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

2. Rough Outline Main assumptions in cosmology Acceleration: how robust?
What can we know without GR?
Main aim: lay out the logic/assumptions behind what we do
20 July 2018
4th Cosmology School, UJ

3. Expt. Physics vs Cosmology
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4th Cosmology School, UJ

4. Making Sense of the Sky We need a metric, 𝑔 𝜇𝜈
𝑑 𝑠 2 = 𝑔 𝜇𝜈 𝑑 𝑥 𝜇 𝑑 𝑥 𝜈
We need a theory for light: Maxwell
−𝑔 𝐹 𝜇𝜈 𝐹 𝜇𝜈
𝐹 𝜇𝜈 = 𝜕 𝜇 𝐴 𝜈 − 𝜕 𝜈 𝐴 𝜇 𝐹 0𝑖 = 𝐸 𝑖 𝐹 𝑖𝑗 = 𝜖 𝑖𝑗𝑘 𝐵 𝑘
Maxwell equations: ∇ 𝜇 𝐹 𝜇𝜈 =0
20 July 2018
4th Cosmology School, UJ

5. Making Sense of the Sky Maxwell: 𝛻 𝜇 𝐹 𝜈 𝜇 =0
Eikonal: 𝐴 𝜇 = 𝐴 𝜇 (𝑥) 𝑒 𝑖𝑆/𝜖
ray vector 𝑘 𝜇 ≡ 𝜕 𝜇 𝑆
rays are null 𝑘 𝜇 𝑘 𝜇 =0
rays are geodsics 𝑘 𝜈 𝛻 𝜈 𝑘 𝜇 =0
Geodesic deviation:
D 2 𝜉 𝜇 d 𝜆 2 = 𝑅 𝜈𝛼𝛽 𝜇 𝜉 𝛽 𝑘 𝜈 𝑘 𝛼
20 July 2018
4th Cosmology School, UJ

6. Making Sense of the Sky Redshift determined by emitter/obs velocity
Distances: corrected by gravitational field
Luminosity 𝐹=𝐿/4𝜋 𝑑 L 2
Angular diameter 𝜃=ℓ/ 𝑑 A
If photons conserved:
𝑢 em 𝜇 𝑘 𝜇 𝑢 obs 𝜇 𝑘 𝜇 = 𝜔 em 𝜔 obs ≡1+𝑧
Redshift
𝑑 L = 1+𝑧 2 𝑑 A
20 July 2018
4th Cosmology School, UJ

7. Ordering by redshift Dark energy is irrelevant by redshift 2-3
Image credit: Ivo Labbé
Dark energy is irrelevant by redshift 2-3
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4th Cosmology School, UJ

8. Use Standard Candles to Map
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4th Cosmology School, UJ

9. SN Ia: Universal Objects
Image credit: D. Butler
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4th Cosmology School, UJ

10. Use Standard Candles to Map
Broader is intrinsically brighter
Standardise to some (unknown) intrinsic luminosity
Obtain luminosity distance as function of redshift 𝑑 L (𝑧)
To interpret need a model
Universe (ICs)
Composition
Gravity
20 July 2018
4th Cosmology School, UJ

11. First Attempt at a Model
The Copernican Principle: here is not special
Isotropy Homogeneity
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4th Cosmology School, UJ

12. First Attempt at a Model
The Copernican Principle: here is not special
Isotropy Homogeneity
Planck (2015)
WiggleZ: Scrimgeur et al. (2012)
20 July 2018
4th Cosmology School, UJ

13. First Attempt at a Model
ICs: The Copernican Principle: here is not special (Friedmann-Robertson-Walker metric)
Redshift for light emitted and absorbed by comoving observers is ratio of 𝑎’s:
1+𝑧= 𝑎 obs / 𝑎 em
Distance-redshift relation a function of 𝐻(𝑧) only
d 𝑠 2 =d 𝑡 2 − 𝑎 2 𝑡 𝛿 𝑖𝑗 d 𝑥 𝑖 d 𝑥 𝑗
𝑢 com 𝜇 = 𝑑 𝑥 𝜇 𝑑𝜏 = 1 𝟎
𝑑 L = 1+𝑧 d𝑧 𝐻 𝑧
𝐻 𝑎 = 𝑑 ln 𝑎 𝑑𝑡
Hubble
parameter
20 July 2018
4th Cosmology School, UJ

14. First Attempt at a Model
Composition: slowly-moving or virialised objects
Pressure ~ K.E. ~ 𝜌 𝜎 2 ~ 𝑘𝑇∼𝜌 Φ 𝑁
Mass conservation
𝜌∝ 𝑎 −3
𝑤= 𝑝 𝜌 ∼ Φ 𝑁 ∼ 10 −(9÷5)
Late universe comprises mostly non-relativistic matter sources (dust)
20 July 2018
4th Cosmology School, UJ

15. First Attempt at a Model
Theory of gravity: General Relativity
Monica Witt
𝐻 2 ≡ 𝑎 𝑎 = 8𝜋 𝐺 N 3 𝜌
𝐺 𝜇𝜈 = 𝑅 𝜇𝜈 − 1 2 𝑔 𝜇𝜈 𝑅= 𝑇 𝜇𝜈
20 July 2018
4th Cosmology School, UJ

