1. CMB physics
Zong-Kuan Guo
《现代宇宙学》

2. Outline §Background § CMB anisotropy theory § CMB experiments
Modern cosmology
Cosmic Microwave Background (CMB) radiation
§ CMB anisotropy theory
§ CMB experiments
§ Cosmological implications
Constraints on cosmological parameters
Some anomalies

3. §Background – modern cosmology
What is the Universe?

4.  4.5 Gpc  15 Gly

5. Cosmological Ladders Object Mass (M⊙) Size stars 1 7× 10 5 km
star clusters
10 4  10 7
100 pc
galaxies
10 8 
1  50 kpc
galaxy groups
10 14
1 Mpc
galaxy clusters
10 15
3 Mpc
superclusters
10 16
10 Mpc
LSS
10 17
10  100 Mpc
voids
uncertain
observable Universe
10 23
4 Gpc
1pc = 1AU/1arcsec = × m = 3.26 ly, 1M⊙= 2× kg

6. Fundamental assumptions
Einstein’s gravitation
Cosmological principle

7. Cosmological principle
The Universe is homogeneous and isotropic on large scales.
Homogeneity means that the Universe looks the same at each point.
Isotropy means that the Universe looks the same in all directions.
𝑑𝑠 2 = 𝑔 𝜇𝜈 𝑑𝑥 𝜇 𝑑𝑥 𝜈 =− 𝑑𝑡 2 + 𝑎 2 (𝑡) 𝑑𝑟 2 1−𝑘 𝑟 2 + 𝑟 2 𝑑𝜃 2 + 𝑟 2 sin 2 𝜃 𝑑𝜙 2
This is called the Friedmann-Robertson-Walker (FRW) metric.
-+++ metric signature!

8. Einstein’s gravitation
Newton’s law of universal gravitation (1687)
𝐹=𝐺 𝑚 1 𝑚 2 𝑟 2
Einstein’s gravitation (1915, 228 years later)
𝐺 𝜇𝜈 =8𝜋𝐺 𝑇 𝜇𝜈
+/- signs!
geometry of space-time
matter and its motion
“It is Probably the most Beautiful of all Existing Theories”

9. 𝑅 𝜌 𝜎𝜇𝜈 = Γ 𝜌 𝜎𝜈,𝜇 − Γ 𝜌 𝜎𝜇,𝜈 + Γ 𝜆 𝜎𝜈 Γ 𝜌 𝜇𝜆 − Γ 𝜆 𝜎𝜇 Γ 𝜌 𝜈𝜆
Einstein tensor, Ricci tensor, Ricci scalar, Riemann tensor and affine connection are defined as
𝐺 𝜇𝜈 = 𝑅 𝜇𝜈 − 𝑔 𝜇𝜈 𝑅
𝑅 𝜇𝜈 = 𝑅 𝜆 𝜇𝜆𝜈
𝑅= 𝑔 𝜇𝜈 𝑅 𝜇𝜈
𝑅 𝜌 𝜎𝜇𝜈 = Γ 𝜌 𝜎𝜈,𝜇 − Γ 𝜌 𝜎𝜇,𝜈 + Γ 𝜆 𝜎𝜈 Γ 𝜌 𝜇𝜆 − Γ 𝜆 𝜎𝜇 Γ 𝜌 𝜈𝜆
+/- signs!
Γ 𝜌 𝜇𝜈 = 𝑔 𝜌𝜎 𝑔 𝜎𝜈,𝜇 + 𝑔 𝜎𝜇,𝜈 − 𝑔 𝜇𝜈,𝜎

10. Geometry of the Universe
(1) Euclidean geometry (k = 0, a flat Universe): an infinite space (2) spherical geometry (k > 0, a closed Universe): a finite size but no boundary (3) hyperbolic geometry (k < 0, a open Universe)
The angles of a triangle add up to (more than, less than) 180°.

