# Symmetries of Differential Equations in Cosmology

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## Abstract

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## 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$$\mathbf{X}=\left(\right)open="("\; close=")">\frac{1}{2}{d}_{1}{a}_{1}t+{d}_{2}$$$${L}_{Y}{V}^{,i}+{d}_{1}{V}^{,i}=0.$$
- 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$$\mathbf{X}=\left(\right)open="("\; close=")">-{c}_{1}\sqrt{\psi k}cos\left(\right)open="("\; close=")">2\sqrt{\frac{\psi}{k}}t$$
- 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)$${\mathbf{X}}_{J}=\left(\right)open="("\; close=")">C\left(t\right){S}_{J}+D\left(t\right)$$$${D}_{,t}=\frac{1}{2}{d}_{1}T\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{4pt}{0ex}}{T}_{,tt}={a}_{1}T\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{T}_{,t}={c}_{2}C\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{D}_{,tt}={d}_{c}C\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{C}_{,t}={a}_{0}T$$$$\begin{array}{cc}\hfill {\mathcal{L}}_{Y}{V}^{,i}+2{a}_{0}S{V}^{,i}+{d}_{1}{V}^{,i}+{a}_{1}{Y}^{i}& =0,\hfill \end{array}$$$$\begin{array}{c}\hfill \left(\right)open="("\; close=")">{S}_{,k}{\delta}_{j}^{i}+2S{,}_{j}{\delta}_{k}^{i}{V}^{,k}+\left(\right)open="("\; close=")">2{Y}^{i}{}_{;\phantom{\rule{3.33333pt}{0ex}}j}-{a}_{0}S{\delta}_{j}^{i}& {c}_{2}-{d}_{c}{\delta}_{j}^{i}\end{array}$$
- 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$${X}_{J}=\left(\right)open="("\; close=")">C\left(t\right){S}_{J}+{d}_{1}$$$${T}_{,tt}+2{C}_{,t}={\lambda}_{1}T\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{T}_{,t}={\lambda}_{2}C\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{C}_{,t}={a}_{0}T$$$$\begin{array}{cc}\hfill {\mathcal{L}}_{{Y}_{J}}{V}^{,i}+{\lambda}_{1}{S}_{J}{V}^{,i}& =0,\hfill \end{array}$$$$\begin{array}{c}\hfill C\left(\right)open="("\; close=")">{\lambda}_{1}{S}_{J}{\delta}_{j}^{i}+2{S}_{J}{,}_{j}{V}^{,i}+{\lambda}_{2}\left(\right)open="("\; close=")">2\lambda {S}_{J,j}{V}^{.i}+\left(\right)open="("\; close=")">2\lambda {S}_{J}-{a}_{0}{S}_{J}& {\delta}_{j}^{i}\end{array}=0.\hfill $$

#### 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$${V}_{,k}{Y}^{,k}+2{\psi}_{Y}V={c}_{2}Y+d.$$$$\mathbf{X}=2{\psi}_{Y}\int T\left(t\right)dt{\partial}_{t}+T\left(t\right){S}_{J}^{,i}{\partial}_{i}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}f\left(\right)open="("\; close=")">t,{x}^{k}\phantom{\rule{3.33333pt}{0ex}}+d\int Tdt,$$$${T}_{,tt}={c}_{2}T\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{K}_{,t}=d\int Tdt+\mathrm{constant},$$$$\overline{\mathsf{\Phi}}={\psi}_{Y}E\int Tdt\phantom{\rule{3.33333pt}{0ex}}-T{g}_{ij}{H}^{i}{\dot{x}}^{j}+{T}_{,t}H+d\int Tdt.$$

