In this paper we will analyse the constraints on a sub-Planckian excursion of a single inflaton field , which would yield a large tensor to scalar ratio , while explaining the temperature anisotropy of the cosmic microwave background ( CMB ) radiation .
In particular , our attempt will be to reconstruct the inflationary potential by constraining , V ( \phi _ { 0 } ) ,~ { } V ^ { \prime } ( \phi _ { 0 } ) ,~ { } V ^ { \prime \prime } ( \phi _ { 0 } ) ,~ { } V ^ { \prime% \prime \prime } ( \phi _ { 0 } ) and V ^ { \prime \prime \prime \prime } ( \phi _ { 0 } ) , in the vicinity of the field , \phi _ { 0 } \ll M _ { p } , and the field displacement , \Delta \phi \ll M _ { p } , where M _ { p } is the reduced Planck mass .
We will provide , for the first time , a set of new consistency relationships for sub-Planckian excursion of the inflaton field , which would help us to differentiate sub-versus-super Planckian models of inflation .
For a generic single field inflationary potential , we will be able to put a stringent bound on the potential energy density : 2.07 \times 10 ^ { 16 } ~ { } { GeV } \leq \sqrt [ 4 ] { V _ { \star } } \leq 2.40 \times 10 ^ { 16 } ~ { % } { GeV } , where inflation can occur on the flat potential within , 0.066 \leq \frac { \left| \Delta \phi \right| } { M _ { p } } \leq 0.092 , for the following observational constraints : ( Planck+WMAP-9+high L+BICEP2 ) .
We then provide a prediction for the spectral tilt ( n _ { T } ) , running ( \alpha _ { T } ) and running of running ( \kappa _ { T } ) of the tensor modes within the window , -0.019 < n _ { T } < -0.033 , -2.97 \times 10 ^ { -4 } < \alpha _ { T } ( = dn _ { T } / d \ln k ) < 2.86 \times 10 ^ { -5 } , and -0.11 \times 10 ^ { -4 } < \kappa _ { T } ( = d ^ { 2 } n _ { T } / d \ln k ^ { 2 } ) < -3.58 \times 10 ^ { -4 } , in a model independent way .
We also provide a simple example of an inflection-point model of inflation and reconstruct the potential in a model independent way to match the current observations .