Dynamical system analysis at background and perturbation levels: Quintessence in severe disadvantage comparing to $\Lambda$CDM

TL;DR

This study compares the stability and physical viability of $ ext{Λ}$CDM and quintessence cosmologies by analyzing background and perturbation equations, revealing significant disadvantages of quintessence due to instabilities and unphysical behaviors.

Contribution

First dynamical system analysis of both background and perturbation equations for $ ext{Λ}$CDM and quintessence with exponential potential, highlighting stability differences.

Findings

Perturbations do not affect $ ext{Λ}$CDM stability, which remains consistent with observations.

Quintessence exhibits instabilities and unphysical matter clustering when perturbations are included.

$ ext{Λ}$CDM is more physically viable than quintessence due to stability and observational consistency.

Abstract

We perform for the first time a dynamical system analysis of both the background and perturbation equations, of $\Lambda$CDM cosmology and quintessence scenario with an exponential potential. In the former case the perturbations do not change the stability of the late-time attractor of the background equations, and the system still results in the dark-energy dominated, de Sitter solution, having passed from the correct dark-matter era with $\gamma\approx6/11$. However, in the case of quintessence the incorporation of perturbations changes the stability and properties of the background evolution, and the only conditionally stable points present either an exponentially increasing matter clustering not favored by observations, or Laplacian instabilities, and thus not physically interesting. This result is a severe disadvantage of quintessence cosmology comparing to $\Lambda$CDM paradigm.

Tables

Table 1. Table 1: The critical points, their stability conditions (the corresponding eigenvalues are given in Copeland:1997et ), and the values of Ω d subscript Ω 𝑑 \Omega_{d} and w d subscript 𝑤 𝑑 w_{d} , for the quintessence scenario with exponential potential, with γ m ≡ w m + 1 subscript 𝛾 𝑚 subscript 𝑤 𝑚 1 \gamma_{m}\equiv w_{m}+1 .

C.P.	$x$	$y$	Existence	Stability	${\Omega }_d$	$w_d$
A	0	0	Always	Saddle for	0	Undefined
B	1	0	Always	Unstable node for	1	1
				Saddle for $\lambda >\sqrt{6}$
C	-1	0	Always	Unstable node for $\lambda >-\sqrt{6}$	1	1
				Saddle for
D	$\lambda /\sqrt{6}$	${\left[1-{\lambda }^2/6\right]}^{1/2}$		Stable node for	1	$\frac{{\lambda }^2}{3}-1$
				Saddle for
E	${\left(3/2\right)}^{1/2}{\gamma }_m/\lambda$	${\left[3\left(2-{\gamma }_m\right){\gamma }_m/2{\lambda }^2\right]}^{1/2}$	${\lambda }^2>3{\gamma }_m$	Stable node for	$3{\gamma }_m/{\lambda }^2$	$w_m$
				Stable spiral for ${\lambda }^2>24{\gamma }_m^2/\left(9{\gamma }_m-2\right)$

Table 2. Table 2: The critical points and their stability conditions, of both background and perturbation equations, in the case of Λ Λ \Lambda CDM paradigm.

C.P.	${\Omega }_m$	$U_m$	Existence	Eigenvalues	Stability
$P_1$	0	$-2$	Always	$\{-3,2\}$	Saddle
$P_2$	1	$-\frac{3}{2}$	Always	$\{3,\frac{5}{2}\}$	Unstable
$P_3$	0	0	Always	$\{-3,-2\}$	Stable
$P_4$	1	1	Always	$\{3,-\frac{5}{2}\}$	Saddle

Table 3. Table 3: The physical (real with 0 ≤ Ω m ≤ 1 0 subscript Ω 𝑚 1 0\leq\Omega_{m}\leq 1 and expanding) critical points, their stability conditions, and their properties, of both background and perturbation equations, in the case of quintessence with exponential potential. The stability conditions arise from the examination of the sign of the eigenvalues of the involved perturbation matrix.