16. First Attempt at Model: FAIL
From D. Huterer
Dimmer
Flat and yellow lines parallel after z>1.5: DE is not around anymore
20 July 2018
4th Cosmology School, UJ

17. Conservation of Energy
Energy Momentum Tensor for continuous media
𝑇 𝜇𝜈 =𝜌 𝑢 𝜇 𝑢 𝜈 +𝑝( 𝑔 𝜇𝜈 + 𝑢 𝜇 𝑢 𝜈 ) + impf
Conservation of EMT from coordinate invariance (not a property of gravity)
∇ 𝜇 𝑇 𝜈 𝜇 =0
On FRW: 𝜌 +3𝐻 𝜌+𝑝 =0
If 𝑤= 𝑝 𝜌 =const 𝜌∝ 𝑎 −3 1+𝑤
W=-1 => constant
20 July 2018
4th Cosmology School, UJ

18. How to increase distances?
Data need acceleration! ⇒ Dark Energy
𝑑 L = 1+𝑧 d𝑧 𝐻 𝑧
𝑎 𝑎 =− 𝜌+3𝑝 >0
𝑤<− 1 3
𝜌 +3𝐻 𝜌+𝑝 =0
3 𝐻 2 =𝜌
20 July 2018
4th Cosmology School, UJ

19. Second Attempt: ΛCDM ⇒ NOBEL ‘12
Planck 2018
𝑤 Λ =−1
𝜌 Λ =const
(stars 0.4%, gas 4.6%)
20 July 2018
4th Cosmology School, UJ

20. Is the Universe fooling us?
Light dims/SNe evolve (“tired light”)
𝒅 𝑨 matches 𝒅 𝑳 : photons conserved (BAO)
Non-linear “averaging problem”
Averaging isolated halos does not give background with 𝑤=0?
True, but no prescription gets more than 1% effect
Here is special (“inhomogenous universe”)
20 July 2018
4th Cosmology School, UJ

21. The FLRW Lightcone 𝜒 𝑧 𝑡
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4th Cosmology School, UJ

22. The inhomogeneous alternative
Only the lightcone need be homogeneous
Lemaître-Tolman-Bondi universe d 𝑠 2 =−d 𝑡 2 + 𝑎 ∥ 2 𝑡,𝑟 d 𝑟 2 + 𝑎 ⊥ 2 𝑡,𝑟 𝑟 2 d Ω 2
This is the metric of a spherical density profile
Observer at 𝑟=0, sees lightcone as homogeneous and isotropic
Construct arbitrary 𝑑(𝑧) by picking a density profile 𝜌(𝑟,𝑡)
But can see inside lightcone! Break degeneracy since far-away observers live in anisotropic universe (kinetic Sunyaev-Zel’dovich effect)
20 July 2018
4th Cosmology School, UJ

23. Is the Universe fooling us?
Acceleration detected at 64σ
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4th Cosmology School, UJ

24. Summary 1: Acceleration
Distances purely cosmography:
Need EM and metric
No theory of gravity
Assumption 1: sources do not evolve in time
Copernican principle: we are not in a special place
This is enough to show there is acceleration
To predict/interpret need to models
Composition: late universe is made of slowly moving stuff
Theory of gravity to link stuff to metric
20 July 2018
4th Cosmology School, UJ

25. So, what is it?
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26. The cosmological constant
Adding one number ( Ω Λ =0.69), essentially makes all observations consistent
Represents a constant energy density of the vacuum 𝜌 Λ ∼ 10 −3 eV 4
𝜌 =−3𝐻(𝜌+𝑝)
GR knows about absolute energy, not just differences (normal-ordering in QFT)
What do we expect?
Cut-off: 𝑀 Pl = eV.
This is an example of a hierarchy problem (similar to Higgs mass)
𝑇 𝜇𝜈 =Λ 𝑔 𝜇𝜈 with Λ∼∫ 𝑑 3 𝑘 𝜔 𝑘 2 ∼ 𝑘 max 4 c.f. 𝑛 ℏ𝜔
20 July 2018
4th Cosmology School, UJ

27. The cosmological constant
e.g. Jerome Martin (2012)
Why is this a problem?
There are lots of infinities in QFT: just add counter-terms
In QED, you have 𝑚 bare + 𝑚 rad , replace with measurement
But GR non-renormalizable: every order needs new measurements
Any solutions?
With “symmetry” certain operators do not get generated!
Smarter calculation Λ∼ 𝑚 4 log 𝑚 𝜇 (if everything is massless then ok)
But: setting Λ=0 seems as difficult as setting Λ= 10 −12 eV 4
20 July 2018
4th Cosmology School, UJ

28. Maybe it’s not stuff, but gravity?
Baker, Psaltis, Skordis (2015)
Binary PSR
Solar System
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
20 July 2018
4th Cosmology School, UJ