11. Evolution of the Universe

12. For a perfect fluid, the energy-momentum tensor is
𝑇 𝜇𝜈 = 𝜌+𝑃 𝑈 𝜇 𝑈 𝜈 +𝑃 𝑔 𝜇𝜈
𝐻 2 = 8𝜋𝐺 3 𝜌− 𝑘 𝑎 2 + Λ 3
The field equations:
1 𝑎 𝑑 2 𝑎 𝑑𝑡 2 =− 4𝜋𝐺 3 (𝜌+3𝑃)+ Λ 3 ,
𝑑𝜌 𝑑𝑡 +3𝐻 𝜌+𝑃 =0
radiation (𝛾), 𝑤= 𝜌 𝛾 ∝ 𝑎 − 𝑎(𝑡)∝ 𝑡 1/2
matter (b, c), 𝑤=0 𝜌 𝑏 ∝ 𝑎 − 𝑎(𝑡)∝ 𝑡 2/3
vacuum energy (Λ), 𝑤=−1 𝜌 Λ ∝ 𝑎 𝑎(𝑡)∝ 𝑒 𝐻𝑡

13. Energy density evolution

14. Expansion and redshift
A light propagating radially (𝑑𝜃=𝑑𝜙=0) from 𝑟 𝑒 to 𝑟 𝑟 obeys the geodesic equation 𝑑𝑠=0.
𝑡 𝑒 𝑡 𝑟 𝑑𝑡 𝑎(𝑡) = 𝑟 𝑒 𝑟 𝑟 𝑑𝑟 1−𝑘 𝑟 2
Consider a light ray emitted a short time interval later at the same coordinates, so the emission time is 𝑡 𝑒 +𝑑 𝑡 𝑒 and reception time is 𝑡 𝑟 +𝑑 𝑡 𝑟 .
𝑡 𝑒 + 𝑑𝑡 𝑒 𝑡 𝑟 +𝑑 𝑡 𝑟 𝑑𝑡 𝑎(𝑡) = 𝑟 𝑒 𝑟 𝑟 𝑑𝑟 1−𝑘 𝑟 2
𝑡 𝑒 𝑡 𝑒 +𝑑𝑡 𝑑𝑡 𝑎(𝑡) = 𝑡 𝑟 𝑡 𝑟 +𝑑 𝑡 𝑟 𝑑𝑡 𝑎(𝑡) ⟹ 𝑑 𝑡 𝑒 𝑎 𝑡 𝑒 = 𝑑 𝑡 𝑟 𝑎( 𝑡 𝑟 )
As the wavelength is proportional to the time between crests, λ∝𝑑𝑡∝𝑎(𝑡), so
1+𝑧= 𝜆 𝑟 𝜆 𝑒 = 𝑎( 𝑡 0 ) 𝑎( 𝑡 𝑒 )
𝑡 𝑟 is identified with 𝑡 0 , to describe epochs of the Universe and the distances to objects.

15. Use this to define the parameter 𝜆:
4-momentum:
𝑃 𝜇 =(𝐸, 𝑃 )
𝑃 𝜇 ≡ 𝑑 𝑥 𝜇 𝑑𝜆
Use this to define the parameter 𝜆:
The zeroth component of the geodesic equation becomes
𝑑 2 𝑥 𝜇 𝑑 𝜆 2 =− Γ 𝛼𝛽 𝜇 𝑑 𝑥 𝛼 𝑑𝜆 𝑑 𝑥 𝛽 𝑑𝜆
𝐸 𝑑𝐸 𝑑𝑡 =− 𝛿 𝑖𝑗 𝑎 𝑎 𝑃 𝑖 𝑃 𝑗
A massless particle:
𝑔 𝜇𝜈 𝑃 𝜇 𝑃 𝜈 =0 ⇒
− 𝐸 2 + 𝛿 𝑖𝑗 𝑎 2 𝑃 𝑖 𝑃 𝑗 =0
𝑑𝐸 𝑑𝑡 + 𝑎 𝑎 𝐸=0 ⇒
𝐸∝ 1 𝑎

16. 1 MeV 3300 1100 10 energy redshift BBN radiation-matter equality
redshift
BBN
radiation-matter equality
recombination
reionization
now

17. Structure formation
Gravitational instability: Gravity pulls material towards the denser regions, enhancing any initial irregularities. An irregular distribution of matter is therefore unstable under the influence of gravity, becoming more and more irregular as time goes by.
please watch a video