$C.P.$	$\{x,y\}$	$U_m$	${\Omega }_m$	$w_d$	Existence	Eigenvalues	Stability
$A_1$	$\{0,0\}$	$-\frac{3}{2}$	$1$	Undefined	Always	$\frac{5}{2},-\frac{3}{2},\frac{3}{2}$	Saddle
$A_2$	$\{0,0\}$	$1$	$1$	Undefined	Always	$-\frac{5}{2},-\frac{3}{2},\frac{3}{2}$	Saddle
$B_1$	$\{1,0\}$	$0$	$0$	$1$	Always	$3,1,3-\sqrt{\frac{3}{2}}\lambda$	Unstable for
							Saddle for $\lambda >\sqrt{6}$
$B_2$	$\{1,0\}$	$1$	$0$	$1$	Always	$3,-1,3-\sqrt{\frac{3}{2}}\lambda$	Saddle
$C_1$	$\{-1,0\}$	$0$	$0$	$1$	Always	$3,1,3+\sqrt{\frac{3}{2}}\lambda$	Unstable for $\lambda >-\sqrt{6}$
							Saddle for
$C_2$	$\{-1,0\}$	$1$	$0$	$1$	Always	$3,-1,\sqrt{\frac{3}{2}}\lambda +3$	Saddle
$D_1$	$\{\frac{\lambda }{\sqrt{6}},\sqrt{1-\frac{{\lambda }^2}{6}}\}$	$0$	$0$	$-1+\frac{{\lambda }^2}{3}$	${\lambda }^2\le 6$	${\lambda }^2-3,\frac{1}{2}\left({\lambda }^2-6\right),\frac{1}{2}\left({\lambda }^2-4\right)$	Stable for
							Saddle for ${\lambda }^2>3$
$D_2$	$\{\frac{\lambda }{\sqrt{6}},\sqrt{1-\frac{{\lambda }^2}{6}}\}$	$\frac{{\lambda }^2}{2}-2$	$0$	$-1+\frac{{\lambda }^2}{3}$	${\lambda }^2\le 6$	${\lambda }^2-3,\frac{1}{2}\left({\lambda }^2-6\right),-\frac{1}{2}\left({\lambda }^2-4\right)$	Saddle
$E_1$	$\{\frac{\sqrt{\frac{3}{2}}}{\lambda },\frac{\sqrt{\frac{3}{2}}}{\lambda }\}$	$-\frac{1}{4}\left(1-\sqrt{25-\frac{72}{{\lambda }^2}}\right)$	$1-\frac{3}{{\lambda }^2}$	0	${\lambda }^2\ge 3$	$-\frac{1}{2}\sqrt{25-\frac{72}{{\lambda }^2}},-\frac{3}{4}\left(1\pm \sqrt{\frac{24}{{\lambda }^2}-7}\right)$	Stable for ${\lambda }^2>3$
$E_2$	$\{\frac{\sqrt{\frac{3}{2}}}{\lambda },\frac{\sqrt{\frac{3}{2}}}{\lambda }\}$	$-\frac{1}{4}\left(1+\sqrt{25-\frac{72}{{\lambda }^2}}\right)$	$1-\frac{3}{{\lambda }^2}$	$0$	${\lambda }^2\ge 3$	$\frac{1}{2}\sqrt{25-\frac{72}{{\lambda }^2}},-\frac{3}{4}\left(1\pm \sqrt{\frac{24}{{\lambda }^2}-7}\right)$	Saddle

Equations

$$H^2=\frac{{\kappa }^2}{3}({\rho }_m+{\rho }_d),$$

$$\dot{H}=-\frac{{\kappa }^2}{2}({\rho }_m+p_m+{\rho }_d+p_d),$$

$${\dot{\rho }}_m+3H(1+w_m){\rho }_m=0,$$

$${\dot{\rho }}_d+3H(1+w_d){\rho }_d=0,$$

$${\Omega }_m^{\prime }=3({\Omega }_m-1){\Omega }_m,$$

$${\Omega }_m=\frac{{\Omega }_{m0}}{e^{3(1+w_m)\tau }(1-{\Omega }_{m0})+{\Omega }_{m0}},$$

$$x\equiv \frac{\kappa \dot{\varphi }}{\sqrt{6}H},y\equiv \frac{\kappa \sqrt{V}}{\sqrt{3}H},$$

$$x^{\prime }=\frac{3}{2}x\left[2x^2+{\gamma }_m(1-x^2-y^2)\right]-3x+\sqrt{\frac{3}{2}}\lambda y^2,$$

