Anisotropic exponential cosmological solutions in 4th and 5th orders of Lovelock gravity

TL;DR

This paper derives anisotropic exponential cosmological solutions in higher-order Lovelock gravity theories, providing conditions on Hubble parameters for arbitrary dimensions and orders.

Contribution

It introduces a generalized framework for anisotropic exponential solutions in Lovelock gravity across arbitrary orders and dimensions.

Findings

Solutions expressed as conditions on Hubble parameters.

Extension of solutions to arbitrary Lovelock order.

Applicable to spaces of arbitrary dimension.

Abstract

Anisotropic exponential cosmological solutions for a space of arbitrary dimension filled with ordinary matter in the 4th and 5th orders of Lovelock gravity are obtained. Also we have supposed a generalization of such solutions on an arbitrary order. All the solutions are represented as a set of conditions on Hubble parameters.

Equations

$$G_{\nu }^{\mu }=\sum _{p=1}^{m-1}{\alpha }_p{G}_{\nu }^{(p)\mu }+\Lambda {\delta }_{\nu }^{\mu },$$

$$m=\frac{1}{2}n,\qquad\mbox{if $n$ is even,}$$

$$m=\frac{1}{2}(n+1),\qquad\mbox{if $n$ is odd,}$$

$$G_{\nu }^{(p)\mu }={\delta }_{\nu {\sigma }_1{\sigma }_2\cdots {\sigma }_{2p}}^{\mu {\lambda }_1{\lambda }_2\cdots {\lambda }_{2p}}{R}_{{\lambda }_1{\lambda }_2}^{{\sigma }_1{\sigma }_2}{R}_{{\lambda }_3{\lambda }_4}^{{\sigma }_3{\sigma }_4}\cdots R_{{\lambda }_{2p-1}{\lambda }_{2p}}^{{\sigma }_{2p-1}{\sigma }_{2p}},$$

$$G_{\mu \nu }=\alpha (R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu })+\beta g_{\mu \nu }$$

$$g_{\mu \nu }=\mathrm{diag}\{-1,t^{2p_1},\dots ,t^{2p_n}\},$$

$$\sum _i{p}_i=1,\sum _{i<j}{p}_i{p}_j=0.$$

$$w=1,\sum _i{p}_i=1,\sum _{i<j}{p}_i{p}_j=-\frac{1}{4{\alpha }_1}\cdot \frac{8\pi G}{c^4}{\epsilon }_0,$$

$$\sum _i{p}_i=3,\sum _{i<j<k<l}{p}_i{p}_j{p}_k{p}_l=0,$$

$$w=1/3,\sum _i{p}_i=3,\sum _{i<j<k<l}{p}_i{p}_j{p}_k{p}_l=-\frac{1}{96{\alpha }_2}\cdot \frac{8\pi G}{c^4}{\epsilon }_0,$$

$$\sum _i{p}_i=2p-1,\sum _{i_1<i_2<\dots <i_{2p}}{p}_{i_1}{p}_{i_2}\cdots p_{i_{2p}}=0,$$

$$\sum _i{p}_i=2p-1,\sum _{i_1<i_2<\dots <i_{2p}}{p}_{i_1}{p}_{i_2}\cdots p_{i_{2p}}=-\frac{1}{{\alpha }_p{2}^p(2p)!}\cdot \frac{8\pi G}{c^4}{\epsilon }_0.$$

$$g_{\mu \nu }=\mathrm{diag}\{-1,e^{2H_{1}t},\dots ,e^{2H_{n}t}\},$$

$${G^{(1)}}_{\mu \nu }+{\alpha }_2{G^{(2)}}_{\mu \nu }=\varkappa T_{\mu \nu }$$

$$\begin{array}[]{l}\sum_{i}H_{i}=0,\\ \sum_{i}H_{i}^{2}=(1-3w)\varkappa_{0},\vphantom{\dfrac{1}{2}}\\ \sum_{i}H_{i}^{4}=\dfrac{w-1}{2\alpha_{2}}\varkappa_{0}+\dfrac{(1-3w)^{2}}{2}\varkappa_{0}^{2}\end{array}$$

