# Symmetries of Differential Equations in Cosmology

^{1}

^{2}

^{3}

^{4}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Point Transformations

#### 2.1. Prolongation of Point Transformations

#### 2.2. Invariance of Functions

## 3. Symmetries of Differential Equations

## 4. The Conservative Holonomic Dynamical System

## 5. Types of Symmetries

#### 5.1. Lie Symmetries

**Definition**

**1.**

#### 5.2. Lie Point Symmetries

#### 5.3. Lie Point Symmetries and First Integrals

**Proposition**

**1.**

#### 5.4. Noether Point Symmetries

**Definition**

**2.**

#### 5.5. First Integral Defined by a Noether Symmetry

**Proposition**

**2.**

## 6. Generalized Killing Equations

#### 6.1. How to Solve the Generalized Killing Equations

## 7. The Inverse Noether Theorem

## 8. Symmetries of SODEs in Flat Space

#### 8.1. The Case of sl(3, R) Algebra

#### 8.2. The Case of the sl (2, R) Algebra

## 9. Symmetries of SODEs and the Geometry of the Underline Space

#### 9.1. Collineations

## 10. Motion and Symmetries in a Riemannian Space

#### 10.1. Lie Point Symmetries of (75)

- Case I Lie point symmetries due to the affine algebra. The resulting Lie point symmetries are$$\mathbf{X}=\left(\frac{1}{2}{d}_{1}{a}_{1}t+{d}_{2}\right){\partial}_{t}+{a}_{1}{Y}^{i}{\partial}_{i},$$$${L}_{Y}{V}^{,i}+{d}_{1}{V}^{,i}=0.$$
- Case IIa The Lie point symmetries are generated by the gradient homothetic algebra and ${Y}^{i}\ne {V}^{,i}$. The Lie point symmetries are$$\mathbf{X}=2\psi \int T\left(t\right)dt{\partial}_{t}+T\left(t\right){Y}^{i}{\partial}_{i},$$$${\mathcal{L}}_{\mathbf{Y}}{V}^{,i}+4\psi {V}^{,i}+{a}_{1}{Y}^{i}=0.$$
- Case IIb The Lie point symmetries are generated by the gradient HV ${Y}^{i}=\kappa {V}^{,i},$ where $\kappa $ is a constant. In this case, the potential is the function generating the gradient HV and the Lie symmetry vectors are$$\mathbf{X}=\left(-{c}_{1}\sqrt{\psi k}cos\left(2\sqrt{\frac{\psi}{k}}t\right)+{c}_{2}\sqrt{\psi k}sin\left(2\sqrt{\frac{\psi}{k}}t\right)\right){\partial}_{t}+\left({c}_{1}sin\left(2\sqrt{\frac{\psi}{k}}t\right)+{c}_{2}cos\left(2\sqrt{\frac{\psi}{k}}t\right)\right){H}^{,i}{\partial}_{i}.$$
- Case IIIa The Lie point symmetries are due to the proper special projective algebra. In this case, the Lie symmetry vectors are (the index J counts the gradient KVs)$${\mathbf{X}}_{J}=\left(C\left(t\right){S}_{J}+D\left(t\right)\right){\partial}_{t}+T\left(t\right){Y}^{i}{\partial}_{i},$$$${D}_{,t}=\frac{1}{2}{d}_{1}T\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{4pt}{0ex}}{T}_{,tt}={a}_{1}T\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{T}_{,t}={c}_{2}C\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{D}_{,tt}={d}_{c}C\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{C}_{,t}={a}_{0}T$$$$\begin{array}{cc}\hfill {\mathcal{L}}_{Y}{V}^{,i}+2{a}_{0}S{V}^{,i}+{d}_{1}{V}^{,i}+{a}_{1}{Y}^{i}& =0,\hfill \end{array}$$$$\begin{array}{cc}\hfill \left({S}_{,k}{\delta}_{j}^{i}+2S{,}_{j}{\delta}_{k}^{i}\right){V}^{,k}+\left(2{Y}^{i}{}_{;\phantom{\rule{3.33333pt}{0ex}}j}-{a}_{0}S{\delta}_{j}^{i}\right){c}_{2}-{d}_{c}{\delta}_{j}^{i}& =0.\hfill \end{array}$$
- Case IIIb Lie point symmetries are due to the proper special projective algebra and ${Y}_{J}^{i}=\lambda {S}_{J}{V}^{,i},$ in which ${V}^{,i}$ is a gradient HV, and ${S}_{J}^{,i}$ is a gradient KV. The Lie point symmetry vectors are$${X}_{J}=\left(C\left(t\right){S}_{J}+{d}_{1}\right){\partial}_{t}+T\left(t\right)\lambda {S}_{J}{V}^{,i}{\partial}_{i},$$$${T}_{,tt}+2{C}_{,t}={\lambda}_{1}T\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{T}_{,t}={\lambda}_{2}C\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{C}_{,t}={a}_{0}T$$$$\begin{array}{cc}\hfill {\mathcal{L}}_{{Y}_{J}}{V}^{,i}+{\lambda}_{1}{S}_{J}{V}^{,i}& =0,\hfill \end{array}$$$$\begin{array}{cc}\hfill C\left({\lambda}_{1}{S}_{J}{\delta}_{j}^{i}+2{S}_{J}{,}_{j}{V}^{,i}\right)+{\lambda}_{2}\left(2\lambda {S}_{J,j}{V}^{.i}+\left(2\lambda {S}_{J}-{a}_{0}{S}_{J}\right){\delta}_{j}^{i}\right)& =0.\hfill \end{array}$$

#### 10.2. Noether Point Symmetries of (75)

- Case I. The HV satisfies the condition:$${V}_{,k}{Y}^{k}+2{\psi}_{Y}V+{c}_{1}=0.$$$$\mathbf{X}=2{\psi}_{Y}t{\partial}_{t}+{Y}^{i}{\partial}_{i},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}f={c}_{1}t,$$$$\mathsf{\Phi}=2{\psi}_{Y}tE-{g}_{ij}{Y}^{i}{\dot{x}}^{j}+{c}_{1}t.$$
- Case II. The metric admits the gradient KVs ${S}_{J}$, the gradient HV ${H}^{,i}$ and the potential satisfies the condition$${V}_{,k}{Y}^{,k}+2{\psi}_{Y}V={c}_{2}Y+d.$$$$\mathbf{X}=2{\psi}_{Y}\int T\left(t\right)dt{\partial}_{t}+T\left(t\right){S}_{J}^{,i}{\partial}_{i}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}f\left(t,{x}^{k}\right)={T}_{,t}{S}_{J}\left({x}^{k}\right)\phantom{\rule{3.33333pt}{0ex}}+d\int Tdt,$$$${T}_{,tt}={c}_{2}T\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{K}_{,t}=d\int Tdt+\mathrm{constant},$$$$\overline{\mathsf{\Phi}}={\psi}_{Y}E\int Tdt\phantom{\rule{3.33333pt}{0ex}}-T{g}_{ij}{H}^{i}{\dot{x}}^{j}+{T}_{,t}H+d\int Tdt.$$