#### 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 ${V}_{eff}\left(\right)open="("\; close=")">x,y$, Lagrangian (120) admits an extra Noether point symmetry provided by the proper HV of the two-dimensional flat space,$${X}_{\left(\varphi \right)1}=2t{\partial}_{t}+\left(\right)open="("\; close=")">x+\frac{4}{d}y{\partial}_{y},$$$${\mathsf{\Phi}}_{\left(\varphi \right)1}=\left(\right)open="("\; close=")">x+\frac{4}{d}y\dot{y}$$$${\mathsf{\Phi}}_{\left(\varphi \right)2}=\dot{x}-\dot{y}.$$
- Finally, when ${V}_{eff}(x,y)=\frac{1}{2}\left(\right)open="("\; close=")">{\omega}_{1}{x}^{2}-{\omega}_{2}{y}^{2}$, that is, $\tilde{V}\left(\theta \right)=\frac{1}{2}\left(\right)open="("\; close=")">{\omega}_{1}{cosh}^{2}\left(\theta \right)-{\omega}_{2}{sinh}^{2}\left(\theta \right)$, the dynamical system admits four extra Noether point symmetries$$\begin{array}{cc}\hfill {X}_{{\left(\varphi \right)}_{2}}& =sinh\left(\right)open="("\; close=")">\sqrt{{\omega}_{1}}t{\partial}_{x}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{X}_{{\left(\varphi \right)}_{3}}=cosh\left(\right)open="("\; close=")">\sqrt{{\omega}_{1}}t\hfill & {\partial}_{x}\phantom{\rule{3.33333pt}{0ex}},\end{array}\hfill {X}_{{\left(\varphi \right)}_{4}}& =sinh\left(\right)open="("\; close=")">\sqrt{{\omega}_{2}}t{\partial}_{y}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{X}_{{\left(\varphi \right)}_{5}}=cosh\left(\right)open="("\; close=")">\sqrt{{\omega}_{2}}t\hfill & {\partial}_{y}\phantom{\rule{3.33333pt}{0ex}},$$$$\begin{array}{cc}\hfill {I}_{{n}_{2}}& =sinh\left(\right)open="("\; close=")">\sqrt{{\omega}_{1}}t\dot{x}-\sqrt{{\omega}_{1}}cosh\left(\right)open="("\; close=")">\sqrt{{\omega}_{1}}t\hfill & x,\end{array}\hfill {I}_{{n}_{3}}& =cosh\left(\right)open="("\; close=")">\sqrt{{\omega}_{1}}t\dot{x}-\sqrt{{\omega}_{1}}sinh\left(\right)open="("\; close=")">\sqrt{{\omega}_{1}}t\hfill & x,$$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$${I}_{1}=\frac{d}{dt}\left(\right)open="("\; close=")">\frac{{r}^{1+k}{e}^{\left(\right)open="("\; close=")">1+k}}{}\left(\right)open="("\; close=")">k+1$$
- For $V\left(\theta \right)={V}_{0}{e}^{2\theta}-\frac{m{N}_{0}^{2}}{2\left(\right)open="("\; close=")">{k}^{2}-1}$ there are two additional Noether point symmetries ${e}^{\pm \sqrt{m}t}{K}^{1}$, with corresponding first integrals$${I}_{\pm}^{\prime}={e}^{\pm \sqrt{m}t}\left(\right)open="["\; close="]">\frac{d}{dt}\left(\right)open="("\; close=")">\frac{{r}^{1+k}{e}^{\left(\right)open="("\; close=")">1+k}}{}\left(\right)open="("\; close=")">k+1$$
- 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$${J}_{1}=\frac{d}{dt}\left(\right)open="("\; close=")">\frac{{r}^{1-k}{e}^{-\left(\right)open="("\; close=")">1-k}}{}k-1\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{J}_{2}=t\frac{d}{dt}\left(\right)open="("\; close=")">\frac{{r}^{1-k}{e}^{-\left(\right)open="("\; close=")">1-k}}{}k-1$$
- For $V\left(\theta \right)={V}_{0}{e}^{-2\theta}-\frac{m{N}_{0}^{2}}{2\left(\right)open="("\; close=")">{k}^{2}-1}$, we have the extra Noether symmetries ${e}^{\pm \sqrt{m}t}{K}^{2}$ $m=$ constant, with first integrals$${J}_{\pm}^{{}^{\prime}}={e}^{\pm \sqrt{m}t}\left(\right)open="("\; close=")">\frac{d}{dt}\left(\right)open="("\; close=")">\frac{{r}^{1-k}{e}^{-\left(\right)open="("\; close=")">1-k}}{}k-1\mp \sqrt{m}\frac{{r}^{1-k}{e}^{-\left(\right)open="("\; close=")">1-k}}{}k-1$$
- For the potential $V\left(\theta \right)={V}_{0}{e}^{2k\theta},$ the additional symmetry is the vector field ${K}^{3}$ with first integrals$${I}_{3}=\frac{r{e}^{2k\theta}}{k}\left(\right)open="("\; close=")">k\dot{r}+r\dot{\theta}$$
- For $V\left(\theta \right)={V}_{0}{e}^{-2\frac{\left(\right)}{{k}^{2}}k}$ 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$${I}_{{H}_{1}}=-\frac{d}{dt}\left(\right)open="("\; close=")">\frac{1}{2}\frac{{r}^{2}{e}^{2k\theta}}{{k}^{2}-1}+\frac{1}{2}\frac{{r}^{2}{e}^{2k\theta}}{{k}^{2}-1}.$$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 $V\left(\theta \right)={V}_{0}{e}^{-2\frac{\left(\right)}{{k}^{2}}k}$ , ${k}^{2}-2\ne 0,$ 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$${I}_{+,-}={e}^{\pm 2\sqrt{m}t}\left(\right)open="("\; close=")">\mp \frac{d}{dt}\left(\right)open="("\; close=")">\frac{1}{2}\frac{{r}^{2}{e}^{2k\theta}}{{k}^{2}-1}$$For this potential, the Noether point symmetries form the $sl\left(\right)open="("\; close=")">2,R$ 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$${I}_{2}^{\prime}=\frac{d}{dt}\left(\right)open="("\; close=")">\theta -lnr$$
- 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$${I}_{1}^{\prime}=\frac{d}{dt}\left(\right)open="("\; close=")">\theta -lnr$$
- 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$${K}_{1}=2t{\partial}_{t}+\frac{4}{3}\phantom{\rule{3.33333pt}{0ex}}{\partial}_{a}-\frac{9}{2}\frac{{f}^{\prime}}{{f}^{\u2033}}{\partial}_{R},$$$${K}_{2}=\frac{1}{a}{\partial}_{a}-\frac{1}{{a}^{2}}\frac{{f}^{\prime}}{{f}^{\u2033}}{\partial}_{R}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{K}_{2}^{\ast}=t\left(\right)open="("\; close=")">\frac{1}{a}{\partial}_{a}-\frac{1}{{a}^{2}}\frac{{f}^{\prime}}{{f}^{\u2033}}{\partial}_{R}$$$${\mathsf{\Phi}}_{1}=6{a}^{2}\dot{a}\sqrt{R}+6\frac{{a}^{3}}{\sqrt{R}}\dot{R},$$$${\mathsf{\Phi}}_{2}=\frac{d}{dt}\left(\right)open="("\; close=")">a\sqrt{R}-a\sqrt{R}.$$
- For $f\left(R\right)={R}^{\frac{7}{8}}$ and $K=0$, the theory admits the additional Noether point symmetries$${K}_{3}=2t{\partial}_{t}+\frac{a}{2}\phantom{\rule{3.33333pt}{0ex}}{\partial}_{a}+\frac{1}{2}\frac{{f}^{\prime}}{{f}^{\u2033}}{\partial}_{R}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{K}_{3}^{\ast}={t}^{2}{\partial}_{t}+t\left(\right)open="("\; close=")">\frac{a}{2}\phantom{\rule{3.33333pt}{0ex}}{\partial}_{a}+\frac{1}{2}\frac{{f}^{\prime}}{{f}^{\u2033}}{\partial}_{R}$$$${\mathsf{\Phi}}_{3}=\frac{d}{dt}\left(\right)open="("\; close=")">{a}^{3}{R}^{-\frac{1}{8}}-{a}^{3}{R}^{-\frac{1}{8}}.$$
- 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$${K}_{1}^{\ast}=2t{\partial}_{t}+\left(\right)open="("\; close=")">\frac{4n}{3}-\frac{2}{3}$$$${\mathsf{\Phi}}_{1}^{\ast}={a}^{2}{R}^{n-1}\dot{a}\left(\right)open="("\; close=")">2-n\left(\right)open="("\; close=")">n-1$$
- For $f\left(R\right)={(R-2\Lambda )}^{3/2}$ the extra Noether point symmetries are$${K}_{\left(\pm \right)2}={e}^{\pm \sqrt{m}t}\left(\right)open="("\; close=")">\frac{1}{a}{\partial}_{a}-\frac{1}{{a}^{2}}\frac{{f}^{\prime}}{{f}^{\u2033}}{\partial}_{R}$$$${\mathsf{\Phi}}_{\left(\pm \right)2}={e}^{\pm \sqrt{m}t}\left(\right)open="("\; close=")">\frac{d}{dt}\left(\right)open="("\; close=")">a\sqrt{R-2L}.$$
- Finally, when $f\left(R\right)={(R-2\Lambda )}^{7/8},$ the field equations admit the Noether point symmetries$${K}_{\left(\pm \right)4}=\pm \frac{1}{\sqrt{m}}{e}^{\pm 2\sqrt{m}t}{\partial}_{t}+{e}^{\pm 2\sqrt{m}t}\left(\right)open="("\; close=")">\frac{a}{2}\phantom{\rule{3.33333pt}{0ex}}{\partial}_{a}+\frac{1}{2}\frac{{f}^{\prime}}{{f}^{\prime \prime}}{\partial}_{R}$$$${\mathsf{\Phi}}_{\left(\pm \right)4}=\frac{d}{dt}\left(\right)open="("\; close=")">{a}^{3}{\left(\right)}^{R}-\frac{1}{8}\mp \frac{1}{2}\sqrt{m}{a}^{3}{\left(\right)}^{R}-\frac{1}{8}$$

