# Gravity with Higher Derivatives in D-Dimensions

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## Abstract

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## 1. Introduction

- Reduction of action to the scalar tensor gravity. This way is effective for the action in the form$$\begin{array}{ccc}\hfill & & S=\frac{{m}_{D}^{D-2}}{2}\int {d}^{D}Z\sqrt{|{g}_{D}|}f\left(R\right).\hfill \end{array}$$
- Direct solution to equations of motion.
- Derivation of approximate equations provided that a system contains a small parameter.
- Method of trial functions.

## 2. Reduction of Action to the Scalar-Tensor Gravity

#### 2.1. Conformal Transformations in D Dimensions

#### 2.2. The Starobinsky Model

## 3. Direct Solution to Equations of Motion

- $f\left(R\right)$ gravity,
- $f\left(R\right)$ + Gauss–Bonnet gravity.

#### 3.1. $f\left(R\right)$ Gravity

#### 3.2. Starobinsky Model, Direct Calculation

#### 3.3. $f\left(R\right)$ + Gauss–Bonnet Gravity

## 4. Approximate Method

#### 4.1. Basic Idea

#### 4.2. Extension of the Model: Low Energies

#### 4.3. Extension of the Model: Moderate Energies

## 5. Method of Trial Functions

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Numerical solution of (19), (20) and (22) for ${d}_{1}={d}_{2}=3,{\beta}_{1}\left(0\right)=0.5,\phantom{\rule{4pt}{0ex}}{\dot{\beta}}_{1}\left(0\right)=0,\phantom{\rule{4pt}{0ex}}{\beta}_{2}\left(0\right)=0.1,\phantom{\rule{4pt}{0ex}}{\dot{\beta}}_{2}\left(0\right)=0.5,\phantom{\rule{4pt}{0ex}}\dot{R}\left(0\right)=0$. $R\left(0\right)\simeq 12.67452$ is found from Equation (23).

**Figure 4.**Numerical solution to the system of Equations (63)–(65) for initial conditions $\alpha \left(0\right)=7,$$\phantom{\rule{4pt}{0ex}}\dot{\alpha}\left(0\right)=1,\phantom{\rule{4pt}{0ex}}\beta \left(0\right)=4,\phantom{\rule{4pt}{0ex}}\dot{\beta}\left(0\right)=0,\phantom{\rule{4pt}{0ex}}\dot{R}\left(0\right)=0$. The initial condition $R\left(0\right)\simeq 11.999$ is found from Equation (66). The Lagrangian parameters are $a=200,c=-0.001,k=500$.

**Figure 5.**Numerical solution to the system of Equations (63)–(65) for initial conditions $\alpha \left(0\right)=15,$$\phantom{\rule{4pt}{0ex}}\beta \left(0\right)=2,\phantom{\rule{4pt}{0ex}}\dot{\alpha}\left(0\right)\simeq 0.404667,\phantom{\rule{4pt}{0ex}}\dot{\beta}\left(0\right)=0,\phantom{\rule{4pt}{0ex}}\dot{R}\left(0\right)=0$. $R\left(0\right)\simeq 2.09126$ is found from Equation (66). The Lagrangian parameters are $a=-2.77,c=-0.49,k=-2.98$.

**Figure 6.**Numerical solution to the system of Equations (63)–(65) for initial conditions $\alpha \left(0\right)=15,$$\phantom{\rule{4pt}{0ex}}\dot{\alpha}\left(0\right)\simeq 0.40467,\phantom{\rule{4pt}{0ex}}\beta \left(0\right)={b}_{c}\simeq 1.99303$, $\dot{\beta}\left(0\right)=0,\phantom{\rule{4pt}{0ex}}\dot{R}\left(0\right)=0$. $R\left(0\right)\simeq 2.0765$ is found from Equation (66). For the found numerical solution, $a=-2.77,c=-0.49,k=-2.98$.

**Figure 7.**The form of the potential (left) and kinetic term (right) for the parameters $n=2,b=1,a\phantom{\rule{3.33333pt}{0ex}}=-2,{c}_{V}=-8,{c}_{K}=15000.$ The potential minimum is in the point ${\varphi}_{m}\simeq 0.083$, ${m}_{D}=1.$

**Figure 8.**Auxiliary function $\mathsf{\Omega}$ vs. the minimization parameter ${\xi}_{1}$ for ${\xi}_{2}^{*}=9.57,{\xi}_{3}^{*}=1.85$ and $f\left(R\right)$-parameters $a=-2,b=1,c=-0.02$. The minimum of $\mathsf{\Omega}$ corresponds to ${\xi}_{1}^{*}=9.96$. $\mathsf{\Omega}({\xi}_{1}^{*}=9.96,{\xi}_{3}^{*}=1.85)=5.3\times {10}^{11}$.

**Figure 9.**Radii of two-dimensional subspaces vs. the Schwarzschild radial coordinate u for parameter values $a=-2,b=1,c=-0.02$, ${\xi}_{3}=1.85$.

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Rubin, S.G.; Popov, A.; Petriakova, P.M.
Gravity with Higher Derivatives in D-Dimensions. *Universe* **2020**, *6*, 187.
https://doi.org/10.3390/universe6100187

**AMA Style**

Rubin SG, Popov A, Petriakova PM.
Gravity with Higher Derivatives in D-Dimensions. *Universe*. 2020; 6(10):187.
https://doi.org/10.3390/universe6100187

**Chicago/Turabian Style**

Rubin, Sergey G., Arkadiy Popov, and Polina M. Petriakova.
2020. "Gravity with Higher Derivatives in D-Dimensions" *Universe* 6, no. 10: 187.
https://doi.org/10.3390/universe6100187