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.

The 27th International Symposium on Lepton Photon Interactions at High Energies, Ljubljana, Slovenia, August 21, 2015

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= \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)

Approaches

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

Compactification

= c \int \text{d}^4 x \sqrt{-g} \sum_{n=0} b_n \Lambda^4 \left( \frac{R}{\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

LARGE

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

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= -\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,

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,

Take

then,

Take

|b_1 | \ll 1.

|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=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

V_{\text{Starobinsky}}=\frac{3}{4}m^2\left(1-e^{-\sqrt{2/3}\phi }\right)^2

where

b= b_1 b_3.

b= b_1 b_3.