#### 10.3. Point Symmetries of Constrained Lagrangians

#### 10.3.1. Lie Point Symmetries of (101)

#### 10.3.2. Noether Point Symmetries of (101)

## 11. Symmetries in Cosmology

#### 11.1. FRW Spacetime and the $\Lambda $CDM Cosmological Model

#### 11.2. Scalar-Field Cosmology

- For constant potential $V\left(\theta \right)={V}_{0},$ the system admits the extra Noether point symmetry $x{\partial}_{y}-y{\partial}_{x}$ with first integral the angular momentum ${r}^{2}\dot{\theta}=const.$
- For the exponential potential ${V}_{eff}\left(x,y\right)={r}^{2}{e}^{-d\theta}$, Lagrangian (120) admits an extra Noether point symmetry provided by the proper HV of the two-dimensional flat space,$${X}_{\left(\varphi \right)1}=2t{\partial}_{t}+\left(x+\frac{4}{d}y\right){\partial}_{x}+\left(y+\frac{4}{d}x\right){\partial}_{y},$$$${\mathsf{\Phi}}_{\left(\varphi \right)1}=\left(x+\frac{4}{d}y\right)\dot{x}-\left(y+\frac{4}{d}x\right)\dot{y}$$$${\mathsf{\Phi}}_{\left(\varphi \right)2}=\dot{x}-\dot{y}.$$
- Finally, when ${V}_{eff}(x,y)=\frac{1}{2}\left({\omega}_{1}{x}^{2}-{\omega}_{2}{y}^{2}\right)$, that is, $\tilde{V}\left(\theta \right)=\frac{1}{2}\left({\omega}_{1}{cosh}^{2}\left(\theta \right)-{\omega}_{2}{sinh}^{2}\left(\theta \right)\right)$, the dynamical system admits four extra Noether point symmetries$$\begin{array}{cc}\hfill {X}_{{\left(\varphi \right)}_{2}}& =sinh\left(\sqrt{{\omega}_{1}}t\right){\partial}_{x}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{X}_{{\left(\varphi \right)}_{3}}=cosh\left(\sqrt{{\omega}_{1}}t\right){\partial}_{x}\phantom{\rule{3.33333pt}{0ex}},\hfill \\ \hfill {X}_{{\left(\varphi \right)}_{4}}& =sinh\left(\sqrt{{\omega}_{2}}t\right){\partial}_{y}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{X}_{{\left(\varphi \right)}_{5}}=cosh\left(\sqrt{{\omega}_{2}}t\right){\partial}_{y}\phantom{\rule{3.33333pt}{0ex}},\hfill \end{array}$$$$\begin{array}{cc}\hfill {I}_{{n}_{2}}& =sinh\left(\sqrt{{\omega}_{1}}t\right)\dot{x}-\sqrt{{\omega}_{1}}cosh\left(\sqrt{{\omega}_{1}}t\right)x,\hfill \\ \hfill {I}_{{n}_{3}}& =cosh\left(\sqrt{{\omega}_{1}}t\right)\dot{x}-\sqrt{{\omega}_{1}}sinh\left(\sqrt{{\omega}_{1}}t\right)x,\hfill \\ \hfill {I}_{{n}_{4}}& =sinh\left(\sqrt{{\omega}_{2}}t\right)\dot{y}-\sqrt{{\omega}_{2}}cosh\left(\sqrt{{\omega}_{2}}t\right)y,\hfill \\ \hfill {I}_{{n}_{5}}& =cosh\left(\sqrt{{\omega}_{2}}t\right)\dot{y}-\sqrt{{\omega}_{2}}sinh\left(\sqrt{{\omega}_{2}}t\right)y.\hfill \end{array}$$The latter dynamical hyperbolic dynamical system reduces to that of the unified dark matter potential (UDM) when ${\omega}_{1}=2{\omega}_{2}$ [125]. The amount of information one receives by the direct application of the geometric symmetries of the kinetic metric is noticeable.

#### 11.3. Brans–Dicke Cosmology

#### 11.3.1. Case $\left|k\right|\ne 1$.

- For arbitrary potential $V\left(\theta \right)$, the dynamical system admits the Noether point symmetry ${\partial}_{t}$.
- For $V\left(\theta \right)={V}_{0}{e}^{2\theta}$ there are two additional Noether point symmetries ${K}^{1},\phantom{\rule{3.33333pt}{0ex}}t{K}^{1}$ with first integrals$${I}_{1}=\frac{d}{dt}\left(\frac{{r}^{1+k}{e}^{\left(1+k\right)\theta}}{\left(k+1\right)}\right)\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{I}_{2}=t\frac{d}{dt}\left(\frac{{r}^{1+k}{e}^{\left(1+k\right)\theta}}{\left(k+1\right)}\right)-\left(\frac{{r}^{1+k}{e}^{\left(1+k\right)\theta}}{\left(k+1\right)}\right).$$
- For $V\left(\theta \right)={V}_{0}{e}^{2\theta}-\frac{m{N}_{0}^{2}}{2\left({k}^{2}-1\right)}{e}^{2k\theta},$ there are two additional Noether point symmetries ${e}^{\pm \sqrt{m}t}{K}^{1}$, with corresponding first integrals$${I}_{\pm}^{\prime}={e}^{\pm \sqrt{m}t}\left[\frac{d}{dt}\left(\frac{{r}^{1+k}{e}^{\left(1+k\right)\theta}}{\left(k+1\right)}\right)\mp \sqrt{m}\left(\frac{{r}^{1+k}{e}^{\left(1+k\right)\theta}}{\left(k+1\right)}\right)\right].$$
- For $V\left(\theta \right)={V}_{0}{e}^{-2\theta}$, we have the extra Noether point symmetries ${K}^{2},\phantom{\rule{3.33333pt}{0ex}}t{K}^{2}$ with Noether first integrals$${J}_{1}=\frac{d}{dt}\left(\frac{{r}^{1-k}{e}^{-\left(1-k\right)\theta}}{k-1}\right)\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{J}_{2}=t\frac{d}{dt}\left(\frac{{r}^{1-k}{e}^{-\left(1-k\right)\theta}}{k-1}\right)-\frac{{r}^{1-k}{e}^{-\left(1-k\right)\theta}}{k-1}.$$
- For $V\left(\theta \right)={V}_{0}{e}^{-2\theta}-\frac{m{N}_{0}^{2}}{2\left({k}^{2}-1\right)}{e}^{2k\theta}$, we have the extra Noether symmetries ${e}^{\pm \sqrt{m}t}{K}^{2}$ $m=$ constant, with first integrals$${J}_{\pm}^{{}^{\prime}}={e}^{\pm \sqrt{m}t}\left(\frac{d}{dt}\left(\frac{{r}^{1-k}{e}^{-\left(1-k\right)\theta}}{k-1}\right)\mp \sqrt{m}\frac{{r}^{1-k}{e}^{-\left(1-k\right)\theta}}{k-1}\right).$$
- For the potential $V\left(\theta \right)={V}_{0}{e}^{2k\theta},$ the additional symmetry is the vector field ${K}^{3}$ with first integrals$${I}_{3}=\frac{r{e}^{2k\theta}}{k}\left(k\dot{r}+r\dot{\theta}\right).$$
- For $V\left(\theta \right)={V}_{0}{e}^{-2\frac{\left({k}^{2}-2\right)}{k}\theta},{k}^{2}-2\ne 0,$ the extra Noether point symmetries are $2t{\partial}_{t}+{H}^{i}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{t}^{2}{\partial}_{t}+t{H}^{i}$ with first integrals$${I}_{{H}_{1}}=-\frac{d}{dt}\left(\frac{1}{2}\frac{{r}^{2}{e}^{2k\theta}}{{k}^{2}-1}\right)\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{I}_{{H}_{2}}=-t\frac{d}{dt}\left(\frac{1}{2}\frac{{r}^{2}{e}^{2k\theta}}{{k}^{2}-1}\right)+\frac{1}{2}\frac{{r}^{2}{e}^{2k\theta}}{{k}^{2}-1}.$$We note that in this case the system is the Ermakov–Pinney dynamical system (because it admits the Noether point symmetry algebra the $sl(2,R),$ hence the Lie symmetry algebra is at least $sl(2,R))$ .
- For $V\left(\theta \right)={V}_{0}{e}^{-2\frac{\left({k}^{2}-2\right)}{k}\theta}-\frac{{N}_{0}^{2}m}{{k}^{2}-1}{e}^{2k\theta}\phantom{\rule{3.33333pt}{0ex}}$ , ${k}^{2}-2\ne 0,$ we have the Noether point symmetries $\frac{2}{\sqrt{m}}{e}^{\pm \sqrt{m}t}{\partial}_{t}\pm {e}^{\pm \sqrt{m}t}{H}^{i}$ , $m=$ constant with corresponding first integral$${I}_{+,-}={e}^{\pm 2\sqrt{m}t}\left(\mp \frac{d}{dt}\left(\frac{1}{2}\frac{{r}^{2}{e}^{2k\theta}}{{k}^{2}-1}\right)+2\sqrt{m}\left(\frac{1}{2}\frac{{r}^{2}{e}^{2k\theta}}{{k}^{2}-1}\right)\right).$$For this potential, the Noether point symmetries form the $sl\left(2,R\right)$ Lie algebra, i.e., the dynamical system is the two-dimensional Kepler–Ermakov system.
- The case $V\left(\theta \right)=0$ corresponds the two-dimensional free particle in flat space and the dynamical system admits seven additional Noether point symmetries.

