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Alan Guth, Cosmic Microwave Background (CMB) Spectrum and the Cosmological Constant, 8.286 Lecture 18, November 14, 2013, p. 1. Blak-Body Radiation Summary of Leture 17: 8.286 Leture 18 November 14, 2013 2 kT 4 ( ) = c 2 Energy

  1. Alan Guth, Cosmic Microwave Background (CMB) Spectrum and the Cosmological Constant, 8.286 Lecture 18, November 14, 2013, p. 1. Bla k-Body Radiation Summary of Le ture 17: 8.286 Le ture 18 November 14, 2013 π 2 kT 4 ( ) = ρc 2 Energy Density: u = g , 30 (¯ hc ) 3 COSMIC MICROWAVE 1 Pressure: p = u . 3 BACKGROUND SPECTRUM ) 3 ζ (3) ( kT Number density: n = g ∗ , AND THE π 2 (¯ hc ) 3 COSMOLOGICAL CONSTANT 2 4 3 2 π k T Entropy density: s = g . hc ) 3 45 (¯ Alan Guth Massa husetts Institute of T e hnology –1– 8.286 Le ture 18, November 14 Summary of Le ture 17 Summary of Le ture 17 Meaning of g and g ∗ and g ∗ for Neutrinos g For photons, g = g ∗ = 2. The correct values are given by pretending that neutrinos are massless, and have only one spin state: all ν ’s are left-handed But neutrinos also contribute, as do e + e − pairs when kT ≫ m e c 2 , ( � p = − 1 J · ˆ 2 ¯ h ) and all ν ¯’s are right-handed. and other particles at higher temperatures. In general, 7 21 � 1 � g ν = 3 2 1 = . × × × (boso s n ) 8 4 g = × number of particle “types” 7 (fermions) � �� � � �� � � �� � � �� � 8 Fe rmion f actor 3 sp eci es Particle / an tiparticle Spi n s tates � 1 ν e ,ν µ ,ν τ � (bosons) g ∗ = × num er of particle “types” b 3 9 3 ∗ = (f ermions ) g ν 3 2 1 = . × × × 4 4 2 � �� � � �� � � �� � � �� � Fermion factor 3 species Particle / antiparticle Spin states By “type,” we mean a complete specification of species, particle ν e ,ν µ ,ν τ vs. antiparticle, and spin state. Alan Guth Alan Guth Massa husetts Institute of T e hnology Massa husetts Institute of T e hnology –2– –3– 8.286 Le ture 18, November 14 8.286 Le ture 18, November 14

  2. Alan Guth, Cosmic Microwave Background (CMB) Spectrum and the Cosmological Constant, 8.286 Lecture 18, November 14, 2013, p. 2. Summary of Le ture 17 Summary of Le ture 17: Radiation Density of the Present Universe for e + e − and g ∗ Pairs g When e + e − pairs disappear from the thermal equilibrium mix, as kT falls below m e c 2 , they give all their entropy to the photons, and none to the neutrinos. Consequently (as you will show on 7 7 Problem Set 7), g e + e − = 1 2 2 = . × × × 8 2 � �� � � �� � � �� � � �� � � 4 � 1 / 3 Fe rmion f actor Sp ecies / an Spi n s tates Particle tiparticle T ν = T γ . 3 11 g e + e − = ∗ × 1 × 2 × 2 = 3 . 4 � �� � � �� � � �� � � �� � Then Species Spin states Fermion factor Particle / antiparticle � 4 � 4 / 3 � � π 2 ( γ ) 4 21 kT = 7 01 × 10 − 14 J/m 3 . u rad , 0 = 2 + . 4 11 30 (¯ hc ) 3 Alan Guth Massa husetts Institute of T e hnology –4– –5– 8.286 Le ture 18, November 14 Summary of Le ture 17 Summary of Le ture 17: The Real Story of Neutrino Masses There are two possibilities: Neutrinos have been observed to “oscillate” from one species to another, which is not allowed unless neutrinos have a nonzero Dirac Mass: Right-handed neutrino would be a new as-yet mass: unseen type of particle. But it would interact so weakly that it would not have been produced in significant 21 c 4 = (7 . 50 ± 0 . 20) × 10 − 5 eV 2 , ∆ m 2 numbers during the big bang. 23 c 4 = 2 . 32 +0 . 12 � 0 . 08 × 10 − 3 eV 2 . � ∆ m 2 Majorana Mass: If lepton number is not conserved (which − seems likely), so the neutrino is absolutely neutral, then For a massive particle with spin J , all spin states the right-handed neutrino could be the particle that we call the anti-neutrino. J z / ¯ h = − J, − J + 1 , . . . , J must exist. In particular, there must be right-handed neutrinos and left-handed antineutrinos. Alan Guth Alan Guth Massa husetts Institute of T e hnology Massa husetts Institute of T e hnology –6– –7– 8.286 Le ture 18, November 14 8.286 Le ture 18, November 14

  3. Alan Guth, Cosmic Microwave Background (CMB) Spectrum and the Cosmological Constant, 8.286 Lecture 18, November 14, 2013, p. 3. Summary of Le ture 17: Thermal History of the Universe 0 . 860 MeV For 0 . 511 MeV ≪ kT ≪ 106 MeV, kT = . � t (in sec) Conservation of entropy implies that s ∝ 1 /a 3 . When g is constant, this implies T ∝ 1 /a . At the densities found in the early universe, the hydrogen plasma becomes neutral atoms (hydrogen “recombines”) at 4,000 K, and becomes transparent to photons (“photon de- coupling”) at 3,000 K. We estimated T decoupling ≈ 380 , 000 yr. CMB Data in 1975 Alan Guth Massa husetts Institute of T e hnology –8– –9– 8.286 Le ture 18, November 14 A Preliminary Measurement of the Cosmic Microwave Background Spectrum by the Cosmic Background Explorer (COBE) Satellite J. C. Mather, E. S. Cheng, et al. Data from Berkeley-Nagoya Rocket Flight, 1987 Cover Page of Original Preprint of the COBE Measurement of the CMB Spectrum, 1990 Alan Guth Massa husetts Institute of T e hnology –10– –11– 8.286 Le ture 18, November 14

  4. Alan Guth, Cosmic Microwave Background (CMB) Spectrum and the Cosmological Constant, 8.286 Lecture 18, November 14, 2013, p. 4. Original COBE Measurement of the CMB Spectrum, Jan 1990. Energy density is in units of electron volts per cubic meter per gigahertz. Alan Guth –12– Massa husetts Institute of T e hnology 8.286 Le ture 18, November 14

  5. MIT OpenCourseWare http://ocw.mit.edu 8.286 The Early Universe Fall 2013 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.

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