$${\alpha }_1{G^{(1)}}_{\mu \nu }+{\alpha }_2{G^{(2)}}_{\mu \nu }+{\alpha }_3{G^{(3)}}_{\mu \nu }=\varkappa T_{\mu \nu }$$

$$\begin{array}[]{l}\sigma_{1}=0,\\ \sigma_{4}=\dfrac{1}{2}\sigma_{2}^{2}-\dfrac{\alpha_{1}}{2\alpha_{2}}\sigma_{2}-\dfrac{1-5w}{16\alpha_{2}}\varkappa_{0},\vphantom{\dfrac{\dfrac{1}{2}}{\dfrac{1}{2}}}\\ \sigma_{6}=\dfrac{1}{3}\sigma_{3}^{2}+\dfrac{1}{4}\sigma_{2}^{3}-\dfrac{3\alpha_{1}}{8\alpha_{2}}\sigma_{2}^{2}+\dfrac{1}{96}\left(\dfrac{\alpha_{1}}{\alpha_{2}}-\dfrac{9(1-5w)}{2\alpha_{2}}\varkappa_{0}\right)\sigma_{2}+\dfrac{1-3w}{384\alpha_{3}}\varkappa_{0}\end{array}$$

$${\sigma }_s\equiv \sum _i{H}_i^s,$$

$${\zeta }_p\equiv \sum _{i_1}{H}_{i_1}\sum _{i_2\ne i_1}{H}_{i_2}\sum _{i_3\ne i_1,i_2}{H}_{i_3}\cdots \sum _{i_p\ne i_1,i_2,\dots i_{p-1}}{H}_{i_p},$$

$${\zeta }_{p(j)}\equiv \sum _{i_1\ne j}{H}_{i_1}\sum _{i_2\ne j,i_1}{H}_{i_2}\sum _{i_3\ne j,i_1,i_2}{H}_{i_3}\cdots \sum _{i_p\ne j,i_1,i_2,\dots i_{p-1}}{H}_{i_p},$$

$${\eta }_{p(j)}\equiv \sum _{i_1\ne j}{H}_{i_1}^2\sum _{i_2\ne j,i_1}{H}_{i_2}\sum _{i_3\ne j,i_1,i_2}{H}_{i_3}\cdots \sum _{i_p\ne j,i_1,i_2,\dots i_{p-1}}{H}_{i_p},$$

$$\begin{array}[]{l}\displaystyle\zeta_{p}=\sum_{i_{1}}H_{i_{1}}\cdots\sum_{i_{p-1}\neq\ldots}H_{i_{p-1}}(\sigma_{1}-H_{i_{1}}-H_{i_{2}}-\cdots-H_{i_{p-1}})=\\ \displaystyle\quad{}=\sigma_{1}\zeta_{p-1}-(p-1)\sum_{i_{1}}H_{i_{1}}\cdots\sum_{i_{p-2}\neq\ldots}H_{i_{p-2}}\sum_{i_{p-1}\neq\ldots}H_{i_{p-1}}^{2}=\\ \displaystyle\quad{}=\sigma_{1}\zeta_{p-1}-(p-1)\sum_{i_{1}}H_{i_{1}}\cdots\sum_{i_{p-2}\neq\ldots}H_{i_{p-2}}(\sigma_{2}-H_{i_{1}}^{2}-\cdots-H_{i_{p-2}})=\\ \displaystyle\quad{}=\sigma_{1}\zeta_{p-1}-(p-1)\sigma_{2}\zeta_{p-2}+(p-1)(p-2)\sum_{i_{1}}H_{i_{1}}\cdots\sum_{i_{p-3}\neq\ldots}H_{i_{p-3}}\sum_{i_{p-2}\neq\ldots}H_{i_{p-2}}^{3}=\\ \displaystyle\quad{}=\cdots=\sum_{k=1}^{p-2}(-1)^{k-1}\displaystyle\frac{(p-1)!}{(p-k)!}\sigma_{k}\zeta_{p-k}+(-1)^{p-2}(p-1)(p-2)\cdots 4\cdot 3\cdot 2\times\\ \displaystyle\quad{}\times\sum_{i_{1}}H_{i_{1}}(\sigma_{p-1}-H_{i_{1}}^{p-1})=\sum_{k=1}^{p}(-1)^{k-1}\displaystyle\frac{(p-1)!}{(p-k)!}\sigma_{k}\zeta_{p-k},\end{array}$$

