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• The flat direction VEV of a curvaton must not break all of the SM gauge symmetry. This will affect the inflaton decay products to thermalize quickly, both kinetic and chemical equilibrium of MSSM degrees of freedom will be delayed, and furthermore, the curvaton oscillations will not be able to dominate the energy density [122, 123, 136]. 

Let us denote the flat direction superfield by ϕ . In the case of u$_{i}$d$_{j}$d$_{k}$ (for i, j, k = 1 , 2 , 3 , j ̸ = k ) as a flat direction, ϕ would represent the VEV. The MSSM superpotential can be rewritten as: 

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where ϕ$_{1}$$_{,}$$_{2}$$_{,}$$_{3}$ are MSSM superfields, see Eq. (103), L is denoted by L$_{1}$, L$_{2}$ or L$_{3}$ , and λ$_{1}$, λ$_{2}$, λ$_{3}$ are the Yukawas. In general m$_{inf}$ ≤ H$_{inf}$ , which is true for high scale models of inflation, and ϕ$_{inf}$ ∼ 10 $^{5}$H$_{inf}$ . Note that the VEV of the flat direction induces VEV dependent SUSY preserving masses to the MSSM particles, ∼ λ$_{1}$$_{,}$$_{2}$$_{,}$$_{3}$ 〈 ϕ$_{inf}$ 〉 . Therefore, for the inflaton decay into MSSM quanta to be kinematically allowed, we require: 

if the Φ H$_{u}$H$_{d}$ coupling is allowed by R-parity, and 

if the Φ H$_{u}$L coupling is allowed, where Φ is the inflaton superfield. 

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The above conditions considerably restrict the curvaton candidates within the MSSM, as only the first generations of (s)quarks and (s)leptons have Yukawa coupling < ∼ 10 − $^{5}$, while first two generations have Yukawas ≥ 10 − $^{3}$. 

Given these constraints an acceptable flat directions should include only one Q or one u . The reason is that D - and F -flatness conditions for directions which involve two or more Q and/or u require them to be of different flavors (for details, see [407]). The only flat directions with λ$_{1}$ ≤ 10 − 5 are as follows: 

