Inflationary scenario in the supersymmetric economical 3-3-1 model
  We construct the supersymmetric economical 3-3-1 model which contains inflationary scenario and avoids the monopole puzzle. Based on the spontaneous symmetry breaking pattern (with three steps), the $F$-term inflation is derived. The slow-roll parameters $\ep$ and $\eta$ are calculated. By imposing as experimental five-year WMAP data on the spectral index $n$, we have derived a constraint on the number of e-folding $N_Q$ to be in the range from 25 to 50. The scenario for large-scale structure formation implied by the model is a mixed scenario for inflation and cosmic string, and the contribution to the CMBR temperature anisotropy depends on the ratio $\frac{M_X}{M_P}$. From the COBE data, we have obtained the constraint on the $M_X$ to be $M_X\in[1.22 \times 10^{16} - 0.98 \times 10^{17}]$ GeV. The upper value $M_X\simeq 10^{17}$ GeV is a result of the analysis in which the inflationary contribution to the temperature fluctuations measured by the COBE is 90%. The coupling $\alpha$ varies in the range: $ 10^{-7} - 10^{-1} $. This value is not so small, and it is a common characteristics of the supersymmetric unified models with the inflationary scenario. The spectral index $n$ is a little bit smaller than 0.98. The SUGRA corrections are slightly different from the previous consideration. When $\xi \ll 1$ and $ \al $ lies in the above range, the spectral index gets the value consistent with the experimental five-year WMAP data. Comparing with string theory, one gets $\xi < 10^{-8}$. Numerical analysis shows that $ \al \approx 10^{-6}$. To get inflation contribution to the CMBR temperature anisotropy $\approx 90%$, the mass scale $M_X \sim 3.5 \times 10^{14}$ GeV. 
Do T. Huong and Hoang N. Long