Reinterpretation of

the Starobinsky model

Takahiro Terada

(JSPS fellow; The University of Tokyo, DESY)

Based on arXiv:1507.04344 [hep-th] with T. Asaka, S. Iso, H. Kawai, K. Kohri, T. Noumi.

P. A. R. Ade et al. [Planck Collaboration],  "Planck 2015 results. XX. Constraints on inflation", arXiv:1502.02114 [astro-ph.CO].

Peculiar Form of the Action

S= \int \text{d}^4 x \sqrt{-g} \left(-\frac{1}{2} M_{\text{P}}^2 R + \frac{M_{\text{P}}^2}{12 m^2}R^2 \right)
S=d4xg(12MP2R+MP212m2R2)S= \int \text{d}^4 x \sqrt{-g} \left(-\frac{1}{2} M_{\text{P}}^2 R + \frac{M_{\text{P}}^2}{12 m^2}R^2 \right)


1.  Scale invariance




2.  Extra dimensions

Higher dimensional action

S= \Lambda^{D} \int \text{d}^D x \sqrt{-g} \sum_{n=0} b_n \left( \frac{R_D}{\Lambda^{2}} \right)^n
S=ΛDdDxgn=0bn(RDΛ2)nS= \Lambda^{D} \int \text{d}^D x \sqrt{-g} \sum_{n=0} b_n \left( \frac{R_D}{\Lambda^{2}} \right)^n


= c \int \text{d}^4 x \sqrt{-g} \sum_{n=0} b_n \Lambda^4 \left( \frac{R}{\Lambda^{2}} \right)^n
=cd4xgn=0bnΛ4(RΛ2)n = c \int \text{d}^4 x \sqrt{-g} \sum_{n=0} b_n \Lambda^4 \left( \frac{R}{\Lambda^{2}} \right)^n


c=(L \Lambda)^{D-4} \simeq 5 \times 10^8 \quad \text{for} \, \, L=30/\Lambda.
c=(LΛ)D45×108forL=30/Λ.c=(L \Lambda)^{D-4} \simeq 5 \times 10^8 \quad \text{for} \, \, L=30/\Lambda.

Tuning one parameter

(in addition to the cosmological constant)

L= -\frac{1}{2}M_{\text{P}}^2 R + \frac{M_{\text{P}}^2}{12m^2}\left( R^2 + \sum_{n=3}^{\infty} b_{n} \left(- \frac{6 m^2}{b_1}\right)^{2-n} R^n \right)
L=12MP2R+MP212m2(R2+n=3bn(6m2b1)2nRn)L= -\frac{1}{2}M_{\text{P}}^2 R + \frac{M_{\text{P}}^2}{12m^2}\left( R^2 + \sum_{n=3}^{\infty} b_{n} \left(- \frac{6 m^2}{b_1}\right)^{2-n} R^n \right)
c b_1 \Lambda^2 =- \frac{M_{\text{P}}^2}{2}, \qquad c= \frac{M_{\text{P}}^2}{12m^2}\simeq 5 \times 10^8,
cb1Λ2=MP22,c=MP212m25×108, c b_1 \Lambda^2 =- \frac{M_{\text{P}}^2}{2}, \qquad c= \frac{M_{\text{P}}^2}{12m^2}\simeq 5 \times 10^8,




|b_1 | \ll 1.
b11.|b_1 | \ll 1.

Modified potential

V=V_{\text{Starobinsky}}\times \left( 1-\frac{b}{2}e^{\sqrt{2/3}\phi}\left(1-e^{-\sqrt{2/3}\phi} \right) \right) + O(b^2)
V=VStarobinsky×(1b2e2/3ϕ(1e2/3ϕ))+O(b2)V=V_{\text{Starobinsky}}\times \left( 1-\frac{b}{2}e^{\sqrt{2/3}\phi}\left(1-e^{-\sqrt{2/3}\phi} \right) \right) + O(b^2)
V_{\text{Starobinsky}}=\frac{3}{4}m^2\left(1-e^{-\sqrt{2/3}\phi }\right)^2
VStarobinsky=34m2(1e2/3ϕ)2V_{\text{Starobinsky}}=\frac{3}{4}m^2\left(1-e^{-\sqrt{2/3}\phi }\right)^2


b= b_1 b_3.
b=b1b3.b= b_1 b_3.


  • Starobinsky model has a peculiar form of the action.
\Lambda \simeq 5 \times 10^{16} \text{GeV} \left( \frac{2\times 10^{-4}}{|b_1|} \right)^{\frac{1}{2}}.
Λ5×1016GeV(2×104b1)12.\Lambda \simeq 5 \times 10^{16} \text{GeV} \left( \frac{2\times 10^{-4}}{|b_1|} \right)^{\frac{1}{2}}.

If the deviation is observed, it implies the fundamental scale,

  • It may be originated from extra dimensions.
  • One tuning is needed to be consistent with data.

Reinterpretation of the Starobinsky model

By Takahiro Terada

Reinterpretation of the Starobinsky model

Slides for a 15 min presentation in Lepton Photon 2015.

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