$$y^{\prime }=\frac{3}{2}y\left[2x^2+{\gamma }_m(1-x^2-y^2)\right]-\sqrt{\frac{3}{2}}\lambda xy,$$

$${\dot{\delta }}_m+\frac{{\theta }_m}{a}=0,$$

$${\dot{\delta }}_d+(1+w_d)\frac{{\theta }_d}{a}+3H(c_{eff}^2-w_d){\delta }_d=0,$$

$${\dot{\theta }}_m+H{\theta }_m-\frac{k^2\psi }{a}=0,$$

$${\dot{\theta }}_d+H{\theta }_d-\frac{k^2{c}_{\mathrm{eff}}^2{\delta }_d}{(1+w_d)a}-\frac{k^2\psi }{a}=0,$$

$$-\frac{k^2}{a^2}\psi =\frac{3}{2}{H}^2[{\Omega }_m{\delta }_m+(1+3c_{\mathrm{eff}}^2){\Omega }_d{\delta }_d].$$

$$U_m\equiv \frac{{\delta }_m^{\prime }}{{\delta }_m}.$$

$${\Omega }_m^{\prime }=3({\Omega }_m-1){\Omega }_m,$$

$$U_m^{\prime }=\frac{3}{2}(U_m+1){\Omega }_m-U_m(U_m+2).$$

$$3\gamma ({\Omega }_m-1){\Omega }_m^{\gamma }+\left({\Omega }_m^{\gamma }+2\right){\Omega }_m^{\gamma }-\frac{3}{2}{\Omega }_m\left({\Omega }_m^{\gamma }+1\right)=0,$$

$$-\left(\frac{11\gamma }{2}-3\right)(1-{\Omega }_m)+O\left((1-{\Omega }_m{)}^2\right)=0.$$

$${\Omega }_m(\tau )=\frac{{\Omega }_{m0}}{e^{3\tau }(1-{\Omega }_{m0})+{\Omega }_{m0}},$$

$$\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!U_{m}(\tau)=\Big{\{}2\Omega_{m}(2U_{m0}+3\Omega_{m0})\big{(}\Omega_{m0}^{2/3}g_{0}-\Omega_{m}^{2/3}g\big{)}$$

$$\displaystyle\ \ \ \ \ +8(1-\Omega_{m0})^{5/6}\Omega_{m0}^{2/3}\Big{\}}^{-1}$$

$$\displaystyle\ \ \,\cdot\Big{\{}3\Omega_{m}(2U_{m0}+3\Omega_{m0})\big{(}\Omega_{m}^{2/3}g-\Omega_{m0}^{2/3}g_{0}\big{)}$$

$$\displaystyle\ \ \ \ \ +4\Omega_{m}^{2/3}\Big{[}(2U_{m0}+3\Omega_{m0})\left(1-\Omega_{m}\right)^{5/6}$$

$$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -3\Omega_{m0}^{2/3}\Omega_{m}^{1/3}(1-\Omega_{m0})^{5/6}\Big{]}\Big{\}},$$

$$U_m=\frac{{\delta }_m^{\prime }}{{\delta }_m}.$$

$$U_m^{\prime }=-U_m^2-\frac{U_m}{2}\left(1-3x^2+3y^2\right)+\frac{3}{2}\left(1-x^2-y^2\right),$$

Full text

I Introduction

Dynamical system approach is a powerful tool that allows to extract information on the evolution of a cosmological model, independently of the initial conditions or its specific behavior at intermediate times REZA . In particular, although a general cosmological scenario may exhibit an infinite number of possible evolutions, its asymptotic behavior, namely its behavior at late times, can be classified in a few different classes, which correspond to the stable critical points of the autonomous-form transformed cosmological equations. Thus, through such an analysis one obtains information of the late-time universe, bypassing the complications of the cosmological equations, which prevent complete analytical treatments, as well as the ambiguity of the initial conditions.

The dynamical system approach has been applied to numerous cosmological scenarios since the late 90’s (see Bahamonde:2017ize and references therein), nevertheless up to now it remained only at the background level, namely examining the behavior of the background equations and calculating at the critical points the values of background-related quantities such as the density parameters, the equation-of-state parameter etc. Although this analysis was important and adequate for the earlier cosmology advance, the significantly advancing cosmological progresses and especially the huge amount of data related to perturbations (such as the growth index and the Large Scale Structure), leads to the need to extend the dynamical system approach in order to investigate cosmological scenarios at both the background and perturbation levels.