#### 11.3.2. Case $\left|k\right|=1$

- For arbitrary potential $V\left(\theta \right)$, the dynamical system admits the Noether point symmetry ${\partial}_{t}$.All the rest of cases admit additional symmetries.
- If $V\left(\theta \right)={V}_{0}{e}^{2\theta},$ we have the extra Noether point symmetries ${K}^{1}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}t{K}^{1}$ with first integrals (134) with $k=1$.
- If $V\left(\theta \right)={V}_{0}{e}^{2\theta}-\frac{m}{2}\theta {e}^{2\theta},$ we have the Noether point symmetries ${e}^{\pm \sqrt{m}t}{K}^{1}$ with first integrals the (135) with $k=1$.
- Noether point symmetries generated by the KV ${K}^{2}$.
- If $V\left(\theta \right)={V}_{0}{e}^{-2\theta},$ then we have the Noether point symmetries ${K}^{2}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}t{K}^{2}$ with first integrals$${I}_{2}^{\prime}=\frac{d}{dt}\left(\theta -lnr\right)\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{I}_{2}^{\prime}=t\left[\frac{d}{dt}\left(\theta -lnr\right)\right]-\left(\theta -lnr\right).$$
- If $V\left(\theta \right)={V}_{0}{e}^{-2\theta}-\frac{1}{4}p{e}^{2\theta}\phantom{\rule{0.166667em}{0ex}}$ then we have the Noether point symmetries ${K}^{2}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}t{K}^{2}$ with first integrals$${I}_{1}^{\prime}=\frac{d}{dt}\left(\theta -lnr\right)-pt\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{I}_{2}^{\prime}=t\left[\frac{d}{dt}\left(\theta -lnr\right)\right]-\left(\theta -lnr\right)-\frac{1}{2}p{t}^{2}.$$
- If $V\left(\theta \right)=0,$ then the system becomes the free particle and admits seven extra Noether point symmetries.

#### 11.4. f (R)-Gravity

- For arbitrary function $f\left(R\right)$, there exists the autonomous symmetry ${\partial}_{t}$, which derives the constraint equation.
- For $f\left(R\right)={R}^{\frac{3}{2}}$, the theory admits the additional Noether point symmetries$${K}_{1}=2t{\partial}_{t}+\frac{4}{3}\phantom{\rule{3.33333pt}{0ex}}{\partial}_{a}-\frac{9}{2}\frac{{f}^{\prime}}{{f}^{\u2033}}{\partial}_{R},$$$${K}_{2}=\frac{1}{a}{\partial}_{a}-\frac{1}{{a}^{2}}\frac{{f}^{\prime}}{{f}^{\u2033}}{\partial}_{R}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{K}_{2}^{\ast}=t\left(\frac{1}{a}{\partial}_{a}-\frac{1}{{a}^{2}}\frac{{f}^{\prime}}{{f}^{\u2033}}{\partial}_{R}\right),$$$${\mathsf{\Phi}}_{1}=6{a}^{2}\dot{a}\sqrt{R}+6\frac{{a}^{3}}{\sqrt{R}}\dot{R},$$$${\mathsf{\Phi}}_{2}=\frac{d}{dt}\left(a\sqrt{R}\right)\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{\mathsf{\Phi}}_{2}^{\ast}=t\frac{d}{dt}\left(a\sqrt{R}\right)-a\sqrt{R}.$$
- For $f\left(R\right)={R}^{\frac{7}{8}}$ and $K=0$, the theory admits the additional Noether point symmetries$${K}_{3}=2t{\partial}_{t}+\frac{a}{2}\phantom{\rule{3.33333pt}{0ex}}{\partial}_{a}+\frac{1}{2}\frac{{f}^{\prime}}{{f}^{\u2033}}{\partial}_{R}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{K}_{3}^{\ast}={t}^{2}{\partial}_{t}+t\left(\frac{a}{2}\phantom{\rule{3.33333pt}{0ex}}{\partial}_{a}+\frac{1}{2}\frac{{f}^{\prime}}{{f}^{\u2033}}{\partial}_{R}\right),$$$${\mathsf{\Phi}}_{3}=\frac{d}{dt}\left({a}^{3}{R}^{-\frac{1}{8}}\right)\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{\mathsf{\Phi}}_{3}^{\ast}=t\frac{d}{dt}\left({a}^{3}{R}^{-\frac{1}{8}}\right)-{a}^{3}{R}^{-\frac{1}{8}}.$$
- The power-law theory $f\left(R\right)={R}^{n}$ (with $n\ne 0,1,\frac{3}{2},\frac{7}{8}$) and for $K=0,$ or with K arbitrary and $n=2$, the system admits the extra Noether point symmetry$${K}_{1}^{\ast}=2t{\partial}_{t}+\left(\frac{4n}{3}-\frac{2}{3}\right)a{\partial}_{t}-3\frac{{f}^{\prime}}{{f}^{\u2033}}{\partial}_{R},$$$${\mathsf{\Phi}}_{1}^{\ast}={a}^{2}{R}^{n-1}\dot{a}\left(2-n\right)+\frac{1}{2}{a}^{3}{R}^{n-2}\dot{R}\left(2n-1\right)\left(n-1\right).$$
- For $f\left(R\right)={(R-2\Lambda )}^{3/2}$ the extra Noether point symmetries are$${K}_{\left(\pm \right)2}={e}^{\pm \sqrt{m}t}\left(\frac{1}{a}{\partial}_{a}-\frac{1}{{a}^{2}}\frac{{f}^{\prime}}{{f}^{\u2033}}{\partial}_{R}\right),$$$${\mathsf{\Phi}}_{\left(\pm \right)2}={e}^{\pm \sqrt{m}t}\left(\frac{d}{dt}\left(a\sqrt{R-2L}\right)\mp 9\sqrt{m}a\sqrt{R-2\Lambda}\right).$$
- Finally, when $f\left(R\right)={(R-2\Lambda )}^{7/8},$ the field equations admit the Noether point symmetries$${K}_{\left(\pm \right)4}=\pm \frac{1}{\sqrt{m}}{e}^{\pm 2\sqrt{m}t}{\partial}_{t}+{e}^{\pm 2\sqrt{m}t}\left(\frac{a}{2}\phantom{\rule{3.33333pt}{0ex}}{\partial}_{a}+\frac{1}{2}\frac{{f}^{\prime}}{{f}^{\prime \prime}}{\partial}_{R}\right),$$$${\mathsf{\Phi}}_{\left(\pm \right)4}=\frac{d}{dt}\left({a}^{3}{\left(R-2\Lambda \right)}^{-\frac{1}{8}}\right)\mp \frac{1}{2}\sqrt{m}{a}^{3}{\left(R-2\Lambda \right)}^{-\frac{1}{8}}.$$