#### 11.5. Two-Scalar Field Cosmology

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

#### 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

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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 | ${A}_{1}{\otimes}_{s}\left(\right)open="("\; close=")">2{A}_{1}$ | ${\partial}_{t},\phantom{\rule{3.33333pt}{0ex}}{K}^{1},\phantom{\rule{3.33333pt}{0ex}}t{K}^{1}$ |

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

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

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

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

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

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

**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 | ${A}_{1}{\otimes}_{s}\left(\right)open="("\; close=")">2{A}_{1}$ | ${\partial}_{t},\phantom{\rule{3.33333pt}{0ex}}{K}^{1},\phantom{\rule{3.33333pt}{0ex}}t{K}^{1}$ |

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

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

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

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 | $\left(\right)open="("\; close=")">2{A}_{1}$ | ${\partial}_{t},\phantom{\rule{3.33333pt}{0ex}}{K}_{1},\phantom{\rule{3.33333pt}{0ex}}{K}_{2},\phantom{\rule{3.33333pt}{0ex}}{K}_{2}^{\ast}$ |

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

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

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

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

**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 $\mathit{f}\left(\mathit{R}\right)$ |
---|---|

${V}_{I}\left(\varphi \right)$ | ${\left(\right)}^{R}$ |

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

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

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

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

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

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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.
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**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