$$\begin{array}[]{l}\zeta_{0}\equiv 1,\\ \displaystyle\zeta_{p}=\sum_{k=1}^{p}(-1)^{k-1}\displaystyle\frac{(p-1)!}{(p-k)!}\sigma_{k}\zeta_{p-k}.\end{array}$$

$$\begin{array}[]{l}\zeta_{0}=1,\qquad\zeta_{1}=0,\qquad\zeta_{2}=-\sigma_{2},\qquad\zeta_{3}=2\sigma_{3},\qquad\zeta_{4}=3\sigma_{2}^{2}-6\sigma_{4},\\ \zeta_{5}=-20\sigma_{3}\sigma_{2}+24\sigma_{5},\qquad\zeta_{6}=-15\sigma_{2}^{3}+90\sigma_{4}\sigma_{2}+40\sigma_{3}^{2}-120\sigma_{6},\vphantom{\dfrac{1}{2}}\\ \zeta_{7}=210\sigma_{2}^{2}\sigma_{3}-504\sigma_{2}\sigma_{5}-420\sigma_{3}\sigma_{4}+720\sigma_{7},\\ \zeta_{8}=105\sigma_{2}^{4}-1260\sigma_{4}\sigma_{2}^{2}-1120\sigma_{3}^{2}\sigma_{2}+3360\sigma_{6}\sigma_{2}+2688\sigma_{5}\sigma_{3}+1260\sigma_{4}^{2}-5040\sigma_{8},\vphantom{\dfrac{1}{2}}\\ \zeta_{9}=-2520\sigma_{3}\sigma_{2}^{3}+9072\sigma_{5}\sigma_{2}^{2}+12096\sigma_{4}\sigma_{3}\sigma_{2}-25920\sigma_{7}\sigma_{2}+2240\sigma_{3}^{3}+20160\sigma_{6}\sigma_{3}-\\ \qquad\quad{}\vphantom{\dfrac{1}{2}}-18144\sigma_{5}\sigma_{4}+40320\sigma_{9},\\ \zeta_{10}=-945\sigma_{2}^{5}+18900\sigma_{4}\sigma_{2}^{3}+25200\sigma_{3}^{2}\sigma_{2}^{2}-75600\sigma_{6}\sigma_{2}^{2}-120960\sigma_{5}\sigma_{3}\sigma_{2}-56700\sigma_{4}^{2}\sigma_{2}+\\ \qquad\quad{}+\vphantom{\dfrac{1}{2}}226800\sigma_{8}\sigma_{2}-50400\sigma_{4}\sigma_{3}^{2}+172800\sigma_{7}\sigma_{3}+151200\sigma_{6}\sigma_{4}+72576\sigma_{5}^{2}-362880\sigma_{10}.\end{array}$$

$${\zeta }_{p(j)}=\sum _{k=1}^p(-1{)}^{k-1}\frac{(p-1)!}{(p-k)!}({\sigma }_k-H_j^k){\zeta }_{p-k(j)},$$

$${\eta }_{p(j)}=\sum _{k=1}^p(-1{)}^{k-1}\frac{(p-1)!}{(p-k)!}({\sigma }_{k+1}-H_j^{k+1}){\zeta }_{p-k(j)}.$$