#### 11.5. Two-Scalar Field Cosmology

- For arbitrary potential $V\left(\varphi ,\psi \right)$, the field equations admit the Noether point symmetry ${\partial}_{t}$ which provides the constraint equation of General Relativity.
- For $V\left(\varphi ,\psi \right)=0$, the dynamical system is maximally symmetric and admits in total twelve Noether point symmetries.
- For $V\left(\varphi ,\psi \right)=V\left(\varphi \right)$, there exists the additional Noether point symmetry, the vector field ${X}_{12}$, with conservation law the angular momentum on the two-dimensional sphere, that is,$${\mathsf{\Phi}}_{12}={e}^{2\varphi}\dot{\psi}.$$
- For ${V}_{A}\left(\varphi ,\psi \right)=\frac{{\omega}_{0}^{2}}{2}{u}^{2}+\frac{{\mu}^{2}}{2\left(1-{a}_{0}^{2}\right)}{\left({S}_{\left(\mu \right)}+{a}_{0}{S}_{\left(\nu \right)}\right)}^{2}-\frac{{\omega}_{3}^{2}}{2}{S}_{\left(\sigma \right)}^{2}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{a}_{0}\ne 1$, the system admits six additional Noether point symmetries given by the vector fields$${T}_{1}\left(t\right){K}^{1}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{T}_{2}\left(t\right){K}^{2}\phantom{\rule{3.33333pt}{0ex}},{T}_{3}\left(t\right){K}^{3},$$$${T}_{,tt}^{A}={\omega}_{\phantom{\rule{3.33333pt}{0ex}}\delta}^{\gamma}{T}^{\delta}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{\omega}_{\phantom{\rule{3.33333pt}{0ex}}\delta}^{\gamma}=diag\left({\left({\omega}_{1}\right)}^{2},{\left({\omega}_{2}\right)}^{2},{\left({\omega}_{3}\right)}^{3}\right)$$$${I}_{C}^{\gamma}={T}_{\gamma}\frac{d}{dt}{S}_{\left(\gamma \right)}-{T}_{\gamma ,t}{S}_{\left(\gamma \right)}.$$$$X={x}^{\nu}{\partial}_{\mu}-\epsilon {x}^{\mu}{\partial}_{\nu},$$
- For the potential being ${V}_{B}\left(\varphi ,\psi \right)=\frac{{\omega}_{0}^{2}}{2}{u}^{2}+\frac{{\mu}^{2}}{2\left(1-{a}_{0}^{2}\right)}{\left({S}_{\left(\mu \right)}+{a}_{0}{S}_{\left(\nu \right)}\right)}^{2}-\frac{{\omega}_{3}^{2}}{2}{S}_{\left(\sigma \right)}^{2}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{a}_{0}\ne 1$, the dynamical system admits the extra Noether symmetries$$\overline{T}\left(t\right)\left({K}^{\mu}+{a}_{0}{K}^{\nu}\right)\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{T}^{\prime}\left(t\right){K}^{\sigma}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{T}^{\ast}\left(t\right)\left({a}_{0}{K}^{\mu}+{K}^{\nu}\right),$$$${\overline{T}}_{,tt}=\left({\mu}^{2}+{\omega}_{0}^{2}\right)\overline{T}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{T}_{\sigma ,tt}=\left({\omega}_{3}^{2}+{\omega}_{0}^{2}\right){T}_{\sigma}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{T}_{,tt}^{\ast}={\omega}_{0}^{2}\overline{T},$$$${\mathsf{\Phi}}_{1a2}=\overline{T}\frac{d}{dt}\left({S}_{\left(\mu \right)}+{a}_{0}{S}_{\left(\nu \right)}\right)-{\overline{T}}_{,t}\left({S}_{\left(\mu \right)}+{a}_{0}{S}_{\left(\nu \right)}\right),$$$${\mathsf{\Phi}}_{3}={T}_{\sigma}\frac{d}{dt}{S}_{\left(\sigma \right)}-{T}_{\sigma ,t}{S}_{\left(\sigma \right)},$$$${\mathsf{\Phi}}_{a12}={T}^{\ast}\frac{d}{dt}\left({a}_{0}{S}_{\left(\mu \right)}+{S}_{\left(\nu \right)}\right)-{T}_{,t}^{\ast}\left({a}_{0}{S}_{\left(\mu \right)}+{S}_{\left(\nu \right)}\right).$$

