# the Starobinsky model

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

# 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.
$b= b_1 b_3.$

# Summary

• 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}}.
$\Lambda \simeq 5 \times 10^{16} \text{GeV} \left( \frac{2\times 10^{-4}}{|b_1|} \right)^{\frac{1}{2}}.$

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.