18. Thermal history of the Universe

19. in thermal equilibrium (natural units ℏ=𝑐= 𝑘 𝐵 =1)
𝑛= 𝑔 (2𝜋) 𝑑 3 𝑝𝑓 𝑝
𝜌= 𝑔 (2𝜋) 𝑑 3 𝑝𝐸(𝑝)𝑓 𝑝
𝑃= 𝑔 (2𝜋) 𝑑 3 𝑝 𝑝 2 3𝐸(𝑝) 𝑓 𝑝
distribution function
𝑓 𝑝 = 1 𝑒 (𝐸−𝜇)/𝑇 ±1 , 𝐸 2 = 𝑝 2 + 𝑚 2

20. In the relativistic limit (𝑇≫𝑚) and 𝑇≫𝜇
𝑛= 𝜁(3) 𝜋 2 𝑔 𝑇 3
𝜌= 𝜋 𝑔 𝑇 4
𝑃= 𝜌 3
𝑛= 3 4 𝜁(3) 𝜋 2 𝑔 𝑇 3
𝜌= 𝜋 𝑔 𝑇 4
𝑃= 𝜌 3
Bose
Fermi
In the non-relativistic limit (𝑚≫𝑇)
𝑛=𝑔 𝑚𝑇 2𝜋 3/2 𝑒 −(𝑚−𝜇)/𝑇
𝜌=𝑛𝑚
𝑃=𝑛𝑇

21. the temperature of radiation scales like
relativistic species
𝜌 𝑅 = 𝜋 𝑔 ∗ 𝑇 4
𝑃 𝑅 = 𝜌 𝑅 /3
𝑔 ∗ = 𝑖=Bosons 𝑔 𝑖 𝑇 𝑖 𝑇 𝑖=Fermions 𝑔 𝑖 𝑇 𝑖 𝑇 4
the temperature of radiation scales like
𝑇∝ 𝑎 −1

22. thermodynamic relation 𝑑𝐸=𝑇𝑑𝑆−𝑃𝑑𝑉
entropy density for photons
𝑠 𝛾 ≡ 𝑑𝑆 𝑑𝑉 = 1 𝑇 𝑑𝐸 𝑑𝑉 +𝑃 = 4 𝜌 𝛾 3𝑇 = 4 𝜋 𝑇 3
𝑛 𝑏 = 𝜌 𝑏 𝑚 𝑏 ∝ 𝑎 −3 ∝ 𝑇 3
The entropy per baryon is a constant.
𝜎≡ 𝑠 𝛾 𝑛 𝑏 = 4 𝜋 2 𝑇 0 3 /45 Ω 𝑏 𝜌 𝑐𝑟 / 𝑚 𝑝 ≈1.4× Ω 𝑏 ℎ 2 −1 ~ 10 10

23. Three stages of cosmology

24. 1. Hot Big Bang cosmology (1920s-1970s)
Cosmic expansion (Hubble, 1929)
Big Bang Nucleosynthesis (BBN)
CMB black-body spectrum (COBE, 1989)
Hubble’s Law:
All galaxies are receding from us, the velocity of recession is proportional to the distance of an object from us.
𝑣 = 𝐻 0 𝑟
Riess et al 1996

26. 2. Standard cosmology (1980s-2000s)
inflation (Guth, 1981)
dark energy (SCP and High-Z, 1998)
cold dark matter (Fritz Zwicky, 1933)
 inflation+CDM model
rotation curve
N-body simulation
bullet clusters
lensing
Fritz Zwicky

27. 3. Precision cosmology (2000s-now)
CMB anisotropies (WMAP, Planck)
LSS (BAO, GC, WL), SNIa (complement)
21cm (promising)
SKA
FAST
LOFAR
𝑃 𝑙
64 antennas
We know much but understand little.

28. 1997-2002, 2-degree-Field Galaxy Redshift Survey (2dFGRS)
2000-now, Sloan Digital Sky Survey (SDSS), (SDSS-I), (SDSS-II), (SDSS-III), (SDSS-IV)

29. Observational windows

30. Electromagnetic waves Gravitational waves (LIGO, Virgo, LISA)
Cosmic neutrino background (1.95 K)
Cosmic ray (PAMELA, Fermi, AMS-02) …
positon excess?