#### 11.6. Galilean Cosmology

## 12. Higher-Order Symmetries in Cosmology

#### 12.1. Scalar-Field Cosmology from Contact Symmetries

#### 12.2. $f\left(R\right)$-Gravity from Contact Symmetries

## 13. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Hall, G.S. Symmetries and Curvature Structure in General Relativity; World Scientific Lecture Notes in Physics 46; World Scientific: Singapore, 2004. [Google Scholar]
- Lie, S. Theorie der Transformationsgruppen I; B. G. Teubner: Leipzig, Germany, 1888. [Google Scholar]
- Lie, S. Theorie der Transformationsgruppen II; B. G. Teubner: Leipzig, Germany, 1888. [Google Scholar]
- Lie, S. Theorie der Transformationsgruppen III; B. G. Teubner: Leipzig, Germany, 1888. [Google Scholar]
- Ovsiannikov, L.V. Group Analysis of Differential Equations; Academic Press: New York, NY, USA, 1982. [Google Scholar]
- Bluman, G.W.; Kumei, S. Symmetries and Differential Equations; Springer: New York, NY, USA, 1989. [Google Scholar]
- Ibragimov, N.H. CRC Handbook of Lie Group Analysis of Differential Equations, Volume I: Symmetries, Exact Solutions, and Conservation Laws; CRS Press LLC: Boca Raton, FL, USA, 2000. [Google Scholar]
- Olver, P.J. Applications of Lie Groups to Differential Equations; Springer: New York, NY, USA, 1993. [Google Scholar]
- Crampin, M. Hidden symmetries and Killing tensors. Rep. Math. Phys.
**1984**, 20, 31–40. [Google Scholar] [CrossRef] - Kalotas, T.M.; Wybourne, B.G. Dynamical noether symmetries. J. Phys. A Math. Gen.
**1982**, 15, 2077. [Google Scholar] [CrossRef] - Prince, G.E.; Eliezer, C.J. On the Lie symmetries of the classical Kepler problem. J. Phys. A Math. Gen.
**1981**, 14, 587. [Google Scholar] [CrossRef] - Lutzky, M. Symmetry groups and conserved quantities for the harmonic oscillator. J. Phys. A Math. Gen.
**1978**, 11, 249. [Google Scholar] [CrossRef] - Govinder, K.S.; Leach, P.G.L. On the determination of non-local symmetries. J. Phys. A Math. Gen.
**1995**, 28, 5349. [Google Scholar] [CrossRef] - Sarlet, W.; Cantrijin, F. Generalizations of Noether’s theorem in classical mechanics. SIAM Rev.
**1981**, 23, 467–494. [Google Scholar] [CrossRef] - Mahomed, F.M. Symmetry group classification of ordinary differential equations: Survey of some results. Math. Meth. Appl. Sci.
**2007**, 30, 1995–2012. [Google Scholar] [CrossRef] [Green Version] - Nucci, M.C. The role of symmetries in solving differential equations. Math. Comp. Mod.
**1997**, 25, 181–193. [Google Scholar] [CrossRef] - Harrison, B.K. The differential form method for finding symmetries. Sigma
**2005**, 1, 001. [Google Scholar] [CrossRef] - Abraham-Shrauner, B.; Guo, A. Hidden symmetries associated with the projective group of nonlinear first-order ordinary differential equations. J. Phys. A Math. Gen.
**1992**, 25, 5597. [Google Scholar] [CrossRef] - Popovych, R.O.J. On Lie reduction of the Navier-Stokes equations. Nonlinear Math. Phys.
**1995**, 2, 301–311. [Google Scholar] [CrossRef] - Katzin, G.H.; Levine, J. Geodesic first integrals with explicit path-parameter dependence in Riemannian space-times. J. Math. Phys.
**1981**, 22, 1878–1891. [Google Scholar] [CrossRef] - Noether, E. Invariante variationsprobleme, Nachr. d. König. Gesellsch. d. Wiss. zu Göttingen. Math.-Phys. Klasse
**1918**, 235–257. [Google Scholar] - Singer, S.F. Symmetry in Mechanics; Birkhauser Boston: Boston, MA, USA, 2004. [Google Scholar]
- Costa, G.; Fogli, G. Symmetry and Group Theory in Particle Physics; Lecture Notes in Physics; Springer: Berlin, Germany, 2012. [Google Scholar]
- Sundermeyer, K. Symmetries in Fundamental Physics; Springer: Heidelberg, Germany, 2014. [Google Scholar]
- Trautman, A. Conservation Laws in General Relativity, in Gravitation, and Introduction to Current Research; Witten, L., Ed.; Willey: New York, NY, USA, 1962. [Google Scholar]
- Tsamparlis, M.; Paliathanasis, A. Lie symmetries of geodesic equations and projective collineations. Nonlinear Dyn.
**2010**, 62, 203–214. [Google Scholar] [CrossRef] [Green Version] - Tsamparlis, M.; Paliathanasis, A. Lie and Noether symmetries of geodesic equations and collineations. Gen. Relativ. Gravit.
**2010**, 42, 2957–2980. [Google Scholar] [CrossRef] [Green Version] - Paliathanasis, A.; Tsamparlis, M. Lie point symmetries of a general class of PDEs: The heat equation. J. Geom. Phys.
**2012**, 62, 2443–2456. [Google Scholar] [CrossRef] [Green Version] - Paliathanasis, A.; Tsamparlis, M. Lie and Noether point symmetries of a class of quasilinear systems of second-order differential equations. J. Geom. Phys.
**2016**, 107, 45–59. [Google Scholar] [CrossRef] [Green Version] - Tsamparlis, M.; Paliathanasis, A. Two-dimensional dynamical systems which admit Lie and Noether symmetries. J. Phys. A Math. Theor.
**2011**, 44, 175202. [Google Scholar] [CrossRef] [Green Version] - Hojman, S.A. A new conservation law constructed without using either Lagrangians or Hamiltonians. J. Phys. A Math. Gen.
**1992**, 25, L291. [Google Scholar] [CrossRef] - Djukic, D.C. A procedure for finding first integrals of mechanical systems with gauge-variant Lagrangians. Int. J. Non-Linear Mech.
**1973**, 8, 479–488. [Google Scholar] [CrossRef] - Nucci, M.C.; Leach, P.G.L. Jacobi’s last multiplier and the complete symmetry group of the Euler–Poinsot system. J. Non. Math. Phys.
**2002**, 9, 110–121. [Google Scholar] [CrossRef] - Nucci, M.C.; Leach, P.G.L. The Jacobi Last Multiplier and its applications in mechanics. Phys. Scr.
**2008**, 78, 065011. [Google Scholar] [CrossRef] - Nucci, M.C.; Tamazhmani, K.M. Lagrangians for dissipative nonlinear oscillators: The method of Jacobi last multiplier. J. Non. Math. Phys.
**2010**, 17, 167–178. [Google Scholar] [CrossRef] - Mahomed, F.M.; Leach, P.G.L. Lie algebras associated with scalar second-order ordinary differential equations. J. Math. Phys.
**1989**, 30, 2770–2777. [Google Scholar] [CrossRef] - Mahomed, F.M.; Leach, P.G.L. Symmetry Lie algebras of nth order ordinary differential equations. J. Math. Anal. Appl.
**1990**, 151, 80–107. [Google Scholar] [CrossRef] - Wulfman, C.E.; Wybourne, B.G. The Lie group of Newton’s and Lagrange’s equations for the harmonic oscillator. J. Phys. A Math. Gen.
**1976**, 9, 507. [Google Scholar] [CrossRef] - Levy-Leblond, J.M. Conservation laws for gauge-variant Lagrangians in classical mechanics. Am. J. Phys.
**1971**, 39, 502–506. [Google Scholar] [CrossRef] - Werner, A.C.J.; Eliezer, J. The lengthening pendulum. Aust. Math. Soc.
**1969**, 9, 331–336. [Google Scholar] [CrossRef] - Lewis, H.; Ralph, J. Classical and quantum systems with time-dependent harmonic-oscillator-type Hamiltonians. Phys. Rev. Lett.
**1967**, 18, 510. [Google Scholar] [CrossRef] - Ermakov, V. Second order differential equations. Conditions of complete integrability. Universita Izvestia Kiev
**1880**, 9, 1–25. [Google Scholar] [CrossRef] - Pinney, E. The nonlinear differential equation y″ + p(x)y + cy
^{−3}= 0. Proc. Am. Math. Soc.**1950**, 1, 681. [Google Scholar] [CrossRef] - Rogers, C.; Hoenselaers, C.; Ray, J.R. On (2 + 1)-dimensional Ermakov systems. J. Phys. A Math. Gen.
**1993**, 26, 2625. [Google Scholar] [CrossRef] - Schief, W.K.; Rogerts, C.; Bassom, A.P. Ermakov systems of arbitrary order and dimension: Structure and linearization. J. Phys. A Math. Gen.
**1996**, 29, 903. [Google Scholar] [CrossRef] - Leach, P.G.L.; Andriopoulos, K. The Ermakov equation: A commentary. Appl. Anal. Discrete Math.
**2008**, 2, 146–157. [Google Scholar] [CrossRef] [Green Version] - Tsamparlis, M.; Paliathanasis, A. Generalizing the autonomous Kepler–Ermakov system in a Riemannian space. A note on the construction of the Ermakov–Lewis invariant. J. Phys. A Math. Theor.
**2012**, 45, 275202. [Google Scholar] [CrossRef] - Moyo, S.; Leach, P.G.L. A note on the construction of the Ermakov–Lewis invariant. J. Phys. A Math. Gen.
**2002**, 35, 5333. [Google Scholar] [CrossRef] - Katzin, G.H.; Levine, J.; Davis, R.W. Curvature Collineations: A Fundamental Symmetry Property of the Space-Times of General Relativity Defined by the Vanishing Lie Derivative of the Riemann Curvature Tensor. J. Math. Phys.
**1969**, 10, 617–629. [Google Scholar] [CrossRef] - Hall, G.S.; Roy, I.M. Some remarks on special conformal and special projective symmetries in general relativity. Gen. Relativ. Gravit.
**1997**, 29, 827–835. [Google Scholar] [CrossRef] - Karpathopoulos, L.; Paliathanasis, A.; Tsamparlis, M. Lie and Noether point symmetries for a class of nonautonomous dynamical systems. Math. Phys.
**2017**, 58, 082901. [Google Scholar] [CrossRef] [Green Version] - Christodoulakis, T.; Dimakis, N.; Terzis, P.A. Lie point and variational symmetries in minisuperspace Einstein gravity. J. Phys. A Math. Theor.
**2014**, 47, 095202. [Google Scholar] [CrossRef] [Green Version] - Gordon, T.J. On the symmetries and invariants of the harmonic oscillator. J. Phys. A Math. Theor.
**1986**, 19, 183. [Google Scholar] [CrossRef] - Weinberg, S. The cosmological constant problem. Rev. Mod. Phys.
**1989**, 61, 1. [Google Scholar] [CrossRef] - Peebles, P.J.; Ratra, B. The cosmological constant and dark energy. Rev. Mod. Phys.
**2003**, 75, 559. [Google Scholar] [CrossRef] - Padmanabhan, T. Cosmological constant—The weight of the vacuum. Phys. Rept.
**2003**, 380, 235–320. [Google Scholar] [CrossRef] - Ratra, B.; Peebles, P.J.E. Cosmological consequences of a rolling homogeneous scalar field. Phys. Rev D
**1988**, 37, 3406. [Google Scholar] [CrossRef] - Overduin, J.M.; Cooperstock, F.I. Evolution of the scale factor with a variable cosmological term. Phys. Rev. D
**1998**, 58, 043506. [Google Scholar] [CrossRef] - Basilakos, S.; Plionis, M.; Solà, J. Hubble expansion and structure formation in time varying vacuum models. Phys. Rev. D
**2009**, 80, 3511. [Google Scholar] [CrossRef] - Starobinsky, A.A. A new type of isotropic cosmological models without singularity. Phys. Lett. B
**1980**, 91, 99–102. [Google Scholar] [CrossRef] - Gorini, V.; Kamenshchik, A.; Moshella, U. Can the Chaplygin gas be a plausible model for dark energy? Phys. Rev. D
**2003**, 67, 063509. [Google Scholar] [CrossRef] - Barrow, J.D.; Saich, P. Scalar-field cosmologies. Class. Quantum Grav.
**1993**, 10, 279. [Google Scholar] [CrossRef] - Rubano, C.; Barrow, J.D. Scaling solutions and reconstruction of scalar field potentials. Phys. Rev. D
**2001**, 64, 127301. [Google Scholar] [CrossRef] - Gariel, J.; le Denmat, G. Matter creation and bulk viscosity in early cosmology. Phys. Lett. A
**1995**, 200, 11–16. [Google Scholar] [CrossRef] - Clifton, T.; Ferreira, P.G.; Padilla, A.; Skordis, C. Modified gravity and cosmology. Phys. Rep.
**2012**, 513, 1–189. [Google Scholar] [CrossRef] - Joyce, A.; Lombriser, L.; Schmidt, F. Dark energy versus modified gravity. Annu. Rev. Nucl. Part. Sci.
**2016**, 66, 95–122. [Google Scholar] [CrossRef] - Amendola, L.; Polarski, D.; Tsujikawa, S. Are f(R)Dark Energy Models Cosmologically Viable? Phys. Rev. Lett.
**2007**, 98, 131302. [Google Scholar] [CrossRef] [PubMed] - Battye, R.A.; Bolliet, B.; Pearson, J.A. f(R) gravity as a dark energy fluid. Phys. Rev. D
**2016**, 93, 044026. [Google Scholar] [CrossRef] - Basilakos, S.; Kouretsis, A.P.; Saridakis, E.N.; Stavrinos, P. Resembling dark energy and modified gravity with Finsler-Randers cosmology. Phys. Rev. D
**2013**, 88, 123510. [Google Scholar] [CrossRef] - Geng, C.-Q.; Lee, C.-C.; Saridakis, E.N.; Wu, Y.P. “Teleparallel” dark energy. Phys. Lett. B
**2011**, 704, 384–387. [Google Scholar] [CrossRef] - De Ritis, R.; Marmo, G.; Platania, G.; Rubano, C.; Scudellaro, P.; Stornaiolo, C. New approach to find exact solutions for cosmological models with a scalar field. Phys. Rev. D.
**1990**, 42, 1091. [Google Scholar] [CrossRef] - Rosquist, K.; Uggla, C. Killing tensors in two-dimensional space-times with applications to cosmology. J. Math. Phys.
**1991**, 32, 3412–3422. [Google Scholar] [CrossRef] - Bianchi, L. On the spaces of three dimensions that admit a continuous group of movements. Soc. Ita. Mem. di Mat.
**1898**, 11, 268. [Google Scholar] - Rayan, M.P., Jr.; Shepley, L.C. Homogeneous Relativistic Cosmologies; Princeton University Press: Princeton, NJ, USA, 1975. [Google Scholar]
- Kasner, E. Geometrical theorems on Einstein’s cosmological equations. Am. J. Math.