II Dynamical analysis at the background level

Let us briefly review the phase space analysis of $\Lambda$CDM paradigm, as well as of the basic dynamical dark energy scenario, namely the quintessence one with an exponential potential, which is the archetype quintessence scenario due to the well-posed theoretical justification of exponential potentials. Considering a flat Friedmann-Robertson-Walker (FRW) metric $ds^{2}=dt^{2}-a^{2}(t)\,\delta_{ij}dx^{i}dx^{j}$, the equations of a general cosmological scenario read as

[TABLE]

with $\kappa^{2}=8\pi G$, and where $\rho_{m}$, $p_{m}$ are respectively the energy density and pressure of the matter fluid, while $\rho_{d}$, $p_{d}$ are the energy density and pressure of the (effective) dark energy fluid. Finally, assuming that interactions do not take place among the cosmic fluid components, the system of equations closes with the conservation equations

[TABLE]

where we have introduced the equation-of-state parameters $w_{i}\equiv p_{i}/\rho_{i}$. Note that only three out of four equations (1)-(4) are independent.

The above framework provides $\Lambda$CDM cosmology for $\rho_{d}=-p_{d}=\Lambda/\kappa^{2}$, with $\Lambda$ the cosmological constant, and in this case Eq. (4) becomes trivial. Additionally, for the case of the basic quintessence scenario, in which a scalar field $\phi$ is introduced, we have $\rho_{d}=\dot{\phi}^{2}/2+V$ and $p_{d}=\dot{\phi}^{2}/2-V$, with $V(\phi)$ its potential, and then Eq. (4) becomes the Klein-Gordon equation $\ddot{\phi}+3H\dot{\phi}+V^{\prime}=0$, with $V^{\prime}(\phi)\equiv\partial V/\partial\phi$.

The essence of the dynamical system approach is to transform the equations into an autonomous system, using $\tau\equiv\ln a$ as the dynamical variable, extract its critical points, perturbing around them, and investigate their stability by examining the eigenvalues of the involved perturbation matrix REZA ; Bahamonde:2017ize .

For $\Lambda$CDM cosmology the cosmological equations can be transformed into an autonomous form by simply using the matter density parameter $\Omega_{m}\equiv\kappa^{2}\rho_{m}/(3H^{2})$ as the auxiliary variable. Thus, Eqs. (1) and (3) give rise to the one-dimensional system

[TABLE]

where primes denote derivatives with respect to $\tau$. The system has two critical points, characterized by $\Omega_{m}=1$ and $\Omega_{m}=0$, and one can see that the former is unstable while the latter stable. Therefore, for $\Lambda$CDM cosmology, the cosmological-constant dominated ($\Omega_{m}=0$ according to (1) implies that $\Omega_{d}\equiv(\kappa^{2}\rho_{d}/3H^{2})=1$), de-Sitter solution is the stable late-time attractor, and thus the universe will result to it independently of the initial conditions and its evolutions at intermediate times. We mention that actually the dynamical system analysis is not needed in this scenario, since the equations are integrable, with the solution

[TABLE]

with $\Omega_{m0}$ the value of $\Omega_{m}$ at $a=1$. Hence, we can immediately see that at late times the system always reaches the de-Sitter solution (for matter sectors that do not violate the null energy condition).

In the case of quintessence scenario, and focusing on the basic model where an exponential potential $V=V_{0}e^{-\lambda\kappa\phi}$ for the scalar field is imposed, introducing the auxiliary variables Copeland:1997et

[TABLE]

we result to the dynamical system

[TABLE]

with $\gamma_{m}\equiv w_{m}+1$, in terms of which the various density parameters are expressed as $\Omega_{d}=x^{2}+y^{2}$, $\Omega_{m}=1-\Omega_{d}$, while $w_{d}=\frac{x^{2}-y^{2}}{x^{2}+y^{2}}$. The critical points of the system (8)-(9), along with their stability conditions and the corresponding values of $\Omega_{d}$ and $w_{d}$ are shown in Table 1. As we observe, the scenario possesses two stable late-time attractors, with the scalar field dominated solution $D$ being the most physically interesting.