$$\begin{array}[]{l}\zeta_{1(j)}=-H_{j},\qquad\zeta_{2(j)}=2H_{j}^{2}-\sigma_{2},\qquad\zeta_{3(j)}=-6H_{j}^{3}+3\sigma_{2}H_{j}+2\sigma_{3},\vphantom{\displaystyle\frac{1}{2}}\\ \zeta_{4(j)}=24H_{j}^{4}-12\sigma_{2}H_{j}^{2}-8\sigma_{3}H_{j}+3\sigma_{2}^{2}-6\sigma_{4},\vphantom{\displaystyle\frac{1}{2}}\\ \zeta_{5(j)}=-120H_{j}^{5}+60\sigma_{2}H_{j}^{3}+40\sigma_{3}H_{j}^{2}-15\sigma_{2}^{2}H_{j}+30\sigma_{4}H_{j}-20\sigma_{3}\sigma_{2}+24\sigma_{5},\vphantom{\displaystyle\frac{1}{2}}\\ \zeta_{6(j)}=720H_{j}^{6}-360\sigma_{2}H_{j}^{4}-240\sigma_{3}H_{j}^{3}+90\sigma_{2}^{2}H_{j}^{2}-180\sigma_{4}H_{j}^{2}+120\sigma_{3}\sigma_{2}H_{j}-\vphantom{\displaystyle\frac{1}{2}}\\ \qquad\quad{}-144\sigma_{5}H_{j}-15\sigma_{2}^{3}+90\sigma_{4}\sigma_{2}+40\sigma_{3}^{2}-120\sigma_{6},\vphantom{\displaystyle\frac{1}{2}}\\ \zeta_{7(j)}=-5040H_{j}^{7}+2520\sigma_{2}H_{j}^{5}+1680\sigma_{3}H_{j}^{4}-630\sigma_{2}^{2}H_{j}^{3}+1260\sigma_{4}H_{j}^{3}-840\sigma_{3}\sigma_{2}H_{j}^{2}+\vphantom{\displaystyle\frac{1}{2}}\\ \qquad\quad{}+1008\sigma_{5}H_{j}^{2}+105\sigma_{2}^{3}H_{j}-630\sigma_{4}\sigma_{2}H_{j}-280\sigma_{3}^{2}H_{j}+840\sigma_{6}H_{j}+210\sigma_{3}\sigma_{2}^{2}-\vphantom{\displaystyle\frac{1}{2}}\\ \qquad\quad{}-504\sigma_{5}\sigma_{2}-420\sigma_{4}\sigma_{3}+720\sigma_{7},\vphantom{\displaystyle\frac{1}{2}}\\ \zeta_{8(j)}=40320H_{j}^{8}-20160\sigma_{2}H_{j}^{6}-13440\sigma_{3}H_{j}^{5}+5040\sigma_{2}^{2}H_{j}^{4}-10080\sigma_{4}H_{j}^{4}+6720\sigma_{3}\sigma_{2}H_{j}^{3}-\vphantom{\displaystyle\frac{1}{2}}\\ \qquad\quad{}\vphantom{\dfrac{1}{2}}-8064\sigma_{5}H_{j}^{3}-840\sigma_{2}^{3}H_{j}^{2}+5040\sigma_{4}\sigma_{2}H_{j}^{2}+2240\sigma_{3}^{2}H_{j}^{2}-6720\sigma_{6}H_{j}^{2}-1680\sigma_{3}\sigma_{2}^{2}H_{j}+\vphantom{\displaystyle\frac{1}{2}}\\ \qquad\quad{}+4032\sigma_{5}\sigma_{2}H_{j}+3360\sigma_{4}\sigma_{3}H_{j}-5760\sigma_{7}H_{j}+105\sigma_{2}^{4}-1260\sigma_{4}\sigma_{2}^{2}-1120\sigma_{3}^{2}\sigma_{2}+\vphantom{\displaystyle\frac{1}{2}}\\ \qquad\quad{}+3360\sigma_{6}\sigma_{2}+2688\sigma_{5}\sigma_{3}+1260\sigma_{4}^{2}-5040\sigma_{8},\vphantom{\dfrac{1}{2}}\vphantom{\displaystyle\frac{1}{2}}\\ \displaystyle\zeta_{9(j)}=\sum_{s_{1}l_{1}+\ldots+s_{k}l_{k}+m=9}(-1)^{l_{1}+\ldots+l_{k}+9}\dfrac{9!}{s_{1}^{l_{1}}\cdot\ldots\cdot s_{k}^{l_{k}}l_{1}!\cdot\ldots\cdot l_{k}!}\sigma_{s_{1}}^{l_{1}}\cdot\ldots\cdot\sigma_{s_{k}}^{l_{k}}H_{j}^{m},\vphantom{\displaystyle\frac{1}{2}}\end{array}$$