**1921**, 43, 217–221. [Google Scholar] [CrossRef] - Moussiaux, A.; Tombal, P.; Demaret, J. Exact solution for vacuum Bianchi type III model with a cosmological constant. J. Phys. A Math. Gen.
**1981**, 14, L277. [Google Scholar] [CrossRef] - Christodoulakis, T.; Terzis, P.A. The general solution of Bianchi type III vacuum cosmology. Class. Quantum Grav.
**2007**, 24, 875. [Google Scholar] [CrossRef] - Terzis, P.A.; Christodoulakis, T. The general solution of Bianchi type VII h vacuum cosmology. Gen. Relat. Gravit.
**2009**, 41, 469–495. [Google Scholar] [CrossRef] - Harvey, A.; Tsoubelis, D. Exact Bianchi IV cosmological model. Phys. Rev. D
**1977**, 15, 2734. [Google Scholar] [CrossRef] - Cotsakis, S.; Leach, P.G.L. Painlevé analysis of the Mixmaster universe. J. Phys. A Math. Gen.
**1994**, 27, 1625. [Google Scholar] [CrossRef] - Maciejewski, A.; Szydlowski, M. On the integrability of Bianchi cosmological models. J. Phys. A Math. Gen.
**1998**, 31, 2031. [Google Scholar] [CrossRef] - Libre, J.; Valls, C.J. Integrability of the Bianchi IX system. Math. Phys.
**2005**, 46, 0742901. [Google Scholar] [CrossRef] - Libre, J.; Valls, C.J. Formal and analytical integrability of the Bianchi IX system. Math. Phys.
**2006**, 47, 022704. [Google Scholar] [CrossRef] - Christiansen, F.; Rugh, H.H.; Rugh, S.E. Non-integrability of the mixmaster universe. J. Phys. A Math. Gen.
**1995**, 28, 657. [Google Scholar] [CrossRef] - Capozziello, S.; de Ritis, R. Nöther’s symmetries in fourth-order cosmologies. Nuovo Cimento B
**1994**, 109, 795–802. [Google Scholar] [CrossRef] - Capozziello, S.; de Ritis, R.; Marino, A.A. Conformal equivalence and Noether symmetries in cosmology. Class. Quantum Grav.
**1997**, 14, 3259. [Google Scholar] [CrossRef] - Modak, B.; Kamilya, S.; Biswas, S. Evolution of dynamical coupling in scalar tensor theory from Noether symmetry. Gen. Relativ. Gravit.
**2000**, 32, 1615–1626. [Google Scholar] [CrossRef] [Green Version] - Sanyal, A.K. Noether and some other dynamical symmetries in Kantowski-Sachs model. Phys. Lett. B
**2002**, 524, 177–184. [Google Scholar] [CrossRef] - Motavali, H.; Golshani, M. Exact solutions for cosmological models with a scalar field. IJMPA
**2002**, 17, 375–381. [Google Scholar] [CrossRef] - Kamilya, S.; Modak, B. Beyond Einstein gravity: A Survey of gravitational theories for cosmology and astrophysics. Gen. Relativ. Gravit.
**2004**, 36, 676. [Google Scholar] - Bonanno, A.; Esposito, G.; Rubano, C. Noether symmetry approach in matter-dominated cosmology with variable G and Λ. Gen. Relativ. Gravit.
**2007**, 39, 189–209. [Google Scholar] [CrossRef] [Green Version] - Camci, U.; Kucukakca, Y. Noether symmetries of Bianchi I, Bianchi III, and Kantowski-Sachs spacetimes in scalar-coupled gravity theories. Phys. Rev. D
**2007**, 76, 084023. [Google Scholar] [CrossRef] - Capozziello, S.; Nesseris, S.; Perivolaropoulos, L. Reconstruction of the scalar–tensor Lagrangian from a ΛCDM background and Noether symmetry. JCAP
**2007**, 12, 009. [Google Scholar] [CrossRef] - Vakili, B. Noether symmetry in f (R) cosmology. Phys. Lett. B
**2008**, 664, 16–20. [Google Scholar] [CrossRef] - Capozziello, S.; Felice, A. f (R) cosmology from Noether’s symmetry. JCAP
**2008**, 8, 16. [Google Scholar] [CrossRef] - Capozziello, S.; Martin-Moruno, P.; Rubano, C. Dark energy and dust matter phases from an exact f (R)-cosmology model. Phys. Lett. B
**2008**, 664, 12–15. [Google Scholar] [CrossRef] - Capozziello, S.; Piedipalumbo, E.; Rubano, C.; Scudellaro, P. Noether symmetry approach in phantom quintessence cosmology. Phys. Rev. D
**2009**, 80, 104030. [Google Scholar] [CrossRef] - Mubasher, J.; Mahomed, F.M.; Momeni, D. Noether symmetry approach in f (R)-tachyon model. Phys. Lett. B
**2011**, 702, 315–319. [Google Scholar] - Cotsakis, S.; Leach, P.G.L.; Pantazi, C. Symmetries of homogeneous cosmologies. Gravit. Cosmol.
**1998**, 4, 314. [Google Scholar] - Dimakis, N.; Christodoulakis, T.; Terzis, P.A. FLRW metric f (R) cosmology with a perfect fluid by generating integrals of motion. J. Geom. Phys.
**2012**, 77, 97. [Google Scholar] [CrossRef] - Paliathanasis, A.; Tsamparlis, M.; Basilakos, S.; Barrow, J.D. Classical and quantum solutions in Brans-Dicke cosmology with a perfect fluid. Phys. Rev. D
**2016**, 93, 043528. [Google Scholar] [CrossRef] [Green Version] - Paliathanasis, A.; Tsamparlis, M.; Basilakos, S.; Barrow, J.D. Dynamical analysis in scalar field cosmology. Phys. Rev. D
**2015**, 91, 123535. [Google Scholar] [CrossRef] [Green Version] - Paliathanasis, A.; Vakili, B. Closed-form solutions of the Wheeler–DeWitt equation in a scalar-vector field cosmological model by Lie symmetries. Gen. Relativ. Gravit.
**2016**, 48, 13. [Google Scholar] [CrossRef] - Dimakis, N.; Giacomini, A.; Jamal, S.; Leon, G.; Paliathanasis, A. Noether symmetries and stability of ideal gas solutions in Galileon cosmology. Phys. Rev. D
**2017**, 95, 064031. [Google Scholar] [CrossRef] [Green Version] - Paliathanasis, A.; Tsamparlis, M. Exact solution of the Einstein-Skyrme model in a Kantowski-Sachs spacetime. J. Geom. Phys.
**2017**, 114, 1–11. [Google Scholar] [CrossRef] - Paliathanasis, A. Symmetries of Differential Equations and Applications in Relativistic Physics. Ph.D. Thesis, University of Athens, Athens, Greece, 2014. [Google Scholar]
- Belinchon, J.A.; Harko, T.; Mak, M.K. Exact Scalar-Tensor Cosmological Solutions via Noether Symmetry. Astrophys. Space Sci.
**2016**, 361, 52. [Google Scholar] [CrossRef] - Christodoulakis, T.; Dimakis, N.; Terzis, P.A.; Vakili, B.; Melas, E.; Grammenos, T. Minisuperspace canonical quantization of the Reissner-Nordström black hole via conditional symmetries. Phys. Rev. D
**2014**, 89, 044031. [Google Scholar] [CrossRef] - Zampeli, A.; Pailas, T.; Terzis, P.A.; Christodoulakis, T. Conditional symmetries in axisymmetric quantum cosmologies with scalar fields and the fate of the classical singularities. JCAP
**2016**, 16, 066. [Google Scholar] [CrossRef] - Capozziello, S.; de Rittis, R. Minisuperspace and Wheeler-DeWitt equation for string dilaton cosmology. IJMPD
**1993**, 2, 373. [Google Scholar] [CrossRef] - Capozziello, S.; Lambiase, G. Selection rules in minisuperspace quantum cosmology. Gen. Relativ. Gravit.
**2000**, 32, 673–696. [Google Scholar] [CrossRef] [Green Version] - Capozziello, S.; Lambiase, G. Higher-order corrections to the effective gravitational action from Noether symmetry approach. Gen. Relativ. Gravit.
**2000**, 32, 295–311. [Google Scholar] [CrossRef] [Green Version] - Paliathanasis, A.; Capozziello, S. Noether symmetries and duality transformations in cosmology. MPLA
**2016**, 31, 1650183. [Google Scholar] [CrossRef] [Green Version] - Gionti, G.; Paliathanasis, A. Duality transformation and conformal equivalent scalar-tensor theories. MPLA
**2018**, 33, 1850093. [Google Scholar] [CrossRef] - Paliathanasis, A.; Tsamparlis, M.; Basilakos, S.; Barrow, J.D. Dynamical analysis in scalar field cosmology. Phys. Rev. D
**2015**, 91, 123535. [Google Scholar] [CrossRef] [Green Version] - Paliathanasis, A. Dust fluid component from Lie symmetries in scalar field cosmology. MPLA
**2017**, 32, 1750206. [Google Scholar] [CrossRef] [Green Version] - Motavali, H.; Capozziello, S.; Rorwshan, M.; Jog, A. Scalar-tensor cosmology with R- 1 curvature correction by Noether symmetry. Phys. Lett. B
**2008**, 666, 10–15. [Google Scholar] [CrossRef] - Basilakos, S.; Tsamparlis, M.; Paliathanasis, A. Using the Noether symmetry approach to probe the nature of dark energy. Phys. Rev. D
**2011**, 83, 103512. [Google Scholar] [CrossRef] - Paliathanasis, A.; Tsamparlis, M.; Basilakos, S. Constraints and analytical solutions of theories of gravity using Noether symmetries. Phys. Rev. D
**2011**, 84, 123514. [Google Scholar] [CrossRef] - Tsamparlis, M.; Paliathanasis, A. Three-fluid cosmological model using Lie and Noether symmetries. Class. Quantum Grav.
**2012**, 29, 015006. [Google Scholar] [CrossRef] - Paliathanasis, A.; Tsamparlis, M.; Basilakos, S.; Capozziello, S. New Schwarzschild-like solutions in f(T) gravity through Noether symmetries. Phys. Rev. D
**2014**, 89, 104042. [Google Scholar] [CrossRef] - Paliathanasis, A.; Tsamparlis, M. Two scalar field cosmology: Conservation laws and exact solutions. Phys. Rev. D
**2014**, 90, 043529. [Google Scholar] [CrossRef] - Mahomed, F.M.; Kara, A.H.; Leach, P.G.L. Lie and Noether counting theorems for one-dimensional systems. J. Math. An. Appl.
**1993**, 178, 116–129. [Google Scholar] [CrossRef] - Paliathanasis, A.; Leach, P.G.L.; Capozziello, S. On the Hojman conservation quantities in Cosmology. Phys. Lett. B
**2016**, 755, 8–12. [Google Scholar] [CrossRef] - Bertacca, D.; Matarrese, S.; Pietroni, M. Unified dark matter in scalar field cosmologies. Mod. Phys. Lett. A
**2007**, 22, 2893–2907. [Google Scholar] [CrossRef] - Brans, C.; Dicke, R.H. Mach’s principle and a relativistic theory of gravitation. Phys. Rev.
**1961**, 124, 195. [Google Scholar] [CrossRef] - Mubarakzyanov, G.M. On solvable Lie algebras. Izv. Vyss. Uchebn Zavendeniĭ Mat.
**1963**, 32, 114. [Google Scholar] - Mubarakzyanov, G.M. Classification of real structures of Lie algebras of fifth order. Izv. Vyss. Uchebn Zavendeniĭ Mat.
**1963**, 34, 99. [Google Scholar] - Mubarakzyanov, G.M. Classification of solvable Lie algebras of sixth order with a non-nilpotent basis element. Izv. Vyss. Uchebn Zavendeniĭ Mat.
**1963**, 35, 104. [Google Scholar] - Sotiriou, T.P.; Faraoni, V. f(R) theories of gravity. Rev. Mod. Phys.
**2010**, 82, 451. [Google Scholar] [CrossRef] - Deffayet, C.; Esposito-Farese, G.; Vikman, A. Covariant galileon. Phys. Rev. D
**2009**, 79, 084003. [Google Scholar] [CrossRef] - Paliathanasis, A.; Tsamparlis, M.; Basilakos, S. Dynamical symmetries and observational constraints in scalar field cosmology. Phys. Rev. D
**2014**, 10, 103524. [Google Scholar] [CrossRef] - Paliathanasis, A. f (R)-gravity from Killing tensors. Class. Quantum Gravit.
**2016**, 33, 075012. [Google Scholar] [CrossRef] - Papagiannopoulos, G.; Barrow, J.D.; Basilakos, S.; Giacomini, A.; Paliathanasis, A. Dynamical symmetries in Brans-Dicke cosmology. Phys. Rev. D
**2017**, 95, 024021. [Google Scholar] [CrossRef] [Green Version] - Sadjadi, H.M. Generalized Noether symmetry in f (T) gravity. Phys. Lett. B
**2012**, 718, 270–275. [Google Scholar] [CrossRef]

