CMB physics Zong-Kuan Guo 《现代宇宙学》 2017.5.27

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

§Background – modern cosmology What is the Universe?

 4.5 Gpc  15 Gly

Cosmological Ladders Object Mass (M⊙) Size stars 1 7× 10 5 km star clusters 10 4  10 7 100 pc galaxies 10 8  10 13 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 = 3.086×10 16 m = 3.26 ly, 1M⊙= 2×10 30 kg

Fundamental assumptions Einstein’s gravitation Cosmological principle

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!

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”

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 − 1 2 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 = 1 2 𝑔 𝜌𝜎 𝑔 𝜎𝜈,𝜇 + 𝑔 𝜎𝜇,𝜈 − 𝑔 𝜇𝜈,𝜎

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°.

Evolution of the Universe

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= 1 3 𝜌 𝛾 ∝ 𝑎 −4 𝑎(𝑡)∝ 𝑡 1/2 matter (b, c), 𝑤=0 𝜌 𝑏 ∝ 𝑎 −3 𝑎(𝑡)∝ 𝑡 2/3 vacuum energy (Λ), 𝑤=−1 𝜌 Λ ∝ 𝑎 0 𝑎(𝑡)∝ 𝑒 𝐻𝑡

Energy density evolution

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.

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

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

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

Thermal history of the Universe

in thermal equilibrium (natural units ℏ=𝑐= 𝑘 𝐵 =1) 𝑛= 𝑔 (2𝜋) 3 𝑑 3 𝑝𝑓 𝑝 𝜌= 𝑔 (2𝜋) 3 𝑑 3 𝑝𝐸(𝑝)𝑓 𝑝 𝑃= 𝑔 (2𝜋) 3 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

In the relativistic limit (𝑇≫𝑚) and 𝑇≫𝜇 𝑛= 𝜁(3) 𝜋 2 𝑔 𝑇 3 𝜌= 𝜋 2 30 𝑔 𝑇 4 𝑃= 𝜌 3 𝑛= 3 4 𝜁(3) 𝜋 2 𝑔 𝑇 3 𝜌= 7 8 𝜋 2 30 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

the temperature of radiation scales like relativistic species 𝜌 𝑅 = 𝜋 2 30 𝑔 ∗ 𝑇 4 𝑃 𝑅 = 𝜌 𝑅 /3 𝑔 ∗ = 𝑖=Bosons 𝑔 𝑖 𝑇 𝑖 𝑇 4 + 7 8 𝑖=Fermions 𝑔 𝑖 𝑇 𝑖 𝑇 4 the temperature of radiation scales like 𝑇∝ 𝑎 −1

thermodynamic relation 𝑑𝐸=𝑇𝑑𝑆−𝑃𝑑𝑉 entropy density for photons 𝑠 𝛾 ≡ 𝑑𝑆 𝑑𝑉 = 1 𝑇 𝑑𝐸 𝑑𝑉 +𝑃 = 4 𝜌 𝛾 3𝑇 = 4 𝜋 2 45 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× 10 8 Ω 𝑏 ℎ 2 −1 ~ 10 10

Three stages of cosmology

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

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

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.

1997-2002, 2-degree-Field Galaxy Redshift Survey (2dFGRS) 2000-now, Sloan Digital Sky Survey (SDSS), 2000-2005(SDSS-I), 2005-2008(SDSS-II), 2008-2014(SDSS-III), 2014-2020(SDSS-IV)

Observational windows

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

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

Astrophysical powers of Electromagnetic Radiation Gravitational Rotational Nuclear Magnetic

30  300 GHz

LIGO Virgo please watch a video LISA PTA 2030~2034

引力波

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). http://chandra.harvard.edu/photo/2011/arp147/more.html

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). http://chandra.harvard.edu/photo/2006/cartwheel/more.html

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

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

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

define the recombination temperature 𝑇 𝑟𝑒𝑐 when 𝑥 𝑒 =1/2 𝑥 𝑒 2 1−𝑥 𝑒 = 45𝜎 4 𝜋 2 𝑚 𝑒 2𝜋𝑇 3/2 𝑒 −∆/𝑇 (Saha equation) 𝑠 𝛾 = 4 𝜋 2 45 𝑇 3 𝜎≡ 𝑠 𝛾 𝑛 𝑏 ≈1.4× 10 8 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

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 5 1 𝑒 ℎ𝑐/𝜆𝑘𝑇 −1

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 0 3 𝐻≈3× 10 −23 cm −1 Ω 𝑚 ℎ 2 1/2 𝑇 1 eV 3/2

the freeze-out temperature Γ(𝑇 𝑔 )=𝐻( 𝑇 𝑔 ) 𝑇 𝑔 1 eV 1/4 𝑒 −∆/2 𝑇 𝑔 =1.2× 10 −13 Ω 𝑚 Ω 𝑏 1/2 For Ω 𝑚 =7 Ω 𝑏 , 𝑇 𝑔 =0.24 eV, 𝑧 𝑔 =1010 𝑇 𝑑𝑒𝑐 𝑇 𝑔

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

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

Discovery of the CMB

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 1964-1965. 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

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。”

Thanks for your attention!