Modern Physics Lecture 3: Modern Cosmology
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5. Hubble’s Expanding Universe Theory
a. Cosmos
1. Ancient Cosmology
⑴ Greek Geometric Model
① Predicting solar and lunar eclipses
② Periodicity of lunar eclipses : Saros Cycle
⑵ Aristotle’s Model
① Spherical universe, theory of four elements, aether
② Prejudice that Earth is the center and celestial objects must be perfect
○ Solar system follows circular motion (geocentric model)
○ Other stars are believed to be motionless
③ Difficult to explain retrograde motion of Mars
⑶ Ptolemaic Model or Epicycloid Model
① Establishing geocentric or geostatic theory
② Eccentricity
○ Center of circular motion is not Earth
○ Explaining retrograde motion of inner and outer planets
③ Equant: Ptolemaic model consists of primary circular motion of a large circle and secondary circular motion of a smaller circle (equant)
○ Center of equant for inner planets lies on a straight line connecting Earth and Sun
○ Reason: Explains why inner planets are not observed at midnight
④ Limitation: Retrograde motion should occur frequently
2. Medieval Cosmology
⑴ Nicolaus Copernicus’s Model
① Proposing heliocentric or heliostatic theory
② Explaining retrograde motion without epicycloid
⑵ Galileo’s Model
① 1st. Galileo visits a lens maker to create a telescope
② 2nd. Observing sunspots, realizing the imperfection of the Sun
③ 3rd. 1543, observing shadows on the Sun and distinguishing between inner and outer planets
④ 4th. Inferring the heliostatic theory through retrograde motion of planets and shadows on the Sun
⑤ 5th. Rejecting geocentric theory through observed phase changes of inner and outer planets using a telescope
⑶ Tycho Brahe’s Model
① 1st. Establishing the heliostatic theory through precise observation of planetary positions
② 2nd. Accurate naked-eye observations of planetary motion
③ 3rd. Proposing a model that compromises between heliostatic and geostatic theories
⑷ Kepler’s Model
① Kepler believes he can formulate laws based on observations by astronomers including Galileo
② Starting to analyze mathematically with Earth at the center, shifting from kinematics to dynamics
○ Kepler’s First Law: Law of elliptical orbits
○ Kepler’s Second Law: Law of equal areas
○ Kepler’s Third Law: Law of harmonies
⑸ Newton’s Model
① Newton generalizes Kepler’s laws, deriving the law of universal gravitation
② Gaining the idea of differentiation while reverse-calculating Kepler’s laws
3. Einstein’s Cosmology: Asserting a Stationary Universe
⑴ General Theory of Relativity
① Overview: Curvature of spacetime is determined by energy-matter distribution, and objects move along geodesics
② Key Concept 1: Einstein’s field equations (cosmic equations)
○ Left-hand side represents space, right-hand side represents mass and energy
○ Sometimes represented as Gμv on the left-hand side
○ Einstein adds the Λgμv term to assume a non-contracting universe
○ Λ is referred to as the cosmological constant
○ Later, Einstein admitted introducing Λ was a mistake
○ Subsequently, with the discovery of the universe’s accelerating expansion, introducing Λ became valid
③ Key Concept 2: Geodesic equation: Determining straight paths
④ Key Concept 3: Friedmann’s equation
⑵ Einstein’s Cosmology
① Applying the cosmological principle to Einstein’s field equations
○ Homogeneous: Meaning uniformity of space
○ Isotropic: Meaning rotational symmetry
② FLRW Metric
③ Tμν = (ρ, -p, -p, -p)
⑶ Differences from General Theory of Relativity
① ρ represents energy density, not mass density
② k is the scalar curvature
③ Additional term ρΛ
⑷ Relationship between Universe Expansion and k
① Formulation
② Expansion and contraction of the universe
Figure. 1. Universe Expansion and Contraction
Figure. 2. Universe Curvature
③ Open Universe: ρ ＜ ρc. Universe with negative curvature resembling a saddle
④ Closed Universe: ρ ＞ ρc
⑤ Flat Universe: ρ = ρc
4. Friedmann’s Cosmology: Asserting Universe Expansion
⑴ Also known as Lemaître’s Cosmology
⑵ Total energy of objects in motion in a gravitational field
⑶ Substituting (4/3)πR3ρ into M
5. Hubble’s Expanding Universe Theory
⑴ Olbers’ Paradox: Why the night sky is dark
① Claiming that if the universe is infinitely large, there should be infinitely many stars, making the night sky bright