Collineation | A | B |
---|---|---|

Killing vector (KV) | ${g}_{ij}$ | 0 |

Homothetic vector (HV) | ${g}_{ij}$ | $\psi {g}_{ij},{\psi}_{,i}=0$ |

Conformal Killing vector (CKV) | ${g}_{ij}$ | $\psi {g}_{ij},\psi {,}_{i}\ne 0$ |

Affine collineation | ${\mathsf{\Gamma}}_{jk}^{i}$ | 0 |

Projective collineation (PC) | ${\mathsf{\Gamma}}_{jk}^{i}$ | $2{\varphi}_{(,j}{\delta}_{k)}^{i},$$\varphi {,}_{i}\ne 0$ |

Special Projective collineation (SPC) | ${\mathsf{\Gamma}}_{jk}^{i}$ | $2{\varphi}_{(,j}{\delta}_{k)}^{i},$$\varphi {,}_{i}\ne 0$ and $\varphi {,}_{jk}=0$ |

Collineation | Gradient | Non-Gradient |
---|---|---|

Killing vectors (KV) | ${\mathbf{S}}_{I}={\delta}_{I}^{i}{\partial}_{i}$ | ${\mathbf{X}}_{IJ}={\delta}_{[I}^{j}{\delta}_{j]}^{i}{x}_{j}{\partial}_{i}$ |

Homothetic vector (HV) | $\mathbf{H}={x}^{i}{\partial}_{i}\phantom{\rule{3.33333pt}{0ex}}$ | |

Affine Collineation (AC) | ${\mathbf{A}}_{II}={x}_{I}{\delta}_{I}^{i}{\partial}_{i}\phantom{\rule{3.33333pt}{0ex}}$ | ${\mathbf{A}}_{IJ}={x}_{J}{\delta}_{I}^{i}{\partial}_{i}\phantom{\rule{3.33333pt}{0ex}}$ |

Special Projective collineation (SPC) | ${\mathbf{P}}_{I}={S}_{I}\mathbf{H}.\phantom{\rule{3.33333pt}{0ex}}$ |

**Table 3.**Noether symmetry classification for the Brans–Dicke action in a spatially flat Friedmann–Lemaître–Robertson–Walker metric FLRW spacetime (I).

