We show a primordial non-linear ( “ f _ { \mathrm { NL } } ” ) term may produce unphysically large CMB anisotropy for a red-tilted primordial power spectrum ( n < 1 ) , because of coupling to primordial fluctuation on the largest scale .
We consider a primordial power spectrum models of a running spectral index , and a transition at very low wavenumbers .
We find that only negative running spectral index models are allowed , provided that there is no transition at a low wavenumbers ( i.e . k \ll 1 ) .
For models of a constant spectral index , we find \log ( k _ { c } / k _ { 0 } ) \gtrsim - 184 , at 1 \sigma level , on the transition scale of sharp cut-off models , using recent CMB and SDSS data .