31. Radio waves: VLA, SKA, FAST, LOFAR
Cosmic microwave background: COBE, WMAP, Planck
Infrared: WISE, Spitzer, JWST(Wide Field InfraRed Survey Telescope, NASA, ESA, CSA, space-based, 2018 ), WFIRST (Wide Field InfraRed Survey Telescope, NASA, space-based, mid-2020)
Optical: Hubble (1990 ), 2dFGRS (1997  2002), SDSS (2000  2020), Euclid (ESA, space-based, 2020 ), LSST (Large Synoptic Survey Telescope, 2022 )
Ultraviolet: GALEX
X-rays: Chandra, XMM-Newton
Gamma-ray: Fermi LAT (0.02  300 GeV), H.E.S.S. (0.01  10 TeV)
Wider, Faster, Deeper

32. Astrophysical powers of Electromagnetic Radiation
Gravitational
Rotational
Nuclear
Magnetic

33. 30  300 GHz

34. LIGO
Virgo
please watch a video
LISA
PTA
2030~2034

37. 引力波

38. X-ray
ultraviolet
optical
infrared
composite
The composite image of Arp 147: Chandra X-ray data (pink), Hubble optical data (red, green and blue), ultraviolet GALEX data (green) and infrared Spitzer data (red).

39. X-ray
ultraviolet
optical
infrared
composite
The composite image of the Cartwheel Galaxy: Chandra X-ray data (purple), ultraviolet GALEX data (blue), Hubble optical data (green) and infrared Spitzer data (red).

40. Origin of the CMB radiation
§ Background – CMB
Origin of the CMB radiation
100 GeV
100 MeV
MeV eV
meV

41. The physics of Recombination
the epoch at which charged electrons and protons formed neutral hydrogen.
𝑒 − +𝑝↔𝐻+𝛾 (13.6 eV)
in thermal equilibrium (𝐸≈𝑚+ 𝑝 2 2𝑚 , 0 ∞ 𝑑𝑥4𝜋 𝑥 2 𝑒 − 𝑥 2 = 𝜋 3/2 )
𝑛 𝑒 = 2 (2𝜋) 3 exp − 𝑚 𝑒 − 𝜇 𝑒 𝑇 2𝜋 𝑚 𝑒 𝑇 3/2
𝑛 𝑝 = 2 (2𝜋) 3 exp − 𝑚 𝑝 − 𝜇 𝑝 𝑇 2𝜋 𝑚 𝑝 𝑇 3/2
𝑛 𝐻 = 4 (2𝜋) 3 exp − 𝑚 𝐻 − 𝜇 𝐻 𝑇 2𝜋 𝑚 𝐻 𝑇 3/2

42. in chemical equilibrium ( 𝜇 𝛾 =0)
𝜇 𝑒 + 𝜇 𝑝 − 𝜇 𝐻 =0
The Universe is globally neutral.
𝑛 𝑒 = 𝑛 𝑝
the binding energy of hydrogen ∆≡ 𝑚 𝑒 + 𝑚 𝑝 − 𝑚 𝐻
𝑛 𝑒 2 𝑛 𝐻 = 𝑚 𝑒 𝑇 2𝜋 3/2 𝑒 −∆/𝑇
the ionization fraction 𝑥 𝑒 ≡ 𝑛 𝑒 /( 𝑛 𝑒 + 𝑛 𝐻 ), 𝑛 𝑏 ≡ 𝑛 𝑝 + 𝑛 𝐻
𝑥 𝑒 2 1−𝑥 𝑒 = 1 𝑛 𝑏 𝑚 𝑒 𝑇 2𝜋 3/2 𝑒 −∆/𝑇

