4.3 φ Decay Rate, Spectral Index and Tensor Fraction 58 verse is matter-dominated (ignoring dark energy). Therefore we have Therefore, from the Friedmann equation, we have This gives We now obtain $^{N}$∗ as where we have used $^{g}$∗ ≈ 10 2 as the number of spin states (effective relativistic degrees of freedom) of all of the particles in the thermal bath, at the time of $^{N}$∗ ≈ ln ( (1 . 5637 × 10 38 GeV − $^{1}$) $^{H}$∗ 0 . 002 ) + 2 3 ln ( 9Γ$_{ϕ}$ 10 π $^{2}$H $_{∗}$) + 1 4 ln ( 10 π $^{2}$V$_{0}$ 9 M 2 $_{P}$H 2 $_{∗}$) + 2 3 ln ( 9Γ 10 π $^{2}$H$_{TI}$ ) + 1 4 ln ( 10 π $^{2}$(9 . 830 × 10 − 13 GeV ) 4 9 M 2 $_{P}$Γ 2 ) − N$_{TI}$ (4.48) 3 M 2 $_{P}$H 2 ∝ T 3 (4.44) ρ ∝ a − 3 (4.42) ∝ T 3 (4.43) H 2 piv H 2 0 = T 3 piv T 3 0 (4.45) T$_{piv}$ = ( 0 . 002 2 Mpc − 2 T 3 0 H 2 0 ) 1 3 (4.46) = 9 . 830 × $_{10}$− 13 GeV (4 s.f.) (4.47)