III Dynamical analysis at the perturbation level

The investigation of scalar perturbations is crucial in every cosmological scenario, since they are connected to perturbation-related observables such as the growth index $\gamma$ and the $\sigma_{8}$ Bahcall:1998ur . From now on, and for calculation convenience, we focus on the most interesting case of dust matter, namely we set $\gamma_{m}=1$ ($w_{m}=0$), since a non-zero $w_{m}$ does not qualitatively affect our results.

In a general non-interacting scenario, which includes dust matter and dynamical dark energy, the scalar perturbations in the Newtonian gauge are determined by the equations Ma:1995ey

[TABLE]

where $k$ is the wavenumber of Fourier modes, and $\psi$ the scalar metric perturbation assuming zero anisotropic stress. Additionally, $\delta_{i}\equiv\delta\rho_{i}/\rho_{i}$ are the densities perturbations and $\theta_{i}$ are the velocity perturbations Ma:1995ey . Furthermore, $c_{\mathrm{eff}}^{2}$ is the effective sound speed of the dark energy perturbations (the corresponding quantity for matter is zero in the dust case), which determines the amount of dark-energy clustering. Note that the above equations can be simplified by considering the Poisson equation, which in sub-horizon scales becomes Ma:1995ey :

[TABLE]

Finally, we mention that the above perturbation equations must be considered alongside the background evolution equations (1)-(4).

In general, the fact that $\Lambda$ does not change in space and time implies that the cosmological constant can not cluster like dark matter. On the other hand, dynamical dark energy may cluster and the amount of clustering is affected by the effective sound speed. Specifically, in the case of $c_{\mathrm{eff}}^{2}=1$, pressure suppresses any dark energy fluctuation at sub-horizon scales. Therefore, for homogeneous dark energy the quantities $\delta_{d}$ and $\theta_{d}$ are vanished. On the other hand, for $c_{\mathrm{eff}}^{2}=0$ dark energy clusters similar to that of dark matter and perturbations will grow with time. The clustering of dark energy modifies the evolution of dark matter fluctuations perturbations, hence it affects the structure formation rate of the universe (for more discussion see Mehrabi:2015hva and references therein).

Let us first investigate the case of $\Lambda$CDM paradigm, which is obtained by the above general framework for $w_{d}=-1$ and $\rho_{d}=\Lambda/\kappa^{2}$, alongside $\delta_{d}=0$ and $\theta_{d}=0$ (i.e. dark energy is not clustering and thus its perturbation equations can be completely ignored). As auxiliary variables we introduce $\Omega_{m}$, as well as the variable

[TABLE]

Hence, in terms of $\Omega_{m}$, ${U}_{m}$, the equations (1)-(4) and (10)-(13) become

[TABLE]

The critical points of the system (16)-(17), along with the corresponding eigenvalues and their stability conditions are presented in Table 2.

The system admits four critical points, with $P_{3}$ being the stable one. It corresponds to the cosmological-constant dominated, de Sitter solution, which moreover has $\delta_{m}=\text{const.}$ (since $U_{m}=0$). Similarly, one can observe the saddle point $P_{4}$, which is a matter dominated universe in which the perturbations increase as $\delta_{m}\propto e^{\tau}=a$ exactly at the critical point. Thus, for $\Lambda$CDM cosmology the incorporation of perturbations does not change the late-time attractor of the background evolution.

For completeness we must examine the possibility of critical points that exist at “infinity” and hence that are missed through the above basic analysis. Introducing the transformation $\{\Omega_{m},U_{m}\}\rightarrow\{\Omega_{m},\bar{U}_{m}\}$ with $\bar{U}_{m}=\frac{2}{\pi}\arctan(U_{m})$, we find that such critical points at infinity do not exist, since $\bar{U}_{m}^{\prime}|_{\bar{U}_{m}=\pm 1}=-2/\pi\neq 0$.

