produced more copiously during inflaton oscillations $^{76}$. D. Particle creation and thermalization There are distinct phases of particle creation in this model, here we briefly summarize them below $^{77}$. • Tachyonic preheating: Right after the end of inflation, the second derivative is negative in the case of inflection point inflation. The inflaton fluctuations with a physical momentum k < ∼ m$_{φ}$ will have a tachyonic instability, see Sec. VI B 3. Moreover V ′′ < 0 only at field values which are ∼ φ$_{0}$ ∼ 10 14 GeV. Tachyonic effects are therefore expected to be negligible since, unlike the case in [725], the homogeneous mode has a VEV which is hierarchically larger than m$_{φ}$ , and oscillates at a frequency ω ∼ m$_{φ}$ . Further note fields which are coupled to the inflaton acquire a very large mass ∼ hφ$_{0}$ from the homogeneous piece which suppresses non-perturbative production of their quanta at large inflaton VEVs. Therefore tachyonic effects, although genuinely present, do not 76 The flat direction is coupled to the scalar fields through gauge and Yukawa interactions. For instance, in the LLe case since the lepton Yukawas are ≤ 10 − 2 we can safely neglect them. The gauge cou- plings arise from the D -term part of the scalar potential. The D -terms corresponding to SU (2)$_{W}$ and U (1)$_{Y}$ symmetries follow: V$_{D}$ ⊃ 1 $_{2}$g$^{2}$[ ∑ 3 i =1 ( ˜ $_{L}$† $_{1}$T i ˜ L$_{1}$ + ˜ $_{L}$† $_{2}$T i ˜ L$_{2}$ ) $^{2}$] + 1 $_{2}$g ′ $^{2}$[( | ˜ e ∗ 3 | 2 − 1 2 | ˜ L$_{1}$ | 2 − 1 2 | ˜ L$_{2}$ | $^{2}$)]. Here T $^{1}$, T $^{2}$, T 3 are the SU (2) generators (i.e. 1 / 2 times the Pauli matrices) and g, g ′ are the gauge couplings of SU (2)$_{W}$ , U (1)$_{Y}$ , respectively. Similarly, couplings of the flat direction to the gauge fields are obtained from the flat direction ki- netic terms: L ⊃ ( D µ ˜ L$_{1}$ ) † ( D$_{µ}$ ˜ L$_{1}$ ) + ( D µ ˜ L$_{2}$ ) † ( D$_{µ}$ ˜ L$_{2}$ ) + ( D $^{µ}$˜ e$_{3}$ ) † ( D$_{µ}$ ˜ e$_{3}$ ), where D$_{µ}$ ˜ L$_{1}$ = ( ∂$_{µ}$ + i $_{2}$g ′ B$_{µ}$ − igW$_{1}$$_{,µ}$T 1 − igW$_{2}$$_{,µ}$T 2 − igW$_{3}$$_{,µ}$T $^{3}$) ˜ L$_{1}$ , D$_{µ}$ ˜ L$_{2}$ = ( ∂$_{µ}$ + i $_{2}$g ′ B$_{µ}$ − igW$_{1}$$_{,µ}$T 1 − igW$_{2}$$_{,µ}$T 2 − igW$_{3}$$_{,µ}$T $^{3}$) ˜ L$_{2}$ , and D$_{µ}$ ˜ e$_{3}$ = ( ∂$_{µ}$ − ig ′ B$_{µ}$ )˜ e$_{3}$ , where W$_{1}$$_{,µ}$, W$_{2}$$_{,µ}$, W$_{3}$$_{,µ}$ and B$_{µ}$ are the gauge fields of SU (2)$_{W}$ and U (1)$_{Y}$ , respectively. The flat direction couplings to the fermions are found from the following part of the Lagrangian: L ⊃ $^{√$_{2}$$_{g}$}$∑ 3 i $_{=1}$[ ˜ $_{L}$† 1 ˜ W t $_{i}$T $^{i}$( iσ$_{2}$L$_{1}$ ) + ˜ $_{L}$† 2 ˜ W t $_{i}$T $^{i}$( iσ$_{2}$L$_{2}$ $_{)]+}$√$_{2}$$_{g}$$_{′}$$_{[˜}$ $_{e}$† 3 ˜ B $^{t}$( iσ$_{2}$e$_{3}$ ) − 1 2 ˜ $_{L}$† 1 ˜ B $^{t}$( iσ$_{2}$L$_{1}$ ) − 1 2 ˜ $_{L}$† 2 ˜ B $^{t}$( iσ$_{2}$L$_{2}$ )]+h . c . , where ˜ W$_{1}$, ˜ W$_{2}$, ˜ W$_{3}$ and ˜ B are the gauginos of SU (2)$_{W}$ and U (1)$_{Y}$ respectively. Superscript t de- notes transposition, and σ$_{2}$ is the second Pauli matrix. The field content of L$_{1}$, L$_{2}$, e$_{3}$ are given by: L$_{1}$ = ( ψ$_{1}$ ψ$_{2}$ ) , L$_{2}$ = ( ψ$_{3}$ ψ$_{4}$ ) , e$_{3}$ = ψ$_{5}$ , where ψ$_{i}$ are left-handed Wyle spinors. Similarly, one can also work out all the relevant couplings of udd . 77 Readers may wish to revisit this section after reading the following Secs. VI B 3, VI B 2, VI C, VI D 4, and VI D 5. 159