿 𝑚 WEP: 𝑣 𝑖 𝑔 = 𝑣 𝑖 𝑚 = 𝜕 𝑖 Ψ Energy conservation: 𝛿 𝑚 + 𝜃 𝑚 ≈0 𝛿 𝑔 𝑧 𝛿 𝑔 𝑧 = 𝑏 2 − 1 𝐻 𝜇 2 𝑏𝑓+ 1 𝐻 2 𝜇 4 𝑓 2 𝜎 8 2 Extract 𝑓 𝜎 8 (which actually is 𝜃 𝑔 )

Redshift-Space Distortions Real space Redshift space Samushia et al. (2013)/BOSS SDSS/BOSS DR10

Growth Rate, 𝑓 𝜎 8 Planck 2018

Caveat Emptor Bose et al. (2017)

Massless Probes: Cosmic Shear

Weak Lensing d 𝑠 2 =−(1+2Ψ)d 𝑡 2 + 𝑎 2 (𝑡) (1−2Φ)(d 𝜒 2 + 𝜒 2 d Ω 2 ) Following Bernardeau, Bonvin, Vernizzi (2009) d 𝑠 2 =−(1+2Ψ)d 𝑡 2 + 𝑎 2 (𝑡) (1−2Φ)(d 𝜒 2 + 𝜒 2 d Ω 2 ) 𝜅: convergence 𝛾 𝑖 : shear 𝜔: rotation, 𝒪(2) 𝐽 𝑏 𝑎 = 𝑑 A (1−𝜅)𝛿 𝑏 𝑎 + 𝑑 A − 𝛾 1 − 𝛾 2 −𝜔 − 𝛾 2 +𝜔 𝛾 1 M. White 𝛾 1 = d𝜒 𝜒 S −𝜒 𝜒 𝜕 𝑖 𝜕 𝑗 Φ+Ψ 𝑒 1 𝑖 𝑒 1 𝑗 − 𝑒 2 𝑖 𝑒 2 𝑗 𝛾 2 =2 d𝜒 𝜒 S −𝜒 𝜒 𝜕 𝑖 𝜕 𝑗 Φ+Ψ 𝑒 1 𝑖 𝑒 2 𝑗 Alwau 𝑑 A 𝜅=−2 d𝜒 𝜒 S −𝜒 𝜒 𝛻 ⊥ 2 Φ+Ψ −2 Φ+Ψ −2 Ψ S 𝜒 S + 𝑑 𝐴 ( 𝜅 v + 𝜅 ISW ) Only depends on lensing potential Φ+Ψ

Weak Lensing Kilbinger (2014)

WL Sensitivity Kilbinger (2014)

WL: Current Status Troxel et al. (2018) 𝜎 8 Φ+Ψ

Latest Constraints on MG params

Can we remove dependence on ICs? Amendola, Kunz, Motta, Saltas, IS (2013) Both lensing shear are growth rate as integrated quantities dependent on ICs 𝐴 𝑖𝑗 ∼ 𝑑𝜒 𝐾 𝜒 ∇ 2 (Φ+Ψ) 𝑓 𝜎 8 ∼∫ ∇ 2 Ψ In GR, all 𝛿 𝑚 , Φ and Ψ related through constraint. One random variable In general – you don’t know. Take derivatives Lensing tomography Something similar with growth rate Form ratios of Φ and Ψ: measure 𝜂 in a model-independent manner

The Nearing Future

Projected DESI Expansion Rate

Measuring shear in next generation wide field cosmic shear surveys

The Takeaway Dark energy is not going away ΛCDM fits, but if you are optimistic, there may be some tensions 𝐻 0 local vs global Growth rates? It could well end up being other physics Massive neutrinos can have similar effects Caveat emptor: All cosmological probes sensitive only to gravity; cannot say anything direct about composition Only in GR is DM overdensity the same as grav. potential