Physical Foundations of Cosmology

80 The hot universe particle having energy within the interval fromto+. Let us first consider a particle with no internal de ...

3.3 Rudiments of thermodynamics 81 ∆σ ∆Ω ∆t n R v Fig. 3.3. where the minus sign applies to bosons and the plus to fermions. The ...

82 The hot universe Note that massless particles (m=0) always have an ultra-relativistic equation of state, p= ε 3 , (3.30) inde ...

3.3 Rudiments of thermodynamics 83 where α≡ m T ,β≡ μ T . In particular, the total energy density of particles(p)and antiparticl ...

84 The hot universe of Integrals, Series, and Products(San Diego: Academic Press, 1994). The result for purely imaginaryβcan be ...

3.3 Rudiments of thermodynamics 85 whereO≡O ( α^2 ,β^2 ) .Similarly we obtain J∓(^3 )= ⎧ ⎪⎪ ⎨ ⎪⎪ ⎩ 2 15 π^4 + 1 2 π^2 ( 2 β^2 −α ...

86 The hot universe To estimate the number density of ultra-relativistic bosons we setm=μb=0in (3.27) and then obtain nb ζ( 3 ) ...

3.3 Rudiments of thermodynamics 87 over antifermions is nf−nf ̄= gT^3 6 β [ 1 + β^2 π^2 ] . (3.54) Substituting the expressions ...

88 The hot universe 3.3.5 Nonrelativistic particles If the temperature is smaller than the rest mass and in addition m−μ T  1 , ...

3.4 Lepton era 89 corresponding formulae are the standard ones found in any book on statistical physics. Having completed our br ...

90 The hot universe Theτ-lepton also decays, for example intoe−,ν ̄e,ντ,therefore μτ=μe−−μνe+μντ. (3.67) Finally, from the react ...

3.4 Lepton era 91 established from observations and is of orderB 10 −^10 − 10 −^9 .This means that the entropy per one baryon o ...

92 The hot universe baryon–antibaryon pairs becomes small compared to the baryon excess at temper- atures below 40 MeV while pos ...

3.4 Lepton era 93 As we found, atT<40 MeV, antibaryons can be neglected and, therefore, np,nnp,n.The conservation law of th ...

94 The hot universe This is not surprising because we need only a small excess of electrons to compen- sate the electric charge ...

3.4 Lepton era 95 Converting from Planckian units, we can rewrite this relation in the following useful form: tsec=tPl ( 3 32 πκ ...

96 The hot universe we have σeνO( 1 ) α^2 w MW^4 ,Z (p 1 +p 2 )^2 , (3.90) whereαw 1 /29 is the weak fine structure constant a ...

3.5 Nucleosynthesis 97 remains constant. Taking into account thatsγ∝Tγ^3 andsν∝Tν^3 ,we have ( Tγ Tν ) 3 ( 1 + se± sγ ) =C, (3.9 ...

98 The hot universe following estimate for the luminosity-to-mass ratio: L Mbar  1 4 1. 1 × 10 −^5 erg (1. 7 × 10 −^24 gm)×(3. ...

3.5 Nucleosynthesis 99 p e− n ν Fig. 3.5. where GF= παw √ 2 MW^2  1. 17 × 10 −^5 GeV−^2 is the Fermi coupling constant and (pi· ...