$$\begin{array}[]{l}\eta_{3(j)}=3\sigma_{2}H_{j}^{2}-\sigma_{2}^{2}-6H_{j}^{4}+2\sigma_{3}H_{j}+3\sigma_{4},\vphantom{\displaystyle\frac{1}{2}}\\ \eta_{5(j)}=-120H_{j}^{6}+60\sigma_{2}H_{j}^{4}-15\sigma_{2}^{2}H_{j}^{2}-20\sigma_{3}\sigma_{2}H_{j}+40\sigma_{3}H_{j}^{3}+30\sigma_{4}H_{j}^{2}+\vphantom{\displaystyle\frac{1}{2}}\\ \qquad\qquad{}+3\sigma_{2}^{3}-18\sigma_{4}\sigma_{2}-8\sigma_{3}^{2}+24\sigma_{5}H_{j}+24\sigma_{6},\vphantom{\displaystyle\frac{1}{2}}\\ \eta_{7(j)}=2520\sigma_{2}H_{j}^{6}-630\sigma_{2}^{2}H_{j}^{4}-840\sigma_{3}\sigma_{2}H_{j}^{3}+105\sigma_{2}^{3}H_{j}^{2}-630\sigma_{4}\sigma_{2}H_{j}^{2}+\vphantom{\displaystyle\frac{1}{2}}\\ \qquad\quad{}+210\sigma_{3}\sigma_{2}^{2}H_{j}-504\sigma_{5}\sigma_{2}H_{j}-15\sigma_{2}^{4}+180\sigma_{4}\sigma_{2}^{2}+160\sigma_{3}^{2}\sigma_{2}-\vphantom{\displaystyle\frac{1}{2}}\\ \qquad\quad{}-480\sigma_{6}\sigma_{2}-5040H_{j}^{8}+1680\sigma_{3}H_{j}^{5}+1260\sigma_{4}H_{j}^{4}+1008\sigma_{5}H_{j}^{3}-280\sigma_{3}^{2}H_{j}^{2}+\vphantom{\displaystyle\frac{1}{2}}\\ \qquad\quad{}+840\sigma_{6}H_{j}^{2}-420\sigma_{4}\sigma_{3}H_{j}-384\sigma_{5}\sigma_{3}-180\sigma_{4}^{2}+720\sigma_{7}H_{j}+720\sigma_{8},\vphantom{\displaystyle\frac{1}{2}}\\ \eta_{9(j)}=H_{j}\zeta_{9(j)}+105\sigma_{2}^{5}-2100\sigma_{4}\sigma_{2}^{3}-2800\sigma_{3}^{2}\sigma_{2}^{2}+8\cdot 7\cdot 6\cdot 5\cdot 5\sigma_{6}\sigma_{2}^{2}+10\cdot 8\cdot 7\cdot 6\cdot 4\sigma_{5}\sigma_{3}\sigma_{2}+\vphantom{\displaystyle\frac{1}{2}}\\ \qquad\quad{}+10\cdot 9\cdot 7\cdot 10\sigma_{4}^{2}\sigma_{2}-7!\cdot 5\sigma_{8}\sigma_{2}+10\cdot 8\cdot 7\cdot 5\cdot 2\sigma_{4}\sigma_{3}^{2}-10\cdot 8\cdot 6\cdot 5\cdot 4\cdot 2\sigma_{7}\sigma_{3}-\vphantom{\dfrac{1}{2}}\\ \qquad\quad{}-10\cdot 8\cdot 7\cdot 6\cdot 5\sigma_{6}\sigma_{4}+8!\sigma_{10}-8\cdot 7\cdot 6\cdot 4\cdot 3\cdot 2\sigma_{5}^{2},\end{array}$$