43. define the recombination temperature 𝑇 𝑟𝑒𝑐 when 𝑥 𝑒 =1/2
𝑥 𝑒 2 1−𝑥 𝑒 = 45𝜎 4 𝜋 𝑚 𝑒 2𝜋𝑇 3/2 𝑒 −∆/𝑇
(Saha equation)
𝑠 𝛾 = 4 𝜋 𝑇 3
𝜎≡ 𝑠 𝛾 𝑛 𝑏 ≈1.4× Ω 𝑏 ℎ 2 −1 ~ 10 10
define the recombination temperature 𝑇 𝑟𝑒𝑐 when 𝑥 𝑒 =1/2
For Ω 𝑏 ℎ 2 =0.02,
𝑇 𝑟𝑒𝑐 =3757 K=0.32 eV,
𝑧 𝑟𝑒𝑐 =1376

44. Why is the recombination temperature much lower than the binding energy of hydrogen? 𝑇 𝑟𝑒𝑐 ≪13.6 eV
Since there are so many more photons than baryons in the Universe, even at a temperature much below ∆=13.6 eV there are still enough photons in the high-energy tail of the Planck distribution to keep the Universe ionized.
𝜂≡ 𝑛 𝑏 𝑛 𝛾 ~ 10 −10
𝐼 𝜆 = 8𝜋ℎ𝑐 𝜆 𝑒 ℎ𝑐/𝜆𝑘𝑇 −1

45. Freeze-out temperature of recombination
at which recombination froze out.
𝑝+ 𝑒 − ↔𝐻+𝛾
the cross section of the reaction
𝜎 𝑅 𝑣 ≈4.7× 10 −24 𝑇 1 eV −1/2 cm 2
the reaction rate
Γ 𝑅 = 𝑛 𝑝 𝜎 𝑅 𝑣 = 𝑥 𝑒 𝑛 𝑏 𝜎 𝑅 𝑣
≈2.4× 10 −10 cm −1 Ω 𝑏 ℎ 2 1/2 𝑇 1 eV 7/4 𝑒 −∆/2𝑇
the expansion rate
𝐻 2 = 8𝜋𝐺 3 Ω 𝑚 𝜌 𝑐 𝑇 𝑇
𝐻≈3× 10 −23 cm −1 Ω 𝑚 ℎ 2 1/2 𝑇 1 eV 3/2

46. the freeze-out temperature
Γ(𝑇 𝑔 )=𝐻( 𝑇 𝑔 )
𝑇 𝑔 1 eV 1/4 𝑒 −∆/2 𝑇 𝑔 =1.2× 10 − Ω 𝑚 Ω 𝑏 1/2
For Ω 𝑚 =7 Ω 𝑏 ,
𝑇 𝑔 =0.24 eV,
𝑧 𝑔 =1010
𝑇 𝑑𝑒𝑐
𝑇 𝑔

47. Freeze-out temperature of Thomson scattering
𝛾+ 𝑒 − ↔𝛾+ 𝑒 −
the cross section
𝜎 𝑇 ≈6.65× 10 −25 cm 2
the reaction rate
Γ 𝑇 = 𝑛 𝑒 𝜎 𝑇 = 𝑥 𝑒 𝑛 𝑏 𝜎 𝑇
≈3.6× 10 −11 cm −1 Ω 𝑏 ℎ 2 1/2 𝑇 1 eV 9/4 𝑒 −∆/2𝑇
the freeze-out temperature
Γ(𝑇 𝑑𝑒𝑐 )=𝐻( 𝑇 𝑑𝑒𝑐 )
For Ω 𝑚 =7 Ω 𝑏 ,
𝑇 𝑑𝑒𝑐 =0.26 eV,
𝑧 𝑑𝑒𝑐 =1100

48. a hot-plasma soup 400 cm −3 now 𝑻 𝒓𝒆𝒄 =𝟎.𝟑𝟐eV 𝑻 𝒈 =𝟎.𝟐𝟒eV

50. Discovery of the CMB

51. The CMB was first predicted
by G. Gamow, R. Alpher and
R. Herman in 1948.
T~5 K
The first discovery of the
CMB radiation by A.A. Penzias
and R.W. Wilson in
It is interpreted by R. Dicke, R. Wilkinson, J. Peebles, et. al.
in 1965.
The Nobel Prize in Physics 1978:
A.A. Penzias and R.W. Wilson

52. 1%来自CMB
任何方向
任何地点
任何时间
“世界上怕就怕‘认真’二字，共产党就最讲认真。”

53. Thanks for your attention!