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

2. 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

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

4. 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

5. 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

6. 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

7. 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

8. 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

9. The Perturbed FLRW Lightcone𝜒 𝑧 𝑡

10. 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

11. 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

12. “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

13. Observables in the Late Universe

14. 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

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

16. But bias
𝛿 𝑔 𝛿 𝑔 = 𝑏 2 𝛿 𝑚 𝛿 𝑚 +⋯

17. 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

18. 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

19. 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 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

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

21. 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 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 )

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

23. Growth Rate, 𝑓 𝜎 8
Planck 2018

24. Caveat Emptor
Bose et al. (2017)

25. Massless Probes: Cosmic Shear

26. 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

27. Weak Lensing
Kilbinger (2014)

28. WL Sensitivity
Kilbinger (2014)

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

30. Latest Constraints on MG params

31. 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

32. The Nearing Future

34. Projected DESI Expansion Rate

35. Measuring shear in next generation wide field cosmic shear surveys

37. 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