$$g_{\mu \nu }=\mathrm{diag}\{-1,e^{2H_{1}t},\dots ,e^{2H_{n}t}\},$$

$${G^{(p)0}}_0=2^p(2p)!\sum _{i_1<i_2<\cdots <i_{2p}}{H}_{i_1}{H}_{i_2}\cdots H_{i_{2p}}=2^p{\zeta }_{2p},$$

Full text

Introduction

A huge amount of investigations in cosmology during the last quarter of a sentury motivate theoreticians to develop new gravitational theories. Many of these theories are modifications of General Relativity (GR).

Any modification of GR includes additional fields (such as scalar field, torsion, second metric tensor etc.) or higher derivatives in field equations or extra spatial dimensions. It is impossible to avoid all above-mentioned features.

Lovelock gravity [1] has no additional fieds and no higher derivatives. It is based on

**Lovelock theorem

**If in $n$-dimensional riemannian space one needs tensor $G_{\mu\nu}$ (gravity field tensor) with the following features:

$G_{\mu\nu}$ is symmetric: $G_{\mu\nu}=G_{\nu\mu}$, 2. 2.

$G_{\mu\nu}$ is divergence free: $\nabla^{\mu}G_{\mu\nu}=0$, 3. 3.

$G_{\mu\nu}$ is a concomitant of the metric tensor and its first two derivatives:

$G_{\mu\nu}=G_{\mu\nu}(g_{\mu\nu},\partial_{\alpha}g_{\mu\nu},\partial_{\alpha}\partial_{\beta}g_{\mu\nu})$,

then general expression for $G_{\mu\nu}$ is

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

$\alpha_{p}$, $\Lambda$ are arbitrary constants, $\delta^{\mu_{1}\cdots\mu_{k}}_{\nu_{1}\cdots\nu_{k}}$ is multidimensional delta-symbol, which equals to one, if $\nu_{1}\cdots\nu_{k}$ is even permutation of $\mu_{1}\cdots\mu_{k}$, equals to minus one, if odd, and equals to zero in other cases. We will call tensor (2) a $p$-th order Lovelock tensor.

It is easy to understand that in 4-dimensional spacetime only 0-th and 1-st Lovelock tensors are nonzero, so

[TABLE]

and we have ordinary Hilbert-Einstein equations with cosmological term.

Hence if we need new results in Lovelock gravity then we should consider spacetimes with 5 dimensions or more. But in such a case we should explain an invisibility of extra dimensions. We can do this by means of Kluza-Klein approach: extra spacial dimensions are considered as closed and small.

But such an approach means that space is anisotropic. So we should look for anisotropic solutions of gravity field equations. And it is interesting to consider maximally anisotropic space: it might arise isotropization of 3-dimensional visible space or invisible dimensions might behave in different ways.