$\left|\mathit{k}\right|$ | Potential | # Symmetries | Lie Algebra | Symmetries |
---|---|---|---|---|

$V\left(\theta \right)$ | 1 | ${A}_{1}$ | ${\partial}_{t}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}$ | |

≠1 | ${V}_{0}{e}^{2\theta}$ | 3 | ${A}_{1}{\otimes}_{s}\left(2{A}_{1}\right)$ | ${\partial}_{t},\phantom{\rule{3.33333pt}{0ex}}{K}^{1},\phantom{\rule{3.33333pt}{0ex}}t{K}^{1}$ |

≠1 | ${V}_{0}{e}^{2\theta}-\frac{m{N}_{0}^{2}}{2\left({k}^{2}-1\right)}{e}^{2k\theta}$ | 3 | ${A}_{1}{\otimes}_{s}\left(2{A}_{1}\right)$ | ${\partial}_{t},\phantom{\rule{3.33333pt}{0ex}}{e}^{\pm \sqrt{m}t}{K}^{1}$ |

≠1 | ${V}_{0}{e}^{-2\theta}$ | 3 | ${A}_{1}{\otimes}_{s}\left(2{A}_{1}\right)$ | ${\partial}_{t},\phantom{\rule{3.33333pt}{0ex}}{K}^{2},\phantom{\rule{3.33333pt}{0ex}}t{K}^{2}$ |

≠1 | ${V}_{0}{e}^{-2\theta}-\frac{m{N}_{0}^{2}}{2\left({k}^{2}-1\right)}{e}^{2k\theta}$ | 3 | ${A}_{1}{\otimes}_{s}\left(2{A}_{1}\right)$ | ${\partial}_{t},\phantom{\rule{3.33333pt}{0ex}}{e}^{\pm \sqrt{m}t}{K}^{21}$ |

≠1 | ${V}_{0}{e}^{2k\theta}$ | 2 | $2{A}_{1}$ | ${\partial}_{t},\phantom{\rule{3.33333pt}{0ex}}{K}^{3}$ |

≠1 | ${V}_{0}{e}^{-2\frac{\left({k}^{2}-2\right)}{k}\theta},{k}^{2}-2$ | 3 | $Sl\left(2,R\right)$ | ${\partial}_{t},\phantom{\rule{3.33333pt}{0ex}}2t{\partial}_{t}+{H}^{i}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{t}^{2}{\partial}_{t}+t{H}^{i}$ |

≠1 | ${V}_{0}{e}^{-2\frac{\left({k}^{2}-2\right)}{k}\theta}-\frac{{N}_{0}^{2}m}{{k}^{2}-1}{e}^{2k\theta}\phantom{\rule{3.33333pt}{0ex}}$ , ${k}^{2}-2\ne 0$ | 3 | $Sl\left(2,R\right)$ | ${\partial}_{t},\phantom{\rule{3.33333pt}{0ex}}\frac{2}{\sqrt{m}}{e}^{\pm \sqrt{m}t}{\partial}_{t}\pm {e}^{\pm \sqrt{m}t}{H}^{i}$ |

**Table 4.**Noether symmetry classification for the Brans–Dicke action in a spatially flat FLRW spacetime (II).

$\left|\mathit{k}\right|$ | Potential | # Symmetries | Lie Algebra | Symmetries |
---|---|---|---|---|

=1 | ${V}_{0}{e}^{2\theta}$ | 3 | ${A}_{1}{\otimes}_{s}\left(2{A}_{1}\right)$ | ${\partial}_{t},\phantom{\rule{3.33333pt}{0ex}}{K}^{1},\phantom{\rule{3.33333pt}{0ex}}t{K}^{1}$ |

=1 | ${V}_{0}{e}^{2\theta}-\frac{m}{2}\theta {e}^{2\theta}$ | 3 | ${A}_{1}{\otimes}_{s}\left(2{A}_{1}\right)$ | ${\partial}_{t},\phantom{\rule{3.33333pt}{0ex}}{e}^{\pm \sqrt{m}t}{K}^{1}$ |

=1 | ${V}_{0}{e}^{-2\theta}$ | 3 | ${A}_{1}{\otimes}_{s}\left(2{A}_{1}\right)$ | ${\partial}_{t},\phantom{\rule{3.33333pt}{0ex}}{K}^{2},\phantom{\rule{3.33333pt}{0ex}}t{K}^{2}$ |

=1 | ${V}_{0}{e}^{-2\theta}-\frac{1}{4}p{e}^{2\theta}\phantom{\rule{0.166667em}{0ex}}$ | 3 | ${A}_{1}{\otimes}_{s}\left(2{A}_{1}\right)$ | ${\partial}_{t},\phantom{\rule{3.33333pt}{0ex}}{K}^{2},\phantom{\rule{3.33333pt}{0ex}}t{K}^{2}$ |

Sp. Curv. K | $\mathit{f}\left(\mathit{R}\right)$ | # Symmetries | Lie Algebra | Symmetries |
---|---|---|---|---|

$=0,\pm 1$ | Arbitrary | 1 | ${A}_{1}$ | ${\partial}_{t}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}$ |

$=0,\pm 1$ | ${R}^{\frac{3}{2}}$ | 4 | $\left(2{A}_{1}\right){\otimes}_{s}\left(2{A}_{1}\right)$ | ${\partial}_{t},\phantom{\rule{3.33333pt}{0ex}}{K}_{1},\phantom{\rule{3.33333pt}{0ex}}{K}_{2},\phantom{\rule{3.33333pt}{0ex}}{K}_{2}^{\ast}$ |

$=0$ | ${R}^{\frac{7}{8}}$ | 3 | $Sl\left(2,R\right)$ | ${\partial}_{t},\phantom{\rule{3.33333pt}{0ex}}{K}_{3},\phantom{\rule{3.33333pt}{0ex}}{K}_{3}^{\ast}$ |

$=0$ | ${R}^{n}$ (with $n\ne 0,1,\frac{3}{2},\frac{7}{8}$) | 2 | $2{A}_{1}$ | ${\partial}_{t},\phantom{\rule{3.33333pt}{0ex}}{K}_{1}^{\ast}$ |

$=1$ | ${R}^{2}$ | 2 | ${A}_{1}{\otimes}_{s}\left(2{A}_{1}\right)$ | ${\partial}_{t},\phantom{\rule{3.33333pt}{0ex}}{K}_{1\left(n=2\right)}^{\ast}$ |

$=0,\pm 1$ | ${(R-2\Lambda )}^{3/2}$ | 3 | ${A}_{1}{\otimes}_{s}\left(2{A}_{1}\right)$ | ${\partial}_{t},\phantom{\rule{3.33333pt}{0ex}}{K}_{\left(\pm \right)2}$ |

$=0$ | ${(R-2\Lambda )}^{7/8}$ | 3 | $Sl\left(2,R\right)$ | ${\partial}_{t},\phantom{\rule{3.33333pt}{0ex}}{K}_{\left(\pm \right)4}$ |

**Table 6.**Analytic forms of $f\left(R\right)$ theory where the field equations admit contact symmetries.

Potential $\mathit{V}\left(\mathit{\varphi}\right)$ | Function $\mathit{f}\left(\mathit{R}\right)$ |
---|---|

${V}_{I}\left(\varphi \right)$ | ${\left(R-{V}_{1}\right)}^{\frac{3}{2}}$ |

${V}_{II}\left(\varphi \right)$ | ${\left(R-{V}_{1}\right)}^{\frac{7}{8}},$ |

${V}_{III}\left(\varphi \right)$ | ${R}^{\frac{1}{3}}-{V}_{1}$ |

${V}_{IV}\left(\varphi \right)\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{V}_{1}=0$ | ${R}^{4}$ |

${V}_{IV}\left(\varphi \right)\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{V}_{2}=0$ | ${R}^{\frac{3}{2}}$ |

${V}_{V}\left(\varphi \right),\phantom{\rule{3.33333pt}{0ex}}{V}_{1}=\pm 4{V}_{2}\sqrt{\beta}$ | $\mp \sqrt{\beta}R+{R}^{\frac{4}{3}}$ |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tsamparlis, M.; Paliathanasis, A.
Symmetries of Differential Equations in Cosmology. *Symmetry* **2018**, *10*, 233.
https://doi.org/10.3390/sym10070233

**AMA Style**

Tsamparlis M, Paliathanasis A.
Symmetries of Differential Equations in Cosmology. *Symmetry*. 2018; 10(7):233.
https://doi.org/10.3390/sym10070233

**Chicago/Turabian Style**

Tsamparlis, Michael, and Andronikos Paliathanasis.
2018. "Symmetries of Differential Equations in Cosmology" *Symmetry* 10, no. 7: 233.
https://doi.org/10.3390/sym10070233