Sub-Planckian inflation & large tensor to scalar ratio with r ≥ 0.1 Sayantan Choudhury 1 and Anupam Mazumdar 2 1 Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata 700 108, INDIA and 2 Consortium for Fundamental Physics, Physics Department, Lancaster University, LA1 4YB, UK We categorically point out why the analysis of Ref. [3] is incorrect. Here we explicitly show why the sub-Planckian field excursion of the inflaton field can yield large observable tensor-to-scalar ratio, which satisfies both Planck and BICEP constraints. We have shown in Refs. [1] and [2] that the sub- Planckian excursion of the inflaton field can generate large value of tensor-to-scalar ratio as observed by BI- CEP2 and also satisfies the constraints obtained from the Planck after foreground subtractions 1 . However, re- cently it was claimed in Ref. [3] that for a single field inflationary model with sub-Planckian field excursion it is not possible to generate the observed large tensor-to- scalar ratio. Unfortunately, the validity of this claim is completely wrong. In this short report our prime ob- jective is to explicitly show why the claim in Ref. [3] is wrong while providing explicitly the steps which the au- thors completely ignored. Here we will refute the points raised in Ref. [3], while clarifying the analytics explicitly: • Step 1 : In Refs. [1, 2], we considered a generic po- tential, which is expanded in a Taylor series around the sub-Planckian VEV, φ 0 <M P as: V (φ)= V (φ 0 )+ V ′ (φ 0 )(φ − φ 0 )+ V ′′ (φ 0 ) 2 (φ − φ 0 ) 2 + V ′′′ (φ 0 ) 6 (φ − φ 0 ) 3 + V ′′′′ (φ 0 ) 24 (φ − φ 0 ) 4 (1) where we have truncated the Taylor expansion as: V (φ 0 ) >V ′ (φ 0 ) >V ′′ (φ 0 ) >V ′′′′ (φ 0 ) (in the Planckian unit) , which is also the necessary condi- tion for the convergence of the Taylor series. Note that φ 0 denotes the VEV where inflation occurs in its vicinity. • Step 2 : We can derive a simple expression for the tensor-to-scalar ratio, r, as, see [1, 2, 4]: r = 8 M 2 p (1 − ǫ V ) 2 [1 − (C E + 1)ǫ V ] 2 [1 − (3C E + 1)ǫ V + C E η V ] 2  dφ dln k  2 + ··· , (2) where C E = 4(ln 2 + γ E ) − 5 with γ E =0.5772 is the Euler-Mascheroni constant, ǫ V ,η V are slow roll parameters 2 , there are higher order terms in 1 An explicit example was provided earlier in the context of high scale MSSM inflation in presence of Hubble induced K¨ahler cor- rections in Ref. [4]. 2 We use the standard notations and for details readers can see Refs. [1, 2]. slow roll parameters, of order O(ǫ 2 V ), O(η 2 V ) ··· , which will give negligible contributions and would not alter the results of our discussion. We can now derive a bound on r(k) in terms of the momentum scale:       k⋆ ke dk k  r(k) 8      ≈ |Δφ| M p    1+ ....  <<1    ≈ |Δφ| M p , (3) where Δφ = φ ⋆ − φ e and we have neglected the contributions from the ....  terms as they are small compared to the leading order term due to the con- vergence of the series mentioned in Eq (1). Here φ e denotes the inflaton VEV at the end of inflation, and φ ⋆ denote the field VEV when the correspond- ing mode k ⋆ is leaving the Hubble patch during inflation. • Step 3 : In order to perform the momentum in- tegration in the left hand side of Eq (3) analyti- cally, we have used the following parameterization of r(k), which can be expressed as 3 : r(k)= r(k ⋆ )  k k ⋆  a+ b 2 ln( k k⋆ )+ c 6 ln 2 ( k k⋆ ) , (4) where a = n T − n S +1, b =(α T − α S ) , c =(κ T − κ S ) . (5) which are defined at the scale k ⋆ . These parameter- ization characterizes the spectral indices, n S ,n T , running of the spectral indices, α S ,α T , and run- ning of the running of the spectral indices, κ S ,κ T . Here the subscripts, (S, T ), represent the scalar and tensor modes. 3 Note that in the following expression, Eq. (4), we have taken running and running of the spectrum, while in Eq. (2) we have only taken the leading order contribution which mainly involves ǫ V , η V . The procedure is perfectly correct, since the higher order corrections are sub-leading. This is precisely by virtue of the Taylor expansion of the potential in the vicinity of φ 0 where inflation occurs.