1 Earlier-obtained anisotropic cosmological solutions

1.1 Power-law solutions

Consider metric tensor with power-law scale factors:

[TABLE]

where $p_{i},$ $i=1,\ldots,n$ are constant values (power-law parameters). In the first order (i. e. in ordinary GR) we have Kasner solution for an empty space [2]:

[TABLE]

and Jacobs solution for maximally stiff fluid ($p=w\rho$) [3]:

[TABLE]

where $\varepsilon_{0}$ is initial matter density. In the second order equations $\alpha_{2}{G^{(2)}}_{\mu\nu}=\varkappa T_{\mu\nu}$ have an analogue of Kasner solution

[TABLE]

discovered by N. Deruelle [4] and rediscovered by A. Toporensky and P. Tretyakov [5] and an analogue of Jacobs solution

[TABLE]

discovered by author [6].

For an arbitrary $p$-th order ($\alpha_{p}{G^{(p)}}_{\mu\nu}=\varkappa T_{\mu\nu}$, hereinafter $\varkappa\equiv 8\pi G/c^{4}$) Kasner solution has been generalized by S. Pavluchenko [7] and, independently, by author [8]:

[TABLE]

Jacobs solution has been generalized by author [8]:

[TABLE]

Unfortunately all these solutions involve only one order of Lovelock gravity (without involving lower orders) and, secondly, solutions with matter order specific value of EoS parameter $w$.

1.2 Exponential solutions

Such shortcomings are absent for exponential solutions:

[TABLE]

where $H_{i},$ $i=1,\ldots,n$ are constant values (Hubble parameters). In the second order equations

[TABLE]

have solution [9]

[TABLE]

(hereinafter $\varkappa_{0}\equiv\dfrac{8\pi G}{c^{4}}\varepsilon_{0}$). In the third order equations

[TABLE]

have solution [8]

[TABLE]

(hereinafter $\sigma_{s}\equiv\sum_{i}H_{i}^{s},$ $s=1,2,\ldots$).

2 New solutions in 4th and 5th orders

In this paper we will obtain exponential solutions in 4th and 5th orders of Lovelock gravity. Firstly define the notations:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $s$, $p$ are arbitrary integers, the summation is over all the spatial dimensions. And find a relation between these values:

[TABLE]

where it is noted: $\zeta_{0}\equiv 1$. Thus,

[TABLE]

Now express $\zeta_{p}$ (we use condition $\sigma_{1}=0$ obtained in [9]):

[TABLE]

Acting as in (20) we obtain

[TABLE]

[TABLE]

Using these equations one can obtain

[TABLE]

[TABLE]

Now consider field equations. Taking metrics in the form

[TABLE]

we have

[TABLE]

[TABLE]

Using (22), (25), (26), one can obtain

[TABLE]

Thus equations

[TABLE]

take form

[TABLE]

where

[TABLE]

In the 3rd order of Lovelock gravity ($\alpha_{4}=\alpha_{5}=0$) it is easy to obtain

[TABLE]

From these equations we have solution (15).

In the 4th order ($\alpha_{5}=0$) one can obtain

[TABLE]

so solution is

[TABLE]

In the 5th order equations (32) imply

[TABLE]

from what we have

[TABLE]

3 Exponential solution in an arbitrary order (supposition)

Generalizing these equalities we may suppose that in an arbitrary order equations

[TABLE]

take form

[TABLE]

where

[TABLE]

so

[TABLE]

from what we have

[TABLE]

Unfortunately, it is only supposition, but I hope to prove it in the next work.

Conclusions

Anisotropic exponential cosmological solutions for a space of arbitrary dimension filled with ordinary matter in the 4th and 5th orders of Lovelock gravity were obtained. Also we have supposed a generalization of such solutions on an arbitrary order.

All the solutions are represented as a set of conditions on Hubble parameters. Unfortunately, it is the problem to write down every Hubble parameter as a fuction of parameters $n$, $\alpha_{i}$, $w$ and $\varkappa_{0}$. Moreover, such conditions may be uncompatible. For the second order of Lovelock gravity this problem was investigated in [10].

Acknowledgements

The author is grateful to Alexey V. Toporensky and Alexander A. Reshetnyak for helpful discussions. The research was fulfilled within the RFBR Project No. 17-02-01333.