② Reason 1: Stars are not eternal
③ Reason 2: There is still undelivered light
④ Reason 3: The universe is expanding
⑵ Hubble’s Law
① Hubble observes external galaxies, finding evidence of universal expansion
Figure. 3. Hubble’s Observational Data
② Nearby and distant galaxies
○ Nearby galaxies are bluish
○ Distant galaxies are consistently redshifted
③ Galaxy’s redshift velocity ∝ distance
○ Redshift formula: Speed of light c, original absorption wavelength λ0, change in wavelength of absorption line Δλ,
v = c Δλ / λ0
○ Hubble constant: 50 kms-1Mpc-1
v = H × r (where r is distance)
④ Conclusion: The universe is expanding
⑶ (Reference) Types of Redshift
① Type 1: Redshift due to the ordinary Doppler effect
○ vs: Velocity of the source (e.g., emitter)
○ vd: Velocity of the detector (e.g., observer)
○ v0: Velocity of the wave (e.g., sound wave)
○ f0: Frequency of the wave
○ f: Observed frequency
② Type 2: Redshift due to relativistic Doppler effect
○ 1st. Gravity warps spacetime → Causes expansion of space
○ 2nd. As space expands, wavelengths of light increase
○ 3rd. Visible light wavelengths shift toward the red end of the spectrum
③ Type 3: Redshift due to cosmological expansion
○ 1st. Universe is expanding
○ 2nd. As space expands, wavelengths of light increase
○ 3rd. Visible light wavelengths shift toward the red end of the spectrum
6. Big Bang Theory
⑴ Overview
① Definition: Theory proposing that the universe originated from an extremely hot, dense point in a Big Bang around 13.8 billion years ago
② (Note) The idea of the Big Bang originated from reversing time in Hubble’s expanding universe theory
③ Before the Big Bang, the concept of time did not exist.
⑵ 1st. Generation of Basic Particles
① Basic particles like quarks and electrons are generated as the temperature of the universe rapidly decreases.
⑶ 2nd. Generation of Protons and Neutrons
① Three quarks combine to form protons and neutrons.
② The ratio of protons to neutrons changes from about 1:1 initially to about 7:1.
③ Neutrons convert to protons through a nuclear reaction due to the stability difference.
⑷ 3rd. Around 3 Minutes After the Big Bang
① With the universe’s temperature at about 1 billion K, two protons and two neutrons combine to form helium nuclei.
② Among 16 nuclei, 14 are protons and 2 are neutrons.
③ 12 protons form 12 hydrogen nuclei, while 2 protons and 2 neutrons form 1 helium nucleus.
⑸ 4th. Around 380,000 Years After the Big Bang
① As the universe’s temperature drops to about 3000 K, atomic nuclei and electrons combine.
② Hydrogen and helium atoms are formed.
③ Transition from opaque universe to transparent universe:
○ Opaque Universe: Light collides with free-roaming electrons, becoming trapped.
○ Transparent Universe: Light can travel straight due to electrons being bound to atomic nuclei, making space transparent.
④ Photons escape the plasma state and propagate through space as cosmic microwave background radiation.
⑹ Evidence 1. Cosmic Microwave Background Radiation
Figure. 4. Cosmic Microwave Background Radiation observed by the WMAP Satellite
① Definition: Light emitted in the early universe.
② When atoms formed at a temperature of 3000 K, light spread throughout the universe.
③ The expansion of the universe causes longer wavelengths, observed as radiation energy at about 2.7 K.
○ In the form of microwaves.
○ The value 2.7 K is calculated using Wien’s displacement law under the assumption of black body radiation.
④ Predicted by Gamow.
⑤ Later discovered by Penzias and Wilson in 1965.
⑺ Evidence 2. Distribution of Elements in the Universe
① Ratio of Hydrogen Nuclei to Helium Nuclei = 12:1
② Mass of 1 Hydrogen Nucleus : Mass of 1 Helium Nucleus = 1:4
③ Mass ratio of all hydrogen nuclei to all helium nuclei = 3:1
④ Actual mass distribution of hydrogen and helium in the universe is approximately 3:1.
7. Modern Cosmology
⑴ Inflationary Cosmology
① Definition: Theory that the universe underwent inflation after the Big Bang, meeting conditions for a stable universe.
⑵ Accelerating Expansion Cosmology
Figure. 5. Observational data of Type Ia supernovae and models explaining cosmic expansion
① Redshift-distance graph shows larger distance indices for the same z value as the universe expands more.
