(Minkowski functional) has lead so far a negatively preferred value for f$_{NL}$ ∼ − 60 ± 60. This discrepancy has not yet been fully resolved [13]. The future observations from PLANCK and the galaxy distribution should be able to constrain a deviation up to | f$_{NL}$ | > ∼ 5 [273–276]. F. Dynamical challenges for inflation Inflation has several dynamical challenges which have been discussed in the literature. 1. Initial conditions for inflation The question of initial condition is a worrisome factor. universe could have started either cold or hot. Whether universe began hot or cold, once vacuum energy density takes over it would always yield a cold universe. However there are nontrivial initial conditions to be satisfied. • Homogeneity problem: • Chaotic initial conditions: In an Einstein gravity inflation does not solve the homogeneity problem, instead infla- tion requires an initial patch of the universe, r , to be sufficiently homogeneous on scales larger than the Hubble patch, r ≫ H − $^{1}$, before inflation could begin see Refs. [277– 282]. Initial conditions if set at the Planckian scale do not suffer through this problem as shown by Refs. [4, 145, 283]. Low scale models of inflation require earlier phases of inflation in order to set the initial conditions. For sufficiently flat potential the only constraint is given by: (1 / 2) ˙ φ 2 + (1 / 2)( ∂$_{i}$φ ) 2 + V ( φ ) ≤ M 4 $_{P}$, see Refs. [4, 145, 187, 283]. The initial conditions are set by: (1 / 2) ˙ φ 2 ∼ (1 / 2)( ∂$_{i}$φ ) 2 ∼ V ( φ ) ∼ O ( M$_{P}$ ). If by any chance (1 / 2) ˙ φ 2 + (1 / 2)( ∂$_{i}$φ ) 2 ≤ V ( φ ) in a particular domain, the inflation begins and within a Planck time the poten- tial energy density, V ( φ ), starts dominating over kinetic term. In domains where (1 / 2) ˙ φ 2 + (1 / 2)( ∂$_{i}$φ ) 2 > V ( φ ), inflation does not take place and do not exist classi- cally. The above mentioned conditions are naturally satisfied when φ ≥ M$_{P}$ for a sim- ple chaotic type potential, V ∼ m $^{2}$φ$^{2}$, where there exists a window, O (100 − 10) M$_{P}$ > φ > O ( M$_{P}$ ), where universe enters a process of eternal self-reproduction [145, 283]. 33