The supergravity ( SUGRA ) theories with exact global U ( 1 ) symmetry or shift symmetry in Kähler potential provide the natural frameworks for inflation .
However , the quadratic inflation is disfavoured by the new results on primordial tensor fluctuations from the Planck Collaboration .
To be consistent with the new Planck data , we point out that the explicit symmetry breaking is needed , and study these two SUGRA inflation in details .
For the SUGRA inflation with global U ( 1 ) symmetry , the symmetry breaking term leads to a trigonometric modulation on inflaton potential .
The coefficient of the U ( 1 ) symmetry breaking term is of the order 10 ^ { -2 } , which is sufficient large to improve the inflationary predictions while its higher order corrections are negligible .
Such models predict sizeable tensor fluctuations and highly agree with the Planck results .
In particular , the model with a linear U ( 1 ) symmetry breaking term predicts the tensor-to-scalar ratio around \textbf { r } \sim 0.01 and running spectral index \alpha _ { s } \sim - 0.004 , which comfortably fit with the Planck observations .
For the SUGRA inflation with breaking shift symmetry , the inflaton potential is modulated by an exponential factor .
The modulated linear and quadratic models are consistent with the Planck observations .
In both kinds of models the tensor-to-scalar ratio can be of the order 10 ^ { -2 } , which will be tested by the near future observations .