29. Tenets of Modifying Gravity
Accept Copernican Principle: universe is FRW with small fluctuations
Let go of Einstein equations: replace them with what?
Accept standard inflationary initial conditions: why is this here?
20 July 2018
4th Cosmology School, UJ

30. Initial Conditions vs Transfer Fns
Einstein + Boltzmann eqs
Verde & Peiris (2008)
Planck (2015)
Pratten (2016)
Δ 𝜁 and 𝑛 𝑠
Einstein + Boltzmann eqs
20 July 2018
4th Cosmology School, UJ

31. Tenets of Modifying Gravity
Accept Copernican Principle: universe is FRW with small fluctuations
Let go of Einstein equations: replace them with what?
Accept standard inflationary initial conditions: why is this here?
20 July 2018
4th Cosmology School, UJ

32. Modify Friedmann Eq
The most general thing you can write down? 3 𝐻 2 = 𝜌 𝑚 + 𝜌 𝑋 + conservation (coordinate invariance) 𝜌 𝑋 +3𝐻 𝜌 𝑋 1+𝑤 =0
This is not a model…
20 July 2018
4th Cosmology School, UJ

33. Measure Distances: Cosmography
Supernovae
𝑑= d𝑧 𝐻 0 𝐻 𝑧
𝜇(𝑧)=5 log 𝑑 L (𝑧) + 𝜇 0 +𝐾
𝐻(𝑧)/ 𝐻 0
Image: HSR
Baryon Acoustic Oscillations
𝑑 A 𝑧 = 𝐻 0 𝑟 BAO Θ(𝑧)
𝐻(𝑧)/ 𝐻 0
Image: Z. Rostomian, LBNL
20 July 2018
4th Cosmology School, UJ

34. So we constrain the EoS. Or?
𝐻 2 = 𝐻 Ω m0 𝑎 −3 + Ω DE 𝑎 −3 1+𝑤
Planck: Ade et al. (2013)
Planck: Ade et al. (2013)
Planck + WP
Planck + WP + BAO w
𝑤≡ 𝑤 0 + 𝑤 𝑎 (1−𝑎)
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4th Cosmology School, UJ

35. But w is not an observable
Kunz (2007)
But w is not an observable
Distances only depend on 𝑑= d𝑧 𝐻 0 𝐻 𝑧
We measure geometry only
DM/DE split is ambiguous
Background info not enough to measure Ω m0
Need to use inhomogeneities, i.e. assume model for DE
Ω m0 =0
Ω m0 =0.2
𝐻 2 = 𝐻 Ω m0 𝑎 −3 + Ω Λ0
ΛCDM Ω m0 =0.3
𝐻 2 = 𝐻 Ω m0 𝑎 −3 + Ω DE0 𝑎 −3 1+𝑤
Ω m0 =0.4
20 July 2018
4th Cosmology School, UJ

36. Andrew Liddle
20 July 2018
4th Cosmology School, UJ

37. Dark Matter Dark Energy 20 July 2018 4th Cosmology School, UJ
Andrew Liddle
Dark Matter
Dark Energy
20 July 2018
4th Cosmology School, UJ

38. Dark Matter Dark Matter Dark Energy Clusters Unclustered 20 July 2018
Andrew Liddle
Dark Matter
Dark Matter
Dark Energy
Clusters
Unclustered
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4th Cosmology School, UJ

39. Dark Matter Dark Energy Clusters Clustering 20 July 2018
Andrew Liddle
Dark Matter
Dark Energy
Clusters
Clustering
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4th Cosmology School, UJ

40. Distances consistent with ΛCDM
Zhao et al. (2018) BOSS DR14
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4th Cosmology School, UJ

41. Distances consistent with ΛCDM
Planck (2018)
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4th Cosmology School, UJ

42. Measuring the Hubble rate today
W. Freedman
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4th Cosmology School, UJ

43. X Mexican School on Gravitation
H0 is not observable
Marra et al. (2013)
H0 is the expansion rate today in the homogeneous FRLW universe
Puts physical scale on all quantities
In FRLW, nearby comoving sources have 𝑑 L ≈ 𝑧 𝐻 0
Only depend on local expansion rate
Map nearby (𝑧<0.1) SNe to get 𝐻 0
But, we live in some local configuration of potential and have some velocity 𝛿 𝑧 O =− 𝒏 ⋅ 𝒗 ​ O − Φ O − Ψ O
Same effect on CMB temperature etc.
Measurements of H0/km s--1Mpc--1
Local (Riess et al): 73.8±2.4
CMB (Planck 2018): ±0.54
1--5 December 2014
X Mexican School on Gravitation

44. Summary 2: c.c. and background
The c.c. works as dark energy, but it’s hard to deal with in QFT (no c.c. similar)
DE background props: Ω m and 𝑤(𝑧) completely describe what you can do
But degenerate with DM
Distances only know about 𝐻(𝑧)
Constraints come from parameterisation and assumption of behaviour
𝐻 0 measurement in tension
Change in v. late universe?
Is here special?
20 July 2018
4th Cosmology School, UJ