② Type Ia supernovae appear darker than predicted by Model B due to larger distance indices.
③ Model A, which explains accelerating expansion, is adopted.
④ Saul Perlmutter awarded Nobel Prize in Physics in 2011 for accelerating expansion.
⑶ Λ-CDM Cosmology
① Overview: Λ stands for dark energy, C for cold, D for dark, and M for matter.
② Ordinary matter (4%) affects cosmic contraction.
③ Dark matter (23%) affects cosmic contraction.
○ Definition: Matter with mass that doesn’t emit light, making it unobservable.
○ (Note) Inferred speeds of planets in the solar system at different distances from the Sun.
○ Inferred galaxy rotation speeds based on surface brightness and actual rotation speeds.
Figure. 6. Inferred galaxy rotation speeds based on surface brightness and actual rotation speeds
○ (Footnote) Inferred rotation speeds exhibit profiles similar to planet speeds at different distances from the Sun.
○ In 1933, Fritz Zwicky concluded there is 400 times more unseen mass in galaxy clusters than observed.
○ 90% of our galaxy’s mass is dark matter.
○ Dark Matter Candidates: Brown dwarfs, white dwarfs, black holes, neutron stars, axions, WIMPs, sterile neutrinos.
○ Indirect confirmation through gravitational lensing.
④ Dark Energy (73%)
○ In a universe with ordinary and dark matter, contraction would prevail.
○ Dark energy is the force driving cosmic expansion.
○ Einstein’s discarded cosmological constant may be related to this.
○ Dark Energy Candidates: Quantum vacuum.
⑤ Dependency on Physical Quantities
○ Light energy ∝ 1/R^4 due to wavelength effect, not 1/R^3.
○ Matter ∝ 1/R^3.
○ Dark Energy Λ ∝ 1.
⑷ Particle Physics
① Standard Model of Particles: Total of 17 particles.
○ Basic Particles: Smallest, indivisible particles. Composed of leptons and quarks.
○ Classification 1: Quarks - up, down, charm, strange, top, bottom.
○ Classification 2: Leptons - electron, muon, tau, electron neutrino, muon neutrino, tau neutrino.
○ Classification 3: Gauge Bosons - gluon, photon, Z boson, W boson.
○ Gluon mediates the strong nuclear force.
○ Photon mediates the electromagnetic force.
○ Z and W bosons mediate the weak nuclear force.
○ All gauge bosons have a spin of 1.
○ Classification 4: Higgs Boson.
○ Mass
○ Stable universe if Higgs boson mass > 130 GeV.
○ Unstable universe if Higgs boson mass < 130 GeV.
○ Currently measured Higgs boson mass is about 125 GeV, indicating a slightly unstable universe.
○ If Higgs boson mass > Planck scale, electron’s gravity competes with electromagnetism, causing discrepancies.
○ Spin: 0 for Higgs boson, 2 for graviton.
○ Interaction: Higgs boson can interact with various particles like photons.
○ Decay: Higgs boson is a fundamental particle but short-lived. Its decay releases gamma rays, detectable at LHC.
○ Relevant Equations (Simplified Version).
○ Relevant Equations (Complex Version).
Figure. 7. Equations related to the Standard Model of Particles (Complex Version)
② Antiparticles: Antiquarks, positrons, etc.
③ Composition of Matter
○ Proton: Formed by combining 2 up quarks and 1 down quark. Carries positive charge.
○ Neutron: Formed by combining 1 up quark and 2 down quarks. Electrically neutral.
○ Atomic Nucleus: Composed of protons and neutrons.
○ Atom: Consists of atomic nucleus and electrons.
④ Pair Production and Annihilation
○ Formation and annihilation of particle-antiparticle pairs.
⑤ Symmetry Breaking
○ Quantum Arrow of Time: According to Copenhagen interpretation, Schrödinger equation, the main equation of quantum mechanics, loses time symmetry due to particle state collapse, though this relation isn’t exact and symmetry is lost upon measurement.
○ CPT Violation: Particles may have time reversal symmetry violations with extremely low probability.
⑸ Unified Theories
⑹ Superstring Theory
① Particles are considered resonant strings, explaining wave nature.
② Intrinsic dimension of superstrings is 6.
⑺ Multiverse Theory
① Imaginary time theory by Stephen Hawking constructs an infinite framework for the universe.
Input : 2019.04.15 23:54