Finally, we note that in the literature it is standard to consider that in the matter-dominated phase, in which the large scale structure builds up due to the increase of matter perturbations, we have the relation $d\ln\!\delta_{m}/d\ln\!a\simeq\Omega_{m}^{\gamma}$ where $\gamma$ is the growth index Peebles:1994xt , which in our notation becomes just $U_{m}\simeq\Omega_{m}^{\gamma}$. Inserting it into (17) we obtain

[TABLE]

which expanded around $\Omega_{m}=1$ leads to

[TABLE]

As expected the asymptotic value of the growth index is $\gamma=\frac{6}{11}$. The curve (18) is depicted in Fig. 1 with a thick (brown) line, and as we observe it coincides with the unstable manifold of the matter dominated solution $P_{4}$.

We mention here that the dynamical system analysis is not needed for $\Lambda$CDM cosmology, since even including the perturbations the system remains integrable. In particular, the general solution reads

[TABLE]

where $g(\tau)=\!\,_{2}F_{1}\left[\frac{1}{6},\frac{2}{3},\frac{5}{3},\Omega_{m}(\tau)\right]$ and $g_{0}=\!\,_{2}F_{1}\left[\frac{1}{6},\frac{2}{3},\frac{5}{3},\Omega_{m0}\right]$, with $\Omega_{m0}$ and $U_{m0}$ the values of $\Omega_{m}$ and $U_{m}$ at $\tau=0$ (i.e. at $a=1$). From the analytical solutions (20),(21) we can easily see that for $\tau\rightarrow\infty$ we have $\Omega_{m}\rightarrow 0$ and $U_{m}\rightarrow 0$, i.e the system results to the de Sitter point $P_{3}$.

We now proceed to the investigation of perturbations in quintessence with exponential potential. As we mentioned above, this simple dark energy scenario has $c_{\mathrm{eff}}^{2}=1$, which implies that dark energy is non-clustering, and hence one should consider only the perturbation equations (10) and (12), alongside the background (1)-(4) ones. In order to transform them into autonomous form we use the variables $x,y$ of (7), as well as the additional variable

[TABLE]

Therefore, the autonomous dynamical system consists of Eqs. (8), (9) and

[TABLE]

i.e. it is now 3-dimensional in contrast to the 2-dimensional one of the background equations. Since the first two equations are decoupled from the third one, the system admits the five critical points of the background analysis of Table 1, each of which is now split into two points due to the additional variable $U_{m}$. The physical critical points and their stability conditions are presented in Table 3. Finally, the analysis at infinity shows that stable critical points do not exist.

The crucial feature, which lies in the center of the analysis of this work, is that the stability and properties of the points changes, due to the existence of extra dimensions (reflecting the incorporation of perturbation equations) in the phase space. In particular, we can see that the only two points that can be conditionally stable are $D_{1}$ and $E_{1}$. For $E_{1}$ we have $U_{m}>0$, which implies that $\delta_{m}$ increases exponentially in an expanding universe, and hence is not physically interesting. For $D_{1}$, although we obtain $U_{m}=0$ this point is not physically interesting since it has $c_{d}^{2}\equiv\frac{dp_{d}}{d\rho_{d}}=\frac{p_{d}^{\prime}}{\rho_{d}^{\prime}}=1+\sqrt{\frac{2}{3}}\lambda x(w_{d}-1)/(w_{d}+1)=-1+\frac{\lambda^{2}}{3}<0$ in its stability region, resulting to Laplacian instabilities. Therefore, the incorporation of perturbation ruins the dark-energy dominated, de-Sitter solution, which is the physically interesting late-time attractor of the background equations, since it induces to it Laplacian instabilities.

IV Conclusions

We performed for the first time a dynamical system analysis of both the background and perturbation equations, of $\Lambda$CDM cosmology and quintessence scenario with exponential potential. In the former case, the incorporation of perturbations does not change the stability of the late-time attractor of the background equations, and the system still results in the dark-energy dominated, de Sitter solution, having passed from the correct dark-matter era with $\gamma\approx 6/11$ (actually in this scenario one extracts analytical solutions). However, in the case of quintessence, the incorporation of perturbation changes the stability and properties of the background evolution, and the only conditionally stable points present either an exponentially increasing matter clustering not favored by observations, or Laplacian instabilities, and thus not physically interesting. In summary, the above results are a severe disadvantage of quintessence with exponential potential, (which is the archetype scenario due to the well-posed theoretical justification of exponential potentials) comparing to $\Lambda$CDM paradigm.